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Local assessment of transient stability for generator tripping Mihirig, Ali Mohamed 1984

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LOCAL  A S S E S S M E N T OF FOR  TRANSIENT  GENERATOR  STABILITY  TRIPPING  by  ALI B.Sc,  A THESIS  MOHAMED  University  SUBMITTED  THE  MIHIRIG of Tripoli,  IN P A R T I A L  REQUIREMENTS MASTER OF  FOR  FULFILLMENT  THE  APPLIED  1978  DEGREE  OF  SCIENCE  in THE  FACULTY  OF  DEPARTMENT OF  We  accept to  THE  GRADUATE  ELECTRICAL  this  thesis  the required  UNIVERSITY  OF  March ©  A l i Mohamed  as  STUDIES ENGINEERING  conforming  standard  BRITISH  COLUMBIA  1984 Mihirig,  1984  OF  In p r e s e n t i n g  this  requirements  f o r an  of  British  it  freely  available  in partial  advanced  Columbia,  agree t h a t for  thesis  understood for  that  financial  shall  for reference  and  study.  I  for extensive be  her  copying or shall  copying of  g r a n t e d by  not  be  of  Zl^ct^cd  The U n i v e r s i t y o f B r i t i s h 1956 Main Mall V a n c o u v e r , Canada V6T 1Y3  Date  DE-6  (3/81)  S^fl• r«j*A<* Columbia  make  further this  thesis  head o f  this  my  It is thesis  a l l o w e d w i t h o u t my  permission.  Department o f  the  representatives. publication  the  University  Library  h i s or  gain  the  the  s c h o l a r l y p u r p o s e s may by  degree at  I agree that  permission  department or  f u l f i l m e n t of  written  ii  ABSTRACT  The gated  l o c a l v a r i a b l e s of a synchronous generator are  as p o s s i b l e i n d i c a t o r s o f the t r a n s i e n t s t a b i l i t y  generator.  A number o f c a s e  t e s t systems t o f i n d c a n be u s e d .  of  s t u d i e s a r e c a r r i e d o u t on  out which  generator v a r i a b l e or  By c o m p a r i n g t h e c o n v e n t i o n a l s w i n g  investithe  three  variables  curve of each  g e n e r a t o r w i t h i t s l o c a l a c c e l e r a t i o n c u r v e d u r i n g t h e same t r a n s i e n t p e r i o d , a new  criterion  assessment i s developed  to replace e x i s t i n g generator  schemes t h a t r e l y new  criterion  the generator  upon p r e - d e t e r m i n e d  contingency  stability tripping  studies.  i s b a s e d upon t h e b e h a v i o r o f t h e a c c e l e r a t i o n f o l l o w i n g a d i s t u r b a n c e t o the system.  measurements from to  for transient  any o t h e r m a c h i n e i n t h e s y s t e m  a s s e s s the s t a b i l i t y  of the  generator.  The of  No  are r e q u i r e d  iii  TABLE OF  CONTENTS Page  ABSTRACT  i  TABLE OF CONTENTS  i i  L I S T OF TABLES  iv  L I S T OF ILLUSTRATIONS ACKNOWLEDGEMENT  v v i i  CHAPTER 1  INTRODUCTION  1  CHAPTER 2  TRANSIENT S T A B I L I T Y ANALYSIS  5  2.1 2.2 2.2.1 2.2.2 2.2.3 2.2.4 2.3 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.3.6 CHAPTER 3  SYSTEM MODELS AND TEST SYSTEMS 3.1 3.2 3.3 3.3.1 3.3.2 3.3.3  CHAPTER 4  Introduction Assessment o f T r a n s i e n t S t a b i l i t y The C o n v e n t i o n a l Method The E q u a l A r e a C r i t e r i o n E n e r g y F u n c t i o n Method L o c a l Assessment S t a b i l i t y Controls Dynamic B r a k i n g H i g h Speed C i r c u i t B r e a k e r R e c l o s i n g Series Capacitors Fast Valve A c t i o n Independent P o l e O p e r a t i o n o f C i r c u i t Breakers Generator Tripping  C l a s s i c a l Model o f Synchronous Machines System R e p r e s e n t a t i o n The T h r e e T e s t S y s t e m s The T h r e e M a c h i n e S y s t e m The F o u r M a c h i n e S y s t e m The F i v e M a c h i n e S y s t e m  ..  5 7 7 8 13 19 20 21 22 23 23 24 25 27 27 28 30 30 30 34  STUDIES OF GENERATOR VARIABLES A V A I L A B L E FOR LOCAL S T A B I L I T Y ASSESSMENT  37  4.1 4.2 4.3 4.4  37 40 41 47  , A c c e l e r a t i n g Power Speed a n d A c c e l e r a t i o n Acceleration S t a b i l i t y Limits Case S t u d i e s  iv  Page 4.4.1 4.4.2 4.4.3 4.5 4.6 CHAPTER 5 REFERENCES  The T h r e e M a c h i n e S y s t e m The F o u r M a c h i n e S y s t e m Case S t u d i e s .. The F i v e M a c h i n e S y s t e m Case S t u d i e s Case S t u d i e s D i s c u s s i o n Transient S t a b i l i t y Criterion Based on A c c e l e r a t o n Measurements ...  CONCLUSION AND FUTURE WORK  47 64 64 68 70 74 77  V  L I S T OF  TABLES P  Table  Table Table  Table  3.1  3.2 3.3  3.4  a  9  G e n e r a t o r d a t a and i n i t i a l c o n d i t i o n s of the three-machine system  32  G e n e r a t o r d a t a and i n i t i a l of the four-machine system  32  conditions  T r a n s m i s s i o n l i n e p a r a m e t e r s and of the five-machine system G e n e r a t o r d a t a and i n i t i a l of t h e f i v e - m a c h i n e s y s t e m  loads 36  conditions 36  e  vi  LIST OF ILLUSTRATIONS P a  9  F i g u r e 2.1  S i n g l e machine i n f i n i t e bus  11  F i g u r e 2.2  The e q u a l a r e a c r i t e r i o n  11  F i g u r e 2.3  Comparison o f e q u a l a r e a c r i t e r i o n and energy method  18  F i g u r e 3.1  C l a s s i c a l model o f synchronous machine  28  F i g u r e 3.2  The three-machine system  31  F i g u r e 3.3  The four-machine system  33  F i g u r e 3.4  The f i v e - m a c h i n e system  35  F i g u r e 4.1  A c c e l e r a t i n g power and a c c e l e r a t i o n ........  39  F i g u r e 4.2  Speed d e v i a t i o n o f machine 4 f o r different fault locations  42  F i g u r e 4.3  Swing c u r v e s and a c c e l e r a t i o n  43  F i g u r e 4.4  A c c e l e r a t i o n o f machine 4 f o r d i f f e r e n t fault locations A c c e l e r a t i o n o f machine 2 f o r d i f f e r e n t fault locations A c c e l e r a t i o n o f machine 3 f o r d i f f e r e n t  F i g u r e 4.5 F i g u r e 4.6  44 46  fault locations  46  F i g u r e 4.7  The three-machine system case s t u d i e s  48  F i g u r e 4.8  Swing for a Swing for a Swing for a Swing for a  F i g u r e 4.9 F i g u r e 4.10 F i g u r e 4.11 F i g u r e 4.12  curve fault curve fault curve fault curve fault  and a c c e l e r a t i o n a t bus 4 and a c c e l e r a t i o n a t bus 4 and a c c e l e r a t i o n a t bus 4 and a c c e l e r a i t o n a t bus 7  o f machine 1 49 o f machine 2 50 o f machine 3 •>•  51  o f machine 2  Swing c u r v e and a c c e l e r a t i o n o f machine 3 f o r a f a u l t a t bus 7  53 54  e  vii  Page F i g u r e 4.13 F i g u r e 4.14 F i g u r e 4.15 F i g u r e 4.16 F i g u r e 4.17 F i g u r e 4.18 F i g u r e 4.19 F i g u r e 4.20 F i g u r e 4.21 F i g u r e 4.22 F i g u r e 4.23 F i g u r e 4.24 F i g u r e 4.25  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 9  o f machine  2  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 9  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 5  o f machine •  2  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 5  of machine  3  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 6  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 6  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 8  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t bus 8  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t b u s 10  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t b u s 10  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t b u s 10  o f machine  S w i n g c u r v e and a c c e l e r a t i o n f o r a f a u l t a t b u s 11  o f machine  55 3 56 58 59 2 60 2 61 2 62 3 63 4 65 2 66 3 67  Rate o f change o f a c c e l e r a t i o n  5 69 73  ACKNOWLEDGEMENT  I  wish  supervisor standing  t o express  during  in  discussions  with  on  the problem  .  proof  am a l s o reading  I  am  the  Thanks  suggesting  and under-  thesis.  the topic  D r . M. D. Wvong  t o my  o f B.C.  of t h i s  thesis  , and f o r i n f o r m a t i o n  colleague  B. G a r r e t f o r  manuscript. t o t h e Department  f o rthe teaching a r e d u e t o my  encouragement  of t h i s  t o my  i s d u e t o M r . K. P o k r a n d t ,  grateful  indebted  Engineering  f o ra l l h i shelp  the preparation  , fororiginally  I  sincerest gratitude  D r . M. D. W v o n g  Acknowledgement Hydro  my  throughout  Electrical  assistantship provided.  wife my  of  Najwa  graduate  f o rher program.  continuous  1 CHAPTER 1 INTRODUCTION  When power s y s t e m e n g i n e e r s mean t h e p r o p e r t y w h i c h e n s u r e s remain i n e q u i l i b r i u m through conditions. stability  use the term  'stability*  t h a t t h e power s y s t e m s  they  will  n o r m a l and a b n o r m a l o p e r a t i n g  As p o w e r s y s t e m s g r o w l a r g e r and more c o m p l e x ,  s t u d i e s become v e r y  important.  With  the ever  i n c r e a s i n g demand f o r e l e c t r i c a l e n e r g y and d e p e n d e n c e on an u n i n t e r r u p t e d s u p p l y , the a s s o c i a t e d requirement reliability stability  d i c t a t e s t h a t power s y s t e m s be d e s i g n e d  affects  to maintain  under s p e c i f i c d i s t u r b a n c e s , c o n s i s t e n t w i t h  The p r o b l e m o f s t a b i l i t y disturbed.  o f high  a r i s e s when t h e s y s t e m i s  The n a t u r e o r m a g n i t u d e o f t h e d i s t u r b a n c e  the s t a b i l i t y  economy.  o f t h e power s y s t e m .  greatly  I f the d i s t u r b a n c e  i s l a r g e , then the o s c i l l a t o r y t r a n s i e n t s t h a t occur w i l l be l a r g e .  The q u e s t i o n o f w h e t h e r t h e power s y s t e m w i l l  t o a new s t a b l e o p e r a t i n g s t a t e , o r w h e t h e r i t w i l l synchronism  t h e n becomes  transient stability Generators  important.  settle  lose  T h i s i s known as t h e  problem.  that are l i k e l y  t o l o s e synchronism  c e r t a i n d i s t u r b a n c e s must be d i s c o n n e c t e d to prevent  also  f r o m t h e power  s y s t e m b r e a k u p o r g e n e r a t o r damage.  complete a n a l y s i s ' o f the s t a b i l i t y g e n e r a t o r s m u s t be d i s c o n n e c t e d c e r t a i n disturbance i n order  under system  T h i s needs a  p r o b l e m t o know w h i c h  from t h e power s y s t e m f o r a  to maintain  stability.  2  The  c o n v e n t i o n a l method u s e d t o a n a l y z e  the t r a n s i e n t  s t a b i l i t y problem i s the time s o l u t i o n o f the equations  using a d i g i t a l computer.  differential  T h e n , b a s e d on  considera-  t i o n o f v a r i o u s l a r g e d i s t u r b a n c e s on  t h e power s y s t e m ,  t r a n s i e n t s t a b i l i t y problem i s solved  ( s p e c i f i c a l l y ) and  r e s u l t s c a n be  used to d e s i g n  S u i t a b l e a c t i o n c a n be  because i t i s o n l y v a l i d  b a s e d upon p r e - d e t e r m i n e d cases  s t u d i e s f o r a power central  T h i s method i s v e r y  f o r s p e c i f i c d i s t u r b a n c e s and  results.  to  t a k e n a u t o m a t i c a l l y when  of the s p e c i f i e d d i s t u r b a n c e s o c c u r s .  rigid  and  disturbances.  be s t o r e d i n an o n - l i n e c o m p u t e r a t t h e  c o n t r o l room. any  specific  r e s u l t s of such t r a n s i e n t s t a b i l i t y  system can  the  t h e power s y s t e m s w i t c h g e a r  a d j u s t the p r o t e c t i v e r e l a y s f o r these The  the  U s u a l l y the most  is  severe  o f d i s t u r b a n c e s o n l y are c o n s i d e r e d , so i n c o r r e c t  d e c i s i o n s may of generator The  be made i n some i n s t a n c e s . tripping  Present  day p r a c t i c e  i s b a s e d upon t h e s e c o n t i n g e n c y  studies.  c o n v e n t i o n a l method u s e d i n s t a b i l i t y s t u d i e s i s  expensive  and  time  c o n s u m i n g b e c a u s e a l a r g e number o f  d i f f e r e n t i a l equations considered. instability  must be  solved f o r each  disturbance  A l s o , the c o n v e n t i o n a l p r o t e c t i o n system r e l i e s h e a v i l y on p r e - d e t e r m i n e d  results  against and  r e q u i r e s e x t e n s i v e h i g h - r e l i a b i l i t y communications equipment to t r a n s m i t t h e commands f o r t r i p p i n g l i n e s , g e n e r a t o r s , There have been a t t e m p t s  a t s o l v i n g the t r a n s i e n t s t a b i l i t y  p r o b l e m d i r e c t l y u s i n g L y a p u n o v f u n c t i o n s and [1,2]  , but  etc.  the r e s u l t s o b t a i n e d  control theories  are u s u a l l y too c o n s e r v a t i v e .  3 A s i m p l e and l e s s e x p e n s i v e s o l u t i o n i s s t i l l b e s t s o l u t i o n t o t h i s problem  needed.  The  i s p r o b a b l y t o r e l y upon the  measurements o f v a r i a b l e s a v a i l a b l e l o c a l l y a t each g e n e r a t o r and t o a s s e s s i t s t r a n s i e n t s t a b i l i t y based upon these l o c a l measurements.  T h i s w i l l not r e q u i r e p r i o r d i g i t a l computer  s t u d i e s and the e x p e n s i v e d e d i c a t e d communication equipment presently  used.  There are two g e n e r a t o r t r i p p i n g schemes t h a t do not the c o n v e n t i o n a l t e c h n i q u e .  The  f i r s t one  use  i s a power swing  r e l a y i n s t a l l e d a t the n o r t h e r n t e r m i n a l o f the O n t a r i o Hydro 500 KV t r a n s m i s s i o n l i n e  [3].  The b a s i c i d e a o f t h i s r e l a y i s  t o use measurements o f the i n s t a n t a n e o u s a c c e l e r a t i n g power w i t h the e q u a l a r e a c r i t e r i o n t o e s t i m a t e the power a n g l e and  to  compare i t w i t h the c r i t i c a l c l e a r i n g a n g l e o f the p r o t e c t e d generator.  A t r i p s i g n a l i s produced  i f t h i s angle i s exceeded.  Of c o u r s e , t h i s scheme depends on the assumption m a c h i n e - i n f i n i t e - b u s system which may cases.  of a  one-  be v a l i d o n l y f o r s p e c i a l  However, the t e c h n i q u e does l o o k a t l o c a l v a r i a b l e s o f  an i n d i v i d u a l g e n e r a t o r . The second g e n e r a t i o n t r i p p i n g scheme i s t h a t used a t C o l d s t r i p - Montana [ 4 ] .  The scheme r e l i e s on speed  and  a c c e l e r a t i o n measurements t o p r e d i c t the t r a n s i e n t s t a b i l i t y o f the p r o t e c t e d g e n e r a t o r .  Mini-computers  are used t o p r o c e s s  measured i n f o r m a t i o n i n s t a n t a n e o u s l y and the t r i p s i g n a l i s i s s u e d f o r a s p e c i a l c o n f i g u r a t i o n o f the a c c e l e r a t i o n curve only.  T h i s scheme was d e s i g n e d f o r a l i m i t e d p r o t e c t e d area  the  4 of  t h e Montana s y s t e m  and f o r two g e n e r a t o r s i n t h e s y s t e m  The scheme c a n n o t be a p p l i e d for  transient  stability  and  speed m e a s u r e m e n t s .  generally  i s developed based  The g o a l o f t h i s t h e s i s p r o j e c t a l t e r n a t i v e method using this  u n l e s s a new  to solve  l o c a l measurements.  criterion  on t h e a c c e l e r a t i o n  h a s b e e n t o s e a r c h f o r an  the t r a n s i e n t A proposed  only.  stability  approach  problem  i s presented i n  thesis. Chapter 2 w i l l  problem  of transient  method o f l o c a l  r e v i e w t h e p r e s e n t methods u s e d stability  assessment  m o d e l s and t e s t s y s t e m s presented  i n C h a p t e r 3.  and w i l l  of transient  used  introduce stability.  to solve the  the proposed The  i n t h e p r o p o s e d method w i l l  of d i f f e r e n t transient  cases to  the  measurements a v a i l a b l e  for stability  ments. future  local  be  Chapter 4 p r e s e n t s the r e s u l t s o f a  number o f s t u d i e s various  system  investigate assess-  C h a p t e r 5 summarizes t h e r e s u l t s and s u g g e s t i o n s f o r work.  5 CHAPTER 2 TRANSIENT S T A B I L I T Y  2.1  ANALYSIS  Introduction Stability  of a generator  i t will  synchronism with  the  stability  t o the amount o f power t h a t c a n  with  refers  stability  disturbance switching  r e s t of  implies that  when t h e power s y s t e m  s u c h as  f a u l t s with  Such a l a r g e d i s t u r b a n c e demand  i n the  transmitted  to a  circuit  c r e a t e s a power  system.  actually  p l a c e a t each generator  between t h e m e c h a n i c a l power i n p u t and  large  generation,  subsequent  and  output  be  i s subjected  between s u p p l y takes  Transient  a sudden c h a n g e i n l o a d o r  operations, or  isolation.  t h e power s y s t e m .  remain in  This  imbalance  imbalance  shaft.  The  mismatch  the e l e c t r i c a l  (neglecting losses) accelerates or decelerates  power the  generator [ 5 ] . In  the  case  beginning  of  pre-fault  steady  period  there  kinetic is  o f a f a u l t on  the d i s t u r b a n c e  i s excess  energy, causing  absorbed o r  to maintain capable continue  the g e n e r a t o r  During  m e c h a n i c a l power w h i c h  transformed  into potential  t h i s k i n e t i c energy  energy  the  then  a c c e l e r a t i n g (or d e c e l e r a t i n g ) u n t i l  fault  When t h e  energy produced during  system a f t e r  their  i s converted  t h e r o t o r s t o s p e e d up.  I f the  the  at  the  the k i n e t i c  stability.  of absorbing  side, at  r o t o r s are  state operating condition.  c l e a r e d , most o f  must be  the h i g h v o l t a g e  fault  the  to fault fault  i n order i s not  the r o t o r s w i l l they  lose  6  synchronism. complicated  The s i t u a t i o n  d u r i n g and a f t e r  the f a u l t  i n t h e c a s e o f a l a r g e i n t e r c o n n e c t e d power s y s t e m ,  b e c a u s e t h e power generators w i l l  imbalance  i n v o l v e s groups o f g e n e r a t o r s .  a c c e l e r a t e and some w i l l  different  r a t e s d e p e n d i n g upon t h e i r  operating  c o n d i t i o n s , i . e . , there  between a c c e l e r a t i n g  inertia  c o n s t a n t s and  i s an e n e r g y  interchange  and d e c e l e r a t i n g r o t o r s a f t e r  The e x c h a n g e o f e n e r g y  continues u n t i l  operating  c o n d i t i o n i s reached.  Otherwise  to lose  g e n e r a t o r s may p u l l - o u t  synchronism.  they p u l l - o u t  some o t h e r g e n e r a t o r s  the f a u l t i s  a new  stable  some h i g h l y  and c a u s e o t h e r  These g e n e r a t o r s  Some  decelerate, at  cleared.  accelerated  i s more  generators  have t o be t r i p p e d  before  and c a u s e t h e s y s t e m t o  break up. Loss because  o f synchronism  must be p r e v e n t e d  i t has a d i s t u r b i n g  effect  or controlled,  on v o l t a g e s , f r e q u e n c y and  power, and i t may c a u s e s e r i o u s damage t o g e n e r a t o r s w h i c h a r e t h e most e x p e n s i v e g e n e r a t o r s which  tend  subsequenty brought damage o c c u r s . water  e l e m e n t i n any power s y s t e m t o l o s e synchronism  back t o synchronism  While  this  Loss o f synchronism  lines.  s h o u l d be t r i p p e d and  b e f o r e any s e r i o u s  turbine generators  r e q u i r e many  steam s o t h a t t h e o p e r a t o r h a s t o shed  compensate f o r l o s s o f t h e s e  to operate  The  c a n be done r e a d i l y w i t h g a s and  t u r b i n e g e n e r a t o r s , steam  hours t o r e b u i l d  [ 6 ] .  falsely  generators.  may a l s o c a u s e some p r o t e c t i v e  and t r i p  load to  the c i r c u i t  breakers  relays  of unfaulted  7 2.2  Assessment o f T r a n s i e n t Since  and  from b e i n g  completely  An e a s i l y  complex when f u l l y  hopeless  solved  But the problem i s  i n s p i t e o f the v a s t  reached c o n c l u s i o n treated  used, d e t a i l e d s t a b i l i t y  i s t h a t the problem i s  t h a t an e x a c t  answer  that the d i g i t a l  studies are very  time consuming  i n v o l v e s many n o n - l i n e a r  equations.  Some o f t h e methods t h a t have been used  transient  stability  will  The C o n v e n t i p n a l T h i s method  transient  behavior  transient  Method on t h e d i g i t a l  o f the interconnected  c o n d i t i o n and c o n t i n u e s  power s y s t e m .  or lose  stability  synchronous machines.  until  the p r e -  each synchronous machine  synchronism.  s t u d i e s a r e r o u t i n e l y conducted on a  ( o r planned) switchgear  arrangements a r e adequate f o r the system s e t of disturbances  differential  s t a r t i n g with  The m a j o r o b j e c t i v e o f e a c h s t u d y  whether the e x i s t i n g  scribed  computer t o a n a l y z e t h e  by a s e t o f  A time s o l u t i o n i s o b t a i n e d ,  Transient  i n assessing  now be d i s c u s s e d .  relies  be shown t o m a i n t a i n  since  and d e t a i l e d m a c h i n e  synchronous machines a r e d e s c r i b e d  equations.  can  i s almost  computer i s b e i n g  the problem  The  amount o f  to obtain.  In s p i t e o f t h e f a c t  2.2.1  recognized  have worked c o n t i n u o u s l y t o  the problem o f t r a n s i e n t s t a b i l i t y .  work done. so  power s y s t e m s have b e e n  e s t a b l i s h e d , power e n g i n e e r s  solve far  interconnected  Stability;  without  i s to ascertain  and n e t w o r k  to withstand  a  l o s s o f synchronism.  preThe  8  r e s u l t s o f the t r a n s i e n t setting  stability  the p r o t e c t i o n equipment  s y n c h r o n o u s machine from The reliable  s t u d i e s are  in order  losing  has  i n d u s t r y , but  been w i d e l y  used  there are c e r t a i n  and  the  transient  like  due  of  reliable  i n the system.  decision.  equations  thus  expensive  time  for  Equal Area  The  equal area c r i t e r i o n  But  this  infinite-bus t o two  to d e c i d e  These  and m u l t i p l e  need a q u i c k  case  [7].  and  technique  A number o f  Hence, i n terms  computer,  of  t h i s method i s  i s a valuable conceptual  transient s t a b i l i t y /  c a n o n l y be  a p p l i e d to a  system o r to l a r g e r  e q u i v a l e n t machines  [5,6].  systems  tool  It is particularly and  t h e s y n c h r o n o u s machine d u r i n g  criterion  day  Criterion  f o r the study o f of  to  consuming.  The  the b e h a v i o r  has  failure  t h e a n a l y s i s o f power s y s t e m s t a b i l i t y .  useful  power  a l a r g e number o f  required.  c o s t u s i n g the d i g i t a l  2.2.2  the  circumstances,  the c o n v e n t i o n a l  computational and  unforseen  f o r each f a u l t  repeat simulations are  operator  breaker  consists of numerically integrating differential  by  i n t h e day  These s i t u a t i o n s  Furthermore  i s very  the machines.  to c e r t a i n  e q u i p m e n t breakdown, c i r c u i t  disturbances  analysis  situations  stability  situations could arise  to p r o t e c t each  accepted  o p e r a t i o n o f t h e power s y s t e m , where an quickly  for  synchronism.  n u m e r i c a l method o f s t a b i l i t y  and  t h e main t o o l  in visualizing  the  transient.  single-machine-  t h a t have been  reduced  For each  M  .Jl  at  generator  - P - P » P m e a  the f o l l o w i n g  swing  e q u a t i o n may  pu  be  (2  r  2H where M • — » Inertia constant  «»> = r a t e d  synchronous  r  6  =  generator rotor  angle  P  m  = mechanical  power i n p u t  P  e  = electrical  power  P  fl  = accelerating  Multiplying  both  d 5 dS 2 * dt dt  M  1  2  M M  j x  m  2  6  power. (2.1) by d6_ we g e t dt  d6 a dt  p  d(d6/dt) „ dt  multiplying  output  s i d e s o f equation  2  u  Now  speed  d5 a dt  p  both  s i d e s by d t and  integrating  2P  • / TT  "  O  dt  /  J  —  D  5  o where * o i s t h e r o t o r a n g l e b e f o r e t h e d i s t u r b a n c e . after state.  t h e d i s t u r b a n c e d6_ = 0 s i n c e t h e s y s t e m dt  B e f o r e and  i s i n steady  10 Now infinite  c o n s i d e r the case o f a s i n g l e machine connected  followed at  i n Fig. 2.1.  b u s a s shown  Consider  f a u l t on t h e t r a n s m i s s i o n l i n e CD  a three-phase  by t h e s i m u l t a n e o u s  steady  t o an  opening  of c i r c u i t  C and D  breakers  state  E »v e " ~ i q ~  8  l  n  6  o  (2.3)  - P sin 5 max o Where  E  =  generator  V  =  infinite steady  Xl  reactance  Pe = The  where  effective The in  inductance  power d e l i v e r e d  Figure  (2 L\ K*-'*)  i s cleared i s (2.5)  6 v  r ^ and r  2  r e p r e s e n t t h e change i n t h e  between t h e m a c h i n e and t h e i n f i n i t e - b u s . the f a u l t curves  t h e m a c h i n e t o be s t a b l e  a r e shown  i n t h i s case, equation  (2.2)  i . e . , t h e a l g e b r a i c sum o f t h e a r e a s A l and  A2 has t o be zero,' o r A l must be l e s s  generator  the g e n e r a t o r .  2.2.  t o be s a t i s f i e d ,  The  from  6  power b e f o r e , d u r i n g , and a f t e r  For has  between E and V  the f a u l t  - r_ P sin 2 max  the c o n s t a n t s  position  d u r i n g the f a u l t i s  - r 1. Pmax s i n  t h e power a f t e r  e  state rotor  electrical  power d e l i v e r e d e  and  bus v o l t a g e  *0 = =  voltage  than o r e q u a l  area A l r e p r e s e n t s the k i n e t i c rotor during  the f a u l t ,  to A2.  energy gained  by t h e  and a r e a A2 r e p r e s e n t s t h e  7  11  A  e -CH  hO-  • i n f i n i t e bus  K3c  D  :  F i g 2 . 1 Single machine Infinite-bus  Power  Angle curve 1 pre-fault power 'curve 2 post-fault power curve 3 during-fault power Fig 2 . 2 The equal area c r i t e r i o n  12 potential at § .  energy  I f a l l the k i n e t i c energy  c  p o t e n t i a l energy be  t h a t c a n be s t o r e d a f t e r  f o ra rotor  stable, otherwise  This brief that  equates of  i s useful  s t a b i l t y , given  The  criterion  the product  6 and c a u s i n g  and v e r y  i s fundamentally  the c r i t e r i o n  varies, error  however, t h e speed  i s however v e r y  r e q u i r e s a second system  [4]  behavior  60 Hz.  i n the s t a b i l i t y  I f t h e speed  the t r a n s i e n t  i s varying.  interchange.  the a n g l e  be e x a c t l y c o n s t a n t .  The e r r o r  The  generators  t o the r e s t o f the angular  velocity  i s t h e r e f o r e maximum o f 1 to other  Although  about t r a n s i e n t  o f each generator  to analyze  Since  Even t h e most dynamic  study.  to the multi-machine  the energy  because i t  t o energy.  i s s m a l l when compared  very valuable r e s u l t s applied  small.  shows  state condition.  i . e . , t h e y a r e t u r n i n g a t an a v e r a g e  i n 60 w h i c h  introduced  in  steady  t o a d v a n c e 360° r e l a t i v e  61 Hz r a t h e r t h a n  part  cannot  up t h e  to carry out f o r  would be e x a c t , b u t i t i s u s e d  m a c h i n e p e r f o r m a n c e when t h e a n g l e  will  instability.  incorrect  o f power and a n g l e  build  area c r i t e r i o n  simple  the p r e - f a u l t  the system  will  t h e s y n c h r o n o u s m a c h i n e were c o n s t a n t d u r i n g  period,  of  energy  presentation of the equal  the c r i t e r i o n  transient  a n g l e 6_< 6max, t h e n  increasing  i s cleared  c a n be c o n v e r t e d t o  the e x t r a k i n e t i c  rotor's acceleration  the f a u l t  errors  the c r i t e r i o n  stability,  i t cannot  gives be  power s y s t e m s t o a n a l y z e t h e  b e c a u s e many g e n e r a t o r s  are involved  13 2.2.3  Energy  Function  Method  T h i s method has been d e v e l o p e d o n energy  f u n c t i o n as a Lyapunov  region  f o r t h e power s y s t e m  been  devoted  analysis has  t o Lyapunov  i n the l a s t  involved  decade.  in  space o r i s v a l i d  potential  The  stability  functions,  f o r more complex function  system k i n e t i c  energy, associated  e n e r g y o f network  always d e f i n e d the energy  system energy which  with a c e r t a i n system  critical  this c r i t i c a l  [10]  system.  i s that  analysis  i s equivalent  potential  energy.  rotors  because  are involved  terms o f an  inertial  the  For a  by c o m p a r i n g  o f the f a c t  i n the energy  of  can, the  period,  two-machine and  t o the e q u a l a r e a c r i t e r i o n . the d i r e c t  idea  stability  the f a u l t - o n  i s uniquely defined  s y s t e m w i t h t h r e e o r more m a c h i n e s more d i f f i c u l t ,  with  The p r i n c i p a l  transient  i s gained during  energy  w i t h the  independent o f  f o r a g i v e n c o n t i n g e n c y , be d e t e r m i n e d d i r e c t l y total  and  e l e m e n t s and m a c h i n e r o t o r s , i s  f o r the p o s t f a u l t  f u n c t i o n method  stability  energy, associated  the n e t w o r k . potential  i.e.,a  contains both k i n e t i c  i s formally  system p o t e n t i a l  effort  system models [ 9 ] .  r e l a t i v e m o t i o n o f machine r o t o r s , The  has  A s u b s t a n t i a l p a r t o f the Lyapunov  the  stability  that either gives l a r g e r regions of  t r a n s i e n t energy terms.  the  Considerable e f f o r t  the s e a r c h f o r b e t t e r  function  The  [2,8].  to find  methods f o r power s y s t e m  Lyapunov state  function  the b a s i s o f using  the  direct  For a  analysis  becomes  t h a t most o f t h e m a c h i n e  interchange.  c e n t r e , o r sometimes c a l l e d  a n g l e , overcomes the problem o f r e f e r e n c e .  Formulation i n centre-of-  In the  inertial  14 centre  f o r m u l a t i o n , the e q u a t i o n s  d e s c r i b i n g the m o t i o n o f  synchronous machines are formulated inertial in  centre  clearly  [11].  The  f o c u s i n g on  more g e n e r a t o r s  from  Consider  importance  the m o t i o n  the r e s t o f  of  this  t h a t tends the  fictitious  formulation  lies  t o s e p a r a t e one  or  system.  t h e c l a s s i c a l model o f  e q u a t i o n o f m o t i o n o f any d 6.  with respect to a  the  machine u n i t  t h e power s y s t e m .  The  i n a power s y s t e m i s :  2  m  M  dt  where P  P  e i  - P  1  e  - j  I " ml P  Pm^ G  p  2  E  "  =  =  E^,Ej  <'>  G  real  2  the  8  power i n p u t  p a r t o f the d r i v i n g internal  transfer =  (2.7)  i l i  for Y^j  cos<  ± j  mechanical =  i i  Ej Y  ±  E  (2.6) i  point  admittance  g e n e r a t o r node  admittance  between node  constant voltage behind  i and  j  transient  reactance  W  r  Equations  =  generator  rotor  =  moment o f  inertia  =  r a t e d synchronous  2.3  synchronously  and  2.4  rotating  angle constant  =  2Hi/Wr  speed.  were w r i t t e n w i t h  r e s p e c t t o an  frame o f r e f e r e n c e .  arbitrary  15  For  the i n e r t i a l  centre,  define:  n 6  o - IT I , i i t i»i ,  0)  •  6  t " I , i i=i  M  <'>  M  2  9  n  M~ I \ " t i-1  ° then  n M  «  (2.10)  ±  the motion o f the i n e r t i a l  centre  i s g i v e n by  n ' I . i=l  Vo  where <$ , C D  a  Q  M t fc  centre  to  e  fc  ^  P  constant  angles  o f the i n e r t i a l  and s p e e d s w i t h  respect  centre,  center.  The  t o the i n e r t i a l  a r e d e f i n e d by:  equation  o f motion o f the i n d i v i d u a l  the i n e r t i a l M  A  Multiply  -  P  centre  become  i" ei ' V V P  (  P  Integrating  [\\  - Pi  +  \t  (2.14) w i t h  t = t s , where  ( 2  P  coi 1  i  »t ' V i - ! . ' 6  s  +  t  h  i  S  ( 2  t o time u s i n g  6 is c o n d i t i o n ) , we g e t t h e e n e r g y  Vl'i  e  s  t  e  a  <  3  v  as a lower  state condition  / i8  *  1 4 )  limit  (pre-  function V i  E  E  6  1  +  de  S  e. 6  1 3  }.. i 3 hi I «'(V i V t 3*1  + (M./M )  ' >  and f o r m t h e sum t h e n  respect  6(ts)=  respect  col  <W  +  machines with  [11].  e q u a t i o n (2ol3) by  I i=l  fault  (2.11)  P  p o s i t i o n and s p e e d o f i n e r t i a l  e  i s the i n e r t i a  generators'  the  r  0  i " e i " coA  P  o is) P  C Q l  d9  1  ,  1 - 1, 2,  n  16 The e n e r g y  function  explained 1.  as  (2.15) c a n be  physically  follows:  The f i r s t and  of equation  term r e p r e s e n t s  potential  e n e r g y due  the change  in kinetic  energy  t o the motion o f the r o t o r  of  machine i 2.  The s e c o n d due  term r e p r e s e n t s  t o t h e change i n r o t o r  t h e change position  in potential  between e i  s  energy  and  ei 3.  The  third  due  t o t h e power f l o w from node  to 4.  term r e p r e s e n t s  fourth  term r e p r e s e n t s  e n e r g y due  =  the  total  - v  j  KEi  v - I v 1=1  in potential t o the  (COI).  as  k i n e t i c energy + p o t e n t i a l  energy  + V PEi  energy  ±  (2.16)  function  - I  ( y  o f the system i s  m+  v  R  1  )  (2.17)  i-1  for s t a b i l i t y n  ° l  t h e change  o f the c e n t r e o f i n e r t i a  (2.15) c a n be w r i t t e n  v  energy  i t o j and f r o m node  t o the i t h machine c o n t r i b u t i o n  acceleration  V i  in potential  i  The  Equation  t h e change  » zero n  r  ml  Total  V  KE1  "  l  ml  V  PE1  k i n e t i c energy  =  total potential  energy.  17 Analogy with Equal Area The  energy  criterion. potential  Criterion  f u n c t i o n method  and a f t e r  two p l o t s w i t h t h e same a b s c i s s a illustrates  clearing  and A2.  The l o w e r p l o t  w h i c h c a n be used  the f a u l t . [11] .  angle i s defined illustrates  to specify  and k i n e t i c  shows  plot  i n which the  by t h e e q u a l i t y o f a r e a s A l t h e t r a n s i e n t e n e r g y method  the c r i t i c a l  e n e r g y a s shown  a n g l e i n terms o f  i n Figure  ( 6 ) i s t h e maximum v a l u e o f p o t e n t i a l  at  6 .  of  t h e s y s t e m , and i s c a l l e d  U  (2.3).  energy  that occurs  I t p r o v i d e s a measure o f t h e e n e r g y - a b s o r b i n g c a p a c i t y  U  transient  the c r i t i c a l  to i n s t a b i l i t y  the p o t e n t i a l  This gives clearing  during  In the  the f a u l t - o n p e r i o d ,  energy a t the c o r r e s p o n d i n g angle  the t o t a l  energy a t c l e a r i n g .  The t o t a l  i s compared w i t h t h e v a l u e o f c r i t i c a l  i s s t a b l e when t h e t o t a l  critical  energy.  when t h e t o t a l critical  energy  The c r i t i c a l  energy a t c l e a r i n g  i s less  clearing just  i s added  coordinate. energy a t  energy.  The  than o r equal t o  angle i s defined  becomes e q u a l t o t h e  energy.  For  a s y s t e m w i t h t h r e e o r more m a c h i n e s , t h e e n e r g y  f u n c t i o n method becomes more d i f f i c u l t critical  energy.  e n e r g y method, t h e e x c e s s k i n e t i c e n e r g y , w h i c h  contributes  the  F i g u r e 2.3  The u p p e r  PE  system  energy with the  the f a m i l i a r equal area c r i t e r i o n  critical  to  t o the equal area  B o t h methods compare t h e k i n e t i c energy d u r i n g  potential  i s similar  e n e r g y c a n n o t be d e f i n e d .  has b e e n done b a s e d on t h e f a c t  that  t o use because the  However, r e c e n t work [12] f o r a l a r g e power  system  P i g 2 . 3 Comparison of equal area c r i t e r i o n and energy method  19 o n l y a few  machines are  location.  The  successfully stability. applied  to those But  overall  The  f u n c t i o n method c a n critical  f o r many f a u l t  energy  transient  kinetic  critical  2.2.4  f o r any  particular  machines to assess locations,  f u n c t i o n method c a n  stability  and  fault  t h e n be a p p l i e d their  t h i s method c a n n o t  b e c a u s e many m a c h i n e s c o n t r i b u t e t o the  exchange.  total  energy  important  be  fault  used  energy  to assess  o f t h e power s y s t e m s by  using  the  the  p o t e n t i a l e n e r g i e s o f a l l the machines, o r  group o f  be  a  machines.  L o c a l Assessment Local  monitoring that w i l l  assessment of t r a n s i e n t  o f some i m p o r t a n t help to indicate  stability  refers  to  quantities of a generator  the t r a n s i e n t  stability  of  the  locally that  generator. Montana Power Company d e v e l o p e d controlled of  this device  n o t and is  device  then  unstable.  f o r generator  of  and  time.  going device  issue a t r i p The  signal  counting I t was  t h e r e has  t o shed  d e v i c e r e l i e s on  i n June 1979.  speed  and  false  trips.  or  generator's  t e e t h which pass  to t r i p . i t  idea  acceleration  i n some p e r i o d  a t the r i g h t  time.  b e e n f u n c t i o n i n g v e r y w e l l s i n c e i t was been no  i s stable  p o s s i b l e t o p r e d i c t when the g e n e r a t o r and  The  t h e g e n e r a t o r when i t  wheel b o l t e d to the  t h e number o f  to be.unstable has  tripping  digitally  i s t o p r e d i c t whether the g e n e r a t o r  measurements u s i n g a t o o t h e d rotor  [4] a  was The  installed  However, t h e c r i t e r i o n  used  and to  20 determine s t a b i l i t y  i s very  special  t o t h e Montana Power System  b e c a u s e two u n i t s o f t h e s y s t e m were t o be c o n t r o l l e d t h e i r output two of  power and t h e s y s t e m i s a r a d i a l  u n i t s connected  through the t r a n s m i s s i o n  synchronous machines d u r i n g the  possibility  t o the r e s t  the behavior  of the  stability.  i n p u t and e l e c t r i c a l  First, output  which  r o t o r undergoes instantaneous the occurrence  ity,  pole  slipping  Therefore, generator  of a fault.  there  takes  power t a k e s  Second,  of i n s t a b i l -  p l a c e between r o t o r and s t a t o r  v a r i a b l e s regarding  place  acceleration or deceleration  T h i r d , i n the case  i s significant  suggest  an i m b a l a n c e  t h e a i r g a p between r o t o r and s t a t o r o f t h e m a c h i n e .  the  of  lines  transient disturbances  of assessing  between m e c h a n i c a l  on  system, with the  t h e l a r g e r w e s t e r n U.S. s y s t e m . T h e r e a r e some f a c t s a b o u t  in  to l i m i t  information  imbedded  locally.  i n the  the disturbance  and t h e s t a b i l i t y  a group o f  the generator. For  the case  o f steady  state s t a b i l i t y  Japanese s c i e n t i s t s  were a b l e  t o m o n i t o r and measure t h e a i r g a p  flux  designed  [ 1 3 ] , and t h e y  tion device  (ASPAC)  In t h i s  a local  [14]. This device  t h e s i s we w i l l  to get s i g n i f i c a n t  automatic  information  stability  predic-  has f u n c t i o n e d v e r y  use the generator  well.  local variables  for transient s t a b i l i t y  assess-  ment.  2.3  Stability This  Controls  section i s a quick  survey  o f some methods used i n  21 the  power i n d u s t r y  to control  and improve  T h e s e methods a r e known i n t h e l i t e r a t u r e controls of transient and  stability  continuous c o n t r o l s  excitation applied  system  o n l y under c e r t a i n  stability.  them f r o m  controls are  severe conditions to maintain  o f p r i m a r y and s u p p l e m e n t a r y  successfully controls.  Dynamic B r a k i n g  t o a b s o r b power.  elements  Generally located  that  are switched  near g e n e r a t o r s ,  they a r e s w i t c h e d on i n case o f f a u l t s  to dissipate  r o t o r energy.  i s consumed by t h e  braking having  Since the excess energy  resistor,  the t r a n s m i s s i o n system  t o t r a n s m i t the energy  acceleration  of the generator.  successfully  used  some l i m i t a t i o n s following  1.  t o the load  excess  i s relieved  from  and s o i t r e d u c e s t h e  T h i s method h a s been  i n some power s y s t e m s .  However, t h e r e a r e  t o t h e use o f dynamic b r a k e s such as t h e  [15,16].  Picking  the proper s i z e o f the r e s i s t a n c e  disturbance 2.  primary  g o v e r n e r and t h e  [ 1 5 ] . Supplementary  Dynamic b r a k e s a r e r e s i s t i v e on-line  stability.  supplementary  Most t u r b o g e n e r a t o r s a r e c o n t r o l l e d  with a combination  2.3.1  as  to distinguish  such as t h e speed  control  transient  f o r several  c o n t i n g e n c i e s c a n be a p r o b l e m [ 1 7 ] .  The maximum s i z e o f t h e b r a k i n g r e s i s t a n c e when t h e r e s i s t a n c e reactance.  i s e q u a l t o t h e machine  i s realized transient  22 3.  The  braking resistance  and  the g e n e r a t o r  conditions  is accelerating.  the g e n e r a t o r ' s  when t h e b r a k e  2.3.2  i s most needed when <? i s l a r g e ,  i s applied  H i g h _Speed C i r c u i t  However, u n d e r  these  t e r m i n a l v o l t a g e i s low, the v o l t a g e goes even  Breaker  and  lower.  Reclosing  When a f a u l t o c c u r s a transmision l i n e , the c i r c u i t breakers from  a t each  end  the system,  reclose.  returns  remain  I f the  the c i r c u i t  o f the l i n e  breakers  The  remain  closed  o f opening  generator  clearing  T h i s method  of  transmission l i n e  is,  i f the l i n e  will  the  approximately  cycles.  it  the  The  fault  then  been c l e a r e d ,  the t r a n s m i s i o n still  then  system exists,  lockout.  and  final  closing reaches  must  take  its critical  because approximatley  transient  i n nature  f o r a s h o r t time  i n t e g r i t y o f the  Typically,  the  t i m e , and  I f the f a u l t  i s used  f a u l t s are  re-established. 12  to i s o l a t e  has  i n the system  i s de-energized  d e i o n i z e and  and  open and  p l a c e b e f o r e any angle.  fault  condition.  breakers w i l l procedure  open  open f o r a s p e c i f i e d  transmission line  to i t s p r e f a u l t  the c i r c u i t  will  fault  insulation  [18].  the f a u l t system  80% That arc  will  be  a r c can d e i o n i z e i n  a d v a n t a g e o f t h i s method  i s that  keeps g e n e r a t i n g u n i t s o n - l i n e f o r t r a n s i e n t t r a n s m i s s i o n  line  faults  and  i t a l s o minimizes  t h e number o f o u t a g e s  the  generator experiences  in i t s lifetime.  serious disadvantages  as a c o n s e q u e n c e o f u n s u c c e s s f u l r e c l o s u r e  [15,19,20] , as follows ;  However, i t has  that some  23 1.  A second major t r a n s i e n t c o u l d before  the i n i t i a l  oscillations  be a p p l i e d  t o the s h a f t  have damped o u t , and t h i s  may damage t h e s h a f t . 2.  Existence  o f two s u c c e s s i v e  3.  Increased  d u t y on c i r c u i t  breakers.  4.  Increased  damage a t f a u l t  locations.  5.  Possible  2.3.3  Series  Capacitors  lines.  transient  to increase  t h e power t r a n s f e r  f o r the i n d u c t i v e  Series capacitors  steady s t a t e s t a b i l i t y ,  1.  a r e used  [21] by c o m p e n s a t i n g  transmission  dips.  instability.  Series capacitors capacity  voltage  a r e good  but i t can p r e s e n t  reactance o f the f o r improving  some p r o b l e m s  under  conditions:  When a f a u l t  occurs,  series capacitors removing  them f r o m  protective devices  may  bypass  i n t h e f a u l t e d and n e a r b y l i n e s  thus  service.  2.  S u b s y n c h r o n o u s r e s o n a n c e may damage machine s h a f t s [ 2 2 ] .  3.  Series of  2.3.4  capacitors  improving  transient  Fast Valve  Action  Fast valve  action  g e n e r a t o r ' s steam v a l v e s Mechanical reduce  input  a r e e x p e n s i v e compared  methods  stability.  [23] i s t h e r a p i d c l o s i n g o f t h e following  t o the generator  the a c c e l e r a t i o n .  to other  a transient  disturbance.  i s t h u s r e d u c e d and t h i s  T h i s method c a n m a i n t a i n  will  transient  24 stability  i n many c a s e s ,  p o s t - f a u l t network level loss  i n order  o f power, o t h e r w i s e of synchronism.  pressure boiler  and  the  reopening valving  10  and  to prevent  instability,  valve  s t r e s s e s , and  the p r e s s u r e . steady  t e s t s may  and  o r opened  of  one  any  remaining  [24].  Circuit  breaker  [25]. any  phase f a u l t ,  the  operation  by  t h e same r e l a y i n g  scheme.  mechanically  independent, such  t h a t the m e c h a n i c a l  any  one  poles  t h r e e phases are s i m u l t a n e o u s l y  p o l e does not p r e v e n t  [15].  operation of  T h i s method h e l p s m a i n t a i n  and  Breakers refers  The of  failure the  two  three  activated for  three phases  the  to  are  However, f o r a  The  to  second  shaft  the b r e a k e r  of each other  operation.  Further-  systems.  phase does not a u t o m a t i c a l l y p r e v e n t phases from proper  fast  necessary  temperature,  three phases of  independently  either  e f f e c t on:  Independent p o l e o p e r a t i o n o f a c i r c u i t t h e mechanism by w h i c h t h e  increase  Although  be  t r a n s i e n t s i n steam s u p p l y  of  The  state instability, i t  to determine t h e i r  steam p r e s s u r e  cause  the  action is required,  field  Independent P o l e Operation  closed  and  increase.  temperature  some power c o m p a n i e s  r e q u i r e d and  f a s t v a l v i n g and  and  the  certain  occur  boiler will  this pressure  used by  swing  t r a n s i e n t may  i n the  m i n u t e s , so q u i c k  more, s t u d i e s a r e  2.3.5  a second  the v a l v e s o r b y p a s s i n g  acceptable  of  to reopen the v a l v e s a t a  temperature  c a n do n o t h i n g  evaluate  i t needs good p r e d i c t i o n  A l s o when t h e v a l v e s a r e c l o s e d  cannot withstand  f o r more t h a n  is  but  are  failure  remaining  system s t a b i l i t y  by  of  25 quickly clearing  or reducing  Independent p o l e o p e r a t i o n criterion with a  i s to guard  breaker  failed  line-to-ground to a double stuck).  easy  equipped  fast  ( i f one  to ground  to f i v e  to i n s t a l l ;  a separate  2.3.6  two  trip  with  i s u s e d a t l o c a t i o n s where t h e  reduce a t h r e e phase f a u l t  T h i s method c a n  as much a s is  line  fault(if  f o r each p o l e  separate  pole  Tripping^  Generator  tripping  i t improves both t h e method was  generator  generation.  Generator  locally  i s stuck), or  poles of  the b r e a k e r  clearing  Independent p o l e complexity  (most EHV  are  time  by  operation  i s to  breakers  If generator  t r a n s i e n t and confined  tripping  scheme, a r r a n g i n g trip  two  single  provide  are  i s a f o r m o f e n e r g y c o n t r o l much  means o f p r o v i d i n g t r a n s i e n t Recently  design  mechanisms).  v a l v i n g o r dynamic b r a k i n g .  Originally  local  [26].  to  the b r e a k e r  i n c r e a s e the c r i t i c a l  cycles  Generator  applied  trip  pole of  the only a d d i t i o n a l  coil  faults.  S u c c e s s f u l independent pole o p e r a t i o n o f  will  fault  s e v e r i t y of multiphase  a g a i n s t a t h r e e phase f a u l t c o i n c i d e n t  failure.  breaker  the  c a n be  the b r e a k e r s  scheme by m e a s u r i n g  is  stability. and  was  to c e r t a i n  initiated  from a  thermal transfer  a t t h e power p l a n t o r by  a  the g e n e r a t o r q u a n t i t i e s  [4]. Schemes have b e e n used where t h e g e n e r a t o r  maintained  connected  t o the u n i t  after  a  f o r remote g e n e r a t i o n .  been e x t e n d e d  tripping  state  to hydro generators  stability has  steady  tripping  like  tripping  load i s  and  the u n i t  is  26 r a p i d l y reloaded  after  be r e s y n c h r o n i z e d 15-30  the d i s t u r b a n c e .  t o t h e s y s t e m and f u l l  Generally  the u n i t can  load r e s t o r e d i n about  minutes [15]. The m o s t common t e c h n i q u e  used t o t r i p g e n e r a t o r s  o f - s t e p r e l a y i n g w h i c h i s b a s i c a l l y a d i s t a n c e scheme.  i s outThe  trip  s i g n a l d e p e n d s on t h e v o l t a g e m e a s u r e m e n t s a t t h e i n v o l v e d l o c a t i o n s and  t h e n t h e s i g n a l has t o be t r a n s m i t t e d  generator  l o c a t i o n to t r i p  technique  cannot provide  every  the s y n c h r o n i z i n g breakers.  t o t a l p r o t e c t i o n unless  s i g n i f i c a n t s t a t i o n are transmitted.  flows.  The  These s i g n a l s would  as t h e power systems t o  total protection.  However, t h e s i g n i f i c a n t  generator  i s a l r e a d y a t the generator  will  o u t p u t and  r e s u l t i n g l o g i c and c o m m u n i c a t i o n s y s t e m u s e d  w o u l d h a v e t o be a s c o m p l e x  imbedded  This  s i g n a l s from  h a v e t o be p e r m i s s i v e l y c o n t r o l l e d by t h e g e n e r a t o r line  to the  tripping  i n the generator  be f u r t h e r d i s c u s s e d  provide  information f o r location.  It is  s p e e d , a c c e l e r a t i o n and a n g l e .  These  i n Chapter 4 .  27  CHAPTER 3 SYSTEM MODELS AND TEST SYSTEMS  3.1  Classical The  Model o f S y n c h r o n o u s  induced  voltage  Machines  i n the s t a t o r windings o f a  s y n c h r o n o u s m a c h i n e c a n be c o n s i d e r e d a component E ^ t h a t c o r r e s p o n d s field  to the flux  w i n d i n g , and a component E  corresponds  E  (until  hence E ^ c a n n o t c h a n g e .  rotor  Hence o t h e r  circuits  Currents  flux  a r m a t u r e and r o t o r c u r r e n t s w i l l  coupled  change  i n t h e network, t h e f l u x  will  currents w i l l  t o keep t h i s  components a s r e q u i r e d  because i t  the e x c i t e r a c t s ) .  i n the a f f e c t e d synchronous machine w i l l  armature.  t h e main  i n the armature, b u t E^ cannot  When a s u d d e n change o c c u r s linkage  linking  c a n change i n s t a n t a n e o u s l y  2  to currents  instantaneously  two components;  t h a t c o u n t e r a c t s the  2  armature r e a c t i o n .  as having  n o t change and  be p r o d u c e d  be i n d u c e d  i n the  i n the v a r i o u s  linkage constant.  Both the  u s u a l l y have AC and DC  t o match t h e ampere-turns o f v a r i o u s  coils.  The f l u x w i l l  decay a c c o r d i n g  t o the e f f e c t i v e  time c o n s t a n t .  Under no-load  conditions  i s on t h e o r d e r o f  several  seconds while  under l o a d  this  i t i s on t h e o r d e r  o f one  second. From t h e a b o v e d i s c u s s i o n we c a n s e e t h a t f o r a p e r i o d o f one  second  constant. period  the f l u x  l i n k a g e and h e n c e  c a n be c o n s i d e r e d  U s u a l l y e x c i t e r s do n o t r e s p o n d  o f one s e c o n d o r l e s s .  This period  f a s t enough i n a i s often  considered  28 adequate  f o r d e t e r m i n i n g the t r a n s i e n t  synchronous machine.  The m a i n f i e l d  same a s a f i c t i t i o u s machine d i r e c t  axis  flux  model o f s y n c h r o n o u s  the  direct  transient  winding  T h i s model  machine.  o f the  flux  t h a t would c r e a t e EMF  reactance.  cal  stability  i s almost the  behind the  i s called  the  classi-  I t i s a voltage source behind  r e a c t a n c e X'^  a s shown  t  i n F i g 3.1  d '7JTP X  E  and 6  c a n be  Q  determined V  from the i n i t i a l c o n d i t i o n s i.e., pretransient  IA  *• |  (  ]  V  while  o  the magnitude E i s c o n s i d e r e d c o n s t a n t  t h e a n g l e 6 i s c o n s i d e r e d a s t h e a n g l e between and  the t e r m i n a l  the  the r o t o r p o s i t i o n machine  p o i n t where change the  kinetic  terminals.  However,  i t i s sensed  masses.  energy  be a f f e c t e d  by t h e change  and any c h a n g e s i n t h e impedance s e e n  The  until  and c o r r e c t e d  i n t h e o u t p u t power w i l l  rotating  the r o t o r  voltage.  The m a c h i n e o u t p u t power w i l l in  ' °  F i g 3.1  the t r a n s i e n t  position  0  0  conditions During  |  the speed changes t o the by t h e g o v e r n o r , t h e  come f r o m  the s t o r e d  important parameters  i n MW.S/MW u s u a l l y  called  energy i n  here are the  H, o r t h e m a c h i n e  inertia constant.  3.2  System R e p r e s e n t a t i o n S i n c e we  the  first  swing,  a r e concerned about  by  the t r a n s i e n t s t a b i l i t y i n  i t i s reasonable to consider  the c l a s s i c a l  29 model o f t h e s y n c h r o n o u s machine and o f t h e power s y s t e m can  be summarized  1.  Constant  i n the f o l l o w i n g  v o l t a g e behind  assumptions:  direct  Q f o r the synchronous machine first 2.  which  transient  i svalid  reactance  model  i n the period o f the  swing.  The m e c h a n i c a l  rotor angle  coincides with  the angle  transient  ( r o t o r p o s i t i o n ) o f a machine  o f the v o l t a g e behind  the d i r e c t  reactance.  3.  Mechanical  power  input t o each generator  4.  The t r a n s m i s s i o n n e t w o r k i s m o d e l e d  i s constant.  by s t e a d y  state  equations. 5.  Damping o r s y n c h r o n i z i n g power i s n e g l i g i b l e .  6.  Loads a r e r e p r e s e n t e d the c l a s s i c a l motion  i i  1  2  P =  The e q u a t i o n s o f  i n a power s y s t e m a r e : (3.1)  ei M  n  G  gi  model b e i n g c o n s i d e r e d .  f o r e a c h machine  M..u » P. - P  Where  by c o n s t a n t p a s s i v e impedances f o r  ± ±  (B s i n ^ - f i ^ - K ^ c o s t f ^ ))  + Z  J#ip  . = mechanical  G^JL = d r i v i n g E^,Ej e  i»j  BJLJ  G  power i n p u t  p o i n t conductance  = constant voltage behind =  generator  = transfer  j = transfer  rotor  susceptance  transient  reactance  angle between nodes i and j  c o n d u c t a n c e between nodes i and j  30 oj = s p e e d d e v i a t i o n w i t h r e s p e c t r  = inertia  c o n s t a n t (M^ =  s p e e d OJ  to synchronous  r  ) r  3.3  The T h r e e T e s t  Systems  In t h i s p r o j e c t  a detailed  m a c h i n e speed and a c c e l e r a t i o n faults  in different  were s t u d i e d : and  often  Machine  This  system  test  three-load  referred The  i n each system.  i s t h e w e l l known n i n e - b u s ,  s y s t e m w i d e l y used  i s shown i n F i g u r e  turbogenerators with rated  3.2.  except  A l l the  Machines  three-  impedances  2 and 3 a r e steam  s p e e d o f 3600 r e v / m i n , w h i l e  g e n e r a t o r d a t a and t h e i n i t i a l  s p e e d o f 180 r e v / m i n . operating  conditions  i n T a b l e 3.1.  The F o u r M a c h i n e This  system  i n t h e l i t e r a t u r e and  machine 1 i s a hydro g e n e r a t o r w i t h r a t e d  3.3.2  systems  System  i n p e r u n i t on a 100-MVA b a s e .  are g i v e n  Three  phase  t o a s t h e V7SCC s y s t e m .  system  The  out f o r three  system.  The T h r e e  machine,  i s carried  individual  a t h r e e - g e n e r a t o r system, a f o u r - g e n e r a t o r  a five-generator  3.3.1  are  locations  investigation of  system  System  i s t h e same a s t h e t h r e e - m a c h i n e  f o r the f o l l o w i n g  changes:  system  32 Table  3.1  G e n e r a t o r d a t a and i n i t i a l c o n d i t i o n s of t h e three-machine system  Generator data Gen. No.  H (MW/MVA)  Initial  conditions  'd (pu)  P *mo (pu)  E (pu)  x  x  (degree)  1  23.64  .0608  2.269  1.096  6.95  2  6.40  .1198  1.6  1.102  13.49  3  3.01  .1813  1.0  1.024  8.21  T a b l e 3.2  G e n e r a t o r d a t a and i n i t i a l o f the four-machine system  conditions i n i t i a l " condiiIons  Generator data H (MV7/MVA)  ^ d (pu)  p *mo. (pu)  E (pu)  1  23.64  .0608  2.269  1.0967  2  6.4  .1198  1.6  1.102  3  3.01  .1813  1.0  1.1125  4  6.4  .1198  1.6  1.074  Gen. No.  (degree) 6.95 13.49 8.21 24.9  33  34 The  rating  o f t h e t r a n s m i s s i o n n e t w o r k was changed  230 KV t o 161 KV; t h e R and X v a l u e s the  i n pu r e m a i n  same.  -  A fourth generator  through line  (161 KV) t o b u s 8.  initial  3.3.3  was c o n n e c t e d  a step-up transformer  as g e n e r a t o r  2.  operating  l i n e s and l o a d d a t a  circuit  i n Table  network  transmission  h a s t h e same  i s shown i n F i g u r e  rating  3.3 and t h e 3.2.  System  i s shown i n F i g u r e are given  p e r u n i t on t h e 10,000 MVA  initial  and a d o u b l e  conditions are given  The F i v e - M a c h i n e  t o the o r i g i n a l  The new g e n e r a t o r  The s y s t e m  T h i s system  in  o f the l i n e s  from  i n Table  3.4. 3.3.  The t r a n s m i s s i o n The d a t a  and 500 KV b a s e s .  c o n d i t i o n s and p a r a m e t e r s a r e g i v e n  are given  The m a c h i n e  i n Table  3.4.  35  36 Table  3.3  T r a n s m i s s i o n l i n e p a r a m e t e r s and l o a d s of the five-machine t e s t system.  Transmission  Load  l i n e (impedances)  (Admittance)  pu  R(pu)  X(pu)  Load No  G(pu)  7-8  .12  2.397  LI  .0098  -.0049  7-6  .03  .597  L2  .0192  -.0092  9-6  .032  .639  L3  Line  10-11  .15  2.996  L4  10-6  .186  3.71  L5  11-6  .24  4.79  Table  3.3  Generator Gen No. 1  G e n e r a t o r d a t a and i n i t i a l the f i v e - m a c h i n e system.  1.088 .6598 1.269  Initial  .46  'd (pu)  X  P m  -.5271 -.3195 -.6144  conditions of  data M (pu)  B (pu)  ? u) P  conditions  E (pu)  3.2  .0498  1.6  .069  1.032  41.24  1.138  1.171  40.9  1.397  37.6  2.08  31.8  2  1.1  3  7.41  4  .28  .799  5  .32'  .89  .25  .6998 1.269  .981  (degree) 37.6  37 CHAPTER 4 STUDIES OF  In to for  GENERATOR VARIABLES AVAILABLE LOCAL STABILITY ASSESSMENT  t h i s c h a p t e r we  investigate assessing The  the  discuss  suitability  transient  a n g l e swing  between s t a b l e  will  and  of  the  studies  a v a i l a b l e machine  curves w i l l  be  used  variables  unstable c a s e s , then a comparison w i l l the  stability.  local  be  variables  angle, speed, a c c e l e r a t i o n  and  that w i l l rate of  be  transient investigated  change o f  are  acceleration.  A c c e 1 e r a t i n g Powe r Accelerating  power i s t h e  m e c h a n i c a l power i n p u t the  out  to d i s t i n g u i s h  v a r i a b l e curves to assess  4.1  carried  stability.  made w i t h l o c a l The  FOR  transient  the  DT  i s the  the  2  swing m  inertia  and  P  accelerating  important q u a n t i t i e s at transients. indication condition  Because the of  of  power o u t p u t  during  the the  equation  constant,  m e c h a n i c a l power i n p u t , i s the  the  e  the  a  electrical  between  period.  Recalling  Where M  and  difference  the  P  e  S i s the  i s the  power.  P  generator  value of P  s e v e r i t y of generator.  a  the  a  rotor angle, P  electrical i s one  of  terminal can  fault  give and  m  power o u t p u t the  most  during a  the  is  valuable stability  38 Since M i s approximately undergoing  s m a l l speed d e v i a t i o n s ,  power i s d i r e c t l y  proportional  both  power and  at  constant  accelerating  the g e n e r a t o r  acceleration  site  generator  t h e r e f o r e the  t o the  accelerating  acceleration.  acceleration  [3,4],  have t h e  f o r any  A l s o both  can  be measured  accelerating  same waveshape, t h e o n l y  be  be measured m e c h a n i c a l l y measured e l e c t r i c a l l y Figure  for  4.1  a three-phase  fault  cleared  accelerating  power c u r v e  at  .1  of  represents  accelerates  F i g 4.1  the machine r o t o r .  after clearing  after  fault  I t can  system, or Therefore 4.1  the  be  can  critical  be  power and  f o r m a c h i n e 4.  can  represent  under the during negative  The  the  curve  the  kinetic before  fault  area  system  which  under by  A  2  synchronism  the  the  system  the  for a fault  at  as a s i n g l e - m a c h i n e - i n f i n i t e - b u s  machine  and  compared  can  acceleration  four-machine  t h e power a b s o r b e d  lose  considered  t h e a r e a A^  power  Machine 4 i s t h e o n l y m a c h i n e i n  f o u r - m a c h i n e s y s t e m t h a t may bus-10.  The  represents  clearing.  the  second,  t h e power g a i n e d  curve  Fig  10 o f  p o t e n t i a l e n e r g y , where t h e a r e a  clearing  and  acceleration  the a c c e l e r a t i n g  accelerating  a t bus  3)  and  power  [3] .  shows b o t h  (see Chapter  and  [4]  locally  difference  between them i s t h e manner o f measurement, where can  Practically  [11]  for a fault  at  under the a c c e l e r a t i n g  to find  the  bus-10. power o f  s t a b i l i t y o f machine 4 ( f o r  4*  m a c h i n e 4 t o be I f we hard  to apply  stable  move t h e  Aj^ must be fault  the e q u a l  l e s s than or equal  f r o m bus-10 t o bus-8 t h e n  area c r i t e r i o n ,  to  A ). 2  i t i s very  because machine 4 i s not  39  i  i  4-Nach.systeni(f a t 10) i  i  l  I  I  I  I  I  L  Time (sec) (a)  n.D  i 0.4  0.8  1  r U  Time (sec) (o) F i g 4.1  ( a ) a c c e l e r a t i n g pwer , (b) a c c e l e r a t i o n machine A f o r a f a u l t  a t bus 10 c l e a r e d  t  0  f  a t .Is  40 the  only  machine  t h a t may l o s e  synchronism.  m a c h i n e 3 and m a c h i n e 2 may a l s o area  criterion  system an  c a n be a p p l i e d  stability  machine first  loses  system.  first  i s to search  tracking  4'2  system t o f i n d  global  t o a s i n g l e machine i n  and w h i c h m a c h i n e we s h o u l d  system s t a b i l i t y  c a n n o t be answered when u s i n g  and  t o t h e whole  The e q u a l  T h e r e f o r e we need t o know w h i c h  synchronism  to maintain  answer  lose synchronism.  b u t i t c a n n o t be a p p l i e d  interconnected  In t h i s case  i fpossible.  a global  These  stability  f o r another c r i t e r i o n  trip  questions  criterion.  b a s e d upon  The  analyzing  each machine v a r i a b l e i n d i v i d u a l l y .  Speed and A c c e l e r a t i o n It  has b e e n f o u n d  energy c o n t r i b u t i n g machine. certain  Critical fault  be p r e d i c t e d two-machine  [10] t h a t a l l o f t h e f a u l t k i n e t i c  to i n s t a b i l i t y  i s imbedded  machines a r e easy t o d e f i n e  i n the unstable [10] f o r a  i n any power s y s t e m , t h e n t r a n s i e n t s t a b i l i t y c a n f o r those c r i t i c a l  equivalent  m a c h i n e s by u s i n g t h e  technique  (2.15) f o r e n e r g y f u n c t i o n  [12].  Recalling  equation  o f an i n d i v i d u a l m a c h i n e  (2.15)  For  the c r i t i c a l  equation the  machine o f a c e r t a i n f a u l t  the f i r s t  term o f  (2.15) i s t h e most e f f e c t i v e [10] t e r m when c a l c u l a t i n g  i n d i v i d u a l m a c h i n e e n e r g y V*.  This  shows t h e i m p o r t a n c e  41 of  speed  deviation  i n transient  stability  m a c h i n e d u r i n g and a f t e r a f a u l t . (2.15)  during  injected  t o the system  represents Tracking  the speed  deviations  and  and a f t e r t h e f a u l t  t o the f a u l t  The  acceleration  [4]  instantaneously  applied.  takes place t h e swing  first  peak o f t h e f i r s t  depending in  Stability  the  i t i s noticed  a maximum f i r s t  first  limit  t h e speed  o f time.  The shows  a t bus-10, the  o c c u r s much e a r l i e r  than t h e  "Limits" v e r s u s time f o r d i f f e r e n t  that  the f i r s t  peak o f t h e  c u r v e s i s d i f f e r e n t f r o m one l o c a t i o n t o a n o t h e r  on f a u l t  acceleration  up from  swing.  When p l o t t i n g a c c e l e r a t i o n  acceleration  locations.  and f o r a n y m i s m a t c h  c u r v e o f machine-4 f o r a f a u l t  locations  speed  F i g 4 . 3 shows t h e a c c e l e r a t i o n  response of the a c c e l e r a t i o n  fault  shows a  machine f o r a c e r t a i n f a u l t  i n the system,  Acceleration  locations  i n a very short period  rapid  4.3  energy  by t h e s y s t e m .  F i g 4 . 2 shows  s i g n a l c a n be p i c k e d  o f the c r i t i c a l  i n equation  i s cleared i t  curves f o r d i f f e r e n t f a u l t  i n s t a n t a n e o u s r e s p o n s e f o r any f a u l t  that  term  r e p r e s e n t s the k i n e t i c  f o r machine 4 f o r d i f f e r e n t f a u l t  acceleration an  The f i r s t  the p o t e n t i a l energy being absorbed  slow response  signal  the f a u l t period  o f the c r i t i c a l  severity,  i . e . , t h e most s e v e r e f a u l t  locations  peak o f t h e a c c e l e r a t i o n  i t i s found  F i g 4 . 4 shows t h e  peak o f a c c e l e r a t i o n .  for'different fault  that  exceeds  t h e machine l o s e s  results  f o r machine 4 .  When  a c e r t a i n maximum  s y n c h r o n i s m r i g h t away.  42  4-flach. system I  i  \  \  I  I  I  1  1  L  Timelsec) Fig F i g 4.2  speed d e v i a t i o n  4.2  of machine A f o r d i f f e r e n t f a u l t  locations.  i  4-HfiCH.SYSTEH(F RT I0JC=.1S) i  CD  i  i  i  i—i—i—i—i—i—i  D(4-1) D(3-t)  D(2-l)  0.0  i—i 0.41 1 0. 183 r nU— 5r U5  1.66  TinE(SEC) (a) J  I  — -  in.  I  I  L  J  L  J  L  d(4-l) ficce.4  tn  i  I  0.0  1  1—l 1  0.16  0.33  1  aS  J  Timelsec) (b)  1 0.66  I  i  i „  0.83 1JJ  Fig 4.3 (a) swing curve of machines 4»3 &nd 2 , (b) swing curve and acceleration of machine 4. For afault at bus 10 cleared at .Is.  44  »  I  4-Mach. system I J I 1 J  D.26  0.4  Time Fig  L  0.53  (sec) 4.4  A c c e l e r a t i o n of machine 4 f o r d i f f e r e n t f a u l t l o c a t i o n s .  45 Also  f o r some f a u l t  found  that  sustained  locations  f a r away f r o m  t h e machine never l o s e s fault,  acceleration  i . e . , there  (kinetic  energy) t h a t  maximum v a l u e o f t h e f i r s t  first  never l o s e For  two  important as the  f o r any f a u l t f o r  maintain synchronism,  i s l e s s than t h i s  the machine w i l l  c a n be d e f i n e d  lose  as the could  peak o f the a c c e l e r a t i o n  minimum v a l u e ,  the machine  will  synchronism. any f a u l t  some c o n s t a n t  first  l o c a t i o n i t i s found peak o f a c c e l e r a t i o n  duration,  that  operating  conditions.  i s because t h i s  for different  lowest  of the f i r s t  level  that  each machine has  regardless  o f the f a u l t  peak d e p e n d s on t h e m a c h i n e  F i g 4.4 shows t h e d i f f e r e n t  acceleraiton  fault  levels  peak o f a c c e l e r a t i o n  t h e minimum l i m i t , w h i l e t h e maximum l e v e l  of  acceleraion  i s t h e maximum l i m i t  c a n be c o n s i d e r e d o f the f i r s t  f o r machine 4.  4.6 show t h e a c c e l e r a t i o n  limits  for  different  i n t h e 4-machine s y s t e m .  locations  Since the a c c e l e r a t i o n severity before  peak  F i g 4.5 and  Fig  fault  of  l o c a t i o n s f o r m a c h i n e 4, t h e  as  fault  i.e.,i f  peak above w h i c h t h e m a c h i n e  synchronism, i . e . , i f the f i r s t any f a u l t  to i n s t a b i l i t y .  c a n be d e f i n e d  S e c o n d , a minimum l i m i t ,  minimum v a l u e o f t h e f i r s t  for  we c a n d e f i n e  peak e x c e e d s t h e maximum l i m i t  synchronism.  lose  contributes  peak o f a c c e l e r a t o n  which the machine can c r i t i c a l l y the  synchronism even f o r a  A maximum l i m i t  limits.  i t is  i s a minimum v a l u e o f t h e  From t h e a b o v e d i s c u s s i o n acceleration  the machine  f o r m a c h i n e 2 and machine 3  h a s t h e p r o p e r t y o f showing t h e  the f a u l t  i s cleared,  therefore  i t will  46  4-dach.  JL  0.26  system  0.4  Time (sec) f i g 4.5  oin t 0D  "  o'.2S  1  0.5  0.75  W  I.2S  1.5  Time(sec) Acceleration  F i g 4.6 f o r d i f f e r e n t f a u l t l o c a t i o n s o f m a c h i n e s 2 and  3  .  47 likely is  reveal  cleared.  the  machine  Several  information  Case  4.4.1  The  about the  stability  were d e t e r m i n e d  plotted  Case  the  each f a u l t  with  with  disturbed  demonstrate  valuable  machines.  the  i n the  and  the  swing  fault  acceleration of and  as  each machine  unstable  cases  curves. critical  stable  s w i n g c u r v e on  unstable  three-phase  three-machine system  stable  l o c a t i o n the  critically  with a  the  c l e a r i n g time  curves of  was  a c c e l e r a t i o n were  same f i g u r e .  Also  the  c u r v e s were p l o t t e d  f o r each machine.  phase f a u l t  5-4  1:  critical  c l e a r i n g time  i s .3  disconnecting  line  c u r v e and  a c c e l e r a t i o n of  at  the  c a s e s were s t u d i e d  from  With a three  and  of  Swing c u r v e s and  then the  critically  t o come up  to  T h r e e - m a c h i n e System  e a c h c a s e were p l o t t e d  f o u n d and  indicate  not.  and  at d i f f e r e n t locations  For  a c c e l e r a t i o n may  fault  Studies  shown i n F i g 4.7. for  power s y s t e m a f t e r the  have b e e n c a r r i e d o u t  acceleration  A number o f applied  the  i s stable or  case s t u d i e s  importance of  4.4  of  In o t h e r w o r d s , t h e  whether the  the  strength  the  unstable .3  cases.  s e c o n d s , and  5-4.  For  on  sec,  Figures  the  f o r the  the  line the  4.8  near bus-4,  fault  being  t o 4.10  m a c h i n e s 1,  the  cleared  show t h e  by  swing  2 and  3 for  stable  stable  cases  the  fault  is  cleared  unstable  cases  the  fault  is cleared  48  8-  J  i  3-riach.system(f i  i  i  i  i —  at  49  4)  i i i Angle D i f f .  — —  Acceleration  IS).  u u CE  I  CM in ro. i  -I 0.33  0.0  1  1  1 — I  1  0.66  1JD  1  1  1  1—  1.33  1.66  L  J _JL  2.0  Time(sec) (a)  8-  J  I  L  J  I  L  J  Angle  Diff.  Acceleration  in  CE  in. to  i T3  in  8 . i 0.0  I  1  1  0.33  1  0.66  1  1  1  1JJ Time(sec) (b)  1  1.33  I  I i 1.66  2.0  F i g 4.8 A n g l e d i f f . a n d a c c e l e r a t i o n o f m a c h i n e 1 f o r a f a u l t a t b u s 4; ( a ) s t a b l e c a s e (Tc=.30 s e c ) . ( b ) u n s t a b l e c a s e (Tc=.31 s e c ) .  g'_l  '  3-nach.systemCf  »  '  I  I  1  I  al  1  4)  1  50 L  Angle D i f f . — — — Acceleration  v ^  \ in.  o> ( N  o  u  QZ  a  «-> o . n  in I  ~T— 0.33  i  i  0.0  8-  J  I I  Angle D i f f . — —  in  Acceleration  INI  rg •  fN " cj u CC  i*> in. to -a  r-7-  in  5?. i 0.0  —|  1 0.33  1  1 0.66  1  1 1.0  1  1 1.33  1  1  I  1.66  2.0  Timelsec) (b) F i g 4.9 A n g l e d i f f . and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 4 ; (a) s t a b l e case (Tc=.30 s e c ) . (b) u n s t a b l e case (Tc=,31 s e c ) .  51  3-riach.systent(f  i  at  4)  *^ Angle  Diff.  (a) 8-  — Angle D i f f . Acceleration  in no «_> w> in_J 10  l  in  8i 0.0  F i g 4.10  <U3  '  0 £ 6  1-0  -  «-33  1.66  2.0  Time(sec) (b) Angle d i f f . and acceleration of machine 3 f o r a fault at btfs 4 ; (a) stable case(Tc=.30 sec). (b) unstable case (Tc=.31 sec).  52 at  .31 s e c o n d s .  The peak o f t h e a c c e l e r a t i o n  after clearing is  l o w e r t h a n t h e f i r s t peak b e f o r e c l e a r i n g f o r t h e s t a b l e at  t  c  noted  = .3 s e c o n d s  f o r a l l the three machines.  t h a t machine 1, w h i c h  is  not the c r i t i c a l  to  this fault  generated  I t c a n be  i s t h e c l o s e s t machine  machine,  because  cases  t o the f a u l t ,  i t accelerated  very slowly  and a l s o most o f t h e k i n e t i c e n e r g y has been  by m a c h i n e s  2 and 3.  for  such a c a s e .  the  peak o f t h e a c c e l e r a t i o n  So m a c h i n e 1 w i l l  For the unstable case a t t after clearing  peak b e f o r e c l e a r i n g f o r a l l t h e m a c h i n e s , three machines  c  n o t be  tripped  = .31 s e c o n d s  i s h i g h e r than the i t means t h a t  a l l the  a r e a c c e l e r a t i n g more a f t e r c l e a r i n g and t h e  s y s t e m b r e a k s up.  Case  2: A t h r e e - p h a s e f a u l t on l i n e  .11  seconds  case.  f o r t h e s t a b l e c a s e and .12 s e c o n d s  f o r the u n s t a b l e  F i g 4.11 and F i g 4.12 show t h e s w i n g c u r v e and a c c e l e r a -  tion  f o r machines  Case  3:  2 and 3 f o r t h e s t a b l e  A t h r e e - p h a s e f a u l t on l i n e cleared for  7-8 n e a r bus 7, c l e a r e d a t  by d i s c o n n e c t i n g  this fault  line  9-8.  and u n s t a b l e c a s e s .  9-8 n e a r b u s - 9 . The c r i t i c a l  The  fault  clearing  time  i s .41 s e c o n d s .  F i g u r e s 4*-13 and 4.14 show t h e s w i n g c u r v e and acceleration  f o r machines  2 and 3.  For the s t a b l e  case the  53  3-nach.system(f  R-  CN u u cr  i  /  o. CD  /  I  •  I  s  I  \  /  I  at t  I  7)  Angle v  1  I  L  Diff.  • Acceleration  \ \  / /  \ \  /  .  /  o in I  ro. 0.0  I  1  0.33  1  1  1— i  D.6B  1  1.0  1  1  1.33  Time(sec)  1 1.66  r  2.0  (a) J  J  L  J  L  I  Angle  I  L  Diff.  Acceleration CM  u u CE  in  U) fM*_J I CM  in ^. ro.  in i  0.0  1  1  0.33  1  1  0.66  1  1  Ul  1  1  1.33  1  1 1  1.66  2.0  Time (sec) F i g 4.11 Angle d i f f . and a c c e l e r a t i o n o f machine 2 f o r a f a u l t at bus 7 ; (a) s t a b l e case ( T c = . l l s e c ) . (b) u n s t a b l e case (Tc=.12 s e c ) .  1  R-|  1  3-f1ach.system(f I  1  1  I  at  I  t  t  7)  i  i  - Angle D i f f . Accleration  2.4  (a)  5-  J  I  L  Angle D i f f . Acceleration  cn u u CX  IO  T  "  CO  "° = (M  m/  o I  — l  0.0  1 0.4  1  1 0.8  1  1 U  1  1 1.6  1  1  1—  2.0  2.4  Time(sec) P i g 4.12  (b) Angle d i f f . and a c c e l e r a t i o n o f machine 2 f o r f a u l t at bus "7 ; (a) s t a b l e case ( T c = . l l s e c ) . (b) u n s t a b l e case (Tc=.12 s e c ) .  55  3 - n a c h . s y s t e m ( f a t 9) j i i i i i i i i' - ^^  Angle D i f f . Acceleration  o o o-  \  \  U  u CC  \ o  u> coin I  CN TO  m.  i 0.33  Q.D  1  1  0.66  r  n  r  IJ)  Time(sec3  i — i — i  1.66 1.33  r  2.0  (a)  S-  J  L  J  L  Angle D i f f . •  Acceleration  in CM u  u  CC  0.0  F i g 4.13  i  r  1  0.33  ~I—I—I—I—I—I—I—I— 0.66 101 1.33 1.66 Time(sec) (b)  2.0  Angle d i f f . and a c c e l e r a t i o n o f machine 2 f o r a f a u l t at b u s 9 » (a) s t a b l e case (Tc=.41 s e c ) . ( b ) u n s t a b l e case (Tc=.42 s e c ) .  3-Mach.systein(f  j  8-  i  i  i  i  i  at  t  9) i  56  i  i  ---- Angle D i f f . • co u  u  H  cc T ro  Acceleration \  a m  8•  0.0  i—i—r  0.33  »  i  1  l  0.66  i  i  1  1  10  1  Time (secj (a) 1 I L /  1  1  I  I  1.33  1  1—  I  L_  1.66  2.0  Angle D i f f . '-  Acceleration  IDCO " CJ  u CE  o u) o'. CD  I CO  8 0.0  Pig 4.14  —i  1 0.4  1  1 0.8  1  1 12  r  ~i—i—i—r~ 1.6  2.0  2.4  Time(sec) (b) Angle,diff. and acceleration of machine 3 f o r a fault at bus 9 ; (a) stable case (Tc=.41 s e c ) . (b) unstable case (Tc=.42 s e c ) .  57 fault .42  i s cleared  at  .41  disconnecting  line  are seen  fault  7-5.  case t  c  =  i n the f o l l o w i n g  cases:  on l i n e  See  7-5  F i g 4.15  near bus-5 i s c l e a r e d  and  Fig  by  4.16.  5: A three-phase  disconnecting  Case  f o r the u n s t a b l e  4: A three-phase  Case  and  seconds. Similar results  Case  seconds  line  fault  9-6.  on  See  line  9-6  F i g 4.17  near bus-6, and  Fig  i s cleared  by  i s cleared  by  4.18.  6: A three-phase f a u l t  disconnecting  line  7-8.  on  See  line  7-8  F i g 4.19  near bus-8, and  4.20.  58 j  0.0  3-nach.system(f  [  i  i  1  0.33  l  1  l  1  0.66  I  I  D  i  I  5)  1  1  L  i—i—r  r—T  1.0  Time(sec) (a)  a-  at  1  1.33  J  L  2.0  1.66  I  L  Angle D i f f . — —  in  i  0.0  1  1  0.33  „!*  1  '  , i  '  Acceleration  U3  '  U S  2.0  Time(sec) (b) Fig 4.15 Angle j i i f f . and acceleration of machine 2 f o r a f a u l t at bus 5 ; (a) stable case (Tc=.21 s e c ) . (b) unstable case (Tc=.22 s e c ) .  59 1  gU  1  3-nach.system(f 1  L  J  1  I  — —  in  r-"  at  5)  L  J  L  Angle D i f f . Acceleration  «T> t_>  u CE  in'. ID  i ro ' " a in IN.  r i  l  00  1  1  0.33  J  I  1— 0.66  I  I  L  (a) i  in  J  — -  I  I  Angle D i f f . Acceleration  CO o  ID  i  co "  "O in r\J. rN  I  0.0  —i  i 1 0.33  1  0.66  1  1  UD  1  1 1.33  1  1  r—  1.66  2.0  Time(sec) (b) Fig 4.16 Angle diff.and acceleration of machine 3 f o r a fault at bus 5 ; (a) stable case (Tc=.21 s e c ) . (b) unstable case (Tc=.22 s e c ) .  60 3-nach.systemlf  0.0  0.0  F  i  g  4.17  JL33  0£6  at  6)  UJ  Timelsec)  0.33  0.66  U  1.33  1.66  2.0  Timelsec)  (b) Angle'diff. and acceleration of machine 2 f o r a fault at bus 6 » (a) stable case (Tc=.51 s e c ) . (b) unstable case (Tc=.52 sec)  J  3-ttach.system(f  I  I  i—i  1  «  1  1—i—i—i—i  0.33  J  8-  I  0.66  I  I  1  L  61  6)  r 1.33  UO  Timelsec) (a) I L J  I  1  J L Angle D i f f . Acceleration  tn  r-'  i  at  0.0  l  1  0.33  1  1  0.66  1  1  UO  1  1  1.33  1  1  1.66  I  2.0  Timelsec) (b) Fig 4.18 Angle d i f f . and acceleration of machine 3 f o r a fault' at bus 6 ; (a) stable case (Tc=.51 sec). (b) unstable case (Tc=.52 sec).  i  0.0  Pig 4.19  3-nach.system(f i  0.33  i  i  DJ56  at  i — i — i — J —  t0  1.33  62  8)  1  —  L  1.66  Tiae(sec)  2.0  (b) Angle d i f f . and acceleration of machine 2 f o r a fault at bus 8 ; (a) stable case(Tc=.29 sec ). (b) unstable case (Tc=.30 sec)  g I  I  I  3-Mach.system(f I  I  0.66  L_  I  I  I  t0  Tine(sec) (a)  i — i 0.0  0.33  1 — i — i — i — i 036  10  at  1  t  i r 1.33  8)  63 I  «  n 1.66  r  2.0  1—i—i—i—r 1.33  1.66  2.0  Time(sec) (b) Fig 4.20 Angle d i f f . and acceleration of machine 3 f o r a fault at bus 8 ; (a) stable case (Tc=.29 s e c ) . (b) unstable case (Tc=.30 s e c ) .  64  4.4.2  system  The F o u r - m a c h i n e  System'Case S t u d i e s  The  system  four-machine  e x c e p t f o r a f o u r t h machine c o n n e c t e d  double c i r c u i t out  i s t h e same a s t h e t h r e e - m a c h i n e  to find  transmission l i n e .  the i n i t i a l  operating  A load  t o bus-4 t h r o u g h a  f l o w s t u d y was  carried  c o n d i t i o n s o f the machines.  A t h r e e - p h a s e f a u l t was c o n s i d e r e d a t e a c h bus o f t h e s y s t e m , and  by l o o k i n g that  a t t h e a c c e l e r a t i o n r e s p o n s e o f e a c h machine i t  is  clear  the a c c e l e r a t i o n o f a machine i n d i c a t e s whether i t  is  stable or not. A number o f s t u d i e s were c a r r i e d  typical  case o f a three-phase f a u l t  considered faulted in  here.  the system  never l o s e  3 i s less  be  their stability  The o t h e r t h r e e m a c h i n e s even  i f machine-4  T h i s means t h a t most o f t h e k i n e t i c e n e r g y  case, a l s o ,  the f i r s t  A  i s c l e a r e d by d i s c o n n e c t i n g t h e  t h e m a c h i n e r o t o r s came from  Fig  n e a r bus-10 w i l l  l i n e between bus-8 and b u s - 1 0 .  stability. by  The f a u l t  o u t on t h e s y s t e m .  loses  produced  the r o t o r o f machine-4.  In t h i s  peak o f a c c e l e r a t i o n o f m a c h i n e s 1, 2 and  t h a n t h e minimum l i m i t o f a c c e l e r a t i o n .  F i g 4.21 t o  4.23 show t h e b e h a v i o r o f m a c h i n e s 4, 3 and 2 f o r s t a b l e and  u n s t a b l e c a s e s o f m a c h i n e 4.  4.4.3  The F i v e M a c h i n e System  Case S t u d i e s  Two c a s e s ' a r e c o n s i d e r e d f o r t h i s s y s t e m Case fault  1; a t h r e e - p h a s e f a u l t near bus-6.  near bus-11,  The c r i t i c a l  Case  (see F i g 3.3),  2; a t h r e e - p h a s e  machine f o r both c a s e s i s  i  i  4-Mach.systen(F i  I  I  I  I  at  65  10)  L 1 1 L Angle D i f f . Acceleration  Tiae(sec) (a) 8-1  I  i  L  _i  i  '  '  i  '  '  Angle D i f f . — —  Acceleration  u u CC  I ro.  I  —I  1  1  1 1 ! 1 1 1 1 1 0.0 0.33 D.66 10 1.33 1.66 2.0 Tine(sec) (b) Fig 4.21 Angle d i f f . and acceleration of machine 4 f o r a fault at bus 10 ; (a) stable case(Tc=.45 sec). (b) unstable case (Tc = .46 sec).  •  a.  i  4-nach.system(f i  i  i  i  at  i  10)  i  i  Angle  diff.  Acceleration  N  fN u u  i  ro  d  I  CM CO.  I  IO.  0.D  1  i  D.5  0.25  r  0.75  i  1.0  1  r  1.25  1.5  Time(sec) (a) l  to  I  I  I  I  I  L  J  Angle  L  diff.  Acceleration fN o CC  «  fN ro. I  10 •  — r 0.0  1  0.25  1  1  0.5  1  1  0.75  1  i 1.0  i  i 1.25  i 1.5  Time(sec) (b) F i g 4.22  Angle d i f f . and a c c e l e r a t i o n o f machine 2 f o r a f a u l t a t bus 10; (a)  s t a b l e (Tc=.45 s e c ) .  (b) s t a b l e (Tc=.46 s e c ) . Note that the a c c e l e r a t i o n i s lower than the min. l i m i t .  67  I  4-Mach.systeni(F '  l  l  J  '  at I  I  10)  Angle  L  diff.  Acceleration u CC  I C3 I  o to. D.O  I  0.25  0.5  i  1 0.75  r  1.0  r  i  1.5  1.25  Timelsec) J  to  1  f  I  (a)  J  I  L  J  L  Angle  diff.  Accelertion  m o u  CC  £2  rft. i  0.0  —i  1  0.25  1  1  0.5  1  1  D.75  1  1  1.0  1  l  1.25  l  1.5  Time (sec) (b) Fig  4.23 A n g l e d i f f . at  and a c c e l e r a t i o n  o f machine 2 f o r a f a u l *  bus 10 ; ( a ) a n d ( b ) a r e s t a b l e  f o r Tc=.45t.46 sec.  68 m a c h i n e 5, t h e o t h e r response  to both  loaded).  4*5  f a u l t s (because each machine i s l o c a l l y  The b e h a v i o r o f m a c h i n e 5 i s shown i n F i g 4.24.  Case S t u d i e s From  found  that  previously then  Discussion  the cases s t u d i e d , i f the f i r s t defined  f o r any f a u l t  peak o f a c c e l e r a t i o n  considered lies  maximum and minimum l e v e l s o f  t h e machine may  system  f o u r m a c h i n e s d i d n o t show any s i g n f i i c a n t  lose  synchronism  i s capable of handling  unless  t h e power  i tis  between  the  acceleration,  the r e s t o f t h e  flow a f t e r  fault  clearing. When we examine c l o s e l y t h e a c c e l e r a t i o n stable fault  c a s e s we n o t i c e clearing  i s less  that  o t h e r hand we  the  peak o f t h e a c c e l e r a t i o n  higher  than For  see the o p p o s i t e  any d i s t u r b e d i s higher  after fault  clearing  before  fault  machine i n t h e t e s t e d  than  after  peak b e f o r e c l e a r i n g .  f o r the u n s t a b l e  t h e peak o f a c c e l e r a i t o n  acceleration that  t h e peak o f t h e a c c e l e r a t i o n  than the f i r s t  the  curves f o r the  t h e minimum l i m i t ,  t h e m a c h i n e i s one o f t h e c r i t i c a l  cases,  On where  i s always clearing.  systems, then  i f the  i t indicates  machines f o r a s p e c i f i c  disturbance. The a c c e l e r a t i o n in  a power  response  s y s t e m when i t i s s u b j e c t e d  nachines are accelerating machines which a r e c l o s e and  i s d i f f e r e n t f o r each machine to a f a u l t .  and some d e c e l e r a t i n g . t o the f a u l t  Some Usually  the  l o c a t i o n are a c c e l e r a t i n g  most p r o b a b l y one o r more o f t h e s e m a c h i n e s may  lose  j  »  5-Machine  i  i  i  i  system  i  i  i  i  i  Angle D i f f . /  Acceleration  N  c u (_i u ^ = u> g". QJ  cn CT o  o  in.0.0 i  -i—r  0.26  0.53  I  T  0.8  Timelsec)  -i—i  1  1.06  1.33  r  1.6  (b) J  m.  »  L  »  »  I  i  '  CM  Angle D i f f . Acceleration  m  r—1 QJ  i-1 U  o UJ  g _ |  cu  cn c CC o LO _ (VI  LO  0.0  I  1  0.26  1  1  0.53  1  1  0.8  1  1  , 1.06  1  1  1.33  1—  1.6  Time (sec) (b) 4.24  Angle difference and acceleration of machine 5 f o r a fai at bus 11 ;  (a) stable case ( T c « . 4 7 s e c ) . (b) unstable case (Tea .48 s e c ) .  70  synchronism.  Also  when t h e f a u l t  a c c e l e r a t i n g machines d e c e l e r a t e configuration after fault a machine should machine  i s stable  is  higher  upon t h e s y s t e m  c l e a r i n g and t h e r e m a i n i n g  that before  i s converging  unstable  (depending  t h e n t h e maximum  be l e s s t h a n  an  i s c l e a r e d , most o f t h e  loads).  If  acceleration after clearing  c l e a r i n g , w h i c h means t h a t  toward a new s t a b l e o p e r a t i n g  this  point.  For  m a c h i n e , t h e peak o f t h e a c c e l e r a t i o n a f t e r c l e a r i n g  than  that before  c l e a r i n g w h i c h means t h a t  machine i s a c c e l e r a t i n g h i g h e r  this  a f t e r c l e a r i n g and d i v e r g i n g  from  stability. Sometimes c l e a r i n g t h e f a u l t disturbance  t h a t may b r e a k up t h e s y s t e m .  machines c o n t i n u e synchronism. acceleration system meeting  4.6  This  c a n be m o n i t o r e d  curves.  This  by l o o k i n g  happens o n l y  i s not capable o f absorbing  This  until  severe case the they  at their  i f the r e s t o f the  conditions.  Stability Criterion  criterion  Measurements  r e l i e s o n t h e i n s t a n t a n e o u s measurements  t h e a c c e l e r a t i o n o f t h e s y n c h r o n o u s machine d u r i n g  transient  period.  the  The i n s t a n t a n e o u s a c c e l e r a t i o n c a n be  measured by k e e p i n g monitoring  t r a c k o f t i m e and s e n s i n g  speed by  t h e p a s s a g e o f t e e t h on a t o o t h e d w h e e l b o l t e d  machine's r o t o r  lose  t h e f a u l t k i n e t i c e n e r g y and  t h e demand o f t h e new o p e r a t i n g  Transient  In t h i s  to accelerate or decelerate  B a s e d on A c c e l e r a t i o n  of  causes a second  [4].  to the  Then t h e speed and t h e a c c e l e r a t i o n can be  71 calculated  by t h e u s e o f s p e c i a l l y d e s i g n e d  V e r y good measurements have b e e n a c h i e v e d the  using  computers.  such a system by  Montana Power Company. The  this  procedure  criterion  1.  Find  indicate  f o r transient  i s outlined  (b)  serious  has o c c u r r e d ,  release  immediately a t r i p  ready  the generator  going  t o the r e l a y  acceleration  passes  t h e maximum  clearing  peak i s  nothing i s stable.  to the r e l a y t o  i f the f i r s t  peak o f t h e  limit.  a tripping signal f o r i f the f i r s t  peak o f t h e b u t does n o t  limit.  i s cleared  after clearing  ( t h i s i s c l e a r l y i n d i c a t e d by and t h e peak o f t h e  i s higher  s i g n a l must be i s s u e d  the f a u l t  signal  t h e minimum l i m i t  o f change o f a c c e l e r a t i o n )  t o be u n s t a b l e .  because  e x c e e d s t h e maximum  (but not release)  V7hen t h e f a u l t  i f the f i r s t  i . e . , t h e machine  transmission  exceed  then a t r i p  steps:  signal  t h a n t h e minimum l i m i t ,  acceleration  acceleration  using  f o r e a c h machine i n t h e power  the a c c e l e r a t i o n  disconnect  (c)  assessment  the t r i p p i n g system t o :  ignore less  rate  i n the f o l l o w i n g  the s e v e r i t y o f the f a u l t  (a)  2.  stability  t h e maximum and minimum l i m i t s o f a c c e l e r a i t o n t o  system, then a d j u s t  the  micro-  than  to trip  that  before  clearing,  the generator as i t i s  B u t i f t h e peak o f t h e a c c e l e r a t i o n  i s lower than t h a t b e f o r e  the f a u l t  after  clearing  then  the g e n e r a t o r  i s stable  and  the  trip  s i g n a l must  be  blocked. The decide  criterion  needs t h r e e i m p o r t a n t measurements  the t r a n s i e n t s t a b i l i t y :  clearing  i n d i c a t i o n , and  first  to  a c c e l e r a t i o n peak,  a c c e l e r a t i o n peak a f t e r  fault  fault  clearing. The one  second  total  f o r case  the d e c i s i o n the f a u l t  to issue a t r i p  ( c ) o f s t e p 1.  to issue  a trip  s h o u l d be o f t h e o r d e r o f  But  f o r cases  (a) and  (b)  s i g n a l must o c c u r i m m e d i a t e l y  after  occurs.  The indicate  time  r a t e o f change o f a c c e l e r a t i o n c a n be u s e d  t h e o c c u r r e n c e and  shows two  c l e a r i n g o f any  s h a r p jumps f o r t h e s e two  fault  to because  it  i n c i d e n t s , a s shown i n F i g  4.25. There faulted  a r e some s p e c i a l c a s e s where some m a c h i n e s  system  criterion  are decelerating  i s applicable  machines a r e not u s u a l l y do  instead  i n the  of accelerating.  to these cases a l s o , but u s u a l l y  the c r i t i c a l  not c o n t r i b u t e  machines f o r such  cases,  t o i n s t a b i l i t y o f the power  The those  and systems.  73  I  4-machine I  I  I  system I  I  L  Time (sec) Fig  4.25  Rate of change of acceleration of machine 4 f o r a f a u l t at bus 8, the two jerks at f a u l t and c l e a r i n g can be used to detect both events.  74 CHAPTER 5 CONCLUSION AND FUTURE WORK  Transient studied  many y e a r s  programs.  to other  machines  is  the l o c a l  The  respect  needed  and d e f i n i t e  to assess  the t r a n s i e n t  The i n f o r m a t i o n  provided  i t s transient  f o r the t e s t e d  by t h e stability  systems, j u s t  l i k e the  machines.  transient stability  s i m p l e and p r a c t i c a l  in  a c c e l e r a t i o n o f each machine c a n g i v e  a c c e l e r a t i o n o f a machine, about  a n g l e d i f f e r e n c e between  is  the i n v e s t i g a t i o n s d i s c u s s e d  o f t h a t machine.  sufficient  the t r a n s i e n t  i s i t s angle d i f f e r e n c e s with  e s s e n t i a l information  stability  i n deciding  computer  i n the system.  i s c l e a r from  Chapter 4 that  i s conventionally  i n advance by t h e use o f d i g i t a l  o f any machine  It  local  o f power s y s t e m s  The m a i n f a c t o r used  stability  the  stability  c r i t e r i o n developed  forlocally  assessing  i n Chapter 4  transient  stability. Generally summarized 1.  t h e b e n e f i t s o f t h i s new c r i t e r i o n  c a n be  as f o l l o w s :  The c r i t e r i o n stability capacitors  c a n be used  f o r automatic switching  c o n t r o l s s u c h a s dynamic b r a k e s ,  o f some  shunt  and f a s t v a l v i n g , where t h e i n s t a n t a n e o u s  acceleration provides  valuable  information  about the  s e v e r i t y o f the f a u l t . 2.  A great  deal  o f time and money c a n be s a v e d s i n c e  large  75 s y s t e m s t u d i e s need n o t be made f o r d i f f e r e n t contingencies. 3.  The c r i t e r i o n works r e g a r d l e s s o f how many d i s t u r b a n c e s a r e t a k i n g p l a c e and what c h a n g e s have o c c u r r e d i n t h e t r a n s m i s s i o n s y s t e m a t t h e same t i m e , b e c a u s e i t d e p e n d s o n l y upon l o c a l a c c e l e r a t i o n m e a s u r e d a t e a c h m a c h i n e .  4.  S i n c e no i n f o r m a t i o n i s needed f r o m o t h e r m a c h i n e s i n t h e system t o a s s e s s the s t a b i l i t y o f a p a r t i c u l a r machine, c o n t r o l o f t h e m a c h i n e s c a n be d e c e n t r a l i z e d .  5.  The c r i t e r i o n d e a l s w i t h r e a l q u a n t i t i e s and hence w i t h the r e a l s i t u a t i o n o f any machine i n t h e system transients.  during  T h i s w i l l prevent any e r r o r s i n the c h o i c e  o f t r i p p i n g . Only the unstable machine o r machines w i l l be t r i p p e d . F u t u r e Work B a s e d o n t h i s new c r i t e r i o n o f t r a n s i e n t s t a b i l i t y we c a n t h i n k o f new a u t o m a t i c p r o t e c t i o n s y s t e m s f o r l a r g e g e n e r a t o r s t o p r e v e n t i n s t a b i l i t y o f power s y s t e m s .  T h i s p r o t e c t i o n system  c a n be used t o s w i t c h o n some s t a b i l i t y c o n t r o l s t o m a i n t a i n s t a b i l i t y o r t o d i s c o n n e c t t h e u n s t a b l e g e n e r a t o r s from t h e r e s t o f t h e power  system.  F u t u r e work c o u l d i n v e s t i g a t e t h e f o l l o w i n g : 1.  D e s i g n o f ' a d e v i c e t o measure t h e i n s t a n t a n e o u s a c c e l e r a t i o n q u i c k l y and a c c u r a t e l y .  2.  Design o f a micro-computer that can process the  i n f o r m a t i o n f r o m t h e m e a s u r i n g d e v i c e and p r o v i d e t h e r i g h t d e c i s i o n i n a reasonable time about the s t a t e o f s t a b i l i t y o f the d i s t u r b e d machine. 3.  Design o f a r e l a y i n g system, w i t h b l o c k i n g a b i l i t y , t o ensure system  stability.  S u c h a p r o t e c t i o n s y s t e m w i l l go a l o n g way t o s o l v e the  t r a n s i e n t s t a b i l i t y p r o b l e m , and t o g i v e more r e l i a b l e and  c o n t i n u o u s power s e r v i c e .  77 References  1.  G l e s s , G.E. " D i r e c t method o f l y a p u n o v a p p l i e d t o t r a n s i e n t power s y s t e m s t a b i l i t y " I E E E T r a n s . PAS-85, F e b . 1966, p p . 164-179.  2.  P a i , M.A., Mohan, M.A. and Rao, J . G . "Power s y s t e m t r a n s i e n t s t a b i l i t y r e g i o n s u s i n g p o p o v ' s method" I E E E T r a n s . PAS-89, May/June 1970, p p . 788-794.  3.  Brown, R.D., McClymont, K.R. "A power swing r e l a y f o r p r e d i c t i n g g e n e r a t i o n i n s t a b i l i t y " I E E E T r a n s . PAS, March 1965, p p . 219-224.  4.  J.F. J o l l e y " D i g i t a l l y controlled generator tripping e n c h a n c e s s t a b i l i t y p e r f o r m a n c e o f Montana Power T r a n s m i s s i o n S y s t e m " Montana Power and L i g h t .  5.  C r a n y , S.B. Power S y s t e m S t a b i l i t y . V o l . I. V o l . I I . York: J o h n W i l e y & S o n s , I n c . , 1948.  6.  K i m b a r k , E.W. Power S y s t e m S t a b i l i t y . J o h n W i l e y & S o n s , I n c . , 1948.  7.  A n d e r s o n , P.M. and Fouad, A.A. Power System C o n t r o l and Stability. V o l . I . Ames, Iowa, Iowa S t a t e U n i v e r s i t y P r e s s , 1977.  8.  E l - A b i a d , A.H. and Nagappan, K. "Transient s t a b i l i t y r e g i o n s o f m u l t i m a c h i n e power s y s t e m s " I E E E . T r a n s . PAS-85, F e b . 1966, p p . 158-168.  9.  A t h a y , T., Podmore, R. and V i r m a n i , S. "A p r a c t i c a l method for d i r e c t analysis of transient s t a b i l i t y " . IEEE. Trans. PAS-98, M a r c h / A p r i l 1979, p p . 573-584.  10.  S t a n t o n , S.E., Mamandur, K.R. and K r u e m p e l , K.C. "Contingency a n a l y s i s using the t r a n s i e n t energy m a r g i n t e c h n i q u e " . P a p e r 81 SM 397-9. IEEE PES Summer M e e t i n g , P o r t l a n d , 1981.  11.  V i t t a l , V. "Power s y s t e m t r a n s i e n t s t a b i l i t y u s i n g t h e c r i t i c a l e n e r g y o f i n d i v i d u a l m a c h i n e s " , Ph.D. T h e s i s , Iowa S t a t e U n i v e r s i t y , Ames, Iowa, 1982.  12.  F o u a d , A o A . , V i t t a l , V. "Power s y s t e m r e s p o n s e t o a l a r g e d i s t u r b a n c e energy a s s o c i a t e d w i t h system s e p a r a t i o n " IEEE PES Summer M e e t i n g , 1983.  V o l . I.  New  New  York:  Fouad,A.A.,  78 13.  Uemosono, M a t s u k i , Okada, " A n a l y s i s o f the s t e p - o u t p r o c e s s of a t h r e e phase synchronous machine by a i r gap f l u x " J I E E B100-1, 1980, J a p a n .  14.  Uemosono, Okada, M a t s u k i , Yamada, Yokokawa, M o r i y a s u , "Development and t e s t i n g o f an a u t o m a t i c s t a b i l i t y p r e d i c t i o n and c o n t r o l (ASPAC) f o r a synchronous g e n e r a t o r by a i r gap f l u x " , IEEE T r a n s . PAS. Paper 81SM 462-1, J u l y 1981.  15.  IEEE. Committee R e p o r t "A d e s c r i p t i o n o f d i s c r e t e supplementary c o n t r o l s f o r s t a b i l i t y " , IEEE. T r a n s . PAS-97, Jan/Feb 1978, pp. 149-165.  16.  Kimbark, E.W. "Improvement o f power system s t a b i l i t y by changes i n network", IEEE T r a n s . , V o l . PAS-88, May 1969, pp. 773-781.  17.  S h e l t o n , M i t t e s l s l a d t , Winkleman, B e l l e r b y " B o n n e v i l l e power a d m i n s t r a t i o n 1400 MW b r a k i n g r e s i s t o r " , IEEE. T r a n s . V o l . PAS-94, March 1975, pp. 602-611.  18.  Westinghouse, E l e c t r i c a l T r a n s m i s s i o n and R e f e r e n c e Book, U.S.A.  19.  C o g s w e l l , S.S. and o t h e r s " G e n e r a t o r s h a f t t o r q u e s r e s u l t i n g from o p e r a t i o n o f EHV b r e a k e r s " , IEEE Paper C-74-087-3, W i n t e r Power M e e t i n g , N.Y., J a n . 1974.  20.  A b o l i n s , A. and o t h e r s " E f f e c t o f c l e a r i n g s h o r t c i r c u i t s and a u t o m a t i c r e c l o s i n g on t o r s i o n a l and l i f e e x p e n d i t u r e s of t u r b i n e - g e n e r a t o r s h a f t s " , IEEE T r a n s . , V o l . PAS-95, Jan. 1976, pp. 14-25.  21.  Kimbark, E.W. "Improvement o f power system s t a b i l i t y by s w i t c h e d s e r i e s c a p a c i t o r s " , IEEE. T r a n s . V o l . PAS-85, Feb. 1966, pp. 180-188.  22.  IEEE. Committee R e p o r t , "A b i b l i o g r a p h y f o r the s t u d y o f subsynchronous resonance between r o t a t i n g machines and power systems" IEEE T r a n s . , V o l . PAS-95, J a n . 1976, pp. 216-218.  23.  C u s h i n g , E.W. and o t h e r s " F a s t v a l v i n g as an a i d t o power system t r a n s i e n t s t a b i l i t y and prompt r e s y n c h r o n i z i n g and r a p i d r e l o a d a f t e r f u l l l o a d r e j e c t i o n " , IEEE. T r a n s . V o l . PAS-91, Aug.' 1972, pp. 1624-1636.  24.  P a r k , R.H. " F a s t t u r b i n g v a l v i n g " , IEEE T r a n s . , V o l . PAS-92, June 1973, pp. 1065-1073.  Distribution  79 25.  Kimbark, E.W., " B i b l i o g r a p h y on s i n g l e p o l e s w i t c h i n g " , IEEE. T r a n s . V o l . PAS-94, May 1 9 7 5 , pp. 1072-1078.  26.  H o l l a n d , D.R. and o t h e r s "Conemaugh p r o j e c t : new c o n c e p t f o r 500 KV system p r o t e c t i o n " , IEEE. T r a n s . V o l . PAS-90, J a n . 1971, pp. 1-10.  

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