Power System Stability: Modelling, Analysis and Control

IET POWER AND ENERGY SERIES 76

Power SystemStability

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Power SystemStability

Modelling, analysis and control

Abdelhay A. Sallam and Om P. Malik

The Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United Kingdom

The Institution of Engineering and Technology is registered as a Charity in England &Wales (no. 211014) and Scotland (no. SC038698).

† The Institution of Engineering and Technology 2015

First published 2015

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The Institution of Engineering and TechnologyMichael Faraday HouseSix Hills Way, StevenageHerts, SG1 2AY, United Kingdom

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The moral rights of the authors to be identified as authors of this work have beenasserted by them in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication DataA catalogue record for this product is available from the British Library

ISBN 978-1-84919-944-5 (hardback)ISBN 978-1-84919-945-2 (PDF)

Typeset in India by MPS LimitedPrinted in the UK by CPI Group (UK) Ltd, Croydon

To our wives Hanzada Sallam and Margareta Malik

Contents

Preface xiii

1 Power system stability overview 11.1 General 11.2 Understanding power system stability 11.3 Classification of power system stability 3

1.3.1 Small signal stability 31.3.2 Transient stability 6

1.4 Need for modelling 81.5 Stability margin increase 9References 10

Part I Modelling 11

2 Modelling of the synchronous machine 132.1 Introduction 132.2 Synchronous machine equations 14

2.2.1 Flux linkage equations 142.2.2 Voltage equations 152.2.3 Torque equation 16

2.3 Park’s transformation 172.4 Transformation of synchronous machine equations 18

2.4.1 Transformation of flux linkage equations 182.4.2 Transformation of stator voltage equations 192.4.3 Transformation of the torque equation 25

2.5 Machine parameters in per unit values 262.5.1 Torque and power equations 30

2.6 Synchronous machine equivalent circuits 322.7 Flux linkage state space model 34

2.7.1 Modelling without saturation 342.7.2 Modelling with saturation 40

2.8 The current state space model 42References 44

3 Synchronous machine connected to a power system 473.1 Synchronous machine connected to an infinite bus 47

3.1.1 Flux linkage state space model 493.1.2 Current state space model 55

3.2 Synchronous machine connected to an integrated power system 573.3 Synchronous machine parameters in different operating modes 583.4 Synchronous machine-simplified models 62

3.4.1 The classical model 623.4.2 The E0

q model 643.5 Excitation system 67

3.5.1 Excitation system modelling 683.6 Modelling of prime mover control system 74

3.6.1 Hydraulic turbines 753.6.2 Steam turbines 77

References 79

4 Modelling of transformers, transmission lines and loads 814.1 Transformers 81

4.1.1 Modelling of two-winding transformers 814.1.2 Modelling of phase-shifting transformers 91

4.2 Transmission lines 934.2.1 Voltage and current relationship of a line 944.2.2 Modelling of transmission lines 95

4.3 Loads 974.3.1 Static load models 994.3.2 Dynamic load models 101

4.4 Remarks on load modelling for stability andpower flow studies 103

References 104

Part II Power flow 107

5 Power flow analysis 1095.1 General concepts 1095.2 Newton–Raphson method 111

5.2.1 Power flow solution with polar coordinate system 1135.2.2 Power flow solution with rectangular coordinate

system 1145.3 Gauss�Seidel method 1215.4 Decoupling method 123

5.4.1 Fast-decoupled method 125References 129

6 Optimal power flow 1316.1 Problem formulation 1316.2 Problem solution 1326.3 OPF with dynamic security constraint 137References 142

viii Power system stability: modelling, analysis and control

Part III Stability analysis 145

7 Small signal stability 1477.1 Basic concepts 147

7.1.1 Equilibrium points 1497.1.2 Stability of equilibrium point 1507.1.3 Phasor diagrams of synchronous machines 152

7.2 Small signal stability 1547.2.1 Forced state variable equation 162

7.3 Linearised current state space model of a synchronous generator 1647.4 Linearised flux linkage state space model of a

synchronous generator 1727.5 Small signal stability of multi-machine systems 177References 183

8 Transient stability 1858.1 Synchronous machine model 1868.2 Numerical integration techniques 1928.3 Transient stability assessment of a simple power system 1938.4 Transient stability analysis of a multi-machine power system 201References 218

9 Transient energy function methods 2219.1 Definitions of stability concepts 221

9.1.1 Positive definite function 2229.1.2 Negative definite function 2229.1.3 Lemma 2229.1.4 Stability regions 2239.1.5 Lyapunov function theorem 223

9.2 Stability of single-machine infinite-bus system 2259.3 Stability of multi-machine power system 234

9.3.1 Energy balance approach 2359.3.2 TEF method 241

References 250

Part IV Stability enhancement and control 251

10 Artificial intelligence techniques 25310.1 Artificial neural networks 25310.2 Neural network topologies 255

10.2.1 Single-layer feed-forward architecture 25510.2.2 Multi-layer feed-forward architecture 25510.2.3 Recurrent networks 25610.2.4 Back-propagation learning algorithm 256

10.3 Fuzzy logic systems 259

Contents ix

10.3.1 Fuzzy set theory 26010.3.2 Linguistic variables 26110.3.3 Fuzzy IF–THEN rules 26110.3.4 Structure of an FL system 261

10.4 Neuro-fuzzy systems 26310.4.1 Adaptive neuro-fuzzy inference system 26310.4.2 Structure of the NFC 26510.4.3 Online adaptation technique 267

10.5 Adaptive simplified NFC 26910.5.1 Simplification of the rule-base structure 269

10.6 Control system design of the proposed ASNFC 272References 274

11 Power system stabiliser 27711.1 Conventional PSS 278

11.1.1 Configuration of common PSS 27811.1.2 PSS input signals 27911.1.3 Characteristics of common PSS 280

11.2 Adaptive control-based PSS 28111.2.1 Direct adaptive control 28211.2.2 Indirect adaptive control 28311.2.3 Indirect adaptive control strategies 286

11.3 PS control-based APSS 28711.3.1 Self-adjusting PS control strategy 28711.3.2 Performance studies with pole-shifting control PSS 290

11.4 AI-based APSS 29111.4.1 APSS with NN predictor and NN controller 29211.4.2 Adaptive network-based FLC 293

11.5 Amalgamated analytical and AI-based PSS 29711.5.1 APSS with neuro identifier and PS control 29711.5.2 APSS with fuzzy logic identifier and PS controller 29911.5.3 APSS with RLS identifier and fuzzy logic control 301

11.6 APSS based on recurrent adaptive control 30111.7 Concluding Remarks 307References 307

12 Series compensation 31112.1 Definitions of transmission line parameters 31112.2 Compensation of lossless transmission line 313

12.2.1 Determination of amount of series compensation 31312.2.2 Transient stability improvement for lossless

compensated line 31612.3 Long transmission lines 319

12.3.1 Series compensation for long transmission lines 32112.4 Enhancement of multi-machine power system

transient stability 329

x Power system stability: modelling, analysis and control

12.5 Investigation of transmission power transfer capacity 33312.6 Improvement of small signal stability 33512.7 Sub-synchronous resonance 340

12.7.1 The mechanical system 34112.7.2 The electrical network 343

References 348

13 Shunt compensation 35113.1 Shunt compensation of lossless transmission lines 351

13.1.1 Shunt-compensated line parameters 35113.1.2 Transient stability enhancement for

shunt-compensated lossless lines 35313.2 Long transmission lines 35713.3 Static var compensators 360

13.3.1 Characteristics of FC-TCR compensators 36213.3.2 Modelling of FC-TCR compensators 362

13.4 Static synchronous compensator (STATCOM) 36913.5 Application of ASNFC to shunt-compensated power systems 374

13.5.1 Simulation studies 37513.5.2 Three-phase to ground short circuit test 376

References 377

14 Compensation devices 37914.1 Introduction 37914.2 Flexible AC transmission system 380

14.2.1 Thyristor-controlled series capacitor 38014.2.2 Static synchronous series compensator 38214.2.3 Static var compensator 38414.2.4 Static synchronous compensator 38714.2.5 Phase-shifting transformer 38914.2.6 Unified power flow controller 389

References 392

15 Recent technologies 39515.1 Energy storage systems 395

15.1.1 Chemical energy storage systems (batteries) 39715.1.2 Flywheel energy storage 39715.1.3 Compressed air energy storage 39715.1.4 Pumped hydroelectric energy storage 39815.1.5 Super capacitors 40015.1.6 Superconducting energy storage 400

15.2 Superconductivity applications 40915.2.1 Superconducting synchronous generators 40915.2.2 Superconducting transmission cables 41115.2.3 Superconducting transformers 41115.2.4 Superconducting fault current limiters 412

Contents xi

15.2.5 SMES applications 41615.2.6 Features of storage systems 418

15.3 Phasor measurement units 42015.3.1 Structure of WAMS 42015.3.2 Benefits of WAMS 42115.3.3 Case studies 422

References 423

Appendix I Calculation of synchronous machine parametersin per unit/normalised form 427

Appendix II Nine-bus test system 437

Appendix III Numerical integration techniques 439

Appendix IV 15-bus, 4-generator system data 445

Index 449

xii Power system stability: modelling, analysis and control

Preface

Modern day large power systems are essentially dynamic systems with stringentrequirements of high reliability for the continuous availability of electricity.Reliability is contingent on the power system retaining stable operation duringsteady-state operation and also following disturbances. The subject of power sys-tem stability has been studied for many decades. With new developments, and therehave been many over the past couple of decades, new concerns and problems arisethat need to be studied and analysed. The objective of this book is a step in thatdirection though not ignoring the conventional and well-established approaches.

To ensure stable operation of the power system, it is necessary to analyse thepower system performance under various operating conditions. Analysis includesstudies such as power flow and both steady-state and transient stabilities. To per-form such studies requires knowledge about the models to represent the variouscomponents that constitute an integrated power system. In situations where there isa risk of loss of stability, it is necessary to apply controls that can ensure stable anduninterrupted supply of electricity following a disturbance.

The subject of stability thus encompasses modelling, computation of load flowin the transmission grid, stability analysis under both steady-state and disturbedconditions and appropriate controls to enhance stability. All these topics are cov-ered in this book in that order to provide a fairly comprehensive treatment of theoverall subject of stability of power systems. The subject matter is covered at alevel that is suitable for students, scientists and engineers involved in the study,design, analysis and control of power systems.

The stage for a study of power system stability is set in Chapter 1 where anoverview of the problem of power system stability is provided. The followingpart of the book is divided into four parts, each part dedicated to a specific topic,i.e. Modelling, Power Flow, Stability Analysis, and Stability Enhancement andControl.

Part I Modelling consists of three chapters. A comprehensive description ofmodelling synchronous machines is provided, which is followed by the models fortransformers, transmission lines and loads.

Part II Power Flow consists of two chapters. Description of the general conceptof power flow is provided, followed by a description of the various commonly usedtechniques for load flow and optimal load flow.

Part III Stability Analysis consists of three chapters. Small signal stability andconventional methods of transient stability assessment are covered in Chapters 7and 8, respectively. Description of transient stability calculation using transient

energy function methods is discussed in Chapter 9. These methods are useful foronline assessment of transient stability of large power systems. They can providecontinuous assessment of the state of the power system so that measures can betaken in advance, in case a gradual degradation in the power system secure stateoperation is noted.

Part IV Stability Enhancement and Control consists of six chapters. Variousmeasures for stability enhancement are described in this part. Not only the conven-tional techniques but also the newly emerging techniques for power system stabilityenhancement and control are described. Power system stabilisers, initially developedin the 1950s, are the most common devices used in the power systems to providedamping following disturbances. Advances making use of adaptive control and arti-ficial intelligence (AI) techniques have taken place in the development of newalgorithms for the power system stabiliser. A brief introduction to AI techniques isgiven in Chapter 10. The conventional power system stabiliser as well as the devel-opments in adaptive and AI-based power system stabiliser is described in Chapter 11.Use of power electronics-based compensation, series and shunt compensation, isdescribed in Chapters 12 and 13, respectively. These devices, commonly known asFACTS devices, are described in Chapter 14.

With the deployment of satellites and the commensurate developments incommunication technologies, such as GPS, investigations in many additionaldirections are taking place to take advantage of these new technologies inimproving power system stability and reliability. In addition, the significant movetowards generation of electricity from renewable sources has resulted in many newdevelopments, in particular energy storage. These technologies are in the initialstate of development, and a brief introduction to the newly developing technologiesis given in Chapter 15.

Certain supporting material is described in Appendices I–IV.The subject matter of this book covers a broad spectrum of topics. The material

covered in the book includes only a very limited part of the authors’ own researchover the last 40 years. Similarly, because of the broad scope, it is possible to covercertain topics only briefly. However, a good set of references is included asresource material so that a reader interested in pursuing a specific topic in moredetail can do so by going to these references.

These days no man is unto self. It is the cooperative effort of many. Eventhough not individually named, the authors wish to acknowledge the work and helpof innumerable graduate students, colleagues and others whose association over themany years has helped them to compile this book.

To conclude, the authors hope that the readers will derive benefit from readingthe book and wish them all success in their endeavours.

xiv Power system stability: modelling, analysis and control

Chapter 1

Power system stability overview

1.1 General

A dynamic system, in general, would necessarily entail a detailed study of someconcepts interrelated to each other. Particularly, in system planning these conceptsare system reliability, security and stability. Definitions of these concepts may helpin understanding the relationships and differences between them [1, 2].

System reliability is defined as the probability of a system’s ability to provide adesired function under specific operating conditions during its lifetime.

System security refers to the degree of risk in its ability to withstand con-tingencies without interrupting the system function. It pertains to system robustnessto contingencies. Thus, it depends on the system operating condition and prob-ability of contingency occurrence.

System stability is defined as the ability of the system to continue its intactoperation and remain stable following a disturbance. Consequently, it depends onthe operating condition and the nature of the physical disturbance.

A power system is similar to any dynamic system. Its function is to provideelectricity to loads at a desired quality with as few interruptions as possible. Thepower system is commonly subjected to disturbances during operation. Accordingto the three concepts defined above reliability can be seen as the primary objectivein power system design and operation. To attain system reliability the system mustbe secure most of the time, during and post fault periods. This necessitates that thesystem must be stable. Therefore, the aspects of security and stability are time-varying attributes that can be judged by analysing the power system performanceunder a specified set of conditions. On the other hand, reliability is determined interms of the time average performance of the power system and can be judged bystudying the behaviour of the system over a period of time.

1.2 Understanding power system stability

Synchronous generators in an interconnected power system are the main source ofproducing electrical power. A necessary condition for the transmission and powerexchange is that all generators must rotate in synchronism, that is, the averageelectrical speed of all generators must remain the same anywhere in the system.Each generator is driven by a prime mover. The prime mover applies mechanical

power to the generator that in turn delivers electrical power into the connectedsystem. In a steady-state operation, the input mechanical power to the generatorbalances the output electrical power. Both the input mechanical power and theoutput electrical power produce mechanical and electrical torques, respectively,when applied to the shaft. The mechanical torque is in the direction of rotation,whereas the electrical torque is in a direction opposite of rotation.

If a fault occurs in the system, the output electrical power changes rapidly at arate faster than the input mechanical power. This is because the excitation systemof the generator has a fast response whereas the prime mover controller has arelatively slow response. Accordingly, a temporary imbalance of power existscausing a difference in torque applied to the shaft. This results in a change of rotorspeed (increase or decrease) and, thus, the relative rotor angle changes. The rotorangle d (also called torque or power angle) is the angle between the rotor mmf andthe resultant of the rotor and stator mmfs (Figure 1.1) [3].

If the change of rotor speed continues perhaps beyond the limits of generatorsynchronous operation, the protective relaying system operates to isolate the gen-erator from the rest of the system. Then the remaining system is disturbed due tothe loss of generation. This disturbance may result in additional units trippingoffline, and potentially a cascading outage.

Therefore, the concept of power system stability relates to the ability ofgenerators on a system to maintain synchronism and the tendency to return to asteady-state operation point following a system disturbance [4].

Direction

of

rotation

ω

Roto

r mm

f

Stator mmf

Resultant

d

x

x

x

a

b′

a′

b

c

c′

Figure 1.1 Rotor angle and resultant of stator and rotor mmfs

2 Power system stability: modelling, analysis and control

1.3 Classification of power system stability

Classification of power system stability is based on the type of disturbance. Dis-turbances can be divided into two types: small and large. A small disturbanceaffects the system by small changes in its behaviour, e.g. small changes in the loadand tripping a line carrying insignificant power. The system dynamics can beanalysed using linearised equations, known as small signal analysis. A largedisturbance results in a sudden big change in some of the system parameters. Thesystem dynamics is studied using non-linear equations. For instance, a suddenchange in the load, loss of generation, switching out overloaded transmission lines,symmetrical and unsymmetrical faults and lightning strokes can be consideredlarge disturbances.

Accordingly, the system stability for the purpose of analysis can be classifiedinto two classes: small signal stability and transient stability.

1.3.1 Small signal stabilityThe synchronous machine in an interconnected power system can be simplyrepresented by an internal voltage source, Eg, behind the generator reactance, Xg,which is equal to synchronous reactance, Xd , for steady-state analysis. Moreexplanation of generator reactance and its variation is given in Chapter 2, Part I.The output electrical power, Pe, on a steady-state basis can approximately beexpressed as

Pe ¼ Pmax sin d ¼ EgEt

Xgsin d ð1:1Þ

where Et is the machine terminal voltage and d is the power angle (angle betweenmachine terminal voltage and machine internal voltage). Pmax ¼ (EgEt/Xg), calledsteady-state stability limit, equals the output power at d¼ 90�. Equation (1.1) istypically plotted as shown in Figure 1.2.

The synchronous generator is assumed to be in steady-state operation at point◙ as shown in Figure 1.2, where the input mechanical power, Pm, equals the outputelectrical power, Pe, at power angle do. When a small temporary disturbanceoccurs, e.g. a small reduction in the load, resulting in a decrease of the output powerto Pe1 (point #1) for a short period, the rotor will accelerate as Pm is greater than Pe.Thus, the rotor speed initially increases to absorb the excess energy in the rotorinertia and the increase of angle continues until point #2. From point ◙ to point #2,Pm is less than Pe and the rotor decelerates and tries to overcome the effect ofinertia that vanishes at point #2. Then, due to excess output Pe2 than input Pm therotor decelerates, the power angle decreases and the operating point moves againtowards the point ◙. The oscillation of operating point between points #1 and #2continues. It is to be noted that the rate of power angle change at points #1 and #2 iszero. If the amplitude of oscillation decays with time (damped oscillation) and theoperating point stands at point ◙ or at a neighbouring equilibrium point, systemstability is attained. Conversely, in the case of increasing oscillation excursions

Power system stability overview 3

(un-damped oscillation) a balanced operating point cannot be achieved and thesystem is unstable. The capability of the power system to damp the oscillations isaffected by a number of factors such as the generator design, the strength of themachine interconnection to the network and the setting of the excitation system.Power systems generally have efficient damping of oscillations at normal operatingconditions. Under special circumstances they may experience a significant reduc-tion of damping capability following disturbances, and in the worst circumstancethe damping may become negative. Thus, the oscillations grow and eventually thesynchronism is lost. This form of instability is referred to as small signal instability.Therefore, small signal stability is defined as the ability of the power system toremain stable in the presence of small disturbances.

Loss of small signal stability results in one or more of the types of oscillationsthat have been experienced with large interconnected power system involving rotorswings. Rotor swing may grow without bound or take a long time to dampen. Threemain types of oscillations are dealt with for small signal stability analysis. First,local mode oscillations that involve one or more synchronous generators at a powerplant swinging together against a comparatively large power system or load centre.Their frequency is in the range of 0.7–2 Hz. Second, inter-unit oscillations thatinvolve two or more synchronous generators at a power plant or nearby powerplants swinging against each other with a frequency range of 1.5–3 Hz. Third,inter-area oscillations that usually involve a group of generators on one part of apower system swinging against another group in another part of that system. Thefrequency of this type of oscillations normally is in the range of less than 0.7 Hz.

Damping of generator oscillations has a prominent role in small signal stabilityanalysis. The power system contains inherent damping effects that tend to damp outdynamic oscillations. The natural damping of the system is represented by the

Pe2

Pe1

Pm = Peo

Rotor angle dd1 do d2

2

1

00

Elec

trica

l out

put

pow

er P

e

Figure 1.2 Power–angle curve illustrating machine oscillations


positive term D in the swing (1.2). It is generally sufficient to prevent any sustainedoscillations unless a source of negative damping is introduced.

2H

ws

d2ddt2

þ D

ws

dddt

þ KDd ¼ 0 ð1:2Þ

where

H ¼D rotor inertia constant (MW.s/MVA)ws ¼D synchronous speed (elec.rad/s)D ¼D damping coefficient (pu power/pu freq. change)K ¼D synchronising coefficient (pu DP/rad) ¼ the slope of power – angle curve

at the particular steady-state operating pointDd¼D rotor angle deviation from the steady-state operating point (rad)

A major source of negative damping is the high gain voltage regulator with fastexcitation system. The main function of the voltage regulator is to continuallyadjust the generator excitation level in response to changes in generator terminalvoltage. It acts to accurately maintain a desired generator voltage and change theexcitation level in response to disturbances on the system. It is found thatincreasing forcing capability and decreasing response time of the excitation systemprovide tremendous benefits to transient stability (as explained in Section 1.3.2).On the contrary, it can contribute a significant amount of negative damping tooscillations because it can reduce damping torque. Thus, an excitation system hasthe potential to contribute to small signal instability of power systems. On the otherhand, it is recognised that the normal feedback control actions of voltage regulatorsand speed governors on generating units have the potential of contributing negativedamping that can cause un-damped modes of dynamic oscillations.

Further understanding of both positive and negative effects of high-performancevoltage regulator-excitation systems can be described by the phase relationship ofthe rotor torque components.

The output electrical power of a synchronous generator, Pe, is the product ofelectrical torque, Te, and the angular speed, w. Following a disturbance the changein electrical torque, DTe, can be expressed as a sum of two components: synchro-nising component KsDdð ) in phase with the rotor angle change and dampingcomponent (KDDw) in phase with the speed change.

DTe ¼ KsDdþ KDDw ð1:3Þwhere

Ks ¼D synchronising coefficient (pu DT/rad)Dd¼D change of rotor angle (rad)Dw¼D change of rotor angular speed (elec.rad/s)KD ¼D damping coefficient (pu DT.s/rad)

It can be found that for a positive value of Ks the synchronising torque componentopposes changes in rotor angle from the equilibrium point. This means that an


increase in rotor angle leads to a net decelerating torque resulting in the unit to slowdown relative to the power system. The slowdown continues until the rotor angle isrestored to its equilibrium point and the change of rotor angle vanishes. Similarly,when KD has positive values the damping torque component opposes the change in therotor speed from the initial steady operating point. Therefore, the generator remains ina stable state when sufficient positive synchronising and damping torques are actingon the rotor for all operating conditions. Figures 1.3 and 1.4 depict the relationbetween the two torque components and the corresponding state of the power system.The calculation methods of small signal stability are explained in Chapter 7, Part III.

1.3.2 Transient stabilityTransient stability is concerned with the generator stability for the first swing when alarge disturbance occurs in the system, e.g. transmission line faults. As in (1.3) thechange of electrical torque is resolved into two components acting on each generatorin the system: the synchronising torque and the damping torque. If the synchronisingtorque is insufficient to oppose the change of rotor angle, the generator may lose itssynchronism. This can be treated by developing sufficient magnetic flux that can be

t

Synchronisingaxis

Dam

ping

ax

is

Ks ∆d

KD ∆w ∆Te ∆d

Figure 1.3 Two torque components positive with damped oscillations‘stable state’

KD ∆w

∆Te

Synchronisingaxis

Dam

ping

ax

is

∆dKs

t

∆d

Figure 1.4 Positive synchronising torque and negative damping torquecomponents with un-damped oscillations ‘unstable state’


provided by an excitation system having fast response and sufficient positive andnegative forcing capability to resist acceleration or deceleration of the rotor. Whenthe mechanical torque is higher than the electrical torque, the rotor accelerates withrespect to the stator flux and its angle increases. The exciter system must increaseexcitation by applying as quickly as possible a high positive voltage to the generatorfield. On the other hand, when the mechanical torque is less than the electrical torquethe rotor decelerates and its angle decreases. Thus, the excitation system must rapidlyapply a high negative voltage to the generator field circuit.

Referring to Figure 1.5, a generator connected with a grid is supposed tooperate steadily at the operating point ◙ at which Pm equals Peo. A largedisturbance (transient disturbance) in the transmission network very close to thegenerator will result in a reduction of output electrical power from Peo to zero. Thisreduction leads to the rotor accelerating with respect to the system and increasingthe power angle from do to d1 at which the fault is cleared. The electrical power isrestored to a level corresponding to the appropriate point on the power–angle curveafter the fault (point #1). This curve is lower than that before the fault as the systemmay become weaker, high impedance transmission network, due to the isolation offaulted line. After clearing the fault the electrical power is higher than themechanical power causing the generator to decelerate reducing the momentumthe rotor had gained during the fault period (point #3). If a sufficient retardingtorque exists, the generator moves back towards its operating point, and on the firstswing it will be transiently stable. If the retarding torque is insufficient, the powerangle continues to increase until the generator loses synchronism with the powersystem. Determination of generator stability in this case depends to a large extent

1

2

3

4

Before the fault

After the fault

Insufficient retarding torque;generator will

lose synchronism

Fast excitation system and sufficient retarding torque and generator recovers

for first swing

00

Elec

trica

l out

put p

ower

Pe

Pm = Peo

do d1 Rotor angle d

Figure 1.5 Power–angle curve for transient disturbance


on the time of fault clearance. If the fault is cleared earlier, the probability ofgenerator stability becomes higher.

With the effect of excitation system, it is found that maintaining systemstability depends also on excitation behaviour and response speed. Thus, increasingthe forcing capability and decreasing the response time help in restoring the power–angle curve to that before the fault, i.e. substituting the weakness of transmissionnetwork. In this case points #2 and #4 correspond to points #1 and #3, respectively,and generator stability on the first swing is much improved. More details are pre-sented in Chapters 8 and 9, Part III.

1.4 Need for modelling

As described in Section 1.1, assessment of both security and stability of powersystem is essentially required to avoid catastrophic consequences of system dis-turbances such as blackouts. Accurate assessment of security and stability is basedon accurate modelling of power system components. This is a great challenge andmore attention must be paid to it as the power systems today are becoming morecomplicated along with the increased complexity of operation and control. Modelsof power system components are the basis of methods to analyse. They arecomposed of mathematical relations established from the physical behaviour ofcomponents. For the steady-state analysis, the models mainly are network structureand distribution of generators and loads; while for the dynamic calculations themodels also include the parameters of generators and the dynamic characteristics ofloads besides their static parameters.

For instance, load modelling is difficult due to the random behaviour of theload and the accumulation of large volumes of measurement data. It involves twomajor issues: modelling and parameter identification. The measurement data areused for parameter identification of the composite load on the load bus. Thedynamic part of the composite load is sometimes represented by an inductionmachine. Power system stability is affected by the sum of dynamic motor loadsconnected to the system and the line loading level, so load behaviour differs underdifferent system conditions. This indicates that choosing an accurate inductionmachine model is important for the accuracy of system stability analysis [5].

Methods of power system analysis and simulation are based on a proper designof adequate models of system components for the purpose of a study. They,generally, are in time domain and include the calculation of system behaviour in thepast, present and future [6, 7]. The system analysis for the past time focuses on theanalysis of the historical data, summarising the experience, recognising the dis-turbances occurred and studying the intrinsic characteristics that help improve theoperating conditions. The present time system analysis means real-time calculations[8, 9] that include state estimation to eliminate bad data from the measurementsystem in addition to estimating the un-measured information [10, 11]. Also, theyinclude power flow analysis to calculate the real-time power flow distribution of thesystem. The analysis for future time analyses the supposed system by simulation toprovide decision for system development plan, system operation and emergency


control strategy. All these calculations are important, in general, for power systemstudy and, in particular, for power system stability. Their accuracy can be measuredby the degree of conformation between the results of calculations and the real system.Selecting appropriate models of system components will contribute to a large extentto improving the accuracy.

1.5 Stability margin increase

Operation of power systems at operating conditions close to stability boundary hasincreased the importance of increasing stability margin to maintain system stability,small signal stability and transient stability. Small signal stability can be enhancedby increasing the system damping to damp the oscillations that may happen due tosmall disturbances or following severe disturbances. Unfortunately, the voltageregulator is a major source of negative damping but its use is inevitable. Removingvoltage regulators from service is not a realistic solution of the problem because ofthe need of their beneficial features. The problem of negative damping effect of thevoltage regulator, fortunately, has been solved by providing supplementary controlsto contribute positive damping for oscillatory angle stabilisation. These controls areknown as power system stabilisers (PSSs) (Figure 1.6). More details are given inChapter 11, Part IV.

Increasing the stability limit improves the system stability and increases thestability margin as well. Referring to (1.1) the stability limit (Pmax) can beincreased by modifying the bus voltage or modifying the line reactance. Shuntcompensation can be used for modifying the voltage at the compensator bus whileseries compensation is required to modify the line reactance.

Thyristor switched capacitor, thyristor-controlled reactor, static var compensa-tor and static synchronous compensator can be used for shunt compensation. Also,fixed series capacitor, thyristor-protected series capacitor, static synchronous series

ExciterVoltage regulator Generator

Power system stabiliser (PSS)

Power systemTe

rmin

al b

us

Ref

Vs

∆Vt

∆e

∆f

+

–

∆w

or ∆P

Figure 1.6 A synchronous generator with exciter, voltage regulator and PSS(Vs ¼ output voltage of PSS; Dw¼ change of shaft speed; Df ¼ changeof generator; electrical frequency; DP ¼ change of electrical power;DVt ¼ change of terminal voltage, De ¼ voltage error)


compensator, unified power flow controller, interline power flow controller andinterphase power controller can be used as series compensators. Part IV deals withmore details about the compensation in power systems related to system stability,and in addition, the flexible AC transmission system devices are described.

References

1. Kundur P., Paserba J., Viter S. (eds.). ‘Overview on definition and classifi-cation of power system stability’. Quality and Security of Electric PowerDelivery Systems 2003. CIGRE/PES 2003. CIGRE/IEEE PES InternationalSymposium; Montreal, Canada, Oct 2003. pp. 1–4

2. Kundur P., Paserba J., Ajjarapu V., Andersson G. ‘Definition and classifica-tion of power system stability’. IEEE Transactions on Power Systems. 2004;19(3):1387–401

3. Basler M.J., Schaefer R.C. ‘Understanding power system stability’. IEEETransactions on Industry Applications. 2008;44(2):463–74

4. IEEE Task Force on Power System Stabilizers (eds.). ‘Overview of powersystem stability concepts’. Proceedings of IEEE PES General Meeting; Toronto,Canada, Jul 2003, vol. 3. pp. 1–7

5. Dahal S., Attaviriyanupap P., Kataoka Y., Saha T. (eds.). ‘Effects of inductionmachines dynamics on power system stability’. Power Engineering Conference,AUPEC 2009, Australasian Universities; Adelaide, SA, Sept 2009. pp. 1–6

6. Anjia M., Zhizhong G. (eds.). ‘The influence of model mismatch to powersystem calculation, Part II: On the stability calculation’. Proceedings of PowerEngineering Conference, IPEC 2005, The 7th International; Singapore,Nov/Dec 2005, vol. 2. pp. 1127–32

7. Avramenko V.N. (eds.). ‘Power system stability assessment for current statesof the system’. Power Tech Conference, IEEE; St. Petersburg, Russia, Jun2005. pp. 1–6

8. Fishov A.G., Toutoundaeva D.V. (eds.). ‘Power system stability standardi-zation under present-day conditions’. Strategic Technology, IFOST 2007,International Forum on; Ulaanbaatar, Mongolia, Oct 2007. pp. 411–5

9. Shirai Y., Nitta T. (eds.). ‘On-line evaluation of power system stability by useof SMES’. Proceedings of IEEE Power Engineering Society Winter Meeting;New York, USA, Jan 2002, vol. 2. pp. 900–5

10. Youfang X. (ed.). ‘Measures to ensure the security and stability of the centralChina power system’. Power System Technology, 1998. Proceedings. POW-ERCON ’98. 1998 International Conference on; Beijing, China, Aug 1998,vol. 2. pp. 1374–7

11. Dai Y., Zhao T., Tian Y., Gao L. (eds.). ‘Research on the influence of primaryfrequency control distribution on power system security and stability’.Industrial Electronics and Applications, ICIEA 2007, 2nd IEEE Conferenceon; Harbin, China, May 2007. pp. 222–6


Part I

Modelling

Chapter 2

Modelling of the synchronous machine

2.1 Introduction

The synchronous machine, as one of the very important power system components,must be modelled mathematically in an adequate manner for dynamic and stabilitystudies. Two models based on the state space formulation of the machine equationshave been developed depending on using either the currents or the flux linkages asstate variables [1, 2].

The synchronous machine considered in this chapter has six magneticallycoupled windings: three stator ‘armature’ windings and three rotor windings, ‘onefor the field circuit and two for the damper circuits’ as demonstrated in Figure 2.1.The field circuit and one of the two damper circuits are located on the same axiscalled the direct or d-axis. The second damper circuit is located on an axis laggingthe d-axis by 90� elec. and is called the quadrature or q-axis. The d-axis defines therotor position in space at some instant of time to be at angle q elec. with respect to afixed reference position. In the case of a larger number of damper windings, the

w

d-axis

q-axis

a-axis

b-axis

KD

FKQ

Directionof

rotation

Va

Vb

Vc

ia

ic

ib

θ

Figure 2.1 Schematic representation of a synchronous machine

same methodology of derivation, as derived here for one damper winding on eachof the two axes, can be applied to model the synchronous machine. The modellingprocess is based on considering a uniformly distributed sinusoidal mmf in the airgap and without harmonics. It commences, for simplicity, with neglecting themagnetic saturation that will be represented later.

2.2 Synchronous machine equations

2.2.1 Flux linkage equationsAs depicted in Figure 2.1, the synchronous machine consists of three-phase statorwindings a, b and c and three rotor windings: F denotes the field winding and KDand KQ denote the damper windings. Vectors and matrices are designated by bold,italic symbols. The symbols for the stator are subscripted by ‘s’ and for the rotor by‘r’. Equation for the flux linkage, Y, can be written in matrix form as

Ys

Yr

� �¼ Lss Lsr

Lrs Lrr

� �isir

� �ð2:1Þ

where

Ys ¼Ya

Yb

Yc

264

375 Yr ¼

Yf

Ykd

Ykq

264

375 is ¼

iaibic

264

375 ir ¼

if

ikd

ikq

264

375 Lrs ¼ Lt

sr

Lss ¼Laa Lab Lac

Lba Lbb Lbc

Lca Lcb Lcc

264

375

Ljk ¼ Lkj ≜ mutual inductance between the stator windings if j 6¼ k and is definedas the self-inductance ‘Ljj’ of the jth winding. On the other hand, Lss representing theself and mutual inductances of stator phase windings can be expressed as

Lss ¼Ls Ms Ms

Ms Ls Ms

Ms Ms Ls

264

375þ Lm

�

cos 2q cos 2q� 2p3

� �cos 2qþ 2p

3

� �

cos 2q� 2p3

� �cos 2qþ 2p

3

� �cos 2q

cos 2qþ 2p3

� �cos 2q cos 2q� 2p

3

� �

2666666664

3777777775

ð2:2Þ

Lm, Ls and Ms are constants.


Lsr ¼ matrix of mutual inductances of stator-to-rotor windings

¼Maf Makd Makq

Mbf Mbkd Mbkq

Mcf Mckd Mckq

264

375

¼

Mf cos q Mkd cos q Mkq sinq

Mf cos q� 2p3

� �Mkd cos q� 2p

3

� �Mkq sin q� 2p

3

� �

Mf cos qþ 2p3

� �Mkd cos qþ 2p

3

� �Mkq sin qþ 2p

3

� �

26666664

37777775

ð2:3Þ

Lrr ¼Lf Lfkd 0

Lfkd Lkd 0

0 0 Lkq

264

375 ¼ constant matrix ð2:4Þ

where Lf, Lkd and Lkq are the self-inductances of the field winding, F; dampercircuit in d-axis, KD; and damper circuit in q-axis, KQ, respectively. Lfkd is themutual inductance between windings F and KD.

It can be observed from (2.2) and (2.3) that both Lss (if Lm 6¼ 0Þ and Lsr

are time varying and functions of the rotor position angle q. In (2.4) the matrixelements representing the mutual inductance between the winding KQ and thewinding F or KD are 0 as the angle between them is 90� elec.

2.2.2 Voltage equationsThe voltage equations of stator and rotor windings, considering the generatorconvention as positive, are given by

vs

vr

� �¼ � _Ys

_Yr

� �� Rs 0

0 Rr

� �isir

� �¼ � _Ys

_Yr

� �� R

isir

� �ð2:5Þ

where

vs ¼va

vb

vc

24

35; vr ¼

�vf

0

0

24

35;

R ¼ Rs 0

0 Rr

� �; Rs ¼

Ra 0 0

0 Rb 0

0 0 Rc

24

35; Rr ¼

Rf 0 0

0 Rkd 0

0 0 Rkq

264

375

and a dot on a symbol represents the differential.Usually Ra ¼ Rb ¼ Rc. Hence, Rs ¼ RaU3, where U3 is a 3 � 3 unit matrix.

Modelling of the synchronous machine 15

In the case of defining the neutral voltage contribution to the stator voltages va,vb and vc, the voltage equations become

vs

vr

� �¼ �

_Ys

_Yr

" #� R½ � is

ir

� �þ vn

0

� �ð2:6Þ

where

vn ¼ �Rn

1 1 1

1 1 1

1 1 1

2664

3775

ia

ib

ic

2664

3775� Ln

1 1 1

1 1 1

1 1 1

2664

3775

dia=dt

dib=dt

dic=dt

2664

3775

¼ �Rnis � Lnpis ð2:7Þwhere p is the operator d/dt. Rn and Ln are the resistance and inductance of theneutral to earth, respectively.

It is to be noted that the positive convention is stator currents flowing out of themachine terminals, i.e. the machine operating as a generator.

2.2.3 Torque equationThe rotor equation of motion is given by J€qm þ D _qm ¼ Tm � Te or

2p

J€q þ D _q� � ¼ Tm � Te ¼ Ta ð2:8Þ

where

p ≜ the number of polesD ≜ the damping coefficientQ ≜ the rotor angle in electrical radiansJ ≜ the moment of inertia in kg � m2

Qm ≜ the rotor angle in mechanical radians with respect to a fixed frame ¼(2/p)Q

Tm ≜ the mechanical torque in the direction of rotation in N � mTe ≜ the electrical torque opposing the mechanical torque in N � mTa ≜ the accelerating torque in N � m

Te ¼ � @W

@qm¼ � p

2@W

@q¼ p

2T 0

e ð2:9Þ

T 0e ¼�@W

@q ≜ the electrical torque of the equivalent two pole machine and W is theco-energy expressed as

W ¼ 12

its itr� Lss Lsr

Lrs Lrr

� �isir

� �ð2:10Þ


Therefore; T 0e ¼ � 1

2its

@Lss

@q

� �is þ 2its

@Lsr

@q

� �ir

� �ð2:11Þ

From (2.8) and (2.9), the equation of motion can be rewritten as

2p

� �2

J€q þ D _q� � ¼ 2

pTm � T 0

e ð2:12Þ

It is observed that (2.12) represents the transformation of a p-pole machine to atwo-pole machine. In expressing the equations in per unit ‘pu’ see Appendix I. It isfound that there is no loss of generality in assuming that the machine has two polesas explained in Section 2.6.

2.3 Park’s transformation

Equation (2.5) can be written as

_Y ¼� R½ �i � v ð2:13Þ

where

_Y ¼ _Ys_Yr

� �; i ¼ is

ir

� �¼ L�1Y; v ¼ vs

vr

� �

Also

_Y ¼ ddt

Lið Þ ¼ Ldidt

þ i@L

@qdqdt

ð2:14Þ

Equating the RHS of (2.13) and (2.14) gives

didt

¼ L�1 �Ri � i@L

@qdqdt

� v

�ð2:15Þ

It is difficult to solve (2.15) as the inductances are time varying. Using Park’stransformation [3], the equations are simplified by referring all quantities to a rotorframe of reference constituting time invariant rather than time variant equations.Consequently, this attains the simplification of both steady-state and transientcalculations.

Defining fabc as the voltage or current or flux linkage of the stator windings ina-b-c frame of reference and fdqo as the same quantities in the d-q-o frame ofreference, Park’s transformation ‘P’ is expressed as [3]

fabc ¼ Pfdqo ð2:16Þ


where

f abc ¼fa

fb

fc

24

35; f dqo ¼

fd

fq

fo

24

35

P ¼ffiffiffi23

rcos q sin q

1ffiffiffi2

p

cos q� 2p3

� �sin q� 2p

3

� �1ffiffiffi2

p

cos qþ 2p3

� �sin qþ 2p

3

� �1ffiffiffi2

p

2666666664

3777777775

Thus,

fdqo ¼ P�1fabc ð2:17Þ

where

P�1 ¼ffiffiffi23

r cos q cos q� 2p3

� �cos qþ 2p

3

� �

sin q sin q� 2p3

� �sin qþ 2p

3

� �1ffiffiffi2

p 1ffiffiffi2

p 1ffiffiffi2

p

2666666664

3777777775

2.4 Transformation of synchronous machine equations

2.4.1 Transformation of flux linkage equationsApplying Park’s transformation as in (2.16) to the flux linkages gives

Ys

Yr

� �¼ P 0

0 U3

� �Ydqo

Yr

� �ð2:18Þ

Equation (2.18) can be transformed to become

Ys

Yr

� �¼ Lss Lsr

Lrs Lrr

� �P 00 U3

� �idqo

ir

� �ð2:19Þ

Equating the RHS of (2.18) and (2.19) gives

Ydqo

Yr

� �¼ L0

ss L0sr

L0rs Lrr

� �idqo

ir

� �ð2:20Þ


where

L0ss ¼ P�1LssP ¼

Ld 0 00 Lq 00 0 Lo

24

35 ð2:21Þ

Ld ¼ Ls � Ms þ 32

Lm; Lq ¼ Ls � Ms � 32

Lm; Lo ¼ Ls þ 2Ms

L0sr ¼ P�1Lsr ¼

ffiffiffi32

r Mf Mkd 00 0 Mkq

0 0 0

24

35 ð2:22Þ

L0rs ¼ LrsP ¼

ffiffiffi32

r Mf 0 0Mkd 0 0

0 Mkq 0

24

35 ð2:23Þ

Equations (2.22) and (2.23) show that L0sr ¼ L0t

rs.The matrix Lrr is a constant matrix as in (2.4) representing the inductance

of rotor windings as there is no transformation of rotor currents and fluxlinkages.

From (2.20) it can be concluded that:

● Stator windings a, b and c are replaced by virtual windings d, q and o byapplying the Park’s transformation.

● The ‘o’ winding can be neglected under balanced conditions as there is nocoupling with rotor windings and no flow of zero sequence current, io.

● d and q windings rotate at the same speed as the rotor because the transformedmutual inductance terms between them are constants.

● Mutual inductance between the d-winding and the rotor windings on theq-axis, and the mutual inductance between the q-winding and the rotor wind-ings on the d-axis are zero. Consequently, the d-winding is aligned with thed-axis and q-winding is aligned with the q-axis.

● The time-varying coefficients are removed from the machine equations.

Therefore, the synchronous machine can be represented as shown inFigure 2.2.

2.4.2 Transformation of stator voltage equationsThe stator voltage (2.6) can be rewritten, by applying Park’s transformation,as below:

Pvdqo ¼ � ddt

PYdqo

� �� RsPidqo þ Pvn dqoð Þ ð2:24Þ


The first term on the RHS is derived as

�ddt

PYdqo

� � ¼ � _qdPdq

Ydqo � PdYdqo

dtð2:25Þ

where

dPdq

¼ffiffiffi23

r �sin q cos q 0

�sin q� 2p3

� �cos q� 2p

3

� �0

�sin qþ 2p3

� �cos qþ 2p

3

� �0

2666664

3777775 ¼ PP1 ð2:26Þ

and

P1 ¼0 1 0

�1 0 00 0 0

24

35

Substituting (2.25) and (2.26) into (2.24) gives the following:

Pvdqo ¼ �wPP1Ydqo � PdYdqo

dt� RsPidqo þ Pvn dqoð Þ

Thus,

vdqo ¼ �wP1Ydqo � dYdqo

dt� P�1RsPidqo þ vn dqoð Þ ð2:27Þ

where w ¼ dqdt is the rotor angular speed in elec. rad/s.

The term vn dqoð Þ can be derived as below.

d-axisq-a

xis

KD

FKQ

Directionof

rotation

θ

id

iq

w

Figure 2.2 Synchronous machine representation by transformed windings


Applying Park’s transformation to (2.7):

vn dqoð Þ ¼ �P�1RnPidqo � P�1LnPdidqo

dt¼ �

00

3Rnio

24

35�

00

3Lnio

24

35 ð2:28Þ

The rotor voltage equations are unchanged. Then, the combined voltageequations of stator and rotor can be expressed as

vdqo

vr

� �¼ � _Ydqo

_Yr

� �� wP1Ydqo

0

� �� Rs 0

0 Rr

� �idqo

ir

� �þ vn dqoð Þ

0

� �ð2:29Þ

where

Rs ¼ RaU3.

Equation (2.29) determines the synchronous machine voltages, for both stator androtor, in terms of flux linkages and currents as state variables in the d-q-o frame ofreference. Using the relation between flux linkages and currents given by (2.20),the machine voltages can be expressed in terms of either currents only or fluxlinkages only. The formulation of machine voltages in terms of currents as statevariables, xt ¼ [id, iq, io, if, ikd, ikq] can be derived as below.

Substituting (2.28) into (2.29), the following set of equations in the expandedform is obtained – it is noted that vr is unchanged due to Park’s transformation:

vd ¼ � _Yd � wYq � Raid

vq ¼ � _Yq þ wYd � Raiq

vo ¼ � _Yo � 3Lnpio � 3Rnio

vf ¼ _Yf þ Rf if

vkd ¼ 0 ¼ _Ykd þ Rkdikd

vkq ¼ 0 ¼ _Ykq þ Rkqikq

9>>>>>>>>>>=>>>>>>>>>>;

ð2:30Þ

Equation (2.20) can be written in expanded form to give a set of equations as

Yd ¼ Ldid þ kMf if þ kMkdikd

Yq ¼ Lqiq þ kMkqikq

Yo ¼ Loio

Yf ¼ kMf id þ Lf if þ Lfkdikd

Ykd ¼ kMkdid þ Lkdf if þ Lkdikd

Ykq ¼ kMkqiq þ Lkqikq

9>>>>>>>>>=>>>>>>>>>;

ð2:31Þ

where k ¼ffiffi32

q


From the two sets of (2.30) and (2.31), the machine voltages in terms of cur-rents are given by

Interesting points observed from the set of (2.32) are:

● Stator, i.e. armature, equations except that of vo include the speed voltageterms, wLi and wMi, the speed voltage providing a difference from those of apassive network.

● The speed voltage terms in the d-axis equation are only due to q-axis currentsand the q-axis speed voltages are due to d-axis currents.* The zero-sequence voltage vo depends only on io and its first derivative,

i.e. its equation can be solved separately by knowing the initial conditionson io.

* Under balanced condition, vo ¼ 0, i.e. the row of vo and the correspondingcolumn are omitted. Thus, the set of equations can be depicted schema-tically by the equivalent circuits shown in Figure 2.3.

● It is noted that the self-inductances of stator windings in d and q axes are Ld

and Lq, respectively. The mutual inductance between the stator winding andany of rotor windings, e.g. ith winding, is denoted by the symbol kMi. The self-inductance of the ith rotor winding and the mutual inductance between the ithand jth rotor windings are denoted by Li and Lij, respectively. The equivalentcircuits in d and q axes include the speed voltage terms as controlled sources.

● Regardless of the angular velocity, w, all other terms in the coefficient matricesare constants, i.e. not time varying unlike the coefficients in (2.5) in a-b-cframe of reference. Under the situation of the variation of w with time, (2.32)becomes non-linear and the formulation of state space equations is in the form:

_x ¼ f x; u; tð Þ ð2:33Þ


where f is a set of non-linear functions, x a vector of the state variables and uthe system-driving functions.

On the other hand, if w is assumed to be a constant ‘it is an accepted approx-imation in the steady state’; (2.32) is a linear time invariant and the formulation ofstate space equations as a set of first-order differential equations is in the form

_x ¼ Axþ Bu ð2:34ÞIn the steady state, if the synchronous generator is unloaded all currents

(id, iq, ikd, ikq) are zero except the field current if(NL) ¼ vf(NL)/Rf, ‘the subscript‘‘NL’’ denotes the value at unloaded conditions’. Therefore, from (2.31) the fluxlinkages are

Yd NLð Þ ¼ kMf vf NLð Þ=Rf ; Yq NLð Þ ¼ Yo NLð Þ ¼ 0;

Yf NLð Þ ¼ Lf vf NLð Þ=Rf ; Ykd NLð Þ ¼ Lkdf vf NLð Þ=Rf ; Ykq NLð Þ ¼ 0ð2:35Þ

Similarly, from (2.32), the stator voltages in d-q frame of reference are

vd NLð Þ ¼ 0 and vq NLð Þ ¼ wokMf vf NLð Þ=Rf ð2:36Þ

Rf

Rkd

ikq

ikd

if

iq

vq

ωΨd

+

+

+

+

+

+

+

–

–

–

–

– –

–

vf

vkd = 0

Lkq kMkq

Lfkd

Lq

Lf

Lkd

Rkq

vkq = 0

Ra

ωΨq

vd

Ra id

Ld

kMkd

kMf

Figure 2.3 d–q equivalent circuit of synchronous generator


Consequently, the machine terminal voltages at no-load conditions equal theinduced voltages in stator windings that are given from (2.16), considering vo ¼ io ¼ 0and q¼wot þ d ‘wo is the rated angular speed’ as

va ¼ffiffiffi23

rvd NLð Þcos wot þ dð Þ þ vq NLð Þsin wot þ dð Þ�

¼ffiffiffi23

rvq NLð Þsin wot þ dð Þ

vb ¼ffiffiffi23

rvd NLð Þcos wot þ d� 2p

3

� �þ vq NLð Þsin wot þ d� 2p

3

� ��

¼ffiffiffi23

rvq NLð Þsin wot þ d� 2p

3

� �

vc ¼ffiffiffi23

rvd NLð Þcos wot þ dþ 2p

3

� �þ vq NLð Þsin wot þ dþ 2p

3

� ��

¼ffiffiffi23

rvq NLð Þsin wot þ dþ 2p

3

� �

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

ð2:37Þ

Under no-load conditions ‘d¼ 0’. Therefore, d in the machine terminalvoltages (2.37) should be equal to zero.

For loaded generator, the synchronous machine delivers electric power deter-mined by the prime mover output. Thus, the currents and flux linkages in themachine are functions of the mechanical torque ‘Tm’ and the rms value ofthe sinusoidal line-to-line terminal voltage V.

Therefore, the voltage sources at generator terminals can be assumed as

va ¼ffiffiffi23

rV sin wotð Þ

vb ¼ffiffiffi23

rV sin wot � 2p

3

� �

vc ¼ffiffiffi23

rV sin wot þ 2p

3

� �

9>>>>>>>>>=>>>>>>>>>;

ð2:38Þ

and the d-q axis components of terminal voltages can be calculated as

vdo ¼ �V sin d; vqo ¼ V cos d ð2:39ÞThe subscript ‘o’ denotes the rated value. Neglecting the armature resistance

and considering no currents flowing in the damper windings in the steady state, it isobserved from (2.30) that

vdo ¼ �woYqo; vqo ¼ woYdo ð2:40Þ


Under the same conditions, from (2.32), (2.39) and (2.40), it is found that

vdo ¼ �woLqiqo i:e: iqo ¼ �vdo= woLq

� �and

vqo ¼ woLdido þ wokMf ifo i:e: ido ¼ 1=woLdð Þ vqo � wokMf vfo=Rf

� ð2:41Þ

2.4.3 Transformation of the torque equationApplying Park’s transformation to (2.11), the transformed electrical torque is

T 0e ¼ � 1

2itdqoPt @Lss

@q

� �Pidqo þ 2itdqoPt @Lsr

@q

� �ir

� �ð2:42Þ

where

@Lss

@q¼ �2Lm

sin 2q sin 2q� 2p3

� �sin 2qþ 2p

3

� �

sin 2q� 2p3

� �sin 2qþ 2p

3

� �sin 2q

sin 2qþ 2p3

� �sin 2q sin 2q� 2p

3

� �

2666666664

3777777775

ð2:43Þ

@Lsr

@q¼

�Mf sin q �Mkd sin q Mkq cos q

�Mf sin q� 2p3

� ��Mkd sin q� 2p

3

� �Mkq cos q� 2p

3

� �

�Mf sin 2qþ 2p3

� ��Mkd sin 2qþ 2p

3

� �Mkq cos qþ 2p

3

� �

26666664

37777775

ð2:44Þ

Assuming

P2 ¼0 1 0

1 0 0

0 0 0

24

35 ð2:45Þ

and substituting (2.43), (2.44) and (2.45) into (2.42), it can be proved that theelectrical torque is given by

T 0e ¼

ffiffiffi32

riq Mf if þ Mkdikd þ

ffiffiffi32

rLmid

!�

ffiffiffi32

rid Mkqikq �

ffiffiffi32

rLmiq

!ð2:46Þ


From (2.20), it can be observed that

Yd ¼ Ldid þffiffiffi32

rMf if þ

ffiffiffi32

rMkdikd and Yq ¼ Lqiq þ

ffiffiffi32

rMkqikq ð2:47Þ

From the definition of Ld and Lq in (2.21), it is found that

Ld � 32

Lm ¼ Lq þ 32

Lm ¼ Ls � Ms ð2:48Þ

From (2.46), (2.47) and (2.48), the electrical torque can be expressed as

T 0e ¼ iqYd � idYq ð2:49Þ

In the steady-state and unloaded generator, the electrical torque is zero asthe currents are zero. For loaded generator, the electrical torque in (2.49) can bewritten as

T 0eo ¼ iqoYdo � idoYqo ð2:50Þ

Substituting (2.39), (2.40) and (2.41) into (2.50), the electrical torque becomes

T 0eo ¼ Efdo

woxdV sin dþ xd � xq

� �2woxdxq

V 2 sin 2d ð2:51Þ

where

xd ¼ woLd ; xq ¼ woLq; Efdo ¼ xfdovfo=Rf ; xdfo ¼ wokMf

2.5 Machine parameters in per unit values

All machine variables in the preceding sections, such as voltage, current, power,flux linkage, inductance, are mainly based on three quantities: volt (V), ampere (A)and time ‘t’ (s). The difficulty that the power engineers meet is that the quantities inphysical units pertaining to the machine stator are in a much higher range thanthose for the machine rotor, e.g. the stator voltage may be in kilovolts and the fieldvoltage of a much smaller value. Consequently, their magnitudes are very differentinvolving inadequacy for engineering use.

Therefore, the equations used to calculate the machine variables are normal-ised by applying a convenient base to obviate this problem and the quantities arethen expressed in percent of the base (per unit ‘pu’ values) [4].

Fixed base quantities must be chosen in such a way that all three quantities V,A and t are involved. Further information on per unit/normalised form is given inAppendix I.


Example 2.1 A two-pole, three-phase synchronous generator has the data givenbelow. Find the per unit values of the machine parameters that can be calculatedfrom these data.

Frequency ¼ 60 Hz, line-to-line voltage ¼ 24 kV, rating ¼ 555 MV, powerfactor ¼ 0.9, Ls ¼ 3.2758 mH, Lm ¼ 0.0458 mH, Ms ¼�1.6379, Mf ¼ 32.653 mH

The stator leakage inductances ‘d ¼ ‘q ≜ ‘a ¼ 0.4129 mHLf ¼ 576.92 mH

Solution:

From (2.20), Ld and Lq are defined as

Ld ¼ Ls � Ms þ 32

Lm; Lq ¼ Ls � Ms � 32

Lm

Hence,

Ld ¼ 3:2758 þ 1:6379 þ 32� 0:0458 ¼ 4:9824 mH

Lq ¼ 3:2758 þ 1:6379 � 32� 0:0458 ¼ 4:845 mH

As in (I.7) and (I.9) of Appendix I, Lmd and Lmq can be calculated by

Lmd ¼ Ld � ‘a ¼ 4:9824 � 0:4129 ¼ 4:5695 mH

Lmq ¼ Lq � ‘a ¼ 4:845 � 0:4129 ¼ 4:4321 mH

kMf ¼ffiffiffi32

r� 32:653 ¼ 40:0 mH

Base quantities:

i. For the stator:

SB ¼ three-phase rating ¼ 555 MVAVB ¼ line-to-line voltage ¼ 24 kVwB ¼ 120p¼ 377 elec. rad/stB ¼ (1/wB) ¼ 2.65258 � 10�3 s

IB ¼ 55524

� 103 ¼ 23:125 kA

ZB ¼ VB

IB¼ 24

23:125¼ 1:0378 W

YB ¼ VB

wB¼ 24 � 103

377¼ 63:66 Wb turn

LB ¼ YB

IB¼ ZB

wB¼ 1:0378

377¼ 2:75 mH


ii. For the rotor:

As in (I.6) and (I.7) of Appendix I, it can be found that

IfB ¼ Lmd

kMfIB ¼ 4:5695

40:0� 23:125 ¼ 2:64 kA

MfB ¼ kMf

LmdLB ¼ 40:0

4:5695� 2:75 � 10�3 ¼ 24:07 mH

VfB ¼ SB

IfB¼ 555 � 106

2:64 � 103 ¼ 210:23 kV

ZfB ¼ 210:232:64

¼ 79:6325 W

LfB ¼ ZfB

wB¼ 79:6325

377� 103 ¼ 211:227 mH

Per unit values of machine parameters:Applying the rule: per unit value ¼ actual value

base value ; the parameters in pu areobtained as below:

Ld ¼ 4.9824/2.75 ¼ 1.81Lf ¼ 576.92/211.227 ¼ 2.73‘d ¼ ‘q ¼ ‘a ¼ 0.4129/2.75 ¼ 0.15Lq ¼ 4.845/2.75 ¼ 1.76

Lmd ¼ 4.5695/2.75 ¼ 1.66Lmq ¼ 4.4321/2.75 ¼ 1.61kMf ¼ kMkd ¼ Ld � ‘d ¼ 1.81 � 0.15 ¼ 1.66

Ra ¼ 0.0031/1.0378 ¼ 2.99 � 10�3

Rf ¼ 0.0715/79.6325 ¼ 0.898 � 10�3

Example 2.2 For the machine in Example 2.1 considering the following para-meters ‘in pu values’ calculate the coefficient matrices to obtain the vector di/dt interms of the vectors i and v.

kMkq ¼ 1.59, Lkd ¼ 0.1713, Lkq ¼ 0.7252, Rkd ¼ 0.0284, Rkq ¼ 0.00619 andassuming kMf ¼ Lfkd ¼ 1.66


Solution:

Equation I.23 derived in Appendix I is written below:

vd

�vf

0

vq

0

266666664

377777775¼ �

Ra 0 0 wLq wkMkq

0 Rf 0 0 0

0 0 Rkd 0 0

�wLd �wkMf �wkMkd Ra 0

0 0 0 0 Rkq

266666664

377777775

id

if

ikd

iq

ikq

266666664

377777775

�

Ld kMf kMkd 0 0

kMf Lf Lfkd 0 0

kMkd Lfkd Lkd 0 0

0 0 0 Lq kMkq

0 0 0 kMkq Lkq

266666664

377777775

pid

pif

pikd

piq

pikq

266666664

377777775

ð2:52Þ

Therefore, substituting the pu machine parameters calculated in Example 2.1as well as the parameters given above, (2.52) becomes

vd

�vf

0

vq

0

266666664

377777775¼ �

0:00299 0 0 1:7618w 1:59w

0 0:00898 0 0 0

0 0 0:0284 0 0

�1:812w �1:66w �1:66w 0:00299 0

0 0 0 0 0:00619

266666664

377777775

id

if

ikd

iq

ikq

266666664

377777775

�

1:812 1:66 1:66 0 0

1:66 2:73 1:66 0 0

1:66 1:66 0:1713 0 0

0 0 0 1:7618 1:59

0 0 0 1:59 0:7252

266666664

377777775

pid

pif

pikd

piq

pikq

266666664

377777775

Thus, the vector di/dt can be written as

didt

¼ �B�12 B1i � B�1

2 v


where

B1 ¼

0:00299 0 0 1:7618w 1:59w0 0:00898 0 0 0

0 0 0:0284 0 0

�1:812w �1:66w �1:66w 0:00299 0

0 0 0 0 0:00619

26666664

37777775

B2 ¼

1:812 1:66 1:66 0 0

1:66 2:73 1:66 0 0

1:66 1:66 0:1713 0 0

0 0 0 1:7618 1:59

0 0 0 1:59 0:7252

26666664

37777775

Hence,

B�12 ¼

0:763 �0:825 0:594 0 0

�0:825 0:817 0:084 0 0

0:594 0:084 �0:732 0 0

0 0 0 �0:581 1:272

0 0 0 1:274 �1:41

26666664

37777775

and

B�12 B1 ¼ 10�3

2:289 7:425 16:632 1344:4w 1487:8w2:475 7:353 2:352 �1453:6w �1311:7w1:782 0:765 �0:020 1046:6w 944:5w

1052:8w 964:5w 964:5w �1:743 7:632

�2308:5w �2114:8w �2114:8w 3:822 �8:460

26666664

37777775

Based on per unit system and normalising voltage equations as explained inAppendix I, the torque, power and swing equations can be derived as below.

2.5.1 Torque and power equationsAccording to (2.8), J€qm ¼ Ta when the damping term is neglected. It is convenientto express qm as qm ¼ (wot þ a) þ dm as the angular reference may be chosenrelative to a synchronously rotating reference frame moving with constant velocitywo, and a ‘a constant’ expresses the angle between the rotor position and theangular reference frame, while dm is the mechanical torque angle in radians. Theelectrical (torque) angle d¼ (p/2)dm. Thus, this equation may be written as

J€dm ¼ J _wm ¼ Ta or 2J=pð Þ€d ¼ 2J=pð Þ�w ¼ Ta or 2=pð ÞM _w ¼ Pa ð2:53Þwhere M is the angular momentum ¼ Jw


The base quantity of the torque ‘TB’ is equal to the rated torque at ratedspeed, i.e.

TB ¼ SB=wmo ¼ 60SB=2pno ð2:54Þwhere SB is the three-phase stator-rated power (VA rms) and no is the rated shaftspeed in rpm.

Thus, dividing (I.24) by (I.25) and substituting p ¼ 120fo/no gives the puquantity of torque, Tau, as

Ta=TB ¼ Jp2n2o

900SBwo_w ¼ Tau pu ð2:55Þ

The quantity H ‘called the inertia constant’ is defined as the ratio ofkinetic energy in megajoules to the rating in MVA, i.e.H ¼ kinetic energy in megajoules

Rating in MVA ¼ Jw2m

� �= 2SBð Þ and has the dimension of time in sec-

onds. Consequently,

Ta=TB ¼ J _wm

SB

wmo

¼ 2Hwmo

_wm ¼ 2H=wBð Þ _w ¼ Tau pu ð2:56Þ

where w is commonly given in the units of electrical rad/s as it is the angularvelocity of revolving magnetic field in the air gap and, therefore, pertains directlyto the network voltages and currents. The value of rated angular speed ‘wo’ is takenas the base quantity ‘wB’. The form of (2.56) is called the swing equation and hasbeen adapted for machines with any number of poles as all machines are incorpo-rated in the same system and synchronised to the same rated angular velocity.

The swing equation may be written in another approximate form that is con-venient for use with the classical model of the synchronous machine. This form isbased on considering that the angular speed is nearly constant, which in turn yieldsthat the numerical value of the pu accelerating torque is nearly equal to the accel-erating power Pa. Thus, the approximated form becomes

2H=wBð Þ _w � Pau pu ð2:57ÞReferring to the definition of basic pu quantities t ¼ tutB, w¼wuwB and

wB ¼ 1/tB, the following relations may be written as

1dt

¼ wB1

dtuand dw ¼ wBdwu ð2:58Þ

Based on which of the terms in (2.56) are given in pu, different forms of swingequation are obtained by incorporating (2.58) as below:

● T in pu, t in second, w in electrical rad/s

2H=wBð Þ dwdt

¼ Tau ð2:59Þ


If w is given in electrical degree/s, it should be multiplied by (p/180) and theswing equation becomes

H

180fB

dwdt

¼ Tau ð2:60Þ

● T and t are in pu and w is in electrical rad/s

2Hdwdtu

¼ Tau ð2:61Þ

If w is given in electrical degree/s, then the swing equation is in the form

pH

90dwdtu

¼ Tau ð2:62Þ

● T, t and w are in pu

2HwBð Þ dwu

dtu¼ Tau ð2:63Þ

2.6 Synchronous machine equivalent circuits

The synchronous machine can be represented by two equivalent circuits: one cor-responding to the d-axis and the other corresponding to the q-axis. The normalisedflux linkages under balanced conditions ‘Yo is omitted’ given in (2.20) can berewritten as

Yd ¼ Ld � ‘að Þ þ ‘a½ �id þ kMf if þ kMkdikd

Yf ¼ kMf id þ Lf � ‘f

� �þ ‘f

� if þ Lfkdikd

Ykd ¼ kMkdid þ Lfkdif þ Lkd � ‘kdð Þ þ ‘kd½ �ikd

9>=>; ð2:64Þ

where ‘a; ‘f and ‘kd are the leakage inductances of the coupled circuits on thed-axis, armature d-circuit, field circuit ‘f ’ and damper circuit ‘KD’, respectively.

Also,

Yq ¼ Lq � ‘a

� �þ ‘a

� iq þ kMkqikq

Ykq ¼ kMkqikq þ Lkq � ‘kq

� �þ ‘kq

� ikq

)ð2:65Þ

where ‘a and ‘kq are the leakage inductances of the coupled circuits on the q-axis,armature q-circuit and damper circuit ‘KQ’, respectively.

For the d-axis: If if ¼ ikd ¼ 0, the d-axis flux linkage mutually coupled to theother circuits is (Ld � ‘a)id or Lmdid. In this case, the flux linkages in the f and KDwindings are given by Yf ¼ kMf id and Ykd ¼ kMkdid , respectively.


As the choice of the base rotor current is based on giving equal mutual flux, thepu values of Lmdid, Yf and Ykd must be equal. Then,

Ld � ‘a ¼ kMf ¼ kMkd ¼ Lmd pu ð2:66ÞIt can be proved that

Ld � ‘a ¼ Lf � ‘f ¼ Lkd � ‘kd ¼ kMf ¼ kMkd ≜ Lmd pu ð2:67ÞSubtracting the pu leakage flux linkage in each circuit results in the equality of

the remaining flux linkages for all other coupled circuits. Therefore,

Yd � ‘aid ¼ Yf � ‘f if ¼ Ykd � ‘kdikd ≜YAd pu ð2:68Þand the pu d-axis mutual flux linkage, YAd , is given by

YAd ¼ Ld � ‘að Þid þ kMf if þ kMkdikd ¼ Lmd id þ if þ ikd

� � ð2:69ÞFrom (2.30) and (2.64), the voltage equations are

vd ¼ � _Yd � wYq � Raid

¼ �‘apid � Lmdpid þ kMf pif þ kMkdpikd

�� Raia � wYq

Thus,

vd ¼ �‘apid � Lmd pid þ pif þ pikd

� �� Raia � wYq ð2:70ÞSimilarly,

�vf ¼ �‘f pif � Lmdðpid þ pif þ pikdÞ � Rf if ð2:71Þvkd ¼ 0 ¼ �‘kdpikd � Lmdðpid þ pif þ pikdÞ � Rkdikd ð2:72Þ

Applying the procedure used in developing the equivalent circuit of transfor-mers to represent the relations given in (2.66) through (2.69) and to satisfy thevoltage equations (2.70) through (2.72) the equivalent circuit for the d-axis isshown in Figure 2.4. The d-axis circuits (D, F and KD) are coupled through the

vdvf

ωΨq+

+

+–

––

id + if + ikd

idif

ikd

Lmd

Rf

Rkd

Raℓf ℓa

ℓkd

Figure 2.4 Equivalent circuit for the d-axis


common magnetising inductance Lmd ¼ (Ld � ‘a) that carries the sum of currents id,if and ikd. The equivalent circuit contains a controlled voltage source wYq.

For the q-axis, following the same procedure as used above, the pu q-axismutual flux linkage, Yaq, and voltage equations are given by

YAq ¼ Lmqiq þ kMkqikq ¼ Lmq iq þ ikq

� � ð2:73Þ

vq ¼ �‘apiq � Lmq piq þ pikq

� �� Raiq þ wYd ð2:74Þ

vkq ¼ 0 ¼ �‘kqpikq � Lmq piq þ pikq

� �� Rkqikq ð2:75Þwhere Lmq is defined as

Lmq ¼ Lq � ‘a ¼ Lkq � ‘kq ¼ kMkq pu

The equivalent circuit for the q-axis satisfying these relations is shown inFigure 2.5. It is to be noted that it contains a controlled voltage source wYd .

2.7 Flux linkage state space model

2.7.1 Modelling without saturationThe relations in Section 2.6 can be used to develop an alternative state space modelbased on choosing Yd, Yf, Ykd, Ykq and Yq as state variables. From (2.68) thed-axis currents are given by

id ¼ 1‘a

Yd �YAdð Þ; if ¼ 1‘f

Yf �YAd

� �; ikd ¼ 1

‘kdYkd �YAdð Þ ð2:76Þ

Incorporating (2.69), where YAd ¼ Lmd(id þ if þ ikd) into (2.76) gives

YAd1

Lmdþ 1‘a

þ 1‘f

þ 1‘kd

� �¼ Yd

‘aþYf

‘fþYkd

‘kdð2:77Þ

vkq = 0

+

+

+–

––

ikq Rkq ℓkq ℓa Ra iq

iq + ikq

Lmq vq

ωΨd

Figure 2.5 Equivalent circuit for the q-axis


If LMd is defined as

1LMd

≜1

Lmdþ 1‘a

þ 1‘f

þ 1‘kd

then

YAd ¼ LMd

‘aYd þ LMd

‘fYf þ LMd

‘kdYkd ð2:78Þ

Similarly, it can be found that

YAq ¼ LMq

‘aYq þ LMq

‘kqYkq ð2:79Þ

where LMq is defined as

1LMq

≜1

Lmqþ 1‘a

þ 1‘kq

ð2:80Þ

and the q-axis currents are given by

iq ¼ 1‘a

Yq �YAq

� �; ikq ¼ 1

‘kqYkq �YAq

� � ð2:81Þ

Equations (2.76) and (2.81) can be rewritten in the matrix form as

Ψd

Ψf

Ψkd

ΨAd

Ψq

Ψkq

ΨAq


Incorporate the currents in (2.82) into voltage equations (2.30) to get thederivatives of flux linkages as

_Yd ¼ �Ra

‘aYd þ Ra

‘aYAd � wYq � vd

_Yf ¼ �Rf

‘fYf þ Rf

‘fYAd � �vf

� �_Ykd ¼ �Rkd

‘kdYkd þ Rkd

‘kdYAd

_Yq ¼ �Ra

‘aYq þ Ra

‘aYAq � wYd � vq

_Ykq ¼ �Rkq

‘kqYg þ Rkq

‘kqYAq

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

ð2:83Þ

Also, substituting the currents in (2.82) in (2.49), where the electrical torqueTe ¼ iqYd � idYq, gives

Te ¼ �YqYd �YAd

‘a

� �þYd

Yq �YAq

‘a

� �

Thus,

Te ¼ � 1‘aYdYAq þ 1

‘aYqYAd þ 1

‘a� 1‘a

� �YdYq ð2:84Þ

Considering the electrical torque, time and angular speed in pu values, theswing equation, Section 2.5.2, is

2HwBð Þ dwu

dtu¼ Tau ð2:85Þ

Substituting (2.73) in (2.74) and ignoring the damping coefficient to give _w as

_w ¼ 12HwB

Tm �YAd

‘aYq þYAq

‘aYd

� �ð2:86Þ

If the damping is considered, the term (�D/2HwB)w is added to the RHS.The equation of electrical torque angle in pu is given by

_d ¼ w� 1 ð2:87ÞTherefore, (2.83), (2.86) and (2.87) are in state space form: _x ¼ f ðx; u; tÞ,

where x≜ the state variable ¼ Yd ;Yf ;Ykd ;Yq;Ykq;w; d�

and u≜ the forcing func-tion is ½vd ; vq, vf �, and Tm. Equations (2.67) and (2.68) are used to calculate YAd andYAq. It is to be noted that this form of equations is adequate for the analysis whensaturation is considered, as the terms YAd and YAq are affected by saturation.

If the saturation is neglected, the terms Lmd and Lmq are constant. This impliesthat LMd and LMq are constant as well. Consequently, the relationships of the


magnetising flux linkages YAd and YAq to the state variables (2.78) and (2.79) areconstant and can be omitted from the machine equations. The d-axis currents in(2.76) can be rewritten by substituting the value of YAd given in (2.78) as

id ¼ 1 � LMd

‘a

� �Yd

‘a� LMd

‘a

Yf

‘f� LMd

‘a

Ykd

‘kd

if ¼ �LMd

‘f

Yd

‘aþ 1 � LMd

‘f

� �Yf

‘f� LMd

‘f

Ykd

‘kd

ikd ¼ � LMd

‘kd

Yd

‘a� LMd

‘kd

Yf

‘fþ 1 � LMd

‘kd

� �Ykd

‘kd

9>>>>>>>>>>=>>>>>>>>>>;

ð2:88Þ

Applying the same procedure using (2.79) and (2.85) to obtain the q-axis cur-rents and then substituting all currents in d and q axes in voltage equation (2.30) gives

_Yd ¼ �Ra 1 � LMd

‘a

� �Yd

‘aþ Ra

LMd

‘a

Yf

‘fþ Ra

LMd

‘a

Ykd

‘kd� wYq � vd

_Yf ¼ RfLMd

‘f

Yd

‘a� Rf 1 � LMd

‘f

� �Yf

‘fþ Rf

LMd

‘f

Ykd

‘kdþ vf

_Ykd ¼ RkdLMd

‘kd

Yd

‘aþ Rkd

LMd

‘kd

Yf

‘fRkd 1 � LMd

‘kd

� �Ykd

‘kd

_Yq ¼ �Ra 1 � LMq

‘a

� �Yq

‘aþ Ra

LMq

‘a

Ykq

‘kqþ wYd � vq

_Ykq ¼ RkqLMq

‘kq

Yq

‘a� Rkq 1 � LMq

‘kq

� �Ykq

‘kq

9>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>;

ð2:89ÞUsing (2.78), (2.79) and (2.84) gives the electrical torque as

Te ¼ YdYqLMd � LMq

‘2a

!�YdYkq

LMq

‘a‘kqþYqYf

LMd

‘a‘fþYqYkd

LMd

‘a‘kdð2:90Þ

Incorporating (2.90) into (2.85):

_w ¼ 12HwB

� YdYqLMd � Lmq

‘2a

!�YdYkq

LMq

‘a‘kqþYqYf

LMd

‘a‘fþYaYkd

LMd

‘a‘kd

" #( )

ð2:91ÞAgain, the equation of electrical torque angle in pu is given by

_d ¼ w� 1 ð2:92Þ


Therefore, (2.89), (2.91) and (2.92) form the state space model describing thesystem in terms of vd and vq that are functions of the currents demanded by theexternal loads. In matrix form the state space model can be written as

_Yd

_Y f

_Ykd

_Yq

_Ykq

_w_d

2666666666666664

3777777777777775

¼

�Ra

‘a1 � LMd

‘a

� �Ra

‘a

LMd

‘f

Ra

‘a

LMd

‘kd�w 0 0 0

Rf

‘f

LMd

‘a�Rf

‘f1 � LMd

‘f

� �Rf

‘f

LMd

‘kd0 0 0 0

Rkd

‘kd

LMd

‘a

Rkd

‘kd

LMd

‘f�Rkd

‘kd1 � LMd

‘kd

� �0 0 0 0

w 0 0 �Ra

‘a1 � LMq

‘a

� �Ra

‘q

LMq

‘kq0 0

0 0 0Rkq

‘kq

LMq

‘a�Rkq

‘kq1 � LMq

‘kq

� �0 0

� LMd

2HwB‘2a

Yq � LMd

2HwB‘a‘fYq � LMd

2HwB‘a‘kdYq

LMq

2HwB‘2a

YdLMq

2HwB‘a‘kqYd �Đ 0

0 0 0 0 0 1 0

26666666666666666666666666666666664

37777777777777777777777777777777775

�

Yd

Yf

Ykd

Yq

Ykq

w

d

2666666666666664

3777777777777775

þ

�vd

vf

0

�vq

0

Tm

2HwB

�1

26666666666666664

37777777777777775

ð2:93Þ

where Đ¼ D/(2HwB)

Example 2.3 Write the coefficient matrix of the flux linkage model (2.93) forthe machine data given in Examples 2.1 and 2.2 considering the inertia constantH ¼ 3.5 s, wo ¼ 3600 rpm and the damping coefficient is neglected.

Solution:

‘f ¼ Lf � kMf ¼ 2:73 � 1:66 ¼ 1:07; ‘kd ¼ Ld � kMkd ¼ 1:81 � 1:66 ¼ 0:15

Lmd ¼ kMkd ¼ kMf ¼ 1:66; ‘kq ¼ Lq � kMkq ¼ 1:76 � 1:59 ¼ 0:17

Lmq ¼ Lq � ‘a ¼ 1:76 � 0:15 ¼ 1:61

1LMd

¼ 1Lmd

þ 1‘a

þ 1‘f

þ 1‘kd

¼ 11:66

þ 10:15

þ 11:07

þ 10:15

¼ 14:87

LMd ¼ 0:067

(2.93)


1LMq

¼ 1Lmq

þ 1‘a

þ 1‘kq

¼ 11:61

þ 10:15

þ 10:17

¼ 13:17

LMq ¼ 0:076

Ra

‘d1 � LMd

‘a

� �¼ 2:99

0:15� 10�3 1 � 0:067

0:15

� �¼ 10:96 � 10�3

Ra

‘a

LMd

‘f¼ 19:93 � 10�3 � 0:0626 ¼ 1:25 � 10�3

Ra

‘a

LMd

‘kd¼ 19:93 � 10�3 � 0:45 ¼ 8:90 � 10�3

Rf

‘f

LMd

‘a¼ 0:898 � 10�3 � 0:067� �

= 1:07 � 0:15ð Þ ¼ 0:375 � 10�3

Rf

‘f1 � LMd

‘f

� �¼ 0:786 � 10�3

Rf

‘f

LMd

‘kd¼ 0:375 � 10�3

Rkd

‘kd

LMd

‘a¼ 28:4 � 10�3 � 0:067� �

= 0:15 � 0:15ð Þ ¼ 84:57 � 10�3

Rkd

‘kd

LMd

‘f¼ 28:4 � 10�3 � 0:067� �

= 0:15 � 1:07ð Þ ¼ 11:86 � 10�3

Rkd

‘kd1 � LMd

‘kd

� �¼ 104:13 � 10�3

Ra

‘a1 � LMq

‘a

� �¼ 9:827 � 10�3

Ra

‘a

LMq

‘kq¼ 8:911 � 10�3;

Rkq

‘kq

LMq

‘a¼ 18:45 � 10�3

Rkq

‘kq1 � LMq

‘kq

� �¼ 20:135 � 10�3

LMd

2HwB‘2a

¼ 1:13 � 10�3 LMd

2HwB‘a‘f¼ 0:158 � 10�3

LMd

2HwB‘a‘kd¼ 1:132 � 10�3 LMq

2HwB‘2a

¼ 1:283 � 10�3

LMq

2HwB‘a‘kq¼ 1:132 � 10�3

w ¼ 2pP

2rpm60

¼ 120p ¼ wB and then the per unit value of w equals 1:


Thus, the coefficient matrix is

�10:96 1:25 8:90 �1 0 0 0

0:375 �0:786 0:375 0 0 0 0

84:57 11:86 �104:13 0 0 0 0

1 0 0 �9:827 8:911 0 0

0 0 0 18:45 �20:135 0 0

�1:13Yq �0:158Yq �1:132Yq 1:283Yd 1:132Yd 0 0

0 0 0 0 0 1 0

26666666666666666664

37777777777777777775

� 10�3

2.7.2 Modelling with saturationThe stator and rotor of the synchronous machine exhibit saturation because theycontain magnetic materials. Equations (2.83) through (2.87) form the flux linkagestate space model that can be used in the case of considering the effect of satura-tion. This implies that the machine mutual inductances Lmd and Lmq are not constantand depend on the levels of magnetising flux linkages YAd and YAq that have non-linear characteristics. So, the terms Lmd and Lmq should be corrected for saturation.The exact analysis of saturation may present some difficulty in parameter identi-fication [5–7] and may require finite element analysis [8].

Several approaches have been proposed for modelling the saturated machinesuch as (i) representing the saturation function as an arctangent function in termsof the initial constant slope and the final constant slope of saturation character-istics [9, 10], (ii) by a polynomial series or (iii) incorporating the saturationcharacteristics as a lookup table function. Other approaches introduce machinemodels that replace the rotor equivalent circuits by arbitrary linear networks toallow for elimination of parameter identification procedure of the equivalentcircuit [11–13] or by determining an intermediate axis saturation characteristic[14]. A comparative study of saturation models used in stability analysis has beenpresented in [15]. It concludes that the complication of modifying the induc-tances at every solution step may not be essential particularly for large-scalesystem studies.

The predominant method proposed in the literature mostly considers that themutual inductances Lmd and Lmq in a machine are affected by saturationand should be corrected to be Lmds and Lmqs ‘the subscript s denotes saturation’given by

Lmds ¼ SdLmd and Lmqs ¼ SqLmq ð2:94Þ

where Sd and Sq are non-linear factors that depend on the flux levels.


Experimentally, the q-axis saturation is neglected and this is sufficiently accu-rate for calculating power system transient stability involving salient pole machines.Therefore, for salient pole machines, Sq ¼ 1 and Sd is a function of YAd while forround rotor machines, Sd ¼ Sq and is a function of linkages YAd and YAq. Thus, forsalient pole machines

Lmds ¼ SdLmd ; Lmqs ¼ Lmq and Sd ¼ f YAdð Þ ð2:95Þand for round rotor machines

LAds ¼ SdLAd ; LAdqs ¼ SqLAq; Sd ¼ Sq ¼ f Yð Þ and Y ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY2

Ad þY2Aq

qð2:96Þ

The factor Sd is normally derived from the saturation curve of the machine(Figure 2.6), indicating the relation between the flux linkage YAd and the magne-tising current, isum, the sum of currents id þ if þ ikd as in (2.69).

This relation is linear for both unsaturated and highly saturated conditions,although the slopes and intercepts of the two regions are different. The slope of thischaracteristic is, therefore, initially constant, undergoes a transition and finallybecomes constant again.

For a given value of YAd the unsaturated magnetising current is i(sum)o corre-sponding to Lmdo, while the saturated value is i(sum)s. The saturation factor Sd is afunction of this magnetising current, i.e. is a function of YAd as well. Thus, thevalue of unsaturated inductance Lmd is given by

Lmd ¼ A3A4

oA4and the saturated inductance Lmds ¼ A2A4

oA4

Ψ

io

ΨAd

ΨAdT

i(sum)T i(sum)O i(sum)S

Δi A4

A1A2

A3

B3

B1

B2

oB1 = unsaturated regionB2B3 = highly saturated region

Figure 2.6 Magnetic saturation curve


Then

Lmds ¼ LmdA2A4

A3A4¼ SdLmd

where

Sd ¼ A2A4

A3A4¼ i sumð Þo

i sumð Þsð2:97Þ

The value i(sum)s can be calculated by adding the current increment Di to theunsaturated magnetising current i(sum)o. So, Di should be calculated first whenthe flux linkage lies in the saturation region (B1B2 in Figure 2.6). This can beobtained by applying the approximate relation:

Di ¼ As exp Bs YAd �YAdTð Þ½ � and YAd > YAdT ð2:98Þwhere As and Bs are constants and determined from the saturation curve of themachine. YAdT is the flux linkage at the transition point from the initial constantslope region to the saturated region. Then, i(sum)s is calculated for a given YAd as

i sumð Þs ¼ i sumð Þo þ Di ð2:99ÞAccordingly, Sd is determined by using (2.97). The solution is achieved

through an iterative process that is terminated when YAdSd ¼LAdoi sumð Þs is satisfied.

2.8 The current state space model

In matrix notation, (2.52) is written as

v ¼ �B1i � B2didt

ð2:100Þ

where the two matrices B1 and B2 are

B1 ¼

Ra 0 0 wLq wkMkq

0 Rf 0 0 0

0 0 Rh 0 0


0 0 0 0 Rkq

2666666664

3777777775

B2 ¼

Ld kMf kMkd 0 0

kMf Lf Lfkd 0 0

kMkd Lfkd Lkd 0 0

0 0 0 Lq kMkq

0 0 0 kMkq lkq

26666664

37777775


Thus,

didt

¼ �B�12 B1i � B�1

2 v ð2:101Þ

From (2.85) and taking the damping term ‘Td’ into account, it is observed that

2HwBð Þ _w ¼ Ta ¼ Tm � Te � Td

where Te ¼ iqYd � idYq as given by (2.49)

Yd ¼ Ldid þ kMf if þ kMkdikd ;Yq ¼ Lqiq þ kMkqikq as given by (2.20) andTd ¼ Dw

Thus,

_w ¼ 12HwB

Tm � Dwð Þ

� 12HwB

iq Ldid þ kMf if þ kMkdikd

� �� id Lqiq þ kMkqikq

� �� ð2:102Þ

and

_d ¼ w� 1 ð2:103Þ

Incorporating (2.101) through (2.103), the current state space model can bewritten as

Example 2.4 Referring to Example 2.2, find the current state space model.

Solution:

The current state space model is expressed by (2.104). The matrices B1, B2,B2�1 and B2

�1B1 have been calculated in Example 2.2. Therefore, the last two


rows in (2.104) should be calculated to form the current state space model asbelow:

Ldiq

2HwB¼ 0:688 � 10�3iq

kMf iq2HwB

¼ 0:631 � 10�3iq

kMkdiq

2HwB¼ 0:631 � 10�3iq

Lqid2HwB

¼ 0:669 � 10�3id

kMkqid2HwB

¼ 0:604 � 10�3id

Thus, the current state space model is

References

1. Anderson P.M., Fouad A.A. Power System Control and Stability. 2nd edn.United States: IEEE – John Wiley & Sons, Inc.; 2003

2. Kundur P. Power System Stability and Control. United States: McGraw-Hill,Inc.; 1994

3. Park R.H. ‘Two reaction theory of synchronous machine, generalized methodof analysis – Part I’. Transactions on AIEE. 1929;48(3):716–30


4. Padiar K.R. Power System Dynamics Stability and Control. 2nd edn. India:BS Publications; 2008

5. Keyhani A., Tsai H. ‘Identification of high-order synchronous generatormodels from SSFR test data’. IEEE Transactions on Energy Conversion.1994;9(3):593–603

6. Sanchez Gasca J.J., Bridenbaugh C.J., Bowler C.E.J., Edmonds J.S. ‘Trajec-tory sensitivity based identification of synchronous generator and excitationsystem parameters’. IEEE Transactions on Power Systems. 1998;3(4):1814–22

7. Martinez J.A., Johnson B., Grande-Moran C. ‘Parameter determination formodeling system transients-Part IV: Rotating machines’. IEEE Transactionson Power Delivery. 2005;20(3):2063–72

8. Minnich S.H., Schulz R.P., Baker D.H., Sharma D.K., Farmer R.G., Fish J.H.‘Saturation functions for synchronous generators from finite elements’. IEEETrans. Energy Conversion. 1987;2(4):680–92

9. Corzine K.A., Kuhn B.T., Sudhoff S.D., Hegner H.J. ‘An improved methodfor incorporating saturation in the Q-D synchronous machine model’. IEEETransactions on Energy Conversion. 1998;13(3):270–5

10. Pekarek S.D., Walters E.A., Kuhn B.T. ‘An efficient and accurate method ofrepresenting magnetic saturation in physical-variable models of synchronousmachines’. IEEE Transactions on Energy Conversion. 1999;14(1):72–9

11. Aliprantis D.C., Sudhoff S.D., Kuhn B.T. ‘A synchronous machine model andarbitrary rotor network representation’. IEEE Transactions on Energy Con-version. 2005;20(3):584–94

12. Aliprantis D.C., Sudhoff S.D., Kuhn B.T. ‘Experimental characterizationprocedure for a synchronous machine model with saturation and arbitraryrotor network representation’. IEEE Transactions on Energy Conversion.2005;20(3):595–603

13. Aliprantis D.C., Wasynczuk O., Valdez C.D.R. ‘A voltage-behind-reactancesynchronous machine model with saturation and arbitrary rotor networkrepresentation’. IEEE Transactions on Energy Conversion. 2008;23(2):499–508

14. El-Serafi A.M., Kar N.C. ‘Methods for determining the intermediate-axissaturation characteristics of salient-pole synchronous machines from themeasured D-axis characteristics’. IEEE Transactions on Energy Conversion.2005;20(1):88–97

15. Harley R.J., Limebeer D.J., Chirricozzi E. ‘Comparative study of saturationmethods in synchronous machine models’. IEE Proceedings. 1980;127(1)Pt B:1–7


Chapter 3

Synchronous machine connectedto a power system

As stated in Section 2.7.1, (2.93) formulates the flux linkage state space modelin a general form: _x ¼ f x; u; tð Þ; where x≜ the vector of state variables ¼Yd ;Yf ;Ykd ;Yq;Ykq;w; d� �

and u≜ the forcing functions vd ; vq, vf and Tm. Thesame is applied to the current state space model developed in Section 2.8 andrepresented by (2.104) where x≜ the vector of state variables ¼ [id, if, ikd, iq, ikq,w; d]. To describe the machine completely, the forcing functions must be known. Inthe case of vf and Tm being known, the two functions of vd ; vq must be identified byrelations added to the model equations. This necessitates identifying the machineterminal conditions by describing and modelling its load. The load modelling isexplained in the forthcoming chapters. However, the load is connected to themachine through a network that is either simple, one machine-infinite bus, orintegrated, multi-machine system.

Moreover, each synchronous machine in the power system is equipped with anexcitation control system and its prime mover is controlled by a governor controlsystem. These controllers decide the values of vf and Tm and may be involved in themachine equations that are written in this chapter in pu using base phase quantities(the alternative per unit/normalising system is given in Appendix I, Section I.3).These features are presented in this chapter.

3.1 Synchronous machine connected to an infinite bus

Figure 3.1 depicts a simple power system, one machine connected to an infinite busthrough a transmission line and its single-line equivalent circuit where the transmis-sion line is represented by an external impedance, resistance Re and inductance Le.Ignoring the mutual coupling between phases a, b and c, and consideringthe generation as positive convention, the three phase to neutral voltages can beexpressed by (3.1):

vta ¼ va1 þ Reia þ Lepia

vtb ¼ vb1 þ Reib þ Lepib

vtc ¼ vc1 þ Reic þ Lepic

9>>=>>; ð3:1Þ

In matrix form, (3.1) can be rewritten as

& vta

vtb

vtc

’¼

va1vb1vc1

24

35þ ReU

iaibic

24

35þ LeU

pia

pib

pic

24

35 ð3:2Þ

or

vabc ¼ vabc1 þ ReUiabc þ LeUpiabc ð3:3Þ

where U is a unit matrix.

Applying Park’s transformation to obtain the voltages in dqo frame ofreference gives

vdqo ¼ P�1vabc ¼ vdqo1 þ Reidqo þ LeP�1piabc ð3:4Þ

vdqo1 ¼ P�1vabc1, idqo ¼ P�1iabc andvabc?≙ a set of balanced three-phase voltages. The last term in the RHS of

(3.4) can be determined as below:

idqo ¼ P�1iabc and by taking the derivative of both sides gives

didqo

dt¼ P�1piabc þ dP�1

dtiabc

Then,

P�1piabc ¼ pidqo � dP�1

dtPidqo ð3:5Þ

ReG

Terminalbus

Infinitebus

(a) (b)

V∞

+

–Transmission line∞

i iLe

LeRe

VtVph

Figure 3.1 One machine–infinite bus system: (a) one-line diagram and(b) equivalent circuit


By the definition of P in Section 2.3 (Chapter 2), and assuming q¼wot þ d, itis found that

dP�1

dtP ¼ w

0 �1 01 0 00 0 0

24

35 ð3:6Þ

Incorporating (3.6) and (3.5) into (3.4) gives

vdqo ¼ vdqo1 þ Reidqo þ Lepidqo � wLe

�iq

id

0

2664

3775 ð3:7Þ

where vdqo1 can be determined in terms of vabc? as below:

vdqo1 ¼ P�1ffiffiffi2

pV1

cos wt þ að Þcos wt þ a� 120ð Þcos wt þ aþ 120ð Þ

2664

3775 and V1 is the rms phase voltage

ð3:8Þ

Substituting (3.8) into (3.7) gives

vdqo ¼ffiffiffi3

pV1

�sin d� að Þcos d� að Þ

0

2664

3775þ Reidqo þ Le

didqo

dt� wLe

�iq

id

0

2664

3775 ð3:9Þ

The set of (3.9) gives the two non-linear relations of vd and vq to be added tothe machine equations in order to completely describe the machine by either fluxlinkage state space model or current state space model. For readers more interestedin machine equations, more details considering mutual effects, cross magnetising,saturation, etc., are discussed in [1–5].

3.1.1 Flux linkage state space modelFrom (2.79) and (2.81) iq in terms of flux linkages is given by

iq ¼ 1‘q

1 � LMq

‘q

� �Yq � LMq

‘q‘kqYkq ð3:10Þ

Synchronous machine connected to a power system 49

and id is given by (2.88). Substitute the currents id and iq into the two relations ofvd and vq (3.9) to obtain

vd ¼ � ffiffiffi3

pV1 sin d� að Þ þ Re

‘d1 � LMd

‘a

� �Yd � ReLMd

‘a‘fYf � ReLMd

‘a‘kdYkd

þ wLe

‘a1 � LMq

‘a

� �Yq � wLeLMq

‘a‘kqYkq þ Le

‘a1 � LMd

‘a

� �_Yd

� LeLMd

‘a‘f

_Yf � LeLMd

‘a‘kd

_Ykd ð3:11Þ

vq ¼ � ffiffiffi3

pV1 cos d� að Þ þ Re

‘a1 � LMq

‘a

� �Yq � ReLMq

‘a‘kqYkq

�wLe

‘a1 � LMd

‘a

� �Yd þ wLe

‘a‘fYf þ wLe

‘a‘kdYkd

þ Le

‘a1 � LMq

‘a

� �_Yq � LeLMq

‘a‘kq

_Ykq ð3:12Þ

Substituting vd and vq as in (3.11) and (3.12) into (2.93) to compose the fluxlinkage state space model in the form

A _x ¼ Bxþ C ð3:13Þ

Equation (3.13) can be expressed in the general form _x ¼ f x; u; tð Þ as

_x ¼ A�1Bxþ A�1C ð3:14Þ

where xt ¼ ½Yd ;Yf ;Ykd ;Yq;Ykq;w; d�



Example 3.1 Use the synchronous generator data given in Examples 2.1–2.4 tocompute the flux linkage state space model when the machine is connected to aninfinite bus through a transmission line represented by a resistance Re ¼ 0.05 puand an inductance Le ¼ 0.35 pu.

Solution (all values are in pu, otherwise will be cited):

R ¼ 0:00299 þ 0:05 ¼ 0:053

Ld and Lq should be modified to be Ld ¼ Ld þ Le and Lq ¼ Lq þ Le, respec-tively. Hence,

Ld ¼ 1:81 þ 0:35 ¼ 2:16 and Lq ¼ 1:76 þ 0:35 ¼ 2:11

Lmd ¼ Ld � ‘a ¼ 2:16 � 0:15 ¼ 2:01 and Lmq ¼ Lq � ‘a ¼ 2:11 � 0:15

¼ 1:96

1LMd

¼ 1Lmd

þ 1‘a

þ 1‘f

þ 1‘kd

¼ 14:765

LMd ¼ 0:068

1LMq

¼ 1Lmq

þ 1‘a

þ 1‘kq

¼ 13:023

LMq ¼ 0:077


Hence,


Substituting into (3.14) gives the flux state space model as


3.1.2 Current state space modelEquation (2.100) can be rewritten as �B2

didt ¼ B1i þ v. Substituting the value of v

from (3.9) gives

�B2didt

¼ B1i þ

ffiffiffi3

pV1 sin d� að Þ þ Reid þ Le

diddt

þ wLeiq

�vf

0

ffiffiffi3

pV1 cos d� að Þ þ Reiq þ Le

diqdt

þ wLeid

0

2666666666666664

3777777777777775

ð3:15Þ

Using R ¼ Ra þ Re; Ld ¼ Ld þ Le and Lq ¼ Lq þ Le in (3.15) gives the cor-responding matrices B1 and B2. Then adding the relations of _w and _d gives thecurrent state space model as

(3.16)

–12 1 0


Example 3.2 Repeat Example 3.1 for current state space model.

Solution:

Referring to (2.100) the coefficient matrices are

B1 ¼

R 0 0 wLq wkMkq

0 Rf 0 0 0

0 0 Rkd 0 0

�wLd �wkMf �wkMkd R 0

0 0 0 0 Rkq

266666664

377777775

¼

0:05299 0 0 2:11w 1:59w

0 0:00898 0 0 0

0 0 0:0284 0 0

�2:16w �1:66w �1:66w 0:05299 0

0 0 0 0 0:00619

266666664

377777775

B2 ¼

Ld kMf kMkd 0 0

kMf Lf Lfkd 0 0

kMkd Lfh Lkd 0 0

0 0 0 Lq kMkq

0 0 0 kMkq Lkq

266666664

377777775

¼

2:16 1:66 1:66 0 0

1:66 2:73 1:66 0 0

1:66 1:66 0:1713 0 0

0 0 0 2:11 1:59

0 0 0 1:59 0:7252

266666664

377777775

B�12 ¼

0:604 �0:652 0:467

�0:652 0:629 0:219

0:468

0

0

0:219

0

0

�0:829

0

0

0 0

0 0

0

1:674

�1:597

0

�1:596

2:119

26666664

37777775

B�12 B1 ¼

0:032 �0:006 0:013

�0:034 0:006 0:006

0:025

�3:616w

3:449w

0:002

�2:779w

2:651w

�0:023

�2:779w

2:651w

1:270w 0:960w�1:376w �1:034w0:987w0:089

�0:085

0:744w0:010

�0:009

26666664

37777775


Thus, the current state space model can be expressed as

3.2 Synchronous machine connected to anintegrated power system

Terminal or load conditions of the synchronous machine determined by vd and vq

can be achieved by direct relations when the machine is connected to an infinitebus. In the case of multi-machine system, each machine is connected to thepower system that comprises numerous static elements, e.g. transmission lines,static loads, shunt capacitors, transformers and dynamic elements, e.g. generatorsand their control systems and dynamic loads. Therefore, the determination ofvd and vq representing the machine terminal conditions is more complicatedbecause of the system non-linearity, the complexity of modelling, the dynamicinteraction between some of system components, the large and complex size ofthe system and the operating mode (normal or contingencies). Load flow tech-niques, more details in Part III, are commonly used to determine the terminalconditions of each machine in the system and to know the values vd and vq. Then,these values are substituted into (2.89) or (2.100) to construct the flux or currentstate space model. Load flow results depend on the representation of the systemcomponents according to the operating mode. So, to accurately model the

–12 1 0

–12


synchronous machine, its parameters, inductances and time constants mustbe known in different operating modes (steady state, transient state and sub-transient state).

3.3 Synchronous machine parameters in differentoperating modes

In sub-transient, transient and steady-state operating conditions, both inductancesand time constants must be identified (the values of inductances are numericallyequal to the corresponding reactances in pu with the synchronous speed as base,i.e. wB ¼ 2pf rad/s). Different testing and measuring methods such as standstillfrequency response and rotating time domain response can be applied to determinethe machine parameters and synthesise their models from test data [6–10]. Whenbalanced three-phase voltages are suddenly applied to the stator terminals while therotor circuits are short circuited, the flux linkage in the d-axis frame of reference,Yd , depends initially on the sub-transient inductances and then on the transientinductances after a few cycles. Assuming the three-phase balanced voltages, vabc,suddenly applied to the stator terminals are expressed as

va

vb

vc

264

375 ¼

ffiffiffi2

pVrms

cos qcos q� 120ð Þcos qþ 120ð Þ

264

375u tð Þ ð3:17Þ

where Vrms ¼ rms phase voltage and u(t) is a unit step function.Applying Park’s transformation gives vdqo as

vd

vq

vo

264

375 ¼

ffiffiffi3

pVrms u tð Þ

0

0

264

375 ð3:18Þ

The flux linking the field circuit, Yf , and the damper winding, Ykd , remainzero at the instant voltage is applied, as flux cannot change instantaneously. Thus,at that instant tþo and using (2.31) it is seen that

Yf ¼ 0 ¼ kMf id þ Lf if þ Lfkdikd

Ykd ¼ 0 ¼ kMkdid þ Lkdikd þ Lfkdif

)ð3:19Þ

Thus,

if ¼ � kMf Lkd � kMkdLfkd

Lf Lkd � L2fkd

id

ikd ¼ � kMkdLf � kMf Lfkd

Lf Lkd � L2fkd

id

9>>>=>>>;

ð3:20Þ


Using (3.20) and substituting in (2.20) Yd can be written as a function of id as

Yd ¼ Ld �k2M2

f Lkd þ Lf k2M2kd � 2kMf kMkdLfkd

Lf Lkd � L2fkd

!id ð3:21Þ

By definition: Yd ≙ L00did ð3:22Þ

where L00d is the d-axis sub-transient inductance.

Comparing (3.21) and (3.22) it is found that

L00d ¼ Ld �

k2M2f Lkd þ Lf k2M2

kd � 2kMf kMkdLfkd

Lf Lkd � L2fkd

!

and by using the definition of Lmd as in (2.67) L00d can be written as

L00d ¼ Ld � Lf þ Lkd � 2Lmd

Lf Lkd

L2md

� �� 1

ð3:23Þ

The d-axis transient inductance, L0d , is obtained when a balanced three-phase

voltage is suddenly applied to a machine without damper windings. Applying thesame procedure gives

if ¼ � kMf

Lfid ð3:24Þ

Yd ¼ Ld �kMf

� �2

Lf

" #id ≙ L0

did ð3:25Þ

Therefore,

L0d ¼ Ld � ðkMf Þ2

Lf¼ Ld � L2

md

Lfð3:26Þ

It is to be noted that after a few cycles from the start of transients in a machinewith damper windings, the damper winding current decays rapidly to zero and thestator inductance is the transient inductance.

To calculate the inductances in the q-axis frame of reference, the suddenlyapplied three-phase voltages are shifted by 90� to be expressed as

va

vb

vc

264

375 ¼

ffiffiffi2

pVrms

sin qsin q� 120ð Þsin qþ 120ð Þ

264

375u tð Þ ð3:27Þ

Applying Park’s transformation gives vdqo as

vd

vq

vo

264

375 ¼

0ffiffiffi3

pVrms u tð Þ

0

264

375 ð3:28Þ


For salient pole machines with damper windings when the initial sub-transientdecays to zero, the stator flux linkage is determined by the same circuit of thesteady-state q-axis flux linkage. Therefore, the q-axis transient inductance can beconsidered the same as the q-axis steady-state inductance, L0

q ¼ Lq. The sameprocedure above is applied for q-axis circuits to determine L00

q as below:

Ykd ¼ kMkqiq þ Lkqikq ¼ 0

Thus,

ikq ¼ � kMkq

Lkqð3:29Þ

Substituting (3.29) in the relation Yq ¼ Lqiq þ kMkqikq gives

Yq ¼ Lq � ðkMkqÞ2

Lkq

" #iq ≙ L00

qiq ð3:30Þ

Hence,

L0q ¼ Lq � ðkMkqÞ2

Lkq

" #¼ Lq �

L2mq

Lkqð3:31Þ

For round rotor machines, multiple paths of eddy currents are provided by thesolid iron rotor and act as equivalent circuits during the sub-transient and transientperiods. Therefore, the q-axis sub-transient and transient inductances are deter-mined by the q-axis rotor circuits resulting in q-axis transient inductance muchsmaller than q-axis steady-state inductance, L00

q � L0q � Lq. This can be verified by

considering two q-axis rotor damper circuits [11].On the other hand, to determine the time constants of a salient pole machine,

as in (3.30) the voltage equations of vf and vkd, when the stator circuits are opencircuited and a step voltage Vf u(t) is applied to the field circuit, become

Vf u tð Þ ¼ _Yf þ Rf if vkd ¼ 0 ¼ _Ykd þ Rkdikd ð3:32ÞFrom (2.31) the flux linkages Yf and Ykd immediately after the sub-transient

can be rewritten as

Yf ¼ Lf if þ Lfkdikd Ykd ¼ Lfkdif þ Lkdikd ¼ 0 ð3:33Þwhere id ¼ 0 as the stator circuits are open.

Hence; if ¼ � Lkd

Lfkdikd ð3:34Þ

From (3.32) and using (3.33) the relations below can be written as

Vf

Lf¼ Rf

Lfif þ pif þ Lfkd

Lfpikd

0 ¼ Rkd

Lfkdikd þ pif þ Lkd

Lfkdpikd

9>>>=>>>;

ð3:35Þ


Solving the relations (3.35) using (3.34) gives a relation of ikd as

pikd þ RkdLf þ Rf Lkd

Lf Lkd � L2fkd

ikd ¼ �VfLfkd

lf Lkd � L2fkd

ð3:36Þ

This relation can be approximated by considering that Rkd � Rf and Lf and Lkd

in per unit are almost equal in magnitude, as

pikd þ Rkd

Lkd � L2fkd=Lf

ikd ¼ �VfLfkd=Lf

Lkd � L2fkd=Lf

ð3:37Þ

It is, therefore, seen that ikd decays with a time constant T00do, where

T00do ¼ Lkd � ðL2

fkd=Lf ÞRkd

≙ d-axis open circuit sub-transient time constant

ð3:38ÞAfter the sub-transient current decays, i.e. in the transient period, the field

current is only affected by the field circuit parameters. Then, the relation below canbe written as

Rf if þ Lf pif ¼ Vf u tð Þ ð3:39ÞTherefore, the time constant of (3.39) denoted by T0

do is called d-axis transientopen circuit time constant and is given by

T0do ¼ Lf

Rfð3:40Þ

The time constants when the stator is short circuited can be calculated usingthe following approximate relations [12].

In the d-axis:T00

d ¼ T00doL00

d=L0d

T0d ¼ T0

doL0d=Ld

)ð3:41Þ

Similarly; in the q-axis:T00

qo ¼ Lkq=Rkq

T00q ¼ T00

qoL00q=Lq

)ð3:42Þ

For the time constants of round rotor machines, a time constant, Tc, is added tothe sub-transient and transient time constants. It is associated with the rate of changeof stator direct current or with the alternating currents enveloped in the fieldwindings in the case of exposing the machine to a three-phase short circuit. Tc isgiven by [12]

Tc ¼ L2

Raand L2 ¼ L0

d þ Lq

2≙ the negative-sequence inductance ð3:43Þ


As explained in this chapter the synchronous machine is modelled by eithercurrent or flux linkage state space model. Both include five currents or flux linkagestate variables (may be increased as the rotor circuits increase) plus two state vari-ables of angular speed and rotor angle. The state variables are presented by non-linear first-order differential equations. Electric power systems are composed of alarge number of components, equipment and control devices, interacting with eachother and exhibiting non-linear dynamic behaviour with a wide range of timescales.For instance, equations describing network constraints, loads, excitation system andprime movers must be included in the mathematical model to study the response of alarge number of synchronous machines to a given disturbance. Therefore, a completemathematical description of the system yields more complexity and more compu-tation time. So, some simplifications are needed such as simplified machine modelsthat are described in Section 3.4. The choice between those models depends on thetype of study and the extent of its adequacy to that study.

Example 3.3 Calculate the transient and sub-transient inductances of a cylind-rical rotor generator having the parameters in per unit values as below:

Ld ¼ 1:81; Lq ¼ 1:76; Lmd ¼ 1:66; Lmq ¼ 1:61; Lf ¼ 2:73; Lkd ¼ 1:601; Lkq ¼ 1:72;Lfkd ¼ 1:66;Rf ¼ 0:898 � 10�3;Rkq ¼ 6:19 � 10�3;Rkd ¼ 0:0284

Solution:

The direct and quadrature axis sub-transient and transient inductances are

By ð3:23Þ; L00d ¼ 1:81 � 2:73 þ 1:67 � 2 � 1:66

2:73 � 1:61:66 � 1:66

� �� 1

¼ 0:125

From ð3:26Þ; L0d ¼ 1:81 � 1:66 � 1:66

2:73¼ 0:8006

Referring to ð3:31Þ; L00q ¼ 1:76 � 1:61 � 1:61

1:72¼ 0:253

L0q ¼ Lq ¼ 1:76

3.4 Synchronous machine-simplified models

3.4.1 The classical modelThis model is relevant to determine the system stability in the first swing of therotor angle in the order of one second or less. The machine is represented by aconstant voltage ‘E’ at initial angle d, behind a direct axis transient reactance, X 0

d .Whether the machine is connected to an infinite bus or a multi-machine system, themodel is based on the following assumptions:

● constant input mechanical power● damping is negligible


● synchronous machine is represented by a constant voltage behind transientreactance

● mechanical rotor angle of the machine coincides with the angle of the voltagebehind the transient reactance

● loads at the machine terminals are represented by a constant impedance oradmittance

3.4.1.1 Classical model for one machine connected to an infinite busThe general configuration of the system consisting of one machine connected to theinfinite bus through a transmission line with impedance ZTL and a shunt impedanceZs connected to the machine terminals (it may represent a load) is shown inFigure 3.2(a). The terminal voltage of the machine is denoted by Vt and the voltageof the infinite bus is V? ff0� (it is used as reference). Its equivalent circuit isdepicted in Figure 3.2(b) where the node representing the machine terminal inFigure 3.2(a) can be eliminated by a Y–D transformation.

The transient stability of the machine is determined by solving the swingequation, (2.85), and therefore the power delivered by the machine, P1, is needed tocalculate the accelerating power. As in the equivalent circuit (Figure 3.2(b)), it canbe found by network theory that

P1 ¼ Re EI1� � ¼ E2Y11 cos q11 þ EV1Y12 cos q12 � dð Þ

where

Y11 ffq11 ¼ y12 þ y1o and Y12 ffq12 ¼ �y12. It is to be noted that y2o is notneeded.

By defining G11 ≜ Y11 cos q11 and g ¼ q12 � p=2, the power P1 can beexpressed as

P1 ¼ E2G11 þ EV1Y12 sin d� gð Þ ð3:44Þ

3.4.1.2 Classical model for multi-machine systemThe multi-machine system (Figure 3.3) comprises a number of generators (n)feeding different loads (m) through a transmission network. Each machine isrepresented by a constant internal voltage source behind its direct axis transientreactance and the loads are represented by constant impedances. The electric output

E

Xd ′ V∞ 0°

E++

– –

21

d ° d °

Vt ZTL

Zs V∞ 0°

I1 y12I2

y1o

(a) (b)

Figure 3.2 One machine to an infinite bus system


power of each generator can be obtained by solving the set of non-linear algebraicequations that represent the relations between currents and voltages, and load flowtechniques to obtain the initial values of system parameters that are needed to solvethe equations of motion of the generators. Then, the system stability is determined.Detailed explanation for load flow techniques is provided in Part II.

3.4.2 The E0q model

In this model the effect of the damper circuits in the d-axis is neglected and conse-quently ikd is omitted from the current state space model (Section 2.8) orYkd is omittedfrom the flux linkages state space model (Section 2.7.1). The effect of damper circuitsin the q-axis also can be neglected, but, in particular, for solid round rotor machinesand in the absence of damper circuits, the rotor acts as a q-axis damper winding.However, this effect is small enough to be neglected or can be included by increasingthe damping coefficient D in the torque equation. Thus, ikq in the current state spacemodel is omitted and Ykq is omitted from the flux linkages state space model.

Under the condition of neglecting the effect of damper circuits, an alternativemachine model containing the well-known machine parameters can be deduced asbelow.

Using (2.76), (2.78), (2.79) and (2.81) with the KD and KQ circuits omitted gives

idif

iq

24

35 ¼

ð‘d � LMdÞ‘2

d

�LMd

‘d‘f0

�LMd

‘d‘f

ð‘f � LMdÞ‘2

f

0

0 01Lq

2666666664

3777777775

Yd

Yf

Yq

24

35 ð3:45Þ

R1 x′d1

Rn x’dn

R2 x′d2

Z1

Zm

Transmission network

Reference node ‘O’

Ign

Ig2

Ig1

I-ldl

I-ldm

Vt1

Vtn

E2,d 2

E1,d 1

En,d n

Vt2

Figure 3.3 Multi-machine power system


The elements of the matrix in (3.45) can be written in terms of L0d , Lmd and Lf as

id

if

iq

264

375 ¼

1L0

d

�Lmd

L0dLf

0

�Lmd

L0dLf

Ld

L0dLf

0

0 01Lq

266666664

377777775

Yd

Yf

Yq

264

375 ð3:46Þ

From the set of (2.30) the first equation is rewritten as_Yd ¼ �Raid � wYq � vd and using (3.46) gives

_Yd ¼ � Ra

L0d

� �Yd þ RaLmd

L0dLf

� �Yf � Raid � wYq � vd ð3:47Þ

Equation (I.11), Appendix I: E0q ¼ wokMf Yf =Lf can be converted into pu

values, it gives

E0q ¼ LmdYf =Lf ð3:48Þ

Substitute (3.48) in (3.47) to obtain

_Yd ¼ � Ra

L0d

� �Yd þ Ra

L0d

� �E0

q � wYq � vd ð3:49Þ

Similarly, apply the same procedure for Yq as in (2.30) to write_Yq ¼ � Ra=Lq

� �Yq þ wYd � vq and vf ¼ _Yf þ Rf if and use (3.45) to obtain

vf ¼ Rf � Lmd

L0dLf

� �Yd þ Ld

L0dLf

� �Yf

þ _Yf ð3:50Þ

The definition given in (I.12), Appendix I, states that Efd ¼ vf =Rf

� �wokMf . It

can be converted into pu values as

Efd ¼ Lmdvf

Rfð3:51Þ

Combining (3.48), (3.50) and (3.51) gives

Rf

LmdEfd ¼ �Lmd

L0d

Rf

lfYd þ Ld

L0d

Rf

LmdE0

q þ Lf

Lmd

_E0q ð3:52Þ

SubstitutingL2

mdLf

¼ Ld � L0d and T 0

do ¼ Lf

Rf

_E0q ¼ 1

T 0do

Efd � Ld

L0d

E0q þ Ld � L0

d

L0d

Yd

� �ð3:53Þ


It is to be noted that the above equations include all variables in pu as well asthe voltages vd, vq, Efd and E0

q are considered as line-to-line pu values.The variables id and iq in (3.46) are substituted in the torque equation

Te ¼ iqYd � idYq and (3.48) is used as well to express the torque as

Te ¼ 1L0

dE0

qYq � 1L0

d� 1

Lq

� �YdYq ð3:54Þ

The swing equation including the damping term is

_w ¼ 12HwB

½Tm � Te � Dw�_d ¼ w� 1

9=; ð3:55Þ

Thus, the E0q model, described by (3.49), (3.50), (3.53) and (3.55) in time-

domain, is a fifth-order system and can be represented in the s domain by the blockdiagram shown in Figure 3.4.

∑Ra/Ld′

π

π

+

++

++

++

+

++

–

–

ωΨq

Vq

ω

π π

1/Ld′

(1/Ld′) – (1/Lq)

1/s

Te

ω

1.0

+ + ++

+

+

+–

–

–

(Ld′/Ra)/[1 + s (Ld′/Ra)]

ωΨd

∑

∑ (Lq/Ra)/[1 + s(Lq/Ra)]

(Ld′/Ld)/[1 + s(Tdo′Ld′/Ld)]

Ψd

Ψd

Ψq

Vd

(Ld – Ld′)/Ld′

Efd Eq′∑

∑

TaTm 1/[D + s(2HωB)]∑

∑ d

Figure 3.4 Block diagram representation of E0q model in s domain


Taking the saturated operating conditions into consideration necessitatesadding additional field current, Di, to substitute the saturation effect as explained inSection 2.7.2. Equation (3.46) implies that id ¼ 1

L0dYd � LAd

L0d lfYf . Substituting this

value of id in equation Yd ¼ Ldid þ Lmdif gives

Lmdif ¼ 1 � Ld

L0d

� �� Yd

þ LdLAd

L0dLf

� � !

Yf

" #ð3:56Þ

It is given from ‘(I.10)’, Appendix I, that wokMf if ¼ Lmdif ¼ EI. It is incorpo-rated with (3.48) and (3.56) to write

EI ¼ Ld

L0d

E0q � Ld � L0

d

L0d

Yd ð3:57Þ

Equations (3.53) and (3.57) show that

_E0q ¼ 1

T0do

Efd � EI

� � ð3:58Þ

Assuming DE is a component that must be added to (3.57) corresponding to Dito obtain the same EMF on the no-load saturation curve, it can be written as

EI ¼ Ld

L0d

E0q � Ld � L0

d

L0d

Yd þ DE ð3:59Þ

Equations (3.58) and (3.59) should be represented by a block diagram to beadded to that depicted in Figure 3.4 when considering the effect of machinesaturation, thus giving the block diagram shown in Figure 3.5.

Each generating unit is individually provided with an excitation control systemand a prime mover control system. Therefore, modelling of both the controllersshould be considered as a supplementary part of the synchronous machine model,in particular, when studying the stability of a large-scale power system.

3.5 Excitation system

The regulation of synchronous machine terminal voltage is the main function of theexcitation system that controls the field current. As the field circuit has ‘to someextent’ a high time constant in order of a few seconds, field forcing is required forfast control of the field current. Consequently, the exciter should have a highceiling voltage to be able to operate transiently with voltage levels of the order ofthree to four times the normal in addition to changing the voltage at a fast rate.

The excitation system must be modelled in such a way that the model isrelevant for use in large-scale power system stability studies. The order of themodel should be chosen in a condition of adequacy to the aim of the study,avoiding the complexity of analysis and keeping the accuracy of results. Therefore,


based on these requirements, the model can be a reduced order model and may notrepresent the system in more detail than that required for stability studies.

3.5.1 Excitation system modellingRegardless of the type of excitation system, the main parts that comprise theexcitation system, as shown in Figure 3.6, are voltage transducer and load

∑

∑

∑

+

++

++

++

+

–

–

Ψd

∑

∑

∑ 1/s

ω

1.0 δ

+ + ++

+

+

+–

–

–

(Ld – Ld′)/Ld′ ∑

Sd = f(E)

+

+∑

+ –

–

–

1

∑

E

ωΨq

ω

∆ E

(Ld′/Ra)/[1 + s(Ld′/Ra)]Ra/Ld′

Vd

Vq (Lq/Ra)/[1 + s(Lq/Ra)]

Ld /Ld′

Efd 1/(sTdo′) Eq′

1/Ld′

(1/Ld′) – (1/Lq)

Tm

Ta

Te

1/[D + s(2HωB)]

Ψq

Ψd

ωΨd

Ψd

π

π

π π

Figure 3.5 Block diagram representation of E0q model with saturation in s domain


compensator, excitation control elements, an exciter, excitation system stabiliser(ESS) and commonly a power system stabiliser (PSS).

(I) Terminal voltage transducer and load compensator

The objective of the load compensation is to provide an output voltage, Vc, equal tothe machine terminal voltage plus the voltage drop in an impedance (Rc þ jXc). Theimpedance or range of adjustment should be specified. Both voltage and currentphasors at the machine terminal are sensed and used to compute Vc, which is thencompared with a reference voltage representing the desired terminal voltage set-ting. Without load compensation, both Rc and Xc are zero; the excitation systemattempts to maintain a terminal voltage determined by the reference signal withinits regulation characteristics. The function of the load compensator is to regulatethe voltage at some point other than the machine terminals in two ways: First, toregulate the voltage at a point internal to the generator by the sharing of reactivepower among units connected to the same bus with zero impedance in between.Rc and Xc are in this case positive values. Second, to regulate the voltage at a pointbeyond the generator terminals when the generating units are operating in parallelthrough unit transformers, a compensation of a portion of transformer impedance isrequired. In this case, Rc and Xc are negative values. Generally, in practice, theresistive component of compensation is neglected when generators are synchro-nised to a large grid over high-voltage interconnections. Thus, Rc is assumed to bezero to simplify the analysis. The terminal voltage transducer and load compensatorcan be represented by the block diagram shown in Figure 3.7 where a single timeconstant, TR, is used for the combined voltage sensing and compensation signal.

1/(1 + sTR)VcVc1

Vt

It

Vc1 = |Vt + (Rc + jXc)It|

Figure 3.7 Elements of terminal voltage transducer and load compensator

ExciterSynchronousmachine andpower system

Excitation control

elements∑ ∑

Excitationsystem

stabilizer (ESS)

Power systemstabilizer (PSS)

Voltage transducerand

load compensator

VcVt

IfdVREF

VOELVUEL

++

+–

–

VERROR

It

Efd

VF

VR

VSVs1

Figure 3.6 Basic functional block diagram of excitation control system


(II) Excitation control elements

They include the functions of both regulating and stabilising the excitation. Theterms ESS and transient gain reduction (TGR) are used for increasing thestable region of operation of the excitation system and permit higher regulatorgains. It is to be noted that feedback control systems (of which the excitationsystem is an example) often require lead/lag compensation or derivative, rate,feedback. The feedback transfer function for ESS is shown in Figure 3.8.

A typical value of the time constant is taken as one second. The feedbackcompensation for ESS can be replaced by using a series-connected lead/lag circuitas shown in Figure 3.9 where T1 is commonly less than T2. Consequently, thismeans of stabilisation is termed as TGR. The main goal of TGR is to reduce thetransient gain or gain at higher frequencies, thereby minimising the negative con-tribution of the regulator to system damping and accordingly the system damping isenhanced. Therefore, if PSS is specifically used to enhance system damping, theTGR may not be required. A typical value of the TGR factor (T2/T1) is 10.

Recently, modelling of field current limiters has become increasingly important,resulting in the addition to this over-excitation and under-excitation limiters, OELsand UELs, respectively. Output of the UEL may be received as an input to theexcitation system (VUEL) at various locations, either as a summing input or as a gatedinput, but for any one application of the model, only one of these inputs would beused. For the OEL some models provide a gate through which the output of the over-excitation limiter or terminal voltage limiter (VOEL) could enter the regulator loop.

(III) Power system stabiliser

The stabilisation provided by PSS differs from that provided by ESS. ESS provideseffective voltage regulation under open- or short-circuit conditions while thefunction of PSS is to provide damping of the rotor oscillations at the occurrence oftransient disturbances. The damping of these oscillations can be impaired by theprovision of high gain AVR, particularly at high loading conditions when a gen-erator is connected through high external impedance. Detailed discussion of PSS ispresented in Chapter 11. It is to be noted that the input signal for PSS is derivedfrom speed/frequency, accelerating power or a combination of these signals.

sKF/(1 + sTF)Efd VF

Figure 3.8 Transfer function of excitation system stabiliser

(1 + sT1)/(1 + sT2)

Figure 3.9 Transfer function of transient gain reduction


Several rotor oscillation frequencies must be considered when designing the PSS ina multi-machine system. However, the stabiliser is designed to have zero output insteady state. Also the output is limited in order not to adversely affect the voltagecontrol. The stabiliser output ‘Vs’ is added to the terminal voltage error signal asshown in Figure 3.6.

(IV) Types of excitation system

Excitation systems are classified into three types based on the source of excitationpower: DC excitation systems, AC excitation systems and static ST excitationsystems. Modelling of excitation system is essential for stability studies [13–16]and identification of model parameters has a prominent role, in particular,identifying non-linearity such as limits and saturation as discussed in [17–21].Commonly, the excitation systems have under-excitation and over-excitationlimiters, thus their modelling must be considered as well [22, 23]. In this section,each type is presented through an example indicating the model in s domainassociated with sample data. It is to be noted that for DC and AC types, the excitersaturation and loading effects should be accounted for as below.

The increase in excitation requirements due to saturation is represented by anexciter saturation function SE(Efd) defined as a multiplier of pu exciter voltage. Thisfunction at a given exciter output voltage, representing the load saturation asdepicted in Figure 3.10, can be calculated by (3.60):

SE Efd

� � ¼ A � B

Bð3:60Þ

Exci

ter o

utpu

t vol

tage

(pu)

Exciter field current (pu)

A

BC

Air-gap line

No-load saturation

Constant resistance load

saturation

Figure 3.10 Exciter saturation characteristics


where

A ≜ the excitation required to produce the exciter output voltage on theconstant-load–saturation curve

B ≜ the excitation required to produce the exciter output voltage on the air-gapline

C ≜ the excitation required to produce the exciter output voltage on theno-load saturation curve

For some alternator-rectifier exciters, the no-load saturation curve is used indefining SE(VE) as

SE VEð Þ ¼ C � B

Bð3:61Þ

because the exciter regulation effects are accounted for by inclusion of both ademagnetising factor and commutating reactance voltage drop in the model.

In general, the saturation function can be specified adequately by two points: thefirst is at the voltage E1 near the exciter ceiling voltage for DC types or V1 nearthe exciter open circuit ceiling voltage for the AC types. The second is at the voltageE2 at a lower value, commonly near 75 per cent of E1 for DC types or near 75 per centof V1 for AC types. Computer programs based on these higher and lower voltagesalong with the corresponding saturation data as inputs have been designed to repre-sent the exciter saturation characteristics with different mathematical expressions.

(a) Type DC excitation systems

For this type, a direct current generator with a commutator is used as the source ofexcitation power. The block diagram of a typical model is shown in Figure 3.11 andsample data is given in Table 3.1 [24]. It includes a proportional, integral anddifferential generator voltage regulator (AVR). An alternative feedback loop (KF,TF) is designed for stabilisation if the derivative term is not included in the AVR.

HVGATE

∑ 1/sTE∑

∑

VT*VRMAX

VUEL

ALTERNATE OEL INPUTS

ALTERNATE UEL INPUTS

++

+

+

++

– –

––

VUEL

VOEL

LVGATE

VT

VS

VREF VRMIN/KA

VRMAX/KA

KP + (KI/s) + sKD/(1 + sTD)

VT*VRMIN

KA/(1 + sTA)

VEMIN

VR

VX

KE

VX = VESE(Efd)

sKF/(1 + sTF)

VF

π

Figure 3.11 A typical example of DC excitation systems (Type DC4B)� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEERecommended Practice for Excitation System Models for Power System Stability Studies’


(b) Type AC excitation systems

These excitation systems use an a.c. alternator and either stationary or rotatingrectifiers to produce the d.c. field requirements. Loading effects on such excitersare significant, and the use of generator field current as an input to the modelsallows these effects to be represented accurately.

An example of this type is shown in Figure 3.12 and its sample data aresummarised in Table 3.2.

Table 3.1 Sample data for excitation system (Type DC4B)

Description Parameter Value Units

Regulator proportional gain KP 80.0 puRegulator integral gain KI 20.0 puRegulator derivative gain KD 20.0 puRegulator derivative filter time constant TD 0.01 sRegulator output gain KA 1.0 puRegulator output time constant TA 0.2 SMax controller output VRMAX 2.7 puExciter field time constant TE 0.8 puExciter field proportional constant KE 1.0 puExciter minimum output voltage VEMIN 0.0 puExciter flux at SE1 E1 1.75 puSaturation factor at E1 SE1 0.08Exciter flux at SE2 E2 2.33 puSaturation factor at E2 SE2 0.27Rate feedback gain KF 0.0 puRate feedback time constant TF 0.0 s

� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEE Recommended Practicefor Excitation System Models for Power System Stability Studies’

1/sTE π

IN = KCIfd/VE

VX = VESE[VE]

+

+ ++

+

+

+

+

++

–

–

VREF KPR

KIR/S

VRMAX

VR

VFE

VX

VEMIN

KA/(1 + sTA)

VFEMAX – KDIfdKE + SE[VE]

VE Efd

FEXVRMIN

VC

VS

KE

KD Ifd

IN

FEX = f [IN]

Σ Σ Σ

Σ Σ

sKDR/(1 + sTDR)

Figure 3.12 Type AC8B – Alternator-rectifier excitation system� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEERecommended Practice for Excitation System Models for Power System Stability Studies’


(c) Type ST excitation systems

In these excitation systems, voltage (and also current in compounded systems) istransformed to an appropriate level. Rectifiers, either controlled or non-controlled,provide the necessary direct current for the generator field. An example of this typeis shown in Figure 3.13 and its sample data is summarised in Table 3.3.

3.6 Modelling of prime mover control system

The prime mover is responsible for providing the synchronous generator with theinput mechanical power required. The power demand from the generator changes

Table 3.2 Sample data of excitation system-type AC8B

KPR ¼ 80 VRMAX ¼ 35.0 SE(E1) ¼ 0.3KIR ¼ 5 VRMIN ¼ 00.0 E1 ¼ 6.5KDR ¼ 10 KE ¼ 1.0 SE(E2) ¼ 3.0TDR ¼ 0.1 TE ¼ 1.2 E2 ¼ 9.0VFEMAX ¼ 6.0 KC ¼ 0.55 KD ¼ 1.1

� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEE RecommendedPractice for Excitation System Models for Power System Stability Studies’

Σ

LVGateΣΣ πKPA + (KIA/s)ΣHV

GateΣ

VOELALTERNATE OEL

INPUTS

+

+

+

+++

+

–

–

– –

–

VOELILR KC1 KLR

IfdVRMIN

KFF

KM

VRMAXVAMAX

VA

VRMIN

VG

VAMIN

VREF VS

VUEL

VR

VB

Efd

KG/(1 + sTG)

Figure 3.13 Type ST6B – Static potential-source excitation system with fieldcurrent limiter� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEERecommended Practice for Excitation System Models for Power System Stability Studies’

Table 3.3 Sample data of excitation system-type ST6B

KPA ¼ 18.038 TG ¼ 0.02 s VAMIN ¼�3.85KIA ¼ 45.094 s–1 TR ¼ 0.012 s KLR ¼ 17.33KFF ¼ 1.0 VAMAX ¼ 4.81 ILR ¼ 4.164KM ¼ 1.0 VAMIN ¼�3.85 VRMAX ¼ 4.81KG ¼ 1.0 VRMIN ¼�3.85

� 2009 IEEE. Reprinted with permission from IEEE Std 421.5TM-2005. ‘IEEERecommended Practice for Excitation System Models for Power System Stability Studies’


corresponding to load characteristics, and the operating conditions of the powersystem vary continuously. The input mechanical power to the generator must vary inresponse to these variations to maintain the frequency constant. So, to regulate thepower system frequency, speed control of the prime mover using a governor isrequired. For stability studies, relevant models of the prime mover and the associatedspeed control system should be developed to verify accurate performance. The mostpopular type of prime mover used in power systems is the turbine, either hydraulic orsteam, while the speed-governing systems are either mechanical-hydraulic or electro-hydraulic. In both types of speed-governing systems the valve position or gatecontrolling water or steam flow are determined by using hydraulic motors. The speedsensing for mechanical hydraulic governors is carried out by mechanical components,but for electro-hydraulic type it is implemented using electronic circuits. However,both types have similar dynamic performance. The following description of turbinesand governing systems is based on the IEEE definitions and standards [25].

3.6.1 Hydraulic turbinesThe hydraulic turbine is simply represented in a suitable manner for stability stu-dies by the block diagram depicted in Figure 3.14.

The transfer function is expressed as

Pm ¼ 1 � sTW

1 þ 0:5 sTWPGV ð3:62Þ

where

TW ≜ water starting time constant ¼ (L � V)/(HT � g) ‘typical value around1.0 s’

L ≜ length of penstockV ≜ water velocity

HT ≜ total headg ≜ acceleration due to gravity

PGV ≜ gate opening expressed in pu and provided by speed governor

Construction of a governing system for hydro-turbines is based on the mainparts shown in Figure 3.15, and a typical non-linear model is given in Figure 3.16as a block diagram with its data summarised in Table 3.4. A dashpot feedback isemployed for stability performance due to the effect of water inertia [26].

For stability studies, the speed governing system is commonly simplified asrepresented by the block diagram shown in Figure 3.17. Its parameters can becalculated in terms of those defined in Figure 3.15 by using the relations below:

T1; T3 ¼ TB

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiT2

B

4� TA

rð3:63Þ

1 – sTW

1 + 0.5 sTW

PmPGV

Figure 3.14 Model of a hydro-turbine


1/(1 + sTP) 1/s1/TG

ωref

δsTR

1 + sTR

Non-linearfunction

σ

ω

Gateposition

+ +

+

+

––

Rate limiter

Pilot valveand

Servomotor Distributor valve

andServomotor

ΣΣ

Σ

Figure 3.16 Model of speed governing system for hydro-turbines� 2009 IEEE. Reprinted with permission from IEEE Committee Report, ‘Dynamicmodels for steam and hydro turbine in power system studies’, IEEE Transactions onPower Apparatus and Systems. Nov/Dec 1973, vol. PAS-92, pp. 1904–15

Pilot valve andServo-meter

Distributor valve andgate Servo-meter

Governorcontrolled gates

Dashpot

Speedgovernor

Speed

Speed changerposition Gate

position

Speed control mechanism

Figure 3.15 Functional block diagram for hydro-turbines governing system� 2009 IEEE. Reprinted with permission from IEEE Committee Report, ‘Dynamicmodels for steam and hydro turbine in power system studies’, IEEE Transactions onPower Apparatus and Systems. Nov/Dec 1973, vol. PAS-92, pp. 1904–15

Table 3.4 Data of the model in Figure 3.16

Parameter Typical value Range

TR 5.0 2.5–25.0TG 0.2 0.2–0.4TP 0.04 0.03–0.05d 0.3 0.2–1.0s 0.05 0.03–0.06

TR ¼ 5TW and d¼ 1.25TW/H, generator inertia constant.


where

TA ¼ 1s

� �TRTG; TB ¼ 1

s

� �sþ dð ÞTR þ TG½ �

K ¼ 1s

� �and T2 ¼ 0

Po ≜ the initial power (load reference)

3.6.2 Steam turbinesAs an example, the turbine system shown in Figure 3.18 (tandem compound-singlereheat) is one of the various types of turbine systems, such as tandem compound-double reheat, cross compound-single reheat with one or two low-pressure (LP)turbines and cross compound-double reheat. It has one shaft on which all turbines;high pressure (HP), intermediate pressure (IP) and LP turbines are mounted. It canbe represented by the block diagram in Figure 3.19 where governor control valvesare used at the inlet to HP turbine to control the steam flow. The steam chest,re-heater and crossover piping introduce delays that are represented by time con-stants TCH, TRH and TCO, respectively. The total power developed in HP, IP and LPturbines is represented by the fractions FHP, FIP and FLP, respectively.

Controlvalves.Steamchest

Reheater Cross over

HP IP LPLP Shaft Valve

position

Figure 3.18 Configuration of steam system – tandem compound, single reheat

K(1 + sT1)

(1 + sT2)(1 + sT3)Δω

ΔP

+–

PMIN

PMAX

PGV

Po

Σ

Figure 3.17 General simplified model of speed governing system for hydro-turbines� 2009 IEEE. Reprinted with permission from IEEE Committee Report, ‘Dynamicmodels for steam and hydro turbine in power system studies’, IEEE Transactions onPower Apparatus and Systems. Nov/Dec 1973, vol. PAS-92, pp. 1904–15


Speedcontrol

Loadcontrol

Pressure/positioncontrol

Servo motorGovernorcontrolled

valves

Non-linearfeedback

Speedtransducer

Speedreference

Speed

Load ref Steam flowfeedback

Valve position

Figure 3.20 Functional block diagram of electro-hydraulic speed governingsystem for steam turbines� 2009 IEEE. Reprinted with permission from � 2009 IEEE. Reprinted withpermission from IEEE Committee Report, ‘Dynamic models for steam and hydroturbine in power system studies’, IEEE Transactions on Power Apparatus and Systems.Nov/Dec 1973, vol. PAS-92, pp. 1904–15

1 1 1

+

+

+

+

PGV

FHP FIP FLP

Pm

1 + sTCH 1 + TRH 1 + TCO

Σ Σ

Figure 3.19 Block diagram of Tandem compound, single reheat (typical values:TCH ¼ 0.1–0.4 s, TRH ¼ 4–11 s, TCO ¼ 0.3–0.5 s, FHP ¼ 0.3, FIP ¼ 0.3,FLP ¼ 0.4)� 2009 IEEE. Reprinted with permission from IEEE Committee Report, ‘Dynamicmodels for steam and hydro turbine in power system studies’, IEEE Transactions onPower Apparatus and Systems. Nov/Dec 1973, vol. PAS-92, pp. 1904–15

K(1 + sT2)

1 + sT1Σ 1/T3 1/s

Δω+

–

–

.

Pdown. PMIN

PMAX

PGV

Pup

Po

Figure 3.21 General simplified model for speed governor for steam turbine� 2009 IEEE. Reprinted with permission from � 2009 IEEE. Reprinted withpermission from IEEE Committee Report, ‘Dynamic models for steam and hydroturbine in power system studies’, IEEE Transactions on Power Apparatus and Systems.Nov/Dec 1973, vol. PAS-92, pp. 1904–15


The main construction of the governing system (Figure 3.20) includes afeedback from steam flow or pressure in the first stage turbine as well as a servo-motor feedback loop to improve linearity. A simplified model of governor controlsystem for steam turbines is shown by the block diagram in Figure 3.21. Typicalvalues of time constants in seconds are

Electro-hydraulic governor: T1 ¼ T2 ¼ T3 ¼ 0.025 � 0.15 sMechanical-hydraulic governor: T1 ¼ 0.2 � 0.3, T2 ¼ 0, T3 ¼ 0.1 s

References

1. De Oliveira S.E.M. ‘Modeling of synchronous machines for dynamicstudies with different mutual couplings between direct axis windings’. IEEETransactions on Energy Conversion. 1989;4(4):591–9

2. El-Serafi A.M., Abdallah A.S. ‘Effect of saturation on the steady-statestability of a synchronous machine connected to an infinite bus system’. IEEETransactions on Energy Conversion. 1991;6(3):514–21

3. Xu W.W., Dommel H.W., Marti J.R. ‘A synchronous machine model forthree-phase harmonic analysis and EMTP initialization’. IEEE Transactionson Power Systems. 1991;6(4):1530–8

4. Wang L., Jatskevich J. ‘A voltage-behind-reactance synchronous machinemodel for the EMTP-type solution’. IEEE Transactions on Power Systems.2006;21(4):1539–49

5. Wang L., Jatskevich J., Domme H.W. ‘Re-examination of synchronousmachine modeling techniques for electromagnetic transient simulations’.IEEE Transactions on Power Systems. 2007;22(3):1221–30

6. Sriharan S., Hiong K.W. ‘Synchronous machine modeling by standstillfrequency response tests’. IEEE Transactions on Energy Conversion. 1987;EC-2(2):239–45

7. Rusche P.E.A., Brock G.J., Hannett L.N., Willis J.R. ‘Test and simulation ofnetwork dynamic response using SSFR and RTDR derived synchronousmachine models’. IEEE Transactions on Energy Conversion. 1990;5(1):145–55

8. Verbeeck J., Pintelon R., Lataire P. ‘Relationships between parameter sets ofequivalent synchronous machine models’. IEEE Transactions on EnergyConversion. 1999;14(4):1075–80

9. Verbeeck J., Pintelon R., Lataire P. ‘Influence of saturation on estimatedsynchronous machine parameters in standstill frequency response tests’. IEEETransactions on Energy Conversion. 2000;15(3):277–83

10. Dedene N., Pintelon R., Lataire P. ‘Estimation of a global synchronousmachine model using a multiple-input multiple-output estimator’. IEEETransactions on Energy Conversion. 2003;18(1):11–6

11. Jackson W.B., Winchester R.L. ‘Direct and quadrature axis equivalent circuitsfor solid-rotor turbine generators’. IEEE Transactions on Power Apparatus andSystems. 1969;PAS-88(7):1121–36


12. Anderson P.M. Analysis of Faulted Power Systems. Ames, IA, US: Iowa StateUniv. Press; 1973

13. IEEE working group on computer modelling of excitation systems ‘Excitationsystem models for power system stability studies’. IEEE Transactions onPower Apparatus and Systems. 1981;PAS-100(2):494–509

14. Ruuskanen V., Niemel€a M., Pyrhonen J., Kanerva S., Kaukonen J. ‘Modellingthe brushless excitation system for a synchronous machine’. IET ElectricPower Applications. 2009;3(3):231–9

15. The Digital Excitation Task Force of the Equipment Working Group ‘Com-puter models for representation of digital based excitation systems’. IEEETransactions on Energy Conversion. 1996;11(3):607–15

16. IEEE Committee Report ‘Computer representation of excitation systems’.IEEE Transactions on Power Apparatus and Systems. 1968;PAS-87(6):1460–4

17. Wang J.C., Chiang H.D., Haung C.T., Chen Y.T., Chang C.L., Huang C.Y.‘Identification of excitation system models based on on-line digital mea-surements’. IEEE Transactions on Power Systems. 1995;10(3):1286–93

18. Benchluch S.M., Chow J.H. ‘A trajectory sensitivity method for the identifi-cation of nonlinear excitation system models’. IEEE Transactions on EnergyConversion. 1993;8(2):159–64

19. Puma J.Q., Colome D.C. ‘Parameters identification of excitation system modelsusing genetic algorithms’. IET Generation, Transmission & Distribution.2008;2(3):456–67

20. Liu C.S., Yuan-Yih H., Jeng L.H., Lin C.J., Huang C.T., Liu A.H., Li T.H.‘Identification of exciter constants using a coherence function based weightedleast squares approach’. IEEE Transactions on Energy Conversion. 1993;8(3):460–7

21. IEEE Committee Report (eds.). ‘Excitation system dynamic characteristics’.Proceedings of Power and Energy Society PES Summer Meeting;San-Francisco, CA, US, 1972. pp. 64–75

22. IEEE Task Force on Excitation Limiters. ‘Under-excitation limiter models forpower system stability studies’. IEEE Transactions on Energy Conversion.1995;10(3):524–31

23. IEEE Task Force on Excitation Limiters. ‘Recommended models for over-excitation limiting devices’. IEEE Transactions on Energy Conversion.1995;10(4):706–13

24. IEEE Std 421.5TM-2005. ‘IEEE Recommended Practice for ExcitationSystem Models for Power System Stability Studies’.

25. IEEE Std 421.1TM-2007. ‘IEEE Standard Definitions for Excitation Systemsfor Synchronous Machines’.

26. IEEE Working Group on Prime Mover and Energy Supply Models for SystemDynamic Performance Studies. ‘Hydraulic turbine and turbine control modelsfor system dynamic studies’. Transactions on Power Systems. 1992;7(1):167–79

27. IEEE Committee Report, ‘Dynamic models for steam and hydro turbine inpower system studies’, IEEE Transactions on Power Apparatus and Systems.Nov/Dec 1973, vol. PAS-92, pp. 1904–15.


Chapter 4

Modelling of transformers, transmissionlines and loads

Generation, transmission and distribution are the main three parts that comprise apower system. Each part has its own function for electric power: generation part togenerate the electric power commonly at medium voltage level by using synchro-nous generators, transmission part to transmit this power through high voltage orextra-high voltage transmission lines and finally distribution part to distribute theelectric power to feed the consumers’ loads at medium voltage or low voltagelevels. The interconnections between these three parts that operate at differentvoltages necessitate use of transformers, e.g. step-up/step-down power transfor-mers, distribution transformers, autotransformers. Accordingly, to start the stabilitystudies it is essential to model all elements – generators, transformers, transmissionlines and loads – in a manner that is convenient for this purpose. Modelling ofsynchronous generator has been explained in Chapters 2 and 3, and this chapterdeals with the modelling of the rest of the elements.

4.1 Transformers

Transformers in power systems are used not only to enable power transfer fromgenerator sending end to consumers at the receiving end but also to control thevoltage and reactive power flow using taps on transformer windings that may beused to change the transformer turns ratio. The most commonly used is either asingle-phase or a three-phase two-winding transformer. In some applications thetransformer may have a third winding, called the tertiary winding. Auto-transformers may be used to regulate the voltage when the transformation ratio issmall. In some applications, such as controlling the power circulation and pre-venting overloading of lines, phase-shift transformers are used.

4.1.1 Modelling of two-winding transformersThe equivalent circuit of a two-winding transformer is shown in Figure 4.1. Thesaturation effect can be neglected as the magnetising reactance Xm1 is large andthe quantities are in physical units. The relations below can be deduced from theequivalent circuit. It is to be noted that the parameters written in bold font are

vectors, the subscript ‘1’ denotes the primary side and subscript ‘2’ denotes thesecondary side of the transformer.

v1 ¼ Z1i1 þ n1

n2v2 � n1

n2Z2i2

v2 ¼ n2

n1v1 � n2

n1Z1i1 þ Z2i2

9>>=>>; ð4:1Þ

where

Zi ¼ Ri þ jXi and i ¼ 1 for primary winding, i ¼ 2 for secondary winding.Ri, Xi ¼ resistance and leakage reactance of ith winding, respectively.n1, n2 ¼ number of turns of primary and secondary windings, respectively.

For power system stability analysis, (4.1) should be written in per unit values.This requires a proper choice of primary and secondary base quantities. They usuallyare chosen based on the nominal turns ratio. Assume that the nominal numbers ofturns of primary and secondary sides are n1o and n2o, respectively. Accordingly,Zio ¼ Zi at the nominal ith side tap position. Therefore, in terms of these nominalvalues, and considering that the impedance is proportional to the square of thenumber of turns, this consideration is accepted as R << X and (ni � nio) is not verylarge; (4.1) can be rewritten as

v1 ¼ n1

n1o

� �2

Z1oi1 þ n1

n2v2 � n1

n2

n2

n2o

� �2

Z2oi2

v2 ¼ n2

n1v1 � n2

n1

n1

n1o

� �2

Z1oi1 þ n2

n2o

� �2

Z2oi2

9>>>>=>>>>;

ð4:2Þ

The transformer connection is assumed to be Y–Y. The nominal number ofturns is related to the base voltages by

n1o

n2o¼ v1 baseð Þ

v2 baseð Þ; v1 baseð Þ ¼ Z1 baseð Þi1 baseð Þ and v2 baseð Þ ¼ Z2 baseð Þi2 baseð Þ ð4:3Þ

21 Z1 Z2

Xm1

n1 : n2

i2i1

v1 v2

Figure 4.1 Basic equivalent circuit of a two-winding transformer


Using (4.3), (4.2) in per unit form becomes

v1 puð Þ ¼ n1 puð Þ2Z1o puð Þi1 puð Þ þn1 puð Þn2 puð Þ

v2 puð Þ �n1 puð Þn2 puð Þ

n2 puð Þ2Z2o puð Þi2 puð Þ

v2 puð Þ ¼n2 puð Þn1 puð Þ

v1 puð Þ �n2 puð Þn1 puð Þ

n1 puð Þ2Z1o puð Þi1 puð Þ þ n2 puð Þ2Z2o puð Þi2 puð Þ

9>>=>>; ð4:4Þ

where

n1 puð Þ ¼ n1

n1oand n2 puð Þ ¼ n2

n2o

Thus, (4.4) can be represented by the per unit equivalent circuit shown inFigure 4.2.

The per unit turns ratio n(pu) is expressed as

n puð Þ ¼n1 puð Þn2 puð Þ

¼ n1n2o

n1on2≜ off-nominal ratio ðONRÞ ð4:5Þ

and the equivalent impedance, Ze, is given by

Ze puð Þ ¼ n22 puð Þ Z1o puð Þ þ Z2o puð Þ

� � ¼ n2

n2o

� �2

Z1o puð Þ þ Z2o puð Þ� � ð4:6Þ

Applying (4.5) and (4.6) to the parameters of Figure 4.2 the standard form ofthe equivalent circuit in per unit values for a two-winding transformer can be drawnas in Figure 4.3. This form is valid to be used for representing transformersequipped with tap changers (off-load or on-load tap changers). It needs only tocalculate the corresponding values of ONR and Ze. On the other hand, this is notadequate for stability and load flow studies, and hence, it is converted to anequivalent p circuit. Assuming that the transformer is connected between bus, p,and bus, q, in the power network (Figure 4.4(a)), the p-equivalent circuit in generalform is as shown in Figure 4.4(b). The deduction of parameters in p circuit form isas below [1].

21 n1(pu)2Z1o(pu)

Xmi-pu

i1(pu)

n1(pu) : n2(pu)

v1(pu)

i2(pu)

v2(pu)

n2(pu)2Z2o(pu)

Figure 4.2 Equivalent circuit in per unit

Modelling of transformers, transmission lines and loads 83

From Figure 4.3, replacing the subscripts 1 and 2 by p and q, respectively, andassuming Ye(pu) ¼ 1/Ze(pu), the current at bus p, ip(pu), is given by

ip puð Þ ¼ vt puð Þ � vq puð Þ� � Ye

n puð Þ¼ vp puð Þ

n puð Þ� vq puð Þ

� �Ye

n puð Þ

¼ vp puð Þ � n puð Þvq puð Þ� � Ye puð Þ

n2puð Þ

ð4:7Þ

The current at bus q can be obtained by the same procedure as

iq puð Þ ¼ n puð Þvq puð Þ � vp puð Þ� �YeðpuÞ

nðpuÞð4:8Þ

Meanwhile, the currents at buses p and q for the p circuit (Figure 4.3(b)) arecalculated by

ip puð Þ ¼ y1 vp puð Þ � vq puð Þ� �þ y2vp puð Þ ð4:9Þ

iq puð Þ ¼ y1 vq puð Þ � vp puð Þ� �þ y3vq puð Þ ð4:10Þ

i2(pu)i1(pu)

21n(pu) : 1 Ze(pu)

v2(pu)v1(pu) vt(pu)

Figure 4.3 Standard per unit equivalent circuit for a two-winding idealtransformer

qp

Transformer

y2

q

vp(pu)

(a) (b)

ip(pu)

py1

y3 vq(pu)

iq(pu)

Figure 4.4 (a) Transformer connected in power network and (b) general form ofp circuit


Equating the admittance terms in (4.7) and (4.9) and also in (4.8) and (4.10),the parameters of the p circuit are found to be

y1 ¼ 1n puð Þ

Ye puð Þ; y2 ¼ 1n puð Þ

1n puð Þ

� 1

� �Ye puð Þ;

y3 ¼ 1 � 1n puð Þ

� �Ye puð Þ

9>>>>=>>>>;

ð4:11Þ

The p-equivalent circuit is shown in Figure 4.5 where its parameters are givenin terms of ONR and the leakage impedance of the transformer.

It is to be noted that the standard equivalent circuit (Figure 4.3) represents thesingle-phase equivalent of a three-phase transformer. To be taken into considerationis the value of the nominal turns ratio (n1o/n2o) equal to the ratio of line-to-line basevoltages on the primary and secondary sides regardless of winding connections, Y–Yor D–D. For Y–D connection, a factor of H3 should be accounted for but the phaseshift of 30� provided by this connection can be neglected as it has no effect on thestability studies.

Example 4.1 Find the model parameters of a three-phase, 60-Hz, two-windingtransformer with the data given below:

Transformer rating ¼ 500 MVANominal voltage of primary side and secondary side ¼ 400 kV and 10.5 kV,

respectivelyWinding connection: Y/YOff-load tap changer on primary side: 4 steps, 2.5% kV eachOn-load tap changer on secondary side: �10% kV in 8 stepsR10(pu) þ R20(pu) ¼ 0.003 pu on rating/phase, X10(pu) þ X20(pu) ¼ 0.12 pu on

rating/phase

qy1 = (1/n(pu))Ye(pu)

(1/n(pu))[(1/n(pu)) – 1]Ye(pu) [1 – (1/n(pu))]Ye(pu)

p

Figure 4.5 Transformer p-equivalent circuit


Solution:

As initial operating condition, it is assumed that the secondary winding is at itsnominal position and the primary winding is set one step above its nominal position(410 kV). The parameters of the equivalent circuit (Figure 4.3) calculated in pu oftransformer rated values with the ONR on the secondary side are

The initial ONR is n puð Þ ¼ 400410

10:510:5

¼ 0:976 by ð4:5Þ

The pu equivalent impedance referred to secondary side Ze(pu) ¼ (1/0.976)2 �(0.003 þ j0.12) ¼ 0.003152 þ j0.1261

The step value of tap changer on the primary side is 10 kV and on thesecondary side is 0.13125 kV.

Maximum pu turns ratio; nmax puð Þ ¼ 400410

10:6312510:5

¼ 0:9878

Minimum pu turns ratio; nmin puð Þ ¼ 400410

10:3687510:5

¼ 0:9634

pu turns ratio step; Dn puð Þ ¼ 1:058 � 10:5

400410

¼ 0:01219

Per unit parameters can be recalculated according to the system voltage andMVA base values. For instance, assuming primary system voltage base ¼ 410 kV,secondary system base voltage ¼ 10.5 kV and system MVA base ¼ 100 MVA; thecorresponding pu parameters are

the initial ONR; n puð Þ ¼ 0:976410400

10:510:5

¼ 0:9994

pu equivalent impedance; Ze puð Þ ¼ 0:003152 þ j0:1261ð Þ 400410

� �2 100500

¼ 0:0006 þ j0:024

minimum pu turns ratio; nmin puð Þ ¼ 0:9878410400

10:510:5

¼ 1:0125

minimum pu turns ratio; nmin puð Þ ¼ 0:9634410400

10:510:5

¼ 0:9875

pu turns ratio step; Dn puð Þ ¼ 0:01219410400

10:510:5

¼ 0:01249


Referring to Figure 4.5 and (4.11) the parameters of transformer p-equivalentcircuit representing the initial tap position are

y1 ¼ 1n puð Þ

Ye puð Þ ¼ 10:9994

10:0006 þ j0:024

¼ 1:04157 � j41:6627

y2 ¼ 1n puð Þ

1n puð Þ

� 1

� �Ye puð Þ ¼ 1

0:99941

0:9994� 1

� �1

0:0006 þ j0:024

¼ 0:00062 � j0:00249

y3 ¼ 1 � 1n puð Þ

� �Ye puð Þ ¼ 1 � 1

0:9994

� �1

0:0006 þ j0:024

¼ �0:00062 þ j0:0249

Example 4.2 Find the model parameters for a three-phase, three-winding, 60-Hztransformer with the data below:

Rating ¼ 500 MVA, high/low/tertiary nominal voltages ¼ 400/240/10.5 kVWinding connection (H/L/T): Y/Y/D

The positive-sequence impedances in pu on transformer MVA rating andnominal voltages tap position are given as

Zps ¼ 0:0016 þ j0:1392; Zst ¼ 0 þ j0:1633; Zpt ¼ 0 þ j0:4741

The on-load tap changer at high-voltage side: 400 � 40 kV in 20 steps.

Solution:

Modelling of three-winding transformer is based on measuring the impedancesdefined below from short-circuit tests and then obtaining the equivalent impedanceof each of the three windings (primary, secondary and tertiary) in pu on the sameMVA base. It should be taken into account that the primary, secondary and tertiarywindings may or may not have the same MVA rating. Under balanced conditionsand neglecting the effect of magnetising reactance, the single-phase equivalentcircuit of three-winding transformer is represented by three impedances connectedin a Y Figure 4.6. It is to be noted that the common star point is fictitious, i.e. it isnot related to the system neutral. It is most convenient if the impedances areexpressed in pu. The pu values must be calculated on the same MVA base even incase of having different ratings for the three windings. The same procedure of usingONR for two-winding transformers is applied to three-winding transformers as wellto consider the difference between the actual turns ratio and base voltages.


If short-circuit tests are carried out on a three-winding transformer, the follow-ing impedances, composed of resistance and leakage reactance, may be measured:

Zps ¼ impedance measured in the primary circuit with the secondary short-circuited and the tertiary open.

Zpt ¼ impedance measured in the primary circuit with the tertiary short-circuited and the secondary open.

Zst ¼ impedance measured in the secondary circuit with the tertiary short-circuited and the primary open.

If the above ohmic impedances are referred to the same voltage base and rat-ing, then their values in terms of the equivalent impedances of the three separatewindings Zp, Zs, Zt are given by

Zps ¼ Zp þ Zs

Zpt ¼ Zp þ Zt

Zst ¼ Zs þ Zt

9>>=>>; ð4:12Þ

Thus,

Zp ¼ 12

Zps þ Zpt � Zst

� �Zs ¼ 1

2Zps þ Zst � Zpt

� �Zt ¼ 1

2Zpt þ Zst � Zps

� �

9>>>>>>>=>>>>>>>;

ð4:13Þ

By substituting the data given above, it is found that

Zp ¼ 0:0008 þ j0:225; Zs ¼ 0:0008 � j0:0858; Zt ¼ �0:0008 þ j0:2491

p s

t

ONR

ONR

Zp Zs

Zt

Figure 4.6 Three-winding transformer representation


The equivalent circuit parameters, in D form, in pu on transformer MVA ratingand nominal voltages are (the subscripts have been written in upper case letters todenote D form)

ZPS ¼ ZpZs þ ZpZt þ ZsZt

Zt¼ �0:01537 þ j0:00035

�0:0008 þ j0:2491¼ 0:0016 þ j0:061

ZST ¼ ZpZs þ ZpZt þ ZsZt

Zp¼ �0:01537 þ j0:00035

0:0008 þ j0:225¼ 0:0016 þ j0:0684

ZPT ¼ ZpZs þ ZpZt þ ZsZt

Zs¼ �0:01537 þ j0:00035

0:0008 � j0:0858¼ �0:0057 � j0:1793

The corresponding parameters of equivalent D circuit, superscripted by 0, in puon system MVA base of 100 MVA and voltage bases, P/S/T, of 400/220/12.47 kVcan be calculated as below:

p s

T

Z ′PS

Z ′STZ ′PT

n ′PS : 1

n ′ST : 1n ′PT : 1

T

ZPS

ZSTZPT

sp


Z0PS ¼ ZPS

240220

� �2 100500

¼ 0:0004 þ j0:0145

Z0ST ¼ ZST

10:512:47

� �2 100500

¼ 0:0002 þ j0:0097

Z0PT ¼ ZPT

10:512:47

� �2 100500

¼ �0:0008 � j0:0254

n0PS ¼ 400400

220240

¼ 0:9167

n0ST ¼ 240220

12:4710:5

¼1:2956

n0PT ¼ 400400

12:4710:5

¼ 1:1876

Data of on-load tap changer:

n0PSmax ¼ 440400

400400

220240

¼ 1:0083

n0PSmin ¼ 360

400400400

220240

¼ 0:825

Dn0PS ¼ 1:0083 � 0:82520

¼ 0:0092

n0PTmax ¼ 440400

400400

12:4710:5

¼ 1:3064

n0PTmin ¼ 360

400400400

12:4710:5

¼ 1:0688

Dn0PT ¼ 1:3064 � 1:068820

¼ 0:01188

Each branch in equivalent D circuit can be represented by an equivalent pcircuit shown in Figure 4.5. Referring to (4.11) the parameters of p circuit are

PS branch:

y1 ¼ 1n0

PS

y0PS ¼ 10:9167

10:0004 þ j0:0145

¼ 2:0762 � j75:2619

y2 ¼ 1n0

PS

1n0PS

� 1

� �y0PS ¼ 0:1868 � j6:7736

y3 ¼ 1 � 1n0PS

� �y0PS ¼ 1 � 1

0:9167

� �1

0:0004 þ j0:0145¼ �0:1714 þ j6:2143


ST branch:

y1 ¼ 1n0ST

y0ST ¼ 11:2956

10:0002 þ j0:0097

¼ 1:6421 � j79:6432

y2 ¼ 1n0ST

1n0ST

� 1

� �y0ST ¼ �0:3747 þ j18:1746

y3 ¼ 1 � 1n0

ST

� �y0ST ¼ 0:4855 � j23:5483

PT branch:

y1 ¼ 1n0PT

y0PT ¼ 11:1876

1�0:0008 � j0:0254

¼ �1:0443 þ j33:1578

y2 ¼ 1n0PT

1n0PT

� 1

� �y0PT ¼ 0:1649 � j5:2389

y3 ¼ 1 � 1n0

PT

� �y0PT ¼ �0:1959 þ j6:222

4.1.2 Modelling of phase-shifting transformersThe phase-shifting transformer can be represented by a series admittance with anideal transformer connected between bus p and bus q and having a complex turnsratio, n ¼ n ffa (Figure 4.7), where a is the phase shift from bus p to bus q. In caseof load flow and transient stability studies, it is reasonable to consider equal phaseangle step size at different tap positions.

p s T p Ty1 = 1.6421 – j79.6432y1 = 2.0762 – j75.2619

y2 = –0.5747+ j18.1746

y1 = –1.0443 + j33.1578

y2 = 0.1868– j6.7736

y2 = 0.1649– j5.2389

y3 = 0.4855– j23.5483

y3 = –0.1714+ j6.2143

y3 = –0.1959+ j6.222

s

qp

Ideal transformer

Ye

s

isip

αn : 1

Figure 4.7 Representation of phase-shifting transformer


The turns ratio n ¼ n ffa as a complex quantity comprises real and imaginarycomponents and, therefore, it can be mathematically written as

n ffa ¼ vp

vq¼ as þ jbs ¼ n cos aþ j sin að Þ ð4:14Þ

Considering the transformer as ideal (no power loss) and considering a as apositive shift-angle, i.e. vp leads vq, the rated power on the primary side is related tothat on the secondary side by the relation:

vpi�p ¼ �vqi

�s ð4:15Þ

From (4.14) and (4.15) the current on the primary side, at bus p, is given by

ip ¼ � 1as � jbs

is ¼ Ye

as � jbsvq � vs

� � ¼ Ye

as � jbs

1as þ jbs

vp � vs

� �

¼ Ye

a2s þ b2

s

vp � as þ jbsð Þvs

� � ð4:16Þ

Similarly,

is ¼ Ye

as þ jbsas þ jbsð Þvs � vp

� � ð4:17Þ

The relations of the currents ip and is in terms of the voltages vp and vs given in(4.16) and (4.17) can be rewritten in matrix form as

ip

is

" #¼

Ye

a2s þ b2

s

�Ye

as � jbs

�Ye

as þ jbsYe

2664

3775 vp

vs

" #ð4:18Þ

It is found from (4.18) that the transfer admittance from bus p to bus s is not thesame as that from bus s to bus p as the admittance matrix is not symmetrical.Consequently, the model cannot be expressed by p-equivalent circuit. It is to benoted that the model of the ideal transformer given by the p circuit in Figure 4.5can be verified by (4.18) when substituting a by n and b by zero.

Example 4.3 Data for a two-winding phase-shifting transformer are given below.Neglecting the resistance per phase, find the elements of admittance matrix in(4.18) when a¼ 0� and 10� ‘at 8th step’.

MVA rating ¼ 42 MVAPrimary/secondary base voltage 110/110 kVLeakage reactance per phase, Xe ¼ 0.1633 puPhase-shift range and steps ¼ 30�, 24 stepsSystem voltage base ¼ 110/115 kVSystem MVA base ¼ 100 MVA


Solution:

Xe in pu on system voltage and MVA base: Xe ¼ 0:1633 110115

� �2 10042 ¼ 0:3557 pu

ONR: n ¼ 110110

115110

¼ 1:04545

The phase shift angle is in a range between amax ¼ 30� and amin ¼�30�

For a¼ 0�

Ye ¼ 1j0:3557

¼ �j2:81136 pu

The turns ratio ¼ as þ jbs ¼ n(cos aþ j sin a) ¼ 1.04545 þ j0

Ye

a2s þ b2

s

¼ �j2:5722�Ye

as � jbs¼ j2:6891

�Y e

as þ jbs¼ j2:6891

Therefore, the admittance matrix is

Y s ¼ j�2:5722 2:6891

2:6891 �2:81136

" #

For a¼ 10�

As the impedance changes with the phase-shift angle, the manufacturer providesan impedance multiplier ‘m’ at each desired angle. Thus, in general, Ye ¼ m/Ze

Ye ¼ �jm2:81136; the turns ratio ¼ as þ jbs ¼ n cos 10� þ j sin 10�ð Þ¼ 1:0295 þ j0:1815; and

Y s ¼ m�j2:5728 �0:5103 þ j2:6487

0:5103 þ j2:6487 �j2:81136

" #

4.2 Transmission lines

Transmission lines are characterised by distributed parameters: (i) series impedanceZ comprising conductor resistance R and inductance L, (ii) shunt conductance G dueto leakage currents between phases and ground and (iii) shunt capacitance C due tothe electric field between conductors.

The transmission line can be modelled by a p-equivalent circuit with lumpedparameters or a number of cascaded p circuits. This depends on the nature of studyand the length of the line. However, p-equivalent circuit is an adequate modelfor studying power system stability. The model is designed based on assumptionssuch as (i) the line is transposed to consider that the three phases of the line are


symmetric, i.e. equal self-impedances of all phases and equal mutual impedancesbetween any two phases as well, and (ii) the line parameters are constant. Inaddition, the relationship between voltage and current in terms of line parametersshould be defined as below [2].

4.2.1 Voltage and current relationship of a lineA transmission line of length l between receiving and sending ends can be repre-sented as shown in Figure 4.8, noting that the voltages and currents are phasorsrepresenting time-varying quantities.

Considering a differential section of length ds at a distance s from the receivingend, the differential voltage across the incremental length ds is given by

dv ¼ i zdsð ÞHence,

dvds

¼ zi ð4:19Þ

The differential current flowing into the shunt admittance is

di ¼ v ydsð ÞHence,

dids

¼ yv ð4:20Þ

Differentiate (4.19) and (4.20) with respect to s to get

d2v

ds2¼ z

dids

andd2i

ds2¼ y

dvds

ð4:21Þ

To solve these two second-order differential equations, the initial conditionsneed to be considered.

dVI + dI IIS

VS VV + dV

sds

l

yds

zdsRdsωLds

Receivingend

Sendingend

IR

VR

Figure 4.8 Distributed parameter line (z and y are the line impedance andadmittance per unit length, respectively)


At s ¼ 0: the voltage v ¼ VR ffF1 and the current i ¼ IR ffF2 where the voltageand current at the receiving end are assumed to be known. The general solutiongiving the voltage and current at a point distance s from the receiving end, asphasors (VS and IS), is given by

VS ¼ VR þZCIR

2egs þ VR � ZCIR

2e�gs

IS ¼ V=ZCð Þ þ IR

2egs � VR=ZCð Þ � IR

2e�gs

9>>=>>; ð4:22Þ

where

ZC ≜ the characteristic impedance ¼ffiffizy

qg ≜ the propagation constant ¼ ffiffiffiffiffi

zyp ¼ aþ jb

a ≜ the attenuation constantb ≜ the phase constant

The exponential terms may be written in the expanded form as

egs ¼ e aþjbð Þs ¼ easðcos bs þ j sinbsÞ and

e�gs ¼ e� aþjbð Þs ¼ e�asðcosbs � j sinbsÞ

In (4.22) both voltage and current consist of two terms: the first term is definedas the incident component and the second is called the reflected component.

Equation (4.22) can be re-arranged to be written in the form

VS ¼ VRegs þ e�gs

2þ ZCIR

egs � e�gs

2¼ VR cosh gsð Þ þ ZCIR sinh gsð Þ

Similarly; IS ¼ 1ZC

VR sinh gsð Þ þ IR cosh gsð Þ

9>>=>>; ð4:23Þ

4.2.2 Modelling of transmission linesA transmission line from the sending to the receiving ends can be represented byp-equivalent circuit as shown in Figure 4.9. It includes series equivalent impedanceZe and two equivalent shunt admittances, Ye/2 each. The voltage at the sending end,VS, in terms of the receiving end voltage, VR, is calculated by

VS ¼ Ze IR þ Ye

2VR

� �þ VR ¼ ZeYe

2þ 1

� �VR þ ZeIR ð4:24Þ

Voltage at the sending end can be obtained from (4.23) by substituting s ¼ l:

VS ¼ VR cosh glð Þ þ ZCIR sinh glð Þ ð4:25Þ


Equating (4.24) and (4.25) gives

Ze ¼ ZC sinh glð Þ andZeYe

2þ 1

� �¼ cosh glð Þ

Therefore,

Ze ¼ ZC sinh glð ÞYe

2¼ 1

ZC

cosh glð Þ � 1sinh glð Þ ¼ 1

ZCtanh

gl

2

� �9>=>; ð4:26Þ

It is to be noted that if gl � 1 then

Ze ¼ ZC sinhðglÞ ZCðlÞ zl ¼ Z and

Ye

2¼ 1

ZCtanh

gl

2

� � 1

ZC

gl

2 yl

2¼ Y

2

In this case, the parameters of p-equivalent circuit are the total impedance andtotal admittance of the line. The equivalent circuit is called the ‘nominalp-equivalent circuit’, which is accepted for medium-length overhead lines (usuallyused for high and extra high voltage networks) of length in the range of 80–200 km.

In general, short overhead lines, l < 80 km, may be represented by their seriesimpedance by ignoring the shunt admittance. Medium-length overhead lines, 80 <l < 200 km, may be represented by a nominal p-equivalent circuit. Long overheadlines, l > 200 km, can be divided into a number of cascaded medium length sec-tions, each section represented by a nominal p-equivalent circuit to partially takeinto consideration the effect of distributed nature of the line parameters.

Example 4.4 Parameters of a typical 230-kV overhead transmission line of 100 kmlength are x ¼ 0.488 W/km, r ¼ 0.05 W/km, y ¼ 3.371 mmho/km. Calculate thecharacteristic impedance, ZC, and propagation constant ‘g’. Find the p-equivalentcircuit in pu of ZC.

S R

VS VR

IS IR

Ze

Ye/2 Ye/2

Figure 4.9 p-equivalent circuit for transmission line representation


Solution:

ZC ¼ffiffiffiz

y

r¼

ffiffiffiffiffiffiffiffiffiffiffiffir þ jx

y

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:05 þ j0:488

j3:371 � 10�6

s 380W

g ¼ ffiffiffiffiffizy

p ¼ aþ jb ¼ jffiffiffiffiffixy

p1 � j

r

2x

¼ j1:2826 � 10�3 1 � j

0:050:976

� �

¼ 0:0000657 þ j00128

Hence,

a¼ 0.0000657 nepers/km and b¼ 00128 rad/kmNeglecting the resistance, XL ¼ 0.488 � 100 ¼ 48.8 WYL ¼ 3.371 � 10�6 � 100 ¼ 0.000371 mhoIn pu of ZC, the values of XL and YL areXL ¼ 48.8/380 ¼ 0.128 puYL ¼ 0.000371 � 380 ¼ 0.141 pu

4.3 Loads

In general, to perform power system analysis, models must be developed for allpertinent system components. Inadequate modelling causing under/over-building ofthe system or degrading reliability may prove to be costly. For stability studies ofthe power system a balance between generated power and demand power by loadsmust be maintained to keep the system continuously in stable operation. Therefore,load characteristics are of crucial importance to be employed in system analysis asthey have a significant effect on system performance and highly impact the stabilityresults. To achieve that, load modelling must be determined in such a way that themodel is relevant to the nature of study and helps obtain useful and, to a largeextent, accurate results. Accurate modelling of loads is a difficult task as, forinstance, the power system includes a huge number of diverse load components indifferent locations with different characteristics and their composition changesfrom time to time. In addition, lack of data regarding the loads all over the systemand lack of a tool to develop models on a large-scale basis makes load modelling aformidable task.

XL = 0.128

YL/2 = 0.07 YL/2 = 0.07


Two main approaches to load model development have been considered byutilities: measurement-based and component-based [3]. The first approach involvesmonitors used at various load points to determine the load sensitivity (active andreactive power) to voltage and frequency variations to be used directly or to iden-tify parameters for a more detailed load model. This approach has the advantage ofproducing load model parameters directly in the form needed for power flow andtransient stability programmes through the direct monitoring of the true load. Onthe other hand, its cost is high because of acquiring and installing the measurementequipment and the need to monitor all system loads. As well, the measurementsmust be repeated as the load changes.

The second approach, component-based approach, implies building the loadmodel from information on its constituent parts as shown in Figure 4.10. It requiresthree sets of data: (i) load class mix data that describe the percentage contributionof each of several load classes to the total active power load at the bus, (ii) loadcomposition data that describe the percentage contribution of each of severalload components to the active power consumption of a particular load class and(iii) load characteristics data that describe the electrical characteristics of each ofthe load components. This approach has the advantage of not requiring systemmeasurements and consequently being more readily put into use, and the possibilityof using standard model for each component as well. It is to be noted that the loadclass mix data need to be prepared for each bus or area, and updated for changes inthe system load [4].

Bus Load

Industrial

Commercial

Residential

ResistanceHeating

Room AirConditioner

Lighting

WaterHeating

Component Characteristics

dVdP

dVdQ

dfdP

df

dQpf

0.82 0.5 2.5 0.6 –2.8

1.0 1.54 0.0 0.0 0.0

1.0 2.0 0.0 0.0 0.0

MotorParameters

1.0 2.0 0.0 0.0 0.0 -

-

-

etc

ClassCompositionLoad Mix

Figure 4.10 Terminology of component-based load modelling� 2009 IEEE. Reprinted with permission from Price W.W., Chiang H.D., Clark H.K.,Concordia C., Lee D.C., Hsu J.C. et al. ‘Load representation for dynamic performanceanalysis’. IEEE Transactions on Power Systems. 1993;8(2):472–82


Several efforts have been made to develop methods for constructing improvedload models [5–9]. The basic load models can be divided into two classes: staticand dynamic.

4.3.1 Static load modelsStatic load models express the active and reactive powers at any instant of time asfunctions of the bus voltage magnitude and frequency at the same instant. Thesemodels are used both for essentially static load components, e.g. resistive andlighting load, and as an approximation for dynamic load components, e.g. motor-driven loads. Polynomial and exponential representations are the two forms thatcan be used to perform static load models [10].

i. Polynomial representation

The power relationship to voltage magnitude is represented as a polynomialequation in the form below.

P ¼ Po ao þ a1V

Vo

� �þ a2

V

Vo

� �2" #

ð4:27Þ

Q ¼ Qo bo þ b1V

Vo

� �þ b2

V

Vo

� �2" #

ð4:28Þ

where Vo, Po and Qo are the initial values of voltage, power and reactive power,respectively (the initial system operating condition for study), when representing abus load. If this model is used for representing a specific load device, Vo should bethe rated voltage of the device and Po and Qo should be the power consumed atrated voltage. The model in this case is composed of sum of three terms; each termrepresents a model as (i) constant impedance, Z, model, where the load powervaries directly with the square of the voltage magnitude. It may be also called aconstant admittance model, (ii) constant current, I, model, where the load powervaries directly with the voltage magnitude. It has been accepted that, in the absenceof data, composite load can be approximated using a constant current load modeland (iii) constant power, P, model, where the load power does not vary withchanges in the voltage magnitude. It may be also called a constant MVA model.This type of load draws higher current under low-voltage conditions to maintainconstant power. So, it has a problem of non-applicability for cases involving severevoltage drops. The coefficients ao, a1, a2 and bo, b1, b2 are the fractions of theconstant power, constant current and constant impedance components in the activeand reactive power loads, respectively. They have the relations as described below:

ao þ a1 þ a2 ¼ 1

bo þ b1 þ b2 ¼ 1

�ð4:29Þ

The composite load model represented by (4.27) and (4.28) is sometimesreferred to as ZIP model and its parameters are the coefficients ao, a1, a2 and bo, b1,b2 and the power factor of the load.


ii. Exponential representation

The power relationship to voltage magnitude is represented as an exponentialequation in the form below:

P ¼ PoV

Vo

� �np

and Q ¼ QoV

Vo

� �nq

ð4:30Þ

The parameters of this model are the exponents np and nq. By setting theseexponents to 0, 1 or 2, the load can be represented by constant power, constantcurrent or constant impedance models, respectively. Other exponents can be used torepresent the aggregate effect of different types of load components where expo-nents greater than 2 or less than 0 may be appropriate for some types of loads. Twoor more terms with different exponents are sometimes included in the two relationsof (4.30). For instance, when a bus in the power system is chosen as a load bus, toinclude both voltage dependence and the effect of frequency variations, the activepower can be expressed as

P ¼ Po C1V

Vo

� �np1

1 þ kpDf� �þ 1 � C1ð Þ V

Vo

� �np2" #

ð4:31Þ

where

C1 ≜ the frequency dependent fraction of active power loadnp1 ≜ the voltage exponent for frequency-dependent component of active

power loadnp2 ≜ the voltage exponent for frequency-independent component of active

power loadDf ≜ the per unit frequency deviation from nominalkp ≜ frequency sensitivity coefficient for the active power load

To add the effect of load compensation, the reactive power is expressed as

Q ¼ Po C2V

V o

� �nq1

1 þ kq1Df� �þ Qo

Po� C2

� �V

Vo

� �nq2

1 þ kq2Df� �" #

ð4:32Þ

where

C2 ≜ the reactive load coefficient-ratio of initial uncompensated reactive loadto total initial active power load Po

nq1 ≜ the voltage exponent for the uncompensated reactive loadnq2 ≜ the voltage exponent for the reactive compensation termkq1 ≜ the frequency sensitivity coefficient for the uncompensated reactive

power loadkq2 ≜ the frequency sensitivity coefficient for reactive compensation


The reactive power is normalised to Po rather than Qo to avoid difficulties whenQo equals zero due to the cancellation of the load reactive consumption and reactivelosses by shunt capacitance. The first term includes the reactive power consumptionof all of the load components, and it is built up using the power factors of theindividual load components. The second term, to a first approximation, representsthe effect of reactive losses and compensation in the sub-transmission and dis-tribution system between the bus and the various loads. The two terms includefrequency sensitivity.

4.3.2 Dynamic load modelsStatic models explained above may be accepted for application to composite loadshaving fast response to voltage/frequency changes and reaching the steady staterapidly. Some cases necessitate considering the dynamics of load components suchas discharge lamps, protective relays, thermostatic controlled loads, transformerswith LTCs and motors. Motors represent the major portion of load componentsregardless of the class of the load (industrial, commercial or residential), and thissection, thus, focuses on the dynamic modelling of motors, in particular, inductionmotors [11–15].

4.3.2.1 Induction motor model

The equivalent circuit of an induction machine in steady state can be in one of thetwo forms shown in Figure 4.11(a, b). The only difference between the two circuitsis that the rotor power is represented by its two components, resistance loss andshaft power, in Figure 4.11(b). All quantities in the equivalent circuit are referred tothe stator side. In motor operation the slip is positive and the directions of currentsshown are positive. More details of equivalent circuit for double-cage inductionmachines with saturation effects and deep-bar induction machines as well can befound in [16, 17].

For stability studies, the DC component in the stator transient currents isneglected permitting representation of only fundamental frequency components.Neglecting stator transients and the rotor windings shortened, the per unit inductionmotor electrical equations of the simplest model can be written as below.

Rs Xs Xr

Xm

Is Ir

Rr(1 – S)/S

P(rotor)P(rotor)

Rs Xs Xr

Xm

Is Ir

Rr /S

Rr

P(shaft)

Vs Vs

(a) (b)

Figure 4.11 Induction machine equivalent circuit: (a) representation of totalrotor power and (b) representation of rotor power components


Dynamics of the rotor inertia is described by

dwr

dt¼ 1

2HTe � Tmð Þ ð4:33Þ

where wr is the per unit motor speedTm is the per unit mechanical torque and is a function of wr as given by the

relation:

Tm ¼ Tmo Awr2 þ Bwr þ C

� � ð4:34Þ

Te is the per unit electrical torque and is a function of the motor slip. It iscomputed from the steady-state equivalent circuit shown in Figure 4.11 as

Te ¼ I2r R

Sð4:35Þ

where H is the motor inertia constantTypical data of induction motors, coefficients A, B and C, and parameters of

the equivalent circuit in different installations can be found in [18].By including the rotor transients, the simplified equivalent circuit for stability

studies is shown in Figure 4.12, where E0 is a complex voltage source behindtransient impedance, X 0

s , and is defined by

dE0

dt¼ �j2pfSE0 � 1

ToE0 � j X � X 0

s

� �It

� � ð4:36Þ

where f is the operating frequency

To ¼ Xr þ Xm

2pfRr; It ¼ V � E0

t

Rs þ jX 0s

¼ iq þ jid

X ¼ Xs þ Xm and X 0s ¼ Xs þ XmXr

Xm þ Xr

ð4:37Þ

+

Vt

Rs

It

X ′ s

E ′

Figure 4.12 Transient-equivalent circuit of induction machine (Vt is the statorterminal voltage, E0 is the voltage behind transient impedance)


Also, E0 in (4.36) can be expressed by two real values E0d and E0

q in the d–qframe of reference as

dE0d

dt¼ �ðws � wrÞE0

q þ1To

ðX � X 0sÞiq �

1To

E0d

dE0q

dt¼ ðws � wrÞE0

d �1To

ðX � X 0sÞid � 1

ToE0

q

9>>>=>>>;

ð4:38Þ

where ws ¼ 2pfThe electrical torque can be calculated by using the relation: Te ¼ E0

did þ E0qiq

4.4 Remarks on load modelling for stability and powerflow studies

● The common practice is to represent the composite load characteristics as seenfrom bulk power delivery points (Figure 4.13).

● To ensure accuracy, stability studies should employ good dynamic load modelsincluding the effect of motor rotor flux transients, discharge lighting dis-continuities, effect of LTCs on load magnitude after a disturbance, saturationeffects in transformers and motors and similar phenomena.

● Static load models may be adequate in such cases that give the same results asmore detailed dynamic models. So, comparison of static load models anddetailed dynamic models using typical data in both cases should be imple-mented to decide which model is considered in the study [19].

● Data gathering is essential to represent composite load characteristics. Twoapproaches can be applied to obtain the data: (i) by direct measurement of thevoltage and frequency sensitivity of load P and Q at representative substationsand feeders, (ii) by building up a composite load model from knowledge of themix of load classes served by a substation, the composition of each class andtypical characteristic of each load component. In general, both should be usedas they are complementary and desirable to understand and predict load char-acteristics under varying conditions.

TransmissionZone

substation SubtransmissionDistributionsubstation

Primary feeders

Secondary feeders

Individualloads

Industrialloads

Powerdelivery

point

Figure 4.13 Illustration of a bulk power delivery point in a part of power system


References

1. Stagg G.W., El-Abiad A.H. Computer Methods in Power System Analysis.New York, USA: McGraw-Hill; 1968

2. Westinghouse Electric Corporation. Electrical Transmission and DistributionReference Book. East Pittsburgh, PA, USA; 1964

3. Price W.W., Wirgua K.A., Murdoch A., Mitsche J.V., Vaahedi E., El-Kady M.‘Load modeling for power flow and transient stability computer studies’.IEEE Transactions on Power Systems. 1988;3(1):180–7

4. Shi J.H., Renmu H. (eds.). ‘Measurement-based load modeling – modelstructure’. IEEE Bologna Power Tech Conference Proceedings, 2003 IEEEBologna; Italy, vol. 2, June 2003

5. Pai M.A., Sauer P.W., Lesieutre B.C. ‘Static and dynamic nonlinear loads andstructural stability in power systems’. Proceedings of the IEEE. 1995;83(11):1562–72

6. Wang J.C., Ciang H.D., Chang C.L., Liu A.H. ‘Development of a frequency-dependent composite load model using the measurement approach’. IEEETransactions on Power Systems. 1994;9(3):1546–56

7. Song Y.H., Dang D.Y. (eds.). ‘Load modeling in commercial power systemsusing neural networks’. Industrial and Commercial Power Systems TechnicalConference, 1994. Conference Record, Papers Presented at the 1994 AnnualMeeting, 1994 IEEE; Irvine, CA, USA, May 1994. pp. 1–6

8. Hsu C.T. ‘Transient stability study of the large synchronous motors startingand operating for the isolated integrated steel-making facility’. IEEE Trans-actions on Industry Applications. 2003;39(5):1436–41

9. Shimada T., Agematsu S., Shoji T., Funabashi T., Otoguro H., Ametani A.(eds.). ‘Combining power system load models at a busbar’. IEEE PowerEngineering Society Summer Meeting, 2000 IEEE; Seattle, WA, USA, 2000.pp. 383–88

10. Coker M.L., Kgasoane H. (eds.). ‘Load modeling’. 5th Africon Conference inAfrica, 1999 IEEE Africon; Cape Town, South Africa, Sep/Oct 1999, vol. 2.pp. 663–8

11. Kao W.S., Huang C.T., Chiou C.Y. ‘Dynamic load modeling in Taipowersystem stability studies’. IEEE Transactions on Power Systems. 1995;10(2):907–14

12. Houlian C., Shande S., Shouzhen Z. (eds.). ‘Radial basis function networksfor power system dynamic load modeling’. TENCON ’93, Proceedings;Computer, Communication, Control and Power Engineering. 1993 IEEERegion 10 Conference on; Beijing, China, Oct 1993. pp. 179–82

13. Karlsson D., Hill D.J. ‘Modeling and identification of nonlinear dynamicloads in power systems’. IEEE Transactions on Power Systems. 1994;9(1):157–66

14. Zhu S.Z., Dong Z.Y., Wong K.P., Wang Z.H. (eds.). ‘Power system dynamicload identification and stability’. Power System Technology, 2000.


Proceedings. PowerCon 2000. International Conference on; Perth, WA,USA, Dec 2000, vol. 1. pp. 13–18

15. Vaahedi E., Zein El-Din H.M.Z., Price W.W. ‘Dynamic load modeling inlarge scale stability studies’. IEEE Transactions on Power Systems. 1988;3(3):1039–45

16. Hung R., Dommel H.W. ‘Synchronous machine models for simulation ofinduction motor transients’. IEEE Transactions on Power Systems. 1996;11(2):833–8

17. Price W.W., Chiang H.D., Clark H.K., Concordia C., Lee D.C., Hsu J.C. et al.‘Load representation for dynamic performance analysis’. IEEE Transactionson Power Systems. 1993;8(2):472–82

18. IEEE Task Force on Load Representation for Dynamic Performance. ‘Standardload models for power flow and dynamic performance simulation’. IEEETransactions on Power Systems. 1995;10(3):1302–13

19. Kao W.S., Lin C.J., Huang C.T., Chen Y.T., Chiou C.Y. ‘Comparison ofsimulated power system dynamics applying various load models with actualrecorded data’. IEEE Transactions on Power Systems. 1994;9(1):248–54


Part II

Power flow

Chapter 5

Power flow analysis

Models of the individual components of the electric power system are described inPart I. The purpose of this chapter is to study mathematical relations betweenindividual components to develop a model for the overall power system network,which is made of an interconnection of the various components. This modelshowing the currents, voltages, real power and reactive power flows at each bus inthe network is known as Power Flow or Load Flow model. It is found that allrelationships – between voltage and current at each bus, between the real andreactive power demand at a load bus or the generated real power and scheduledvoltage magnitude at a generator bus – are non-linear. Therefore, power flow cal-culation implies the solution of a set of non-linear equations to give the electricalresponse of the transmission system to a particular set of loads and generator poweroutputs. In practice, the distribution system is not represented in power flow studiesof bulk transmission systems and the loads are represented at substation levels. Inaddition, some assumptions regarding component modelling are made dependingon the operating condition, whether it is in steady state or under a contingency, andshould be consistent with the time period and purpose of study. A single-phaseequivalent representation of the power network is used in power flow studies as thesystem is generally assumed to be balanced. Section 5.1 is focused on the generalconcepts of AC power flow calculation methods using bus admittance matrixbecause of its relevance to the needs of the stability studies. In particular, Newton–Raphson and fast-decoupled methods have been explained with illustrative exam-ples because of their accuracy and fast convergence.

5.1 General concepts

For a network with n independent buses, applying Kirchhoff’s law at each bus thefollowing n equations can be written [1] as

Y11V1 þ Y12V2 þ � � � þ Y1nVn ¼ I1

Y21V1 þ Y22V2 þ � � � þ Y2nVn ¼ I2

..

.

Yn1V1 þ Yn2V2 þ � � � þ YnnVn ¼ In

9>>>>=>>>>;

ð5:1Þ

Equation (5.1) is expressed in matrix form as

Y11 Y12

Y21 Y22� � � Y1n

Y2n

..

. . .. ..

.

Yn1 Yn2 � � � Ynn

26664

37775

V1

V2

..

.

Vn

26664

37775 ¼

I1

I2

..

.

In

26664

37775 or dYe V½ � ¼ I½ � ð5:2Þ

where

I ≜ bus current injection vectorV ≜ bus voltage vectorY ≜ bus admittance matrix

Yii ≜ the diagonal element of the bus admittance matrix, called the ‘self-admittance of bus i’. It equals the sum of all branch admittances con-necting to bus i ‘yio þ yi2 þ � � � þ yin’

yio ≜ the total capacitive susceptance at bus iYij ≜ the off-diagonal element of the bus admittance matrix, called ‘mutual-

admittance of bus i’. It equals the negative of branch admittance betweenbuses i and j.

It is noted that the off-diagonal element Yij is zero if there is no line betweenbuses i and j. Also, the bus admittance matrix is, in general, a sparse matrix.

The bus current in terms of bus voltage and power can be written as

I i ¼ S�i

V�i

¼ P netð Þi � jQ netð Þi� �

V�i

ð5:3Þ

where

S ≜ the complex power injection vector, the superscript ‘*’ denotes theconjugate vector

P(net)i ≜ the net real power injected to bus i ¼ PGi � PLi

Q(net)I ≜ the net reactive power injected to bus i ¼ QGi � QLi

PGi ≜ the real power output of the generator connected to bus iPLi ≜ the real power demand of the load connected to bus iQGi ≜ the reactive power output of the generator connected to bus iQLi ≜ the reactive power demand of the load connected to bus i

From (5.1) and (5.3) the following relation can be obtained:

P netð Þi � jQ netð ÞiV�

i

¼ Yi1V1 þ Yi2V2 þ � � � þ YinVn; i ¼ 1; 2; . . . ; n ð5:4Þ

or

P netð Þi þ jQ netð Þi ¼ V i

Xn

j¼1

Y �ijV

�j ; i ¼ 1; 2; . . . ; n ð5:5Þ


By equating real and imaginary parts in (5.5), two equations for each bus areobtained in terms of four variables P, Q, V and angle q. Thus, two of these variablesat each bus should be specified to solve the power flow equations and determine theother two variables. According to the known variables of the bus and the operatingconditions of the power system as well, the buses are classified into three types:

Type 1 – PV bus: The real power P and voltage magnitude |V| are known whilethe reactive power Q and voltage angle q are unknown. The bus connected to thegenerator is usually a PV bus.

Type 2 – PQ bus: The real and reactive power, P and Q, respectively, areknown and voltage magnitude and angle, V and q, are unknown. The bus connectedto a load is commonly a PQ bus as well as the bus at which a generator of constantor un-adjustable output power is connected.

Type 3 – Slack bus: It is also called Swing bus or Reference bus. During thepower flow calculations the power loss of the network is unknown till the end ofthe power flow solution. So, a generator bus – called slack bus – is selected. Thevoltage magnitude and phase angle at this bus are specified so that the unknownpower losses are also assigned to this bus in addition to the balance of generation.Traditionally, there is only one slack bus in the power flow calculations. As thevoltage of the slack bus is given, only n � 1 bus voltages need to be calculated andaccordingly the number of power flow equations is 2(n � 1).

The conventional methods used to solve the power flow equations are pre-sented in Sections 5.2 through 5.4. These methods have some common features asthey are iterative computational methods because of the non-linearity of equationsand start with guessing an initial solution.

5.2 Newton–Raphson method

The general form of a set of non-linear equations with n variables is

f1 x1; x2; . . . ; xnð Þ ¼ 0

f2 x1; x2; . . . ; xnð Þ ¼ 0

..

.

fn x1; x2; . . . ; xnð Þ ¼ 0

9>>>>=>>>>;

ð5:6Þ

To solve this set of non-linear equations, an initial solution, xoi , i ¼ 1, 2, . . . , n,

is selected. The difference between the initial value xoi and the final solution x will

be Dxo, i.e. x ¼ xo þ Dxo is the solution of non-linear (5.6). Thus,

f1 x01 þ Dxo

1; xo2 þ Dxo

2; . . . ; xon þ Dxo

n

� � ¼ 0

f2 x01 þ Dxo

1; xo2 þ Dxo

2; . . . ; xon þ Dxo

n

� � ¼ 0

..

.

fn x01 þ Dxo

1; xo2 þ Dxo

2; . . . ; xon þ Dxo

n

� � ¼ 0

9>>>>=>>>>;

ð5:7Þ

Power flow analysis 111

Applying Taylor series expansion to (5.7) and ignoring the second and higherderivatives gives

f1 xo1; x

o2; . . . ; xo

n

� � þ @f1

@x1

��xo

1

Dx01 þ � � � þ @f1

@xn

��xo

n

Dx0n ¼ 0

f2 xo1; x

o2; . . . ; xo

n

� � þ @f2

@x1

��xo

1

Dx01 þ � � � þ @f2

@xn

��xo

n

Dx0n ¼ 0

..

.

fn xo1; x

o2; . . . ; xo

n

� � þ @fn

@x1

��xo

1

Dx01 þ � � � þ @fn

@xn

��xo

n

Dx0n ¼ 0

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð5:8Þ

and in matrix form

f1 xo1; xo

2; . . . ; xon

� �f2 xo

1; xo2; . . . ; x

on

� �...

fn xo1; xo

2; . . . ; xon

� �

266666666

377777777¼ �

@f1

@x1

��xo

1

@f1

@x2

��xo

2

@f2

@x1

��xo

1

@f2

@x2

��xo

2

. . .

@f1@xn

��xo

n

@f2@xn

��xo

n

..

. ...

@fn

@x1

��xo

1

@fn@x2

��xo

2

� � �

..

.

@fn

@xn

��xo

n

266666666666664

377777777777775

Dxo1

Dxo2

..

.

Dxon

266664

377775 ð5:9Þ

Thus, DXo ¼ ½Dxo1;Dxo

1; . . .;Dxo1�t can be calculated from (5.9) and, therefore,

the new solution is obtained. This solution is an approximate solution as the higher-order derivative terms of Taylor series are neglected. Therefore, the solution is anapproximation as well, i.e. not the real solution. Consequently, further iterations arerequired. The iteration equations can be expressed as

f1 xk1; xk

2; . . . ; xkn

� �f2 xk

1; xk2; . . . ; x

kn

� �...

fn xk1; xk

2; . . . ; xkn

� �

266666666

377777777¼ �

@f1

@x1

��xk

1

@f1

@x2

��xk

2

@f2

@x1

��xk

1

@f2

@x2

��xk

2

. . .

@f1@xn

��xk

n

@f2@xn

��xk

n

..

. ...

@fn

@x1

��xk

1

@fn@x2

��xk

2

� � �

..

.

@fn

@xn

��xk

n

266666666666664

377777777777775

Dxk1

Dxk2

..

.

Dxkn

2666664

3777775 ð5:10Þ

and

xkþ1i ¼ xk

i þ Dxki ; i ¼ 1; 2; . . . ; n 5:11

The iteration can be stopped if maxjDxij � e; i ¼ 1; 2; . . . ; n, where e is a smallpositive number that represents the permitted convergence precision.


In matrix form, (5.10) and (5.11) are written as

F X k� � ¼ �J kDX k

X kþ1 ¼ X k þ DX k

)ð5:12Þ

where J ≜ n � n Jacobian matrixThe mathematical principles explained above can be applied to solve the non-

linear power flow equations expressed in polar coordinate system or rectangularcoordinate system [2].

5.2.1 Power flow solution with polar coordinate systemThe complex voltage and real and reactive powers (5.5) can be expressed in polarcoordinates as

V i ¼ Vi cos di þ j sin dj

� � ð5:13Þ

Pi ¼ Vi

Xn

j¼1

VjðGij cos dij þ Bij sin dijÞ ð5:14Þ

Qi ¼ Vi

Xn

j¼1

VjðGij sin dij � Bij cos dijÞ ð5:15Þ

where dij ¼ di � dj ≜ the angle difference between bus i and bus j.For a network with n buses and according to the types of buses explained in

Section 5.1, it is assumed that the network is composed of PQ buses (1?m), PV buses(m þ 1 ? n� 1) and the nth bus is the slack bus. Therefore, the magnitude of voltagesVmþ1 ? Vn�1 are given as well as the voltage magnitude and angle at the slack bus, Vn

and dn, are known. This results in the unknown variables being the voltage angle atn � 1 buses and the voltage magnitude at m buses. For each bus in the network, thedifference between the scheduled and produced real power, Psch and Pi, respectively,is given by

DPi ¼ Psch � Pi ¼ Psch � Vi

Xn�1

j¼1

VjðGij cos dij þ Bij sin dijÞ ð5:16Þ

Similarly, for each PQ bus the difference of reactive power is

DQi ¼ Qsch � Qi ¼ Qsch � Vi

Xm

j¼1

VjðGij sin dij � Bij cos dijÞ ð5:17Þ

Applying (5.12) the following equation can be obtained:

DPDQ

� �¼ �J

DdDV

V

" #or

DPDQ

� �¼ � H N

M L

� �Dd

V�1D DV

� �ð5:18Þ


where

DP ¼DP1

DP2

..

.

DPn�1

26664

37775; DQ ¼

DQ1

DQ2

..

.

DQm

26664

37775; Dd ¼

Dd1

Dd2

..

.

Ddn�1

26664

37775; DV ¼

DV1

DV2

..

.

DVm

26664

37775;

VD ¼V1

V2

. ..

Vm

26664

37775

H ≜ (n � 1) � (n � 1) matrix and Hij ¼ @DPi/@ dj

N ≜ (n � 1) � m matrix and Nij ¼ Vj(@DPi/@ Vj)M ≜ m � (n � 1) matrix and Mij ¼ @DQi/@dj

L ≜ m � m matrix and Lij ¼ Vj (@ DQi/@ Vj)

By definition, the off-diagonal elements ‘i 6¼ j’ of the Jacobian matrix can becomputed using the relations

Hij ¼ �ViVj Gij sin dij � Bij cos dij

� �Nij ¼ �ViVj Gij cos dij � Bij sin dij

� �Mij ¼ ViVj Gij cos dij � Bij sin dij

� �Lij ¼ �ViVj Gij sin dij � Bij cos dij

� �

9>>>=>>>;

ð5:19Þ

Similarly, the relations of diagonal elements ‘i ¼ j’ of the Jacobian matrix are

Hii ¼ V 2i Bii þ Qi

Nii ¼ �V 2i Gii � Pi

Mii ¼ V 2i Gii � Pi

Lii ¼ V 2i Bii � Qi

9>>>=>>>;

ð5:20Þ

The flow chart shown in Figure 5.1 depicts the steps of the Newton–Raphsonpower flow solution with polar coordinate system.

5.2.2 Power flow solution with rectangular coordinate systemFrom (5.5) the voltage and real and reactive power can be expressed in the rec-tangular coordinate system as

Vi ¼ ei þ jfi

Pi ¼ ei

Xn

j¼1

Gijej � Bij fj� �þ fi

Xn

j¼1

Gijfj þ Bijej

� �

Qi ¼ fi

Xn

j¼1

Gijej � Bij fj

� �� ei

Xn

j¼1

Gijfj þ Bijej

� �

9>>>>>>>=>>>>>>>;

ð5:21Þ


Input: system data; line admittance,

Psch at each bus, V at PV buses & Qsch

at PQ buses

Define the numbers of both PV & PQ busesSpecify the slack bus ‘s’, Vs, δs

Assume initial bus voltages Vi

’o’ i = 1, 2, ... n and i ≠ s

Form bus admittance matrixYbus

Set bus number i → 1

Is i = s?

Calculate for bus i∆Pi = Psch – Vi Σ Vj (Gij cos δij+Bij sin δij)

j = 1, 2, … n

Isi = number of

PQ bus?

No

Yes

Calculate for bus i∆Qi = Qsch – Vi Σ Vj (Gij sin δij – Bij cos δij)

j = 1, 2, ...n

Aremax |∆P| and

max |∆Q| ≤ e ?

No Isi = n?

No

i = i + 1

No

Yes

Yes

Set iterationK → 0

K→K + 1

Calculate Jacobian matrix elements and obtain ∆Vi and ∆δi by solving

|∆P| |∆δ|

|∆Q| |∆V/V|= – j

Perform new V and δ asFor PQ buses: |V| K+1 = |V| K+∆V| K

And for PV and PQ buses: δK+1 = δK+∆δK

Calculate lines power flowand power at slack bus

Figure 5.1 Flow chart for Newton–Raphson power flow solution with polarcoordinates


For each PQ bus in the network, the differences between the scheduled andproduced real and reactive power are given by

DPi ¼ Psch � PI ¼ Psch � ei

Xn

j¼1

Gijej � Bij fj

� �� fi

Xn

j¼1

Gij fj þ Bijej

� �

DQi ¼ Qsch � Qi ¼ Qsch � fi

Xn

j¼1

Gijej � Bij fj

� �þ ei

Xn

j¼1

Gij fj þ Bijej

� �

9>>>>>=>>>>>;

ð5:22Þ

Similarly, for each PV bus the following relations can be written:

DPi ¼ Psch � PI ¼ Psch � ei

Xn

j¼1

Gijej � Bij fj

� �� fi

Xn

j¼1

Gij fj þ Bijej

� �DV 2

i ¼ V 2sch � V 2

i ¼ V 2sch � e2

i þ f 2i

� �9>>=>>; ð5:23Þ

Equations (5.22) and (5.23) include 2(n � 1) equations: (n � 1) equationsrepresent the real power at all buses excluding the slack bus and the other (n � 1)equations comprise m equations representing the reactive power for PQ buses and(n � m � 1) equations representing DV 2

i for PV buses. Expanding the equations intoTaylor series to be written in a linearised form, neglecting the derivative terms ofsecond and higher order, and according to Newton–Raphson method the equationin the form DF ¼ �JDV can be written as

DP1

DP2

..

.

DPn�1

DQ1

DQ2

..

.

DQm

DV 2mþ1

..

.

DV 2n�1

266666666666666666664

377777777777777777775

¼ �

@DP1

@e1� � � @DP1

@en�1

..

. � � � ...

@DPn�1

@e1� � � @DPn�1

@en�1

@DP1

@f1� � � @DPn�1

@fn�1

..

. � � � ...

@DPn�1

@f1� � � @DPn�1

@fn�1

@DQ1

@e1� � � @DQ1

@en�1

..

. � � � ...

@DQm

@e1� � � @DQm

@en�1

@DQ1

@f1� � � @DQ1

@fn�1

..

. � � � ...

@DQm

@f1� � � @DQm

@fn�1

@DV 2mþ1

@e1� � � @DV 2

mþ1

@en�1

..

. � � � ...

@DV 2n�1

@e1� � � @DV 2

n�1

@en�1

@DV 2mþ1

@f1� � � @DV 2

mþ1

@fn�1

..

. � � � ...

@DV 2n�1

@f1� � � @DV 2

n�1

@fn�1

26666666666666666666666666666666664

37777777777777777777777777777777775

De1

De2

..

.

Den�1

Df1

..

.

Dfmþ1

..

.

Dfn�1

26666666666666664

37777777777777775

ð5:24Þ


Equation (5.24) may be written as

DPDQDV 2

24

35 ¼

J1 J2J3 J4J5 J6

24

35 De

Df

� �ð5:25Þ

where

For all network buses: (1 ? n � 1) except the slack bus, nth bus:

DP ¼ DP1;DP2; . . .;DPn�1½ �t;De ¼ De1;De2; . . .;Den�1½ �t;Df ¼ Df1;Df2; . . .;Dfn�1½ �t

J1 ≜ (n � 1) � (n � 1) matrix with elements obtained by the expression:

@DPi

@ej¼ �

Xn

j¼1

Gijej � Bij fj� �� Giiei � Bii fi; for i ¼ j

¼ � Gijei þ Bij fi� �

; for i 6¼ j

9>=>; ð5:26Þ

J2 ≜ (n � 1) � (n � 1) matrix and its elements are given by

@DPi

@fj¼ �

Xn

j¼1

Gijej � Bij fj� �� Giiei � Bii fi; for i ¼ j

¼ � Gij fi � Bijei

� �; for i 6¼ j

9>=>; ð5:27Þ

For PQ buses: (1?m)

DQ ¼ DQ1;DQ2; . . .;DQm½ �t

J3 ≜ m � (n � 1) matrix with elements given by

@DQi

@ej¼

Xn

j¼1

Gij fj þ Bijej

� �� Gii fi þ Biiei; for i ¼ j

¼ Gij fi � Bijei

� �; for i 6¼ j

9>=>; ð5:28Þ

J4 ≜ m � (n � 1) matrix with elements given by

@DQi

@fj¼ �

Xn

j¼1

Gijej � Bij fj

� �þ Giiei þ Bii fi; for i ¼ j

¼ Gijei þ Bij fi

� �; for i 6¼ j

9>=>; ð5:29Þ

For PV buses: (m þ 1?n � 1)

DV 2 ¼ DV 2mþ1;DV 2

mþ2; . . .;DV 2n�1

� �t

J5 ≜ (n � m � 1) � (n � 1) matrix with elements given by

@V 2i

@ej¼ �2ei; for i ¼ j

¼ 0; for i 6¼ j

9=; ð5:30Þ


J6 ≜ (n � m � 1) � (n � 1) matrix with elements given by

@V 2i

@fj¼ �2fi; for i ¼ j

¼ 0; for i 6¼ j

9=; ð5:31Þ

The steps of solution are the same as that applied when using polar coordinatesin Section 5.2.1.

Example 5.1 Using Newton–Raphson method, find the power flow solution of thethree generators–nine bus system shown in Figure 5.2 [3]. The system data are givenin Appendix II.

Solution:

The system buses are classified as bus #1 is a slack bus, buses #2, 3 are PV busesand buses #4 ? 9 are PQ buses. Power and voltage set-points on the system base of100 MVA are summarised in Table 5.1.

Table 5.1 Power and voltage set points

Bus no. 1 2 3 4 5 6 7 8 9

Gen/Load G2 G3 – L5 L6 – L8 –

P (pu) Slack 1.63 0.85 0 1.25 0.9 0 1.0 0

Q (pu) bus – – 0 0 0 0 0 0

V (pu) 0� 1.025 1.025 1.02 – – – – –

GG

G

1

2 3

4

5 6

7 8 9

Figure 5.2 Three generators–nine bus system [3]


By writing the program and downloading the data in PSAT Toolbox, v2.1.6 [4],to solve the iterative (5.18) in polar coordinates according to the flow chart(Figure 5.1) it is found that:

The program run terminated at the second iteration. The maximum con-vergence error for first iteration is 8.8336 � 10�4 and for the second iteration is7.7519 � 10�7.

The elements of bus admittance matrix, Ybus ¼

Results: The power and voltage at each bus are summarised in Table 5.2 andline flows are summarised in Table 5.3. The real power and reactive power losses ineach line are illustrated in Table 5.4.

0 �j17.361

0 0 0 þ j17.361 0 0 0 0 0

0 0 �j14.388

0 0 0 0 0 þ j14.388 0 0

0 0 0 �j17.065

0 0 0 0 0 0 þj17.065

0 þj17.361

0 0 2.9253 �j27.621

�0.98314 þj0.082739

�1.9422 þj10.511

0 0 0

0 0 0 �0.98314 þj0.082739

2.2015 �j5.5629

0 �1.2184 þj5.9622

0 0

0 0 0 �1.9422 þj10.511

0 3.2242 �j15.583

0 0 �1.282 þj5.5882

0 0 þj14.388

0 0 �1.2184 þj5.9622

0 2.8355 �j33.593

�1.6171 þj13.698

0

0 0 0 0 0 0 �1.6171 þj13.698

2.7722 �j23.124

�1.1551 þj9.7843

0 0 0 þj17.065

0 0 �1.282 þj5.5882

�1.1551 þj9.7843

2.4371 �j31.87

Table 5.2 Bus power and voltage

Bus no. |V | (pu) Voltage angle (deg.) P (pu) Q (pu)

1 1.04 0 0.71641 0.716412 1.025 9.28 1.63 1.633 1.025 4.6648 0.85 0.854 1.0258 �2.2168 0 05 0.99563 �3.9888 �1.25 �1.256 1.0127 �3.6874 �0.9 �0.97 1.0258 3.7197 0 08 1.0159 0.72754 �1 �19 1.0324 1.9667 0 0


Tab

le5.

3L

ine

flow

s

Bus

no.

12

34

56

78

9

10

00

0.71

641þ

j0.2

7046

00

00

0

20

00

00

01.

63þ

j0.0

6654

00

30

00

00

00

00.

85�

j0.1

086

4�0

.716

41�

j0.2

3923

00

00.

4093

7þ

j0.2

2893

0.30

704þ

j010

30

00

50

00

�0.4

068�

j0.3

8687

00

�0.8

432�

j0.1

1313

00

60

00

�0.3

0537

�j0

.165

430

00

0�0

.594

63�

j0.1

3457

70

�1.6

3þ

j0.0

9178

00

0.86

62�

j0.0

8381

00

0.76

38�

j0.0

0797

0

80

00

00

0�0

.759

05�

j0.1

0704

0�0

.240

95�

j0.2

4296

90

0�0

.85þ

j0.1

4955

00

0.60

817�

j0.1

8075

00.

2418

3þ

j0.0

312

0

Line(ij) ≜ the line connecting bus i to bus jPlosses(ij) ≜ real power losses in line ijQlosses(ij) ≜ reactive power losses in line ijIt is to be noted that Plosses(ij) ¼ Plosses(ij) and Qlosses(ij) ¼ Qlosses(ij)

Total real power generation, PG(total) ¼ 3.1964 puTotal reactive power generation, QG(total) ¼ 0.2284 puTotal load, real power, PL(total) ¼ 3.15 puTotal load, reactive power, QL(total) ¼ 1.15 puTotal real power losses, Plosses ¼ 0.04641 puTotal reactive power losses, Qlosses ¼�0.9216 pu

5.3 Gauss�Seidel method

Again, a system of n non-linear equations in n unknown variables can be describedgenerally by (5.32).

f1 x1; x2; . . . ; xnð Þ ¼ 0

f2 x1; x2; . . . ; xnð Þ ¼ 0

..

.

fn x1; x2; . . . ; xnð Þ ¼ 0

9>>>>=>>>>;

ð5:32Þ

Its solution can be formulated as

x1 ¼ g1 x1; x2; . . . ; xnð Þx2 ¼ g2 x1; x2; . . . ; xnð Þ

..

.

xn ¼ gn x1; x2; . . . ; xnð Þ

9>>>>=>>>>;

ð5:33Þ

At first iteration, the initial solution is assumed and substituted into the RHS of(5.33) to get a new solution that is considered as a primary solution for the nextiteration. Thus, at the kth iteration, the new solution is expressed as

xkþ11 ¼ g1 xk

1; xk2; . . . ; xk

n

� �xkþ1

2 ¼ g2 xk1; x

k2; . . . ; xk

n

� �...

xkþ1n ¼ gn xk

1; xk2; . . . ; xk

n

� �

9>>>>>=>>>>>;

ð5:34Þ

Table 5.4 Line power losses in pu

Line(ij) 1–4 2–7 3–9 4–5 4–6 5–7 6–9 7–8 8–9

Plosses(ij) 0 0 0 0.00258 0.00166 0.023 0.01354 0.00475 0.00088Qlosses(ij) 0.031 0.158 0.041 �0.158 �0.155 �0.196 �0.315 �0.115 �0.212


The iterative solution is terminated when the convergence condition (5.35) issatisfied.

maxjxkþ1i � xk

i j � e; i ¼ 1; 2; . . . ; n ð5:35ÞThe solution by following the above steps is called Gauss method. This method

has been modified to be called Gauss–Seidel method. The modification is mainlyconcerned with speeding up the convergence and requires smaller number ofiterations, i.e. less computation time is required. It is based on substituting imme-diately the new values of variables in the calculation of the next variable (in thesame iteration) rather than waiting until the next iteration. Therefore, the modifiedformula of the iteration calculation is

xkþ11 ¼ g1 xk

1; xk2; . . . ; x

kn

� �xkþ1

2 ¼ g2 xkþ11 ; xk

2; . . . ; xkn

� �...

xkþ1n ¼ gn xkþ1

1 ; xkþ12 ; . . . ; xkþ1

n�1; xkn

� �

9>>>>>=>>>>>;

ð5:36Þ

or

xkþ1i ¼ gi xkþ1

1 ; xkþ12 ; . . .; xkþ1

i�1 ; xki ; . . .; x

kn

� � ð5:37ÞApplying this method to a network with n-buses ordered as buses 1?m are PQ

buses; buses m?n � 1 are PV buses and the slack bus is the nth bus. From (5.5) thevoltage at each bus can be written as

V i ¼ 1Yii

Pi � jQi

V�i

�Xn

j¼1j 6¼i

YijV j

264

375; i ¼ 1; 2; . . . ; n � 1 ð5:38Þ

Thus, by Gauss–Seidel method, (5.38) at the kth iteration is

Vkþ1i ¼ 1

Yii

Pi � jQi

V�ki

�Xi�1

j¼1

YijVkþ1j �

Xn

j¼iþ1

YijVkj

" #; i ¼ 1; 2; . . . ; n � 1

ð5:39ÞFor the PQ bus: The real and reactive powers are known. Equation (5.39) can

be used to perform iteration calculations as the initial bus voltages are assumed.For the PV bus: The bus real power and voltage magnitude are known. Thus,

the voltage magnitude must be fixed at the scheduled value and its angle is com-puted from the estimated voltage as below.

The voltage as a complex quantity is expressed as Vi ¼ ei þ jfi.Thus, the relation |Vi(sch)|

2 ¼ |ei|2 þ | fi|

2 must be satisfied. This necessitates thatthe phase angle of the scheduled voltage is equal to that of the estimated voltage di.


Also, the components of the estimated voltage are adjusted accordingly. At the kthiteration di is given by

dki ¼ tan�1 f k

i

eki

� �ð5:40Þ

Then, the adjusted components of estimated voltage are

eki adjð Þ ¼ jVi schð Þj cos dk

i

f ki adjð Þ ¼ jVi schð Þj sin dk

i

)ð5:41Þ

and the corresponding reactive power is calculated by

Qki ¼ Im Vk

i I�ki

� � ¼ Im Vki

Xi�1

j¼1

Y �ijV

�kþ1j þ

Xn

j¼i

Y �ijV

�kj

" #ð5:42Þ

Using the adjusted voltage components (5.41) and the corresponding reactivepower (5.42) the new estimated voltage Vi

kþ1 can be computed.It is to be noted that the limits of the reactive power source must be considered

in such a way that in case of violating the limits the value of the reactive power isfixed at the violated limit. The PV bus is treated as a PQ bus ignoring the desiredvoltage magnitude.

Finally, when the convergence is satisfied, and the iteration is terminated, thevalues of voltage (magnitude and angle) and real and reactive power at all buses areobtained. Then, the power flow into a line with a shunt admittance yio and anadmittance yij between bus i and bus j can be calculated by

Sij ¼ Pij þ jQij ¼ V iI�ij ¼ V2

i yio þ V i V�i � V�

j

y�ij ð5:43Þ

and the power at the slack bus is given by

Pn þ jQn ¼ Vn

Xn

j¼1

Y �njV

�j ð5:44Þ

The steps of Gauss–Seidel program are depicted by the flow chart shown inFigure 5.3.

5.4 Decoupling method

This method is based on some simplifications to make power flow iteration veryeasy, to expedite solution computation and satisfying accepted accuracy of results.For instance, in Newton–Raphson power flow calculation, to obtain a solution withhigh accuracy, there is no simplification as well as the elements of Jacobian matrixmust be recalculated each iteration. So, more iteration, more storage and morecomputation time are required.

The simplification used in the decoupling method mainly depends on theavailability of neglecting the impact of both voltage magnitude change on the real


Input: system data;line admittance,

Psch at each bus, Vat PV buses & Qsch

at PQ buses

Define the numbers of both PV & PQ busesSpecify the slack bus ‘s’, Vs, δs

Assume initial bus voltagesVi

’o’, i = 1, 2, ... n and i ≠ s

Form bus admittance matrixYbus

Set bus numberi →1

Is i = s?

Calculate for bus voltage Vik+1

Using Eq. (5.39)

Isi = number of PV

bus?

No

Yes

– Calculate δik by Eq. (5.39)– Adjust the voltage components, Eq. (5.40)– Calculate the reactive power, Eq. (5.41)

Aremax│∆V│ ≤ e?

No

Noi = i + 1

No

Yes

Set iterationK → 0

K→K + 1

Calculate linespower flow and

power at slack bus

– Q = Q-lower limit if Q < Q-lower limit– Q = Q-upper limit if Q > Q-upper limit– Calculate the new bus voltage Vi

k+1

Yes

Calculate max│∆V│= max│Vik+1–Vi

k│

Is i = n ?

Figure 5.3 Flow chart for Gauss–Seidel power flow method using Ybus


power and angle change on the reactive power. This assumption, to some extent, isreasonable as in practical power systems the branches have much higher reactancethan the resistance, i.e. the coupling between the real power and voltage magnitudeis weak and the same is for the coupling between the reactive power and voltageangle. That is weak as well [5].

Neglecting weak couplings implies that @DPi@Vi

0 and @DQi

@di 0, consequently,

the sub-matrices N and M in (5.18) vanish. Thus,

DPDQ

� �¼ � H 0

0 L

� �Dd

V�1D DV

� �

Hence,

DP ¼ �HDd and DQ ¼ �LV�1D DV ð5:45Þ

5.4.1 Fast-decoupled methodFurther simplification can be applied to (5.45) as the difference between the voltageangles at the two ends of a line ij is small. Then, cos dij ¼ cos(di – dj) ffi 1 andGij sin dij << Bij. Assuming that Qi << V 2

i Bii, the elements of the two sub-matrices H and L are given by

Hij ¼ � @Pi

@dj¼ ViVjBij i; j ¼ 1;2; . . . ; n � 1

Lij ¼ � @Qi

@V j

Vj

� � ¼ ViVjBij i; j ¼ 1;2; . . . ;m

9>>>>>=>>>>>;

ð5:46Þ

Thus, the matrices [H] and [L] can be formulated as

H½ � ¼

V1

V2

. ..

Vn�1

2666664

3777775

B11 B12 � � � B1;n�1

B21 B22 � � � B2;n�1

..

.

Bn�1;1

..

.

Bn�1;2

. ..

� � �...

Bn�1;n�1

2666664

3777775

V1

V2

. ..

Vn�1

2666664

3777775

¼ VB0V

and

L½ � ¼

V1

V2

. ..

Vm

2666664

3777775

B11 B12 � � � B1;m

B21 B22 � � � B2;m

..

.

Bm;1

..

.

Bm;2

. ..

� � �...

Bm;m

2666664

3777775

V1

V2

. ..

Vm

2666664

3777775

¼ VB00V

ð5:47Þ


where

½B0 � ¼B11 B12 � � � B1;n�1

B21 B22 � � � B2;n�1

..

.

Bn�1;1

..

.

Bn�1;2

. ..

� � �...

Bn�1;n�1

26664

37775 and ½B00 � ¼

B11 B12 � � � B1;m

B21 B22 � � � B2;m

..

.

Bm;1

..

.

Bm;2

. ..

� � �...

Bm;m

26664

37775

Substitute (5.47) into (5.45) to obtain

DP

V¼ �B

0VDd

DQ

V¼ �B

00DV

9>>=>>; ð5:48Þ

Equation (5.48) can be written in matrix form as

DP1

V1

DP2

V2

..

.

DPn�1

Vn�1

2666666666664

3777777777775¼ �

B11 B12 � � � B1;n�1

B21 B22 � � � B2;n�1

..

.

Bn�1;1

..

.

Bn�1;2

. ..

� � �...

Bn�1;n�1

2666664

3777775

V1Dd1

V2Dd2

..

.

Vn�1Ddn�1

2666664

3777775

DQ1

V1

DQ2

V2

..

.

DQm

Vm

26666666666664

37777777777775¼ �

B11 B12 � � � B1;m

B21 B22 � � � B2;m

..

.

Bm;1

..

.

Bm;2

. ..

� � �...

Bm;m

2666664

3777775

DV1

DV2

..

.

DVm

2666664

3777775

9>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>;

ð5:49Þ

It is found that the elements of matrices �B0 and �B00 in (5.49) are the imaginarypart of the corresponding elements of bus admittance matrix. Therefore, for a spe-cific configuration of a power system, matrices B0 and B00 are constant, symmetrical,real and sparse matrices. In addition, they need to be triangularised only once at thebeginning of the study. Thus, it is called the ‘fast decoupled power flow model’.

The fast decoupled power flow solution requires more iterations than theNewton–Raphson method, but requires considerably less time per iteration and apower flow is obtained very rapidly. This technique is very useful in contingencyanalysis where a power flow solution is required for online control [6].

An illustrative example to show the application of power flow methods to thepower system and a comparison between them is given below.


Example 5.2 Repeat Example 5.1 using fast-decoupled method.

Solution:

The program terminated at the fourth iteration with maximum convergence error asbelow:

Iteration ¼ 1: Maximum Convergence Error ¼ 0.017302Iteration ¼ 2: Maximum Convergence Error ¼ 0.00046404Iteration ¼ 3: Maximum Convergence Error ¼ 1.8107 � 10�5

Iteration ¼ 4: Maximum Convergence Error ¼ 5.9507 � 10�7

Power and voltage set-points on the system base of 100MVA are the same asused in Example 5.1 and Table 5.1.

Results: The power and voltage at each bus are summarised in Table 5.5 andline flows are summarised in Table 5.6. The real power and reactive power losses ineach line are illustrated in Table 5.7.

line(ij) ≜ the line connecting bus i to bus jPlosses(ij) ≜ real power losses in line ijQlosses(ij) ≜ reactive power losses in line ij

It is to be noted that Plosses(ij) ¼ Plosses( ji) and Qlosses(ij) ¼ Qlosses( ji)

Total real power generation, PG(total) ¼ 3.1964 puTotal reactive power generation, QG(total) ¼ 0.2284 puTotal load, real power, PL(total) ¼ 3.15puTotal load, reactive power, QL(total) ¼ 1.15 puTotal real power losses, Plosses ¼ 0.04641 puTotal reactive power losses, Qlosses ¼�0.9216 pu

It is found that the results obtained from Examples 5.1 and 5.2 are very close toeach other. On the other hand, the fast-decoupled power flow for Example 5.2 hastaken four iterations with maximum convergence error of 5.9507 � 10�7 comparedto the Newton–Raphson method (Example 5.1) that took only two iterations with

Table 5.5 Bus power and voltage

Bus no. |V| (pu) Voltage angle (deg.) P (pu) Q (pu)

1 1.04 0 0.71641 0.270462 1.025 9.28 1.63 0.066543 1.025 4.6648 0.85 �1.10864 1.0258 �2.2168 0 05 0.99563 �3.9888 �1.25 �0.56 1.0127 �3.6874 �0.9 �0.37 1.0258 3.7197 0 08 1.0159 0.72754 �1 �0.359 1.0324 1.9667 0 0


Tab

le5.

6L

ine

flow

s

Bus

no.

12

34

56

78

9

10

00

0.71

641þ

j0.2

7046

00

00

0

20

00

00

01.

63þ

j0.0

6654

00

30

00

00

00

00.

85�

j0.1

086

4�0

.716

41�

j0.2

3923

00

00.

4093

7þ

j0.2

2893

0.30

704þ

j010

30

00

50

00

�0.4

068�

j0.3

8687

00

�0.8

432�

j0.1

1313

00

60

00

�0.3

0537

�j0

.165

430

00

0�0

.594

63�

j0.1

3457

70

�1.6

3þ

j0.0

9178

00

0.86

62�

j0.0

8381

00

0.76

38�

j0.0

0797

0

80

00

00

0�0

.759

05�

j0.1

0704

0�0

.240

95�

j0.2

4296

90

0�0

.85þ

j0.1

4955

00

0.60

817�

j0.1

8075

00.

2418

3þ

j0.0

312

0

the maximum convergence error of 7.7519 � 10�7. Regarding the computationtime, it is found that with running the program on a PC, Processor INSPIRON I5RN5110 Core i7, the time for the two iterations of Newton–Raphson and the fouriterations of fast-decoupled method is 24 ms and 39 ms, respectively. This illus-trates that the fast-decoupled solution requires less time per iteration than thatrequired by Newton–Raphson solution.

References

1. El-Hawary M.E., Christensen G.S. Optimal Economic Operation of ElectricPower Systems. New York, NY, US: Academic Press; 1979

2. Murty P.S.R. Operation and Control in Power Systems. Hyderabad, India: BSPublications; 2008

3. Anderson P.M., Fouad A.A. Power System Control and Stability. 2nd edn.Hoboken, NJ, US: Wiley-IEEE Press; 2003

4. Milano F. Power System Analysis Toolbox PSAT [online]. 2014. Available fromhttp://www.power.uwaterloo.ca/~fmilano/psat.htm [Accessed 12 Jul 2014]

5. Zhu J. Optimization of Power System Operation. Hoboken, NJ, US: Wiley-IEEE Press; 2009

6. Saadat H., Power System Analysis. 3rd edn. New York, NY, US: McGraw-Hill;2010

Table 5.7 Line power losses in pu

Line(ij) 1–4 2–7 3–9 4–5 4–6 5–7 6–9 7–8 8–9

Plosses(ij) 0 0 0 0.0026 0.0017 0.023 0.014 0.005 0.001Qlosses(ij) 0.031 0.158 0.041 �0.158 �0.155 �0.197 �0.315 �0.115 �0.212


Chapter 6

Optimal power flow

One of the important tools for power system planning and operation is the optimalpower flow (OPF). It is a power flow problem in which some control variables areadjusted to minimise or maximise an objective function, while satisfying physicaland operating limits as constraints on various controls, dependent variables andfunctions of variables. In power system analysis, the control variables that must beaccommodated with OPF are active and reactive power generation from powerplants, generator terminal voltage, reactive power compensation, LTC of transfor-mers and phase shift angles. The objective function includes minimisation of lossesor costs in the case of studying economic power dispatch. As the controls includereactive power devices, the problem is characterised by a non-separable objectivefunction and consequently the problem solution gets more difficult [1]. The con-straints are either equality constraints such as power balance or inequality con-straints, e.g. the generated power must be within a maximum and minimumpermissible output power of the generator. However, the problem expressed inmathematical notation can be compactly formulated as

Minimise f x; uð Þ ð6:1ÞSubject to h x; uð Þ ¼ 0 ð6:2Þ

and gðx; uÞ � 0 ð6:3Þ

where x and u are the dependent and control variable vectors, respectively.

6.1 Problem formulation

The OPF problem is mainly formulated as an optimisation problem. The formulationis based on three elements: objective function, control variables and constraints.

The conventional OPF model is described below where the control variablesinclude real and reactive power generation and control voltage setting.

minimise Ploss ¼Xn

i¼1

jVijXn

j¼1

jVjj Gij cos dij þ Bij sin dij

� � ð6:4Þ

subject to

equality constraints

PGi � PDi � Vi

Xn

j¼1


� � ¼ 0

QGi � QDi � Vi

Xn

j¼1

jVjj Gij sin dij � Bij cos dij

� � ¼ 0

8>>>>>><>>>>>>:

ð6:5Þ

i, j ¼ 1, 2, . . . , n

and inequality constraintsð Þ

PminGi � PGi � Pmax

Gi ði 2 SGÞQmin

Ri � QRi � QmaxRi ði 2 SRÞ

V mini � Vi � V max

i ði 2 SBÞSmin

Li � SLi � SmaxLi ði 2 SLÞ

8>>>><>>>>:

ð6:6Þ

where

PGi ≜ active power output of the ith generator in the controllable generator set SG

QRi ≜ reactive output of the ith reactive source in the reactive source set SR

Vi ≜ voltage magnitude of the ith bus in the bus set SB

PDi ≜ load of the ith busGij and Bij ≜ conductance and susceptance between the ith and the jth bus,

respectivelySLi ≜ apparent power across the ith branch in the branch set SL

n ≜ number of busesGij þ jBij ≜ element of nodal admittance matrixdij ≜ voltage angle difference at the two ends of the tie-line ij

Depending on the type of study, other constraints may be considered such asboiler constraints, running reserve constraints, phase shift transformer constraints,line outage or security constraints.

6.2 Problem solution

The solution is based on two steps: First, applying Newton’s method the normalstatic load flow as a feasible solution is calculated. Second, using the gradientmethod and Lagrange multiplier the optimum solution can be obtained [2, 3].

● Equation (6.5) for convenience can be rewritten as

PiðnetÞ � Pi V ; dð Þ ¼ 0

QiðnetÞ � Qi V ; dð Þ ¼ 0

)i ¼ 1; 2; . . .; n ð6:7Þ


where

PiðnetÞ ¼ PGi � PDi;QiðnetÞ ¼ QGi � QDi;PGi and QGi have positive sign whenentering and PDi and QDi have positive sign when leaving the ith bus

Pi V ; dð Þ ¼ Vi

Xn

j¼1


� �

Qi V ; dð Þ ¼ Vi

Xn

j¼1

jVjj Gij sin dij � Bij cos dij

� �

9>>>>>>=>>>>>>;

ð6:8Þ

It is seen that each bus in the power system is characterised by four variables:the net real and reactive power entering the bus and the bus voltage (magnitude andphase angle), which are P(net), Q(net), V and d, respectively. Thus, to solve (6.7) thatis a set of 2n non-linear equations, two of the four variables at each bus must bespecified. Determination of which two variables are specified depends on the typeof the bus as summarised in Table 6.1. Usually, the phase angle ds of the voltage Vs

at the slack bus is taken as zero and considered as reference. The control variablesare regarded as specified variables in the power flow solution. P(net) and Q(net) canbe calculated directly from (6.7), but if V and/or d are unknown the solution is notdirect and is explained below.

Assume X and Y are the vectors of unknown (V and d) and known variables,respectively. Thus, Xt ¼ [V1, . . . ,Vm, d1, . . . , dm, dmþ1, . . . , dn�1] and Yt ¼ [P1, . . . ,Pn�1, Q1, . . . , Qm, Vmþ1, . . . , Vn, ds].

The vector X includes n þ m� 1 elements (unknown). Then, a set of n þ m � 1relations must be selected from (6.7) that has 2n relations to form a vector of func-tions in terms of X and Y elements, h(x, y).

Using Newton’s method described in Chapter 5 and assuming the initialsolution X(o) is improved successively by DX, the solution can be found by solvingthe set of equations:

@h

@xxk ; y� ��

DX½ � ¼ � h xk ; y� �� ð6:9Þ

where @h=@xð Þ is the Jacobian matrix.

Table 6.1 Variables characterising network buses

Bus type Specified variables Unknown variables Bus no.

PQ-bus P(net), Q(net) V, d 1? m

PV-bus P(net), V Q(net), d m þ 1? n � 1

Slack-bus V, d P(net), Q(net) N

Optimal power flow 133

The solution obtained (as a first step) represents the static feasible non-OPFsolution. Therefore, the second step is to optimise that solution in two cases:without and with inequality constraints.

OPF without inequality constraints: The objective function to be minimised is thetotal system losses given by (6.4). It is a function of the independent and dependentvariables and can generally be expressed as f (x, y).

Using the classical Lagrange-multipliers method to minimise f(x, y) subjectto the equality constraints h(x, y) ¼ 0 as in (6.7), the solution is found by introducingthe Lagrangian function

L x; yð Þ ¼ f x; yð Þ þ l½ �thðx; yÞ ð6:10Þwhere the elements of [l] are called Lagrangian multipliers, and satisfy the fol-lowing three necessary conditions for a minimum:

@L@x

� �¼ @f

@x

� �þ @h

@x

� �t

l½ � ¼ 0

@L@y

� �¼ @f

@y

� �þ @h

@y

� �t

l½ � ¼ 0

@L@l

� �¼ h x; yð Þ ¼ 0

9>>>>>>>>=>>>>>>>>;

ð6:11Þ

Any feasible solution, such as the solution obtained by the Newton’s method inthe first step, satisfies the third condition while the first condition can be verified bycalculating l as

l½ � ¼ � @h

@x

� �t�1

@f

@x

� �ð6:12Þ

It is noted that the Jacobian @h=@xð Þ has already been calculated in (6.9). Tosatisfy the second condition, (@L=@y) is determined by using the gradient methodwith incremental change of controls ‘y’ given by

yikþ1 ¼ yi

k þ c@L@yi

¼ yik � crf ð6:13Þ

where c is a scalar factor and k is the number of iterations.The computation is repeated iteratively until the minimum is reached.

OPF with inequality constraints: Actually, the control variables have permissiblevalues such as given in (6.6), and then they cannot be assumed to have any value.These inequality constraints can easily be treated by setting the control variables attheir limits if (6.13) gives violating values, i.e. beyond the permissible limits. Thus,

ykþ1i ¼

ymaxi ; if yk

i þ Dyi > ymaxi

ymini ; if yk

i þ Dyi < ymini

yki þ Dyi; otherwise

8><>: ð6:14Þ


It is to be noted that when a control variable has reached its limit its componentin the gradient vector must be computed in the following cycles because it mighteventually back off from the limit. At the minimum, Kuhn–Tucker theorem provesthat the components (@f/@yi) will be as in (6.15) as necessary conditions.

@f

@yi¼ 0; if ymin

i < yi < ymaxi

@f

@yi� 0; if yi ¼ ymax

i

@f

@yi� 0; if yi ¼ ymin

i

9>>>>>>>=>>>>>>>;

ð6:15Þ

Example 6.1 Find the OPF in the nine-bus system given in Appendix II. Formulatethe objective function as minimising the generation cost subject to the constraints.

Solution:

The objective function is considered as

F ¼XNG

i¼1

FiðPGiÞ

where F is the total generated power fuel cost, FiðPGiÞ is the ith generating unit fuelcost that is a function of the active power generation output, PGi, in MW of the ithgenerator. NG is the total number of generating units.

The fuel cost curve is considered as a quadratic cost curve (Fq) as

Fqi PGið Þ ¼ ai þ biPGi þ ciP2Gi

where the cost coefficients are summarised in Table 6.2.The constraints are the equations of active and reactive power balance

equality constraints

PGi � PDi � Vi

Xn

j¼1

jVjj Gij cos qij þ Bij sin qij

� � ¼ 0

QGi � QDi � Vi

Xn

j¼1

jVjj Gij sin qij � Bij cos qij

� � ¼ 0

8>>>>>><>>>>>>:

Table 6.2 Cost coefficients

Generator 1 Generator 2 Generator 3

a ($/hr) b ($/MW/hr) c ($/MW2/hr) a ($/hr) b ($/MW/hr) c ($/MW2/hr) a ($/hr) b ($/MW/hr) c ($/MW2/hr)

150 5 0.11 600 1.2 0.085 335 1 0.1225


and the limits of active and reactive power, voltage, transformer tap setting and lineloading.

inequality constraints

PminGi � PGi � Pmax

Gi ði ¼ 1; 2; . . . ;NGÞQmin

Gi � QGi � QmaxGi ði ¼ 1; 2; . . . ;NGÞ

V mini � Vi � V max

i i ¼ 1; 2; . . . ;Nð ÞTi;min � Ti � Ti;max i ¼ 1; 2; . . . ;NTð Þ

SLij j � SmaxLi ði ¼ 1; 2; . . . ;NÞ

8>>>>>>><>>>>>>>:

where

PGi and QGi are the total active and reactive power generation at bus i. PDi andQDi are the total active and reactive power demands at bus i. Vi and Vj arethe voltage magnitudes at buses i and j

Gij and Bij are the real and imaginary parts of the ijth element of admittancematrix (Ybus); qij is the difference of voltage angles between buses i and j

PGi,max and PGi,min are the upper and lower limits of active power output of theith generator

QGi,max and QGi,min are the upper and lower limits of reactive power output ofthe ith generator

Vi,max and Vi,min are the upper and lower limits of voltage magnitude at bus iTi is the tap setting of the ith transformer; Ti,max and Ti,min are the lower and

upper limits of the tap setting of the ith transformer|SLi| is the line loading in MVA at line iSLi,max is the line loading limit in MVA at line i (the line flow when the system

is fully loaded)N, NL and NT are the total number of buses, lines and transformers,

respectively.

The limits of P, Q, V and T at generation buses are summarised in Table 6.3.Applying MATPOWER Version 4.1, 14-Dec-2011-AC Optimal Power Flow,

MATLAB� Interior Point Solver – MIPS, Version 1.0 and using PC-Intel�

CoreTM i5-2430M, [email protected] GHz, RAM: 8 GB, System type: 64-bit OS, it is

Table 6.3 Limits of parameters: P, Q, V and T

Generatorno.

Active powergeneration

Reactive powergeneration

Voltage at all buses (pu)

Min(MW)

Max(MW)

Min(MVAr)

Max(MVAr)

Vmin ¼ 0.9 Vmax ¼ 1.1

1 10 250 �300 300 The tap setting of transformersranges from 0 to 12 10 300 �300 300

3 10 270 �300 300


found that the program is converged in 0.07 s and the objective function value ¼5353.58 $/hr. The voltage, power and cost at system buses as well as the powerflow in system lines are obtained as summarised in Tables 6.4 and 6.5, respectively.

6.3 OPF with dynamic security constraint

There is increasing interest and need to take into account the dynamic securityconstraints in the OPF computations to protect the system against disturbances. Asthe issue of system stability, small signal stability and transient stability is of anincreasing concern in relation to the security of system operation, the conventionalOPF should be adapted to find an optimal solution that minimises a cost function

Table 6.4 Voltage, power and cost at system buses

Busno.

Voltage Generation Load Lambda($/MVA-hr)

Mag(pu)

Ang(deg)

P(MW)

Q(MVAr)

P(MW)

Q(MVAr)

P Q

1 1.087 0 87.05 �22.20 24.1502 1.1 0.044 136.16 6.68 24.3473 1.1 �6.212 97.45 20.89 24.8754 1.1 �2.403 24.1505 1.098 �4.248 90 30 24.384 �0.0026 1.011 �17.468 100 35 26.976 0.8487 1.099 �3.993 24.350 0.0458 1.076 �8.530 125 50 24.842 0.1669 1.090 �8.941 24.880 0.120Total 320.65 5.37 315 115

Table 6.5 Power flow in the lines

Line From bus no. To bus no. From bus injection Losses

P (MW) Q (MVAr) P (MW) Q (MVAr)

1 1 4 87.05 �22.20 0 3.932 4 5 41.49 �13.82 0.244 1.323 6 9 �100.00 �35.00 3.918 17.084 3 9 97.45 20.89 0 4.815 9 8 �6.47 3.58 0.03 0.256 8 7 �131.50 �22.15 1.283 10.877 7 2 �136.16 2.92 0 9.608 7 5 3.37 �18.31 0.003 0.029 5 4 �45.39 �7.47 0.172 1.46Total 5.650 49.34


while maintaining certain stability criteria. This may need to modify the objectivefunction or add some constraints to the problem formulated in Section 6.1.

Small signal stability, as is known, is defined as the ability of a power systemto maintain the generators in synchronism when subjected to small disturbances.To analyse power system small signal stability, the state equation describing thesystem should be linearised at the operating point [4].

Assuming zero input, the power system can be described by

_X ¼ f ðXÞ ð6:16Þwhere X is the state vector of the power system, f is a set of non-linear functionsand the derivative is w.r.t. time.

By Taylor’s series expansion and assuming a small deviation of the statevector, DX, the linearised form of (6.16) is

dDXdt

¼ A½ �DX ð6:17Þ

where [A] is the state matrix. The small signal stability is determined by theeigenvalues of matrix A, which can be written in a general form as

l ¼ s� jw ð6:18Þwhere the real and imaginary components give the damping and frequency ofthe corresponding mode, respectively. Accordingly, determination of stability isdetermined as explained below:

● The system is stable if all eigenvalues have negative real parts.● The system is unstable if at least one eigenvalue has positive real part.● The system is oscillatory if at least one eigenvalue has zero real part.

The decay rate of oscillation can be defined by a common index (damping ratioz) deduced in terms of eigenvalues components by (6.19). The system is consideredto have wider stability margin as z gets larger.

z ¼ �sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ w2

p ð6:19Þ

Therefore, to consider small signal stability in the OPF, mode analysis describedabove should be conducted for each candidate operating point so that the eigenvaluereal parts or damping ratios can be achieved to compare these operating points. Theseindices can be involved in the objective function or constraints in the OPF process.

Transient stability means the ability of the power system to maintain its gen-erators in synchronism when subjected to a large severe disturbance. To keep thesystem operating satisfactorily it is of importance to develop the OPF calculationsby incorporating transient stability constraints in addition to static security con-straints. Different approaches have been proposed for this purpose. However, theseapproaches can be categorised into three categories [5]. The first category includesthe methods based on transient energy function or energy balance. They have


limitations pertaining to power system modelling with the difficulty in forming thefunction expressing the stability margin in terms of generator active power. Thesecond category comprises the methods that express the transient stability boundaryby an approximated non-linear function in terms of the control variables. Thesemethods are exposed to the difficulty of determining the required non-linearfunction. In the third category the methods simulate the power system in time-domain with detailed dynamic models. Then, stability constraints based on relativerotor angles at each time interval and a set of algebraic equations for all timeintervals are included in the OPF. Of course, high accuracy and robustness may beachieved, but, on the other hand, the practical implementation is time consuming aswell as the computation requires high memory storage. This is due to implyinglarge numbers of additional variables associated with power system dynamicalmodel and constraints in various time intervals in the OPF formulation.

Power system dynamics is described by a set of differential-algebraic equa-tions as [6]:

_X ¼ f1½XðtÞ;BðtÞ;Y �0 ¼ f2½XðtÞ;BðtÞ;Y �

)ð6:20Þ

where X(t) is the state variables vector including generator rotor angles and speeds,and B(t) is the vector of algebraic variables including network-related variablessuch as bus voltages and angles. Y represents a set of control variables such asgenerator active power output, size of capacitor banks installed at a bus and so onthat is generally time independent for transient stability analysis and can be viewedas parameters of (6.20).

The initial values of ith generator rotor angles doi and emf E0

i are obtained fromthe system pre-fault steady-state conditions as below:

E0iVt sin do

i � dt

� �X 0

di

� PGi ¼ 0

E0iVt cos do

i � dt

� �X 0

di

� QGi ¼ 0

9>>>>=>>>>;

ð6:21Þ

where the generator is represented by a constant voltage source E0 behind direct-axistransient reactance X 0

d , and dt and Vt represent the voltage angle and magnitude atgenerator terminal bus, respectively. In addition, as the pre-fault is a steady-statecondition the initial angular speed

ðwoi Þ ¼ 1 pu ð6:22Þ

The swing equations for the generator as given in Chapter 2 are represented by

_di ¼ wi � 1

_wi ¼ 12HiwB

ðPmi � PeiÞ

9>=>; ð6:23Þ


and must be discretised using the trapezoidal rule to be converted into algebraicequations. Hence, generator rotor angles and speeds for a generic time interval(k þ 1) are defined by the following equations:

dkþ1i � dk

i �Dt

2wkþ1

i þ wki

� � ¼ 0

wkþ1i � wk

i �Dt

21

2HiwBPkþ1

ai � Pkai

� � ¼ 0

9>>>=>>>;

ð6:24Þ

where Pkai ¼ Pmi � Pk

ei ¼ accelerating power at time interval k.The dynamical constraints for the transient stability can be considered as the

angle deviation between any two generators is less than the appointed value dmax inthe whole system trajectory.

jdki tð Þ � djðtÞj < dmax t 2 0; T½ � ð6:25Þ

where T denotes the concerned time range.Additional constraints may be added, such as

�p � di � p

dref ¼ 0

)ð6:26Þ

Therefore, the formulation of stability-constrained OPF problem can be sum-marised as below [7].

Minimise (6.4)subject to

● power flow (6.5)● the limits in (6.6)● initial values of generator rotor angles and emf’s, (6.21) and (6.22)● discrete swing (6.24)● transient stability limit (6.25)● additional constraints (6.26)

Several studies are reported in the literature to solve the stability-constrainedOPF problem aiming to minimise the costs of generating power output as anobjective function [8] or considering various constraints [9, 10]. Different methods,e.g. differential evolution, inexact Newton method and primal-dual interior, havebeen proposed to formulate various OPF problems [11–19] as well as attemptsto reduce the computation time of solution [20] using appropriate algorithms [21, 22].Numerical discretisation of dynamic security constraints should be implementedappropriately [23]. On the other hand, it has been found that the considerationof stability constraints (steady-state and/or transient stability constraints) hasremarkable impact on security pricing and solution results [24–26].


Example 6.2 Solve the problem in Example 6.1 taking into account the transientstability constraints.

Solution:

The difference between the bus voltage angle qf at the ‘from end’ of a branch andthe angle qt at the ‘to end’ can be bounded above and below to act as a proxy for atransient stability limit. MATPOWER creates the corresponding constraints on thevoltage angle variables. So, using MATPOWER Version 4.1, AC Optimal PowerFlow – MATLAB Interior Point Solver – MIPS, Version 1.0, it is seen that theprogram is converged in 0.12 s and the objective function value is 6095$/hr. Thevoltage, power and cost at system buses as well as the power flow in system linesare obtained as summarised in Tables 6.6 and 6.7, respectively.

Table 6.6 Voltage, power and cost at system buses

Busno.

Voltage Generation Load Lambda($/MVA-hr)

Mag(pu)

Ang(deg)

P(MW)

Q(MVAr)

P(MW)

Q(MVAr)

P Q

1 1.089 0.000 90.00 �18.61 17.0352 1.096 �1.673 87.76 9.42 24.0353 1.073 �1.120 142.62 14.83 15.0004 1.100 �2.480 23.773 �0.2925 1.096 �4.380 90 30 23.999 �0.2476 0.985 �14.227 100 35 16.312 0.4777 1.092 �4.300 24.0358 1.063 �7.255 125 50 24.365 0.2059 1.068 �5.303 15.000Total 320.38 5.64 315 115

Table 6.7 Power flow in the lines

Line From bus To bus From bus injection Losses

P (MW) Q (MVAr) P (MW) Q (MVAr)

1 1 4 90.00 �18.61 0.000 4.102 4 5 42.95 �12.23 0.260 1.413 6 9 �100.00 �35.83 4.147 18.084 3 9 142.62 14.83 0.000 10.465 9 8 38.47 �10.94 0.155 1.316 8 7 �86.68 �38.53 0.633 5.367 7 2 �87.76 �5.37 0.000 4.058 7 5 0.45 �21.22 0.002 0.019 5 4 �46.86 �9.19 0.183 1.55Total 5.380 46.34


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Part III

Stability analysis

Chapter 7

Small signal stability

7.1 Basic concepts

Power systems are subjected to either small or large disturbances. Small disturbances,e.g. load changes, occur continually. Thus, the system must have the ability towithstand the effect of such disturbances that implies change of conditions andrestoration of the normal operating conditions. On the other hand, large disturbances(disturbances of a severe nature) such as loss of a large generator or short circuit on atransmission line may lead to structural changes as the faulted elements are isolatedby actuating the protective relays. Some generators and loads may be disconnected topreserve the continuity of operation of bulk of the system. Interconnected systems,for certain severe disturbances, may also be intentionally split into a number ofislands to preserve as much of the generation and load as possible, permitting theactions of automatic controls to eventually restore the system to a normal state.Otherwise, the system may become unstable, particularly, if the normal operation ofthe system could not be restored. In this case, a progressive increase in angulardeviation between generator rotors or a progressive decrease in bus voltages (speed-up and speed-down situation) may occur. The instability condition of a power systemcould lead to cascading outages and a shutdown of a major portion of the system.

Analysing the power system with a goal of determining its stability is based onmodels of system components encompassing adequate assumptions to formulate anappropriate mathematical model in the time scale that properly describes the phe-nomenon under study. Then, by selecting an analytical method the stability of thepower system can be determined when a specific disturbance occurs. The resultscan be examined to test the adequacy of model assumptions.

Referring to the equation of motion of the rotor, swing equation, discussed inSections 2.5.1 and 2.5.2, Chapter 2, and considering the damping term it can bewritten as

2HwB

€d þ D _d ¼ Pm � Pe ð7:1Þ

where t is the time in seconds, H is the inertia constant (s), w is the angular speed inelec. rad/s, D is the damping coefficient, Pm is the mechanical input power and Pe

is the electrical output power. The difference, Pm � Pe, is called the acceleratingpower, Pa. The damping coefficient D is assumed to be small and positive.

Thus, the damping term D _d can be ignored when solving the equation for a shortperiod following a disturbance. Consequently, D can be neglected in transientstability but should be considered in small disturbance analysis.

Solution of (7.1) as a second-order non-linear ordinary differential equationgives the change of angle d versus time to decide whether the generator is stable.As there is no analytic solution in general, numerical methods have to be used. Thisentails rewriting (7.1) in the form _x ¼ f x; uð Þ as below:

_d ¼ w_w ¼ wB

2HPm � Pe � Dw½ �

9>=>; ð7:2Þ

The variables in (7.2) depend on machine parameters, network topology,nature of disturbance and characteristics of prime movers and automatic voltageregulators. Therefore, the response of the power system to a disturbance mayinvolve much of the equipment. For example, a fault on a critical element followedby its isolation will cause variations in power flow, network bus voltages andmachine rotor speeds. The voltage variations will actuate both generator andtransmission network voltage regulators while the generator speed variations willactuate prime mover governors. As well the voltage and frequency variations willaffect the system loads to varying degrees depending on their individual char-acteristics. Further, devices used to protect individual equipment may respond tovariations in system variables and cause tripping of the equipment, thereby weak-ening the system and possibly leading to system instability.

Accordingly, (7.2) will be incorporated with mathematical equations represent-ing the dynamic characteristics of both voltage regulators and excitation system todetermine the electrical power delivered by generators as well as the characteristics ofprime mover governors to calculate the input mechanical power. Additional equa-tions have been included to consider some constraints such as excitation limits andvoltage limits. This leads to modelling each generator in the power system by asystem of algebraic-differential equations. The order of the model gets higher as moredetailed models of involved devices are considered. Thus, a typical power system is ahigh-order non-linear multi-variable process whose dynamic response is influencedby a wide array of devices with different characteristics and response rates. It operatesin a constantly changing environment (loads, generator output, operating parameterschange continually). The system stability when subjected to a disturbance depends oninitial operating conditions and the nature of disturbance as well.

It is seen that the stability problem is a high dimensional and complicatedproblem. So, some simplifying assumptions may help analysing specific types ofproblems with a condition of using appropriate degree of detail of system repre-sentation and adequate analytical techniques.

For instance, the simplified second-order model of a generator used in theclassical methods of analysis is expressed by an equivalent voltage source, Eg,behind impedance, Xg, with considering some assumptions: (i) manual excitationcontrol is used with absence of voltage regulators, i.e. in steady state the magnitude


of the voltage source is determined by the field current, which is constant;(ii) damper circuits are neglected; (iii) transient stability is judged by the firstswing; (iv) saliency has little effect and can be neglected; (v) flux decay in the fieldcircuit is neglected for short period of time less than the field time constant;(vi) losses are neglected as the impedance is purely reactive; and (vii) response ofprime mover governor is neglected, i.e. the input mechanical power to the generatoris constant, Thus, (7.2) becomes

_d ¼ w

_w ¼ wB

2HPm � Pmax sin d½ �

9=; ð7:3Þ

where Pmax ¼ EgEt

Xgas given in (1.1) and Et is the terminal machine voltage.

If the generator is connected to an infinite bus, the external reactance is addedto Xg and Et is replaced by the infinite bus voltage to get the angle deviations withrespect to the infinite bus voltage. On the other hand, if the generator is connectedto a multi-machine system, the electrical output power delivered by the generator iscalculated by applying load flow techniques that may contain some constraints [1].

7.1.1 Equilibrium pointsStability is a condition of equilibrium between opposing forces. For an electricpower system, it is a property of the system motion around the initial operatingcondition when the system is subjected to a disturbance, small or large [2]. In thecase of a small disturbance, at the equilibrium points, different opposing forces thatexist in the system are equal instantaneously or over a cycle as in the case of slowcyclic variations. For large disturbances, it is impractical and uneconomical tosatisfy equilibrium points at which the various sets of opposing forces are balancedfor every possible disturbance. Therefore, design contingencies are selected on thebasis that they have a reasonable probability of occurrence. Hence, large dis-turbance stability refers to a specified disturbance scenario. On the other hand, thevarious sets of opposing forces may experience sustained imbalance leading todifferent forms of instability, depending on system operating condition, type ofdisturbance and network topology.

Various equilibrium points may get involved in the stability analysis of powersystems. In the case of large disturbances, the perturbations of interest are specifiedand all post-disturbance equilibrium points relevant for a given pre-disturbanceequilibrium are assumed to be determined. Numerical methods are usually used tosolve a large non-linear set of differential equations in the form:

_x ¼ f x; u; tð Þ ð7:4Þwhere x is the state vector and a function of time t of dimension n, _x is its deriva-tive, f is differentiable function of dimension n with a domain including the origin(called vector field) and u can be viewed, generally, as control input vector ofdimension r.

Small signal stability 149

As the state vector x is a function of time, the explicit writing of time argumentt can be omitted and commonly (7.4) can be rewritten as

_x ¼ f x; uð Þ ð7:5ÞFor small disturbances, the perturbation is characterised by size (small signalanalysis) and linearised system equations are valid for such analysis. The sameequilibrium point typically characterises the system pre and post disturbance. Thelinearised form of (7.5) is

_x ¼ Ax þ Bu ð7:6Þand the stability analysis is based on studying its characteristic equation asexplained in the forthcoming sections.

The system described by (7.4) is said to be autonomous if u is a constantvector. Otherwise, if the elements of u are explicit functions of time t, the system issaid to be non-autonomous.

For autonomous systems, the solution of (7.5) for a specified initial conditionx(to) ¼ xo can be expressed as Ft(xo) to illustrate explicitly the dependence oninitial condition. Ft (xo) is called the trajectory through xo while Ft(x) where x [ Rn

is called the flow. The vector u is not explicitly written as it is constant and can beconsidered a parameter. Meanwhile, for non-autonomous systems the trajectory isalso a function of time t and can be expressed as Ft(xo, to) to illustrate that thesolution passes through xo at to.

As the power system can be modelled as an autonomous system, the solution of(7.5) for autonomous systems is of interest taking some consideration into accountsuch as (i) the presence of solution for all t, (ii) with respect to the initial conditionthe derivative of a trajectory exists and is non-singular, i.e. Ft(xo) is continuouswith respect to initial state xo and (iii) Ft(x)¼Ft(y) at any time t if and only ifFt1þt2 ¼Ft1Ft2. Consequently, a trajectory of an autonomous system is uniquelyspecified by its initial condition and that distinct trajectories do not intersect.

Therefore, an equilibrium point xeq of an autonomous system is a constantsolution Ft(xeq) and satisfies the relation

0 ¼ f ðxeq; uÞ ð7:7ÞIt is to be noted that the real solutions of (7.7) give several equilibrium points.

7.1.2 Stability of equilibrium pointThe equilibrium point that satisfies (7.7) is said to be an asymptoticallystable equilibrium point (SEP) if all trajectories lying initially in a sufficientlysmall spherical neighbourhood of radius r can be forced for all time t � to to lieentirely in a given cylinder of radius Ɛ. Figure 7.1 depicts the asymptotic stabilityfor the case of a two-dimensional system in the space of real variables [3]. Thus, foreach Ɛ> 0, r¼ r(Ɛ, to), such that

x toð Þj jj j < r ) x tð Þj jj j < Ɛ 8 t � to > 0 ð7:8Þ


and h toð Þ > 0, such that

x toð Þj jj j < h toð Þ ) x tð Þ ! 0 as t ! 1 ð7:9ÞThe stability of an equilibrium point should be decided [4]. For small

disturbances, it can be determined by the solution of the linearised differentialequations describing the system behaviour at xeq. Assuming small changes in sys-tem quantities, linearisation can be found by making a Taylor series expansionabout xeq and neglecting higher-order terms as below.

Assuming x ¼ xeq þ Dx and substituting in (7.5) gives

_x ¼ _xeq þ D _x ¼ f xeq; u� �þ @f x; uð Þ

@x

� �x¼xeq

Dx ð7:10Þ

From (7.7) and (7.10) the following relation can be obtained:

D _x ¼ A xeq; u� ��

Dx ð7:11Þ

where A is a matrix of dimension n � n. Its elements are functions of xeq and u. Theijth element is given by

Aij xeq; u� � ¼ @fi

@xjxeq; u� � ð7:12Þ

The matrix A is a constant matrix for a given xeq and u. The solution of (7.11) canbe found as

Dx tð Þ ¼ eA t�toð ÞDx toð Þ ¼ c1el1tv1 þ c2el2tv2 þ � � � þ cnelntvn ð7:13Þwhere c1; c2; . . .; cn are constants depending on the initial conditions. li and vi arethe ith eigenvalue and corresponding eigenvector of matrix A, respectively.

ε

η

ρx(t)

t

Figure 7.1 Asymptotic stable trajectories


By examining (7.13) it is found that (i) if <e(li) < 0 for all li, i ¼ 1, 2, . . . , n,then for any sufficiently small disturbance from the equilibrium point xeq, thetrajectories tend to xeq as t ??. Thus, the equilibrium point xeq is said to beasymptotically stable; (ii) if <e(li) > 0 for all li, then any disturbance leads to atrajectory leaving the neighbourhood of xeq and the equilibrium point is unstable;(iii) the equilibrium point is a saddle point if there are two eigenvalues li and lj

such that <e(li) < 0 and <e(lj) > 0.It is concluded that an equilibrium point is asymptotically stable if all nearby

trajectories approach xeq as t ??. It will be an unstable equilibrium point (UEP) ifno trajectories in proximity remain nearby. Of course, an UEP is asymptoticallystable in reverse time, i.e. as t ?�?. An equilibrium point is unstable and called a‘saddle point’ if there is a nearby trajectory that approaches xeq as t ?? andanother trajectory approaches xeq as t ?�?.

7.1.3 Phasor diagrams of synchronous machinesThe stability study concerns with the examination of machine behaviour when thesystem is subjected to a disturbance and its state is investigated (stable or not)after clearing the disturbance. This can be achieved by solving system equations[5]. As explained in Chapters 2 and 3 and in Section 7.1, the system includingsynchronous generators is represented by a set of differential equations todescribe its behaviour as a function of time. Solution of these equations entailsdetermination of the initial conditions that can be considered the steady-statecondition prior to the disturbance. Consequently, phasor diagrams representingphasor equations help determine the initial conditions as in steady state the dif-ferential equations are not needed where the variables are constants or varyingsinusoidaly with time.

At steady state, the currents in (3.16) are constants.Thus,

pid ¼ piq ¼ pif ¼ pikd ¼ pikq ¼ 0 ð7:14Þand

Rkdikd ¼ Rkqikq ¼ 0 ðas ikd ¼ ikq ¼ 0 at steady stateÞ ð7:15Þ

Therefore, the voltages vd and vq from (3.16) can be written as

vd ¼ �Raid � wLqiq and vq ¼ �Raiq þ wLdid þ kMf wif ð7:16Þ

Using (2.16) and substituting vo ¼ 0 at balanced conditions, the phase voltageva is

va ¼ffiffiffi23

rvd cos qþ vq sin q� � ð7:17Þ

where q is defined as wot þ dþ p/2


From (7.16) and (7.17) the equation below can be obtained:

va ¼ffiffiffi23

r h� Raid þ wLqiq� �

cos wot þ dþ p2

þ �Raiq þ wLdid þ kMf wif

� �sin wot þ dþ p

2

iHence,

va ¼ffiffiffi23

r h� Raid þ wLqiq� �

cos wot þ dþ p2

þ �Raiq þ wLdid þ kMf wif

� �cos wot þ dð Þ

ið7:18Þ

At steady state, it is to be noted that (i) the angular speed w is constant andequals wo; (ii) the direct and quadrature axis reactances ‘Xd and Xq’ equal wLd andwLq, respectively; (iii) wkMf if ¼H3E, where E is the rms of stator EMF as a line-to-neutral value corresponding to the field current if as given in (I.11), Appendix I.Accordingly, the rms voltage phasor Va can be found as

Va ¼ �Raidffiffiffi

3p ff dþ p

2

þ iqffiffiffi

3p ffd

� �� Xq

iqffiffiffi3

p ff dþ p2

þ Xd

idffiffiffi3

p ffdþ Effd

ð7:19ÞUsing the relation j ¼ 1 ff p/2, (7.19) becomes

Va ¼ �Raiqffiffiffi3

p ffdþ jidffiffiffi

3p ffd

� �� jXq

iqffiffiffi3

p ffdþ Xdidffiffiffi

3p ffdþ Effd ð7:20Þ

where Va and E are stator rms phase voltages in pu, using the rated phase quantitiesas base values (Appendix I); id and iq are the currents obtained from the Park’stransformation. The rms equivalent d and q axes currents are defined as

Id ≜idffiffiffi

3p and Iq ≜

iqffiffiffi3

p ð7:21Þ

and the stator current ia as a phasor will have the two rectangular components Id

and Iq. Thus, the relation below can be written if the q-axis is a phasor reference:

Ia ¼ Iq þ jId

� �e jd ð7:22Þ

Incorporating (7.21) and (7.22) into (7.20) gives

EI ¼ Va þ RaIa þ jXqIq þ jXdId ð7:23Þ

where EI ¼ EIffd, Iq ¼ Iqffd, Id ¼ jIdffd


From (7.16) the rms stator voltages can be computed as

Vd ≜vdffiffiffi

3p ¼ �RaId � XqIq and Vq ≜

vqffiffiffi3

p ¼ �RaIq þ XdId þ E ð7:24Þ

Equation (7.23) can be represented by the phase diagram shown in Figure 7.2.It is noted from Figure 7.2 that the components of voltages and currents in thedirection of d-axis are negative quantities as the current Ia lags the voltage Va andthe direct axis leads the quadrature axis by 90 electrical degrees. The position of thequadrature axis can be determined by calculating a fictitious voltage located on thisaxis. The voltage E0 is defined as the internal machine voltage behind transientdirect axis reactance, X 0

d . Its projection in q-axis direction is known as E0q. The

difference between Eq and X 0q is given by j(Xq � X 0

d )Id. Also, the differencebetween EI ‘equals Efd in steady state’ and Eq is j(Xd � Xq)Id.

EI is a voltage proportional to field current, Efd is a term representing the fieldvoltage acting along the quadrature axis, Eq is the voltage behind quadrature axissynchronous reactance and X 0

q is the voltage proportional to the field flux linkagesresulting from the combined effect of the field and armature currents.

7.2 Small signal stability

Regarding the rotor angle stability, small signal stability is the ability of powersystem to maintain synchronism under small disturbances. Some analyses have

Quadrature a

xis

Direct axis

Reference axisId

Vd

Va

EI

δ

β φ

E′

Ia

Iq

Vq

E′q

j(xq – x′d)Id

j(xd – xq)Id

Eq

jxqIq

jxdIdjxqIa

RaIa

Jx′ dI

a

Figure 7.2 Phasor diagram of synchronous machine to determine steady stateparameters


been done to determine this type of system stability [6–8]. The disturbances areconsidered to be sufficiently small so that linearisation of system equations isallowed for analysis.

To mathematically explain the small signal stability problem, the dynamicbehaviour of a machine connected to an infinite bus is considered. The equation ofmotion, swing equation, of the synchronous machine (7.25) is a non-linear functionof the power angle d.

2HwB

€d ¼ Pm � Pe

¼ Pm � Pmax sin d

9>=>; ð7:25Þ

For small disturbances, linearisation of (7.14) can be applied with acceptableaccuracy.

Assuming Dd is the small deviations in power angle from its initial operatingpoint do. Then, (7.25) becomes

2HwB

d2 do þ Ddð Þdt2

¼ Pm � Pmax sin do þ Ddð Þ

i.e.

2HwB

d2do

dt2þ 2HwB

d2Dddt2

¼ Pm � Pmax sin do cos Ddþ cos do sin Ddð Þ

Substituting sin Dd ffi Dd and cos Dd ffi 1 as Dd is small, gives

2HwB

d2do

dt2þ 2HwB

d2Dddt2

¼ Pm � Pmax sin do � Pmax cos doDd ð7:26Þ

At the initial state

2HwB

d2do

dt2¼ Pm � Pmax sin do

Thus, (7.26) in the linearised form and written as

2HwB

d2Dddt2

þ Pmax cos doDd ¼ 0 ð7:27Þ

is the power–angle curve. At do, i.e. dPedd do ;j its value is known as the ‘synchronising

power coefficient’ Ps. This coefficient has a prominent role in system stabilitydetermination.


Equation (7.27) can be rewritten as

2HwB

d2Dddt2

þ PsDd ¼ 0 ð7:28Þ

and has the solution depending on the roots of its characteristic equation2HwB

s2 þ Ps ¼ 0. The roots are given by

s ¼ j

ffiffiffiffiffiffiffiffiffiffiffiPswB

2H

rð7:29Þ

In case of a negative value of Ps, the response is exponentially increasingand stability is lost as one root lies in the right-half of the s-plane. When Ps ispositive the two roots lie on the imaginary axis and the motion is un-dampedoscillatory. The system is marginally stable with a natural frequency of oscilla-tion, wn, given by

wn ¼ffiffiffiffiffiffiffiffiffiffiffiPswB

2H

rð7:30Þ

It can be seen that the synchronising power coefficient, Ps ¼ dPe/dt, is positivewhen d lies between 0o and 90o with a maximum value at do ¼ 0o.

The damping torque is a component of electrical torque. It is proportional tothe speed change and will be set up on the rotor tending to minimise the differencebetween the rotor angular velocity and the angular velocity of the resultant rotatingair gap field. The damping power ‘Pd’ is approximately proportional to the speeddeviation and can be expressed as

Pd ¼ Ddddt

ð7:31Þ

where D is the damping coefficient in pu.The damping coefficient D has a small value and can be neglected in transient

analysis as the solution of swing equation is required for a short period, e.g. 1–2 sfollowing a disturbance. However, in small signal stability analysis it should beconsidered as the damping power oscillations may damp out eventually when Ps ispositive and the operation at the equilibrium angle will be restored. Therefore, adamping power term is added to (7.28) to become

2HwB

d2Dddt2

þ DdDddt

þPsDd ¼ 0 ð7:32Þ

and the roots of its characteristic equation determine the system response.Further mathematical explanation is given below by writing (7.32) in state

space form to make it possible to extend the analysis to multi-machine systems.


Equation (7.32) is rewritten as

d2Dddt2

þ wB

2HD

dDddt

þ wB

2HPsDd ¼ 0 ð7:33Þ

Assuming x1 ¼ Dd and x2 ¼ Dw ¼ D _d, then

_x1 ¼ x2

_x2 ¼ �wB

2HPsx1 � wB

2HDx2

9=; ð7:34Þ

and in matrix form, (7.34) is written as

_x1

_x2

� �¼

0 1

�Ps

Ĥ� D

Ĥ

24

35 x1

x2

� �ð7:35Þ

where Ĥ ¼ 2HwB

or

_XðtÞ ¼ AXðtÞ ð7:36Þwhere

A ¼0 1

�Ps

Ĥ� D

Ĥ

24

35

Equation (7.36) is a homogeneous state equation, unforced state variableequation, as it is assumed that the disturbances causing the changes disappear.When the state variables are the desired response the output vector y(t) is defined asy(t) ¼ Cx(t), where C is a unit matrix of dimension 2 � 2.

Applying Laplace transform gives

sX sð Þ � x oð Þ ¼ AX sð Þ

or

X sð Þ ¼ ðsI � AÞ�1x oð Þ ð7:37Þwhere

sI � Að Þ ¼s �1

�Ps

Ĥs þ D

Ĥ

24

35


Hence,

X sð Þ ¼

s þ D

Ĥ1

�Ps

Ĥs

2664

3775x oð Þ

s2 þ D

Ĥs þ Ps

ĤIf the rotor is suddenly disturbed by a small angle Ddo, the state variablesx1 oð Þ ¼ Ddo and x2 oð Þ ¼ Dwo ¼ 0. Then

Dd sð Þ ¼s þ D

MĤ

� �Ddo

s2 þ D

Ĥs þ Ps

Ĥ

and Dw sð Þ ¼Ps

ĤDdo

s2 þ D

Ĥs þ Ps

Ĥ

ð7:38Þ

The eigenvalues are the roots of the characteristic equation s2 þ DĤ

s þ Ps

Ĥ¼ 0 and

given by

l ¼ � D

2Ĥ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2

4Ĥ2 �

Ps

Ĥ

sð7:39Þ

It can be seen that both eigenvalues have negative real parts when Ps is posi-tive. If Ps is negative one of the eigenvalues is positive real. For small D and Ps > 0the eigenvalues are complex and given by

l ¼ �s jwd ð7:40Þwhere

s ¼ D=2Ĥ� �

and wd ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPs

Ĥ� D2

4Ĥ2

q damping frequency of oscillation

Then, (7.38) can be rewritten as

Dd sð Þ ¼ s þ 2sð ÞDdo

s2 þ 2ss þ w2n

and Dw sð Þ ¼ w2nDdo

s2 þ 2ss þ w2n

ð7:41Þ

Taking inverse Laplace transform results in zero-input response:

Dd ¼ wn

wdDdoe�stsin wdt þ qð Þ and Dw ¼ w2

n

wdDdoe�stsin wdt ð7:42Þ

where q¼ cos�1(s/wn)Dd and Dw are added to do and wo, respectively, to give the rotor angle and

angular frequency.The response time constant, t, and the damping ratio, DR, are defined as

t ¼ 1=s and DR ¼ swn

¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffis2 þ w2

d

q ¼ffiffiffiffiffiffiffiffiffiffiffi

D2

4ĤPs

sð7:43Þ


Therefore, for stability of an equilibrium point, the necessary condition Ps > 0must be satisfied. This is illustrated by the power–angle curve, shown in Figure 7.3.When Pm < Pmax there are two values of d corresponding to a specified value of Pm

and �180� < d< 180�. Thus, there are two equilibrium points as

xs ¼ ds; 0ð Þ ≜ SEP at Ps > 0

xu ¼ du; 0ð Þ ≜ UEP at Ps < 0

)ð7:44Þ

where the subscripts s and u denote stable and unstable states, respectively.The maximum electrical power output, Pmax, at d¼ 90� is called the ‘steady-

state stability limit’. This stability limit is the boundary at which the systembecomes unstable if any small disturbance occurs.

Example 7.1 Referring to (7.40) find the eigenvalues loci in the s-plane as Pm isvaried.

Solution:

At a specific value of Pm there are two equilibrium points: one of them is SEP andthe other is UEP. These two equilibrium points come closer as Pm is increased.

As shown in Figure 7.4(a), at SEP and 0 < Pm < Pmax: It is found that Ps ispositive and the eigenvalues are complex quantities with negative real parts (points#1 and #2). As Pm increases, Ps decreases and both eigenvalues reach the Re-axisat �s when Ps

Ĥ¼ D2

4Ĥ2 (point #3). Continuing the increase in Pm, Ps continues to

δ

Pmax

δs δu 90o

Pm

Pe

Ps > 0 Ps < 0

Figure 7.3 Power–angle curve with stable and unstable equilibrium points


decrease until being zero at Pm ¼ Pmax and then one of the eigenvalues reaches theorigin while the other approaches �D/H0, points #4 and #5, respectively.

At UEP, Figure 7.4(b), Ps is negative and both eigenvalues are real and movetowards the origin (points #1 and #2). When Pm ¼ Pmax one of the values is zero,i.e. the eigenvalue reaches the origin, point #3, and the other equals (�D/H), point #4.

Example 7.2 A synchronous generator is connected to an infinite bus through atransformer and a transmission line with data in pu as shown in Figure 7.5 anddelivers a power of 0.8 pu. Find the equilibrium points when (i) neglecting thesystem resistance and (ii) considering the system resistance.

ImIm

Re Re

X

X

1

2

X35 4

(a) (b)

XX12 3

Figure 7.4 Eigenvalues loci: (a) at SEP, (b) at UEP

V∞ = 1.0 0Vs = 1.01 θ sE δ

x′d = 0.2R = 0.003

xtr = 0.1

xTL = 0.35RTL = 0.05

(a)

#1 #2

(b)

R = 0.003 x′d = 0.2 xtr = 0.1 RTL = 0.05 xTL = 0.35

Vs = 1.01 θV∞ = 1.0 0E δ

Figure 7.5 Machine connected to an infinite bus: (a) single-line diagram, (b) theequivalent circuit


Solution:

The power delivered by the machine, Pe, can be calculated by (3.45), that is,

Pe ¼ E2G11 þ EV1Y12 sin d� gð Þwhere

G11 ≜ Y11 cos q11; g ¼ q12 � p=2; Y11ffq11 ¼ y12 þ y1o

Y12ffq12 ¼ �y12 and g ¼ q12 � p=2

This relation indicates that the power–angle curve of the synchronous gen-erator is a sine curve shifted from the origin vertically by an amount E2G11 andhorizontally by the angle ‘g’.

(i) The system resistance is neglected: In this case, the transmission network isreactive.

y1o ¼ 0 as there is no shunt admittance at the sending bus; y12 ¼ 1=j0:65 ¼�j1:538; Y11 ¼ �j1:538, Y12 ¼ j1:538 , q11 ¼ �p=2, q12 ¼ p=2

It can be seen that both E2G11 and g are equal to zero. Thus, the power at thesending bus, P1, is P1 ¼ 0:8 ¼ VsV1YTL sin qs ¼ 1:01=0:35ð Þ sin qs

Then qs ¼ 16.09�

The current flow, I, can be calculated as

I ¼ V s � Vað Þ=ZTL ¼ 1:01ff16:09� � 1:0ff0�ð Þ=j0:35 ¼ 0:804ff1:88�

The internal machine voltage is

Effd ¼ 1:01ff16:09� þ ð0:804ff1:88�Þð0:3ff90�Þ ¼ 1:09ff28:4�

Therefore, the power delivered by the machine to the infinite bus, Pe, isgiven by

Pe ¼ ð1:09 � 1:0Þ=0:65½ �sin d ¼ 1:677 sin d

Hence, at Pe ¼ 0.8, the power angle at the two equilibrium points is 26.986�

and 153.014�. The first represents SEP while the second is UEP, i.e. ds ¼26.986� and du ¼ 153.014�

(ii) Considering the system resistance: From the sending bus to the infinite bus itis found that

Ys1 ¼ �1= 0:05 þ j0:35ð Þ ¼ �0:4 þ j2:8 ¼ 2:828ff98:13�

Yss ¼ 0:4 � j2:8 ¼ 2:828ff�81:87�

qs1 ¼ 98:13�; qss ¼ �81:87�; g ¼ 98:13 � 90 ¼ 8:13�

Gss ¼ 2:828 � cosð�81:87Þ ¼ 0:4

The power delivered at the sending bus, P1 ¼ 0.8 ¼ 0.4(1.01)2 þ 1.01 � 2.828sin(qs � 8.13)


Hence, qs ¼ 16.02�

The current, I, is given by

I ¼ (1.01ff16.02� � 1.0ff0�)/(0.05 þ j0.35) ¼ 0.768 þ j0.192 ¼ 0.79ff14.04�

The internal machine voltage Effd¼ 1.01ff16.02� þ 0.79ff14.04� (0.003 þj0.3) ¼ 0:915 þ j0:509 ¼ 1:04ff29:08�

For the system shown by its equivalent circuit (Figure 7.4(b)):

Y12 ¼ �1= 0:035 þ j0:65ð Þ ¼ �0:0813 þ j0:951 ¼ 0:9545ff94:9�

Y11ffq11 ¼ 0:0813 � j0:951 ¼ 0:9545ff � 85:1�

q12 ¼ 94.9�, q11 ¼�85.1�, g¼ 94.9 � p/2 ¼ 4.9� and

G11 ¼ 0.9545 � 0.085 ¼ 0.081

The power delivered from the machine to the infinite bus, Pe, is given by

Pe ¼ 0:09 þ 1:04=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:0532 þ 0:652

p sin d� gð Þ

¼ 0:09 þ 1:6 sinðd� 4:9�ÞAt Pe ¼ 0.8 ¼ 0.09 þ 1.6 sin(d – 4.9�)

Therefore, the rotor angle at SEP, ds ¼ 29�, and at UEP, du ¼ 151�.

7.2.1 Forced state variable equationOn the other hand, if the system is subjected to an increase of power input by asmall amount DP the system response can be determined by a linearised forcedswing equation that becomes

d2Dddt2

þ wB

2HD

dDddt

þ wB

2HPsDd ¼ wB

2HDP ð7:45Þ

Incorporating (7.30), (7.40) and (7.45) gets

d2Dddt2

þ 2sdDddt

þ w2nDd ¼ Du ð7:46Þ

where

Du ¼ wB

2HDP

Assuming x1 ¼ Dd and x2 ¼ Dw ¼ _Dd, (7.46) is given in the state space form as

_x1 ¼ x2

_x2 ¼ �w2nx1 � 2sx2 þ Du

9=; ð7:47Þ


and in matrix form

_x1

_x2

� �¼ 0 1

�w2n �2s

� �x1

x2

� �þ 0

1

� �Du ð7:48Þ

or

_X tð Þ ¼ AX tð Þ þ BDu tð Þ ð7:49ÞEquation (7.49) represents a forced state variable equation with x1 and x2 thedesired response, and the output vector y(t) is given by y(t) ¼ Cx(t), where C is aunit matrix of dimension 2 � 2.

The Laplace transform of (7.49) is

sX sð Þ ¼ AX sð Þ þ BDU sð Þor

X sð Þ ¼ sI � Að Þ�1BDU sð Þ ð7:50Þwhere DU sð Þ ¼ Du=s

By substituting for (sI � A)�1 and B in (7.49) obtain

X sð Þ ¼

s þ 2s 1

�w2n s

" #0

1

" #Du

s

s2 þ 2ss þ w2n

Hence,

Dd sð Þ ¼ Du

s s2 þ 2ss þ w2n

� � and Dw sð Þ ¼ Du

s2 þ 2ss þ w2n

In the step response inverse Laplace transform gives

Dd ¼ Du

w2n

1 � wn

wde�stsin wdt þ qð Þ

� �and Dw ¼ Du

wde�stsin wdt ð7:51Þ

where q¼ cos�1(s/wn)In (7.46) Du is defined as Du ¼ wB

2H DP ¼ DPĤ

. Substituting its value into (7.51) toobtain Dd and Dw, then adding to do and wo, respectively, gives the rotor angle inelectrical radians and rotor angular frequency in radians per second as

d ¼ do þ DP

Ĥw2n

1 � wn

wde�stsin wdt þ qð Þ

� �

w ¼ w0 þ DP

Ĥwde�stsin wdt

9>>>>=>>>>;

ð7:52Þ

The criterion of stability, Ps < 0, explained above is a simple algebraic relationand does not need the computation of eigenvalues. It is derived from dynamic analysis


based on some assumptions. Therefore, as the complexity of system dynamicsincreases, a detailed linearised model of the synchronous generator is required and theassumptions should be avoided. Two models, current and flux linkage state spacemodels, have been described in Chapter 2. Their linearisation is explained as below.

7.3 Linearised current state space model ofa synchronous generator

To linearise the current model given in (2.104), the state space vector x is assumedto have an initial state xo ¼ x(to) that is known and constant for a specific dynamicstudy [9]. Hence,

xto ¼ ido; ifo; ikdo; iqo; ikqo;wo; do

� � ð7:53ÞWhen a small disturbance occurs, the states will change slightly from their initialvalues and become

x ¼ xo þ Dx ð7:54ÞThe general form of the state space model is

_x ¼ f x; tð Þ ð7:55Þand by substituting ‘(7.54)’ it gives

_xo þ D _x ¼ f xo þ Dx; tð Þ ð7:56ÞBy Taylor series expansion and neglecting the second- and higher-order terms, it isfound that

f xo þ Dx; tð Þ ¼ f xo; tð Þ þ @f@x1

xo

Dx1 þ @f@x2

xo

Dx2 þ � � � þ @f@xn

xo

Dxn ð7:57Þ

Incorporating ‘(7.55)’ at xo, ‘(7.56)’ and ‘(7.57)’ obtain

D _x ¼ A xoð ÞDx ð7:58Þwhere

A xoð Þ ¼ @f@x1

@f@x2

� � � @f@xn

� �xo

ð7:59Þ

Equation (2.53) can be used in the expanded form to deduce the change of eachstate as below:

vdo þ Dvd ¼ �Ra ido þ Didð Þ � wo þ Dwð ÞLq iqo þ Diq

� �� wo þ Dwð ÞkMkq ikqo þ Dikq

� �� Ld pido þ pDidð Þ� kMf pifo þ pDif

� �� kMkd pikdo þ pDikdð Þ


Dropping the second-order terms, e.g. DxiDxj, as they are very small, it is seen that

vdo þ Dvd ¼ �Raido � woLqiqo � wokMkqikqo � Ldpido � kMf pifo � kMkdpikdo

� �� RaDid � woLqDiq � iqoLqDw� wokMkqDikq � ikqokMkqDw� LdpDid � kMf pDif � kMkdpDikd

It is noted that vdo is equal to the quantity between parentheses on the RHS, thus,

Dvd ¼ �RaDid � woLqDiq � wokMkqDikq � Lqiqo þ kMkqikqo

� �Dw

� LdpDid � kMf pDif � kMkdpDikd ð7:60ÞAs Yqo ¼ Lqiqo þ kMkqikqo, ‘(7.60)’ can be rewritten as

Dvd ¼ �RaDid � woLqDiq � wokMkqDikq �YqoDw� LdpDid

� kMf pDif � kMkdpDikd ð7:61ÞSimilarly, the q-axis voltage change Dvq can be found as

Dvq ¼ woLdDid þ wokMf Dif þ wokMkdDikd þ idoLd þ ifokMf þ ikdokMkd

� �Dw

� RaDiq � LqpDiq � kMkqpDikq

Thus,

Dvq ¼ woLdDid þ wokMf Dif þ wokMkdDikd

þYdoDw� RaDiq � LqpDiq � kMkqpDikq ð7:62Þand the change of the field voltage, Dvf, can be deduced as

�Dvf ¼ �Rf Dif � kMf pDid � Lf pDif � LfkdpDikd ð7:63ÞFor the d- and q-axis damper windings ‘KD and KQ’, respectively, the linearisedequations are

0 ¼ �RkdDikd � kMkdpDid � LfkdpDif � LkdpDikd

0 ¼ �RkqDikq � kMkqpDiq � LkqpDikq

)ð7:64Þ

To compute the linearised torque equation, referring to the relations in Section 2.8,the following equations can be written again for convenience.

_w ¼ 1

ĤTm � Te � Td½ �

Te ¼ iqYd � idYq

Yd ¼ Ldid þ kMf if þ kMkdikd

Yq ¼ Lqiq þ kMkqikq

Td ¼ Dw

9>>>>>>>>>=>>>>>>>>>;

ð7:65Þ


Hence,

_w ¼ 1

ĤTm � Dwð Þ � 1

Ĥiq Ldid þ kMf if þ kMkdikd

� �� id Lqiq þ kMkqikq

� �� ð7:66Þ

and the linearised form can be computed as

D _w ¼ 1

Ĥ

hLdiqoDid � LdidoDiq � kMf iqoDif � kMf ifoDiq � kMkdiqoDikd � kMkdikdoDiq

þ LqidoDiq þ LqiqoDid þ kMkqidoDikq þ kMkqikqoDid � DDwþ DTm

ið7:67Þ

In terms of flux linkages in d- and q-axis, ‘(7.67)’ can be written as

D _w ¼ 1

Ĥ½DTm � Ldiqo �Yqo

� �Did � Ydo � Lqido

� �Diq

� kMf iqoDif � kMkdiqoDikd þ kMkqidoDikq � DDw� ð7:68ÞThe linearised form of torque angle (2.103) may be written as

D _d ¼ Dw ð7:69ÞEquations ‘(7.61)’ through ‘(7.69)’ form the linearised system equations for asynchronous machine excluding the load equation. In matrix form and dropping‘D’, as the variables are considered as small changes, the equations can be writtenas below:

(7.70)


or

v ¼ �L1x � L2 _x ð7:71ÞTherefore, the state equation can be derived from ‘(7.71)’ to be in the linear form:

_x ¼ Ax þ Bu ð7:72Þwhere

A½ � ¼ �L�12 L1; B½ � ¼ �L�1

2 and u ¼ v

To include the load equation, the relation of both vd and vq must be linearisedto obtain Dvd and Dvq and then substituting their values into ‘(7.70)’ obtains thecurrent model of one machine problem as below.

Referring to (3.9) the voltages vd and vq can be computed by

vd ¼ �p3V1 sin d� að Þ þ Reid þ Lepid þ wLeiq

vq ¼ p3V1 cos d� að Þ þ Reiq þ Lepiq � wLeid

ð7:73Þ

The trigonometric non-linearities are treated as

sin(do þ Dd) ¼ sin do cos Ddþ cos do sin Ddffi sin do þ (cos do)Dd

where

cos Dd ffi 1 and sin Dd ffi DdThen; the incremental change in sin d ≜ sin do þ Ddð Þ � sin do ffi cos doð ÞDd

ð7:74ÞSimilarly, cos(do þ Dd) ¼ cos do cos Dd� sin do sin Ddffi cos do � (sin do)Dd

Thus; the incremental change in cos d ≜ cos do þ Ddð Þ � cos do ¼ � sin doð ÞDdð7:75Þ

Linearisation of ‘(7.73)’ leads to

Dvd ¼ � ffiffiffi3

pV1 cos do � að ÞDdþ ReDid þ woLeDiq þ iqoLeDwþ LepDid

Dvq ¼ � ffiffiffi3

pV1 sin do � að ÞDdþ ReDiq � woLeDid � idoLeDwþ LepDiq

)

ð7:76ÞSubstituting ‘(7.76)’ into ‘(7.61)’ and ‘(7.62)’

� ffiffiffi3

pV1 cos do � að ÞDdþ ReDid þ woLeDiq þ iqoLeDwþ LepDid

¼ �RaDid � woLqDiq � wokMkqDikq �YqoDw� LdpDid � kMf pDif

� kMkdpDikd �ffiffiffi3

pV1 sin do � að ÞDdþ ReDiq � woLeDid � idoLeDwþ LepDiq

¼ woLdDid þ wokMf Dif þ wokMkdDikd þYdoDw

� RaDiq � LqpDiq � kMkqpDikq

ð7:77Þ


Assuming R ¼Ra þRe;Lq ¼ Lq þLe;Ld ¼ Ld þLe;Yd ¼Yd þLeid ;Yq ¼Yq þLeiqand rearranging ‘(7.77)’ by dropping D, as explained above, gives

0 ¼�Rid �woLqiq �wokMkqikq �Yqow

þ ffiffiffi3

pV1 cos do �að Þd�Ldpid � kMf pif � kMkdpikd

0 ¼�Riq þwoLdid þwokMf if þwokMkdikd

þYdowþ ffiffiffi3

pV1 sin do �að Þd�Lqpiq � kMkqpikq

ð7:78Þ

Incorporating ‘(7.63)’, ‘(7.64)’, ‘(7.68)’, ‘(7.69)’ and ‘(7.78)’, the linearised set ofsystem equations with constant coefficients is

0

�vf

0

0

0

Tm

0

266666666666666664

377777777777777775

¼ �

R 0 0 woLq wokMkq Yqo cosðdo � aÞ0 Rf 0 0 0 0 0

0 0 Rkd 0 0 0 0

�woLd �wokMf �wokMkd R 0 �Ydo sinðdo � aÞ0 0 0 0 Rkq 0 0

Yqo � Ldiqo �kMf iqo �kMkdiqo �Ydo þ Lqido kMkqido �D 0

0 0 0 0 0 �1 0

266666666666666664

377777777777777775

�

id

ifikd

iqikq

wd

2666666666664

3777777777775�

Ld kMf kMkd 0 0 0 0

kMf Lf Lfkd 0 0 0 0

kMkd Lfkd Lkd 0 0 0 0

0 0 0 Lq kMkq 0 0

0 0 0 kMkq Lkq 0 0

0 0 0 0 0 �H 0

0 0 0 0 0 0 1

2666666666664

3777777777775

pidpif

pikd

piq

pikq

pwpd

2666666666664

3777777777775

ð7:79Þ

where

¼ �ffiffiffi3

pV1

and in matrix form it can be written as

v ¼ �L3x � L4 _x

Therefore, the state equations can be expressed by the general linear form as

_x ¼ Ax þ Bu ð7:80Þwhere

A ¼ �L�14 L3 and B ¼ �L�1

4 v


Example 7.3 For machine data given below find:

● The matrices L1 and L2 of the linearised current model for unloaded machine.● The linearised current model for loaded machine that is connected to an infi-

nite bus through a transmission line with resistance, Re ¼ 0.05, and reactance,Xe ¼ 0.35. Determine the machine stability when the system is subjected to asmall disturbance (H ¼ 3.5 s and D ¼ 0). The power delivered at machineterminal is 0.8 pu at power factor ¼ 0.85 (lagging)

Ld ¼ 1.81, Lq ¼ 1.76, Lf ¼ 1.75, Lmd ¼ 1.66, Lmq ¼ 1.61, ld ¼ lq ¼ la ¼ 0.15,Lkd ¼ 1.72, Lkq ¼ 1.63, kMf ¼ kMkd ¼ Ld � ld ¼ 1.66, Lfkd ¼ kMf ¼ 1.66,kMkq ¼ 1.59, Ra ¼ 0.003, Rf ¼ 0.009, Rkd ¼ 0.0284, Rkq ¼ 0.006, lf ¼ Lf �kMf ¼ 0.09, lkd ¼ Lkd � kMkd ¼ 0.06, lkq ¼ Lkq � kMkq ¼ 0.03

Using (2.77) and (2.80), LMd and LMq can be obtained as LMd ¼ 0.05, LMq ¼ 0.042.

Solution:

(i) From ‘(7.70)’ and using the power per phase as a base quantity, it can be foundthat

It is to be noted that H ¼ 2Hwo ¼ 2637.6 as the form of (2.36) is used andwo ¼wB. The numeric quantities in matrix L1 are constants while the other non-numeric values depend on the load data as explained in case (ii).

(ii) As a first step the steady-state operating conditions are calculated. The phasordiagram of the machine is depicted in Figure 7.6: if V?¼ 1ff0� is taken as areference, then a¼ 0�. The current components in the direction of reference axis


‘in phase with V?’ and its perpendicular are Ir and Ix, respectively. As anapproximation, the power loss in the transmission line is estimated based onassumed current of 1 pu. Then the power loss equals (1.0)2Re ¼ 0.05 pu and P?¼0.75 pu. Therefore, Ir ¼ 0.75 pu.

F¼ cos�1 0.85 ¼ 31.788� ¼ bþ q

tan F ¼ tan bþ tan q1 � tan b tan q

; tan b ¼ XeIr þ ReIx

V1 � XeIx þ ReIrand tan q ¼ 1:333Ix

0:62 ¼ �1:333Ix 1:002 � 0:35Ixð Þ þ 0:263 þ 0:05Ixð Þ1:002 � 0:35Ixð Þ þ 1:333Ix 0:263 þ 0:05Ixð Þ

From which Ix ¼�0.256 pu

tan q¼ 1.333Ix and then q¼ 18.84� ‘lagging V?’

and tan b¼ (0.25)/(1.092) ¼ 0.2289, then, b¼ 12.89�

Vt ¼ V1 � XeIx þ ReIrð Þ þ j XeIr þ ReIxð Þ ¼ 1:127 þ j0:25 ¼ 1:154ff12:5�

d ¼ tan�1 Xq þ Xe

� �Ir þ Ra þ Reð ÞIx

V1 � Xq þ Xe

� �Ix þ Ra þ Reð ÞIr

¼ tan�10:81 ¼ 39�

Ia ¼ 0.75 � j0.256 ¼ 0.793 ff�18.84�

It can be seen that Ido ¼�Ia sin(do � bþF) ¼�0.793 sin 57.9 ¼�0.672 andIqo ¼ Ia sin(do � bþF) ¼ 0.421

Using (7.21) to get ido ¼ p3Ido ¼ �1:164 pu and iqo ¼ p

3Iqo ¼ 0:729 pu

Vd ¼ �Va sinðd� bÞ ¼ �1:154 sinð39 � 12:5Þ ¼ �0:515 pu

Vd ¼ Va cosðd� bÞ ¼ 1:154 cosð39 � 12:5Þ ¼ 1:033 pu

q-axis

d-axis

Reference axisV∞

Va = Vt

δ

θ φ β

jxeIa

ReIaIa

Id

Ix

Iq

Ir

Figure 7.6 Phasor diagram of a synchronous machine connected to an infinite bus


Applying (7.24) to obtain

vdo ¼ p3Vdo ¼ 0:892 and vqo ¼ p

3Vqo ¼ 1:789 pu

E ¼ Vq þ RaIq � XdId ¼ 1:789 þ 0:001 þ 1:21 ¼ 2:999ð¼EfdÞ

ifo ¼ ðp3EÞ=kMf ¼ p3 � 2:999=1:66 ¼ 3:129 pu

Ydo ¼ Ldido þ kMf ifo ¼ 3:087; Yqo ¼ Lqiqo ¼ 1:283

Yfo ¼ kMf ido þ Lf ifo ¼ 3:544; Ykdo ¼ kMkdido þ Lfkdifo ¼ 3:262

Ykqo ¼ kMkqiqo ¼ 1:159

Lq ¼ Lq þ Le ¼ 1:76 þ 0:35 ¼ 2:11; Ld ¼ Ld þ Le ¼ 1:81 þ 0:35 ¼ 2:16

R ¼ Ra þ Re ¼ 0:053

Ydo ¼ Ydo þ Leido ¼ 1:411 þ 0:35 ��0:22 ¼ 2:68

Yqo ¼ Yqo þ Leiqo ¼ 1:355 þ 0:35 � 0:77 ¼ 1:538

p3V1 cos do ¼ 1:346

p3V1 sin do ¼ 1:09

Thus,


A ¼ �L�14 L3

¼

�0:1005 0:0052 0:0362 �4:002 �3:0157 �2:9171 2:5529

0:0304 �0:0713 0:2009 1:2112 0:9127 0:8828 �0:7726

0:0685 0:0646 �0:248 2:7251 2:0535 1:9864 �1:7384

�1:5686 �1:2055 �1:2055 0:0385 �1:1547 �1:9462 �0:7916

3:4401 2:6438 2:6438 �0:0844 1:5323 4:2682 1:7360

�0:0000 �0:0002 �0:0002 �0:0006 �0:0002 0 0

0 0 0 0 0 1:0000 0

2666666666664

3777777777775

The computation of eigenvalues of matrix A using MATLAB� 2012a gives thevalues

�0:0535 j0:9841; �0:0039 j0:0265; �0:3043; �0:0084; 1:5785

It is to be noted that the system is unstable as one of the eigenvalues has a positivereal component.

7.4 Linearised flux linkage state space modelof a synchronous generator

The same procedure used to linearise the current model can be applied to obtain thelinear flux model of a synchronous machine. The linearised form of (2.89) and(2.90) can be written as (7.81) and (7.82), respectively.

D _Yd ¼ �Ra 1 � LMd

‘a

� �DYd

‘aþ Ra

LMd

‘a

DYf

‘fþ Ra

LMd

‘a

DYkd

‘kd

� woDYq �YqoDw� Dvd

D _Yf ¼ RfLMd

‘f

DYd

‘a� Rf 1 � LMd

‘f

� �DYf

‘fþ Rf

LMd

‘f

DYkd

‘kdþ Dvf

D _Ykd ¼ RkdLMd

‘kd

DYd

‘aþ Rkd

LMd

‘kd

DYf

‘f� Rkd 1 � LMd

‘kd

� �DYkd

‘kd

D _Yq ¼ �Ra 1 � LMq

‘a

� �DYq

‘aþ Ra

LMq

‘a

DYkq

‘kqþ woDYd þYdoDw� Dvq

D _Ykq ¼ RkqLMq

‘kq

DYq

‘a� Rkq 1 � LMq

‘q

� �DYkq

‘kq

9>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>;ð7:81Þ


DTe ¼ LMd � LMq

‘2a

Yqo � LMq

‘a‘kqYkqo

� �DYd þ LMd

‘a‘fYqoDYf þ LMd

‘a‘kdYqoDYkd

þ LMd � LMq

‘2a

Yqo þ LMd

‘a‘fYfo þ LMd

‘a‘kdYkdo

� �DYq � LMq

‘a‘kqYdoDYkq

ð7:82ÞThe angular speed equation becomes

D _w ¼ 1

ĤLMq

‘a‘kqYqo � LMd � LMq

‘2a

Yqo

� �DYd � LMd

‘a‘fYqo

� �DYf

�

� LMd

‘a‘kdYqo

� �DYkd � LMd � LMq

‘2a

Ydo þ LMd

‘a‘fYfo þ LMd

‘a‘kdYkdo

� �DYq

þ LMq

‘a‘kqYdo

� �DYkq � DDwþ DTm

�ð7:83Þ

and the torque angle equation is

D _d ¼ Dw ð7:84ÞIf the machine is connected to a simple power system, Chapter 3, Section 3.1, thevoltages vd and vq can be computed by (3.11) and (3.12), respectively. Then, sub-stituting these voltages into (2.89) gives the load equations as

1 þ Le

‘a1 � LMd

‘a

� �� _Yd � LeLMd

‘a‘f

_Yf � LeLMd

‘a‘kd

_Ykd

¼ � R

‘a1 � LMd

‘a

� �Yd þ RLMd

‘a‘fYf þ RLMd

‘a‘kdYkd

� w 1 þ Le

‘a1 � LMq

‘a

� �� Yq þ wLeLMq

‘a‘kqYkq þ

ffiffiffi3

pV1 sin d� að Þ ð7:85Þ

1 þ Le

‘a1 � LMq

‘a

� �� _Yq � LeLMq

‘a‘kq

_Ykq

¼ � R

‘a1 � LMq

‘a

� �Yq þ RLMq

‘a‘kqYkq þ w 1 þ Le

‘a1 � LMd

‘a

� �� Yd

� wLeLMd

‘a‘fYf � wLeLMd

‘a‘kdYkd �

ffiffiffi3

pV1 cos d� að Þ ð7:86Þ


Equations (7.85) and (7.86) are then linearised to obtain

1 þ Le

‘a1 � LMd

‘a

� �� D _Yd � LeLMd

‘a‘fD _Yf � LeLMd

‘a‘kdD _Ykd

¼� R

‘a1 � LMd

‘a

� �DYd þ RLMd

‘a‘fDYf þ RLMd

‘a‘kdDYkd

þ woLeLMq

‘a‘kqDYkq � wo 1 þ Le

‘a1 � LMq

‘a

� �� DYq

�YqoDwþ ffiffiffi3

pV1 cos d� að ÞDd ð7:87Þ

and

1 þ Le

‘a1 � LMq

‘a

� �� D _Yq � LeLMq

‘a‘kqD _Ykq

¼ wo 1 þ Le

‘a1 � LMd

‘a

� �DYd

� �� wo

LeLMd

‘a‘fDYf

� woLeLMd

‘a‘kdDYkd � R

‘a1 � LMq

‘a

� �DYq þ RLMq

‘a‘kqDYkq

þ YdoDwþ ffiffiffi3

pV1 sin d� að ÞDd ð7:88Þ

where

R ¼ Ra þ Re

Yqo ¼ 1 þ Le

‘a1 � LMq

‘a

� �� Yqo � LeLMq

‘a‘kqYkqo

Ydo ¼ 1 þ Le

‘a1 � LMd

‘a

� �� Ydo � LeLMd

‘a‘fYfo � LeLMd

‘a‘kdYkdo

Therefore, the linearised flux linkage model can be derived from (7.81) through(7.84) and (7.87) through (7.88) in the matrix form C1 _x ¼ C2x þ u as in (7.89).


(7.8

9)


where

To conform to the form _x ¼ Ax þ Bu, it can be found that

A ¼ C�11 C2 and B ¼ C�1

1

Example 7.4 Derive the matrices C1 and C2 of the linearised flux linkage statespace model for the system in Example 7.3 and determine its stability

Solution:

The computation of the elements of matrix C1 gives

From (2.78) and (2.79), YAdo and YAqo are

YAdo ¼ LMd

‘aYdo þ LMd

‘fYfo þ LMd

‘kdYkdo ¼ 5:716

YAqo ¼ LMq

‘aYqo þ LMq

‘kqYkqo ¼ 1:982


The matrix C2 can be derived and written as below:

From the relation A ¼ C�11 C2, matrix A can be computed as

A ¼

�0:0771 0:2747 0:0293 �1:0485 1:2782 �0:6017 0:5266

0:0060 �0:0040 0:0080 0 0 0 0

0:0160 0:2630 �0:1180 0 0 0 0

0:9537 �0:4836 �0:7254 �0:0265 0:1408 1:0000 0:4067

0 0 0 0:0560 �0:0360 0 0

0:0013 �0:0006 �0:0009 �0:0040 0:0036 0 0

0 0 0 0 0 1:0000 0

0BBBBBBBBBBB@

1CCCCCCCCCCCA

and the corresponding eigenvalues are

�0:0938 j0:9987; 0:0070 j0:0481; �0:1319; 0:0258; 0:0182

Therefore, the system is unstable since some of the eigenvalues have positive realcomponents.

7.5 Small signal stability of multi-machine systems

The multi-machine system can be viewed as a number of synchronous generators indifferent locations connected to a transmission network to feed various loads.Therefore, to study the small signal stability of such a system the desired mathe-matical relations describing the interconnected machines must be obtained. Also,the impact of other system components is involved. In this section, without loss ofgenerality, the loads are represented by constant impedances. The transmissionnetwork can be represented by its impedance or admittance matrix. Each machineis modelled by using detailed or classic model as a unit connected to the rest ofsystem components. Then, all generators in the system as interconnectedmachines can be completely described by mathematical relations in terms of sys-tem parameters. Finally, these relations are linearised to be in the form


_x ¼ Ax þ Bu, and consequently the system stability can be determined by exam-ining the eigenvalues of matrix A [10].

It is essential to note that, to obtain the relationships of system components,their various quantities to the same reference as a common frame of reference isdesirable.

Network and load representation

Figure 7.7(a) depicts a schematic diagram of a transmission network with n generatorsfeeding m loads. The loads are represented by constant impedances that can becalculated from pre-fault conditions in the system. Therefore, the network has onlyn active sources and can be reduced to n-node network as shown in Figure 7.7(b).The phasor currents and phasor terminal voltages are denoted by I1, I2, . . . , In andV1, V2, . . . , Vn, respectively. These phasors are expressed in terms of frames ofreference that are different for each generator node.

Thus, the currents Ii and voltages Vi, i ¼ 1, 2, . . . , n can be converted to pha-sors to a common frame of reference, I i and V i at steady state as below:

I ¼ YV ð7:90Þwhere

I ≜

I 1

I 2

..

.

I n

2666664

3777775; V ¼

V 1

V 2

..

.

V n

2666664

3777775

Transmissionnetwork

1

2

n

1

0

r

I1

Transmissionnetwork

1

n

0

I2V2

V1

Vn

In

0

IL1

ILr

V2

V1

2 I1

I2

In

Vn

(a) (b)

Figure 7.7 Representation of multi-machine system: (a) transmission system with ngenerators and m equivalent load impedances, (b) the reduced network


and Y is the short circuit admittance matrix of the reduced network that comprises anumber of branches ‘k ¼ 1, 2, . . . , b’ between any two nodes in the network. Moredetails of its calculation are found in References 11 and 12.

Both Vi and Ii can be converted to V i and I i, respectively, as below.Assume di–qi is the frame of reference of machine node i and D–Q is the com-

mon frame of reference rotating at synchronous speed. The phasor Vi ¼ Vqi þ jVdi

where the reference is q-axis of rotor i located at angle di (Figure 7.8). Thus, thisphasor to the common frame of reference can be expressed as V i ¼ VQi þ jVDi. Byinspection of Figure 7.8 it can be found that

V i ¼ VQi þ jVDi ¼ Vqi cos di � Vdi sin di

� �þ j Vqi sin di þ Vdi cos di

� �¼ V ie jdi ð7:91Þ

Similarly,

I i ¼ I iejdi ð7:92Þ

Machine representation

As explained in Chapter 2, the synchronous machine can be represented by eitherdetailed model or classical model. The detailed model is presented by current statespace model (2.104) or flux state space model (2.93). It is to be noted that thegeneral form of the models can be written as

_x ¼ f x; u;Tm; tð Þ ð7:93Þwhere x is a vector of state variables (currents or flux linkages), w and d; u is avector of voltages (vd, vq and vf); and Tm is the mechanical torque. The value of vf is

Q-axis

q-axis

D-axis

d-axis

Vi = Vi

VqiVdi

VQi

VDi

δi

Figure 7.8 Phasor quantities to two frames of reference (d–q and D–Q)


determined by presentation of excitation system mathematical model, i.e. addi-tional state variables are added to x [13]. In this analysis, this presentation is notincluded and vf is assumed to be known. Consequently, (7.93) can be expanded as aset of seven first-order differential equations for each machine in nine unknownvariable; five currents or flux linkages, w, and d in addition to two voltages vd andvq. If the system comprises n machines, then a set of 7n differential equations with9n unknowns is obtained. Thus, a set of 2n additional equations is required tocompletely describe the system. This additional set can be obtained by deriving thealgebraic relations between machine terminal voltages, currents and angles for nmachines interconnected to the network and loads.

Each machine in the reduced network (Figure 7.7(b)) is represented by aninternal node at a voltage V connected to the network through the machineequivalent impedance. Hence, the vector of terminal voltages of machines to d–qframe of reference of each machine is given by

V ¼

Vq1 þ jV d1

Vq2 þ jV d2

..

.

Vqn þ jV dn

26666664

37777775

ð7:94Þ

and can be transformed to a common frame of reference D–Q moving at synchro-nous speed as

V ¼

VQ1 þ jV D1

VQ2 þ jV D2

..

.

VQn þ jV Dn

26666664

37777775

ð7:95Þ

satisfying the relation

V ¼ TV ð7:96Þwhere

T ¼

e jd1 0 � � � 0

0 e jd2 � � � 0

..

. . .. ..

.

0 0 � � � e jdn

2666664

3777775 ð7:97Þ

Similarly for the node currents, the relation below is given as

I ¼ TI ð7:98Þ


Applying (7.90) by using (7.96) and (7.98) the relation between machine currents Iand voltages V can be found as below:

TI ¼ YTV ð7:99ÞPre-multiplying (7.99) by T�1 to obtain

I ¼ ðT�1YTÞV ¼ MV ð7:100Þwhere

M ¼ T�1YT ð7:101ÞHence,

V ¼ M�1I ðassuming M�1 existsÞ ð7:102ÞIt can be seen from (7.97) that

T�1 ¼

e�jd1 0 � � � 0

0 e�jd2 � � � 0

..

. ... . .

. ...

0 0 � � � e�jdn

2666664

3777775 ð7:103Þ

The form of network matrix Y can be written as

Y ¼

Y11e jq11 Y12e jq12 � � � Y1ne jq1n

Y21e jq21 Y22e jq22 � � � Y2ne jq2n

..

.

Yn1e jqn1

..

.

Yn2e jqn2

. ..

� � �...

Ynne jqnn

2666664

3777775 ð7:104Þ

Using (7.97), (7.101) and (7.103), the matrix M is given by

M ≜

Y11e jq11 Y12e j q12�d12ð Þ � � � Y1ne j q1n�d1nð Þ

Y21e j q21�d21ð Þ Y22e jq22 � � � Y2ne j q2n�d2nð Þ

..

. ... . .

. ...

Yn1e j qn1�dn1ð Þ Yn2e j qn2�dn2ð Þ � � � Ynne jqnn

2666664

3777775 ð7:105Þ

It is to be noted that the off-diagonal elements mij can be calculated by

mij ¼ Yijejðqij�dijÞ ¼ ðGij cos dij þ Bij sin dijÞ þ jðBij cos dij � Gij sin dijÞ ð7:106Þ

where

Gij ¼ Yij cos qij and Bij ¼ Yij sin qij


Thus, (7.100) can be rewritten in expanded form as

Iq1 þ jId1

Iq2 þ jId2

..

.

Iqn þ jIdn

2666664

3777775¼

Y11e jq11 Y12e j q12�d12ð Þ � � � Y1ne j q1n�d1nð Þ

Y21e j q21�d21ð Þ Y22e jq22 � � � Y2ne j q2n�d2nð Þ

..

.

Yn1e j qn1�dn1ð Þ

..

.

Yn2e j qn2�dn2ð Þ

. ..

� � �...

Ynne jqnn

26666664

37777775

Vq1 þ jV d1

Vq2 þ jV d2

..

.

Vqn þ jV dn

2666664

3777775

ð7:107Þ

Equation (7.107) gives a set of 2n real algebraic relations that are needed to beincorporated with (2.104) to complete the description of the system of n inter-connected machines by 9n relations with 9n unknowns.

Linearisation of these 9n relations is required to study the small signal stability.The set of 7n differential equations given by (2.104) have been linearised asexplained in Section 7.3 and the rest of 2n algebraic relations given by (7.100) or(7.107) can be linearised as below.

Linearisation of (7.100) gives

DI ¼ MoDV þ DMVo ð7:108Þ

where Mo is calculated at the initial angles dio, i ¼ 1, 2, . . . , n, and Vo is the initialvalue of the vector V. Assuming di ¼ dio þ Ddi. The matrix M becomes

M ¼

Y11e jq11

Y21e j q21�d21o�Dd21ð Þ

..

.

Yn1e j qn1�dn1o�Ddn1ð Þ

Y12e j q12�d12o�Dd12ð Þ

Y22ejq22

..

.

Yn2e j qn2�dn2o�Ddnoð Þ

� � �� . ..

� � �

Y1ne j q1n�d1no�Dd1nð Þ

Y2ne j q2n�d2no�Dd2nð Þ

..

.

Ynne jqnn

2666664

3777775

ð7:109Þ

Thus, Yijej qij�dijo�Ddijð Þ ¼ mij ≜ the general term of matrix M can be written as

mij ¼ Yijej qij�dijoð Þe�jDdij and considering cos Ddij ffi 1, sin Ddij ffi Ddij it becomes

mij ffi Yijej qij�dijoð Þ 1 � jDdij

� � ð7:110Þ

Consequently, the general term in the matrix DM is given by

Dmij ffi �jYijej qij�dijoð ÞDdij for i 6¼ j

ffi 0 for i ¼ jð7:111Þ

i.e. the matrix DM has off-diagonal elements only, with all diagonal elements equalto zero.


The second term of RHS in (7.108) is

DMVo ¼ �j

0 � � � Yijej qij�dijoð ÞDdij

Yijej qij�dijoð ÞDdij � � � Yije

j qij�dijoð ÞDdij

..

. . .. ..

.

Yijej qij�dijoð ÞDdij � � � 0

266666664

377777775

V1o

V2o

� � �Vno

266664

377775

¼ �j

Xn

k¼1

Y1ke j q1k�d1koð ÞVkoDd1k

Xn

k¼1

Y2ke j q2k�d2koð ÞVkoDd2k

..

.

Xn

k¼1

Ynke j qnk�dnkoð ÞVkoDDnk

266666666666664

377777777777775

ð7:112Þ

Then substituting into (7.108) gives the linearised equation as

DI1

DI2

..

.

DIn

2666664

3777775 ¼

Y11e jq11 � � � Y1ne j q1n�d1noð Þ

Y21e j q21�d21oð Þ � � � Y2ne j q2n�d2noð Þ

..

. . .. ..

.

Yn1e j qn1�dn1oð Þ � � � Ynnejqnn

2666664

3777775

DV1

DV2

..

.

DVn

2666664

3777775

� j

Pnk¼1 Y1ke j q1k�d1koð ÞVkoDd1kPnk¼1 Y2ke j q2k�d2koð ÞVkoDd2k

..

.

Pnk¼1 Ynke j qnk�dnkoð ÞVkoDDnk

26666664

37777775

ð7:113Þ

In (7.113) D can be dropped for convenience. Then substituting into (7.70)obtains the linearised set of equations in the form _x ¼ Ax þ Bu. By examining theeigenvalues of A, the system stability can be determined.

References

1. Condren J., Gedra T.W. ‘Expected-security-cost optimal power flow withsmall-signal stability constraints’. IEEE Transactions on Power Systems.2006;21(4):1736–43

2. Tayora C.J., Smith O.J.M. ‘Equilibrium analysis of power systems’. IEEETransactions on Power Apparatus and Systems. 1972;PAS-91(3):1131–7


3. Kundur P., Paserba J., Ajjarapu V., Anderson G. ‘Definition and classificationof power system stability IEEE/CIGRE joint task force on stability terms anddefinitions’. IEEE Transactions on Power Systems. 2004;19(3):1387–401

4. Chen L., Min Y., Xu F., Wang K.P. ‘A continuation-based method to com-pute the relevant unstable equilibrium points for power system transientstability analysis’. IEEE Transactions on Power Systems. 2009;24(1):165–72

5. Rueda J.L., Colome D.G., Erlich I. ‘Assessment and enhancement of smallsignal stability considering uncertainties’. IEEE Transactions on Power Sys-tems. 2009;24(1):198–207

6. Byerly R.T., Sheman D.E., McLain D.K. ‘Normal modes and mode shapesapplied to dynamic stability analysis’. Transactions on Power Apparatus andSystems. 1975;94(2):224–9

7. Gross G., Imparato C.F., Look P.M. ‘A tool for the comprehensive analysis ofpower system dynamic stability’. IEEE Transactions on Power Apparatusand Systems. 1982;101(1):226–34

8. Ewart D.N., Demello F.P. ‘A digital computer program for the automaticdetermination of dynamic stability limits’. IEEE Transactions on PowerApparatus and Systems. 1967;PAS-86(7):867–75

9. Anderson P.M., Fouad A.A. Power System Control and Stability. 2nd edn.Piscataway, NJ, US: IEEE Press; 2003

10. Ma J., Dong Z.Y., Zhang P. ‘Comparison of BR and QR eigenvalue algo-rithms for power system small signal stability analysis’. IEEE Transactionson Power Systems. 2006;21(4):1848–55

11. Stagg and El-Abiad A. Computer Methods in Power System Analysis.New York, NY, US: McGraw-Hill; 1968

12. Anderson P.M. Analysis of Faulted Power Systems. Ames, IA, US: Iowa StateUniversity Press; 1973

13. Arcidiacono V., Ferrari E., Saccomanno F. ‘Studies on damping ofelectromechanical oscillations in multimachine systems with longitudinalstructure’. IEEE Transactions on Power Apparatus and Systems. 1976;95(2):450–60


Chapter 8

Transient stability

The objective of transient stability study is to determine whether the system gen-erators remain in synchronism when subjected to large disturbances. The transientstability is evaluated by studying the system dynamic response during the transientperiod that usually lasts up to a few seconds taking into account the rapid changeof electrical variables, including relative swinging between generators. A longertransient period may be covered in the study when the behaviour of some controlsis of interest.

Because of the nature of transient disturbances, the non-linear system equa-tions cannot be linearised and must be solved in stability evaluation. Significantsimplifications are required to obtain analytical solutions. Therefore, numericalintegration techniques are applied.

To form the system equations, adequate models of system components areneeded to implement stability study with the desired accuracy. Models of systemcomponents, such as synchronous generators with associated controls, excitationsystem and prime mover, transformers, transmission lines and loads [1, 2], havebeen discussed in Part I. It is to be noted that the most important component is thesynchronous generator with its associated controls. On the other hand, in stabilityanalysis, load frequency controllers and prime mover models are often neglectedwithout loss of accuracy. Per unit equations of current or flux linkage models givenin Chapter 3 completely describe the dynamic performance of a synchronousmachine. However, these equations cannot be used directly for system transientstability studies. Some simplifications and approximations are required to representthe synchronous machine in stability studies. Therefore, the performance equationsof a synchronous machine are developed based on the following assumptions:

● Effect of mutual inductance between stator and rotor is considered assumingsinusoidal distribution of the stator windings around the air gap.

● Effect of stator slots on rotor inductances with rotor position is neglected.● Magnetic hysteresis is negligible.● Magnetic saturation effects are negligible.● Stator transients are ignored.

As reported in [3], the number of rotor windings and corresponding statevariables can vary from one to six depending on the degree of detail used. Thesuggested models are defined based on the degree of complexity and are denoted by

Model x.y, where x and y are the number of rotor windings on d and q axes,respectively. Thus,

● Model 0.0: damper circuits and field flux decay are neglected, i.e. all statevariables for rotor coils are ignored.

● Model 1.0: a field circuit only is considered on the d-axis.● Model 1.1: a field circuit and one damper on q-axis● Model 2.1: a field circuit and one damper on the d-axis plus one damper on the

q-axis are considered.● Model 2.2: a field circuit and one damper on the d-axis plus two dampers on

the q-axis are considered.● Model 3.2: a field circuit and two dampers on the d-axis plus two dampers on

the q-axis are considered.● Model 3.3: a field circuit and two dampers on the d-axis plus three dampers on

the q-axis are considered.

As an application, Model 2.1 (Figure 8.1) is used in the following to represent thesynchronous machine.

8.1 Synchronous machine model

The stator equations expressed in per unit are given by (I.4). As all quantities are inper unit, the subscript u can be dropped and the equation is rewritten as

vd ¼ � 1wB

dYd

dt� wwB

Yq � Raid

vq ¼ � 1wB

dYq

dtþ wwB

Yd � Raiq

9>>>=>>>;

ð8:1Þ

assuming that the zero sequence current in the stator does not exist, i.e. vo ¼ 0.In transient stability studies, it is common to neglect the transformer voltage

q-axis

d-axis

vf

KQ

KD

θAxis of phase ‘a’ va

vb

vc

Ψa

Ψb

Ψc

a

b

c

F

Figure 8.1 Synchronous machine Model 2.1. Stator phase windings: a, b, c: androtor windings F, KD on d-axis, KQ on q-axis


terms dYddt and dYq

dt ; and the effect of speed variations as well. Accordingly (8.1)becomes

vd ¼ �Yq � Raid

vq ¼ Yd � Raiq

)ð8:2Þ

Similarly, the voltage equations for F, KD and KQ rotor windings, (I.21)–(I.23),become:

vf � Rf if ¼ 1wB

dYf

dtð8:3Þ

�Rkdikd ¼ 1wB

dYkd

dtð8:4Þ

�Rkqikq ¼ 1wB

dYkq

dtð8:5Þ

It is noted that neglecting stator transients, the stator equations become algebraic.Consequently, it is not possible to choose stator currents id and iq as state variables,as these currents can be discontinuous functions due to any sudden changes in thenetwork. On the other hand, the flux linkages of rotor windings, field and damperscannot change suddenly. This implies that with sudden changes of id, the field anddamper currents also change suddenly in order to maintain the field and damperflux linkages continuous, i.e. immediately after a disturbance flux linkages remainconstant at the value just prior to the disturbance. Accordingly, rotor windingcurrents cannot be treated as state variables. Hence, rotor flux linkages can bechosen as state variables.

From (I.5) the stator flux linkages on d–q axes are

Yd ¼ Ldid þ kMf if þ kMkdikd ¼ Xdid þ Xadðif þ ikdÞ ð8:6Þ

Yq ¼ Lqiq þ kMkqikq ¼ Xqiq þ Xaqikq ð8:7Þ

and the rotor circuits flux linkages are

Yf ¼ Lf if þ kMf id þ Lfkdikd ¼ Xf if þ Xadid þ Xfkdikd ð8:8ÞYkd ¼ Lkdikd þ kMkdid þ Lfkdif ¼ Xkdikd þ Xadid þ Xfkdif ð8:9Þ

Ykq ¼ Lkqikq þ kMkqikq ¼ Xkqikq þ Xaqiq ð8:10Þ

where kMf ¼ kMkd ¼ Xad and kMkq ¼ Xaq

It is to be noted that by choosing angular frequency, wB; as the base, the perunit values Ld, Lq, Lf, Lkd, Lkq equal Xd, Xq, Xf, Xkd, Xkq, respectively.

Transient stability 187

Solving (8.8)–(8.10) gives

if ¼ Yf

Xf� Xad

Xfid � Xfkd

Xfikd ð8:11Þ

ikd ¼ Ykd

Xkd� Xad

Xkdid � Xfkd

Xkdif ð8:12Þ

ikq ¼ Ykq

Xkq� Xaq

Xkqiq ð8:13Þ

Then by substitution in (8.6) and (8.7) obtains

Yd ¼ X 0did þ E0

q ð8:14ÞYq ¼ X 0

qiq � E0d ð8:15Þ

where

X 0d ¼ Xd � X 2

ad

xfð8:16Þ

X 0q ¼ Xq �

X 2aq

Xkqð8:17Þ

E0q ¼ Xad

XfdYf þ ðLf � LfkdÞikde ¼ xad

XfYf

¼ Xad

XfYf when the second term is neglected ð8:18Þ

E0d ¼ �XaqYkq

Xkqð8:19Þ

Substituting (8.11) and (8.18) into (8.3) gives

1wB

Xf

Xad

dE0q

dt¼ �Rf E0

q

Xadþ Rf Xad

Xfid þ vf ð8:20Þ

Hence,

dE0q

dt¼ wBRf

Xf�E0

q þX 2

ad

Xfid þ Xad

Rfvf

� �ð8:21Þ

or

dE0q

dt¼ 1

T0do

�E0q þ Xd � X 0

d

� �id þ Efd

h ið8:22Þ


where

Efd ¼ Xad

Rfvf ð8:23Þ

T0do ¼ X

wBRfð8:24Þ

Substituting (8.13) and (8.19) into (8.5) the relation below can be found:

dE0d

dt¼ 1

T 0qo

�E0d � Xq � X 0

q

� �iq

h ið8:25Þ

where

T0qo ¼ Xkq

wBRkqð8:26Þ

When considering the stator voltage and torque equations, it is convenient to definethe equivalent voltage sources E

0d and E

0q as state variables rather than rotor flux

linkages.Substituting (8.14) and (8.15) into (8.2) gives

vq ¼ E0q þ X

0did � Raiq

vd ¼ E0d � X 0

qiq � Raid

)ð8:27Þ

Assuming X 0d ¼ X 0

q ¼ X0

in case of neglecting transient saliency, (8.27) can bewritten as

vq þ jvd ¼ E0q þ jE0

d

� �� Ra þ jX

0� �

iq þ jid

� � ð8:28Þ

In vector notation, (8.28) can be expressed as

V t ¼ E0 � Ra þ jX

0� �

I t ð8:29Þ

where

V t ≜ machine terminal voltage ¼ vq þ jvd

E0 ≜ voltage behind transient reactance ¼ E0q þ jE0

dI t ≜ machine terminal current ¼ iq þ jid

Ra ≜ armature resistanceX 0 ≜ transient reactance

The equivalent circuit of the stator corresponding to (8.29) is shown in Figure 8.2.It shows a voltage source E0 behind equivalent impedance Ra þ jx0ð Þ.


The rotor mechanical (2.59), i.e. swing equation, can be expressed as two first-order differential equations as

_d ¼ w� wo

_w ¼ wB

2HTm � Te � Dw½ �

9=; ð8:30Þ

The electrical torque Te is given by Te ¼Ydiq �Yqid. Substituting from (8.14) and(8.15) gives

Te ¼ E0did þ E0

qiq þ X 0d � X 0

q

� �idiq ð8:31Þ

The third term in (8.31) is zero if the transient saliency is ignored as X 0d ¼ X 0

q.The variables id and iq can be obtained from the stator algebraic (8.27) and thenetwork equations or power flow solution.

Therefore, for synchronous machine Model 2.1 in addition to rotor (8.30) and(8.31) the following stator equations are used to represent the machine.

vq ¼ E0q þ X

0did � Raiq

vd ¼ E0d � X

0qiq � Raid

dE0q

dt¼ 1

T0do

½�E0q þ ðXd � X

0dÞid þ Efd �

dE0d

dt¼ 1

T 0qo

½�E0d � ðXq � X

0qÞiq�

9>>>>>>>>>>=>>>>>>>>>>;

ð8:32Þ

It is noted that the d–q axis transient effects require differential equations ‘sE 0q

and sE 0d’. A block diagram representation is shown in Figure 8.3 [4].

The field voltage Efd and the mechanical power Pm (�Tm in pu) can be heldconstant in the transient calculations for a period of analysis less than one second asthe effects of the exciter and governor control systems on power system responseare neglected. When a more detailed evaluation of system response is required or

E ′

+

Vt

Ra X ′

It

Figure 8.2 Stator equivalent circuit


the period of analysis extends beyond 1 s, it is important to consider the effects ofthe exciter and governor systems.

The exciter control system provides the proper field voltage to maintain adesired system voltage. An important characteristic of an exciter control system isits ability to respond rapidly to voltage deviations during both normal and emer-gency system operation. Different types of excitation systems and their block dia-grams that relate the input and output variables through transfer functions havebeen explained in Chapter 3. Thus, the differential equations relating the input andoutput variables of excitation system components must be solved simultaneouslywith the stator and rotor equations.

Similarly, the effects of the speed governor control that provide the mechanicalpower Pm during transient periods can be taken into account by using the repre-sentation of the selected governor control system as described in Chapter 3. Thisrepresentation includes a transfer function describing the system components.The differential equations relating the input and output variables of these transferfunctions are solved simultaneously with the stator and rotor equations.

As explained above, in transient stability analysis, the synchronous machinewith its associated controllers can be represented by a set of equations that com-prises a combination of algebraic and non-linear differential equations. To solvethese equations simultaneously the differential equations are converted to algebraicequations and one of the numerical integration methods is applied to obtain thesolution step by step.

∑

∑

x ′d

∑

∑

Xd – x ′d

id

iq

Efd

Vq

1sT ′qo

E ′q

+

+

+

++

+

–

1∑Xq – x ′q

x ′q

Ed+

+–sT ′do

Vd

Figure 8.3 Block diagram representation for Model 2.1


8.2 Numerical integration techniques

Many integration techniques have been applied to the power system transient sta-bility analysis such as trapezoidal method, Euler’s method, modified Euler–Cauchymethod and Runge–Kutta methods [5–8]. More details about these methods aregiven in Appendix III and a summary is given below.

Assuming, for instance, two simultaneous ordinary differential equations (ODEs)of the form

_x ¼ f1 t; x; y; hð Þ_y ¼ f2 t; x; y; hð Þ

where x and y are the state variables, e.g. d and w, that are calculated numericallyversus time t. Considering the step size h equals the time increment Dt, the rules ofsome numerical solution methods are summarised in Table 8.1.

Table 8.1 Summary of some numerical integration methods formulae

Method Rules

Euler’s method x;iþ1 ¼ xi þ hf1ðxi; yiÞy;iþ1 ¼ yi þ hf2ðxi; yiÞ

Modified Euler–Cauchy method xiþ1 ¼ xi þ hf1 ti þ h

2; xi þ h

2f1ðxi; yiÞ; yi þ h

2f2ðxi; yiÞ

� �

yiþ1 ¼ yi þ hf2 ti þ h

2; xi þ h

2f1ðxi; yiÞ; yi þ h

2f2ðxi; yiÞ

� �

Trapezoidal method xiþ1 ¼ xi þ h

2½ f1ðti; yiÞ

þ f1ðtiþ1; xi þ hf1ðxi; yiÞ; yi þ hf2ðxi; yiÞÞ�yiþ1 ¼ yi þ h

2½ f ðti; yiÞ

þ f ðtiþ1; xi þ hf1ðxi; yiÞ; yi þ hf2ðxi; yiÞÞ�

Second-order Runge–Kutta method xiþ1 ¼ xi þ h

2½K11 þ K21�

yiþ1 ¼ yi þ h

2½K12 þ K22�

Third-order Runge–Kutta method xiþ1 ¼ xi þ h

6½K11 þ 4K21 þ K31�

yiþ1 ¼ yi þ h

6½K12 þ 4K22 þ K32�

Fourth-order Runge–Kutta method xiþ1 ¼ x þ h

6½K11 þ 2K21 þ 2K31 þ K41�

yiþ1 ¼ yi þ h

6½K12 þ 2K22 þ 2K32 þ K42�

The coefficients Kij of Runge–Kutta methods are given in Appendix III.


8.3 Transient stability assessment of a simple power system

A single machine connected to an infinite bus through a transmission line is definedas a simple power system. For example, a remote power station connected to a loadthrough a long transmission line can be represented by a simple system comprisingone machine that is equivalent to all generators in the power station. This isacceptable for disturbances external to the power station and it needs to model thesystem elements – generator, transmission line and infinite bus – for studying thetransient stability.

Two main points should be taken into account when writing the systemequations. First, the system equations must be referred to a common frame ofreference. Second, the non-state variables must be eliminated from the systemequations and expressed as parameters and/or in terms of state variables.

The machine can be represented by (8.29)–(8.31). The transmission line isrepresented by its p-equivalent circuit as explained in Chapter 4. The transmissionline is considered as a two-port external network: one port is connected to thegenerator terminals and the second port is connected to the infinite bus. The infinitebus, representing a large stiff system, may be modelled by a voltage source ofconstant magnitude and phase angle Eb ff q. The angle q is usually assumed as zerowhere the bus is taken as a common reference (Figure 8.4).

Assuming that the simple system consists of only series impedance Ze ¼ Re þjXe and taking the system axis shown in Figure 8.5 as a common frame of reference,it can be seen that

vq þ jvd

� �e jd ¼ Re þ jXeð Þ iq þ jid

� �ejd þ Eb ð8:33Þ

Multiplying both sides by e�jd and equating real and imaginary parts gives

vq ¼ Reiq � Xeid þ Eb cos dvd ¼ Xeiq þ Reid � Eb sin d

)ð8:34Þ

As a further simplification, Re is assumed to be zero, thus,

vq ¼ �Xeid þ Eb cos dvd ¼ Xeiq � Eb sin d

)ð8:35Þ

Generator model Two-port external network Infinite bus

E′ Eb θ

Ze

y1 y2

Ra x′

Figure 8.4 Equivalent circuit of a simple power system


where id and iq are non-state variables and must be eliminated. This can be obtainedby expressing vd and vq ‘machine terminal voltages’ as written in (8.32) andassuming Ra ¼ 0 as

vq ¼ E0q þ X 0

did

vd ¼ E0d � X 0

qiq

)ð8:36Þ

From (8.35) and (8.36), id and iq can be obtained by

id ¼ Eb cos d� E0q

Xe þ X 0d

� �iq ¼ Eb sin dþ E0

d

Xe þ X 0q

� �

9>>>>=>>>>;

ð8:37Þ

Then, substitute these values of id and iq, in (8.31) and (8.32) fordE

0q

dt anddE0

ddt to get

the system equations in the form _X ¼ f x; uð Þ as

dE0q

dt¼ 1

T0do

�E0q þ Xr1E1 þ Efd

h idE0

d

dt¼ 1

T 0qo

�E0d � Xr2Er2

� _d ¼ w� wo

_w ¼ wB

2HTm � Te � Dw½ �

9>>>>>>>>>>>>=>>>>>>>>>>>>;

ð8:38Þ

δSystem real axis

System imaginaryaxis

Machine q-axisMachine d-axis

Vt

Vq

Vd

Figure 8.5 Synchronous machine and power system frames of reference


in addition to the algebraic equation

Te ¼ 1X1X2

E0dE1X2 þ E0

qE2X1 þ E1E2 X 0d � X 0

q

� �h ið8:39Þ

where

Xr1 ¼ Xd � X 0d

� �Xe þ X 0

d

� � ; Xr2 ¼Xq � X 0

q

� �Xe þ X 0

q

� �

X1 ¼ Xe þ X 0d

� �; X2 ¼ Xe þ X 0

q

� �; E1 ¼ Eb cos d� E0

q; E2 ¼ Eb sin dþ E0d

It is concluded that the simple power system can be represented by a setof differential-algebraic (8.38) and (8.39). It is noted that Eb is treated as aparameter. Efd and Tm are inputs from the excitation and governor control systems,respectively. They are treated as parameters if the dynamics of the controllers areignored. Otherwise, the dynamics of the controllers represented by differentialequations are to be appended to (8.38) to determine their outputs Efd and Tm.

The set of equations representing the power system – machine stator equations,rotor mechanical equations and network equations – can be solved simultaneouslyby applying one of the numerical integration methods to obtain the change of statevariables versus time, and then the stability can be determined. The initial condi-tions required to solve the ODEs are calculated from system behaviour at steadystate prior to the disturbance as explained in Section 7.1.3.

Derivation of stator equations depends on the degree of complexity required tomodel the machine. For instance,

● Representing the machine as a constant voltage magnitude behind d-axistransient reactance X 0

d requires no differential equations. Only the followingalgebraic equation is used:

E0 ¼ Vt þ RaIt þ jX 0

dIt ð8:40Þ

● If d-axis transient effects are considered one differential equation is requiredand the set of stator equations is

E0q ¼ vq � X 0

did þ Raiq

E0d ¼ vd þ X 0

qiq þ Raid

dE0q

dt¼ 1

T0do


d

� �id þ Efd

h i

9>>>>>>=>>>>>>;

ð8:41Þ


● Representation of d- and q-axis sub-transient effects requires three differentialequations as below:

E00q ¼ vq � X

00d id þ Raiq

E00d ¼ vd þ X

00q iq þ Raid

dE0q

dt¼ 1

T0do

�E0q þ Xd � X

0d

� �id þ Efd

h i

dE00q

dt¼ 1

T00do

E0q � E

00q þ X

0d � X

00d

� �id

h idE

00q

dt¼ 1

T 00qo

E0d � E

00q � X

0q � X

00d

� �iq

h i

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

ð8:42Þ

● Representation of d- and q-axis sub-transient effects by four differentialequations is as below:

E00q ¼ vq � X

00d id þ Raiq

E00d ¼ vd þ X

00q iq þ Raid

dE0q

dt¼ 1

T0do


d

� �id þ Efd

h idE0

d

dt¼ 1

T 0qo

�E0d � Xq � X 0

q

� �iq

h i

dE00q

dt¼ 1

T00do

E0q � E

00q þ X 0

d � X00d

� �id

h idE

00d

dt¼ 1

T 00qo

E0d � E

00d � X 0

q � X00q

� �iq

h i

9>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>;

ð8:43Þ

The steps to assess the transient stability of a simple power system are sum-marised below:

● Derive machine equations as a model adequate to the degree of complexityrequired for system study.

● Derive network equations.● Form system differential-algebraic equations: stator equations, rotor swing

equation and network equations.● Calculate initial conditions for the system at steady state prior to the disturbance.● Calculate power delivered from the machine to the infinite bus during and after

the fault period. This entails determination of an equivalent circuit between thesending and receiving ends during each period or using power flow analysis.

● Select a numerical integration method to solve the system algebraic-differentialequations and plot the torque angle and machine speed versus time.


Example 8.1 A synchronous generator is connected to an infinite bus through atransformer and two parallel identical transmission lines with data in pu as shownin Figure 8.6 and delivers a power of 0.8 pu. A three-phase to earth fault occurs at apoint, F, near the beginning of a TL and is cleared at 0.08 s by isolating the faultyTL. Neglecting all resistances and velocity damping coefficient, compute the var-iation of d and w of the machine versus time by solving the problem using anumerical integration method with a step size of 0.02 s when:

(i) The machine is represented by a voltage source E0 ‘constant magnitude andvarying d’ behind the transient reactance X 0

d .


dIt

(ii) The machine is represented by a voltage source Eq behind Xq with systemequations as

E0q ¼ vq � X 0

did þ Raiq

E0d ¼ vd þ X 0

qiq þ Raid

dE0q

dt¼ 1

T0do

Efd � EI

� �

¼ 1

T0do


d

� �id þ Efd

h i

Eq ¼ Vt þ RaIt þ jXqIt

EI ¼ Vt þ RaIt þ jXdId þ jxqIq

E0q ¼ Eq � j Xq � X 0

d

� �Id

Other machine data are given below:

T0do ¼ 0:4; Xd ¼ 1:9; Xq ¼ 1:75; X

0q ¼ 0:24; H ¼ 3:5

V∞ = 1.0 0Vs = 1.01 θsE δ

R = 0.003xd’ = 0.2

xtr = 0.1

xTL = 0.35RTL = 0.05Vt θt

F

Figure 8.6 Simple system for Example 8.1


Solution:

(i) The machine is represented by a voltage source E0 ‘constant magnitude andvarying d’ behind the transient reactance X 0

d .

The initial values of system parameters are

ZL pre-fault ¼ (RL þ jXL)/2 ¼ 0.0250 þ j0.1750ZL during fault ¼?

ZL post-fault ¼ (RL þ jXL) ¼ 0.0500 þ j0.35qo ¼ 18�

wo ¼ 314.1593 elec. rad/sqs ¼ sin�1[(Pm abs(ZL pre))/(VsV?)] ¼ 8.05�

Vs ¼ Vs(cos(qs) þ j sin(qs)) ¼ 1.0000 þ j0.1414Igen ¼ (Vs � V?)/ZL pre-fault ¼ 0.7920 þ j0.1129

S ¼ (Pm/pf )(exp(�i cos�1( pf ))) ¼ 0.8000 � j0.4958

jVtj ¼ jSj=jIgenj ¼ 1:1765

qt ¼ sin�1[(Pm(abs(ZL pre) þ 0.1)/(VtV?))] ¼ 10.85�

Vt ¼ Vteqtj ¼ 1:1554 þ j0:2214

E0 ¼ Vt þ RaIgen þ jX0dIgen ¼ 1:1329 þ j0:3798

Clearing time ¼ 0.08: Using the initial values of system parameters to applyPSAT/MATLAB� toolbox to solve the swing equation, the variations ofpower angle and machine speed versus time are shown in Figure 8.7(a and b),respectively.

If the clearing time is decreased to be 0.02 s or increased to be 0.3 s, thevariations of d and w versus time are depicted in Figures 8.8 and 8.9,respectively.

Time (s)Time (s)(a)0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.50

–4

–2

0

2

4

5

15

25

35

45

(b)

Om

ega

(ele

c. ra

d/s)

Del

ta (d

egre

e)

Figure 8.7 (a) Variation of d versus time and (b) variation of w versus time withfault clearing time 0.08 s


(ii) The machine is represented by a voltage source Eq behind Xq with systemparameters as

T0do ¼ 0:4; Xd ¼ 1:9; Xq ¼ 1:75; X

0q ¼ 0:24; H ¼ 3:5

ZL pre-fault ¼ 0.0250 þ j0.1750ZL during fault ¼? ZL post-fault ¼ 0.0500 þ j0.3500

qo ¼ 73.1134�, wo ¼ 314.1593 elec. rad/sEfdo ¼ 0.4258 þ j2.4262Edo ¼ 0.8963 þ j1.0621Eqo ¼ 0.6891 þ j2.2699

By solving the swing equation the variations of d and w versus time atdifferent values of clearing time are shown in Figures 8.10–8.12.

0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.4 0.5

0

1

2

–1

–218

22

26

30

34

Time (s)Time (s)(a) (b)

Om

ega

(ele

c. ra

d/s)

Del

ta (d

egre

e)


0

4

8

12

–8

–4

0

100

200

400

300

0 0.1 0.2 0.3 0.4 0.50 0.1 0.2 0.3 0.50.4Time (s)Time (s)(a) (b)

Om

ega

(ele

c. ra

d/s)

Del

ta (d

egre

e)



It is seen that the system is stable when the fault clearing time is 0.02 s or0.08 s while it is unstable for fault clearing time of 0.3 s. Thus, as the faultclearing time decreases the stability is better and the system is more secure.

65

75

85

95

0 0.1 0.2 0.3 0.50.4Time (s)(a)

0 0.1 0.2 0.3 0.4 0.5Time (s)(b)

Om

ega

(ele

c. ra

d/s)

0

2

–2

–4

Del

ta (d

egre

e)


0

1

–1

–2

–320

40

60

80

0.50.40.30.20.10

Time (s)(b)Time (s)(a)0.50.40.30.20.10

Om

ega

(ele

c. ra

d/s)

Del

ta (d

egre

e)


10

0

20

50

150

250

350

0.50.40.30.20.10Time (s)(b)Time (s)(a)

0.50.40.30.20.10

Om

ega

(ele

c. ra

d/s)

Del

ta (d

egre

e)



8.4 Transient stability analysis of a multi-machinepower system

A multi-machine power system encompasses interconnected generators to feedloads with electrical power through a network (Figure 8.13). The first step tostudy transient stability is to model the system by incorporating the equationsrepresenting each component in the system. Stability studies involve short periodsof analysis of the order of a second or less. Thus, the synchronous machine can berepresented by a voltage source behind transient reactance. As a simplifiedrepresentation, the voltage source is assumed to be constant in magnitude andvaries in angular position as the saturation and saliency effects are neglected aswell as constant flux linkages and small speed change are assumed. There is noneed to involve stator differential equations and the voltage source is denoted byE0 that is given by (8.40), repeated below:


dIt ð8:44Þ

where

E0 ¼ voltage behind transient reactance

Vt ¼ machine terminal voltageIt ¼ machine terminal current

Ra ¼ armature resistanceX 0

d ¼ transient reactance

Accordingly, the synchronous machine representation used for networksolution is shown in Figure 8.14(a) and its phasor diagram is depicted inFigure 8.14(b).

When the effects of saliency and changes in field flux linkages need to betaken into account in a study, the synchronous machine can be represented by avoltage Eq behind quadrature-axis synchronous reactance Xq and is determinedfrom

Eq ¼ Vt þ RaIt þ jXqIt ð8:45Þ

The machine representation used for network solution and its phasor dia-gram are shown in Figure 8.15(a and b), respectively. The field current actingalong d-axis produces a sinusoidal flux. The induced voltage EI lags this flux by90� and acts along the q-axis. The voltage EI is determined by the summation ofterminal voltage Vt, the voltage drop across the armature resistance and the voltagedrops across Xd and Xq, representing the demagnetising effects Figure 8.15(b).

Thus,

EI ¼ Vt þ RaIt þ jXdId þ jXqIq ð8:46Þ


As explained in Section 7.1.3 and illustrated in Figure 7.2, the relation betweenEq and E0

q can be written as

E0q ¼ Eq � j Xq � X 0

d

� �Id ð8:47Þ

And, as given by (8.21) the rate of change of E0q is

dE0q

dt¼ 1

T0do

Efd � EI

� � ¼ 1

T0do


d

� �id þ Efd

h ið8:48Þ

Generators Network

Loads

Figure 8.13 Schematic diagram of a multi-machine power system

E ′

Ra X ′d

It

E ′

RaIt

X ′dIt

Vt

It

(a) (b)

δq-axis

d-axis

Com

mon

refe

renc

eim

agin

ary

axis

Commonreference realaxis

′

Vt

Figure 8.14 Machine simplified representation and phasor diagram: (a) machinerepresentation, (b) the phasor diagram


Therefore, (8.45)–(8.48) can be incorporated to give the stator equations. Then,adding the rotor swing equation to obtain the synchronous machine modeldescribes the dynamic behaviour when considering the effects of saliency andchange of field flux linkages.

Network equations

To develop equations representing the network, models of power system loads mustbe determined. Motors as loads are represented by equivalent circuits; otherwise,the loads during the transient period can commonly be represented by staticimpedance or admittance to ground, constant current at fixed power factor, constantreal and reactive power or a combination of these representations as explained inChapter 4, Section 4.3.1.

The constant power load is either equal to the scheduled real and reactive busload or a percentage of specified values in case of a combined representation. For aconstant current load representation, the current is calculated as the initial valuefrom the scheduled bus loads and voltages obtained from the load flow solutionfor the power system prior to a disturbance. Thus, at bus i the load current Iio isobtained from

Iio ¼ PLi � jQLi

E�i

ð8:49Þ

where PLi and QLi are the scheduled real and reactive bus loads, respectively. Ei isthe calculated bus voltage. The current Iio flows from bus i to ground. Its magnitudeand power factor remain constant.

(a) (b)

Ra Xq

Eq

ItVt

Commonreference realaxis

d-axis

q-axisIt

δ

Eq

EI

jXqIq

jXqIt

jXdIdVt

RaIt

Com

mon

refe

renc

eim

agin

ary

axis

Figure 8.15 Machine representations by Eq: (a) machine representation, (b) thephasor diagram


The static admittance yio representing the load at bus i can be obtained from

yio ¼ Iio

Ep¼ gio � jbio ð8:50Þ

where the ground voltage is zero and

gio ¼ PLi

e2i þ f 2

i

and bio ¼ QLi

e2i þ f 2

i

ei and fi are the real and imaginary components of the bus voltage Ei.The network equations used for load flow calculations, explained in Chapter 5,

can be applied to describe the performance of the network during a transientperiod. The calculation of bus admittance matrix, Ybus, at different states – priorto a disturbance, instant of fault occurrence and post-fault clearance – isrequired.

Elements of Ybus are identified as diagonal elements ≜ Yii ¼ sum of all theadmittances connected to bus i and off-diagonal elements ≜ Yij ¼ the negative ofthe admittance between bus i and bus j. The matrix Ybus can be reduced by con-sidering only the internal generator buses and eliminating all other buses in thenetwork. The reduced matrix can be obtained by recalling that all buses have zeroinjection currents except for the internal generator buses. Accordingly, the relationsbelow can be written as

I ¼ YV ð8:51Þ

where

I ¼ Ig

0

�

Thus, both the matrix Y and vector V in (8.51) are partitioned to get

Igg

0

�¼ Ygg Ygb

Ybg Ybb

�Vg

Vb

�: ð8:52Þ

where the subscript g denotes the internal generator buses and subscript b denotesthe other network buses. The vectors Vg and Vb have the dimensions ng � 1 andnb � 1, respectively.

Equation (8.52) can be rewritten in expanded form as

Ig ¼ YggVg þ YgbVb and 0 ¼ YbgVg þ YbbVb

Thus, Vb can be eliminated to get

Ig ¼ Ygg � YgbY�1bb Ybg

� �Vg ¼ YreducedVg ð8:53Þ


where Yreduced is the required reduced matrix. It has the dimensions ng � ng ‘ng isthe number of generators’. It is noted that this procedure of reduction is only viablewhen the loads are treated as constant shunt admittance.

Changes in system parameters that simulate the disturbance must be specified.The initial values of system parameters such as bus voltages and power can beobtained by applying load flow techniques to the system at steady state. Theinternal machine bus voltages are calculated according to the model used. Then, thesystem differential equations are solved numerically along with the algebraicequations and load flow calculations at each step. Details of the solution procedureare described by an example below.

Example 8.2 For a nine-bus test system shown in Figure 8.16, evaluate the sys-tem transient stability when subjected to a three-phase short circuit at bus no. 7 ‘atthe beginning of line 7-5’ for durations of 0.08 s and 0.20 s. The fault is cleared byisolating line 7-5. The system data are given in Appendix II.

Solution:

It is assumed that the system is operating at steady state for 1 s, then the fault occursfor a duration of 0.08 s, Case (I), and 0.20 s, Case (II). Elements of matrix Ybus (inpu) as calculated at steady state are summarised in Table 8.2. Bus voltages, gen-eration and loads and line flow in pu at steady state are calculated by load flowanalysis and are given in Tables 8.2–8.4.

GG

G

1

2 3

4

5 6

7 8 9

F

Figure 8.16 Nine-bus test system


Tab

le8.

2B

usad

mit

tanc

em

atri

xin

stea

dyst

ate

Bus

12

34

56

78

9

10� j4

.845

90

00þ

j8.4

459

00

00

0

20

0� j5

.485

50

00

00þ

j5.4

855

00

30

00� j4

.168

40

00

00

0þ

j4.1

684

40þ j8

.445

90

03.

3074

�j3

0.39

37�1

.365

2þ

j11.

6041

�1.9

422þ

j10.

5107

00

0

50

00

�1.3

652þ

j11.

6041

2.55

28�

j17.

3382

0�1

.187

6þ

j5.9

751

00

60

00

�1.3

652þ

j11.

6041

03.

2242

�j1

5.84

090

0�1

.282

0þ

j5.5

882

70

0þ j5

.485

50

0�1

.187

6þ

j5.9

751

02.

8047

�j2

4.93

11�1

.617

1þ

j13.

6980

0

80

00

00

0�1

.617

1þ

j13.

6980

2.77

22�

j23.

3032

�1.1

551þ

j9.7

843

90

00þ j4

.168

40

0�1

.282

0þ

j5.5

882

0�1

.155

1þ

j9.7

843

2.43

71�

j19.

2574

At the instant of fault occurrence:Each machine is represented by a voltage source ‘constant magnitude’ E0 behinddirect-axis transient reactance X 0

d . Each load is represented by constant shuntadmittance yL (8.50). Based on steady-state load flow results, the calculated valuesof E0 and yL for each generator and load, respectively, are given below:

E01 ¼ 1:0565ff19:55�; E

02 ¼ 1:0264ff12:93�; E

03 ¼ 1:0319ff2:37�

yL5 ¼ 1:261 � j0:5044; yL6 ¼ 0:877 � j0:2926; yL8 ¼ 0:969 � j0:3391

The system admittance matrix for transient stability study can be formulated assummarised in Table 8.5. The admittance of each load is considered as a shuntelement to the corresponding bus, i.e. it will be added to all admittances connectedto that bus to calculate its self-admittance element.

The values of system state variables and parameters at steady state are takenas initial conditions for numerical integration. The second-order Runge–Kutta

Table 8.3 Line flow from load flow calculation prior to fault

Bus no. Bus voltage (Ep) Generation Load

V Phase angle P Q P Q

1 1.04 0 0.71641 0.27046 0 02 1.025 0.16197 1.63 0.06654 0 03 1.025 0.08142 0.85 �0.1086 0 04 1.026 �0.0387 0 0 0 05 0.996 �0.0696 0 0 1.25 0.56 1.013 �0.0644 0 0 0.9 0.37 1.026 0.06492 0 0 0 08 1.016 0.0127 0 0 1 0.359 1.032 0.03433 0 0 0 0

Table 8.4 Line flow from load flow calculation prior to fault

From bus To bus P flow Q flow P loss Q loss

4 1 �0.71641 �0.23923 0 0.031237 2 �1.63 0.09178 0 0.158329 3 �0.85 0.14955 0 0.040967 8 0.7638 �0.00797 0.00475 �0.115029 8 0.24183 0.0312 0.00088 �0.211767 5 0.8662 �0.08381 0.023 �0.196949 6 0.60817 �0.18075 0.01354 �0.315315 4 �0.4068 �0.38687 0.00258 �0.157946 4 �0.30537 �0.16543 0.00166 �0.15513


Tab

le8.

5B

usad

mit

tanc

em

atri

xfo

rtr

ansi

ent

stud

y

Bus

12

34

56

78

9

10� j8

.445

90

00þ

j8.4

459

00

00

0

20

0� j5

.485

50

00

00þ

j5.4

855

00

30

00� j4

.168

40

00

00

0þ

j4.1

684

40þ j8

.445

90

03.

3074

�j3

0.39

37�1

.365

2þ

j11.

6041

�1.9

422þ

j10.

5107

00

0

50

00

�1.3

652þ

j11.

6041

3.81

3�

j17.

826

0�1

.187

6þ

j5.9

751

00

60

00

�1.9

422þ

j10.

5107

04.

019�

j16.

1355

00

�1.2

820þ

j5.5

882

70

0þ j5

.485

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.617

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0

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00

00

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.617

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method and PSAT/MATLAB� toolbox are used for the transient analysis as below.The time interval is taken as 0.02 s. At the instant of fault occurrence bus voltages,generation, loads, line flow and state variables are summarised in Tables 8.6–8.8.

It is to be noted that in all tables, d is given in elec. rad.; w in elec. rad./s; andV, P and Q in pu.

Table 8.6 Bus voltages, generation and loads at the instant of fault occurrence

Bus code (p) Bus Voltage (Ep) Generation Load

V Phase P Q P Q

1 0.83441 �0.00472 0.7169 0.1843 0 02 0.36933 0.33849 1.63 0.1214 0 03 0.65588 0.80722 0.85 �0.0548 0 04 0.62722 0.11775 0 0 0 05 0.64996 �0.07416 0 0 1.25 0.56 0.58798 �0.09371 0 0 0.9 0.37 0 0 0 0 0 08 0.21255 0.01174 0 0 1 0.359 0.50141 0.04652 0 0 0 0

Table 8.7 Line flow at the instant of fault occurrence


4 1 �0.65326 �2.0587 0 0.636097 2 �0.00807 �0.06205 0 2.05649 3 �0.38196 �1.0629 0 0.297367 8 0.00287 �0.03025 0.0659 0.554869 8 0.20363 �1.3872 0.09653 0.786647 5 0 0 0 09 6 0.17834 �0.32423 0.01703 �0.032665 4 �0.42051 �1.0905 0.07583 0.591936 4 �0.14917 �0.39506 0.00774 �0.01877

Table 8.8 State variables (d and w) and algebraic variables ( P and Q) at theinstant of fault occurrence


d w P Q d w P Q d w P Q

0.34097 1 0.0080 2.1185 0.2255 1 0.3819 1.360 0.0414 1 0.653 2.694


Case (I) Fault duration ¼ 0.08 s

At each interval the load flow calculation is applied two times for the second-orderRunge–Kutta ‘in general k times for kth-order Runge–Kutta’. The first time is usingthe load flow for calculating the coefficients K11 and K21 at the beginning of theinterval and the second time for calculating K12 and K22 at the middle of interval.

For instance, at the first interval just after fault occurrence the results of loadflow in Tables 8.6–8.8 are used to calculate K11 and K21, and then applying loadflow analysis at the middle of this interval to calculate K12 and K22. The results aretabulated in Tables 8.9–8.11.

Table 8.9 Bus voltages, generation and loads at the middle of first interval offault occurrence

Bus code (p) Bus voltage Generation Load

V Phase P Q P Q

1 0.8344 �0.0051 0.7169 0.18434 0 02 0.3693 0.3399 1.63 0.12138 0 03 0.6272 0.1182 0.85 �0.0548 0 04 0.6499 �0.0745 0 0 0 05 0.4192 �0.1663 0 0 1.25 0.56 0.5879 �0.0939 0 0 0.9 0.37 0.0109 0.2137 0 0 0 08 0.2125 0.0121 0 0 1.00 0.39 0.5014 0.0469 0 0 0 0

Table 8.10 Line flow middle of first interval of fault occurrence


4 1 �0.65242 �2.0589 0 0.636067 2 �0.00811 �0.06204 0 2.05649 3 �0.38273 �1.0628 0 0.297457 8 0.00288 �0.03024 0.0659 0.554819 8 0.20359 1.3871 0.09652 0.786577 5 0 0 0 09 6 0.17914 �0.32425 0.01708 �0.032445 4 �0.42046 �1.0905 0.07583 0.59196 4 �0.14838 �0.39529 0.00774 �0.01878

Table 8.11 State variables (d and w) and algebraic variables (P and Q) middle offirst interval of fault occurrence



0.34245 1.001 0.0081 2.1185 0.226 1 0.382 1.360 0.040 1 0.652 2.6949


The angles and speeds of the generators at the end of the interval are calcu-lated in terms of the average values of K coefficients. Consequently, the new busvoltages and line flow are calculated to express the initial values for the nextinterval Tables 8.12–8.14.

Table 8.12 Bus voltages, generation and loads as initial values for secondinterval of fault occurrence

Bus code (p) Bus voltage Generation Load

V Phase P Q P Q

1 0.83439 �0.00645 0.7169 0.18434 0 02 0.36932 0.34436 1.63 0.12138 0 03 0.62711 0.1197 0.85 �0.05481 0 04 0.64988 �0.07553 0 0 0 05 0.41914 �0.16736 0 0 1.25 06 0.58784 �0.0943 0 0 0.9 07 0.0109 0.21607 0 0 0 08 0.21251 0.01314 0 0 1 09 0.50131 0.04787 0 0 0 0

Table 8.13 Initial values of line flow for second interval of fault occurrence


4 1 �0.64989 �2.0593 0 0.635967 2 �0.00824 �0.062 0 2.05659 3 �0.38503 �1.0624 0 0.297737 8 0.00292 �0.03022 0.06588 0.554679 8 0.2035 1.3867 0.09649 0.786377 5 0 0 0 09 6 0.18153 �0.32431 0.01722 �0.031775 4 �0.4203 �1.0903 0.07582 0.591816 4 �0.14603 �0.39598 0.00774 �0.0188

Table 8.14 State variables (d and w) and algebraic variables (P and Q) forsecond interval of fault occurrence



0.34689 1.0025 0.0082 2.1185 0.228 1.0016 0.3850 1.3601 0.0394 1 0.6498 2.6953


By repeating the same procedure for the next intervals until reaching the lastinterval of a total period of 5 s, the results obtained at the end of the period aresummarised in Tables 8.15–8.17.

It is important to note that at the instant of fault clearance, the bus admittancematrix is changed as the line 7-5 is isolated. This admittance matrix is calculated assummarised in Table 8.18. The reduced matrices Yreduced pre, during and post thefault are summarised in Table 8.19.

Table 8.15 Bus voltages, generation and loads at the last interval after faultclearing

Bus code (p) Bus voltage (Ep) Generation Load

V Phase P Q V Q

1 0.9504 �0.22136 0.7169 0.18434 0 02 0.95206 0.91582 1.63 0.12138 0 03 0.91456 0.53713 0.85 �0.05481 0 04 0.87361 �0.19968 0 0 0 05 0.82835 �0.29988 0 0 1.25 06 0.81676 �0.0214 0 0 0.9 07 0.92093 0.77827 0 0 0 08 0.885 0.61164 0 0 1 09 0.88667 0.46614 0 0 0 0

Table 8.16 Line flow at the last interval after fault clearing


4 1 0.31256 �1.1613 0 0.109167 2 �1.9235 �0.32619 0 0.28059 3 �0.98157 �0.38716 0 0.082997 8 1.9235 0.32619 0.0386 0.205449 8 �1.1019 0.14489 0.01916 �0.001727 5 0 0 0 09 6 2.0835 0.24227 0.22261 0.71025 4 �0.88627 �0.35451 0.01271 �0.019526 4 1.2617 �0.66763 0.05021 0.15871

Table 8.17 State variables (d and w) and algebraic variables (P and Q) at the lastinterval after fault clearing



1.15 1.026 1.923 0.607 0.728 1.025 0.982 0.470 �0.240 1.023 �0.312 1.270


Tab

le8.

18B

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ter

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t

Bus

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56

78

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The changes of d and w for each generator are shown in Figures 8.17 through8.19. It is clear that the power angles of all machines in the system increase andthen decrease. Therefore, the system is stable.

Case (II) Fault duration ¼ 0.20 s

In this case, the fault clearing time becomes 0.20 s rather than 0.08 s. The sameprocedures as above are carried out. The initial conditions are the same in bothcases. The results obtained are plotted as shown in Figures 8.20–8.22. It is found

Table 8.19 Reduced matrices Yreduced

Network type Generator bus 1 2 3

Pre-fault 1 0.846 � j2.988 0.287 þ j1.513 0.210 þ j1.2262 0.287 þ j1.513 0.420 � j2.724 0.213 þ j1.0883 0.210 þ j1.226 0.213 þ j1.088 0.277 � j2.368

During fault 1 0.657 � j3.816 0.000 þ j0.000 0.070 þ j0.6312 0.000 þ j0.000 0.000 � j5.486 0.000 þ j0.0003 0.070 þ j0.631 0.000 þ j0.000 0.174 � j2.796

Post-fault 1 1.181 � j2.229 0.138 þ j0.726 0.191 þ j1.0792 0.138 þ j0.726 0.389 � j1.953 0.199 þ j1.2293 0.191 þ j1.079 0.199 þ j1.229 0.273 � j2.342

0 1 2Time (s)

Time (s)

3 4 5

0 1 2 3 4 5

10

30

50

70

(a)

(b)

0.9951

1.0051.01

1.02

1.031.035

1.025

1.015

δ 1°

ω1 (

elec

. rad

/sec

)

Figure 8.17 (a) Power angle versus time and (b) angular speed versus time formachine 1 for fault clearing time 0.08 s


0 1 2 3 4 5

0 1 2 3 4 5

(a)

0

5

–5

–10

–15

1

1.01

1.02

δ 3°

ω3 (

elec

. rad

/sec

)

Time (s)

Time (s)(b)


0 1 2 3 4 5

0 1 2 3

Time (s)

Time (s)(a)

(b)4 5

5

15

25

35

45

1

1.01

1.02

1.03

δ 2°

ω2 (

elec

. rad

/sec

)



that the power angles of the machines in the system are continuously increasing ordecreasing. Thus, the system is unstable because of delayed fault clearing.

From the figures of Case (I) and Case (II), it is found that the system isstable when the fault duration is 0.08 s. On the contrary, the system is unstable ifthe fault duration is 0.20 s. Therefore, the duration of fault clearing must beinvestigated carefully. It depends on several parameters such as system topology,selected machine model, type of fault, location of fault and characteristics of pro-tective gears.

It can be concluded that the main steps to determine the transient stability of amulti-machine power system can be summarised as below:

● Define the data of each element in the power system.● Define the period of study.● Specify the fault: type, location and, clearing time and how the fault is cleared.● Select an adequate model for each machine ‘it may differ from one machine to

another’.

8000

0 1 2 3 4 5

0 1 2 3 4 5

(a)

(b)

0

2000

4000

6000

1

1.1

1.2

1.3

1.4

δ 1°

ω1 (

elec

. rad

/sec

)

Time (s)

Time (s)

Figure 8.20 (a) Power angle versus time and (b) angular speed versus time forfault clearing time ¼ 0.20 s


● Model the other components of the system ‘transmission lines,transformers, etc.’.

● Define the frame of reference to which all parameters of component modelsare referred.

● Construct the bus-admittance matrix, Ybus, and apply the load flow analysis toobtain the system parameters at steady state and calculate the initial values ofparameters necessary to solve the swing equation.

● Calculate the elements of Ybus for transient analysis as well as for post-faultoperation.

● Calculate the reduced matrices Yreduced pre, during and post fault.● Select the numerical integration method to solve the system differential

equations.● Define the time interval and start solving the system equations.● During the numerical solution load flow analysis is applied whenever machine

angles are changed. This is to calculate the power delivered from eachmachine, which is involved in the swing equation.

8000

0 1 2 3 4 5

0 1 2 3 4 5

(a)

(b)

01000

3000

6000

1

1.1

1.2

1.3

1.4

δ 2°

ω2 (

elec

. rad

/sec

)

Time (s)

Time (s)

Figure 8.21 (a) Power angle versus time and (b) angular speed versus time formachine 2 at fault clearing time ¼ 0.20 s


● When the clearing time is reached the admittance matrix is switched to Ybus

post fault.● The numerical solution is continued until reaching the end of period of study.● The system is stable if the change of power angle with time for each machine

increases and then decreases. Otherwise the system is unstable.

References

1. Roderick J, Podmere F.R., Waldron M. ‘Synthesis of dynamic load modelsfor stability studies’. IEEE Transactions on Power Apparatus and Systems.Jan 1982;101(1):127–35

2. Price W.W., Wirgau K.A., Murdoch A., Mitsche J.V., Vaahedi E, El-KadyM.A. ‘Load modelling for power flow and transient stability computer studies’.IEEE Transactions on Power Systems. Feb 1988;3(1):180–8

0 1 2 3 4 5

0 1 2 3 4 5

(a)

(b)

0

500

–500

–1000

–2000

–3000

1

1.002

1.004

1.006

1.01

1.014

Time (s)

Time (s)

δ 3°

ω3 (

elec

. rad

/sec

)

Figure 8.22 (a) Power angle versus time and (b) angular speed versus time formachine 3 at fault clearing time ¼ 0.20 s


3. IEEE Task Force. ‘Current usage and suggested practices in power systemstability simulations for synchronous machines’. IEEE Transactions on EnergyConversion. 1986;1(1):77–93

4. Momoh J.A., El-Hawary M.E. Electric Systems. Dynamics, and Stability withArtificial Intelligence Applications. Germany: Marcel Dekker; 2000

5. Nandakumar K. Numerical Solutions of Engineering Problems. Edmonton,Alberta, Canada: University of Alberta; 1998

6. Harder D.W., Khoury R. Numerical Analysis for Engineering. Saskatoon,Saskatchewan, Canada: University of Waterloo; 2010

7. Awrwjcewiz J. Numerical Analysis: Theory and Application. Rijeka, Croatia:InTech; 2011

8. Moursund D.G., Duris C.S. Elementary Theory & Application of NumericalAnalysis. New York, US: McGraw-Hill; 1967


Chapter 9

Transient energy function methods

The power system is described mathematically, as explained in Chapter 8, by a setof differential-algebraic equations. The equations are large in numbers and non-linear in nature. A typical number of equations is in the order of hundreds, if notthousands, for a network of moderate size. The solution of system equations isrequired for time simulation to obtain generator power angles and other systemparameters of interest at different time instants. This helps in examining the systembehaviour and deciding whether its dynamic response will result in anacceptable performance. The solution is repeated for different scenarios such aschanging fault type, fault location, network topology and considering variouscontrol devices. Consequently, the solution is time consuming. Therefore, timesimulation techniques are not appropriate for online stability monitoring. Forinstance, transient stability studies for a typical system with detailed modelling fora 500-bus, 100-machine system may take up to an hour [1].

Accordingly, power engineers and system operators are motivated to look foran alternative method by which system stability can be determined directly and hasthe desired specification for online applications. The key idea that the alternativemethod is based on is to specify a certain function by which the system transientenergy at the end of the disturbance period can be calculated. The calculated valueis compared with a critical energy value to assess the transient stability, as thedifference between the two values gives an indication of stability.

9.1 Definitions of stability concepts

As explained in Chapter 7, Sections 7.1.1 and 7.1.2, the power system can be modelledas an autonomous system and described by the ordinary differential equation:

_x ¼ f xð Þ ð9:1Þthat is assumed to have an equilibrium point at the origin, i.e. _x ¼ f 0ð Þ ¼ 0

The equilibrium point is either stable (SEP) or unstable (UEP). In the sense ofLyapunov, the equilibrium x ¼ 0 is stable if for any given e> 0, there exists ar � e such that ||xo|| < r satisfies ||x(t)|| < e for all t where xo ¼ x(to) ≜ initial state.This concept is pictorially described in Figure 9.1. It is shown that the initial statexo has a magnitude less than r and the trajectory of x remains within a cylinder of

radius e. It is noted that the stability concept in the sense of Lyapunov is a localconcept as it does not specify how small r had to be chosen in the definition [2].

The origin x ¼ 0 at time to is called unstable if it is not stable at to. Thus, theUEP x ¼ 0 implies that for some e> 0 there is no r> 0 satisfying the condition||x(t)|| < e for all t � to when ||xo|| < r. The trajectory of x eventually leaves thecylinder of radius e. Hence, intuitively speaking, an equilibrium point is stable ifnearby trajectories stay nearby [3].

As discussed in Section 7.1.2, the equilibrium point x ¼ 0 is asymptoticallystable at to if it is stable at t ? to and also if given ||xo||< r satisfies x ? 0 as t ??(Figure 7.1). In this case, the requirement of ‘nearby trajectories stay nearby’ is notsufficient; it should be ‘nearby trajectories stay nearby and all converge to the equi-librium point’.

Again, this concept is local as the region containing all initial conditions thatconverge to the equilibrium is some portion of the state space. Asymptotic stabilityis global if the equilibrium x ¼ 0 is stable and satisfies the condition ||xo|| ? 0 ast ?? for any xo in the whole space.

9.1.1 Positive definite functionA scalar and continuous function v(x) is said to be positive definite in a region R ifv(x) > 0 for x 6¼ 0 and v(0) ¼ 0. All x satisfying v(x) ¼ C form a corresponding spacesurface called contour. With varying C different un-intersected contours are obtained.

9.1.2 Negative definite functionv is negative definite if �v is positive definite.

9.1.3 LemmaIt states that there exists a sphere defined by ||x||¼ N in which v(x) increases mono-tonically along radial vectors emanating from the origin. That is, v(bu) increasesmonotonically with b in 0 � b � N for any unit vector u started from the origin.

This can be illustrated by using the assumption of positive definiteness wherev(x) > 0 for x 6¼ 0 and v(0) ¼ 0. Assume that v(bu) increases monotonically with b in

to

x(t)

ε

ρ

ρ t

ε

Figure 9.1 Stability concepts in the sense of Lyapunov


an interval 0 � b � bu and begins to decrease after b¼ bu. For a given u, there existsan associated bu, which may be unbounded bu (¼?). If w is among all the u’s that hasthe smallest bw, then ||bww||¼ bw||w||¼ bw � bu. As v(bu) increases monotonicallywithb in the interval 0 � b � bw � bu; the positive number N ¼ bw can be identified.

9.1.4 Stability regionsFor a SEP, xs, a number r> 0 exists such that every point in the set ||xo � xs|| < rimplies that the trajectory starting from the initial point xo converges to the SEP, xs,i.e. Fi x0ð Þ ! x as t??. If r is arbitrarily large, then x is called a global SEP.There are many systems containing SEPs but not globally SEPs. For these systemsthe stability region is also called the region of attraction. The stability region of aSEP, xs, is the set of all points x such that limt!1;t xð Þ! xs.

9.1.5 Lyapunov function theoremThe main concept of Lyapunov function theorem is to derive stability properties ofthe equilibrium point without numerically solving the system differential equations,i.e. without time-domain simulation. Assuming _v xð Þ is the total derivative of v(x)on the trajectory specified by (9.1),

_v xð Þ ¼ dv

dt¼

Xn

i¼1

@v xð Þ@x

fi xð Þ

¼ rv xð ÞT f xð Þ ð9:2Þwhere rv xð Þ is the vector formed by the partial derivatives of v(x) and can be per-formed without knowledge of the system trajectory; n is the dimension of the space.

Assuming regions R, R1, R2 such that R2 � R1 � R and all regions contain theorigin as an interior point, the theorem states that:

If v(x) be a positive definite function with continuous partial derivatives in aregion R, then

● the origin of the system described by (9.1) is stable if _v xð Þ � 0 in a sub-regionR1 � R.

● The system is asymptotically stable in the region if it is stable and _v xð Þ ¼ 0takes place only in a sub-region R2 � R1.

● The origin is globally asymptotically stable if the system is asymptoticallystable: R2 is the whole space and v xð Þ!1 as jjxjj! 0.

Example 9.1 The dynamics of a pendulum is chosen to illustrate how to use Lya-punov theorem for studying its stability. The motion of a pendulum can be described by

J€q þ D q_ þ a sin q ¼ 0

where J and D are inertia and damping constant, respectively. a is a constant interms of pendulum mass and q is the angle. It is noted that this equation is

Transient energy function methods 223

analogous to the equation that describes the rotor motion. Multiplying by _q, thenintegrating with respect to t obtains

½ J _q2 þ D

ð_q2

dt þ ½að�cos qÞ þ a� ¼ C

The sum of kinetic and potential energy of the pendulum is the total energy andis given by

GT ¼ ½ J _q2 þ að1 � cos qÞ ¼ C � D

ð_q2

dt > 0

Differentiation with respect to t yields

_GT ¼ _qðJ€q þ a sin qÞ ¼ �D _q2 � 0

The Lyapunov function, v(x), is proposed to be the total energy per unit inertiaof the pendulum system.

Hence; vðxÞ ¼ E

J¼ ½ _q2 þ að1 � cos qÞ ¼ ½ x2

2 þ að1 � cos x1Þ

where x1 ¼ q and _x1 ¼ x2

The region R is defined by R ¼ �2p < x < 2p; free x2f g.From (9.2) the derivative of v xð Þ is

_v xð Þ ¼ rv xð ÞT f xð Þ ¼ a sin x1; x2½ � x2

�Dx2 � a sin x1

� �¼ �Dx2

2

where f xð Þ in (9.1) can be obtained by writing the equation of motion in the form

_x1

_x2

� �¼ x2

�Dx2 � asin x1

� �¼ f xð Þ

It is important to note that _GT � _vðxÞ. This means that the Lyapunov functionis actually the sum of kinetic and potential energy per unit inertia.

By examining the functions v xð Þ and _v xð Þ and applying Lyapunov theorem, it isseen that:

● The origin of the system is stable as _v xð Þ � 0 in R1, which equals the wholespace R ¼ �2p < x < 2p; free x2f g.

● The system is asymptotically stable in the region R2 � R1, whereR2 ¼ �2p < x < 2p; free x2f g. This can be proved as below. Verification ofthis condition means that _v xð Þ ¼ �Dx2

2 ¼ 0. It implies that x2 ¼ 0, then_x2 ¼ 0 ¼ �Dx2 � asin x1 ¼ 0. Consequently, sin x1 ¼ 0 or x1 ¼ np. Thus,_v xð Þ¼ 0 only at the origin by choosing R2 ¼ �p < x < p; free x2f g.

● Part iii of Lyapunov theorem is not satisfied as R2 is not the whole space and ifkxk ! 0, the function v xð Þ ¼ a 1 � cos x1ð Þ � 2a, i.e. v(x) does not approach?.


9.2 Stability of single-machine infinite-bus system

Transient energy-based stability assessment of a single machine connected to aninfinite bus system leads to a method called the ‘equal area criterion, EAC’ asexplained below.

The first step is to select the machine model in order to deduce the relation ofelectric power calculation. The machine model 0.0 ‘where only two stator algebraicequations are only used since there is no need to use differential equations and allstate variables for rotor coils are ignored’ is chosen to illustrate the idea behind EAC.

The machine is represented by a voltage source, E0 ¼ E0q þ jE0

d , with constantmagnitude behind d-axis transient reactance, x0d as shown in Figure 8.2. From(8.27) the two components of E0 are expressed as

E0q ¼ Vq þ RaIq � IdX 0

d

E0d ¼ Vd þ RaId þ IqX 0

q

)ð9:3Þ

More simplifications are used such as (i) the armature resistance Ra is neglectedand (ii) a direct axis rotor winding is only considered. Thus, E0

d ¼ 0;E0 ¼ E0q and

(9.3) becomes

E0 ¼ Vq � IdX 0

d

0 ¼ Vd þ IqX 0q

)ð9:4Þ

Hence, the current components in d–q axis can be found as

Id ¼ ðVq � E0 Þ

X 0d

Iq ¼ �Vd

X 0q

9>>>=>>>;

ð9:5Þ

The machine electric output power ‘Pe’ can be calculated by

Pe ¼ VdId þ VqIq ð9:6ÞSubstituting (9.5) in (9.6) gives

Pe ¼ VdVq � E

0

X 0d

� VqVd

X 0q

¼ VdVq

X 0d � X 0

q

X 0dX 0

q

� E0Vd

X 0d

ð9:7Þ

As shown in the phasor diagram (Figure 8.5), the terminal voltage is assumedto be taken as a reference as well as E0 leads Vt by an angle d. Thus, the followingrelations can be written as

Vq ¼ Vt cos dVd ¼ �Vt sin d

)ð9:8Þ


From (9.7) and (9.8) Pe can be expressed as

Pe ¼ E0Vt

X 0d

sin dþ V 2t

X 0d � X 0

q

2X 0dX 0

q

sin 2d ð9:9Þ

The second term in the RHS of (9.9) represents the saliency effect for salientpole machines where X 0

d 6¼ X 0q. On the other hand, X 0

d and X 0q are equal for round

rotor machines, and then

Pe ¼ E0Vt

X 0d

sin d ð9:10Þ

According to (9.9) and (9.10) the power–angle curve for salient pole and roundrotor machines can be drawn as shown in Figure 9.2(a and b), respectively. It is tobe noted that the P–d curve for salient pole machines is not pure sinusoidal curvebecause of the presence of saliency effect.

In transient stability studies, the input mechanical power can be assumed as aconstant. This assumption is accepted as the electric changes involved are muchfaster than the resulting mechanical changes produced by the generator/turbinespeed control, e.g. the time constant of excitation control loop is much less than thatof the governor control loop.

Considering a round rotor machine connected to an infinite bus through a trans-mission network with external reactance ‘Xe’ the reactance in (9.10) is replaced by theequivalent reactance, Xeq ¼ X 0

d þ Xeq and the electrical power is calculated by

Pe ¼ E0V1Xeq

sin d ð9:11Þ

where V1 ≜ the infinite bus voltage at zero angle ‘taken as a reference’d ≜ the angle between E0 and V1

Pe Pe

d d

(a) (b)

E¢Vt/X¢d

Figure 9.2 Power–angle curves for synchronous machines: (a) p–d curve forsalient pole machines and (b) p–d curve for round rotor machines


Equation (9.11) with constant mechanical power input Pm can be depicted asshown in Figure 9.3. The intersection between Pm and the power–angle curve givestwo equilibrium points, do and p� do. The first is SEP while the second is UEP asdefined by (7.44) and illustrated by Figure 7.3. Physically, the stability andinstability of equilibrium points can be interpreted by assuming that a small dis-turbance causes a change of d by a small amount Dd. This change results in Pe > Pm

and dw/dt becomes negative according to the swing equation. Consequently, d isdecreased until the system reaches its initial stable equilibrium point at do. On theother hand, if this change occurs when the system is operating at p� do, theincrease of d continues as Pm > Pe and dw/dt is positive, i.e. d moves further fromp� do. That is why p� do is called UEP.

As shown in Figure 9.4 the system is operating at an equilibrium state (do, Peo)where Peo ¼ Pm and is subjected to a sudden change of Pe from Peo to Pe1 at whichthe power angle is d1. As Pm is greater than Pe1 the rotor kinetic energy isincreased. The accelerating power Pa ¼ Pm – Pe and dw/dt are positive and dincreases until reaching the point (do, Peo) where both accelerating power and dw/dtare zero. Because of the rotor inertia d continues to increase beyond do where rotorretardation starts until reaching the point (ds, Pes) from which point the retardationwill bring d down. At point ‘ds, Pes’ the areas A1 and A2 (Figure 9.4) are equal. Theprocess continues on in the form of oscillations around the equilibrium point(do, Peo). If damping is present, the oscillations decrease, the system is stable andcontinues to operate at the equilibrium point.

The same oscillations may occur if the input mechanical power is changedsuddenly. Assume Pm is increased at a fast rate from Pmo at initial equilibrium state(do, Peo) to Pm1 (Figure 9.5). Accordingly, the angle d will increase to d1 as the

δo p/2 p – δo p δ

Pe

Pm

Pmax

SEP UEP

Figure 9.3 Power–angle curve illustrating stable and unstable equilibrium points


accelerating power is positive and the rotor kinetic energy is increased as well.At (d1, Pe1) the accelerating power Pa is zero but the speed deviation from thesynchronous speed is not zero because of rotor inertia. Even though rotor retarda-tion sets in at (d1, Pe1), the angle d1 continues to increase until reaching ds at whichpoint the speed deviation is zero and the areas A1 and A2 (Figure 9.5) are equal.Rotor retardation brings d down and the process continues on in the form ofoscillations. If damping is present, the oscillations decrease and stable operationresults at new equilibrium point (d1, Pe1).

Pe Pmax

Pm

Pe1

Peo

Pes

A1

A2

δp – δoδo δSδ1 p/2 p

Figure 9.4 Power–angle curve in response to changing the electrical power

PePmax

Pmo

Pm1Pe1

Peo

Pes

A1

A2

δδo δS p – δ1δ1 p/2 p – δo p

Figure 9.5 Power–angle curve with sudden change in mechanical input power


If the pre-fault, during fault and post-fault system configurations are notidentical, then each case is represented by a power–angle curve corresponding tothe parameters of the system operating at each specific state (Figure 9.6).

Assuming that the system is operating at initial steady state (do, Peo) a faultoccurs. During the fault, the power–angle curve can be determined by calculating thetransfer impedance between the internal machine voltage source and the infinite bus.

Accordingly, the electrical power, Peo, is decreased and this results in positiveaccelerating power and positive dw/dt. Thus, the angle d increases from do to dc atwhich the fault is cleared and the corresponding post-fault transmitted electricalpower is Pec > Pmo. Both Pa and dw/dt are negative causing rotor retardation butthe speed error is not zero. Thus, d continues to increase until reaching the point(dm, Pes) at which point the areas A1 and A2 (Figure 9.6) are equal: the rotor willmomentarily stop and retardation will bring d down. The process continues in theform of oscillations around the new equilibrium point (A) as shown in Figure 9.6.The system will be stable if the oscillations are damped.

Rotor oscillation when the system is subjected to sudden change of electricalpower, sudden change of mechanical power and occurrence of a fault resulting inchange of network configuration is shown in Figures 9.4–9.6, respectively. It canbe concluded that in all cases the oscillation comprises two areas: A1 below Pm lineand A2 above this line. A1 represents the energy absorbed by the rotor in the form ofkinetic energy causing rotor speed up and an increase in angle as Pm > Pe. A2

represents the energy delivered from the rotor causing rotor slow down and adecrease in angle as Pe > Pm. Therefore, the stability condition in terms of A1 andA2 can be derived by going back to the swing equation as below.

Post-fault

Pre-fault

Duringfault

δmaxδcδo δm δ

A1

A2 AfPmo Peo

Pec

Pes

Pe

δs

A

Figure 9.6 Power–angle curves for pre-fault, during fault and post-faultconditions


The swing equation for the machine connected to the infinite bus is

M€d ¼ Pa ð9:12ÞSubstituting _d ¼ w gives the alternative form

wdw ¼ Pa

Mdd

By definition, do is the rotor angle when the machine is operated synchro-nously before the disturbance occurs, at which time dd/dt ¼ 0. Thus, integratingboth sides

w2 ¼ 2M

ðdm

do

Padd

Hence, the relative speed (w¼ dd/dt) of the machine with respect to a frame ofreference moving at a constant speed is given by

dddt

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2M

ðdm

do

Padd

sð9:13Þ

The angle d will cease to change and the machine will again be operating atsynchronous speed after a disturbance when dd/dt ¼ 0. From (9.13) the conditionfor stability can be expressed asðdm

do

Padd ¼ 0 ð9:14Þ

When (9.14) is satisfied, the maximum value of d is reached and dd/dt ¼ 0. Thearea A1 below Pm line in Figure 9.6 (the same procedure can be followed forFigures 9.4 and 9.5) is

A1 ¼ðdc

do

Padd ¼ðdc

do

Pm � Peð Þdd ð9:15Þ

Similarly, area A2 is

A2 ¼ðdm

dc

Padd ¼ðdm

dc

Pe � Pmð Þdd ð9:16Þ

Thus,

A1 � A2 ¼ðdc

do

Pm � Peð Þdd�ðdm

dc

Pe � Pmð Þdd ¼ðdm

do

Padd ð9:17Þ

It is found from (9.14) and (9.17) that A1 � A2 ¼ 0, i.e. A1 ¼ A2. The maximumangle of oscillations dm is located graphically so as to make A2 equal to A1.


Therefore, it is not necessary to assess stability by inspecting the swing curves.Stability can be determined by integrating the difference between the power anglecurve and the constant mechanical power. This integral is interpreted as the areabetween Pe curve and Pm line. The area must equal zero as a condition for stability.So, the area must consist of two equal portions: one portion is positive, A1, andthe second is negative, A2. This is the reason to call this method ‘Equal AreaCriterion, EAC’.

Based on EAC, the area A1 represents the energy converted to rotor kineticenergy at clearance. This entails the existence of area A2 of opposite sign withmagnitude ‘at least’ equal to A1 as a condition for stability. Determination of afunction called ‘transient energy function, TEF’ to decide the capability of satis-fying this condition can be used to directly assess system stability. The TEF can bederived as below.

Equation (9.10) can be rewritten as

Pe ¼ A xð Þ sin d ð9:18Þwhere A xð Þ ¼ E0V1

Xeq

Again for convenience, the alternative form of (9.12) is

Mwdw ¼ Pm � Peð Þdd ð9:19ÞAssuming the states of the fault ‘as depicted graphically in Figure 9.6’ areSo ¼ (do, 0) ≜ initial state, Sc ¼ (dc, wc) ≜ fault clearing state, Sm ¼ (dm, 0) ≜

maximum state, S ¼ (d, w) ≜ any state on the power–angle curve generated by xeq,integration of (9.19) from S to Sc gives

12

Mw2 � 12

Mw2c ¼ Pm d� dcð Þ � A xð Þ cos dc � cos dð Þ ð9:20Þ

Hence, the rotor kinetic energy at clearance, Xeq ¼ Xc, for any (d, w), is

12

Mw2 ¼ G� ð9:21Þ

where

G ¼ 12

Mw2c þ Pm d� dcð Þ and ¼ A xcð Þ cos dc � cos dð Þ

and d is substituted in radians.It is to be noted that

(i) During the fault: xeq ¼ xf and for S ¼ So, (9.20) becomes

12

Mw2c ¼ Pm do � dcð Þ � A xf

� �cos dc � cos doð Þ ¼ A1 ð9:22Þ


(ii) Post-fault clearance, xeq ¼ xc and for S ¼ Sm, the relation below can beobtained from (9.20).

12

Mw2c ¼ A xcð Þ cos dc � cos dmð Þ � Pm dm � dcð Þ ¼ A2 ð9:23Þ

As explained above, according to EAC (9.22) and (9.23) show that therelation A1 ¼ A2 can be solved to obtain dm to judge stability.

The RHS of (9.21), (G� ) is the TEF that expresses the difference A1 � A2max

(A2max ¼ A2 þ Af), i.e. the total area above Pm line from dc to dmax as shown inFigure 9.6. It can be directly used to assess system stability as follows:

(i) Evaluate TEF at the angle dmax, ‘the intersection of Pm line with post-faultpower–angle curve at UEP’ (Figure 9.6).

(ii) The system in transient state is stable if G�ð Þ < 0; large magnitude yieldslarge margin from stability boundary, better stability and more securesystem.

(iii) The system in transient state is unstable if G�ð Þ� 0.

Example 9.2 A single machine is connected to an infinite bus system through adouble-circuit transmission line as shown in Figure 9.7. The system is delivering anapparent power of 1.1 pu at 0.8 power factor lagging. All reactances are given in puon the machine rating as base. Find the source voltage and do. If a three-phase shortcircuit occurs at the beginning of one circuit of the transmission line determinewhether the system will be stable when the fault is cleared at dc ¼ 45� by isolatingthe faulty circuit.

Solution:

The equivalent reactance xeq ¼ 0.2 þ 0.1 þ 0.35/2 ¼ 0.475 puThe power delivered Peo ¼ 1.1 0.8 ¼ 0.88 pu ¼ Pm

The current flow in the circuit I ¼ 1:1ff�36:87�

The source voltage E¼ V þ jXI¼ 1 ff0� þ (0.475 ff90�)(1.1 ff�36.87�)¼ 1.314þj0.418 ¼ 1.38 ff17.65�

The maximum transmitted power ¼ EV/xeq ¼ 1.38 1/0.475 ¼ 2.9 puThe angle do ¼ 17.65�

The power–angle relation is Pe ¼ 2.9 sin d

Fxtr = 0.1xd’ = 0.2

E xTL = 0.35V� = 1.0 0

d

Figure 9.7 System for Example 9.2


When the fault occurs near the generator bus, the delivered power is zero.Thus, the horizontal axis represents the power–angle curve during the fault.

Post-fault clearance: xeq ¼ 0.2 þ 0.1 þ 0.35 ¼ 0.65 puThe maximum delivered power, Pmax( pf ) ¼ 1.38/0.65 ¼ 2.12 puThe power–angle relation is Pe ¼ 2.12 sin dTo determine the stability:

A1 ¼ Pm dc � doð Þp=180 ¼ 0:88 45 � 17:65ð Þp=180 ¼ 0:42 pu

A2 ¼ðdm

dc

Pmax pfð Þsin ddd� Pm dm � dcð Þ

¼ 2:12 cos dc � cos dmð Þ � 0:88 dm � 0:785ð Þ

A1 ¼ A2 if the system is stable, thus

0:42 ¼ 2:12 cos 45� � cos dmð Þ � 0:88 dm � 0:785ð Þ

which in turn yields 2.41 cos dm þ ds ¼ 1.88Solution of this non-linear equation by trial and error yields dm � 78�. There-

fore, the system is stable as dm has a value less than dmax. A schematic diagram forthe solution is depicted in Figure 9.8.

Post-fault

Pre-fault

During fault

P e p

u

δo

Peo = Pm= 0.88

2.12

2.9

17.65δo

45δc δm

78δmax

155.5

AfA2

A1

Figure 9.8 Power–angle relations: pre, during and post fault for Example 9.2


Example 9.3 Find the solution of Example 9.2 by using the TEF. Investigate theeffect of fault clearing angle on system stability.

Solution:

The TEF is given by (9.21) as TEF ¼ G�At dc ¼ 45� : G ¼ A1 þ Pm dmax � dcð Þ and ¼ AðxcÞ cos dc � cos dmaxð ÞThus,

G� ¼ A1 � A xcð Þ cos dc � cos dmaxð Þ � Pm dmax � dcð Þ½ �¼ A1 � A2max

and A xcð Þ ¼ 2:12 pu and A1 ¼ 0.42 as calculated in Example 9.2dmax ¼ sin�1(Pm/Pmax(pf)) ¼ sin�1(0.415) ¼ 180 � 24.5 ¼ 155.5�

G� ¼ 0:42 � 2:12 cos 45� � cos 155:5�ð Þ � 0:88 2:713 � 0:785ð Þ½ �¼ �1:311 pu

It is noted that the TEF is negative, i.e. the system is stable.Following the same procedure at different values of clearing angle dc the

results can be obtained as

It is seen that as the fault clearance is more delayed the magnitude of negativeTEF decreases. This means that the system approaches the stability boundary withsmaller margins. At dc ¼ 120� the system is unstable where the TEF is positive. Thecritical clearing angle at which the TEF is zero lies between 110o and 120o.

9.3 Stability of multi-machine power system

In a multi-machine power system, the generators and loads are connected through atransmission network. Each generator, Gi, is connected to a certain number ofnetwork buses, j, through its terminal bus i as illustrated in Figure 9.9.

The following assumptions are made to model the system in transientconditions:

● The turbine dynamics are neglected, and the input mechanical power isconstant.

● The loads are represented by constant impedances.

d�c 45 80 110 120

TEF �1.311 �0.719 �0.089 0.097System state Stable Stable Stable Unstable


● The model is adequate only to examine the first swing stability, and so thedamping torque is neglected.

● Each synchronous machine is represented by a voltage behind quadraturereactance to take into account the changes in field flux linkages when the busvoltage distribution is calculated by load flow techniques.

● The bus voltage distribution during and after the fault are time invariant.● Resistances of the transmission elements are neglected in comparison to their

reactances.

9.3.1 Energy balance approachAs explained in the former sections, the synchronism of a disturbed machine in apower system is maintained when a balance between the energies exists. The equalarea criterion is based on this approach; it equates the change of kinetic energyproduced during the fault to the post-fault change of potential energy as a conditionfor stability. From the instant of fault occurrence to that of fault clearing, a changeof kinetic energy is created representing the total transient kinetic energy. If themachine is to survive the first swing, the transient kinetic energy must be totallyconverted into potential energy.

The energy balance approach can be extended to multi-machine power systems[4]. It necessitates an analytical justification. The basic idea of this justification isto study the individual machine stability. For each machine, the electrical poweroutput ‘potential energy’ is derived as a function of

● the rotor angle d referred to the common synchronously rotating reference axisof the network

● the bus voltage distribution

The electrical output power Pei of generator Gi is

Pei ¼ Re I iE�i

� � ð9:24Þ

1

2jjXn

jxq EiEq

Terminalbus

Generator Gi

Transmissionnetwork

Load

Figure 9.9 Representation of a generator Gi connected to an integrated powersystem


whereIi ≜ generator current flowing into terminal bus

¼ Eq � Ei

� �=jXq

Ei ≜ terminal bus voltage

¼ �1=Y iið ÞXN

j¼16¼i

Y ijEi

andN, number of network busesnb, plus the internal machine buses, ng

Yij, off-diagonal element of the admittance matrixEq, machine internal voltage source behind quadrature reactanceTherefore,

Ei ¼ 1

1=jXnð Þ þ PNj¼16¼i

1=jXij

� �XN

j¼16¼i

Ej=jXij

� � ð9:25Þ

Assuming the real and imaginary components of the voltage E to be e and f,respectively, substitute (9.24) into (9.23) to get

Pei ¼ Reeq þ jfq

jXq

1

1jXn

þ PNj¼16¼i

1=jXij

� �XN

j¼16¼i

ej þ jfj

jXij

0BBBBBBBB@

1CCCCCCCCA

�

� 1jXq

e2i þ f 2

i

� �2666666664

3777777775

i.e.

Pei ¼1=Xq

� �1

Xnþ PN

j¼16¼i

1=Xij

� � fqi

XN

j¼16¼i

ej=Xij

� �� eqi

XN

j¼16¼i

fj=Xij

� �266664

377775 ð9:26Þ


Let

gi ¼ 1=Xq

� �1=Xn þ

XN

j¼16¼i

1=Xij

� �0BBB@

1CCCA

s1i ¼XN

j¼16¼i

ej=Xij

� �

s2i ¼XN

j¼16¼i

fj=Xij

� �

L1i ¼ giEqs1i

L2i ¼ giEqs2i

9>>>>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>>>>;

ð9:27Þ

Then (9.26) becomes

Pei ¼ L1i sin di � L2i cos di ð9:28Þwhere di ¼ tan�1 fqi=eqi

� �i≜ machine power angle referred to the common refer-

ence axis of the system.It is seen from (9.28) that the generator output power is calculated in terms of

the bus voltage distribution, the internal voltage source of the machine and the rotorangle. To determine the delivered power of each machine in a disturbed system, the

δo δc δmax

PmPE

900 180

Pe

δ°

KE

Figure 9.10 Power–angle curves at the three-fault periodsbefore the fault Pe ¼ L1B sin d� L2B cos d ( )during the fault Pe ¼ L1D sin d� L2D cos d ( )after the fault Pe ¼ L1A sin d� L2A cos d ( )


coefficients L1 and L2 must be determined at the three states: pre-, during andpost-fault (Figure 9.10). According to (9.27), these factors are dependent on the busvoltage distribution and the system configuration.

In practice, the bus voltages can be measured continually but, theoretically, theload flow techniques taking into account the assumptions mentioned above are usedto compute these voltages, whatever the type and location of the fault. Equations(9.14) and (9.17) are used to examine the machine stability. The machine is stable ifthe kinetic energy generated during the fault is less than, or equal (totally con-verted) to, the potential energy during the post-fault period. The equality of bothenergies takes place in the critical case.

Example 9.4 Determine the system stability for 15-bus, 4-generator test systemshown in Figure 9.11 with the data given in Appendix IV. A three-phase short circuitoccurs at bus no. 15. At fault clearance, the generator G2 is assumed to be dis-connected and bus no. 15 is completely isolated from the rest of the system. The faultis cleared at dc1 ¼ 55�, dc3 ¼ 57� and dc4 ¼ 60� for G1, G3 and G4, respectively. Theinput mechanical power for each generator equals its rated active power.

Solution:

The bus voltage distribution, pre-, during and post-fault, is calculated by the loadflow technique as explained in Chapter 5. The results are summarised in Table 9.1.Accordingly, the power delivered from each generator at states pre-, during andpost-fault are calculated using (9.28) and are summarised in Table 9.2.

To judge the stability for each generator, the transient kinetic energy, KE,represented by the area under Pm line and the potential energy, PE, represented bythe area above Pm line must be calculated. The machine is stable when the differ-ence (KE � PE) is a negative value, i.e. KE is fully converted into potential energy.

The intersection of Pm line with power–angle curve before the fault and after thefault gives the values of do and dmax, respectively. The parameters required tocalculate KE and PE are summarised in Table 9.3.

The KE is given by

KE ¼ Pm dc � doð Þ �ðdc

do

L1D sin d� L2D cos dð Þ

¼ Pm dc � doð Þ � L1D cos do � cos dcð Þ � L2D sin do � sin dcð Þ ð9:29Þand the potential energy is

PE ¼ðdmax

dc

L1A sin d� L2A cos dð Þdd� Pm dmax � dcð Þ

¼ L1A cos dc � cos dmaxð Þ þ L2A sin dmax � sin dcð Þ � Pm dmax � dcð Þð9:30Þ


By the data summarised in Tables 9.2 and 9.3 and using (9.29) and (9.30) theresults of KE, PE and the difference between them are summarised in Table 9.4.It is seen that the difference ‘KE � PE’ is negative, which means that KE is fullyconverted into PE and all generators are stable.

Equating KE and PE the critical clearing angle ‘dcr’ can be obtained from(9.29) and (9.30) as summarised in Table 9.4. The corresponding power–anglecurves for generators G1, G3 and G4 are shown in Figure 9.12.

SVC1

SVC2SVC3

G1

G2

G3

G4

3

9

13

12

15

14

6

10

2

8

4

7

1

5

11

Figure 9.11 Test system for Example 9.4


Table 9.1 Bus voltage distribution of the test system pre-, during and post-faultoccurrence

Fault period Bus code Voltage Bus code Voltage

mag. (pu) angle (�) mag. (pu) angle (�)

Before the fault 1 1.0065 �2.4 9 1.0059 4.62* 1.0000 0.0 10 1.0009 �7.13 1.0082 8.3 11 0.9971 �7.64 1.0068 �8.8 12 1.0062 �1.45 1.0030 �11.7 13 1.0037 3.46 1.0066 �7.6 14 1.0101 1.07 1.0020 �6.5 15 1.0072 4.68 1.0066 �2.8

During the fault 1 0.6913 1.05 9 0.3414 7.52 0.6521 5.08 10 0.5534 �4.73 0.5025 13.5 11 0.5649 �5.04 0.4981 �7.05 12 0.3650 �0.75 0.4849 �10.59 13 0.2928 5.46 0.4775 �6.16 14 0.1897 0.87 0.5792 �4.00 15 0.0000 0.08 0.5108 0.05

After the fault 1 0.9844 �10.5 9 0.9797 �5.62* 1.0000 0.0 10 0.9792 �15.43 0.9818 �1.7 11 0.9740 �15.94 0.9842 �16.7 12 0.9814 �11.05 0.9767 �21.0 13 0.9776 �6.86 0.9876 �15.8 14 0.9961 �11.47 0.9797 �14.8 15 Isolated8 0.9943 �7.9

*Bus no. 2 is the slack bus.

Table 9.2 Electrical output power in terms of rotor angle for each generator inoperation, pre-, during and post-fault period

Generator Pe (pre) Pe (during) Pe (post)

G1 1.0615 sin dþ 0.12 cos d 0.756 sin dþ 0.05 cos d 1.009 sin dþ 0.26 cos dG3 0.906 sin d� 0.07 cos d 0.432 sin d� 0.06 cos d 0.880 sin dþ 0.08 cos dG4 0.956 sin dþ 0.05 cos d 0.840 sin d� 0.02 cos d 0.936 sin dþ 0.13 cos d

Table 9.3 Parameters for calculating KE and PE

Generator do dc dmax Pm (pu)

G1 46� 55� 111.5� 0.85G3 53� 57� 125.0� 0.68G4 45.7� 60� 122.5� 0.80


9.3.2 TEF method

9.3.2.1 Formulation of centre of inertiaThe equation of motion of the ith generator in a power system is

Mi€di þ Di

_di ¼ Pmi � Pei ð9:31Þwhere

Mi ≜ inertia constant of ith generator ¼ 2Hi/wr_di ¼ wi � wr; i ¼ 1; 2; . . . ; ng

wi ≜ generator speedwr ≜ reference speed

Pei ¼ E2i Gii þ

Xng

j¼16¼i

EiEjYij cos qij � di þ dj

� � ð9:32Þ

Yijqij ¼ Gij þ jBij ¼ transfer admittance between nodes i and j

It is to be noted that the network includes only generator nodes; all other nodes areeliminated and loads are represented by constant impedance, i.e. included intransfer conductances.

The generator angle and speed in (9.31) are given with respect to a synchro-nous frame of reference. They can be referred to as centre of inertia (COI) coor-dinates that are defined by satisfying

do ¼ 1MT

Xng

i¼1

Midi and _do ¼ 1MT

Xng

i¼1

Mi_di ð9:33Þ

Table 9.4 Transient kinetic energy, potential energy, difference and dcr

Generator KE PE KE � PE dcr

G1 0.0369 0.0819 �0.0450 60.5G3 0.0247 0.1790 �0.1543 62.1G4 0.0360 0.1016 �0.0656 64.4

PE

90 1800–0.3

0

0.45

Pin

1.2

G1

Pe

δc δmδo

δ

PE

90 1800–0.3

0

0.45

Pin

1.2

G3

Pe

δc δm

KE

δo

δ

PE

90 1800–0.3

0

0.45

Pin

1.2

G4

Pe

δc δm

KE

δo

δ

KE

Figure 9.12 Power–angle curves at pre-, during and post-fault occurrence


where

MT ≜Xng

i¼1

Mi

The equation of motion of COI is given by

MT p þ Dð Þ _do ¼Xng

i¼1

Pmi �Xng

i¼1

Pei ≜ PCOI ð9:34Þ

where

Xng

i¼1

Pei ¼Xnb

j¼1

Plj þ Ploss

For lossless network and constant active power loads, PCOI is constant as themechanical input power Pmi is considered as constant. Thus, (9.34) becomes

MT p þ Dð Þ _do ¼ PCOI ð9:35Þ

where

PCOI ¼Xng

i¼1

Pmi �Xnb

j¼1

Plj and p ¼ d=dt

Relative to COI, all generators have phase angles:

qi ¼ di � do ð9:36Þ

Hence, (9.31) incorporated with (9.35) and (9.36) gives

Mi p þ Dið Þ _qi þ Mi p þ Dið Þ _do ¼ Pmi � Pei

i.e.

Mi p þ Dið Þ _qi ¼ Pmi � Pei � Mi

MTPCOI ð9:37Þ

It is noted that the variables of COI satisfy the constraints

Xng

i¼1

Mi_qi ¼

Xng

i¼1

Mi_di � _do

¼

Xng

i¼1

Mi_di � MT

_do ¼ 0


9.3.2.2 Derivation of the TEFThe dynamics of the post-disturbance system are represented by (9.37). The energyfunction for this system can be derived as below:

(i) Neglecting damping and multiplying (9.37) by _qi; get the sum for all gen-erators in the system as

Xng

i¼1

Mi€qi � Pmi þ Pei þ Mi

MTPCOI

� �_qi ð9:38Þ

(ii) Equation (9.32) can be rewritten as

Pei ¼ E2i Gii þ

Xng

j¼16¼i

EiEj Bij sin di � dj

� �þ Gij cos di � dj

� ��

Pei ¼ E2i Gii þ

Xng

j¼16¼i

Bij sin di � dj

� �þ Gij cos di � dj

� ��

where

Bij ¼ EiEjBij and Gij ¼ EiEjGij

(iii) Substitute Pei to get the sum in the form

Xng

i¼1

Mi€qi � Pi þ

Xng

j¼16¼i

Bij sin qij þ Gij cos qij

� �26664

37775 _qi ð9:39Þ

where

Pi ¼ Pmi � E2i Gij; qij ¼ qi � qj ¼ di � dj

(iv) In the expression (9.39), as Bij ¼ Bji and Gij ¼ Gji, it is seen that

Xng

i¼1

Xng

j¼16¼i

Bij sin qij_qi ¼

Xng�1

i¼1

Xng

j¼iþ1

Bij sin qij_qij

Xng

i¼1

Xng

j¼16¼i

Gij cos qij_qi ¼

Xng�1

i¼1

Xng

j¼iþ1

Gij cos qij_qij

9>>>>>>>>>=>>>>>>>>>;

ð9:40Þ


(v) Substituting expression (9.40) into (9.39) and integrating the resultingexpression with respect to time, from t ¼ ts at which _q(ts) ¼ 0 and q(ts) ¼ qs,the energy function V describing the total system transient energy for thepost-disturbance system is given by

V ¼ 12

Xng

i¼1

Mi_q2

i �Xng

i¼1

Pi qi � qisð Þ

�Xng�1

i¼1

Xng

j¼iþ1

Bij cos qij � cos qsij

�

ðqiþqj

qsiþqs

j

Gij cos qij d qi þ qj

� �264

375 ð9:41Þ

where

qis ¼ the angle of bus i at the post-disturbance SEP.

The TEF (9.41) consists of the following four terms:

(i) ½Xng

i¼1

Mi_q2

i ¼ ½Xng

i¼1

Mi_di � _do

2

¼ ½Xng

i¼1

Mi_d

2i �

Xng

i¼1

Mi_di_do þ ½

Xng

i¼1

Mi_d

2o

¼ ½Xng

i¼1

Mi_d

2i � MT

_do

_do þ ½ _d

2o

Xng

i¼1

Mi

¼ ½Xng

i¼1

Mi_d

2i � MT

_d2o þ ½ MT

_d2o

¼ ½Xng

i¼1

Mi_d

2i � ½ MT

_d2o

¼ total change in KE of all rotors in the COI frame of reference

This change equals the change in KE of all generator rotors minus thechange in PE associated with the COI.

(ii)Xng

i¼1

Pi qi � qisð Þ ¼Xng

i¼1

Pi di � disð Þ � do � dosð ÞXng

i¼1

Pi

¼ change in PE of all rotors relative to the COI

This change equals the change in PE of all generator rotors minus the changein PE associated with the COI.

(iii)Xng�1

i¼1

Xng

j¼iþ1

Bij cos qij � cos qijs

� �¼ the change in stored magnetic energy of all

branches. It is independent of the path of integration


(iv)Xng�1

i¼1

Xng

j¼iþ1

ðqiþqj

qsiþqs

j

Gij cos qij d qiþqj

� �¼ the change in dissipated energy

of all branches. It depends on the path of qi.

The first term is called the kinetic energy, Gke, and is a function of only thegenerator speeds. The sum of the second, third and fourth terms is called thepotential energy ‘Gpe’ and is a function of only the generator angles.

Therefore, in a multi-machine power system, the energy function V describingthe total system transient energy for the post-disturbance system is given by

V ¼ Gke � Gpe ð9:42Þ

where

Gke ¼ ½Xng

i¼1

Mi_q2

i

Gpe ¼Xng

i¼1

Pi qi � qisð Þ

þXng�1

i¼1

Xng

j¼iþ1

Bij cos qij � cos qijs

� �� ðqiþqj

qsiþqs

j

Gij cos qijd qi þ qj

� �264

375

To assess the system stability, both the critical energy function Vcr and systemenergy at the instant of fault clearing Vc are calculated. The difference, DV ¼Vcr � Vc,is defined as stability index or stability margin, which is positive when the systemis stable. Otherwise, the system is unstable.

To calculate Vc: The angles and speeds of all generators in the system at theinstant of fault clearing are required. They can be obtained by running up thesimulation in time-domain. Vcr is defined as the potential energy at the controllingunstable equilibrium point ‘UEP’ for a particular disturbance under study.

The integral term in (9.41) representing the dissipated energy is difficult to beevaluated as the system trajectory is unknown. So, a linear angle trajectory isassumed. It has been found that this assumption is acceptable for a first swingtransient [5, 6]. It can be derived as below.

Assume qi(t) and qj(t) are the angular trajectories of machines i and j withrespect to time, respectively. qic and qiu denote the angles of ith generator at theclearance and UEP state. Thus, qic and qiu represent the initial and final vectors ofangular positions of the ng generators. The linear angle trajectory between theinitial state ‘at t ¼ 0, qi ¼ qic’ and the final state ‘at t ¼ 1, qi ¼ qiu’ is expressed by

qi ¼ qic þ qiu � qicð Þt 0 � t � 1 i ¼ 1; 2; . . .; n ð9:43Þ


Differentiating (9.43)

dqi ¼ qiu � qicð Þdtdqj ¼ qju � qjc

� �dt

ð9:44Þ

Add the two equations to give

d qi þ qj

� � ¼ qiu � qic þ qju � qjc

� �dt ð9:45Þ

Subtract dqj from dqi to give

d qi � qj

� � ¼ dqij ¼ qiu � qic � qju þ qjc

� �dt ð9:46Þ

Using (9.45) and (9.46) eliminates dt as below:

d qi þ qj

� � ¼ qiu � qic þ qju � qjc

qiju � qijcdqij ð9:47Þ

Thus, substituting d qiþqj

� �from (9.47) into the expression representing the

dissipated energy,Ð qiþqj

qsiþqs

jGij cos qijd qiþqj

� �gives an expression that can be inte-

grated with respect to qij between any two points as

Iij ¼ Gijqiu � qic þ qju � qja

qiju � qijcsin qiju � sin qijc

� � ð9:48Þ

Therefore, by using the dissipated energy expressed by (9.48) between theconditions at the clearance of the disturbance and the controlling UEP, and thensubstituting for Vcr and Vc from (9.41) the stability index is given by

DV ¼ � 12

Xng

i¼1

Mi_q2

ic �Xng

i¼1

Pi qiu � qicð Þ

�Xng�1

i¼1

Xng

j¼iþ1

Bij cos qiju � cos qijc

� �� Gijqiu � qic þ qju � qjc


� ��

ð9:49Þwhere

qc; _qc

� �≜ the conditions at the clearance of disturbance

qu; 0ð Þ ≜ the conditions at controlling UEP

The main steps of transient stability assessment using TEF method for multi-machine power system are summarised in the flowchart shown in Figure 9.13.

9.3.2.3 Calculation of critical energyThe critical energy, Vcr, represents the boundary of stability region. It is the mostdifficult step to calculate Vcr when using the TEF method for stability assessment.


The calculation depends mainly on computing the UEP that may be made by one ofthe following approaches.

The closest UEP approachAt different initial values of bus angles, the steady state equations of the post-disturbance system are solved to determine all unstable equilibrium points (UEPs).They can be obtained by computing the set of generator’s angles that satisfy:

fi ¼ Pmi � Pei � Mi

MTPCOI ¼ 0 i ¼ 1; 2; . . .; ng ð9:50Þ

For a multi-machine power system with ng generators, there are 2ng�1 solu-tions. Each solution gives a value of potential energy. The chosen UEP is the onethat results in the minimum potential energy. It is noted that the results are to someextent pessimistic and usually of little practical value as this approach implies theassumption of worst fault location. In addition, it is found that the trajectory ofseverely disturbed generators passes close to a UEP different from that having theminimum potential energy. This can mostly be avoided by applying the approachdescribed next.

The controlling UEP approachThe system trajectories for all critically stable cases get close to UEPs (calledcontrolling UEPs) that are closely related to the boundary of system separation.This approach is based on using the disturbed trajectory to determine its intersec-tion with the post-disturbance principle singular surface, qss. At this intersection, adirection vector h is formed and along this direction a one-dimensional minimisa-tion problem can be solved to minimise

F qð Þ ¼Xng

i¼1

f 2i qð Þ ¼

Xng

i¼1

Pmi � Pei � Mi

MTPCOI

� �2

ð9:51Þ

and obtain qu that is considered as a starting point to apply a suitable numericaltechnique to achieve the controlling UEP. It is noticed that there are two aspectsto determining the controlling UEP: (i) the effect of the different generators and (ii) theeffect of the post-disturbance network, in particular, its energy-absorbing capacity.

These two aspects must be considered when determining the controlling UEPas the more severely disturbed generators may or may not lose synchronism withthe rest of the system. It depends on whether the potential energy-absorbingcapacity of the network is relevant to convert the kinetic energy at clearing thedisturbance into potential energy.

The boundary of stability region based on controlling unstableequilibrium point (BCU) approachIf the starting point, qu, for the UEP is not sufficiently close to the exact UEP theconvergence problem may take place, in particular, when the system is highly


stressed or highly unstressed. In this case BCU approach can be used to overcomesome of these problems. The algorithm for determining the stability boundary canbe found in Chapter 3 of Reference 3. Moreover, the BCU approach is based on therelationship between the boundary of stability region of a power system and that ofa reduced system [7, 8]. Some of the other work is concerned with determining theUEP for detailed generator models rather than the classic model [9].

At the desired calculated controlling UEP, the change in potential energy,DVPE, can be obtained. It is preferred to be normalised with respect to the kineticenergy at the end of disturbance, DVPEn, to give a reliable indication of the degreeof stress on different generators.

If the disturbance is large enough, the post-disturbance trajectory approachesthe controlling UEP that has in this case the lowest normalised potential energyindex at the instant of clearing the disturbance. Thus,

DVPEn ¼ DVPE

VKEð9:52Þ

DVPE ¼ VPEu � VPEc

¼ �Xng

i¼1

Pi qiu � qicð Þ �Xng�1

i¼1

Xng

j¼iþ1

hBijðcos qiju � cos qijcÞ

�Gijqiu � qic þ qju � qjc


� �i ð9:53Þ

Based on the former explanation, the computational steps of stability analysisusing the TEF for a multi-machine power system can be outlined as describedbelow:

● Collect the input data, applying steady-state power flow analysis, the syn-chronous machine parameters and specification of disturbance. The initialvalues of machine internal bus voltages and rotor angles are calculated. Themachines are represented by classical model.

● According to the specification of the disturbance and system topology, thesystem admittance matrix and its reduced form are built.

● The conditions at disturbance clearance are determined, and then the systemadmittance matrix and its reduced form at the end of disturbance are computed.

● The relevant mode of disturbance is determined by identifying the mostaffected generators.

● The UEP is calculated.● The total energy at clearing time and the critical transient energy as well as the

stability index can be computed to decide whether the system is stable.

The flowchart in Figure 9.13 presents the main steps outlined above.


Start

Input System data

Pre-disturbance power flow solution

Specify the disturbance

– Compute during fault admittance matrix– Eliminate the fixed buses and compute

the reduced Y matrix

– Compute post-fault admittance matrix– Eliminate the fixed buses and compute

the reduced Y matrix

Compute post-fault SEP

Calculate total energy at clearing time VcCalculate critical energy Vcr

Compute stability indexΔV = Vcr – Vc

IsΔV > 0

Stable Unstable

End

– Identify the mode of disturbance– Compute the UEP

Figure 9.13 Flowchart of transient stability assessment using TEF


References

1. Momoh J.A., El-Hawary M.E. Electric Systems, Dynamics, and Stability withArtificial Intelligence Applications. New York, NY, US: Marcel Dekker; 2000

2. Fouad A.A., Vittal V. Power System Transient Stability Analysis Using theTransient Energy Function Method. Upper Saddle River, NJ, US: PrenticeHall; 1992

3. Chiang H.D. Direct Methods for Stability Analysis of Electric Power Systems.Hoboken, NJ, US: John Wiley & Sons; 2011

4. Sallam A.A. ‘Power systems transient stability assessment using catastrophetheory’. IEE Proceedings. 1989;136(2) Pt C:108–14

5. Uyemura, K., Matsuki J., Yamada J., Tsuji T. ‘Approximation of an energyfunction in transient stability analysis of power systems’. Electrical Engi-neering in Japan. 1972;92(4):96–100

6. Athay, T., Sherkat V.R., Podmore R., Virmani S., Puech C. ‘Transient energystability analysis’. System engineering for power. Emergency operating statecontrol-Section IV. U.S. Department of Energy Publication No. CONF-790904-PL, 1979

7. Chiang H.D., Wu F.F., Varaiya P.P. ‘A BCU method for direct analysis ofpower system transient stability’. IEEE Transactions on Power Systems.1994;9(3):1194–208

8. Chu C.C., Chiang H.D. (eds.). ‘Boundary properties of the BCU methodfor power system transient stability assessment’. International Symposiumon Circuits and Systems ISCAS 2010, IEEE; Paris, France, May/Jun 2010.pp. 3453–6

9. Chen L., Min Y., Xu F., Wang K.P. ‘A continuation-based method to computethe relevant unstable equilibrium points for power system transient stabilityanalysis’. IEEE Transactions on Power Systems. 2009;24(1):165–72


Part IV

Stability enhancement and control

Chapter 10

Artificial intelligence techniques

Traditional analytic and time analysis approaches may not easily handle onlinereal-time applications for large systems due to computational time requirements. Inparticular, the power systems being non-linear and time varying, application oftraditional approaches to a power system for the purpose of identifying its para-meters, controlling the operation to maintain stability and damping oscillationsfollowing disturbances is not suitable for online monitoring. They are moresuitable for offline design and investigations.

Advent of artificial intelligence (AI) techniques based on logic mathematicshas encouraged power system engineers, planners and designers to employthese techniques with the goal of reducing computation time and designingfast algorithms that are adequate for power system online applications. Many AIand computational intelligence techniques, such as artificial neural network(ANN), fuzzy logic (FL), neuro-FL (NFL), particle swarm optimisation (PSO),genetic algorithms, exist. The basics of ANN, FL and NFL as well as theadaptive neuro-fuzzy control (ANFC) are presented in this chapter as theyare used, in addition to the time analysis techniques, for some applications(e.g. power system stabilisers and static var compensators) to power systems inthe subsequent chapters.

10.1 Artificial neural networks

ANNs are biologically inspired computational models that consist of processingelements (called neurons) interconnected together to constitute the network struc-ture. They are essentially non-linear function approximations that utilise processinputs to estimate process outputs. An important feature of the ANNs is the abilityto adjust their connections through an adaptive learning process called Learning.Learning can be accomplished using a series of examples and patterns. Informationobtained through learning is retained and represented by a set of connectionweights within the neural network structure [1, 2].

A simple neuron model consists of two main parts: a linear combiner and anonlinear activation function. Typically, the neuron has more than one input andcan be mathematically modelled as shown in Figure 10.1.

The input signals, x1, x2, . . . , xn, are multiplied by weights w1, w2, . . . , wn andadded together to produce the net input to the activation function. The output signalof the neuron, y, can be expressed as

y ¼ fXn

k¼1

wkxk þ b

!ð10:1Þ

It is worth mentioning that the weights are the most important coefficientsin determining the output of the neural network. They are used to adjust therelative importance of the connections between the neurons, according to amodification rule.

It can be noted from (10.1) that the effect of the bias, b, is to increase ordecrease the input to the activation function. The activation function is utilised totransform the activity level of the neuron into the output signal. Many activationfunctions such as a hard-limit, sigmoid, Gaussian and hyperbolic tangent functionshave been used successfully to build neural networks [1–4]. The choice of theactivation function relies on the applications where the neural network is used. Themost common activation functions used in multi-layer networks are the sigmoidand hyperbolic tangent functions.

The outputs of sigmoid and hyperbolic tangent functions are described using(10.2) and (10.3), respectively.

f xð Þ ¼ 11 þ e�x

ð10:2Þ

f xð Þ ¼ ex � e�x

ex þ e�xð10:3Þ

Inputsignals Output

signals

Activationfunction

f (f)

w1

w2

wn

x1

x2

xnb

Bias

∑....

.

.

.

.

y

Figure 10.1 Simple model of an artificial neuron


10.2 Neural network topologies

The neurons themselves are not very powerful in terms of computation or repre-sentation. However, their interconnection allows one to encode relations betweenthe variables and gives powerful processing capabilities. The way the neurons areconnected within the neural network and the type of activation function used toconstruct the network yield to different network architectures. In general, threedifferent types of network architectures can be identified.

10.2.1 Single-layer feed-forward architectureA feed-forward network has a layered structure. A single-layer network (Figure 10.2)consists of multi-input and multi-output signals. The input signals are connected toeach of the neurons in the network. The sum of the products of the weights andthe inputs is calculated in each node. The input layer is not accounted for as nocomputation has taken place there.

Regardless of how many neurons the network has or what kind of activationfunction is chosen, the limitation of this type of network can only approximate alinear function. The common approach of approximating a non-linear function canbe obtained by using a multi-layer perceptron.

10.2.2 Multi-layer feed-forward architectureIn this type of network, two or more single-layer networks are connected togetherto form one network. Each layer consists of neurons that receive their inputs fromthe neurons located in the layer directly before them and send their outputs to theneurons located in the subsequent layer. The layer whose output is the networkoutput is called an output layer, while other layers are called hidden layers.

Input layer Output layer

Figure 10.2 Single-layer feed-forward network

Artificial intelligence techniques 255

A multi-layer perceptron network ‘often known as an MLP network’ with onehidden and one output layer is shown in Figure 10.3.

10.2.3 Recurrent networksA recurrent neural network (RNN) is another class of neural network that containsfeedback connections between the outputs and inputs of the network. Using thiskind of network structure will allow signal flow in both forward and backwarddirections, providing the network with a dynamic memory and is useful to mimicdynamic systems [5]. Compared with the MLP, RNN is more difficult to train dueto the feedback connections. Configuration of an RNN is shown in Figure 10.4.

10.2.4 Back-propagation learning algorithmAs previously mentioned, by adjusting the weights of the neural network, the out-put of the network will be altered. The weight modifications can be achieved byapplying a suitable learning algorithm, which leads the network to converge to the

Input layer Output layerHidden layer

Figure 10.3 Multi-layer perceptron network

Input layer Output layer

D

D

D

Figure 10.4 Recurrent neural network


desired value. Back-propagation is a learning algorithm, which has been widelyutilised to train the multi-layer neural networks [6, 7]. The learning technique isbased on the gradient method, which, using the derivatives of error, minimises theerror between the actual and the desired outputs. By passing the derivative of theerror from the output layer of the network back towards the input layer, the weightsof the network can be adjusted in a suitable manner.

The error function E for a given training data set p can be described in the formof a well-known error function called the sum of squared errors:

E ¼ 12

Xp

Xk

dpk � ypk

� �2

!ð10:4Þ

where dpk is the desired output at instant time k for pattern p and ypk is the actualoutput at instant time k for pattern p. The objective is to reduce the error function Eto zero so that the output of the network is equal to the desired value. It is assumedthat by minimising the error of each pattern individually, E will also be minimised.Therefore, notation p can be neglected assuming there is only one pattern to beconsidered.

A three-layer network (Figure 10.5) is considered as an example in order toexplain the back-propagation algorithm. The output of neuron j ‘located in thehidden layer’ is given as

Oj ¼ j netj

� � ð10:5Þ

netj netk

OiOj

Ok

Wji

Wkj

Xi

i j k

Figure 10.5 Three-layer neural network


where j represents the activation function used for the output of neuron j. netj iscalled the local field and it is given by

netj ¼X

i

Wji Oi ð10:6Þ

where Wji and Oi are the weight and the input signal to neuron j, respectively.Similarly, the output of neuron k, which represents the output of the neural

network, can be expressed as

Ok ¼ j netkð Þ ð10:7ÞThe local field netk is given by

netk ¼X

j

Wkj Oj ð10:8Þ

where Wkj is the weight related to neuron k and Oj is the output of neuron j.The network (Figure 10.5) is capable of calculating the total error E for a given

training set. Typically, the weights are the only parameters of the network that canbe iteratively modified to make the error function as low as possible. The relationbetween the error function E and the weights of the network can be defined as aquadratic function as illustrated in Figure 10.6. If the slope is positive, the weightsshould be decreased by a small amount to lower the error. On the contrary, if theslope is negative, the weights of the network should be increased.

By applying the chain rule method, the partial derivative of the error functionE with regard to the weights Wkj, leading to an output unit change, can be calcu-lated as

@E

@Wkj¼ @E

@Ok

@Ok

@netk

@netk

@Wkjð10:9Þ

Erro

r

Weight

Figure 10.6 Error versus weights of network


The weights can be updated using the gradient descent method as follows:

Wkj n þ 1ð Þ ¼ Wkj nð Þ þ h@E

@Wkjð10:10Þ

where h is the learning rate of the back-propagation algorithm.It can be noted from Figure 10.5 that changing Wkj will only affect the output

of neuron k, while changing Wji will affect the output of neurons j and k. Therefore,the change in E with regard to Wji can be written as the sum of the changes to eachof the output units. The adaptation of weights between hidden j, and input i, layerscan be expressed as follows:

@E

@Wji¼ @E

@Oj

@Oj

@netj

@netj

@Wjið10:11Þ

The term @E@Oj

can be calculated as

@E

@Oj¼ @E

@Ok

@Ok

@netk

@netk

@Ojð10:12Þ

Hence, the adaptation of the hidden layer weights can be written as

@E

@Wji¼X

k

@E

@Ok

@Ok

@netk

@netk

@Oj

@Oj

@netj

@netj

@Wjið10:13Þ

Wji n þ 1ð Þ ¼ Wji nð Þ þ h@E

@Wjið10:14Þ

The back-propagation algorithm requires a large number of training examplesin order to provide an acceptable level of accuracy. It is also important to care-fully select the learning rate to ensure the convergence of the network, as a largevalue of h might lead to network instability and a small value will cause a veryslow convergence.

10.3 Fuzzy logic systems

In the real world, there are a lot of imprecise conditions that defy a simple, true orfalse statement as a description of their state. A computer system and its binarylogic are incapable of adequately representing these vague (yet understandable)states and conditions. FL, which was developed in the mid-1960s by L.A. Zadeh, isa branch of mathematics that deals with vague and linguistic representations of datathat mimic human understanding or intuition [8]. It expands the reach of traditionalbinary logic by allowing for the use of analog values as inputs and outputs in logiccalculations. FL was developed based on the concept of Fuzzy Set Theory. It isconsidered a valuable tool, which can be used to solve highly complex problemswhere a mathematical model is too difficult or impossible to create.


The applications of FL can be found in many engineering and scientific works.FL has been successfully used in numerous applications, such as control systemsengineering, image processing, power system engineering, industrial automation,robotics, consumer electronics, optimisation, medical diagnosis and treatmentplans, as well as stock trading [9]. Regarding the power system, FL has also beenuseful in the application of parameter identification. A fuzzy identifier has beenused to track the parameters of the power system and update the adaptive con-troller. Based on the knowledge of the plant, the input signals to the fuzzy systemare fuzzified: a rule table is constructed and the output signals are finally defuzzi-fied. Parameters of the fuzzy identifier are updated in real time by minimising adefined cost function using the gradient descent method. More description of fuzzytheory is presented in Section 10.3.1.

10.3.1 Fuzzy set theoryFuzzy set can be defined by changing the usual definition of the characteristicfunction of a crisp set and introducing degree of membership. A fuzzy set A in areference set X (called the universe of discourse) is defined by a mapping function(called the Membership Function, MF) that takes values in the range between 0 and 1,which can be mathematically written as mA: X ? [0, 1]. The MF is a curve thatdefines how each point in the input space is mapped to a membership valuebetween 0 and 1. The higher the membership X has in the fuzzy set A, the truer thatX is A [10]. Many MFs, such as triangular, trapezoidal, bell and Gaussian, are usedin FL; however, triangular and trapezoidal are the most common MFs. Unfortu-nately, there are no general rules or guidelines for selecting the appropriate shape ofthe MFs. The fact that trapezoidal and triangular shapes are the MFs most used inthe literature is because they produce good results for most input variables in var-ious applications.

FL has operators defined in a similar way to the classical Boolean logic. TheAND operator can be evaluated, e.g. using min while the max operator representsthe OR operator and the NOT is replaced by 1 � A in FL. Let X be a fuzzy set, and Aand B two fuzzy sets with the MFs mA(x) and mB(x), respectively. Then the union,intersection and complement of fuzzy sets can be respectively defined as

mA [ mB xð Þ ¼ max mA xð Þ; mB xð Þð Þ ð10:15ÞmA \ mB xð Þ ¼ min mA xð Þ; mB xð Þð Þ ð10:16Þ

�mA xð Þ ¼ 1 � mA xð Þ ð10:17ÞSome other definitions are also available in the literature. For instance, the

intersection operator ‘also known as the T-norm operator’ could also be describedas the algebraic product of two fuzzy sets:

mA \ mB xð Þ ¼ mA xð Þ � mB xð Þ ð10:18ÞThe choice of the fuzzy operator ‘in the end’ depends on the expert knowledge

and implementation feasibility.


10.3.2 Linguistic variablesAs FL deals with events and situations with subjectively defined attributes, a propo-sition in FL does not have to be either true or false. For example, a room temperaturecan be described as cold, cool, comfortable, warm or hot, as opposed to only cold orhot. The descriptions mentioned previously are known as the linguistic variables in FLterminology. The range of possible values of a linguistic variable is called the universeof discourse. In the case of the temperature example, the universe of discourse can bewithin the interval [10�C, 35�C]. However, for simplicity, a common practice is tonormalise or scale the values to be in the range of [�1, þ1] [9].

10.3.3 Fuzzy IF–THEN rulesA single fuzzy IF–THEN statement can be explained as follows:

IFðx is AÞ ! THENðy is BÞwhere x and y are the input and output variables, respectively. A and B are thelinguistic values defined by fuzzy sets on the ranges x and y. The IF part is calledthe antecedent while THEN part is called the consequent. The IF–THEN rule can beinterpreted in such a way that if the antecedent is a fuzzy statement that is true tosome degree of membership, then the consequent is also true to that same degree.

10.3.4 Structure of an FL systemThe basic structure of an FL system is illustrated in Figure 10.7. It can be seen thatin order to design an FL system, four steps are required to be considered:

(1) FuzzificationFuzzification is the process of mapping the input data into corresponding universesof discourses and converting the input into suitable linguistic values. The task ofthe fuzzification process can be summarised as follows:

● measure the values of input variables● map the values of input variables to a corresponding universe of discourse● convert the input data into appropriate linguist values

Knowledge

DefuzzificationFuzzification

Inference

Input Output

Figure 10.7 Structure of a fuzzy system


(2) Knowledge baseThe purpose of the knowledge base or rule base step is to provide definitions thatexpress the relation between the input and output fuzzy variables. It defines thecontrol goals by means of a set of linguistic control rules. The rule base is oftenexpressed in the form of IF–THEN rules.

(3) Fuzzy inferenceThe fuzzy inference is considered the core of any FL system. The fuzzy inferencemechanism is a process by which the input values for each of the fuzzy variablesin the antecedent are matched with all rules in the fuzzy rule base and an inferredfuzzy set is obtained. The membership values obtained in the fuzzification stepare combined through a specific fuzzy operator to obtain the firing strength ofeach rule. Based on the firing strength, the consequent part of each qualified ruleis produced.

There are two methods that are used in fuzzy inference known as Mamdani andSugeno inference systems [9]. The difference between the two methods resides inthe consequent part. Mamdani fuzzy inference expects the output MF to be fuzzysets while Sugeno fuzzy inference method treats the consequent parts as eitherlinear polynomials or constants in the form of single spikes. The output of each ruleis weighted by the firing strength of the rule and the final output is the weightedaverage of all rule outputs.

(4) DefuzzificationThe process by which a non-fuzzy (crisp) output is obtained from the fuzzy set iscalled defuzzification. Several defuzzification methods – centre of area (COA),mean of maximum (MOM), smallest of maximum (SOM) and largest of maximum(LOM) – are used in the defuzzification process of a fuzzy system. The twoapproaches most commonly used are the COA and MOM methods. The COA(also known as the centre of gravity, COG) calculates the centre of gravity of thedistribution of the membership degrees under the curve. This method can beexpressed in the discrete form as

COA ¼Pn

k¼1 xmA xð ÞPnk¼1 mA xð Þ ð10:19Þ

where mA(x) is the MF of a fuzzy set A defined in the universe x and n is the numberof quantisation levels of the output.

The MOM method calculates the output value by averaging only the part of theinferred fuzzy set whose MFs reach the maximum. The output of using this methodcan be described in the discrete form as

MOM ¼Xl

i¼1

xi

lð10:20Þ

where l is the number of elements, xi, with membership equal to the maximumvalue. Figure 10.8 shows different types of defuzzification methods for a fuzzyfunction.


10.4 Neuro-fuzzy systems

The neuro-fuzzy system is an AI approach, resulting from the merging of an FLsystem and a neural network structure. The basic idea of the integrated system(called neuro-fuzzy system) is to model an FL system by a neural network andapply the learning algorithms developed in the field of neural networks to adapt theparameters of the fuzzy system. The motive for combining FL with neutral net-works is to take advantage of their strengths and overcome the shortcomings. Infact, FL and neural networks can be treated as complementary technologies. In aneuro-fuzzy system, the fuzzy system can be provided by an automatic tuningmechanism without altering its functionality. It has the advantage of tuning therules of the fuzzy system using learning algorithms applied to neural networks. Inreturn, the neural network can improve the transparency by having the rule-basedfuzzy reasoning considered in its construction [11].

Many neuro-fuzzy network structures have been presented in the literature.Some of these networks are Fuzzy Adaptive Learning Control Network [12],Adaptive Neuro-Fuzzy Inference System (ANFIS) [13], Fuzzy Net (FUN) [14] andothers [15]. However, one of the most well-known networks that has been reportedin many publications and applied to various applications is the ANFIS.

Neuro-fuzzy control (NFC) has been widely used in many control systemapplications [16–20]. It represents a control approach where FL and ANNs arecombined. An NFC can be defined as a multi-layer network that has the elementsand functions of typical FL control systems, with additional capability to adjust itsparameters through learning techniques [21].

A background about NFC and the online adaptation technique used to adjustthe parameters of the NFC controller are described in Section 10.4.1.

10.4.1 Adaptive neuro-fuzzy inference systemANFIS was first introduced by Takagi and Sugeno in 1985 and further developedby Jang [12]. The network is built to have the capability of ANNs in adaptingand learning, together with the merit of approximate reasoning offered by FL.

μ

A

x

Smallest of maxMean of max Centre of area

Largest of max

Figure 10.8 Defuzzification methods for a fuzzy function


Unlike neural networks, the weights of the connections between nodes located inone layer and the nodes in the subsequent layer are constant and have values of one.

There are two main ANFIS structures known as the first-order or zero-orderSugeno models. A typical rule in the first-order Sugeno model has the form of

IF input 1 ¼ x1 and input 2 ¼ x2 ! THEN output is y ¼ ax1 þ bx2 þ c

where {a,b,c} is a parameter set.For the zero-order Sugeno model, the output y is considered a constant and

does not depend on the inputs to the network. The fuzzy IF–THEN rule for this typecan be written as

IF input 1 ¼ x1 and input 2 ¼ x2 ! THEN output is y ¼ c

where a ¼ b ¼ 0.The basic structure of a first-order ANFIS with two inputs and one output is

depicted in Figure 10.9 [22].The function of each layer can be described as follows:

● Layer 1: input membership layer

The first layer represents the MFs and contains adaptive nodes. The membershipvalue specifying the degree to which an input value belongs to a fuzzy set isdetermined in this layer. The output of the nodes in this layer can be defined by

O1;i ¼ mAi x1ð Þ for i ¼ 1; 2O1;i ¼ mBi�2 x2ð Þ for i ¼ 3; 4

�ð10:21Þ

Assuming that the MFs are triangular functions, the output from node Ai can begiven as

O1;i ¼ mAi x1ð Þ ¼ max minx1 � a

b � a;c � x1

c � b

� �; 0

� �for i ¼ 1; 2 ð10:22Þ

A1

A2

B1

B2

N

N

Π

Π

∑

X1 X2

X1 X2

X2

X1

Figure 10.9 ANFIS structure with two inputs and one output


where x1 is the input of node i, Ai is linguistic label associated with this node and{a, b, c} is the parameter set of the triangular MF.

● Layer 2: firing strength layer

The output of each node in this layer is the product of all incoming signals andrepresents the firing strength of a rule. Each node in this layer is a fixed node,which performs the fuzzy AND operation using the algebraic product. The output ofeach node in this layer is given by

O2;i ¼ wi ¼ mAi x1ð ÞmBi x2ð Þ for i ¼ 1; 2 ð10:23Þ

● Layer 3: normalised firing strength layer

The nodes in this layer are fixed nodes and they calculate the normalised firingstrength for each rule, which is given by

O3;i ¼ wi ¼ wiPni¼1 wi

for i ¼ 1; 2 ð10:24Þ

● Layer 4: consequent layer

The output of each node in this layer is adaptive and represents the weightedconsequent part of the rule table. The output of each node can be expressed as

O4;i ¼ fi ¼ wi pix1 þ qix2 þ rið Þ for i ¼ 1; 2 ð10:25ÞThe parameter sets {pi, qi, ri} are called consequent parameters (CPs).

● Layer 5: defuzzification layer

This layer is the output layer and acts as a defuzzifier. The single node in this layeris a fixed node, which computes the overall output as the summation of allincoming signals. The output of this layer is given by

O5;i ¼ y ¼Xn

i¼1

fi ð10:26Þ

Like any other neural network, a set of parameters in ANFIS is required to beupdated in order for the network to be adaptive. These parameters are the MFsrepresented by Layer 1 and the CPs represented by Layer 4. The common adapta-tion technique is based on a gradient descent method [13].

10.4.2 Structure of the NFCThe structure of a typical NFC is shown in Figure 10.10. The two inputs to thecontroller are usually the error signal and change-of-error. The error signal e(t)represents the difference between the actual output of the plant and a desired set-point while the change-of-error De(t) is the difference between the error e(t) and theprevious error value e(t � 1). A negative sign of e(t) means that the output of the


plant y(t) has a value above the desired value yd as e(t) ¼ yd � y(t), while a positivesign of e(t) indicates that output of the plant is below the desired value. Further-more, a negative sign of De(t) suggests that the plant output has increased whencompared to its previous value y(t � 1) while a positive De(t) means the opposite.

The input scaling factors, K1 and K2, are typically used to map the real input to thenormalised input space in which the MFs are defined. In general, the normalisationrange can be in the range of [�1, þ1] for the universe of discourse. It is noted that theinput scaling factors influence the sensitivity of the NFC and affect its performance[9, 23, 24]. On the other hand, the output scaling factor K3 is used to map the output ofthe fuzzy inference system to the real output. The value of K3 should be appropriatelyselected so the output range of the NFC will not exceed a certain boundary, where aphysical limitation is violated. It is clear that the output scaling factor has the mostinfluence on system stability and oscillation tendency [9].

Considering there are seven MFs located in the first layer of the NFC and havetriangular shapes, seven triangular MFs are associated with fuzzy linguistic sets usedfor each input to the controller. The input MFs of the NFC are depicted in Figure 10.11.

As shown in Figure 10.11, the centres of the MFs are distributed evenly alongthe normalised input space, which is a common technique used in most fuzzycontrol applications. Given that the peak value of a MF is equal to 1, the cross point

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

k1

k2

k3

Layer 5Layer 4Layer 3Layer 2Layer 1

u(t)

μA

w w__

f

Figure 10.10 Architecture of an adaptive neuro-fuzzy controller


between two MFs is at 0.5. The linguistic terms used for the MFs are big positive,medium positive, small positive, zero (ZO), small negative, medium negative andbig negative. As the NFC has two inputs of which each has seven MFs, the rule-base table associated with the controller will contain 49 rules. The Sugeno-typerule-base table with 49 rules is summarised in Table 10.1.

10.4.3 Online adaptation techniqueTo make the NFC adaptive, two set of parameters are required to be adjusted. Thiscan be achieved through network learning of which the MFs of linguistic terms andthe CPs can be modified, using certain adaptation techniques [25]. One of these

–1.5 –1.0 –0.5 0 0.5 1.0 1.5

Mem

bers

hip

degr

ee

Normalised input variable

0.2

0.4

0.6

0.8

1.0NB NM NS ZO PS PM PB

Figure 10.11 Triangular membership functions of the inputs

Table 10.1 Sugeno-type rule-base table with 49 rules

De e

PB PM PS ZO NS NM NB

NB 0.0 �0.333 �0.666 �1.0 �1.0 �1.0 �1.0NM 0.333 0.0 �0.333 �0.666 �1.0 �1.0 �1.0NS 0.666 0.333 0.0 �0.333 �0.666 �0.666 �1.0ZO �1.0 0.666 0.333 0.0 �0.333 �0.666 �1.0PS �1.0 �1.0 0.666 0.333 0.0 �0.333 �0.666PM �1.0 �1.0 �1.0 0.666 0.333 0.0 �0.333PB �1.0 �1.0 �1.0 �1.0 0.666 0.333 0.0

PB, Big positive; PM, medium positive; PS, small positive; ZO, zero; NS, small negative; NM, mediumnegative; NB, Big negative.


adaptation schemes is the back-propagation algorithm. The back-propagationalgorithm can be employed to adjust the centres of the MFs and the CPs of theNFC. Given the cost function, described in (10.27), the centres of the MFs and CPscan be updated online using the gradient descent method [22, 26].

Jc kð Þ ¼ 12

ec½ k þ 1ð Þ�2 ð10:27Þ

where ec k þ 1ð Þ is the error signal between the estimated output, provided by theidentifier, and the desired system output at time step k þ 1ð Þ.

Assuming q is an arbitrary parameter in the NFC, updating the MFs and CPscan be achieved using the gradient optimisation method given as [27]

q k þ 1ð Þ ¼ q kð Þ � h@Jc

@qð10:28Þ

where

rqJc kð Þ ¼ ec k þ 1ð Þ @ec k þ 1ð Þ@u kð Þ

� �@u kð Þ@q

� �ð10:29Þ

@u kð Þ@q

¼XO��S

@u kð Þ@O�

@O�

@qð10:30Þ

where h and u(k) are the learning rate and output of the NFC, respectively. S and O*are the set of nodes whose outputs depend on q and the output of nodes belongingto S, respectively.

For the output node, @u kð Þ@O� is given as

@u kð Þ@O� ¼ K3 ð10:31Þ

For an internal node, @u kð Þ@O� is given as

@u kð Þ@Ol

i

¼ K3

XP

n¼1

@u kð Þ@On

lþ1

@Onlþ1

@Oli

ð10:32Þ

where Oli is the output of the ith node of the lth layer and P is the number of nodes

in the (l þ 1) layer. The updating of the membership centres and CPs of the NFCtakes place in every sampling period.

The numbers of neurons (Figure 10.10) in the adaptive NFC are 14, 49, 49, 7and 1 for layers L1, L2, L3, L4 and L5, respectively. This means that 14 centre pointsin the MFs (seven for each input) and seven CPs need to be updated online. Anetwork with this number of parameters to update is considered relatively complexand computationally expensive, especially for real-time applications [27].

The purpose of the adaptive controller is to be applied to an SVC device(explained in Chapters 13 and 14) in order to damp power system oscillationsand enhance system stability. As the SVC is an electronic device, designed


based on high-speed power electronic components, it is desirable to design acontroller that possesses the characteristic of having a fast response time.Therefore, the objective is to design a simplified version of the ANFC, described inSection 10.4.2, that requires less computation time, and apply it to the SVC device.It is necessary for the simplified controller to provide similar performance ascompared to the ANFC.

10.5 Adaptive simplified NFC

The essence of designing an NFC is to be able to transform an FL controller (FLC)and represent it in a neural network structure. This implies that it is imperative thatthe FLC is the main part taken into consideration when designing an NFC. If anFLC is constructed, an NFC can be designed and represented by a neural network,accordingly. Therefore, in order to develop a simplified version of an NFC, asimplified FLC is first required.

In general, it is always desirable to design the simplest control system thatperforms the expected task it is built for, as long as its accuracy is not compro-mised. In fact, in designing FLCs, interpretability and accuracy are the mostimportant aspects to be considered [28].

Although, accuracy and interpretability represent contradictory objectives, anoptimum fuzzy control system design should satisfy both criteria, to a certaindegree. For instance, a complex FLC might successfully control a high-order non-linear system accurately; nonetheless, the drawback would be the difficulty inexpressing the behaviour of the controller in an understandable way. On the con-trary, a simple FLC can be easily understood but, its performance is not satisfac-tory. Therefore, a trade-off between the interpretability and accuracy should alwaysbe considered.

10.5.1 Simplification of the rule-base structureThere have been several publications proposing the design of simplified FL con-trollers using different simplification approaches. One of these techniques isreducing the size of the fuzzy rule tables [29–31]. The principle of designing theproposed adaptive simplified NFC (ASNFC) is based on the concept of reducingthe fuzzy rule-base table (Table 10.1). It can be seen from Table 10.1 that the rule-base table can be viewed as a Toeplitz structure with zero diagonal line. Havingsuch a structure provides the advantage of using the symmetrical property of thetable to construct a one-dimensional fuzzy-rule table.

A new variable, called the signed distance, can be introduced to build a sim-plified FL controller (SFLC). This new variable represents the distance, d, to anactual state from the main diagonal line, called the switching line. The distance canbe positive or negative, depending on the position of the actual state in the rule-basetable, illustrated in Figure 10.12 [32].

As shown in Figure 10.12, a control action can be proportionally related to theperpendicular distance from any consequent in the table to the switching line. Three


distances, d1, d2 and d3, can be obtained in both the upper half-plane and the lowerhalf-plane. These distances will have negative signs if they are located in the upperhalf-plane and positive signs if they are in the lower half-plane. This can clearly beillustrated in Figure 10.13.

The switching line (Figure 10.13) can be represented by a general straight lineequation given by

Ae þ BDe þ C ¼ 0 ð10:33Þ

where A, B and C are constants and are equal to A ¼ B ¼�1, C ¼ 0.

Switching line

Δe

ed1

d1

d2

d2

d3

d3

Figure 10.13 Illustration of d1, d2 and d3 between the switching line and actualstates in the phase-plane

–d

+dSwitching

line

NBNB

NM

NM

NS

NS

ZO

ZO

e

PS

PS

PM

PM

PB

PB0.0

0.333–0.333

–0.333–0.333

–0.333–0.333

–0.333

–0.666–0.666

–0.666–0.666

–0.666

0.333

0.3330.333

0.333

0.333

0.00.0

0.00.0

0.00.0

0.6660.666

0.6660.666

0.666

1.01.01.01.0

1.01.01.0

1.01.0

–1.0 –1.0–1.0 –1.0

–1.0–1.0–1.0–1.0

–1.0 –1.0

1.0

Δe

Figure 10.12 The distance, d, between the switching line and actual state


The perpendicular distance between the switching line and a given point P(e, De),located on the phase-plane, can be shown in Figure 10.14 and expressed as [33]:

d ¼ Ae þ BDe þ Cj jffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA2 þ B2

p ð10:34Þ

Substituting A, B and C into (10.34) yields

d ¼ f1 e; Deð Þ ¼ � e þ Deð Þj jffiffiffi2

p ð10:35Þ

Four symmetrical triangular MFs with 50 per cent overlap in the range of[0, þ1] are chosen to be the input to the SFLC. This range is considered, as opposedto the range of [�1, þ1] used in the typical FLC, as d1 can be located in the upperhalf-plane or the lower half-plane (Figure 10.13), with negative or positive signs,respectively. Therefore, a calculation of only one distance from the switching lineis required and a positive or negative sign can be associated, based on the locationof the actual state in the phase-plane.

The values of d1, d2 and d3 for the upper and lower half-plane represent thecentres of the triangular MFs. The rule-base table is reduced to one dimension withthe fuzzy linguistic terms, ZO, small (S), medium (M) and big (B) for the distance,d, and fuzzy singletons for the control signal. The control signal for any point in thephase-plane is given by [29]

u ¼ K3Suuus ð10:36Þwhere K3 is the output scaling factor, uus is the unsigned control action and Su isgiven by

Su ¼ f2 e; Deð Þ ¼ 1 when e þ De � 0�1 otherwise

ð10:37Þ

P(e, Δe)Δe

e

Ae + BΔe + C = 0

d

Figure 10.14 Perpendicular distance between point P(e, De) and the switchingline in the phase-plane


The reduced rule-base table and input MFs are summarised in Table 10.2 andillustrated in Figure 10.15, respectively.

10.6 Control system design of the proposed ASNFC

The reduced rule-base table summarised in Table 10.2 is used for the design of theproposed ASNFC. As the objective is to design a simple NFC with a reducednumber of parameters to update, a zero-order Sugeno-type fuzzy controller-basedANFIS is employed to construct the proposed controller. The overall structure ofthe proposed ASNFC is illustrated in Figure 10.16.

The ASNFC comprises an ANFIS network, with a reduced number of layersand nodes, and two function blocks, f1 and f2, given by (10.35) and (10.37),respectively. As shown in Figure 10.16, the input to the ANFIS network is thedistance in the phase-plane, d, while the output is the unsigned control action, uus.The cost function considered to update the centres of the MFs and the CPs of theproposed controller is defined in (10.38).

J kð Þ ¼ 12

ec k þ 1ð Þ2 ¼ 12

hDP^svc k þ 1ð Þ � DPd k þ 1ð Þ

i2ð10:38Þ

ZO

0

1.0

0.5

Normalised input variable

Mem

bers

hip

degr

ee

1.0dn

S M B

Figure 10.15 Input membership functions for the SFLC

Table 10.2 Reduced rule-base table

d

ZO S M B

uus 0.0 0.33 0.66 1.0


where DP^svc k þ 1ð Þ and DPd k þ 1ð Þ are the estimated power deviation and thedesired value at time step k þ 1, respectively. As the desired value at time step k þ 1is always zero, (10.38) can be written as

J kð Þ ¼ 12

ec k þ 1ð Þ2 ¼ 12

hDP^svc k þ 1ð Þ

i2ð10:39Þ

The update of the MF centres and CPs is taking place at every sampling time,employing (10.28) and (10.29). Equation (10.29) can be expressed in terms ofDP^svc k þ 1ð Þ and the control signal u(k) as

rqJc kð Þ ¼ DP^svc k þ 1ð Þ @DP^svc k þ 1ð Þ@u kð Þ

� �@u kð Þ@q

� �ð10:40Þ

Three terms need to be calculated in (10.40). The terms DP^svc k þ 1ð Þ and@DP^ svc kþ1ð Þ

@u kð Þ are the estimated output and the Jacobian of the plant, respectively. Theycan be obtained from the neuro-identifier. The term @u kð Þ

@q can be calculated using theback-propagation algorithm, as expressed in (10.30).

The new ANFIS network consists of four layers and has 4, 4, 4 and 1 neuronsfor layers 1, 2, 3 and 4, respectively. As the rule-base table is reduced to a one-dimensional rule-table instead of a two-dimensional rule-table, the second layer

k1

k2

k3

f1(.)

f2(.)

dn

Su

u(k)Plant

ei

ec

ANIIdentifier

uus

D

D

D

D

D

D

ANFIS

ASNFC

∆Psvc

∆Psvc(k+1)

∆Pd(k+1)

∏+–

+

+

–

–

Figure 10.16 Overall control system structure of ASNFC


described in Section 10.4.1 will not be applicable in the proposed design. It isobvious that the number of layers and neurons of the new ANFIS network is sig-nificantly reduced, which yields to a simple type of ANFIS network. In addition,the control parameters required to be updated online are reduced from twenty-one(fourteen centre points in the MFs plus seven CPs) to eight (four centre points in theMFs plus four CPs). This will reduce the overall computation time of the controller.

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25. Ramirez-Gonzalez M., Malik O.P. ‘Power system stabilizer design using anonline adaptive neurofuzzy controller with adaptive input link weights’. IEEETransactions on Power Systems. 2008;23(3):914–22

26. Lee S.J., Ouyang C.S. ‘A neuro-fuzzy modeling with self-constructing rulegeneration and hybrid SVD-based learning’. Transactions on Fuzzy Systems.2003;11(3):341–53

27. Yao W., Wen J.J., Wu Q.H. ‘Wide-area damping controller for FACTSdevices for inter-area oscillations considering communication time delays’.IEEE Transactions on Power Systems. 2014;29(1):318–29

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Chapter 11

Power system stabiliser

In an AC-interconnected power system, all synchronous generators rotate at thesame speed, i.e. synchronous speed, under steady-state conditions. With all syn-chronous generators operating in synchronism, the power system is said to bestable. The ability of all generators on a system to maintain synchronism andto return to a stable operating point following a system disturbance leads to theconcept of power system stability.

Power systems are often subjected to a variety of disturbances, such as suddenchanges in load, short circuit faults on transmission lines and loss of transmissionlines. This can result in the lack of balance between the mechanical input and theelectrical output of a generating unit resulting in deviations in generator speed fromthe synchronous speed. This leads to individual generating units oscillating againsteach other. In other cases, particularly under heavily loaded conditions or loss ofone or more transmission lines, the natural oscillation frequency of the system maynot be adequately damped. In that case, even small disturbances, e.g. normal loadfluctuations, can cause generating unit shaft oscillations of increasing magnituderesulting in angular instability.

In an interconnected power system, two distinct types of oscillations can exitsimultaneously. In one type, called the local mode, a generator swings against therest of the system with an oscillation frequency generally in the range of 0.8–2.0 Hz.In the other oscillation mode, called the inter-area mode, a number of generators inone part of the interconnected system (area 1) swing against machines in another partof the system (area 2) in a frequency range of 0.4–0.8 Hz. Depending on the systemcharacteristics, there also can be a small overlap in the local and inter-area modes ofoscillations.

Continuously acting automatic voltage regulators (AVRs) are employed on allsynchronous generators. It is widely recognised that although AVRs are essential tomaintain a proper voltage at the generator terminals and in the system, high gainfast-acting AVRs have the potential of introducing negative damping in the exci-tation control system [1, 2].

The local and inter-area modes of oscillations can be damped by introducinga supplementary signal through the synchronous generator excitation system.This was recognised in the 1950s. A lot of successful experience gained sincethen has shown that a supplementary control signal, properly derived from anappropriately selected feedback signal, acting through the AVR can significantly

enhance damping of rotor oscillations. The device used to generate the supple-mentary control signal is called a power system stabiliser (PSS).

11.1 Conventional PSS

A schematic block diagram of the generator excitation system is shown in Figure 11.1.The AVR output is based on the voltage error difference between the generatorterminal voltage reference set point and the actual terminal voltage magnitude. Thelarge inductance of the field winding causes a delay in the machine flux and hence inthe terminal voltage response. This delay, which can be interpreted as a phase lag,causes the un-damping effect and may cause oscillatory stability problems.

11.1.1 Configuration of common PSSThe negative damping effect of the high gain fast-acting AVRs and the time delayin the excitation circuit can be reduced by employing a PSS. The PSS outputmodulates the generator excitation so as to develop a torque in phase with the rotorspeed deviations and adds damping to the characteristic electromechanical oscil-lations [3, 4]. Innumerable studies and tests on all types of utility-scale generatorshave proven that power system stability can be improved even beyond the classicalsteady-state stability limit and overall damping increased by using a properly tunedand tested PSS.

The most commonly used PSS (CPSS) provides phase compensation forthe phase difference between the AVR input and the generator shaft speed(Figure 11.1) through adjustable lead-lag compensation functions (T1–T4)(Figure 11.2) over a dynamic frequency of interest, usually 0.4–2.0 Hz. The PSSgain, Ks, is determined to be the highest within the constraints of the PSS controlloop stability. The high-frequency filters allow for the suppression of potentiallyunstable torsional oscillations or other sources of torsional noise. The wash outfilter is a high-pass filter to remove any DC signals. It generally has a long timeconstant (Tw) of 5–10 s. It is common to have an output limiter to limit the PSSoutput from overwhelming the AVR forcing during transient conditions.

Commonly used PSS input signals are change in generator shaft speed Dw,electrical frequency deviation Df, variation in electrical power output DPe and

PSS

Exciter ~

∆f

or ∆P∆ω

GeneratorXe+

–

EO

VoltageRegulator

Manual control

EFD

VT's

Ref. ∑

Figure 11.1 Generator excitation system block diagram


accelerating power. In some cases even a combination of these is used. Dependingon the feedback signal used, alternative forms of PSS have been developed [5].

11.1.2 PSS input signalsThe early designs of PSSs employed a direct measurement of shaft speed [2]. Asthis requires the use of a torsional filter to attenuate torsional components, anadditional phase lag is introduced that limits the allowable stabiliser gain. Place-ment of the speed pick-up transducers on the generating unit shaft requires specialcare so that undesirable frequencies are not picked up [6].

Another input signal that has been employed successfully is the terminal fre-quency. Frequency has been used directly. Also, in some cases the terminal voltageand current inputs were combined to generate a signal to approximate the generatorshaft speed, called as ‘compensated’ frequency.

Frequency signal is more sensitive to inter-area mode of oscillations than thelocal mode. It can thus provide better damping of the inter-area mode [3]. Frequencysignal also needs to be filtered for torsional components. In addition, changes inpower system configuration or noise caused by large industrial loads may producelarge frequency transients that can affect the generator field voltage [7].

It is simple to measure electric power, and it can be related to the generatorspeed through the torque equation:

Accelerating torque ¼ Input mechanicalð Þ torque � Output electricalð Þ torque

ð11:1ÞIt can be written as

2H

wd2ddt2

¼ Tm � Te ð11:2Þwhere

H is the inertia constant (s)w is the frequency (elec. rad/s)Tm is the mechanical input torque (N-m) andTe is the electrical output torque (N-m)

VS

VSMAX

(1 + A5S + A6S2)(1 + A3S + A4S2)(1 + A1S + A2S2)

VSMIN

VSI

1 + T2S

1 + T1S

1 + T4S

T5S1 + T3S

1 + T5SKS

High-frequency filters

Figure 11.2 Power system stabiliser structure

Power system stabiliser 279

Considering that the mechanical time constant is much larger than the elec-trical time constant, ignoring the variations in the mechanical torque, the shaftacceleration (which leads the speed by 90�) may be considered as a scaled versionof electrical power. The stabilising signal, derived from the deviations in electricalpower in combination with high-pass and low-pass filters, can thus provide puredamping torque. This power-based PSS has been used as the basis for a number ofPSSs. Such a PSS can provide pure damping at only one frequency, and also,unwanted PSS output is produced whenever mechanical power changes. This canput severe limits on the gain and output of the PSS.

With a number of limitations manifest in deploying PSSs with any of thespeed, frequency or power signals as input, efforts were made to directly measurethe accelerating power of the generator [8–10]. As these methods involved sig-nificant complexity in the design, an indirect method of deriving the acceleratingpower, PSS2A of [5], was developed. The principle of this PSS is based on derivingthe integral-of-accelerating power signal from shaft speed and electrical powersignals by integrating and manipulating (11.1) into the form

ðDPa

2Hdt ! �DPeðsÞ

2Hsþ G sð Þ DPe sð Þ

2HsþDw

� �ð11:3Þ

where

Pa is the accelerating powerPe is the electrical power andG(s) is the transfer function of a low-pass filter

The two input signals pass through high-pass filters and are processed inindividual channels before being added to form one input signal to the stabilisergain and lead/lag stage. This PSS does not require a torsional filter in the pathinvolving the electrical power signal. However, as it employs two inputs, speed andactive power, it is sensitive to the relationship between the two inputs and, there-fore, is critical to match the two signal paths in terms of gain and filter constants.

11.1.3 Characteristics of common PSSSuccessful experience has been gained with the injection of a supplementaryfeedback signal through the generator excitation system to enhance damping ofgenerator rotor oscillations. As described in Section 11.1.2, various input signalshave been used as input to a PSS that, in general, consists of a second-order phaselead/lag network with a gain. For proper damping action, appropriate PSS settingsare determined by adjusting the lead, lag and gain of the stabiliser. The conven-tional PSS, adopted by most electric utilities, is designed offline using linear con-trol theory and is based on a model of the power system with a fixed configurationlinearised for one operating condition. It is simple in structure, has flexibility and iseasy to implement. It has made significant contribution in enhancing the quality ofelectrical supply.


Power systems, in general, are complex and non-linear systems. Their para-meters not only depend on the operating condition, but both their configuration andparameters can change with time. This may create discrepancies between themathematical model and the physical conditions. Therefore, with the conventionallinear control theory-based PSS it is difficult to realise the desired control perfor-mance over wide operating conditions of the power plant. To further improve theperformance and stability of the power system, various other approaches usinglinear quadratic (LQ) optimal control, H-infinity, variable structure, rule-based andartificial intelligence (AI) technologies [3, 11–18] have been proposed in theliterature to design a fixed parameter PSS. One common feature of all fixedparameter controllers is that the design is done offline. To yield satisfactory controlperformance, it is desirable to develop a controller that considers the non-linearnature of the plant and has the ability to adjust its parameters online according tothe environment in which it is working, i.e. track the plant operating conditions.

The conventional stabiliser parameters have to be designed for each applica-tion. Its parameters, once designed, tuned and implemented, are fixed. They can beset to contribute optimal damping at only one oscillation frequency. A powersystem is subject to multi-modes of oscillations. As the parameters of a conven-tional PSS are tuned for one set of operating conditions, the selected parameters area compromise between the local and inter-area mode oscillations. Therefore, thefixed parameter PSS generally cannot maintain the same quality of performanceunder all conditions of operation.

11.2 Adaptive control-based PSS

The common procedure in process control is to compare the actual measured valuesof the output with the desired values and the difference, the error, is fed as input tothe process through a regulator and an actuator. Various criteria are available forthe computation of the control to minimise the error. Using this technique, thedesired control law is obtained as

uðtÞ ¼ f qsðtÞ; yðtÞ; uðt � TÞ½ � ð11:4Þ

where

qs(t) is the system parameter vectory(t) is the output vector [y(t) y(t � T ) . . . ]t

u(t � T ) is the control vector [u(t � T ) u(t � 2T ) . . . ]t

t superscript denotes the transposeT is the sampling period andf [.] denotes function

If the parameter vector is known, control to meet specific performance criterioncan be computed directly. However, the dynamics of a complex non-linear systemvary with time depending on the operating conditions, disturbances and so on.


An adaptive controller has the ability to modify its behaviour depending on theperformance of the closed-loop system. The basic functions of the adaptivecontroller may be described as

● identification of unknown parameters, or measurement of a performance index● decision of the control strategy● online modification of the controller parameters

Depending on how these functions are synthesised, different types of adaptivecontrollers are obtained. Various adaptive control techniques have been proposedfor excitation control since the mid-1970s. A brief review of the adaptive controltechniques from the excitation control aspect is presented in this section.

Two distinct approaches – direct adaptive control and indirect adaptive control –can be used to control a plant adaptively. In the direct control, the parameters of thecontroller are directly adjusted to reduce some norm of the output error. In theindirect control, the parameters of the plant are estimated as the elements of a vectorat any instant k, and the parameters’ vector of the controller is adapted based on theestimated plant vector.

11.2.1 Direct adaptive controlA very common form of direct adaptive control is the model reference adaptivecontrol (MRAC). The objective of an MRAC system is to update the controllerparameters such that the closed-loop system maintains a performance specified by areference model. It requires a suitable model, an adaptive mechanism and acontroller.

The structure of a MRAC system is shown in Figure 11.3. In MRAC, the actualsystem performance is measured against a desired closed-loop performance specifiedby a reference model that is driven by the same input as the controlled system. Theobjective is to minimise the error, the difference between the actual system outputand the reference model output. The ‘adaptation mechanism’ block in Figure 11.3 isused to update the parameters of the controller. Various methods are available tominimise the error function.

The most important feature in ensuring the success of MRAC is the selectionof a proper reference model and its parameters. The selected parameters must be

Reference model

Systemucu

Adaptation mechanism

e

yyr

– +Controller

Figure 11.3 MRAC structure


such that the system is capable of following the reference model output and that thecontrol signal remains within the physical control limits. A systematic method todetermine a proper reference model for the plant is described in [19].

Application of an adaptive PSS (APSS) based on the MRAC principle is shownin Figure 11.4. A fuzzy logic controller (FLC) with self-learning capability is usedto adapt the system performance to track the reference model. Two inputs, gen-erator speed deviation and its derivative, and the supplementary control output,each have seven membership functions. The FLC uses the Mamdani-type fuzzyproportional derivative (PD) rule base [20]. Updating the centre points of thecontroller input membership functions, i.e. the weights of the fuzzy controller,using the steepest descent algorithm provides it with a self-learning capability. Itcan thus adapt the system performance to track the reference model.

Results of a number of studies show that this APSS provides good dampingover a wide operating range and improves the performance of the system. Anillustrative example showing the system response to a three phase to ground fault atthe middle of one transmission line and successful reclosure with the self-learningMRAC-based FLC and a fixed centre FLC is given in Figure 11.5.

11.2.2 Indirect adaptive controlA general configuration of the indirect adaptive control as a self-tuning controller isshown in Figure 11.6. At each sampling instant, the input and output of the gen-erating unit are sampled and a plant model to represent the dynamic behaviour ofthe generating unit at that instant in time is obtained by some online identificationalgorithm. It is expected that the model obtained at each sampling instant can trackthe system operating conditions.

The required control signal is computed based on the identified model. Variouscontrol techniques can be used to compute the control. All control algorithmsassume that the identified model is the true mathematical description of the con-trolled system.

Field Vt Vg

Transmission lines

Referencemodel

FLC

Δe

–

+yr +– y

yc

u

eZ –1

Generating unitAVR

&exciter

Figure 11.4 System configuration with MRAC-based APSS


In the analytical approach to the design of an adaptive controller, sampled datadesign techniques are used to compute the control. The indirect adaptive controlprocedure involves:

● Selection of a sampling frequency, fs, about ten times the normal frequency ofoscillation to be damped.

● Updating of the system model parameters (coefficients of system transferfunction in the z-domain) each sampling interval T (¼1/fs) using an identifi-cation technique suitable for real-time application. A number of identification

MRAFCFLC PSS

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0 1 2 3 4 5Time (s)

Pow

er a

ngle

(rad

)

6 7

Figure 11.5 Three-phase to ground fault with APSS (MRAFC) and fixed FLC PSS(P ¼ 0.95 pu, 0.9 pf lag)

Plant

YU

Plant model

Controller

Model parameteridentification

Figure 11.6 Block diagram of a self-tuning controller� 2009 IEEE. Reprinted with permission from Malik O.P., ‘Adaptive and artificialintelligence based PSS’, Proceedings, IEEE PES 2003 General Meeting, Vol. 3,pp. 1792–7


routines, in recursive form, e.g. recursive least squares (RLS), recursiveextended least squares (RELS), can be used to determine the transfer functionof the controlled plant in the discrete domain.

● Use the updated estimates of the parameters to compute the control outputbased on the control strategy chosen. Various control strategies, among themoptimal, minimum variance (MV), pole-zero assignment, pole assignment andpole shift (PS), have been proposed.

11.2.2.1 System modelThe generating unit is described by a discrete ARMAX model of the form

Aðz�1Þy tð Þ ¼ Bðz�1Þu tð Þ þ e tð Þ ð11:5Þwhere

A(z�1) and B(z�1) polynomials in the delay operator z�1 are of the form

Aðz�1Þ ¼ 1 þ a1 z�1 þ � � � þ ai z�i þ � � � þ ana z�na ð11:6ÞBðz�1Þ ¼ b1 z�1 þ � � � þ bi z�i þ � � � þ bnb z�nb ð11:7Þ

na � nb

The variables y(t) and u(t) are the system output and system input, respectively,and e(t) is assumed to be a sequence of independent random variables with zero mean.

11.2.2.2 System parameter estimationThe control is computed based on the identified model parameters, ai and bi. Thus,to compute the control appropriate to the varying conditions the system parametershave to be estimated online. The correctness of the identification determines thepreciseness of the identified model that tries to reflect the true system. For a time-varying system the tracking ability of the identification method is very important.

An online estimate of the system parameters is obtained by providing in theregulator a mathematical model having a desired structure describing the actualprocess. Such a model may be expressed as

yðtÞ ¼ g qm; xðtÞ½ � ð11:8Þwhere

yðtÞ is the predicted (estimated) value of the system outputqm is the model parameter vector andx(t) is the information known at the time of prediction

The model parameter vector may either be constant, qm, or be a function oftime, qm(t). For the model to track the system dynamics, i.e. tune itself to thesystem, its parameters must be updated continuously at an interval that is consistentwith the time constants of the system.

Several methods can be used to obtain an estimate for the model parametervector, qm(t) [21]. A commonly used technique of achieving a continuous tracking


of the system behaviour is the RLS parameter estimation technique. It minimisesthe square of the error between the actual system output and the model output, andthe estimated parameter vector qmðtÞ is given by

qm tð Þ ¼ h qmðt � TÞ;PðtÞ; xðtÞ� � ð11:9Þwhere

P(t) is the covariance matrix of the error of estimates. In general terms itcontains the entire history of the process.

To enhance the ability of the identifier to track the operating conditions of theactual system, a forgetting factor is used to discount the importance of the olderdata. It can be chosen as a constant or a variable. A variable forgetting factor,employed to improve the tracking ability especially under large disturbances, iscalculated online every sampling interval [22].

11.2.3 Indirect adaptive control strategiesFour control strategies that need explicit clarification are described below.

11.2.3.1 LQ controlIn the LQ control algorithm the objective is to minimise a performance index [23].The performance is chosen so that the system output error is minimised withrespect to the system input. The LQ controller has the advantage that it will alwaysresult in a stable closed-loop system provided that the parameter estimates areexact. However, the achievement of this characteristic imposes heavy computa-tional burden because it requires the solution of a matrix Riccati equation. Also,this controller is designed in the state space form and a common identificationtechnique estimates the system parameters in the input/output form. Thus, anobserver is required to convert the system parameters into a canonical form.

11.2.3.2 MV controlIn this control strategy, the objective is to minimise the variance of the output [24].Output error at the next sampling instant for zero control is predicted first. Thecontrol that will drive this predicted error to zero is then computed. Although thiscontrol strategy has nice properties, it has characteristics that make it difficult touse for excitation control.

In this strategy, the controller poles are obtained directly from the identifiedsystem zeros. The closed-loop system will be unstable if the dynamics of thesampled system are non-minimum phase, i.e. the system has a zero on or outside theunit circle in the z-domain. This might cause an unstable control computation ifidentified zeros are not cancelled exactly with the system zeros. When the cancel-lation of large parameter errors is not possible within one sample due to the limitson the control signal, the MV controller will produce an oscillatory response. Theexcitation signal is band limited, and the use of MV controller will result inexcessive control and a poor control action. These problems associated with the MVcontroller can be avoided by using a pole-zero or pole-assigned (PA) controller.


11.2.3.3 Pole-zero and PA controlIn the pole-zero assignment (PZA) controller the poles and zeros in the closed-loopare pre-specified by the designer [25]. Whereas, in the MV case all poles are shiftedtowards the centre of the unit circle in the z-domain, poles and zeros in the PZAcase are shifted to locations that produce the desired closed-loop characteristics.This permits a trade-off between performance and control effort. Although thiscontroller does not suffer from the problems of non-minimum phase and bandlimited output associated with the MV controller, the designer has to know thesystem characteristics to achieve the desired characteristics. In this respect, thisalgorithm can be compared to MRAC.

Pre-selection of the locations of poles and zeros is difficult for non-deterministiccase and their poor choice may lead to unstable control computations.

In the PA controller only poles, instead of both poles and zeros, are assigned[26]. Otherwise, it is exactly the same as the PZA controller.

11.2.3.4 PS controlThe PS controller is in essence the PA controller, but the closed-loop poles areobtained by shifting the open-loop poles radially towards the centre of the unitcircle in the z-domain. Shifting the poles towards the centre is directly related toincreased damping. This approach has the advantage of producing astable controller. Detailed description of the PS control algorithm and its applica-tion as an APSS is given in Section 11.3.

11.3 PS control-based APSS

Extensive amount of work has been done to develop and implement an APSS basedon the PS strategy. Such a PSS can adjust its parameters online according to theenvironment in which it works and can provide good damping over a wide range ofoperating conditions of the power system.

11.3.1 Self-adjusting PS control strategyIn the PS control strategy, in closed-loop (with PSS) the poles of the controlledsystem are shifted from their open-loop (without PSS) locations towards thecentre in the z-plane by a factor less than one. This factor, called the ‘pole shiftingfactor’, is varied online to always produce maximum damping contributionwithout exceeding the control limits. To determine the desired control, such asystem may be modelled by a linear low-order discrete model with time-varyingparameters.

The parameters of the system model of a given structure, estimated as inSection 11.2.2.2, are used in the control algorithm to compute the updated control.A block diagram of the regulator is shown in Figure 11.6. Because the control isbased on the estimated model parameter vector, qmðtÞ, (11.4) now becomes

uðtÞ ¼ f qmðtÞ; yðtÞ;Uðt � TÞ� � ð11:10Þ


For the system modelled by (11.5), assume that the feedback loop has the form(cf. Figure 11.7(a))

uðtÞyðtÞ ¼ �Gðz�1Þ

Fðz�1Þ ð11:11Þ

From (11.5) and (11.11) the closed loop characteristic polynomial T(z�1) can bederived as

Aðz�1ÞFðz�1Þ þ Bðz�1ÞGðz�1Þ ¼ Tðz�1Þ ð11:12ÞUnlike the pole-assignment algorithm in which T(z–1) is prescribed [26], the PSalgorithm makes T(z�1) take the form of A(z�1) but the pole locations are shifted bya factor a, i.e.

Tðz�1Þ ¼ Aðaz�1Þ ð11:13ÞIn the PS algorithm, a, a scalar, is the only parameter to be determined and

its value reflects the stability of the closed-loop system. Supposing l is the absolutevalue of the largest characteristic root of A(z–1), then al is the largest characteristicroot of T(z–1). To guarantee the stability of the closed-loop system, a ought to satisfythe following inequality (stability constraint):

� 1l< a >

1l

ð11:14Þ

The PS process is presented schematically in Figure 11.7(b). It can be seen thatonce T(z�1) is specified, F(z–1) and G(z–1) can be determined by (11.12), and thusthe control signal u(t) can be calculated from (11.11).

To consider the time domain performance of the controlled system, a perfor-mance index J is formed to measure the difference between the predicted systemoutput, yðt þ 1Þ, and its reference, yr(t þ 1):

J ¼ E½ yðt þ 1Þ � yrðt þ 1Þ�2 ð11:15Þ

Unit circle Z-plane

(a) (b)

ŷ(l+1)

u(t)

uref

–1 0

A(z–1)/A(αz–1)–j1

j1+

Identified modelB(z –1)

A(z –1)

ControllerG(z –1)

F(z –1)

Figure 11.7 (a) Closed-loop system block diagram and (b) pole-shifting process


E is the expectation operator. yðt þ 1Þ is determined by system parameter poly-nomials A(z–1), B(z–1) and past y(t) and u(t) signal sequences. Considering thatu(t) is a function of the pole-shifting factor a, the performance index J becomes

min aJ ¼ f Aðz�1Þ;Bðz�1Þ; u tð Þ; y tð Þ;a; yrðt þ 1Þ� � ð11:16ÞThe pole-shifting factor a is the only unknown variable in (11.16) and thus can

be determined by minimising J.

Constraints:

When minimising J(t þ 1, a), it should be noted that a will be subject to thefollowing constraints:

● The stabiliser must keep the closed-loop system stable. It implies that all rootsof the closed-loop characteristic polynomial A(z–1) must lie within the unitcircle in the z-plane (cf. (11.14)).

● The control limit should be taken into account in the stabiliser design to avoidservo saturation or equipment damage. The optimal solution of a should alsosatisfy the following inequality (control constraint):

umin � uðt;aÞ � umax ð11:17ÞPole patterns of T(z–1) for a 50-ms three-phase to ground fault at the middle of

one line of a double-circuit transmission line connecting a generator to a constantvoltage bus (Figure 11.8) are shown in Figure 11.9. The pole pattern before theapplication of control is shown in Figure 11.9(a). As two poles map outside the unitcircle, the closed-loop system is in an unstable state. The pole pattern after the PScontrol is applied is shown in Figure 11.9(b). As all the poles lie within the unit circle,

Generation unit

ControlSamples (system output)

Transmission lines

Pow

er g

rid

ARMAparameters

Samples (control signal)

Field

Vt Vg

Constant input AVR&

exciter

PS control

Online identifier(ADALINE network)

Figure 11.8 Power system with adaptive PSS


the closed-loop system is stable. It shows that the PS control assures the stabilityof the closed-loop system and also optimises the performance given by (11.16).

11.3.2 Performance studies with pole-shifting control PSSPerformance of the self-tuning adaptive controller based on the pole-shifting con-trol algorithm has been investigated by conducting simulation studies on singlemachine [22, 27] and multi-machine system [28], on a single machine [29] and on amulti-machine physical model [30] in the laboratory, and on a 400-MW thermalmachine under fully loaded conditions connected to the system [31].

The single machine power system consists of a synchronous generator con-nected to a constant voltage bus through two transmission lines (Figure 11.8).A non-linear seventh-order model is used to simulate the dynamic behaviour of thissystem. The differential equations used to simulate the synchronous generator andthe parameters used in simulation studies are given in [22, 27]. The generatorhas an IEEE Standard 421.5, Type ST1A AVR and Exciter. An IEEE Standard421.5, PSS1A Type CPSS [32] is used for comparative studies.

The system output is sampled at the rate of 20 Hz for parameter identificationand control computation. Studies performed with various sampling rates showthat the performance is practically the same for a sampling rate in the range of20–100 Hz. Sampling frequencies above 100 Hz are of no practical benefit and theperformance deteriorates for sampling rate under 20 Hz. A sampling rate of 20 Hzis chosen to make sure that there is enough time available for updating the para-meters and control computation. In most studies, deviation of electrical poweroutput is used as the input to the PSS. The control output is limited to 0.1 pu.

Results of a simulation study to demonstrate the effect of the APSS on the tran-sient stability margin are summarised in Table 11.1. With the single machine infinitebus system initially operating at 0.95 pu power, 0.9 pf lag, a three phase to groundfault was applied near the sending end of one transmission line. It can be observedfrom Table 11.1 that the APSS provides the largest maximum clearance time.

2.52

1.51

0.50

–0.5–1

–1.5–2

–2.5–2 –1 0

Real axis

Imag

e ax

is

Unit circle

z-plane

1 2

2.52

1.51

0.50

–0.5–1

–1.5–2

–2.5–2 –1 0

Real axis

Imag

e ax

is

Unit circle

z-plane

(b) With control(a) Without control

1 2

Figure 11.9 Pole patterns for T(z–1) (a) with and (b) without pole-shift control


This APSS was implemented on a microprocessor and tested in real time on aphysical model of a single-machine infinite bus system. With the system operatingat a stable operating point, the APSS was applied and the torque referenceincreased gradually to the level, P ¼ 1.307 pu, pf ¼ 0.95 lead, vt ¼ 0.950 pu. At thisload, the system was still stable with the APSS.

At 5 s (Figure 11.10) the APSS was replaced by the CPSS. After the switchover, the system began to oscillate and diverge, which means that the CPSS isunable to keep the system stable at this load level. At about 25 s, the APSS wasswitched back to control the unstable system and the system came under controlvery quickly as shown in Figure 11.10. This test demonstrates that the ASPSS canprovide a larger dynamic stability margin than the CPSS. Also, more power can betransmitted with the help of the APSS if an overload operation is necessary undercertain circumstances.

11.4 AI-based APSS

Various approaches using analytical and/or AI-based algorithms can be used todesign an adaptive controller. It is also possible that the analytical and AI

Table 11.1 Transient stability margin results

Without PSS With CPSS With APSS

Maximum clearing time (ms) 120 150 165

CPSS

Time (sec)0

–0.05

–0.04

–0.03

–0.02

–0.01

0

Act

ive

pow

er d

evia

tion

(pu)

0.01

0.02

0.03

0.04

5 10 15 20 25 30

APSSAPSS

Figure 11.10 Dynamic stability improvement by the APSS� 2009 IEEE. Reprinted with permission from IEEE. Reprinted with permissionfrom Malik O.P., ‘Adaptive and artificial intelligence based PSS’, Proceedings, IEEEPES 2003 General Meeting, Vol. 3, pp. 1792–7


techniques be integrated such that some functions are performed using analyticalapproach while the others are performed using AI techniques. Successful imple-mentation of purely AI and integrated approaches is illustrated by application as anAPSS to improve damping and stability of an electric generating unit.

11.4.1 APSS with NN predictor and NN controllerIdentification of the power plant model using an online recursive identificationtechnique is a computationally extensive task. Neural networks (NNs) offer thealternative of a model-free method. An adaptive NN-based controller using indirectadaptive control method has been developed. It combines the advantages of NNswith the good performance of the adaptive control. In this controller, the learningability of the NNs is employed in the adaptation process by training the NN inreal-time each sampling period.

The controller consists of two sub-networks as shown in Figure 11.11. Onenetwork is an adaptive neuro-identifier (ANI) that identifies the power plant in termsof its internal weights and predicts the dynamic characteristics of the plant. It isbased on the inputs and outputs of the plant and does not need the states of the plant.

ANC

uPlant

ANID

D

D

D

D

D

D

D

D

D

Δwd = 0

Δw

Δw

ΔPe

+

+

–

–

ei

ec

Figure 11.11 Controller structure for single-machine study� 2009 IEEE. Reprinted with permission from IEEE. Reprinted with permissionfrom Malik O.P., ‘Adaptive and artificial intelligence based PSS’, Proceedings, IEEEPES 2003 General Meeting, Vol. 3, pp. 1792–7


The second sub-network is an adaptive neuro-controller (ANC) that provides thenecessary control action to damp the oscillations of the power plant.

The success of the control algorithm depends on the accuracy of the identifierin predicting the dynamic behaviour of the plant. The ANI and ANC are initiallytrained offline over a wide range of operating conditions and a wide spectrum ofpossible disturbances. After the offline training stage, the controller is hooked up inthe system. Further updating of the weights of the ANI and ANC is done online forevery sampling period. Online training enables the controller to track the plantvariations as they occur and to provide control signal accordingly.

Employing a feed-forward multi-layer network in each of the two sub-networks,a NN-based APSS (NAPSS) has been built [33]. The two networks are trainedfurther in each sampling period using an online version of the back-propagationalgorithm. The errors used to train the ANI and ANC are both scalar, and thelearning is done only once in each sampling period for each of the two sub-networks.This simplifies the training algorithm in terms of the computation time.

Performance of the adaptive network-based APSS was also tested on a fivemachine interconnected power system shown in Figure 11.12. Generating units in thefive machine power system without infinite bus are modelled by fifth-order differ-ential equations [34]. Results for a three-phase to ground fault on one circuit of thedouble-circuit transmission line between bus nos. 3 and 6 are shown in Figure 11.13.

The adaptive NN-based PSSs were installed on two generators and CPSSswere installed on the other three generators. It can be seen that both the local modeand the inter-area mode oscillations are damped effectively.

11.4.2 Adaptive network-based FLCThe characteristics of fuzzy logic and NNs complement each other in respectof their prospects and concepts. That offers the possibility of using a hybrid

6 7 1

8

4

3

2

5G5

G2

G3

G4

G1

L1

L2

L3

Figure 11.12 Five machine power system� 2009 IEEE. Reprinted with permission from Shamsollahi P., Malik O.P.Application of neural adaptive power system stabilizer in a multi-machine powersystem, IEEE Transactions on Energy Conversion. 1999;14(3):731–6


neuro-fuzzy approach in the form of an adaptive network-based FLC whereby it ispossible to take advantage of the positive features of both fuzzy logic and neuralnetworks. Such a system can automatically find an appropriate set of rules andmembership functions [35].

11.4.2.1 ArchitectureIn the neuro-fuzzy controller, the system is implemented in the framework of net-work architecture. Considering the functional form of the FLC (Figure 11.14), itbecomes apparent that the FLC can be represented as a five-layer feed-forwardnetwork, in which each layer corresponds to one specific function with the nodefunctions in each layer being of the same type. With this network representation ofthe fuzzy logic system, it is straightforward to apply the back-propagation or asimilar method to adjust the parameters of the membership functions and inferencerules.

In this network, the links between the nodes from one layer to the next layeronly indicate the direction of flow of signals and part or all of the nodes contain theadjustable parameters. These parameters are specified by the learning algorithmand should be updated according to the given training data and a gradient-basedlearning procedure to achieve a desired input/output mapping. It can be used as an

0.08

0.04

–0.00

–0.04

–0.08

–0.34

–0.17

–0.00

0 2 4 6 8 10Time [s]

12 14 16 18 20

0.17

0.34

Δω2–

Δω3 [

rad/

s]Δω

1–Δω

2 [ra

d/s]

NAPSS+CPSSOPEN

NAPSS+CPSSOPEN

Figure 11.13 System response with NAPSS installed on generators G1 and G3 andCPSS on G2, G4 and G5

� 2009 IEEE. Reprinted with permission from Malik O.P. ‘Adaptive and artificialintelligence based PSS’, Proceedings, IEEE PES 2003 General Meeting, Vol. 3,pp. 1792–7


identifier for non-linear dynamic systems or as a non-linear controller withadjustable parameters.

11.4.2.2 Training and performanceBecause the neuro-fuzzy controller has the property of learning, fuzzy rules andmembership functions of the controller can be tuned automatically by the learningalgorithm. Learning is based on the error in the controller output. Thus, it isnecessary to know the error that can be evaluated by comparing the output of theneuro-fuzzy controller and a desired controller.

To train this controller as an adaptive network-based fuzzy PSS (ANF PSS),training data were obtained from a self-optimising pole-shifting APSS. Trainingwas performed over a wide range of operating conditions of the generating unitincluding various types of disturbances. Based on earlier experience, seven lin-guistic variables for each input variable were used to get the desired performance.

Extensive simulation [36] and experimental studies with the ANF PSS showthat it can provide good performance over a wide operating range and can sig-nificantly improve the dynamic performance of the system over that with a fixedparameter CPSS.

11.4.2.3 Self-learning ANF PSSIn the above case the ANF PSS was trained by data obtained from a desired con-troller. However, in a general situation, the desired controller may not be available.In that case, the neuro-fuzzy controller can be trained using a self-learningapproach [37].

In the self-learning approach two neuro-fuzzy systems are used in a mannersimilar to Figure 11.11: one acting as the controller and the other acting as thepredictor. The plant identifier can compute the derivative of the plant’s output withrespect to the plant’s input by means of the back-propagation process illustrated bythe line passing through the forward identifier and continuing back through theneuro-fuzzy controller that uses it to learn the control rule.

Inference mechanism Fuzzi-fication

Defuzzi-fication

ReferenceInput

FLC

++

System Output

Knowledge base

Figure 11.14 Basic structure of fuzzy logic controller


The self-learning ANF PSS was initially trained offline on a power systemsimulation model over a wide range of operating conditions and disturbances.Electric power deviation and its integral were used as the input to the stabiliser. TheANF PSS, with the parameters, membership functions and inference rules obtainedfrom the offline training procedure, was implemented on a DSP mounted on a PCand its performance was evaluated on a physical model of a power system in thelaboratory. A digital CPSS was also implemented in the same environment on theDSP board for comparative studies.

Out of the various tests, results for a 0.25-pu step decrease in the input torquereference applied at 1 s and removed at 9 s with the generator operating at 0.9 pupower, 0.85 pf lag and 1.10 pu Vt are shown in Figure 11.15. The ANF PSSprovides a consistently good performance for either of the two disturbances.

Simulation studies on a single machine connected to a constant voltage bus andon a multi-machine power system [34] and experimental studies on a physicalmodel of a power system have demonstrated the effectiveness of the ANN PSS inimproving the performance of a power system over a wide operating range and abroad spectrum of disturbances.

11.4.2.4 Neuro-fuzzy controller architecture optimisationAdaptive fuzzy systems offer a potential solution to the knowledge elicitationproblem. The controller structure, expressed in terms of the number of membership

–0.210 3 6 9 12 15

–0.14

0

0.07

Time (s)

–0.07

Act

ive

pow

er d

evia

tion

(pu)

0.14

0.21

OPENCPSSANF PSS

Figure 11.15 Comparison of ANF PSS and CPSS responses to a 0.25 pu steptorque disturbance (P ¼ 0.9 pu, 0.85 pf lag)� 2009 IEEE. Reprinted with permission from Malik O.P. ‘Adaptive and artificialintelligence based PSS’, Proceedings, IEEE PES 2003 General Meeting, Vol. 3,pp. 1792–7


functions and the number of inference rules, is usually derived by trial and error.The number of inference rules has to be determined from the standpoint of overalllearning capability and generalisation capability.

The above problem can be resolved by employing a genetic algorithm todetermine the structure of the adaptive fuzzy controller. By employing bothgenetic algorithm and adaptive fuzzy controller, the inference rules parameterscan be tuned and the number of membership functions can be optimised at thesame time.

11.5 Amalgamated analytical and AI-based PSS

11.5.1 APSS with neuro identifier and PS controlA self-tuning APSS described above can improve the dynamic performance of thesynchronous generator by allowing the parameters of the PSS to adjust as theoperating conditions change. However, proper care needs to be taken in the designof the RLS algorithm for identification to make it stable, especially under largedisturbances.

It is possible to make the identification more robust by using an NN foridentifying the system model parameters. An analytical technique, such as the PScontrol, can be retained to compute the control signal. One approach, using a radialbasis function (RBF) network for model parameter identification, is describedbelow [38]. The APSS shown in Figure 11.6 now consists of an ANN identifier andthe pole-shifting control algorithm described above.

……

……

……

…

y(t – 1)

y(t – 2)

y(t – 3)

u(t – 1)

u(t – 3)

u(t – 2)

RBF CentresPast samples Linear combiner

Weights

Θ′(t) = [System parameters]

y(t) = f(y(t – 1), u(t – 1))

Centres: Adjusted using u-means clustering (offline)Weights: Adjusted using recursive least-square algorithm (online)

Figure 11.16 Radial basis function network model� 2009 IEEE. Reprinted with permission from Malik O.P. ‘Adaptive and artificialintelligence based PSS’, Proceedings, IEEE PES 2003 General Meeting, Vol. 3,pp. 1792–7


The RBF network (Figure 11.16) is used to identify the system model para-meters, ai, bi, (11.6) and (11.7). The network consists of three layers: input, hiddenand output. The input vector is

VðtÞ ¼ ½DPeðt � TÞ; DPeðt � 2TÞ;DPeðt � 3TÞ; uðt � TÞ;uðt � 2TÞ; uðt � 3TÞ�

ð11:18Þ

Each of the six input variables is assigned to an individual node in the inputlayer and passes directly to the hidden layer without weights. The hidden nodes,called the RBF centres, calculate the Euclidean distance between the centres andthe network input vector. The result is passed through a widely used Gaussianfunction characterised by a response that has a maximum value of 1 when thedistance between the input vector and the centre is 0. Thus, a radial basis neuronacts as a detector that produces ‘1’ whenever the input vector is identical to thecentre (active neuron). The other neurons with centres quite different from the inputvector will have outputs near 0 (non-active neurons).

The connections between the hidden neurons and the output node are linearweighted sums as described by the equation:

y ¼Xnh

i¼1

qtexp �kp � cik2

s2

!ð11:19Þ

where

ci, s, qt and nh are the centres, widths, weights and the number of hidden layerneurons, respectively.

To make the proposed RBF identifier faster for online applications, the hiddenlayer is created as a competitive layer wherein the centre closest to the input vectorbecomes the winner and all the other non-active centres are deactivated. Also, thescalar weights are modified as a vector qt whose size equals the size of the inputvector. The weight vector is given by

q0ðtÞ ¼ a01a02a0

3b01b02b03� � ð11:20Þ

Linearising the output of the RBF, y(t) ¼ f [y(t � 1), u(t � 1)], by Taylor seriesexpansion at each sampling instant, a one-to-one relationship between the weightvector q0 and the system model parameters qmðtÞ, (11.9), can be obtained. Theseparameters are then used in computing the control signal.

The RBF identifier was first trained offline to choose appropriate centres usingdata collected at a number of operating points for various disturbances. Then-means clustering algorithm used for training yielded 15 centres for the RBFmodel. After the offline training, the weights (system parameters) were updatedonline to obtain the appropriate control signal using the pole-shifting controller.A 100-ms sampling period was chosen for digital implementation.


Results of an experimental study for a 0.10-pu decrease in torque referenceapplied at 10 s and removed at 20 s, with the generator operating at 0.6 pu power,0.92 pf lead and Vt of 0.99 pu are shown in Figure 11.17. It can be seen that theAPSS can provide a well-damped response.

11.5.2 APSS with fuzzy logic identifier and PS controllerTakagi–Sugeno (TS) fuzzy systems have been successfully employed in the designof stabilisation control of non-linear systems.

A non-linear plant can be represented by a set of linear models interpolated bymembership functions of a TS fuzzy model. Although the TS system identifier isa NARMAX model, at each sample an average linear discrete auto-regressivemoving average (ARMA) model can be determined to identify the controlled plantaccording to the current active rules. This ARMA model can be used to determinethe control signal by the pole-shifting control strategy. Using this approach, a self-tuning adaptive controller has been developed and applied as a PSS [39].

The proposed single-input single-output TS model used for the identificationof dynamic systems is composed of fuzzy rules, the consequent part of which

Time (s)

ΔPe

(pu)

10 15 20 25 30–0.20

–0.15

–0.10

–0.05

0

0.05

0.10

0.15

50

OPENCPSSAPSS3

Figure 11.17 DPe response for 0.1 pu input torque reference step change withAPSS� 2009 IEEE. Reprinted with permission from Malik O.P. ‘Adaptive and artificialintelligence based PSS’, Proceedings, IEEE PES 2003 General Meeting, Vol. 3,pp. 1792–7


provides the rule output at time k based on the past inputs and past outputs withfuzzy sets designed in the universe of discourse. The consequent part of the rulethen identifies the parameters of a desired order discrete model of the plant. Twoparallel online learning procedures, one each for the identification of premise andconsequent parameters, are used to track the plant in real time [40].

In the proposed TS system for generating unit identification, two input signals,the past control input, u(k � 1), and the past generator speed output, y(k � 1), areused to identify a third-order model of the plant. The output at sample k is theestimated generator speed output, yðkÞ. The TS system is trained by usingthe steepest descent algorithm for the premise parameters and RLS algorithm forthe consequent parameters using the error of the system output and the estimatedTS output. Initially a set of three equally spaced membership functions, over thenormalised universe, are used for the inputs of the system.

The response of the system with the TS system-based identifier and PS con-troller-based APSS has been studied for various disturbances at different operatingconditions. One illustrative result for a three-phase to ground fault is shown inFigure 11.18.

0.8

0.85

0.9

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.40 1 2 3 4 5

Time (s)

Pow

er a

ngle

(rad

)

APSSCPSS

6 7 8 9 10

Figure 11.18 Three phase to ground fault at the middle of one transmission lineand successful reclosure (P ¼ 0.95 pu, 0.9 pf lag)� 2009 IEEE. Reprinted with permission from Abdelazim T., Malik O.P. ‘Fuzzylogic based identifier and pole-shifting controller for PSS application’. Proceedingsof Power Engineering Society General Meeting, 2003, IEEE; Toronto, Canada, July2003. pp. 1680–85


11.5.3 APSS with RLS identifier and fuzzy logic controlFLCs have attracted considerable attention as candidates for novel computational sys-tems because of the advantages they offer over the conventional computationalsystems. They have been successfully applied to the control of non-linear dynamicsystems, especially in the field of adaptive control, by making use of online training.

A self-learning adaptive FLC has been developed. Only the inputs and outputsof the plant are measured and there is no need to determine the states of the plant.Using online training by the steepest descent method and the identified systemmodel, the adaptive FLC is able to track the plant variations as they occur andcompute the control.

In the proposed controller, a discrete model of the plant is first identified usingthe RLS parameter identification method. This allows a continuous tracking of thesystem behaviour.

The control learning is based on the prediction of the identified model. Theidentified model output is used as input to the Mamdani-type PD controller [20].The centre points of the controller inputs are updated [40] by treating them exactlythe same as the weights of an NN and by using the steepest descent algorithm withchain rule.

The proposed adaptive FLC has been applied as an adaptive fuzzy PSS(AFPSS) [41]. For the AFPSS, the generating unit is identified as a third-ordermodel. The controller has two input signals, the generator speed deviation and itsderivative, with an initial set of seven equally spaced membership functions overthe normalised universe of discourse. The output, the supplementary control signal,also having seven membership functions, is added to the AVR summing junction.A number of simulation studies have been performed for various disturbances atdifferent operating conditions. An illustrative result for a 0.05-pu increase in torqueand return to initial condition, shown in Figure 11.19, demonstrates the perfor-mance of this AFPSS.

11.6 APSS based on recurrent adaptive control

For nonlinear systems with a general form, unless the reference model is welldefined, the traditional MRAC may cause system oscillations. This problem arisesbecause the connection between the current system state and the controller para-meters is ignored by the MRAC. The focus of recurrent adaptive control (RAC) ison optimising a certain objective function in which the lost connection is picked up.

Development of RAC is inspired on observing the similarity between theadaptive control system and the recurrent NNs (RNNs) [42]. Because of that, amodified version of the back propagation through time (BPTT) [43], a learningalgorithm of RNNs, can be exploited in RAC. A new control algorithm for RAC,named recursive gradient (RG), which improves the performance of the originaland truncated BPTT algorithms, has been developed and an APSS has beendeveloped based on the RG algorithm.


The system in Figure 11.3 can also be expressed by the non-linear equations

X ðk þ 1Þ ¼ FðX ðkÞ;UðkÞÞUðkÞ ¼ GðX ðkÞ; qÞ

(ð11:21Þ

where

X(k), X(k þ 1)2 Rp, U(k) 2 Rq and q 2 Rr. p, q and r are the number of systemstates, system inputs and controller parameters, respectively. At each discretetime k, the controller parameters q are updated online to minimise a predefinedobjective function J(X(k þ 1)), (11.22), that is used to evaluate the controlperformance.

minq

JðX ðk þ 1ÞÞ ð11:22ÞIn many cases, the performance index J(X(k þ 1)) is not directly related to the

state but the output of the system. The output of the system, Y(k), can be a subset ofthe state X(k) or, in general way, a function of the state. Many non-linear optimi-sation methods can be utilised to minimise the performance index J(X(k þ 1)).

100.62

0.63

Pow

er a

ngle

(rad

)

0.64

0.65

0.66

0.67

0.68

0.69

2 3 4Time (s)

APSSCPSSNo PSS

5 6 7

Figure 11.19 Response to a 0.05 pu step increase in torque and return to initialcondition (P0 ¼ 0.95 pu, 0.9 pf lag)� 2009 IEEE. Reprinted with permission from Abdelazim T., Malik O.P. (eds.).‘An adaptive power system stabilizer using on-line self-learning fuzzy system’.Proceedings of Power Engineering Society General Meeting, 2003, IEEE; Toronto,Canada, July 2003. pp. 1715–20


One of the most popular local optimisation methods is the gradient descentalgorithm [44] with one-step or multi-step optimisation control. In these schemes itis also assumed that the current state X(k) is independent of the controller parameterq and, therefore,

@X ðkÞ@q

��q¼qðkÞ

¼ 0 ð11:23Þ

In this approach, the feedback loop, which makes the adaptive control systemto be a recurrent system, is ignored. However, the real control system is recurrent.

By using the rule given by (11.24):

qðk þ 1Þ ¼ qðkÞ � a � @X ðk þ 1Þ@q q¼qðkÞ

� @JðX ðk þ 1ÞÞ@X ðk þ 1Þ

�� ð11:24Þ

wherea is the step size and

@X ðk þ 1Þ@q

��q¼qðkÞ

¼ @X ðkÞ@q

��q¼qðkÞ

@X ðk þ 1Þ@X ðkÞ þ @UðkÞ

@X ðkÞ@X ðk þ 1Þ@UðkÞ

� �

þ @UðkÞ@q

��q¼qðkÞ

@X ðk þ 1Þ@UðkÞ

ð11:25Þ

to update the controller parameters in RAC, the assumption (11.23) can be removed.Control algorithm (11.24) can be solved using the BPTT or the truncated

BPTT algorithm. Both versions of BPTT control algorithm require extensivecomputation. This problem can be overcome using the RG algorithm in which thecontroller parameters q are updated by the following rule:

qðk þ 1Þ ¼ qðkÞ � a �Xk

m¼0

lm � @X ðk þ 1Þ@q

��q¼qðkÞ;X ðk�mÞ

" #� @JðX ðk þ 1ÞÞ

@X ðk þ 1Þ ð11:26Þ

where 1 > l> 0 and a is the step size.

Although the RG control algorithm has been developed for RAC control applica-tions, it can also be used to train RNNs.

To design an APSS based on the RAC, a model that tracks the dynamics of thesynchronous machine has to be built first. A second-order operating conditiondependent (OC-dependent) ARMA model [45] has been used in this application.By rewriting the A and B coefficients in the widely used model (2) as functions ofthe operating conditions represented by the easily measured active power output,Pe, and reactive power, Qe, the OC-dependent ARMA model is given by

Dw ðk þ 1Þ ¼ a1ðPe;QeÞDwðkÞ þ a2ðPe;QeÞDwðk � 1Þþ b1ðPe;QeÞðuðkÞ � uðk � 1ÞÞ ð11:27Þ


where

a1ðPe;QÞ ¼XN

i¼1

ai1riðPe;QeÞ

a2ðPe;QeÞ ¼XN

i¼1

ai2riðPe;QeÞ

b1ðPe;QeÞ ¼XN

i¼1

bi1riðPe;QeÞ

The operating region vector is f¼ [Pe Qe]t and the region function ri(f) is com-monly chosen as a normalised Gaussian function.

In this model, one set of parameters can work for various operating conditionswithout updating [36]. The model (11.27) can be realised by N local model net-works (LNNs) [46] that are a general form of RBF networks [47]. Each local modelcan be interpreted as a good approximation to the desired function in a regiondefined by the region function.

The synchronous generator with the OC-dependent linear controller can bewritten as

X ðk þ 1Þ ¼ AX ðkÞ þ BuðkÞuðkÞ ¼ HðX T ðkÞFcqLCÞ

(ð11:28Þ

where

A ¼a1ðPe;QeÞ a2ðPe;QeÞ �b1ðPe;QeÞ

1 0 0

0 1 0

2664

3775

B ¼ b1ðPe;QeÞ 0 1½ �T

Fc ¼r1ðfÞ 0 0 rN ðfÞ 0 0

0 r1ðfÞ 0 � � � 0 rN ðfÞ 0

0 0 r1ðfÞ 0 0 rN ðfÞ

264

375

qLC ¼ ½g11; g12; h11; . . . ; gN 1; gN 2; hN 1�T

and H function involves hard constraints.Using the objective function:

JðX ðk þ 1ÞÞ ¼ 12

Dw2ðk þ 1Þ þ bu2ðkÞ� �¼ 1

2X T ðk þ 1ÞQX ðk þ 1Þ ð11:29Þ


where

b is the weight on energy expense in the objective function and

Q ¼1 0 00 0 00 0 b

24

35

the RG control algorithm for the OC-dependent linear PSS, given in (11.30), isobtained as

q k þ 1ð Þ ¼ q kð Þ � a0 � D0 kð Þ � Q � X k þ 1ð Þ ð11:30Þwhere

D0 kð Þ ¼ ð1 � lÞ � R kð Þ þ lD0ðk � 1ÞT kð Þ

RðkÞ ¼ FcT X ðkÞBT umin � uðkÞ � umax

0 umin > uðkÞ or uðkÞ > umax

(

TðkÞ ¼ A þFCqLCðkÞBT umin � uðkÞ � umax

0 umin > uðkÞ or uðkÞ > umax

(

D0(� 1) ¼ 0 (initial condition).

5Time (s)

Pow

er a

ngle

(rad

)

6 7 8 9 1000.4

0.42

0.44

0.46

0.48

0.5

0.52

0.54

1 2 3 4

CPSSPSPSSRGPSS

Figure 11.20 Response to a 50 ms three-phase short circuit fault at the middle ofone transmission line at 1 s (Pe ¼ 1.0 pu, pf ¼ 0.85 lag)� 2009 IEEE. Reprinted with permission from Zhao P., Malik O.P. Design of anadaptive PSS based on Recursive Adaptive Control theory. IEEE Transactions onEnergy Conversion. 2009;24(4):884–92


0–1.5

Spee

d di

ffere

nce

(pu)

Spee

d di

ffere

nce

(pu)

Spee

d di

ffere

nce

(pu)

–1

–0.5

0

0.5

1

1.5� 10–4

� 10–4

� 10–4

2 4 6 8 10Time (s)(a)

12 14 16 18 20

0–8

–6

–4

–2

0

2

4

6

8

–8

–6

–4

–2

0

2

4

6

8

2 4 6 8 10

Time (s)

No PSSCPSSRGPSS

No PSSCPSSRGPSS

No PSSCPSSRGPSS

(b)

12 14 16 18 20

0 2 4 6 8 10Time (s)(c)

12 14 16 18 20

Figure 11.21 System response to 0.10 pu step increase in the mechanical torquechange on the generator G3 in the five-machine power system. PSSsare installed on G1, G2 and G3 (a) Speed different between thegenerator G1 and G2 (b) Speed different between the generatorG2 and G3 (c) Speed different between the generator G1 and G3

� 2009 IEEE. Reprinted with permission from Zhao P., Malik O.P. Design of anadaptive PSS based on Recursive Adaptive Control theory. IEEE Transactions onEnergy Conversion. 2009;24(4):884–92


The OC-dependent ARMA model was first trained offline on a single machineconstant voltage bus system described in Section 11.3.2 with white noise signal asthe input. Based on this sufficiently trained model, the APSS is trained offline bythe RG control algorithm over a wide range of operating conditions varying from0.1 pu to 1.0 pu power and 0.6 lag to 0.8 lead power factor.

Performance of the synchronous machine operating at 1.0 pu power, 0.85 pflag without PSS, with a CPSS and with the PSS based on the RG algorithm for ashort-circuit at the middle of one line at 1 s, opening of the line after 50 mms andline reconnection at 6 s is shown in Figure 11.20. This APSS was also tested on thefive machine power system (Figure 11.12). Response for a 0.1-pu increase in inputtorque reference of G3 at 1 s is shown in Figure 11.21. The system returns to itsinitial conditions at 11 s.

It can be seen that the multi-modal oscillations are also damped out moreeffectively than with a CPSS.

11.7 Concluding remarks

PSSs based on the control algorithms described above have been studied exten-sively in simulation. They have also been implemented and tested in real time onphysical models in the laboratory with very encouraging results. The pole-shiftingcontrol algorithm-based APSS has also been tested on a multi-machine physicalmodel [30], on a 400-MW thermal machine under fully loaded conditions con-nected to the system [31], and is now in regular service in a hydro power stationafter extensive testing in the field [48]. These studies have shown clearly theadvantages of the advanced control techniques and intelligent systems.

Very satisfactory adaptive controllers can be developed and implementedusing a number of approaches, i.e. purely analytical, purely AI techniques or byamalgamating the analytical and AI approaches. Which approach to use depends onthe expertise of the designer and the developer of the controller, and the confidencethat they or the client have in a particular technology.

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Chapter 12

Series compensation

The transmission network connects scattered clusters of loads to different sets ofgenerating units in the power system. Such networks are often required to becompensated for reactive power flow to maintain proper voltage levels. Compen-sation can be provided by inserting elements that are capable of delivering orabsorbing reactive power resulting in changes of reactive power flow in the trans-mission network. Thus, the reactive power flow can be controlled to make anoptimal use of the transmission system in a manner that the voltage along thetransmission lines and at the network nodes is kept within desired values as well asincreasing the power transfer capacity of the transmission lines. Consequently, thesystem stability may be improved.

The compensation highly pertains to the transmission system and is providedby installing capacitors and/or reactors at different locations in the transmissionnetwork. Capacitors are installed either in series with the transmission lines called‘series capacitive compensation’ or in shunt at specific points, commonly, near theloads called ‘shunt capacitive compensation’. Reactors can be used as shunt reac-tive compensators connected to the transmission lines or the transmission nodes atlocations determined by the study and its objectives.

This chapter focuses on the series compensation of transmission network andits benefits, in particular, the system stability improvement. This entails explana-tion and discussion of some definitions and some basic concepts as below.

12.1 Definitions of transmission line parameters

Parameters ABCD:

As explained in Chapter 4, the voltage and current at the sending end, VS and IS,respectively, can be deduced from (4.23) by substituting x ¼ l (the total length ofline), thus

VS ¼ AVR þ BIR

IS ¼ CVR þDIR

)ð12:1Þ

and in matrix form, (12.1) can be written as

VS

IS

� �¼ A B

C D

� �VR

IR

� �ð12:2Þ

where

A ¼ cosh gl; B ¼ ZC sinh gl; C ¼ 1=ZCð Þsinh gl; D ¼ A ¼ cosh gl ð12:3Þand

ZC ¼ffiffiffiz

y

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffirL þ jxL

gc þ jbc

sð12:4Þ


p ¼ aþ jb ð12:5ÞThe transmission line is described by complex parameters, ABCD, which relate

the voltage and current at the sending end to those at the receiving end. The para-meters A and D are dimensionless, B has dimensions ofW and C has dimensions of ℧.

The characteristic impedance, ZC:

It is the square root of the ratio of the line series impedance per unit length to theline shunt admittance per unit length. It is given by (12.4). The reciprocal of ZC isdefined as the characteristic admittance, YC.

The propagation constant, g:

It is the square root of the product of the line series impedance per unit length andthe line shunt admittance per unit length as given by (12.5). Thus, g is a complexnumber. The real part, a, is defined as the attenuation constant and the imaginarypart, b, is called ‘the phase constant’.

The natural power, Pn:

It is the power transmitted by the line that is terminated by its characteristicimpedance. Natural power is often used to indicate the nominal capability of theline, sometimes called the ‘characteristic impedance loading’ or ‘surge impedanceloading, SIL’ of the transmission line. It is defined as

Pn ≜ SIL ¼ V 2

Z�C

ð12:6Þ

where V is the line-to-line voltage and Pn is a three-phase power. It includes bothactive and reactive power. If V is the line-to-neutral voltage, the value of Pn is perphase.

The total line angle, q:

q ¼ Im glð Þ ¼ bl ¼ 2pl

lrad ð12:7Þ

where l is the wavelength of the line. For fixed line parameters, the line angle is aconstant.

Based on the definitions of the transmission line parameters given above,some concepts are discussed for two cases in the next two forthcoming sections.


The first case is the lossless case where the real part of both z and y are assumedto be zero. The second is a real case in which the resistance of the system isconsidered [1].

12.2 Compensation of lossless transmission line

A lossless transmission line uniformly compensated through its length is con-sidered. Although this case is not real and cannot be seen in practice, it gives a goodunderstanding of reactive compensation impact through simplified mathematics.

From (12.4) and (12.5), the characteristic impedance and propagation con-stant are

ZC ¼ffiffiffiffiffixL

bc

r¼ RC ð12:8Þ

g ¼ jbl ¼ jlffiffiffiffiffiffiffiffiffixLbc

pð12:9Þ

It is noted that ZC is a real number, RC, and g becomes purely imaginary.Consequently, the sinh function is purely imaginary. Accordingly and from (12.3),the following relations can be written:

ImB ¼ RC Im sinh glð ÞImC ¼ 1

RCIm sinh glð Þ ¼ ImB

R2C

9=; ð12:10Þ

Thus,

ImB

ImC¼ R2

C ¼ xL

bcð12:11Þ

The line angle as defined by (12.7) is computed for lossless line as

q ¼ bl ¼ l Imffiffiffiffiffizy

p� � ¼ lffiffiffiffiffiffiffiffiffixLbc

pð12:12Þ

From (12.6), the three-phase natural power is

Pn ¼ V 2

RCMW ð12:13Þ

where V is the line-to-line voltage in kV and RC is the characteristic impedance inohms. It is noted that this relation does not include reactive power and the voltageprofile across the line when operating at its natural loading is flat.

12.2.1 Determination of amount of series compensationIt has been shown that it is worthy to operate the line at its natural loading.Satisfying this condition for all lines in the system is not practical as the line

Series compensation 313

loadings depend on the load, generation and system configuration. So it is betterto attempt to change the natural loading, i.e. SIL of each line, by using seriescompensation to conform to the flow on that line where the inductive reactance ofthe line is varied by varying the amount of series compensation. This can beillustrated by the relations below that determine the compensated line parametersas a ratio to the uncompensated case. The subscript o is added to denote theuncompensated case.

The characteristic impedance ratio is

RC

RCo¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixL

bco

bco

xLo¼

s ffiffiffiffiffiffixL

xLo

rð12:14Þ

The ratio of line angle is computed as

qqo

¼ffiffiffiffiffiffiffiffiffiffiffiffixLbco

xLobco

s¼

ffiffiffiffiffiffixL

xLo

rð12:15Þ

and the ratio of natural power is given by

Pn

Pno¼ RCo

RC¼

ffiffiffiffiffiffixLo

xL

rð12:16Þ

It is to be noted that in (12.14) through (12.16) the capacitive susceptance, bco, isunchanged for series compensated line. In addition, the subscript C as an uppercase letter refers to the characteristic impedance while the subscript c as a lower caseletter refers to the capacitive susceptance.

The total reactance of series compensation, XC, is commonly expressed as apercentage of the total line inductive reactance, which is called the ‘degree of seriescompensation’. As it is assumed that the lossless line is uniformly compensated, thenet line reactance per unit length is written as

xL ¼ xLo � XC

lð12:17Þ

and the degree of series compensation, k, is defined as

k ¼ XC

ImBoð12:18Þ

Another definition of the degree of series compensation, which is referred to asthe nominal degree of series compensation, knom, is expressed as

knom ¼ XC

xLolð12:19Þ


Both of the above definitions may be used in accordance with (12.18) and(12.19). In terms of knom and using (12.17) the ratios of the characteristic impe-dance, line angle and natural power (12.14–12.16) can be rewritten as

RC

RCo¼ q

qo¼

ffiffiffiffiffiffixL

xLo

r¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � knom

pð12:20Þ

Pn

Pno¼ 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1 � knomp ð12:21Þ

Therefore, increasing the degree of series compensation increases the naturalpower and decreases the characteristic impedance and line angle as well.

The increase of natural power results from the amount of reactive power sup-plied by the series compensation. To determine this amount, it is supposed that theseries capacitors are added uniformly along the length of the lossless line in order toincrease the natural power to times its uncompensated value, > 1. Thus,(12.21) gives

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � knom

p ¼ and then

knom ¼ 1 � 12 ð12:22Þ

Incorporating (12.19) and (12.22) obtains

XC ¼ 1 � 12

� �lxLo ð12:23Þ

Hence, the three-phase reactive power supplied by series compensation, DQC, canbe computed as

DQC ¼ 3XCð InoÞ2 ð12:24ÞSubstituting XC from (12.23) to (12.24) gives

DQC ¼ ð 2 � 1ÞlPno

ffiffiffiffiffiffiffiffiffiffiffiffixLobco

pð12:25Þ

The reactive power can be expressed as a ratio to the uncompensated power as

DQC

Pno¼ ð 2 � 1Þ l

ffiffiffiffiffiffiffiffiffiffiffiffixLobco

pð12:26Þ

From (12.20) and (12.22) the ratio of line angle is

qqo

¼ 1 ð12:27Þ


Example 12.1 Typical data of an overhead transmission line are given as: nom-inal voltage 345 kV, rL ¼ 0.037 W/km, XL ¼ 0.367 W/km, bc ¼ 4.518 ms/km, and thetotal length of the line is 160 km. The line is bundled conductors and rL, xL, bc areper phase values.

Considering the line as a lossless line (neglecting the resistance), find theelectric line parameters. If the natural power, SIL, is required to be 1.5 times thenominal value by using series compensation, calculate the degree of compensationand the amount of reactive compensation.

Solution:

The characteristic impedance ZC ¼ffiffiffiffixLbc

q¼ RC ¼ 103

ffiffiffiffiffiffiffiffi0:3674:518

q¼ 285W

The propagation constant g ¼ jb ¼ jffiffiffiffiffiffiffiffiffixLbc

pwhere b ¼ 10�3

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi0:367 � 4:518

p ¼0:00129 rad=km ¼ 0:074�=km

The line angle q¼ bl ¼ 0.00129 � 160 ¼ 0.2064 rad ¼ 11.83�

Pn ≜ SIL ¼ V 2

RC¼ 417:63 MW three phaseð Þ

When the line is series compensated to increase Pn to be 1.5 � 417.63 MW, therequired degree of series compensation ‘knom’ can be obtained using (12.22). Thus,

knom ¼ 1 � 4=9ð Þ ¼ 55:6%

Using (12.25), the amount of reactive compensation, DQC ¼ 107.55 MVAR andusing (12.27), the line angle ¼ 0.2064/1.5 ¼ 0.1376 rad ¼ 7.89�

12.2.2 Transient stability improvement for losslesscompensated line

The series compensation supplies the system a reactive power that increases thenatural power of the transmission line, which in turn increases its power transfercapacity. This results in an improvement of the system transient stability. Theamount of reactive compensation can be determined by examining the method ofcompensation as well as the system operation when subjected to different faults.

Usually, a three-phase fault is considered as the most severe fault. The analysisbelow illustrates this concept through studying a simple uncompensated and com-pensated power system (one machine to an infinite bus).

For uncompensated system, the voltages at the two ends of the transmissionline, sending and receiving, are VS and VR, respectively. The receiving end voltage,VR, at the infinite bus is taken as a reference. The total reactance between thesending and receiving ends is denoted by XL and the angle between VS and VR isdenoted by d. The equivalent circuit is shown in Figure 12.1.

It is seen that VS ¼ VSejd and VR ¼ VR ff0 ð12:28Þ


The current flow from sending to receiving end, I, is given by

I ¼ VS � VR

jXL¼ VS sin d

XL� j

VS cos d� VR

XLð12:29Þ

The complex power phasor at both the sending and the receiving ends can becomputed as

PS þ jQS ¼ VSI�

¼ VSVR sin dXL

� �þ j

V 2S � VSVR cos d

XL

� �9>=>; ð12:30Þ

PR þ jQR ¼ VRI�

¼ VSVR sin dXL

� �� j

V 2R � VSVR cos d

XL

� �9>=>; ð12:31Þ

For compensated system:The voltages at sending and receiving end are obtained by the same (12.28).Assuming the reactance of the series capacitors is XC, the relations to calculate thepower at the two ends are the same as above but replacing the reactance XL by X.From (12.17) and (12.19), it can be found that

X ¼ XL 1 � knomð Þ ð12:32Þ

To calculate the reactive power of the capacitors, QC, using (12.29), first, it is seenthat

I2 ¼ 1X 2

V 2S þ V 2

R � 2VSVR cos d� � ð12:33Þ

Using (12.32) the value (XC/X2) can be computed as

XC

X¼ XC

XL 1 � knomð Þ ¼knom

1 � knomð Þ ð12:34Þ

jXL

+ +

_ _

I

VR 0°VS δ°

Figure 12.1 Equivalent circuit of a simple power system


Hence,

XC

X 2¼ 1

X

XC

X¼ 1

X

knom

1 � knom¼ knom

XL 1 � knomð Þ2 ð12:35Þ

Thus, from (12.33) and (12.35)

QC ¼ I2XC ¼ knom

XL 1 � knomð Þ2 V 2S þ V 2

R � 2VSVR cos d� � ð12:36Þ

Example 12.2 The parameters of the equivalent circuit (Figure 12.1) have the perunit values as

VS ¼ 1:2ff11�; VR ¼ 1:0ff0�; XL ¼ 0:275

Find the power at both ends when the system is uncompensated as well as com-pensated at 50 per cent degree of compensation.

Solution:

It is noted that the subscript o is added to the parameters in case of uncompensatedsystem. Using (12.30) and (12.31), the powers at the two ends of the transmissionline are

PSo þ jQSo ¼ 1:2 sin 11�

0:275þ j

1:44 � 1:2 cos 11�

0:275¼ 0:833 þ j0:945 pu

PRo þ jQRo ¼ 1:2 sin 11�

0:275þ j

1 � 1:2 cos 11�

0:275¼ 0:833 þ j0:6545 pu

It is seen that PSo and PRo are equal as the system is lossless while QSo and QRo

differ by the reactive power absorbed by the system reactance.

In case of compensation, X ¼ XL(1 � knom) ¼ 0.275(1 � 0.5) ¼ 0.1375 pu

Similarly, PS þ jQS ¼ 1.665þ j1.891 and PR þ jQR ¼ 1.665þ j1.309 pu

Therefore, the power transfer capacity of the line at the same power angle, d, isdoubled by the series compensation. It means that by referring to equal area cri-terion, the area between the P–d curve and the input power line is increasedresulting in more marginal stability and security.

As computed by (12.36) the reactive power of the capacitors is

QC ¼ knom

XL 1 � knomð Þ2 V 2S þ V 2

R � 2VSVR cos d� �

¼ 0:5

0:275 1 � 0:5ð Þ2 1:44 þ 1 � 2:4 cos 11�ð Þ ¼ 0:582 pu


Example 12.3 For the system in Example 8.1-i, find the change of d versus timeat fault clearing time 0.08 s when the system is uncompensated and compensatedwith 30 per cent and 50 per cent, degree of compensation.

Solution:

Using the initial values of system parameters obtained in Example 8.1, andapplying PSAT/MATLAB� toolbox to solve the swing equation, the variation ofpower angle versus time is shown in Figure 12.2. It is seen that the stability of thesystem is improved by compensation.

For high-voltage lines, the resistance is small compared to the reactance. Inaddition, for overhead lines, the real part of the line admittance is zero. Thus, theselines may be approximated with a little error as lossless lines. For long transmissionlines, further detailed analysis to describe the transmission lines ‘in a more realisticmanner’ is given in Section 12.3.

12.3 Long transmission lines

As explained in Chapter 4, Section 4.2.1, voltage and current, as phasors, Vs and Is,at a point distance s from the receiving end, are calculated using (4.22). For con-venience they are written again as

V s ¼ VR þ ZcIR

2egs þ VR � ZcIR

2e�gs

I s ¼ VR=ZCð Þ þ IR

2egs � VR=ZCð Þ � IR

2e�gs

9>>=>>; ð12:37Þ

The first term in each equation is called the ‘incident component’ and thesecond term is the ‘reflected component’. If the line is terminated at the receiving

0 0.1 0.2 0.3 0.4 0.5

Time (s)

45

55

65

75

85

95

δ°

Uncompensated

50% compensation

30% compensation

Figure 12.2 Power angle variation versus time


end by a constant impedance load ZLd, then the voltage and current at the receivingend must satisfy the relation:

VR ¼ ZLdIR ð12:38Þ

Therefore, (12.37) can be expressed as a function of ZLd as below:

V s ¼ VR

21 þ ZC

ZLd

� �egs þ VR

21 � ZC

ZLd

� �e�gs

I s ¼ VR

2ZC1 þ ZC

ZLd

� �egs � VR

2ZC1 � ZC

ZLd

� �e�gs

9>>>=>>>;

ð12:39Þ

A complex constant ‘L’ is defined as the ‘reflection coefficient’ and written as

L ¼ ZLd � ZC

ZLd þ ZCð12:40Þ

Assuming the impedance ratio ZCLd ¼ ZC/ZLd, (12.40) can be written in the form

V s ¼ VR

21 þ ZCLdð Þegs þ VR

21 � ZCLdð Þe�gs

I s ¼ VR

2ZC1 þ ZCLdð Þegs � VR

2ZC1 � ZCLdð Þe�gs

9>>=>>; ð12:41Þ

where

ZCLd ¼ ZC

ZLd¼ ZCejqC

ZLdejqLd¼ ZCLdej qC�qLdð Þ ð12:42Þ

By examining the case at which ZC ¼ ZLd, i.e. the line is terminated at itsreceiving end by impedance equal to the line characteristic impedance, it is foundthat (12.38) becomes VR ¼ ZCIR, and the voltage and current along the line as afunction of the distance x can be obtained from (12.41) as

V x ¼ VRegx

I x ¼ VR

ZCegx

9=; ð12:43Þ

It is noted that both the reflection component and the reflection coefficientequal zero, and the driving point impedance at any distance x from the receivingend equals ZC.

In addition, the voltage and current magnitudes increase from the receiving endtowards the sending end by the amount of attenuation constant, and the phaseangles move linearly in the leading direction. On the other hand, the attenuationconstant is usually small; consequently, the voltage magnitude is almost constantalong the line and is exactly constant for the lossless line.


As defined by (12.6), the SIL is given by

SIL ¼ V 2

Z�C

¼ V 2ðRC þ jXCÞjZCj2

ð12:44Þ

The parameter SIL provides a useful measure of the line nominal capability aswell as the condition at which the line absorbs or delivers reactive power. As thereflected component of the voltage or current vanishes when the line is terminatedby its characteristic impedance, the reactive power generated by the distributed linesusceptance is exactly the reactive I2X losses in the line. Therefore, at loadingsgreater than SIL and to maintain the voltage within normal limits it is necessary tosupply VArs to the line. Conversely, the line generates excess VArs at loadings lessthan SIL.

It is concluded that to guarantee the network voltages at the different nodes tobe nearly constant, the lines terminated by these nodes should be loaded with SIL.

The active and reactive power phasor power at the receiving end can be cal-culated as below.

SR ¼ VRI�R VA=phase ð12:45Þ

Substituting: IR ¼ (VS � AVR)/B, VR ¼ VRe j0, VS ¼ VSe jd

A ¼ Ae jm; B ¼ Be jz

Then,

SR ¼ VRI�R ¼ PR þ jQR ¼ VSVR

Bcos d� zð Þ � AV 2

R

Bcos m� zð Þ

� �

� jVSVR

Bsin d� zð Þ � AV 2

R

Bcos m� zð Þ

� �ð12:46Þ

The angle m � z and for lossless line z¼ 90�, VS and VR are phase voltages in kV,thus

PR ¼ VSVR

Xsin d ð12:47Þ

The maximum active power at receiving end, PRmax, can be computed by using(12.46) and substituting d� ¼ z� to obtain

PRmax ¼ VSVR

B� AV 2

R

Bcos m� zð Þ MW=phase ð12:48Þ

12.3.1 Series compensation for long transmission linesIn practice, series compensation is carried out by inserting capacitor banks in serieswith the transmission line located somewhere along the line or at its two ends. So,the compensation is not uniformly distributed along the line as it is assumed in the


former analysis. To describe the line in this case, the simplest way is to derive theline equations in terms of the equivalent ABCD parameters as below.

Consider the compensated line shown in Figure 12.3(a), where the seriescapacitors are located at a specific distance from the receiving end dividing the lineinto two parts described by A1B1C1D1 and A2B2C2D2 parameters. The system canbe represented as shown in Figure 12.3(b). The compensation can be described byAcomBcomCcomDcom where Acom ¼ Dcom ¼ 1, Ccom ¼ 0, Bcom ¼�jXC and the rela-tion of VS and IS as a function of VR and IR is

VS

IS

" #¼ A1

C1

B1

D1

" #1 �jXC

0 1

" #A2 B2

C2 D2

" #VR

IR

" #

¼ A1A2 � jXCA1C2 þ B1C2 A1B2 � jXCD2A1 þ B1D2

C1A2 � jXCC1C2 þD1C2 C1B2 � jXCC1D2 þD1D2

" #VR

IR

" #

ð12:49ÞIf the line is uncompensated and comprises of two parts in cascade described

by A1B1C1D1 and A2B2C2D2, respectively, then the parameters of the total line are

Ao ¼ A1A2 þ B1C2Bo ¼ A1B2 þ B1D2

Co ¼ C1A2 þD1C2Do ¼ C1B2 þD1D2

It is noted that the subscript o denotes the uncompensated case. Equation (12.49)can be rewritten in the form

VS

IS

� �¼ Aeq

Ceq

Beq

Deq

� �VR

IR

� �ð12:50Þ

–jXC

VS

IS

VR

IR

(a)

A1 A2AcomB1 Bcom B2

C2 D2C1 D1 Ccom Dcom

VS VR

IS IR

Part I of the line Part II of the line

(b)

Figure 12.3 (a) Compensated transmission line and (b) representation ofcompensated line


where

Aeq ¼ Ao � jXCA1C2Beq ¼ Bo � jXCA1D2 ¼ Bo þ DB

Ceq ¼ Co � jXCC1C2Deq ¼ Do � jXCC1D2

The same procedure can be applied to a line compensated at the sending andreceiving end by equal capacitance (�jXC/2) (Figure 12.4).

The equivalent ABCD parameters are given by

Aeq Beq

Ceq Deq

" #¼ 1 �jXC=2

0 1

" #A B

C D

" #1 �jX C=2

0 1

" #

¼ A� jCXC=2 B � CX 2

C

4� j AþDð ÞXC

2C D� jCXC=2

24

35 ð12:51Þ

It is to be noted that the location of the series capacitors along the transmissionline, e.g. lumped at midpoint, equally divided at line ends or two equal capacitors atequal distances, affects the value of equivalent ABCD parameters, which in turnchanges the transmission power transfer [2, 3]. From (12.51) the value of parameterB becomes

Beq ¼ B þ DB where DB ¼ �CX 2

C

4� j AþDð ÞXC

2

It is seen that DB represents the change of B due to compensation. It can beconsidered an index to indicate the impact of compensation on the line reactance,

–jXC/2 –jXC/2

VS

IS

VR

IR

IS IR

(a)

AA1com B1com B

C DC1com D1com

A2com B2com

C2com D2com

VS VR

The line

(b)

Figure 12.4 (a) Transmission line with series compensation at both ends and(b) equivalent representation


i.e. the effectiveness of compensation [4]. As shown in (12.50) and (12.51) thevalue of DB differs, and thus the effectiveness of series compensation variesaccording to its location.

Example 12.4 A single machine represented by an internal voltage source,E ¼ 1.4ff17� behind a reactance is connected to an infinite bus through a transfor-mer and two identical parallel transmission lines as shown in Figure 12.5. The dataof the transmission line are given below.

r ¼ 0.028 W/km, xL ¼ 0.325 W/km, bc ¼ 5.2 ms/km, l ¼ 160 km, the nominalvoltage ¼ 500 kV. Find the transmission network parameters ‘ABCD’ that relatethe machine voltage and current to the voltage and current at the receiving end‘infinite bus’ for two cases: (i) uncompensated transmission lines and (ii) each lineis compensated by series capacitors with 50 per cent degree of compensationlocated at the midpoint of the line.

If a fault occurs at point F (Figure 12.5) and cleared at power angle d¼ 50� byisolating the faulty line, examine the transient stability of the system to indicatehow much the stability is enhanced.

Solution:

For each transmission line

ZC ¼ffiffiffiz

y

r¼ 250W


p ¼ aþ jb ffi jffiffiffiffiffixy

p1 � j

r

2x

a ¼ 0:000057 nepers=km; b ¼ 0:0013 rad=km

XL ¼ 0:325 � 160=250 ¼ 0:208 pu of ZC

Bc ¼ 5:2 � 10�6 � 160 � 250 ¼ 0:208 pu of ZC

bl ¼ 0:208 rad ¼ 11:9�

V∞ = 1.0 0Vs θsE δ

R = 0.003xd ′ = 0.2

xtr = 0.1

Vt θt

F

Figure 12.5 System of Example 12.4


As a � b it is assumed that g¼ jb to simplify the calculations without loss ofgenerality.

Applying (12.3), the ABCD parameters of the line are

AL ¼ 0:978ff0�; BL ¼ j0:206 pu; CL ¼ j0:206 pu; DL ¼ 0:978ff0� ð12:52Þ

For uncompensated transmission lines:

*When one of the two lines is isolated (Figure 12.6), the system ABCD parameters are

A ¼ AL þ j0:3CL ¼ 0:916ff0�

B ¼ BL þ j0:3DL ¼ j0:499 pu

C ¼ CL ¼ j0:206 pu

D ¼ DL ¼ 0:978ff0�

Using (12.46), the active power transfer capacity at the receiving end is obtained as

PR ¼ 1:4=0:499ð Þsin 17� ¼ 0:82 pu=phase

*When the two lines are connected to the system in parallel (Figure 12.7), thesystem ABCD parameters are calculated as below

AL BL

CL DL

E VR

Ig IgIR IR1 j0.3

0 1

E VR

A B

C D

Figure 12.6 Equivalent ABCD parameters of the system ‘from machine voltagesource to the infinite bus’ with one line in operation

AL

A

BL

B

CL DL

AL BL

CL DL

C D

E VR

VR

VR

Ig IR Ig

Ig

IR

IR

1 j0.3

0 1

AT BT

CT DT

1 j0.3

0 1

E

E.

IR ′IS ′

IS

IS

VS

′ ′ IR ′ ′

Figure 12.7 Equivalent ABCD parameters of the system (from machine voltagesource to the infinite bus) with two parallel lines in operation


ABCD parameters are obtained by two steps: the first is to get the equivalentATBTCTDT parameters for the two parallel lines:

IS ¼ I 0S þ I 00S and IR ¼ I 0R þ I 00R ð12:53Þ

VS ¼ ALVR þ BLI0R

VS ¼ ALVR þ BLI00

R

)ð12:54Þ

Summation of the two equations, (12.54) gives

VS ¼ ALVR þ BL

2IR ð12:55Þ

Similarly,

I 0S ¼ CLVR þDLI0R

I 00S ¼ CLVR þDLI00R

)ð12:56Þ

Hence,

IS ¼ 2CLVR þDLIR ð12:57Þ

From (12.55) and (12.57), it is observed that

AT ¼ AL; BT ¼ BL

2; CT ¼ 2CL; DT ¼ DL ð12:58Þ

The second step is to get the system ABCD parameters as below.Using (12.52) and (12.58) to obtain

A ¼ AT þ j0:3CT ¼ 0:854ff0�; B ¼ BT þ j0:3DT ¼ j0:396 pu

C ¼ CT ¼ j0:412 pu; D ¼ DT ¼ 0:854ff0�

The active power transfer capacity at the receiving end is

PR ¼ 1:4=0:396ð Þsin 17� ¼ 1:034 pu=phase

For compensated transmission lines:

The capacitors are located at the midpoint of the transmission line. So, the line isdivided into two equal parts of length 80 km and equal AL1BL1CL1DL1 parameters.Thus,

AL1 ¼ 0:995ff0�; BL1 ¼ j0:194 pu; CL1 ¼ j0:104 pu; DL1 ¼ 0:995ff0�


*When one of the two lines is isolated:

XC ¼ 50%XL ¼ 0:104 pu

Applying the parameters as written in (12.50) and taking into account thatAoBoCoDo are the parameters of the uncompensated line given by (12.52), obtainALBLCLDL for the compensated line as

AL ¼ 0:978 � j0:104 � 0:995 � j0:104 ¼ 0:989ff0�

BL ¼ j0:206 � j0:104 � 0:995 � 0:995 ¼ j0:103 pu

CL ¼ j0:206 � j0:104 � j0:104 � j0:104 ¼ j0:207 pu

DL ¼ 0:978 � j0:104 � j0:104 � 0:995 ¼ 0:989ff0�

9>>>>>=>>>>>;

ð12:59Þ

Consequently, the system ABCD parameters are

A ¼ 0:989 þ j0:3 � j0:207 ¼ 0:927ff0�

B ¼ j0:103 þ j0:3 � 0:989 ¼ j0:4 pu

C ¼ CL ¼ j0:207 pu

D ¼ DL ¼ 0:989ff0�

Accordingly, the active power transfer capacity at receiving end is

PR ¼ 1:4=0:4ð Þsin 17� ¼ 1:023 pu=phase

*When the two compensated lines are connected to the system in parallel:

The same procedure is implemented as in the case of uncompensated lines buttaking into account the values of parameters with compensation. Using (12.59), it isobserved that

AT ¼ AL ¼ 0:989ff0�; BT ¼ BL=2 ¼ j0:052 pu

CT ¼ 2CL ¼ j0:414 pu; DT ¼ DL ¼ 0:989ff0�

Therefore, the system ABCD parameters are

A ¼ 0:989 þ j0:3 � j0:414 ¼ 0:865ff0�

B ¼ j0:052 þ j0:3 � 0:989 ¼ j0:349 pu

C ¼ j0:414 pu

D ¼ 0:989ff0�

The active power transfer capacity at receiving end is

PR ¼ 1:4=0:349ð Þsin 17� ¼ 1:173 pu=phase


Examination of transient stability

It is of interest to evaluate the benefits that may be gained from the series com-pensation. As explained in Chapter 9, the equal area criterion can be used to checkthe system stability and to know how much the system is secure. Assuming that theactive power received is 0.8 pu, the system analysis when the lines are eitheruncompensated or compensated is given below.

For the system with uncompensated lines:

Before the fault: 0.8 ¼ (1.4/0.396)sin do and do ¼ 13.1�

After the fault: 0.8 ¼ (1.4/0.499)sin dm and dm ¼ 163.4�

The area between the input power line and P versus d curve during the fault

A1 ¼ 0.8(50 � 13.1)p/180 ¼ 0.515 pu, where the electric power is zero

The area between the input power line and P versus d curve after the fault:

A2 ¼ EVR=Bð Þ cos dc � cos dmð Þ � 0:8 dm � dcð Þp=180

¼ 1:4=0:499ð Þ 0:643 þ 0:958ð Þ � 1:58 ¼ 2:92 pu

For the system with compensated lines:

Before the fault: 0.8 ¼ (1.4/0.349)sin do and do ¼ 11.5�

After the fault: 0.8 ¼ (1.4/0.4)sin dm and dm ¼ 166.8�

A1 ¼ 0:8 50 � 11:5ð Þp=180 ¼ 0:537 pu

A2 ¼ 1:4=0:4ð Þ 0:643 þ 0:974ð Þ � 0:8 166:8 � 50ð Þp=180 ¼ 4:03 pu

It is concluded that:

● As in Table 12.1, the active power transfer capacity of the system at a specificpower angle is increased by compensating the lines with series capacitors.

● The marginal stability increases with line compensation and the system isgetting more secure. This can be noted by the ratio, A2=A1 (Table 12.2), wherethe increase due to compensation is about 32.27 per cent.

Table 12.1 Summary of results

System state Connected lines Aff0� B (pu) C (pu) Dff0� PR (pu/ph)

Uncompensated One line 0.916 j0.499 j0.206 0.978 0.82Two lines 0.854 j0.396 j0.412 0.854 1.034

Compensated One line 0.927 j0.400 j0.207 0.989 1.023Two lines 0.865 j0.349 j0.414 0.989 1.173


● The power transfer from the sending to the receiving end of the transmissionline is approximately inversely proportional to the series reactance of the line,(12.47). Its maximum value (steady-state stability limit) is obtained at d equals90�. So, decreasing the series reactance by series capacitive compensationresults in an increase of stability limit and improves system stability.

12.4 Enhancement of multi-machine power systemtransient stability

One of the major reasons to apply series compensation to the transmission lines inpower systems is to improve the system transient stability. This can clearly beshown by the example below.

Example 12.5 As in Example 8.2, the nine-bus test system (Figure 12.8) is sub-jected to a three-phase short circuit at bus no. 7 for durations of 0.08 s and 0.20 s.Investigate the system transient stability when the system is uncompensated,

GG

G

1

2 3

4

5 6

7 8 9

F

Figure 12.8 Nine-bus test system

Table 12.2 Results of transient stability study

System state A1 (pu) A2 (pu) A2=A1

Uncompensated 0.515 2.92 5.67Compensated 0.537 4.03 7.50


30 per cent series capacitive compensation, 50 per cent series capacitive compen-sation and 90 per cent series capacitive compensation for all lines in the system.The system data are given in Appendix II.

Solution:

At the same initial operating conditions as calculated in Example 8.2, the second-order Runge–Kutta method and PSAT/MATLAB� toolbox are used for the transientanalysis. The variation of rotor angle and speed of each generator are plotted asbelow (Figures 12.9–12.14). For all figures, it is to be noted that the solid curvesrepresent the uncompensated case, the curves crossed by black circles are for30 per cent compensation, the doted curves for 50 per cent compensation and thecurves crossed by white circles represent 90 per cent compensation.

At fault duration 0.08 s

0–10

10203040506070

δ 1°

Time (s)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1.04

1.02

1

0.98

ω1

(ele

c. ra

d/s)

0.96

0.940 0.5 1 1.5 2 2.5

Time (s)3 3.5 4 4.5 5

(a)

(b)

Figure 12.9 (a) d�1 versus time (s) and (b) w1 (elec. rad/s) versus time (s)


45403530252015105

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5

δ 2°

1.03

1.021.01

0.990.98

0.970.960.95

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5

1

ω2

(ele

c. ra

d/s)

(a)

(b)


10

5

0

–5

–10

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5–15

δ 3°

1.04

1.021.03

11.01

1

0.980.99

0.97

ω3

(ele

c. ra

d/s)

0.960 0.5 1 1.5 2 2.5 3

Time (s)3.5 4 4.5 5

(a)

(b)



At fault duration 0.20 s

500

400

300

200

100

0

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5–100

δ 1°

(a)

1.5

1.4

1.31.2

1.1

1

0.90.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

ω1

(ele

c. ra

d/s)

Time (s)(b)


500

400

300

200

100

0

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5–100

δ 2°

(a)

1.4

1.3

1.2

1.1

1

0.90 0.5 1 1.5 2 2.5

Time (s)3 3.5 4 4.5 5

ω2

(ele

c. ra

d/s)

(b)



12.5 Investigation of transmission power transfer capacity

It seems that increasing the degree of series compensation increases the improve-ment of system performance and system stability with no limit to the degree ofcompensation. Actually, this is not true as the power transfer capacity depends onvarious parameters that are not guaranteed to be simultaneously at the desired valuesgiving the maximum power transfer. This can be realised from Example 12.5 wherethe improvement of system stability at 90 per cent compensation is less than that at30 per cent. This concept is further discussed below.

As shown in Figure 12.15, a transmission line connects the node i as a sendingend to the node j as a receiving end. The line is represented by a series impedanceZL ¼ R þ jXL and compensated by series capacitors of reactance XC. So, the netseries reactance X is given by X ¼ XL � XC ¼ XL(1 � knom), where knom is thedegree of compensation.

500

–50–100

–150

–200

–250–300

0 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5

δ 3°

(a)

1.04

1.02

1

0.98

0.96

0.94

0.92

0.90 0.5 1 1.5 2 2.5Time (s)

3 3.5 4 4.5 5

ω3

(ele

c. ra

d/s)

(b)



The net series impedance is Z ¼ R þ jX. The voltage at the receiving end, VR,is taken as a reference while the voltage at the sending end is defined as VS withangle d. Thus,

VS ¼ VSe jd ¼ VS cos dþ j sin dð Þ ð12:60ÞHence, the line current is given by

I ¼ VS � VR

Z¼ VR

Z2R

VS

VRcos d� 1

� �þ X

VS

VRsin d

� ��

þ j RVS

VRsin d

� �� X

VS

VRcos d� 1

� �� ð12:61Þ

The apparent power at both sending and receiving ends can be computed as

PS þ jQS ¼ VSI� ¼ VSVR

Z2R

VS

VR� R cos dþ X sin d

�

þ j XVS

VR� X cos d� R sin d

� ��ð12:62Þ

and

PR þ jQR ¼ VRI� ¼ V 2

R

Z2R

VS

VRcos d� 1

� �þ X

VS

VRsin d

� ��

� j RVS

VRsin d

� �� X

VS

VRcos d� 1

� �� ð12:63Þ

Equation (12.63) can be rewritten as a function of the line impedance ‘f’ as

PR þ jQR ¼ VR

ZVS cos f� dð Þ � VR cosf½ þ j VS sin f� dð Þ � VR sinf½ f g

ð12:64Þwhere

f ¼ tan�1 X

R¼ tan�1 XL 1 � knomð Þ

Rð12:65Þ

i j

VR 0°VS δ°RL XL XC

Figure 12.15 Series compensated line between node i and node j


The real part of both (12.63) and (12.64) represents the value of PR, i.e.

PR ¼ VR

ZVS cos f� dð Þ � VR cosf½

¼ V 2R

Z2R

VS

VRcos d� 1

� �þ X

VS

VRsin d

� �� ð12:66Þ

Therefore, PR is a function of the parameters; the ratio VS/VR, the line impe-dance and the power angle d. It is impractical to decide the values of these para-meters that maximise the power transfer at the receiving end. This is due to the factthat it may need a large value of d to close to ∅ (12.66) or a voltage ratio out of theregulated value. In addition, for a multi-machine system that comprises many linesconnecting different types of buses, the angles and magnitudes of bus voltages aswell as the power flow into the lines are determined by load flow calculationsdepending on the system topology. So, no guarantee to achieve the maximum powertransfers for each line when using load flow results. Thus, based on these facts it canbe concluded that by continuous increase of the degree of compensation that affectsboth VS/VR and d it is not guaranteed to obtain the maximum power transfer. It maybe increased with increasing the degree of compensation until a certain limit, reach aplateau with no benefit and the gain may decrease beyond this limit. The sameconclusion is applied to system stability improvement as increasing the transmissionpower transfer leads to an enlarged area between the input power line and the powerversus angle curve, Equal Area Criterion, i.e. more marginal stability. This can beobserved from Example 12.5, where the stability improvement of 50 per cent com-pensated system is larger than that of 90 per cent compensation. However, severalstudies have been done with a goal of achieving the optimal improvement of bothpower capability and system stability [5–10] as well as the schemes of applying andcontrolling the series compensation [11, 12].

12.6 Improvement of small signal stability

It has been shown that the transient stability of the power system is enhanced whenapplying series capacitors. An additional benefit may be obtained as well where thesmall signal stability is improved [13]. This can be demonstrated by the examplebelow.

Example 12.6 Examine the small signal stability of the system given inExample 7.1 and shown in Figure 12.16 in terms of perturbed values of flux lin-kages and currents. Find the effect of degree of compensation of 30 per cent,50 per cent and 90 per cent on the system performance.

Solution:

The exciting system is on manual control, constant Efd. The linear flux linkage statespace generator model, Chapter 7, Section 7.4, with the effect of governor included


in the mechanical torque is shown as a block diagram in Figure 12.17 [1]. Themodel is characterised by the following assumptions: (i) amortisseur effects andstator resistance are neglected, (ii) terms related to the derivative of stator fluxlinkages are neglected, (iii) balanced load conditions and (iv) constant speed in thespeed voltage wy terms. The change of both Tm and Efd depends on the prime-mover and excitation controls. So, with constant mechanical input torque, DTm ¼ 0;with constant exciter output voltage, DEfd ¼ 0.

The model constants are related to the synchronous machine and transmissionline parameters [1] as

K1 ¼ V1R2

E þ ðXq þ XEÞðX 0d þ XEÞ

nEqo RE sin do þ ðx0d þ XEÞcos do

�

þ IqoðXq � X 0dÞðXq þ XEÞsin do � RE cos do

oð12:67Þ

K2 ¼ 1

R2E þ Xq þ XE

� �X 0

d þ XE

� � REEqo þ Iqo R2E þ Xq þ XE

� �2h in o

ð12:68Þ

K3 ¼ 1

1 þ KA Xd � X 0d

� �Xq þ XE

� � ð12:69Þ

K4 ¼ V1KA Xd � X 0d

� �Xq þ XE

� �sin do � RE cos do

� ð12:70Þ

V∞ = 1.0 0Vs = 1.01 θsE δ

R = 0.003

xtr = 0.1

xTL = 0.35RTL = 0.05

xd ′ = 0.2

Figure 12.16 System for Example 11.6

K1

K4

ΔTm

K2

ωB

s2Hs + KD

1

1 + sT3

K3+

+

+ΔEfd = 0 ∑∑ ∑ ΔδΔωΔEq ′

Figure 12.17 Representation of a single-machine infinite bus with lineargenerator model


where

KA ¼ 1

R2E þ Xq þ XE

� �X 0

d þ XE

� � ð12:71Þ

RE ¼ 0:05; XE ¼ 0:45; H ¼ 3:5 MW:s=MVA

By using PSAT/MATLAB� toolbox the constants associated with block diagram(Figure 12.17) for uncompensated and compensated systems are calculated as sum-marised in Table 12.3. It is to be noted that this toolbox uses the concept of partici-pation factors (PFs) that can be interpreted as the sensitivity of a given eigenvalue withrespect to a corresponding entry of the system dynamic matrix and that the entries ofthe system matrix are a linear combination of PFs and eigenvalues. PFs have beenintroduced to analyse the connection between a system variable and some modes andto quantify the corresponding participation degree of a system variable in a mode andvice versa. Therefore, PFs can be used to detect the states most involved in a modeevolution. It is evident that once the modes of interest have been identified, PFs mighthelp to obtain a reduced order model of the system that still preserves the dynamic ofinterests, selective modal analysis. More details about PFs can be found in [14].

In addition, the incremental saturation associated with perturbed values of fluxlinkages and currents is taken into account. Values of Ld and Lq are modified [15] tobe Lds and Lqs that given by Lsd ¼ KsdLd and Lqs ¼ KsdLq, where

Ksd ¼ 1

1 þ BsatAsateBsat yto�yT1ð Þ

The numerical values used in the program are Asat ¼ 0.031, Bsat ¼ 6.93, YT1¼ 0.8

For uncompensated system:

State matrix A is obtained as

A ¼0 �0:2051 �0:1771

376:9911 0 00 �5:5497 �11:6616

24

35

Eigen values are

l1 ¼ �1:114; l2 ¼ �1:114; l3 ¼ �9:434

Table 12.3 Values of the constants associated with block diagram

System state K1 K2 K3 K4 T3

Uncompensated 1.436 1.240 0.228 2.089 0.086Compensated 30% 1.679 1.384 0.209 2.341 0.079

50% 1.866 1.511 0.196 2.535 0.07490% 4.551 0.000 0.062 0.000 0.024


Eigen vector matrix V:

V ¼0:0027 � j0:0181 0:0027 þ j0:0181 0:0093

�0:9184 �0:9184 �0:3724

0:3232 � j0:2274 0:3232 þ j0:2274 0:9280

24

35

Participation matrix P is:

P ¼0:5845 þ j0:0197 0: 5845 � j0:0197 �0:1690

0:5845 þ j0:0197 0:5845 � j0:0197 �0:1690

�0:1690 � j0:0394 �0:1690 � j0:0394 1:338

24

35

For compensated system at 30 per cent degree of compensation:

Eigen values:

l1 ¼ �1:175; l2 ¼ �1:175; l3 ¼ �10:328

Eigen vector matrix V is:

V ¼0:0029 � j0:0195 0:0029 þ j0:0195 0:0097

�0:9142 �0:9142 �0:3537

0:3317 � j0:2321 0:3317 þ j0:2321 0:9353

24

35

Participation matrix P is:

P ¼0:5817 þ j0:0199 0:5817 � j0:0199 �0:1635

0:5817 þ j0:0199 0:5817 � j0:0199 �0:1635

�0:1635 � j0:0398 �0:1635 þ j0:0398 1:3269

24

35


Eigen values:

l1 ¼ �1:221; l2 ¼ �1:221; l3 ¼ �11:133

Eigen vector matrix:

V ¼�0:0030 þ j0:0205 0:0027 þ j0:0181 0:0101

0:9119 0:9119 �0:3409

�0:3380 þ j0:2318 �0:3380 � j0:2318 0:9401

24

35

Participation matrix:

P ¼0:5799 þ j0:0214 0:5799 � j0:0214 �0:1598

0:5799 þ j0:0214 0:5799 � j0:0214 �0:1598

�0:1598 � j0:0429 �0:1598 þ j0:0429 1:3196

24

35



Eigen values:

l1 ¼ �0:000; l2 ¼ �0:000; l3 ¼ �42:499

Eigen vector matrix:

V ¼�0:0000 � j0:0415 0:0027 þ j0:0415 0:000

�0:9991 �0:9991 �0:0000 0 1:000

24

35

Participation matrix:

P ¼0:5000 0:5000 00:5000 0:5000 0

0 0 1:0000

24

35

Summary of results: The eigenvalues and some coefficients are tabulated inTable 12.4.

It is seen that the magnitude of the negative real part of eigenvalues becomelarger in case of compensation, particularly, at 50 per cent degree of compensation.It is noted that the increase of marginal stability does not continue as the degree ofcompensation increases where at 90 per cent compensation the system becomescritically stable: two eigenvalues are zero.

It is to be noted that, in the above analysis, the series capacitors as passivecompensators consist of fixed or switchable susceptances. They help in modifyingtransmission system reactance, increasing power transfer and stabilisation andcontrolling the reactive power. The behaviour of voltage and reactive power on anuncompensated transmission system is influenced by system impedance, variableactive and reactive load characteristics as a function of supply voltage and loadingon synchronous machines. Therefore, series capacitors as series compensationdevice must be equipped with controllers to effectively allow continuous control ofreactive power that flows through the system and improving the performance‘steady and dynamic’ of the overall power system. However, series compensation

Table 12.4 Eigenvalues and coefficients

System state Eigenvalues KS Ksrf Kdrf wn EP

l1 l2 l3

Uncompensated �1.114 �1.114 �9.434 0.846 1.016 13.573 7.398 0.13130% compensation �1.175 �1.175 �10.328 1.001 1.196 14.383 8.024 0.12850% compensation �1.221 �1.221 �11.133 1.116 1.327 14.976 8.452 0.12790% compensation �0.000 �0.000 �42.499 4.551 4.551 0.000 15.656 0.000

KS: Steady-state synchronising torque coefficientKsrf: Synchronising torque coefficient at rotor oscillating frequencyKdrf: Damping coefficient at rotor oscillating frequencywn: Un-damped natural frequency of the oscillatory modeEP: Damping ratio of the oscillatory mode


is implemented not only with series capacitors but also with current or voltagesource devices. Some configuration of such devices is explained in Chapter 14.

12.7 Sub-synchronous resonance

It has been shown that applying series capacitors to long transmission linesimproves the system stability as well as increases the power transfer capacity.On the other hand, series capacitors may cause self-excited oscillations at lowfrequencies because of the low X/R ratio or sub-synchronous frequencies due toinduction generator effect.

The problem of self-excited torsional frequency due to torsional interaction is aserious problem since it may cause damage to the turbine-generator shaft [16–19].The problem of oscillations at frequencies lower than the synchronous frequency istermed as ‘sub-synchronous resonance’ (SSR).

The formal definition of SSR as reported in [20] is SSR is an electric powersystem condition where the electric network exchanges energy with a turbine-generator at one or more of the natural frequencies of the combined system belowthe synchronous frequency of the system.

Therefore, the stability limit and hence the operational capability of longtransmission lines are greatly improved by the use of series capacitor compensation,but unfortunate experiences of SSR with generator shaft torques have demonstratedthe need for caution in the use of such series compensation [21]. Resonance canoccur between the natural frequencies of oscillation inherent in the rotating massesof synchronous generators and prime movers coupled by shafts that are elastic, andthe natural frequencies of the electric system to which the generator is connected.

Sudden change of torque to the main turbine-generator coupling, produced by atransient variation of electric power, can excite torsional natural resonant frequencies.When series capacitors are used to compensate the reactance of the transmissionsystem the torsional natural resonant oscillations in the turbine-generator shafts maybe excited by the power system natural frequency. Self-sustaining torsional oscilla-tions can shorten shaft life.

The effects of long transmission line systems connecting generation at aremote point to the main power system have been studied. It is necessary to cal-culate the electrical natural frequencies, ENFs, to be able to locate the zone oftorsional interaction, which is around the points of peak of resonance (the points ofcoincidence of ENFs with mechanical natural frequencies). Simply, for seriescapacitor compensated transmission lines their series LC combinations have naturalfrequencies wn that are defined by the equation:

wn ¼ffiffiffiffiffiffiffi1

LC

r¼ wB

ffiffiffiffiffiffiXC

XL

rð12:72Þ

where wB is the system base frequency, and XL and XC are the inductive and capacitivereactances, respectively. It is crucial to know which one of these sub-synchronous fre-quencies may interact with one of the natural torsional modes of the turbine-generatorshaft. For a multi-machine power system, the problem of identifying the electric naturalfrequencies is difficult as the transmission network topology is more complicated.


12.7.1 The mechanical systemIn modelling the system for SSR analysis, it is useful to consider the entire systemas multi-subsystems connected to each other through the transmission network. Inthe mechanical system, it is sufficient to model the turbine-generator components,e.g. it is not important to model the boiler for SSR analysis. Torsional modes of theshaft oscillation can be obtained from the turbine-generator manufacturer or cal-culated by expressing the turbine-generator model as a spring-mass model whoseparameters are known. The matrices expressing the dynamic equation of motioninclude both the inertia matrix and the velocity damping matrix, and when coupledby the stiffness matrix to the applied torque vector permit the derivation of eigen-values representing the mechanical modes of natural oscillations as well as thederivation of the eigenvectors of the mode shape. The system is assumed to berepresented by the spring-mass model shown in Figure 12.18 with a rotating exciterand assumes an unforced and un-damped mechanical system [22, 23]. It is to benoted that in the case of using a static exciter ‘as it is common these days’ themasses are reduced to only four: M1, M2, M3 and M4.

It can be described by a set of second-order differential equations as

1wB

H€d þ Kd ¼ 0 ð12:73Þ

where all quantities are in pu, except time in seconds and d in radians, and

H: moment of inertia matrix ¼ diag[2H1, 2H2, . . . , 2H5], d: angular displace-ment matrix.

K: shaft-stiffness matrix

¼

K12 �K12

�K12 K12 þ K23 �K23

�K23 K23 þ K34 �K34

�K34 K34 þ K45 �K45

�K45 K45

266664

377775

Kij: spring constant of shaft between mass i and mass j

M1 M2 M3 M4 M5

d1 d2 d3 d4 d5

HP IP LP Gen. Exc.

Tm1 Tm2 Tm3 Te

K12 K23 K34 K45

Figure 12.18 Spring-mass model for the shaft of a steam turbine unit


At resonance, all masses oscillate at the same frequency, wm, such that

di ¼ disin wmt þ að Þ ð12:74ÞSubstituting (12.74) into (12.73) gives

w2m

wBH d � K d¼ 0 ð12:75Þ

Assuming, M ¼ H�1K and lm ¼ w2m

wB, (12.75) can be rewritten as

M d ¼ lmd ð12:76ÞTherefore, by definition the eigenvalues of the matrix M, ‘lm’ provide themechanical natural frequencies, where

wm ¼ffiffiffiffiffiffiffiffiffiffiffilmwB

pð12:77Þ

To each lm corresponds an eigenvector Qm that gives the mode shape ofoscillations.

Example 12.7 Typical data of the turbine shaft are given as

The stiffness: K12 ¼ 29.437, K23 ¼ 62.241, K34 ¼ 73.906, K45 ¼ 5.306

Turbine inertia constant(s): HP-stage¼ 0.0649, IP-stage ¼ 0.2552, LP-stage ¼ 1.539

Generator rotor inertia constant¼ 0.98 s, Exciter rotor inertia constant ¼ 0.0298 s

Find the mechanical natural frequencies and the mode shapes.

Solution:

H ¼ diag 0:1298 0:5104 3:078 1:96 0:0596½

K ¼

29:437 �29:437 0 0 0�29:437 91:678 �62:241 0 0

0 �62:241 136:147 �73:906 00 0 �73:906 79:212 �5:3060 0 0 �5:306 5:306

2666664

3777775

H�1 ¼

7:7042 0 0 0 00 1:9592 0 0 00 0 0:3249 0 00 0 0 0:5102 00 0 0 0 16:7785

2666664

3777775

M ¼

226:7874 �226:7874 0 0 0�57:6744 179:6199 �121:9455 0 0

0 �20:2212 44:2323 �24:0110 00 0 �37:7071 40:4143 �2:70710 0 0 �89:0268 89:0268

2666664

3777775


Using MATLAB, the eigenvalues in ascending order are �0.0000, 51.2133,93.1688, 112.0765, 323.6221.

Thus, the corresponding mechanical natural frequencies (12.77) are

fmo ¼ 0; fm1 ¼ 22:12; fm2 ¼ 29:8; fm3 ¼ 32:72; fm4 ¼ 55:6; cps

wmo ¼ 0, wm1 ¼ 138.91, wm2 ¼ 187.36, wm3 ¼ 205.5, wm4 ¼ 349.2 rad/s and theeigenvector matrix, Qm, is:

Qm ¼

0:4472 0:4299 0:1200 �0:8118 0:91930:4472 0:3328 0:0707 �0:4106 �0:39250:4472 0:1471 �0:0066 0:1565 0:02870:4472 �0:3230 �0:0460 �0:0964 �0:00380:4472 �0:7606 0:9892 0:3724 0:0015

266664

377775

So, the mode shapes of the mechanical system are as shown in Figure 12.19.

12.7.2 The electrical networkFor a simple power system, for example, a generator connected to an infinite bus, itis easy to model the line by its series impedance and then applying (12.72) calcu-lates the ENFs. On the other hand, to study the generator dynamic performance, thelinearised generator model in d–q axes used for stability analysis can be used forSSR analysis as well, Chapter 7. In case of a multi-machine power system, thegeneralised form of a ring, which can be a part or whole of the power system, is

0

1

–1

0

1

–1

Mode 00.0 Hz

Mode 122.12 Hz

Mode 229.8 Hz

Mode 332.72 Hz

Mode 455.6 Hz

δ

δ 0

1

–1

δ

0

1

–1

δ

0

1

–1

δ

Figure 12.19 Mode shapes of the mechanical system of Example 12.7


shown in Figure 12.20(a). The possibility of series compensation in every inter-connecting transmission line is considered but of course such compensation can beset at zero for as many lines as desired or at any required value whose resonanceeffect is to be determined.

The ENFs can be calculated by the following steps [24]:

(i) calculation of the single-phase equivalent circuit as shown in Figure 12.20(b).(ii) calculation of the inductance matrix L, which is given by

L ¼

L11 L12 L1n

L21 L22 L33 ... ..

. . .. ..

.

Ln1 Lnn

2666664

3777775 ð12:78Þ

whereLij is the summation of inductances, which are included in the loop iLij is the inductance of the mutual link between loops i and j

(iii) calculation of the capacitance matrix C, assuming (1/ci) ¼ SiLij, i and j ¼1, 2, . . . , n

whereci ¼ capacitance inserted in loop iSi ¼ degree of compensation for that loopLij ¼ inductance between the two directly linked buses i and j

Then the capacitive reactance matrix F for Figure 12.20(b) is

F ¼

1=c1 1=c1

1=c2 0 1=c21=c3 1=c3

0 . ..

..

.

1=c1 1=c2 1=cn

26666664

37777775

ð12:79Þ

C1

C2

C3C4

C5

Cn

Load Gen.(a)

c1 c2

cn

L1 L2

Ln

(b)

Figure 12.20 Generalised form of a ring system: (a) General loop configuration foran interconnected system and (b) the equivalent L-C loops of the system


Thus, the capacitance matrix is

C ¼ F�1 ð12:80Þ(iv) the ENF’s of a network are those that arise from its configuration, without

applied emf [25]. They can be found as the square roots of the reciprocal ofthe eigenvalues of the V matrix where

V ¼ LC ð12:81Þand those values are viewed from the stator side. Alternatively, they can beviewed from the rotor side

fr ¼ fn � fs ð12:82Þwhere fr is the rotor frequency, fn is the natural frequency, fs is thesynchronous frequency, positive sign for super-synchronous frequency andnegative sign for sub-synchronous frequency.

Steps (i)–(iv) can be applied for two or more coupled rings, but, of course, as thenetwork gets more complex, the representation also gets more difficult. Another tech-nique, called the ‘state space approach’ can be used as it is suitable for digital computersolution [21]. However, several computer programs, e.g. PSCAD, EMTP, are availableto analyse the power system in steady state, transient state or ‘in general’ the dynamicperformance of the power system.

Example 12.8 Reactance diagram of the nine-bus test system is shown inFigure 12.21. Find the ENFs when all lines in the system are compensated by seriescapacitors at 50 per cent degree of compensation. Investigate the torsional inter-action for one of the system generators that has the turbine-shaft system given inExample 12.7.

GG

G

1

2 3

4

5 6

7 8 9

xd ′ = j0.1198 xd’ = j0.1813

j0.0576

j0.085

j0.092

j0.17

j0.161

j0.0625j0.072 j0.1008

j0.0586

j0.274

j0.322

j0.35

xd ′ = j0.0608

Figure 12.21 Reactance diagram of nine-bus test system


Solution:The equivalent loop configuration is shown in Figure 12.22.

The inductance matrix is:

L ¼

0:477 0:118 0:000 0:000 0:000 0:085

0:118 0:543 0:350 0:000 0:000 0:092

0:000 0:350 0:76 0:240 0:000 0:170

0:000 0:000 0:240 0:660 0:322 0:101

0:000 0:000 0:000 0:322 0:576 0:072

0:085 0:092 0:170 0:101 0:072 0:681

2666666664

3777777775

The capacitive reactance matrix is:

F ¼

0:0425 0:0425

0:0460 0:0460

0:0850 0:0850

0:0504 0:0504

0:0360 0:0360

0:0425 0:0460 0:0850 0:0504 0:0360 0:0805

2666666664

3777777775

The capacitance matrix is:

C ¼

17:9522 �5:5772 �5:5772 �5:5662 �5:5772 5:5772

�5:5772 16:1619 �5:5772 �5:5662 �5:5772 5:5772

�5:5772 �5:5772 6:1875 �5:5662 �5:5772 5:5772

�5:5662 �5:5662 �5:5662 14:2468 �5:5662 5:5662

�5:5772 �5:5772 �5:5772 �5:5662 22:2005 5:5772

5:5772 5:5772 5:5772 5:5662 5:5772 �5:5772

2666666664

3777777775

c1 c2

c6

0.085 0.092

0.161

c3 0.17 c4 0.1008 0.072c5

1 2 3 4 5

6

0.274

0.118

0.35

0.24

0.322

0.182

Figure 12.22 Equivalent loop configuration


Then,

¥ ¼ L C ¼

8:3791 0:2792 �2:8444 �2:8388 �2:8444 2:8444

�2:3490 6:6789 �1:0078 �5:1153 �5:1255 5:1255

�6:5785 1:0302 2:3627 �1:8130 �6:5785 6:5785

�6:2448 �6:2448 �3:4213 6:8369 2:6996 6:2448

�4:6032 �4:6032 �4:6032 1:7821 11:3968 4:6032

2:8991 2:8991 2:8991 2:8973 2:8991 �0:8991

266666664

377777775

Using MATLAB, the eigenvalues are obtained as �11.8538, 19.3129, 11.1760,2.8122, 5.6125, 7.6955

The corresponding ENFs ‘omitting the negative value’ are 13.18, 17.2, 20.6,22.9, 34.4 Hz.

The mechanical modes of oscillations and ENFs are plotted in Figure 12.23.The horizontal lines represent the mechanical modes of oscillation. The black-filledcircles represent the ENFs that are close to one of the mechanical modes while thewhite circles are those apart from mechanical modes.

Therefore, at modes #1, 2 and 3, the torsional natural resonant oscillations inthe turbine generator shafts may be excited by the power system natural frequencycausing shaft damage.

The modes of oscillations may move upwards or downwards for other turbineshafts in the system, with different values of inertia constants and different shaft

10

10

20

20 30

30

40

40

50

60

Electrical natural frequencies

Mec

hani

cal n

atur

al fr

eque

ncie

s

Mode 1

Mode 2Mode 3

Mode 4

Figure 12.23 Mechanical modes of oscillations and electrical natural frequencies(50 per cent degree of compensation for all lines in the system)


stiffness, giving different modes. The inductance matrix changes with load whilethe capacitance matrix depends on the degree of compensation. Thus, changing theloads and/or degree of compensation yields a change of ENFs.

References

1. Anderson P.M., Farmer R.G. The Series Compensation of Power Systems.Encinitas, CA, US: PBLSH; 1996

2. Sallam A.A., Khafaga A.M. (eds.). ‘Optimal parameters of series capacitorsin compensated power systems’. International Power Engineering Con-ference IPEC’93; Singapore, Mar 1993

3. Sallam A.A., Khafaga A.M. (eds.). ‘Optimal series compensation in powersystems by using complex method’. Proceedings of 3rd InternationalSymposium of Electricity Distribution and Energy Management ISEDEM’93;Singapore, Oct 1993. pp. 215–19

4. Belur S., Kumar A., Parthasarathy K., Prabhakara F.S., Khincha H.P.‘Effectiveness of series capacitors in long distance transmission lines’. IEEETransactions on Power Apparatus and Systems. 1970;89(5):941–51

5. Leonidaki E.A., Georgiadis P., Hatziargyriou N.D. ‘Decision trees for determi-nation of optimal location and rate of series compensation to increase powersystem loading margin’. IEEE Transactions on Power Systems. 2003;21(3):1303–10

6. Hedin R., Jalali S., Weiss S., Cope L., Johnson B., Mah D. et al. (eds.).‘Improving system stability using an advanced series compensation scheme todamp power swings’. Sixth International Conference on AC and DC PowerTransmission, IET Conf. Publ. No. 423, Apr/May 1996. pp. 311–14

7. Chen X.R., Pahalawaththa N.C., Annakkage U.D., Kumble C.S. ‘Controlledseries compensation for improving the stability of multi-machine power systems’.IEE Proceedings – Generations, Transmission and Distribution. 1995;142(4):361–66

8. Crary S.B., Saline L.E. ‘Location of series capacitors in high-voltage trans-mission systems’. IEEE Transactions on Power Apparatus and Systems,Part III Transactions of the AIEE. 1953;72(2):1140–51

9. Kosterev D.N., Mittalstadt W.A., Mohler R.R., Kolodziej W.J. ‘An applicationstudy for sizing and rating controlled and conventional series compensation’.IEEE Transactions on Power Delivery. 1996;11(2):1105–11

10. de Oliveira S.E.M., Gardos I., Fonseca E.P. ‘Representation of series capaci-tors in electric power system stability studies’. IEEE Transactions on PowerSystems. 1991;6(3):1119–25

11. Yu-Jen L. (ed.). ‘Power systems transient stability preventive control incorpor-ating network series compensation with the aid of if-then rules extracted from amulti-layer perceptron artificial neural network’. 4th International Conferenceon Electric Utility Deregulation and Restructuring and Power Technologies(DRPT), 2011; Weihai, China, Jul 2011. US: IEEE; 2011. pp. 31–8


12. Fernandes A.A. (ed.). ‘Series compensation using variable structure andLyapunov function controls for stabilization multi-machine power system’.16th IEEE International Conference on Control Applications, Part of IEEEMulti-Conference on Systems and Control; Singapore, Oct 2007. pp. 1091–6

13. Lie T.T., Li G.J., Shrestha G.B., Lo K.L. (eds.). ‘Coordinated decentralizedoptimal control of inter-area oscillations in power systems’. InternationalConference on Energy Management and Power Delivery (EMPD’98);Singapore, Mar 1998, vol. 1. pp. 97–102

14. Garofalo F., Iannelli L., Vsca F. (eds.). ‘Participation factors and theirconnections to residues and relative gain array’. IFAC, 15th Triennial WorldCongress; Barcelona, Spain, 2002, vol. 15, part 1. pp. 180–5

15. Kundur P. Power System Stability and Control. New York, NY, US:McGraw-Hill, Inc.; 1994. chapter 12

16. Kumar R., Harada A., Merkle M., Miri A.M. (eds.). ‘Investigation of theinfluence of series compensation in AC transmission systems on bus connectedparallel generating units with respect to sub-synchronous resonance (SSR)’.IEEE Bologna Power Tech Conference; Bologna, Italy, Jun 2003. pp. 23–6

17. de Oliveira A.L.P., Moraes M. (eds.). ‘Sub-synchronous resonance analysisafter Barra do Peixe 230 kV fixed series compensations installation at 230 kVMatoGrosso transmission system (Brazil)’. Transmission and DistributionConference and Exposition; Latin America, 2008. pp. 1–6

18. de Oliveira A.L.P. (eds.). ‘The main aspects of fixed series compensationdimensioning at Brazilian 230 kV transmission system’. Transmission andDistribution Conference and Exposition. Latin America; 2008. pp. 1–9

19. Jowder F.A.L. ‘Influence of mode of operation of the SSSC on the smalldisturbance and transient stability of a radial power system’. IEEE Transac-tions on Power Systems. 2005;20(2):935–42

20. IEEE SSR Working Group. ‘Proposed terms and definitions forsub-synchronous resonance’. IEEE Symposium on Countermeasures forSub-synchronous Resonance, IEEE Pub. 81TH0086-9-PWR, 1981. pp. 92–7

21. Anderson P.M., Agrawal B.L., Van Ness J.E. Sub-Synchronous Resonance inPower Systems. New York, NY, US: IEEE Press; 1990

22. Fouad A.A., Khu K.T. ‘Damping of torsional oscillations in power systemswith series-compensated lines’. IEEE Transactions on Power Apparatus andSystems. 1978;97(3):744–53

23. Fouad A.A., Khu K.T. ‘Sub-synchronous resonance zones in the IEEE‘‘BENCHMARK’’ power system’. IEEE Transactions on Power Apparatusand Systems. 1978;97(3):754–62

24. Sallam A.A., Dineley J.L. (eds.). ‘Sub-synchronous problems in an integratedpower system’. 19th Universities Power Engineering Conference; Aberdeen,UK, Apr 1984

25. Kimbark E.W. ‘How to improve system stability without risking sub-synchronous resonance’. IEEE Transactions on Power Apparatus and Systems.Sept/Oct 1977;96(5):1608–18


Chapter 13

Shunt compensation

Improvement of AC power systems’ performance is an important issue for powersystem planning and operation engineers. Application of series compensation oftransmission system to achieve this goal is explained in Chapter 12. Anothermethod is to compensate the transmission system by shunt compensators. In bothmethods the line reactance is controlled to modify the natural electrical character-istics of AC power systems. Consequently, the reactive power that flows throughthe system can effectively be controlled improving the system performance, inparticular, increasing power transfer capacity, controlling steady state and dynamicvoltage, controlling reactive power of dynamic loads, damping of power systemoscillations and improving system stability [1–3]. For instance, if the shunt com-pensator injects reactive power near the load, the transmission line current can bereduced resulting in reduction of power losses, improved voltage regulation at loadterminals and increased power transfer capacity.

Shunt compensation is applied by using shunt capacitors and shunt reactors thatare permanently connected to the network or switched on and off according tooperating conditions. Shunt capacitors help increase the system load ability [4] andreduce the voltage drop in the line by improving the power factor. Shunt reactors areused to limit voltage rise under both open line and light load conditions. Principlesillustrating the impact of shunt compensation on transmission system parameters aswell as its benefits are discussed in the next sections.

13.1 Shunt compensation of lossless transmission lines

For simplicity of analysis consider the line is lossless and is uniformly compensatedthroughout its length. The line parameters are affected as given below.

13.1.1 Shunt-compensated line parametersBased on the definitions given in Chapter 12, Section 12.1, the parameters char-acteristic impedance, line angle and natural power of shunt compensated line canbe given as a ratio of the uncompensated line, denoted by subscript o as

The characteristic impedance ratio:

ZC

ZCo¼ RC

RCo¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffixLo

bC

bCo

xLo

s¼

ffiffiffiffiffiffiffibCo

bC

sð13:1Þ

The ratio of line angle:

qqo

¼ffiffiffiffiffiffiffiffiffiffiffiffiffixLobC

xLobCo

s¼

ffiffiffiffiffiffiffibC

bCo

sð13:2Þ

and the ratio of natural power:

Pn

Pno¼ RCo

RC¼

ffiffiffiffiffiffiffibC

bCo

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffibCo þ DbC

bCo

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 þ DbC

bCo

sð13:3Þ

where DbC ≜ change of bC due to shunt compensationThus, the degree of shunt compensation ‘ⅆ’ is defined as

ⅆ ¼ DbC

bCo¼ Pn

Pno

� �2

� 1 ð13:4Þ

It is found that by adding capacitive shunt compensation to the line, the line angleis increased: the characteristic impedance decreases while the natural power increa-ses. It means that more power can be carried by the line while maintaining a flow thatcorresponds to the characteristic impedance loading. On the other hand, if the shuntcompensation is inductive the relations above can be applied with a negative DbC.

The amount of reactive power delivered by the shunt capacitive compensator‘DQC’ to increase the natural power from Pno to Pn can be computed as below.

Assuming that the natural power, Pn, is required to be R times the originalvalue, Pno, (13.3) and (13.4) give

Pn

Pno¼ R ¼ ffiffiffiffiffiffiffiffiffiffiffi

1 þ dp ð13:5Þ

and

ⅆ ¼ R2 � 1 ð13:6Þ

It is shown that the amount of shunt susceptance added is (R2 � 1) times theoriginal value. Thus, the reactive power supplied by the added shunt compensation‘DQC’ is given by

DQC ¼ DbC‘V2 ¼ ðR2 � 1ÞbCo‘V

2 ð13:7ÞAlso, DQC as a ratio of Pno is obtained by

DQC

Pno¼ R

2 � 1� �

bCo‘V2

� �= V 2p bCo=xLoð Þ� � ¼ ‘ R

2 � 1� � ffiffiffiffiffiffiffiffiffiffiffiffiffi

bCoxLo

pð13:8Þ

From (13.2) and (13.3) the ratio of line angle is

qqo

¼ R ð13:9Þ


Example 13.1 Repeat Example 12.1 with using shunt compensation rather thanseries compensation.

Solution:

The original values of ZC , b and q are 285W, 0.00129 rad/km and �11.83�,respectively. They are the same as calculated in Example 11.1. Also, SIL¼ 417.63 MW‘three-phase’.

When the line is shunt compensated to increase Pn to 1.5 � 417.63 MW, therequired degree of shunt compensation ‘ⅆ ’ can be obtained using (12.6). Thus,

ⅆ ¼ R2 � 1 ¼ 1:5ð Þ2 � 1 ¼ 1:25

Using (13.7) the amount of reactive compensation is

DQC ¼ 1:25 � 160 � 4:518 � 10�6 � 3452 ¼ 107:55 MVA

By (13.9) the line angle ¼ 0.2064 � 1.5 ¼ 0.3096 rad ¼ 17.75�

Comparing the results obtained in Example 13.1 with the results of Example 12.1,it is noted that (i) the reactive power added to the system to get a given natural poweris the same whether added by series or shunt compensation and (ii) the line angle forshunt compensation case is R2 ‘equals 2.25’ times larger than the series compensa-tion case.

13.1.2 Transient stability enhancement for shunt-compensatedlossless lines

Considering the system shown in Figure 13.1, it comprises an ideal shuntcompensator with unlimited capacity to control the voltage and fast response at themidpoint of the transmission line. The shunt compensation is assumed to becapable of holding the voltage at midpoint constant at all times.

VS VRVM

IS IR

IM

jXL/2 jXL/2

Idea

l shu

ntco

mpe

nsat

or

Figure 13.1 Shunt-compensated transmission line

Shunt compensation 353

As shown in Figure 13.1 the voltages can be expressed as

VS ¼ VSeqS , VM ¼ VMeqR and VR ¼ VReqR ; where the voltage angles arereferred to an arbitrary reference.

The currents (IS and IR) and powers (PS, QS and PR, QR) at the sending andreceiving ends, respectively, are computed using the relations below:

IS ¼ VS � VM

jXL=2¼ 2 VS sin qS � VM sin qMð Þ

XLþ j

2 VM cos qM � VS cos qSð ÞXL

IR ¼ VM � VR

jXL=2¼ 2 VM sin qM � VR sin qRð Þ

XLþ j

2 VR cos qR � VM cos qMð ÞXL

9>>>=>>>;

ð13:10Þ

PS þ jQS ¼ VSI�S ¼ 2VSVM

XLsin qS � qMð Þ þ j

2XL

V 2S � VSVM cos qS � qMð Þ� �

PR þ jQR ¼ VRI�R ¼ 2VM VR

XLsin qM � qRð Þ � j

2XL

V 2R � VM VR cos qM � qRð Þ� �

9>>>=>>>;

ð13:11Þ

The reactive power of the compensator, Qcomp, can be defined as the differencebetween the generated reactive power at the sending end, QS , and the summation ofreactive power losses, Qloss, and reactive power at receiving end, QR. Using thecurrent directions shown in Figure 13.1, it is considered a positive term as input tothe compensator. Thus,

Qcomp ¼ QS �X

Qloss þ QRð Þ ð13:12Þ

Qloss is calculated as the reactive power losses in the two sections of the line, i.e.

Qloss ¼ QlossðSÞ þ QlossðRÞ ð13:13Þ

where

Qloss Sð Þ ¼ I2S XL=2ð Þ and Qloss Rð Þ ¼ I2

R XL=2ð Þ

Using (13.10) gives

Qloss Sð Þ ¼ 2V 2S

XLþ 2V 2

M

XL� 4VSVM

XLcos qS � qMð Þ

Qloss Rð Þ ¼ 2V 2M

XLþ 2V 2

R

XL� 4VM VR

XLcos qM � qRð Þ

9>>>=>>>;

ð13:14Þ


and QS is given using (13.11). Hence,

Qcomp ¼ 2VSVM

XLcos qS � qMð Þ þ 2VM VR

XLcos qM � qRð Þ � 4V 2

M

XLð13:15Þ

Therefore, by specifying the desired value of voltage at the point at which thecompensator is connected to the system, the required reactive power supplied bythe compensator can be determined using (13.15) as well as the active power at thesending and receiving ends is obtained using (13.11). It is found that the powertransfer capability of the transmission line is increased when using shunt com-pensator, which in turn enhances both the system stability and system security.

Example 13.2 The parameters of the equivalent circuit (Figure 13.1) have the perunit values:

VS ¼ 1:0ff14:15�; VR ¼ 0:9ff0�; XL ¼ 0:275

Find the power at both ends when the system is uncompensated delivering a powerof 0.8 pu and when midpoint shunt is compensated to hold the magnitude of VR at1.0ff 0�. Examine the system stability.

Solution:

For uncompensated system, the apparent power at sending and receiving ends(12.30) and (12.31) are

PSo þ jQSo ¼ 0:80 þ j0:463 pu and PRo þ jQRo ¼ 0:8 þ j 0.228 pu

The relation of PS versus d is given by

PS ¼ 3.27 sin d (plotted in Figure 13.2)

In case of shunt compensation, taking the voltage at receiving end, VR, as a refer-ence the angles and voltages in the set of (13.11) through (13.15) are qR ¼ 0�, qS ≜d¼ 14.15�, VS ¼ VR ¼ V ‘to simplify the relations’. Consequently, qM ¼ (d/2) andthe set of relations becomes

PS þ jQS ¼ 2VVM

XLsin d=2ð Þ þ j

2XL

V 2 � VVM cos d=2ð Þ� �PR þ jQR ¼ 2VM V

XLsin d=2ð Þ � j

2XL

V 2 � VV M cos d=2ð Þ� �9>>=>>; ð13:16Þ

Qloss Sð Þ ¼ Qloss Rð Þ ¼ 2V 2

XLþ 2V 2

M

XL� 4VVM

XLcos d=2ð Þ ð13:17Þ

Qcomp ¼ 2VVM

XLcos d=2ð Þ þ 2VVM

XLcos d=2ð Þ � 4V 2

M

XLð13:18Þ


To plot the variables in the set of relations versus the angle d, VM is assumed toequal V as a further simplification to generally examine the system stability. If thesystem voltages are VS ¼ VM ¼ VR ¼ V pu, the set of relations, again, becomes ofthe form:

PS þ jQS ¼ 2V 2

XLsin d=2ð Þ þ j

2V 2

XL1 � cos d=2ð Þð Þ

PR þ jQR ¼ 2V 2

XLsin d=2ð Þ � j

2V 2

XL1 � cos d=2ð Þð Þ

9>>>=>>>;

ð13:19Þ

Qloss Sð Þ ¼ Qloss Rð Þ ¼ 4V 2

XL1 � cos d=2ð Þð Þ ð13:20Þ

Qcomp ¼ 4V 2

XLcos d=2ð Þ � 1ð Þ ð13:21Þ

Phasor diagram of the system in this special case is shown in Figure 13.3.The compensation reactive power, QC, is the negative of the absorbed reactive

power, i.e. QC ¼�Qcomp

It is found from (13.19) that PS ¼ (2/0.275) sin 14.15 ¼ 0.896 pu and the rela-tion of PS versus d for the compensated system is given by

PS ¼ 7.27 sin(d/2) (plotted in Figure 13.2)

Qloss Sð Þ ¼ Qloss Rð Þ ¼ 0:11 pu; QM ¼ �0:11 pu and QC ¼ 0:11 pu

Power angle δ in elec. rad

Act

ive

and

reac

tive

pow

er in

pu

π

QC

Po

PS (compensated)

PS (uncompensated)

10.500

5

10

15

1.5 2 2.5 3

Figure 13.2 Variation of QC, PS for compensated and uncompensated systems


As depicted in Figure 13.2, the area under PS versus d curve for the compen-sated case is much larger than that of the un-compensated case. Thus, the systemhas more stability margin.

13.2 Long transmission lines

It is appropriate to describe the long transmission line by ABCD parameters. It isfound that the values of these parameters depend on the location of shunt com-pensation. In case of locating the compensators at a point somewhere along theline the equivalent ABCD parameters (Figure 13.4) can be calculated using therelations below.

Aeq ¼ A1A2 þ B1C2 � jXCA2B1

Beq ¼ A1B2 þ B1D2 � jXCB1B2

Ceq ¼ A2C1 þ C2D1 � jXCA2D1

Deq ¼ B2C1 þD1D2 � jXCD2D1

9>>>>>>>=>>>>>>>;

ð13:22Þ

VR

VM

VS

δ

δ/2

Figure 13.3 System phasor diagram with shunt compensation

VS

IS

VR

IR

(a)

A1 B1

C1 D1

A1B1C1D1 A2B2C2D2

jXC

A2 B2

C2 D2

VS VR

IS IR

1 0

–jXC 1

(b)

Figure 13.4 (a) Transmission line with shunt compensation at midpoint and( b) equivalent representation


Similarly, when the line is shunt compensated at both sending and receiving ends(Figure 13.5) ABCD parameters are

Aeq ¼ A� jXC2B

Beq ¼ B

Ceq ¼ C � jXC1A� jXC2D� XC1XC2B

Deq ¼ D� jXC1B

9>>=>>; ð13:23Þ

Example 13.3 The system shown in Figure 13.6 comprises a transmission linewith the data given in Example 12.4. The internal machine and receiving endvoltages are given as 1.2ff17� pu and 0.9ff0� pu, respectively. Find the powertransfer capacity. If the system is shunt capacitive compensated at the receivingend, calculate the power transfer capacity and the compensated reactive powernecessary to raise the receiving end voltage up to 1ff0�.

Solution:

For uncompensated system the system ABCD parameters as calculated inExample 12.4 are

A ¼ AL þ j0:3CL ¼ 0:916ff0�

B ¼ BL þ j0:3DL ¼ j0:499 puC ¼ CL ¼ j0:206 pu

D ¼ DL ¼ 0:978

VS

IS

VR

IR

(a)

1 0

1

A B C D

jXC jXC

A B

C D

VS VR

IS IR

1 0

1

(b)

–jXC –jXC

Figure 13.5 (a) Transmission line with shunt compensation at both the sendingand receiving ends and ( b) equivalent representation

jXL

I

VsVt

δ°E

VR 0°

Com

pens

ator

xtr = 0.1xd′ = 0.2

R = 0.003

QC

Figure 13.6 System studied in Example 12.3


E ¼ Eejd ¼ 1:2e j17� and VR ¼ VRe j0� ¼ 0.9ff0� (VR is taken as a reference)

PS þ jQS ¼ EI� ¼ EVR

Xsin dþ j

E2 � EVR cos dð ÞX

ð13:24Þ

PR þ jQR ¼ VRI� ¼ EVR

Xsin d� j

V 2R � EVR cos d

� �X

ð13:25Þ

Qloss ¼ 1X

E2 þ V 2R � 2EVR cos d

� � ð13:26Þ

Thus, the following parameters can be calculated as:

PR ¼ ð1:2 � 0:9=0:499Þ sin 17� ¼ 2:16 sin 17� ¼ 0:63 pu

QR ¼ 1=0:499ð Þð1:2 � 0:9 cos 17� � 0:81Þ ¼ 0:446 pu

Qloss ¼ 1=0:499ð Þð1:44 þ 0:81 � 2 � 1:2 � 0:9 cos 17�Þ ¼ 0:37 pu

For compensated system the compensation is required to increase the voltage atreceiving end, VR ¼ 1ff0�. In this case it is found that

PR ¼ ð1:2 � 1:0=0:499Þ sin 17� ¼ 2:4 sin 17� ¼ 0:70 pu

QR ¼ 1=0:499ð Þð1:2 � 1:0 cos 17� � 1:0Þ ¼ 0:296 pu

Qloss ¼ 1=0:499ð Þð1:44 þ 1:0 � 2 � 1:2 � 1:0 cos 17�Þ ¼ 0:29 pu

It is seen that the reactive power flow into the line at the receiving end isreduced from 0.446 to 0.296. So, the difference ‘QC’ must be provided by the shuntcompensation, i.e. QC ¼ 0.15 pu.

The reactive power losses are reduced by a percentage of 27.6.P versus d curves for both the uncompensated and compensated cases are

shown in Figure 13.7. It is found that the area under P–d curve for the compensatedsystem is increased by the hatched area. Therefore, the compensation provides thesystem more marginal stability.

00

0.5

0.5

1

1

1.5

1.5

2

2 2.5

2.5

3 π

Uncompensatedsystem

Compensated system

Power angle in elec. degree (δ◦)

Rea

l pow

er (p

u)

Input power line

Figure 13.7 Power–angle curves for compensated and uncompensated systems


13.3 Static var compensators

It has been shown that the shunt capacitive compensation improves power systemstability by controlling both steady-state voltage and reactive power. It suppliespartially or fully the load with the required reactive power. Thus, it increases thepower system load-ability and reduces both the line current and the voltage drop inthe line. In the case of system operation at light loads or under open line conditionsthe voltage may increase. So, shunt inductive compensation is needed to limit thevoltage rise by continuously controlling the reactive power, which in turn willreduce the transmission losses and increase the transmission capacity of activepower. Consequently, the shunt compensation is used to compensate for the effectsof stressed and light load conditions of the transmission system. It should be acombination of fast controlled capacitances and inductances.

The static var compensator (SVC) is defined as a shunt device connected at a properlocation in the transmission system [5]. It is built up by static components (capacitorsand inductors), which may be controlled very fast by semiconductors, thyristors [6].

SVCs have different configurations as explained in Chapter 14. The basicconfiguration ‘FC-TCR’ is depicted in Figure 13.8. It is a combination of fixedcapacitor, FC, branch and thyristor-controlled reactor (TCR), branch as well as afilter for low-order harmonics for each of the three phases [7].

A continuous range of reactive power consumption is obtained by using thyr-istor firing angle, a, control. However, odd harmonic current components during thecontrol process are generated in the reactor current. Full conduction is obtained witha and equals 90�. The fundamental component of the reactor current is reduced byincreasing a, i.e. the inductance is increased and the reactive power absorbed by thereactor is reduced. It is to be noted that the change in the reactor current may onlytake place at discrete points of time, i.e. adjustments cannot be made more fre-quently than once per half-cycle. Static compensators of TCR type are characterisedby the ability to perform continuous control, maximum delay of one half-cycle and

HV

LV

FCTCR Filtercircuit

Controller

Figure 13.8 FC-TCR configurations


no transients. The main drawbacks of this configuration are the generation of low-frequency harmonic current components and higher losses when working in theinductive region [8, 9]. However, SVCs have a wide range of applications [10–12].

Using Fourier transform, the fundamental component of the reactor current, I1,in terms of the firing angle a, is given by

I1 ¼ Vrms

wL

2p� 2aþ sin 2að Þð Þp

¼ Vrms

XL

s� sinsp

ð13:27Þ

where

s ≜ the conduction angle ¼ 2(p – sin s)

The amplitude of each harmonic component is defined by

Ih ¼ 4Vrms

pXL

sin h þ 1ð Þa2 h þ 1ð Þ þ sin h � 1ð Þa

2 h � 1ð Þ � cos að Þ sin hað Þh

ð13:28Þ

where

Vrms is rms value of the voltage applied to the compensatorh is harmonic ordera is thyristor firing angle

To eliminate low-frequency current harmonics, 3rd, 5th, 7th, etc., deltaconnections for triplet harmonics and passive filters for the others may be usedas shown in Figure 13.9. The fixed capacitors may be connected in series withcurrent limiting reactors.

Filtercircuit

TCR

FC

VT

HV

Figure 13.9 TCR-delta-connection with FC and tuned filter for harmonicelimination


13.3.1 Characteristics of FC-TCR compensatorsThe amount of reactive power interchanged with the system, QS, depends on theapplied voltage, VT. The steady-state QS � VT characteristic of FC-TCR compen-sator is shown in Figure 13.10, indicating the amount of reactive power generatedor absorbed (QC or QL, respectively) by the compensator as a function of theapplied voltage. At the rated voltage the characteristic is linear and limited bythe rated power of both the capacitor and reactor. Between the voltage limits of thelinear part, Vmax and Vmin, the control range is defined. When the rated voltage isapplied to the compensator there is no interchange of reactive power between thecompensator and the power system. This rated voltage is taken as a referencevoltage, Vref, for the control process. The slope of the linear characteristic reflects achange in voltage with compensator rating and, therefore, can be considered a slopereactance resulting in the SVC response to the voltage variation. The slope reac-tance, XSL, is given by

XSL ¼ DV 2Cmax

QCmax¼ DV 2

Lmax

QLmaxð13:29Þ

Out of the control range, the VT � QS characteristic is non-linear. Assuming thecompensator is ideal, the change of QS versus VT out of the control range can beconsidered a linear relationship as an approximation.

13.3.2 Modelling of FC-TCR compensatorsFor power flow analysis and stability study of power systems, a model of the SVCis required to express its performance by mathematical relations that introduceaccurate representation [12]. Assuming an ideal SVC, the steady-state V–I char-acteristic is as shown in Figure 13.11.

FC TCR QS

VT

ΔVLmaxΔVCmax Vref

Vmax

Vmin

QLmaxQCmax

σ = 130o

σ = 150o

σ = 180oσ = 0o

QC = BC.V2

Q(α) = [BC – BL(α)].V2

XL

XC

VT

Figure 13.10 VT–QS characteristic of FC-TCR compensator


For power flow analysis: referring to the schematic diagram (Figure 13.8) andV–I characteristic (Figure 13.11), it is seen that the SVC has three operation modes:

(i) The normal mode of operation: it is in the control range and the SVC isequivalent to a voltage source, Vref, behind the slope reactance, XSL, con-nected to HV-bus (Figure 13.12(a)).

(ii) The capacitive mode of operation: when the SVC operation reaches thecapacitive limit it becomes a fixed capacitive susceptance, BC, connected tothe LV-bus, which is connected to HV-bus through the transformer leakagereactance, XT (Figure 13.12(b)).

(iii) The inductive mode of operation: in this mode the SVC is representedby fixed inductive susceptance, BL, when it reaches the inductive limit(Figure 13.12(c)). The SVC current is limited to ISmax.

In the control range where Imin < ISVC < Imax and Vmin < V < Vmax, the SVC isrepresented as PV-node at an auxiliary or phantom bus with P ¼ 0 and V ¼ Vref. Theslope reactance is added between the auxiliary bus and HV-bus (node of coupling tothe system). The HV-bus is a PQ bus with P ¼ 0 and Q ¼ 0 (Figure 13.13(a)). If theSVC transformer is represented, the reactance from the HV-bus to the auxiliary busis a portion of the transformer leakage reactance. The LV-bus is a PV bus remotelyregulating the auxiliary bus voltage to equal Vref (Figure 13.13(b)).

VHV

IS

ISmax

Inductivelimit

Vref

Capacitivelimit

Slope XSL

Vmin

Vmax

Figure 13.11 Steady-state V–I characteristic

Vref

HV-bus HV-busLV-bus HV-busLV-bus

IS IS IS

XTXTXSL BLBC

(a) (b) (c)

Figure 13.12 SVC modes of operation: (a) normal mode; ( b) capacitive mode;and (c) inductive mode


In the case of operating the SVC outside the control range, it is represented as ashunt element with susceptance B, defined as

B ¼ 1=XCð Þ if V < Vmin and B ¼ 1=XLð Þ if V > Vmax ð13:30ÞFor stability study: The general approach to the modelling of an FC-TCR-type SVC isrepresented by the functional block diagram shown in Figure 13.14. It includes ameasuring circuit block, voltage regulator block and TCR block. The measuring cir-cuit block comprises instrument transformers, A/D converters and rectifiers. It con-tains a transport delay and very small time constants. Thus, the measuring circuit maybe represented by a simple time constant and a unit gain. The voltage regulator blockrepresents a proportional type regulator that could be with lead/lag circuit (integraltype may be used as well). The TCR block represents the variation of thyristor firingangle. Accordingly, the simplified model of an FC-TCR compensator is shown inFigure 13.15. A voltage control signal can be modulated by a fast power oscillationdamping (POD) control in case of severe system stability problems after system faults.

HV-bus

Auxiliary busRemotelycontrolled

XSL

Vref

XT – XSL

LV-bus

IS

SVC

HV-bus

Auxiliary(PV) bus

XSL

Vref

IS

SVC

(a) (b)

Figure 13.13 SVC representations in load flow analysis: (a) SVC as a node at theauxiliary bus and (b) SVC connected to a regulated auxiliary busthrough transformer

+

+_V VM

Othersignals

Vref

Measuring circuit Voltage regulator TCRα

Figure 13.14 FC-TCR functional block diagram


The SVC-regulator model takes into account the firing angle a as an outputassuming a balanced fundamental frequency operation. Thus, the model can bedeveloped with respect to a sinusoidal voltage. It can be represented by two differ-ential equations of vM and a deduced from Figure 13.15 and an algebraic equation tocalculate the reactive power ‘Q’ [13]. So, the representation can be written as below:

_vM ¼ KM V � vMð ÞTM

_a ¼�KDaþ K

T1

T2TMvM � KM Vð Þ þ K Vref þ VPOD � vM

� �� T2

Q ¼ �2a� sin 2a� p 2 � XL

XC

� �pXL

V 2¼ �bSVC að ÞV 2

9>>>>>>>>>>=>>>>>>>>>>;

ð13:31Þ

The thyristor firing angle a is allowed to vary within upper and lower limits.The SVC state variables are initialised after the power flow solution. To impose thedesired voltages at the compensated bus, a PV generator bus with zero active powershould be used in the power flow solution. After the power flow solution the PVbus is removed and the SVC equations are used. During the state variable initi-alisation a check for SVC limits is performed.

Example 13.4 The system transient stability has been evaluated in Example 8.1for a nine-bus test system, when subjected to a three-phase short circuit at bus no. 7‘at the beginning of line 7-5’ for durations of 0.08 s and 0.20 s. The fault is clearedby isolating line #7-5. It is required to investigate the system stability improvementusing FC-TCR compensator. The system data are given in Appendix II.

Solution:

SAT/MATLAB� toolbox and typical control system parameters of the model inFigure 13.15 and Table 13.1 are used to check the transient stability when usingFC-TCR shunt compensator at different locations. It is found that the best resultsare obtained when connecting the compensator at bus no. 7. The change of powerangle and angular speed of each generator for fault durations of 0.08 s and 0.2 s areshown in Figures 13.16–13.18 and Figures 13.19–13.21, respectively.

+

+_V VM

VPOD

Vref

1α

αmin

αmax

KM

TMs + 1K(1 + T1s)KD + T2s

Figure 13.15 Simplified model of an FC-TCR-type SVC


Fault duration 0.08 s

Table 13.1 Control system parameters

Variable Description Unit Value

S Power rating MVA 100V Voltage rating KV 230f Frequency rating Hz 60T2 Regulator time constant S 10K Regulator gain pu/pu 100Vref Reference voltage pu 1afmax Maximum firing angle Rad 1afmin Minimum firing angle Rad �1KD Integral deviation pu 0.001T1 Transient regulator time constant S 0KM Measure gain pu/pu 1TM Measure time delay S 0.01xL Reactance (inductive) pu 0.2xC Reactance (capacitive) pu 0.1

0 2 4 6 8 10 12 14 16 18 20Time (s)

55

60

65707580859095

δ 1°

0.990.995

11.0051.01

1.0151.02

1.0251.03

1.035

ω1 (

elec

. rad

/s)

0 2 4 6 8 10 12 14 16 18 20Time (s)(b)

(a)

Figure 13.16 Generator 1: (a) d versus time and ( b) w versus time,fault duration ¼ 0.08 s


Time (s)

60708090

δ 2°

110100

50403020

20181614121086420

20181614121086420

Time (s)

0.98

0.99

1

1.01

1.02

1.03

1.04

ω2

(ele

c. ra

d/s)

(a)

(b)


4

2

0

–10

–8

–6

–4

–2

δ 3°

(a)

(b)

Time (s)20181614121086420

Time (s)20181614121086420

0.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

ω3 (

elec

. rad

/s)




1101009080706050

δ 1°

120130140150

(a) Time (s)20181614121086420

(b) Time (s)

20181614121086420

0.9950.99

0.985

1

0.98

1.0051.01

1.0151.02

1.025

ω1 (

elec

. rad

/s)


Time (s)

Time (s)

0.980.99

11.01

1.05

1.021.03

ω2 (

elec

. rad

/s)

0.970.960.95

1.04

(a)

(b)

100

50

0

–50

150

200

20181614121086420

86420 201816141210

δ 2°



13.4 Static synchronous compensator (STATCOM)

A static compensator, STATCOM, acts as a solid-state synchronous voltage sourcein analogy with a synchronous machine generating a balanced set of three sinu-soidal voltages at the fundamental frequency with controllable amplitude and phaseangle. This device, however, has no inertia [14].

Principle of operation: The static compensator consists of a voltage source con-verter, a coupling transformer and controls. In this application the DC energysource device can be replaced by a DC capacitor, so that the steady-state powerexchange between the static compensator and the AC system can only be reactive,as illustrated in Figure 13.22, where Iq is the converter output current, in quadraturewith the converter voltage Vi. The magnitude of the converter voltage, and thus thereactive power output of the converter, is controllable. If Vi is greater than theterminal voltage ‘VHV’ the static compensator will supply reactive power to the ACsystem. If Vi is smaller than VHV, the static compensator absorbs reactive power.

STATCOM model: The STATCOM is modelled as a current injection model. TheSTATCOM current is always kept in quadrature in relation to the bus voltage so

–5

–10

–15

–20

δ 3°

0

5

(a) Time (s)

20181614121086420

(b) Time (s)

20181614121086420

0.995

0.99

1

1.005

1.01

1.015

1.02

1.025

ω3 (

elec

. rad

/s)



that only reactive power is exchanged between the a.c. system and the STATCOM.The dynamic model is shown in Figure 13.23 where it can be seen that theSTATCOM assumes a single time constant regulator role like the SVC. The dif-ferential equation and the reactive power injected at the STATCOM node aregiven, respectively, by

_I q ¼ Kr Vref þ VPOD � VHV

� �� Iq

� �Tr

Q ¼ IqVHV

9=; ð13:32Þ

where

Kr: regulator gainVref : reference voltage of the STATCOM regulator.

VPOD: additional stabilising signal, which is the output of the power oscillationdamper

Tr: regulator time constant

VSC

VDC

Vi

VHV

T

Iq

+ _

HV-bus

Iq

VDC

Vi > VHV

Vi < VHV

Suppliesreactive power

Absorbsreactive power

Figure 13.22 Static compensator comprising voltage source converter (VSC),coupling transformer T and control

imax

imin

Iq

VPOD

Vref

VHV

+

+

_ Kr

1 + Trs

Figure 13.23 Block diagram of STATCOM simplified model


Example 13.5 Repeat Example 13.4 using STATCOM instead of SVC.

Solution:

A typical data of control parameters (Figure 13.23) is summarised in Table 13.2.Using PSAT/MATLAB� toolbox and connecting the STATCOM as a shunt

compensator at different locations in the system, the best results are obtainedwith the compensator connected in shunt at bus no. 8. The results, variation of dand w with time, are shown in Figures 13.24–13.26 for fault duration 0.08 s andFigures 13.27–13.29 for fault duration 0.2 s.


Table 13.2 Control system parameters

Variable Description Value Unit

S Power rating 100 MVAV Voltage rating 13.8 KVf Frequency rating 60 HzTr Regulator time constant 50 SKr Regulator gain 0.1 pu/puImax Maximum current 0.2 puImin Minimum current �0.2 pu

(a) Time (s)20181614121086420

(b) Time (s)

201816141210864200.995

1

1.005

1.01

1.015

1.02

1.0251.03

1.035

ω1 (

elec

. rad

/s)

5560

65707580859095

100

δ 1°



(a) Time (s)20181614121086420

(b) Time (s)

201816141210864200.995

1

1.005

1.01

1.015

1.02

1.025

1.03

1.035

ω2 (

elec

. rad

/s)

55

50

45

60

65

70

75

80

δ 2°


–10

–2

0

2

46

δ 3°

–8

–4

–6

(a) Time (s)20181614121086420

(b) Time (s)20181614121086420

1.005

1

1.01

1.015

1.02

1.025

1.03

1.035

ω3 (

elec

. rad

/s)




(a) Time (s)

20181614121086420

(b) Time (s)

20181614121086420

20

40

60

80

140

100

120

δ 1°

0.99

1

1.01

1.02

1.03

1.04

1.05

ω1 (

elec

. rad

/s)


(a) Time (s)18 201614121086420

(b) Time (s)18 201614121086420

0.9950.99

0.985

11.0051.01

1.0151.02

1.0251.03

1.035

ω2 (

elec

. rad

/s)

5040302010

60708090

110100

δ 2°



It is to be noted that the system without compensation is unstable when subjectedto a fault of duration 0.2 s (Example 8.2). From the results obtained above,Examples 13.4 and 13.5, it is concluded that:

● The system is stabilised when using either SVC or STATCOM compensatorsin the case of fault duration of 0.2 s.

● For the stable state at which the fault duration is 0.08 s, damping of theoscillations is increased when using shunt compensation.

● The preferred location as well as the effectiveness of compensation depends onthe type of compensation. It is seen that the STATCOM is connected at busno. 8 while the SVC is connected at bus no. 7.

In power system stability studies, a combination of both series and shuntcompensation may be applied simultaneously to enhance the system stability,increase power transfer capability and modify the voltage profile at different buses.Studies should be performed to determine the cost of each compensator beforefinalising the design.

13.5 Application of ASNFC to shunt-compensatedpower systems

In the preceding sections the analytic time analysis has been used to study thedynamic performance of the power system. On the other hand, the adaptive simpli-fied neuro-fuzzy controller, ASNFC, has been applied for designing the PSS as a

(a) Time (s)20181614121086420

(b) Time (s)

201816141210864201

1.005

1.01

1.015

1.02

1.025

1.03

1.035

ω3 (

elec

. rad

/s)

5040302010

60708090

110100

δ 3°



supplementary controller of the excitation system to damp the generator oscillationsas explained in Chapter 10, Section 10.5. In this section, the application of a pro-posed ASNFC as an artificial intelligent controller is used to control the SVCoperation as a shunt compensator. The system model considered for studying theperformance of the proposed adaptive controller is depicted in Figure 13.30. Thesimulation study has been performed on a single machine connected to an infinitebus, SMIB, through a long transmission line, with the SVC at the middle of the line.As shown in Figure 13.30, the SVC is used rather than the ideal shunt compensator.It is modelled as a susceptance that varies within a limit, depending on the controlprovided by the regulator of the SVC [15, 16].

The general structure of ASNFC system given in Chapter 10, Section 10.6, isapplied to the SVC to damp power system oscillations. The control structure hastwo steps: estimation of the system parameters online, using artificial neural-identifier, ANI, and calculation of the controller parameters using the gradientdescent method. If the system parameters vary, the identifier will provide an esti-mate of these parameters and the adaptation mechanism will subsequently tune thecontroller parameters. Inputs to the ASNFC are the power deviation at the SVCbus, DPSVC(k), and its derivative, D _Psvc; while the output of the controller is u(k).The input scaling blocks, K1 and K2, map the real input to the normalised inputspace in which the membership functions are defined. The output block K3 is usedto map the output of the fuzzy inference system to the real output.

13.5.1 Simulation studiesAn SVC device connected at the middle of an SMIB system is used to test theperformance of the proposed ASNFC. An adaptive neuro-identifier is used to trackthe behaviour of the plant online and update the ASNFC. The proposed controller

ANI

ASNFC

SVCDevice

Infinite-Bus

G

Vref

Vm

u(t)

+_

∆PSVC

Figure 13.30 ASNFC for an SVC device in an SMIB system


is designed based on a zero-order Sugeno-type fuzzy controller with one input tothe ANFIS network.

The performance of the ASNFC is compared with the ANFC and SVC con-ventional power system stabiliser (SCPSS). The SCPSS was carefully tuned for thebest possible performance at the nominal load of 0.70 pu power and 0.85 pf lag.The parameters of the SVC were then kept unchanged for the test performed. Asampling frequency of 25 Hz is used in the simulation and the learning rate is set tobe 0.08. The absolute limits for the control output are limited to �0.1 pu and theabsolute limits for the SVC output are limited to �0.15 pu. The performance of theproposed controller has been verified for a three-phase to ground short circuit [15].

The parameters of (i) the seventh-order model differential equations used tosimulate the generating unit, (ii) the transmission line and (iii) the transfer functionsof the exciter, the governor, the SVC device, the SCPSS and the generator con-ventional power system stabiliser (GCPSS) are provided in Appendix V [17, 18].

13.5.2 Three-phase to ground short circuit testThe GCPSS is applied to the generator during the three-phase to ground shortcircuit test. The behaviour of the system with the proposed ASNFC is investigatedin this test. The generator speed deviation response, under the normal operatingcondition, to a three-phase to ground short circuit at the middle of one of thetransmission lines connecting the generator to the middle bus is illustrated inFigure 13.31. The fault occurs at 1.0 s and is cleared 80 ms later by disconnection

3Time (s)

Gen

erat

or sp

eed

devi

atio

n (r

ad/s

)

4 5 6 7210–2.5

–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

2.5NO SCPSSSCPSSANFCASNFC

Figure 13.31 Generator speed deviation in response to a three-phase fault at themiddle of a transmission line connecting the generator to the middlebus with a successful re-closure


of the faulted line and successful re-closure at 5.0 s. The results show that theASNFC reduces the speed deviation after the fault and helps the system to reachthe new operating point quickly. More investigation and further explanation areprovided in [19].

References

1. Happ H.H., Wirgau K.A. ‘Static and dynamic VAR compensation in systemplanning’. IEEE Transactions on Power Apparatus and Systems. Sept/Oct1978;97(5):1564–78

2. Diaz U.A.R., Hermadez J.H.T. (eds.). ‘Reactive shunt compensation planningby optimal power flows and linear sensitivities’. 2009 Electronics, Roboticsand Automotive Mechanics Conference; Cuernavaca, Mexico, Sept 2009.pp. 326–31

3. Mahdavian M., Shahgholian G., Shafaghi P., Bayati-Poudeh M. (eds.). ‘Effectof static shunt compensation on power system dynamic performance’. IEEEInternational Symposium on Industrial Electronics (ISIE), 2011; Gdansk,Poland, Jun 2011. pp. 1029–32

4. Edris A.A. ‘Controllable VAR compensator: a potential solution to load-ability problem of low capacity power systems’. IEEE Transactions on PowerSystems. 1987;2(3):561–7

5. Goh S.H., Saha T.K., Dong Z.Y. (eds.). ‘Optimal reactive power allocationfor power transfer capability assessment’. IEEE PES General Meeting;Montreal, Canada, Jun 2006. pp. 1–7

6. Sahadat M.N., Al-Masood N., Hossain M.S., Rashid G., Chowdhury A.H.(eds.). ‘Real power transfer capability enhancement of transmission linesusing SVC’. Power and Energy Engineering Conference (APEEC), 2011 AsiaPacific; Wuhan, China, Mar 2011. pp. 104–7

7. Tyll H.K., Schettler F. (eds.). ‘Historical overview on dynamic reactive powercompensation solutions from the begin of AC power transmission towardspresent applications’. Power Systems Conference and Exposition PSCE’09;Seattle, WA, US, Mar 2009. pp. 1–7

8. Dixon J., Rodriguez J. ‘Reactive power compensation technologies: state-of-art review’. IEEE/JPROC. 2005;93(2):2144–64

9. Hauth R.L., Miske S.A., Nozari F. ‘The role and benefits of static VARsystems in high voltage power system applications’. IEEE Transactions onPower Apparatus and Systems. 1982;101(10):3761–70

10. Lajoie L.G., Larsen E.V. ‘Hydro-Quebec multiple SVC application controlstability study’. IEEE Transactions on Power Delivery. 1990;5(3):1543–51

11. Bronfeld J.D. (ed.). ‘Utility application of static VAR compensation’.Southern Tier Technical Conference, 1987, Proceedings of the 1987 IEEE;Binghamton, NY, US, Apr 1987. IEEE; 1987. pp. 53–63

12. Jayabarathi R., Sindhu M.R., Devarajan N., Nambiar T.N.P. (eds.). ‘Devel-opment of a laboratory model of hybrid static compensator’. Power India


Conference; New Delhi, India, Apr 2006. IEEE: Curran Associates; 2007.pp. 377–82

13. Talebi N., Ehsan M., Bathaee S.M.T. (eds.). ‘Effects of SVC and TCSCcontrol strategies on static voltage collapse phenomena’. IEEE Proceedings,Southeast Conference; Greensboro, NC, US, Mar 2004. pp. 161–8

14. Tan Y.L. ‘Analysis of line compensation by shunt connected FACTS con-trollers: A comparison between SVC and STATCOM’. Power EngineeringReview, IEEE. 1999;19(8):57–8

15. Albakkar A. ‘Adaptive simplified neuro-fuzzy controller as supplementarystabilizer for SVC’. PhD Thesis. Alberta, Canada: University of Calgary;2014

16. Albakkar A., Malik O.P. (eds.). ‘Adaptive neuro-fuzzy controller based onsimplified ANFIS network’. IEEE Power Engineering Society GeneralMeeting. San Diego, CA, US, Jul 2012. pp. 1–6

17. Gokaraj R. ‘Beyond gain-type scheduling controllers: new tools of identifi-cation and control for adaptive PSS’. PhD Dissertation. Alberta, Canada:Department of Electrical and Computer Engineering, University of Calgary;May 2000

18. IEEE Excitation System Model Working Group. ‘Excitation system modelsfor power system stability studies’. IEEE Standard 421.5. IEEE, 1992

19. Albakkar A.M. ‘Adaptive simplified neuro-fuzzy system as a supplementarycontroller for an SVC device’. PhD Thesis. Alberta, Canada: University ofCalgary; Sept 2014


Chapter 14

Compensation devices

14.1 Introduction

With the increase in demand and limitations on increasing power transfer capacity,power systems become more stressed and may be exposed to the risk of losingstability following a disturbance. One of the solutions to reducing this risk is tooptimise the utilisation of power system components by maximising their per-formance. The transmission network, as one of the major components in powersystems, attracts the power engineers to enhance its performance in both steady-state and transient conditions. As described in Chapters 12 and 13, the performanceof a transmission network can be improved by using either series or shunt com-pensation or a combination of both. The basic configurations of compensationmethods are described as well.

New compensation devices based on power electronics, which are very effi-cient for better utilisation of the existing transmission networks without sacrificingthe desired stability margin, have been developed. The transmission networkequipped with such devices is called ‘flexible AC transmission system, FACTS’.Different configurations of FACTS devices, such as controllers of network com-pensation based on what is described in Chapters 12 and 13, are described in thischapter [1].

Voltage instability refers to system voltage collapse, which makes the systemvoltage decay to a level from which the system is unable to recover. Voltage col-lapse occurs when a system is loaded beyond its maximum load-carrying limit. Theconsequence of voltage collapse may lead to a partial or full power interruption inthe system [2]. The only way to save the system from voltage collapse is to reducethe reactive power load or add additional reactive power prior to reaching the pointof voltage collapse. Introducing sources of reactive power, i.e. shunt capacitorsand/or FACTS controllers at appropriate location(s) in the system, is the mosteffective way for electric utilities to improve voltage stability of the system. Recentdevelopment and the use of FACTS controllers in power transmission system haveled to many applications of these controllers not only to improve the voltagestability of the existing power network but also to provide operating flexibility tothe power system [3].

14.2 Flexible AC transmission system

FACTS devices have been defined by IEEE as ‘alternating current transmissionsystem incorporating power electronic based and other static controllers to enhancecontrollability and increase power transfer capability’ [4]. There are six well-known FACTS devices utilised by the utilities for this purpose. These FACTSdevices are thyristor-controlled series capacitor (TCSC), synchronous series com-pensator (SSSC), static var compensator (SVC), static synchronous compensator(STATCOM), phase-shifting transformer (PST) and static and unified power flowcontroller (UPFC) [5]. Each of these devices has its unique characteristics andlimitations. From the utility’s perspective, the objective is to achieve voltagestability with the help of the most beneficial FACTS device.

14.2.1 Thyristor-controlled series capacitor

14.2.1.1 Principle of operationBasic configuration of a TCSC comprises controlled reactors in parallel with sec-tions of a capacitor bank as shown in Figure 14.1. This combination allows smoothcontrol of the fundamental frequency capacitive reactance over a wide range. Thecapacitor bank for each phase is mounted on a platform to ensure full insulation toground. The valve contains a string of series-connected high-power thyristors. Theinductor is of the air-core type. A metal-oxide varistor is connected across thecapacitor to prevent over-voltages.

The characteristic of the TCSC main circuit depends on the relative reactancesof the capacitor bank, XC ¼ 1/wnC, and the thyristor branch, XV ¼wnL, where wn isthe fundamental angular frequency in rad/s, C is the capacitance of the capacitorbank in F and L is the inductance of the parallel reactor in H.

MOV

Capacitor banks

Earthing switchEarthing switch

Platformdisconnect

Platformdisconnect

ThyristorvalveTCSC reactor

Bypass CBDamping circuit

Bypass disconnect

+ –VC

Figure 14.1 Basic structure of TCSC


The TCSC can operate in several different modes with varying values ofapparent reactance, Xapp. In this context, Xapp is defined simply as the imaginarypart of the quotient given below, in which the phasors represent the fundamentalvalue of the capacitor voltage, VC1, and the line current, IL1, at rated frequency.

Xapp ¼ ImVC1

IL1

� �ð14:1Þ

The characteristic of the TCSC depends on the relative reactances of thecapacitor bank and thyristor branch. The resonance frequency, wr, at which thecapacitive reactance, XC, equals the inductive reactance, XL, is expressed as [6]

XC ¼ � 1wnC

XL ¼ wnL

9>=>; ð14:2Þ

Thus,

wr ¼ 1ffiffiffiffiffiffiffiLC

p ¼ wn

ffiffiffiffiffiffiffiffiffiffi�XC

XL

rð14:3Þ

It is also practical to define a boost factor, KB, as the quotient of the apparent andphysical reactance, XC, of the TCSC:

KB ¼ Xapp

XCð14:4Þ

Blocking mode: When the thyristor valve is not triggered and the thyristorsremain non-conducting the TCSC will operate in blocking mode. Line currentpasses through the capacitor bank only. The capacitor phasor voltage, VC, is givenin terms of the line phasor current, IL. In this mode the TCSC performs in the sameway as a fixed series capacitor with a boost factor equal to 1.

Bypass mode: If the thyristor valve is triggered continuously it will remainconducting all the time and the TCSC will behave like a parallel connection of theseries capacitor bank and the inductor of the thyristor valve branch.

In this mode the capacitor voltage at a given line current is much lower than inthe blocking mode. The bypass mode is therefore used to reduce the capacitor stressduring faults.

Capacitive boost mode: If a trigger pulse is supplied to the thyristor withforward voltage just before the capacitor voltage crosses the zero line, a capacitordischarge current pulse will circulate through the parallel inductive branch. Thedischarge current pulse is added to the line current through the capacitor bank andcauses a capacitor voltage, which is added to the voltage caused by the line current(Figure 14.2). The capacitor peak voltage will thus be increased in proportion to thecharge passing through the thyristor branch. The fundamental voltage also increa-ses almost in proportion to the charge.

Compensation devices 381

The TCSC has the means to control the angle of conduction, b, as well as tosynchronise the triggering of the thyristors with the line current.

14.2.1.2 Application of TCSC for damping electromechanicaloscillations

The basic power flow equation shows that modulating the voltage and reactanceinfluences the flow of active power through the transmission line. In principle, aTCSC is capable of fast control of the active power through a transmission line.The possible control of transmittable power points to this device as being used todamp electromechanical oscillations in the power system. Features of this dampingeffect are:

● The effectiveness of the TCSC for controlling power swings increases withhigher levels of power transfer.

● The damping effect of a TCSC on an intertie is unaffected by the location ofthe TCSC.

● The damping effect is insensitive to the load characteristic.● When a TCSC is designed to damp inter-area modes, it does not excite any

local modes.

14.2.2 Static synchronous series compensatorA voltage source converter (VSC) can be used in series in a power transmissionsystem. Such a device is referred to as a static SSSC.

0 10 20 30 40 50 60 8070 90–3

–2

–1

0

1

2

3

4

Capacitive boost

Inductive boost

KB

β °

Figure 14.2 Boost factor, KB, versus conduction angle, b, for a TCSC


14.2.2.1 Principle of operationA VSC connected in series with a transmission line through a transformer is shownin Figure 14.3. A source of energy is needed to provide the DC voltage across thecapacitor and make up for the losses of the VSC.

In principle, an SSSC is capable of interchanging active and reactive powerwith the power system. However, if only reactive power compensation is inten-ded, the size of the energy source could be quite small. The injected voltage canbe controlled in terms of magnitude and phase if there is a sufficiently largeenergy source. With reactive power compensation, only the magnitude of thevoltage is controllable as the vector of the inserted voltage is perpendicular to theline current.

In this case the series injected voltage can either lead or lag the line current by90�. This means that the SSSC can be smoothly controlled at any value leading orlagging within the operating range of the VSC. Thus, an SSSC can behave in asimilar way to a controllable series capacitor and a controllable series reactor. Thebasic difference is that the voltage injected by an SSSC is not related to the linecurrent and can be independently controlled. This important characteristic meansthat the SSSC can be used with great effect for both low and high loading [7].

14.2.2.2 ApplicationsThe general application of a controllable series capacitor applies also to the SSSC,dynamic power flow control and voltage plus angle stability enhancement. The factthat an SSSC can induce both capacitive and inductive voltage on a line widens theoperating region of the device. For power flow control, an SSSC can be used toboth increase and reduce the flow. In the stability area it offers more potential fordamping electromechanical oscillations. However, the inclusion of a high-voltagetransformer in the scheme means that, compared with controllable series capacitors,it is at a cost disadvantage. The transformer also reduces the performance ofthe SSSC due to an extra reactance being introduced. This shortcoming may be

Energy source

VSC

+VC –V1 θ1 Vi θi Vj θ j V2 θ2+jXL1 +jXL2

Iq

Figure 14.3 Basic configuration of a static synchronous series compensator(SSSC)


overcome in the future by introducing transformerless SSSCs. The scheme alsocalls for a protective device that bypasses the SSSC in the event of high faultcurrents on the line.

14.2.3 Static var compensatorOver the years SVCs of many different designs have been built. Nevertheless, themajority of them have similar controllable elements. The most common ones are

● thyristor-controlled reactor (TCR)● thyristor-switched capacitor (TSC)● thyristor-switched reactor (TSR)● mechanically switched capacitor (MSC)

14.2.3.1 Principle of operationIn the case of the TCR a fixed reactor, typically an air-core type, is connected inseries with a bidirectional thyristor valve. The fundamental frequency current isvaried by phase control of the thyristor valve. A TSC comprises a capacitor inseries with a bidirectional thyristor valve and a damping reactor. The function ofthe thyristor switch is to connect or disconnect the capacitor for an integral numberof half-cycles of the applied voltage. The capacitor is not phase-controlled, beingsimply on or off. The reactor in the TSC circuit serves to limit current underabnormal conditions as well as to tune the circuit to a desired frequency [8].

The impedances of the reactors, capacitors and power transformer define theoperating range of the SVC. The corresponding V–I diagram has two differentoperating regions. Inside the control range, voltage is controllable with an accuracyset by the slope. Outside the control range the characteristic is that of a capacitivereactance for low voltages and that of a constant current for high voltages. The low-voltage performance can be easily improved by adding an extra TSC bank (for useunder low-voltage conditions only) [9].

The TSR is a TCR without phase control of the current, being switched in orout like a TSC. The advantage of this device over the TCR is that no harmoniccurrents are generated. The MSC is a tuned branch comprising a capacitor bank anda reactor. It is designed to be switched no more than a few times a day as theswitching is performed by circuit-breakers. The purpose of the MSC is to meetsteady-state reactive power demand [7].

14.2.3.2 SVC configurationsControlled reactive power compensation is usually achieved in electric powersystems by means of the SVC configurations shown in Figure 14.4.

A further variation of SVC configuration is achieved by using multi-levelconverters that have less harmonic generation and higher voltage capabilitybecause of serial connection of bridges or semiconductors. The arrangement ofthree-level converters is the most popular arrangement (Figure 14.5). Due to reducedharmonic interaction with the surrounding system multi-level converter-based SVCs


need fewer components and are easier to integrate in power systems than othertypes of static compensators. In addition, power losses of a multi-level converterare considerably lower than those of other SVCs of the same power rating but stillslightly higher than those of the thyristor-based SVCs.

QrefQref Qref

MSCFilters FiltersTSR TSC TCR TSC TCR

Figure 14.4 SVC configurations used to control reactive power compensation inelectric power systems

Load

Pow

er sy

stem

Controller

00 0

Figure 14.5 Three-level converter-based SVC


14.2.3.3 SVC applicationsSVCs are installed to perform the following functions:

● dynamic voltage stabilisation: increased power transfer capability, reducedvoltage variation

● synchronous stability improvements: increased transient stability, improvedpower system damping

● dynamic load balancing● steady-state voltage support

Typically, SVCs are rated such that they are able to vary the system voltageby at least �5%. This means that the dynamic operating range is normally about10–20 per cent of the short-circuit power at the point of common connection.Three different locations are suitable for the SVC. One is close to major loadcentres, such as large urban areas, another is in critical substations, normally inremote grid locations, and the third is at the in feeds to large industrial or tractionloads.

The two most popular configurations of this type of shunt controller are thefixed capacitor (FC) with a TCR and the TSC with TCR. Among these two setups,the second, TSC-TCR, minimises standby losses; however, from a steady-statepoint of view, this is equivalent to the FC-TCR. In this chapter, the FC-TCRstructure is used for the analysis of the SVC, which is shown in Figure 14.6.

The TCR consists of a fixed reactor of inductance L and a bidirectional thyr-istor valve fired symmetrically in an angle control range of 90–180�, with respect tothe SVC voltage.

Assuming controller voltage equal to the bus voltage and performing a Fourierseries analysis on the inductor current wave form, at fundamental frequency theTCR, can be considered to act like variable inductance given by [9]:

XV ¼ XLp

2 p� að Þ þ sin 2að14:5Þ

XC

X1

i1(t)

iC(t)

i(t)

Figure 14.6 Equivalent FC-TCR circuit of SVC


where XL is the reactance caused by the fundamental frequency without thyristorcontrol and a is the firing angle. Hence, the total equivalent impedance of thecontroller can be represented as

Xeq ¼ XC

pk

sin 2a� 2aþ p 2 � 1k

� � ð14:6Þ

where k ¼ XC/XL. The limits of the controller are given by the firing angle limits,which are fixed by design.

The typical steady-state control law of an SVC used here is depicted inFigure 14.7 and may be represented by the following voltage–current characteristic:

V ¼ Vref þ XSLI ð14:7Þwhere V and I stand for the total controller rms voltage and current magnitudes,respectively, and Vref represents a reference voltage. Typical values for the slope XSL

are in the range of 2–5 per cent, with respect to the SVC base; this is needed to avoidhitting limits for small variations of the bus voltage. A typical value for the con-trolled voltage range is �5 per cent about Vref [7]. At the firing angle limits, the SVCis transformed into a fixed reactance. Of course, changing the reactance of the FCbanks (from XC to XC1 or to XC2) will change the capacitive region accordingly.

14.2.4 Static synchronous compensatorThe static compensator is based on a solid-state synchronous voltage source inanalogy with a synchronous machine generating balanced set of (three) sinusoidalvoltages at the fundamental frequency with controllable amplitude and phase angle.This device, however, has no inertia [10].

αmax

αminVref(αo)

XC

XL

XSL

V

I

XC1 XC2

Capacitive Inductive

Currentlimiting

Figure 14.7 Typical steady state V–I characteristic of a SVC


14.2.4.1 Principle of operationA static compensator consists of a VSC, a coupling transformer and controls. In thisapplication the DC energy source device can be replaced by a DC capacitor, so thatthe steady-state power exchange between the static compensator and the AC systemcan only be reactive, as illustrated in Figure 14.8. Iq is the converter output current,perpendicular to the converter voltage Vi. The magnitude of the converter voltage,and thus the reactive output of the converter, is controllable. If Vi is greater than theterminal voltage, Vt, the static compensator will supply reactive power to the ACsystem. If Vi is smaller than Vt, the static compensator absorbs reactive power.

The AC circuit is considered in steady state, whereas the DC circuit isdescribed by the following differential equation, in terms of the voltage Vdc on thecapacitor [11]. The power injection at the AC bus has the form

P ¼ V 2G � kVdcVG cos q� að Þ � kVdcVB sin q� að ÞQ ¼ �V 2B þ kVdcVB cos q� að Þ � kVdcVG sin q� að Þ

)ð14:8Þ

where k ¼ ffiffiffiffiffiffiffiffi3=8

pm

14.2.4.2 ApplicationsThe functions performed by STATCOMs are [12–14]

● dynamic voltage stabilisation: increased power transfer capability, reducedvoltage variations

● synchronous stability improvements: increased transient stability, improvedpower system damping, damping of SSR

● dynamic load balancing● power quality improvement● steady-state voltage support

VSC

Vi

VHV

T

Iq

+ _

HV-bus

Iq

Vdc

Vdc

Vi > VHV

Vi < VHV

Suppliesreactive power

Absorbsreactive power

m : 1

Figure 14.8 Static compensator, comprising VSC, coupling transformer T, andcontrol


14.2.5 Phase-shifting transformerPhase angle regulating transformers, phase shifters, are used to control the flow ofelectric power over transmission lines. Both the magnitude and the direction of thepower flow can be controlled by varying the phase shift across the series trans-former [15] (Figure 14.9).

14.2.5.1 Principle of operationThe phase shift is obtained by extracting the line-to-ground voltage of one phaseand injecting a portion of it in series with another phase. This is accomplished byusing two transformers: the regulating ‘or magnetising’ transformer, which isconnected in shunt, and the series transformer.

The star–star and star–delta connections used are such that the series voltagebeing injected is in quadrature with the line-to-ground voltage.

A portion of the line voltage is selected by the switching network and insertedin series with the line voltage. The added voltage is in quadrature with the linevoltage, e.g. the added voltage on phase ‘a’ (DVa) is perpendicular to Vbc. The angleof a phase shifter is normally adjusted by on-load tap-changing (LTC) devices. Theseries voltage can be varied by the LTC in steps determined by the taps on theregulating winding. Progress in the field of high-power electronics has made itpossible for thyristors to be used in the switching network.

14.2.6 Unified power flow controllerThe UPFC consists of two switching converters operated from a common DC linkas shown in Figure 14.10. At times it is also referred to as a combination ofSTATCOM (shunt) and SSSC (series) compensators.

14.2.6.1 Principle of operationIn Figure 14.10 converter 2 performs the main function of the UPFC by injecting,through a series transformer, an a.c. voltage with controllable magnitude and phaseangle in series with the transmission line. The basic function of converter 1 is tosupply or absorb the real power demanded by converter 2 at the common d.c. link.

1

2

3

Vai

Vbi

Vci

Vao

Vao Vao

Vao

Vao

VbiVci

Vai

ΔVa

Φ

Figure 14.9 Phase shifter with quadrature voltage injection


It can also generate or absorb controllable reactive power and provide independentshunt reactive compensation for the line. Converter 2 supplies or absorbs therequired reactive power locally and exchanges the active power as a result of theseries injection voltage [1].

The UPFC model is shown in Figure 14.11. According to this figure, someparameters can be adjusted for keeping voltage level and power flow of the net-work. The model can be obtained from the models of the associated shunt com-pensator, e.g. STATCOM, and series compensator, e.g. SSSC, where the d.c.

Converter 1 Converter 2

Seriestransformer

Shunttransformer

Bus i Bus j

Vi θi Vj θjP, Q

Figure 14.10 Basic circuit arrangement of the unified power flow controller(UPFC)

Vk θK Vm θm

Ik Im

Pk + jQK msh : 1

Psh + jQsh

Pse + jQsemshIsh

Rsh + jXsh Rse + jXse

mseIse

mse : 1Pm + jQm

++

+Vdc

–

RCC

Pdc

RT + jXT+ v θ

kshVdc s 0 kseVdc s β

Figure 14.11 UPFC model


voltage Vdc is common for the two inverters. The related formulae of the powerflow are described below [16–18]:

Pk ¼ Psh þ ReðVkI�mÞ

Qk ¼ Qsh þ ImðVkIkmÞ

)ð14:9Þ

Pm ¼ �ReðVmI�mÞ

Qm ¼ �ImðVmI�mÞ

)ð14:10Þ

The power Psh and Qsh absorbed by the shunt side are

Psh ¼ V 2Gsh � kshVdcVkGsh cos qk � að Þ � kshVdcVkBsh sin qk � að ÞQsh ¼ V 2Bsh � kshVdcVkBsh cos qk � að Þ � kshVdcVkGsh sin qk � að Þ

)ð14:11Þ

The current, Im, and the voltage, V, produced due to series compensation aregiven by

_Im ¼ ð1 � a1ÞðVm � VÞ � a2V1

RT þ jXT

_V ¼ a1ðVm � VÞ þ a2V1

9>=>; ð14:12Þ

where

V1 ¼ kseVdcejb

a1 ¼ � Rse þ jXse

RT � Rseð Þ þ j XT � Xseð Þ

a2 ¼ � Rsh þ jXsh

RT � Rshð Þ þ j XT � Xshð Þand

ksh ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=8msh

pand kse ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi3=8mse

pThe DC circuit is modelled by the following differential equation:

_V dc ¼ Psh

CVdcþ Re VI�m

� �CVdc

� Vdc

RCC� Rsh P1

sh þ Q1sh

� �CVdcV

2k

� RseI1m

CVdcð14:13Þ

14.2.6.2 ApplicationsA UPFC can regulate the active and reactive power simultaneously. In general, ithas three control variables and can be operated in different modes. The shunt-connected converter regulates the voltage of bus i in Figure 14.10, and the series-connected converter regulates the active and reactive power or active power and thevoltage at the series-connected node. In principle, a UPFC is able to perform thefunctions of the other FACTS devices, which have been described, namely voltagesupport, power flow control and improved stability [19–24].


References

1. Hingorani N., Gyugyi L. Understanding FACTS: Concepts and Technology ofFlexible AC Transmission Systems. Piscataway, NJ, US: Wiley-IEEE Press;2000

2. Dobson I., Chiang H.D. ‘Towards a theory of voltage collapse in electricpower systems’. Systems & Control Letters. 1989;13:253–62

3. Canizares C.A., Alvarado F.L. ‘Point of collapse and continuation methods forlarge AD/DC systems’. IEEE Transactions on Power Systems. 1993;7(1):1–8

4. IEEE-PES and CIGRE. ‘Facts overview’. IEEE Cat. #95TP108, 19955. Canizares C.A. (ed.). ‘Power flow and transient stability models of FACTS

controllers for voltage and angle stability studies’. Proceedings of the 2000IEEE/PES Winter Meeting; Singapore, Jan 2000. pp. 1–8

6. Acharya N., Sode-Yome A., Mithulananthan N. ‘Comparison of shunt capa-citor, SVC and STATCOM in static voltage stability margin enhancement’.International Journal of Electrical Engineering Education, UMIST. 2004;41(3):1–6

7. Sode-Yome A., Mithulananthan N., Lee K.Y. (eds.). ‘Static voltage stabilitymargin enhancement using STATCOM, TCSC and SSSC’. IEEE/PESTransmission and Distribution Conference & Exhibition, Asia and Pacific;Dalian, China, 2005. pp. 1–6

8. Canizares C.A., Faur Z.T. ‘Analysis SVC and TCSC controllers in voltagecollapse’. IEEE Transactions on Power Systems. 1999;14(1):158–65

9. Boonpirom N., Paitoonwattanakij K. (eds.). ‘Static voltage stability enhance-ment using FACTS’. The 7th International Power Engineering ConferenceIPEC/IEEE; Singapore, Nov/Dec 2005, vol. 2. pp. 711–15

10. Tyll H.K., Schettler F. (eds.). ‘Historical overview on dynamic reactive powercompensation solutions from the begin of AC power transmission towardspresent applications’. IEEE/PES Power Systems Conference and Exposition,2009, PSCE’09; Seattle, Washington, US, Mar 2009. pp. 1–7

11. Natesan R., Radman G. (eds.). ‘Effects of STATCOM, SSSC and UPFC onvoltage stability’. Proceedings of the System Theory Thirty – Sixth South-eastern Symposium; Atlanta, GA, US, 2004. pp. 546–50

12. Talebi N., Ehsan M., Bathaee S.M.T. (eds.). ‘Effects of SVC and TCSCcontrol strategies on static voltage collapse phenomena’. IEEE Proceedings,SoutheastCon; Greensboro, NC, US, Mar 2004. pp. 161–8

13. Kazemi A., Vahidinasab V., Mosallanejad A. (eds.). ‘Study of STATCOMand UPFC controllers for voltage stability evaluated by saddle-node bifurca-tion analysis’. First International Power and Energy Conference PECon,IEEE; Putrajaya, Malaysia, Nov 2006. pp. 191–5

14. Verboomen J., Hertem D.V., Schavemaker P.H., Kling W.L., Belmans R.(eds.). ‘Phase shifting transformers: principles and applications’. Interna-tional Conference on Future Power Systems, 2005; Amsterdam, Holland, Nov2005. pp. 1–6


15. Mathur R., Varma R. Thyristor-Based FACTS Controllers for ElectricalTransmission Systems. NJ, US: Wiley-IEEE Press; 2002

16. Sun H., Luo C. (eds.). ‘A novel method of power flow analysis with unifiedpower flow controller (UPFC)’. PES Winter Meeting, IEEE; Singapore, Jan2000, vol. 4. pp. 2800–5

17. Kawkabani B., Pannatier Y., Simond J.J. (eds.). ‘Modeling and transientsimulation of unified power flow controllers (UPFC) in power system stu-dies’. IEEE Power Tech Conference 2007; Lausanne, Jul 2007. pp. 1–5

18. Shu-jun Y., Xiao-yan S., Yu-xin Y., Zhi Y. (eds.). ‘Research on dynamiccharacteristics of unified power flow controller (UPFC)’. Electric UtilityDeregulation and Restructuring and Power Technologies (DRPT), 2011, 4thInt. Conference on; Weihai, Shandong, China, Jul 2011. pp. 490–3

19. Ande S., Kothari M.L. (eds.). ‘Optimization of unified power flow controllers(UPFC) using GEA’. IEEE Power Engineering Conference, IPEC 2007;Singapore, Dec 2007. pp. 53–8

20. Saied E.M., El-Shibini M.A. (eds.). ‘Fast reliable unified power flow con-troller (UPFC) algorithm’. 7th International Conference on IET, AC-DCPower Transmission, 2001; Nov 2001. London: IET; 2001. pp. 321–6

21. Sedraoui K., Al-haddad K., Chandra A. (eds.). ‘Versatile control strategy ofthe unified power flow controller (UPFC)’. IEEE, Electrical and ComputerEngineering, 2000 Canadian Conference on; Halifax, NS, Canada, May2000, vol. 1. pp. 142–7

22. Balakrishnan F.G., Sreedharan S.K., Michael J. (eds.). ‘Transient stabilityimprovement in power system using unified power flow controller (UPFC)’.4th International Conference on Computing, Communications, and NetworkingTechnologies (ICCNT) 2013; Tiruchengode, India, Jul 2013. Piscataway, NJ,US: IEEE; 2013. pp. 1–6

23. Sen K.K., Stacey E.J. ‘UPFC-unified power flow controller: theory, modeling,and applications’. IEEE Transactions on Power Delivery. 1998;13(4):1453–60

24. Sharma N.K., Jagtap P.P. (eds.). ‘Modeling and application of unified powerflow controller (UPFC)’. 3rd International Conference on Energy Trendsin Engineering and Technology (ICETET), 2010; Goa, India, Nov 2010.pp. 350–5


Chapter 15

Recent technologies

Modern power systems have been growing in size and complexity. They arecharacterised by long distance bulk power transmission and wide area inter-connections. Such networks have a chance to produce transmission congestion dueto load increase (active and reactive power), particularly at peak periods, and alsoun-damped low-frequency power swings. This may cause severe problems such asreduction of power transfer capability of transmission lines, increased line losses,loss of generator synchronism. The system becomes stressed and has the risk oflosing stability following a disturbance. Considerable progress has been made toovercome such problems by (i) controlling the active power of both the generatorsand loads; (ii) controlling the reactive power using compensators, SSSC, SVC,STATCOM, UPFC, etc.; and (iii) using fast-response excitation control and gov-ernor control on the generating units [1, 2].

The interest in applying new technologies in electric power systems is directlyrelated to the expectation of improved performance, stability and efficiency. Someof the recently developed technologies, energy storage systems and phasor mea-surement devices, are presented in this chapter. Examples of the actual imple-mentation, in particular those from the perspective of power system stability, andthe trends in current research are discussed. Possible applications of energy storagein utility systems include transmission enhancement, power oscillation damping(POD), dynamic voltage stability, tie-line control, short-term spinning reserve, loadlevelling, reducing the need for under-frequency load shedding, allowing lessstringent time limits for circuit breaker reclosing, sub-synchronous resonancedamping and power quality improvement.

15.1 Energy storage systems

Electrical energy cannot be stored directly. It is possible to convert electricalenergy to another form that can be stored. The stored energy then can be convertedback to electricity when desired. There are a wide variety of possible forms inwhich the energy can be stored. Common examples include chemical energy(batteries), kinetic energy (flywheels or compressed air), gravitational potentialenergy (pumped hydroelectric) and energy in the form of electrical (capacitors) andmagnetic field. These energy storage methods act as loads while energy is beingstored and sources of electricity when the energy is returned to the system.

General configuration of an energy storage system is shown in Figure 15.1.It mainly comprises four components: (i) the energy storage medium, which setsthe basic system storage capability limits, (ii) the charging system, which takespower from the utility system (Grid) and converts it into the form that can be storedin the storage medium, (iii) the discharging system, which takes energy from thestorage and converts it back to electricity and then delivers to the utility grid and(iv) the control system, which is used to monitor the performance of the unit as wellas to control the how and when of flow of electrical energy between the storagesystem and the grid.

A limited amount of bulk energy storage, mainly in the form of pumpedhydroelectric storage (PHES), has long played a role in the electric power grid, andstorage continues to grow in importance as a component of the electric powerinfrastructure [3]. Advances in storage technologies and the needs of the electricpower grid enable energy storage to become a more substantial component of theelectric power grid of the future. Several primary drivers described below haveincreased interest in energy storage:

● The increase in peak demand and the need to respond quickly and efficiently tochanges in demand given constraints on generation and transmission capacity.

● The need to integrate distributed and intermittent renewable energy resourcesinto the electricity supply system.

● The need for investments in transmission and distribution systems that areexperiencing increasing congestion.

Storage medium unit

Chargingunit

Dischargingunit

Controlunit

Energy storage system

Utility grid

Figure 15.1 General configuration of energy storage system


● The need to provide grid ancillary services critical to the efficient and reliableoperation of the grid.

● The increase in the need for high-quality, reliable power as a result ofincreased use of consumer power electronics and information and commu-nication systems that are highly sensitive to power fluctuations.

15.1.1 Chemical energy storage systems (batteries)Batteries have the potential to span a broad range of energy storage applicationsdue in part to their portability, ease of use and variable storage capacity. In parti-cular, they can stabilise electrical systems by rapidly providing extra power and bysmoothing out ripples in voltage and frequency. Currently, numerous batteriesincluding lead-acid, flow, sodium-sulphur and lithium-ion have commercialapplications. However, many battery types have only limited market penetration,are expensive or have short lifetime in terms of charge/discharge cycles [4]. Effortsto develop battery technologies to improve their power and energy density char-acteristics, life cycle and costs are in progress on various fronts. These efforts mayresult in better storage options in the future.

15.1.2 Flywheel energy storageFlywheel energy storage (FES) converts electricity to rotational kinetic energy inthe form of the momentum of a spinning mass. Simply, a flywheel is a disc with acertain amount of mass that spins, holding kinetic energy. The disc is attached to arotor in an upright position to prevent the influence of gravity. The spinning mass,rotor and disc, rests on bearings that facilitate its rotation and altogether are con-tained in a sealed housing designed to reduce friction between them and the sur-rounding environment as well as to provide a safeguard against hazardous failuremodes. Reducing the friction increases the efficiency. Therefore, the spin massspins in a vacuum, i.e. no air friction, and has electromagnetic bearings [3, 5].

FES has several advantages: low maintenance cost, fast access to the storedenergy, no need for toxic resources and no carbon emissions. On the other hand,FES has the disadvantages of high cost and low capacity compared to systems suchas the pumped hydro-storage.

As flywheels can be charged and discharged quickly and frequently, they canbe used to maintain power quality and reliability of power systems by regulatingfrequency and providing protection against transient interruptions in the powersupply.

The flywheel model was initially developed and supplied by Beacon PowerCorporation [6]. The model incorporates charging and discharging losses, floatinglosses and auxiliary power as shown in Figure 15.2.

15.1.3 Compressed air energy storageCompressed air energy storage (CAES) system is a hybrid generation/storagetechnology in which electricity is used to inject air at high pressure into under-ground geologic formations ‘cavern’. When demand for electricity is high, the

Recent technologies 397

high-pressure air is released from the underground cavern and used to help powernatural gas-fired turbines (Figure 15.3). The pressurised air allows the turbines toprovide a generator with the kinetic energy necessary to generate electricity usingsignificantly less natural gas. CAES system is appropriate for load levelling as itcan be constructed in capacities of a few hundred megawatts and can be dischargedover long (4–24 hours) period of time.

15.1.4 Pumped hydroelectric energy storagePHES uses low-cost electricity generated during periods of low demand to pumpwater from a lower level reservoir to a higher elevation reservoir. During periods ofhigh electricity demand, the water is released to flow back down to the lowerreservoir while turning turbines to generate electricity, similar to conventional

Previousenergy

EnergyNet poweroutput

Limit

∆t_

+

Flywheelstate

One-wayefficiency

Auxiliarypower

Flywheelstate

Floatinglosses

+

+

+

Figure 15.2 Flywheel model [6]

M G

Utility system

Compressed aircarven

Fuel

combustorHigh-

pressureturbine

Low-pressure turbine

Compressor

Air

Motor Generator

Discharging cycleCharging cycle

Exhaust

Figure 15.3 Schematic diagram of compressed air energy storage system


hydropower plants. PHES is appropriate for load levelling as it can be constructedfor large capacities of hundreds to thousands of megawatts and discharge over longperiods of time (4–10 hours).

PHES is considered to be one of the possible ways to store energy in a largeamount while maintaining high efficiency, being economical and having fastresponse. A schematic illustration of a pumped storage plant is shown inFigure 15.4. It basically contains two water reservoirs at different elevations: one atlow level and the other at high level. The water is pumped from the lower reservoirinto the higher reservoir storing electricity in the form of potential energy of water.When needed, e.g. on peak demand or transmission congestion, water can bereleased flowing down the pipes again and back through the turbine, which thengenerates electricity. The output power, P, is given as

P ¼ Q � h � h� g � r ð15:1Þwhere

h is the water headQ is the volume flow rate passing the turbineh is the turbine efficiencyr is the water densityg is the gravitational acceleration

High-levelreservoir

Low-levelreservoir

Direction ofpumped water atlow electricity

demand

Direction of waterflow to generate

electricity

Water head

Water pipe

G Turbine

Figure 15.4 Schematic diagram of a pumped hydroelectric storage


Kaplan or Francis turbines are commonly used to maximise efficiency. Theyare reversible and capable of handling both the pumping and generating process.Similarly, synchronous machine can be operated as motor during pumping and as agenerator to generate electricity.

The developed hydro power plant model is shown in Figure 15.5. It includesdelay block simulating the delay in the plant’s response to the changing regulationsignal, dead band element, first-order plant response model, error range simulatingdeviations of the actual plant response from the load setting and limiting elementrestricting the maximum and minimum regulation output provided by the plant [6].

15.1.5 Super capacitorsSuper capacitors, like traditional dielectric capacitors, store energy by increasingthe electric charge accumulation on the metal plates and discharge energy when theelectric charges are released by the metal plates. The energy density of supercapacitor is, however, much higher than that of traditional capacitors and could beused to improve power quality as they can rapidly provide short bursts of energy(less than a second) and store energy for a few minutes. The current applicationstypically take place in combination with batteries or other storage or power sup-plies, in situations where a low average, but high pulse, power is needed.

15.1.6 Superconducting energy storageFirst, before dealing with superconducting energy storage, it is of crucial importanceto understand what superconductors (SCs) are and how superconductivity works? Itcan be said that superconductivity is a phenomenon that occurs in certain materialsand is characterised by the absence of electrical resistivity. Superconductivity is theproperty of complete disappearance of electrical resistance in substances when theyare cooled below a characteristic temperature (Figure 15.6) [7]. This temperature iscalled transition temperature or critical temperature (TC). So, for a substance to besuperconductive it must be cooled to below its TC. TC varies with the substanceused: this means that for a superconductive substance, once a current is set up in aclosed circuit comprising only superconductive wires, a current will flow forever.Superconductivity is essentially a macroscopic quantum phenomenon.

15.1.6.1 Types of SCsSCs are divided into two types depending on their characteristic behaviour in thepresence of a magnetic field. Type I SCs comprise pure metals, whereas Type II

Error

Limit

Power

First-order modelDead bandDelay

Load setting K1 + sT

τd

Figure 15.5 Hydro power plant model [6]


SCs comprise primarily alloys or intermetallic compounds. Both, however, haveone common feature: below a TC their resistance vanishes.

Type I SCs: They were the first to be discovered and generally are pure metalsor metal alloys. These are often referred to as conventional SCs and are also knownas low-temperature SCs (LTS) due to the fact that the highest TC of a type I SC isonly 23.2 K. They are operated in liquid helium (LH2). Type I SCs have beenexplained using BCS theory by Leon Cooper, John Bardeen and Robert Schrieffer[7]. The theory states that electrons pair up in what is known as (Cooper pairs). In atypical metal at room temperature, electrons are able to move throughout the latticestructure of metals, giving metals their conductive properties. However, due to thetemperature, vibrations occur inside the lattice and this causes collisions betweenelectrons and the lattice causing resistance and a loss of energy. However, when ametal is super cooled, the lattice gets to a point (TC) where the lattice effectivelystops vibrating and the Cooper pairs of electrons work together to overcome anyremaining obstacles and avoid collisions. These two electrons work together tocreate a slipstream in much the same way that a car will be dragged along ahighway by a semi-trailer in front. Table 15.1 gives the transition temperature ofsome elements of type 1 superconductive metals.

Type II SCs: They are known as high-temperature SCs (HTS) and arecommonly made from ceramic compounds. The first and most common type ofhigh-temperature SC is the YBCO (YBa2Cu3O7) SC and has a TC of around 92 K.This type is not a metal as type I, and then it does not contain a lattice structure thatwould allow the Cooper pairs to flow. HTS SCs provide new impetus for pursuingsuperconducting applications in different fields because of the prospect for highertemperature operation at liquid nitrogen (LN2) (77 K) temperatures or above. A list

temperatureTC

Elec

trica

l res

ista

nce

0

Non-superconductivemetal

Superconductor

Figure 15.6 Electrical resistance versus temperature


of some high-TC SCs with their respective transition temperature and the number ofCu–O layers present in unit cell, n, is summarised in Table 15.2. Transition tem-perature has been found to increase as the number of Cu–O layers increases to threein Bi–Sr–Ca–Cu–O, TI–Ba–Ca–Cu–O and Hg–Ba–Ca–Cu–O compounds [8].

15.1.6.2 Magnetic properties of superconductive materialsThe experiment described below can be implemented to recognise the magneticproperties of SCs [9]. The procedures are:

1. Preparing a ring made of superconductive material.2. Applying an external magnetic field to the ring in its normal state, i.e. at

T > TC, it is found that the magnetic field penetrates the ring (Figure 15.7(a)).3. Reducing the temperature to be T < TC, then removing the external magnetic

field.4. It is seen that the magnetic field applied in Step 2 remains there although it is

turned off in Step 3. Thus, the magnetic flux remains trapped in the ringopening (Figure 15.7(b)).

Table 15.1 Values of TC for different LTS materials [7]

Lead (Pb) 7.196 KLanthanum (La) 4.88 KTantalum (Ta) 4.47 KMercury (Hg) 4.15 KTin (Sn) 3.72 KIndium (In) 3.41 KThallium (Tl) 2.38 KRhenium (Re) 1.697 KProtactinium (Pa) 1.40 KThorium (Th) 1.38 KAluminium (Al) 1.175 KGallium (Ga) 1.083 KMolybdenum (Mo) 0.915 KZinc (Zn) 0.85 KOsmium (Os) 0.66 KZirconium (Zr) 0.61 KAmericium (Am) 0.60 KCadmium (Cd) 0.517 KRuthenium (Ru) 0.49 KTitanium (Ti) 0.40 KUranium (U) 0.20 KHafnium (Hf) 0.128 KIridium (Ir) 0.1125 KBeryllium (Be) 0.023 K (SRM 768)Tungsten (W) 0.0154 KLithium (Li) 0.0004 KRhodium (Rh) 0.000325 K


Referring to Faraday’s law of induction:þE:dI ¼ � df

dtð15:2Þ

where E is the electric field along the closed loop, f is the magnetic flux throughthe opening of the ring ¼ B*(area); it is found that before turning the external

Table 15.2 Critical temperature of some high-TC SCs [8]

High-TC SCs TC (K) n

Formula Notation

La1.6Ba0.4CuO4 214 30 1La2�xSrxCuO4 214 38 1YBa2Cu3O7 123 92 2YBa2Cu4O8 124 80 2Y2Ba2Cu7O14 247 80 2Bi2Sr2CuO6 Bi-2201 20 1Bi2Sr2CaCu2O8 Bi-2212 85 2Bi2Sr2Ca2Cu3O10 Bi-2223 110 3TIBa2CuO5 TI-1201 25 1TIBa2CaCu2O7 TI-1212 90 2TIBa2Ca2Cu3O9 TI-1223 110 3TIBa2Ca3Cu4O11 TI-1234 122 4TI2Ba2CuO6 TI-2201 80 1TI2Ba2CaCu2O8 TI-2212 108 2TI2Ba2Ca2Cu3O10 TI-2223 125 3HgBa2CuO4 Hg-1201 94 1HgBa2CaCu2O6 Hg-1212 128 2HgBa2Ca2Cu3O8 HG-1223 134 3(Nd2�xCex)CuO4 T 30 1

(a) (b)

I

Figure 15.7 Ring of superconductive material (a) at T> TC under externalmagnetic field and (b) at T < TC and removed external magnetic field


magnetic field off there was a magnetic flux through the ring. When the tempera-ture of the SC is decreased to be below TC its resistivity becomes zero and conse-quently the electric field inside the SC is zero as well. Thus,

þE:dI ¼ 0 ð15:3Þ

Hence, (15.2) gives

dfdt

¼ 0; i:e: f ¼ B� areað Þ ¼ constant ð15:4Þ

Therefore, the magnetic flux, f; through the ring must remain constant. So, itremains trapped in the opening of the ring after turning the external magnetic fieldoff. In addition, when the external magnetic field is turned off a current is inducedin the ring resulting in trapped magnetic field passing through the ring (internalfield). This induced current is called the persistent current, and it does not decay asthe resistance of the ring is zero.

15.1.6.3 Meissner effectIt is a phenomenon of magnetic flux expulsion during the transition from normalconductor to a superconductor. The Meissner effect is an effect whereby themagnetic field created in a superconductor will repel all other magnetic fields,regardless of whether they change. This is because the creation of magnetic field ina superconductor results in the creation of poles to repel all fields.

Expulsion of magnetic field from the interior of the superconductor: Con-sidering a sphere made out of superconductive material, at T > TC the material isin normal state. When the external magnetic field is turned on, the externalmagnetic field penetrates through the material. On the basis of Faraday’s law(15.2), it is expected that at T < TC the magnetic field would remain trapped in thematerial after the external field has been turned off. Trapping of the magneticfield actually does not happen because of the Meissner effect as shown inFigure 15.8.

The magnetic field is expelled from the interior of the superconductor by set-ting up electric current at the surface. The surface current creates a magnetic fieldthat exactly cancels the external magnetic field. It appears at T < TC in order thatB ¼ 0 inside the superconductor and is distributed in the surface layer. As the layercarrying the electric current has a finite thickness, l, the external magnetic fieldpartially penetrates into the interior of the superconductor with a value given by(15.5) and as depicted in Figure 15.9(a).

B xð Þ ¼ Bexternale�x

l ð15:5Þ


wherex is the distance from the surface inside the superconductorl is given and defined as the penetration distance at temperature T

The value of l can be calculated in terms of the penetration distance at temperatureT¼ 0, lo, by

l ¼ loffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 � T

TC

� �4r ð15:6Þ

lo ranges from 30 to 130 nm, depending on the superconductor material. Accord-ingly, l varies with time as shown in Figure 15.9(b).

Current

InducedSurface

B = 0

Bexternal T < TC

Figure 15.8 Superconductor in magnetic field

B

Bexternal

+xInside Outside

0T

TC(a) (b)

λ

λo

Figure 15.9 (a) magnetic field versus distance and (b) penetration distance versustemperature


From what is mentioned above, it is seen that in addition to the loss ofresistance, superconductors prevent external magnetic field from penetrating theinterior of the superconductor. This expulsion of external magnetic fields takesplace for magnetic fields that are less than the critical field, BC. External magneticfields greater than BC destroy the superconductive state. Therefore, the criticalfield can be defined as the maximum field that can be applied to a superconductorat a particular temperature and still maintain superconductivity. It can be con-cluded that the ability of superconductivity depends on meeting three conditions:(i) temperature, TC; (ii) magnetic field strength, BC; and (iii) current density, JC,where [10]

TC is the critical temperature in the absence of external magnetic field and withno current flowing in the sample

BC is the critical magnetic field strength with no current flowing at T ¼ 0JC is the critical current density at T ¼ 0 with no external magnetic field

strength

The variation of T, B and J represents a manifold in three-dimensional spaceseparating the superconductivity and normal state as shown in Figure 15.10.

The critical magnetic field as a function of temperature is given by

BC Tð Þ ¼ BCo 1 � T

TC

� �2" #

ð15:7Þ

where BCo is the critical magnetic field at T ¼ 0The variation of BCo against temperature change (15.7) variation of resistivity

r versus external magnetic field and internal magnetic field versus external fieldare shown in Figure 15.11(a–c), respectively.

Superconducting

Normal

BBC

J

JC

TC

Figure 15.10 Three-dimensional manifold comprising superconducting space


The maximum superconductive current, IC, carried by the superconductorcorresponds to the critical magnetic field, BC, at its surface where IC in terms of BC

is obtained as

IC ¼ 2p � R � BC

m0ð15:8Þ

Consequently, the current density, magnetic field and TC are all inter-dependent. By increasing any of these parameters to a sufficiently high value,superconductivity can be destroyed and the conductor will revert to a normal,non-superconducting state.

Type I and Type II superconductors exhibit different response to externalmagnetic field. A type I superconductor completely excludes the magnetic fieldfrom the interior for B < BC. Type II superconductors, however, permit the field topartially penetrate through the material in quantised amounts of flux for BC1 < B <BC2, whereas for B < BC the magnetic field is completely expelled, the same as typeI superconductor, and for B > BC2 the superconductor reverts to its normal state,i.e. full penetration by external magnetic field (Figure 15.12).

Values of critical current, IC, and current density, JC, for some HTS materialsprovided in Table 15.2 are summarised in Table 15.3.

Based on the study of superconducting and magnetic properties explainedabove, electricity can be stored in the form of d.c. magnetic field as described inSection 15.1.6.4.

15.1.6.4 Superconducting magnetic energy storageSuperconducting magnetic energy storage (SMES) is an energy storage device thatstores energy in the form of direct current that is the source of a d.c. magnetic field.The conductor operates at cryogenic temperatures where it is a superconductor andthus has virtually no resistive losses as it produces the magnetic field. Conse-quently, the energy can be stored in a persistent mode until required.

Bint

Bint = 0

B int = B ext

ρ

ρ = 0

BCO

B

BextBCTTC0 0BextBC0(a) (b) (c)

Figure 15.11 (a) B versus T; (b) r versus Bext; (c) Bint versus Bext


The core element of an SMES unit is a superconducting coil of high inductance(Lcoil in Henrys). It stores energy in the magnetic field generated by a d.c. current(Icoil in amperes) flowing through the coil. Therefore, the inductively stored energy(E in joules) and the rated power (P in watts) are the common specifications forSMES devices. They are expressed as

E ¼ 12

LI2coil ð15:9Þ

P ¼ dE

dt¼ LIcoil

dIcoil

dt¼ VcoilIcoil ð15:10Þ

As the energy is stored as circulating current, energy can be discharged from,or stored in, a SMES unit with almost instantaneous response over periods rangingfrom a fraction of a second to several hours [12].

The entire SMES unit consists of four parts: (i) large superconducting coil withthe magnet (SCM) at the desired cold temperature; (ii) power conditioning system(PCS) to interface the AC utility and SCM. Through the PCS the power is con-verted from a.c. to d.c. or inversely; (iii) the cryogenic system (CS) that is requiredto cool the SCM and keep it at the operating temperature; and (iv) the controller

Table 15.3 Values of IC and JC for some HTS materials [11]

Material IC (A) JC (kA/cm2) Condition

Bi-2223/Ag sheath 70 27.8 77 K, 114 (m)-lengthBi-2212/Ag sheath 500 490 4.2 K, 50 (m)-lengthBi-2212 coated 130 100 4.2 K, 450 (m)-lengthTI-1223 coated 18 8 77 K, 0.02 (m)-length

External field,

BC1 BC2BC Bext

Inte

rnal

fiel

d, B

int

Full expulsion

Full expulsion

Normal stateType Isuperconductor

Type IIsuperconductor

Partial expulsion Normal state

Type I

Type II

Figure 15.12 Bext versus Bint for type I and type II superconductors


that is the essential part of SMES system as it performs various functions: con-trolling the modes of charge/discharge/standby by controlling the voltage across themagnet coil, controlling CS and controlling the power conversion unit. A typicalsystem is shown in Figure 15.13.

Because of refrigeration needs, SMES technology is costly compared with otherenergy storage technologies, but with the development of HTS, SMES has become abit more cost effective. Electric utilities pay more attention towards SMES becauseof its fast response capability, flexibility, reliability and high efficiency.

15.2 Superconductivity applications

15.2.1 Superconducting synchronous generatorsThe use of superconductors, LTS or HTS, in a.c. machines is primarily motivatedby achieving higher current densities that allow overall reduction in cross-sectionand field winding volume compared to ordinary copper-wound rotors. Reduction ofwinding volumes leads to a reduction in the size and weight of the entire machine.During the 1970s a stationary room-temperature armature with a rotating SC fieldwinding was used for the design of an ac machine. This design faced problemsconcerning transferring cryogen into a rotating vacuum-insulated container [11].

In a synchronous machine, a.c. current is supplied to the armature to provide aflux component that rotates in synchronism with the flux component produced bythe rotating field winding. The rotor is thus phase-locked at the synchronous speedand under balanced load conditions will see essentially a d.c. field from thearmature. The armature, however, is connected to the electric power system, whichexperiences load-related electrical disturbances under steady-state and transientconditions. These electrical disturbances are reflected back into the armature andproduce non-synchronous AC effects that impact the rotor. Rapid changes in theDC excitation also occur as a result of load changes. The primary armature

Powerconversion

unit

Superconductingmagnet coil

Cyrogenicsystem (CS)

Coi

l pro

tect

ionTransformer

Controller

a.c.utilitygrid

Byp

ass s

witc

h

SCMPCS

Figure 15.13 Typical SMES unit components


disturbance is caused by unbalanced loads, which gives rise to negative sequencecurrents, proportional to the load and degree of unbalance, that counter-rotate attwice the synchronous speed.

Transient events caused by system faults provide the other major source fornon-synchronous a.c. effects on the rotor. These time-varying fields in thearmature, at frequencies different from the synchronous frequency, inducecompensating currents to flow in the rotor and produce heating in the d.c.superconducting winding and structural support. This a.c. influence can effec-tively be minimised by incorporating two concentric electromagnetic shields,called as inner cold damper shield and outer warmer cold damper shield, betweenthe two windings that attenuate these non-synchronous a.c. fields. The outerdamper shield acts as a damper and provides damping for the mechanical oscil-lations of the rotor related to the phase of the system. In addition to preventing themagnetic flux with high frequencies, it also helps to shield the entire cryogeniczone. The inner cold damper shield helps in shielding the superconducting fieldwinding from time-varying magnetic field. Therefore, the shielding must becarefully designed, taking into account the limitation of the thermal margin forthe superconductors, to minimise AC heating of the field winding, to preventdegradation of the superconductor JC and magnetic field capability, and possiblenormalisation during extreme transient conditions [13, 14].

The important aspect of the superconducting synchronous generator (SSG)is to keep the field winding at low temperature that is enhanced by therefrigerator. The refrigeration unit has four parts: cryostat, cryogenic pump, heatexchanger and liquid coupling junction. Depending on the type of super-conductors (LTS or HTS) liquid cryogen such as helium or nitrogen is chosenand the most used cryo-cooler is Gifford–McMahon cryo-cooler. At present allknown superconductors have to be operated at cryogenic temperature between4 K and 80 K [15, 16].

15.2.1.1 Benefits of SSGs● Steady-state stability improvement: The SSG is characterised by low synchronous

reactance that helps to enhance the power transfer capacity of a transmission line.It is represented as a voltage source, E, connected to its terminal bus. Whenthe machine is connected to an infinite bus through an external reactance Xe themaximum power transmitted is given, as known, by

Pmax ¼ EV

Xg þ Xeð15:11Þ

where V is the infinite bus voltageThus, as the machine reactance is low the maximum power is high. Com-

pared with a conventional synchronous generator of the same rating, the perunit synchronous reactance of the SSG ranges from one-third to one-quarter ofthat of the conventional one. This means that the maximum power transmittedis getting higher and the system has more marginal stability.


● Improved voltage regulation (VR): The field current of SSG between no loadand full load is much smaller than the conventional generator. Consequently,the SSG can be operated at any power factor within its MVA rating. In addi-tion, the short circuit ratio of SSG is much higher than the conventional gen-erator due to the low synchronous impedance [17]. This results in providingstable operation by SSG for any type of load. The VR of SSG is calculated as

VRSSG ¼ VNL � VFL

VFL� 100 ð15:12Þ

where VNL and VFL are the no-load and full-load terminal voltages, respectively.● Higher current density: This permits higher magnetic fields and allows a

reduction in weight and size. In addition, the machine efficiency is increasedbecause of the elimination of I2R heating in the field winding.

15.2.2 Superconducting transmission cablesOne of the fields that will benefit most from superconductor technology is trans-mission. When it comes to the transmission of electricity through the power grid, thisis where superconductors can have arguably the biggest impact. If superconductingtechnology is implemented into the power grid right now, the LN2-cooled cablescould be placed underground in place of copper cables. These superconductorcables are 7000 per cent more space efficient than their copper rivals. These newpower lines effectively have negligible energy losses, reducing the need for boostingof voltage at substations. By using superconducting electrical cables as opposed tocopper, the cost of the transmission is reduced and very large current densities areable to be transmitted with three to five times the current of regular wires. The onlyconcern with this, as mentioned previously, is the practicality of cooling kilometres ofunderground cables [11].

Sumitomo Cable has been working with TEPCO in Japan, since 1995, fordeveloping HTS power transmission cables. The best performance has beendemonstrated in both fundamental materials development and cable construction.A typical design is configured as shown schematically in Figure 15.14 for a cross-section liquid hydrogen (LH2) transmission pipeline with HTS electrical powercable [18]. The pipeline has the multi-layer insulator and LN2 channel as thermalshield. HTS tapes are used for both current transmission and shielding the externalpipe from the magnetic fields generated by the tapes transmitting the power.

15.2.3 Superconducting transformersIn a conventional power transformer, under load condition: Joule heating (I2R losses)of the copper coil adds considerably to the amount of lost energy. Almost 80 per centof load losses represent I2R losses and the remaining 20 per cent consist of stray andeddy current losses. Therefore, power engineers pay more attention towards thereduction of load losses. Unlike copper and aluminium, superconductors present noresistance to the flow of electricity, with the consequence that I2R losses becomeessentially zero, yielding a dramatic reduction in overall losses. Previously developed


LTS required cooling by LH2 to about 4.2 K, with advanced cryogenic technologythat is expensive in terms of both cost and refrigeration power expended per unit ofheat power removed from the cryostat. The HTS, based on LN2 at temperature up to78 K, is simpler, cheaper and a reduced ratio of refrigeration power is used to removeheat. Even with the added cost of refrigeration, HTS-based transformers of rating10 MVA and higher range are projected to be substantially more efficient and lessexpensive than their conventional counterparts. Furthermore, due to the increasedcapacity of HTS transformers, it is easy to replace the existing oil transformers in thegrid by HTS transformers of the same size to meet the growth in power demand [19].

15.2.4 Superconducting fault current limitersThe electric power demand has been continuously growing and, as a consequence,the power system must be expanded to meet this growth in demand. Systemexpansion may need replacement of transformers and addition of new generators toincrease the power capacity of the system. This results in an increase of fault current,but the existing buses and switchgear are not rated for the new fault current. Othersources can increase the fault current such as addition of distributed generation tothe system and using parallel feeder to increase system reliability. Therefore, the

Former(Al corrugated pipe)

Liquid HydrogenLH2 channel

Protector

Multi-layer insulator(MLI)

HTS conductorand insulator

Liquid nitrogenLN2 channel

Figure 15.14 Configuration of 7-m HTS cable� 2009 IEEE. Reprinted with permission from Nakayama T., Yagai T., Tsuda M.,Hamajima T. ‘Micro power grid system with SMES and superconducting cablemodules cooled by liquid hydrogen’. IEEE Transactions on AppliedSuperconductivity, Vol. 19(3), June 2009, pp. 2062–5


trade-off between fault current control and bus capacity is a problem facing powersystem engineers.

The fault current must be controlled in a manner that limits the investmentsrequired for equipment replacement or addition. Different solutions can be applied,for instance, (i) using high impedance transformers but degradation of VR arises forthe loads on the bus; (ii) expanding the bus by adding new sections through bus-tiecircuit breakers and supplying each section by a small transformer; and (iii) addingreactors in the path of fault current.

An alternative solution that overcomes the effects resulting from the classicsolutions of fault current control is using SFCL. The SFCL has the advantage ofinserting high resistance in the system in the event of a fault while at the normaloperation state its resistance vanishes. Thus, the SFCL acts as a non-linear resistancethat varies according to the operating conditions. The earliest designs used LTSmaterials cooled with LH2, but with the development of superconductive technology,HTS materials are used with LN2 for cooling to decrease the expenses of SFCLs.

Two approaches of applying SFCLs can be presented in power systems: series-resistive limiter and inductor limiter. In the first approach, series-resistive limiter,the SFCL is inserted in the circuit with a critical current of two or three times thefull load current. The fault current, during the fault, causes the SFCL to be in aresistive state and resistance R appears in the circuit. To limit the energy absorbedby the SFCL, a shunt coil called ‘trigger coil’ can be used allowing the bulk of faultcurrent to flow through the paralleled resistor and inductor (Figure 15.15(a)). It isnoted that the SFCL at the normal state is a short circuit across the copper-inductiveelement. In addition, limiting the energy absorbed by the current limiter allows thepower system designers to use it for transmission line applications.

The second approach, inductive limiter, uses a transformer. Its primary copperwinding is connected in series with the circuit and the secondary is connected witha resistive HTS limiter as shown in Figure 15.15(b). At normal operation the limiteris in its steady state with impedance nearly zero, which in turn leads to reflection ofzero impedance of secondary HTS winding to the primary. During the fault, a largecurrent is induced in the secondary because of the large current flow in the circuitand then the SFCL loses its superconductivity. Consequently, a resistance isdeveloped in the HTS winding and reflected to the primary limiting the fault

SFCL

Z

Line

Load Load

LinePrimarywinding

Secondarywinding

HTS winding(a) (b)

Figure 15.15 (a) Resistive current limiter approach and (b) inductive current limiter


current. This approach is appropriate for high current circuits. Whatever theapproach used, it can be seen that with SFCLs, the utilities can provide a lowimpedance system with a low fault current level.

As an illustration of the benefits of applying SFCLs to the power system, thepossible locations of inserting the limiters in substations are investigated. When thelimiter is inserted in the main feeder feeding the substation (e.g. location #1(marked inside a circle), Figure 15.16(a)), the fault current in case of faulted bus islimited. So, a larger low impedance transformer can be used to meet increaseddemand and maintain VR at the new power level without replacing or upgrading theswitchgear. In addition, as the fault current is limited, damage of transformer due toI2R is avoided and the voltage dip on the upstream high-voltage bus during the faultis minimised as well. For limiter located at the outgoing feeder (location #2,Figure 15.16(a)), less-expensive limiters can be used to protect the existing over-stressed equipment without replacement. Another possible location is at bus-tieposition (location #3, and Figure 15.16(b)). In this case the limiter requires a smallload current rating. During a fault on one of the two sections of the bus, a largevoltage drop across the limiter helps to maintain the voltage level on the un-faultedsection. In contingencies, the two sections of the bus can be tied together without alarge increase in the fault duty on both sections.

Use of SFCL in power systems not only allows more capacity of generationand transformation equipment but also improves the system transient stability [20].As explained in Chapter 9, Section 9.2, and considering the system shown inFigure 15.17 where each circuit breaker is provided with FSCL, when a three-phaseshort-circuit occurs at the beginning of one of the two transmission lines and iscleared by disconnecting the faulty line, it is found that the P versus d curves (pre,during and post fault) are as given in Figure 15.18. Neglecting the resistance of

(a) (b)

Source 1Source 2Source

Section #1 Section #23

1

2

Figure 15.16 Locations of SFLC in a substation at (a) outgoing or incomingfeeder and (b) bus tie


system elements, the power delivered from the generator to the infinite bus in termsof the power angle, is expressed as

Pe ¼ EV1Xeq

sin d ð15:13Þ

where E is the internal machine voltage, X 0d is the transient direct axis reactance and

Xeq is the equivalent transfer reactance between the generator and the infinite bus.Before the fault and after the fault, the power–angle curves of the system with

and without SFCL are the same as the system has not any added impedance due toSFCLs. The value Xeq is derived as below:

Xeq pre-faultð Þ ¼ X 0d þ Xtr þ XTL

2and Xeq post-faultð Þ ¼ X 0

d þ Xtr þ XTL ð15:14Þ

V∞E δ

Xd¢Xtr F

SFCL SFCL

SFCL SFCL

XTL

XTL

Figure 15.17 One machine to an infinite bus system (circuit breakers areprovided with SFCL)

Post-fault

Pre-faultDuringfault

with SFCLsDuringfault

withoutSFCLs

δmaxδCδo δm δ

A1

A2 AfPmPeo

Pec

Pes

Pe

Figure 15.18 Power–angle curves for pre, during and post fault: the shadowedareas and the dark-shadowed areas represent the energy balancewithout and with SMES, respectively


During the fault and without using SFCLs, the output power of the generator iszero. On the other hand, with SFCLs the electric power has a specific value cor-responding to the equivalent reactance that can be deduced as below.

The reactance diagram of the system is shown in Figure 15.19 from which theequivalent reactance, Xeq, can be obtained by Y–D transformation as below:

Xeq ¼ 1XTL

½XTLXFCL þ XFCLðX 0d þ XtrÞ þ ðX 0

d þ XtrÞXTL�

¼ XF þ X 0d þ Xtr þ XF

XTLðX 0

d þ XtrÞ 6¼ 1 ð15:15Þ

Accordingly, and as depicted in Figure 15.18, the accelerating area of machinerotor that is the area below the input power line is decreased when using SFCLs andbecomes A1 (dark shadowed). Consequently, the required decelerating energyrepresented by the area above the input power line is also decreased to achieve theenergy balance and the system in this case has more marginal stability. Therefore, itis concluded that the use of SFCLs in power system substations improves thesystem stability in addition to the other mentioned benefits.

15.2.5 SMES applicationsPower systems during the transient period following a disturbance such as com-ponent failure, line switching, load changes and fault clearance should be equippedwith the devices that provide adequate damping of oscillations in the system. So,countermeasures such as power system stabilisers (PSSs), turbine governor controlsystem and phase shifters have been used to prevent system collapse due to loss ofsynchronism or voltage instability.

SMES offers a feasible application to improve transmission capacity bydamping inter-area modal oscillations. It can actively dampen these system oscil-lations through modulation of both active and reactive power. The first fullcommercial application of superconducting power grid is SMES in BonnevillePower Administration 1981. It is American superconductor’s SMES systemfor power quality and grid stability and was located along 500 kV Pacific Intertiethat interconnects California and the Northwest.

E

Xd¢ Xtr XTL

XTL

XFCLXFCL

+V∞V∞ E Xshunt1Xshunt2

Xeq

(a) (b)

Figure 15.19 (a) Reactance diagram and (b) equivalent reactance diagram


Transient stability improvement by using SMES can be demonstrated byconsidering the power system shown in Figure 15.20 [21, 22]. The system is sub-jected to a three-phase short circuit at the beginning of one of the transmissionlines. The SMES is shunt connected at the generator terminal node. At the normalstate ‘before the fault event’ the SMES operates in the energy output mode(Figure 15.20(a)). Thus, the power balance relation is

PB ¼ Pg þ PSMES ð15:16Þ

At the instant of fault occurrence the control system of the SMES detects thefault and switches the SMES to go into the energy input mode after reverseswitching time trev. The SMES acts as an additional load to the generator, which isnecessary to keep the system stability and prevent the loss of synchronism. Thus,the power balance at generator terminal node (Figure 15.20(b)) is:

PD ¼ Pg � PSMES ð15:17Þ

The SMES power versus time can be expressed by

PSMESðtÞ ¼ PSMESe�ðtþtrev=TSMESÞ ð15:18Þ

where PSMES is the power of SMES at t ¼ 0 when released instantly to the powersystem and TSMES is the time constant of SMES model.

V∞

V∞

E δ

Xd¢Xtr

Xtr

XTL

XTL

FSMES

Pg PB

PSMES

Xd¢

XTL

XTL

FSMES

Pg PD

PSMES

(a)

(b)

E δ

Figure 15.20 Power system with SMES connection: (a) SMES in energy outputmode and (b) SMES in energy input mode


The SMES operates in energy input mode during the accelerating power to therotor and is disconnected from the system when the generator rotor starts decel-eration. Therefore, the system stability condition can be determined by getting theaccelerating power (absorbed by the rotor) and decelerating power (delivered fromthe rotor) in balance, i.e.

ðdrev

do

PBddþðddsc

drev

PDdd ¼ðdm

ddsc

Pmax Að Þ sin d� PD

� �dd ð15:19Þ

where Pmax(A) is the maximum power delivered to the transmission system after thefault clearance and can be calculated by (15.11). do, drev, ddsc and dm are the rotor’srelative angles in normal operating mode, at the instant of SMES connection, at theinstant of clearing the fault and maximum desired angle of rotor oscillation,respectively.

Equation (15.19) can be solved to give

PD ¼ Pmax Að Þ cos ddsc � cos dmð Þ � PB drev � doð Þdm � drevð Þ ð15:20Þ

Thus, the angle at which the SMES is disconnected and the fault is cleared canbe obtained by

cos ddsc ¼PD dm � drevð Þ þ PB drev � doð Þ þ Pmax Að Þ cos dm

Pmax Að Þð15:21Þ

Subtracting (15.17) from (15.16) to give

PSMES ¼ 0:5 PB � PDð Þ ð15:22Þ

The SMES parameters (trev, ddsc, PSMES, etc.) can be decided by using theaforementioned relations to keep the system stability when the system is subjectedto transient disturbances. SMES systems have some prominent performances suchas rapid response (millisecond), high power, high efficiency and four-quadrantcontrol due to the advantages in both superconducting technologies and powerelectronics. Thus, SMES systems offer flexible, reliable and fast-acting powercompensation. Superconducting technology is a promising technology in the future.Developing this technology is very attractive for power engineers and researchersto work on it. Paying more attention and more research to developing the super-conducting technology will lead to more applications in real practices.

15.2.6 Features of storage systemsSome features of aforementioned storage systems are listed in Table 15.4 [6].


Tab

le15

.4So

me

feat

ures

ofst

orag

esy

stem

s[6

]

Stor

age

syst

emA

dvan

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15.3 Phasor measurement units

The continuous load growth necessitates an increase in transmission networks capa-cities. Otherwise, the transmission lines are congested and the power system isstressed as well as it may be forced to operate closer to its stability limit. Furthermore,penetration of renewable energy sources into power systems is inevitable and addsmore uncertainty that requires more severe operation. To achieve flexible operation,power system monitoring should be considered [23]. It needs installation of excessivenumber of measuring instruments to monitor active power, reactive power, bus vol-tage and frequency at different points in a wide-area power system. Wide-area meansdifferent adjacent areas belonging to different utilities. Supervisory control and dataacquisition (SCADA) system, two-to-four second measurements, has been applied tolocally monitor and control these areas separately. Tremendous efforts have beenmade to design controllers such as PSSs for damping electromechanical oscillationsas explained in Chapter 11. They provide supplementary control action through thegenerator excitation systems to improve the power system stability [24].

The wide-area measurement system (WAMS) based on rms values of themeasured parameters can be associated with a state estimator (SE) that is respon-sible for estimating the phase angles of voltages and currents. Of course, it will bemore advanced to measure both the voltage/current magnitudes and their angles asestimation is less than measurement in accuracy. In addition, the measured voltagesand their angles at all measuring points must be determined by time synchronisa-tion of a fraction of milliseconds accuracy [25]. Fortunately, the recent globalpositioning system (GPS) and development of communication systems for fast andlarge data transmission satisfy the time synchronisation of measured parameters atdifferent remote locations and help the controllers for decision making on real time.With the aid of GPS and communication systems, the phasor measurement units(PMUs) that are able not only to measure the phasor voltages but also to determinethe phase reference effectively help the WAMS to be a promising technique forpower system monitoring.

15.3.1 Structure of WAMSPMUs are located at different measurement points in the wide-area power system.By offline analysis the optimal placement and number of PMUs can be determined[26–30]. The data are measured by each PMU and then collected by phasor dataconcentrators (PDCs) through narrow-band channel communication network. EachPDC receives data from multiple PMUs and sorts its frames. The concentrated dataare locally stored and can be exchanged with other PDCs belonging to other uti-lities by using standard format including the time stamp of the synchronised GPStime. An efficient server is used to receive all concentrated data through wide-bandcommunication network for processing, analysis and applications such as mon-itoring, state estimation, protection, real time control, contingency analysis, oscil-lation detection and stability estimation [23, 31].

As shown in Figure 15.21, the wide-area comprises adjacent areas in the powersystem belonging to different utilities (area #1, 2, . . . , n). Each area has a number of


PMUs inserted at predetermined measurement points. The phasor measurementsmay be sampled at rates of 30 times a second or more. They can also be time-stamped and synchronised with very high accuracy and millisecond resolutionacross a wide geographic area [32]. All the data measured by PMUs in that area aregathered and sent to a PDC in compliance with the standard data format. Thesehuge data sorted by the PDCs of the different areas are collected at a server throughwide-band channels communication network for processing to give the informationrequired for system performance improvement. Exchange of information betweenthe PDCs of the different areas can be implemented. GPS satellite is used for timesynchronisation between PDCs and determining the time reference. For instance,measurement of frequency, voltage and current magnitude and angles, and voltage/current phase difference by PMUs pre, during and post-fault can be processed andsent to the real-time controller for online decision-making to keep the system sta-bility. Exchange of information between the PDCs helps to decide which areas arecontrollable and which areas are observable.

15.3.2 Benefits of WAMSThe traditional security assessment of power systems based on offline analysisand SCADA data becomes increasingly unreliable for real-time operation, in

PMU PMU PMU PMU PMU PMU……….……….……….

Ethernet data communicationEthernet data communication Ethernet data communication

Wide-area

Utility #1Utility #n Utility #2 Area #1Area #n Area #2

GPSSatellite

……...

Communication network,wide-band channels

Server

StorageStorageStorage

……….………. ……….Communication network narrow-band channels

PDCPDCPDC

Figure 15.21 Schematic diagram of PMUs and wide-area measurement structure


particular, for wide-area power systems. WAMS and PMUs allow monitoring,assessing and taking the control action in real time to prevent or alleviate powersystem problems. The capability of synchronised measurements of system para-meters in magnitude and angle at different locations in a wide-area provides thebenefits below:

● The possibility of directly measuring the system state rather than estimating itby state estimation techniques.

● Real-time monitoring gives the system operators real-time information thathelp increasing their operational efficiency under normal conditions andenabling them to detect problems during abnormal conditions.

● Avoiding large area disturbances.● Maximising exploitation of present networks.● Increasing power transmission capability.● Damping of electromechanical oscillations when the system is subjected to

disturbances.● Improving transient and voltage stability since adding and networking MPUs

gives the utilities the capability to more closely monitor the stability of the bulkelectric system and control remedial actions with their neighbours [33, 34].

● Avoiding system congestion.● Increasing the accuracy of SEs, which in turn provides a valid best estimate of

a consistent network model that can be used as a starting point for real-timeapplications, e.g. VAr optimisation, constrained re-dispatch and contingencyanalysis [35].

● Enhancing relays performance and protection schemes, because use of syn-chronised phasor measurement, certain relays (adaptive relays) could be madeto adapt to the prevailing system conditions [36].

15.3.3 Case studiesIn Finland, a WAMS project was launched in 2006 by Fingrid, the Finish trans-mission system owner and operator (TSO), to obtain real-time information aboutdamping of 0.3 Hz electromechanical inter-area oscillations, which typically is themain factor limiting power transfer capability from South Finland to North andfurther down to South Scandinavia. Then the project has been developed and in2011 the system deployed 12 PMUs and a PDC. The measurements of PMUs arealso streamed from and to wide-area measurements in Norway and Denmark.PMUs in addition to WAMS have been successfully applied for other power systemplanning, analysis and control purposes. For instance, two PMUs are installed as anintegral part of SVC POD controls to provide local frequency measurement andpositive sequence voltage signals for the POD controls. More details about thisproject can be found in [31].

In Brazil, a large-scale blackout of the Brazilian interconnected power system(BIPS) with a 40 per cent load loss occurred due to a fault at the AC Itaipu trans-mission system. This event was recorded at the low voltage level by a synchronisedphasor measurement prototype, the LVPMS, with PMUs installed in nine


universities throughout Brazil. The recorded data contain relevant information onthe events leading to the blackout and the BIPS restoration [37]. Many otherinstallations of PMU prototype are presented in different countries such as Austria,China, Canada, USA, Sweden and Switzerland.

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Appendix I

Calculation of synchronous machineparameters in per unit/normalised form

I.1 Per unit values

I.1.1 Base quantities for statorIt is common to choose the following three base quantities for armature (stator)variables:

● SB ≜ Base power ¼ three-phase stator rated power (VA rms)● VB ≜ Base voltage ¼ stator rated line-to-line voltage, VL�L (V rms)● tB ≜ Base time (s)

The other base quantities can be accordingly determined as

● IB ≜ Base current ¼ SBVB

¼ ffiffiffi3

p � rated line current, IL

● wB ≜ Generator rated speed (wo) ¼ 1tB

(elec. rad/s)

● ZB ≜ Base impedance ¼ VBIB¼ VL�Lffiffi

3p

IL

● YB ≜ Base flux linkage ¼ LBIB ¼ VBtB ¼ VBwB

● LB ≜ Base inductance ¼ YBIB

¼ ZBwB

To ensure the validity of base quantities identified above, the total power in thethree stator phases, Pabc, must be the same as the power in the d–q circuits. This canbe proved as below:

Pabc ¼ vaia þ vbib þ vcic ¼ vtabciabc ðI:1Þ

Applying Park’s transformation: vtabc ¼ vt

dqoPt and iabc ¼ Pidqo

As P is an orthogonal matrix (power invariant), i.e. Pt ¼ P�1 and assuming thezero sequence power is zero, (I.1) becomes

Pabc ¼ vtdqidq ¼ vd id þ vqiq ðI:2Þ

The d–q voltages can be written as

vd ¼ V sin d and vq ¼ V cos d, where V is the line-to-line voltage, and in pu valuesvdu ¼ Vu sin d and vqu ¼ Vu cos d, where the subscript u indicates the pu values

Thus,

v2du þ v2

qu ¼ V 2u ðI:3Þ

Equation (I.3) shows that the d–q axis voltages are numerically equal to theline-to-line voltages.

Similarly, the d–q axis currents are

id ¼ I sin g and iq ¼ I cos g (where I is the line current),

and in pu values:

idu ¼ Iu sin g and iqu ¼ Iu cos g

Substituting the pu values of d–q axis voltages and currents into (I.2), thethree-phase stator power is given by

Pabc ¼ VuIu sin d sin gþ cos d cos gð Þ¼ VuIu cos d� gð Þ

In terms of the base quantities, the d–q stator voltages (2.30) can be written inpu values as

vdu ¼ � 1wB

dYdu

dt� wwB

Yqu � Rauidu

vqu ¼ � 1wB

dYqu

dtþ wwB

Ydu � Rauiqu

9>>=>>; ðI:4Þ

To normalise any quantity, it is divided by the base quantity of the samedimension, e.g.

iqu ¼ iq

IB; idu ¼ id

IB; vdu ¼ vd

VB; Ydu ¼ Yd

YB

More details about normalisation of equations are given in Section I.2.

I.1.2 Base quantities for rotorThe base power, SB, of the armature is based on its rating and the time base is fixedby the rated radian frequency. These base quantities must be the same for rotorcircuits as the rotor and stator circuits are coupled electromagnetically. To satisfythis condition, the numeric value of the rotor quantities in per unit is small, becausethe stator base power is much larger than the rated power of the rotor circuits.

Therefore, it is necessary to decide on a suitable base quantity in the rotor thatgives the correct base quantity in the stator. Equality of mutual flux linkages is themain concept on which the choice of base quantity in the rotor is based. Thisimplies that the base currents in rotor circuits in the d-axis (base field current orbase damper current) are chosen in such a way that they produce the same spacefundamental of air gap flux as produced by the base stator current flowing in thed-axis stator windings.


Again, (2.20) is rewritten below in an expanded form as

Yd

Yq

Yo

Yf

Ykd

Ykq

2666666664

3777777775¼

Ld 0 0 kMf kMkd 0

0 Lq 0 0 0 kMkq

0 0 Lo 0 0 0

kMf 0 0 Lf Lfkd 0

kMkd 0 0 Lfkd Lkd 0

0 kMkq 0 0 0 Lkq

2666666664

3777777775¼

id

iqioif

ikd

ikq

2666666664

3777777775

ðI:5Þ

In (I.5), assume that the currents id ¼ IB, if ¼ If B and ikd ¼ IkdB are applied oneat a time with other currents set to zero. Then equating the mutual flux linkages ineach d-axis winding (Ymd, Ymf, Ymkd), the equations below are obtained.

Ld � ‘dð ÞIB ¼ kMf If B ¼ kMkdIkdB ðI:6Þwhere

‘d is the leakage inductance of the d-axis armature windingIf B and IkdB are the base currents in the rotor field and damper windings,

respectively

Hence,

If B ¼ Lmd

kMfIB and IkdB ¼ Lmd

kMkdIB ðI:7Þ

where Lmd ¼ Ld � ‘d

The base flux linkages for rotor circuits are chosen such that

YBIB ¼ Yf BIf B ¼ YkdBIkdB; Yf B ¼ IB

If BYB and YkdB ¼ IB

IkdBYB ðI:8Þ

Similarly, for the q-axis rotor circuits (KQ coil), the base current and fluxlinkages are given by

IkqB ¼ Lmq

kMkqIB and YkqB ¼ IB

IkqBYB ðI:9Þ

where Lmq ¼ Lq � ‘q and ‘q ≜ the leakage inductance of the q-axis armaturewinding.

Commonly, ‘d ¼ ‘q and can be written as ‘a.As the base quantity SB for the stator must be equal to SB for the rotor, the

relations below can be computed.

Vf B

VB¼ IB

If B¼ kMf

Lmd

VkdB

VB¼ IB

If B¼ kMkd

Lmd

VkqB

VB¼ IB

If B¼ kMkq

Lmq

9>>>>>>>=>>>>>>>;

ðI:10Þ

Calculation of synchronous machine parameters in per unit/normalised form 429

I.1.3 Conversion of rotor quantities to equivalent stator EMFIn synchronous machine equations, it is preferable to convert the rotor current, fluxlinkage and voltage to an equivalent stator EMF as below.

In steady state and at open circuit conditions, the field current if corresponds toa peak stator EMF of (woMf if). Then, woMf if ¼H2E, where E is the rms of statorEMF as a line-to-neutral value. As the coupling between the d-axis rotor and statorwindings involves the factor k ¼H(3/2) as formerly explained, this relation can bewritten as

wokMf if ¼ p3E or wokMf if ¼ EI ; where EI is the line-to-line rms value

ðI:11Þin compliance with the American National Standard Institute (ANSI) or wokMf if ¼ Eq

in compliance with the International Electro-technical Commission (IEC) notations. Itis to be noted that the field current if corresponds to a given EMF by a scaling factorwhere wo and Mf are constants for a given machine. Therefore, EI in pu corresponds toif in pu.

The flux linkage Yf can also be converted to a corresponding stator EMF. Insteady-state and at open circuit conditions if ¼Yf /Lf. Multiplying the field currentby wokMf to give the d-axis stator EMF, E 0

q (as line-to-line rms value) corre-sponding to the flux linkage Yf is

wokMf Yf =Lf ¼ E0q ðI:12Þ

where E 0q is represented by the quadrature component of stator voltage behind the

transient reactance.Similarly, at steady state, the field voltage vf corresponds to a field current if ¼

vf /Rf. Consequently, it corresponds to a peak stator EMF as if woMf ¼ (vf /Rf)woMf.If its line-to-line rms value is denoted by Efd, the d-axis stator EMF corresponds toa field voltage vf given by

vf =Rf

� �wokMf ¼ Efd ðI:13Þ

It is to be noted that the values of stator EMF in (I.11)–(I.13) are considered asline-to-line rms values as the base value of the voltage VB is taken as line-to-linerms value (Section I.1.1). In some literature, VB is taken as line-to neutral rmsvoltage and that leads to different equations using a factor of H3 as described inSection I.3.

I.2 Normalising the synchronous machine voltage equations

Based on a choice of appropriate base values, the voltage equations can benormalised so that they are easier to deal with, as the numerical values of voltageand current in the normalised form will be of the same order of magnitude.The subscript u is added to all pu quantities and is omitted later when all values


are normalised. As the actual value of any quantity ¼ the pu value � base value,(2.32) is rewritten as

(I.14)

This system of equations can be written in expanded form by substitutingw¼wuwo (the rated angular speed wo is taken as the base value wB) as below:

vdu ¼ �RaIB

VBidu � wuwoLq

IB

VBiqu � wuwokMkq

IkqB

VBikqu � Ld

IB

VBpidu

� kMfIf B

VBpifu � kMkd

IkdB

VBpikdu pu ðI:15Þ

Hence,

vdu ¼ � Ra

RBidu � wu

Lq

LBiqu � wu

woIkqB

VBkMkqikqu� Ld

woLBpidu � kMf

wo

woIf B

VBpifu

� kMkd

wo

woIkdB

VBpikdu

ðI:16Þ

By definition

Ru ¼ Ra=RB; Ldu ¼ Ld=LB; Mfu ¼ Mf woIf B=VB; Lkdu ¼ Lkd=LB;

Mkdu ¼ MkdwoIkdB=VB; Mkqu ¼ MkqwoIkqB=VB


Then, by substituting into (I.16), it gives

vdu¼�Ruidu � wuLquiqu � wukMkquikqu � Ldu

wopidu � k

Mfu

wopifu � k

Mkdu

wopikdu

ðI:17ÞSimilarly, applying similar analysis, the q-axis voltage equation in pu is

vqu ¼ �Ruiqu þ wuLduidu þ wukMfuifu þ wukMkduikdu � Lqu

wopiqu

� kMkqu

wopikqu pu ðI:18Þ

The equation of vou can be written as below, and it is noted that it vanishesunder balanced conditions:

vou ¼ �Ra þ 3Rn

RBiou � Lo þ 3Ln

woLBpiou

Thus,

vou ¼ � Ra þ 3Rnð Þuiou � 1wo

ðLo þ 3LnÞupiou pu ðI:19Þ

The rotor equations are normalised on the rotor base values. The pu fieldvoltage is

vfu ¼ RfIf B

Vf Bifu þ k

Mf

wo

woIB

Vf Bpidu þ Lf

wo

woIf B

Vf Bpifu þ Lfkd

wo

woIkdB

Vf Bpikdupu

ðI:20ÞThe last two terms are normalised by incorporating the base rotor

inductance as

Lfu ¼ Lf =Lf B and Lfkdu ¼ Lfkd=LfkdB

Thus, the normalised field voltage equation is

vfu ¼ Rfuifu þ kMfu

wopidu þ Lfu

wopifu þ Lfkdu

wopikdu ðI:21Þ

Applying the same procedure to damper winding equations for circuits KD andKQ, the following normalised equations are obtained:

vkdu ¼ 0 ¼ Rkduikdu þ kMkdu

wopidu þ Lfkdu

wopifu þ Lkdu

wopikdu ðI:22Þ

vkqu ¼ 0 ¼ Rkquikqu þ kMkqu

wopiqu þ Lkqu

wopikqu ðI:23Þ


Under balanced conditions and incorporating the normalised equations in amatrix form, where the first three rows express the voltage relations in the d-axis,the fourth and fifth rows express the voltage relations in the q-axis; the followingform can be written as

vd

�vf

0

vq

0

26666664

37777775¼ �

Ra 0 0 wLq wkMkq

0 Rf 0 0 0

0 0 Rkd 0 0


0 0 0 0 Rkq

26666664

37777775

id

if

ikd

iq

ikq

26666664

37777775

�

Ld kMf kMkd 0 0

kMf Lf Lfkd 0 0

kMkd Lfkd Lkd 0 0

0 0 0 Lq kMkq

0 0 0 kMkq Lkq

26666664

37777775

pid

pif

pikd

piq

pikq

26666664

37777775

ðI:24Þ

In (I.24), the subscript u is dropped as all values are in pu as well as this form isadequate to analyse the system in time domain (time in seconds).

I.3 Alternative per unit/normalising systems

In some literature phase quantities are used as base quantities. Therefore, the statorbase quantities are chosen as below.

I.3.1 Base quantities for stator● SB ≜ Base power ¼ stator rated power/phase (VA rms)

● VB ≜ Base voltage ¼ stator rated line-to-neutral voltage, VL–N (V rms)

● tB ≜ Base time (s)

The other base quantities can accordingly be determined as

● IB ≜ Base current ¼ SBVB

¼ rated phase current, or in star connection linecurrent, IL

● wB ≜ Generator rated speed woð Þ ¼ 1tB

(elec. rad/s)

● ZB ≜ Base impedance ¼ VBIB¼ VL�N

IL

● YB ≜ Base flux linkage ¼ LBIB ¼ VBtB ¼ VL–NtB ¼ VL�NwB

● LB ≜ Base inductance ¼ YBIB

¼ ZBwB

Under balanced conditions, the d–q axis pu quantities, such as vdqo, idqo and thetotal power in the three stator phases, Pabc, can be obtained as below.


Assuming the stator voltages in the form

va ¼ Vmax sin dþ að Þ ¼ ffiffiffi2

pV sin dþ að Þ

vb ¼ ffiffiffi2

pV sin dþ a� 2p

3

� �

vc ¼ ffiffiffi2

pV sin dþ aþ 2p

3

� �

9>>>>>=>>>>>;

ðI:25Þ

where Vffa is the rms phase voltage.Applying Park’s transformation to give vdqo as

vd

vq

vo

24

35¼

ffiffiffi3

pV sinaffiffiffi

3p

V cosa0

264

375 ðI:26Þ

Thus, the pu voltages in d–q frame of reference are

vdu ¼ vd=VB ¼ ffiffiffi3

pV=VBð Þsina ¼ ffiffiffi

3p

Vu sinavqu ¼ vq=VB ¼ ffiffiffi

3p

V=VBð Þcosa ¼ ffiffiffi3

pVu cosa

)ðI:27Þ

Hence,

v2du þ v2

qu ¼ 3V 2u ðI:28Þ

Equation (I.27) illustrates that the d- and q-axis voltages are numerically equalto

ffiffiffi3

ptimes the pu voltages.

Similarly, assuming the rms phase current is Iffg. The stator currents in d–qframe of reference are

idiqio

24

35¼

ffiffiffi3

pI sin gffiffiffi

3p

I cos g0

264

375 ðI:29Þ

Thus, the pu currents can be obtained by

idu ¼ffiffiffi3

pIu sin g; iqu ¼

ffiffiffi3

pIu cos g ðI:30Þ

Using (I.27) and (I.30), the total power in the three stator phases, Pabc, is given by

Pabc ¼ iduvdu þ iquvqu ¼ 3IuVu sina sin gþ cosa cos gð Þ ¼ 3IuVucos a� gð Þpu

ðI:31ÞIt is seen that (I.31) validates the equality of the power in d–q circuits and the

power in the three phases of the stator.


I.3.2 Base quantities for rotorBased on the concept of equating the mutual flux linkages in each d-axis winding(Ymd, Ymf, Ymkd), the same relations obtained in Section I.2.2 to calculate the baserotor quantities If B, IkdB and IkqB, (I.7), (I.9) and (I.10) can be applied.

It has been shown that in the per unit system explained in Section I.1, the statorbase quantities are the three-phase rated power and rated line-to-line voltage,whereas in the alternative per unit system, Section I.3, the rated power/phase andrated line-to-neutral voltage are used as stator base quantities. The other basequantities, calculated for each system accordingly, are summarized in Table I.1.

As summarised in Table I.1, it is noted that:

● The base rating in system I.1 is three times its value in system I.3.● The stator base voltage and base current in system I.1 is

ffiffiffi3

ptimes its value in

system I.3.● The stator base impedance and inductance are the same for the two pu systems.● The base value of stator flux linkage is

ffiffiffi3

ptimes its value in system I.3.

● The d–q base currents, flux linkages and voltages of rotor circuits areffiffiffi3

ptimes

their values in system I.3.● The pu value of v2

du þ v2qu equals V 2

u in system I.1 while it equals 3 V 2u in

system I.3.● The pu three-phase stator power in system I.1 equals VuIu cos(d� g) and in

system I.3 equals 3IuVucos a� gð Þ.

Table I.1 Base quantities for per unit systems I.1 and I.3

Base quantity Per unit system

System I.1 System I.3

Base quantities for statorSB

ffiffiffi3

pVL–LIL VL–NIL

VB VL–L VL–N

IB (SB/VB) ¼ ffiffiffi3

pIL IL

ZB (VB/IB) ¼ VL–L/(ffiffiffi3

pIL) (VB/IB) ¼ VL–N/(IL)

LB ZBtB ZBtBYB VBtB VBtB

Base quantities for rotorIfB (Lmd/kMf)IB (Lmd/kMf)IB

IkdB (Lmd/kMkd)IB (Lmd/kMkd)IB

IkqB (Lmq/kMkq)IB (Lmq/kMkq)IB

YfB (IB/IfB)YB (IB/IfB)YB

YkdB (IB/IkdB)YB (IB/IkdB)YB

YkqB (IB/IkqB)YB (IB/IkqB)YB

VfB (kMf /Lmd)VB (kMf /Lmd)VB

VkdB (kMkd/Lmd)VB (kMkd/Lmd)VB

VkqB (kMkq/Lmd)VB (kMkq/Lmd)VB


Appendix II

Nine-bus test system

Single-line diagram

System data

GG

G

1

2 3

4

5 6

7 8 9

Table II.1 Transmission line data on 100 MVA base

From busnumber

To busnumber

Series resistance(Rs) pu

Series resistance(Xs) pu

Shunt susceptance(B) pu

1 4 0 0.0576 04 6 0.0170 0.0920 0.15806 9 0.0390 0.17 0.35809 3 0 0.0586 09 8 0.0119 0.1008 0.20908 7 0.0085 0.0720 0.14907 2 0 0.0625 07 5 0.0329 0.1610 0.30605 4 0.0100 0.0850 0.1760

Table II.2 Bus data of the system

Bus no. Bus type Generation (pu) Load (pu) Voltage magnitude

PG QG PL QL

1 Swing – – 0 0 1.042 PV 1.63 – 0 0 1.0253 PV 0.85 – 0 0 1.0254 PQ 0 0 0 0 –5 PQ 0 0 1.25 0 –6 PQ 0 0 0.9 0 –7 PQ 0 0 0 0 –8 PQ 0 0 1 0 –9 PQ 0 0 0 0 –

Table II.3 Generator data

Generator 1 2 3

Rated MVA 247.5 192 128KV 16.5 18 13.8Power factor 1 0.85 0.85Type Hydro Steam SteamSpeed 180 r/min 3600 r/min 3600 r/minXd 0.1460 0.8958 1.3125Xd0 0.0608 0.1198 0.1813Xq 0.0969 0.8645 1.2578Xq0 0.0969 0.1969 0.25Xl (leakage) 0.0336 0.0521 0.0742T 0

d0 8.96 6 5.89T 0

q0 0 0.535 0.6Stored energy at rated speed 2364 MW�s 640 MW�s 301 MW�sH (MW�s/MVA) 9.55 3.33 2.35


Appendix III

Numerical integration techniques

Consider a first-order non-linear ordinary differential equation (ODE), y 0 ¼ f (x,y),y(xo) ¼ yo. It has a unique solution y ¼ u(x) on the interval I ¼ [xo, b]. The solutionu(x) is a function at each point of I. The task of finding an approximate solutionto the ODE is, therefore, one of approximating the (usually unknown) functiony ¼ u(x) on I.

In general, the approximation of u(x) at any one point will involve a number ofarithmetic operations. Because there are an infinite number of points in I, it is notproposed to calculate an approximate value of u(x) at each individual point.Therefore, the task is to find approximate values of u(x) on a certain finite subset of I.The points of this subset will be denoted by xo, x1, . . . , xm. While it is not necessarythat the points be equally spaced, it is more convenient computationally to have themso. Thus, it is assumed to approximate u(x) at points xi ¼ xo þ ih(i ¼ 0, 1, . . . , m).The quantity h is called the step size. The integer m is such that xm � b, whilexm þ h > b.

In the literature on differential equations, the exact solution at a point xi isusually denoted by y(xi), whereas an approximation to this is denoted by yi. Thus,the objective is to look for y1, y2, . . . , ym, which approximate y(x1), y(x2), . . . , y(xm).Various methods described in the following sections can be used to obtain thenumerical solution of ODE.

III.1 Euler’s method

This method is perhaps the simplest of all numerical methods. Studying its appli-cation helps in understanding the basic ideas involved in the numerical solutionof ODE.

Assuming that f (x, y), xo, yo, h, and m are given, the numbers x1, x2, . . . , xm andy1, y2, . . . , ym are formed by the rules:

xiþ1 ¼ xi þ hyiþ1 ¼ yi þ hf xi; yið Þ for i ¼ 1; 2; . . .;m � 1

A geometric picture of Euler’s method is depicted in Figure III.1. An initialpoint (xo, yo) on the solution curve is given. The slope of the solution curve at thispoint is given by f (xo, yo). Thus, the tangent line to the solution curve at the initial

point can be determined. Euler’s method consists in approximating the solutionfunction by this tangent line.

tan q ¼ f xo; yoð Þ ¼ y1 � yo

h

This is easily solved for y1 giving:

y1 ¼ yo þ hf xo; yoð ÞThe error equals y1 � y x1ð Þ:This is one step of the Euler algorithm. If (x1, y1) is considered the initial point

and the entire process is repeated, the result will be (x2, y2) by the rules. An error,i.e. the difference between the calculated value y1 and the actual value y x1ð Þ, arisesfrom the first step. Then, each subsequent step is made using an incorrect value ofthe slope and moving from an incorrect point under the incorrect assumption thatthe solution curve is a linear function.

Euler’s algorithm can be extended to a system of simultaneous first-orderODEs. Suppose that the equations are given in the form

y01 ¼ f1 x; y1; y2; . . . ; ynð Þ y1 xoð Þ ¼ y1o

y02 ¼ f2 x; y1; y2; . . . ; ynð Þ y2 xoð Þ ¼ y2o

..

.

y0n ¼ fn x; y1; y2; . . . ; ynð Þ yn xoð Þ ¼ yno

The problem is to find approximate values for the unknown functions y1 xð Þ,y2 xð Þ, . . . , yn xð Þ at the points x1, x2, . . . , xm. For a particular function, say yj(x), theexact solution is denoted by yj(xo), yj(x1), . . . , yj(xm) whereas an approximatesolution is denoted by yjo, yj1, yj2, . . . , yjm.

xo x1h

q

y1

y(x1)

yo

Error

Figure III.1 One step of Euler’s method


Assuming that xo, y1o, y2o, . . . , yno, h and m are given, the numbers yji ( j ¼ 1,2, . . . , m) can be formed by the rules: xi þ 1 ¼ xi þ h

y1;iþ1 ¼ y1i þ hf1 xi; y1i; y2i; . . . ; ynið Þy2;iþ1 ¼ y2i þ hf2 xi; y1i; y2i; . . . ; ynið Þ

..

.

yn;iþ1 ¼ yni þ hfn xi; y1i; y2i; . . . ; ynið Þ for i ¼ 0; 1; . . . ;m � 1

This algorithm has been modified to what is called modified Euler–Cauchymethod by using the rule:

yiþ1 ¼ yi þ hf xi þ h

2; yi þ h

2f xi; yið Þ

� �

In general, if the solution to an ODE is known to possess many derivatives,then the more involved methods will provide more accurate approximations to thesolution as explained in the next sections.

III.2 Trapezoidal method

This method is attributed to Heun and sometimes called ‘Heun’s method’. Itsatisfies more accurate approximation to the solution of an ODE than that obtainedby Euler’s method. A differential equation in the form y 0 ¼ f x; yð Þ; y xoð Þ ¼ yo isgiven. If xi þ 1 ¼ xi þ h, by integrating each side of the equationðx1

xo

y 0 xð Þdx ¼ðx1

xo

f x; y xð Þð Þdx

The left side may be simplified to obtain

y x1ð Þ ¼ y xoð Þ þðx1

xo

f x; y xð Þð Þdx

If the integral on the right side is approximated by the trapezoidal rule, then

y x1ð Þ ¼ y xoð Þ þ h

2f xo; y xoð Þð Þ þ f x1; y x1ð Þð Þ½ � þ remainder

Finally, if the quantity y x1ð Þ in the right side is approximated by the use of theEuler’s method and all remainder terms are ignored, the result is

y1 ¼ yo þ h

2f xo; yoð Þ þ f x1; yo þ hf xo; yoð Þð Þ½ �

The result can be stated as an algorithm as below.

Numerical integration techniques 441

III.2.1 The algorithmAssuming that f (x, y), xo, yo, h and m are given, the numbers x1, x2, . . . , xm andy1, y2, . . . , ym are formed by the rules

xiþ1 ¼ xi þ h

yiþ1 ¼ yi þ h

2f xi; yið Þ þ f xiþ1; yi þ hf xi; yið Þð Þ½ � for i ¼ 0; 1; . . . ;m � 1

It is noted that each step involves two evaluations of the function f (x,y). To getyiþ1, it is necessary to evaluate f (x,y) at (xi, yi) and at f xiþ1; yi þ hf xi; yið Þð Þ. It isassumed that the value computed in the first evaluation will be stored so that it neednot be recomputed in the second evaluation.

III.3 Runge–Kutta Methods

Different algorithms are used to numerically solve the ODE:

y0 ¼ f x; yð Þ; y xoð Þ ¼ yo for x 2 xo; b½ �to obtain the points yiþ1 as an approximation to y(xiþ1) for i ¼ 0, 1, 2, . . . , m � 1 byapplying the form:

yiþ1 ¼ yi þ h; xi; yi; hð Þ

III.3.1 Second-Order Runge–Kutta MethodThe following form is used:

yiþ1 ¼ yi þ h

2K1 þ K2½ �

where

K1 ¼ f xi; yið Þ and K2 ¼ f xi þ h

2; yi þ h

2K1

� �

A geometric picture of second-order Runge–Kutta method is depicted inFigure III.2. An initial point (xo, yo) on the solution curve is given. The slope of thesolution curve at this point is given by K1 ¼ f (xo, yo). Then, a tangent line at thepoint (xo þ h/2, yo þ K1h/2) is drawn to determine the slope of the solution curveK2. Thus, the tangent line to the solution curve at the initial point can be determinedas (K1 þ K2)/2.

The Runge–Kutta methods of higher orders have the same basic idea, but theydiffer in calculating the slope of the tangent line at the initial point of each step asstated below.


III.3.2 Third-Order Runge–Kutta MethodThe form used is

yiþ1 ¼ yi þ h

6K1 þ 4K2 þ K3½ �

where

K1 ¼ f xi; yið Þ; K2 ¼ f xi þ h

2; yi þ h

2K1

� �and K3 ¼ f xi þ h; yi þ hK1ð Þ

III.3.3 Fourth-Order Runge–Kutta MethodThe form used is

yiþ1 ¼ yi þ h

6K1 þ 2K2 þ 2K3 þ K4½ �

where

K1 ¼ f xi; yið Þ; K2 ¼ f xi þ h

2; yi þ h

2K1

� �;

K3 ¼ f xi þ h

2; yi þ h

2K2

� �; K4 ¼ f xi þ h; yi þ hK3ð Þ

h

Error

h/2

Line with slope K1

Line with slope K2

Line with slope (K1 + K2)/2

x1

y(x1)y1

yo

xo

Figure 3.2 A geometric picture of one step in second-order Runge–Kuttamethod

Numerical integration techniques 443

This method can be extended to solve a system of simultaneous first-orderODEs. For instance, assume the two equations below are given.

y01 ¼ f1 x; y1; y2ð Þy1 xoð Þ ¼ y1o

y02 ¼ f2 x; y1; y2ð Þy2 xoð Þ ¼ y2o

for x 2 [xo, b]. Using the notation xi ¼ xo þ ih, and y1,i, y2,i as the numericalapproximations to y1(xi), y2(xi) gives

y1;iþ1 ¼ y1;i þ hf1 xi; y1;i; y2;i; h� �

y2;iþ1 ¼ y2;i þ hf2 xi; y1;i; y2;i; h� �

where

f1 xi; y1;i; y2;i; h� � ¼ 1

6K11 þ 2K12 þ 2K13 þ K14ð Þ

f2 xi; y1;i; y2;i; h� � ¼ 1

6K21 þ 2K22 þ 2K23 þ K24ð Þ

and

K11 ¼ f xi; y1;i; y2;i

� �K21 ¼ f xi; y1;i; y2;i

� �K12 ¼ f1 xi þ h

2; y1;i þ h

2K11; y2;i þ h

2K21

� �

K22 ¼ f1 xi þ h

2; y1;i þ h

2K11; y2;i þ h

2K21

� �

K13 ¼ f1 xi þ h

2; y1;i þ h

2K12; y2;i þ h

2K22

� �

K23 ¼ f1 xi þ h

2; y1;i þ h

2K12; y2;i þ h

2K22

� �

K14 ¼ f1 xi þ h; y1;i þ hK13; y2;i þ hK23

� �K24 ¼ f2 xi þ h; y1;i þ hK13; y2;i þ hK23

� �


Appendix IV

15-bus, 4-generator system data

Single-line diagram

SVC2SVC3

G1

G2

G3

G4

3

9

13

12

15

14

6

10

2

8

4

7

1

5

11SVC1

System data

Table IV.1 Impedance and line-charging data (400 MVA base)

Linedesignation

R (pu) X (pu) Line-charging*(pu)

1-7 0 0.085 02-8 0 0.110 03-9 0 0.095 04-7 0.0364 0.3925 0.02654-8 0.0276 0.2983 0.02014-10 0.0334 0.3611 0.02434-12 0.0290 0.3140 0.02125-11 0.0262 0.2826 0.01905-12 0.0378 0.4082 0.02756-8 0.0233 0.2512 0.01696-10 0.0116 0.1256 06-14 0.0349 0.3768 0.02547-10 0.0029 0.0314 07-11 0.0035 0.0377 08-14 0.0262 0.2826 0.01909-13 0.0026 0.0283 012-13 0 0.1000 013-15 0.0116 0.1256 014-15 0 0.1000 0

*Line-charging one-half of total charging of line.

Table IV.2 Static capacitor data (400 MVA base)

Bus no. Susceptance (pu)

4 0.4635 0.2956 0.419

Table IV.3 Operating conditions (400 MVA base)

Bus no. Generation (pu) Impedance load (pu)

Real power Reactive power Real power Reactive power

1 0.850 0.080 0 02 0.720 0.050 0 03 0.680 0.039 0 04 0 0 0.950 0.4005 0 0 0.700 0.2006 0 0 0.800 0.32010 0 0 0.180 0.09011 0 0 0.210 0.11015 0.800 0.011 0 0


Table IV.4 Generator data (400 MVA base)

Generator H (s) D pu (MWs/rad) Xd Xq Xmd X0d Tdo

pu pu pu pu pu

1 3 0.0121 1.73 1.73 1.61 0.26 7.02 3.3 0.0110 1.82 1.82 1.71 0.27 6.43 2.95 0.0117 1.80 1.80 1.68 0.31 6.04 3.10 0.0113 1.75 1.75 1.70 0.29 6.6

15-bus, 4-generator system data 447

Index

accelerating power 63, 147, 227, 228, 280AC excitation systems 73adaptive control-based power system

stabiliser 281–2direct adaptive control 282–3indirect adaptive control 283–5

strategies 286system model 285system parameter estimation 285

adaptive fuzzy power system stabiliser(AFPSS) 301

adaptive network-based fuzzy logiccontroller 293

architecture 294neuro-fuzzy controller architecture

optimisation 296–7self-learning ANF PSS 295training and performance 295

adaptive network-based fuzzy powersystem stabiliser (ANF PSS) 295

adaptive neuro-controller (ANC) 293Adaptive Neuro-Fuzzy Inference System

(ANFIS) 263–5, 272adaptive neuro-identifier (ANI) 292, 293adaptive power system stabiliser

(APSS) 283artificial intelligence based 291based on recurrent adaptive control

301–7with fuzzy logic identifier and PS

controller 299–300with neuro identifier and PS control

297–9with NN predictor and NN controller

292–3PS control-based 287with RLS identifier and fuzzy logic

control 301

adaptive simplified neuro-fuzzy control(ASNFC) 269

application of, to shunt-compensatedpower systems 374–7

simulation studies 375–6three-phase to ground short circuit

test 376–7control system design of 272–4simplification of rule-base structure

269–72alternator-rectifier excitation system 73amalgamated analytical and AI-based PSS

297angular trajectories 245arbitrary reference 354ARMAX model 285, 299artificial intelligence (AI)-based APSS 291

adaptive network-based FLC 293architecture 294neuro-fuzzy controller architecture

optimisation 296–7self-learning ANF PSS 295training and performance 295

APSS with NN predictor and NNcontroller 292–3

artificial intelligence (AI) techniques 253adaptive simplified NFC (ASNFC) 269

simplification of rule-base structure269–72

artificial neural networks (ANN)253–4

control system design of the proposedASNFC 272–4

fuzzy logic (FL) systems 259Fuzzy IF–THEN rules 261Fuzzy set theory 260linguistic variables 261structure of 261–3

neural network topologies 255back-propagation learning algorithm

256–9multi-layer feed-forward

architecture 255–6recurrent neural network (RNN)

256single-layer feed-forward

architecture 255neuro-fuzzy system 263

adaptive neuro-fuzzy inferencesystem (ANFIS) 263–5

online adaptation technique 267–9structure of NFC 265–7

artificial neural networks (ANN) 253–4asymptotic stable trajectories 151automatic voltage regulators (AVRs) 277auto-regressive moving average (ARMA)

model 299autotransformers 81

back-propagation learning algorithm256–9

back propagation through time (BPTT)301, 303

base quantitiesfor rotor

alternative per unit/normalisingsystems 435

per unit values 428–9for stator

alternative per unit/normalisingsystems 433–5

per unit values 427–8batteries (lead-acid) 419Brazilian interconnected power system

(BIPS) 422, 423bus power and voltage 119, 127

capacitance matrix 344–5capacitive reactance matrix 344centre of area (COA) 262centre of gravity (COG) 262centre of inertia (COI) 241–2chain rule method 258chemical energy storage systems 397classification of power system stability 3

small signal stability 3–6transient stability 6–8

commonly used power system stabiliser(CPSS) 278, 291,

characteristics of 280–1configuration of 278–9

compensated system 317power–angle curves for 359transient stability study results 329

compensation devices 379flexible AC transmission system 380–91phase-shifting transformer 389static synchronous compensator

(STATCOM) 387–8static synchronous series compensator

(SSSC) 382–4static var compensator (SVC) 384–7thyristor-controlled series capacitor

(TCSC) 380–2unified power flow controller (UPFC)

389–91component-based load modelling 98compressed air energy storage (CAES)

397–8, 419computational intelligence techniques 253constant magnitude 207control system parameters 366conventional power system stabiliser 278

common PSScharacteristics of 280–1configuration of 278–9

input signals 279–80critical energy, calculation of 246–9current state space model 42–4, 55–7

damper circuits 13, 64damping coefficient 156damping power 156damping torque 6, 156d-axis 13, 22, 32d-axis transient open circuit time constant

61DC excitation systems 71, 72–3decoupling method 123

fast-decoupled method 125–9defuzzification 262–3degree of series compensation 314, 333design contingencies 149direct adaptive control 282–3d–q equivalent circuit of synchronous

generator 23, 24


dynamic load models 101induction motor model 101–3

dynamic security constraint, optimal powerflow with 137–41

dynamic system 1

E0q model 64–8

eigenvalues and coefficients 339electrical torque 2, 5, 26, 190energy balance approach 235–41energy storage systems 395

chemical energy storage systems 397compressed air energy storage (CAES)

397–8flywheel energy storage 397magnetic properties of 402

Meissner effect 404–7superconducting magnetic energy

storage (SMES) 407–9phasor measurement units 420

case studies 422–3wide-area measurement system

(WAMS) 420–2pumped hydroelectric energy storage

398–400super capacitors 400superconductivity applications 409

SMES applications 416–18storage systems, features of 418–19superconducting fault current

limiters 412–16superconducting synchronous

generators (SSGs) 409–11superconducting transformers 411–12superconducting transmission cables

411superconductors (SCs) 400

type I 401type II 401–2

equal area criterion (EAC) 225, 231, 335equilibrium points 149–50

stability of 150–2, 159stable 150, 221unstable 152, 221, 247–8

equivalent loop configuration 346equivalent reactance diagram 416equivalent voltage sources 189Euler’s method 192, 439–41excitation control elements 70

excitation control system 67, 69excitation system 67

excitation control elements 70power system stabiliser 70–1sample data for 73terminal voltage transducer and load

compensator 69types of 71–2

type AC excitation systems 73type DC excitation systems 72–3type ST excitation systems 74

excitation system stabiliser (ESS) 69, 70transfer function of 70

exciter control system 191exciter saturation characteristics 71exciter saturation function 71exponential representation 100–1

fast-decoupled method 125–9FC-TCR configurations 360feed-forward network 25515-bus system data

operating conditions 446single-line diagram 445static capacitor data 446system data 446

five machine power system 293fixed series capacitor 9flexible AC transmission system (FACTS)

379, 380–91phase-shifting transformer 389static synchronous compensator

(STATCOM) 387–8static synchronous series compensator

(SSSC) 382–4static var compensator (SVC) 384–7thyristor-controlled series capacitor

(TCSC) 380–2unified power flow controller (UPFC)

389–91flux linkage equations 14–15

transformation of 18–19flux linkage state space model 34, 49–54

modelling without saturation 34–40modelling with saturation 40–2

flywheel energy storage (FES) 397, 419forced state variable equation 162–44-generator system data 445

generator data 447

Index 451

Fourth-order Runge–Kutta method 192,443–4

frequency variations 148fuzzification 261Fuzzy Adaptive Learning Control

Network 263fuzzy inference 262Fuzzy logic (FL) systems 259

Fuzzy IF–THEN rules 261Fuzzy set theory 260linguistic variables 261structure of 261–3

fuzzy logic controller (FLC) 269, 283, 294,295, 301

Fuzzy Net (FUN) 263

Gauss–Seidel method 121–3, 124generator conventional power system

stabiliser (GCPSS) 376generator data

4-generator system data 447of nine-bus test system 438

generator excitation system block diagram278

global positioning system (GPS) 420

high-temperature SCs (HTS) 401hydraulic turbines 75–7hydro power plant model 400hydro-turbines, model of speed governing

system for 76–7hyperbolic tangent functions 254

incident component’ 319indirect adaptive control 282, 283–5

strategies 286LQ control 286MV control 286pole-zero and PA control 287PS control 287

system model 285system parameter estimation 285

inductance matrix 346, 348induction motor model 101–3inductive limiter 413inertia constant 31infinite bus

classical model for one machineconnected to 63

synchronous machine connected to 47current state space model 55–7flux linkage state space model

49–54integrated power system, synchronous

machine connected to 57–8inter-area mode 277, 279inter-area oscillations 4interline power flow controller 10interphase power controller 10intersection operator 260inter-unit oscillations 4

Kirchhoff ’s law 109knowledge base 262Kuhn–Tucker theorem 135

large disturbance stability 149largest of maximum (LOM) 262Learning 253lemma 222–3linear quadratic (LQ) control 281, 286linguistic variables 261Load Flow model 109load flow techniques 57, 64, 238load modelling 8

for stability and power flow studies103

loads 97dynamic load models 101

induction motor model 101–3static load models 99

exponential representation 100–1polynomial representation 99

local model networks 304local mode oscillations 4long transmission lines 319–21, 357–9

series compensation for 321–9lossless transmission line

compensation of 313amount determination of series

compensation 313–16transient stability 316–19

shunt-compensated line parameters 351transient stability enhancement for

353–7low-temperature SCs (LTS) 401, 402, 409,

410, 413Lyapunov function theorem 223–4


machine parameters in per unit values 26torque and power equations 30–2

machine stator equations 195magnetic properties of superconductive

materials 402–4magnetic saturation curve 41Mamdani and Sugeno inference systems

262mean of maximum (MOM) 262mechanically switched capacitor (MSC)

384mechanical torque 2, 7Meissner effect 404–7Membership Function (MF) 260, 266minimum variance (MV) control 286modeling, need for 8–9model reference adaptive control (MRAC)

282, 283modified Euler–Cauchy method 192, 441multi-layer feed-forward network 255–6multi-layer perceptron (MLP) network 256multi-machine power system 63–4

stability of 234energy balance approach 235–41TEF method 241–9

transient stability, enhancement of329–33

transient stability analysis of 201–18multi-machine systems, small signal

stability of 177machine representation 179–83network and load representation 178–9

NARMAX model 299negative definite function 222network buses, variables characterising

133network equations 195, 203–18neural network (NN)-based APSS

(NAPSS) 293, 294neural network (NN) topologies 255

back-propagation learning algorithm256–9

multi-layer feed-forward architecture255–6

recurrent neural network (RNN) 256single-layer feed-forward architecture 255

neuro-fuzzy control (NFC) 263structure of 265–7

neuro-fuzzy system 263adaptive neuro-fuzzy inference system

(ANFIS) 263–5online adaptation technique 267–9structure of neuro-fuzzy control 265–7

Newton–Raphson method 111, 123, 126,127, 129

power flow solutionwith polar coordinate system 113–14with rectangular coordinate system

114–21nine-bus test system 205, 329, 437

bus data 438generator data 438reactance diagram of 345single line diagram 437system data 437

non-linear system equations 185numerical integration techniques 192, 439

Euler’s method 439–41Runge–Kutta methods 442

fourth-order 443–4second-order Runge–Kutta method

442–3third-order 443

trapezoidal method 441–2

one machine–infinite bus system 48, 63online adaptation technique 267–9optimal power flow (OPF) 131

with dynamic security constraint137–41

without inequality constraints 134problem formulation 131–2problem solution 132–7

ordinary differential equations (ODEs)192

Park’s transformation 17–18per unit equivalent circuit 83

for a two-winding ideal transformer 84per unit turns ratio 83per unit values, machine parameters in 26

torque and power equations 30–2phase-shifting transformer 389

modelling of 91–3phase-shift transformers 81phasor data concentrators (PDCs) 420,

421

Index 453

phasor measurement units (PMUs) 420case studies 422–3wide-area measurement system

(WAMS)benefits of 421–2structure of 420–1

p-equivalent circuit 85, 96transformer 85for transmission line representation 96

pole shift (PS) control 287pole shift (PS) control-based APSS 287

pole-shifting control PSS 290–1self-adjusting PS control strategy 287–90

pole-zero and PA control 287pole-zero assignment (PZA) controller 287polynomial representation 99positive definite function 222potential energy 235, 238–9power–angle curve 7, 8, 155

for compensated and uncompensatedsystems 359

illustrating machine oscillations 4illustrating stable and unstable

equilibrium points 227for pre-fault, during fault and post-fault

conditions 229in response to changing the electrical

power 228with stable and unstable equilibrium

points 159with sudden change in mechanical input

power 228of synchronous generator 161for synchronous machines 226at three-fault periods 237, 241, 415for transient disturbance 7

power flow analysis 8, 109, 362, 363decoupling method 123

fast-decoupled method 125–9Gauss–Seidel method 121–3general concepts 109–11Newton–Raphson method 111

power flow solution with polarcoordinate system 113–14

power flow solution with rectangularcoordinate system 114–21

power flow in the lines 137, 141Power Flow model 109power oscillation damping (POD) control

364

power system, function of 1power system dynamics 139power system stabiliser (PSS) 9, 69, 70–1,

277, 278, 416adaptive control-based PSS 281–2

direct adaptive control 282–3indirect adaptive control 283–5indirect adaptive control strategies 286

amalgamated analytical and AI-basedPSS 297

with fuzzy logic identifier and PScontroller 299–300

with neuro identifier and PS control297–9

with RLS identifier and fuzzy logiccontrol 301

APSS based on recurrent adaptivecontrol 301–7

artificial intelligence (AI)-based APSS291

adaptive network-based FLC 293with NN predictor and NN controller

292–3conventional PSS 278

characteristics of common PSS 280–1configuration of common PSS 278–9PSS input signals 279–80

PS control-based APSS 287pole-shifting control PSS,

performance studies with 290–1self-adjusting PS control strategy

287–90PQ bus 111, 113, 116, 122, 123, 363prime mover control system, modelling of

74hydraulic turbines 75–7steam turbines 77–9

PSAT/MATLAB� toolbox 209pumped hydroelectric energy storage

(PHES) 396, 398–400, 419PV bus 111, 116, 118, 122, 123, 363, 365

q-axis 13, 19, 22, 32, 34, 41, 60, 64, 153,179, 196, 429, 432, 433

quadratic function 258quadrature 13, 154, 389

radial basis function (RBF) network 297–8,304

reactance diagram 345, 416


recurrent adaptive control (RAC) 301, 303recurrent neural network (RNN) 256, 301recursive extended least squares (RELS)

285recursive gradient (RG) 301, 303, 305recursive least squares (RLS) 285, 286,

297, 300, 301Reference bus 111reflected component 319, 321reflection coefficient 320region of attraction 223resistive current limiter approach 413retarding torque 7rotor angle 2, 6, 235

stability 154rotor equation of motion 16rotor flux linkages 187, 189rotor mechanical equations 195Runge–Kutta method 192, 442

fourth-order 443–4second-order 442–3third-order 443

saddle point 152saturation factor 41second-order model of generator 148second-order non-linear ordinary

differential equation 148Second-order Runge–Kutta method 192,

207, 442–3self-inductances 14, 22series capacitive compensation 311,

329–30series compensation 311

long transmission lines 319–21series compensation for 321–9

lossless transmission line 313amount determination of series

compensation 313–16transient stability 316–19

multi-machine power system transientstability 329–33

small signal stability 335–40sub-synchronous resonance 340

electrical network 343–8mechanical system 341–3

transmission line parameters, definitionsof 311–13

transmission power transfer capacity,investigation of 333–5

series-resistive limiter 413shunt capacitive compensation 311, 360shunt compensation 9, 351

ASNFC application to 374–5simulation studies 375–6three-phase to ground short circuit

test 376–7long transmission lines 357–9of lossless transmission lines

shunt-compensated line parameters351–7

static synchronous compensator(STATCOM) 369–74

static var compensators (SVC) 360FC-TCR compensators

shunt reactors 351sigmoid functions 254simple power system 47, 193, 343

equivalent circuit of 193, 317transient stability assessment of 193–200

simplified FL controller (SFLC) 269, 271,272

Single-layer feed-forward network 255single line diagram

15-bus system data 445of nine-bus test system 437

single-machine infinite-bus systemwith linear generator model 336stability of 225–34

slack bus 111, 113small disturbances 3, 4, 9, 138, 147–8,

149–52, 164, 277smallest of maximum (SOM) 262small signal instability 4, 5small signal stability 3–6, 138, 147, 154–62

equilibrium point, stability of 150–2equilibrium points 149–50forced state variable equation 162–4improvement of 335–40of multi-machine systems 177

machine representation 179–83network and load representation

178–9synchronous generator

linearised current state space modelof 164–72

linearised flux linkage state spacemodel of 172–7

synchronous machines, phasor diagramsof 152–4

Index 455

speed-governing systems 75general simplified model of, for

hydro-turbines 77speed governor control 191stability 221

definition of 149lemma 222–3Lyapunov function theorem 223–4negative definite function 222positive definite function 222stability regions 223

stability index 245, 246, 248stability margin 245

increase of 9–10stability regions 223, 246, 248stable equilibrium point (SEP) 150, 221,

223, 227state space approach 345state space model 34, 40, 42–4, 47, 164

flux linkage 49–57, 62, 64in matrix form 38

static capacitor data 446static load models 99, 103

exponential representation 100–1polynomial representation 99

static synchronous compensator(STATCOM) 9, 369–74, 387–8,389, 390

static synchronous series compensator(SSSC) 9–10, 382–4

static var compensator (SVC) 9, 253, 360,384–7

FC-TCR compensatorscharacteristics of 362modelling of 362–9

modes of operationcapacitive mode 363inductive mode 363normal mode 363

SVC conventional power systemstabiliser (SCPSS) 376

stator equations 186, 187, 190, 195, 203stator voltage equations, transformation of

19–25steady-state stability limit 3, 159, 278,

329steam turbines 77–9ST excitation systems 74

sub-synchronous resonance (SSR) 340electrical network 343–8mechanical system 341–3

sum of squared errors 257super capacitors 400, 419superconducting magnetic energy storage

(SMES) 407–9, 419applications 416–18

superconductive materials, magneticproperties of 402–4

superconductivity applications 409storage systems, features of 418–19superconducting fault current limiters

412–16superconducting synchronous generators

(SSGs) 409–10benefits of 410–11

superconducting transformers 411–12superconducting transmission cables

411superconductors (SCs) 400, 404, 409–11

in magnetic field 405type II SCs 401–2, 407type I SCs 401, 407

supervisory control and data acquisition(SCADA) system 420, 421

surge impedance loading (SIL) 312, 321Swing bus 111swing equation 31–2, 66, 147, 155, 190,

198, 230synchronising power coefficient 155, 156synchronising torque 5, 6synchronous generator 1, 3, 4, 5, 9

d–q equivalent circuit of 23linearised current state space model of

164–72linearised flux linkage state space model

of 172–7superconducting 409, 410

synchronous machine 3, 13, 47, 201, 339,400, 409

connected to an infinite bus 47current state space model 54–7flux linkage state space model

49–54connected to an integrated power

system 57–8current state space model 42–4


equations 14flux linkage equations 14–15flux linkage equations, transformation

of 18–19stator voltage equations,

transformation of 19–25torque equation 16–17torque equation, transformation of

25–6voltage equations 15–16

equivalent circuits 32–4excitation system modelling 68

excitation control elements 70power system stabiliser 70–1terminal voltage transducer and load

compensator 69type AC excitation systems 73type DC excitation systems 72–3type ST excitation systems 74

flux linkage state space model 34modelling without saturation 34–40modelling with saturation 40–2

machine parameters in per unit values26

torque and power equations 30–2parameters, in different operating

modes 58–62Park’s transformation 17performance equations of 185phasor diagrams of 152–4, 170and power system frames of reference

194prime mover control system, modelling

of 74hydraulic turbines 75–7steam turbines 77–9

representation, by transformedwindings 20

schematic representation of 13synchronous machine model 186–91synchronous machine parameters

alternative per unit/normalising systemsbase quantities for rotor 435base quantities for stator 433–5

calculation of 427per unit values 427

base quantities for rotor 428–9base quantities for stator 427–8

conversion of rotor quantities toequivalent stator EMF 430

synchronous machine voltage equations430–3

synchronous machine representation 20, 201synchronous machine-simplified models

62classical model 62

for multi-machine system 63–4for one machine connected to an

infinite bus 63E0

q model 64–8system admittance matrix for transient

stability study 207, 208system buses 118

voltage, power and cost at 137, 141system data

15-bus system data 446of nine-bus test system 437

system equations 152, 193, 221system phasor diagram with shunt

compensation 357system reliability 1, 412system security 1system stability, defined 1system trajectories 247

Takagi–Sugeno (TS) fuzzy systems 299tandem compound, single reheat 77, 78Taylor series 112, 116, 151, 164, 298terminal voltage transducer and load

compensator 69tertiary winding 81, 87Third-order Runge–Kutta method 192, 443three-layer neural network 257three-winding transformer 87–8thyristor-controlled reactor (TCR) 9, 360,

384thyristor-controlled series capacitor

(TCSC) 380–2thyristor-protected series capacitor 9thyristor-switched capacitor (TSC) 9, 384thyristor-switched reactor (TSR) 384T-norm operator 260torque 2

damping 5, 6, 156, 280electrical 2, 5mechanical 2

Index 457

and power equations 30–2relation between two torque

components 6retarding 7synchronising 6

torque equation 16–17transformation of 25–6

trajectory 150transfer function 70, 75, 191transformed electrical torque 25transformers 81

phase-shifting 91–3, 389superconducting 411two-winding 81–91p-equivalent circuit 85

transient energy function (TEF) methods221, 231, 241–9

critical energy, calculation of 246–9derivation of 243–6formulation of centre of inertia 241–2multi-machine power system, stability

of 234energy balance approach 235–41TEF method 241–9

single-machine infinite-bus system,stability of 225–34

stability concepts, definitions of 221lemma 222–3Lyapunov function theorem 223–4negative definite function 222positive definite function 222stability regions 223

transient-equivalent circuit of inductionmachine 102

transient gain reduction (TGR) 70transfer function of 70

transient kinetic energy (KE) 235, 238–9transient stability 6–8, 138, 185, 316–19

multi-machine power system, analysisof 201–18

numerical integration techniques 192simple power system, assessment of

193–200synchronous machine model 186–91

transmission line parameters, definition of311–13

transmission lines 57, 81, 93, 323modelling of 95–7voltage and current relationship of a

line 94–5transmission network 7, 177, 311,

379transmission power transfer capacity,

investigation of 333–5trapezoidal method 192, 441–2trigger coil 413two-winding transformers, modelling of

81–7typical power system 148

uncompensated system 316–17power–angle curves for 359transient stability study results

329unified power flow controller (UPFC) 10,

389–91universe of discourse 260, 261, 266, 300,

301unstable equilibrium point (UEP) 152, 159,

221, 227, 245, 247

voltage and current relationship of a line94–5

voltage equations 15–16, 430voltage instability 379, 416voltage source converter (VSC) 369, 370,

382, 383, 388voltage transducer and load compensator

69voltage variations 148, 362

wide-area measurement system (WAMS)420

benefits of 421–2structure of 420–1

zero-order Sugeno model 264zero-sequence voltage 22


Documents

Power System Stability: Modelling, Analysis and Control