Analytical and Intelligent Techniques for Dynamically

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Analytical and Intelligent Techniques forDynamically Secure DispatchesAftab AlamClemson University, [email protected]

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ii

ANALYTICAL AND INTELLIGENT TECHNIQUES FOR DYNAMCALLY SECURE DISPATCHES

A DissertationPresented to

the Graduate School ofClemson University

In Partial Fulfillmentof the Requirements for the Degree

Doctor of PhilosophyElectrical Engineering.

byAftab AlamMay 2007

Accepted by:Dr. Elham B. Makram, Committee Chair

Dr. Adly. A. GirgisDr. Ian WalkerDr. Hyesuk Lee

iii

ABSTRACT

The NERC August 14th Blackout report brought out by the task force cited

‘failure to ensure operation within secure limits’ as one of the main reasons.

Many of the numerous recommendations focused on the need for better real-time

tools for operators and reliability coordinators. In the absence of such tools the

operators are limited to operating in conservative secure operating regions

established using offline studies. At the same time, with the fast inception of

deregulation, the need to ensure a reliable and secure power system has become

all the more vital. The success of a competitive market is dependent upon a

reliable and secure transmission system at all times. This dissertation investigates

ideas for analytical and intelligent techniques to obtain generation dispatches that

would be first-swing stable for a certain set of credible contingencies (three-phase

faults) with their respective fault clearing times.

In a deregulated environment, fair operation is dependent upon maximum

utilization of available resources. Conventionally the maximum utilization has

been made possible by use of the optimal power flow where certain objective(s)

are maximized/minimized subject to certain constraints through various

mathematical programming techniques. Due to the extremely large computation

times required to carry out dynamic security assessment of large disturbances,

constraints to ensure dynamic stability, more specifically transient stability have

been ignored from state of the art optimal power flow routines today. Rather, such

limits are established using offline studies. Since, these limits are established from

iv

forecasted data, they are kept on the conservative side to ensure that the power

system would be transiently stable for a certain set of credible contingencies. One

of the major obstacles of including dynamic security assessment subroutines

within the optimal power flow method for real-time operations is the heavy

computational burden since it requires solution of differential equations. Another

is the extremely high analytical and programming complexity involved with

inclusion of constraints to ensure that the ‘solution of the differential equations is

within certain bounds’.

The presented research proposes to explore analytical and intelligent

techniques of dynamic security assessment which would aid in the development

of software based subroutines to be included in the optimal power flow. This

would enhance the already existing state of the art dispatch routines within energy

managements systems by allowing operation closer to stability limits. It would

also aid in the successful and fair operation of a competitive market. On a more

general note, the research work carried out would help in improving power

system reliability and operation. It would also allow maximum utilization of

available resources, Transmission and Distribution assets by having operating

closer to ‘real-time’ steady-state and dynamic stability limits. On the whole, it

would aid in the move towards a deregulated environment leading to long term

economic benefits for both generating companies and consumers.

iii

DEDICATION

This dissertation is dedicated to My Parents, Shadab & Sadaf for their loving

support and encouragement, Chinar, my partner for life and friends without

whom this would not have been possible.

iv

v

ACKNOWLEDGMENTS

I would like to express my sincere appreciation to my advisor Dr. Elham

B. Makram for her guidance throughout this research. I am grateful for her

support and patience during the entire period of this research. I would like to

thank Dr. Adly. A. Girgis, Dr. Ian D. Walker and Dr. Hyesuk Lee for serving as

my graduate committee members.

The financial support of Clemson University Electric Power Research

Association (CUEPRA), and the Electrical and Computer Engineering (ECE)

Department at Clemson University are greatly appreciated.

I would like to thank all my colleagues in the power group for their

professional and friendly relationship.

Finally I would like to thank my parents for their patience and their

emotional and financial support throughout my life.

vi

vii

TABLE OF CONTENTS

Page

TITLE PAGE.............................................................................................. i

ABSTRACT................................................................................................ iii

DEDICATION............................................................................................ v

ACKNOWLEDGEMENTS........................................................................ vii

LIST OF TABLES...................................................................................... xi

LIST OF FIGURES .................................................................................... xii

CHAPTER

1. INTRODUCTION ....................................................................... 1

1.1 Motivation…………....................................................... 11.2 Transient Stability Assessment. ...................................... 51.3 Research Objectives/Contributions................................. 10

2. TRANSIENT STABILITY CONSTRAINEDOPTIMAL POWER FLOW .................................................. 13

2.1 Background..................................................................... 132.2 A Computationally Efficient Method to

Obtain Rotor Angles ...................................................... 172.2.1 Review of Equations for Classical Transient Stability Analysis.................................. 182.2.2 Solution Using Taylor Series

Expansion.............................................................. 222.2.3 Results with the IEEE 9-Bus and 39-Bus Systems..................................................... 242.2.4 Discussion ............................................................. 27

2.3 The Transient Stability Constraint .................................. 272.4 Optimal Power Flow with Transient

Stability Constraints Formulation .................................. 292.4.2 Objective Function................................................ 292.4.3 Equality Constraints.............................................. 302.4.4 Inequality Constraints ........................................... 30

viii

Table of Contents (Continued)

Page

2.5 Results with the IEEE 9-Bus System.............................. 34 2.5.1 75% Loading, BOPF............................................. 342.5.2 75% Loading, TSCOPF ........................................ 372.5.3 100% Loading, BOPF........................................... 382.5.4 100% Loading, TSCOPF ...................................... 402.5.5 125% Loading, BOPF........................................... 422.5.6 125% Loading, TSCOPF ...................................... 44

2.6 Results with the IEEE 39-Bus System............................ 462.6.1 75% Loading, BOPF............................................. 472.6.2 75% Loading, TSCOPF ........................................ 492.6.3 100% Loading, BOPF........................................... 502.6.4 100% Loading, TSCOPF ...................................... 532.6.5 125% Loading, BOPF........................................... 542.6.6 125% Loading, TSCOPF ...................................... 57

2.7 Discussion ....................................................................... 592.8 Conclusions..................................................................... 70

3. APPLICATION OF SINGLE MACHINE EQUIVALENT METHOD AND NEURAL NETWORKS FOR ESTIMATION OF CRITICAL CLEARING TIME ............................................. 73

3.1 Single Machine Equivalent Method ............................... 733.1.1 Methodology......................................................... 75

3.2 Results with the IEEE 9-Bus System.............................. 813.2.1 Fault near Bus 5 on Line 5-7 for 0.28s ................. 823.2.2 Fault near Bus 5 on Line 5-7 for 0.33s ................. 843.2.3 Fault near Bus 7 on Line 7-5 for 0.083s ............... 863.2.4 Fault near Bus 7 on Line 5-7 for 0.17s ................. 883.2.5 Fault near Bus 6 on Line 6-9 for 0.17s ................. 903.2.6 Fault near Bus 6 on Line 6-9 for 0.14s ................. 92

3.3 Results with the IEEE 39-Bus Systems .......................... 943.3.1 Fault near Bus 3 on Line 3-4 for 0.28s ................. 943.3.2 Fault near Bus 3 on Line 3-4 for 0.33s ................. 96

3.4 Discussion ....................................................................... 983.5 Fast Determination of Critical Clearing Time

Using SIME Method....................................................... 983.5.1 Background........................................................... 98 3.5.2 Methodology......................................................... 100 3.5.4 Discussion ............................................................. 103

3.6 Application of Feedforward Networks for Critical Clearing Time Estimation .................................. 103

ix

Table of Contents (Continued)

Page

3.6.1 Background........................................................... 103 3.6.1 Methodology......................................................... 1033.6.3 Result with the IEEE 9-Bus System ..................... 1063.6.4 Discussion ............................................................. 108

3.7 Conclusions..................................................................... 111

4. A HYBRID NEURAL NETWORK-OPTIMIZATIONAPPROACH FOR DYNAMIC SECURITYCONSTRAINED OPTIMAL POWER FLOW ..................... 115

4.1 Transient Stability Constraint ......................................... 1154.2 Results with the IEEE 9-Bus System (

single contingency) ......................................................... 1174.3 Discussion ....................................................................... 1204.4 Results with the IEEE 9-Bus System (

multiple contingencies)................................................... 1234.5 Discussion ....................................................................... 1344.5 Conclusions..................................................................... 135

5. SUMMARY AND CONCLUSIONS .......................................... 137

APPENDICES ............................................................................................ 145

A: System Data for the 3-Machine 9-Bus IEEE Test System (Base Case) [46]................................................ 147

B: System Data for the 10-Machine 39-Bus IEEE Test System (Base Case) [46]................................................ 149

REFERENCES ........................................................................................... 153

x

xi

LIST OF TABLES

Table Page

2.1 List of faults considered in the IEEE 9-Bus System ...................... 35



2.4 List of faults considered in the IEEE 39-Bus System .................... 47




4.1 TSCOPF results for fault near Bus 7 on Line 5-7........................... 118




4.5 TSCOPF results for fault near Bus 7 on Line 5-7and near Bus 6 on Line 6-9 ....................................................... 124

4.6 TSCOPF results for fault near Bus 7 on Line 5-7 and near Bus 5 on Line 5-4 ....................................................... 126



4.9 TSCOPF results for fault near Bus 7 on Line 5-7, near Bus 6 on Line 6-9 and near Bus 6 on Line 6-9..................................................................................... 132

xii

List of Tables (Continued)

Table Page

A.1 Generator Data ............................................................................... 147

A.2 Bus Data ......................................................................................... 147

A.3 Branch Data ................................................................................... 148

B.1 Generator Data ............................................................................... 149

B.2 Bus Data ......................................................................................... 150

B.3 Branch Data ................................................................................... 151

B.4 Branch Data (continued) ................................................................ 152

xiii

LIST OF FIGURES

Figure Page

2.1 Sample Power System..................................................................... 21

2.2 Comparison of Taylor series expansion method and 4/5th order Runge-Kutta method for fault nearBus 7 on Line 5-7 for 0.15s ...................................................... 25

2.3 Comparison of Taylor series expansion method and4/5th order Runge-Kutta method for fault nearBus 7on Line 5-7 for 0.17s ....................................................... 25

2.4 Comparison of Taylor series expansion method and 4/5th order Runge-Kutta method for fault nearBus 3 on Line 3-4 for 0.2s ........................................................ 26

2.5 Comparison of Taylor series expansion method and4/5th order Runge-Kutta method for fault nearBus 3 on Line 3-4 for 0.3s ........................................................ 26

2.6 General representation of the transient stabilityconstrained optimal power flow problem ................................. 33

2.6 Rotor angles for fault near Bus 7 on Line 7-5 for afault duration of 0.26s following a dispatch given by the BOPF.............................................................................. 35

2.7 Rotor angles for fault near Bus 9 on Line 9-8 for afault duration of 0.28s following a dispatch givenby the BOPF.............................................................................. 36


2.9 Rotor angles for fault near Bus 9 on Line 9-8 for a fault duration of 0.28s following a dispatch givenby the TSCOPF......................................................................... 37

xiv

List of Figures (Continued)

Figure Page



2.12 Rotor angles for fault near Bus 8 on Line 7-8 for a fault duration of 0.35s following a dispatch givenby the BOPF.............................................................................. 40

2.13 Rotor angles for fault near Bus 7 on Line 7-5 for a fault duration of 0.26s following a dispatch given by the TSCOPF......................................................................... 41


2.15 Rotor angles for fault near Bus 7 on Line 7-5 for a fault duration of 0.26s following a dispatch given by the BOPF.............................................................................. 43





xv


Figure Page



2.22 Rotor angles for fault near Bus 10 on Line 10-13 fora fault duration of 0.35s following a dispatchgiven by the BOPF.................................................................... 48

2.23 Rotor angles for fault near Bus 25 on Line 25-2 for a fault duration of 0.24s following a dispatchgiven by the BOPF.................................................................... 49

2.24 Rotor angles for fault near Bus 25 on Line 25-2 for a fault duration of 0.24s following a dispatch givenby the TSOPF............................................................................ 50

2.25 Rotor angles for fault near Bus 3 on Line 3-4 for a fault duration of 0.2s following a dispatch givenby the BOPF.............................................................................. 51



2.29 Rotor angles for fault near Bus 10 on Line 10-13 for a fault duration of 0.35s following a dispatchgiven by the TSCOPF ............................................................... 53

2.30 Rotor angles for fault near Bus 25 on Line 25-2 for afault duration of 0.24s following a dispatch givenby the TSCOPF......................................................................... 54

xvi


Figure Page


2.32 Rotor angles for fault near Bus 10 on Line 10-13 fora fault duration of 0.35s following a dispatch given by the BOPF.................................................................... 56

2.33 Rotor angles for fault near Bus 25 on Line 25-2 fora fault duration of 0.24s following a dispatch given by the BOPF.................................................................... 56


2.35 Rotor angles for fault near Bus 10 on Line 10-13 fora fault duration of 0.35s following a dispatch given by the TSCOPF ............................................................... 58

2.36 Rotor angles for fault near Bus 25 on Line 25-2 fora fault duration of 0.24s following a dispatch given by the TSCOPF ............................................................... 59

2.37 Comparison of generator active power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 7 on Line 7-5................................................ 60


2.39 Comparison of generator active power outputs obtained by BOPF and TSCOPF with constraintsfor fault near Bus 8 on Line 7-8................................................ 62

2.40 Comparison of generator reactive power outputsobtained by BOPF and TSCOPF with constraintsfor fault near Bus 7 on Line 7-5................................................ 63

xvii


Figure Page

2.41 Comparison of generator reactive power outputs obtained by BOPF and TSCOPF with constraintsfor fault near Bus 9 on Line 8-9................................................ 63

2.42 Comparison of generator reactive power outputsobtained by BOPF and TSCOPF with constraintsfor fault near Bus 8 on Line 7-8................................................ 64

2.43 Comparison of Bus voltage magnitudes obtained by BOPF and TSCOPF with constraints for fault near Bus 7 on Line 7-5...................................................................... 64




2.47 Comparison of generator active power outputs obtained by BOPF and TSCOPF with constraintsfor fault near Bus 10 on Line 10-13.......................................... 66

2.48 Comparison of generator active power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 25 on Line 25-2............................................ 67

2.49 Comparison of generator reactive power outputsobtained by BOPF and TSCOPF with constraints for fault near Bus 3 on Line 3-4................................................ 68

2.50 Comparison of generator reactive power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 10 on Line 10-13.......................................... 68

xviii


Figure Page

2.51 Comparison of generator reactive power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 25 on Line 25-2............................................ 69

3.1 An example of a stable OMIB trajectory........................................ 79

3.2 An example of an unstable OMIB trajectory.................................. 80

3.3 Phase plane plot of OMIB speed vs. the OMIB angle.................... 81

3.4 Plot of OMIB mechanical input, electrical power output, accelerating power and angle and speed trajectory for Case 1.................................................................. 82

3.5 OMIB speed vs. the OMIB angle for Case 1 .................................. 83

3.6 Plot of generator angles for Case 1 ................................................. 83




3.10 Plot of OMIB mechanical input, electrical power output, accelerating power and angle and speedtrajectory for Case 3.................................................................. 86



3.13 Plot of OMIB mechanical input, electrical power output, accelerating power and angle and speedtrajectory for Case 4.................................................................. 88



xix


Figure Page




3.19 Plot of OMIB mechanical input, electrical poweroutput, accelerating power and angle and speed trajectory for Case 6.................................................................. 92



3.22 Plot of OMIB mechanical input, electrical poweroutput, accelerating power and angle and speedtrajectory for Case 7.................................................................. 94






3.28 Plot of stable/unstable margins for a fault near Bus 7 on Line 7-8.............................................................. 101

3.29 Plot of stable/unstable margins for a fault near Bus 7 on Line 7-5.............................................................. 102

3.30 Proposed Feedforward Neural Network for CriticalClearing Time Estimation ......................................................... 107

xx


Figure Page

3.31 Comparison of Critical Clearing Time for a fault nearBus 7 on Line 7-5...................................................................... 109




4.1 Conceptual illustration of the transient stabilityconstrained optimal power flow formulation............................ 117

4.2 Comparison of the critical clearing time with the faultclearing time and the neural network estimate ......................... 122

4.3 Comparison of generator active power outputs .............................. 122

4.4 Comparison of the critical clearing time with the faultclearing time and the neural network estimate forfault near Bus 7 on Line 5-7 .................................................... 125





xxi


Figure Page

4.9 Comparison of the critical clearing time with thefault clearing time and the neural networkestimate for fault near Bus 6 on Line 6-9 ................................ 129



4.12 Comparison of the critical clearing time with thefault clearing time and the neural networkfault Bus 7 on Line 7-8 ............................................................ 133



xxii

1

CHAPTER ONE

INTRODUCTION

1.1 Motivation

The August 14th Blackout affected an area of 50 million people and 61,800

MWs of load in the states of Ohio, Michigan, Pennsylvania, New York, Vermont,

Massachusetts, Connecticut, New Jersey and the Canadian state of Ontario[1].

The total estimated losses were between 4 to 10 billion dollars. One of the main

reasons cited by the task force that led to the blackout was ‘failure to ensure

operation within secure limits’. It has been reiterated in the blackout report that it

could have been prevented. The task force provided an exhaustive list of

recommendations many of which focused on the need for better tools for dynamic

security assessment. Recommendation no. 22 asserts to ‘evaluate and adopt better

real-time tools for operators and reliability coordinators’. In the absence of such

tools the operators are limited to operating in regions of secure operating limits

established using offline studies, since they require a considerable engineering

and computational effort. Also these studies consider every possible credible

contingency which would deem them unusable for real-time applications due to

the large computation time.

The Federal Energy Regulatory Commission (FERC) issued a final rule,

Order No. 888 [2] in response to provisions of the Energy Policy Act (EPACT) of

1992. Order No. 888 opens wholesale electric power sales to competition. It

2

requires utilities that own, control, or operate transmission lines to file non-

discriminatory open access tariffs that offer others the same electricity

transmission service they provide themselves. The second final rule, Order No.

889 [3] issued on the same date, requires a real-time information system to assure

that transmission owners and their affiliates do not have an unfair competitive

advantage in using transmission to sell power. It is expected that Orders No. 888

and No. 889 and other actions taken by State Public Service Commissions to

promote competition in the electric power industry will result in increased

demands for transmission services. With the fast inception of deregulation into the

market the need for a reliable and secure power system has become all the more

vital. The successful operation of a competitive market is based on the

transmission system being reliable and secure at all times [4]. Failure can lead to

huge financial losses and ensuring that the power system is reliable and secure

both in a static and dynamic sense is very important. Utilities in almost every

country over the last century have been regulated utilities which are a natural

monopoly. In such vertically integrated systems, it is easier to ensure security for

various reasons. One of the main reasons is that these are monopolies operating

and managing their own generation, transmission and distribution systems and are

well aware of the load growth in their systems. This allows the regulated utility to

avoid overloading lines and equipment whose failure could lead to severe static

and dynamic system disturbances.

In a deregulated environment large inter-area transactions can occur over the

transmission system provided that the transaction does not violate system

3

operating security limits such as transmission line-flow limits and bus voltage

limits. In the presence of such large transactions, large disturbances such as faults,

loss or acquisition of generation, loss or acquisition of loads etc can lead to power

system instability due to loss of synchronism. So it is very vital that the entity

managing the operation of the transmission system, the Independent System

Operator (ISO), ensure that the execution of the transaction occurs within the

bounds of not only static-security criterion, but also within a similar region which

ensures dynamic or transient stability for the power system.

According to [5], ‘if the oscillatory response of the power system during the

transient period following such disturbances is damped and the system settles in a

finite time to a new steady operating condition, the system is stable’. One of the

biggest problems of building real time online dynamic security assessment

software applications is the heavy computation involved since any dynamic

security assessment involves solution of power system differential equations. This

has caused the system operators considering only static-security criterion for real-

time applications and using corrective procedures established by offline methods

to ensure power system dynamic security [6]. In some markets, offline studies are

used to develop operating limits (e.g. maximum output of a generator,

transmission line flow limits, bus voltage limits) to avoid dynamic stability

problems. These are often conservative and contradictory to the concept of a

competitive deregulated environment. In a deregulated system, the efficient

utilization of the transmission system and maximum utilization of revenue would

require the operation of the power system close to limits of stability. But the lack

4

of fast assessment methods has avoided the incorporation of dynamic security

assessment into online applications. Dynamic security assessment software is

used to carry out transient stability analysis for each condition of a large set of

possible outage conditions that could occur. A transient stability simulation time

frame of usually up to 10 seconds is required [6]. These studies are usually done

offline and take hours to complete. This makes them impossible to use in a real-

time environment, wherein an operator would need to perform real world control

actions within ten minutes after a real-world outage has occurred. The operator

would need to do this to ensure that the outage does not cause the grid to go

unstable due to a voltage instability situation or a cascading outage situation

leading to blackouts.

Dynamic security assessment methods can be broadly divided into corrective

and preventive methods. A "corrective" approach requires immediate action, such

as switching circuits or other actions, after a contingency occurs, so the system

performance will be adequate. Corrective operation is less reliable than preventive

operation, but allows greater power transfers during normal operations. Corrective

measures between systems sometimes become so complex that when a certain

contingency occurs, the system fails. Some corrective methods are used to

reschedule generator dispatches based on limits established by offline studies.

Power transfers between areas are limited to levels determined by offline system

contingency studies. However as mentioned earlier these limits are conservative.

“Preventive” operating procedures mean operating the system in such a way

as to avoid service interruptions as a result of certain component outages. So, the

5

preventive methods eliminate the need for a re-dispatch or corrective actions

following an outage as the dispatch itself would be transiently stable for the set of

faults considered while obtaining the dispatch. It is recognized as good utility

practice and regarded by the North American Electric Reliability Council (NERC)

as the primary means of preventing disturbances in one area from causing service

failures in another. The NERC guidelines recommend making it an operational

requirement that systems be able to handle any single contingency. The ability to

handle multiple contingencies should be an operational requirement when

practical, according to NERC. Hence, the "preventive" operating procedures,

ensures that no action is required in the event of a system contingency other than

clearing the fault. When contingencies arise, the system is capable of responding

without lines overheating, voltage problems, and instability.

1.2 Transient Stability Assessment

Transient stability studies have been part of electric utility guideline for more

than two decades now. Whether a corrective or preventive method is employed,

its effectiveness in a real-time environment is based on upon the method being

extremely fast and reasonable accurate. The early methods used to detect the

transient stability swings were based on the out-of-step relays using the apparent

impedance concept [8, 9]. In the early years of transient stability assessment

studies, time domain simulation methods were used. These basically relied on

using classical integration methods such as the Euler method, the trapezoidal

method, Runge-Kutta methods etc of converting the differential equations into

6

algebraic form. These methods were did not include load modeling and used

classical power system models of constant impedance models. This eliminated the

need to consider the non-generator bus voltages and the prefault, faulted and

postfault admittance matrices used for the transient stability analysis were reduced

to the generator nodes. Inclusion of load models was deemed necessary and this

required the power flow solution at each step of the time domain simulation to

obtain the new bus voltages and angles. This led to the development of explicit

and implicit methods to solve the differential-algebraic set of equations to include

load models [10]. Each of these methods has its own advantages and

disadvantages. [11] proposed semi-implicit numerical integration methods for

solving differential equations which had the advantages of both the explicit and

implicit methods and also allowed a large time step without encountering

numerical instability problems. Pai et al in [12] presented a new trajectory

approximation technique for transient stability analysis where a number of

contingencies can be simulated in a very short period of time. The proposed

method uses piecewise linearization of the nonlinearities combined with

trapezoidal integration of the differential equation to approximate the trajectory of

a multimachine system. But the method is limited to the use of classical power

system models.

In parallel with the development of the time domain simulation methods has

been the development of a class of direct methods derived from Lyapunov’s

stability criterion. These classes of methods are based on calculating the transient

energy margin by using transient energy functions which are free from time

7

domain simulations and then provide an index known as the transient energy

margin that provides both qualitative and quantitative information about the

transient stability of the system [13]. These methods have been explored in great

depth in [14-15] and have been very popular for use in on-line dynamic security

assessment. One of the major problems associated with these methods is the use

of detailed power system models. Another major disadvantage is the accurate

determination of the unstable equilibrium point required to calculate the transient

energy margin. The transient energy function methods were later coupled with

time domain simulation methods to form hybrid methods to further improve the

accuracy of the method and also provided the provision to include transfer

conductances and detailed load models. One such method is provided in [16].

Some of the newer corrective methods are based on employing sensitivity

analysis. Laufenberg et. al in [17] suggested an approach dynamic security

analysis based on sensitivity theory. The trajectory sensitivity is computed with

respect to a pre-selected set of parameters such as generator output using different

contingencies and clearing times. From this the critical machines were found out

for a given clearing time. The biggest disadvantage of this method is the

additional burden on integrating the additional differential equations required

which was hoped to be overcome with faster computers and newer parallel

algorithms. The method is also limited to simple power system models. Another

disadvantage of this method is that it would be helpful in corrective schemes only

and preventive methods cannot employ this method due to the extremely large

computation time.

8

Another class of methods employs the development of equivalent one

machine infinite bus equivalents for multimachine systems and using the well

known equal area criterion for transient stability assessment. In [18] Da-zhong et

al developed a dynamic single machine equivalent system model for on-line first

swing critical clearing time estimation. Assessment of the transient energy

margin, identification of a group of machines called the ‘dominant critical

machines and an interpolation formula for CCT evaluation were proposed to

achieve high speed and accuracy in transient stability assessment. These classes of

methods have provided consistent results to utilities and are not limited to

classical power system models and also provide early termination criterion to

improve computation time for carrying out the transient energy assessment of a

large number of contingencies.

Transient instability is a major concern of system operators because it is the

most common source of instability and because changes in operating conditions

produce the greatest variation in stability constraints. If system limitations can be

calculated for actual conditions rather than off line, the system can be operated

closer to actually needed limitations. These calculations require on-line data that

provide immediate measurements of actual loading, generation, and transmission

system status. Some utilities perform their off-line dynamic security studies every

day based on the operating conditions forecast for the next day. The results of

these studies, which are usually performed overnight, are provided to the control

center for operating the power system the next day. On-line dynamic security

assessment eliminates all conservative assumptions about future operating

9

conditions because actual data on system operating conditions are used. This on-

line assessment can increase the actual transfer capability of a power system. Also

as the restructuring of the electric power industry for increased competition

continues, along with increases of wholesale trade, it is expected that the future

operators of the transmission system, whether they are independent system

operators (ISOs), regional transmission groups (RTOs), power pools, or utilities,

will be interested in increasing the utilization rates of the existing transmission

lines. This increased utilization and also taking into account system security at the

same time can only be provided by preventive methods. Preventive methods

limits the amount of inter-area transfers as compared to considering only static-

security constraints. But there is an increasing interest among utilities to increase

transmission capacity by upgrading the existing lines since it can be done at a

considerably less cost than constructing a new transmission line and with a shorter

lead time. Hence preventive methods definitely have the potential for use in real-

time operations and finally finding their way into energy management systems.

The advent of open access and the competitive market has given the optimal

power flow a new lease on life and respectability as the indispensable tool for

nodal pricing. One of the greatest provisions provided by the optimal power flow

is the inclusion of the security constraints. The state-of-the-art optimal power flow

software today includes static security constraints to ensure precontingency

transmission line flows, thermal limits, bus voltages, generator power outputs etc.

Many of the preventive schemes also include post-contingency transmission line

10

flows and bus voltage limits to ensure that the power system would be stable in a

static sense following credible contingencies.

1.3 Research Objectives / Contributions

Thus, this research aims at developing techniques for assessment of dynamic

security of the power system which can be applicable to both regulated and

deregulated power systems. The primary motives have been to

1. Development of a preventive method that provides generation dispatches

using the optimal power flow that would be ‘transiently stable’ for a set of

credible contingencies

This involves:

a. Investigation of methods to efficiently integrate the differential equations

by converting them to algebraic form

b. Assessing the transient stability assessment using the values of the rotor

angles obtain by the efficient method investigated above leading to the

development of the dynamic security constraint that needs to be included

in the optimal power flow to ensure transient stability for a given fault.

c. Developing a transient stability constrained optimal power flow method

by including the algebraic equations obtained by converting the

differential equations by the above method in the optimal power flow

without drastically increasing both the state-variable set and the constraint

set for a given fault location and fault clearing time.

11

2. Development of an artificial intelligence based method to assess the transient

stability of the power system for a given fault and fault clearing time:

This involves:

a. Investigation of methods to quickly generate the training set required to

train the weights of the artificial neural network.

b. Use of feedforward neural networks trained for each fault location for fast

estimation of critical clearing time with a very high accuracy.

3. Development of a computationally efficient way to implement transient

stability constrained optimal power flow with the consideration of multiple

contingencies:

This involves:

a. Use of the developed neural net in the optimal power flow to further

decrease the computation time required to carry out the transient stability

constrained optimal power flow.

b. Extension of the above method for an efficient implementation of

multiple-contingency transient stability constrained optimal power flow.

c. Study of the impact of unbalanced faults in systems with dispatches

obtained by the transient stability constrained optimal power flow with

constraints for three-phase faults in the same location as the unbalanced

fault to assess the need of inclusion of constraints for unbalanced faults.

12

13

CHAPTER TWO

TRANSIENT STABILITY CONSTRAINED OPTIMAL POWER FLOW

The initial thrust has been on identifying and developing means for quick

assessment of the impact of various contingencies on the power system from the

point of view of transient stability. In this chapter, efforts have been specifically

made to obtain the rotor angles of the generators in the system very quickly which

are the best indicators of power system stability or instability. The value of the

rotor angles with respect to a center of inertia frame of reference are an

industrially accepted indicator of transient stability/instability. This concept is

used to form the transient stability related constraint. The method is then utilized

to develop a transient stability constrained optimal power flow formulation that

enables us to obtain a dispatch that would be transient stable for a given set of

credible contingencies.

2.1. Background

The optimal power flow has been an important tool in power system

operations for the past 4 decades. The development of various efficient techniques

of nonlinear mathematical programming have allowed the implementation of

complex algorithms for the optimal power flow where a certain objective is

optimized while respecting certain physical and operating constraints also. The

state of the art optimal power flow is able to handle static-security considerations

14

also where physical and operating constraints would not be violated following a

credible contingency. Prior to the move of the vertically integrated power system

industry towards a deregulated type, the power systems have been operated in a

conservative manner. With the advent of competition into the power system

industry, the transmission systems would be pushed to near their operating limits

for fair usage. The optimal power flow is set to become a fundamental tool for use

in such deregulated power systems to decide the final dispatch and hence the

operating point. As mentioned earlier, although conventional optimal power flows

are able to take into account static-security considerations, the operating point

decided by the optimal power flow does not guarantee that the power system

would be transiently stable following a fault. With the power systems being

operated near their stability limits transient stability is the main concern in real-

time operations and is the biggest challenges to the optimal power flow [19].

Conventionally this problem has been approached by trial and error methods

where a re-dispatch is carried out after analyzing the transient stability for various

faults. This is a part of the standard online Dynamic Security Assessment

procedures [20] where the transient stability analysis is carried out by: i) A direct

step-by-step integration (SBSI) method which is computationally intensive ii) An

indirect energy function approach which is computationally cheaper. There has

been some recent research in direct incorporation of transient stability based

constraints into the conventional OPF. Deb et al presented a theoretically straight-

forward method in [21-22] where both voltage stability and transient stability

constraints were integrated, at least in theory, into the dispatch/pricing

15

optimization model. The dynamic equations were converted into numerically

equivalent pure algebraic equations so that they can be easily incorporated into

the dispatch optimization as additional equality constraints. This increased the

state variable and constraint set significantly and is the major source of the

computational burden of the method. Singh and David have proposed a transient

energy function based re-dispatch method in [23] where the generators are re-

dispatched if the stability margin of power system, calculated is inadequate. The

re-dispatch was also made sensitive to price signals so that the competing

generators had an input to the re-dispatch. But the method can provide a

transiently stable dispatch for only a single fault at a time and cannot provide a

dispatch that would be transiently stable for any fault among a set of faults at

different locations. This is in general a major drawback of any such re-dispatch

method. Vittal and Gleason have proposed an application of linearized techniques

for the transient energy function method to determine transient stability

constrained line flow limits [24]. A parallel computation method was presented in

[25] where the large computation was distributed among a cluster of workstations

to ensure transient stability for a set of contingencies collectively. Chen et al

attacked the OPF with transient stability constraints (OTS) problem using

functional transformation techniques by converting the infinite dimensional OTS

into a finite dimensional optimization problem [26]. A primal-dual Newton

interior point method was presented by Yuan et al in [27] to solve a multi-

contingency OTS formulation. Another primal-dual interior point method was

used to implement a concept of “the most effective section of the transient-

16

stability constraints” in [28]. Li et al exploited the quadratic convergence of the

inexact Newton method in [29]. Sun et al have proposed a penalty based approach

in [30] where the adjoint equation method is applied to evaluate the gradient of

the penalty term associated with the stability constraints. A functional estimation

technique was used in [31] to form a dynamic security constraint by evaluating

the critical clearing time as a function of the bus voltages and restricting the actual

clearing time. A preventive and an emergency control technique for real-time

transient stability control based on shifting active power generation were

proposed in [32-33]. A trajectory sensitivities based method was suggested by T.

Nguyen et al in [34]. A rescheduling method to improve small-signal stability was

presented by C. Y. Chung et al in [35]. D. Gan et al showed the openness of the

feasible set for a transient stability constrained optimal power flow problem and

hence the uncertainty of an optimal solution in [36]. [37] presented a transient

stability constrained optimal power flow formulation based on expansion of

differential equations using the trapezoidal rule. A nonlinear transformation

technique was used in [38] to transform the power system variables into a higher

dimensional feature space to determine the transient stability boundary. A

lagrangian based method to incorporate voltage stability constraints and allowed

for ‘reserve’ generation to be used to move from the current loading point to the

maximum loading point in [39]. An Eigenvalue and Eigenvector sensitivity based

optimal power flow formulation was used in [40] to find an operating point that is

both economically optimal and stable in the small-signal sense. M. La. Scala et al

17

described a methodology for an online dynamic preventive control scheme using

discretization of the differential algebraic equations of the power system in [41].

This chapter presents the results of a formulation of an OTS where the

solution of the differential equations is obtained using the Taylor series expansion

(TSE) of differential equations method. The TSE method does not require a small

time step as compared to the discretization of the differential equations to

algebraic form using the Trapezoidal rule or the Euler method. This considerably

reduces the number of integration steps needed for the solution of the machine

dynamic equations. The value of the rotor angles in the center of inertia frame of

reference have been used to model the transient stability constraint as described in

[22]. The MATLAB optimization toolbox [42] has been used to implement the

method.

2.2. A Computationally Efficient Method to Obtain Rotor Angles

This section presents a method to obtain the generator rotor angles to evaluate

first swing stability following the occurrence of a fault on the system. A single

switching action is considered here i.e., it is assumed that the fault is cleared by

opening of the line that the fault occurs on. It is also assumed that the fault occurs

close to a bus so that the fault can be simulated by assuming occurrence of the

fault on the bus which makes it to form the faulted and post-fault admittance

matrices. These are for convenience only and do not affect the results in any way.

The classical model representation of power system is used here, where generator

mechanical power is assumed constant, loads are represented by constant

18

impedances, and generators are represented by voltages behind transient

reactances. The traditional methods of solving differential equations like the

Trapezoidal method or the Euler method are avoided here which results in

considerable saving in computation time required for the solution of the

differential equations.

2.2.1 Review of Equations for Classical Transient Stability Analysis

The main equations involved in the carrying out transient stability with the

classical model representation of the power system are repeated here for a quick

review [47]. The passive electrical network is represented by the admittance

matrix Y . The diagonal elements iiY are the driving point admittances for the

respective nodes and are given by Equation (2.1).

iiiiiiiiii jBGYY (2.1)

The off-diagonal elements ijY are the negative of the transfer admittance

between nodes i and j.

ijijijijij jBGYY (2.2)

Let iV represent the complex voltage at Bus i in the network with magnitude

imV and angle i . We assume there are p buses in the network.

19

p1,2,...,ifor VV imi i (2.3)

The system consists of gn generators that output active and reactive powers

given by gP and gQ respectively. These are supplied to loads that demand

active and reactive powers given by dP and dQ respectively. A steady state

solution of the system requires the solution of the following set of equations. This

solution is known as the loadflow solution.

p1,2,....,i ,PPPp

ijjijLDg ii

0,1

(2.4)

p1,2,.....,i ,0QQQp

ij1,jijLDg ii

(2.5)

ggiggi n1,2,....,i ,PPPmaximin

(2.6)

ggiggi n1,2,....,i ,QQQmaximin

(2.7)

Here ijP and ijQ represent the real and reactive power flow between nodes

and are given by Equations (2.8)-(2.9).

jiijijmmij YVVPji

cos (2.8)

jiijijmmij YVVQji

sin (2.9)

20

Any transient stability analysis requires some preliminary calculations which

are summarized below. The internal emf of each generator is calculated by first

calculating the current output of each generator for the instant before the

occurrence of the fault.

*i

gggen

V

jQPI ii

i

(2.4)

The output current is then used along with the corresponding terminal bus

voltage and direct axis reactance 'dx to find the internal emf of each generator.

ii gendii IjxVE ' (2.5)

Fig.2.1 illustrates a simple power system showing the main variables and

constants involved in the various calculations described above.

The equations of motion for the generators are given by:

ii emiii

i P PDdt

dM

(2.6)

grii n1,2,.....,ifor

dt

d

(2.7)

21

Fig.2.1 Sample Power System

Here eiP represents the electrical power output and is given by Equation

(2.8). iM is the inertial constant of generator i . i is the rotor speed and iD is

the damping constant. r is a constant equal to the synchronous speed which is

377 rad/s.

n

ijj

jiijijjiiiiei YEEGEP1

'2 cos (2.8)

G1 G3

G2 G4

'd1

x

'd2

x

'd3

x

'd4

x

1V 3V

2V 4V

5V

6V

7V

1E 2E

3E4E

11, gg QP

22, gg QP

33, gg QP

44, gg QP

55, dd QP

66, dd QP

77, dd QP

G1G1 G3G3

G2G2 G4G4

'd1

x

'd2

x

'd3

x

'd4

x

1V 3V

2V 4V

5V

6V

7V

1E 2E

3E4E

11, gg QP

22, gg QP

33, gg QP

44, gg QP

55, dd QP

66, dd QP

77, dd QP

22

Here 'ijY refers to the admittance matrix of the network that has been reduced

to the internal nodes of the generators obtained by Kron reduction. The

mechanical power output of each generator i is given by its electrical power

output just before the instant the fault occurs, i.e.,

n

ijj

jiijijjiiii0(t

eimi YEEGEPP-

1

00'2) cos (2.9)

Equations (2.6) and (2.7) are solved for two different time periods. One is

from the instant of the fault to the time the fault is cleared. The second is from the

time, the fault is cleared to the time, the simulation is to be carried out. During

these two time periods what differs is the admittance matrix of the network, 'Y .

2.2.2 Solution using Taylor Series Expansion

The idea is based on converting the differential equations to algebraic form by

expanding them using the Taylor series expansion method. The Taylor series

expansion method was used by Haque et al to efficiently identify coherent

generators in [43], determine first swing stability in [44] and for rapid

computation of critical clearing time by utilizing energy functions in [45]. The

rotor angle and speed of the generator can be predicted rapidly by using Equations

(2.10) and (2.11) as shown below:

23

..)!4

()!3

(

)!2

()()()(

4)4(

3)3(

2)2()1(

1

tt

tttt

ii

iinini

(2.10)

...)!3

()!2

()()(3

)4(2

)3()2()1(1

tttt iiiini (2.11)

Here )t( ni is the prefault angle for generator i at time nt .

,..3,2,1m,)m(i is the mth derivative of rotor angle evaluated at the beginning of

each time interval nn ttt 1 and N is the number of integration steps. Here

)t( ni is the rotor speed for generator i at time nt . Detailed expressions for

higher derivatives are available in [43]. After carrying out several runs of

transient stability analysis by the Taylor series expansion method, it was observed

that a time step of 0.05 seconds and calculations of derivatives upto 4th were

enough to approximately simulate the trajectory as given by the time-domain

simulation method. The MATLAB differential equation solver ‘ode45’ which

uses the 4/5th order Runge-Kutta method, was used to solve for the solution of the

differential equations to compare the accuracy of the solution obtained with the

Taylor series expansion method.

24

2.2.3 Results with IEEE 9-Bus and 39-Bus Systems

Fig.2.2 shows a comparison of the rotor angles of the three generators in the

9-Bus system [46] as obtained by the Taylor series method and the 4/5th order

Runge-Kutta method. The comparison is shown for a fault near Bus 7 on Line 5-7

for 0.15s which does not cause transient instability. The generator rotor angles

have been plotted for 2s. The same fault is simulated for 0.17s which causes the

generators to lose synchronism as evident from the rotor angles plot. The

comparison is carried out again for the two methods. The rotor angles obtained by

the two methods are plotted in Fig.2.3. A similar pair of results is shown for two

faults on the 39-Bus system, one stable and one unstable. Fig.2.4 shows a

comparison of the rotor angles obtained by the two methods for a fault near Bus 4

on Line 3-4 for 0.2s. Fig.2.5 shows the comparison for another fault near Bus 10

on Line 10-11 for 0.3s which renders the system unstable.

25

0 0.2 0.4 0.6 0.8 10

50

100

150

200

250

Time (s)

Rot

or A

ngle

s (d

egre

es)

R-K MethodTSE Method

Fig.2.2 Comparison of Taylor series expansion method and 4/5th order Runge-Kutta method for fault near Bus 7 on Line 5-7 for 0.15s

0 0.2 0.4 0.6 0.8 10

100

200

300

400

500

Time (s)

Rot

or A

ngle

s (d

egre

es)



26

0 0.2 0.4 0.6 0.8 1-50

0

50

100

150

200

250

Time (s)

Rot

or A

ngle

s (d

egre

es)



0 0.2 0.4 0.6 0.8 1-200

0

200

400

600

800

1000

Time (s)

Rot

or A

ngle

s (d

egre

es)



27

2.2.4 Discussion

As seen from the comparisons above the Taylor series method provides an

efficient and accurate method to provide the rotor angles of the generators. One of

the main advantages of converting the differential equations to algebraic form by

using the Taylor series expansion method is the use of the large time step. The

large step of 0.05s considerably reduces the number of iterations required to find

out accurately the state of the rotor angles at 1s. In comparison to the trapezoidal

rule method or the Euler method or other conventional differential equation

solving methods which require a very small time-step, the Taylor series method

reduces the total number of iterations. Also it is seen that expansion upto the 4th

derivative provides highly accurate values of the rotor angles. So the rotor angles

to evaluate first-swing stability can be obtained quickly and accurately using the

Taylor series expansion method.

2.3 The Transient Stability Constraint

This section present the formulation of the transient stability related inequality

constraints to be incorporated into the conventional optimal power flow. The

above section described how the rotor angles at 1s would be obtained efficiently

and accurately by using the Taylor series expansion method. The transient

stability or instability state of a power system at any time can be indicated by the

distance of the rotor angles from the center of inertia angle at that time [22] as

shown in Equation (2.12).

28

gicoi n1,2,...,i (2.12)

where

gg n

ii

n

iiicoi MM

11/ (2.13)

Here coi is the center of inertia angle of the generator i . The value of in

Equation (2.12) above varies from system to system and an appropriate value for

a particular system would be learnt from operator experience after intensive

offline transient stability analysis of the system for various credible contingencies

under various loading conditions.

Consider a function ),,( gPV that is a function of the bus voltages and

the power output of each generator. The function calculates the rotor angles using

the TSE method and the center of inertia at 1s and evaluates a vector with the

absolute value of the difference between each rotor angle and the center of inertia.

Hence the transient stability can be enforced for a particular fault by limiting the

value of the rotor angle at time nt using the inequality constraint shown below

0),,(_ gPV (2.14)

Here _ is an gn x1 vector with all components equal to

29

2.4 Optimal Power Flow with Transient Stability Constraints Formulation

This section highlights the details of the formulation of the optimal power

flow with transient stability constraints.

2.4.1 Objective Function

The objective function in a centrally dispatched market is the sum of the cost

functions of the generators and the objective is to minimize the total cost. When

demand bids are also considered the objective is the difference between the sum

of the cost functions and the sum of the demand bids and the objective is to

maximize social welfare. Demand bids have not been considered in this study. So

the objective here is:

g

i

n

igi Pf Min

1 (2.15)

Here igi Pf represents the cost function of each generator and is given by a

quadratic expression as shown in Equation (2.6).

cbPaPPfiii g

2ggi (2.16)

30

2.4.2 Equality constraints

The basic equality constraints consist of the load flow equations which are

required to be satisfied for any optimal power flow and are shown in Equations

(2.17) and (2.18)

p1,2,....,i ,PPPp

ijjijLDg ii

0,1

(2.17)

p1,2,.....,i ,0QQQp

ij1,jijLDg ii

(2.18)

2.4.3 Inequality constraints

The basic inequality constraints arise from limitations on the active and

reactive power output of the generators and also on the voltages at each bus as

shown below in Equations (2.8)-(2.10).


(2.19)


(2.20)

p1,2,....,i ,VVV maxiimini (2.21)

Additional inequality constraints are needed to limit the rotor angles of the

generators in the case of faults. Hence inequality constraints of the form shown in

Equation (2.14) have to be included for each generator for each fault. We are

31

considered with only first-swing stability here and the classical model is

appropriate for dynamic simulation during that period. Hence, the classical model

of the generator has been considered for the dynamic equations in ).,,( gPV

Additional inequality constraints can be included for thermal limits of the lines,

power flow over the lines etc. But these are quite straightforward and are not

included in this study. The bus voltage angles are also included in the state-

variable set.

The above formulation was implemented in MATLAB using the ‘fmincon’

function [42] available in the optimization toolbox. An advantage of using the

‘fmincon’ is that the constraints can be directly evaluated as functions of the state

variables which can be separate modules reducing programming complexity. In

this case for example, the transient stability constraint was formed by having

another function that evaluates and returns a gn x1 matrix with the absolute value

of the difference of the rotor angles from the center of inertia at 1s. The [C]

matrix required by the ‘nonlcon’ [42] is evaluated as shown below.

1

2

1

),,(

xnncoi

coi

coi

g

gg

PVA

(2.22)

32

11

1

1

xng

xB

(2.23)

BAC (2.24)

Hence, the transient stability constraint is given by equation (2.14).

10

0

0

][

gn

C (2.25)

The number of transient stability related constraints and the related state-

variables i.e., the rotor angle for each generator and for each fault would increase

considerably. Obviously this would have a direct effect on the computation time

and also lead to convergence issues. Instead of restricting rotor angles in the

center of inertia frame of reference for each step of the integration, the rotor

angles at 1s have only been considered to control first swing stability. This

reduces the number of additional constraints and state variables needed. Fig.2.6

summarizes the dynamic stability constrained optimization process required to

ensure transient stability along with static-security.

33

Fig.2.6 General representation of the transient stability constrained optimal power flow problem

p

p

g

g

g

g

V

V

Q

Q

P

P

ng

ng

1

1

1

1

p1,2,....,i ,PPPp

ijjijLDg ii

0,1

p1,2,.....,i ,0QQQp

ij1,jijLDg ii



p1,2,....,i ,VVV maxiimini

0

0

0

ε

ε

ε

δδ

δδ

δδ

gncoi

2coi

1coi

),,( gPV

Stea

dy S

tate

Con

stra

ints

Dyn

amic

Con

stra

ints

tosubject

Pf Ming

i

n

igi

1

Set

Variable

tateS

p

p

g

g

g

g

V

V

Q

Q

P

P

ng

ng

1

1

1

1

p1,2,....,i ,PPPp

ijjijLDg ii

0,1

p1,2,.....,i ,0QQQp

ij1,jijLDg ii




0

0

0

ε

ε

ε

δδ

δδ

δδ

gncoi

2coi

1coi

),,( gPV

Stea

dy S

tate

Con

stra

ints

Dyn

amic

Con

stra

ints

p

p

g

g

g

g

V

V

Q

Q

P

P

ng

ng

1

1

1

1

p1,2,....,i ,PPPp

ijjijLDg ii

0,1

p1,2,.....,i ,0QQQp

ij1,jijLDg ii




0

0

0

ε

ε

ε

δδ

δδ

δδ

gncoi

2coi

1coi

),,( gPV

Stea

dy S

tate

Con

stra

ints

Dyn

amic

Con

stra

ints

tosubject

Pf Ming

i

n

igi

1

Set

Variable

tateS

34

2.5 Results with the IEEE 9-Bus System

This section presents the results of the application of the above OTS

formulation on the 9-Bus system. Three different loading situations of 75%, 100%

and 125% have been considered. For each loading scenario, three faults have been

considered. The fault locations have been selected so that represent locations near

a generator, to a load bus and buses where neither a generator nor a load is

connected. Initially a base optimal power flow is carried out to obtain a basic

dispatch that includes steady-state operating constraints on the generator active

and reactive power outputs and the voltages at all buses. The results of the fault

simulations following these dispatches is shown first. The transient stability

constraint for the unstable faults are then included separately and a new dispatch

is obtained. The fault simulations are then carried out again to show the effect of

inclusion of the transient stability constraint. Note that each fault is considered

separately. So the dispatch obtained to restrain the rotor angles for one particular

fault is not the same as the dispatch obtained to restrain the rotor angles for

another fault.

2.5.1 75% Loading, BOPF

A base optimal power flow is first carried out with no transient stability

related constraints. Table 2.1 lists the faults that have been considered and shows

the status of the transient stability of the system for each of the faults taking into

the consideration the base optimal power flow for a 75% loading situation. All

35

plots have been obtained using the Taylor series expansion method with a time-

step of 0.05s. Fig. 2.7-2.9 show the rotor plots obtained for simulating the faults.

Table 2.1 List of faults considered in the IEEE 9-Bus System

Case Fault Near Bus Faulted Line Fault Clearing Time Stable/Unstable

1 7 7-5 0.26 Stable

2 9 9-8 0.28 Unstable

3 8 7-8 0.35 Stable

0 0.5 1 1.5 20

100

200

300

400

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.2.7 Rotor angles for fault near bus 7 on Line 7-5 for a fault duration of 0.26s following a dispatch given by the BOPF

36

0 0.5 1 1.5 20

500

1000

1500

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.2.8 Rotor angles for fault near Bus 9 on Line 9-8 for a fault duration of 0.28s following a dispatch given by the BOPF

0 0.5 1 1.5 20

100

200

300

400

500

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.2.9 Rotor angles for fault near Bus 8 on Line 7-8 for a fault duration of 0.35s following a dispatch given by the base optimal power flow

37

2.5.2 75% Loading, TSCOPF

It is seen from the above results that at 75% loading, the fault near Bus 9 on

Line 9-8 causes instability. So the transient stability constraints for this particular

fault were included in the optimal power flow to obtain a new dispatch. Fig.2.10.

shows the rotor angles for the same fault following the new dispatch. Since the

other two faults did not render the system unstable, they were not considered.

0 0.5 1 1.5 20

100

200

300

400

500

Tme (s)

Rot

or A

ngle

s (d

egre

es)

Fig.2.10 Rotor angles for fault near Bus 9 on Line 9-8 for a fault duration of 0.28s following a dispatch given by the TSCOPF

38







step of 0.05s.



1 7 7-5 0.26 Unstable

2 9 9-8 0.28 Unstable

3 8 7-8 0.35 Stable

Figs.2.11-2.13 show the rotor plots obtained for simulating the above faults

following a dispatch obtained by running the base optimal power flow as

described earlier.

39

0 0.5 1 1.5 20

500

1000

1500

2000

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

500

1000

1500

2000

2500

Time (s)

Rot

or A

ngle

s (d

egre

es)


40

0 0.5 1 1.5 20

200

400

600

800

Time (s)

Rot

or A

ngle

s (d

egre

es)



It is seen from the above results that at 100% loading, the faults near Bus 7 on

Line 7-5 and near Bus 9 on Line 9-8 cause instability. Hence, two different

dispatches were obtained. The first includes the transient stability constraints for

the fault near Bus 7. Fig.2.14 shows the rotor angles for the same fault following

the new dispatch. The second dispatch includes the transient stability constraints

for the fault near Bus 9. Fig.13 shows the rotor angles obtained for the fault near

Bus 9 on Line 9-8 for 0.28s following the new dispatch.

41

0 0.5 1 1.5 20

200

400

600

800

1000

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

200

400

600

800

Time (s)

Rot

or A

ngle

s (d

egre

es)


42

2.5.4. 125% Loading, BOPF

A base optimal power flow is first carried out with no transient stability related

constraints. Table 2.3 lists the faults that have been considered and shows the

status of the transient stability of the system for each of the faults taking into the

consideration the base optimal power flow for a 125% loading situation. All plots

have been obtained using the Taylor series expansion method with a time-step of

0.05s.



1 7 7-5 0.26 Unstable

2 9 9-8 0.28 Unstable

3 8 7-8 0.35 Unstable



described earlier.

43

0 0.5 1 1.5 20

500

1000

1500

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

Time (s)

Rot

or A

ngle

s (d

egre

es)


44

0 0.5 1 1.5 20

1000

2000

3000

4000

Time (s)

Rot

or A

ngle

s (d

egre

es)


2.5.5 125% Loading, Transient Stability Constrained Optimal Power Flow

It is seen from the above results that at 125% loading, all the three faults

considered cause instability. Hence, three different dispatches were obtained. The

first includes the transient stability constraints for the fault near Bus 7. Fig. 2.19

shows the rotor angles for the same fault following the new dispatch. The second

dispatch includes the transient stability constraints for the fault near Bus 9.

Fig.2.20 shows the rotor angles obtained for the fault near Bus 9 on Line 9-8 for

0.27s following the new dispatch. The third dispatch includes the transient

stability constraints for the fault near Bus 8. Fig. 2.21 shows the rotor angles

obtained for the fault near Bus 8 on Line 7-8 for 0.35s following the new

dispatch.

45

0 0.5 1 1.5 20

200

400

600

800

1000

1200

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

200

400

600

800

1000

Time (s)

Rot

or A

ngle

s (d

egre

es)


46

0 0.5 1 1.5 20

200

400

600

800

1000

1200

Time (s)

Rot

or A

ngle

s (d

egre

es)


2.6 Results with the IEEE 39-Bus system

This section presents the results of the application of the above OTS

formulation on the 39-Bus system. Three different loading situations of 75%,

100% and 125% have been considered. For each loading scenario, three faults

have been considered. As with the 9-Bus system, initially a base optimal power

flow is carried out to obtain a basis dispatch that includes steady-state operating

constraints on the generator active and reactive power outputs and the voltages at

all buses. The results of the fault simulations following these dispatches is shown

first. The transient stability constraint for the unstable faults are then included

separately and a new dispatch is obtained, one for each of the unstable faults. The

fault simulations are then carried out again to show the effect of inclusion of the

transient stability constraint. Note that each fault is considered separately. So the

47

dispatch obtained to restrain the rotor angles for one particular fault is not the

same as the dispatch obtained to restrain the rotor angles for another fault.







step of 0.05s.



1 3 3-4 0.2 Stable

2 10 10-13 0.35 Stable

3 25 25-2 0.24 Unstable

Figs. 2.22-2.24 show the rotor plots obtained for simulating the above faults


described earlier.

48

0 0.5 1 1.5 2

0

50

100

150

200

250

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 2

0

100

200

300

400

Time (s)

Rot

or A

ngle

s (d

egre

es)


49

0 0.5 1 1.5 20

200

400

600

800

1000

Time (s)

Rot

or A

ngle

s (d

egre

es)



It is seen from the above results that at 75% loading, the fault near Bus 25 on

Line 25-2 causes instability. So the transient stability constraints for this particular

fault were included in the optimal power flow to obtain a new dispatch. Fig.2.25

shows the rotor angles for the same fault following the new dispatch. Since the

other two faults did not render the system unstable, they were not considered.

50

0 0.5 1 1.5 2

0

50

100

150

200

250

Time (s)

Rot

or A

ngle

s (d

egre

es)








step of 0.05s. Figs.2.26-2.27 show the rotor plots obtained for simulating the

above faults following a dispatch obtained by running the base optimal power

flow as described earlier.

51



1 3 3-4 0.2 Stable

2 10 10-13 0.35 Unstable

3 25 25-2 0.24 Unstable

0 0.5 1 1.5 20

100

200

300

400

Time (s)

Rot

or A

ngle

s (d

egre

es)


52

0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

Time (s)

Rot

or A

ngle

s (d

egre

es)


53


It is seen from the above results that at 100% loading, the faults near Bus 10

on Line 10-13 and near Bus 25 on Line 25-2 cause instability. Hence, two

different dispatches were obtained. The first includes the transient stability

constraints for the fault near Bus 10. Fig.2.29 shows the rotor angles for the same

fault following the new dispatch. The second dispatch includes the transient

stability constraints for the fault near Bus 25. Fig.2.30 shows the rotor angles


dispatch.

0 0.5 1 1.5 20

100

200

300

400

500

600

Time (s)

Rot

or A

ngle

s (d

egre

es)


54

0 0.5 1 1.5 20

100

200

300

400

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.2.30 Rotor angles for fault near Bus 25 on Line 25-2 for a fault duration of 0.24s following a dispatch given by the TSCOF





the consideration the base optimal power flow for a 125% loading situation.



described earlier.

55



1 3 3-4 0.2 Unstable

2 10 10-13 0.35 Unstable

3 25 25-2 0.24 Unstable

0 0.5 1 1.5 20

200

400

600

800

1000

1200

1400

Time (s)

Rot

or A

ngle

s (d

egre

es)


56

0 0.5 1 1.5 20

500

1000

1500

2000

2500

3000

3500

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

1000

2000

3000

4000

Time (s)

Rot

or A

ngle

s (d

egre

es)


57


It is seen from the above results that at 125% loading, all the three faults

considered cause instability. Hence, three different dispatches were obtained. The

first includes the transient stability constraints for the fault near Bus 3. Fig.2.34

shows the rotor angles for the same fault following the new dispatch. The second

dispatch includes the transient stability constraints for the fault near Bus 10.

Fig.2.35 shows the rotor angles obtained for the fault near Bus 10 on Line 10-13

for 0.35s following the new dispatch. The third dispatch includes the transient

stability constraints for the fault near Bus 25. Fig.2.36 shows the rotor angles


dispatch.

0 0.5 1 1.5 2

0

100

200

300

400

500

600

Time (s)

Rot

or A

ngle

s (d

egre

es)


58

0 0.5 1 1.5 20

200

400

600

800

Time (s)

Rot

or A

ngle

s (d

egre

es)


0 0.5 1 1.5 20

100

200

300

400

500

600

Time (s)

Rot

or A

ngle

s (d

egre

es)


59

2.7 Discussion

It is seen from the results that although the base optimal power flow provides

a dispatch that respects the physical and operational limitations, it does not

guarantee the transient stability of the system after a fault has been cleared.

Incorporating the transient stability constraints into the optimal power flow to

limit the value of the rotor angles allows us to ensure that the system would be

transiently stable following the occurrence and clearing of the fault. By using an

appropriate value of , a dispatch can be obtained that would be transiently stable

for the set of faults in consideration. Fig.2.37 shows the comparison of the

generator active power outputs obtained by both the base optimal power flow

(BOPF) and the transient stability constrained optimal power flow (TSCOPF)

with constraints for the fault near Bus 7 on Line 7-5. It is seen that the active

power outputs of generators 1 and 3 is slightly increased and generator 2 is

slightly decreased. At 125% loading, the increase in power output of generator 1

is greater than before and a similar decrease in the power output of generator 2.

The power output of generator 3 does not change appreciably from the base case

value. The variation in the power outputs of the generators depends on a lot of

factors like proximity to the fault, active power output etc. From Fig.2.37 it can be

concluded that the active power output of generator 2 needs to be decreased

60

1 2 30

50

100

150100% Loading

Generator

Pg

(MW

)

1 2 30

50

100

150

200125% Loading

Generator

Pg

(MW

)

BOPFTSCOPF

Fig.2.37 Comparison of generator active power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 7 on Line 7-5

Fig.2.38 shows the comparison of the generator active power outputs obtained

by BOPF and the TSCOPF with constraints for the fault near Bus 9 on Line 8-9. It

is seen that to prevent the fault at near Bus 8 on Line 8-9, the active power outputs

of generators 2 needs to be increased in contrary to the case above to prevent a

fault near Bus 7 on Line 7-5. Also the active power output of generator 3 is

decreased for all loading conditions. The decrease in power output increases with

loading. Hence system loading also has an effect on the ‘redispatch’ required to

prevent a fault from causing instability for a given fault clearing time. Fig.2.39

shows the comparison of the generator active power outputs obtained by BOPF

and TSCOPF with constraints for the fault near Bus 8 on Line 7-8.

61

1 2 30

50

100

75% Loading

Generator

Pg

(M

W)

1 2 30

50

100

150100% Loading

Generator

Pg

(M

W)

1 2 30

100

200125% Loading

Generator

Pg

(MW

) BOPFTSCOPF


1 2 30

50

100

150

200125% Loading

Generator

Pg

(MW

)

BOPFTSCOPF

Fig.2.39 Comparison of generator active power outputs obtained by the TSCOPF with constraints for fault near Bus 8 on Line 7-8

Fig.2.40 shows the comparison of the generator reactive power outputs

obtained BOPF and TSCOPF with constraints for the fault near Bus 7 on Line 7-

62

5. Fig.2.41 shows the comparison of the generator reactive power outputs

obtained by BOPF and TSCOPF with constraints for the fault near Bus 9 on Line

8-9. Fig.2.42 shows the comparison of the generator reactive power outputs


7-8. Fig.2.43 shows the comparison of the bus voltages obtained by BOPF and

TSCOPF with constraints for the fault near Bus 7 on Line 5-7. Fig.2.44 shows the

comparison of the bus voltages obtained by BOPF and TSCOPF with constraints

for the fault near Bus 9 on Line 8-9. Fig.2.45 shows the comparison of the

generator reactive power outputs obtained BOPF and TSCOPF with constraints

for the fault near Bus 8 on Line 7-8.

1 2 3-10

-5

0

5

10100% Loading

Generator

Qg

(MV

AR

)

1 2 3-20

0

20

40

60125% Loading

Generator

Qg

(MV

AR

)

BOPFTSCOPF

Fig.2.40 Comparison of generator reactive power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 7 on Line 7-5

63

1 2 3-30

-20

-10

0

75% Loading

Generator

Qg

(MV

AR

)

1 2 3

-10

0

10

20100% Loading

Generator

Qg

(MV

AR

)

1 2 3

0

20

40

125% Loading

Generator

Qg

(MV

AR

)

BOPFTSCOPF


1 2 3-10

0

10

20

30

40

50125% Loading

Generator

Qg

(MV

AR

)

BOPFTSCOPF


64

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

100% Loading

Buses

Vm

(pu

)

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

125% Loading

Buses

Vm

(pu

)

BOPFTSCOPF

Fig.2.43 Comparison of bus voltage magnitudes obtained by BOPF and TSCOPF with constraints for fault near Bus 7 on Line 5-7

1 2 3 4 5 6 7 8 90

0.5

1

75% Loading

Buses

Vm

(pu

)

1 2 3 4 5 6 7 8 90

0.5

1

100% Loading

Buses

Vm

(pu

)

1 2 3 4 5 6 7 8 90

0.5

1

125% Loading

Buses

Vm

(pu

) BOPFTSCOPF


65

1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

125% Loading

Buses

Vm

(pu

)

BOPFTSCOPF


Fig.2.46 shows the comparison of the generator active power outputs obtained

by BOPF and TSCOPF with constraints for the fault near Bus 3 on Line 3-4.

Fig.2.47 shows the comparison of the generator active power outputs obtained by

BOPF and TSCOPF with constraints for the fault near Bus 10 on Line 10-13.

Fig.2.48 shows the comparison of the generator active power outputs obtained by

BOPF and TSCOPF with constraints for the fault near Bus 25 on Line 25-2.

66

1 2 3 4 5 6 7 8 9 100

500

1000

1500

125% Loading

Generator

Pg

(MW

)

BOPFTSCOPF


1 2 3 4 5 6 7 8 9 100

200

400

600

800

1000

1200100% Loading

Generator

Pg (M

W)

1 2 3 4 5 6 7 8 9 100

500

1000

1500

125% Loading

Generator

Pg (M

W)

BOPFTSCOPF


67

1 2 3 4 5 6 7 8 9 100

500

75% Loading

Generator

Pg

(MW

)

1 2 3 4 5 6 7 8 9 100

500

1000100% Loading

Generator

Pg

(MW

)

1 2 3 4 5 6 7 8 9 100

500

1000

1500125% Loading

Generator

Pg

(MW

) BOPFTSCOPF

Fig.2.48 Comparison of bus generator active power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 25 on Line 25-2

Fig.2.49 shows the comparison of the generator reactive power outputs


3-4. Fig.2.50 shows the comparison of the generator reactive power outputs


10-13. Fig.2.51 shows the comparison of the generator active power outputs


25-2.

68

1 2 3 4 5 6 7 8 9 100

100

200

300

400

100% Loading

Generator

Qg

(MV

AR

)

BOPFTSCOPF

Fig.2.49 Comparison of bus generator reactive power outputs obtained by BOPF and TSCOPF with constraints for fault near Bus 3 on Line 3-4

1 2 3 4 5 6 7 8 9 10-50

150

350

500100% Loading

Generator

Qg

(MV

AR

)

1 2 3 4 5 6 7 8 9 100

200

400

550125% Loading

Generator

Qg

(MV

AR

)

BOPF

TSCOPF


69

1 2 3 4 5 6 7 8 9 10

0

100

200

Generator

Qg

(MV

AR

)

75% Loading

1 2 3 4 5 6 7 8 9 10-200

0

200

400

Generator

Qg

(MV

AR

)

100% Loading

1 2 3 4 5 6 7 8 9 10-200

0

200

400

Generator

Qg

(MV

AR

)

125% Loading

BOPFTSCOPF


In each of the above cases it is seen that the variation of active generator

power outputs obtained from the transient stability constrained optimal power

flow is not very much different than those obtained from the base optimal power

flow. The variation in bus voltages on the other hand was very large. Hence, the

reactive power variation was also very large to maintain the voltages within the

prescribed limits. The large sensitivity of the reactive power to transient stability

constrained optimal power flows also concurs with the results in [18] where it was

concluded that the generator reactive power was an important input for estimating

the critical clearing time and the output was very sensitive to variations in the

reactive power output of each generator.

70

2.6 Conclusions

From the above results we can conclude that

1. With small adjustments of the state-variables in the original optimal power

flow it is possible to ensure the transient stability of the system for a particular

fault. Although the locational marginal prices would increase considering the

increased constraint set, since the variation in active power outputs is not very

large, the increase in cost would not be very large.

2. The functional evaluation of the transient stability of the system for the

particular fault under consideration and its use as inequality constraints in the

OPF allows us to ensure transient stability for faults at a particular fault

location.

3. Incorporating such equations for a set of faults at different locations can

ensure transient stability of the power system for faults occurring at those

locations. Hence a set of faults can be treated simultaneously.

4. It is a well known fact that the numerical integration of the differential

equations by methods such as the trapezoidal rule requires a very small time-

step to be accurate. Using the Taylor series expansion method to form a

functional constraint that allows us to use a much larger time step of

integration can prove to be more efficient.

71

5. Also instead of actually integrating the solution of the differential equations

into the optimal power flow by forming a functional constraint that restricts

the rotor angles at 1s considerably reduces the number of additional

constraints and state variables needed.

6. It has also been observed that using previous converged states as starting

points for obtaining new dispatches with additional constraints such as

transient stability constraints considerably reduces the total computation time

required.

72

73

CHAPTER THREE

APPLICATION OF SINGLE MACHINE EQUIVALENT METHOD AND

NEURAL NETWORKS FOR ESTIMATION OF CRITICAL CLEARING TIME

The analysis of system security from a dynamic point of view by running time

domain simulations seems useful only for offline studies and almost impossible to

use in a real-time operations environment where solutions are needed fast and

within reasonable tolerance as far as accuracy is concerned. This chapter aims at

analyzing a class of One Machine Infinite Bus (OMIB)-based methods, the Single

Machine Equivalent Method which offers a solution towards the goal mentioned

above. The method is employed to quickly generate a large training set of data

which is then utilized to estimate the critical clearing time for a fault with a single

switching action (i.e the faulted line is opened to clear the fault) using

feedforward neural networks which seem to offer promising intelligent system

solutions for real-time operations if trained well i.e. provide accurate estimates for

cases unknown during training, that are suitable for use in real-time operations.

3.1 Single Machine Equivalent Method

The single machine equivalent method belongs to those class of methods that

rely on building a OMIB equivalent of the system. Such OMIB methods are based

on the simple idea of replacing the machines in a system with two sets of

machines, one consisting of the critical machines and the other consisting of the

74

non-critical machines. These two sets of machines are then further reduced to a

OMIB system as is done to analyze the transient stability of a system for a two-

machine system using the Equal Area Criterion. Thus, the OMIB method provides

a way of transforming the large number of multidimensional multimachine

dynamical equations into a single dynamic equation. The OMIB-based methods

to carry out transient stability analysis are divided into “time-invariant”, “time-

varying” and “generalized ones”. A common feature of all OMIB-based methods

is that they rely on the classical and well-known equal area criterion method

which offers a ‘one-shot’ way to assess the transient stability of a system

following a fault without time domain simulations. Simplified power system

modeling and coherency of the machines within each one of the critical and non-

critical sets, so as to ‘freeze’ their relative motion in the fault-on and post-fault

periods, is one of the main properties of time-invariant OMIB-based methods.

[48-52] present a brief history of these methods. Time-varying methods relax

the coherency assumption, but stick to a simplified power system model. In

contrast to the time-varying methods, the generalized OMIB-based methods

consider detailed power system models. Here variation between the accelerating

power and the rotor angle of the OMIB system is no longer sinusoidal as is with

both the time-invariant and time-varying OMIB-based methods. Detailed theory

and derivations of expressions for these methods can be found in [53-54].

The single machine equivalent method is a method to assess the transient

stability and is based on a generalized OMIB method. It is a hybrid method

created by coupling time domain simulation with the direct analysis offered by the

75

equal-area criterion for the time-varying OMIB system developed at each step of

the time domain simulation for the respective decomposition of the machines into

the critical and non-critical sets. Obviously, the good performance of the single

machine equivalent method is based on the right decomposition of the machines

into the critical and non-critical sets. A number of ways are available to sort

machines so as to form the critical and non-critical sets. Rotor angles at fault

initiation, initial accelerations, proximity to fault etc can be used to form the

critical and non-critical sets.

3.1.2 Methodology

A time domain simulation program is run. As soon as the system enters the

post-fault state, the SIME method subroutine starts considering at each time step,

candidate decomposition patterns (i.e., splitting of the machines into the critical

clusters and non-critical clusters). For each of these decompositions, the One

Machine Infinite Bus (OMIB) parameters are computed according to the

expressions shown below. The above procedure is repeated till a particular

“candidate” OMIB reaches the unstable condition as defined later. This candidate

OMIB is declared to be the critical OMIB. For a given set of critical and non-

critical machines, the procedure of forming the OMIB is repeated here for a quick

review.

At each time step of the time domain simulation we have:

1. A set ‘S’ representing the cluster of critical machines [53].

2. A set ‘A’ represent the cluster of non-critical machines [53].

76

3. The OMIB parameters to be calculated are aem PPP M,, ,,, . Here

aP is the accelerating power given by Equation (3.1).

ema PPP (3.1)

Let the inertia constant for set ‘S be given by

Sk

kS MM

(3.2)

The inertia constant for the set ‘A’ is given by

Al

lA MM

(3.3)

Let S , and A denote the Center-Of-Inertia-Angle (COA) for the sets S and

A respectively. They are calculated as shown below

)(1 tMMSk

kksS

(3.4)

)(1 tMMAl

llAA

(3.5)

77

Let the corresponding rotor speeds for each set be given by:

)(1 tMMSk

kkss

(3.6)

)(1 tMMAl

llAA

(3.7)

The rotor angle and speed for the OMIB are then given by:

)()()( ttt AS (3.8)

)()()( ttt AS (3.9)

The OMIB mechanical power is given by

Sk AlmAmSm tPMtPMMtP

lk

)()()( 11 (3.10)

In Equation (3.10) the inertial constant M of the OMIB system is given by:

AS

AS

MM

MMM

* (3.11)

78

The OMIB electrical power is given by

Sk AleAeSm tPMtPMMtP

lk

)()()( 11 (3.12)

The time domain simulation is carried out until we reach the termination

situation as described below:

i) The unstable angle u is found at the crossing of eP and mP curves

during the post-fault stage for an unstable scenario

ii) The stable (or return) angle r , (r for return) represents the maximum

angular excursion of eP during the post-fault stage for a stable

scenario

iii) ru when a maximum angular excursion is reached at the

crossing of eP and mP during the post-fault stage

A trajectory would be classified as stable if eP returns back before

crossing mP . If r is the return angle ( )ur at time rt with

0rt and 0ra tP (3.13)

79

Then, these conditions classify the trajectory to be stable and the time domain

simulation can be terminated. Fig.3.1 illustrates a stable scenario.

10 20 30 40 50 60 70 80 90 1000

50

100

150

200

250

OMIB Angle (degrees)

OM

IB P

m &

Pe

(MW

)

Pm

Pe

r(Point of Return)

Fig.3.1 An example of a stable OMIB trajectory

The conditions of an unstable trajectory are that it reaches the unstable angle

u at a time ut when the accelerating power aP becomes zero and the rate of

change of the accelerating power is positive, i.e.,

0dt

dPtP ,0)(tP

utt

auaua

with 0 for ott (3.14)

80

The above conditions determine the early conditions to terminate the time

domain simulation. These conditions are also used to identify the critical OMIB.

Fig.3.2 illustrates an unstable scenario.

0 20 40 60 80 100 120 140 160 180

100

150

200

250


OM

IB P

m &

Pe

Pm

Pe

u

Fig.3.2 An example of an unstable OMIB trajectory

Another property of a stable OMIB trajectory is the phase-plane plot of the

OMIB speed vs. the OMIB angle. A stable OMIB trajectory always has a

continuously revolving phase-plane plot while an unstable OMIB trajectory

would have a divergent phase-plane plot that showing no decrease in the OMIB

speed as shown below in Fig.3.3

81

0 50 100

-50

0

50


OM

IB S

peed

(0.

1 ra

d/s)

Stable Case

0 10000

100

200

300

400


OM

IB S

peed

(0.

1 ra

d/s)

Unstable Case

Fig.3.3 Phase plane plot of OMIB speed vs. the OMIB angle


This section presents illustrations of the above methodology on various fault

cases in the 9-Bus system. For each case, the set ‘S’ consisting of the critical

machines and the set ‘A’ consisting of the non-critical machines at the terminating

time step are also listed. It has been assumed that no reclosing takes place and the

fault is a permanent one that is cleared by opening the faulted line.

82

3.2.1 Case 1: Fault near Bus 5 on Line 5-7 for 0.28s (S: [2, 3], A: [1])

In this case, a fault near Bus 5 on Line 8 is considered. It is assumed to be

cleared in 0.28s by opening the line. Fig.3.4 shows the OMIB mechanical input,

electrical power output, accelerating power and angle and speed trajectory. From

the conditions in Equation (3.13) we can conclude that the system is stable. The

phase plane plot is shown in Fig.3.5. The generator rotor plots in Fig.3.6 confirm

it as a stable scenario.

-100 -50 0 50 100

100

150

200


OM

IB P

m &

Pe

(MW

)

-100 -50 0 50 100-0.8

-0.6

-0.4

-0.2

0

OMIB Angle (degrees), OMIB Speed (0.1 rad/s), OMIB Pa(MW)

Tim

e (s

)

SpeedPaAngle

PePm

Fig.3.4 Plot of OMIB mechanical input, electrical power output, accelerating power and angle and speed trajectory for Case 1

83

-20 0 20 40 60 80 100

-60

-40

-20

0

20

40

60


OM

IB S

peed

(0.

1 ra

d/s)

Fig.3.5 OMIB speed vs. the OMIB angle for Case 1

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

400

500

600

700

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.3.6 Plot of generator rotor angles for Case 1

84


In this case the same fault is considered as in case 1 but with an increased

fault clearing time of 0.33s. Fig.3.7 shows the OMIB mechanical input, electrical

power output, accelerating power and angle and speed trajectory. From the

conditions in Equation (3.14) we can conclude that the system is unstable. The

phase plane plot is shown in Fig.3.8. The generator rotor plots in Fig.3.9 confirm

it as an unstable scenario.

-50 0 50 100 150

100

150

200


OM

IB P

m &

Pe

(MW

)

-50 0 50 100 150-1

-0.5

0


Tim

e (s

)

PePm

SpeedPaAngle


85

0 200 400 600 800 1000 1200 1400 16000

100

200

300

400

500


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

Time (s)

Rot

or A

ngle

s (d

egre

es)


86


This case illustrates another stable case with the fault on the same line, but

near Bus 7 this time. Fig.3.10 shows the OMIB mechanical input, electrical power

output, accelerating power and angle and speed trajectory. The phase plane plot is

shown in Fig.3.11. The generator rotor plots in Fig.3.12 confirm it as a stable

scenario.

-40 -20 0 20 40 60 800

50

100

150

200


OM

IB P

m &

Pe

(MW

)

-40 -20 0 20 40 60 80

0.4

0.2

0


Tim

e (s

)

PePm

SpeedPaAngle


87

0 10 20 30 40 50 60 70 80-40

-30

-20

-10

0

10

20

30

40


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

300

Time (s)

Rot

or A

ngle

s (d

egre

es)


88


This case illustrates the same fault as above with an increased fault clearing

time of 0.17s which renders the system unstable. Fig.3.13 shows the OMIB

mechanical input, electrical power output, accelerating power and angle and speed

trajectory. The phase plane plot is shown in Fig.3.14. The generator rotor plots in

Fig.3.15 confirm it as an unstable scenario.

-50 0 50 100 1500

50

100

150

200


OM

IB P

m &

Pe

(MW

)

-50 0 50 100 150

0.6

0.4

0.2


Tim

e (s

)

PePm

SpeedPaAngle


89

0 200 400 600 800 1000 1200 1400 1600 18000

100

200

300

400

500


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

Time (s)

Rot

or A

ngle

s (d

egre

es)


90


This case illustrates the methodology for a fault near Bus 6 on Line 6-9 for

0.17s. Fig.3.16 shows the OMIB mechanical input, electrical power output,

accelerating power and angle and speed trajectory. The phase plane plot is shown

in Fig.3.17. The generator rotor plots in Fig.3.18 confirm it as a stable scenario.

-40 -20 0 20 40 60

100

150

200


OM

IB P

m &

Pe

(MW

)

-40 -20 0 20 40 60

-0.6

-0.4

-0.2

0


Tim

e (s

)

PePm

SpeedPaAngle


91

0 10 20 30 40 50-40

-30

-20

-10

0

10

20

30

40


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

50

100

150

200

250

300

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.3.18 Plot of generator rotor angles for the Case 5

92







-50 0 50 100 150

100

150

200

250


OM

IB P

m &

Pe

(MW

)

-50 0 50 100 150-1

-0.5

0


Tim

e (s

)

PePm

SpeedPaAngle


93

0 500 1000 15000

50

100

150

200

250

300

350

400


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

500

1000

1500

2000

Time (s)

Rot

or A

ngle

s (d

egre

es)


94


3.3.1 Case 7: Fault near Bus 3 on Line 3-4 for 0.18s (S: [3, 5, 6, 7, 8, 9], A: [1, 2,

4, 10])

This case illustrates the methodology for a fault near Bus 3 on Line 3-4 for

0.18s. Fig.3.22 shows the OMIB mechanical input, electrical power output,

accelerating power and angle and speed trajectory. The phase plane plot is shown

in Fig.3.23. The generator rotor plots in Fig.3.24 confirm it as a stable scenario.

-20 0 20 40 60 801000

2000

3000

4000


OM

IB P

m &

Pe

(MW

)

-20 0 20 40 60 80-0.8

-0.6

-0.4

-0.2

0


Tim

e (s

)

PePm

SpeedPaAngle


95

-10 0 10 20 30 40 50 60 70 80-30

-20

-10

0

10

20

30


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0

50

100

150

200

250

300

350

Time (s)

Rot

or A

ngle

s (d

egre

es)


96

3.3.2 Case 8: Fault near Bus 3 on Line 3-5 for 0.23s (S: [3, 5, 6, 7, 8, 9], A: [1, 2,

4, 10])






-20 0 20 40 60 80 100 120 140 1601000

2000

3000

4000


OM

IB P

m &

Pe

(MW

)

-20 0 20 40 60 80 100 120 140 160-1

-0.5

0


Tim

e (s

)

PePm

SpeedPaAngle


97

0 100 200 300 400 500 6000

50

100

150


OM

IB S

peed

(0.

1 ra

d/s)


0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

200

400

600

800

1000

1200

Time (s)

Rot

or A

ngle

s (d

egre

es)

Fig.3.27 Plot of generator rotor angles for the Case 8

98

3.4 Discussion

It can be seen from the above results that the single machine equivalent

method provides a faithful picture of the multimachine stability phenomenon. The

consistent results for the 9-Bus and 39-Bus system show that the method is

applicable to both small-scale and large-scale systems and is also independent of

the number of the machines in the system. One of the major advantages of

performing the SIME analysis at each step of the time domain simulation is that

the time domain simulation can be stopped once the OMIB trajectory exhibits the

termination criterion for a stable or unstable scenario.

3.5 Fast Determination of Critical Clearing Time Using SIME Method

3.5.1 Background

The SIME Method can be used for fast determination of critical clearing time

by plotting the stable/unstable margins for a set of clearing times. These points are

then fitted by a polynomial. The roots of the polynomial are found to obtain the

critical clearing time (The margin at critical clearing time is 0). As described

earlier by Equation (3.14), the conditions of an unstable trajectory are that it

reaches the unstable angle u at a time ut when the accelerating power aP

becomes zero and the rate of change of the accelerating power is positive. These

conditions determine the early conditions to terminate the time domain

simulation. These conditions are also used to identify the critical OMIB. Now

aPM (3.15)

99

Multiplying both sides of Equation (3.15) by we obtain

aPM (3.16)

Also 00 . Integrating Equation (3.16) we obtain,

u

dPM au

0

2

2

1 (3.17)

The general stability margin is obtained as

ch u

ch

dPdPAA aaaccdec

0

(3.18)

In Equation (3.15) ch is the angle at which aP changes sign from positive

to negative. Using the Equations (3.14) and (3.15), we obtain the unstable margin

as:

2

2

1uu M (3.19)

100

As described earlier, a trajectory would be classified as stable if eP returns

back before crossing mP . If r is the return angle ( )ur at time rt with

0rt and 0tP ra , then these conditions determine classify the trajectory

to be stable and the time domain simulation can be terminated. Observing that

0 at 0t and rt , we get the stable margin as,

ruaraast PdPdPu

r

u

r

2

1 (3.20)

u is obtained by extrapolating the aP curve as a function of by using

values of aP taken at three successive time steps near the ‘point of return’.

3.5.2 Methodology

The above described procedure is used to calculate the stable/unstable margins

for four different clearing times. These are then fitted using polynomial

interpolation functions. The roots of the polynomial are then found one of which

would be the critical clearing time as the margin at the critical clearing time is 0.

3.5.3 Results with the IEEE 9-Bus System

This section shows some results of the above methodology to quickly obtain

the critical clearing time using a limited number of Single Machine Equivalent

Method simulations which are terminated as soon as a stable or unstable condition

101

is detected. In each of the results shown below, the value of the margin is

calculated for four different fault clearing times. These are then fitted by a

polynomial using MATLAB’s polynomial fitting function. A cubic polynomial

was found to give the best polynomial to obtain the critical clearing time. Fig.3.28

shows the results for a fault near Bus 7 on Line 7-8. In a similar manner SIME

simulations carried out for other fault locations for various fault clearing times

and the critical clearing times were estimated as above. The cases studied have

been listed in Table 3.1. The critical clearing times from time-domain simulations

for the respective cases obtained using time domain simulations have also been

listed

0.2 0.22 0.24 0.26 0.28 0.3-200

-150

-100

-50

0

50

100

Clearing Time

Stab

ility

Mar

gin

CCT

Fig.3.28 Plot of stable/unstable margins for a fault near Bus 7 on Line 7-8

102

0.23 0.24 0.25 0.26 0.27 0.28-100

-80

-60

-40

-20

0

20

40

60

Clearing Time (s)

Stab

ility

Mar

gin

CCT

Fig.3.29 Plot of stable/unstable margins for a fault near Bus 7 on Line 7-5


Case Fault Near

Bus

Fault on

Line

Actual critical

clearing time

Critical clearing time

from TD

1 7 7-8 0.252 0.253

2 7 5-7 0.248 0.249

3 9 6-9 0.242 0.242

4 5 5-7 0.288 0.289

5 6 4-6 0.306 0.306

103

3.5.4 Discussion

From the above results we can see that the error between the estimated critical

clearing time and the actual critical clearing time obtained by time domain

simulations is of the order of 0.001 and negligible as far as real-time operations

are concerned. Thus the single machine equivalent method provides a

computationally efficient technique to obtain the critical clearing time for a fault.

This technique is not only fast as compared to running numerous time domain

simulations to narrow down on the critical clearing time but is also very accurate.

The single machine equivalent method allows further saving in time as it is not

required to run the entire time domain simulation. The simulations are terminated

as soon as a stable or an unstable condition is reached. In this manner the critical

clearing time for a set of cases at various loading levels and for various topology

situations can be found using the SIME method to train a neural network.

3.6 Application of Feedforward Neural Networks for critical clearing time

estimation

3.6.1 Background

Artificial neural networks have been used for dynamic security assessment.

One method used several indices as input to the artificial neural network to

predict system instability (stable/unstable) [55]. Other methods use several key

features as input to the ANN to estimate maximum generator swing angles [56],

and the Transient Energy Margin that are used for contingencies screening and

ranking [57]. Other methods utilize the artificial neural network in transient

104

stability assessment to estimate the critical clearing time, or the stability index

(stable/unstable) for a given clearing time, of a multimachine power system [58].

A. D. Angel et al focused on the development of a technique for the estimation of

generator rotor angle and speed, based on phasor measurement units for transient

stability assessment and control in real-time in [59]. M. Moghavvemi examined

the performance of two nonlinear multilayered ANN models for the estimation of

a stability index to gauge the stability of a power system network in [60].

Dynamic security margin assessment for a both voltage and dynamic stability

using artificial neural networks were explored by A. Sittithumwat in [61].

Eigenvalue prediction of critical stability modes of power systems based on

neural networks was dealth with in [62] to predict the stability condition of the

power system with high accuracy. Pattern recognition techniques using artificial

neural networks and linear classification were used in [63] for the classification of

swing curves generated in a power system transient stability study. Artificial

neural networks have been used for a wide variety of other applications in power

systems. [64] presents a method for topological observability based on multilayer

perceptrons using the back-propogation algorithm for training. Y. Y. Hsu et al

evaluated the dynamic performance of power system by computation of the

dominant eigenvalues for the worst damped electromechanical mode using neural

networks in [65]. A load forecasting technique has been suggested in [66] that

utilizes a swarm optimization algorithm. B. Thukaram et al have dealt with

providing solutions for monitoring and control of voltage stability in the day-to-

day operations of power systems in [67]. An intelligent load shedding scheme

105

using artificial neural networks was developed by D. Nosovel et al in [68]. Some

other applications have been explored in [69-70].

3.6.2 Methodology

A feed-forward artificial neural network was developed using the MATLAB

Neural Network toolbox to predict the critical clearing time for various other

loading conditions and topology situations. The inputs to the neural network were:

1. Mechanical input of each generator:imP

2. Initial rotor angle for each generator with respect to the center of inertia:

n

ii

n

iii

icoiM

M

i

1

1

(3.21)

Here iM is the inertial constant and i is the rotor angle of the generator

3. Initial acceleration of each generator:

i

tem

i M

PPii

0

(3.22)

106

4. Initial acceleration energies of each generator:

i

tem

M

PPii

i

2

2

0

(3.23)

5. Post-fault driving point susceptance of each generator i.e., iiB

6. Reactive power output of each generator i.e., igQ

The output of the ANN is a single quantity i.e., the critical clearing time. Two

layers of hidden neurons were used. The thumb rule of having 2n+1 hidden

neurons was followed here. Since, the 9-Bus, 3-machine system is being used

here, there were a total of 5*3=15 inputs and 31 neurons in each hidden layer.

The log sigmoid transfer function was used and the TRAINLM learning method

was used for training the weights of the neural method. Although the TRAINLM

method requires large memory, it is the fastest weight training method as

compared to the other training algorithms in the MATLAB Neural Network

Toolbox. Fig.3.30 gives a diagrammatic representation of the neural network. The

maximum number of epochs to train was set to 50,000 while the minimum

performance gradient was set to 10-6. Also the learning rate was set to 0.02 while

the performance goal was set to 10-6. A large training set is to be generated to

consider varying generator active power outputs and loads. The initial estimations

of the critical clearing time for various faults showed significant errors even

107

though a large training set covering the entire range of possible loading conditions

and generator active power outputs was considered. Instead of having a single

neural network for the whole system, it was observed that having ‘localized’

neural networks for each fault location i.e., a neural network trained for a

particular fault location worked much better in estimating the critical clearing

time. This is expected because the training set for the ‘localized’ neural networks

would be concentrated on the particular fault location and not be generalized due

to inputs for other fault locations.

Fig.3.30 Proposed Feedforward Neural Network for Critical Clearing Time Estimation

Input Layer

Hidden Layer

CCT

OutputLayer

.

.

.

.

.

.

.

.

.

.

1g

11

21

1

1

m

Q

B

γ

γ

δ

P1

g

gg

g

g

g

gn

n

nn

2n

n

n

m

Q

B

γ

γ

δ

P

Input Layer

Hidden Layer

CCT

OutputLayer

.

.

.

.

.

.

.

.

.

.

1g

11

21

1

1

m

Q

B

γ

γ

δ

P1

g

gg

g

g

g

gn

n

nn

2n

n

n

m

Q

B

γ

γ

δ

P

108

3.6.3 Results with the IEEE 9-Bus system

Four different fault locations have been considered. For each fault, a wide

variety of loading conditions was used. To test the robustness and applicability of

the trained neural networks, different type of fault locations considered i.e. close

to a generator or a load bus etc. The optimal power flow varies the output of each

generator to minimize the net operating cost and also respect the security

constraints. Hence, the training inputs were also varied by having a wide range of

generator active power outputs for each loading condition. As expected a large

training set is generated for each fault location. This is to ensure the best possible

mapping from any given operating state to the critical clearing time, given the

fault location. A performance goal of 10-6 for the testing set was used to

terminate the training. This made sure that the maximum error for testing cases

would be less than 1 cycle time. Training was carried out for four different fault

locations in the 9-Bus system. Hence, four different neural networks were

developed and stored for use later in the optimization process. A comparison of

the critical clearing time estimates obtained using the trained neural networks

(ANN) and the actual critical clearing times obtained by time domain simulations

(TD) for various testing cases are shown in Figs.3.31-3.34. The testing cases were

not known to the neural networks at the time of training.

109

5 10 15 20 25 30 35 40 450.25

0.3

0.35

0.4

0.45

0.5

Testing Cases

Cri

tica

l Cle

arin

g T

ime

(s)

ANNTD

Fig.3.31 Comparison of critical clearing time for a fault near Bus 7 on Line 7-5

5 10 15 20 25 30 35 40 450.2

0.3

0.4

0.5

0.6

0.7

Testing Cases

Cri

tica

l Cle

arin

g T

ime

(s)

ANNTD


110

5 10 15 20 25 30 35 40 450.2

0.3

0.4

0.5

0.6

0.7

Testing Cases

Cri

tica

l Cle

arin

g T

ime

(s)

ANNTD


5 10 15 20 25 30 35 40 450.2

0.3

0.4

0.5

0.6

0.7

Testing Cases

Cri

tica

l Cle

arin

g T

ime

(s)

ANNTD


111

3.6.4 Discussion

It is observed that the critical clearing estimated by using neural networks is

very close and in many cases almost exactly equal to the critical clearing times

obtained by time domain simulations. Also, the time taken by the neural network

to estimate the critical clearing time is negligible. It is extremely fast and very

accurate if trained properly. In each of the above testing cases, the maximum error

was less than one cycle time (0.0167s).

From the above results we can conclude that the artificial neural network

provides a promising computationally efficient and reasonably accurate method to

estimate the critical clearing time for a given input to the neural net.

3.7 Conclusions

1. The idea of converting the multidimensional dynamic equations into a single

machine dynamic equation by decomposition of the machines into critical and

non-critical sets and then further reducing the two sets to a OMIB system is

supportive to elimination of the vast number of calculations required to solve

the dynamic equations in a multimachine system.

2. In a real-time operations scenario, qualitative information would be more

important rather than quantitative information i.e., it would be more vital to

know if the system is stable or not rather than knowing the values of the rotor

angles or other state variables. Provided the critical and non-critical clusters

are formed correctly, the SIME method is not only accurate and fast, but the

112

consistency shown as evident in the results makes it suitable for real-time

dynamic security assessment. The rotor angle plots obtained from running the

entire time domain simulation showed that the results of the single machine

equivalent method were consistent in all cases and we could accurately know

if a particular fault would cause system transient instability or not.

3. Another great qualitative advantage of the single machine equivalent method

is that it provides knowledge of ‘how far’ the system is from instability or

‘how unstable’ is the system. This was shown in the second part of this

chapter. These margins can be utilized to quickly estimate the critical clearing

time. Conventionally this has been done by running various time domain

simulations, till we ‘lock down’ on the critical clearing time. The single

machine equivalent method avoids this need by providing stable/unstable

margins which are then used along with polynomial interpolation to obtain the

critical clearing time with very high accuracy provided the sample fault

clearing times used to estimate the critical clearing time are in close proximity

to the actual critical clearing time. Operator experience would be a big factor

in determining what these sample fault clearing times are. Hence, the single

machine equivalent method can be used to quickly estimate the critical

clearing time.

4. The above was the main incentive for using the SIME method to generate the

training set data to be fed to the feedforward neural networks that would allow

113

a further saving in computation time as far as estimation of critical clearing

time is concerned. With proper training the feedforward neural networks can

be used for estimation of the critical clearing time. It was seen that with

proper training, the neural network can be used to approximately estimate the

critical clearing time.

5. The application of artificial intelligence to establish the complex mapping

between an operating condition to the critical clearing time for a particular

fault leads to a vast saving in computation time and completely eliminates the

need for any time domain simulations needed to assess the transient stability.

6. The accuracy of the artificial neural network output can be improved by

having different neural nets for different fault locations and it is expected that

the estimate can be made even more accurate by further considering different

sets of loading conditions. This would avoid the need for extrapolation by the

neural network and provide better results.

114

115

CHAPTER FOUR

A HYBRID NEURAL NETWORK-OPTIMIZATION APPROACH FOR

DYNAMIC SECURITY CONSTRAINED OPTIMAL POWER FLOW

The chapter presents the formulation of a hybrid neural network-optimization

scheme to carry out transient stability constrained optimal power flow. The

transient stability constraint ensures that the critical clearing time for the

particular three-phase fault under consideration will be more than the fault

clearing time for the respective fault. The neural network developed in chapter 3

has been used here for the estimation of the critical clearing time, in each iteration

of the optimization process. The results show that accurate estimation of the

critical clearing time using intelligent techniques and their incorporation into

conventional optimization schemes can allow the inclusion of dynamic security

constraints for real-time operations.

4.1 Transient Stability Constraint

In addition to the conventional constraints listed out in Chapter 2, an

additional inequality constraint is needed to ensure the transient stability of the

power system with regards to the respective fault under consideration and its fault

clearing time. The estimated critical clearing time given by the neural networks

can be greater than or less than the actual clearing time. Also in practice, the fault

116

clearing time is kept at a distance of at least one cycle time from the critical

clearing time. Hence, a safety factor of one cycle has been considered, i.e., the

constraint ensures that the fault clearing time (FCT) will be at a distance of atleast

one cycle from the critical clearing time (CCT). This factor depends upon the

accuracy of the estimate of the trained neural networks for the testing cases. In

chapter 3, it was seen that the error for the testing cases was much lower than one

cycle time. Also, it is a small value that will burden the minimization process in a

significant manner. At the same time, using it shall ensure that even if the ANN is

estimate is off by a factor less than one cycle time, the safety factor shall ensure

that the critical clearing time from the time-domain simulation is always greater

than the fault clearing time. So the constraint is as given below

0)0167.0( CCTFCT (4.1)

As mentioned earlier, the CCT is obtained by taking into account the

operating state corresponding to the current iteration of the optimization approach

and the fault under consideration. The operating state refers to the set of values of

the state variables. This information is input to a function that calculates the

required inputs for the neural network corresponding to the fault under

consideration. The neural network then returns the critical clearing time

corresponding to the current operating state and the fault under consideration.

Fig.4.1 illustrates conceptually the hybrid neural network-optimization approach

to carry out a transient stability-constrained optimal power flow problem.

117

Fig.4.1 Conceptual illustration of the transient stability constrained optimal power flow formulation

4.2 Results with the IEEE 9-Bus System (Single-Contingency)

This section presents results for the optimization process with the dynamic

constraint included for a single contingency using the 9-Bus system. Five

different fault locations have been considered and the optimization is carried out

with the transient stability constraint for each location taking one at a time. For

each fault location, the optimization is carried out for various loading conditions.

To study the performance of the optimization process under stressed and non-

stressed conditions, the fault clearing time is varied for each loading condition.

For each case the critical clearing time that was estimated by the neural network

for the last iteration of the optimization process has been noted. Also the critical

Evaluation of physical/operating

constraints

Evaluation of Static-security

constraints

NeuralNetwork

FCT – CCT 0

CCT

Current Operating

State

Transient stability constrained optimal power flow

Evaluation of physical/operating

constraints

Evaluation of Static-security

constraints

NeuralNetwork

FCT – CCT 0FCT – CCT 0

CCT

Current Operating

State

Transient stability constrained optimal power flow

118

clearing times from time domain simulations for the corresponding operating

states, obtained after the optimization has converged, have also been listed. Table

5.1 lists the results obtained for including the constraint for a fault near Bus 7 on

Line 5-7. Table 4.2 lists the results obtained for including the constraint for a fault

near Bus 6 on Line 6-9. Table 4.3 lists the results obtained for including the

constraint for a fault near Bus 5 on Line 5-4. Table 4.4 lists the results obtained

for including the constraint for a fault near Bus 5 on Line 5-7. Table 4.5 lists the

results obtained for including the transient stability constraint for a fault near Bus

6 on Line 6-4.

Table.4.1 TSCOPF results with constraint for fault near Bus 7 on Line 5-7

Case Loading FCT ANN TD% CCT CCT

1 120 0.3 0.3167 0.32532 110 0.3 0.3167 0.31743 100 0.3 0.3167 0.31594 90 0.3 0.3167 0.31885 80 0.3 0.3167 0.32556 70 0.3 0.3314 0.34297 120 0.2 0.2167 0.21588 100 0.2 0.2167 0.21479 90 0.2 0.261 0.2688

10 80 0.2 0.2938 0.303811 70 0.2 0.3314 0.343

119



12 120 0.5 0.5167 0.519413 110 0.5 0.5167 0.52114 100 0.5 0.5167 0.521315 90 0.5 0.6409 0.649116 120 0.4 0.4167 0.421717 110 0.4 0.4212 0.427518 100 0.4 0.5094 0.51419 90 0.4 0.6409 0.649120 120 0.3 0.3643 0.364821 110 0.3 0.4212 0.427522 100 0.3 0.5093 0.513923 90 0.3 0.6409 0.6492



24 120 0.6 0.6167 0.618625 110 0.6 0.6167 0.621126 100 0.6 0.6167 0.619427 90 0.6 0.6322 0.640328 120 0.5 0.5167 0.516729 110 0.5 0.5167 0.520930 100 0.5 0.5434 0.545831 90 0.5 0.6322 0.640332 120 0.4 0.4294 0.429533 110 0.4 0.4751 0.479734 100 0.4 0.5434 0.545835 90 0.4 0.6322 0.6403

120



36 120 0.6 0.6167 0.613837 110 0.6 0.6167 0.615938 90 0.6 0.6167 0.615639 120 0.5 0.5167 0.513240 110 0.5 0.5167 0.520241 100 0.5 0.5167 0.518842 90 0.5 0.5754 0.574843 120 0.4 0.4725 0.475744 110 0.4 0.5754 0.574945 120 0.3 0.4756 0.472446 90 0.3 0.5755 0.575

4.3 Discussion

It was seen in chapter 3 earlier that the critical clearing estimated by using

neural networks is very close and in many cases almost exactly equal to the

critical clearing times obtained by time domain simulations. Also, the time taken

by the neural network to estimate the critical clearing time is negligible. It is

extremely fast and was therefore, the main incentive in the hybrid neural network-

optimization approach proposed in this chapter. When used along with the

optimization process to carry out the optimal power flow, it is seen in all cases

that the reformulated transient stability constrained optimal power flow was able

to converge to find a solution even under highly stressed conditions

corresponding to very large loading or long fault clearing times. The estimated

critical clearing time can be more than or less than the actual critical clearing

time. Hence it is possible that in the final iteration of optimization, the estimated

critical clearing time is equal to the fault clearing time thereby respecting the

121

constraint, but the critical clearing time from time domain simulations actually

being less than the estimated value. The use of the safety factor of one cycle time

eliminates any such situation. The value of the safety factor is obviously

dependent on the accuracy of the neural network also. As mentioned earlier, the

neural networks were trained well enough to keep the errors for the testing sets to

much below one cycle time. Fig. 4.2 shows the difference between the critical

clearing time from time domain simulations and the fault clearing time (TD CCT

– FCT, blue solid line) and the difference between the critical clearing time from

time domain simulations and the neural network estimate (TD CCT – ANN CCT,

black dotted line) for the various cases listed in Section 4.2. Two key observations

are to be noted. Firstly, the transient stability constrained optimal power flow was

able to converge even under stressed conditions keeping the critical clearing time

from time domain simulations at a distance of atleast once cycle time from the

fault clearing time (SF limit). In some cases this difference was a little less than

one cycle. These are the cases where the neural network estimate is more than the

actual critical clearing time. But the error, as seen in Fig. 4.2 is negligible. So the

neural network was able to accurately estimate the critical clearing time even for

cases that could not have been generated otherwise i.e., without the optimization

process. Fig. 4.3 shows a comparison of the generator active power outputs

obtained by carrying out the optimization without the transient stability constraint

at 120% loading (Base OPF) and those obtained for case 24. It is seen that with

little variation in the active power outputs of the generators, it is also possible to

ensure transient stability for a given fault.

122

5 10 15 20 25 30 35 40

0.0167

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Cases

Tim

e (s

)

TD CCT - FCTTD CCT - ANN CCT

SFLimit

Fig.4.2 Comparison of the critical clearing time with the fault clearing time and the neural network estimate

1 2 30

20

40

60

80

100

120

140

160

Generators

Pg

(MW

)

BOPFTSCOPF

Fig.4.3. Comparison of generator active power outputs

123

4.4 Results with the IEEE 9-Bus System (Multiple-Contingencies)

In a practical system it would be necessary to consider not one but multiple

contingencies with varying fault clearing times. The proposed method makes this

possible in a very simple manner without any additional comparable

computational burden on the already existing optimization process. This is made

possible by including constraints of the type shown in Equation 4.1 for each

contingency to be considered. The appropriate neural network is used for each

fault location. The application of the method is first shown considering two

contingencies at a time. The first of these scenarios considers a fault near Bus 7

on Line 7-5 and near Bus 6 on Line 6-9. Table 4.6 lists the results for various

stressed conditions, either due to loading or high FCTs. The second scenario

considers a fault near Bus 5 on Line 5-7 and near Bus 6 on Line 6-9. A third one

considers a fault near Bus 5 on Line 5-7 and Bus 6 on Line 6-4. Constraints for

three different fault locations with varying fault clearing times and loading

conditions were studied in the final scenario.

124

Table.4.5 TSCOPF results with constraint for faults near Bus 7 on Line 5-7 and near Bus 6 on Line 6-9

Case Loading Bus Near Line FCT ANN TD% Fault CCT CCT

47 120 7 8 0.15 0.2168 0.21926 3 0.5 0.5167 0.5194

48 110 7 8 0.15 0.2269 0.22876 3 0.5 0.5167 0.521

49 100 7 8 0.15 0.2347 0.23836 3 0.5 0.5167 0.5213

50 120 7 8 0.25 0.2667 0.27736 3 0.5 0.6041 0.6017

51 110 7 8 0.25 0.2667 0.26796 3 0.5 0.593 0.5941

52 100 7 8 0.25 0.2667 0.27036 3 0.5 0.6126 0.616

53 120 7 8 0.25 0.2667 0.26976 3 0.6 0.6453 0.6451

54 110 7 8 0.25 0.2667 0.26836 3 0.6 0.6167 0.6187

55 100 7 8 0.25 0.2667 0.27186 3 0.6 0.6607 0.6713

125

47 48 49 50 51 52 53 54 550

0.0167

0.04

0.06

0.08

0.1

Tim

e (s

)

Cases


SFLimit

Fig.4.4 Comparison of the critical clearing time with the fault clearing time and the neural network estimate for fault near Bus 7 on Line 5-7

47 48 49 50 51 52 53 54 550

0.0167

0.05

0.1

Tim

e (s

)

Cases


SFLimit


126



56 120 7 8 0.2 0.2167 0.21565 9 0.5 0.5167 0.5172

57 110 7 8 0.2 0.224 0.22285 9 0.5 0.5167 0.5209

58 100 7 8 0.2 0.2167 0.2285 9 0.5 0.5252 0.5257

59 120 7 8 0.2 0.2454 0.24555 9 0.6 0.6167 0.6186

60 110 7 8 0.2 0.2516 0.25275 9 0.6 0.6167 0.6211

61 100 7 8 0.2 0.2532 0.25815 9 0.6 0.6167 0.6194

62 120 7 8 0.25 0.2667 0.26975 9 0.6 0.7127 0.7159

63 110 7 8 0.25 0.2667 0.26795 9 0.6 0.6617 0.6648

64 100 7 8 0.25 0.2667 0.27035 9 0.6 0.6509 0.654

127

56 57 58 59 60 61 62 63 64

0

0.010.0167

0.03

0.04

0.05

0.06T

ime

(s)

Cases


SFLimit


56 57 58 59 60 61 62 63 640

0.0167

0.04

0.06

0.08

0.1

0.12

Tim

e (s

)

Cases


SFLimit


128



65 120 5 8 0.3 0.3953 0.39326 3 0.4 0.4167 0.4217

66 110 5 8 0.3 0.4054 0.40386 3 0.4 0.4212 0.4275

67 100 5 8 0.3 0.4724 0.47566 3 0.4 0.5094 0.514

68 120 5 8 0.4 0.4167 0.41656 3 0.4 0.4287 0.4338

69 110 5 8 0.4 0.4167 0.41586 3 0.4 0.4336 0.4397

70 100 5 8 0.4 0.4725 0.47576 3 0.4 0.5094 0.5141

71 120 5 8 0.5 0.5167 0.51326 3 0.5 0.5316 0.533

72 110 5 8 0.5 0.5167 0.52026 3 0.5 0.5513 0.5552

73 100 5 8 0.5 0.5167 0.51886 3 0.5 0.5611 0.5655

129

65 66 67 68 69 70 71 72 730

0.0167

0.05

0.1

0.15T

ime

(s)

Cases


SFLimit


65 66 67 68 69 70 71 72 730

0.0167

0.04

0.06

0.08

0.1

0.12

Tim

e (s

)

Cases


SFLimit


130



74 120 5 8 0.3 0.3481 0.34666 2 0.4 0.4257 0.4274

75 110 5 8 0.3 0.4054 0.40386 2 0.4 0.4839 0.4856

76 100 5 8 0.3 0.4724 0.47566 2 0.4 0.5646 0.5664

77 120 5 8 0.5 0.5167 0.51326 2 0.5 0.5943 0.5919

78 110 5 8 0.5 0.5167 0.52026 2 0.5 0.609 0.6094

79 100 5 8 0.5 0.5167 0.51886 2 0.5 0.6136 0.6161

80 120 5 8 0.5 0.5167 0.51386 2 0.6 0.6167 0.6153

81 110 5 8 0.5 0.5188 0.52196 2 0.6 0.6167 0.6175

82 100 5 8 0.5 0.5167 0.51856 2 0.6 0.6167 0.6193

131

1 2 3 4 5 6 7 8 90

0.0167

0.05

0.1

0.15

Tim

e (s

)

Cases


SFLimit


75 76 77 78 79 80 81 82 830

0.0167

0.05

0.1

0.15

Tim

e (s

)

Cases


SFLimit


132

Table.4.9 TSCOPF results with constraint for faults near Bus 7 on Line 5-7, near Bus 5 on Line 5-9 and near Bus 6 on Line 6-9


83 120 7 8 0.2 0.2167 0.21585 9 0.3 0.5128 0.51526 3 0.4 0.4383 0.4428

84 110 7 8 0.2 0.2167 0.21475 9 0.3 0.4918 0.49686 3 0.4 0.4353 0.4413

85 100 7 8 0.2 0.2336 0.23695 9 0.3 0.5434 0.54586 3 0.4 0.5094 0.514

86 120 7 8 0.25 0.2667 0.27655 9 0.5 0.7178 0.72186 3 0.5 0.5814 0.5808

87 110 7 8 0.25 0.2667 0.26795 9 0.5 0.6617 0.66486 3 0.5 0.5928 0.5939

88 100 7 8 0.25 0.2667 0.27055 9 0.5 0.6545 0.65766 3 0.5 0.6209 0.625

89 120 7 8 0.25 0.2667 0.26915 9 0.6 0.7181 0.72236 3 0.65 0.6667 0.6714

90 110 7 8 0.25 0.2667 0.26935 9 0.6 0.6898 0.69596 3 0.65 0.6667 0.677

91 100 7 8 0.25 0.2822 0.28685 9 0.6 0.7099 0.71996 3 0.65 0.6667 0.6775

133

83 84 85 86 87 88 89 90 91

0

0.01

0.0167

0.03

0.04

Tim

e (s

)

Cases


SFLimit


83 84 85 86 87 88 89 90 91 920

0.0167

0.04

0.06

0.08

0.1

0.12

Tim

e (s

)

Cases


SFLimit


134

83 84 85 86 87 88 89 90 910

0.0167

0.05

0.1

0.15

Tim

e (s

)

Cases


SFLimit


4.5 Discussion

With multiple contingency constraints, the optimization process was able to

converge, where possible. It is evident from the results from that dynamic limits

cannot be established by offline studies, because the ‘distance from instability’ is

always a function of current operating state which seldom matches conditions

used for offline studies. In Case 47, it is seen that at 120% loading, with the fault

clearing time for the fault near Bus 7 on Line 8 at 0.15s, both the constraints are

binding at the time of convergence. In Case 50 however, with the fault clearing

time for the fault near Bus 7 on Line 8 increased to 0.25s, the second constraints

was not binding for the optimal solution. It would be extremely difficult to arrive

at such solutions manually using offline studies and also very time consuming.

Instead we see the advantage of rapid and accurate mapping of an operating state

not known during training to the dynamic stability index i.e. the critical clearing

135

time and how it can be used to ensure dynamic stability the operating state in

question.

4.6 Conclusions

1. It is seen from the results that the trained neural networks were able to predict

the critical clearing times for the various operating conditions established by

the optimal power flow with very high accuracy.

2. The application of artificial intelligence to establish the complex mapping

between an operating condition to the critical clearing time for a particular

fault leads to a vast saving in computation time and completely eliminates the

need for any time domain simulations needed to assess the transient stability.

3. Coupling the neural networks with the already existing optimization

techniques provides a computationally efficient approach to carry out transient

stability constrained optimal power flow. Only one more state variable i.e., the

critical clearing time and constraint are added for each contingency to be

considered in addition to the already existing state variables and constraints.

So no additional burden is posed on the optimization process. Hence with

small adjustments in the active power output of the generators it is not only

possible to respect the conventional static-security constraints, but also

transient stability constraints that are going to be more vital for deregulated

power system operations.

136

4. The use of a safety factor as small as one cycle time ensures that the actual

critical clearing time is always greater than the fault clearing time provided

the neural networks are well trained. Hence, it is possible to carry out real-

time dispatches for generators respecting both the conventional static-security

constraints and also the more important transient stability constraints.

5. Ensuring dynamic security is very vital for a successful competitive market

that depends upon the reliable and secure operation of the power system. The

method also provides a computationally efficient way to carry out the optimal

power flow with transient stability constraints for multiple contingencies.

137

CHAPTER FIVE

SUMMARY AND CONCLUSIONS

An attempt has been made in this dissertation to enhance the usage of the

optimal power flow tool with security constraints as a means to operate in regions

that would not only be secure with respect to pre-contingency and post-

contingency static security constraints, but also be secure with respect to dynamic

stability of the power system. Ideas have been proposed and implemented for both

analytical and intelligent techniques to obtain generation dispatches that would be

first-swing stable for a certain set of credible contingencies (three-phase faults)

with their respective fault clearing times. In the absence of such tools the

operators are limited to operating in conservative secure operating regions

established using offline studies. With the fast inception of deregulation, the need

to ensure a reliable and secure power system and allocate fair generation

dispatches has become very vital. Large computation times have avoided the use

of dynamic security constraints to ensure dynamic stability, more specifically

transient stability in state-of-the-art optimal power flow methods today, an

indispensable tool for nodal pricing in a competitive market. The major

disadvantages of the conservative solutions and the corrective methods are that

although they may be able to withstand the effect of major disturbances on the

transient stability of the system, they may not be the optimal solution required for

a fair operation of a competitive electricity market. The work in this dissertation is

138

an attempt to enhance the already existing state-of-the-art dispatch methods

within energy managements systems by development of computationally efficient

formulations for the evaluation and inclusion of ‘transient stability’ constraints.

This would allow the transient stability constrained optimal power flow

techniques to be adopted for real-time dispatch methods. It would thus aid in

maintaining a reliable and secure power system by prevention of blackouts and

also in the successful and fair operation of a competitive market.

The analytical technique developed initially is focused on ‘bounding the

solution of the differential equations’ to implement a practical preventive

methodology for transient stability constrained optimal power flow. The

industrially accepted stability/instability indicator was formed using the value of

the rotor angles with respect to a center of inertia frame of reference. Putting a

bound on this value would mean that the generator rotor angles are within close

proximity of each other and not diverging indicating an unstable scenario. This

idea is used to form the transient stability related constraint. The other part of

implementing the transient stability constrained optimal power flow formulation

involved including the solution of the differential equations corresponding to the

values of the state variables for the current iteration of the optimization process.

The current values of the state variables decide the initial conditions for the

differential equations. This was implemented by an external function that actually

calculates the solution of the differential equations using the Taylor series

expansion of differential equations. This external function solves the differential

equations and returns a vector containing the distance of the rotor angle of each

139

generator from the center of inertia angle at 1s. This distance is constrained to be

less than a particular value that is learnt from offline stability studies on the power

system. It was seen that with small adjustments of the state-variables in the

original optimal power flow it is possible to ensure the transient stability of the

system for a particular fault. Although the net cost of supplying this power would

increase considering the increased constraint set, but the dynamic stability of the

system with respect to a certain fault that has been considered in the transient

stability constrained optimal power flow would not be uncertain. This was

confirmed by solving the differential equations with the new generator active

power outputs and looking at the generator rotor plots. Incorporating such

constraints for a set of faults at different locations can ensure transient stability of

the power system for faults occurring at those locations. Hence a set of faults can

be treated simultaneously. Also instead of actually integrating the solution of the

differential equations into the optimal power flow, a functional constraint

externally calculates the distance of the rotor angles from their center of inertia at

1s. This distance is restricted by including the respective constraints in the

optimization process. Some of the issues dealing with computation are reviewed

here. It has also been observed that using previous converged states as starting

points for obtaining new dispatches with additional constraints such as transient

stability constraints considerably reduces the total computation time required.

Also, for further improving the performance of this method, the process for the

solution of the differential equations can be carried out on a machine with a much

powerful processor for a faster and practical application.

140

Following this, a methodology has been proposed for the development of an

intelligent technique that would completely eliminate the need for any time

domain simulations. The main thrust of the idea is to establish an excellent

nonlinear mapping of an operating state to the critical clearing time for a

particular fault. A well established mapping would require a large training set

covering the entire range of the possible values for each state variable. Using

time-domain simulations for each possible operating state to manually obtain the

critical clearing time and form the training set does not seem viable. So the initial

work in this direction focused on a method for quick training of neural networks

for the estimation of critical clearing time for a particular fault corresponding to a

given operating state (i.e. the state variable set of the optimization process). The

SIME method was employed for the quick training of the neural networks.

Provided the critical and non-critical clusters are formed correctly, the SIME

method provides for the early termination of the time domain simulations on

detection of a stable or unstable situation. The consistency evident in the results

makes it suitable for real-time dynamic security assessment also in regulated

systems. The qualitative advantage of the SIME method providing margins to

indicate stable/unstable situations can be utilized to quickly estimate the critical

clearing time. These two advantages makes the SIME were the main reasons of its

selection to train the neural networks. It was seen that with proper training, the

neural network can be used to estimate the critical clearing time with very high

accuracy. The accuracy of the artificial neural network output can be improved by

having different neural nets for different fault locations and it is expected that the

141

estimate can be made even more accurate by further considering different sets of

loading conditions. Also, this completely eliminated the need for any time domain

simulations needed to assess the transient stability.

The neural networks trained above are used further in the next section for the

formulation of a hybrid neural network-optimization scheme to carry out transient

stability constrained optimal power flow. Since, the trained neural networks

estimate the critical clearing time, the transient stability constraint in this

formulation ensures that the critical clearing time for the particular three-phase

fault under consideration will be more than the fault clearing time for the

respective fault. It was seen that with small adjustments in the active power output

of the generators it is not only possible to respect the conventional static-security

constraints, but also transient stability constraints that are going to be more vital

for deregulated power system operations. Also, coupling the neural networks with

the already existing optimization techniques provides a highly computationally

efficient approach to carry out transient stability constrained optimal power flow

of multiple contingencies. Only one more state variable i.e., the critical clearing

time and constraint are added for each contingency to be considered in addition to

the already existing state variables and constraints. So no additional burden is

posed on the already existing state-of-the-art optimization processes that provide

static-security constrained optimal power flows.

Ensuring dynamic security is very vital for a successful competitive market

that depends upon the reliable and secure operation of the power system. An

introductory treatise is presented for an efficient implementation of the transient

142

stability constrained optimal power flow problem. The practical application of

such a real-time scheme based on preventive methodology can enable power

system operators to prevent potential stability problems before they occur and

lead to cascading outages.

The faults studied have been assumed to be 3-phase faults although it is well

known that 3-phase faults are the rarest of the faults and 85% of the faults are

single-line-to-ground faults. The dynamic constrained dispatches obtained were

tested for single-line-to-ground faults in the same locations for which the dynamic

constraint for the three-phase was included. None of the cases studied produced

an unstable scenario. But, it is always possible that the fault could be unstable and

practical scenarios would want to consider including constraints for unbalanced

faults also to make sure it is not a binding constraint for the fault location under

question. Also the faults have been assumed to occur near the bus so that the

analysis for bus faults can be used for reasons of simplicity. But faults are a

random phenomenon and may occur anywhere in the line. So a realistic

application should be able to consider any fault location. Also, since the classical

model was used for the transient stability analysis, this does not represent a

realistic model beyond the first swing time period and the effect of excitation

systems, regulators and governors would need to be included if the second and

subsequent stability scenarios need to be considered while obtaining an optimal

dispatch. All the dispatches obtained in this work, assumed that the fault has been

cleared by opening the line. Inadvertently, that assumes a permanent fault. From a

practical power system operations viewpoint, the effect of reclosing should be

143

taken into account and optimal dispatches should be obtained considering a

permanent fault or temporary fault. Future work can also involve further use of

neural networks to remove other nonlinear constraints from the optimization

process and further decrease the computation time for the optimization process.

The transmission system is the heart of a successful competitive market in a

deregulated environment. Keeping it secure and reliable is the most important task

of the entity handling the transmission operations.

144

145

APPENDICES

146

147

Appendix A

System Data for the 3-Machine 9-Bus IEEE Test System (Base Case) [46]

Table A.1 Generator Data

Generator At H x'dNumber Bus (sec) (p.u)

1 1 23.64 0.06082 2 6.4 0.11983 3 3.01 0.1813

MVAbase=100

Table A.2 Bus Data

Bus Number Bus Type V θ(1 - Slack) (p.u) (degree) PL QL PG QG(2 - P-V) (MW) (MVAR) (MW) (MVAR)

(0 - Load)1 1 1.0400 0 0 0 71.641 27.04592 2 1.0250 9.28001 0 0 163 6.653623 2 1.0250 4.66475 0 0 85 -10.85974 0 1.0258 -2.2168 0 0 0 05 0 0.9956 -3.9888 125 50 0 06 0 1.0127 -3.6874 90 30 0 07 0 1.0258 3.7197 0 0 0 08 0 1.0159 0.72754 100 35 0 09 0 1.0324 1.96672 0 0 0 0

Load Generation

MVAbase=100

148

Table A.3 Branch Data

Shunt Branch typeSusceptance (0 - T.L)

From To R X (B/2) (1 - Transf.) Mag Angle(p.u) (p.u) (p.u) (p.u) (degree)

1 4 0 0.0576 0 1 1 04 6 0.017 0.092 0.079 0 0 06 9 0.039 0.17 0.179 0 0 03 9 0 0.0586 0 1 1 08 9 0.0119 0.1008 0.1045 0 0 07 8 0.0085 0.072 0.0745 0 0 07 2 0 0.0625 0 1 1 05 7 0.032 0.161 0.153 0 0 05 4 0.01 0.085 0.088 0 0 0

Impedance TapTransformerBus Series

MVAbase=100

149

Appendix B

System Data for the 10-Machine 39-Bus IEEE Test System (Base Case) [46]

Table B.1 Generator Data

Generator At H x'dNumber Bus (sec) (p.u)

1 30 42 0.0312 31 30.3 0.06973 32 35.8 0.05314 33 38.6 0.04365 34 26 0.1326 35 34.8 0.057 36 26.4 0.0498 37 24.3 0.0579 38 34.5 0.05710 39 500 0.006

MVAbase=100

150

Table B.2 Bus Data

Bus Bus Type V θNumber (1 - Slack) (p.u) (degree) PL QL PG QG

(2 - P-V) (MW) (MVAR) (MW) (MVAR)(0 - Load)

1 0 1.0475 -9.582818 0 0 0 02 0 1.04883 -7.023205 0 0 0 03 0 1.03016 -9.870743 322 2.4 0 04 0 1.00355 -10.66587 500 184 0 05 0 1.0048 -9.477869 0 0 0 06 0 1.00716 -8.775401 0 0 0 07 0 0.99648 -10.9799 233.8 84 0 08 0 0.9955 -11.48621 522 176.6 0 09 0 1.028 -11.31081 0 0 0 0

10 0 1.01683 -6.390371 0 0 0 011 0 1.01233 -7.203946 0 0 0 012 0 0.99978 -7.219655 8.5 88 0 013 0 1.01397 -7.105162 0 0 0 014 0 1.01143 -8.775415 0 0 0 015 0 1.01526 -9.193591 320 153 0 016 0 1.03165 -7.78864 329.4 32.3 0 017 0 1.03349 -8.787632 0 0 0 018 0 1.03091 -9.628719 158 30 0 019 0 1.04979 -3.162591 0 0 0 020 0 0.99084 -4.574777 680 103 0 021 0 1.03128 -5.378162 274 115 0 022 0 1.04898 -0.921697 0 0 0 023 0 1.04322 -1.117214 247.5 84.6 0 024 0 1.03697 -7.668835 308.6 -92.2 0 025 0 1.05741 -5.661638 224 47.2 0 026 0 1.05193 -6.918366 139 17 0 027 0 1.03761 -8.930536 281 75.5 0 028 0 1.05005 -3.406101 206 27.6 0 029 0 1.04989 -0.646634 283.5 26.9 0 030 2 1.0475 -4.603657 0 0 250 103.331 1 0.982 0 9.2 4.6 572.9 170.332 2 0.9831 1.608247 0 0 650 175.933 2 0.9972 2.05504 0 0 632 103.334 2 1.0123 0.615694 0 0 508 164.435 2 1.0493 4.043931 0 0 650 204.836 2 1.059 6.784177 0 0 560 96.937 2 1.0278 1.123671 0 0 540 -4.438 2 1.0265 6.41694 0 0 830 19.439 2 1.03 -11.12007 1104 250 1000 68.5

Load Generation

MVAbase=100

151

Table B.3 Branch Data



1 2 0.0035 0.0411 0.34935 0 0 01 39 0.001 0.025 0.375 0 0 02 3 0.0013 0.0151 0.1286 0 0 02 25 0.007 0.0086 0.073 0 0 03 4 0.0013 0.0213 0.1107 0 0 03 18 0.0011 0.0133 0.1069 0 0 04 5 0.0008 0.0128 0.0671 0 0 04 14 0.0008 0.0129 0.0691 0 0 05 6 0.0002 0.0026 0.0217 0 0 05 8 0.0008 0.0112 0.0738 0 0 06 7 0.0006 0.0092 0.0565 0 0 06 11 0.0007 0.0082 0.06945 0 0 07 8 0.0004 0.0046 0.039 0 0 08 9 0.0023 0.0363 0.1902 0 0 09 39 0.001 0.025 0.6 0 0 0

10 11 0.0004 0.0043 0.03645 0 0 010 13 0.0004 0.0043 0.03645 0 0 013 14 0.0009 0.0101 0.08615 0 0 014 15 0.0018 0.0217 0.183 0 0 015 16 0.0009 0.0094 0.0855 0 0 016 17 0.0007 0.0089 0.0671 0 0 016 19 0.0016 0.0195 0.152 0 0 016 21 0.0008 0.0135 0.1274 0 0 016 24 0.0003 0.0059 0.034 0 0 017 18 0.0007 0.0082 0.06595 0 0 017 27 0.0013 0.0173 0.1608 0 0 021 22 0.0008 0.014 0.12825 0 0 022 23 0.0006 0.0096 0.0923 0 0 023 24 0.0022 0.035 0.1805 0 0 025 26 0.0032 0.0323 0.2565 0 0 026 27 0.0014 0.0147 0.1198 0 0 026 28 0.0043 0.0474 0.3901 0 0 026 29 0.0057 0.0625 0.5145 0 0 0

Bus Series TransformerImpedance Tap

MVAbase=100

152

Table B.4 Branch Data (continued.)



28 29 0.0014 0.0151 0.1245 0 0 012 11 0.0016 0.0435 0 1 1.006 012 13 0.0016 0.0435 0 1 1.006 06 31 0 0.025 0 1 1.07 0

10 32 0 0.02 0 1 1.07 019 33 0.0007 0.0142 0 1 1.07 020 34 0.0009 0.018 0 1 1.009 022 35 0 0.0143 0 1 1.025 023 36 0.0005 0.0272 0 1 1 025 37 0.0006 0.0232 0 1 1.025 02 30 0 0.0181 0 1 1.025 0

29 38 0.0008 0.0156 0 1 1.025 019 20 0.0007 0.0138 0 1 1.06 0

Bus Series TransformerImpedance Tap

MVAbase=100

153

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