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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: cherry@eagle.ca 
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 2222.6 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 inmagnitude. 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. 
G. Rogers, Power System Oscillations
© Kluwer Academic Publishers 2000
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, butstill 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. The Nature of Power System Oscillations 11 
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 
02 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 
2. The Nature of Power System Oscillations 
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 
2. The Nature of Power System Oscillations 
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 
2. The Nature of Power System Oscillations 
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 
2. The Nature of Power System Oscillations 21The 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 
2. The Nature of Power System Oscillations 
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 
2. The Nature of Power System Oscillations 25 
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 
2. The Nature of Power System Oscillations 27 
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 modelhaving 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 
2. The Nature of Power System Oscillations 29 
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 
G. Rogers, Power System Oscillations
© Kluwer Academic Publishers 2000
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 
SystemToolbox [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 
3. Modal Analysis of Power Systems 37 
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. 
3. Modal Analysis of Power Systems 39 
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 realpart 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. 
3. Modal Analysis of Power Systems 41 
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 withoutthe 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 
3. Modal Analysis of Power Systems 
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. 
3. Modal Analysis of Power Systems 47 
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]. 
3. Modal Analysis of Power Systems 
X 10.4 
6 
:::l 1- gen1 
~ 4 
" 
~.' .. ' gen2 
.~ 
'S;: 2 
'" -., -., 
'" 0 '" Q. <n 
-2 
0 2 3 4 5 6 7 8 9 10 
X 10'4 
3 
:::l 1- gen3 
~ 2 --' .. gen4 
0 
.~ 
'S;: 1 
'" --c 
-c 
'" 0 '" Q. <n 
-1 
0 2 3 4 5 6 7 8 9 10