TÀI LIỆU VỀ ỔN ĐỊNH ĐỘNG HỆ THỐNG ĐIỆN VÀ ĐIỀU KHIỂN HỆ THỐNG ĐIỆN TẬP 2 (Power System Dynamics Stability and Control Second Edition)

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 toldus 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 WAMSbased realtime control.

POWER SYSTEM DYNAMICS

Stability and Control

POWER SYSTEM DYNAMICS

POWER SYSTEM DYNAMICS

Stability and Control

This edition first published 2008

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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.

1 Electric power system stability 2 Electric power systems–Control I Bialek, Janusz

W II Bumby, J R (James Richard) III Title.

Typeset in 9/11pt Times New Roman by Aptara Inc., New Delhi, India.

Printed in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire

PART I INTRODUCTION TO POWER SYSTEMS

1.3 Two Pairs of Important Quantities:

Reactive Power/Voltage and Real Power/Frequency 7

2.6 Protection 54

3.3.4 Reactive Power Capability Curve of a Round-Rotor Generator 91

3.3.5 Voltage–Reactive Power Capability Characteristic V(Q) 95

PART II INTRODUCTION TO POWER SYSTEM DYNAMICS

4.2 Three-Phase Short Circuit on a Synchronous Generator 129

4.2.1 Three-Phase Short Circuit with the Generator on No Load and Winding

4.2.3 Armature Flux Paths and the Equivalent Reactances 134

4.2.4 Generator Electromotive Forces and Equivalent Circuits 140

4.2.5 Short-Circuit Currents with the Generator Initially on No Load 146

4.3 Phase-to-Phase Short Circuit 152

4.3.1 Short-Circuit Current and Flux with Winding Resistance Neglected 153

4.5 Short-Circuit in a Network and its Clearing 166

5.4.6 Analysis of Rotor Swings Around the Equilibrium Point 191

5.4.7 Mechanical Analogues of the Generator–Infinite Busbar System 1955.5 Steady-State Stability of the Regulated System 196

5.5.1 Steady-State Power–Angle Characteristic of Regulated Generator 196

5.5.2 Transient Power–Angle Characteristic of the Regulated Generator 200

6.1.1 Fault Cleared Without a Change in the Equivalent Network Impedance 207

6.1.2 Short-Circuit Cleared with/without Auto-Reclosing 212

6.3.5 Lyapunov Direct Method for a Multi-Machine System 230

6.5 Asynchronous Operation and Resynchronization 239

6.6 Out-of-Step Protection Systems 244

6.7 Torsional Oscillations in the Drive Shaft 253

6.7.1 The Torsional Natural Frequencies of the Turbine–Generator Rotor 253

7.4 Induction Generators with Slightly Increased Speed Range via External Rotor

7.5 Induction Generators with Significantly Increased Speed Range: DFIGs 282

7.5.1 Operation with the Injected Voltage in Phase with the Rotor Current 284

7.5.2 Operation with the Injected Voltage out of Phase with the Rotor Current 286

7.6 Fully Rated Converter Systems: Wide Speed Control 290

7.7 Peak Power Tracking of Variable Speed Wind Turbines 293

7.10 Influence of Wind Generators on Power System Stability 296

8.3 Critical Load Demand and Voltage Collapse 310

8.3.3 Influence of the Shape of the Load Characteristics 315

8.5 Dynamic Analysis 321

8.7 Self-Excitation of a Generator Operating on a Capacitive Load 329

8.7.2 Self-Excitation of a Generator with Open-Circuited Field Winding 330

8.7.3 Self-Excitation of a Generator with Closed Field Winding 332

9.5.2 Interconnected Systems and Tie-Line Oscillations 364

9.6.2 State-Variable Control Based on Lyapunov Method 375

9.6.4 Coordination Between AGC and Series FACTS Devices in Tie-Lines 379

10.6 Series Compensators 416

PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS

11.1.2 The Flux Linkage Equations in the Stator Reference Frame 434

11.1.3 The Flux Linkage Equations in the Rotor Reference Frame 436

11.1.5 Generator Reactances in Terms of Circuit Quantities 443

12.2 Steady-State Stability of Unregulated System 514

12.2.3 Including the Voltage Characteristics of the Loads 521

12.3 Steady-State Stability of the Regulated System 523

12.3.2 Including Excitation System Model and Voltage Control 525

13.2.4 Ways of Avoiding Iterations and Multiple Network Solutions 551

14.6 Properties of Coherency-Based Equivalents 573

14.6.1 Electrical Interpretation of Zhukov’s Aggregation 573

14.6.4 Eigenvalues and Eigenvectors of the Equivalent Model 582

About the Authors

Professor Jan Machowski received his MSc and PhD degrees in

Elec-trical Engineering from Warsaw University of Technology in 1974 and

1979, respectively After obtaining field experience in the DispatchingCentre and several power plants, he joined the Electrical Faculty ofWarsaw University of Technology where presently he is employed as aProfessor and Director of the Power Engineering Institute His areas

of interest are electrical power systems, power system protection andcontrol

In 1989–93 Professor Machowski was a Visiting Professor at slautern University in Germany where he carried out two researchprojects on power swing blocking algorithms for distance protectionand optimal control of FACTS devices

Kaiser-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 tional fora He has carried out many projects on electrical power systems, power system stabilityand power system protection commissioned by the Polish Power Grid Company, Electric PowerResearch Institute in the United States, Electrinstitut Milan Vidmar in Slovenia and Ministry ofScience and Higher Education of Poland

interna-Professor Janusz Bialek received his MEng and PhD degrees in

Elec-trical Engineering from Warsaw University of Technology in 1977 and

1981, respectively From 1981 to 1989 he was a lecturer with saw University of Technology In 1989 he moved to the University ofDurham, United Kingdom, and since 2003 he has been at the Univer-sity of Edinburgh where he currently holds the Bert Whittington Chair

War-of Electrical Engineering His main research interests are in able energy systems, security of supply, liberalization of the electricitysupply industry and power system dynamics and control

sustain-Professor Bialek has co-authored two books and over 100 researchpapers He has been a consultant to the Department of Trade andIndustry (DTI) of the UK government, Scottish Executive, Elexon,Polish Power Grid Company, Scottish Power, Enron and Electrical Power Research Institute (EPRI)

He was the Principal Investigator of a number of major research grants funded by the Engineeringand Physical Sciences Research Council and the DTI

Professor Bialek is a member of the Advisory Board of Electricity Policy Research Group,Cambridge University, a member of the Dispute Resolution Panel for the Single Electricity MarketOperator, Ireland, and Honorary Professor of Heriot-Watt University, Scotland

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 De-velopment Company, Newcastle-upon-Tyne, on superconducting ma-chines, hybrid vehicles and sea-wave energy Since 1978 he has worked

in the School of Engineering at Durham University where he is rently Reader in Electrical Engineering He has worked in the area ofelectrical machines and systems for over 30 years, first in industry andthen in academia

cur-Dr Bumby is the author or co-author of over 100 technical papers andtwo books in the general area of electrical machines/power systems andcontrol He has also written numerous technical reports for industrialclients These papers and books have led to the award of a number of national and internationalprizes 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 researchinterests are in novel generator technologies and their associated control for new and renewableenergy systems

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 quicklyrealized that the developments in the power systems industry over the intervening years required alarge 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 Thenew title also reflects a slight shift in focus from solely describing power system dynamics to themeans of dealing with them For example, we believe that the new WAMS technology is likely torevolutionize power system control One of the main obstacles to a wider embrace of WAMS bypower 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 forWAMS-based real-time control

The second reason for adding so much new material is the unprecedented change that has beensweeping the power systems industry since the 1990s In particular the rapid growth of renewablegeneration, driven by global warming concerns, is changing the fundamental characteristics ofthe system Currently wind power is the dominant renewable energy source and wind generatorsusually use induction, rather than synchronous, machines As a significant penetration of suchgeneration will change the system dynamics, the new material in Chapter 7 is devoted entirely towind generation

The third factor to be taken into account is the fallout from a number of highly publicized outs that happened in the early years of the new millennium Of particular concern were the autumn

black-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 blackoutshave exposed a number of critical issues, especially those regarding power system behaviour atdepressed voltages Consequently, the book has been extended to cover these phenomena togetherwith an illustration of some of the blackouts

It is important to emphasize that the new book is based on the same philosophy as the previousone We try to answer some of the concerns about the education of power system engineers Withthe 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 especiallydangerous for students and young researchers who think that simulation is a panacea for everythingand 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 simulationresults It is by no means bad practice to treat the initial results of any computer software with ahealthy pinch of scepticism

Power system dynamics are not easy to understand There are a number of good textbooks whichdeal with this topic and some of these are reviewed in Chapter 1 As the synchronous machineplays a decisive role in determining the dynamic response of the system, many of these books startwith a detailed mathematical treatment of the synchronous generator in order to introduce Park’sequations and produce a mathematical model of the generator However, it is our experience that tobegin a topic with such a detailed mathematical treatment can put many students off further studybecause 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 understandwhat is happening in the system without having to refer continuously to a complicated mathematicalmodel of the generator

Our approach is different We first try to give a qualitative explanation of the underlying physicalphenomena of power system dynamics using a simple model of the generator, coupled with the basicphysical 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 Inthis way we hope that the material is made more accessible to the reader who wishes to understandthe system operation without first tackling Park’s equations

All our considerations are richly illustrated by diagrams We strongly believe in the old adagethat 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) reviewsthe 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 ofthe generator–infinite busbar system The third part (Chapters 11–14) tackles some of the moreadvanced 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 powersystem dynamics and provides a brief historical overview The new material expands on the defini-tions of power system stability and security assessment and introduces some important conceptsused in later chapters Chapter 2 contains a brief description of the major power system compo-nents, including modern FACTS devices The main additions here provide a more comprehensivetreatment of FACTS devices and a whole new section on WAMS Chapter 3 introduces steady-statemodels and their use in analysing the performance of the power system The new material coversenhanced treatment of the generator as the reactive power source introducing voltage–reactivepower capability characteristics We believe that this is a novel treatment of those concepts since wehave not seen it anywhere else The importance of understanding how the generator and its controlsbehave 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 foranalysing the dynamic performance of the synchronous generator Chapter 5 explains the powersystem dynamics following a small disturbance (steady-state stability) while Chapter 6 examinesthe system dynamics following a large disturbance (transient stability) There are new sections onusing the Lyapunov direct method to analyse the stability of a multi-machine power system and onout-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 withsome of the methods used for stability assessment The new material includes examples of powersystem 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 controlincluding 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 automaticgeneration control (AGC) and FACTS devices installed in tie-lines that control the flow of powerbetween systems in an interconnected network Chapter 10 provides an overview of the mainmethods of stability enhancement, both conventional and using FACTS devices The new material

includes the use of braking resistors and a novel generalization of earlier derived stabilizationalgorithms to a multi-machine power system.

Chapter 11 introduces advanced models of the different power system elements The new materialincludes models of the wind turbine and generator and models of FACTS devices Chapter 12contains a largely expanded treatment of the steady-state stability of multi-machine power systemsusing eigenvalue analysis We have added a comprehensive explanation of the meaning of eigenvaluesand eigenvectors including a fuller treatment of the mathematical background As the subjectmatter 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 powersystem 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 themodal 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 year undergraduate and postgraduate students in mind, there are also large parts of material whichconstitute cutting-edge research, some of it never published before This includes the use of theLyapunov 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)

final-J Machowski, final-J.W Bialek and final-J.R Bumby

Warsaw, Edinburgh and Durham

We would like to acknowledge the financial support of Supergen FutureNet (www.supergennetworks.org.uk) Supergen is funded by the Research Councils’ Energy Programme, UnitedKingdom We would also like to acknowledge the financial support of the Ministry of Science andHigher Education of Poland (grant number 3 T10B 010 29) Both grants have made possible thecooperation between the Polish and British co-authors Last but not least, we are grateful as everfor the patience shown by our wives and families during the torturous writing of yet another book

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 sition; j, angular shift by 90◦; a, angular shift by 120◦)

transpo-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 ITRAFO, ILINE)

Subscripts relating to physical quantities or numerical variables are written using italic type (e.g

A i j , x k)

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 operators shifting the angle by 120◦and 240◦, respectively

B µ magnetizing susceptance of a transformer

Bsh susceptance of a shunt element

D damping coefficient

Ek kinetic energy of the rotor relative to the synchronous speed

Ep potential energy of the rotor with respect to the equilibrium point

ef field voltage referred to the fictitious q-axis armature coil

eq steady-state emf induced in the fictitious q-axis armature coil proportional to the field

winding self-flux linkages

ed 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 generatorsonly)

eq transient emf induced in the fictitious q-axis armature coil proportional to the field

winding flux linkages

ed 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)

eq 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)

E steady-state internal emf

Ef excitation emf proportional to the excitation voltage Vf

Efm peak value of the excitation emf

Ed d-axis component of the steady-state internal emf proportional to the rotor

self-linkages due to currents induced in the q-axis solid steel rotor body (round-rotorgenerators only)

Eq 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)

E transient internal emf proportional to the flux linkages of the field winding and

solid steel rotor body (includes armature reaction)

Ed d-axis component of the transient internal emf proportional to flux linkages in

the q-axis solid steel rotor body (round-rotor generators only)

Eq q-axis component of the transient internal emf proportional to the field winding

flux linkages

E subtransient internal emf proportional to the total rotor flux linkages (includes

armature reaction)

Ed d-axis component of the subtransient internal emf proportional to the

to-tal flux linkages in the q-axis damper winding and q-axis solid steel rotorbody

Eq q-axis component of the subtransient internal emf proportional to the total

flux linkages in the d-axis damper winding and the field winding

Er resultant air-gap emf

E r m amplitude of the resultant air-gap emf

EG vector of the generator emfs

fn rated frequency

F magnetomotive force (mmf) due to the field winding

Fa armature reaction mmf

Fa AC AC armature reaction mmf (rotating)

Fa DC DC armature reaction mmf (stationary)

Fad, Faq d- and q-axis components of the armature reaction mmf

Ff resultant mmf

GFe core loss conductance of a transformer

Gsh conductance of a shunt element

H i i , H i j self- and mutual synchronizing power

iA, iB, iC instantaneous currents in phases A, B and C

iA DC, iB DC, iC DC DC component of the current in phases A, B, C

iA AC, iB AC, iC AC AC component of the current in phases A, B, C

id, iq currents flowing in the fictitious d- and q-axis armature coils

iD, iQ instantaneous d- and q-axis damper winding current

if instantaneous field current of a generator

iABC vector of instantaneous phase currents

ifDQ vector of instantaneous currents in the field winding and the d- and q-axis

damper windings

i0dq vector of armature currents in the rotor reference frame

I armature current

Id, Iq d- and q-axis component of the armature current

IS, IR currents at the sending and receiving end of a transmission line

I , I vector of complex current injections to the retained and eliminated nodes

IG, IL vector of complex generator and load currents.

IL vector of load corrective complex currents

j operator shifting the angle by 90◦

kPV, kQV voltage sensitivities of the load (the slopes of the real and reactive power

demand characteristics as a function of voltage)

kPf, kQf frequency sensitivities of the load (the slopes of the real and reactive

power demand characteristics as a function of frequency)

K Eq steady-state synchronizing power coefficient (the slope of the steady-state

power angle curve P Eq(δ)).

K Eq transient synchronizing power coefficient (the slope of the transient power

angle curve P Eq(δ))

K E transient synchronizing power coefficient (the slope of the transient power

angle curve P E(δ))

K i reciprocal of droop for the i th generating unit.

KL frequency sensitivity coefficient of the system real power demand

KT reciprocal of droop for the total system generation characteristic

l length of a transmission line

LAA, LBB, LCC, self-inductances of the windings of the phase windings A, B, C, the field

winding, and the d-and the q-axis damper winding

q d- and q-axis transient and subtransient inductances

LS minimum value of the self-inductance of a phase winding

Lxy 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

LS amplitude of the variable part of the self-inductance of a phase winding

LR submatrix of the rotor self- and mutual inductances

LS submatrix of the stator self- and mutual inductances

LSR , LRS submatrices of the stator-to-rotor and rotor-to-stator mutual inductances

M coefficient of inertia

Mf, MD, MQ amplitude of the mutual inductance between a phase winding and,

re-spectively, the field winding and the d- and the q-axis damper winding

N generally, number of any objects

Pacc accelerating power

Pe electromagnetic air-gap power

P Eq cr critical (pull-out) air-gap power developed by a generator

P Eq(δ), P E(δ), air-gap power curves assuming Eq= constant, E= constant and E

constant

P Eq(δ)

Pg in induction machine, real power supplied from the grid (motoring mode),

or supplied to the grid (generating mode)

PL real power absorbed by a load or total system load

Pm mechanical power supplied by a prime mover to a generator; also

mechan-ical power supplied by a motor to a load (induction machine in motoringmode)

Pn real power demand at rated voltage

PR real power at the receiving end of a transmission line

PrI, PrII, PrIII, PrIV 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

PsI, PsII, PsIII, PsIV contribution of the system to covering the real power imbalance during

the first, second, third and fourth stages of load frequency control

P s stator power of induction machine or power supplied by the system

PS 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

PSIL surge impedance (natural) load

P sEq(δ) curve of real power supplied to an infinite busbar assuming Eq=

constant

PT total power generated in a system

Ptie net tie-line interchange power

P Vg(δ) air-gap power curve assuming Vg= constant

P Vg cr critical value of P Vg(δ).

QL reactive power absorbed by a load

QG reactive power generated by a source (the sum of QL and the reactive

power loss in the network)

Qn reactive power demand at rated voltage

QR reactive power at the receiving end of a transmission line

QS reactive power at the sending end of a transmission line or reactive power

supplied by a source to a load

R resistance of the armature winding of a generator

r total resistance between (and including) a generator and an infinite

busbar

RA, RB, RC, RD, resistances of the phase windings A, B, C, the d- and q-axis damper

winding, and the field winding

RQ, Rf

RABC diagonal matrix of phase winding resistances

RfDQ diagonal matrix of resistances of the field winding and the d- and q-axis

damper windings

s slip of induction motor

scr critical slip of induction motor

Sn rated apparent power

qo open-circuit q-axis transient and subtransient time constants

Ta armature winding time constant

T transformation matrix between network (a, b) and generator (d, q)

coor-dinates

vA, vB, vC, vf instantaneous voltages across phases A, B, C and the field winding

vd, vq voltages across the fictitious d- and q-axis armature coils

vABC vector of instantaneous voltages across phases A, B, C

vfDQ vector of instantaneous voltages across the field winding and the d- and

q-axis damper windings

Vcr critical value of the voltage

Vd, Vq direct- and quadrature-axis component of the generator terminal voltage

Vf voltage applied to the field winding

V voltage at the generator terminals

Vs infinite busbar voltage.

Vsd, Vsq direct- and quadrature-axis component of the infinite busbar voltage

VS, VR voltage at the sending and receiving end of a transmission line

Vsh local voltage at the point of installation of a shunt element

V i = V i  δ i complex voltage at node i

VR, VE vector of complex voltages at the retained and eliminated nodes

W Park’s modified transformation matrix

W , U modal matrices of right and left eigenvectors

Xa armature reaction reactance (round-rotor generator)

XC reactance of a series compensator

XD reactance corresponding to the flux path around the damper winding

d total d-axis synchronous, transient and subtransient reactance between

(and including) a generator and an infinite busbar

xd PRE , x

d F, x

Xf reactance corresponding to the flux path around the field winding

Xl armature leakage reactance of a generator

q total q-axis synchronous, transient and subtransient reactance between

(and including) a generator and an infinite busbar

XSHC short-circuit reactance of a system as seen from a node

YT admittance of a transformer

YGG, YLL, YLG, YLG admittance submatrices where subscript G corresponds to fictitious

gen-erator nodes and subscript L corresponds to all the other nodes (includinggenerator terminal nodes)

Y i j = G i j + jB i j element of the admittance matrix

YRR, YEE, YRE, YER complex admittance submatrices where subscript E refers to eliminated

nodes and subscript R to retained nodes

Zc characteristic impedance of a transmission line

Zs= Rs+ jXs internal impedance of an infinite busbar

ZT= RT+ jXT series impedance of a transformer

β phase constant of a transmission line

γ instantaneous position of the generator d-axis relative to phase A;

prop-agation constant of a transmission line

γ0 position of the generator d-axis at the instant of fault

δ power (or rotor) angle with respect to an infinite busbar

δg power (or rotor) angle with respect to the voltage at the generator

terminals

ˆ

δs stable equilibrium value of the rotor angle

δ transient power (or rotor) angle between Eand Vs

δfr angle between the resultant and field mmfs

ω rotor speed deviation equal to (ω − ωs)

ρ static droop of the turbine–governor characteristic

ρ droop of the total system generation characteristic

τe electromagnetic torque.

τ ω fundamental-frequency subtransient electromagnetic torque

τ2ω double-frequency subtransient electromagnetic torque

τd,τq direct- and quadrature-axis component of the electromagnetic torque

τR,τr subtransient electromagnetic torque due to stator and rotor resistances

ϕg power factor angle at the generator terminals

a armature reaction flux

ad, aq d- and q-axis component of the armature reaction flux

a AC AC armature reaction flux (rotating)

a DC DC armature reaction flux (stationary)

f excitation (field) flux

A, B, C total flux linkage of phases A, B, C

AA, BB, B self-flux linkage of phases A, B, C

a r rotor flux linkages produced by the total armature reaction flux

D, Q total flux linkage of damper windings in axes d and q

d, q total d- and q-axis flux linkages

f total flux linkage of the field winding

fa excitation flux linkage with armature winding

fA, fB, fC excitation flux linkage with phases A, B and C

ABC vector of phase flux linkages

fDQ vector of flux linkages of the field winding and the d- and q-axis damper

windings

0dq vector of armature flux linkages in the rotor reference frame

ω angular velocity of the generator (in electrical radians)

ωs synchronous angular velocity in electrical radians (equal to 2π f ).

ωT rotor speed of wind turbine (in rad/s)

frequency of rotor swings (in rad/s)rotation matrix

d, q reluctance along the direct- and quadrature-axis

Abbreviations

AC alternating current

ACE area control error

AGC Automatic Generation Control

AVR Automatic Voltage Regulator

BEES Battery Energy Storage System

d direct axis of a generator

DC direct current

DFIG Doubly Fed Induction Generator

DFIM Double Fed Induction Machine

DSA Dynamic Security Assessment

emf electro-motive force

EMS Energy Management System

FACTS Flexible AC Transmission Systems

HV high voltage

HAWT Horizontal-Axis Wind Turbine

IGTB insulated gate bipolar transistor

IGTC integrated gate-commutated thyristor

LFC load frequency control

mmf magneto-motive force

MAWS mean annual wind speed

PMU Phasor Measurement Unit

PSS power system stabiliser

SCADA Supervisory Control and Data Acquisition

SIL surge impedance load

SMES superconducting magnetic energy storage

SSSC Static Synchronous Series Compensator

STATCOM static compensator

SVC Static VAR Compensator

TCBR Thyristor Controlled Braking Resistor

TCPAR Thyristor Controlled Phase Angle Regulator

TSO Transmission System Operator

VAWT Vertical-Axis Wind Turbine

UPFC unified power flow controller

WAMS Wide Area Measurement System

WAMPAC Wide Area Measurement, Protection and Control

Part I

Introduction to Power Systems

Introduction

1.1 Stability and Control of a Dynamic System

In engineering, a system is understood to be a set of physical elements acting together and realizing

a common goal An important role in the analysis of the system is played by its mathematical

model It is created using the system structure and fundamental physical laws governing the system

elements In the case of complicated systems, mathematical models usually do not have a sal character but rather reflect some characteristic phenomena which are of interest Because ofmathematical complications, practically used system models are usually a compromise between arequired accuracy of modelling and a degree of complication

univer-When formulating a system model, important terms are the system state and the state variables.

The system state describes the system’s operating conditions The state variables are the minimum set

of variables x1, x2, , x n uniquely defining the system state State variables written as a vector x=

[x1, x2, , x n]Tare referred to as the state vector A normalized space of coordinates corresponding

to the state variables is referred to as the state space In the state space, each system state corresponds

to a point defined by the state vector Hence, a term ‘system state’ often refers also to a point in thestate space

A system may be static, when its state variables x1, x2, , x n are time invariant, or dynamic, when they are functions of time, that is x1(t) , x2(t) , , x n (t).

This book is devoted to the analysis of dynamic systems modelled by ordinary differentialequations of the form

where the first of the equations above describes a nonlinear system and the second describes a linear

system F(x) is just a vector of nonlinear functions and A is a square matrix.

A curve x(t) in the state space containing system states (points) in consecutive time instants is referred to as the system trajectory A trivial one-point trajectory x(t) = ˆx = constant is referred to

as the equilibrium point (state), if in that point all the partial derivatives are zero (no movement), that

is ˙x= 0 According to Equation (1.1), the coordinates of the point satisfy the following equations:

A nonlinear system may have more than one equilibrium point because nonlinear equationsmay have generally more than one solution In the case of linear systems, according to the Cramertheorem concerning linear equations, there exists only one uniquely specified equilibrium point

ˆx = 0 if and only if the matrix A is non-singular (det A
= 0).

C

 2008 John Wiley & Sons, Ltd

3

All the states of a dynamic system, apart from equilibrium states, are dynamic states because

the derivatives ˙x
= 0 for those states are non-zero, which means a movement Disturbance means a

random (usually unintentional) event affecting the system Disturbances affecting dynamic systemsare modelled by changes in their coefficients (parameters) or by non-zero initial conditions ofdifferential equations

Let x1(t) be a trajectory of a dynamic system, see Figure 1.1a, corresponding to some initial conditions The system is considered stable in a Lyapunov sense if for any t0it is possible to choose anumberη such that for all the other initial conditions satisfying the constraint x2(t0)− x1(t0) < η,

the expressionx2(t) − x1(t)  < ε holds for t0≤ t < ∞ In other words, stability means that if the

trajectory x2(t) starts close enough (as defined by η) to the trajectory x1(t) then it remains close to

it (numberε) Moreover, if the trajectory x2(t) tends with time towards the trajectory x1(t), that is

limt→∞x2(t) − x1(t)  = 0, then the dynamic system is asymptotically stable.

The above definition concerns any trajectory of a dynamic system Hence it must also be valid

for a trivial trajectory such as the equilibrium point ˆx In this particular case, see Figure 1.1b, the trajectory x1(t) is a point ˆx and the initial condition x2(t0) of trajectory x2(t) lies in the vicinity of

the point defined byη The dynamic system is stable in the equilibrium point ˆx if for t0≤ t < ∞

the trajectory x2(t) does not leave an area defined by the number ε Moreover, if the trajectory x2(t)

tends with time towards the equilibrium point ˆx, that is lim t→∞x2(t) − ˆx = 0, then the system

is said to be asymptotically stable at the equilibrium point ˆx On the other hand, if the trajectory

x2 (t) tends with time to leave the area defined by ε, then the dynamic system is said to be unstable

at the equilibrium point ˆx.

It can be shown that stability of a linear system does not depend on the size of a disturbance.Hence if a linear system is stable for a small disturbance then it is also globally stable for any largedisturbance

The situation is different with nonlinear systems as their stability generally depends on the size

of a disturbance A nonlinear system may be stable for a small disturbance but unstable for a largedisturbance The largest disturbance for which a nonlinear system is still stable is referred to as a

critical disturbance.

Dynamic systems are designed and constructed with a particular task in mind and assumingthat they will behave in a particular way following a disturbance A purposeful action affecting

a dynamic system which aims to achieve a particular behaviour is referred to as a control The

definition of control is illustrated in Figure 1.2 The following signals have been defined:

control device

z(t)

task

x(t) u(t)

z(t)

Figure 1.2 Illustration of the definition of: (a) open-loop control; (b) closed-loop control

Control can be open loop or closed loop In the case of open-loop control, see Figure 1.2a,control signals are created by a control device which tries to achieve a desired system behaviourwithout obtaining any information about the output signals Such control makes sense only when

it is possible to predict the shape of output signals from the control signals However, if there areadditional disturbances which are not a part of the control, then their action may lead to the controlobjective not being achieved

In the case of closed-loop control, see Figure 1.2b, control signals are chosen based on thecontrol task and knowledge of the system output signals describing whether the control task hasbeen achieved Hence the control is a function of its effects and acts until the control task has beenachieved

Closed-loop control is referred to as feedback control or regulation The control device is then called a regulator and the path connecting the output signals with the control device (regulator) is called the feedback loop.

A nonlinear dynamic system with its control can be generally described by the following set ofalgebraic and differential equations:

˙x = F(x, u) and y = G(x, u), (1.3)while a linear dynamic system model is

˙x = A x + B u and y = C x + D u. (1.4)

It is easy to show that, for small changes in state variables and output and control signals, Equations(1.4) are linear approximations of nonlinear equations (1.3) In other words, linearization of (1.3)leads to the equations

 ˙x = Ax + B  u and  y = C x + D u, (1.5)

where A, B, C, D are the matrices of derivatives of functions F, G with respect to x and u.

1.2 Classification of Power System Dynamics

An electrical power system consists of many individual elements connected together to form alarge, complex and dynamic system capable of generating, transmitting and distributing electricalenergy over a large geographical area Because of this interconnection of elements, a large variety

of dynamic interactions are possible, some of which will only affect some of the elements, others

are fragments of the system, while others may affect the system as a whole As each dynamic effectdisplays certain unique features Power system dynamics can be conveniently divided into groupscharacterized by their cause, consequence, time frame, physical character or the place in the systemwhere they occur.

Of prime concern is the way the power system will respond to both a changing power demand and

to various types of disturbance, the two main causes of power system dynamics A changing powerdemand introduces a wide spectrum of dynamic changes into the system each of which occurs on

a different time scale In this context the fastest dynamics are due to sudden changes in demandand are associated with the transfer of energy between the rotating masses in the generators andthe loads Slightly slower are the voltage and frequency control actions needed to maintain systemoperating conditions until finally the very slow dynamics corresponding to the way in which thegeneration is adjusted to meet the slow daily demand variations take effect Similarly, the way inwhich the system responds to disturbances also covers a wide spectrum of dynamics and associatedtime frames In this case the fastest dynamics are those associated with the very fast wave phenomenathat occur in high-voltage transmission lines These are followed by fast electromagnetic changes

in the electrical machines themselves before the relatively slow electromechanical rotor oscillationsoccur Finally the very slow prime mover and automatic generation control actions take effect.Based on their physical character, the different power system dynamics may be divided into four

groups defined as: wave, electromagnetic, electromechanical and thermodynamic This classification

also corresponds to the time frame involved and is shown in Figure 1.3 Although this broadclassification is convenient, it is by no means absolute, with some of the dynamics belonging to two ormore groups while others lie on the boundary between groups Figure 1.3 shows the fastest dynamics

to be the wave effects, or surges, in high-voltage transmission lines and correspond to the propagation

of electromagnetic waves caused by lightning strikes or switching operations The time frame ofthese dynamics is from microseconds to milliseconds Much slower are the electromagnetic dynamicsthat take place in the machine windings following a disturbance, operation of the protection system

or the interaction between the electrical machines and the network Their time frame is frommilliseconds to a second Slower still are the electromechanical dynamics due to the oscillation ofthe rotating masses of the generators and motors that occur following a disturbance, operation

of the protection system and voltage and prime mover control The time frame of these dynamics

is from seconds to several seconds The slowest dynamics are the thermodynamic changes whichresult from boiler control action in steam power plants as the demands of the automatic generationcontrol are implemented

Careful inspection of Figure 1.3 shows the classification of power system dynamics with respect

to time frame to be closely related to where the dynamics occur within the system For example,moving from the left to right along the time scale in Figure 1.3 corresponds to moving through the

power system from the electrical RLC circuits of the transmission network, through the generator

hours minutes

seconds milliseconds

microseconds

wave phenomena

electromagnetic phenomena electromechanical phenomena

thermodynamic phenomena

Figure 1.3 Time frame of the basic power system dynamic phenomena

armature windings to the field and damper winding, then along the generator rotor to the turbineuntil finally the boiler is reached.

The fast wave phenomena, due to lightning and switching overvoltages, occur almost exclusively inthe network and basically do not propagate beyond the transformer windings The electromagneticphenomena mainly involve the generator armature and damper windings and partly the network.These electromechanical phenomena, namely the rotor oscillations and accompanying networkpower swings, mainly involve the rotor field and damper windings and the rotor inertia As thepower system network connects the generators together, this enables interactions between swinginggenerator rotors to take place An important role is played here by the automatic voltage control andthe prime mover control Slightly slower than the electromechanical phenomena are the frequencyoscillations, in which the rotor dynamics still play an important part, but are influenced to a muchgreater extent by the action of the turbine governing systems and the automatic generation control.Automatic generation control also influences the thermodynamic changes due to boiler controlaction in steam power plants

The fact that the time frame of the dynamic phenomena is closely related to where it occurswithin the power system has important consequences for the modelling of the system elements Inparticular, moving from left to right along Figure 1.3 corresponds to a reduction in the accuracyrequired in the models used to represent the network elements, but an increase in the accuracy in themodels used first to represent the electrical components of the generating unit and then, further tothe right, the mechanical and thermal parts of the unit This important fact is taken into account inthe general structure of this book when later chapters describe the different power system dynamicphenomena

1.3 Two Pairs of Important Quantities: Reactive Power/Voltage

and Real Power/Frequency

This book is devoted to the analysis of electromechanical phenomena and control processes in powersystems The main elements of electrical power networks are transmission lines and transformers

which are usually modelled by four-terminal (two-port) RLC elements Those models are connected

together according to the network configuration to form a network diagram

For further use in this book, some general relationships will be derived below for a two-port

π-equivalent circuit in which the series branch consists of only an inductance and the shunt branch

is completely neglected The equivalent circuit and the phasor diagram of such an element are

shown in Figure 1.4a The voltages V and E are phase voltages while P and Q are single-phase powers The phasor E has been obtained by adding voltage drop jXI, perpendicular to I, to the voltage V The triangles OAD and BAC are similar Analysing triangles BAC and OBC gives

|BC| = XI cos ϕ = E sin δ hence I cos ϕ = E

This equation shows that real power P depends on the product of phase voltages and the sine of the

angleδ between their phasors In power networks, node voltages must be within a small percentage

of their nominal values Hence such small variations cannot influence the value of real power.The conclusion is that large changes of real power, from negative to positive values, correspond to

X P,Q

δ A

B

C D

(b)

E V I

Figure 1.4 A simplified model of a network element: (a) equivalent diagram and phasor diagram;(b) real power and reactive power characteristics

changes in the sine of the angleδ The characteristic P(δ) is therefore sinusoidal1and is referred to

as the power–angle characteristic, while the angle δ is referred to as the power angle or the load angle.

Because of the stability considerations discussed in Chapter 5, the system can operate only in thatpart of the characteristic which is shown by a solid line in Figure 1.4b The smaller the reactance

X , the higher the amplitude of the characteristic.

The per-phase reactive power leaving the element is expressed as Q = VI sin ϕ Substituting (1.7)

into that equation gives

Q= EV

X cosδ − V2

The term cosδ is determined by the value of real power because the relationship between the sine

and cosine is cosδ =√1− sin2δ Using that equation and (1.8) gives

Q=



EV X

2

− P2−V2

The characteristic Q(V ) corresponds to an inverted parabola (Figure 1.4b) Because of the stability

considerations discussed in Chapter 8, the system can operate only in that part of the characteristicwhich is shown by a solid line

The smaller the reactance X , the steeper the parabola, and even small changes in V cause large

changes in reactive power Obviously the inverse relationship also takes place: a change in reactivepower causes a change in voltage

The above analysis points out that Q, V and P, δ form two pairs of strongly connected variables.

Hence one should always remember that voltage control strongly influences reactive power flows

and vice versa Similarly, when talking about real power P one should remember that it is connected

with angleδ That angle is also strongly connected with system frequency f , as discussed later in

the book Hence the pair P, f is also strongly connected and important for understanding power

system operation

1 For a real transmission line or transformer the characteristic will be approximately sinusoidal as discussed

in Chapter 3.

power system stability

rotor anglestability

frequencystability

voltagestability

largedisturbancevoltage stability

transientstability

smalldisturbanceangle stability

smalldisturbancevoltage stability

Figure 1.5 Classification of power system stability (based on CIGRE Report No 325) Reproduced

by permission of CIGRE

1.4 Stability of a Power System

Power system stability is understood as the ability to regain an equilibrium state after being subjected

to a physical disturbance Section 1.3 showed that three quantities are important for power systemoperation: (i) angles of nodal voltagesδ, also called power or load angles; (ii) frequency f ; and (iii)

nodal voltage magnitudes V These quantities are especially important from the point of view of

defining and classifying power system stability Hence power system stability can be divided (Figure1.5) into: (i) rotor (or power) angle stability; (ii) frequency stability; and (iii) voltage stability

As power systems are nonlinear, their stability depends on both the initial conditions and thesize of a disturbance Consequently, angle and voltage stability can be divided into small- andlarge-disturbance stability

Power system stability is mainly connected with electromechanical phenomena – see Figure1.3 However, it is also affected by fast electromagnetic phenomena and slow thermodynamic

phenomena Hence, depending on the type of phenomena, one can refer to short-term stability and

long-term stability All of them will be discussed in detail in this book.

1.5 Security of a Power System

A set of imminent disturbances is referred to as contingencies Power system security is understood as

the ability of the power system to survive plausible contingencies without interruption to customerservice Power system security and power system stability are related terms Stability is an importantfactor of power system security, but security is a wider term than stability Security not only includesstability, but also encompasses the integrity of a power system and assessment of the equilibriumstate from the point of view of overloads, under- or overvoltages and underfrequency

From the point of view of power system security, the operating states may be classified as inFigure 1.6 Most authors credit Dy Liacco (1968) for defining and classifying these states