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Power System Dynamics. Stability and Control
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POWER SYSTEM
DYNAMICS
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POWER SYSTEM
DYNAMICS
Stability and Control
Second Edition
Jan Machowski
Warsaw University of Technology, Poland
Janusz W. Bialek
Durham University, UK
James R. Bumby
Durham University, UK
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Reprinted with corrections September 2012.
This edition first published 2008
C 2008 John Wiley & Sons, Ltd.
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Library of Congress Cataloging-in-Publication Data
Machowski, Jan.
Power system dynamics: stability and control / Jan Machowski, Janusz W. Bialek,
James R. Bumby. – 2nd ed.
p. cm.
Rev. ed. of: Power system dynamics and stability / Jan Machowski, Janusz W. Bialek,
James R. Bumby. 1997.
Includes bibliographical references and index.
ISBN 978-0-470-72558-0 (cloth)
1. Electric power system stability. 2. Electric power systems–Control. I. Bialek, Janusz
W. II. Bumby, J. R. (James Richard) III. Title.
TK1010.M33 2008
621.319 1–dc22
2008032220
A catalogue record for this book is available from the British Library.
ISBN 978-0-470-72558-0
Typeset in 9/11pt Times New Roman by Aptara Inc., New Delhi, India.
Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire
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Contents
About the Authors
xiii
Preface
xv
Acknowledgements
xix
List of Symbols
xxi
PART I
INTRODUCTION TO POWER SYSTEMS
1 Introduction
1.1 Stability and Control of a Dynamic System
1.2 Classification of Power System Dynamics
1.3 Two Pairs of Important Quantities:
Reactive Power/Voltage and Real Power/Frequency
1.4 Stability of a Power System
1.5 Security of a Power System
1.6 Brief Historical Overview
3
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2 Power System Components
2.1 Introduction
2.1.1 Reliability of Supply
2.1.2 Supplying Electrical Energy of Good Quality
2.1.3 Economic Generation and Transmission
2.1.4 Environmental Issues
2.2 Structure of the Electrical Power System
2.2.1 Generation
2.2.2 Transmission
2.2.3 Distribution
2.2.4 Demand
2.3 Generating Units
2.3.1 Synchronous Generators
2.3.2 Exciters and Automatic Voltage Regulators
2.3.3 Turbines and their Governing Systems
2.4 Substations
2.5 Transmission and Distribution Network
2.5.1 Overhead Lines and Underground Cables
2.5.2 Transformers
2.5.3 Shunt and Series Elements
2.5.4 FACTS Devices
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2.6 Protection
2.6.1 Protection of Transmission Lines
2.6.2 Protection of Transformers
2.6.3 Protection of Busbars
2.6.4 Protection of Generating Units
2.7 Wide Area Measurement Systems
2.7.1 WAMS and WAMPAC Based on GPS Signal
2.7.2 Phasors
2.7.3 Phasor Measurement Unit
2.7.4 Structures of WAMS and WAMPAC
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3 The Power System in the Steady State
3.1 Transmission Lines
3.1.1 Line Equations and the π -Equivalent Circuit
3.1.2 Performance of the Transmission Line
3.1.3 Underground Cables
3.2 Transformers
3.2.1 Equivalent Circuit
3.2.2 Off-Nominal Transformation Ratio
3.3 Synchronous Generators
3.3.1 Round-Rotor Machines
3.3.2 Salient-Pole Machines
3.3.3 Synchronous Generator as a Power Source
3.3.4 Reactive Power Capability Curve of a Round-Rotor Generator
3.3.5 Voltage–Reactive Power Capability Characteristic V(Q)
3.3.6 Including the Equivalent Network Impedance
3.4 Power System Loads
3.4.1 Lighting and Heating
3.4.2 Induction Motors
3.4.3 Static Characteristics of the Load
3.4.4 Load Models
3.5 Network Equations
3.6 Power Flows in Transmission Networks
3.6.1 Control of Power Flows
3.6.2 Calculation of Power Flows
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PART II
INTRODUCTION TO POWER SYSTEM DYNAMICS
4 Electromagnetic Phenomena
4.1 Fundamentals
4.2 Three-Phase Short Circuit on a Synchronous Generator
4.2.1 Three-Phase Short Circuit with the Generator on No Load and Winding
Resistance Neglected
4.2.2 Including the Effect of Winding Resistance
4.2.3 Armature Flux Paths and the Equivalent Reactances
4.2.4 Generator Electromotive Forces and Equivalent Circuits
4.2.5 Short-Circuit Currents with the Generator Initially on No Load
4.2.6 Short-Circuit Currents in the Loaded Generator
4.2.7 Subtransient Torque
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4.3 Phase-to-Phase Short Circuit
4.3.1 Short-Circuit Current and Flux with Winding Resistance Neglected
4.3.2 Influence of the Subtransient Saliency
4.3.3 Positive- and Negative-Sequence Reactances
4.3.4 Influence of Winding Resistance
4.3.5 Subtransient Torque
4.4 Synchronization
4.4.1 Currents and Torques
4.5 Short-Circuit in a Network and its Clearing
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5 Electromechanical Dynamics – Small Disturbances
5.1 Swing Equation
5.2 Damping Power
5.2.1 Damping Power at Large Speed Deviations
5.3 Equilibrium Points
5.4 Steady-State Stability of Unregulated System
5.4.1 Pull-Out Power
5.4.2 Transient Power–Angle Characteristics
5.4.3 Rotor Swings and Equal Area Criterion
5.4.4 Effect of Damper Windings
5.4.5 Effect of Rotor Flux Linkage Variation
5.4.6 Analysis of Rotor Swings Around the Equilibrium Point
5.4.7 Mechanical Analogues of the Generator–Infinite Busbar System
5.5 Steady-State Stability of the Regulated System
5.5.1 Steady-State Power–Angle Characteristic of Regulated Generator
5.5.2 Transient Power–Angle Characteristic of the Regulated Generator
5.5.3 Effect of Rotor Flux Linkage Variation
5.5.4 Effect of AVR Action on the Damper Windings
5.5.5 Compensating the Negative Damping Components
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6 Electromechanical Dynamics – Large Disturbances
6.1 Transient Stability
6.1.1 Fault Cleared Without a Change in the Equivalent Network Impedance
6.1.2 Short-Circuit Cleared with/without Auto-Reclosing
6.1.3 Power Swings
6.1.4 Effect of Flux Decrement
6.1.5 Effect of the AVR
6.2 Swings in Multi-Machine Systems
6.3 Direct Method for Stability Assessment
6.3.1 Mathematical Background
6.3.2 Energy-Type Lyapunov Function
6.3.3 Transient Stability Area
6.3.4 Equal Area Criterion
6.3.5 Lyapunov Direct Method for a Multi-Machine System
6.4 Synchronization
6.5 Asynchronous Operation and Resynchronization
6.5.1 Transition to Asynchronous Operation
6.5.2 Asynchronous Operation
6.5.3 Possibility of Resynchronization
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Contents
6.6 Out-of-Step Protection Systems
6.6.1 Impedance Loci During Power Swings
6.6.2 Power Swing Blocking
6.6.3 Pole-Slip Protection of Synchronous Generator
6.6.4 Out-of-Step Tripping in a Network
6.6.5 Example of a Blackout
6.7 Torsional Oscillations in the Drive Shaft
6.7.1 The Torsional Natural Frequencies of the Turbine–Generator Rotor
6.7.2 Effect of System Faults
6.7.3 Subsynchronous Resonance
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Wind Power
7.1 Wind Turbines
7.1.1 Generator Systems
7.2 Induction Machine Equivalent Circuit
7.3 Induction Generator Coupled to the Grid
7.4 Induction Generators with Slightly Increased Speed Range via External Rotor
Resistance
7.5 Induction Generators with Significantly Increased Speed Range: DFIGs
7.5.1 Operation with the Injected Voltage in Phase with the Rotor Current
7.5.2 Operation with the Injected Voltage out of Phase with the Rotor Current
7.5.3 The DFIG as a Synchronous Generator
7.5.4 Control Strategy for a DFIG
7.6 Fully Rated Converter Systems: Wide Speed Control
7.6.1 Machine-Side Inverter
7.6.2 Grid-Side Inverter
7.7 Peak Power Tracking of Variable Speed Wind Turbines
7.8 Connections of Wind Farms
7.9 Fault Behaviour of Induction Generators
7.9.1 Fixed-Speed Induction Generators
7.9.2 Variable-Speed Induction Generators
7.10 Influence of Wind Generators on Power System Stability
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8
Voltage Stability
8.1 Network Feasibility
8.1.1 Ideally Stiff Load
8.1.2 Influence of the Load Characteristics
8.2 Stability Criteria
8.2.1 The dQ/dV Criterion
8.2.2 The dE/dV Criterion
8.2.3 The dQG /dQL Criterion
8.3 Critical Load Demand and Voltage Collapse
8.3.1 Effects of Increasing Demand
8.3.2 Effect of Network Outages
8.3.3 Influence of the Shape of the Load Characteristics
8.3.4 Influence of the Voltage Control
8.4 Static Analysis
8.4.1 Voltage Stability and Load Flow
8.4.2 Voltage Stability Indices
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8.5 Dynamic Analysis
8.5.1 The Dynamics of Voltage Collapse
8.5.2 Examples of Power System Blackouts
8.5.3 Computer Simulation of Voltage Collapse
8.6 Prevention of Voltage Collapse
8.7 Self-Excitation of a Generator Operating on a Capacitive Load
8.7.1 Parametric Resonance in RLC Circuits
8.7.2 Self-Excitation of a Generator with Open-Circuited Field Winding
8.7.3 Self-Excitation of a Generator with Closed Field Winding
8.7.4 Practical Possibility of Self-Excitation
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9
Frequency Stability and Control
9.1 Automatic Generation Control
9.1.1 Generation Characteristic
9.1.2 Primary Control
9.1.3 Secondary Control
9.1.4 Tertiary Control
9.1.5 AGC as a Multi-Level Control
9.1.6 Defence Plan Against Frequency Instability
9.1.7 Quality Assessment of Frequency Control
9.2 Stage I – Rotor Swings in the Generators
9.3 Stage II – Frequency Drop
9.4 Stage III – Primary Control
9.4.1 The Importance of the Spinning Reserve
9.4.2 Frequency Collapse
9.4.3 Underfrequency Load Shedding
9.5 Stage IV – Secondary Control
9.5.1 Islanded Systems
9.5.2 Interconnected Systems and Tie-Line Oscillations
9.6 FACTS Devices in Tie-Lines
9.6.1 Incremental Model of a Multi-Machine System
9.6.2 State-Variable Control Based on Lyapunov Method
9.6.3 Example of Simulation Results
9.6.4 Coordination Between AGC and Series FACTS Devices in Tie-Lines
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10 Stability Enhancement
10.1 Power System Stabilizers
10.1.1 PSS Applied to the Excitation System
10.1.2 PSS Applied to the Turbine Governor
10.2 Fast Valving
10.3 Braking Resistors
10.4 Generator Tripping
10.4.1 Preventive Tripping
10.4.2 Restitutive Tripping
10.5 Shunt FACTS Devices
10.5.1 Power–Angle Characteristic
10.5.2 State-Variable Control
10.5.3 Control Based on Local Measurements
10.5.4 Examples of Controllable Shunt Elements
10.5.5 Generalization to Multi-Machine Systems
10.5.6 Example of Simulation Results
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10.6 Series Compensators
10.6.1 State-Variable Control
10.6.2 Interpretation Using the Equal Area Criterion
10.6.3 Control Strategy Based on the Squared Current
10.6.4 Control Based on Other Local Measurements
10.6.5 Simulation Results
10.7 Unified Power Flow Controller
10.7.1 Power–Angle Characteristic
10.7.2 State-Variable Control
10.7.3 Control Based on Local Measurements
10.7.4 Examples of Simulation Results
PART III
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ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
11 Advanced Power System Modelling
11.1 Synchronous Generator
11.1.1 Assumptions
11.1.2 The Flux Linkage Equations in the Stator Reference Frame
11.1.3 The Flux Linkage Equations in the Rotor Reference Frame
11.1.4 Voltage Equations
11.1.5 Generator Reactances in Terms of Circuit Quantities
11.1.6 Synchronous Generator Equations
11.1.7 Synchronous Generator Models
11.1.8 Saturation Effects
11.2 Excitation Systems
11.2.1 Transducer and Comparator Model
11.2.2 Exciters and Regulators
11.2.3 Power System Stabilizer (PSS)
11.3 Turbines and Turbine Governors
11.3.1 Steam Turbines
11.3.2 Hydraulic Turbines
11.3.3 Wind Turbines
11.4 Dynamic Load Models
11.5 FACTS Devices
11.5.1 Shunt FACTS Devices
11.5.2 Series FACTS Devices
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12 Steady-State Stability of Multi-Machine System
12.1 Mathematical Background
12.1.1 Eigenvalues and Eigenvectors
12.1.2 Diagonalization of a Square Real Matrix
12.1.3 Solution of Matrix Differential Equations
12.1.4 Modal and Sensitivity Analysis
12.1.5 Modal Form of the State Equation with Inputs
12.1.6 Nonlinear System
12.2 Steady-State Stability of Unregulated System
12.2.1 State-Space Equation
12.2.2 Simplified Steady-State Stability Conditions
12.2.3 Including the Voltage Characteristics of the Loads
12.2.4 Transfer Capability of the Network
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12.3 Steady-State Stability of the Regulated System
12.3.1 Generator and Network
12.3.2 Including Excitation System Model and Voltage Control
12.3.3 Linear State Equation of the System
12.3.4 Examples
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13 Power System Dynamic Simulation
13.1 Numerical Integration Methods
13.2 The Partitioned Solution
13.2.1 Partial Matrix Inversion
13.2.2 Matrix Factorization
13.2.3 Newton’s Method
13.2.4 Ways of Avoiding Iterations and Multiple Network Solutions
13.3 The Simultaneous Solution Methods
13.4 Comparison Between the Methods
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14 Power System Model Reduction – Equivalents
14.1 Types of Equivalents
14.2 Network Transformation
14.2.1 Elimination of Nodes
14.2.2 Aggregation of Nodes Using Dimo’s Method
14.2.3 Aggregation of Nodes Using Zhukov’s Method
14.2.4 Coherency
14.3 Aggregation of Generating Units
14.4 Equivalent Model of External Subsystem
14.5 Coherency Recognition
14.6 Properties of Coherency-Based Equivalents
14.6.1 Electrical Interpretation of Zhukov’s Aggregation
14.6.2 Incremental Equivalent Model
14.6.3 Modal Interpretation of Exact Coherency
14.6.4 Eigenvalues and Eigenvectors of the Equivalent Model
14.6.5 Equilibrium Points of the Equivalent Model
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Appendix
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References
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Index
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About the Authors
Professor Jan Machowski received his MSc and PhD degrees in Electrical Engineering from Warsaw University of Technology in 1974 and
1979, respectively. After obtaining field experience in the Dispatching
Centre and several power plants, he joined the Electrical Faculty of
Warsaw University of Technology where presently he is employed as a
Professor and Director of the Power Engineering Institute. His areas
of interest are electrical power systems, power system protection and
control.
In 1989–93 Professor Machowski was a Visiting Professor at Kaiserslautern University in Germany where he carried out two research
projects on power swing blocking algorithms for distance protection
and optimal control of FACTS devices.
Professor Machowski is the co-author of three books published in
Polish: Power System Stability (WNT, 1989), Short Circuits in Power Systems (WNT, 2002) and
Power System Control and Stability (WPW, 2007). He is also a co-author of Power System Dynamics
and Stability published by John Wiley & Sons, Ltd (1997).
Professor Machowski is the author and co-author of 42 papers published in English in international fora. He has carried out many projects on electrical power systems, power system stability
and power system protection commissioned by the Polish Power Grid Company, Electric Power
Research Institute in the United States, Electroinstitut Milan Vidmar in Slovenia and Ministry of
Science and Higher Education of Poland.
Professor Janusz Bialek received his MEng and PhD degrees in Electrical Engineering from Warsaw University of Technology in 1977 and
1981, respectively. From 1981 to 1989 he was a lecturer with Warsaw
University of Technology. Currently he holds the Chair of Electrical
Power and Control at Durham University having previously (2003–
2008) held Bert Whittington Chair of Electrical Engineering at the
University of Edinburg. Janusz is Fellow of Institute of Electrical and
Electronics Engineers (IEEE) and Honorary Professor of Heriot-Watt
University, UK.
His research deals with achieving stable, secure, sustainable and economic supply of electricity while meeting the challenges of reducing
CO2 emissions. His particular expertise is in technical and economic
integration of renewable generation in the power system, in preventing electricity blackouts and in
analysis of power system dynamics. He has published 2 books and about 130 research papers. He has
been a consultant to the UK government, Scottish Government, European Commission, Elexon,
Polish Power Grid Company, Scottish Power and Enron. He has been the Principal Investigator
of a number of major research grants funded by the Engineering and Physical Sciences Research
Council (EPSRC) and Electrical Power Research Institute (EPRI) in USA.
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About the Authors
Dr Jim Bumby received his BSc and PhD degrees in Engineering from
Durham University, United Kingdom, in 1970 and 1974, respectively.
From 1973 to 1978 he worked for the International Research and Development Company, Newcastle-upon-Tyne, on superconducting machines, hybrid vehicles and sea-wave energy. Since 1978 he has worked
in the School of Engineering at Durham University where he is currently Reader in Electrical Engineering. He has worked in the area of
electrical machines and systems for over 30 years, first in industry and
then in academia.
Dr Bumby is the author or co-author of over 100 technical papers and
two books in the general area of electrical machines/power systems and
control. He has also written numerous technical reports for industrial
clients. These papers and books have led to the award of a number of national and international
prizes including the Institute of Measurement and Control prize for the best transactions paper in
1988 for work on hybrid electric vehicles and the IEE Power Division Premium in 1997 for work
on direct drive permanent magnet generators for wind turbine applications. His current research
interests are in novel generator technologies and their associated control for new and renewable
energy systems.
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Preface
In 1997 the authors of this book, J. Machowski, J.W. Bialek and J.R. Bumby, published a book
entitled Power System Dynamics and Stability. That book was well received by readers who told
us that it was used regularly as a standard reference text both in academia and in industry. Some
10 years after publication of that book we started work on a second edition. However, we quickly
realized that the developments in the power systems industry over the intervening years required a
large amount of new material. Consequently the book has been expanded by about a third and the
word Control in the new title, Power System Dynamics: Stability and Control, reflects the fact that
a large part of the new material concerns power system control: flexible AC transmission systems
(FACTS), wide area measurement systems (WAMS), frequency control, voltage control, etc. The
new title also reflects a slight shift in focus from solely describing power system dynamics to the
means of dealing with them. For example, we believe that the new WAMS technology is likely to
revolutionize power system control. One of the main obstacles to a wider embrace of WAMS by
power system operators is an acknowledged lack of algorithms which could be utilized to control
a system in real time. This book tries to fill this gap by developing a number of algorithms for
WAMS-based real-time control.
The second reason for adding so much new material is the unprecedented change that has been
sweeping the power systems industry since the 1990s. In particular the rapid growth of renewable
generation, driven by global warming concerns, is changing the fundamental characteristics of
the system. Currently wind power is the dominant renewable energy source and wind generators
usually use induction, rather than synchronous, machines. As a significant penetration of such
generation will change the system dynamics, the new material in Chapter 7 is devoted entirely to
wind generation.
The third factor to be taken into account is the fallout from a number of highly publicized blackouts that happened in the early years of the new millennium. Of particular concern were the autumn
2003 blackouts in the United States/Canada, Italy, Sweden/Denmark and the United Kingdom,
the 2004 blackout in Athens and the European disturbance on 4 November 2006. These blackouts
have exposed a number of critical issues, especially those regarding power system behaviour at
depressed voltages. Consequently, the book has been extended to cover these phenomena together
with an illustration of some of the blackouts.
It is important to emphasize that the new book is based on the same philosophy as the previous
one. We try to answer some of the concerns about the education of power system engineers. With
the widespread access to powerful computers running evermore sophisticated simulation packages,
there is a tendency to treat simulation as a substitute for understanding. This tendency is especially
dangerous for students and young researchers who think that simulation is a panacea for everything
and always provides a true answer. What they do not realize is that, without a physical understanding
of the underlying principles, they cannot be confident in understanding, or validating, the simulation
results. It is by no means bad practice to treat the initial results of any computer software with a
healthy pinch of scepticism.
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Preface
Power system dynamics are not easy to understand. There are a number of good textbooks which
deal with this topic and some of these are reviewed in Chapter 1. As the synchronous machine
plays a decisive role in determining the dynamic response of the system, many of these books start
with a detailed mathematical treatment of the synchronous generator in order to introduce Park’s
equations and produce a mathematical model of the generator. However, it is our experience that to
begin a topic with such a detailed mathematical treatment can put many students off further study
because they often find it difficult to see any practical relevance for the mathematics. This can be
a major obstacle for those readers who are more practically inclined and who want to understand
what is happening in the system without having to refer continuously to a complicated mathematical
model of the generator.
Our approach is different. We first try to give a qualitative explanation of the underlying physical
phenomena of power system dynamics using a simple model of the generator, coupled with the basic
physical laws of electrical engineering. Having provided the student with a physical understanding
of power system dynamics, we then introduce the full mathematical model of the generator, followed
by more advanced topics such as system reduction, dynamic simulation and eigenvalue analysis. In
this way we hope that the material is made more accessible to the reader who wishes to understand
the system operation without first tackling Park’s equations.
All our considerations are richly illustrated by diagrams. We strongly believe in the old adage
that an illustration is worth a thousand words. In fact, our book contains over 400 diagrams.
The book is conveniently divided into three major parts. The first part (Chapters 1–3) reviews
the background for studying power system dynamics. The second part (Chapters 4–10) attempts
to explain the basic phenomena underlying power system dynamics using the classical model of
the generator–infinite busbar system. The third part (Chapters 11–14) tackles some of the more
advanced topics suitable for the modelling and dynamic simulation of large-scale power systems.
Examining the chapters and the new material added in more detail, Chapter 1 classifies power
system dynamics and provides a brief historical overview. The new material expands on the definitions of power system stability and security assessment and introduces some important concepts
used in later chapters. Chapter 2 contains a brief description of the major power system components, including modern FACTS devices. The main additions here provide a more comprehensive
treatment of FACTS devices and a whole new section on WAMS. Chapter 3 introduces steady-state
models and their use in analysing the performance of the power system. The new material covers
enhanced treatment of the generator as the reactive power source introducing voltage–reactive
power capability characteristics. We believe that this is a novel treatment of those concepts since we
have not seen it anywhere else. The importance of understanding how the generator and its controls
behave under depressed voltages has been emphasized by the wide area blackouts mentioned above.
The chapter also includes a new section on controlling power flows in the network.
Chapter 4 analyses the dynamics following a disturbance and introduces models suitable for
analysing the dynamic performance of the synchronous generator. Chapter 5 explains the power
system dynamics following a small disturbance (steady-state stability) while Chapter 6 examines
the system dynamics following a large disturbance (transient stability). There are new sections on
using the Lyapunov direct method to analyse the stability of a multi-machine power system and on
out-of-step relaying. Chapter 7 is all new and covers the fundamentals of wind power generation.
Chapter 8 has been greatly expanded and provides an explanation of voltage stability together with
some of the methods used for stability assessment. The new material includes examples of power
system blackouts, methods of preventing voltage collapse and a large new section on self-excitation
of the generator. Chapter 9 contains a largely enhanced treatment of frequency stability and control
including defence plans against frequency instability and quality assessment of frequency control.
There is a large new section which covers a novel treatment of interaction between automatic
generation control (AGC) and FACTS devices installed in tie-lines that control the flow of power
between systems in an interconnected network. Chapter 10 provides an overview of the main
methods of stability enhancement, both conventional and using FACTS devices. The new material
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includes the use of braking resistors and a novel generalization of earlier derived stabilization
algorithms to a multi-machine power system.
Chapter 11 introduces advanced models of the different power system elements. The new material
includes models of the wind turbine and generator and models of FACTS devices. Chapter 12
contains a largely expanded treatment of the steady-state stability of multi-machine power systems
using eigenvalue analysis. We have added a comprehensive explanation of the meaning of eigenvalues
and eigenvectors including a fuller treatment of the mathematical background. As the subject
matter is highly mathematical and may be difficult to understand, we have added a large number
of numerical examples. Chapter 13 contains a description of numerical methods used for power
system dynamic simulation. Chapter 14 explains how to reduce the size of the simulation problem
by using equivalents. The chapter has been significantly expanded by adding novel material on the
modal analysis of equivalents and a number of examples.
The Appendix covers the per-unit system and new material on the mathematical fundamentals
of solving ordinary differential equations.
It is important to emphasize that, while most of the book is a teaching textbook written with finalyear undergraduate and postgraduate students in mind, there are also large parts of material which
constitute cutting-edge research, some of it never published before. This includes the use of the
Lyapunov direct method to derive algorithms for the stabilization of a multi-machine power system
(Chapters 6, 9 and 10) and derivation of modal-analysis-based power system dynamic equivalents
(Chapter 14).
J. Machowski, J.W. Bialek and J.R. Bumby
Warsaw, Edinburgh and Durham
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Acknowledgements
We would like to acknowledge the financial support of Supergen FutureNet (www.super
gennetworks.org.uk). Supergen is funded by the Research Councils’ Energy Programme, United
Kingdom. We would also like to acknowledge the financial support of the Ministry of Science and
Higher Education of Poland (grant number 3 T10B 010 29). Both grants have made possible the
cooperation between the Polish and British co-authors. Last but not least, we are grateful as ever
for the patience shown by our wives and families during the torturous writing of yet another book.
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List of Symbols
Notation
Italic type denotes scalar physical quantity (e.g. R, L, C) or numerical variable (e.g. x, y).
Phasor or complex quantity or numerical variable is underlined (e.g. I, V, S).
Italic with arrow on top of a symbol denotes a spatial vector (e.g. F).
Italic boldface denotes a matrix or a vector (e.g. A, B, x, y).
Unit symbols are written using roman type (e.g. Hz, A, kV).
Standard mathematical functions are written using roman type (e.g. e, sin, cos, arctan).
Numbers are written using roman type (e.g. 5, 6).
Mathematical operators are written using roman type (e.g. s, Laplace operator; T, matrix transposition; j, angular shift by 90◦ ; a, angular shift by 120◦ ).
Differentials and partial differentials are written using roman type (e.g. d f/dx, ∂ f/∂ x).
Symbols describing objects are written using roman type (e.g. TRAFO, LINE).
Subscripts relating to objects are written using roman type (e.g. I TRAFO , I LINE ).
Subscripts relating to physical quantities or numerical variables are written using italic type (e.g.
Ai j , xk ).
Subscripts A, B, C refer to the three-phase axes of a generator.
Subscripts d, q refer to the direct- and quadrature-axis components.
Lower case symbols normally denote instantaneous values (e.g. v, i ).
Upper case symbols normally denote rms or peak values (e.g. V, I).
Symbols
a and a2
Bµ
Bsh
D
Ek
Ep
ef
eq
ed
eq
ed
operators shifting the angle by 120◦ and 240◦ , respectively.
magnetizing susceptance of a transformer.
susceptance of a shunt element.
damping coefficient.
kinetic energy of the rotor relative to the synchronous speed.
potential energy of the rotor with respect to the equilibrium point.
field voltage referred to the fictitious q-axis armature coil.
steady-state emf induced in the fictitious q-axis armature coil proportional to the field
winding self-flux linkages.
transient emf induced in the fictitious d-axis armature coil proportional to the flux
linkages of the q-axis coil representing the solid steel rotor body (round-rotor generators
only).
transient emf induced in the fictitious q-axis armature coil proportional to the field
winding flux linkages.
subtransient emf induced in the fictitious d-axis armature coil proportional to the total
q-axis rotor flux linkages (q-axis damper winding and q-axis solid steel rotor body).
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eq
E
Ef
Efm
Ed
Eq
E
Ed
Eq
E Ed
Eq
Er
Er m
EG
f
fn
F
Fa
Fa AC
Fa DC
Fad , Faq
Ff
G Fe
G sh
Hii , Hi j
iA, iB, iC
i A DC , i B DC , i C DC
i A AC , i B AC , i C AC
id, iq
iD, iQ
if
i ABC
i fDQ
i 0dq
I
Id , Iq
I S, I R
I R, I E
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List of Symbols
subtransient emf induced in the fictitious q-axis armature coil proportional to
the total d-axis rotor flux linkages (d-axis damper winding and field winding).
steady-state internal emf.
excitation emf proportional to the excitation voltage Vf .
peak value of the excitation emf.
d-axis component of the steady-state internal emf proportional to the rotor selflinkages due to currents induced in the q-axis solid steel rotor body (round-rotor
generators only).
q-axis component of the steady-state internal emf proportional to the field
winding self-flux linkages (i.e. proportional to the field current itself).
transient internal emf proportional to the flux linkages of the field winding and
solid steel rotor body (includes armature reaction).
d-axis component of the transient internal emf proportional to flux linkages in
the q-axis solid steel rotor body (round-rotor generators only).
q-axis component of the transient internal emf proportional to the field winding
flux linkages.
subtransient internal emf proportional to the total rotor flux linkages (includes
armature reaction).
d-axis component of the subtransient internal emf proportional to the total flux linkages in the q-axis damper winding and q-axis solid steel rotor
body.
q-axis component of the subtransient internal emf proportional to the total
flux linkages in the d-axis damper winding and the field winding.
resultant air-gap emf.
amplitude of the resultant air-gap emf.
vector of the generator emfs.
mains frequency.
rated frequency.
magnetomotive force (mmf) due to the field winding.
armature reaction mmf.
AC armature reaction mmf (rotating).
DC armature reaction mmf (stationary).
d- and q-axis components of the armature reaction mmf.
resultant mmf.
core loss conductance of a transformer.
conductance of a shunt element.
self- and mutual synchronizing power.
instantaneous currents in phases A, B and C.
DC component of the current in phases A, B, C.
AC component of the current in phases A, B, C.
currents flowing in the fictitious d- and q-axis armature coils.
instantaneous d- and q-axis damper winding current.
instantaneous field current of a generator.
vector of instantaneous phase currents.
vector of instantaneous currents in the field winding and the d- and q-axis
damper windings.
vector of armature currents in the rotor reference frame.
armature current.
d- and q-axis component of the armature current.
currents at the sending and receiving end of a transmission line.
vector of complex current injections to the retained and eliminated nodes.
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List of Symbols
I G, I L
I L
J
j
kPV , kQV
kPf , kQf
K Eq
K Eq
K E
Ki
KL
KT
l
LAA , LBB , LCC ,
Lff , LDD , LQQ
Ld , Lq
Ld , Lq , Ld , Lq
LS
Lxy
LS
LR
LS
LSR , LRS
M
Mf , MD , MQ
N
p
Pacc
PD
Pe
PEq cr
PEq (δ), PE (δ ),
PEq (δ )
Pg
PL
Pm
Pn
PR
PrI , PrII , PrIII , PrIV
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vector of complex generator and load currents.
vector of load corrective complex currents.
moment of inertia.
operator shifting the angle by 90◦ .
voltage sensitivities of the load (the slopes of the real and reactive power
demand characteristics as a function of voltage).
frequency sensitivities of the load (the slopes of the real and reactive
power demand characteristics as a function of frequency).
steady-state synchronizing power coefficient (the slope of the steady-state
power angle curve PEq (δ)).
transient synchronizing power coefficient (the slope of the transient power
angle curve PEq (δ )).
transient synchronizing power coefficient (the slope of the transient power
angle curve PE (δ )).
reciprocal of droop for the i th generating unit.
frequency sensitivity coefficient of the system real power demand.
reciprocal of droop for the total system generation characteristic.
length of a transmission line.
self-inductances of the windings of the phase windings A, B, C, the field
winding, and the d-and the q-axis damper winding.
inductances of the fictitious d- and q-axis armature windings.
d- and q-axis transient and subtransient inductances.
minimum value of the self-inductance of a phase winding.
where x, y ∈ {A, B, C, D, Q, f} and x = y, are the mutual inductances
between the windings denoted by the indices as described above.
amplitude of the variable part of the self-inductance of a phase winding.
submatrix of the rotor self- and mutual inductances.
submatrix of the stator self- and mutual inductances.
submatrices of the stator-to-rotor and rotor-to-stator mutual inductances.
coefficient of inertia.
amplitude of the mutual inductance between a phase winding and, respectively, the field winding and the d- and the q-axis damper winding.
generally, number of any objects.
number of poles.
accelerating power.
damping power.
electromagnetic air-gap power.
critical (pull-out) air-gap power developed by a generator.
air-gap power curves assuming Eq = constant, E = constant and Eq =
constant.
in induction machine, real power supplied from the grid (motoring mode),
or supplied to the grid (generating mode).
real power absorbed by a load or total system load.
mechanical power supplied by a prime mover to a generator; also mechanical power supplied by a motor to a load (induction machine in motoring
mode).
real power demand at rated voltage.
real power at the receiving end of a transmission line.
contribution of the generating units remaining in operation to covering
the real power imbalance during the first, second, third and fourth stages
of load frequency control.
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PsI , PsII , PsIII , PsIV
Ps
PS
PSIL
PsEq (δ)
PT
Ptie
PVg (δ)
PVg cr
QL
QG
Qn
QR
QS
R
r
RA , RB , RC , RD ,
RQ , Rf
RABC
RfDQ
s
s
scr
Sn
SSHC
t
Td , Td
, Tdo
Tdo
Tq , Tq
, Tqo
Tqo
Ta
T
vA , vB , vC , vf
vd , vq
vw
vABC
vfDQ
V
Vcr
Vd , Vq
Vf
Vg
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contribution of the system to covering the real power imbalance during
the first, second, third and fourth stages of load frequency control.
stator power of induction machine or power supplied by the system.
real power at the sending end of a transmission line or real power supplied
by a source to a load or real power supplied to an infinite busbar.
surge impedance (natural) load.
curve of real power supplied to an infinite busbar assuming Eq =
constant.
total power generated in a system.
net tie-line interchange power.
air-gap power curve assuming Vg = constant.
critical value of PVg (δ).
reactive power absorbed by a load.
reactive power generated by a source (the sum of QL and the reactive
power loss in the network).
reactive power demand at rated voltage.
reactive power at the receiving end of a transmission line.
reactive power at the sending end of a transmission line or reactive power
supplied by a source to a load.
resistance of the armature winding of a generator.
total resistance between (and including) a generator and an infinite
busbar.
resistances of the phase windings A, B, C, the d- and q-axis damper
winding, and the field winding.
diagonal matrix of phase winding resistances.
diagonal matrix of resistances of the field winding and the d- and q-axis
damper windings.
Laplace operator.
slip of induction motor.
critical slip of induction motor.
rated apparent power.
short-circuit power.
time.
short-circuit d-axis transient and subtransient time constants.
open-circuit d-axis transient and subtransient time constants.
short-circuit q-axis transient and subtransient time constants.
open-circuit q-axis transient and subtransient time constants.
armature winding time constant.
transformation matrix between network (a, b) and generator (d, q) coordinates.
instantaneous voltages across phases A, B, C and the field winding.
voltages across the fictitious d- and q-axis armature coils.
wind speed.
vector of instantaneous voltages across phases A, B, C.
vector of instantaneous voltages across the field winding and the d- and
q-axis damper windings.
Lyapunov function.
critical value of the voltage.
direct- and quadrature-axis component of the generator terminal voltage.
voltage applied to the field winding.
voltage at the generator terminals.
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List of Symbols
Vs
Vsd , Vsq
VS, VR
Vsh
V i = Vi δi
VR, VE
W
W
W, U
Xa
XC
XD
Xd , Xd , Xd
xd , xd , xd
xd PRE , xd F , xd POST
Xf
Xl
Xq , Xq , Xq
xq , xq , xq
XSHC
YT
Y
YGG , YLL , YLG , YLG
Yi j = G i j + jBi j
YRR , YEE , YRE , YER
Zc
Zs = Rs + jXs
ZT = RT + jXT
β
γ
γ0
δ
δg
δ̂s
δ
δ fr
ω
ε
ζ
ϑ
λR
λi = αi + j
ρ
ρT
i
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infinite busbar voltage.
direct- and quadrature-axis component of the infinite busbar voltage.
voltage at the sending and receiving end of a transmission line.
local voltage at the point of installation of a shunt element.
complex voltage at node i .
vector of complex voltages at the retained and eliminated nodes.
work.
Park’s modified transformation matrix.
modal matrices of right and left eigenvectors.
armature reaction reactance (round-rotor generator).
reactance of a series compensator.
reactance corresponding to the flux path around the damper winding.
d-axis synchronous, transient and subtransient reactance.
total d-axis synchronous, transient and subtransient reactance between
(and including) a generator and an infinite busbar.
prefault, fault and postfault value of xd .
reactance corresponding to the flux path around the field winding.
armature leakage reactance of a generator.
q-axis synchronous, transient and subtransient reactance.
total q-axis synchronous, transient and subtransient reactance between
(and including) a generator and an infinite busbar.
short-circuit reactance of a system as seen from a node.
admittance of a transformer.
admittance matrix.
admittance submatrices where subscript G corresponds to fictitious generator nodes and subscript L corresponds to all the other nodes (including
generator terminal nodes).
element of the admittance matrix.
complex admittance submatrices where subscript E refers to eliminated
nodes and subscript R to retained nodes.
characteristic impedance of a transmission line.
internal impedance of an infinite busbar.
series impedance of a transformer.
phase constant of a transmission line.
instantaneous position of the generator d-axis relative to phase A; propagation constant of a transmission line.
position of the generator d-axis at the instant of fault.
power (or rotor) angle with respect to an infinite busbar.
power (or rotor) angle with respect to the voltage at the generator
terminals.
stable equilibrium value of the rotor angle.
transient power (or rotor) angle between E and Vs .
angle between the resultant and field mmfs.
rotor speed deviation equal to (ω − ωs ).
rotor acceleration.
damping ratio.
transformation ratio.
frequency bias factor.
eigenvalue.
static droop of the turbine–governor characteristic.
droop of the total system generation characteristic.
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xxvi
τe
τm
τω
τ 2ω
τ d, τ q
τ R, τ r
ϕg
a
ad , aq
a AC
a DC
f
A , B , C
AA , BB , B
a AC r
a DC r
a r
D , Q
d , q
f
fa
fA , fB , fC
ABC
fDQ
electromagnetic torque.
mechanical torque.
fundamental-frequency subtransient electromagnetic torque.
double-frequency subtransient electromagnetic torque.
direct- and quadrature-axis component of the electromagnetic torque.
subtransient electromagnetic torque due to stator and rotor resistances.
power factor angle at the generator terminals.
armature reaction flux.
d- and q-axis component of the armature reaction flux.
AC armature reaction flux (rotating).
DC armature reaction flux (stationary).
excitation (field) flux.
total flux linkage of phases A, B, C.
self-flux linkage of phases A, B, C.
rotor flux linkages produced by a AC .
rotor flux linkages produced by a DC .
rotor flux linkages produced by the total armature reaction flux.
total flux linkage of damper windings in axes d and q.
total d- and q-axis flux linkages.
total flux linkage of the field winding.
excitation flux linkage with armature winding.
excitation flux linkage with phases A, B and C.
vector of phase flux linkages.
vector of flux linkages of the field winding and the d- and q-axis damper
windings.
vector of armature flux linkages in the rotor reference frame.
angular velocity of the generator (in electrical radians).
synchronous angular velocity in electrical radians (equal to 2π f ).
rotor speed of wind turbine (in rad/s)
frequency of rotor swings (in rad/s)
rotation matrix.
reluctance.
reluctance along the direct- and quadrature-axis.
0dq
ω
ωs
ωT
d , q
Abbreviations
AC
ACE
AGC
AVR
BEES
d
DC
DFIG
DFIM
DSA
emf
EMS
FACTS
HV
HAWT
alternating current
area control error
Automatic Generation Control
Automatic Voltage Regulator
Battery Energy Storage System
direct axis of a generator
direct current
Doubly Fed Induction Generator
Double Fed Induction Machine
Dynamic Security Assessment
electro-motive force
Energy Management System
Flexible AC Transmission Systems
high voltage
Horizontal-Axis Wind Turbine
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List of Symbols
IGTB
IGTC
LFC
mmf
MAWS
PMU
PSS
pu
q
rms
rpm
rhs
SCADA
SIL
SMES
SSSC
STATCOM
SVC
TCBR
TCPAR
TSO
VAWT
UPFC
WAMS
WAMPAC
insulated gate bipolar transistor
integrated gate-commutated thyristor
load frequency control
magneto-motive force
mean annual wind speed
Phasor Measurement Unit
power system stabiliser
per unit
quadrature axis of a generator
root-mean-square
revolutions per minute
right-hand-side
Supervisory Control and Data Acquisition
surge impedance load
superconducting magnetic energy storage
Static Synchronous Series Compensator
static compensator
Static VAR Compensator
Thyristor Controlled Braking Resistor
Thyristor Controlled Phase Angle Regulator
Transmission System Operator
Vertical-Axis Wind Turbine
unified power flow controller
Wide Area Measurement System
Wide Area Measurement, Protection and Control
xxvii
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