power system oscillations - graham rogers

POWER SYSTEM OSCILLATIONS

THE KLUWER INTERNATIONAL SERIES IN ENGINEERING AND COMPUTER SCIENCE

Power Electronics and Power Systems Series Editor

M. A. Pai

Other books in the series:

STATE ESTIMATION IN ELECTRIC POWER SYSTEMS: A Generalized Approach A. Monticelli, ISBN: 0-7923-8519-5

COMPUTATIONAL AUCTION MECHANISMS FOR RESTRUCTURED POWER INDUSTRY OPERATIONS

Gerald B. Sheble, ISBN: 0-7923-8475-X ANALYSIS OF SUB SYNCHRONOUS RESONANCE IN POWER SYSTEMS

K.R. Padiyar, ISBN: 0-7923-8319-2 POWER SYSTEMS RESTRUCTURING: Engineering and Economics

Marija Ilic, Francisco Galiana, and Lester Fink, ISBN: 0-7923-8163-7 CRYOGENIC OPERATION OF SILICON POWER DEVICES

Ranbir Singh and B. Jayant Baliga, ISBN: 0-7923-8157-2 VOLTAGE STABILITY OF ELECTRIC POWER SYSTEMS, Thierry

Van Cutsem and Costas Vournas, ISBN: 0-7923-8139-4 AUTOMATIC LEARNING TECHNIQUES IN POWER SYSTEMS, Louis A.

Wehenkel, ISBN: 0-7923-8068-1 ENERGY FUNCTION ANALYSIS FOR POWER SYSTEM STABILITY,

M. A. Pai, ISBN: 0-7923-9035-0 ELECTROMAGNETIC MODELLING OF POWER ELECTRONIC

CONVERTERS, J. A. Ferreira, ISBN: 0-7923-9034-2 MODERN POWER SYSTEMS CONTROL AND OPERATION, A. S. Debs,

ISBN: 0-89838-265-3 RELIABILITY ASSESSMENT OF LARGE ELECTRIC POWER SYSTEMS,

R. Billington, R. N. Allan, ISBN: 0-89838-266-1 SPOT PRICING OF ELECTRICITY, F. C. Schweppe, M. C. Caramanis, R. D.

Tabors, R. E. Bohn, ISBN: 0-89838-260-2 INDUSTRIAL ENERGY MANAGEMENT: Principles and Applications,

Giovanni Petrecca, ISBN: 0-7923-9305-8 THE FIELD ORIENTATION PRINCIPLE IN CONTROL OF INDUCTION

MOTORS, Andrzej M. Trzynadlowski, ISBN: 0-7923-9420-8 FINITE ELEMENT ANALYSIS OF ELECTRICAL MACHINES, S. J. Salon,

ISBN: 0-7923-9594-8

POWER SYSTEM OSCILLATIONS

Graham Rogers Cherry Tree Scientific Software

~ .

Kluwer Academic Publishers Boston! ILondorv'Dordrecht

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Library of Congress Cataloging-in-Publication Data

A C.I.P. Catalogue record for this book is available from the Library of Congress.

Copyright 2000 by Kluwer Academic Publishers.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, mechanical, photo-copying, recording, or otherwise, without the prior written permission of the publisher, Kluwer Academic Publishers, 101 Philip Drive, Assinippi Park, Norwell, Massachusetts 02061

Printed on acid-free paper.

Notice to Readers

I have prepared a CD_ROM containing the data and results for all examples used in this book. The data are in the form ofMATLAB m-files, and the results are given as MA TLAB binary files.

The CD_ROM is available, at cost. Please contact me at

Cherry Tree Scientific Software RR#5 Colborne, Ontario, KOK ISO CANADA

or by email: [email protected]

Graham Rogers

Dedication

I dedicate this book to Jean, my wife, best friend, and constant companion, who has sustained and supported me in my endeavours for so many years.

Contents

1 Introduction 1 2 The Nature of Power System Oscillations

1 Introduction 7 2 Classical Generator Model 10

2.1 Local Modes 11 2.2 Inter Area Mode 14

2.2.1 Case 1 15 2.2.2 Case 2 16

3 Detailed Generator Model 17 3.1 Local Modes 18 3.2 Inter Area Mode 19

4 Controlled Detailed Generator Model 23 4.1 Local Modes and Inter Area Mode 24

5 Response to System Faults 27 6 Final Discussion and Comments 28

6.1 Classical Generator Model 29 6.2 Detailed Generator Model 29 6.3 Detailed Generator Model with Turbine/Governor Model 30 and Automatic Voltage Control Model

7 References 30 3 Modal Analysis of Power Systems

1 Introduction 31 2 Modal Analysis of Linear Dynamic Systems 32

2.1 Example 34

4

5

Contents

2.1.1 Lag Block 2.1.2 Lead Lag Block 2.1.3 Transient Feedback Block 2.1.4 The Complete State Space Model 2.1.5 Power System Example

2.2 Eigenvectors 2.2.1 Modes of Oscillation 2.2.2 Example 2.2.3 Equal Eigenvalues

2.3 Eigenvalue Sensitivity 2.3.1 Participation Factors

3 Modal Analysis Applied to the Detailed Generator Case with and without Controls 3.1 Detailed Generator Model 3.2 Detailed Generator Model with Controls

3.2.1 Real Eigenvalue 3.2.2 Unstable Complex Mode 3.2.3 Stable Oscillatory Modes

3.3 Step Response 4 Final Comments and Discussion 5 References Modal Analysis for Control 1 Introduction 2 Transfer Functions

2.1 Transfer Function Poles and Zeros 2.1.1 Example

2.2 Controllability and Observability 2.3 Residues

2.3.1 Sensitivity and Residues 2.3.2 Example 2.3.3 Root locus and Residues 2.3.4 Sensitivity to Dynamic Feedback 2.3.5 Example

2.4 Frequency Response 2.4.1 Nyquist's Stability Criterion 2.4.2 Example 2.4.3 Application to Feedback Systems 2.4.4 Power System Example

3 Synchronizing and Damping Torques 4 Summary and Conclusions 5 References Power System Structure and Oscillations 1 Introduction 2 Coherent Generator Groups

2.1 Ideal Coherency in a Multiple Generator Plant 3 Coherency in an Interconnected Power System

3.1 Reference Generators 3.2 Bus Coherency

Vlll

34 35 36 37 38 40 41 42 46 50 50 54

54 58 60 61 64 70 72 73

75 75 76 77 82 82 83 84 88 88 89 90 91 92 93 94 97 100 100

101 102 102 105 106 108

ix

3.3 Example 16 generator 68 bus system 108 4 Tie Line Influence on Inter-area Mode Stability 115

4.1 Response to a fault 116 5 Comments on System Structure 118 6 References 119

6 Generator Controls 1 Introduction 121 2 Speed Governor Controls 122

2.1 Hydraulic Turbine Governors 123 2.2 Thermal Turbine Governors 127 2.3 Turbine Governor Effects on Low Frequency Inter-area 128

Modes

3 Excitation Controls 129 3.1 Open Circuit Stability and Response 129

3.1.1 Dc Exciter 130 3.1.2 Static Exciter 135

4 References 137 7 Power System Stabilizers

1 Introduction 139 2 Power System Stabilizer Basics 140

2.1 Example - Single Generator Infinite Bus 141 2.1.1 Static Exciter 143

2.1.1.1 Compensation Determination using Residue 145 Angle

2.1.1.2 Compensation of the Phase Lag between V ref 148 and Electrical Torque

2.1.2 Rotating DC Exciter 149 3 Stabilization of a Complete System 154

3.1 Power System Stabilizer Placement 154 3.2 Power System Stabilizer Design 156

3.2.1 Generator 11 157 3.2.2 Generator 8 158 3.2.3 The Remaining Generators 159

4 Evaluation of Power System Stabilizer Performance 166 4.1 Small Signal Performance 166 4.2 Transient Stability 167

5 Comments 168 6 References 168

8 Power System Stabilizers - Problems and Solutions 1 Introduction 171 2 Generator Torsional Oscillations 171

2.1 Speed Input Stabilizers 172 2.1.1 Application of Torsional Filters 175

2.2 Power Input Stabilizers 179 2.2.1 Example 181

2.3 DeltaP/Omega Stabilizers 182 2.3.1 Example 184

9

10

Contents

3 Power System Stabilizers at a Plant of Identical Generators 3.1 Solutions

4 Comments 5 References Robust Control

x

185 194 196 197

1 Introduction 199 2 Performance Specifications 200

2.1 System Model Conventions 201 2.1.1 One Degree of Freedom System 204

2.1.1.1 Sensitivity Matrices 204 2.2 Power System Example 207 2.3 Power System Performance in Control Terms 212

2.3.1 Step 1 Normalization 212 2.3.2 Step 2 Performance Specification 215

2.4 System Performance without Controls 216 2.4.1 Excitation System 216 2.4.2 Turbine Control Valve Servo 219

2.5 System Performance with Automatic Voltage Regulator 221 2.5.1 Performance with Power System Stabilizer 222

2.6 System Performance with Turbine Governor 225 2.7 Comments 226

3 Robust Control 227 3.1 Performance Weights 228

4 Robustness of Power System Controls 234 4.1 Nominal Performance 235 4.2 Robust Stability 237 4.3 Robust Performance 240

4.3.1 Structured Singular Value 244 4.4 Decentralized Controls 246

5 Robust Control Design 247 5.1 Coprime Factors and Coprime Factor Uncertainty 248

6 Final Comments 250 7 References 251 Damping by Electronic Power System Devices 1 Introduction 2 System Performance without Electronic Controls

2.1 Contingency Performance 3 Static V Ar Compensators

3.1 SVC Location, Poles and Zeros 3.2 Damping Control Design

3.2.1 Damping Control to Modify Residue at Inter-area Mode

3.2.2 Robust H~ Loop Shaping Control 3.2.2.1 System Reduction

3.3 Control Performance 3.3.1 Small Signal Stability Performance 3.3.2 Transient Stability Performance

3.4 Comments

253 254 258 258 261 262 263

265 266 270 270 273 274

xi

4 Thyristor Controlled Series Capacitor 276 4.1 Performance with an Uncontrolled Series Capacitor 276 4.2 Damping Control Design 277

4.2.1 Residue Based Design 279 4.3 Robust Loop Shaping Design 281 4.4 Transient Performance 284 4.5 Comments on TCSC Damping Control 284

5 High Voltage DC Link Modulation 285 5.1 System Performance without Damping Control 286 5.2 Damping Control Input 286 5.3 Residue Based Damping Control 290 5.4 Robust Damping Control 291 5.5 System Performance with Robust Control 297

5.5.1 Transient Performance with Damping Control 298 5.6 Comments on HVDC Damping Control 298

6 General Comments 299 7 References 299

Al Model Data Formats and Block Diagrams 301 1 Load Flow Data 302 2 Dynamic Data 303

2.1 Generator 303 2.2 Exciter System Data 305 2.3 Turbine/ Governor Data 308

3 HVDCData 309 4 Case Data 311

4.1 Two-Area Test Case 311 4.2 Two-Area Test Case with Series Capacitor 312 4.3 Two-Area System with Parallel HVDC Link 314 4.4 16 Generator System 315 4.5 Single Generator Infinite Bus System 317 4.6 Multiple Generator Infinite Bus System 318

A2 Equal Eigenvalues 1 Nonlinear Divisors 319

1.1 Example 320 1.2 Time Response Calculation with Nonlinear Divisors 322

2 Linear Divisors 323 2.1 Example 324

Index 327

Chapter 1

Introduction

Electric power systems are among the largest structural achievements of man. Some transcend international boundaries, but others supply the local needs of a ship or an aeroplane. The generators within an interconnected power system usually produce alternating current, and are synchronized to operate at the same frequency. In a synchronized system, the power is naturally shared between generators in the ratio of the rating of the generators, but this can be modified by the operator. Systems, which operate at different frequencies, can also be interconnected, either through a frequency converter or through a direct current tie. A direct current tie is also used between systems that, while operating at the same nominal frequency, have difficulty in remaining in synchronism if interconnected.

Alternating current generators remain in synchronism because of the self-regulating properties of their interconnection. If one machine deviates from its synchronous speed, power is transferred from the other generators in the system in such a way as to reduce the speed deviation. The moments of inertia of the generators also come into play, and result in the speed overcorrecting in an analogous manner to a pendulum swinging about its equilibrium; the pendulum inertia is equivalent to the generator inertia, and the torque on the pendulum due to gravity is equivalent to the synchronizing torque between the generators in the power system. However, generators are much more complicated dynamic devices than are pendulums, and one must not be tempted to put too much emphasis on this analogy. However, it is true to say that power system oscillations are as natural as those of pendulums.

G. Rogers, Power System Oscillations Kluwer Academic Publishers 2000

2

An interconnected power system cannot operate without control. This is effected by a combination of manual operator controls and automatic controls. The operators control the power that the generator supplies under normal operating conditions, and the automatic controls come into play to make the fast adjustments necessary to maintain the system voltage and frequency within design limits following sudden changes in the system. Thus, most generators have speed governing systems which automatically adjust the prime mover driving the generator so as to keep the generator speed constant, and voltage regulating systems which adjust the generators' excitation to maintain the generator voltages constant. These controls are necessary for any interconnected power system to supply power of the quality demanded by today's electric power users. However, most automatic controls use high gain negative feedback, which, by its active nature, can cause oscillations to grow in amplitude with time. The automatic controls in power systems must, as with other automatic feedback controls, be designed so that oscillations decay rather than grow.

This then brings us to the reason for this book. It is to discuss the nature of power system oscillations the mathematical analysis techniques necessary to predict system

performance control methods to ensure that oscillations decay with time

Oscillations were observed in power systems as soon as synchronous generators were interconnected to provide more power capacity and more reliability. Originally, the interconnected generators were fairly close to one another, and oscillations were at frequencies of the order of I to 2 Hz. Amortiseur (damper) windings on the generator rotor were used to prevent the oscillations amplitudes increasing. Damper windings act like the squirrel cage winding of an induction motor and produce a torque proportional to the speed deviation of the rotor from synchronous speed. They absorb the energy associated with the system oscillations and so cause their amplitudes to reduce.

As power system reliability became increasingly important, the requirement for a system to be able to recover from a faults cleared by relay action was added to the system design specifications. Rapid automatic voltage control was used to prevent the system's generators loosing synchronism following a system fault. Fast excitation systems, however, tend to reduce the damping of system oscillations. Originally, the oscillations most affected were those between electrically closely coupled generators. Special stabilizing controls (Power System Stabilizers) were designed to damp these oscillations.

In the 1950s and 1960s, electric power utilities found that they could achieve more reliability and economy by interconnecting to other utilities,

1. Introduction 3

often through quite long transmission lines. In some cases, when the utilities connected, low frequency growing oscillations prevented the interconnection from being retained [1]. In some instances, lowering automatic voltage regulator gains was all that was necessary to make the system interconnection successful. However, in other cases the interconnection plans were abandoned until asynchronous HYDC interconnections were technically possible. AC tie lines became more stressed, and low frequency oscillations between some interconnected systems were found to increase in magnitude. In the worst cases, these oscillations caused the interconnection to be lost with consequent inability to supply customer load.

From an operating point of view, oscillations are acceptable as long as they decay. However, oscillations are a characteristic of the system; they are initiated by the normal small changes in the systems load. There is no warning to the operator if a new operating condition causes an oscillation to increase in magnitude. An increase in tie line flow of as little as 10 MW may make the difference between decaying oscillations which are acceptable and increasing oscillations which have the potential to cause system collapse. Of course, a major disturbance may finally result in growing oscillations and system collapse. Such was the case in the August 1996 collapse of the

1800

1600

1400 'h~ .,. 1200

5: :2 1000

~ 0

= 800 :;; ~ 0 "'- 600 Q)

: 400

200

0 I-::

-200 350 400 450 500 550 600 650 700 750 800

time s

Figure 1. Line flow transient - August 10, 1996 western USA/Canada system

4

western US/Canada interconnected system. The progress of this collapse was recorded by the extensive monitoring system, which has been installed [2], and its cause is explained clearly in [3]. A record of the power flow in a major transmission line is shown in Figure 1. The recording starts well before the incident, which triggered the system's collapse, and continues until the line is disconnected. Details of this record in Figure 2 and 3 show the response of the system to the initial fault, and to subsequent smaller disturbances. The system oscillates at about 0.26 Hz and the oscillations decay. Such oscillations, which may last for 30 s, are not noticeable by the system's operators unless they have special instrumentation that detects them. The final collapse was caused by the growing oscillations shown in Figure 4. The decaying oscillations of figures 2 an 3 were turned into growing oscillations by the sequence of faults and protective relay operations. The amplitude of the oscillations eventually caused the system to split into a number of disconnected regions, with the loss of power to a considerable number of customers.

14~.-------------~------------~------------~

1400

3: 1380 2

~ ~ 1360

~ Co Q)

:. 1340

1320

1300 '--____________ -'--____________ --'-____________ ---1 350 400 450 500

time s

Figure 2. Detail of transient showing decaying oscillations following the initial fault

1. Introduction

1420r-------------~------------_.------------~

1400

5 1380 :2

~ ~ 1360

~ 0..

~ 1340

1320

1300~------------~------------~------------~ 500 550 600 650

time s

Figure 3. Detail of oscillations caused by a sequence of small disturbances

1500

1450

1400

1350 s: :::!: 1300

'" 0

~ 1250 '" 0

~ 1200 :

1150

1100

1050

1000 700 750 800 850

time s

Figure 4. Detail oftransient showing growing oscillations

5

6

Even now, it is often difficult to explain why increasing oscillations occur in a specific system. As recently as 20 years ago, the mathematical tools which are needed to analyze power system oscillations and to design successful damping controls were not available. Today, there are no problems with analysis tools, but oscillation dynamics are not always easy to understand. Another area, which is still being addressed, is the provision of accurate dynamic data for use in the analytical models.

I hope that this book will help to increase power system engineers' awareness of oscillations sufficiently to encourage them to treat oscillations seriously and to set up accurate system models. It is based, largely, on my own experience within the planning division of a large utility. I will assume very little prior knowledge of power system oscillations and their control. However, I cannot cover these aspects in detail and, in the same volume, cover the basis and detail of the models used for analysis. Fortunately, other books are available which cover this material in depth. The ones which I have used most, and recommend for their readability and depth of coverage are, 'Power System Stability and Control' by Prabha Kundur [4], and 'Power System Dynamics and Stability' by M.A. Pai and P. Sauer [5].

Data for each of the systems used in this book is given in the Appendix as MATLAB [6] matrices. The results of simulations together with the simulation data may be found on the CD-ROM supplied with this book. All analysis was performed using the Power System Toolbox [7] for MA TLAB.

1 REFERENCES

1. Inter-area Oscillations in Power Systems, IEEE Power Engineering Society, Special Publication 95 TP 101, 1995

2. J.P. Hauer, DJ. Trudinowski, GJ. Rogers, W.A. Mittelstadt, W.H. Litzenburger, and J.M. Johnston, 'Keeping an eye on power system dynamics', IEEE Computer Applications in Power, October 1997, pp. 50-54.

3. Carson W. Taylor, 'Improving grid behavior', IEEE Spectrum, June 1999, pp. 40-45. 4. Prabha Kundur, Power System Stability and Control, McGraw-Hili Inc., New York,

1993. 5. P.W. Sauer and M.A. Pai, Power System Dynamics and Stability, Prentice Hall, New

Jersey, 1997. 6. Using MATLAB, The MathWorks Inc., Natick, 1999. 7. Power System Toolbox, Cherry Tree Scientific Software, Colbome, 1999.

Chapter 2

The Nature of Power System Oscillations

1 INTRODUCTION

Power system oscillations are complex, and they are not straightforward to analyze. Therefore, before going into any detail, I will use an example to show the basic types of oscillations that can occur. The example two-area system is artificial; its model was created for a research report commissioned from Ontario Hydro by the Canadian Electrical Association [1,2] to exhibit the different types of oscillations that occur in both large and small interconnected power systems. A single line diagram of the system is shown in Figure 1. There are two generation and load areas interconnected by transmission lines. Each area has two generators. The generators and their controls are identical. The system is quite heavily stressed; it has 400 MW flowing on the tie lines from area 1 to area 2. In all cases, the active load is modelled as 50% constant current and 50% constant impedance; the reactive load is modelled as constant impedance. Using the two-area system as the basis, I will discuss the different types of oscillations that can occur in this system and, by implication, other interconnected systems.


8

G1 G3

11

10 110 3 101 13

20 120

4 14 2 12

G4 G2

Figure 1. Single Line Diagram of Two-Area System

I will also consider the following different model complexities l :

1. The generators are modelled as 'classical'. Each classical generator model has two dynamic variables:

the angle of the generator's internal voltage the generator's speed deviation from synchronous speed.

2. The generator models are detailed, but with no additional automatic controls. Each detailed generator model has six dynamic variables:

the rotor angle the rotor speed the field flux linkage the direct axis rotor damper winding flux linkage the two flux linkages associated with the quadrature axis damper windings

3. The generator model is detailed and models of the excitation control and speed governor are included. The excitation control is a fast acting thyristor-based system. The turbine is a steam turbine with a HP and LP stage and a fast acting governor. This model has five additional dynamic variables:

the output of the voltage transducer the automatic voltage regulator output three governor/turbine variables

I The models' are described in Appendix I, and data are given on the CD-ROM

2. The Nature of Power System Oscillations 9

The simple model shows the fundamental electromechanical oscillations that are inherent in interconnected power systems. There are three different electromechanical modes2 of oscillation, one less than the number of generators. These are:

two local modes one in which generator 1 oscillates against generator 2, the other in which generator 3 oscillates against generator 4

one inter-area mode in which the generators in area 1 (generators 1 and 2) oscillate against those in area 2 ( generators 3 and 4)

The more detailed models allow exploration the less fundamental, but still very important, effects of the generator and its controls on the system's oscillations. The controls introduce additional oscillations as well as modifying the basic electromechanical modes of oscillation. Checking that there is no detrimental interaction between controls and the interconnected power system is part of a control's design. I will consider this in detail in later chapters.

In this initial study of power system oscillations, I will use nonlinear simulation. Although, as we will see later, much of the information about oscillations may be obtained more directly by applying modal analysis to a linearized system model. Nonlinear simulation is the normal tool used by power system operators and planners to study power system dynamics. In normal usage, nonlinear simulations resolve the question of whether or not a power system will recover successfully following severe faults, for example, a three phase fault cleared by line removal. If the system recovers from the fault, oscillations can often be seen in the simulation of the post fault system. If the simulation is continued for a sufficiently long time, it is possible to determine whether the oscillations decay with time ( they are stable), or continue at a constant amplitude or increase in amplitude (they are unstable).

Here, I will use the nonlinear simulation to give a physical feel for the types of oscillations that occur in power systems, and the way in which they are affected by standard system controls.

2 Mode is the technical term for a specific oscillation pattern, it is discussed in more detail in Chapter 3. It is often used, more loosely, to refer to an oscillation at a specific frequency.

10

2 CLASSICAL GENERATOR MODEL

Synchronously connected generators represented by classical generator models exhibit only electromechanical oscillations. Electromechanical oscillations are those associated with the tendency for the generators to remain in synchronism when interconnected. Using classical models for all four generators will allow me to demonstrate the three modes of electromechanical oscillation in the two-area system. While the details change when the generators and their controls are modelled more accurately, the nature of the electromechanical oscillations remains the same. I will apply small disturbances to the generators' mechanical torques that excite the different modes of oscillation, and examine the resulting responses of the generator speeds and the tie bus voltages.

X 10.4 4

a. 3 1- gen1 c: 0

....... gen2 .~ 2 .;;: Q) -c -c Q)

~ 0 '"

2 3 4 5 6 7 8 9 10

X 10.4 3

:> 1- gen3 ~ 2 .S! ....... gen4 ro .;;: 1 Q) -c -c Q) 0 Q) 0..

'"

-1 0 2 3 4 5 6 7 8 9 10

time 5

Figure 2. Change in Generator speeds - change in torque 0.01 at generator 1, -0.01 at generator 2


2.1 Local Modes

There are two local modes of oscillation, one in each area. Using a disturbance applied to a single generator, it is not possible to excite a local mode without also exciting the inter-area mode. Nevertheless, by using an equal and opposite change in the generator mechanical torque on the generators in one area, the local mode in that area is made dominant.

The responses of the generators' speeds to a step change in the mechanical torque at generators 1 and 2 are shown in Figure 2. The change in mechanical torque at generator 1 is 0.01 pu on the generator base, and at generator 2 it is -0.01 pu. It can be seen that there are oscillations in the speeds. In area 1, the speed changes oscillate at a frequency of about 1.1 7 Hz. The speed changes of generator 1 and generator 2 at this frequency are in antiphase - generator 1 is oscillating against generator 2. In area 2, the generators oscillate at lower amplitude. At the start of the transient, the generators in area 2 move together at a lower frequency (0.53 Hz). This corresponds to the frequency of the inter-area mode. The mode local to area 2 is also excited. It is at the same frequency as the local mode in area 1, but it is 90 out of phase with that mode. The increase in the mean speed over the duration of the transient is caused by a reduction in the system load,

X 10.4 3~~.---.---.---.---.---.---.---.---,----

~ 2 1-- gen1 1 o ...... gen2 '~ iii 1 "'0 ,.' ' -0 g: 0 a.

'"

a 3 c: o .~ 2 .:;:

'" -01

2 3 4 5 time s

6 7

.....

8 9 10

Figure 3. Change in Generator speeds - change in torque 0.01 at generator 3, -0.01 at generator 4

12

which in tum is caused, by a reduction in the average load voltage. Classical generator models have no inherent damping. Thus, the

oscillations are continuous once initiated. In addition, there is no governor to return the speed of the generators back to their original synchronous speed.

The response of the generator speeds to a step change in mechanical torque at generators 3 and 4 is shown in Figure 3. It can be seen that in this case the largest magnitude oscillations are those of the speeds of the generators in area 2. In area 1, the inter-area mode is dominant initially. It has a frequency of about 0.5 Hz. The local mode in area 1 increases in amplitude and is 90 out of phase with the local mode in area 2.

The two local modes and the inter-area mode are the three fundamental modes of oscillation the two area system. They are each due to the electromechanical torques which keep the generators in synchronism. The frequencies of the oscillations depend on the strength of the system and on the moments of inertia of the generator rotors.

The amplitudes of the local modes in the area in which there is no disturbance increase with time. While this is the correct response, it should not in this case be interpreted as indicating a growing mode of oscillation. In this system model, all the oscillations remain at constant amplitude once excited, and the mechanism for the growth of the oscillations is the effect of beating between the two local modes, which have almost identical frequencies. It is this question of interpretation that is answered by the analysis of a linearized system model.

The same oscillations can be observed in other system variables. Figures 4 and 5 show the oscillations in the magnitude of terminal voltage at the tie buses (3 and 13). These buses are at each end of the tie lines between the areas. Again, one can see the local and inter-area modes of oscillation in the responses. Since the frequencies of the local modes are so close, it is not possible to recognize them separately in the tie bus voltage responses.

2. The Nature of Power System Oscillations

" . Q)

'" '" 'E > 0.9855

0.985

0.98450'---'---..:'..2----'3----'4'--5-'---.J..6----'-7----'8--9'-----'10 lime s

Figure 4. Change in tie line bus voltages - change in torque 0.01 at generator I, -0.01 at generator 2

0.98721---.----.-~---,----.--_r_-..----""?======il 1 - bus3 1 0.987

0.9868

0.9866

a 0.9864 Q) C>

~ 0.9862 >

0.986

0.9858

0.9856

mm bus 13

0.9854 '---_1...-_-'--_-'--_-'---_-'--_-'--_-'--_-'--_-'------' o 2 3 4 5 6 7 8 9 10

lime s

Figure 5. Change in tie bus voltages - change in torque 0.01 at generator 3, -0.01 at generator 4

13

14

The voltage at the sending end of the tie (bus 3) contains a large component at the inter-area mode frequency for both disturbances. With the disturbance in area 1, the oscillations at the receiving end tie bus (bus 13) are smaller in amplitude. With the disturbance in area 2, the receiving end bus voltage oscillations are dominated by the local mode in area 2. For both disturbances, the average voltage at the tie bus closest to the disturbance is reduced. This reduction in voltage leads to the reduction in load that causes the generator speeds to increase.

X 10.4 20

5..15 1- gen1 1 c: ...... gen2 0 ~ 10 .s: Q)

"C 5 ,0 .-"C Q) Q) 0 c.. '"

-5 0 2 3 4 5 6 7 8 9 10

X 10.4 20

5.. 15 1- gen31 c: ... --- gen4 0 '~ 10 'S: Q)

"C 5 "C Q) Q) 0 0.. '"

-5 0 2 3 4 5 6 7 8 9 10

time s

Figure 6. Change in generator speeds change in torque 0.01 generator 3, -0.01 generator 1

2.2 Inter Area Mode

The inter-area mode can be excited more directly by changing the mechanical torque at one generator in both areas. I will consider two cases

l. the torque is increased at generator 1 by 0.01 pu and decreased at generator 3 by 0.01 pu

2. the torque is increased at generator 3 by 0.01 pu and decreased at generator 1 by 0.01 pu


0.99 l----r----,-----r----,---,----,--.-------;===::::::;l 1

- bus3 1

" 0.

0.988

0.986

F 0.984 i

0.982

0.98

...... bus 13

0.978 '----''---'---'------'-----'-----'---'-----'-----'------' o

2.2.1 Case 1

2 3 4 5 time s

6 7

Figure 7. Change in tie bus voltages

8

change in torque 0.01 generator I, -0.01 generator 3

9 10

15

The generator speed changes are shown in Figure 6. The inter-area oscillation may be seen in all generator speeds. However, the amplitude of the oscillation is larger in the responses of the area 2 generators (3 and 4). The inter-area oscillation magnitude is less in the responses of the generators in area 1 generators (1 and 2). On the other hand, the local mode is more in evidence in the speeds of the generators in area 1. The frequency of the inter-area oscillation is about 0.53 Hz.

The voltage magnitude response at the tie buses (3 and 13) is shown in Figure 7. Here we can see a marked difference in the responses. The inter-area oscillation is dominant in both bus voltages, but it has a much higher magnitude at bus 3 than at bus 13. There is some evidence of a local mode in the response at bus 13. The difference in response is due to the unbalance caused by the flow on the tie lines. It can be seen that the average voltage magnitude at bus 3 is lowered by the disturbance. This causes the system loads to be reduced, and in turn, causes the generator power to be reduced. Since the torque applied to the generator rotors is constant, the torque imbalance causes the generator rotors to accelerate and the rotor speeds to mcrease.

16

2.2.2 Case 2

The response of the generator speeds is shown in Figure 8. The oscillations are similar to those in Figure 6. However, the speed decreases. This is because, in this case, the average load voltage is increased by the disturbance.

The tie bus voltage responses are shown in Figure 9. These too have a similar response pattern to case 1.

X 10.4 5

'" 1= gen1 1 0.. 0 gen2 " 0 .... '" " .~

-"'0, 'S'

-5 '" "C

"C '.

," -.. '

..

~ -10 .. ," 0..

"' -.....

.. "

-15 0 2 3 4 5 6 7 8 9 10

X 10'4 5

'" 1= gen3 I 0.. 0 " ' '. gen4 " 0 .~ ," "" 'S'

-5 '" "C

"C '" -10 '" 0..

"'

-15 0 2 3 4 5 6 7 8 9 10

time s

Figure 8, Change in generator speeds - change in torque 0.01 at generator 3, -0.01 at generator 1


0.995 i---r-----r-----r----r----.---r--.,..----;:===:::::;-, 1

- bus3 1 0.994

0.993

0.992

~ 0.991 '" "" '" ~ 0.99

0.989

2 3 4 5 time s

6 7

nnn bus 13

B 9 10

Figure 9. Change in tie bus voltages - change in torque 0.01 at generator 3, -0.01 at generator I

3 DETAILED GENERATOR MODEL

17

In this section, I will repeat the simulations of section 2 with identical detailed generator models replacing the classical generator models. The detailed model has one damper winding on the direct axis and two on the quadrature axis. Since there are no generator controls, the field voltages and the rotor mechanical torques are kept constant throughout the simulations. The transmission system model and the disturbances applied are identical to those used in the classical generator simulation.

The generators' damper windings produce torques proportional to speed deviation from synchronous speed. This causes both the local and inter-area modes of oscillation to decay. Because no governor is modelled, the speeds of the generators are not controlled. As in the classical generator case, the generators try to stay in synchronism, but, if the average load voltages increase, the speeds will decrease; if the load voltages decrease, the speeds will increase.

18

3.1 Local modes

The responses of the generator speeds to generator torque changes in area 1 are shown in Figure 10. The amplitude of the local oscillations decays in the first half of the transient. The area 1 local mode is dominant in the area 1 generators' responses and are in antiphase, which indicates that the generators in area 1 are oscillating against one another. The generators in area 2 oscillate together at the inter-area mode frequency and the inter-area mode becomes dominant in the response of all generators as the local mode decays. The overall trend is for the speed to increase, due to the reduction in the average load voltage. The responses of voltages at the tie buses are shown in Figure 11. As in the classical generator case, the inter-area mode is dominant, but the local mode is difficult to detect in the voltage response. As the simulation progresses, the average value of both tie bus voltages reduces: the voltage reduction at the sending end is larger than that at the receiving end.

~ 10 c: o .~ .~ 5 -c -c i 0 -.. ,' '"

I gen1 I ...... gen2

_5L---~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~ __ ~

o 2 3 4 5 6 7 8 9 10

X 10.4 15r---.---.---.---.---.---.---.---.---.---,

'" ~ 10 o .~ .~ 5

1-- gen31 ...... gen4

-c -c

g; 0 a.

'" -5L---~--~--~--~--~--~--~--~--~--~

o 2 3 4 5 time s

6 7 8 9 10

Figure 10. Change in generator speeds - change in torque 0.01 at generator I, -0.01 at generator 2


0.988 i-----,------r--,.---,---,----,--,------;:c::=:::!=::::;-J

1- bus3 1

0.987

0.986

a 0.985 Q) C) 2 ] 0.984

0.983

0.982

...... bus 13

......

.... __ ... '

.~.~._ .0- .....

0.981 '----_-'--_...L.-_....I....-_--L-_--'-_---'-_--'-_--'-_----'_--l o 2 3 4 5

time s 6 7 8 9 10

Figure 11. Change in tie bus voltages - change in torque 0.01 at generator 1. -0.01 at generator 2

19

The average voltage reduction, with this model, is due to the high effective impedance of the uncontrolled generator compared to that of the classical generator model. The internal impedance of a classical generator model is equal to its transient reactance. This makes the classical model closer in performance to that of a synchronous generator with excitation control, than to that of an uncontrolled generator. The voltage reduction is non-oscillatory J(monotonic), and will eventually lead to loss of synchronism between the generators in area 1 and the generators in area 2. As I will show in the next section, the voltage reduction may be prevented by automatic voltage regulators fitted to the generators' exciters.

J The voltage responses also contain significant oscillations that are due to the inter-area mode. It is the average value that decreases monotonically.

20

" ~ 2 o

"-;; .~ 1 "0 "0

~ 0 a. (/)

1- genl ...... gen2

X 10.4 3,--.--.--.--.---.--.--.--.--.--,

5. 2 " o .~

.s;: Q) "0 "0 Q) :,

~1 '-.:" (/)

1- gen3 ...... gen4

_2~--~--~--~--~----~--L---~--~--~--~ o 2 3 4 5

lime s 6 7 8 9 10

Figure 12. Change in generator speeds - change in torque 0.01 at generator 3, -0.01 at generator 4

0,989 r----,---.--.--.---.--.--.--,---,--,

0.9885

0.988

0.9875

5. 0.987 Q) Cl

~ 0.9865 >

0.986

0.9855

0.985

1- bus3 ...... bus 13

' ...... ~ ..

0.9845 '--__ ~ __ _'_ __ ___'___-'-_---J'__ __ '__ __ _'_ _ _'_ __ ___'__ _ __' o 2 3 4 5

lime s 6 7 B 9 10

Figure 13. Change in tie bus voltage - change in torque 0.01 at generator 3, -0.01 at generator 4


The system's responses to a change in torque of 0.01 pu at generator 3 and -0.01 pu at generator 4 are shown in Figures 12 and 13. In this case, the generator speeds reach a maximum and then decrease. They initially rise because the load voltages, and hence the loads, initially decrease. Eventually, as the voltage at the sending end load increases, the load at the sending end of the tie increases correspondingly, and this causes the generator speed to decrease. The increase in voltage in this case has the same cause as the decrease in voltage in the previous case.

3.2 Inter-area modes

The inter-area oscillation is clearly visible in the speed response, which is shown in Figure 14. There is very little evidence of the local modes of oscillation. However, the local modes may be observed at the very beginning of the transient. The speed change amplitudes at the sending end (area I) are lower than those at the receiving end (area 2). The speeds increase because the average voltages of the load buses decrease as shown in Figure 15. There is almost no evidence of the local modes in the tie bus voltage responses.

X 10.3 8

:> 1- genii ~6 0

: ..... gen2 .~

.~ 4 " " ~ 2 g-

o 0 2 3 4 5 6 7 8 9 10

8 X 10.3

5.6 I:::::: ~:~~ I c 0

"'; 4 'j;: '" "

2 " '" ~O en

-2 0 2 3 4 5 6 7 8 9 10

time s

Figure 14. Change in generator speeds - change in torque 0.01 at generator 1, -0.01 at generator 3

22

X 10.4 2

:::> a. c 0 .~ '5O Q) -0 -0 1- gen1 1 ~ 1 a. en ....... gen2 .

-2 0 2 3 4 5 6 7 8 9 10

X 10.5 2

:::> a. c 0 .~ '5O 0 Q) ...,

-0 Q) Q) a. en

-2 0 2 3 4 5 6 7 8 9 10

time s

Figure 16. Change in generator speed - change in torque 0.01 generator 1, -0.01 generator 2

0.9868

,. .. -~ ,', .-"~ :

'. 0.9866 ~ : : '. : : ,.' '.' .. '

0.9864

:::> a.

g, 0.9862 -'5 >

0.986

0.9858

I - bus 3 ...... bus 13 0.9856

0 2 3 4 5 6 7 8 9 10 time s

Figure 17. Change in tie bus voltages - change in torque 0.01 at generator 1, -0.01 at


0.995

0.99 1- bus3 1 ...... bus 13

0985

0.98

i:l. 0 975 '" '" '" ~ 097

0.965

0.96

0.955

0.95 0 2 4 5 6 10

time s

Figure 15. Change in tie bus voltages - change in torque 0.0 I at generator 1, -0.01 at generator 3

4 CONTROLLED DETAILED GENERATOR MODEL

23

This model is representative of most modern power system generators. As far as oscillations are concerned, the response of the system is quite close to that with the classical generator model considered in section 2. However, the speed is held close to synchronous speed by the action of the governors and the local mode oscillations decay. Compared to the detailed generator model without controls, the system voltages are held close to their pre-disturbance level by the action of the automatic voltage regulator. The non-oscillatory decrease in tie bus voltage, observed in the response of the system with detailed generator models with no controls, is eliminated by the automatic voltage regulators. However, we will see that, in this case, the inter-area mode is unstable - the amplitude of the inter-area oscillations increases with time.

The mode of oscillation local to area 1 is induced by applying a step change in mechanical torque of 0.01 pu at generator 1 and an equal and opposite torque change at generator 2. The response of the generator speeds is shown in Figure 16. In area 1, the local mode oscillation is initially dominant and damped. In area 2, the inter-area mode is dominant. After the

24

local mode has decayed in area 1, the inter-area mode in area 1 can be seen to be in antiphase with that in area 2. The amplitude of the inter-area oscillation increases slightly over the time of the transient. Figure 17 shows the response of the tie line bus voltages. The response is quite similar to that in Figure 3, although there is less evidence of the local modes. This is to be expected, since the local modes decay with time, while the amplitude of the inter-area mode increases slowly.

4.1 Local Modes and Inter-area Mode

Disturbing the mechanical torque is rather artificial. When generator controls are modelled, more realistic small disturbances are available. For example, the governor power reference input, or the automatic voltage regulator voltage reference may be changed. The response of the change in generator speeds to a step change in PrefofO.Ol pu at generator 1 is showQ in Figure 18. In this case, the electromechanical oscillations are superimposed on a well-damped slower oscillation that is associated with the turbines and governors. The response of the change in generator field voltages to a step change in Vref of 0.01 at generator 1 is shown in Figure 19. The area 1 local mode and the inter-area mode can be clearly seen in the response. The

OX 10.4

::J - generator 1 ~ -0.5 ...... generator 2 C) c:: co

-1 ..c:: u -0

'0, m ~ -1.5 (J)

-2 0 2 3 4 5 6 7 8 9 10

X 10'4 0

::J - generator 3 ~-0.5 ..... generator 4 C) c:: co

-1 ..c:: u -0 Q)

~-1.5 (J)

-2 0 2 3 4 5 6 7 8 9 10

time s

Figure 18. Generator speed change response to a step change in the governor power reference


responses of the generator terminal bus voltages are shown in Figure 20. The terminal voltage of generator 1 is rapidly set to the value defined by the new voltage reference, while the terminal voltages of the other generators are maintained at their pre-disturbance levels. The inter-area mode is dominant in the responses at the end of the simulation period.

a. 3.5 generator 1 ., -

'" 3 generator 2 co ...... ~ 0 > 2.5

-c a; "" 2 0

','

........ ,. .. , .. - 'h.- ,.' .... ~ 1.5 \} Q) c: .,

(!) 1 0 2 3 4 5 6 7 8 9 10

" a. 2

'" '" .'!! 1.95 0 >

.... , '"' . -c

\,r' ., " : a; 1.9

"" .. ',.' , .. '

0 :': 70 1.85 Q; l)

c: "

'" (!) 1.8 0 2 3 4 5 6 7 8 9 10

time 5

Figure 19. Response of generator field voltages to a step change of 0,01 in Vref of generator 1

26

X 10.3 15

:::> ...... generator 1

:::- 10 - generator 2 en c:

'" .c '-'

(I) 0)

~ 0 ~ -5

0 2 3 4 5 6 7 8 9 10

X 10.4 6

:::> - generator 3

. ...... generator 4

(I) en c: 2 '" .c '-'

(I) 0 en n> .,'

'5 -2 ".

-4 0 2 3 4 5 6 7 8 9 10

time s

Figure 20. Change in generator terminal voltage - step change in Vref at generator 1

1.2

:::> 0.8 "'-

'" OJ ~ 06

0.4

0.2

2 3 4 5 time s

6 7 8 9 10

Figure 21. Tie bus voltage response to a three-phase fault at bus 3 - classical generator model


5 THE RESPONSE TO SYSTEM FAULTS

Transient simulation is normally carried out to investigate whether or not an interconnected power system can survive a fault. To round off this examination of power system oscillations using transient simulation, I will apply a three-phase fault at bus 3. The near end of the line from bus 3 to bus 101 is cleared in 0.05 s and the remote end is cleared after a further 0.05 s in each case. The tie bus voltage responses are shown in Figures 21 to 23

In each case, the inter-area mode is clearly visible in the tie line bus voltage magnitudes. In all three fault cases, the voltage at bus 3 initially recovers when the fault is cleared. With the classical generator model, the subsequent inter-area mode oscillations remain at constant amplitude - the mode has no damping.

" . ....... ~~

0.9 ".

0.8 '. --

0.7

0.6 :::> c.

~0.5 %! 0 >

0.4

0.3

0.2

0.1

0 0 2- 3 4 5 6 7 8 9 10

time s

Figure 22. Tie bus voltage response to three phase fault at bus 3 - detailed generator model no controls

28

1.2

" 0.8 Cl. Q) 0)

~ 06

0.4

0.2

2 3 4 5 time 5

6 7 8 9 10

Figure 23. Tie bus voltage response to three-phase fault at bus 3 - detailed generator model with controls

With the detailed generator model having no controls, the voltage begins to decay after the initial recovery, at 9.5 s the voltage is falling so rapidly that the simulation fails. Finally, with the controlled detailed generator model, the voltage oscillations oscillate at the inter-area mode frequency and,on careful examination, the amplitude of the oscillations can be seen to increase.

After the fault is cleared, the transmission system is more highly stressed, since one branch of the tie is removed in order to clear the fault. Thus, the final oscillations pertain to the more highly stressed, post-fault system.

6 FINAL DISCUSSION AND COMMENTS

In this chapter, I have shown simulations of a small system having two generation areas. The system has two electromechanical modes of oscillation that can be associated with one of the generator areas. These are termed local modes of oscillation. There is another electromechanical mode


in which the machines in both areas take part and which has a lower frequency than the local modes. In it, the speeds of the generators in area 1 are in anti-phase with the speeds of the generators in area 2. The mode is thus termed an inter-area mode of oscillation. The local and inter-area mode frequencies vary only slightly when different generator models are used in the simulation, but the modelling detail affects other aspects of the system model performance.

Even with this small system, it is difficult to identify all the factors that influence power system oscillations and their stability using time simulation alone. In much of the rest of this book, I will use linearized dynamic models of the system to analyze power system oscillations. However, it is important to realize that power systems have nonlinear dynamic characteristics for large disturbances. It is, therefore, good engineering practice to combine nonlinear simulation with linearized analysis techniques in any practical study of power system oscillations.

6.1 Classical Generator Model

With this model, all modes of oscillation are completely undamped. Oscillations once initiated are continuous. While, the system is transiently stable following a three-phase fault, there are continuous tie line bus voltage oscillations, at the inter-area mode frequency after the fault is cleared.

6.2 Detailed Generator Model

With this model, both the inter-area and local modes decay. The oscillations are damped by the action of the generator damper windings. However, following disturbances, the bus voltage magnitude at the sending end of the tie lines decreases considerably. Following a normally cleared three-phase fault the system voltages do not settle, but drift monotonically. This is due to a lack of synchronizing torque between the uncontrolled synchronous interconnected generators.

30

6.3 Detailed Generator Model with Turbine/Governor Model and Automatic Voltage Control Model

This model is the most representative of current power system generation practice. The high gain, fast acting automatic voltage regulators act to keep the generators' voltages close to their nominal values. The speed governors hold the generators' speeds close to the nominal synchronous speed.

The local modes of oscillation are damped and decay following a disturbance. However, the inter-area mode is unstable and the amplitude of the inter-area mode increases as the simulation progresses. The unstable inter-area mode is the price paid for the increase in synchronizing torque provided by the automatic voltage regulators holding the generator terminal voltages close to constant. In this system, additional controls will be necessary to have good voltage regulation, high synchronizing torque and to have stable electromechanical oscillations. Power System Stabilizers are the most common additional damping controls, they act through a generator's automatic voltage regulator. I will discuss their action in more detail in Chapter 7.

7 REFERENCES

1. Canadian Electrical Association Report, " Investigation of Low Frequency Inter-area Oscillation Problems in Large Interconnected Systems", Report of Research Project 294T622, prepared by Ontario Hydro, 1993.

2. M. Klein, OJ. Rogers and P. Kundur, "A fundamental study of inter-area oscillations in Power Systems", IEEE Trans, PWRS-6, 1991, pp. 914- 921.

Chapter 3

Modal Analysis of Power Systems

1 INTRODUCTION

In Chapter 2, I discussed the oscillations that may occur in interconnected power systems. By looking at different models, and with different disturbances, I showed examples of the different types of oscillation that can occur. To do this, I performed a considerable number of 10-second nonlinear simulations. It is apparent that in larger systems the use of transient simulation for the analysis of system oscillations could be very time consuming. To study inter-area oscillations, it is often necessary to run simulations for longer than lOs; 30 s is quite common in practice. Not only is the use of non-linear simulation time consuming, but also it is often difficult to interpret the results. Larger systems may have a number of inter-area modes at very similar frequencies, and it can be quite difficult to separate them from a response in which more than one is excited.

In most of the simulations of Chapter 2, the transients were induced by small disturbances to the system. The resulting oscillations were essentially linear in character. This can be inferred from the fact that for the same system model, while the amplitude of the oscillations varied depending on


32

the type of disturbance and the monitored variable, the damping and frequency of the oscillations remained constant. Even in the three phase fault simulations, the final oscillations of the classical generator model and the detailed generator model with controls were of quite small amplitude about the post fault equilibrium point. This implies that the post fault oscillations in these cases were also essentially linear. The post fault response of the detailed generator model with no control is, in contrast, non-linear. The two areas eventually separate and the resulting changes are large.

This linearity of behaviour is a great help in the analysis of system oscillations. It enables the use of a system model that has been linearized about a steady state operating point. Once we have a linear model, the very powerful methods of modal analysis are open to us. They allow oscillations to be characterised easily, quickly and accurately. In addition, linear models can be used to design controls that damp system oscillations. Of course, we must not forget that the power system is a non-linear system. At the least, this means that controls designed using linear models must be tested using nonlinear simulation of the system under a wide range of operating conditions.

In very rare cases, undamped oscillations may be caused by system nonlinearities. Nonlinear oscillations require careful analysis and simulation. For example, nonlinearity would be detected by the observation of differences in the basic behaviour of the system under different disturbances. An interesting review of non-linear dynamics in power systems is contained in [1].

2 MODAL ANALYSIS OF LINEAR DYNAMIC SYSTEMS

To apply modal analysis, a dynamic system model is put into state space form, i.e., the equations of the system are expressed as a set of coupled first order, linear differential equations4

dx Ax + Ed --=

dt y = ex + Dd 3.1

where x is a vector of length equal to the number of states n

4 The detailed derivation of the linearized power system equations is beyond the scope of this book. Detailed model development can be found in the books by Kundur [2] and Pai and Sauer [3]. Functions to formulate the equations in MATLAB [4] may be found the Power System Toolbox [5].

3. Modal Analysis of Power Systems

A is the n by n state matrix B is the input matrix with dimensions n by the number of inputs nj d is the input disturbance vector of length nj y is the output vector of length no C is the output matrix of dimensions no by n D is the feed forward matrix of dimensions no by nj

33

Since the system is linear, the homogeneous state equation, with d zero, has n solutions of the form

Z i = K i exp( A.i t) i = 1 to n 3.2

The coefficient of t in the exponential is an eigenvalue of the state matrix A.

The eigenvalues satisfy

det( A - A/) = 0 3.3

The choice of variables to use in a state space formulation of a linear dynamic system is not unique. However, the eigenvalues of the system are unique.

v. ref

Efd VI + 1 + sTc K 1 ~ ----

- 1 + S Tr 1 + sTb 1 + sTa ~

sKf

1 + sTf

Figure 1. Block diagram of a simple exciter

34

1 x 1 y --

--

-s T

1 -- I--

T

Figure 2. Two alternative representations of a lag block

2.1 Example

I will derive a state space model from a block diagram model of a simple excitation system. The block diagram is shown in Figure 1.

2.1.1 Lag Block

The block between the terminal voltage magnitude input and the summing junction models the voltage transducer's time constant. Blocks of this type may be redrawn as shown in Figure 2. Two different realisations are shown. For the same input, the values of the output will be identical, but the value of the state (x) in the second realisation will be T times that in the first realisation. The state equations for realisation 1 are

dx I 1 -=--x+-u dt T T y=x

and for realisation 2

dx 1 -=--x+u dt T

1 Y =-x

T

3.4

3.5

3. Modal Analysis of Power Systems 35

The final block in the forward loop represents the automatic voltage regulator gain and the time constant of the exciters power amplifier. It is also modelled by a lag block. The gain may be included in the block at either the input or the output. For example, in the first realisation of the lag block, the gain may be incorporated as

dx 1 K 3.6 = --x + -u dt T T Y = x or as

dx 1 1 3.7 = --x + -u dt T T Y = Kx

The value of the state in the first case is K times the value of the state in the second case.

2.1.2 Lead Lag Block

The first block following the summing junction is a lead/lag block. In an automatic voltage regulator this block normally has a lead time constant smaller than the lag time constant so that it acts to reduce the high frequency

Tc --

Tb

u

-

~ 1 1 x ~ y '--- 1-~ -- --Tb -

Tb s +

Figure 3. Lead-lag block - state space formulation

36

u Kf -. -- ~

Tf

1 1 + Y .~ x ... -- --Tf -- ~ s

Figure 4. Transient feedback block

gain. It is often called transient gain reduction by power system engineers. A modified diagram for state space formulation is shown in Figure 3.

Just as for the lag element, the state space model for the lead/lag element is not unique. However, this representation is the one most often used. The state equations are

(1 - !') dx 1 Tb - = - -x + ---"--u 3.8 dt Tb

T y=x+_cu Tb

2.1.3 Transient Feedback Block

In many exciters, transient output feedback is used to stabilise the exciter when it is controlling an open circuited generator. A block diagram is shown in Figure 4. Transient feedback is seldom used when transient gain reduction is used. I include both in this example to illustrate the state space model formulation. The state space equations for the transient feedback block are

dx dt

1 Kf ---x+--u Tf T}

Kf y=-x+--u T f

3.9


2.1.4 The Complete State Space Model

The state equations of the whole exciter are then formulated by interconnecting the state space equations of the separate blocks to give

dx -=Ax+Bd dt y=Cx+Dd where x = [Xl x2 x3 x4l

0 0 0 Tr

(1 - Tc ) Tc (1 - Tc ) K f (1 - -) Tb Tb Tb

A = Tb Tb T fTb Tb KT c K 1 ( K f KT, 1 KTc

---- - 1 +

TaTb Ta Ta Tb T f TaTb 0 0

K f 1 --

T2 T f f 0

Tr Tc (1- -)

B 0 Tb Tb

0 KTc TaTb

0 0

y = E fd ; C = [0 0 0]; D=O

38

Note: 1. The number of states is equal to the number of integrators in the

model 2. There is no feed forward matrix in this system since the output is

one of the states (X3) Generally, power system state space models are put together

automatically by a computer program. While this reduces tedium and increases accuracy, it is always good practice to understand the formulation process.

2.1.5 Power System Example

The state matrix of the two-area system with classical generator models linearized about the operating point set by a load flow is

0 376.9911 0 0 0 0 0 0 -0.0744 0 0.0676 0 0.0037 0 0.0031 0 0 0 0 376.9911 0 0 0 0 0.0718 0 -0.0865 0 0.0072 0 0.0075 0 0 0 0 0 0 376.9911 0 0 0.0073 0 0.1 070 0 -0.780 0 0.0600 0 0 0 0 0 0 0 0 376.9911 0.0113 0 0.173 0 0.0672 0 -0.0959 0

The states are the changes in rotor angles and speeds, i.e.,

Note: I indicates the transpose - x is a column vector

In this case, the state matrix was obtained, using the Power System Toolbox, directly from a nonlinear simulation model by perturbing each state in turn by a small amount, and finding the corresponding rates of change of all ofthe states. The rates of change of the states divided by the perturbation gives the column of the state matrix corresponding to the disturbed state. This technique is satisfactory if system nonlinearities are avoided ( the perturbations must be very small), and it requires at least double precision calculations ( the normal calculation mode in MATLAB).

The eigenvalues of the state matrix, calculated using the QR calculation algorithm[5], (the eig function in MATLAB) are shown in Table 1.


Table 1 . Eigenvalues of Classical Generator Model

-0.0111 0.0111

-0.0000 - 3.5319i -0.0000 + 3.5319i

-0.0000 - 7.5092i

-0.0000 + 7.5092i -0.0000 - 7.5746i

-0.0000 + 7.5746i

Since there are eight states, there are eight eigenvalues. Both of the two real eigenvalues should be zero. Theoretically, the angle terms in the speed rows of the state matrix should sum to zero, i.e., the state matrix should be singular. This singularity is caused by the fact that an equal change in each of the generator angles has no effect on the power flow in the interconnecting network. Round-off errors in the calculation of the state matrix, and errors in the initial conditions determined by the iterative load flow solution, have made this sum nonzero. Hence, one of the eigenvalues which should be zero has a small negative real value. Also, because the rate of change of rotor angle is proportional to the change in rotor speed, there should be a zero eigenvalue associated with the speed as well as the angle. This is approximated by the other small real eigenvalue. Notice that the two small eigenvalues sum to zero. Indeed, the sum of the eigenvalues is zero. This is because the state matrix has zero diagonal entries. The sum of the diagonal entries of the state matrix is called the trace of the matrix. This can be shown to be equal to the sum of the eigenvalues of the state matrix. The addition of speed governors, or the addition of damping at the generator shafts eliminates the second zero eigenvalue.

The three oscillatory modes are identified by the complex eigenvalues. Since the state matrix is real, the complex eigenvalues occur in complex conjugate pairs.

For a complex conjugate pair of eigenvalues (a .:t im), the corresponding modes have the form

-K e(a+iw)t Z cl - cl

-K e(a-iw)1 Zc2 - c2

Zc2 is the complex conjugate of Zcl and Kc2 is the complex conjugate of Kc/.

40

The output of the system is real, and the complex mode will have the form Kealsin(co t+cp) in any output. The values of K and cp will depend on the magnitude and type of the input and on which output is selected.

The real part of a complex eigenvalue indicates whether an oscillation decays (the real part is negative), remains at a constant amplitude (the real part is zero) or grows (the real part is positive). In this example the real parts of the complex eigenvalues are zero, and once initiated the amplitudes of the oscillations remain constant. The frequencies of the oscillations are found from the imaginary part of the eigenvalue

f = imag (A) / 27r = co / 27r

The frequencies of the oscillatory modes are 0.5621 Hz, 1.1951 Hz, and 1.2055 Hz respectively. These are the inter-area mode, and local mode frequencies that were identified from time simulations in Chapter 2. However, using modal analysis, they have been evaluated more accurately.

2.2 Eigenvectors

While accurate evaluation of the frequency and damping of oscillations is useful, even more information about the nature of the oscillations can be obtained from modal analysis. Using eigenvectors, the way in which each mode contributes to a particular state may be determined. However, first, what are eigenvectors?

Mathematically, there is one eigenvector associated with each eigenvalue. For the ith eigenvalue, the eigenvector Ui satisfies the equation

Au i = Ai u i 3.10 Strictly, Ui should be called the right eigenvector, but if right is not

specified, it is normally implied. Each right eigenvector is a column vector with a length equal to the number of states. The eigenvectors are not unique. Each remains a valid eigenvector when scaled by any constant.

Left eigenvectors are row vectors that satisfy

3.11

Note: Some workers define the left eigenvector as the transpose of Vi . This definition makes the left eigenvector equal to the right eigenvector of the transpose of A. However, the reason for the designation left is hidden with this definition.


Left and right eigenvectors have the special property of being orthogonal, I.e.,

v iU j = kij 3.12 where k .. lj *'

0 i = j kij = 0 i

*' j

It is normal to choose the eigenvalue scaling to make kii = 1. The orthogonal property of eigenvectors allows any vector of length n

(the number of states) to be expanded in terms of the right eigenvectors. In particular, we can expand the state vector in terms of the right eigenvectors.

x 3.13

The coefficient Zk can be found by pre-multiplying 3.13 by the kth left eigenvector. Because the left and right eigenvectors are orthogonal, only the kth term of the resulting summation is nonzero, and if we scale the eigenvectors so that VkUk is unity

3.14

2.2.1 Modes of Oscillation

In dynamic analysis, the state vector varies with time and satisfies the state equation. The coefficients, z, in the expansion of the state vector in terms of the right eigenvectors are defined as the modes of oscillation. To find equations for the modes, we substitute the summation 3.13) into the state equation and then premultiply by the kth left eigenvector as before. This gIves

The n coupled linear differential equations of the state matrix have been transformed to n decoupled linear differential equations. The modes of oscillation are the solutions to these decoupled equations. Each decoupled

42

(modal) equation can be solved independently of the otherss. In general, for any input disturbance d, the time variation of the kth mode is

t

Z k (t) = f exp( Ad t - T)) v k Bd (T) d T o

3.16

The state vector is then assembled by summing all the modes multiplied by their corresponding right eigenvectors as in 3.13.

Physically, the right eigenvector describes how each mode of oscillation is distributed among the systems states. It is sometimes called mode shape. The left eigenvector, together with the input coefficient matrix and the disturbance determines the amplitude of the mode.

2.2.2 Example

For the classical generator system model, the eigenvectors for the real, inter-area mode and local mode eigenvalues are shown in Tables 2, 3 and 4 respectively.

Table 2 . Eigenvectors for real eigenvalues

A= -0.011 A= 0.011 0.5000 -0.5000

-1.4667e-5 -1.4667e-5

0.5000 - 0.5000

-1.4667e-5 -1.4667e-5

0.5000 -0.5000 -1.4667e-5 -1.4667e-5

0.5000 -0.5000

-1.4667e-5 -1.4667e-5

5 If some eigenvalues are equal, the modal equations may not be able to be completely decoupled. The time response calculation must be modified in such a case. The modification is given in Appendix 2.

3. Modal Analysis oj Power Systems 43

Table 3. Eigenvectors for inter-area mode eigenvalues

A,= -3.5319i A,= 3.5319i

-0.4800 -0.4800

0.0045i - 0.0045i -0.3887 -0.3887

0.0036i - 0.0036i 1.0000 1.0000

- 0.0094i 0.0094i

0.8762 0.8762

- 0.0082i 0.0082i

Table 4. Eigenvectors for local mode eigenvalues

A,= -7.5092i A,= 7.5092i A,= -7.5746i A,= 7.5746i

-0.7643 -0.7643 0.4435 0.4435

0.0152i - 0.0152i - 0.0089i 0.0089i

0.8517 0.8517 -0.5141 -0.5141

- 0.0170i + 0.0170i 0.0103i - 0.0103i

-0.8882 -0.8882 -0.7789 -0.7789

O.OI77i - O.OI77i 0.0157i -0.0157i

1.0000 1.0000 1.0000 1.0000

- 0.0199i 0.0199i -0.020Ii 0.0201i

How is this information interpreted? The right eigenvector indicates the relative magnitude of a mode in the state vector. The even rows of the right eigenvector correspond to the changes in generator speed, the odd rows to changes in rotor angle. The eigenvectors of the two real modes are almost equal. They are real, and they are scaled so that the sum of the squares of the vector elements is unity. The complex eigenvectors are scaled so that the value of the largest element is unity. Eigenvectors are not unique and may be multiplied by any scalar quantity and still be a valid eigenvector. However, the ratio between one element and another is unique, provided that the eigenvalues are distinct. The large difference in the magnitude of the angle eigenvector components (the odd rows) and the speed eigenvector components (the even rows) is due to the speed dimensions being per unit in the Power System Toolbox model. That is

elM =01 Am dt 0 3.17

whertmo = 120r jora60Hz system

44

This is fairly arbitrary, and in other power system modelling programs the dimensions may have been chosen differently.

The right eigenvectors show that the two real modes have an almost identical mode shape - this is a

characteristic of interconnected power system models which have no speed control included

in the inter-area mode, the lowest frequency complex mode, the speed oscillations at the generators in area 1, are 1800 degrees out of phase with the speed variations in area 2. Both the angle oscillation and the speed oscillation amplitudes are larger in area 2 than in area 1. The speed eigenvector components are 90 out of phase with the angle eigenvector components.

in the first local mode, the amplitude of the angle oscillations will be almost the same in the two areas. The angle changes of generator 1 are in antiphase with the angle changes at generator 2, and the angle changes at generator 3 are in antiphase with the angle changes at generator 4. The angle change at generator 1 is in phase with that at generator 3. The speed eigenvector components are 90 out of phase with the angle eigenvector components.

in the second local mode, the amplitudes of the angle oscillations in area 1 are lower than those in area 2. The angle change at generator 1 is in antiphase with that at generator 2, and the angle change at generator 3 is in antiphase with that at generator 4. However, the angle change at generator 1 is in antiphase with that at generator 3. The speed eigenvector components are 90 out of phase with the angle eigenvector components.

In the local modes, the speed variations show a similar pattern to the angles. These do not appear to be the local modes observed in the transient simulation of Chapter 2. There, we saw a local oscillation in area 1 or area 2 depending on the disturbance. In the area without the disturbance, the inter-area oscillation was initially dominant, and the amplitude of the local mode appeared to grow with time. Why do we not see this pattern in the local mode eigenvectors? Firstly, the right eigenvectors do not give the whole story. The magnitudes of the modes of oscillation are determined by the left eigenvectors and by the type of disturbance that is applied.

The left eigenvectors for the real, inter-area and local eigenvalues are given in Tables 5, 6, and 7 respectively


Table 5. Left eigenvectors for real eigenvalues

A. /lOI /lOOI /lO2 /lOO2 /lO, /lOO, /lO4 /lOO4 -0.011 0.3556 -1212 0.3282 -II19 0.1716 -5848 0.1446 -4931 o .Oll -0.3556 -1212 -0.3282 -I 119 -0.1716 -5848 -0.1446 -4931

Table 6. Left eigenvectors for inter-area eigenvalues

A. /lOI /lOOI /l~ /lOO2 /lO, /lOO, /lO4 /lOO4 -3.53i -.2002 -21.37' -0.1611 -17.19i 0.1997 21.32i 0.1616 17.25i 3.53i -.2002 21.37i -0.1611 17.19i 0.1997 -21.32i 0.1616 -17.25i

Table 7. Left eigenvectors for local eigenvectors

A. /lOI /lOOI /lO2 /lOO2 /lO, /lOO, A04 /lOO4 - 7.5li -0.188 -9.44Ii 0.192 9.612i -0.104 -5.227i 0.101 5.057i

7.5Ii -0.188 9.441i 0.192 -9.612i -0.104 5.227i 0.101 -5.057i

-7.58i 0.186 9.24 Ii -0.214 -10.67i -0.157 -7.797i 0.185 9.227i

7.58i 0.186 -9.24 Ii -0.214 1O.67i -0.157 7.797i 0.185 -9.227i

X 10.4 4

5. 3 1- gen1 1 " 0

....... gen2 . .~ 2 'j; Q) -0 -0 Q)

~ 0 '"

2 3 4 5 6 7 B 9 10

3 X 10.4

'" 1- gen31 ~ 2 0 ...... gen~ .~

'iii 1 -0 -0 g: 0 0.

'" -1

0 2 3 4 5 6 7 B 9 10 time s

Figure 5. Change in generator speeds - change in torque 0.01 at generator I, -0.01 at generator 2

45

46

The response to a disturbance is calculated using the right and left eigenvectors, the input and output matrices and the characteristics of the disturbance. Provided the linearizing assumptions are valid, we will obtain the same result as observed in the transient simulation. For example, the change in generator speeds for a step change in mechanical torque of 0.01 pu at generator 1 and -0.01 pu at generator 2 is shown in Figure 5. The response is clearly the same as that shown in Figure 2, Chapter 2, which was calculated using nonlinear step-by-step simulation.

In this system, the complex modes have no damping. Once initiated by a step input, their amplitudes remain constant. The growth of the observed local mode oscillations is caused by the two local modes beating. In the area with no disturbance, the two modes gradually move from being initially in antiphase to being in phase, and then in antiphase again at the difference frequency between the modes.

2.2.3 Equal Eigenvalues

The eigenvectors show that in this case there is interaction between the two higher frequency modes. It is brought about by the closeness in frequency of these modes and, in this system it is not strictly correct to call them local. The eigenvalues are close but distinct, and the modal equations are all decoupled. However, when eigenvalues are exactly equal, two possibilities exist

the corresponding right eigenvectors are equal the eigenvectors are not equal but any combination of the

different eigenvectors is itself an eigenvector

In the first case, the equal eigenvalues are called nonlinear divisors. It is impossible to diagonalize the state matrix and decouple all the system differential equations if some eigenvalues are nonlinear divisors. In the second case, the eigenvalues are said to be linear divisors. With linear divisors the state matrix may be diagonalized, but the response associated with the equal eigenvalues must be determined by the sum of their individual responses. The individual responses have no meaning.

In this example of 2.2.2, the two real eigenvalues should both be zero. Because the second zero comes about by the speed change being proportional to the rate of change of the rotor angle change, equal zero eigenvalues are nonlinear divisors [5]. Their eigenvectors obtained using the QR eigenvalue calculation algorithm would be almost exactly identical.


The eigenvalues of the two local modes are pathologically close to being equal, and the mode shapes are distorted because of that. However, the eigenvalues are not exactly equal. Indeed, it is difficult to force them to be equal by reasonable changes to the systems initial conditions. See Appendix 2 for additional discussion of the analysis of systems with equal eigenvalues.

If the inertias of the generators are changed, so that they are not equal, the local mode frequencies are more distinct, and the eigenvector pattern shows that the higher frequency modes are essentially local to an area. In the following example, I have changed the generator inertias to 3.5,4.5,5.5 and 6.5 respectively. All other parameters are unchanged.

The new state matrix is

0 377 0 0 0 0 0 0 -0.1370 0 0.12612 0 0.00605 0 0.00479 0 0 0 0 377 0 0 0 0 0.1057 0 -0.12404 0 0.00962 0 0.00912 0 0 0 0 0 0 377 0 0 0.00905 0 0.0122 0 -0.0921 0 0.07023 0 0 0 0 0 0 0 0 377 0.01209 0 0.0174 0 0.6753 0 -0.09727 0

The modified eigenvalues are given in Table 8.

Table 8. Modified eigenvalues

-0.0132

0.0132

+ 3.9868i

+7.8427i

+9.6292i

It can be seen that the two local mode eigenvalues are now quite distinct. The corresponding right eigenvectors are shown in Table 9.

48

Table 9. Right eigenvectors of modified system

A. = -0.013238 A. = 0.013238 A. = 3.9868i A. = 7.8427i A. = 9.6292i ~Ih 0.5 -0.5 -0.69443 0.0084677 1 ~OOI -1.7425e-005 -1.7425e-005 +0.0073437i 0.00017616i 0.025542i

~O2 0.5 -0.5 -0.62768 -0.0098843 -0.85938 A002 -1.7425e-005 -\. 7425e-005 +0.0066379i +0.00020563i +0.02195i

~O3 0.5 -0.5 1 1 0.0044704 ~003 -1.7425e-005 -1.7425e-005 0.010575i 0.020803i 0.00011418i ~O4 OJ -0.5 0.90182 -0.99927 0.021641 ~004 -1.7 425e-005 -\. 7425e-005 0.009537i +0.020788i 0.00055275i

Note: The complex eigenvalues and eigenvectors occur as complex conjugate pairs and I have compressed the tables.

From the right eigenvectors associated with the lower frequency local mode, we see that the mode is local to area 2. From the eigenvector associated with the higher frequency local mode, we see that the mode is local to area 1. This is to be expected, since the inertia is now lower in area 1 than in area 2. Figures 6 and 7 show the change in generator speeds following a step change in the mechanical torque in area 1, 0.01 pu at generator 1 and -0.01 pu at generator 2. The response in Figure 6 was obtained using modal analysis, while the response in Figure 7 was obtained using a step-by-step nonlinear simulation. The difference between Figure 5 and Figure 6 is that the mode local to area 2 is not excited by the torque change in area 1. However, the inter-area mode is excited, and can be observed at the area 2 generators.

The response is correct whether or not we have equal eigenvalues as long as the eigenvalues are linear divisors. However, the interpretation of eigenvectors associated with equal or almost equal eigenvalues must be treated with caution.

Nonlinear divisors, other than the zero eigenvalues, are rare in power systems. They may occur in a system in which feedback controls have been disabled by using zero gain, and in addition, have two or more elements of the control with the same time constant. In such a case, the eigenvector matrix would be singular, or very close to being singular. In most other cases, equal eigenvalues are likely to be linear divisors. Exceptions may be caused by interactions between controls or between controls and electromechanical modes [6].


X 10.4 6

:::l 1- gen1 ~ 4 "

~.' .. ' gen2 .~ 'S;: 2 '" -.,

-.,

'" 0 '" Q.

50

2.3 Eigenvalue Sensitivity

It is often useful, particularly in control studies, to determine the sensitivity to changes in the elements of the state matrix. In the following, I make the restriction that the eigenvalues are distinct. To start, we differentiate the equation for the ith eigenvector with respect to the element of the state matrix in the rth row and sth column (ars).

oA au au ax --u + A--'- = X __ 1- + __ I_U'

1 I 1 a ars a ars a ars oars

3.18

now pre-multiply both sides by the ith left eigenvector Vi.

oA au oX v--u +v(A-XI)-_I_=V __ ' u 3.19 I ::J 1 I I I I

U a rs a a rs a a rs

The second term on the left hand side is zero (from the definition of the left eigenvector) so

a A a Ai

Vi oars

Ui a A

= = Vi ui 3.20 oars viui oars since viui = 1

2.3.1 Participation Factors

Consider the senSItIVIty of an eigenvalue to a change of a diagonal element of the state matrix arr. The sensitivity of the ith eigenvalue is

ok 1 ----'--- = vir U ir o a rr

3.21

In power system analysis, this is defined as the participation factor of the rth state in the ith mode, i.e., the product of the rth element in the ith left (row) eigenvector and the rth element in the ith right (column) eigenvector.

The states that are used to describe a dynamic system model are not unique; there may be differences in scale, or different combinations of variables may be used as states. For example, the modes z, are alternative state variables, which are linear combinations of the original states x. Not only are the states not unique, but the eigenvectors may also be scaled by an


arbitrary constant. This means that, while the eigenvector is a good indication of the relative activity of the states within a mode, hence its common name 'mode shape', it is not a good indicator of the importance of states to the mode from a control point of view.

The participation factor is quite a good indication of the importance of a state to the mode. In power systems, it is particularly useful as a screen for power system stabiliser placement. If mechanical damping could be applied to the shaft of a generator, it would appear in the state equations as a negative coefficient on the diagonal of the state matrix in the row corresponding to the speed change of the generator. If, for any mode, the corresponding participation factor of the generator speed is zero, we can imply that adding damping at that generator will have no effect on the mode. If the participation factor is real positive, adding damping at that generator will increase the damping of the mode. If the participation factor is real negative, adding damping at that generator will reduce the damping of the mode. Since participation factors are based on eigenvectors, one must be careful when using them with equal eigenvalues. Indeed, the sensitivity that forms the basis of participation factors is ambiguous when some of the system eigenvalues are equal.

Participation Vectors are just the collection of the participation factors for all the diagonal elements in the state matrix. For the original classical generator case with equal inertias, the participation vectors associated with the complex conjugate pair of inter-area mode eigenvalues are

0.096107 0.096107 0.096107 0.096107 0.06261 0.06261 0.06261 0.06261 0.19972 0.19972 0.19972 0.19972 0.14156 0.14156 0.14156 0.14156

All the participation factors in this case are real and positive. The participation factors for the generator angle changes are the same as for the generator speed changes. Physically, it may be possible to add mechanical damping at the shafts of the generators, and model this as a change in the diagonal in the speed change rows of the state matrix. In contrast, the diagonal terms in the rows corresponding to the rotor angles cannot be changed, so that the participation factors associated with the angle changes are of mathematical interest only.

52

The largest participation factors are associated with the area 2 generators (the sending end generators). Therefore, if we put -Ion the diagonal of the speed change row of the state matrix corresponding to generator 3, the participation factor indicates that the real part of the inter-area mode eigenvalue will become -0.19972 and the imaginary part will remain unaltered.. In fact, the inter-area mode eigenvalue changes to -0.20066i3.5232 with this change to the state matrix. A similar entry in the diagonal of the state matrix corresponding to generator 1 results in the inter-area mode eigenvalue changing to -0.09594i3 .3720. These values should be compared with the original eigenvalue of 0i3 .5319. The slight differences in the prediction of the eigenvalue's change are due to the incremental nature of the sensitivity calculation.

The participation vectors for the local modes are

A = - i7.5092 A = i7.5092 A = -i7.5746 A = i7.5746 0.14373 0.14373 0.082355 0.082355 0.14373 0.14373 0.082355 0.082355 0.16306 0.16306 0.11023 0.11023 0.16306 0.16306 0.11023 0.11023

0.092482 0.092482 0.12203 0.12203 0.092482 0.092482 0.12203 0.12203 0.10073 0.10073 0.18539 0.18539 0.10073 0.10073 0.18539 0.18539

Again, the participation factors are all real, and the angle change factors are equal to the speed change factors. This seems to imply that the lower frequency local mode can be controlled more easily from area 1 than from area 2. We can check this out by calculating the sensitivity of both modes to changes in the damping as we did for the inter-area mode.

If the diagonal of the row of the state matrix corresponding to generator 1 is changed to -1, the two local modes change to

-0.22039i7.5108 and -.0.00437i7.5516

These seem to have some connection to the sensitivities, but the connection is not as obvious as for the inter-area mode. If the same element of the state matrix is changed to -0.1, the local mode eigenvalues are changed to -.01459 i 7.5109 and -0.00802 i 7.5727 These changes are more in line with the participation factors.

3. Modal Analysis of Pow

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