Power System Small Signal Stability Analysis and Control

POWER SYSTEM

DebasishFMondal

Abhijit Chakrabarti

Aparajita Sengupta

Power System

Small Signal

Stability Ana ysis

and Contro

Power System

Small Signal

Stability Analysis

and Contro

Debasish Mondal

H A L D IA IN S T IT U T E O F T E C H N O L O G Y , P .O . - H A T IB E R IA ,

H IT , P U R B A M E D IN IP U R , H A L D I A - 7 2 1 6 5 7

Abhijit Chakrabarti

B E N G A L E N G IN E E R IN G & S C IE N C E

U N IV E R S IT Y , P .O . - B . G A R D E N ,

S H IB P U R , H O W R A H - 7 1 1 1 0 3

Aparajita Sengupta

B E N G A L E N G IN E E R IN G A N D S C IE N C E

U N IV E R S IT Y , P .O . - B . G A R D E N ,

S H IB P U R , H O W R A H - 7 1 1 1 0 3

E L SE V IE R

AMSTERDAM • BOSTON • HF.IDtLBERG ■ LONDON

NEW YORK • OXFORD • PARIS • SAN DIEGO

SAN FRANCISCO ■ SINGAPORF. • SYDNEY • TOKYO

AuJrmii ?im is dn imprini ol hlKvicr

A c a d e m ic P r e s s is a n im p r im o f E ls e v ie r

3 2 J a m e s to w n R o a d . L o n d o n N W | 7 B Y . U K

T h e B o u le v a rd . L a n g i o r d L a n e . K id lin g l o n . O x f o r d O X .S I G B . U K

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2 2 5 W y m a n S t r e e t . W a llh a m . M A 0 2 4 5 1 . U S A

5 2 5 B S lre e l. S u ite IKCK). S a n D ie g o . C A 9 2 1 0 1 - 4 4 9 5 . U S A

F irs t e d itio n 2 0 1 4

C o p y r ig h t < 2 0 1 4 E ls e v ie r In c. A ll r ig h ts r e s e rv e d

N o p a rt o f th is p u b lic a t io n m a y b e r e p r o d u c e d , s to r e d in a re tr ie v a l s y s te m o r tr a n s m itte d in a n y f o rm

o r b y a n y m e a n s e le c tr o n ic , m e c h a n ic a l. p h o tc K o p y in g . r e c o r d in g o r o th e r w is e w iih o u t th e p r io r w ritte n

p e r m is s io n o f th e p u b lis h e r P e r m is s i o n s m a y b e s o u g h t d ir e c tly f ro m E ls e v ie r 's S c ie n c e & T e c h n o lo g y

R ig h ts D c p a n m e n t in O x fo r d . U K : p h o n e ( t 4 4 ) (0 ) 1X65 S 4 . W 0 ; fa x ( + 4 4 ) ( 0 ) 1 8 6 5 8 5 3 3 3 .^ ; e m a il:

p e r m i s s io n s @ e l s e v i e r .c o m . A lte r n a tiv e ly y o u c a n s u b m it y o u r r e q u e s t o n lin e b y v is itin g th e E ls e v ie r w e b

s ite a t h ttp ://e ls e v ie r .c o m /l( K :a ie /p e rm is s io n s . a n d s e le c tin g O b ta in in g p e n n is s io n to u s e E ls e v ie r m a te r ia l.

N o tic e

K n o w le d g e a n d b e s t p r a c tic e in t h is f ie ld a re c o n s ta n t ly c h a n g in g . A s n e w r e s e a r c h a n d e x p e r ie n c e

b r o a d e n o u r u n d e r s t a n d in g , c h a n g e s in r e s e a r c h m e th o d s , p r o f e s s io n a l p r a c tic e s , o r m e d ic a l tr e a tm e n t

m a y b e c o m e n e c e s s a r y .

P r a c ti tio n e r s a n d r e s e a r c h e r s m u s t a lw a y s r e ly o n th e ir o w n e x p e r ie n c e a n d k n o w le d g e in e v a lu a tin g

a n d u s in g a n y in f o r m a tio n , m e ih tx ls . c o m p o u n d s , o r e x p e r im e n ts d e s c r ib e d h e re in . In u s in g su c h

in f o r m a tio n o r m e th o d s th e y s h o u ld b e m in d f u l o f th e ir o w n s a fe ty a n d th e s a fe ty o f o th e r s , in c lu d in g

p a r ti e s f o r w h o m th e y h a v e a p r o f e s s io n a l r e s p o n s ib ility .

T o th e f u lle s t e x te n t o f th e la w . n e ith e r th e P u b lis h e r n o r th e a u th o r s , c o n tr i b u to r s , o r e d ito r s , a s s u m e

a n y lia b ility f o r a n y in ju r y a n d /o r d a m a g e to p e rs o n s o r p r o p e r ty a s a m a tte r o f p r tx lu c ts lia b ility ,

n e g lig e n c e o r o th e r w is e , o r f ro m a n y u s e o r o p e r a tio n o f a n y m e th o d s , p n x lu c ts . in s t r u c tio n s , o r id e a s

c o n ta in e d in th e m a te r ia l h e re in .

L i b r a r y o f ( . 'u n g r e s s C 'a l a l o g i n g - i n - P u b l i c a l i o n D a t a

A p p lic a ti o n s u b m itte d

B r i t i s h L i b r a r y ( 'a t a l u g u i n g in P u b l i c a t i o n ] ) a l a

A c a ta lo g u e r e c o rd f o r th is b o tik is a v a ila b le f ro m th e B r itis h L ib r a r y

F o r in f o n n a t io n o n a ll A c a d e m ic P r e s s p u b lic a tio n s

v is it o u r w e b s ite a t s to r e .e ls e v ie r .c o m

P r in te d a n d b o u n d in U S A

14 15 1 6 17 I S 1 0 9 8 7 6 5 4 3 2 1

I S B N : 9 7 8 - 0 - 1 2 -8{X )572-9

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Contents

A ckn ow ledgm ents.................................................................................................................... x iii

A uthor B iography.......................................................................................................................xv

P reface.........................................................................................................................................xvii

CHAPTER 1 C oncepts of S m all-S ig n a l S ta b ility ..........................................1

1.1 Introduction......................................................................................................1

1.2 Swing Equation............................................................................................. 2

1.3 Nature o f O scillations..................................................................................5

1.4 Modes o f O scillations and its Study Procedure...................................6

1.5 Synchronizing Torque and Damping Torque....................................... 7

1.6 Sm all-Signal O scillations in a Synchronous Generator

connected to an Infinite Bus.......................................................................S

1.7 An Illu stra tio n ...............................................................................................11

Exercises...................................................................................................................13

References................................................................................................................ 14

CHAPTER 2 Fund am ental M odels of S ynchronous M a c h in e .........15

2.1 Introductio n...................................................................................................15

2.2 Synchronous M achine Dynam ic M odel in the a -b -c

Reference Fram e..........................................................................................18

2.3 Park's Transform ation and Dynam ic M odel in the d -q -ti

Reference Fram e..........................................................................................21

2.4 Per U nit (PU) Representation and S c a lin g .........................................24

2.5 Physical Significance o f PU System ....................................................26

2.6 Stator Flux-C urren t R elationships........................................................28

2.7 Rotor Dynam ic E quations....................................................................... 28

2.8 Reduced Order M o d e l..............................................................................30

2.9 Equivalent C ircuit o f the Stator Algebraic Equations.................... 32

2 .10 Synchronous M achine E xciter................................................................ 34

2.10.1 IEEE Type 1 E xcite r....................................................................34

2.10.2 Static E xciter..................................................................................38

E.xercises...................................................................................................................39

References...............................................................................................................40

CHAPTER 3 M o d e ls of Pow er N etw ork and R elevant

P o w er E q u ip m en ts........................................................................... 41

3.1 Introductio n...................................................................................................41

3.2 Sim ple M odel o f a Synchronous G enerator....................................... 41

vii

v iii Contents

3.3 Sleady-Slaie M odeling o f Synchronous M achine

(A nalytical A spects)...................................................................................42

3 .4 G overnor M o d e l.......................................................................................... 45

3.5 Turbine M o d e l............................................................................................ 46

3.6 Power N etw ork M o d e l............................................................................. 4S

3.7 M odeling o f Lo a d ........................................................................................5 1

3.8 Power System Stabilizer...........................................................................53

3.9 M odel o f FA C TS Devices........................................................................55

3.9.1 Static V ar Compensator.................................................................55

3.9.2 Static Synchronous Compensator...............................................60

3.9.3 Thyristor-C ontrolled Series Comf>ensator................................64

3.9.4 Static Synchronous Series Compensator.................................. 72

3.9.5 U n ified Power Flow C o n tro lle r...................................................76

E xercises...................................................................................................................82

References................................................................................................................ 83

CHAPTER 4 S m a ll-S ig n a l S ta b ility A nalysis in S M IB

P o w er S y s te m .....................................................................................8 5

4.1 Introductio n...................................................................................................85

4 .2 H e ffro n -P h ilip s M odel o f S M IB Power S ystem ..............................86

4.2.1 Fundamental E quations.................................................................87

4.2.2 Linearization Process and State-Space M o d e l....................... 89

4.2.3 D erivation o f K Constants: K \, K j, K 4, K^,. and K^,....... 93

4.3 Sm all-Signal S tability Analysis Using State-Space M odel

and B lock D iagram .................................................................................... 96

4 .4 An Illu s tra tio n ..............................................................................................98

4 .5 Effect o f Generator F ield........................................................................104

4 .6 Effect o f Excitation S ystem ..................................................................109

4.6.1 Effect o f Excitation System in Torque-A ngle Loop........ 111

4.6.2 C alculation o f Steady-State Synchronizing and

Dam ping Torque............................................................................113

4.6.3 S ynchronizing and Dam ping Torque at Rotor

O scillation Frequency.................................................................. 113

4.7 An illu s tra tio n ............................................................................................ 114

E xercises................................................................................................................ 117

References.............................................................................................................. 118

CHAPTER 5 S m a ll-S ig n a l S ta b ility A nalysis in

M u ltim a c h in e S y s te m ..............................................................1 1 9

5.1 Introduction................................................................................................. 119

5.2 M ultim achine Sm all-Signal M o d e l..................................................... 119

5.2.1 T w o -A x is M odel o f M ultim achine System ..........................120

Contents

5.2.2 Linearization Process and M ultim achine State-Space

M o d e l..............................................................................................123

5.2.3 Reduced-Order Flux-D ecay M o d e l.......................................143

5.3 Com putation o f In itia l C onditions o f the State Variables.......... 144

5.3.1 A n Illu s tra tio n ..................................................................................148

5 .4 Identification o f Electrom echanical Swing M odes.........................154

5.4.1 Participation Factor A n a ly s is .....................................................154

5.4.2 Sw ing M ode and Participation R a tio .......................................157

5 .5 A n Illustration: A Test Case..................................................................157

E xercises................................................................................................................. 159

References...............................................................................................................160

CHAPTER 6 M itig a tio n of S m a ll-S ig n a l S ta b ility P roblem

E m ploying P o w er S ystem S ta b iliz e r............................... 161

6.1 Introductio n..................................................................................................161

6.2 The A pplication o f PSS in an S M IB System ....................................161

6.2.1 Com bined M odel o f S M IB System w ith PSS.........................162

6.2.2 Results and D iscussion................................................................ 163

6.3 M ultim achine Sm all-Signal Stability Im provem ent...................... 164

6.3.1 M ultim achine M odel w ith PSS................................................. 165

6.3.2 An Illustration-C om putation o f Eigenvalues and

Swing M odes.................................................................................. 167

6 .4 Development o f a Location Selection Indicator o f PSS............... 170

6.4.1 Participation Factor....................................................................... 170

6.4.2 S ensitivity o f PSS E ffe c t............................................................. 171

6.4.3 O ptim um PSS Location In d e x ...................................................171

6.4.4 A n Illu s tra tio n .................................................................................172

6.4.5 Im plicatio n o f PSS Gain in SPE and O PLI

C haracteristics.................................................................................174

6.5 E ffect o f lo a d ............................................................................................. 178

6.5.1 Effect o f Type o f L o a d ................................................................178

6.5.2 Effect o f Load on C ritica l Swing M o d e ................................179

6.5.3 Effect on PSS Location Indicators...........................................183

E xercises.................................................................................................................183

References..............................................................................................................184

CHAPTER 7 A p p lic a tio n of FACTS C o n tro lle r.......................................1 8 5

7.1 Introductio n................................................................................................. 185

7.2 FA C TS Technology..................................................................................186

7 .2 .1 Series C om pensation................................................................... 186

7.2.2 Shunt C om pensation.................................................................... 188

C ontents

7.3 A pplication o f SVC in Sm all-Signal S tability Im provem ent.... 190

7.3.1 M odel o f S M IB System w ith S V C ..........................................190

7.3.2 A n Illustration: Sim ulation R e s u lt........................................... 191

7 .4 A pplication o f a TCSC C o ntroller in an S M IB S ystem ................ 192

7.4.1 M odel o f an S M IB System w ith a TCSC C o n tro lle r........ 192

7.4.2 An Illustration: Eigenvalue C om putation and

Performance A n alysis................................................................... 195

7.5 M ultim achine A pplication o f S V C ...................................................... 198

7.5.1 M ultim achine M odel w ith S V C ................................................198

7.5.2 An illu.slration................................................................................ 200

7.6 A pplication o f TCSC in a M ultim achine Power S ystem ..............201

7.6.1 M ultim achine M odel w ith T C S C .............................................203

7.6.2 A n Illu stration: Study o f Sm all-Signal S ta b ility ................. 204

7.7 Voltage Source Converter-Based FA C TS Device

(S T A T C O M )...............................................................................................208

7.7.1 S M IB System w ith the S T A T C O M C o n tro lle r.................. 208

7.7.2 A n Illu s tra tio n ................................................................................ 210

7.7.3 M ultim achine M odel w ith S T A T C O M ...................................211

7.7.4 Sm all-Signal Performance A n alysis........................................212

7 .8 A pplication o f TC SC in a Longitudinal Power S ystem ................215

7.8.1 Description o f the Test System and Base Case Study.......215

7.8.2 im pact o f TCSC in the Face o f Power System

Disturbances....................................................................................217

7.8.3 Sm all-Signal S tability R ank......................................................222

Exercises................................................................................................................ 223

References..............................................................................................................224

CHAPTER 8 O ptim al and R obust C o n tro l................................................... 2 2 7

8.1 Introduction.................................................................................................. 227

8.2 Genetic Algorithm -B ased O ptim ization............................................. 228

8.2.1 O verview o f G A ........................................................................... 228

8.2.2 Parameter O ptim ization A p plying G A ...................................229

8.2.3 An Illustration: GA-Based TCSC C o ntroller.......................233

8.3 Particle Swarm O p tim iz a tio n .................................................................235

8.3.1 O verview o f PSO.......................................................................... 237

8.3.2 O ptim al Placement and Parameter Setting o f SVC

and TC SC Using P S O .................................................................238

8.3.3 Performance Study o f PSO-Based SVC and T C S C ..........243

8 .4 Im plication o f SVC and TCSC Controllers on C ritical Loading 243

8.5 Comparison Between PSO- and G A-Based D esigns......................246

Contents

8.6 / / ^ O ptim al C o n tro l...............................................................................246

8.6.1 Background.................................................................................... 248

8.6.2 A lgorithm s f o r / / ^ Control T h e o ry.........................................248

8.6.3 M ixed-Sensitivity-Based C ontroller: An LM I

Approach.........................................................................................250

8.6.4 Design o f an /7 ^ TCSC C ontroller..........................................255

8.6.5 Performance o f the Closed-Loop C o n tro l...................259

8.7 M ultiarea Closed-Loop C o n tro l........................................................... 262

E xercises............................................................................................................... 265

References............................................................................................................ 266

Nomenclature,

A P P E N D IX A

A P P E N D IX B

In d e x ...............

.........................................................................................................................269

Fundamenlal C oncepts..............................................................................2 7 1

Data Used for Relevant Power Sy.stem C om ponents.................... 283

.........................................................................................................................307

Acknowledgments

Authors express their sincere lhanks and deepest sense o f gratitude to all the

reviewers o f this book fo r their valuable suggestions and coiiim enls to enrich and

upgrade the contents o f this book.

Auihors w ould also wish 10 convey their graliiude to all faculty members o f the

Depanment o f E lectrical Engineering. Bengal Engineering and Science U niversity

(BES U ). West Bengal. India, fo r their wholehearted cooperation to make this work

Him into a reality. Thanks to ihe faculty members o f the Department o f A pplied Elec￾tronics and Instrum entation Engineering, Haldia Institute o f Technology (H IT ).

West Bengal. India, tor their cooperation.

The authors also acknowledge the inicresi and effort o f the entire editorial and

managemcnJ teams o f Elsevier Inc. Science and Technology Books and Energy &

Power Group.

Lust bui not the least, we feel proud fo r the respect and encouragement we had

received from our fa m ily members to carry out this w ork.

The authors cordia lly invite any conslruclive crilicism or com m cni from the

reader about ihe book.

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