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Excitation control of torsional oscillations due to subsynchronous resonance Yan, Andrew 1979

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EXCITATION CONTROL OF TORSIONAL OSCILLATIONS DUE TO SUBSYNCHRONOUS RESONANCE  by  Andrew {^<a  n  B.S.E.E. U n i v e r s i t y o f Texas a t A r l i n g t o n , 1977  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES i n the Department of Electrical  We accept  Engineering  t h i s t h e s i s as conforming t o the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1979 0  Andrew Yan, 1979  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives.  It is understood that copying or publication  of this thesis for financial gain shall not be allowed without my written permission.  ( Andrew Yan )  Department of  Electrical  Engineering  The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5  n.tP ? 9A  3E-6  B P  75-5  1  I E  r11 1979  ABSTRACT  Subsynchronous with  resonance  series-capacitor-compensated  phenomena . i n a p o w e r  system  t r a n s m i s s i o n l i n e s may  cause  damaging t o r s i o n a l o s c i l l a t i o n s i n the erator.  In t h i s  resonance of of  studies  is  the  on t h e  Dynamic performance  synthesized  simultaneously within  transient  capacitor  the  from the  mode  ii  system's  excitation  subsynchronous control  but  is  output  controller  oscillations  range of  compensation,  fault.  for  gen-  The e x c i t a t i o n c o n t r o l  t e s t s of the  n o n l i n e a r m o d e l show t h a t a l l  a wide range of  turbine  d e r i v e d . An e x c i t a t i o n p r o c e d u r e  l i n e a r optimal type  stabilized  of the  a h i g h - o r d e r power s y s t e m model f o r  t o r s i o n a l o s c i l l a t i o n s i s presented.  signals.  for  thesis  shafts  can  be  dynamic s t a b i l i t y not  for a  severe  TABLE OF CONTENTS Page i i  ABSTRACT T A B L E OF CONTENTS  i i i  L I S T OF TABLES LIST  '  OF ILLUSTRATIONS  viii  NOMENCLATURE  2.  3.  4.  5.  v  i  v  ACKNOWLEDGMENT  1.  ,.  ix  INTRODUCTION  1  1.1  Subsynchronous  R e s o n a n c e P r o b l e m s and C o r r e c t i v e M e a s u r e s  1.2  R e c e n t Work o n E x c i t a t i o n  1.3  Scope o f t h e  . . .  ±  C o n t r o l o f SSR  2  Thesis  3  MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE S T U D I E S .  5  . . .  5  2.1  Introduction  2.2  M o d e l l i n g the  M e c h a n i c a l System  2.3  Modelling  Electrical  2.4  L i n e a r i z e d Model  17  2.5  Reduced Order Model  18  2.6  Eigenvalue Analysis  18  the  6  System  10  L I N E A R OPTIMAL E X C I T A T I O N CONTROL DESIGN 3.1  Introduction.  .  3.2  Linear State Regulator Problem  3.3  S o l u t i o n to  3.4  W i d e R a n g e SSR S t a b i l i z e r a n d t h e  2  4  24  ;  the M a t r i x R i c c a t i  24 Equation Choice of Weighting M a t r i c e s .  EFFECT OF E X C I T A T I O N CONTROLLER ON STEADY STATE AND TRANSIENT 4.1  Introduction  4.2  Controller for  4.3  Controller for Transient  2  SSR. .  6  27 32 32  Steady  S t a t e SSR SSR  32 44 56  CONCLUSION i i i  Page REFERENCE  57  APPENDIX I  59  APPENDIX I I  60  iv  LIST OF TABLES  Table 2.1  2.2  2.3  2.4  3.1  3.2  3.3  Page E i g e n v a l u e s o f v a r i o u s o r d e r SSR model at 30% c a p a c i t o r compensation f o r P=0.9 p.u. at 0.9 power f a c t o r l a g g i n g . . . . .  20  E i g e n v a l u e s o f v a r i o u s o r d e r SSR model a t 50% c a p a c i t o r compensation f o r P=0.9 p.u. a t 0.9 power f a c t o r l a g g i n g . . . . .  21  E i g e n v a l u e s o f v a r i o u s o r d e r SSR model at 70% c a p a c i t o r compensation f o r P=0.9 p.u. a t 0.9 power f a c t o r l a g g i n g . . . . .  22  E i g e n v a l u e s o f v a r i o u s o r d e r SSR model at 80% c a p a c i t o r compensation f o r P=0.9 p.u. a t 0.9 power f a c t o r l a g g i n g . . . . .  23  E i g e n v a l u e s o f t h e system . w i t h l i n e a r o p t i m a l c o n t r o l at v a r i o u s degrees o f c a p a c i t o r compensation f o r P=0.9 p.u. power f a c t o r = 0.9 l a g g i n g  29  Shaft modes- o f the system w i t h and. without l i n e a r o p t i m a l c o n t r o l a t v a r i o u s degrees o f c a p a c i t o r compensation f o r P= 1.25 p.u. a t 0.9 power f a c t o r l a g g i n g  30  Shaft modes o f the system w i t h and without l i n e a r o p t i m a l c o n t r o l at v a r i o u s degrees o f c a p a c i t o r compensation f o r P=0.5 p.u. a t 0.9 power f a c t o r l e a d i n g • • •  31  v  L I S T OF ILLUSTRATIONS  Figure  -  Page  2.1  F u n c t i o n a l b l o c k d i a g r a m o f a power s y s t e m f o r  2.2  Model of the  2.3  The g e n e r a t i n g  2.4  M o d e l l i n g of the mass-spring i  steam t u r b i n e  SSR s t u d y .  .  .  system  unit mass-spring  5 6  system  8  system i n the v i c i n i t y  of  the  r o t a t i o n a l mass  8  2.5  The s p e e d  2.6  The t r a n s m i s s i o n s y s t e m  11  2.7  Voltage regulator  12  2.8  A six-winding  2.9  Equivalent c i r c u i t of a five-winding s i x - w i n d i n g model  2.10  governor model  10  and e x c i t e r model  generator  model  14  Armature current responses of the w i n d i n g generator model Q - a x i s damper  current  2.12  Q - a x i s damper  currents  4.1  Dynamic r e s p o n s e s o f the system w i t h o u t s u b j e c t e d t o 10% l o a d c h a n g e  4.3  4.4  4.5  4.6  4.7  of the  14  five-winding  generator  model  .  .  16  six-winding generator  model  .  .  16  c o n t r o l when  Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h subjected to a pulse torque disturbance  33  Electrical resonance  c o n t r o l when  network f o r the  39  generator-exciter  simulation of  35  c o n t r o l when  of the g e n e r a t o r - e x c i t e r control)  T o r s i o n a l o s c i l l a t i o n s of the compensation (with c o n t r o l )  five15  Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t subjected to a p u l s e torque d i s t u r b a n c e  Torsional oscillation compensation (without  model from a  s i x - w i n d i n g and the  2.11  4.2  o f the  generator  shaft  at  80% 43  shaft  at  80% 43  subsynchronous 44  Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t c o n t r o l when s u j e c t e d t o a t h r e e - p h a s e f a u l t a t t h e r e m o t e e n d (X = 0 . 0 1 p . u . )  46  ii  4.8  D y n a m i c r e s p o n s e s o f t h e p o w e r s y s t e m w i t h c o n t r o l when s u b j e c t e d t o a t h r e e - p h a s e f a u l t a t t h e r e m o t e end ( X g = 0 . 0 1 p.u.) vi  50  Figure 4.9  4.10  Page Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t c o n t r o l when s u b j e c t e d t o a t h r e e - p h a s e f a u l t ( X = 0 . 0 6 p . u . )  54  D y n a m i c r e s p o n s e o f t h e p o w e r s y s t e m w i t h c o n t r o l when s u b j e c t e d t o a t h r e e p h a s e f a u l t (X = 0 . 0 6 p . u . )  55  vii  ACKNOWLEDGEMENT  I would l i k e gratitude project,  t o e x p r e s s my m o s t g r a t e f u l t h a n k s  and d e e p e s t  t o D r . Y a o - n a n Yu and D r . M . D . Wvong, s u p e r v i s o r s o f for their  continued i n t e r e s t ,  the r e s e a r c h work and w r i t i n g o f t h i s deep a p p r e c i a t i o n t o M r . E l - S h a r k a w i The  financial  support  encouragement thesis.  and g u i d a n c e d u r i n g  I a l s o w i s h t o e x p r e s s my  for his interest  of the N a t u r a l  and t i m e l y a d v i c e .  S c i e n c e s and E n g i n e e r i n g  R e s e a r c h C o u n c i l o f Canada and t h e U n i v e r s i t y o f B r i t i s h gratefully  this  Columbia  is  acknowledged. I  am g r a t e f u l t o my p a r e n t s ,  F r a n k Wong f o r t h e i r  encouragment  members o f my f a m i l y ,  and M r .  t h r o u g h o u t my u n i v e r s i t y c a r e e r .  viii  NOMENCLATURE General A  system  B  control matrix  u  control vector  x  s t a t e v e c t o r o f the unmeasurable model  y  s t a t e v e c t o r o f the measurable model  H  transformation matrix  Q  symmetric p o s i t i v e s e m i - d e f i n i t e w e i g h t i n g m a t r i x  R  symmetric p o s i t i v e d e f i n i t e w e i g h t i n g m a t r i x  K  R i c c a t i matrix  G  c l o s e d loop system  M  composite  A,X  eigenvalue matrix,eigenvector matrix of M  o  subscript-denoting i n i t i a l  x  time d e r i v a t i v e o f x  t  s u p e r s c r i p t denoting  -1  s u p e r s c r i p t denoting inverse  s  differential  A  p r e f i x denoting a l i n e a r i z e d  j  complex o p e r a t o r , S-T  Mass-spring  matrix  matrix  m a t r i x as d e f i n e d i n (3.14)  condition  transpose  operator  system  M  inertia  coefficient  K  shaft s t i f f n e s s  D  damping  0  rotor  angle  to  rotor  speed  CJ^  synchronous  constant  speed ix  variable  Synchronous  Machine  i  instantaneous  value of  current  V  instantaneous  value of  voltage  T  flux-linkage  R  resistance  X  reactance  6  torque  angle  electric  torque  terminal  current  terminal  voltage  generator  output  d,q  subscript  d e n o t i n g d i r e c t - and q u a d r a t u r e - a x i s  f  subscript  denoting f i e l d  D,Q,S  subscript  d e n o t i n g d i r e c t - and q u a d r a t u r e - a x i s  &,L  subscript  denoting leakage  c  subscript  denoting quantities  a  subscript  denoting armature  i V jQ^  Transmission  power  circuit  impedance  associate  phase  with  transformer  X ,R e' e  reactance  and r e s i s t a n c e  of the  transmission  X  reactance  of  infinite  bus  capacitor voltage  and V o l t a g e R e g u l a t o r regulator  gain  T.  regulator  time  T  exciter  V  L ref  time  reference  constant  constant  voltage  • x  stator  capacitor  Line of the  Excitor  quantities  quanities  and r e s i s t a n c e  o  damper  i n damper a n d  reactance  V  quantities  quantities  X ,R  c  stator  line  G o v e r n o r and Steam T u r b i n e System K  actuator  gain  T^,T2  actuator  time  T^  servomotor  a  actuator  signal  power a t  gate  g  P  constant  time  constant  outlet  VJV  T T  steam chest  LH  reheater  Kri  T  time  time  constant  cross-over time  F  high pressure  constant  constant  t u r b i n e power  fraction  rlr  F^.p  intermediate  pressure  t u r b i n e power  fraction  LPA  low pressure  t u r b i n e A power  fraction  F^pg  low pressure  turbine  fraction  T^p  high pressure  T^p  intermediate  pressure  ^  turbine  F  T  LPA' LPB T  O W  P  r  e  s  s  u  r  e  turbine  B power torque turbine torque  xi  torque  1  1.  1.1  INTRODUCTION  Subsynchronous Resonance P r o b l e m s and C o r r e c t i v e Measures The  generation the  right  i n c r e a s i n g e l e c t r i c power demand,  sites  at  heavy l o a d  f r e q u e n t l y made l o n g  transmission l i n e s necessary series  capacitors  system resonance  oscillation  for  "Subsynchronous Resonance encompassing the  the  capacitor-compensated  D e s p i t e the favor  the use o f s e r i e s  as a n a l t e r n a t i v e ment.  seriously considered u n t i l station  mechanical  the  the  general  phenomemon  series-  system.  p o t e n t i a l l y d a m a g i n g SSR e f f e c t , to increase  the  utilities  power t r a n s f e r  t o overcome the  p r o b l e m s due t o  two s h a f t  as  torsional interaction,  i n d u c t i o n generator  effect,  shaft  term  still capability  t o a d d i t i o n a l t r a n s m i s s i o n l i n e s a n d more c a p i t a l  Hence, i n order  of  two  [ 1 ] . The  when c o u p l e d t o a  h a s b e e n made i n a n a l y z i n g t h e  [ 2  interaction  o f e l e c t r i c a l and m e c h a n i c a l v a r i a b -  turbine-generators  capacitors  of  The h a z a r d s  after  i n 1970 a n d 1 9 7 1  (SSR)" i s used to d e s i g n a t e  transmission  result  mass-spring system.  oscillatory attributes  associated with  the  e l e c t r i c a l s y s t e m and the  turbine-generator  use  c o m p e n s a t i o n may c a u s e e l e c t r i c a l  These o s c i l l a t i o n s a r e  a t Mohave g e n e r a t i n g  environmental  t r a n s f e r r i n g b u l k power. However, the  phenomena i n t h e  of the  i n obtaining  series-capacitor-compensated  transmission line  t h e s e o s c i l l a t i o n s were not  les  for  and damage.  between resonance  failures  and t h e d i f f i c u l t i e s  o f way t o b u i l d new t r a n s m i s s i o n l i n e s due t o  c o n s i d e r a t i o n s have  of  centres,  the u n a v a i l a b i l i t y of  failures.  invest-  SSR, e x t e n s i v e  Problems are  effort  identified  and t r a n s i e n t  torques  ]. SSR phenomenon may o c c u r i n two d i f f e r e n t  steady  state  oscillations  SSR a n d t r a n s i e n t that  are  either  forms: s e l f - e x c i t e d or  S S R . S e l f - e x c i t e d SSR i n v o l v e s  sustained  or increased  spontaneous  i n magnitude w i t h  time.  2  Two m e c h a n i s m s effect  that  produce these o s c i l l a t i o n s are  where the n a g a t i v e  frequencies bilateral  exceeds  the  resistance  resistance  c o u p l i n g known a s  o f the  of the  generator  turbine-generator  to reduce the  i n the  These torques  the  transient result  a n a l y z i n g and i d e n t i f y i n g  have been proposed  [ 2 ], e.g.  torques  mechanical the  induction generator currents  at  effect  critical  on segments  from subsynchronous  the  SSR p r o b l e m s ,  ; static  and f i n a l l y , i n case  a subsynchronous of sustained  p r o p o s a l s have a l r e a d y been put  R e c e n t Work on E x c i t a t i o n  corrective  amortisseur  windings to  block  ; supplementary  excit-  a mode f r e q u e n c y  s h a r p l y reduced  a better  o f ambient  alternative  filter,  in the  o s c i l l a t i o n s . Most o f  the genthese  practice.  s y s t e m , when d e t u n e d ,  i n system frequencies  shortcomings of s t a t i c  torques  r e l a y to p r o t e c t  tuned h i g h - Q b l o c k i n g  the  and c h a n g e s  transient  gap  C o n t r o l o f SSR  adequate damping to  Detuning occurs because  overcurrent  subsynchronous  into  Although a p e r f e c t l y  is  of the  the  operations.  high-Q f i l t e r s  frequencies  of  oscillation  or switching  , additional  scheme t o m i n i m i z e t h e m a g n i t u d e  erating unit  1.2  the  c o n t r o l to p r o v i d e a d d i t i o n a l system damping; a c a p a c i t o r d u a l  flashing shaft;  the  and  o s c i l l a t i o n mode o f  e l e c t r i c a l network caused by f a u l t s  subsynchronous  ation  subsynchronous  t o r s i o n a l i n t e r a c t i o n between  SSR i n v o l v e s t h e  shaft.  After measures  at  generator  network. Transient  currents  induction  transmission system;  modes o f t h e m a s s - s p r i n g s y s t e m a n d t h e n a t u r a l electrical  the  filter  can p r o v i d e  the high-Q f i l t e r  r e s u l t i n g i n reduced damping temperature  resistance  performance.  variation, capacitor  failure,  during swing c o n d i t i o n s . In view of  the  s u p p l e m e n t a l e x c i t a t i o n c o n t r o l may p r o v i d e  to suppress  the  steady  state  SSR p h e n o m o n .  at  3  Saito is  an e x c i t a t i o n  et.al.  controller  Hamdam a n d Hughes o r r e a l power  found  a N e g a t i v e Damping S t a b i l i z e r w h i c h  using reactive  [ 4 ] proposed  feedback. is  [ 3 ] proposed  p o w e r as  a similar  type  In examining t h e i r that  their  aim i s  the  controller  approach  to  control signal  controls,  it  phenomena  w h e r e a s SSR i s much more c o m p l i c a t e d due  using  i n designing  suppress the to  reactive these  electrical  the  ;  resonance  electromechanical  interaction. To a c c o u n t electromechanical  for  deviation w i l l  Wvong and T s e  a controller  be produced  [ 6 ] proposed  measurable s t a t e  electromechanical  system model s h o u l d be used  and Khu [ 5 ] p r o p o s e d speed  the  feedback  as  terminal voltage  excitation voltage the  optimistic  1.3  Scope o f t h e  type  t e s t e d on b o t h  order  power is  Chapter  s i g n a l where  the  pick-up  controlled. Yu,  the  El-Serafi  control signal.  c o n t r o l l e r by u s i n g  signals. limits  In the  are  on t h e  analog  neglected. non-linear  and v o l t a g e  ceiling  Fouad  using  and S h a l t o u t  filtered  and  [ 7 ]  phase  s t u d i e s by Fouad and K h u ,  In the  last  two p a p e r s  power s y s t e m .  Hence,  l i m i t s may h a v e l e d  [6,7],  the to:.  .  Thesis  this  thesis,  linear  an e x c i t a t i o n  at  and n o n l i n e a r  system model i s  reduced  system  and d e s i g n .  optimal controller  using measurable s t a t e v a r i a b l e s  linear  unified  a wide range l i n e a r  verified  testing  u s i n g speed  a  c o n c l u s i o n s ' a b o u t e x c i t a t i o n " c o n t r o l o f " SSR..  In  the  ceiling  r e s u l t s were not  lack of nonlinear  as  in analysis  s o l e l y b y t h e mode t o b e  suggested a m u l t i - l o o p e x c i t a t i o n shifted  interaction,  to  various  for  the  the  optimal and  m o d e l s ,[21], I n C h a p t e r  2, a u n i f i e d  non-  the model i s  controller  linear  feedback  linear  be d e v e l o p e d  degrees of c a p a c i t o r  3, a wide-capacitor-range  of the will  developed;  facilitate  controller  design;  then  l i n e a r i z e d and  and the  compensation  are  optimal controller  eigenvalues  studied. is  its of  In  designed  and  4  its  effectiveness  controller  is  is verified  further  t e s t e d on t h e  developed i n Chapter 2. out  5.  Finally,  In Chapter 4,  the  o r i g i n a l n o n l i n e a r power s y s t e m model  The d y n a m i c r e s p o n s e s o f t h e  c o n t r o l are o b t a i n e d .  given i n Chapter  by e i g e n v a l u e a n a l y s i s .  s y s t e m w i t h and  a summary o f i m p o r t a n t  findings  is  with-  5 2 . MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE STUDIES  2.1  Introduction The c o m p o n e n t r e p r e s e n t a t i o n  resonance  studies  is  quite different  stability  study.  In order  o f a power s y s t e m f o r  from t h a t o f a c o n v e n t i o n a l power  to account  f o r the  t o r s i o n a l resonance  m e c h a n i c a l m a s s - s p r i n g s y s t e m due t o S S R , f a c t o r s in  conventional s t a b i l i t y studies  and t u r b i n e parameters  shaft  I n the  torsional stiffnesses  f i r s t part  mathematical model i s mass-spring system,  governor,  capacitor-compensated model i s  model f o r  U  system's  further  chapter,  steam t u r b i n e  In the  eigenvalues  investigation is  Figure 2.1  P  Q V e, e, t  o f the  torques,  neglected  generator  amortisseur  order  generator  generator,  the  l i n e a r i z e d about examined.  turbine  exciter,  system  the  a nominal operating  Finally  a reduced  Steam Turbine  Governor"  fD  Generator Electromagne t i c Dynamics  and  equations  obtained.  J  the  A f u n c t i o n a l block diagram of  E x c i t e r and Voltage Regulator  Transmission System  a detailed high  second p a r t ,  are  rotor  system  transients.  The m o d e l c o n s i s t s  the h i g h order model are  and t h e  and n e t w o r k  of  usually  namely the  [8], generator  transmission line.  shown i n F i g u r e 2 . 1 .  describing point  of t h i s  derived.  that are  a r e now i m p o r t a n t :  [9.], a r m a t u r e t r a n s i e n t s  subsynchronous  m Mass-spring System  F u n c t i o n a l b l o c k d i a g r a m o f a power f o r SSR s t u d i e s  system  order  6  2.2 M o d e l l i n g the Mechanical Steam t u r b i n e  System  :  The s t e a m t u r b i n e  system model i s  shown i n F i g u r e 2 . 2  . It  t a n d e m c o m p o u n d , s i n g l e r e h e a t s y s t e m . The m o d e l i s m a i n l y b a s e d on t h e committee r e p o r t  is  a  IEEE  [ 10 ] .  1 + sT CH  1 + sT RH  1 + sT CO  HP  T Figure 2.2  IP  T  HP  Model of the  The c o r r e s p o n d i n g e q u a t i o n s  f  steam t u r b i n e  LPA  LPB  T LPB  system  are  CH  G  V  T  CH  H  P  ( 2.1  )  ( 2.2  )  ( 2.3  )  ( 2.4  )  IP  "IP F  LPA  T  IP  HP T  LPA  HP  F  T  RH  H  LPA T I P CO  P  IP  T  RH  co  T  IP  LPA  LPB  LPB F  LPA  L  P  A  7  G e n e r a t i n g U n i t Mass-spring The  System :  generating unit  t o r s i o n a l system c o n s i s t s o f one h i g h  p r e s s u r e t u r b i n e (HP), one i n t e r m e d i a t e p r e s s u r e t u r b i n e ( I P ) , two low p r e s s u r e t u r b i n e s (LPA,LPB), one generator are a l l m e c h a n i c a l l y  coupled  ( G ) , and one e x c i t e r  (EX) which  t o g e t h e r by s h a f t s as shown i n F i g u r e 2.3 .  To make the a n a l y s i s s i m p l e , the f o l l o w i n g assumptions a r e made : (a)  Each mass-spring  element has a lumped mass o f i n e r t i a constant  M.  (b)  The mass o f the s h a f t between any two elements i s n e g l i g i b l e and behaves l i k e a l i n e a r t o r s i o n a l s p r i n g .  (c)  Only mechanical  damping i s considered.'  F i g u r e 2.4 i l l u s t r a t e s by t h e i * " *  1  the v a r i o u s t o r s i o n a l f o r c e s  experienced  element i n the system - a p o s i t i v e t o r s i o n a l torque  0. ) on the l e f t , a n e g a t i v e torque 1 '  -K.,., .( l+l, x  0. x  ±+±('®±+±  0. , ) on the r i g h t , x-1 '  an e x t e r n a l torque T_^ ,a p o s i t i v e a c c e l e r a t i n g torque M^io^ , and a n e g a t i v e damping torque  ~®± ± . A g e n e r a l e q u a t i o n o f motion o f the i <ii  ^ r o t o r i s as  follows M.u>. = T. - D.u. + K. ... ( 6... - 8. ) - K . X X  X  X X  1,1+1  V  1+1  X  ( 6 . - 0 )  1,1-1  X  (2.5)  x-1  where M.  :  i n e r t i a constant o f the i * " ^ r o t o r  6^  :  r o t a t i o n displacement  x  K. . ,.. :  o f the i  th  rotor  t o r s i o n a l s t i f f n e s s constant o f t h e s h a f t between i * " * and 1  x .1+1  ...th l+l rotor  Figure  2.4  M o d e l l i n g of the mass-spring system i n v i c i n i t y o f the i r o t a t i o n a l mass  the  9  Applying equation differential  (2.5)  t o t h e s i x mass t o r s i o n a l  system,  twelve  equations areobtained :  High pressure turbine  L  =  K  12  M  K  a  S7~  2  1  12  e  °1 i " MT  ^ HP i M7" T  w  (2.6)  (2.7) Intermediate pressure turbine  u)„  2  =  K  12  M  1  2  <  ^  *  23 + M,  2  D  2  ^  ^ u  2  +  s  T IP  r  (2.8)  (2.9) Low p r e s s u r e turbine A  (D„  =  23 M  2 3 + 34  K  K  K  34  D  3  ,  T LPA  (2.10)  2  3  (2.11) Low p r e s s u r e turbine B  to.  =  34  K r  4  Generator  OJ.  M  Exciter  w  4  " %  = 0), ( 0)  OJ,  =  K  K M  +  56  D M  M.  56  K 6  W  -  6 "  5  T _ _e  5 "  "  c  M  (2.16) 6  M,  u  o  M,  6  6 (2.17)  }  i nelectrical  r a d i a n which i s  r a d i a n f o r a two p o l e m a c h i n e .  til  03- : s y n c h r o n o u s q  speed  o), : b a s e s p e e d w h i c h b : mechanical  (2.14)  5  56  angular displacement  of the i  (2.12)  4  (2.15)  equal, t o the mechanical  : electric  4  D  6 : electrical  6  T LPB  4  (2.13)  , 45 + 56, . ( ) 6  4  6  D  }  K  5  V  O K : speed  45  4  6  where  K  M,  45  =  K  3  M.  V  3 4 + 45.  rotor i nper unit. which  i sequal to 1 per u n i t .  i s e q u a l t o 377  angular displacement  radian/second, i n radian.  t o r q u e a c r o s s t h e a i r gap i n p e r u n i t ,  10  Speed G o v e r n o r : The s p e e d committee r e p o r t  [10].  combines w i t h  the  P  to  ,,, s u b j e c t  g o v e r n o r m o d e l shown i n F i g u r e 2 . 5  increments the  power P  The i n i t i a l  due t o s p e e d  is  are  P  P  <  P  GV . mm  2.3 M o d e l l i n g the E l e c t r i c a l  <  GV -  GV  transformer  the  servomotor  which total  power,  mechanism.  max  System  System:  t o an i n f i n i t e bus as  across  load reference  IEEE  governor  The t r a n s m i s s i o n s y s t e m i s r e p r e s e n t e d  and r e a c t a n c e  on an  d e v i a t i o n to o b t a i n the  The s p e e d  The c o r r e s p o n d i n g s t a t e e q u a t i o n s  the  the  t i m e l a g , To, i n t r o d u c e d by the  Figure 2.5  Transmission  q  i s based  are  of the  capacitor  shown i n F i g u r e 2 . 6 .  represented  by R  is V  c  , whereas  The r e s i s t a n c e  and X  t r a n s m i s s i o n l i n e are V  by a s i n g l e l i n e  t  respectively.  denoted  by R  represents  the  connected  and r e a c t a n c e The  and X .  of  resistance Voltage  terminal voltage  at  11  the c a p a c i t o r .  f" Gen } - 4 - ^ - A / " \ A ^ ~ R  t  X  t  \  -/yVV  ct  F i g u r e 2.6  The t r a n s m i s s i o n system  The g e n e r a l v o l t a g e e q u a t i o n  [ v ] , _  r  -  t phase  1  J  R 1 [ I  L  J  1, t phase  1  J  o All  L  + [ L ] dt J  t  L  [ I ] phase  [ T ]  Since balance  cos(9 - 120)  cos (9 +  120)  -sine  - s i n ( 9 - 120)  -sin(6 +  120)  1  =  q  =  + X . to  (  £  + V  V  i n t o Park's  (2.21)  1  coordinate  o f the t r a n s m i s s i o n l i n e a r e :  voltage X  V, d  ], phase  o p e r a t i o n i s assumed, the o-component o f the d-q-o  z e r o . Hence the equations  Terminal  c  matrix  cos©  1  + f V  (2.20)  [ 11 ] by a t r a n s f o r m a t i o n  =  '  phase  the q u a n t i t i e s i n phase c o o r d i n a t e are transformed  coordinate  is  f o r the system shown i n F i g u r e 2.6 i s :  (  X  + V  o t  o  ) I, d  - ( X_ + X ) I t e q  +  ( R  t  + R  e  ) I, + V d cd (2.22)  sinfi  +  X  e ) I  cos6  q  + ( X  t  +X  e  ) I j + ( R +R ) I d t e q 7  +V  cq  (2.23)  12  Capacitor  voltage V  cd  =  V cq  +  a) X I , c d  (2.24)  -£SL_ = _•V ,  +  co X I c q  (2.25)  cd  V o l t a g e R e g u l a t o r and The It  Exciter  e x c i t e r and v o l t a g e r e g u l a t o r model i s  i s a c o n t i n u o u s a c t i n g t y p e w h i c h i s based on an IEEE committee  [ 1 2 ] w i t h some s i m p l i f i c a t i o n s the  shown i n F i g u r e  saturation,  and the  the  stabilizing  regulator feedback  input  loop are  filter  time  2.7. report  constant,  neglected.  max J  ref  1 + sT  R  F i g u r e 2.7  where  min  Voltage regulator  the  K  the v o l t a g e r e g u l a t o r  gain  T^ i s  the v o l t a g e  time  T„ i s E  the  is  V. R  is  D  min  supplementary  and e x c i t e r m o d e l  U is E  E^  The  fD  regulator  e x c i t e r time  the  control signal  constant  output v o l t a g e of the  and V. R  are  constant  the  regulator  exciter ceiling  voltage  max  corresponding equations  are:  A —— ( V I, ref A  - V t  K  V  R  =  1  7  Kmin —  "R  —  + U ) V  "R max  -  (2.26)  13  E  f  V  D  t  =  -Yl\  =  / V  The s y n c h r o n o u s  '  -T-hv  +  2  d  V q  •  2  as  shown i n F i g u r e 2 . 8  permanently  short  w i n d i n g s on t h e  (2.28)  , a s i x - w i n d i n g synchronous  , i s assumed.  c i r c u i t e d , the  Since the  question i s  Q - a x i s be r e p r e s e n t e d  : should the  two  shown i n F i g u r e 2 . 9  model i s  The d y n a m i c r e s p o n s e s  F i g u r e 2.12 the of the  same, the  to.  . It  and i  the  is  rest  o f the  found that  of the  0  of the  the  system are armature  representation  is  V  =  q  of the  f. d  -  uf  ¥ q  +  V  =  ¥  -  0  -  %  "  of the  simply the  two m o d e l s  sum o f i  e l e c t r i c torque  same.  to  Q  are  and  i  b  p r o d u c e d by  Hence, a f i v e - w i n d i n g  model  The e l e c t r i c t o r q u e =  circuit  :  ml' d  -  R i a q  (2.30)  R  i  ; R  (2.31)  D S>  V -  q  are  (2.29)  q  ( 2  (2 Q  equation i n per unit  i d  rotor R i a d  Q  e  the  shown i n F i g u r e 2 . 1 0  models  -  o - * T  as  sufficient.  The v o l t a g e e q u a t i o n s V, d  gap i s  impedance,  derived.  currents  f i v e w i n d i n g model i s  the a i r  ? By  s i x - w i n d i n g and f i v e - w i n d i n g  s i x - w i n d i n g model. In other words, the  two m o d e l s a c r o s s  are  damper  b y one e q u i v a l e n t w i n d i n g two Q-damper l e a k a g e  , a f i v e winding-  generator  damper w i n d i n g s  t a k i n g the p a r a l l e l e q u i v a l e n t of the  when c o n n e c t e d  2 7 )  generator  I n p r e v i o u s w o r k [13] model,  '  ( 2  y  i , q  d  '  3 2 )  "  33)  is (2.34)  q-axis  D d-axis  d  f  ( T O  Figure  2.8  A six-winding  generator  model  d-axis  X  • qL  r  i i i  R • a  T  -AA/V  j  V  Si  X  mq  q-axis  ii i Figure  2.9  Equivalent c i r c u i t of a five-winding generator model from a s i x - w i n d i n g model ( s o l i d l i n e damper l i n k a g e impedance o f t h e five-winding model; d o t t e d l i n e - d a m p e r leakage impedances of  the  six-winding  model)  15  Figure  2.10  Armature current response and f i v e - w i n d i n g g e n e r a t o r  o f the model  six-winding  16  Figure  2.12  Q - a x i s damper c u r r e n t s g e n e r a t o r model  o f the  six-winding  17  The f l u x  linkage equation  ' -  is  d  x  X  -X  q =  md  X  -X  X  A  ±  md X  X  md  X  D  f  X  J  2.4 L i n e a r i z e d  md  q  md X  1Q  :  md  i  mq  X  mq  Q  . V  .  Model  imposed on t h e quadratic  voltage  regulator  terms o f the  metric functions  however,  applied  Park's  sufficiently  that  systems  ceiling  Equation as  (2.36)  the  other  electric  transformation,  torque  and the  ceiling  nonlinearities  i f the  deviations  point  the  can be p a r t i t i o n e d  [ 15 ] w h i c h r e s u l t s  [ T4 ] is  X  I  X  II  —  Vi A  II.I  V I I A  II,II  fx I  I  1  X  IIJ  are .  i n a set  subsystems  follows f  be  linearized  (2.36) e l e c t r i c and m e c h a n i c a l  It  could  form o f  ]  into  terms,  systems.  system of equations  :  trigon-  speed v o l t a g e  only a minor effect  as d e s c r i b e d i n  [ A ] [ x  are  from the: e q u i l i b r i u m s t a t e  original  i n the  limits  e q u a t i o n , the  complex f o r n o n l i n e a r  t h e n o n l i n e a r i t y has  =  E x c l u d i n g the  s t a b i l i t y c r i t e r i a f o r l i n e a r systems  state v a r i a b l e equations  [ x ]  i n the  is quite  limits,  a nominal operating  standard  the  s m a l l so t h a t  By n e g l e c t i n g t h e about  analysis,  expected  to n o n l i n e a r  and g o v e r n o r ,  currents  due t o  Stability is,  (2.35)  f  ±  The m o d e l d e r i v e d a b o v e i s n o n l i n e a r .  the  q  (2.37)  of  18  where  2.5  [ x^. ] a r e  the  s t a t e v a r i a b l e s of the mechanical  [ X-J-J] a r e  the  s t a t e v a r i a b l e s o f the  system  e l e c t r i c a l system  Reduced Order Model For system design purposes,  model to effort.  approximate  study,  of  the  retain  to  component  component v a l u e s ,  system.  desirable  a l l the  it  to minimize  and d e g r e e o f c o u p l i n g o f t h e that  the  the  computational  is  is neglected.  turbines  are neglected because of i t s  model i s  reduced  small  of the  to  span  the  rest  high order model.  i n comparison  Furthermore,  large  component  time  lower order model s h o u l d  dominant p r o p e r t i e s  e x c i t e r mass s p r i n g c o n s t a n t  the mass-spring system,  to have a low o r d e r  to n e g l e c t m a i n l y depends on the  H o w e v e r , one p r i n c i p l e i s  a c e r t a i n extent  Since the  is  t h e h i g h o r d e r m o d e l s o as  D e c i d i n g on w h i c h  of  it  the  to  the  governor  time constant.  Thus t h e  and  rest steam  26*"^ o r d e r  th to  a 19  order  model.  2.6 E i g e n v a l u e A n a l y s i s I n modern a n a l y s i s , s y s t e m s differential  equations  in state  space  are form.  u s u a l l y d e s c r i b e d by a s e t Hence, the  s t a b i l i t y of a  t i m e i n v a r i a n t s y s t e m can e a s i l y be d e t e r m i n e d by e x a m i n i n g t h e of  the  s y s t e m m a t r i x . The r e a l p a r t  of  the  system  unstable part  : stable  eigenvalues For  the  change w i t h  confined  to  the  P  = 0.9  q  different  following p.u.  The e i g e n v a l u e s  system's  natural  consideration here,  loadings  conditions ,  have n e g a t i v e  [ 15 ]  stability  real parts  ;  The i m a g i n a r y  frequencies. most o f  the  eigenvalues  eigenvalue analysis  27^  lagging  ,  order model  V  FC  [ 15 ]  = 1.0  is  p.u.  , 26*"^ o r d e r m o d e l ,  th a n d 19  linear  :  P . F . = 0.9 of the  , so t h e  of  eigenvalues  d i s c l o s e s the  have a p o s i t i v e r e a l p a r t .  i n d i c a t e s the  system under  eigenvalues  eigenvalues  i f any o f i t s . . e i g e n v a l u e s  of the  do n o t  i f a l l its  o f the  order model f o r v a r i o u s degrees of compensation are  listed  of  in  19  Table 2.1 natural as  - Table 2.4  frequency  . It  of the  is  found t h a t  the  e l e c t r i c a l mode i s  the degree of compensation i n c r e a s e s ,  s y s t e m i s u n s t a b l e when  the  c l o s e t o a m e c h a n i c a l mode,  two m e c h a n i c a l modes a r e  and  excited  simultanously. From t h e the  27 ' t  1  order  T a b l e s , we c a n s e e  and 2 6 ^ model a r e  that  the  c l o s e , so i t  corresponding eigenvalues  provides further  evidence  of  that  t h 26  order model i s  26 ' t  1  that  order model are the  19  sufficient. retained  Furthermore,  i n the  o r d e r model c a n be u s e d  19^  the  dominant e i g e n v a l u e s  order model. Hence, i t . i s  for s t a b i l i z e r design.  of  the  decided  20  order 27 model t h  shaft modes  Turbine and Governor  Stator and Network  Machine rotor  t h , 26 order model u  reduced 1 9 ^ o r d e r model  -0.1818 ± J298.18  -0.1818 ± J298.18  -0.1818 ±j298.18  +0.1541 ± j 2 0 4 . 3 5  +0.1544 ± j 2 0 4 . 3 6  +0.0652 ± j 2 0 3 . 7 1  -0.2496 ± J160.72  - 0 . 2 4 9 7 ± .1160.72  -0.2397 ± j160.12  -0.6706 ± J127.03  -0.6706 ± J127.03  -0.2877 ± j99.210  -0.2877 ± j99:210  -0.2477 ± j l O l . 1 3  -0.0479 ± j8.4801  -0.0528 ± j8.4800  -0.0849 ± J8.5400  -0.1417  -0.1417  -4.6160  -4.5976  -3.0336  -3.1049  -4.6732 ± jO.6269  -4.6642 ± J0.5978  -7.0224 ± j542.80  -7.0227 ± j542.80  -7.0228 ± J542.80  -6.1984 ± j209.20  -6.1986 ± j209.19  -6.1057 ± J209.32  -8.4404  -8.4702  -8.5206  -31.920  -31.988  -31.985  -1.9830  -2.0218  -2.2591  -499.97  -500.00  -500.00  -101.91  -101.93  -101.93  -25.404  Exciter and Voltage Regulator  Table 2.1  E i g e n v a l u e s o f v a r i o u s o r d e r SSR m o d e l .at 30% c a p a c i t o r compenation f o r P = 0.9 p . u . at 0.9 power f a c t o r l a g g i n g .  "  21  order 27 model t h  Shaft modes  Turbine and Governor  Stator and Network  Machine rotor  26 order model  r e d u c e d 19 ^ order model  -0.1818 + J298.18  -0.1818 ± J298.18  -0.1818 + J298.18  +0.1560 + J202.68  +0.1560 ± J 2 0 2 . 6 8  +0.1513 + J202.16  +0.9100 + J161.42  + 0 . 9 1 0 6 ± J161.42  +0.7977 + J160.86  + J127.08  -0.6798 ± J127.49  -0.3545 + J99.790  -0.3546 ± J99.490  -0.3251 + j l O l . 4 6  -0.2674 + J9.5459  -0.2749 ± J9.550  -0.2958 + J9.6000  -0.1418  -0.1418  -4.0496  -3.8604  -3.3335  -3.5104  -0.6799  -4.7939  + jO.3198  -4.8147 ± J0.2992  -7.0800  + J591.15  -7.0800 ± J591.16  -7.0800 + J591.16  -6.8387 + J161.47  -6.8386 ± J161.47  -6.7059  -8.1277  -8.1746  -8.2293  -32.808  -32.882  -32.879  -1.9070  -1.9447  -2.1700  -499.97  -500.00  -500.00  -101.76  -101.77  -101.77  + J161.41  -25.423  Exciter and Voltage Regulator  Table 2.2  E i g e n v a l u e s o f v a r i o u s o r d e r SSR m o d e l a t 50% c a p a c i t o r compensation f o r P = 0.9 p . u . at 0.9 power f a c t o r l a g g i n g .  22  order 27 model t h  Shaft modes  Turbine and Governor  Stator and Network  Machine rotor  Exciter and Voltage Regulator  „^th 26 order model  reduced 1 9 ^ order model  -0.1818 ± J298.18  -0.1818 ± J298.18  -0.1818 ± J298.18  +0.0386 ± J 2 0 2 . 7 8  +0.0386 ± J 2 0 2 . 7 9  +0.0371 ± J202.26  ^0.0331 ± j160.36  -0.0331 ± J160.37  -0.0326 ± J159.78  -0.2335 ± J126.83  -0.2338 ± J126.83  -0.4532 ± J100.57  -0.4533 ± J100.57  -0.4174 ± J102.85  -0.5712 ± jlO.905  —0.5819 ± j l O . 9 1 0  -0.5931 ± jlO.980  -0.1419  -0.1419  -3.5340 ± jO.4595  -3.5334 ± jO.5266  -4.9325 ± jO.1861  -4.9436 ± jO.1672  -7.1203 + J630.45  -7.1202 ± J630.45  -7.1203 ± J630.45  -5.4951 ± il22.22  -5.4943 ± J122.22  -5.0390 ± J121.63  -7.7154  -7.7791  -7.8397  -34.198  -34.2077  -34.204  -1.8116  -1.8463  -2.0397  -499.97  -500.00  -500.00  -101.56  -101.56  -101.56  -25.455  Figure 2.3  E i g e n v a l u e s o f v a r i o u s o r d e r S.SRimodel a t 70% c a p a c i t o r compensation f o r P = 0.9 p . u . at 0.9 power f a c t o r l a g g i n g .  23  27*"' o r d e r model 1  Shaft modes  Turbine and Governor  Stator and Network  Machine rotor  26 order model  reduced 19^ o r d e r model  -0.1818 ± J298.18  -0.1818 ± j298.18  -0.1818 ± J298.18  +0.0173 i j 2 0 2 . 8 1  +0.0173 ± J 2 0 2 . 8 1  +0.0166 ± J202.29  -0.0812 ± j160.43  -0.0882 1 J160.43  -0.0862 ± J159.84  -0.5931 ± j126.89  -0.5931 ± j126.89  +2.9035 ± J102.28  +2.9061 ± J102.77  +3.8541 ± J103.68  -0.7741 ± j11.751  -0.7872 ± j l l . 7 6 0  -0.7942 ±  -0.1419  -0.1419  -3.4782 ± jO.5514  -3.4783 ± jO.5989  ± jO.1080  -4.9837 ± jO.0801  -7.1365 ± J647.97  -7.1365 ± J647.97  -7.1365  ± J647.69  -7.9760 ± J102.77  -7.9774 ± J102.77  -8.7957  ± J103.69  -7.4474  -7.5284  -7.5924  -35.120  -35.132  -35,128  -1.7450  -1.7780  -1.9464  -499.97  -500.00  -500.00  -101.42  -101.42  -101.42  -4.9759  jll.840  -25.472  Exciter and Voltage Regulator  F i g u r e 2.4  -  E i g e n v a l u e s o f v a r i o u s o r d e r SSR m o d e l a t 80% c a p a c i t o r compensation f o r P = 0.9 p . u . at 0.9 power f a c t o r l a g g i n g .  24  3.  3.1  L I N E A R OPTIMAL E X C I T A T I O N CONTROL DESIGN  Introduction With the  techniques, economic, theory,  of large  digital  some m o d e r n c o n t r o l t h e o r i e s  and o t h e r  known a s  stabilizer  advent  the  d e s i g n to  large  systems.  frequency  system o s c i l l a t i o n  the  l i n e a r optimal c o n t r o l l e r not  but  also  and t h e  SSR [ 6  the  ].  I t was  2, multiple unstable  t h e p o w e r s y s t e m due t o s u b s y n c h r o n o u s  resonance.  the  operation  eigenmodes  design,  to s t a b i l i z e the a reduced  19^  subsynchronous  order model w i t h  a l l measurable  that  system,  [18,19].  Because o f the m e r i t s  resonance i n the  the  may e x i s t  optimal control, a linear optimal excitation controller w i l l  designed  e.g., found  o n l y c a n p r o v i d e good damping t o  system over a wide-power-range  As f o u n d i n C h a p t e r  linear  of linear optimal control  p r o b l e m , has been a p p l i e d to  [16-19]  analysis  electrical,  dynamic r e s p o n s e o f power s y s t e m s ,  low  can s t a b i l i z e the  and n u m e r i c a l  can r e a d i l y be a p p l i e d t o  A certain class  state regulator improve the  computers  system.  in  of  be  For  state variables  the is  th chosen. test  But the  in this  c o n t r o l i s a p p l i e d to  C h a p t e r , where the  various operating  the  26  eigenvalues  order of the  full  model f o r a  c l o s e d loop system  and c o m p e n s a t i o n c o n d i t i o n s w i l l b e  linear at  examined.  3.2 L i n e a r S t a t e R e g u l a t o r Problem The l i n e a r o p t i m a l r e g u l a t o r Consider  F i n d the  the  l i n e a r i z e d system s t a t e  x  Ax  y  Hx  to  the  =  1 9-  /  0  f o r m u l a t e d as  follows:  equations  Bu  (3.1) (3.2)  o p t i m a l performance  J  subject  +  may b e  function  . t t ( y Q y + u R u)  oo  system dynamic c o n t r i a n t  (3.3)  dt  (3.1)  and  (3.2)  25  substituting  (3.2)  J  into  2  p  f c  is  the  t [ x  0  t  t  ( H Q H) x  formed by appending  H = |[ where  gives  °°  1 ^ /  =  A H a m i l t o n i a n was  (3.3)  u R u]  (3.1)  to  x ( H Q H) x + u R u ] t  t  The o p t i m a l c o n t r o l c a n be R  dt  (3.4)  (3.4),  + p  fc  costate vector  u = -  +  C  ( A x + Bu)  (3.5)  or Lagrange m u l t i p l i e r s .  found  from  8f//8u  ,  resulting  B Kx  1  (3.6)  t  or u = - R ^ K where  i n the  particular  H  _ 1  y  (3.7)  LOC d e s i g n i n t h i s  thesis,  H is  a invertable  square  matrix. K of  (3.6)  and  (3.7)  nonlinear matrix algebraic  is  t  (3.8)  , Q is  R i c c a t i matrix which  satifies  the  equation  K A + A K - K B R  In  the  _ 1  B K  = - H Q H  t  (3.8)  fc  a positive semi-definite  m a t r i x and R i s  a positive  definite  matrix. With u decided,  the  closed loop system equation  becomes  x = Gx where For  the  (3.9)  G = (A-B R particular  _ 1  B K)  (3.10)  t  case K  B = [ 0  0  [ Ao^.Ae^  t  0 ] T  x =  A  A  Au> , A 0 , Aw 2  (3.11)  2  A 9 , Ao> , A 6 , Atii , A6 ; 3  4  4  A i , , A i , A i . , A i , A i . , AV , , AV , d' q' f D' Q' cd' cq'  A V , AE R' fD n  J  (3.12) y  =  [ Aw1,Aei,  A w , A 6 , Aw  A 0 ^ , A u ^ , AQ^,  AP ,AQ ,  AV , Ai ,Ai ,  AV , AV^ .AV ,AE  e  e  2  t  2  a  f  c  R  Aco , A6 f l )  ]  t  ;  (3.13)  26  The t r a n s f o r m a t i o n  3.3  S o l u t i o n to  matrix H is  shown i n t h e A p p e n d i x I.  the M a t r i x R i c c a t i  Equation  The n o n l i n e a r a l g e b r a i c m a t r i x e q u a t i o n c o m p o s i t e m a t r i x method  (3.8)  s o l v e d by  the  [20].  For the continuous c o n t r o l system of equation cost  is  function of equation  (3.4),  the  (3.1)  ,  2 n x 2n c o m p o s i t e m a t r i x  (3.2)  , and  [ M ] is  given  by  -B R - V TM  (3.14)  ] -H Q H  -A  t  The  2n  the  right  matrix  eigenvalues and the  of  left  matrix  [ M ]  p a r t s of the  are  symmetrically distributed  complex p l a n e .  L e t the  on  eigenvalue  be ( A. (3.15)  [ A ] =  II and the  corresponding eigenvector  I  [ X ] X  where  [ A^ ]  values  of the  Then  the  matrix  be  I I I  (3.16)  II i v x  constitutes n stable eigenvalues c l o s e d loop system G i n  Riccati matrix  (3.9),  and  [ K ] may b e c o m p u t e d  [ K ] = [ X  T  I  ]  [ X  I  ]"  of  [ M ] which are  [ A  ] = -  [  the ]  eigen-  .  from  (3.17)  27  3 . 4 W i d e Range SSR S t a b i l i z e r a n d t h e The p r o p e r t i e s  of the  chosen w e i g h t i n g m a t r i c e s  information sation  suggests  that  s t a b i l i z e the  operating  conditions.  m a i n l y upon the [18]  c a n be  [ R ] , so t h e  used  to o b t a i n of  up t o  that  [Q] a c c o r d i n g t o  based  70%. W i t h i n t h a t  s t a b i l i t y of the  sation;  t o have  (3.7)  amount  [Q] f o r t h e  g i v e s u f f i c i e n t damping to  there are  a single  i s unique  o n 50% c a p a c i t o r  system  capacitor  c h o s e n as  [R]  = 1  [Q]  =  depend scheme  eigenvalues, [16-18]  c o m p e n s a t i o n i n most  results  Hence, the  from Chapter  the worst  at  utilities  2 disclose  50% c o m p e n -  controller design w i l l  has  suggested  that  a speed  an e l e c t r i c a l v a r i a b l e d e v i a t i o n . A c c o r d i n g l y , are  closed  system.  be  d e v i a t i o n be  w e i g h t e d more t h a n a n a n g l e d e v i a t i o n , a n d a m e c h a n i c a l v a r i a b l e  matrices  of  compensation.  P r e v i o u s work [16-18]  more t h a n  and  for a given set  of p r e v i o u s work  consideration is  modes.  comp-  controller  Although i t e r a t i v e  results  compensation range,  two u n s t a b l e  above  compensation  closed loop  any  always  o f damping a c h i e v e d i n the  prescribed  the  the  for series  system under  for  there w i l l  Although the  g o a l of wide range s t a b i l i z a t i o n of the  Common p r a c t i c e is  desirable  choice of the w e i g h t i n g m a t r i c e s .  c h o i c e of elements may a l s o  is  system.  that  f o r v a r i o u s degrees of c a p a c i t o r  [K] f o r e q u a t i o n  [Q] a n d  l o o p s y s t e m and the  specified type,  s t a b i l i z e the  it  [ M ] suggest  system f o r a wide range o f c a p a c i t o r  The s o l u t i o n weighting matrices  [R] o f t h e  controllers  can a l w a y s be d e s i g n e d ,  that w i l l  composite m a t r i x  [Q] and  be an o p t i m a l c o n t r o l w h i c h w i l l  Choice of Weighting Matrices  the  deviation  final  weighting  follows:  diag[100,10,5000,50,5000,50,50000,50,5000,50; 150,150,1,1,0,1,1,0,0]  (3.15)  28  In the  this  case,  a s p e c i a l weight  low pressure t u r b i n e .  effectively  shifted  to  With this  the  left  choice,  without  With the w e i g h t i n g m a t r i c e s for  50% c o m p e n s a t i o n h a s  For  [x]  -[  R  _ 1  B K]  = [-2.4072,  t  -18.964, 39.714,  -29.103,  -[  [y] t  _ 1  ] = [-2.4072,  eigenvalues  individual  above,  deviation of can  gains.  a controller  designed  :  9.4417,  80.702,  -8.2681,  -18.964, 39.714,  -0.6465,  411.04,  34.312,  -0.0261,  153.82,  -77.163,  492.45, -76.528,  -0.0086]  eigenvalues  of the  ranging  f r o m 10% t o  90% a n d f o r  3.1  controlled However,  and 0 . 5 -  3.3  p.u.),  . It  system i s  (3.16)  examined.  (P  within =0.5  the p.u.)  £  492.45,  -49.630,  66.567,  -0.2729,  -0.0086]  1  at  (3.17)  1  a compensation  (X  /  loading conditions results  = 0.9  prescribed the  153.82,  l i n e a r i z e d 26 "* o r d e r m o d e l - L  the  Typical  for P  411.04,  -0.0261,  three different  found t h a t  stable  for l i g h t load  -52.002,  c o n t r o l l e d system,  are  is  -0.8738,  then a p p l i e d to  and t h e  1.25,  34.380, 9.4417,  173.74;  57.242,  The c o n t r o l i s  Tables  chosen  speed  undesirable  excessive  gain  34.380,  135.27;  13.445,  305.44,  0.9,  the  the  model  R~ B K H 1  following  given to  model  305.44,  For  the  is  p.u.  are  £  (P  ) =  given i n  and 1 . 2 5  range o f  x  p.u.,  the  compensation.  system i s only s t a b l e  up t o  70%  compensation. Thus t h e effective  l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l has  i n s t a b i l i z i n g SSR w i t h i n  system.  But, this  may n o t  optimal  controller w i l l  as d e r i v e d i n C h a p t e r  be v a l i d  be f u r t h e r  the for  proved  linear-operation-range the n o n l i n e a r  t e s t e d on t h e  2 , and p r e s e n t e d i n C h a p t e r  range.  of  to the  So t h e  original nonlinear 4.  be  linear system  be  29  capacitor compensation  Shaft modes  Turbine and Governor  Stator and Network  30% compensation  50% compensation  80% compensation  -0.1818 ± J298.18  - 0 . 1 8 1 8 ± .1298.18  -0.1818 ± J298.18  -0.1064 ± J201.65  -0.6052 ± J202.69  -0.4736 ± J202.62  -0.5481 ± J160.46  -1.0620 ± J161.12  -0.3208 ± J160.71  -0.7134 ± J127.15  -0.7362 ± J127.28  -0.4166 ± J127.06  -0.9626 ± J98.941  -1.1751 ± J98.981  -0.4185 ± J99.963  -3.5559 ± J4.8254  -5.0336 ± J6.8617  -9.6731 ± j l l . 2 7 5  -0.1408  -0.1415  -0.1422  -2.9296  -3.2877  -3.7417 ± jO.8178  -4.7568  -4.7426  -5.1707 ± j2i2117  -4.6489 ± j l . 5 9 1 5  -4.8122 ± jO.2262  -7.0199 ± j542.77  -7.0884 ± J591.16  -7.1523 ± J648.01  -7.9112 ±  J213.14  -11.057 ± j163.70  -44.178 ± j111.40  -70.193 ±  j76.110  -63.552 ±  J71.648  -22.391 ± J63.227  Machine rotor  Exciter and Voltage Regulator  -2.0417  -2.1165  -2.2008  -500.00  -500.00  -500.44  -118.71  -120.54  -130.74  Table 3.1  Eigenvalues of the system w i t h l i n e a r o p t i m a l c o n t r o l at v a r i o u s degrees of c a p a c i t o r compensation . O p e r a t i n g c o n d i t i o n : P=0.9 p . u . , P . F . = 0 . 9 l a g g i n g , V = 1.0 p . u .  30  30% compensation  50% compensation  70% compensation  80% compensation  -0 . 1 8 1 8 ± J 2 9 8 . 1 8  -0.1818 ± J298.18  -0.1818 ± J298.18  -0 .1818 ± J 2 9 8 . 1 8  -0 .2227 ± J 2 0 4 . 6 9  +0.2282 ± J202.69  +0.0719 ± j 2 0 2 . 7 9  +0 . 0 4 2 7 ±  -0 .2965 ± J 1 6 0 . 7 1  +0.8909 ± j 1 6 1 . 1 6  +0.0144 + ' j 1 6 0 . 3 8  -0 .0926 ± J 1 6 0 . 4 6  -0 .6783 ± J127.03  -0.6977 ± J127.08  -0.1727 ± J126.88  -0 .6164 ± J 1 2 6 . 9 4  - 0 .3557 ± J 9 9 . 1 7 8  -0.4657 ± J99.464  -0.7468 ± j100.59  +2 . 9 0 1 3 ± J 1 0 2 . 1 5  +0 . 3 5 4 4 ± J 8 . 1 8 2 1  +0.0760 ± J9.3132  -0.2975 ± j l O . 7 5 8  - 0 .5416 ± j 1 1 . 6 5 6  j202.81  (a)  30% compensation  50% compensation  70% compensation  80% compensation  - 0 .1818 ± J 2 9 8 . 1 8  -0 .1818 ± j 2 9 8 . 1 8  -0 .1818 ± J 2 9 8 . 1 8  -0.1818 ± J298.18  - 0 .5033 ± J 2 0 1 . 6 1  -0 .5054 ± J 2 0 2 . 7 1  -0 .4337 ± J 2 0 2 . 6 0  -0.4306 ± J202.58  - 0 .6497 ± J 1 6 0 . 4 5  -1 .1105 ±  jl63.ll  -0 .2363 ± j l 6 0 . ' 7 7  -0.2927 ± J160.69  -0 .7233 ± J127.17  - 0 .7406 ± J 1 2 7 . 3 2  -0 .2397 ± J 1 2 7 . 2 2  -0.4043 ± J127.04  -1 .1814 ± J 9 8 . 8 0 3  -1 .4867 ± J 9 9 . 0 3 0  -1 .5509 ± j l O O . 4 3  -0.1491 ± jlOO.03  - 6 . 9 8 9 5 ± j 2 . 3612  -6 .6118 ± J 3 . 7 6 5 5  -9 .1417 ± J 6 . 9 3 7 3  -12.162 ± J8.1309  (b)  T a b l e 3.2  (a)  S h a f t modes< o f t h e s y s t e m a t v a r i o u s d e g r e e s o f c o m p e n s a t i o n at P=1.25 p . u . , P . F . = 0 . 9 l a g g i n g V =1.0 p . u . ( w i t h o u t c o n t r o l )  (b)  S h a f t modes o f t h e s y s t e m w i t h l i n e a r o p t i m a l c o n t r o l at v a r i o u s degrees o f compensation at P=1.25 p . u . , P . F . =0.9 l a g g i n g , V =1.0 p . u .  31  30% compensation  50% compensation  70% compensation  80% compensation  J298.18  -0.1818 ± J298.18  - 0 . 1 8 1 8 ± j 2 9 8 .18  -0 .1818 ±  J298.18  +0.8503 ± J204.29  +0.0901 ± j 2 0 2 . 6 1  +0.0064 ± j 2 0 2 .77  -0 .0072 ±  J202.80  -0.1949  ± J160.79  +1.2050 ± J161.67  -0.0759  -0 .1167 ±  j160.41  -0.6608 ± J127.05  -0.6583 ± j l 2 7 . l l  -0.2535 ±  j l 2 6 .73  -0 .6122 ±  J126.87  -0.2187 ± J99.405  -0.2390 ± J99.695  - 0 . 1 2 4 7 ± jlOO .84  +3 . 3 9 5 2 ±  j101.84  -0.4057  -0.5625 ± jlO.756  - 0 . 7 8 8 3 ± j l l . 773  -0 .9451 ±  J12.427  -0.1818 ±  ± J9.9930  ± J160 .33  (a)  30% compensation  50% compensation  70% compensation  - 0 . 1 8 1 8 ± j29.8.18  - 0 .1818 ± J 2 9 8 . 1 8  -0 .1818 ± J 2 9 8 . 1 8  -0.1818 ± J298.18  -0.2251 ± j201.86  -0 .4357 ± j 2 0 2 . 6 1  - 0 .3604 ± J 2 0 2 . 5 5  -0.3525 ± J202.52  -0.5738 ± J160.56  - 0 .5739 ± J 1 6 1 . 2 2  -0 .2447 ± J 1 6 0 . 6 3  -0.3004 ± J160.58  - 0 . 6 9 7 1 ± .1127.19  - 0 .6559 ± J 1 2 7 . 3 3  - 0 .2837 ± J 1 2 7 . 0 8  -0.4683 ± J127.00  -1.0151 ± J99.025  -1 .1638 ± J 9 9 . 2 7 3  -0 .7136 ± j 1 0 0 . 1 9  +0.1498 ± J99.657  -4.6696  -6 .0982 ± J 8 . 9 2 5 4  -9 .1094 ± j l l . 2 3 3  -12.553 ± J12.562  ± J7.4463  80% compensation  (b) Table 3.3  (a)  S h a f t modes f t h e s y s t e m a t v a r i o u s d e g r e e s o f compensation at P=0.5 p . u . , P . F = 0 . 9 l e a d i n g , . V =1.0 p . u . ( w i t h o u t c o n t r o l )  (b)  S h a f t modes o f t h e s y s t e m w i t h l i n e a r o p t i m a l c o n t r o l at v a r i o u s degrees of compensation at P=0.5 p . u . , P . F . = 0 . 9 l e a d i n g , V =1.0 p . u .  G  32  4.  4.1  EFFECT OF E X C I T A T I O N CONTROLLER ON STEADY STATE AND TRANSIENT SSR  Introduction It  h a s b e e n shown t h a t l i n e a r o p t i m a l c o n t r o l l e r d e s i g n e d  Chapter 3 can suppress all  the  SSR p h e n o m e n o n .  t e s t s were p e r f o r m e d on t h e  exciter voltage limits turbances  In t h i s  Chapter,  and w i t h o u t s u p p l e m e n t a r y  will  be i n v e s t i g a t e d . o n  and t h e  Transient  4.2  In turbance  torques  is  initially  ed  as  change;  shafts  of the  generator  with  generator  3,  first  mass-spring  to v a r i o u s  are system be  disturbances.  a power s y s t e m i s  P  frequently  is  subjected  of 0.9  p.u.  F i g u r e 4 . 1 shows t h e  t o a 10% l o a d  and t h e  change;  10% l o a d c h a n g e  follows  =  mo  + O.lt  0 < t -  < 1 -  t  < 1 -  sec.  { T  mo  Figure 4 . 1 , growing shaft  exposed to  s u c h d i s t u r b a n c e may n o t b e a p r o b l e m o f  s y s t e m when i t  operating at  T  system,  and w i t h o u t c o n t r o l w i l l  s e l f - e x c i t e d SSR may a r i s e .  T  In  on the  operation,  load  s t a t e s t a b i l i t y but responses of the  of the  S t e a d y S t a t e SSR  daily  s u c h as  reason-  A l l g o v e r n o r and e x c i t e r p h y s i c a l l i m i t s  system i s subjected  Controller for  dynamic performance  dis-  e x c i t a t i o n c o n t r o l , designed i n Chapter  dynamic responses o f the  e x a m i n e d when t h e  the  t h e n o n l i n e a r s y s t e m m o d e l d e s c r i b e d b y 26  d i f f e r e n t i a l equations.  included.  the  that  considering  e x c i t e r v o l t a g e t o s w i n g w e l l b e y o n d any  with  order  l i n e a r i z e d system without  a n d o t h e r n o n l i n e a r i t i e s . I n some c a s e s ,  have caused the  able c e i l i n g voltage.  H o w e v e r , i t must be n o t e d  in  + 0.1  torques  •• due t o SSR a r e  sec.  observed.  is  dis-  steady  dynamic  the  system  represent-  33  2  0.0  1  1  1.0  1  1  2.0  1  TIME(SEC)  1  3.0  1  1  4.0  1  1 5.0  XT  Figure 4.1  Dynamic r e s p o n s e s o f the system w i t h o u t when s u b j e c t e d t o 10% l o a d c h a n g e .  control  34  For further t o r q u e o f 20% f o r 0 . 2 Typical result pensation  are  no e x c e s s i v e  studies,  second  f o r 0.9  per  is  assumed  unit  shown i n F i g u r e s r e s p o n s e and t h e  a more s e v e r e d i s t u r b a n c e ,  in  the  shows s u s t a i n e d  system without  4.2  and 4 . 3 . W i t h c o n t r o l , t h e  disturbed  Without their as  control, Figure 4.4,  once-in-a-lifetime  shown i n F i g u r e 4 . 5 ,  the  as  to  its  system  has  normal  oper-  shown i n F i g u r e 4 . 3 .  In  is  a p p l i e d to  are  the  contrast, torques  s y s t e m w i t h 80%  shown i n F i g u r e 4 . 4  t o r s i o n a l o s c i l l a t i o n s on t h e  so t h a t  the  system has  the  to  in practice,  the it  d a i l y power s y s t e m o p e r a t i o n . H e n c e ,  system i s  can be t a k e n as  shown i n t h e  to  compensation.  SSR c a n  control.  an  disturbextreme  above  l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l can e f f i c i e n t l y c o n t r o l the  s t a t e SSR p r o b l e m f o r a w i d e r a n g e o f c a p a c i t o r  exceed  However,  a pseudo as  4.5.  shafts  s h a f t s w o u l d be damaged.  linear optimal excitation  torque input  capacitor  and F i g u r e  d a n g e r o u s t o r s i o n a l o s c i l l a t i o n s due  not normally encountered  case i n the the  the  limit  Although a pulse ance,  system r e t u r n s  com-  control.  and t y p i c a l r e s u l t s  b e a l l e v i a t e d when t h e  pulse  conditions.  o s c i l l a t i o n s and g r o w i n g t o r s i o n a l s h a f t  The same d i s t u r b a n c e compensation  operating  a  g e n e r a t o r l o a d i n g and 50% c a p a c i t o r  a t i n g c o n d i t i o n i n a w e l l - d a m p e d manner Figure 4.2  for various  i.e.,  results,  steady  CO  CO  U J a  in. in  0.0  1.0  I 2.0  I  1 3.0  I  4.0  5.0  4.0  5.0  TIME(SEC)  —<o. X —  0_cn  03  I 0.0  1.0  ~1  1  2.0  1— 3.0  TIME(SEC) F i g u r e 4.2  2.0  3.0  TIME(SEC)  Dynamic responses o f t h e power system at 0.9 p.u. generator l o a d i n g , 0.9 power f a c t e r l a g g i n g , and 50% c a p a c i t o r compensation when s u b j e c t e d t o a p u l s e torque d i s t u r b a n c e (without c o n t r o l )  Co  36  in o  0.0  ) .0  2.0  3.0  TIME(SEC) Figure 4.2  ( continued  )  4.0  5.0  37  Figure 4.2  (continued  )  38  in  0.0  1.0  2.0  3.0  TIME(SEC) Figure 4.2  (continued  )  4.0  5.0  Figure 4.3  D y n a m i c r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l a t 0.9 p . u . g e n e r a t o r l o a d i n g , 0 . 9 p o w e r f a c t o r l a g g i n g , a n d 50% c a p a c i t o r c o m p e n s a t i o n w h e n , s u b j e c t e d to a p u l s e torque d i s t u r b a n c e .  CO  TORQUE CLPR-LPB) P.U.  TORQUE (LPB-GEN) P.U. 0.88  o  0.9S  1.04  1.12  1.2  1.28 1.35  0.64  0.72 0.8  0.88 0.9S  1.04 1.12  in  0.0  1 .0  n  2.0  i  I  3.0  TIME(SEC) Figure 4.3  (continued)  4.0  43  Figure  4.5  Torsional oscillations of s h a f t a t 80% c o m p e n s a t i o n  the g e n e r a t o r - e x c i t e r (with control)  44  4 . 3 C o n t r o l l e r f o r T r a n s i e n t SSR The t r a n s i e n t a result  torques  of e l e c t r i c a l transients  considered.  In order  controller,  a simultanous  to e v a l u a t e three  t i m e t=0 a n d t h e n r e m o v e d a f t e r c a p a c i t o r are location,  e x p e r i e n c e d by the  assumed  hence  the  turbogenerator  caused by t r a n s m i s s i o n f a u l t s the  effectiveness  of the  f a u l t was a p p l i e d a t 0.075 second.  to be 0.04 p . u .  The f a u l t  and 0 . 2 8 p . u .  impedance between bus  bus  B and t h e  E l e c t r i c a l network f o r the subsynchronous resonance.  as  a r e now  designed  excitation  B i n Figure 4.6 impedance  and  r e s p e c t i v e l y . The infinite  bus,  also varied.  Figure 4.6  shafts  simulation of  Xg ,  at  series fault is  45  With the f a u l t l o c a t e d at the remote end o f the line, i.e.  , Xg e q u a l to 0 . 0 1 p . u .  transmission  , the responses o f the system w i t h o u t  and w i t h c o n t r o l are shown i n F i g u r e s 4 . 7 and 4 . 8 r e s p e c t i v e l y . F i g u r e 4 . 7 shows c l e a r l y the t o r s i o n a l i n t e r a c t i o n between the e l e c t r i c a l and m e c h a n i c a l s y s t e m . The f o r c i n g frequency o f the e l e c t r i c a l t o r q u e T^ i n p . u . ) i s c l o s e to a t o r s i o n a l mode frequency o f the  ( P  =  mass-spring  system so t h a t the mode i s e x c i t e d , and growing t o r s i o n a l s h a f t  torques  are o b s e r v e d . But no e x c e s s i v e o s c i l l a t i o n s are observed i n F i g u r e 4 . 8 f o r the system w i t h e x c i t a t i o n c o n t r o l . As the f a u l t l o c a t i o n i s moved c l o s e r to the g e n e r a t o r the d i s t u r b a n c e the s t a n d a r d  to the system becomes more s e v e r e .  t e s t as  The system i s s u b j e c t e d  d i s c r i b e d i n the Benchmark Model [ 1 3 ] w i t h X  to 0 . 0 6 p . u . and c a p a c i t o r e q u a l to 0 . 2 8 p . u . i n F i g u r e s 4 . 9 and 4 . 1 0 . The f a u l t e f f e c t case t h a t the s h a f t s  terminal,  . Typical results  equal  are shown  seems more severe than the p r e v i o u s  e x p e r i e n c e l a r g e r t o r s i o n a l s t r e s s and i t t a k e s a l o n g e r  time f o r o s c i l l a t i o n s to decay. F u r t h e r m o r e , when the c o n t r o l i s t e s t e d on the system a t 80% c a p a c i t o r compensation and the system i s s u b j e c t e d same d i s t u r b a n c e ,  i t i s u n s t a b l e and the s h a f t  torque i n c r e a s e s  to  the  as those  shown i n F i g u r e 4 . 4 , because o f the e x c i t a t i o n v o l t a g e c e i l i n g s and o t h e r l i m i t a t i o n s t h a t are not c o n s i d e r e d i n a l i n e a r c o n t r o l l e r . To summarize, the l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l can p r o v i d e sufficient  damping to s t a b i l i z e the system i n most c a s e s .  recommended f o r severe  fault.  to  transient  But i t i s not  s t a b i l i t y c o n t r o l o f a power system w i t h a v e r y  Figure 4.7  Dynamic responses of the power system without control at 0.9 p.u. generator loading, 0.9 power factor lagging,and at 50% capacitor compensation when subjected to a three-phase fault at the remote end (X =0.01 p.u.)  47  F i g u r e 4.7  (continued  )  48  F i g u r e 4.7  (continued )  49  o  • o 1  D_(D  o o H  0.0  1  r • l.O  1  1  1  2.0  1  3.0  TIMECSECJ F i g u r e 4.7  (continued  )  1  i  4.0  i  i  5.0  _  o " I  Xo" ^o 'o CL , .o. LU '  41  i t  J,  !!  ill o Oo. LU LUm 1  " A f i r r y i iiw mi -i  n  1  1—  2.0  3.0  TIME(SEC)  4.0  -1 5.0  C0 ". 0  .i  0.0  ~i— 1.0  1  i 2.0  1— 3.0  4.0  5.0  TIMEtSEC)  O  co  01  o-  •cn  CL,  CC LU Oo' CL.  rsj  U3  o"  0.0  1.0  ~T  1  2.0  1— 3.0  TIMEtSEC) Figure 4.8  I— 4.0  —! 5.0  0.0  n 1.0  1  I 2.0  I  I 3.0  4.0  ~1 5.0  TIMEtSEC) Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l a t 0 . 9 p . u . g e n e r a t o r l o a d i n g , 0 . 9 p o w e r f a c t o r l a g g i n g , a n d 50% c a p a c i t o r c o m p e n s a t i o n when s u b j e c t e d t o a t h r e e - p h a s e f a u l t a t t h e remote end (Xg=0.01 p . u . )  o  51  -JtD " •—4  CL.  CD  oH 0.0  i  i 1.0  F i g u r e 4.8  i  i 2.0  i——i 3.0  TIME(SECJ  (continued)  1  1 4.0  1  1 5.0  52  in  o  tM  o l  0.0  1  1  1.0  1  1  2.0  1  1  3.0  TIME(SEC)  Figure 4.8  (continued)  1  1  4.0  1  1  5.0  53  Figure 4.8  (continued)  54  in C M "  3°. •CVJ  D_  lllii  LU LD I a  "'1 ti'feli  JK ! 31  ijliifliifjlliilj J;?'U'l'jjj,.Vd. " "I''!  !!'"':'!!'!  Q_ _J 'in  iiiiihi! 1  r  LU°"  ZD  O  a  cn -. in  -i 0.0  1  r—  2.0  1.0  5.0  4.0  3.0  TIME(SEC)  CD  •CM  CL  CL I I -< I'lll :!!:, i  'fe  III  III'1  I  I/it  §11  81  ill JJ  ID  i  0.0  1 .0  iijil  I  n 2.0  1 3.0  4.0  5.0  TIME(SEC) F i g u r e 4.9  Dynamic responses o f the power system w i t h o u t c o n t r o l f o r a three-phase f a u l t ( X g 0.06 p.u.) a t 50% compensation. =  55  Figure  4.10  Dynamic responses o f t h e power system w i t h c o n t r o l f o r a three-phase f a u l t (X =0.06 p.u.) a t 50% compensation. B  56  5.  CONCLUSION  A h i g h - o r d e r n o n l i n e a r power s y s t e m model f o r s t u d y i n g t o r s i o n a l o s c i l l a t i o n s due t o s u b s y n c h r o n o u s From e i g e n v a l u e  a n a l y s i s of the  resonance  l i n e a r i z e d model, i t  has been  is  the  developed.  found t h a t  more  t h a n one m e c h a n i c a l mode c a n b e e x c i t e d s i m u l t a n o u s l y f o r a h i g h d e g r e e o f th capacitor revealed  compensation. that  these unstable  mass i s n e g l e c t e d . cannot  E i g e n v a l u e a n a l y s i s o f the  19  order  potential  i n a turbine  dangers of subsynchronous  generator  unit  initially  system for  t e s t e d by e i g e n v a l u e  capacitor  conditions,  but  analysis.  It  not  improves the  exciter. effective-  only stabilizes  c o m p e n s a t i o n r a n g i n g f r o m 10% t o 90% a t  also  exciter  resonance  even w i t h a s t a t i c  A l i n e a r o p t i m a l c o n t r o l l e r has been d e s i g n e d and i t s ness i s  model  m e c h a n i c a l modes r e m a i n e v e n t h o u g h t h e  Hence, the  be n e g l e c t e d  reduced  s t a b i l i t y c o n s i d e r a b l y at  normal other  the  operating  operating  conditions. Dynamic performance further not  show t h a t  o n l y i n the  other  sections  the  tests using a high-order  system without  s e c t i o n between s u c h as  the  the  shafts  n o n l i n e a r model  c o n t r o l e x h i b i t s growing shaft generator  on e i t h e r  and the  e x c i t e r , but  s i d e of the  torques also  intermediate  in  pressure  turbine. With moderate shows damped r e s p o n s e s . three-phase fault controller  disturbance,  system w i t h  H o w e v e r , w i t h more s e v e r e  c l o s e to  i s no l o n g e r  the  the  generator  disturbance,  t e r m i n a l , the  the  to e f f e c t i v e l y s t a b i l i z e the  range of dynamic s t a b i l i t y , but  oscillations  of the  linear  control  s u c h as  a  optimal  effective.  I n summary, an e x c i t a t i o n c o n t r o l l e r o f t h e can be d e s i g n e d  excitation  severe  transient  it  i s not  type.  linear optimal  type  torsional oscillations within recommended  for  torsional  57  REFERENCES  [1]  M.C. H a l l and D.A. Hodges, " E x p e r i e n c e w i t h 500kV Subsynchronous Resonance and R e s u l t i n g T u r b i n e Generator Shaft Damage at Mohave G e n e r a t i n g S t a t i o n " , IEEE P u b l i c a t i o n 76CH1066-0-PWR, pp. 22-25, 1976.  [2]  R.G. Farmer, A.L. Schalb and E l i K a t z , "Navajo P r o j e c t Report on Subsynchronous Resonance A n a l y s i s and S o l u t i o n s " , IEEE P u b l i c a t i o n 76Chl066-0-PWR, pp. 55-58, 1976.  [3]  0. S a i t o , H. Mukae and K. M u r o t a n i , " S u p p r e s s i o n o f S e l f E x c i t e d O s c i l l a t i o n s i n Series-compensated T r a n s m i s s i o n L i n e s By E x c i t a t i o n C o n t r o l o f Synchronous Machine", IEEE Trans, on PAS, V o l . PAS 94, pp. 1777-1788, Sept/Oct. 1975.  [4]  H.M.A. Hamdan and F.M. Hughes, " E x c i t a t i o n C o n t r o l l e r Design For The Damping o f S e l f E x c i t e d O s c i l l a t i o n s i n S e r i e s Compensated L i n e s " , Paper A78 5 6 5 - 4 , IEEE PES Summer M e e t i n g , Los A n g e l e s , J u l y 1978.  [5]  A.A. Fouad and K.T. Khu, "Damping o f T o r s i o n a l O s c i l l a t i o n s i n Power Systems With Series-compensated L i n e s " , I E E E T r a n s , on PAS, V o l . PAS 97, pp. 744-751, May/June 1978.  [6]  Yao-nan Yu, M.D. Wvong and K.K. T s e , "Multi-Mode Wide-Range Subsynchronous Resonance S t a b i l i z a t i o n " , Paper A78 554-8, IEEE PES Summer M e e t i n g , Los A n g e l e s , J u l y 1978.  [7]  A.M. E l - S e r a f i and A.A. S h a l t o u t , " C o n t r o l o f Subsynchronous Resonance O s c i l l a t i o n s By M u l t i - l o o p E x c i t a t i o n C o n t r o l l e r " Paper A79 076-1, IEEE PES Winter Meeting, New York C i t y , Feb. 1979.  [8]  R.A. Hedin, R.C. Dancy and K.B. Stump, "An A n a l y s i s o f the Subsynchronous I n t e r a c t i o n o f Synchronous Machine and T r a n s m i s s i o n Networks", P r o c e e d i n g o f the American Power Conference, V o l . 35, 1973, pp. 1112-1119.  [9]  C. C o n c o r d i a and R.P. S c h u l z , " A p p r o p r i a t e Component R e p r e s e n t a t i o n f o r t h e S i m u l a t i o n o f Power Sysyem Dynamics", IEEE P u b l i c a t i o n 75CH0970-4-PWR, pp. 16-23, 1975.  [10].  IEEE Committee Report, "Dynamic Models f o r Steam and Hydro Turbi n e s i n Power System S t u d i e s " , IEEE Trans, on PAS, V o l . PAS 92, pp. 1904-1915, Nov/Dec. 1973.  [11]  E.W. Kimbark, York 1956.  [12]  IEEE Committee Report, "Computer R e p r e s e n t a t i o n o f E x c i t a t i o n Systems", IEEE T r a n s , on PAS, V o l PAS 87, pp. 1460-1470, J u n e / J u l y 1968.  " Power System S t a b i l i t y " , V o l . I I I . , W i l e y , New  58  [13]  I E E E C o m m i t t e e R e p o r t , " F i r s t Benmark M o d e l f o r C o m p u t e r S i m u l a t i o n o f S u b s y n c h r o n o u s R e n s o n a n c e " , I E E E T r a n s , o n P A S , V o l . PAS 9 6 , p p . 1565-1572, S e p t . / O c t . 1977.  [14]  J . L . W i l l e m s , " S t a b i l i t y Theory of Dynamical Systems", h e d b y J o h n W i l e y & S o n s , I n c . , New Y o r k , 1 9 7 0 .  [15]  K . K . T s e , " H i g h O r d e r S u b s y n c h r o n o u s Resonance M o d e l s and M u l t i Mode S t a b i l i z a t i o n " , M a s t e r T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , Vancouver B . C . , June 1977.  [16]  Y a o - n a n Y u , K . V o n g s u r i y a a n d L . N . Wedman, " A p p l i c a t i o n o f a n O p t i m a l C o n t r o l T h e o r y t o a Power S y s t e m " , I E E E T r a n s , on P A S , V o l . PAS 8 9 , pp. 55-62, J a n . / F e b . 1970.  [17]  Y a o - n a n Y u a n d C . S i g g e r s , " S t a b i l i z a t i o n and O p t i m a l C o n t r o l S i g n a l s F o r a P o w e r S y s t e m " , I E E E T r a n s , o n P A S , V o l . PAS 8 9 , N o . 1 , J a n u a r y 1970.  [18]  H . A . M . Moussa and Y a o - n a n Y u , " O p t i m a l Power S y s t e m S t a b i l i z a t i o n T h rough E x c i t a t i o n a n d / o r G o v e r n o r C o n t r l " , IEEE T r a n s , on P A S , V o l . PAS 9 1 , p p . 1 1 6 6 - 1 1 7 4 , M a y / J u n e 1972  [19]  B . H a b i b u l l a h a n d Y a o - n a n Y u , " P h y s i c a l l y R e a l i z a b l e W i d e P o w e r Range O p t i m a l C o n t r o l l e r s f o r Power S y s t e m s " , IEEE T r a n s , on P A S , V o l . PAS 9 3 , p p . 1 4 9 8 - 1 5 0 6 , S e p t . / O c t . 1 9 7 4 .  [20]  J . E . Potter, "Matrix Quadratic Solutions", SIAM,Journal of M a t h . , V o l . 1 4 , N o . 3 , p p . 4 9 6 - 5 0 6 , May 1 9 6 6 .  [21]  A . Y a n , M . D . Wvong, and Y a o - n a n Y u , " E x c i t a t i o n C o n t r o l o f T o r s i o n a l O s c i l l a t i o n s " , P a p e r s u b m i t t e d t o I E E E PES Summer M e e t i n g , V a n c o u v e r , J u l y 1979.  Book P u b l i s ^  Applied  APPENDIX I  EQUATIONS FOR THE TRANSFORMATION  AP  e  MATRIX H  = (X - X , ) i A i , + [(X - X , ) i , + E._ ] A i + i X , A i _ + i X , A i _ q d qo d q d do fD q qo md f — qo md D - i, X A i do mq... Q  n  AQ e  = ( E ^ - 2XAj ) A i , - 2X i A i + X , i , A i . + X , i , A i + X i A i . fD d do d q qo q md do f md do D mq qo Q  AV  = [(V R + V X ) A i , +(-V, X + RV ) A i + V, AV , + V AV + do qo d do qo q do cd qo cq  x  t  V (V,, cos6 - V s i n 6 )A6 ] / V^ o do o qo o to  Ai  AV  AV  a  = ( 1 , A i , + i A i )/ i do d qo q ao  c  = ( V „ AV + V ' AV ) / V cdo cd cqo cq co  ct  = [ V , AV , + V 1AV + V ( V , cos6 - V sin6 )A6 ] / V cdo cd cqo cq o cdo o cqo o co  where X = X  e  R = R  e  + X t + R t  60  APPENDIX  II  NUMERIAL VALUES OF MODEL I N P . U . SYSTEM  Mass-spring M  l  system  = 0.185794  K  1 2  = 19.303  2 3  = 34.929  M  2  = 0.311178  K  M  3  = 1.717340  K„, = 52.038 34  M  4  - 1.768430  K . _ = 70.858 45  M  5  = 1.736990  K_,  M  6  - 0.068433  D. . = 0.1  Synchronous Machine X  d  X  md  X X  q mq  R  Exciter  a  Parameters X  f  =  1.6999  1.66  X  D  =  1.6657  = 1.71  X  Q  = 1.6845  = 1.58  X  g  =  f  = 0.00105  %  = 0.00371  R  R  1.8250  R  0.00526  Q  s  "  0.01820  = 0.0015  Regulator T„ = 0 . 0 0 2 L  A = 50  Exciter voltage Turbine  i = 1,2,.. ..,6  = 1.79  and V o l t a g e K  = 2.8220  ceiling limits  T  A =  T  2  T  RH=  F  IP=  = ±7.0  and G o v e r n i n g S y s t e m K T T  F  g  = 25  T  = 0.3  T  CH=  ° '  3  F  HP=  ° '  3  F  LP2  3  co= 0.2 LP1  P  GV  = 0.22 = ±0.1  1  = 0.2  =  ° -  • 2  2  "  0.0 7.0  

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