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High order subsynchronous resonance models and multi-mode stabilization Tse, King Kui 1977

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HIGH ORDER SUBSYNCHRONOUS RESONANCE MODELS AND MULTI-MODE  STABILIZATION  by  K i n g K u i Tse B.Sc.  (Hon.)» N o r t h e a s t e r n U n i v e r s i t y ,  A THESIS  1974  SUBMITTED I N P A R T I A L F U L F I L M E N T OF  THE REQUIREMENTS FOR THE DEGREE OF MASTER OF A P P L I E D SCIENCE in THE FACULTY OF GRADUATE STUDIES in t h e D e p a r t m e n t of Electrical  We a c c e p t t h i s  Engineering  thesis  the r e q u i r e d  as c o n f o r m i n g standard  THE UNIVERSITY OF BRITISH COLUMBIA June 1977 0  King Kui Tse, 1977  to  In p r e s e n t i n g t h i s t h e s i s  in p a r t i a l  an advanced degree at the U n i v e r s i t y the L i b r a r y  s h a l l make i t f r e e l y  f u l f i l m e n t o f the requirements of B r i t i s h C o l u m b i a , I agree  a v a i l a b l e for  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e  r e f e r e n c e and copying o f t h i s  It  i s understood that copying or  thesis  permission.  Department of The U n i v e r s i t y  Electrical  o f B r i t i s h Columbia  2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  Date  Engineering  J u T v 6. 1 9 7 7 .  or  publication  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d without my written  that  study.  f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my Department by h i s r e p r e s e n t a t i v e s .  for  ABSTRACT  Subsynchronous  resonance (SSR) o c c u r s i n a s e r i e s - c a p a c i t o r -  compensated power system when a m e c h a n i c a l m a s s - s p r i n g mode c o i n c i d e s w i t h t h a t of the e l e c t r i c a l system.  I n t h i s t h e s i s , a complete h i g h o r d e r  model i n c l u d i n g m a s s - s p r i n g system, s e r i e s - c a p a c i t o r - c o m p e n s a t e d  trans-  m i s s i o n l i n e , synchronous g e n e r a t o r , t u r b i n e s and governors, e x c i t e r and voltage regulator i s derived. effect  E i g e n v a l u e a n a l y s i s i s used t o f i n d t h e  of c a p a c i t o r compensation,  and dampers on SSR.  Finally,  conventional lead-lag s t a b i l i z e r ,  c o n t r o l l e r s a r e designed t o s t a b i l i z e  mode subsynchronous resonance s i m u l t a n e o u s l y over a wide range compensation.  ii  loading multi-  of capacitor  TABLE OF CONTENTS Page ABSTRACT  i i  TABLE OF CONTENTS  i i i  LIST OF TABLES  v  LIST OF ILLUSTRATIONS  vi  ACKNOWLEDGEMENT NOMENCLATURE  1.  2.  vii . .  viii  INTRODUCTION  1  1.1  Subsynchronous Resonance  1.2  The  1  Scope of the T h e s i s  2  A COMPLETE POWER SYSTEM MODEL FOR STUDIES  SUBSYNCHRONOUS RESONANCE •  2.1  Introduction  2.2  The Steam T u r b i n e s and Generator Multi-Mass  4 4  Torsional  System  3.  4.  4  2.3  The T u r b i n e Torques  and Speed Governor  2.4  The Synchronous Generator  2.5  The E x c i t e r and V o l t a g e R e g u l a t o r  2.6  S t a t e E q u a t i o n s f o r the Complete System  EIGENVALUE ANALYSIS OF THE  8 10  . .  14 16  SSR MODEL  1  ,  8  3.1  Introduction  3.2  The E f f e c t of C a p a c i t o r Compensation  I  3.3  The E f f e c t  18  3.4  The E f f e c t of L o a d i n g  I  9  3.5  The E f f e c t of Dampers  I  9  of C o n v e n t i o n a l S t a b i l i z e r  MULTI-MODE TORSIONAL OSCILLATIONS STABILIZATION WITH LINEAR OPTIMAL CONTROL  2  8  9  4.1  S t a t e E q u a t i o n s w i t h Measurable  4.2  S t a t e E q u a t i o n s i n C a n o n i c a l Form  31  4.3  L i n e a r Optimal C o n t r o l Design  34  iii  Variables  18  29  Page 4.4 5.  Stabilization  of SSR . .  35  CONCLUSIONS  42  REFERENCES  43  iv  LIST OF TABLES Table  Page  3-1  Data f o r SSR Model  20  3-2  E i g e n v a l u e s of SSR model a t d i f f e r e n t degrees of c a p a c i t o r compensation w i t h o u t c o n v e n t i o n a l s t a b i l i z e r . . . .  21  3- 3  E f f e c t of damper winding  28  4- 1  E i g e n v a l u e s of o r i g i n a l system and reduced w i t h o u t c o n t r o l l e r a t 30% compensation  4-2  4-3  4-4  f o r zero t o t a l r e a c t a n c e  . . .  o r d e r models 37  E i g e n v a l u e s of reduced 22nd o r d e r model w i t h / w i t h o u t c o n t r o l l e r and o r i g i n a l system w i t h the c o n t r o l l e r a t 30% compensation .  38  E i g e n v a l u e s of reduced 22nd order model w i t h / w i t h o u t c o n t r o l l e r and o r i g i n a l system w i t h the c o n t r o l l e r a t 50% compensation .  39  E i g e n v a l u e s of reduced 19th o r d e r model w i t h / w i t h o u t c o n t r o l l e r and o r i g i n a l system w i t h the c o n t r o l l e r a t 30% compensation  40  v  LIST OF ILLUSTRATIONS Figure 2-1  Page A f u n c t i o n a l block  diagram o f the complete system  f o r subsynchronous resonance s t u d i e s  . . .  6  2-2  M e c h a n i c a l mass and s h a f t system  ^  2-3  Torques o f a mass-shaft system  ^  2-4  A speed governor model f o r the steam t u r b i n e system  2-5  A l i n e a r model o f the steam t u r b i n e system  2-6  A synchronous machine model  2-7  A single l i n e representation  2-8  E x c i t e r and v o l t a g e  2- 9  A supplementary e x c i t a t i o n c o n t r o l  . . .  ^  3- 1  The e f f e c t of c a p a c i t o r compensation w i t h o u t s t a b i l i z e r .  22  3-2  E n l a r g e d p o r t i o n of F i g . 3-1  23  3-3  The e f f e c t of c a p a c i t o r compensation w i t h s t a b i l i z e r  3-4  E n l a r g e d p o r t i o n of F i g . 3-3  25  3-5  The e f f e c t o f l o a d i n g w i t h o u t s t a b i l i z e r  26  3- 6  The e f f e c t o f l o a d i n g w i t h s t a b i l i z e r  27  4- 1  The e f f e c t o f c a p a c i t o r compensation w i t h c o n t r o l l e r . .  41  . .  9 . . . .  o f the t r a n s m i s s i o n  line  ^ .  r e g u l a t o r model  vi  8  ^ ^  . .  24  ACKNOWLEDGEMENT I w i s h t o e x p r e s s my s i n c e r e g r a t i t u d e to my s u p e r v i s o r s , D r . Y . N . Yu and D r . M . D . Wvong, f o r t h e i r p a t i e n c e ,  g u i d a n c e , many hours  o f c o n s u l t a t i o n and v a l u z b l e a d v i c e d u r i n g the c o u r s e o f t h e work and w r i t i n g o f t h i s  research  thesis.  F i n a n c i a l s u p p o r t from a B r i t i s h Columbia Telephone Company S c h o l a r s h i p , a U n i v e r s i t y o f B r i t i s h Columbia Summer Research F e l l o w s h i p and a t e a c h i n g a s s i s t a n t s h i p  i s g r a t e f u l l y acknowledged.  Thanks a r e a l s o due to Mary E l l e n F l a n a g a n and S a n n i f e r L o u i e for  typing this  thesis.  I am g r a t e f u l to my p a r e n t s and M u i Ha i n Hong Kong f o r p a t i e n t u n d e r s t a n d i n g and encouragement  their  throughout my u n i v e r s i t y c a r e e r .  ^NOMENCLATURE General X  s t a t e vector of immeasurable model  Z  s t a t e vector of measurable model  Y  s t a t e vector of canonical  A  system matrix of X-model  F  system matrix of Z-model  F  o  system matrix of Y-model 1  B  c o n t r o l matrix of X-model  G  c o n t r o l matrix of Z-model  G o  c o n t r o l matrix of Y-model  U  c o n t r o l vector  M  transformation matrix f o r Z-model  T  transformation matrix f o r Y-model  X  eigenvalue  j  complex operator,  Mass-Spring System M  i n e r t i a c o e f f i c i e n t = 2H  H  i n e r t i a constant  K  shaft s t i f f n e s s  D  damping  Q  r o t o r angle  a)  r o t o r speed  oj  synchronous speed  Synchronous Machine i  instantaneous value of current  V  instantaneous value of voltage  viii  ib  flux-linkage  R  resistance  X  reactance  6  torque angle, rad.  to  angular v e l o c i t y , rad./s  T  e l e c t r i c a l torque  e  i  terminal voltage  P+jQ  generator output power  Transmission Network X ,R fc  t  reactance and r e s i s t a n c e of transformer  X ,R e e  reactance and r e s i s t a n c e of the l i n e  X  reactance of capacitor  c  V o  i n f i n i t e bus voltage  E x c i t e r and Voltage Regulator regulator gain T^  regulator time constant, s  T E V ref.  e x c i t e r time constant, s reference voltage °  Governor and Turbine System K  actuator gain  g T T r  T  3  a P  GV  T  CH  T RH T CO  2  actuator time constant servomotor time constant change i n actuator s i g n a l power at. gate o u t l e t steam chest time constant reheater time constant cross-over time constant  ix  F  high pressure turbine power f r a c t i o n intermediate pressure turbine power f r a c t i o n  F^p^  low pressure turbine 1 power f r a c t i o n  F^p2  l ° pressure turbine 2 power f r a c t i o n  T  high pressure turbine torque  T-j-p  intermediate pressure turbine torque  w  T p^,T p2 L  L  low pressure turbine torque  Subscripts d,q  d i r e c t - and quadrature-axis s t a t o r q u a n t i t i e s  f  f i e l d c i r c u i t quantities  D,Q,G  d i r e c t - and quadrature-axis damper q u a n t i t i e s  c  q u a n t i t i e s associate w i t h capacitor  a  armature phase q u a n t i t i e s  Superscripts -1  inverse of a matrix  t  transpose of a matrix d i f f e r e n t i a l operator  Prefix linearized quantities d i f f e r e n t i a l operator d i f f e r e n t i a l operator  x  1.  1. 1.1  INTRODUCTION  Subsynchronous Resonance [1] To increase the power t r a n s f e r c a p a b i l i t y of a power system,  the use of series-capacitor-compensated transmission l i n e s i s the best a l t e r n a t i v e to the a d d i t i o n of transmission l i n e s because of environmental considerations and the l i m i t e d a v a i l a b i l i t y of right-of-way.  They are  also more economical than other methods such-as HVDC. However, subsynchronous resonance (SSR) may occur and shaft damage may r e s u l t .  Two  turbine shafts were severely damaged [2] at the Mohave generating s t a t i o n of the Southern^ C a l i f o r n i a Edison Company because of the excessive t o r s i o n a l o s c i l l a t i o n s caused by i n t e r a c t i o n between the e l e c t r i c a l resonance of the series-capacitor-compensated system and the n a t u r a l modes of the multi-mass generator turbine mechanical system. Subsynchronous resonance may occur i n a system i n the steadystate or t r a n s i e n t state due to a system f a u l t switching.  or major  The former may be c a l l e d the steady-state subsynchronous res-  onance and the l a t t e r the transient subsynchronous resonance.  The main  v.  problems are the s e l f e x c i t a t i o n , the t o r s i o n a l i n t e r a c t i o n , and the t r a n s i e n t torques [3]. When SSR occurs, the synchronous machine i s '••'self-' excited and behaves l i k e an induction generator. I f the negative r e s i s - " tance of the machine, as an induction generator, exceeds the t o t a l r e s i s tance of the e x t e r n a l e l e c t r i c a l system, s e l f e x c i t a t i o n of SSR occurs. T o r s i o n a l o s c i l l a t i o n i s due to the mechanical modes of the multi-mass turbine-generator system. the subsynchronous range.  The t o r s i o n a l frequencies are i n  I f the e l e c t r i c a l resonant frequency i s equal  or close to a t o r s i o n a l mode, the r o t o r o s c i l l a t i o n s and the induced voltages w i l l b u i l d up and the i n t e r a c t i o n between the e l e c t r i c a l and  2.  mechanical systems ensues [1,4]. Transient torques are caused by system disturbances on a s e r i e s capacitor-compensated l i n e and the energy stored i n the s e r i e s capacitor produces l a r g e subsynchronous currents i n the l i n e s .  When the frequency  of the current coincides with the n a t u r a l t o r s i o n a l frequency, transient torque r e s u l t s . A f t e r the reported turbine shaft f a i l u r e s [ 3 ] , c o r r e c t i v e measures have been proposed.  Some of them are under s e r i o u s consideration-and others  already put i n t o p r a c t i c e .  Without too much m o d i f i c a t i o n t o the e x i s t i n g  system, the simplesttway to avoid the subsynchronous resonance i s t o reduce the  degree of capacitor compensation.  Another suggestion i s the i n s t a l -  l a t i o n of passive f i l t e r units i n s e r i e s w i t h the generator transformer n e u t r a l at the high voltage side.  Each f i l t e r unit i s a high-Q p a r a l l e l  resonant c i r c u i t tuned t o block the subsynchronous current at a p a r t i c u l a r frequency corresponding to one of the mechanical modes.  A d d i t i o n a l amor-  t i s s e u r windings on the pole faces can reduce the e f f e c t i v e negative resistance [5]. Supplementary e x c i t a t i o n c o n t r o l i s being considered and the  s t a b i l i z i n g s i g n a l s are derived from r o t o r speed.  F i n a l l y a subsyn-  chronous overcurrent r e l a y has been developed f o r the automatic protect i o n of generating units i n case of sustained subsynchronous o s c i l l a t i o n s . 1.2  Scope of the t h e s i s The widely accepted method f o r subsynchronous resonance studies  i n engineering p r a c t i c e consists of a two-step analysis [5]. The e l e c t r i c a l and mechanical modes are determined separately.  The t r a n s i e n t e l e c -  t r i c a l torque from the e l e c t r i c a l system i s c a l c u l a t e d f i r s t and then applied to the mechanical system as a f o r c i n g function.  In t h i s t h e s i s ,  a complete model i n c l u d i n g the e l e c t r i c a l , mechanical and c o n t r o l systems  3.  w i l l  be  ysis, and  developed  the  effect  conventional  onance w i l l mode be  of  w i l l  in be  various  examined  in  presented  4. in  Chapter of  3.  control,  A summary o f Chapter  5.  2.  By  using  compensation,  excitation  Chapter  resonance  Chapter  in  degrees  supplementary  subsynchronous  designed  sions  be  and p r e s e n t e d  control  For  broad-band  linear a l l  on  optimal  important  eigenvalue  loading  anal-  conditions  subsynchronous frequency  res-  multi-  controllers  w i l l  results  conclu-  and  4.  2. 2.1  A COMPLETE POWER SYSTEM MODEL FOR SUBSYNCHRONOUS RESONANCE STUDIES Introduction For any dynamic or t r a n s i e n t s t a b i l i t y study o f a power system,  an accurate model of the system i s required.  In a d d i t i o n t o the i n d i v i d u a l  e f f o r t s [2,7,12], a benchmark model has been proposed by the IEEE Subsynchronous Resonance Working Group f o r SSR studies [20],  In t h i s chapter,  a complete subsynchronous resonance model i s presented, i n c l u d i n g steam turbines and generator multi-mass t o r s i o n a l system, the turbine torques and speed governor, the synchronous generator, the capacitor-compensated transmission l i n e s , and the e x c i t e r and voltage r e g u l a t o r . A f u n c t i o n a l block diagram of the complete system i s shown i n F i g . 2-1. 2.2  The Steam Turbines and Generator Multi-Mass T o r s i o n a l System Assume that the steam turbine-generator set c o n s i s t s of one  high-pressure steam t u r b i n e , one intermediate-pressure t u r b i n e , 'two lowpressure turbines, one generator r o t o r and one e x c i t e r , a l l mechanically coupled on the same shaft as shown i n F i g . 2-2. They comprise a six-mass t o r s i o n a l system.  For the purpose of a n a l y s i s [13], they are considered  to have concentrated masses and to be coupled by shafts of n e g l i g i b l e mass and known t o r s i o n a l s t i f f n e s s .  Each mass i s denoted by a c i r c u l a r d i s c ,  as i n F i g . 2-3, with an i n e r t i a constant M_^, a p o s i t i v e t o r s i o n a l torque on the l e f t and a negative torque  ,) on the l  K.(9.,,  -  right.  There i s an e x t e r n a l torque T^ applied to the mass inaa p o s i t i v e  x  1+1  9.) 1  - K .  .,(8. - 8. 1 i -  i-l  d i r e c t i o n , an a c c e l e r a t i n g torque M ^ O K i n the same d i r e c t i o n and a damping torque D^. OK i n the opposite d i r e c t i o n .  M.u). l  l  =  T.  i  -  D.O). + K . ( e . , ,  i  i i  The net a c c e l e r a t i n g torque becomes  -  i+i  e.)  i  -  K .  .(e.  i - i  - e.  ,)  ii - i  (2-1)  5.  M. 1  where  =  the i n e r t i a c o n s t a n t o f i  th  rotor th  0^  K. .... x,x+l  =  the r o t a t i o n a l displacement  for i  =  damping c o e f f i c i e n t f o r 1 ^  rotor  =  the t o r s i o n a l s t i f f n e s s of the s h a f t between the i r o t o r and the i+l*"*  By a p p l y i n g e q u a t i o n  rotor  th  rotor  1  (2-1) t o the s i x mass t u r b i n e - g e n e r a t o r system,  twelve d i f f e r e n t i a l e q u a t i o n s a r e o b t a i n e d : V  „. . _ Hxgh P r e s s u r e  p ^  =  —  K  6  56  = tog 0)0"  pa,  =  5  =  56  5  ^  a  K  +  56  K  } M B  03.  =  K , . . (K__ .+.K_,) ^ 6, 63  =  O J OJO  p «o =  23 g - 83  Low P r e s s u r e 2 ^  p6  3  ..  T  A  IP  5  (2-4) 5  (2-5)  =  4  °55  Q  Q  °  K  p8,  45  +—  5  45 g —  4  ( K  65  %  34  +  K  g-  45  )  K 9  + 4  4  J  34 M~  6  3  _  °44 MT"  LP1 ^ ~ T  W  4  +  M,  (  2  wo  4  +  ^  K„„ 6  D , _ _ M ^  +  T _LP2  ( 2  ( K  n  =  OJ OJO  po), 1 r  =  12 — —  p6^  =  0)^ OJO  K  6  )  _  8 )  (2-9).  3  p 6  _  (2-7)  K  Exciter  45  5  w  (2-2)  6  =  Low P r e s s u r e 1 po>  HP  + —  %  (2-3)  (K  —  w  - —  T  ,  6  5  p 9  Generator  ^  6  p9g  K  66 .  a  - —  6  Intermedxate Pressure  X)  K  56 .  12  +  23 gK  }  ^  K 6  12 22 M ~ °1 " D  +  Q  Te " "  i n  ,  ( 2  "  1 0 )  (2-11)  12 6 - — — K  °11 6, - — — 1  0).. . 1  (2-12)  (2-13)  K  e ,u , ±  SR Governor  ±  i=3^6  Torsional System  Steam Turbines  6,u  e ,o) 1  Exciter  Generator  i=3^6  FD  Voltage Regulator  Electrical Network 6  o)  1  etc  IL  6  o)  ... e t c  F i g . 2-1 A f u n c t i o n a l block diagram of the complete system f o r subsynchronous resonance studies.  7.  The generator has an e l e c t r i c torque output Te, and the e x c i t e r e l e c t r i c torque i s neglected.  Note that while angles are i n radians, the speed i s  i n p.u.; 03o High pressure 0£  ,  to  A  =  1 p.u. =  Intermediate pressure 6,0),-  377 e l e c t r i c a l radian/second  Low pressure 1 0 ^ , 0 ) ^  Low pressure 2 9^  D.  i i  ,  o)^  .0). l  F i g . 2-3 Torques of a mass-shaft  system  Generator 6 , 0 )  Exciter 0^,0)^  8.  2.3  The Turbine Torques and Speed Governor The steam turbine and speed governor representation i s based on  an IEEE committee report [14]. Usually the speed pressure turbine and the generator r o t o r .  i s , sensed.between the low-  Combined with the speed r e f e r -  ence, the speed d e v i a t i o n or e r r o r s i g n a l i s derived and relayed through the actuator to a c t i v a t e the servomotor, which i n turn opens or closes the steam valves.  A block diagram [14] i s shown i n F i g . 29-4.  Foraa l i n -  ear study, the system equations may be w r i t t e n ; K p A a  1  =  Ato - — 1  P AP  Ato  G V  -  |-  \ '- K. . • : S (1 + S T ^  F i g . 2-4  Aa  (2-14)  1 Aa - |- A P  Aa  + (  \  (2-15)  G V  1 S  A speed governor model f o r the steam turbine system  AP  GV  9.  AT  AT. HP  F PIP  HP  A P  1 + ST  GV  AT. LPl'.  IP  LPl]  1 + ST RH  CH  AT. LP 2  "LP 21  1 + ST CO  F i g . 2-5 AA Linear model of the steam turbine system F i g . 2-5 shows a standard turbine representation f o r s t a b i l i t y studies [14].  The system c o n s i s t s of one high-pressure, one intermediate-  pressure and two low-pressure turbines. by T^-., T , T i , T T rir  TT>  T  i r  Liri  Their output torques are denoted  r e s p e c t i v e l y . There i s a reheater between high-  LarZ  pressure and intermediate-pressure stages, and crossover pipings between intermediate-pressure and low-pressure stages.  The steam i n t o the turbines  flow through the governor-controlled valves a t the i n l e t of the steam chest. The time constants of the steam chest, 's the reheater and the crossover p i p i n g are denoted by T ^ , T ^ a n d T , r e s p e c t i v e l y . F ^ , F , HP I P CO p n  F_ _ . j and F  represent f r a c t i o n s of the t o t a l power developed i n the  0  various stages.  Therefore F,. HP CH  p AT HP  P  A  T p !  T  IP  p A T  A  P  C  =  "IP F, x~T„„ HP "HP RH A T  m  L p l  GV "  =  m  X  T  LPl „c o A T " ~  x  T  A  T  (2-16)  HP  ATT  --T-,  RH  (2-17)  IP  1 I P  T  co '  (2-18) T  l  p  1  10.  A  2.4  LP2  T. PL2  A  T. LP1  (2-19)  The Synchronous Generator The synchronous generator i s assumed to have s i x windings. In  addition to the d and q armature windings on the respective axes, there i s a f i e l d winding f, a damper winding D on the d-axis and two damper windings Q and G on the q-axis.  They are schematically shown i n F i g . 2-6.  d- axis  rqrraXlS  •-fM) +  v.  F i g . 2-6  A synchronous machine model  I'l.  The voltage equations i n the l i n e a r form are  —  A V, = pAip , -  OJOAIJJ  A V q  = vAty q  OrJ O A I J ; , r +  A V  =  f  v  +  r  q  d  pAi|j + R  = pA^ +  A i  0  = pA* + R A i Q  A  Q  +  R  G  A  Ao) - R A i a q  f  0  P *G  IJJ,  do  A i  f  Au - Ra Ad i qo  D  Q  (2-20)  H  where the f l u x linkages are Aijj.  - d x  0  Aip.  D  _1 uo  " ad X  " ad X  0 Ai  r  0  0 -X q 0 0  X  ad  X  ad 0  0 X  X  f  X  ad  ad  -X aq -X aq  Ai .  A  0  0  0  0  X aq  X aq  Ai  0  0  Ai,  0  0  X aq  X aq X G r  (2-21)  Ai, Ai  G  anand the tf/s, X's, R's and i ' s are the per u n i t f l u x l i n k a g e s , reactances, resistances and currents r e s p e c t i v e l y . The s a t u r a t i o n i n the i r o n c i r c u i t i s neglected.  The s t a t o r  t r a n s i e n t voltages p i ^ an P ^ j although normally neglected i n s t a b i l i t y studies [15,16] are retained i n t h i s study because the capacitor compensated transmission l i n e s , to which the armature windings are connected i n s e r i e s , must be described by d i f f e r e n t i a l equations.  12.  R-  X„  ct  Fig.  2-7  A s i n g l e l i n e r e p r e s e n t a t i o n of the t r a n s m i s s i o n  In F i g . 2-7, age at  and  are d-q components o f the t e r m i n a l  i s the v o l t a g e a c r o s s the c a p a c i t o r and V the c a p a c i t o r .  a resistance R  t  The t r a n s f o r m e r  volt-  i s the t e r m i n a l v o l t a g e  i s r e p r e s e n t e d by a r e a c t a n c e X  and the t r a n s m i s s i o n l i n e by a r e a c t a n c e  resistance R . e L e t the t e r m i n a l v o l t a g e e q u a t i o n s  [v ]  line  X  e  and  and a l i n e  i n a-b-c phase c o o r d i n a t e s be  t  a b c ' '  =  [  R  ]  [  t  I  ]  +  a,b,c  [ L ]  dt  [  I  t  ]  +  1  V  a,b,b  c  ]  a,b,c  (2-22)  + [VJ a,b, c  where [R]  =  a resistance matrix:  [L]  =  an i n d u c t a n c e  Let  R^ + R t e X + X matrix: t e 0)O  Park's t r a n s f o r m a t i o n m a t r i x be o cos 6  [T]  =  -sin  1  cos(0-120)  -sin( 6-120.)  1  cos(6+120)  -sin(6+120)  1  (2-23)  13.  and the transformations are [V] , a,b,c L  J  =  [T][V]. d,q,o  =  [ R ] [ I ]  J  and  [I] , a,b,c  =  ^ d f  ^d.q.o  J  [T][I], d,q,o n  (2-24)  n  Then we have  ^d,q,0  d,q,0  +  +  [V!]  c d,q,o ,  ^'^d.q  +  + [v ]  (2-25)  od,q,o ,  Note that  [T]  ^  [T] =  0  -1  0  1  0  0  0  0  1  de dt  (2-26)  The terminal voltage equation i n d-q coordinates when l i n e a r i z e d , becomes AV  R +R t e  -X -X t e  Ai, d  Ai  , +  AV  X +X t e  where V ^ and V t o r , and V  q  Ai  R +R t e  :(x  ~  t V  —  AV  +  :  +  P  —  A<S (2  + AV  Ai  q  V cos 6 o o  cd  cq  -V s i n 6 o o  are the d-q components of the voltage across the capaci-  the i n f i n i t e bus voltage.  The zero component equation i s o r -  thogonal to the other two equations and i s u s u a l l y neglected except f o r asymmetric  loading.  The capacitor equations may be w r i t t e n (2-28) a,b, c  A f t e r transformation, i t becomes [I1  d,q,o  "  [  C  ]  ^  [  which when l i n e a r i z e d , gives AI  c]  +  [C][T]  d,q,o  AV  < " 2  c  29)  (2-30) AV  cq  ^[T][V ]  cq  X  1  _ 1  d,q,o -AV  AV cd to X oc  AI  V  cd  14.  2.5  The E x c i t e r and Voltage  Regulator  The e x c i t e r and voltage regulator model i n t h i s t h e s i s i s based on an IEEE committee report [18] with some s i m p l i f i c a t i o n .  The regulator  input f i l t e r time constant, the s a t u r a t i o n function and the s t a b i l i z i n g feedback loop are neglected.  R  1 + ST  1 + ST,  u;  voltage regulator  E  F i g . 2-8 In F i g . 2-8, V U"  m  exciter  E x c i t e r and Voltage Regulator Model  i s the generator terminal voltage,  the supplementary c o n t r o l ,  E  4a E FD  T  i t s time constant, T  voltage of the e x c i t e r .  .  the voltage regulator gain, T^  the e x c i t e r time constant and E  £  .  p D  a per u n i t output  Although the voltage l i m i t s are shown i n the  f i g u r e , they w i l l be neglected i n l i n e a r a n a l y s i s . Mathematically we have  p AV„ R v  =  P FD = AE  K.  T. A  AV  u - -=• AV„ T. R  A  T l R " TT A V  hi  .  K,  + —— t T.  (2-31)  A  AE  FD  (2-32)  Ci  where the l i n e a r i z e d terminal voltage AV  V,  V AV. + =32. to V to _ d V^_  (2-33)  A V  q  15.  S u b s t i t u t i n g \AV','AV;f rom equation (2-27) i n t o equation (2-33) and the r e s u l t s i n t o equation (2-31), we have V  P  A V  R  "  +  K.V ° AV + T,V cq A to q  m  TT  m  P  A l  .. , W W d V, T ao)o to A o +  m  q  p  A  ^A^do .  TXT  * cd A to V  K „ [V, (R +R ) + V (X +X ) ] A i T. V do t e qo t e' d A to ^ L  v  t  i A to V ^ d o W>  +  +  doWV V T. o)Q to A o  +  V  qo<W  ]  M  q Y7 A +  K V ° [V, cos6 - V sinS ] AS T.V do o qo o A to ^  A  "\  (2-34)  L  F i g . 2-9 shows a supplementary e x c i t a t i o n c o n t r o l of the leadlag compensation type [19].  K ST s 1 + ST  Aoi  F i g . 2-9  b  1 + ST  X  1 + ST y  c  1 + ST  X  1 + ST y  A supplementary e x c i t a t i o n c o n t r o l  Mathematically, K  s  T pAd) - T p  b  T p . b - T p x y -T  p y  u + T p x  =  b  c = c =  (2-35)  \c -  b  \u - • . c  (2-36) (2-37) "~  16.  2.6  State Equations f o r the Complete System The component system equations previously derived can be combined  i n t o a s i n g l e set of s t a t e equations i n the form of X  =  [A] X  (2-38)  where X i s the s t a t e v a r i a b l e vector and [A] the system matrix.  Equation  (2-38) can be conveniently p a r t i t i o n e d  A  X. II  I,  I | I , II A  A. II, I  (2-39) X  II,II  II  where X^ contains the state variables of the mechanical system and  X^  those of the e l e c t r i c a l system; namely Xj.  X  II  =  =  [  [ ±  e  V  i r  d ' \>  6,  o),  ±  V  f  A_  e ,  v  3  V  ±  y  e ,  v  4  e , 5  G' c d ' c q ' V V  e ,-  v  fi  AT  P  G V  ,  T  H  P  ,  T  I  P  ,  T  L  V  V  represents the coupling between the two systems where the  i n t e r a c t i o n occurs through the e l e c t r i c a l torque T  a,  =  e  ( i K i - ib i , ) d q q d  ={ (ij;,  e  Y  per  Since (2-40)  unit  A i + i A^j - ib A i , - i , Aib >}/ u> q qo d qo d do q o  do  T  T  T  = { (X - X , ) i A i , + [(X - X , ) i , + X , i | ] A i + i X , A i ^ q d qo d q d do ad fo q qo ad f 1  + i X , Ai - i , X Ai - i , X A i } /j oo qo ad D do aq Q do aq G o Next, the p a r t i t i o n e d matrices A.^ ^ and A ^  ^  (2-41)  of the e l e c t r i c a l system  s h a l l be f i r s t assembled i n the form of  B X  fl  -  CX 1  1  + C  Z1  X  IT  (2-42)  P  L  ]  17.  Then we have  Ki  x  where  n  -  •  A ^  B _ 1  A  =  c  i  n,i  B '  i  x  x  1  i  4  +  +  B _ 1  A  c  u  I I , I I  ;  n  x  x  A ^  ( 2  -  4 3 )  n  -  B "  1  (2-44)  Thus we have completed the derivation of the state equations for the overall system.  18.  3. 3.1  EIGENVALUE ANALYSIS OF THE SSR MODEL  Introduction Eigenvalue analysis technique is useful in investigating the  stability of systems.  The complex eigenvalues are associated with o s c i l -  latory modes of the system and the real! part of the eigenvalues provide the information on system damping. When an eigenvalue has a positive real part, instability of the system is indicated. In this thesis the effect of capacitor compensation, conventional stabilizer and dampers w i l l be investigated using the data taken from the benchmark model [20] Table 3-1. 3.2  The Effect of Capacitor Compensation Fig. 3-1 and 3-2 show the eigenvalues of the system with various  degrees of capacitor compensation, at a particular loading. The pair of eigenvalues corresponding to A6 and Ato of the synchronous machine have positive real parts when the compensation is 20% or less.  The natural  frequencies of the multi-mass torsional system are approximately 298, 203, 160, 127 and 99 radians/second which correspond to 47.4, 32.3, 25.5, and 16 Hz respectively.  20.2,  By changing the degree of compensation, the nat-  ural oscillation frequency of the transmission system changes. When the frequency of the electrical mode i s closedto a mechanical mode, SSR may occur.  At 50% and 60% compensation, two mechanical modes are excited  simultaneously. 3.3  The Effect of Conventional Stabilizer Fig. 3-3 and 3-4 repeat the study of the effectoof capacitor  compensation, but with the addition of a conventional stabilizer of the lead-lag type.  Whereas the pair of eigenvalues corresponding to A6 and  19.  Au of the synchronous machine were unstable f o r compensation below 30%, F i g . 3-1 shows they are s u b s t a n t i a l l y moved to the l e f t h a l f of the complex plane w i t h the supplementary e x c i t a t i o n c o n t r o l , F i g . 3-3. However, the lowest mechanical mode of 99 rad/sec i s always e x c i t e d and s h i f t e d to the r i g h t - h a l f plane. 22].  This i s i n agreement with other f i n d i n g s [21,  The damping of other mechanical modes i s decreased  slightly.  3.4 The E f f e c t of Loading The e f f e c t of d i f f e r e n t loading with and without s t a b i l i z e r on SSR i s shown i n F i g . 3-5 and 3-6 r e s p e c t i v e l y .  Most of the eigenvalues  do not change except those corresponding to the generator mechanical model Generally the system becomes more unstable w i t h more leading power f a c t o r . Most u t i l i t i e s operate t h e i r systems between 0.9 power f a c t o r lagging and u n i t y power f a c t o r .  For t h i s reason, 0.9 power f a c t o r lagging i s chosen  f o r the studies here. 3.5  The E f f e c t of Dampers As reported [5] the a d d i t i o n of an amortisseur winding can reduce  the p o s s i b i l i t y of SSR. For t h i s i n v e s t i g a t i o n , the a d d i t i o n a l damper e f f e c t i s represented by decreased damper impedance.  When the t o t a l r e -  actance of l i n e and transformer i s zero, SSR occurs.  The r e s u l t i s shown  i n Table 3-2. The e x c i t e d mode i s damped out by decreasing the damper impedance which agrees with previous r e s u l t s [5].  Table 3-1  Numerical Values of Model i n p.u. system  Mass-spring System Parameters l  M  M  2  k  3  M  4  M  M M  5  6  = 0.068433  K  12  =  2.822  = 1.736990  R  23  =  70.858  D  22  = 1.768430  K  34  =  52.038  D  33  = 1.717340  K  45 "  34.929  D  44  = 0.311178  K  56 =  19.303  D  55  D  66  R  f  = 0.185794  ll  D  = 0.1 = 0.1 = 0.1 = 0.1 = 0.1 = 0.1  Synchronous Machine Parameters 1.6999  X, = 1.79 f a X , = 1.66 h ad X = 1.71 Q q X = 1.58 aq R = 0.0015 a E x c i t e r and Voltage Regulator X  X  K, A  = 50  = 0.00105  1.6657  %  = 0.00371  1.6845  \  = 0.00526  1.8250  R„  = 0.01820  = 0.002  T, A  = 0.01  = 0.01  X  = 0.56  Transmission Line Parameters X  =0.14  R  R e  = 0.02  X  c  v a r i e s from 0.056 - 0.56 (10% - 100%)  Governing and Turbine System K  T  =25  g T  =  3  0.3  T  C0  F  LP1= ° '  =  °'  2  22  e  0  =0.2  1  T  CH  F  HP " °'  F  LP2= ° '  =  °-  RH  3  IP  3  7.0 0.26  22  S t a b i l i z e r parameters K  s  T y  =20 = 00.05  T  = 3.0  T x  =0.125  21.  20%  Shaft modes  Stator/Network  30%  50%  -0.1817  +  j298.18  -0.1818  +  J298.18  -0.1818  +  j298.18  -0.2104  +  j203.20  +0.1541  +  J204.35  +0.1560  +  J202.68  -0.2266  +  j160.66  -0.2496  +  J160.72  +0.9100  +  J161.42  -0.6679  +  J127.03  -0.6706  +  J127.03  -0.6799  +  J127.08  -0.2660  +  j 99.13  -0.2877  +  j 99.21  -0.3545  +  j 99.79  -6.9800  +  J512.30  -7.0224  +  J542.80  -7.0800  +  J591.15  -6.0717  +  J241.01  -6.1984  +  j209.20  -6.8387  +  J161.47  +  J9.5459  +  J0.3198  -8.5681  -8.4404  -8.1277  Synchronous  -31.578  -31.920  -32.808  Machine Rotor  -25.397  -25.404  -25.423  -2.0196  -1.9830  -1.9070  E x c i t e r and  -499.98  -499.97  -499.97  Voltage Regulator  -101.97  -101.91  -101.76  X & a)  +0.0415  +  J8.0234  -0.0479  +  J8.4801  -0.2674  -0.1416  -0.1417  -0.1418  Turbine and  -4.6679  -4.6160  -4.0496  Governor  -2.9271  -3.0336  -3.3335  -4.7039  Table 3.2  +  jO.7567  -4.6732  +  J0.6269  -4.7939  Eigenvalues of SSR model at d i f f e r e n t degrees of capacitor compensation without conventional s t a b i l i z e r f o r P = 0.9 p.u. at 0.9 power f a c t o r lagging.  70D  f-600  5 3 2  500  400  •300  250 3  cq  cd  4  h!50 100  ^3> 3 e  50 r  to, 6  25  15 '87 t-  T J  FD  l  -100-70  F i g . 3-1  I  20-10  I"  -9  -8  The e f f e c t of c a p a c i t o r  5 1  "R rzSJ  -600  T HP IP  -1  LPl •I"  'I"  1  2  compensation w i t h o u t s t a b i l i z e r f o r P = 0.9 p.u. a t 0.9 power  factor lagging. (The symbols 1,2,3,...9 r e s p e c t i v e l y  correspond to 10,20,30,...90% compensation)  N3  -300  -250  V ,,V cd cq  2 1  014,6^  43a  !S|§  aa i  4  0  6 •150  2  98  <S3>'3 e  7 6 5 4321  100  h 50  0),5 B -1.0  Fig. 3-2  -o.a  5 -0.6  -0.4  • 4 -0.2  3  2  1 • i • • •»i • • • • i • • • • i 1 2 3 4 5  Enlarged portion of Fig. 3-1. (The symbols 1,2,3,...9 respectively correspond to 10,20,30,...90% compensation) IjO  700  id'  E  FD  V  R  -100-70  J H P '  Q  rB • • • g -3H r— -600  1  -20-10  -8  -S  -4  -3  T  I P  •1  0  "I  2  Ml"  3  rT  T  rr  4  Fig. 3=3' The'effect of capacitor compensation-with stabilizer for P = 0.9 p.u. at 0.9 power factor lagging. (The symbols 1,2,3...9 respectively correspond to 10,20,30...90% compensation)  S3  •300 u  6» 6 6  250  cd  "5 , e  cq  21  !Sg,  2437  6  5  Wit, o it  3}  0  150  m5 6  t&3 ,63  7  h 50 . c,u 1 2 3 4 5 67B9  p-rrr -10  Fig.  9 8 7 6 5  4-321  6  ,td 1 • • 1 " • 1 " " 1 " •' 1 1  -a  3-4  -6  -4 -1.4  -1.2  -1.0  -0.B  E n l a r g e d p o r t i o n o f F i g . 3-3 (The symbols 1,2,3,...9 r e s p e c t i v e l y  -0.6  -0.4  correspond  to  -0.2  1  1  1  2  10,20,30...90% compensation)  3  4  5  •700  d  q  F-500 '  F-MO  -250  V  cd' cq V  obit, 0^  •ISO  0)1,01 100  ^3.03  so zs  IS o) >  6  CD a,P G  -QH»B-| -20 -10  5  0 M..X...|J...M..HJ I  l  V  & CD  T  j .*..i.j;.a6JDi»i l  -4  The e f f e c t o f l o a d i n g w i t h o u t s t a b i l i z e r  P' IP T  H  LP1  L  TTCTT ' -3  /.*  ' 1 i "  ...  (The s y m b o l s © = 0.8 p . f . l e a d i n g , A = 0.9 p . f . l e a d i n g , +.= u n i t y power f a c t o r X = 0.9 p . f . l a g g i n g , <J> = 0.8 p . f . l a g g i n g )  d q F SOO  Pap  L 0  6» 6 6  -250  V  cd' cq V  .05.65^ 0)4, 0^  HlSO  co 1 > 8 j B "3.63  ,  *- SO C-2S  c,u  6 ,co  X  OA  o  5  0 FLD  J  a  R  |B' • -HJO -100  -70  Fig. 3-6  JO-IS  1  iO -g  111  »n 11 -a  ' GV P  +  T  HP' IP D  »j"KlyiD|"i'  T  X  S g n  b  P(  r  1  ' LPl T  ii8mii|iiiiiiiii^niiiiiii^ii  The effect of loading with stabilizer (The symbols CD = 0.8 p.f. leading, A = OV-9 >.f. leading, + =•'unity power factor X = 0.9 p.f. lagging, 0 = 0.8 p.f. lagging)  28. damper impedance x 0.6  damper impedance x 1.5  -0.1818 ± J298.18  -0.1818 ± J298.18  -0.1818 ± J298.18  -0.0288 ± J202.87  -0.0296 ± J202.87  -0.0278 ± J202.87  -0.1536 ± J160.52  -0.1543 ± j160.52  -0.1528 ± J160.52  -0.6521 ± J126.98  -0.6522 ± J126.98  -0.6518 ± J126.98  -0.0238 ± j 98.47  -0.0285 ± j 98.50  -0.0163 i 3 98.42  -7.1913 ± J715.78  -7.1841 ± J718.50  -7.1873 ± J712.71  +0.3604 ± j 37.21  -0.2149 ± j 30.47  +0.9064 ± j 42.09  -5.8109  -5.1370  -7.2552  Synchronous  -43.39  -31.63  -55.88  Machine Rotor  -25.60  -25.51  -25.72  -0.5040  -0.4187  -0.5597  E x c i t e r and  -499.96  -499.98  -499.94  Voltage Regulator  -100.68  -100.43  -100.95  A 6 co  -3.8086 ± j 20.07  -4.1469 ± j 23.81  -3.7030 ± j 18.24  -0.1406  -0.1404  -0.1407  Turbine  -4.1296  -4.8816  -3.8373  Governor  -3.1202  -3.0225  -3.1684  -4.5440 ± J0.1525  -3.7438 ± J-.5651  -4.7916 ± j-,2688  original system  Shaft modes  Stator/Network  Table 3. 3  E f f e c t of Damper Winding i n zero t o t a l reactance and P = 0.9 p.u. at 0.9 power f a c t o r  lagging.  29.  4. MULTI-MODE TORSIONAL OSCILLATIONS STABILIZATION WITH LINEAR OPTIMAL CONTROL Linear optimal control theory has been applied to the stabilizer design of power systems [23,8,24].  For practical applications, the state  variables used in the design must be measurable.  Another problem of opti-  mal control design i s the choice of the weighting matirices Q and R i n the cost index. A simple procedure was proposed [8] which requires thesstate equations i n the canonical form.  The Q/R ratio i n the procedure can be  judiciously chosen. 4.1  State Equations With Measurable Variables The state equations of the system was written in Chapter 2 i n  the form X  = A X + Bu  (4-1)  where A was given as (2-39). For an excitation control, K t B = [0 0 0 ... 0 -^] A as i n (2-31). Let r  Z = M X  (4-3)  where Z is the measurable variable vector. Z = MX  (4-2)  = MA M  or  Z = F Z+ Gu  where  F = MA M  _1  _1  Then  Z + M Bu (4-4)  and  G = MB  Assume that a l l mechanical system variables, such-,as angles and speeds of every turbine rotor and that of generator and exciter; torques output from each stage of turbine and governor system and the electrical system variables such as generator power and current, voltage across the capacitor and to ground (generator side), damper currents and voltage output from voltage regulator and exciter j are-measurable,, then we have  30.  For electrical power AP  = V_, Ai + V Ai + i , AV, + i AV do d qo q do d qo q  (4-5)  By substituting V^, V^ from equation (2-25) AP  = in,. A i , +m._ Ai + m. . AV , + m. _ AV + nx., A6 (4-6) 11 d 12 q 14 cd 15 cq 16  jre m._ = Vj + (R +R ) i , + (X + X j i whei 11 do e t do e t qo ix 12 = qo " ^ e + V ^ o e t qo V  m  14  +  (R  +R  ):L  do  rn,- = i 15 qo = V [ i , cos 6 - i sin 6] 16 o do qo For terminal current m  A i  or where  M  i, _do  =  t  =  m  21 d M  i jo  +  m  22  A ±  i do = — , to  m ^  A  . .  ( 4  _  7 )  q _ -  m  x qo to  For voltage across the capacitor A V  = c t  or  AV  where  m,,  c  l£do V  co  Icqo C d  V  A V  co  ( 4  _  8 )  C q  = m..AAV , +rn,-AV 44 cd 45 cq V , cdo = ^ — co  ,  ^  _ -  V cqo y  co  For voltage at the terminal of the capacitor. As shown i n Fig. 2-7, V  i s the voltage at the generator side  of the capacitor with respect to ground  31.  AV  where V  ctdo ct  V  and V  ctd  AV' c t d, + V  cto  ctqo AV . 'ctq cto  (4-9)  are the d, q components of V  ctq  ct  and can be expressed i n  terms of the v o l t a g e a c r o s s the c a p a c i t o r and i n f i n i t e bus v o l t a g e s . AV AV  ctd  ,  ctq  or  AV  where  m 54  ct  AV  =  AV  =  m . AV , + m AV + m A <S 54 cd 55 cq 56  v  =  c  , + V cos 6 A 6 cd o cq  Other  o  sin 6 A 6 (4-11)  cc  V ^ ctdo  m 55  , cto  [V  ctdo  B e s i d e s m,,, 11 are u n i t y .  - V  c  V  m, 56  (4-10)  =  cto  cos 6 - V ^ s i n 6] — ctqo V  m„„, m.. and m , 22 ra44 55 rc  cto the o t h e r main d i a g o n a l elements  o f f - d i a g o n a l elements are zero except  those a l r e a d y  derived. 4.2  S t a t e E q u a t i o n s i n C a n o n i c a l Form A d e s i g n procedure has been developed  u t i l i s i n g state  equations  i n c a n o n i c a l form [ 8 ] :  0  1  0  0  (4-12) 1. 0  ;  -a Let  Z  =  T Y  1  -a  2  -a  .-a n-1 n (4-13)  32.  we s h a l l have Y = T "  1  where  F = o  T  and  G = o  TG  Z = F Y + G U o o  (4-14)  F T  (4-15)  _ 1  =  [ 0 0 0 ... 1 ] '  (4-16)  The transformation matrix T can be found as f o l l o w s : Since the eigenvalues remain unchanged with s i m i l a r i t y transformation, we s h a l l have | AI - F | =  Q-A^  (A-A ) ... (\-* ) =  IAI - F I = .i o  A + d A n  2  n  0  (4-17)  and n  1  n _ 1  +  a ,A ~ + ... + a, n-1 1 n  2  where A^, X^, ... A are the eigenvalues of the system. n  = U0  (4-18)  The a's can be  determined from (4-16) and (4=17), n  a  =  - I  X.  1=1  a .. = n-1  A.. A„ + A,A„ + ... + A„A„ + ... + A J 12 13 2 3 n-1 n  ot „ = n-2  — A A _ A „ — A A. A „ A . — ... — A _ A .A 12 3 12 4 n-2 n-1 n  a,  =  n ( - l ) ir  A.  n  i-l"  1  (4-19)  1  Let the transformation T matrix be w r i t t e n as T  =  [ T  R  T  2  ,T  3  , ... T ] q  where T., , T , T - , . . . T are the column vectors of T matrix. 1 2 3 n 0  (4-20) From (4-15)  and (4-20), we have T  F = o  F  T  (  33.  [T  or  T, T  r  2  ... T ] F Q  3>  Q  =  F[T  1 5  1^  ?  y  ... T j  (4-21)  From (4-16) and (4-21), we have T  =  n  G  (4-22)  Hence, we can compute T^, T , 1^ ... 2  by using the f o l l o w i n g recursive  formula FT • i-i n-i+1  T . = n-i  A  +  MI G n-i+1  i ==1, 2, 3, ... n-1  (4-23)  and F T  +  :  =  0  (4-24)  The condition of (4-24) may not be met due to the accumulated computation errors.  Let T ^  T  T , ... T^ be the computed r e s u l t s and T^  2>  3  T  2>  T  3>  ... T be correct values n F T  1  F T  + a + o  1  G  =  0  G  =  ee  (4-25)  and the e r r o r T  - T  Then  =  n  =  1  -F  - 1  (4-26)  ±  e  (4-27)  Similarly,  and  F T  2  + a  2  G  =  T  F T  2  + a  2  G  =  T  n n  2 2  =  F- l ^  =  F  (4-28)  1  (4-29)  - 1  r  =  x  r  Therefore, n. l  _ 1  n. . i - l  A.  T. l  =  i  n. + T. l  i = 2, 3, 4, ... n  (4-30)  34.  4.3  Linear Optimal Control Design The system equations i n canonical form were Y  =  F  Y + G  o  U  o  (4-14)  The c h a r a c t e r i s t i c equation of the open loop system i s IXI - F I o 1  =  1  X  n  + ct A n  + a  n _ 1  . X~ n-1 U  2  + . . . + a. • 1 •\  (4-31) y\  ^  Let the desired eigenvalues of the closed-loop system be A^, X^, A^, ... X^ The new c h a r a c t e r i s t i c equation w i l l be ( A - A . ) (A - A ) (A - A J ... (A - A ) 1 z -- 3 n =  A  + a A " n  n  11  1  + a  .A" n-1 n  2  + . . . + a, 1  =  0  (4-32)  Since c h a r a c t e r i s t i c equation of the closed loop system i s |AI - (F - G S )| o o o 1  1  =  A  + (a + 3 ) A n n  n  n - 1  + (a _ + 3 J n-x n-x  A~ n  2  + ... + (a, + B j X l  =0  (4-33)  where U  =  -S  Y  Q  (4-34)  Equating (4-32) to (4-33) gives a. - a. X  and  S  =  o  N  3.  =  X  i = 1, 2, 3, ... n  (4-35)  X  [B-, 3,, 3o, ••• 3 J 1 2. 3J> n  (4-36)  F i n a l l y , the l i n e a r optimal c o n t r o l l e r i n measurable s t a t e v a r i a b l e s U  I  -S  U  =  -S  o  Y  or o  T"  1  Z  (4-37)  35.  4.4  S t a b i l i z a t i o n of SSR Because of the number of state v a r i a b l e s which can be measured,  the 22nd order and 19th reduced order models are used f o r the l i n e a r optimal c o n t r o l design.  Eigenvalue a n a l y s i s shows that a l l the important  mechanical and e l e c t r i c a l eigenvalues are e s s e n t i a l l y unchanged; Table 4-1. Single mechanical mode s t a b i l i z a t i o n At 30% compensation and 0.9 power f a c t o r lagging of the reduced 22nd order system without s t a b i l i z e r , the 204 rad./sec. or 32.5 hertz mechanical mode i s e x c i t e d and has negative damping (eigenvalues with positive real part).  By u t i l i z i n g the design procedure described i n t h i s  chapter, an optimal c o n t r o l l e r can be designed to s h i f t the eigenvalues from +0.1541 ± j204.35 to -6.500 ± J204.35 and another mechanical mode which i s barely stable from -0.08805 ± j8.4938 to -6.000 ± j8.4938. A l l eigenvalues are s t a b i l i z e d as shown i n Table 4-2.  The c o n t r o l l e r i s 7 . "  (7.823Aco , 0.0964AG , -183.005Aco, -8.801A6, 192.487Aco , -3.398A6 , 1  1  3  3  65.534AC0-, 7.448A6.v -1.336A6,-, -47.478Aco^, -2.738A6,, -29.746AP, 4 4 5 6 6 28.645A1 , 1 . 4 5 4 A i , 1.475A1 , -5.931Ai„, -5.929Ai„, -60.094AV , t f D Q G c £  35.987AV  , -0.000260AV,,, -0.00259AE J1 c ' R FD 17  t  S t a b i l i z a t i o n of two mechanical modes simultaneously For the same system but with 50% compensation, two mechanical modes were e x c i t e d +0.1560 ± j202.68 and +0.9101 ± j161.42 were e x c i t e d simultaneously.  Another optimal c o n t r o l l e r i s designed to s h i f t the  two mechanical modes to -6.500 ± J202.68 and -3.500 ± J161.42 as shown i n Table 4-3.  The c o n t r o l l e r i s (1.124Aco , -4.959^^ 1  189.462A6, 13.843Aco  3>  -23.848Aio,  -321.30lAe , 24.286Aco , 147.568A6 , -6.018Aa) 3  4  4  5>  4.467A0 , -9.927Au , -25.064AG , -26.719AP, 29.533Ai , -1.030Ai , -1.007Ai , 5  6  &  - 6 . 7 1 l A i , -6.712Ai„, -22.422AV , 31.169AV n  fc  f  , -0.000278AV , R  D  -0.00264AE„ ). n  36.  Low Order Stabilization Design Although the two controllers designed by the procedure presented in this chapter have been proved to be effective i n stabilizing the system, the damper currents are not directly measurable.  S t i l l another linear  optimal controller i s designed for the system, without.the need for damper currents.  The equations associated with the damper windings are dropped,  resulting i n a 19th order system.  The controller i s (1.84Aio^, l.OlAO^,  -41.51Au), -30.63AS, 54.5lAco , 39.93A6 , 7.37Au , -5.77A0 3  3  4  4>  -6.41AOJ  5>  -3.32A0 , -7!46Au>, -2.16A0 , -1.66AP, 2.63Ai , -0.872Ai , -2.26AV , 5  6  6  t  f  c  -2.35AV. , -0.000295AV^, -0.00274AEO, and the eigenvalues of the system tc K £D with and without the controller are shown in Table 4-4.  Finally the con-  troller i s tested on the original system for various degrees of compensation.  The results are plotted in Fig. 4-1. It i s found that the con-  troller designed for the 19th order model with 30% compensation, not only can stabilize the original 27th order system for 30% compensation but also can stabilize the original system from 10 to 70% compensation. proves the effectiveness of such controller design in wide-rangecompensation multi-mode SSR stabilization.  This  37.  original system  reduced 22nd model  reduced 19th model  -0.1818 ± J298.18  -0.1818 ± j298.18  -0.1818 ± J298.18  +0.1541 ± j204.35  +0.1541 ± J204.35  -0.2290 ± J203.22  -0.2496 ± J160.72  -0.2496 ± J160.72  -0.2273 ± J160.66  -0.6706 ± J127.03  -0.6706 ± j127.03  -0.6677 ± J127.03  -0.2877 ± j 99.21  -0.2877 ± j 99.21  -0.2627 ± j 99.14  -0.0479 ± J8.4801  -0.0881 ± J8.4938  -0.2266 ± j7.9054  -7.0224 ± J542.80  -7.0224 ± J542.80  -4.8208 ± j514.02  -6.1984 ± j209.20  -6.1984 ± j209.20  -3.6580 ± J238.75  -8.4404  -8.4858  -8.0056  Synchronous  -31.920  -31.920  Machine Rotor  -25.404  -25.404  -1.9830  -2.1855  E x c i t e r and  -499.97  -499.97  -499.52  Voltage Regulator  -101.91  -101.91  -93.682  Shaft modes  X  <5 co  Stator/Network  -0.1417 Turbine and  -4.6160  Governor  -3.0336 -4.6732 ± jO.6269  Table 4-1  Eigenvalues of o r i g i n a l system and reduced order models without c o n t r o l l e r at 30% compensation and P = 0.9 p.u. at 0.9 power f a c t o r lagging.  38.  reduced 22nd order model without controller  reduced 22nd order model with controller  original system with controller  -0.1818 ±:j298.18  -0.1818 ± J298.18  -0.1818 ± j298.18  +0.1541 ± j204.35  -6.5000 ± J204.35  -6.5000 ± J204.35  -0.2496 ± J160.72  -3.5000 ± J160.72  -3.5000 ± J160.72  -0.6706 ± J127.03  -0.6706 ± J127.03  -0.6706 ± J127.03  -0.2877 ± j 99.21  -0.2877 ± j 99.21  -0.2877 ± j 99.21  -0.0881 ± j8.4938  -6.0000 ± J8.4938  -6.2367 ± j8.4158  -7.0224 ± j542.80  -7.0224 ± J542.80  -7.0224 ± J542.80  -6.1984 ± j209.20  -6.1984 ± J209.20  -6.1984 ± j209.20  -8.4858  -8.4858  -9.6038  Synchronous  -31.920  -31.920  -31.923  Machine Rotor  -25.404  -25.404  -25.404  -2.1855  -2.1855  -1.5570  E x c i t e r and  -499.97  -499.97  -499.97  Voltage Regulator  -101.91  -200.00  -200.00  Shaft modes  X 6  to  Stator/Network  -0M404 Turbine and  -4.8741  Governor  -2.8538 -3.9883 ± J2.9898  Table 4-2  Eigenvalues of reduced 22nd order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r at 30% compensation and P =0.9 p.u. at 0.9 power factor lagging.  39.  reduced 22nd o r d e r model without controller  reduced 22nd o r d e r model with controller  original system with controller  -0.1818 ± j298.18  -0.1818 ± J298.18  -0.1818 ± J298.18  +0.1560 ± J202.68  -6.5000 ± J202.68  -6.5000 ± J202.68  +0.9101 ± j161.42  -3.5000 ± j161.42  -3.5000 ± J161.42  -0.6799 ± J127.08  -0.6799 ± J127.08  -0.6799 ± j127.08  -0.3545 ± j 99.49  -0.3545 ± j 99.49  -0.3545 ± j 99.49  -0.2958 ± J9.5621  -6.0000 ± J9.5621  -6.4682 ± J9.7544  -7.0799 ± J591.16  -7.0799 ± J591.15  -7.0799 ± J591.15  -6.8387 ± J161.47  -6.8387 ± J161.47  -6.8387 ± J161.47  -8.1783  -8.1783  -10.953  Synchronous  -32.808  -32.808  -32.793  Machine Rotor  -25.423  -25.423  -25.424  -2.1030  -2.1030  -1.0891  E x c i t e r and  -499.97  -499.97  -499.97  Voltage  -101.76  -200.00  -200.00  S h a f t modes  A 6 co  Stator/Network  Regulator  -0.1384 T u r b i n e and  -4.8909  Governor  -3.1454 -2.9767 ± J4.0177  Table  4-3  E i g e n v a l u e s o f reduced 22nd o r d e r model w i t h / w i t h o u t c o n t r o l l e r and o r i g i n a l system w i t h the c o n t r o l l e r a t 50% compensation and P =0.9 p.u. a t 0.9 power f a c t o r l a g g i n g .  40.  reduced 19th order model without controller  Shaft modes  A  6  oo  Stator/Network  reduced 19th order model with controller  original system with controller  -0.1818 + J298.18  -0.1817 + J298.18  -0.1818 + J298.18  -0.2290 + J203.22  -6.5000 + J203.22  -0.4968 + J203.19  -0.2273 + jl60v66  -3.5000 + j160.66  -0.2790 + J160.52  -0.6677 + j127.03  -0.6676 + j127.03  -0.6697 + J127.04  -0.2627 + j 99.14  -0.2628 + j 99.14  -0.2770 + j 99.22  -0.2266 + j7.9054  -6.0000 + j7.9054  -0.1092 + j8.7874  -4.8208 + J514.02  -4.8208 + J514.02  -7.1255 + J542.54  -3.6580 + j238.75  -3.6582 + J238.75  -5.9600 + J209.43  -8.0056  -8.0056  -25.4025  Synchronous  -2.8136 + J0.2572  Machine Rotor  -2.8136 - J0.2572 -1.6473 + J391.91  E x c i t e r and  -499.52  -499.52  -773.59  Voltage Regulator  -93.682  -200.00  -1.6473-,— J391.91  -0.1401 Turbine and  -4.6414  Governor  -0.2592 -4.7914 ± J0.9552  Table 4-4 Eigenvalues of reduced order model with/without c o n t r o l l e r and o r i g i n a l system with the c o n t r o l l e r a t 30% compensation and P = 0.9 p.u. at 0.9 power f a c t o r lagging.  7D0  £-500 4 3 2  500  t-400  •300  w6, 6  [-250  5, 5  u  4  cd cq  6  32S-2D0  %°4  5 6 7  •150  co1,°1 u  3, 3 6  s  50  I  25  t-  5  r-  1  15  co, 6  a,P HP T  rrrTT  ^300- TO  -20-10  -9  4-1-The e f f e c t of capacitor  (SHy^S^S....?  ii  i | i i i ir I I u | i  -7  -6  i  TfrrJJ; 11 i i | i n  -S  -4  ^  !. j -2  111 iTrfr ?!''''"' l ' 0 1 5  |  -3  -1  3  4  compensation w i t h c o n t r o l l e r f o r P = 0.9 p.u. a t 0.9 power respectively correspond to 10,20,30,.. . 70% compensation)  42.  5.  CONCLUSIONS  A high-order power system model f o r subsynchronous resonance studies i s developed.  The model includes mass-spring system, synchronous  machine, s e r i e s capacitor compensated  transmission l i n e s , turbines and  governor, voltage regulator and e x c i t e r . pipq  The t r a n s i e n t terms p^^ and  are included. From eigenvalue a n a l y s i s , i t i s found that by changing the  degree of compensation the frequency of the e l e c t r i c a l mode w i l l be changed and t h a t , i n some cases, even more than one mechanical mode can be e x c i t e d at the same time. When a conventional lead-lag supplementary e x c i t a t i o n c o n t r o l f o r the s t a b i l i z a t i o n of small o s c i l l a t i o n s i s included, i t has an adverse e f f e c t on the other mechanical modes close to the small o s c i l l a t i o n mode. [22],  Such f i n d i n g i s i n agreement with other previous work  When the damper impedance i s decreased, i t does reduce the p o s s i -  b i l i t y of SSR under i d e a l conditions [5]. L i n e a r optimal  coritrollers-.h^^d /upon--aa-'-earlier-.- develpped  method [8] are designed.  Two c o n t r o l l e r s are designed with a reduced  :  :  22nd order model and one with a reduced 19th order model and the l a t t e r controller  not only can s t a b i l i z e the o r i g i n a l 27th order system f o r 30%  compensation, but also can s t a b i l i z e the system f o r wide-range compensat i o n and multi-mode SSR.  43.  REFERENCES [1]  IEEE Task Force, "Analysis and Control of Subsynchronous Resonance", IEEE P u b l i c a t i o n 76CH1066-0-PWR, IEEE, New York, 1976.  [2]  L.A. K i l g o r e , L.C E l l i o t t and E.R. Taylor, "The P r e d i c t i o n and Control of S e l f - E x c i t e d O s c i l l a t i o n s due to Series Capacitors i n Power Systems", IEEE Transactions on Power Apparatus and Systems, V o l . PAS 90, pp. 1305-1311, May/June 1971.  [3]  M.C. H a l l and D.A. Hodges, "Experience w i t h 500 KV Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating 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.  [4]  IEEE Committee, "Proposed Terms and D e f i n i t i o n s f o r Subsynchronous Resonance i n Series Compensated Transmission Systems", IEEE P u b l i c a t i o n 76CH1066-0-PWR, pp.55-58, 1976.  [5]  R.G. Farmer, A.L. Schwalb and E l i Katz, "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 76CH1066-0-PWR, pp.55-58, 1976.  [6]  L.A. K i l g o r e , D.G. Ramey and M.C. H a l l , " S i m p l i f i e d Transmission and Generation S t a t i o n System Analysis Procedures f o r Subsynchronous Resonance Problems", IEEE P u b l i c a t i o n 76CH1066-0-PWR, pp.6-11, 1976.  [7]  C o l i n E.J. Bowler and Donald N. Ewart, " S e l f - E x c i t e d T o r s i o n a l Frequency O s c i l l a t i o n s with Series Capacitors", IEEE Trans, on PAS, Vol. PAS 92, pp. 1688-1695, Sept./Oct. 1973.  [8]  B. Habibullah and Yao-Nan Yu, " P h y s i c a l R e a l i z a b l e Wide Power Range Optimal C o n t r o l l e r s f o r Power Systems", IEEE Trans, on PAS, V o l . 93, pp. 1498-1506, Sept./Oct. 1974.  [9]  IEEE Task Force, "Symposium on Adequacy and Philosophy of Modelling: Dynamic System Performance", IEEE P u b l i c a t i o n 75CH0970-4-PWR, IEEE, New York, 1975.  [10]  F.P. deMello, "Power System Dynamics - overview", IEEE P u b l i c a t i o n 75CH0970-4-PWR, pp. 5-15, 1975.  [11]  Charles Concordia and Richard P. Schulz, "Appropriate Component Representation f o r the Simulation of Power System Dynamics", IEEE P u b l i c a t i o n 75CH0970-4-PWR, pp. 16-23, 1975.  [12]  P.L. Dandeno, " P r a c t i c a l A p p l i c a t i o n of Eigenvalues Techniques i n the Analysis of Power System Dynamic S t a b i l i t y Problems", Canadian E l e c t r i c a l Engineering Journal, V o l . 1, No. 1, pp. 35-46, 1976.  [13]  W.A. Tuplin, "Torsional V i b r a t i o n " , Pitman, England, 1966.  [14]  IEEE Committee Report, "Dynamic Models f o r Steam and Hydro Turbines i n Power System Studies", IEEE Trans, on PAS, V o l . PAS 92, pp. 19041915, Nov./Dec. 1973.  44.  [15] W. Janischewskyj and P. Kunder, "Simulation of the Non-Linear Dynamic Response of the Interconnected Synchronous Machines", IEEE frans^ on PAS, Vol. PAS 91, pp. 2064-2077, Sept./Oct. 1972. c  [16] P.L. Dandeno, P. Kunder and R.P. Schulz, "Recent Trends and Progress in Synchronous Machine Modeling in the Electric U t i l i t y Industry", Proceeding of IEEE, July 1974. [17] E.W. Kimbark, "Power System Stability", (Vol. I l l , pp. 57-60), Wiley, New York, 1956. [18] IEEE Committee Report, "Computer Respresentation of Excitation Systems", IEEE Trans. PAS, Vol. PAS 87, pp. 1460-1470, June/July 1968. [19] F.P. deMello and C. Concordia, "Concepts of Synchronous Machine Stability as Affected by Excitation Control", IEEE Trans. PAS, Vol. PAS 88, pp. 316-329, Mar./Apr. 1969. [20]  IEEE Task Force, "First Benchmark Model for Computer Simulation of Subsynchronous Resonance", IEEE Publication F 77 102-7.  [21] W. Watson and M.E. Coultes, "Static Exciter Stabilizing Signals on Large Generators - Mechanical Problems", IEEE Trans. PAS, Vol. 92, pp. 204-211, Jan./Feb. 1973. [22] V.M.' Raina, W.J. Wilson and J.H. Anderson, "The Control of Rotor Torsional Oscillations Excited by Supplementary Exciter Stabilization", IEEE Publication A76 457-2. 1  [23] H.A.M. Moussa and Y.N. Yu, "Optimal Power System Stabilization Trough Excitation and/or Governor Control", IEEE Trans. PAS, Vol. PAS 91, pp. 1166-1174, May/June 1972. [24] Y.N. Yu, K. Vongsuriya and L.N. Wedman, "Application of an Optimal Control Theory to a Power System", IEEE Trans. PAS, Vol. PAS 89, pp. 55-62, Jan./Feb. 1970. [25] J.A. Anderson and V.M. Raina, "Power System Excitation and Governor Design Using Optimal Control Theory", Int. Journal of Control, Vol.' 12, pp; 289-308,1972. '  

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