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POWER SYSTEM STABILIZER : ANALYSIS &

SIMULATIONS

Technical Report

By

Vihang M. Dholakiya (10MEEE05)

Devendra P. Parmar (10MEEE07)

Under the Guidance of

Dr. S. C. Vora

DEPARTMENT OF ELECTRICAL ENGINEERING

INSTITUTE OF TECHNOLOGY

NIRMA UNIVERSITY

AHMEDABAD 382 481

MAY 2012

Dedicated To

Mentor

Dr. S. C. Vora

CERTIFICATE

This is to certify that the Technical Report entitled “POWER SYSTEM STA-

BILIZER : ANALYSIS & SIMULATIONS” submitted by Mr. Vihang M.

Dholakiya (10MEEE05) and Mr. Devendra P. Parmar (10MEEE07), is the

record of work carried out by them under my supervision and guidance. The work

submitted has in my opinion reached a level required for being accepted. The results

embodied in this project work to the best of my knowledge are satisfactory.

Date:

Project Guide

Dr. S. C. Vora

Professor

Department of Electrical Engineering

Institute of Technology

Nirma University

Ahmedabad

Acknowledgements

We take this opportunity to express our sincere gratitude to our honorable guide

Dr.S.C.Vora for his invaluable guidance. It would have never been possible for

us to wok on this project without his technical support and continuous encourage-

ment. We consider ourself, extremely fortunate for having chance to work under his

guidance. In spite of his hectic schedule, he was always approachable and spent his

precious time to discuss problems. It has been a very learning and enjoyable experi-

ence to work under him.

We would like to acknowledge Mr. A. Ragaveniran et.al., authors of technical

paper titled as “MATLAB/Simulik-Based Modeling and Operation of Power System

Stabilizer” which provides us initial motivation for doing work in the area of imple-

mentation of power system stabilizer.

We would also like to thank all faculty members of Department of Electrical Engi-

neering, who have helped us during this project work. I wish to express my thanks

to other staff members of Electrical Department as well for their regular help and

co-operation during the project work. We heartily thankful staff member of library

for providing technical recourses for project work.

We would be specially thankful to our dear friend Narendra C. Mahavadia for

providing continuous help and motivational support during entire project work. We

never forget the time that we have spent with him during this tenure.

We are thankfull to our classmates for their invaluable help, suggestions and support

during the project work. We would like to thank all who have directly or indirectly

contributed to this project work.

Finally, We would like to thank The Almighty and express my deep sense of rever-

ence gratitude to our Parents and Family Members who have provided support and

blessings without which we wouldn’t have reached at this stage.

- Vihang M. Dholakiya

- Devendra P. Parmar

i

Abstract

The extensive interconnection of power networks by weak tie-lines can restrict the

steady-state power transfer limits due to low frequency electromechanical oscillations.

The low frequency oscillations may result in interruptions in energy supply due to loss

of synchronism among the system generators and affect operational system economics

and security. Further, in order to maintain steady state and transient stability of syn-

chronous generators, high performance excitation systems are essential. The static

exciters with thyristor controllers are generally used for both hydraulic and thermal

units. Such exciters are characterized by high initial response and increased reliabil-

ity due to advances in thyristor controllers and hence have become one of the major

problems in the power system stability area. As a solution to this, the generators are

equipped with Power System Stabilizers(PSSs) that provide supplementary feedback

stabilizing signals which is added to the Automatic Voltage Regulator (AVR). PSSs

augment the power system stability limit and extend the power transfer capability

by enhancing the system damping of low-frequency oscillations in the order of 0.2 to

3.0 Hz.

The report focuses on small signal performance analysis of Single Machine Infinite

Bus(SMIB) as well as of multimachine power system. The dynamic behaviour of

Haffron-Phillips model of excitation system with typical data is evaluated by devel-

oping MATLAB code for eigenvalue tracking analysis. The effect of implementation

of power system stabilizer to SMIB system has been realized by time domain sim-

ulations. In the later part optimal placement of PSS is decided, because from the

economic point of view and to avoid redundancy, it is desired, not to employ PSS

on individual generators to overcome the problem of power system oscillations. The

eigenvalue analysis of the power system for various areas is used to determine the

inter-area and local mode frequencies and participation of the generators. It is also

important to identify the generator that shall be installed with PSS. A simulation

study on well-adopted test system is carried out, with various possibilities, to deter-

mine the optimal placement of the PSS. The observations of the certain analysis are

helpful in determining the PSS placement and are presented in the report.

ii

List of Figures

2.1 Classification of Power System Stability . . . . . . . . . . . . . . . . . 62.2 Phasor Representation of Electrical Torque . . . . . . . . . . . . . . 132.3 Synchronizing & Damping Torque Stability Analysis . . . . . . . . . . 14

3.1 Haffron- Phillips Model of Excitation System . . . . . . . . . . . . . . 183.2 Linearized SMIB Model . . . . . . . . . . . . . . . . . . . . . . . . . 193.3 Block Diagram of Linearized SMIB Model . . . . . . . . . . . . . . . 203.4 Eigenvalue Loci for Variation in AVR Gain . . . . . . . . . . . . . . . 233.5 Basic Arrangement of PSS . . . . . . . . . . . . . . . . . . . . . . . . 243.6 Haffron- Phillips Model of Excitation System with PSS . . . . . . . . 253.7 Operating Principle of PSS . . . . . . . . . . . . . . . . . . . . . . . . 253.8 General Structure of PSS . . . . . . . . . . . . . . . . . . . . . . . . . 263.9 SMIB Model in MATLAB/SIMULINK . . . . . . . . . . . . . . . . . 283.10 Rotor Angle Deviation Before Introducing PSS . . . . . . . . . . . . . 293.11 Speed Deviation Before Introducing PSS . . . . . . . . . . . . . . . . 293.12 Rotor Angle Deviation After Introducing PSS . . . . . . . . . . . . . 303.13 Speed Deviation After Introducing PSS . . . . . . . . . . . . . . . . . 30

4.1 Single Line Diagram of Test System . . . . . . . . . . . . . . . . . . . 334.2 Speed Participation Factor v/s Generator No. For Inter-Area Mode-

0.54 Hz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.3 Participation Factor v/s Generator No. For Local Mode-1.05 Hz . . . 394.4 Eigenvalues of Two Area Test System without PSS . . . . . . . . . . 394.5 Eigenvalues of Two Area Test System with PSS in Area-2 . . . . . . 404.6 Eigenvalues of Two Area Test System with PSS in Both Areas . . . . 404.7 MATLAB/SIMULINK Model of Test System . . . . . . . . . . . . . 414.8 Block Diagram of PSS . . . . . . . . . . . . . . . . . . . . . . . . . . 414.9 Power Flow from Area-1 to Area-2 . . . . . . . . . . . . . . . . . . . 424.10 Per Unit Speed Deviation for Case-1 . . . . . . . . . . . . . . . . . . 434.11 Per Unit Speed Deviation for Case-2 . . . . . . . . . . . . . . . . . . 444.12 Per Unit Speed Deviation for Case-3 . . . . . . . . . . . . . . . . . . 444.13 Per Unit Speed Deviation for Case-4 . . . . . . . . . . . . . . . . . . 45

iii

List of Tables

2.1 Types of Swing Mode of Oscillations . . . . . . . . . . . . . . . . . . 8

4.1 Network Statastics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 Power Flow Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.3 Line Flow Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Summerized Power Flow Analysis . . . . . . . . . . . . . . . . . . . . 354.5 Summerized Power Flow Results . . . . . . . . . . . . . . . . . . . . . 364.6 Effects of PSS with Different Arrangement Scheme . . . . . . . . . . 37

B.1 Machine Data of Two Area Test System . . . . . . . . . . . . . . . . 50B.2 Line Data of Two Area Test System . . . . . . . . . . . . . . . . . . . 51B.3 Load Data of Two Area Test System . . . . . . . . . . . . . . . . . . 51B.4 Exciter & PSS Data of Two Area Test System . . . . . . . . . . . . . 51

iv

Abbreviations

AVR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Automatic Voltage RegulatorCPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous Power FlowGUI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical User InterfaceHVDC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . High Voltage Direct CurrentOPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Optimal Power FlowPSAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Power System Analysis Tool BoxPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Power System StabilizerSMIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single Machine Infinite Bus System

Nomenclature

Gex(s) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exciter Transfer FunctionH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Moment of InertiaKD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damping Co-efficientKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Exciter GainKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Synchronizing Co-efficientKPSS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PSS Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Laplace FunctionTw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Wash out Time Constantωn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Natural Frequency∆ωr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Speed Deviation∆P . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Change in Power Transfer∆δ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Change in Rotor Angleζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Damping Ratioφ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phase

v

Contents

Acknowledgements i

Abstract ii

List of Figures iii

List of Tables iv

Nomenclature/Abbreviations v

Contents vi

1 Introduction 11.1 Problem Identification . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Objective of The Work . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Project Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Outline of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Power System Stability 42.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 Small Signal Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Small Signal Stability Analysis . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Eigenvalue Analysis . . . . . . . . . . . . . . . . . . . . . . . . 92.3.2 Synchronizing and Damping Torque Analysis . . . . . . . . . . 122.3.3 Frequency Response and Residue Analysis . . . . . . . . . . . 152.3.4 Time Domain Solutions . . . . . . . . . . . . . . . . . . . . . 15

2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 SMIB & PSS: Physical Aspects, Implementation & Analysis 173.1 Small Signal Performance of Single Machine Infinite Bus System . . . 17

3.1.1 Formulation of Heffron-Phillips Model of Excitation System forStability Studies . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.1.2 Linearized Model of Single Machine Infinite Bus System . . . 193.1.3 Oscillatory Stability Assessment of SMIB . . . . . . . . . . . . 213.1.4 Introducing PSS in AVR loop of SMIB . . . . . . . . . . . . . 23

3.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

vi

vii

4 Optimal Placement of PSS in Multimachine Power System 324.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.2 Test System Description . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 Load Flow Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Linear Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.1 System Behavior in Different Arrangement of PSS . . . . . . . 364.4.2 Mode Identification and Participation Analysis . . . . . . . . . 374.4.3 S-Domain Plots . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4.4 Time Domain Simulations . . . . . . . . . . . . . . . . . . . . 40

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5 Conclusions & Future Work 465.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

References 48

A IEEE Proceeding 49

B System Data 50

C Introduction to PSAT 2.1.6 52C.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

C.1.1 Useful Features . . . . . . . . . . . . . . . . . . . . . . . . . . 53C.1.2 PSAT Utilities . . . . . . . . . . . . . . . . . . . . . . . . . . 53

C.2 Starting Process of PSAT . . . . . . . . . . . . . . . . . . . . . . . . 53C.3 Useful Links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Index 58

Chapter 1

Introduction

1.1 Problem Identification

Power transactions are increasing day by day in restructured power systems. Re-

structured power system is therefore, expected to be operated at a greater variety of

operating points and closer to their operating constraints. The “low frequency os-

cillations” is one of the operational constraints which limit bulk power transmission

through power network. In such scenario, power system controls plays significant role.

Power system controls can contribute either positive or negative damping. Generation

control and particularly the generator voltage regulation can be significant sources of

negative damping. High gain in the generator voltage regulation can lead to poor or

negative damping of the oscillation. This problem has lead to the implementation of

Power System Stabilizer (PSS) to damp out the oscillations.

1.2 Objective of The Work

The project work is aimed at the implementation of power system stabilizer with

appropriate parameters in single machine infinite bus system. Effect of power system

stabilizer implementation on system damping is also targeted. Further, for multi

machine power system, it is aimed that, the damping of power system oscillations

can be achieved with minimum No.of power system stabilizers located at optimal

locations.

1

CHAPTER 1. INTRODUCTION 2

1.3 Project Planning

• Formation & analysis of Haffron-Phillips model of excitation system.

• Development of program in MATLAB for sweep analysis of exciter gain and its

effect on system response.

• Formation of SMIB in MATLAB c©/SIMULINK.

• Effect of PSS on response of SMIB system.

• Formation of Kundur’s Two Area System in MATLAB c©/SIMULINK.

• S-domain analysis of Kundur’s Two Area System using Power System Analysis

Toolbox(PSAT).

• Primary screening of generators for placement of PSS through participation

factors.

• Time domain simulations of developed Kundur’s Two Area System.

1.4 Scope of Work

The scope of the project work can be broadly outlined as below:

• Realization of Haffron-Philips Model of Excitation System.

• Effect of PSS implementation on behavior of SMIB.

• Eigenvalue analysis of Kundur’s Two Area System in various PSS arrangement.

• Participation factor analysis.

• Response analysis in time domain for Kundur’s Two Area System with typical

different cases.

• Choice of optimal location of PSS based on observations of S-domain analysis

and time-domain simulations.

CHAPTER 1. INTRODUCTION 3

1.5 Outline of Thesis

• Chapter 1 introduces the main problem associated with the low frequency

oscillation damping by optimally placed PSS and the same is considered as the

objective of this work. The project planning and scope of work is also included.

• Chapter 2 gives general background of power system stability. The detail

description of eigenvalue analysis method used for evaluating small signal per-

formance of power system is focused and the other methods are discussed briefly.

• Chapter 3 includes the dynamic analysis of Hafron-Phillips model of excitation

system through eigenvalue tracking method. The effect of implementation of

PSS in SMIB has also been analyzed by performing time domain simulations.

• Chapter 4 discusses about optimal placement of PSS in multimachine power

system. The eigenvalue analysis of for Kundur’s two area system is used to de-

termine the mode of oscillation and participation of the generators. A simulation

study on considered test system is carried out, with various PSS arrangement,

the optimal placement of the PSS is decided.

• Chapter 5 comprises of conclusion and future work.

Chapter 2

Power System Stability

2.1 Background

Modern power system can be characterized by widespread system interconnections.

The interconnected power system is comprised of multiple machines connected by the

transmission network. The supply of reliable and economic electric energy is a major

determinant of industrial progress and consequent rise in the standard of living. In

practical terms this means that both voltage and frequency must be held within allow-

able tolerances so that the consumer’s equipment can operate satisfactorily. Further,

with deregulation of power supply utilities, the power network has become a highway

for transmitting electric power from wherever it is available to places where required,

depending on the pricing that varies with time of the day. In such scenario, the anal-

ysis of dynamic performance and stability of power system has great importance.

The stability problem is concerned with the behavior of the synchronous machines

under perturbed conditions. If the perturbation does not involve any net change in

power, the machines should return to their original state and if an unbalance between

the supply and demand is created by perturbation, a new operating state should be

achieved. When the system changes its operating point from one stable point to the

other,it is mandatory that all interconnected synchronous machines should remain in

synchronism. i.e., they should all remain operating in parallel and at the same speed.

4

CHAPTER 2. POWER SYSTEM STABILITY 5

Thus, Power System Stability may be broadly defined as that property of power

system that enables it to remain in a state of operating equilibrium under normal

operating condition and to regain an acceptable state of equilibrium after being sub-

jected to disturbance[1].

Although, stability of a system is an integral property of the system, for purposes of

the system analysis, it is mainly divided into following categories:

• Steady State Stability relates to ability of synchronous machine to maintain

synchronism followed by small disturbances. e.g. gradually changing load.

• Dynamic or Small Signal Stability concerns with the response of syn-

chronous machine to small perturbations that are oscillating in nature. If these

oscillations are of small amplitude, the system may be considered as small signal

stable, but if the amplitude of oscillations is of growing nature, with the passage

of time the system may lose its stability. Usually, heavy power flow in trans-

mission line or interaction of controller with system frequency is responsible for

small signal instabilities. The phenomenon is concerning with few seconds to

10s of seconds of time period.

• Transient Stability involves response of synchronous machine to large dis-

turbances such as application and clearing of faults, sudden load changes and

inadvertent tripping of transmission lines or generators. Such large disturbances

can create large changes in rotor speeds, power angles and power transfer. The

phenomenon is concerns with time period of 1 second or less.

The detailed classification of power system stability is depicted in the following Fig.2.1

[1].The report focuses on the Small Signal Stability of power system.


Figure 2.1: Classification of Power System Stability


2.2 Small Signal Stability

Small Signal Stability is the ability of the power system to maintain synchronism

when subjected to small disturbances. A power system at a particular operating

state may be large disturbance unstable and still such a system may be operated with

insecurity with proper control and protective actions. But, if the system is small-signal

unstable at a given operating condition, it cannot be operated at all, because small

signal instability may result in steady increase in generator rotor angle due to lack of

synchronizing torque or in rotor oscillations of increasing amplitude due to insufficient

damping torque. Thus, small-signal stability is a fundamental requirement for the

satisfactory operation of power systems.The reasons for the system can become small

signal unstable are enlisted hereunder[3]:

• Use of high gain fast acting exciters

• Heavy power transfer over long transmission lines from remote generating plants

• Power transfer over weak ties between systems which may result due to line

outages.

• Inadequate tuning of controls of equipment such as generator excitation systems,

HVDC-converters and static var compensators.

• Adverse interaction of electrical and mechanical systems causing instabilities of

torsional mode oscillations.

The issue of small signal instability in current scenario is generally because of insuffi-

cient damping of oscillations. In practical power system, the main types of oscillations

associated with small signal stability are as follows:

1) Swing Mode

2) Control Mode

3) Torsional Mode


• Swing Mode of Oscillation

This mode is also referred to as electromechanical oscillations. For an n gen-

erator system, there are (n-1) swing (oscillatory) modes associated with the

generator rotors.The location of generators in the system determines the type

of swing mode.Hence, the swing mode of oscillation can be further sub classified

as shown in following Table 2.1 [3].

Table 2.1: Types of Swing Mode of Oscillations

Local Mode Inter-Unit(Intra-plant) Mode Inter-AreaThese oscillations generally These oscillations These oscillations usuallyinvolve one or more typically involve two or involve combinations of manysynchronous machines at a more synchronous synchronous machines onpower station swinging machines at a power one part of a power systemtogether against a plant swing against each swinging against machinescomparatively large power other. on another part of the system.system or load center.Freqency Range: 0.7 to 2 Hz Freqency Range: 1.5 to 3 Hz Freqency Range: 0.1 to 0.5 Hz

• Control Modes of Oscillations

Control modes are associated with generating units and other controls. Poorly

tuned exciters, speed governors, HVDC converters and static var compensators

are the usual causes of instability of these modes.

• Torsional Mode of Oscillations

These oscillations involve relative angular motion between the rotating elements

(synchronous machine rotor, turbine, and exciter) of a unit, with frequencies

ranging from 4Hz and above. Instability of torsional mode may be caused

by interaction with excitation controls, speed governors and series capacitor

compensated transmission lines.

Of these oscillations, local mode, intra-plant mode, control mode and torsional mode

are generally categorized as local problems as it involves a small part of the system.

Further, inter-area mode oscillations are categorized as global problems and have

widespread effects.


2.3 Small Signal Stability Analysis

There are mainly four techniques which are used to analyze the small signal stability

of power system:

a. Eigenvalue Analysis

b. Synchronizing and Damping Torque Analysis

c. Frequency Response and Residue Based Analysis

d. Time domain Simulations

2.3.1 Eigenvalue Analysis

In this report ,of the above methods, Eigenvalue Analysis is used to study oscillatory

behavior of power systems and hence has been described in detail. The system is

linearized about an operating point and typically involves computation of eigenvalues,

eigenvectors, participation factors and system modes from state-space representation

of power system model. This can also be termed as “Small Signal Stability Analysis”

or “Modal Analysis”. Technique employed in this report for studying oscillatory

modes is also based on eigenvalue analysis. Initially, eigenvalues and eigenvectors

are derived. From this, modes of oscillations and participation factor of particular

generator are found out. It gives preliminary idea about possible location of PSS. The

derivation of eigenvalues and participation factor can be found in [1] and can be briefly

explained as follows: Linear approximation of power systems can be characterized by

the following state-space equations:

∆x = A∆x+B∆u (2.1)

∆y = C∆x+D∆u (2.2)

Where,

∆ x is the state vector of length equal to the number of states n


∆ y is the output vector of length m

∆ u is the input vector of length r

A is the (n × n) state matrix

B is the input matrix of (n × r)

C is the output matrix of (m × n)

D is the feed forward matrix of (m × r)

Eigenvalues of the system state matrix is available from the characteristic equation

of the state matrix A. It can be expressed as

det(A− λI) = 0 (2.3)

For each of the eigenvalues, there are two sets of orthogonal eigenvectors, namely the

left and right eigenvectors, satisfying the following equations:

AΦi = λiΦi (2.4)

ΨiA = λiΨi (2.5)

Where,

λi, is the ith eigenvalue Φi is the right eigenvector corresponding to λi Ψi is the left

eigenvector corresponding to λi

Eigenvalue & Stability of Power System

The time-dependent characteristic of a mode corresponding to an eigenvalue λi is

given by eλit. Therefore, the stability of the system is determined by the eigenvalues

analysis. Real eigenvalues are associated with non-oscillatory modes, whereas the

complex ones, appearing in conjugate pairs, correspond to oscillatory modes - one

mode for each pair. If the eigenvalue of an oscillatory mode is expressed as,

λi = σ ± jω (2.6)


The damping coefficient which gives the rate of decay of amplitude of the oscillation

is given by,

ζ =−σ√σ2 + ω2

(2.7)

and the frequency of oscillation in Hz is determined by,

f =ω

2π(2.8)

A negative real part of the eigenvalue represents positive damping coefficient that

is, decaying oscillation, and the positive real part indicates negative damping, i.e.,

increasing oscillation. The right eigenvector of a mode gives an idea about how this

mode is distributed among different states of the system and hence known as Mode

Shape. Based on this idea, if a mode is found to be distributed among specific state

variable of generating units in different areas, then that mode can be identified as a

local mode or inter-area mode. Typically, rotor speed is used as the test state variable

for mode shape analysis in inter-area oscillation study[1] , [4].

Participation Factor is a measurement of relative participation of any state variable

in any specific mode. It is mathematically expressed as the multiplication of left and

right eigenvectors. For example, participation factor pki of any kth state variable in

any ith mode can be measured as[1],

pki = Φki ×Ψik (2.9)

Where,

Φki is the kth entry of the right eigenvector of ith mode

Ψik is the kth entry of the left eigenvector of ith mode

Thus, participation factors are the sensitivities of the eigenvalues to changes in the

diagonal elements of the state matrix. They indicate possible locations where a

stabilizer may effectively control the mode of concern[8]. Eigenvalue or modal analysis

describes the small-signal behavior of the system about an operating point, and does

not take into account the nonlinear behavior of components such as controller’s limits

at large system perturbations. Further, design and analysis carried out using various


indices such as participation factors, may lead to many alternate options. These

options need to be verified by time-domain simulations.

Advantages of Eigenvalue Analysis

i) Separate identification of modes of oscillations

ii) Root loci plotted with variations in system parameters or operating conditions

provide valuable insight into the dynamic characteristics of the system.

iii) Using eigenvectors coherent groups of generators which participate in a given

swing mode can be identified.

2.3.2 Synchronizing and Damping Torque Analysis

The synchronous operation of generators is generally dealing with balance between the

input mechanical torque and output electrical torque of each machine. The change in

electrical torque of alternator following small perturbation can be illustrated through

following equation:

∆Te = Ts∆δ + TD∆ω (2.10)

Where,

• Ts∆δ is the component of torque change in phase with the rotor angle pertur-

bation and is referred to as the synchronizing torque component.

• Ts is the synchronizing torque coefficient.

• TD∆ω is the component of torque in phase with the speed deviation and is

referred to as the damping torque component.

• TD is the damping torque coefficient.

The nature of system oscillations to small perturbation depends on both the com-

ponents of electrical torque. The response of generator without automatic voltage

regulator (constant field) can result into instability due to lack of sufficient synchro-

nizing torque. Such instability is known as non-oscillatory instability. Further, the


presence of automatic voltage regulator can also result in instability with oscillations

having nature of continuously growing amplitude. Such instability is known as oscilla-

tory instability. Both types of instabilities are illustrated in following Fig.3.1 [1]. This

analysis assumes that the rotor angle and the speed deviations oscillate sinusoidally.

Hence this can be represented by phasors as depicted in Fig.2.2

Figure 2.2: Phasor Representation of Electrical Torque

From the above figure the damping torque component can be written as

∆TeD = ∆Te cosφ (2.11)

And synchronizing torque component can be written as

∆TeS = ∆Te sinφ (2.12)

If either or both damping and synchronizing torques are negative, i.e., if ∆TeD <

0 and/or ∆TeS < 0, then the system is unstable. A negative damping torque im-

plies that the response will be in the form of growing oscillations, and a negative

synchronizing torque implies monotonic instability.


Figure 2.3: Synchronizing & Damping Torque Stability Analysis


2.3.3 Frequency Response and Residue Analysis

Frequency response is characterization of a systems transfer function between a given

input and output. Frequency response methods allow a deeper insight into small-

signal dynamics and have widespread use in the design of power system controllers.

Frequency response can also be measured directly, even in a power system. Residues

give the contribution of a mode to a transfer function. They also give the sensitivity

of the corresponding eigenvalue to a positive feedback between the output of the

transfer function and its input. Thus, residues are useful to get an idea of which

modes will be affected most by feedback. An advantage of using residues in such

analysis is that it takes into account the transfer function structure of the excitation

system unlike participation factors. However, evaluation of residues dependent on the

specific input/output combinations and may be computationally intensive for large

systems.

2.3.4 Time Domain Solutions

Time domain techniques provide an exact determination of stability of non-linear

systems both for small and large disturbances. The choice of disturbance and selection

of variables to be observed in time response are critical. If the input is not chosen

properly, there is possibility that the substantial excitation of the important modes

may not be achieved. The observed response may contain many modes and the

poorly damped modes may not be so dominant. The solution is highly affected by the

modeling of the components. Larger systems may have a number of inter-area modes

of similar frequencies, and it is quite difficult to separate them from a response in

which more than one is excited. Therefore, for a large power system it is not possible

to identify any desired mode and study their characteristics.

Of all these methods, eigenvalue or modal analysis is widely used for analyzing

the small-signal stability of power system due to advantages described earlier in this

section.


2.4 Summary

The chapter provides general introduction to basic concepts of power system stability

including a discussion of classification,definitions of related terms in brief. The details

regarding small signal stability analysis includes eigenvalue analysis along with other

methods to evaluate small signal performance of power system in simplified manner.

Chapter 3

SMIB & PSS: Physical Aspects,

Implementation & Analysis

3.1 Small Signal Performance of Single Machine

Infinite Bus System

In order to analyze the small signal stability of SMIB, following methodology has

been adopted:

A) Formulation of Heffron-Phillips Model of Excitation System for Stability Studies

B) Linearized Model of Single Machine Infinite Bus System

C) Oscillatory Stability Assessment of SMIB

D) Introducing PSS in AVR loop of SMIB

3.1.1 Formulation of Heffron-Phillips Model of Excitation

System for Stability Studies

The third-order model of the synchronous machine can be represented as a block dia-

gram shown in Fig.3.1. The basis for the model presented here, which was originally

proposed by Heffron and Phillips, is the “Single Machine Infinite Bus” (SMIB) setup.

By introducing a number of new constants, a very compact notation is achieved. The

17

CHAPTER 3. SMIB & PSS: PHYSICAL ASPECTS, IMPLEMENTATION &ANALYSIS18

model is useful to directly implement a simplified representation of an SMIB system,

including the mechanical dynamics, field winding, and excitation system. This im-

plementation can be used directly for stability studies. In the present case, a generic

simplified representation of the excitation system is used. Detailed descriptions and

common variants of these systems can be found in [1]. All quantities presented

Figure 3.1: Haffron- Phillips Model of Excitation System

are in per unit. The mechanical system is represented by the system inertia and the

damping constant, where the torque balances ∆Tm −∆Te is considered as an input

and the incremental torque angle ∆δ as an output. The electrical part of the system

consists of three main parts:

a. The composition of the electrical torque (influenced by ∆δ over constant K1

and the internal incremental voltage ∆eq over constant K2),

b. The effect of the field winding (determined by the field winding constant K3

and influenced by ∆δ over constant K4),

c. The effect of the excitation system (influenced by ∆δ over constant K5 and ∆eq

over constant K6).


The excitation system itself is modeled by a first-order transfer function including

the amplification factor KA and the time constant TA.

Interpretation of K-constants

The constants K1 to K6 shown in block diagram describe internal influence factors

within the system and can be found by a comparison of coefficients with the equations

governing the synchronous machine dynamics. While K1 and K2 are derived from the

computation of the electric torque, K3 and K4 have their origin in the field voltage

equation. K5 and K6 come from the equation governing the terminal voltage magni-

tude. The detailed derivation of K-constants is given in chapter 12 of P. Kundur.

K1 - Influence of torque angle on electric torque

K2-Influence of internal Voltage on electric torque

K3- field winding constant

K4-Influence of torque angle on field voltage

K5-Influence of torque angle on terminal voltage

K6-Influence of internal voltage on terminal voltage

3.1.2 Linearized Model of Single Machine Infinite Bus Sys-

tem

Figure 3.2: Linearized SMIB Model

Fig.3.2 shown above is representation of single machine connected to large power

system through transmission line. For any condition, the magnitude of infinite bus

voltage EB remains constant when machine is perturbed. Considering classical model

of synchronous generator and neglecting stator resistance, linearization of system


equations gives state space equation as follows:

d

dt

∆ωr

∆δ

=

KD

2HKS

2H

ω0 0

∆ωr

∆δ

+

12H

0

∆Tm (3.1)

Block diagram representation of linearized system is depicted in following figure3.3:

where,

Figure 3.3: Block Diagram of Linearized SMIB Model

KS=Synchronizing torque coefficient in pu torque/rad

KD=Damping torque coefficient in pu torque/rad

H =Inertia Constant in MW sec/MVA

∆ωr=Speed Deviation in pu

∆δ=Rotor angle deviation in elec. rad

s=Laplace operator

ω0=Rated speed in elec. rad/sec

The characteristic equation for the system can be given as

s2 +KD

2Hs+Ks

ω0

2H= 0 (3.2)


Comparing this with general characteristic equation

s2 + 2ζωns+ ωn2 = 0 (3.3)

Natural frequency of oscillation is

ωn =

√ksω0

2H(3.4)

and damping ratio is

ζ =12KD√

Ks2Hω0

(3.5)

a. As the synchronizing torque coefficient Ks increases, the natural frequency in-

creases and the damping ratio decreases.

b. An increase in damping torque coefficient KD increases the damping ratio

c. An Increase in inertia constant decreases both natural frequency and damping

ratio.

3.1.3 Oscillatory Stability Assessment of SMIB

Dynamic Performance Evaluation of AVR Loop and Stability of SMIB

System

Under heavy loading condition, the modern regulator-exciter system introduces neg-

ative damping and may lead to oscillatory instability. For the sake of small signal

stability analysis, a typical single machine infinite bus bar system given as example

in book titled as “ Power System Dynamics-Stability and Control” by K.R. Padiar

has been considered [2]. The system data considered are as follows:

Pg =1.1 Vt = Eb =1.0 KD=0 T3=6.0 sec KE=200 TE=0.05 sec

K1 = 0.0345 K2 = 2.2571 K3 = 0.4490 K4 = 0.1826 K5 =-0.0649 K6 =1.0613


In order to evaluate dynamic performance of SMIB in small signal instable condition,

Eigenvalue Tracking approach has been adopted.

Eigenvalue Tracking Analysis

As discussed in chapter 1 that the eigenvalues computed from the system state matrix

can be helpful to predict the system stability.From the data shown in the above table,

a MATLAB code has been developed to analyze the behaviour of system against gain

variation from KE=0 to KE=400. It is possible to express the system equations in

the state space form. From the block diagram, shown in Fig.3.1, the following system

equations and state matrix are derived.

x = [A]x+ [B]Vref (3.6)

x = [∆δ ∆ωr ∆ψfd ∆Efd]T (3.7)

[A] =

0 ω0 0 0

−K1

2H−KD

2H−K2

2H0

−K4

T30 −1

T3K3

1T3

−KEK5

T50 −KEK6

TE

−1TE

(3.8)

B = [0 0 0KE

TE]T (3.9)

The damping term KD, is included in the swing equation. The eigenvalues of the

matrix should lie in left half plane in the‘s’ plane for the system to be stable. The effect

of various parameters for example, here effect of variation in KE is examined from

eigenvalue analysis. It is to be noted that the elements of matrix [A] are dependent

on the operating condition.

The loci of the complex critical eigenvalues for the considered case is shown in Fig.3.4.

It is observed that, when K5 < 0, the increase in AVR gain beyond KE=4 results in

oscillatory instability. Although the locus turns around as KE is further increased,but


still it remains in the right hand plane only. Further,there is only one of the critical

eigenvalues is shown in Fig.3.4. The other is the complex conjugate whose locus is

the mirror image about the real axis. Thus, it can be realized from the analysis that

the fast acting AVR with higher gains are creating oscillatory instability in the power

system. In order to avoid such situation, the effect of implementation of PSS in small

signal unstable system is discussed in subsequent sections.

Figure 3.4: Eigenvalue Loci for Variation in AVR Gain

3.1.4 Introducing PSS in AVR loop of SMIB

Basics of Power System Stabilizer

Apart from providing fast control of the terminal voltage, high performance excitation

systems are essential for maintaining steady state and transient stability of modern

synchronous generators. It is observed that fast acting exciters with high gain AVR

can contribute to oscillatory instability in power systems. This type of-instability is

characterized by low frequency (0.2 to 2.0 Hz) oscillations which can persist or even

grow in magnitude for no apparent reason. A cost effective and satisfactory solution

to the problem of oscillatory instability is to provide damping for generator rotor


oscillations. This is conveniently done by providing Power System Stabilizers (PSS)

which are supplementary controllers in the excitation systems. The basic function of

PSS is to add damping to the generator rotor oscillations by controlling its excitation

using auxiliary stabilizing signal.

The basic arrangement of PSS along with the generator controls is depicted in Fig.3.5.

In this arrangement, speed change of generator is taken as input to PSS and its out-

put is added to the reference input of AVR to change the excitation voltage.

Figure 3.5: Basic Arrangement of PSS

The theoretical basis for PSS can be illustrated by extended block diagram of Heffron-

Phillips model of excitation system as depicted in 3.6 As the purpose of PSS is to

introduce damping torque component, a logical signal to use for controlling generator

excitation is the speed deviation ∆ωr.


Figure 3.6: Haffron- Phillips Model of Excitation System with PSS

Power System Stabilizer adds damping to generator rotor oscillations by controlling

its excitation using an auxiliary stabilizing signal. The stabilizer does so by producing

component of electrical torque in phase with the rotor speed deviation. Fig. 3.7 shows

the operating principle of PSS.

Figure 3.7: Operating Principle of PSS


From the above figure it can be clearly visualize that due to stabilizing signal provided

by PSS in phase with speed deviation, the electrical torque is shifted from third to

first quadrant, means becomes positive from negative in magnitude.

General Structure of PSS

The block diagram of basic structure of power system stabilizer is depicted in

Figure 3.8: General Structure of PSS

Fig.3.8.It consists of gain block, washout circuit, dynamic compensator, and lim-

iter. The functions of each of the components of PSS are given in subsequent sections.

1) PSS Gain

Stabilizing gain KPSS determines the amount of damping introduced by PSS.

Ideally, PSS gain is set to get the maximum damping of the oscillatory modes.

However, due to practical considerations, high gain may not be always the best

option and may cause excessive amplification of stabilizer input signal. In general,

the gain value is set such that it results in satisfactory damping of critical system

modes without compromising the stability limits.

2) Washout Circuit

The washout circuit is provided to eliminate steady-state bias in the output of

PSS which will modify the generator terminal voltage. The PSS is expected to

respond only to transient variations in the input signal, say rotor speed and not

to the dc offsets in the signal. The washout circuit acts essentially as a high pass

filter and it must pass all frequencies that are of interest. If only the local modes

are of interest, the time constant Tw can be chosen in the range of 1 to 2. However,


if inter area modes are also to be damped, then Tw must be chosen in the range of

10 to 20. The value of Tw = 10 is necessary to improve damping of the inter area

modes[1]. There is also a noticeable improvement in the first swing stability when

Tw is increased from 1.5 to 10. The higher value of Tw also improved the overall

terminal voltage response during system islanding conditions.

3) Lead-Lag Compensator

Lead-Lag compensator block provides the suitable phase lead to compensate for

the phase lag between the exciter input and generator electrical torque.The dy-

namic compensator, used in practice, is made up of several multiple stages of

lead-lag compensators depending upon the requirement of phase compensation to

be provided.

4) PSS Output Limits

Stabilizer output voltage is limited between typical maximum and minimum val-

ues to restrict the level of generator terminal voltage fluctuation during transient

conditions. Large output limits ensure maximum contribution of stabilizers but

generator terminal voltage may face large fluctuation.The main objective in select-

ing the output limits of PSS is to allow maximum forcing capability of stabilizer,

while maintaining the terminal voltage within desired limits. Most commonly used

value of the maximum limit is between 0.1 to 0.2 p.u. , while minimum limit is

taken between -0.05 and -0.1 p.u.

5) Input of PSS

Many signals, like rotor speed deviation, frequency deviation, change in load angle,

change in electrical power etc are possible to use as input signal to PSS. However,

from practical point of view, the following three types of input signals are most

commonly used as input to power system stabilizer:

• Rotor Speed Deviation (∆ ω)

• Frequency Deviation (∆ f)

• Electrical Power Deviation (∆ P)


Though in practice, the speed deviation signal is used as input to PSS, it is in-

herently sensitive to torsional modes of oscillations in the frequency range of 8

to 20 Hz, which can lead to negative damping for torsional mode. Hence, It also

advisable to use torsional filter-typically a low pass filter- for avoiding interaction

of PSS with torsional mode of oscillations.

Time Domain Simulations of SMIB with PSS

The test system discussed in the previous section has also been used to realize the

impact of PSS installation in SMIB.The SIMULINK model of the test system is

depicted in Fig.3.9. From the Fig3.10. and Fig.3.11 of rotor angle deviation and

speed deviation respectively, it can be inferred that, in absence of sufficient stabilizing

control the system has become oscillatory instable. The oscillations of growing nature

has been found that leads the test system to be small signal instable.

After introducing properly tuned PSS in AVR loop, it can be visualize from Fig.3.12

and Fig.3.13 of rotor angle deviation and speed deviation respectively, that the system

oscillations have been sufficiently damped out in short period of time.

Figure 3.9: SMIB Model in MATLAB/SIMULINK


Figure 3.10: Rotor Angle Deviation Before Introducing PSS

Figure 3.11: Speed Deviation Before Introducing PSS


Figure 3.12: Rotor Angle Deviation After Introducing PSS

Figure 3.13: Speed Deviation After Introducing PSS


3.2 Summary

The chapter provides basic analysis on effect of AVR gain on power system stability

that produces power system oscillations of increasing amplitude.For the purpose of

analysis, typical SMIB system has been linearized to get the eigenvalues for different

values of AVR gain. Further, the damping of low frequency oscillations is achieved

by properly tuned PSS. The chapter provide s platform for implementation of power

system stabilizer in multimachine power system which is discussed in subsequent

chapter.

Chapter 4

Optimal Placement of PSS in

Multimachine Power System

4.1 Background

The enhancement of damping of low frequency oscillations in multimachine power

systems by the application of a PSS has become a matter of great attention. It

is much more significant in current scenario where many large and complex power

systems frequently operate close to their stability limits. Though, there is common

perception that the application of PSS is almost a mandatory requirement on all

generators, the use of high price PSS with each and every generator is constrained by

economical limits. In view of the potentially high cost of using a PSS and to assess its

effectiveness in damping poorly damped oscillatory modes to achieve better stability,

identification of the optimum site of PSS is an important task.

4.2 Test System Description

Fig.4.1 shows the two-area benchmark power system which is developed by P.Kundur

considered for choosing optimal location of PSS.

The system contains eleven buses and two areas, connected by a weak tie between

bus 7 and 9. Each area consists of two generators, each having a rating of 900 MVA

and 20 kV. Inertia constant of area-1 generators is 6.5sec and of area-2 generators is

32

CHAPTER 4. OPTIMAL PLACEMENTOF PSS INMULTIMACHINE POWER SYSTEM33

6.17 5sec. Totally two loads are applied to the system at bus 7 and 9. Two shunt

capacitors are also connected to bus 7 and 9 as shown in the figure below. For the

comparisons of the obtained results with that available in literature, the frequency of

operation is chosen as 60 Hz.

Figure 4.1: Single Line Diagram of Test System

The three step methodology adopted for deciding optimal location of PSS is as

follows. Power System Analysis Toolbox-PSAT is used to carry out the following

analysis:

(i) Load Flow Analysis

(ii) Linear (S-Domain) Analysis

(iii) Time Domain Simulations

4.3 Load Flow Analysis

Power flow analysis provides information regarding active and reactive power along

with other pertinent information such as bus bar voltage levels, effect of inphase

and quadrature boost voltages on system loading, reactive power compensation. The

stability of the power system is related to the deficiency or redundancy in generation


against particular load demand, power flow through transmission lines and various

operating constraints. Before executing eigenvalue analysis, to know the status of the

system, means weather the system is operating closer to operating constraints or not,

load flow analysis is performed on the above described system. The detailed system

data is shown in AppendixB. and results achieved by power flow in PSAT are shown

in table hereunder:

Table 4.1: Network Statastics

Elements No.Buses 11Lines 7

Generators 4Transformers 4

Loads 2

Table 4.2: Power Flow Results

Bus V Phase Phase P gen Q gen P load Q load[p.u.] [rad] [degree] [p.u.] [p.u.] [p.u.] [p.u.]

Bus 01 1.03 0.35252 20.19 7 1.7917 0 0Bus 02 1.01 0.18237 10.44 7 2.2053 0 0Bus 03 1.03 -0.11868 -6.79 7.1871 1.6911 0 0Bus 04 1.01 -0.29607 -16.96 7 1.8571 0 0Bus 05 1.0074 0.23984 13.74 0 0 0 0Bus 06 0.98044 0.06428 3.68 0 0 0 0Bus 07 0.96516 -0.08171 -4.68 0 0 9.67 -1Bus 08 0.9532 -0.32301 -18.5 0 0 0 0Bus 09 0.97616 -0.55941 -32.05 0 0 17.67 -2.5Bus 10 0.98614 -0.41348 -23.69 0 0 0 0Bus 11 1.0094 -0.23416 -13.41 0 0 0 0


Table 4.3: Line Flow Results

From Bus To Bus Line P Flow Q Flow P Loss Q Loss[p.u.] [p.u.] [p.u.] [p.u.]

Bus5 Bus6 1 7 0.97147 0.12314 1.1882Bus6 Bus7 2 13.8769 1.1086 0.20163 1.9997Bus7 Bus8 3 2.0132 0.05628 0.04811 0.304Bus8 Bus9 4 1.9651 -0.24772 0.04706 0.29145Bus11 Bus10 5 7.1871 0.83471 0.12855 1.242Bus9 Bus10 6 -13.8549 1.4263 0.20361 2.0192Bus7 Bus9 7 1.992 0.0526 0.09499 0.58713Bus1 Bus5 8 7 1.7917 0 0.82022Bus2 Bus6 9 7 2.2053 0 0.88004Bus4 Bus10 10 7 1.8571 0 0.85693Bus3 Bus11 11 7.1871 1.6911 0 0.85641

Table 4.4: Summerized Power Flow Analysis

Synchronous Machine P[MW] Q[MW] Et [p.u.] Angle[degree]G1 700 185 1.03 20.2G2 700 235 1.01 10.5G3 719 176 1.03 -6.8G4 700 202 1.01 -17

Observations and interpretation of above results can be enlisted as fol-

lows:

• Load angle variations in generator rotor angles

• Power flows from area-1 to area-2 through two ties lines: (1) connected between

buses 7 and 9 (2) connected as bus 7 to 8 and 8 to 9.

• No power mismatch

• Oscillations can be predicted due to tie line power flow and generator rotor

angle variations.

• Provides base data to develop state matrix of the system.


Table 4.5: Summerized Power Flow Results

Total GenerationReal Power [p.u.] 28.1871

Reactive Power [p.u.] 7.5453Total Load

Real Power [p.u.] 27.34Reactive Power [p.u.] -3.5

Total LossesReal Power [p.u.] 0.84709

Reactive Power [p.u.] 11.0453

4.4 Linear Analysis

After performing load flow analysis, S-domain analysis of the test system is carried out

in typical test conditions.The analysis gives eigenplots for different cases considered

and from the movement of eigenvalues, the stability of the considered power system is

predicted.The system is said to be stable if the eigenvalues are far from the imaginary

axis. The system is considered to be in critically damped conditions if eigenvalues

are situated on imaginary axis itself. The eigenvalues in right half of the S-plane

represents unstable mode of oscillations. This has been realized in subsequent sections

of this chapter.

4.4.1 System Behavior in Different Arrangement of PSS

In order to discuss the impacts of different stabilizer arrangement on the power oscil-

lation damping, the following test conditions have been considered:

(1) No PSS;

(2) Install PSS in area-1 at G1 & G2;

(3) Install PSS in area-2 at G3 & G4;

(4) Install PSS in both areas at all generators.

The observations for the above cases are achieved by small perturbation at bus-8 for

the duration of 0.05s. In the corresponding situations, the small signal stability for


the inter-area oscillation has been analyzed in detail with the eigenvalues analysis

method using PSAT Toolbox. PSAT is an add-on MATLAB toolbox for electric

power system analysis and control. PSAT includes power flow, continuation power

flow, optimal power flow, small-signal stability analysis and time domain simulation.

All operations can be assessed by means of Graphical User Interfaces (GUIs) and a

Simulink-based library provides a user friendly tool for network design[9].The details

of PSAT is given in AppendixC.

4.4.2 Mode Identification and Participation Analysis

As a result of eigenvalue analysis, frequency and mode of oscillations have been iden-

tified using the report of small signal analysis in PSAT. It can be illustrated as

hereunder:

Table 4.6: Effects of PSS with Different Arrangement Scheme

Study case Inter-Area Mode Area-1 Local ModeFrequency (Hz) Damping Ratio Frequency (Hz) Damping Ratio

Case-1 0.59 0.0378 1.085 0.08221.054 0.08232

Case-2 0.55 0.1 1.08 0.0821.16 0.122

Case-3 0.60 0.144 1.19 0.1221.05 0.083

Case-4 0.61 0.103 1.19 0.1221.15 0.122

As shown in Table 4.6, four different cases have been compared to check the suitable

positioning of power system stabilizer in the test system to damp out local as well as

inter area mode of oscillations. The four different cases are as mentioned before in

this section.

The generators participating in the inter-area mode of oscillation, on installation of

PSS, offer different frequency of inter-area oscillation. This can be well observed in

Table 4.6. As the frequency of oscillation increases, the damping of oscillation is more

effective due to presence of PSS. From the values of damping ratio shown in Table


4.6, it can be inferred that, the local & inter area both the modes of oscillations

can be damped definitely by placing PSS in both the areas. For damping of local

mode oscillations, there are three choices. One can follow the either of choice from

case no. 2, 3 and 4. The value of damping ratio that we get for local as well as

inter area mode, in case 1, is comparatively smaller than the other cases. Hence it

is not considered as a suitable choice for installation of PSS. In case no. 4 PSS is

installed at all generators. Although it gives best damping effect, but it will result in

redundancy of no. of PSS used. So, in this report case no. 3 has been considered as

suitable choice for placement of PSS, because it provides sufficient damping to local

as well as inter area mode of oscillations. It is being worth noted that the generators

of Area-2 are participating significantly only in inter area mode for considered cases.

They have negligible contribution to local mode oscillations. Hence, not depicted in

Table 4.6. Further this can also be realized from the plot of speed participation factor

v/s generator no. shown in Fig.4.2 and 4.3.Resembling plots can be possible for other

modes of frequency for various cases.

Figure 4.2: Speed Participation Factor v/s Generator No. For Inter-Area Mode-0.54Hz


Figure 4.3: Participation Factor v/s Generator No. For Local Mode-1.05 Hz

4.4.3 S-Domain Plots

Figures shown below are eigenvalues plots for the two-area test system with different

stabilizer arrangements. From Fig.4.4, it can be seen that as for the open loop system

without any installed stabilizer, there is some margin of stability. By installing the

stabilizers in area-2, oscillation modes have been suppressed, and hence system is

enhanced greatly which is shown in Fig. 4.5. If we install the stabilizers in both the

areas, the inter-area mode and two local modes can attain high damping and lower

oscillation frequency shown in Fig.4.6.

Figure 4.4: Eigenvalues of Two Area Test System without PSS


Figure 4.5: Eigenvalues of Two Area Test System with PSS in Area-2

Figure 4.6: Eigenvalues of Two Area Test System with PSS in Both Areas

4.4.4 Time Domain Simulations

For the verification and realization of concept of small signal oscillations, a two area

model described earlier has been developed using MATLAB c©/SIMULINK software

as shown in 4.7. In this model, standard p.u. model of generator is considered. The

generator parameters in per unit on the rated MVA and kV base and the details of


exciter parameter, Step-up transformer, transmission line & load are as per Kundur’s

Two Area test system .

Figure 4.7: MATLAB/SIMULINK Model of Test System

Details of power system stabilizers employed are as in block diagram in Fig.4.8. The

simulations were carried out and various analysis results are depicted pictorially for

clarity.

Figure 4.8: Block Diagram of PSS

The power system stabilizer model is with single input of the rotor speed deviation.

The damping is mostly determined by the gain KS=20, and the following sub-block

of wash out circuit has the high-pass filtering function to ensure the stabilizer has the

relative better response on the speed deviation. There are also lead-lag transfer func-

tions to compensate the phase lag between the excitation model and the synchronous

machine.


Power Flow Plots

In order to represent the concerned oscillation effects, the time domain analysis for

the test system has been performed. Fig.4.9 shows the simulation results on the line

power flow from area 1 to area 2.

It can be observed that the arrangement on stabilizer installation for every machine

in both areas has the best damping effects on inter-area oscillation, which is in unison

with the dominant eigenvalues analysis results discussed in earlier section.

If there is no stabilizer for machines in both areas, the inter-area oscillation is un-

avoidable. The power transfer for such a case would reach to 800 MW with significant

oscillations. This leads to stresses in a weakly coupled tie line. System will be highly

oscillatory and hence may result into instability.

Figure 4.9: Power Flow from Area-1 to Area-2

Comparative analysis of the oscillations between areas indicate that installing the

stabilizers for G3 and G4 in area-2 is a relative optimal solution to damp the inter-

area oscillations. For this particular location of PSS, the constant active power flow

is identified as 413 MW from area-1 to area-2. Thus by proper placement of PSS and

tuning, constant power flow and damped inter area oscillations is achieved.


Per Unit Speed Deviation Plots

Figure 4.10 to Fig.4.13 shown below are simulation results of per unit speed deviation

of each generator in both areas. The results are achieved for perturbed test system to

analyze the behavior of power system stabilizer at various locations.Cases considered

for analysis are same as mentioned in section IV.

For case-1, it is clearly visible from the Fig.4.10, that both the generators of a par-

ticular area deviate in reasonably in harmony. As expected, generators of both the

areas are oscillating in opposite direction. Further the system does not stabilize even

after long time; generators continue to oscillate around a particular operating point.

Figure 4.10: Per Unit Speed Deviation for Case-1

For cases 2 and 3 as evident from Fig.4.11 and 4.12, generators of specific area are

oscillating in particular direction where as in case-4, Fig.4.13 all the generators are

oscillating in unison.

Further it is observed from Fig.4.11 that choice of area-1 as location of PSS is resulting

in damped oscillation. But constraint with this arrangement of PSS is that PSS is

taking long time of around 25 to 30 seconds to mitigate the oscillation completely.

In case-4, if PSS is introduced in both areas at every generator, it is definite from

the Fig.4.13 that the oscillations must die out within very short duration of 6 to 7



seconds. But at the same time it is not favorable situation because placing PSS at

every generator has a redundancy in the solution.


For better choice which is optimized one is case-3. In this case PSS are installed at

both the generators of area-2. As shown in Fig. 4.12, the oscillations are mitigated

in about 10 sec.



4.5 Summary

This chapter presents the power system stabilizer with the consideration of local

and inter-area mode of oscillations, to damp the potential power oscillation. Based

on this, the eigenvalues analysis method has been adopted to analyze the damping

effects of various arrangement schemes of such stabilizer. The case study on the

typical 4-machines 2-area test system shows that although the best arrangement

scheme that install the stabilizer for every machine and area can obtain the best

oscillation damping effect, it is not the economical solution scheme especial to the

large power networks, and the scheme that arrange stabilizer for one area is the

optimal arrangement with the consideration of economical factor.

Chapter 5

Conclusions & Future Work

5.1 Conclusions

The small perturbation stability characteristics of a single machine supplying an in-

finite bus through external impedance have been explored by means of eigenvalue

tracking analysis giving insights into effects of automatic voltage regulator gain, and

stabilizing functions derived from speed and working through the voltage reference

of the voltage regulator. An attempt has been made to understand basic concepts

that explain the stability phenomena and effect of implementation of PSS in power

system.

Further, the stabilizing signals obtained from the PSS help mitigating the inter-area

and local mode oscillations. Studies on PSS placement has been an interest. Using

eigenvalues analysis and from the participation factors of the generators for different

modes, optimal solution to PSS placement was desired. The simulation studies car-

ried out on a test system indicated specific generators, when supplemented by PSS,

helps mitigating oscillations reasonably fast. The work is done on the test system

with identical PSS and optimal solution for the system is proposed, based on which

further extension to actual systems after the zone and area formation seems feasible.

Thus, it is concluded from the results that by performing S-domain analysis, the op-

timal location of the PSS in multi-machine system can be decided. The time-domain

results in the said cases shall be no different than the S-domain results.

46

CHAPTER 5. CONCLUSIONS & FUTURE WORK 47

5.2 Future Work

• As an extension to the presented work, it is suggested to observe results of

employing PSS with various gains and design to achieve quicker oscillation mit-

igation, avoid speed deviations, which may give better insight.

• The future researches on the arrangement rules with evolutionary algorithm and

the coordinated FACTS device to obtain the better power oscillation damping

effects can be concerned and performed.

References

[1] ] Prabha Kundur, “Power System Stability and Control,” The EPRI Power Sys-

tem Engineering Series, 1994, McGraw-Hill, ISBN 0-07-063515-3.

[2] K.R.Padiyar, “Power System Dynamics Stability and Control,” B.S Publications,

2002, ISBN: 81-7800-024-5.

[3] K.N.Shubhanga, “Manual for a Multimachine Transient Stability Programme,”

Version 1.0.

[4] M. Klein, G.J.Rogers, and P.Kundur, “A Fundamental Study of Inter-Area Os-

cillations,” IEEE Trans. on Power Systems, Vol.6, no-3, Aug. 1991, pp. 914-921.

[5] Liao Quingfin, Liudichen, Zeng Cong, Ying Liming, Cui Xue. “Eigen value sen-

sitivities of excitation system, model and parameters,” 3rd IEEE Conf. On In-

dustrial Electronics and Applications, June-2008, pp. 2239-2243.

[6] F.P.DeMello and C.Concordia, “Concepts of Synchronous Machine Stability as

Affected by Excitation Control,” IEEE Trans. on Power Apparatus and Systems,

PAS-88, Apr. 1969, pp.316-329.

[7] Gurunath Gurrala, “Power System Stabilizer Design for Interconnected power

System,” IEEE Trans. on Power Systems, vol.25, no.2, May 2010, pp. 1042-1051.

[8] M.Klein, G.J.Rogers, S.Moorty, P.Kundur, “Analytical Investigation of Factors

Influencing Power System Stabilizer Performance,” IEEE Trans. on Energy Con-

version , vol.7, no.3, Sept. 1992, pp. 382 - 390.

[9] Federico Milano, “Documentation for Power System Analysis Toolbox (PSAT),”

2004.

48

Appendix A

IEEE Proceeding

[1] Devendra P.Parmar, Vihang M.Dholakiya, and Santosh C.Vora, “Optimal Place-

ments of Power System stabilizers: Simulation studies on Test System”, Presented

at 2nd International conference of Current Trends in Technology (NUiCONE’11) at

Nirma University, Ahmedabad , ISBN No.: 978-1-4577-2169-4 .

49

Appendix B

System Data

The system consist of two similar area connected by a weak tie line. Each area con-

sists of two coupled units, The parameters of system as depicted in below tables.

Machines Data:

Table B.1: Machine Data of Two Area Test SystemGen. MVA Xd Xd’ Xd” Tdo’ Tdo” Xq Xq’ Xq” Tqo’ Tqo”No Rating (p.u.) (p.u.) (p.u.) Sec. Sec. (p.u.) (p.u.) (p.u.) Sec. Sec.1 900 1.8 0.3 0.25 8.0 0.03 1.7 0.55 0.25 0.4 0.052 900 1.8 0.3 0.25 8.0 0.03 1.7 0.55 0.25 0.4 0.053 900 1.8 0.3 0.25 8.0 0.03 1.7 0.55 0.25 0.4 0.054 900 1.8 0.3 0.25 8.0 0.03 1.7 0.55 0.25 0.4 0.05

Inertia constant of Area-1 machines are 6.5 (for G1 & G2), Area-2 machines are 6.125

(for G3 & G4). Each step up transformer has an impedance of 0 + j0.15 p.u. on 900

MVA and 20/230 kV base, and has an off-nominal ratio of 1.0.

Line Data:

The Transmission system nominal voltage is 230 kV. The parameters of the lines in

per unit on 100 MVA, 230 kV base are shown in below Table:

50

APPENDIX B. SYSTEM DATA 51

Table B.2: Line Data of Two Area Test SystemFrom Bus To Bus Length km r pu/km xl pu/km bc pu/km

5 6 25 0.0001 0.001 0.001756 7 10 0.0001 0.001 0.001757 8 110 0.0001 0.001 0.001758 9 110 0.0001 0.001 0.001757 9 220 0.0001 0.001 0.001759 10 10 0.0001 0.001 0.0017510 11 25 0.0001 0.001 0.00175

Load Data:

The load and reactive power supplied by the shunt capacitors at buses 7 and 9 are as

follows:

Table B.3: Load Data of Two Area Test SystemBus No. PL (MW) QL (MVAr) QC (MVAr)

7 967 100 2009 1767 100 350

Exciter and PSS Data:

Table B.4: Exciter & PSS Data of Two Area Test SystemKA = 200 TR = 0.01 KSTAB = 20.0 TW = 10.0T1 = 0.05 T2 = 0.02 T3 = 3.0 T4 = 5.4

Appendix C

Introduction to PSAT 2.1.6

PSAT is free and open Source software particularly suited for analysis of Power Sys-

tems Stability and Control. The PSAT is a MATLAB toolbox for electric power

systems analysis and Simulation. All the operation can be accessed by means of

graphical user interfaces (GUIs) and Simulink based library provides an user friendly

tool for network design.

C.1 Overview

PSAT core is powerflow routines, once the powerflow has been solved, further static

and/or dynamic analysis can be performed.Following analysis performed in PSAT

tool box.

• Optimal Power flow (OPF)

• Continuous Power Flow (CPF) / Voltage Stability Analysis

• Small Signal Stability analysis

• Time domain Simulations

• PMU Placement

• Graphical user Interface (GUIs)

• Graphical Network Construction (CAD)

52

APPENDIX C. INTRODUCTION TO PSAT 2.1.6 53

C.1.1 Useful Features

• Mathematical model & utilities

• Bridges to other Programs

• Data format conversion capability

C.1.2 PSAT Utilities

• Simulink library for drawing network.

• GUIs For settings system and routine parameters.

• GUI for plotting results.

• Filters for converting data to and from other formats.

C.2 Starting Process of PSAT

Prilimary steps of PSAT applications in analysis of Power Systems Problems:

a. PSAT Installation

b. Launching PSAT

c. Loading data

d. Running Power flow program

e. Displaying results

f. Saving results

g. Setting


F PSAT Installation

a. Download PSAT version from site :

http://www.uclm.es/area/gsee/Web/Federico/psat.htm

b. Requirements : >> Matlab 6.5

c. Installation

• Extract Zip file in a good location

• Open Matlab

• Add PSAT’s path to Matlab default path

Launching of PSAT

• Type psat.m in MATLAB command window: >> psat.m

This will create all the structures required by the toolbox , then opens psat GUI.

Loading data

Almost all operations require that a data file is loaded. To load a file, Load only a

data file:

• Menu File/Open/Data File

• Short cut <Ctr-d>

Load a saved system:

• File/Open/Saved System

• Short cut <Ctr-y>


The data file can be either a .m file in PSAT format or a .mdl Simulink model created

with the PSAT library. If the source is in a different format supported by the PSAT

format conversion utility, first perform the conversion in order to create the PSAT

data file.

Running Power flow Program

To run the power flow routine:

• Menu Run/Power Flow

• Short cut <Ctr-p>

It is not necessary to load again the data file every time it is modify, because the last

version of the data file is read each time the power flow is performed. The data are

updated also in case of changes in the Simulink model originally loaded.

Other Analysis

• After solving the first power flow, the program is ready for further analysis,

such as CPF , OPF , ...

• Each of these procedures can be launched from the tool-bar or the menu-bar of

the main window.

Displaying Results

Results can be generally displayed in more than one way, either by means of a graph-

ical user interface in MATLAB or as a ASCII text file.

These routines only create a text report:

• Standard Power flow (in a .txt , .xls or .tex file)


• PMU placement (in a .eig file)

• SSSA (Small Signal Stability analysis) (in a .pmu file)

Saving Results

To save actual System status in a .mat file :

• Menu File/Save/Current System

• Shortcut <Ctrl-a>

All global structures used by PSAT are stored in this file which is placed in the folder

of the current data file. all static computations allow to create a report in a text file

that can be stored and used later.

Settings

The main settings of the system are directly included in the main window:

• Base frequency

• Base Power

• Start & End Simulation Time

• static and dynamic tolerance

• Maximum number of iterations

Other general settings

• Menu Edit/General Settings or <Ctr-k>

Network Design

PSAT allows drawing electrical schemes by means of pictorial blocks. To launch

PSAT library


• click on Simulink icon in the menu-bar,

• the menu Edit/Network/Edit Network or

• the short cut <Ctr-s>.

PSAT comes with variety of procedures for static and dynamic analysis, several mod-

els of standard and unconventional devices, a complete GUI, and a Simulink-based

network editor. These features make PSAT suited for both educational and research

purposes.

C.3 Useful Links

Following links related to power systems:

a. http://www.engin.umich.edu/group/ctm/ : Control Tutorial for MATLAB

b. http://www.dmoz.org/Science/Technology/Software for Engineering/

Electrical Engineering/Power Distribution/ : Open Software for Power Systems

c. http://www.uclm.edu/area/gsee/Web/Federico/psat.htm :PSAT software down-

load

d. http://www.ece.mtu.edu/faculty/ljbohman/peec/Dig Rsor.htm : Opensource

Educational Software (Power Engineer Education Committee)

e. http://www.animations.physics.unsw.edu.au/jw/electricmotors.html : Electri-

cal Motor and Generator Fundamental Tutorial

f. http://www.ee.iitb.ac.in/trivedi/latex.htm : Help on Latex Software

g. http://www.latex-project.org

Index

Classification of Power System Stability, 5

Eigenvalue Analysis Method, 9

Eigenvalue Tracking Analysis, 22

General Structure of PSS, 26

Haffron-Phillips Model of Excitation System,

18

Interpretation of K-constants, 19

Small Signal Stability, 7

Synchronizing and Damping Torque Analy-

sis, 12

58

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