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The dynamic stability of voltage-regulated and speed-governed synchronous machines in power systems. Vongsuriya, Khien 1964

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THE DYNAMIC STABILITY OP VOLTAGE-REGULATED AND SPEEDGOVERNED SYNCHRONOUS MACHINES IN POWER SYSTEMS  by  KHIEN VONGSURIYA B.E., C h u l a l o n g k o r n U n i v e r s i t y , 1960  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  In the Department of E l e c t r i c a l Engineering  We accept t h i s t h e s i s as conforming t o the standards r e q u i r e d from c a n d i d a t e s f o r the degree o f Master o f A p p l i e d Science  Members of the Department of E l e c t r i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h SEPTEMBER 1964  Columbia  In the  presenting  Columbia, I agree  available mission  f o r reference  f o r extensive  representatives.  cation  Department o f  the L i b r a r y  and s t u d y ,  fulfilment of  agree  that  f o r f i n a n c i a l gain  ^-/UAyn<^i^  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  Columbia,  '  that  per-  f o r scholarly  by the Head o f my Department  permission.  V4 <> *t  s h a l l make i t f r e e l y  I further  I t i s understood  of this thesis  w i t h o u t my w r i t t e n  that  copying of t h i s thesis  p u r p o s e s may be g r a n t e d  Date  i n partial  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f  British  his  this thesis  o r by  copying or p u b l i -  shall  n o t be a l l o w e d  ABSTRACT  The constant are  dynamic  stability  i n t e r e s t f o r many y e a r s .  Concordia, Heffron, I n most s t u d i e s ,  e f f e c t were n e g l e c t e d . to  o f a power system has b e e n o f  s t u d y t h e dynamic  including  Phillips,  Among t h e many and M e s s e r l e .  the t i e l i n e  resistance  stability  e f f e c t s s u c h as t i e l i n e  saliency, voltage  and  and g o v e r n o r time c o n s t a n t s  The  important  significant  limit  stability  region  stabilizer and g a i n .  significant  r e s u l t s found  Two d i f f e r e n t namely, t h e  resistance  and s a l i e n c y upon t h e  studies  using  studies  and upon t h e  t h e D - p a r t i t i o n method.  are t h e e f f e c t s of t h e machine s h o r t the e x c i t e r and s t a b i l i z e r  and  coordination.  and t h e i r  constants  i n the study are the  r a t i o - , :"the g o v e r n o r , gains,  time  and t h e D — p a r t i t i o n method.  from Routh-Hurwitz c r i t e r i o n from  resistance,  have b e e n applied„  e f f e c t s of t i e l i n e  stability  Also,  regulator,  stability  Routh-Hurwitz c r i t e r i o n  thesis  o f t h e power system i n t h e s m a l l  reactance,  methods f o r s t u d y i n g  and t h e s a l i e n c y  An a t t e m p t has b e e n made i n t h i s  a l l important  gains,  contributors  i  time  circuit  constants  ACKNOWLEDGEMENT  The a u t h o r w o u l d l i k e  to express h i s sincere  gratitude  to h i s s u p e r v i s i n g p r o f e s s o r , D r . T. N. Yu who g u i d e d a n d i n s p i r e d him throughout t h e course  of the r e s e a r c h .  a l s o w i s h e s t o t h a n k members o f t h e Department Engineering their  e s p e c i a l l y D r . F . Noakes  generous a s s i s t a n c e The encouragement  gratefully  and D r . A. D. Moore f o r  o f D r . B. B i n s o n o f T h a i l a n d  i s also  acknowledged* t o t h e Columbo P l a n  i n Canada f o r t h e s c h o l a r s h i p g r a n t e d and t o t h e N a t i o n a l  financial  of E l e c t r i c a l  and e n c o u r a g e m e n t .  The a u t h o r i s i n d e b t e d  1964  The a u t h o r  support  t o him d u r i n g  Research Council  of the research.  Administration 1962 t o  o f Canada f o r  TABLE OP CONTENTS Page Abstract T ab l e List  *o.oso.eo.eo.o...oo««ooo. o ooooooooooooooe  o f Contents  ..o.oo.eo..•«<>».  Of I l l u s t r a t i o n s  • ,» a *  a  .  «  o  «  1  .  ooooo.«*«o»ee»oooooo.ooo«oo«  v  AcknOWled-gement « « « a a o a o'o o a .«oo.eo«»ooooooooooooooo 1»  Introduction  2.  Review  • a s a a a o o o e•a »*a•*• aa o o o a o o e s a o » o s  of General  Theories  ». *«• • o o. o o o o <>.o a.. :  2#1  Classical  S y n c h r o n o u s Machine  2»2  Stability  o f a D y n a m i e a l System  T h e o r y .... .........  8  ...........  8  2.2.2  S t a b i l i t y o f d system w i t h l i n e a r i z e d d i f f e r e n t i a l equations  8  The R o u t h - H u r w i t z s t a b i l i t y c r i t e r i a «.*aaaa•*aaa.............  9  D - p a r t i t i o n method f o r two r e a l  par am e t e r s ...... ......oooo........  10  D e r i v a t i o n o f System E q u a t i o n s f o r t h e Dynamic Stab i i i t y  Studi e s  .....»«•.«•«..«...........•.  3*1  The System Under  3*2  System E q u a t i o n  Study  . . o . . . . . . . . . . . . .  .o«s.*«eae>aa«o<>o<>...<>....o  3.2.1  The s y n c h r o n o u s machine  3.2.2  The t x e l i n e  3.2.3  The v o l t a g e r e g u l a t o r  15  and t h e  o f The System E q u a t i o n s  3»4  The C h a r a c t e r i s t i c E q u a t i o n o f t h e System  3»5  The R o u t h - H u r w i t z D e t e r m i n a n t s o f t h e  Equation  The D - p a r t i t i o n E q u a t i o n s  System P a r a m e t e r s and I n i t i a l S y n c h r o n o u s Machines  i. i.  14  aaa«a*aaa..o.o.os..oa  Linearization  3«6  13  14  3*3  Characteristic  13  ..........  g O V e r no P ..aaaaa a* a'aaoooooo.ooooaa  4,  5  Stability  2.2.4  i n the small  5  2.2.1  2.2.3  3»  v ii  ...  15 16 17  o . o ........ o  20  *•.............  21  Conditions of  o . . . . . . . i . i.. . . . . . . . . . . . . .  23  1  Page 4.1  System P a r a m e t e r s  •«..••«•«•»« o « . o ... ... « .  Synchronous Machines Voltage  4.2  23  »«««• « . . « . » o <>. . ... . o  Regulator Settings  •.•••»»«.•«•..  23  G o v e r n o r Settings.*••«•••••••«»».»».•««.o  23  I n i t i a l C o n d i t i o n s of Salient--pole S y n c h r o n o u s Machine «•• ....... ...........  24  5. : Dynamic S t a b i l i t y S t u d i e s U s i n g R o u t h - H u r w i t z C r i t e r i o n .............................. » 5«1  Effect  5*3  . . . . . . . . . . . . . . . . . . . . . . . . . . . a . . .  27  E f f e c t of the Short C i r c u i t R a t i o of a S y n c h r o n o u s Machine . . . o . . . . . . . . . . . . . . . . .  29  Effect  of the T i e L i n e  31  5.3.1  Effect  5.3.2  The  Impedance  ........  of the l i n e r e s i s t a n c e  effect  re ac t a n c e  o f the  o «-« .  1  ....  . » . . . « ^ . e . . . . . . . . . . .  Effect  of E x c i t e r  5.5  Effect  o f t h e G o v e r n o r Time C o n s t a n t s on  31  Time C o n s t a n t .........  Dynamic S t a b i l i t y L i m i t  5.5.1  31  line  5.4  the  6.  34  .............  34  G o v e r n o r w i t h one t i m e c o n s t a n t  34  5.5.2 G o v e r n o r w i t h two l i n e c o n s t a n t s .. Dynamic S t a b i l i t y S t u d i e s U s i n g t h e D^ p a r t i t i o n Method «. . . . . « » « . « « . « « . « . o . . . . o . . . . . . 6.1 The E f f e c t o f S a l i e n c y on t h e S t a b i l i t y BoUIlcLclPi GS  6*2  6.3  The E f f e c t Reactance  « a o o o e « o 0 e 9 « » « * « « o o o  « a a  o a a « a  of the T i e L i n e R e s i s t a n c e  o  o  N e g l e c t i n g B o t h the T i e L i n e  The E f f e c t  of Short  6.5  The E f f e c t  of the E x c i t e r  6*6  The E f f e c t Constant * The E f f e c t  ...  C i r c u i t Ratio Time C o n s t a n t .  of the S t a b i l i z e r  40 40 42  Resistance  i n a R o u n d - r p t o r Machine  6*4  37  and  a « a o o o « « « » « « O i « « ' » * a « o « « o « e o « « 0 »  and S a l i e n c y  6*7  27  o f S a l i e n c y on t h e Dynamic  S t ab i i i t y  5.2  23  44 44 44  Time  B « O « O O « « « « O O . « « « * « « « « » O O O O O O O O « O  o f t h e G o v e r n o r Time C o n s t a n t s  4T 47  Page 7.  ..«•«• ...... o ............  Conclusions  APPENDIX I . APPENDIX I I .  Symbols Voltage  and U n i t s Regulator  52 54  a n d Speed  Governor T r a n s f e r F u n c t i o n s  ........  57  APPENDIX I I I . D e r i v a t i o n o f t h e C h a r a c t e r i s t i c Equation APPENDIX I V .  59  D e r i v a t i o n of the D - p a r t i t i o n Equations  REFERENCES  ............ ... .  o « » . . . . . . .  .  .  .  .  .  .  .  .  .  0  .  . . . . . . . . . . . . . . . . o o o . . . . . . . . . . . . . . . . . . . .  iv  .  62 64  L I S T OP  ILLUSTRATIONS  Figure  P  e  Synchronous  3»1  S c h e m a t i c o f System U n d e r S t u d y  3-2  V o l t a g e Phasor Diagram o f t h e I n f i n i t e Bus and a S y n c h r o n o u s Machine  13  S a l i e n c y E f f e c t on Power L i m i t s w i t h o u t Governor ................................  28  S a l i e n c y E f f e c t on Power L i m i t s w i t h a O n e - t i m e - c o n s t a n t G o v e r n o r ..............  28  S a l i e n c y E f f e c t on Power L i m i t s w i t h a Two-time-constant Governor  30  S h o r t C i r c u i t R a t i o E f f e c t on Power L i m i t s ..................................  30  T i e L i n e R e s i s t a n c e E f f e c t on Power L i m i t s ..................................  32  5-2  5-3  5-4  5-5  5—6  Tie Line at  Resistance E f f e c t  U n i t y Power F a c t o r  ..............  g  1—1  5-1  Machine Models  a  ....  on Power  13  Limit  ...................  32 33  5-7  Tie Line  Reactance E f f e c t  o n Power L i m i t s  5—8  Tie Line  Reactance E f f e c t  on Power  5—9  a t U n i t y Power F a c t o r « . . . . . . . . . . . . . . . . . . E x c i t e r Time C o n s t a n t E f f e c t on Power  5—10  5—11  5—12  Limits 33  E x c i t e r Time C o n s t a n t and G a i n E f f e c t on Power L i m i t s .............o . . . . . . . . . . . . . .  35  E f f e c t of the Qne-time-constant Governor on Power L i m x t s . * « • « . . . . . . . . . . . . . . o o o . . .  36  Effect  o f t h e G a i n o f a One-t t i m e - c o n s t a n t  Governor 5—13  6  Effect  of  on Power L i m i t s the  ................ ^  36  T w o - t i m e - i C o n s t a n t Governor  on Power L i m i t s  38  E f f e c t o f t h e G a i n o f a Two—time~*constant G o v e r n o r on Power L i m i t s  38  6- 1  Saliency Effect  .  41  6—2  S a l i e n c y E f f e c t on t h e S t a b i l i t y R e g i o n f o r V a r i o u s Power L i m i t s . . . . . . . . . . . . . . . .  41  5— 14  on t h e S t a b i l i t y R e g i o n  v  Figure 6-3 6-4 6-5 6-6  Page T i e L i n e R e s i s t a n c e E f f e c t on t h e S t a b i l i t y R e g i o n •.«.«••«••• ;  6—8  6-9 6-10 6—11  6-12  43  T i e L i n e Reactance E f f e c t on the S t a b i l i t y R e g i o n ..•.......••••«..«........ a......  43  E f f e c t of Neglecting Tie Line Resistance and S a l i e n c y on t h e S t a b i l i t y R e g i o n ...  45  E f f e c t of Short C i r c u i t Stability  6-7  •»  R a t i o on t h e  Region  45  E x c i t e r Time C o n s t a n t E f f e c t on t h e S t a b i l i t y R e g i o n o f a Round-rotor Machine  46  E x c i t e r Time C o n s t a n t E f f e c t on t h e S t a b i l i t y Region of a Salient-pole Machine . . . . . . . . a o . . . . . . * . . « « . . . . . . . . . . .  46  S t a b i l i z e r Time C o n s t a n t E f f e c t on t h e S t a b i l i t y Region of a Round-rotor Machine  48  S t a b i l i z e r Time C o n s t a n t E f f e c t on t h e S t a b i l i t y Region of a S a l i e n t - p o l e Machine  48  E f f e c t of the Two-time-rCbnstant Governor on t h e S t a b i l i t y R e g i o n o f a R o u n d - r o t o r Machine ........................  49  E f f e c t of the Two-time-constant Governor on t h e S t a b i l i t y R e g i o n o f a S a l i e n t - p o l e M a c h i n e •• .......»«».....  49  6—13  E f f e c t o f t h e O n e - t i m e - c o n s t a n t Governor on. t h e S t a b i l i t y R e g i o n .................... 51  I  V o l t a g e R e g u l a t o r .......................  vi  59  1.  Since  the  i n t r o d u c t i o n of the  r e g u l a t o r , b o t h t h e o r y and regulated a power  INTRODUCTION  g e n e r a t o r has  practice  continuously-acting voltage have shown t h a t  a g r e a t e r steady  a power system  the  steady  state  stability  falling  generator, t h i s  out  of synchronism.  limit (21  parameters  alone.  22 '  i n t r o d u c e d so t h a t cause  '  v  can be  t o a new The  '  Por  an  i n t r o d u c e d by governor. the  the  system  F o r a machine w i t h a c o n t i n u o u s l y governor,  negative  are  feedback  c o n s t a n t l y monitored  called  stability  t h e dynamic  limit  upon the p a r a m e t e r s  which  and  I n t h i s way,  the  state  stability  of a generator  limit.  i n a jpower  i n the feedback  system  loops  c o n t i n u o u s l y — a c t i n g v o l t a g e r e g u l a t o r and  However, the machine p a r a m e t e r s  system  the  i n c r e a s e d beyond the o r d i n a r y steady  p r i m a r i l y depends  limit  unregulated  the power and v o l t a g e d i f f e r e n c e s ,  power l i m i t  dynamic  load  23)  machine i n s t a b i l i t y ,  machine l o a d  of  in  •  c o r r e c t e d b e f o r e the machine becomes u n s t a b l e .  limit  limit  of a generator i n  state  depends on the machine and  a c t i n g v o l t a g e r e g u l a t o r and  may  limit  i s d e f i n e d as i t s maximum s t e a d y  (2lY  is  stability  system.^^^ Normally,  without  state  a voltage-  t o w h i c h t h e machine  and  the-"nature  i s connected  are  also  important. In the  the d e v e l o p m e n t o f a modern power s y s t e m ,  synchronous  generator to operate  i n t h e dynamic  r e g i o n becomes more and more i m p o r t a n t . t h a t many s y n c h r o n o u s power f a c t o r  T h i s i s due  the  ability  stability t o the  g e n e r a t o r s have t o o p e r a t e w i t h a l m o s t  o r e v e n l e a d i n g power f a c t o r s .  The  of  steady  fact unity  state  2 s t a b i l i t y however, region  l i m i t is  and  small  with  soundness  of  the  of  an  unless  aid  such  under—excited  of  an  i t  a  is  synchronous  operated  voltage  operation  i n  the  regulator.  has  been  generator, dynamic  The  s t a b i l i t y  technical  confirmed  i n  both  theory  p r a c t i c e . ^ » ^ » 1 3 ) To  the  regulated This  generator,  requires  d i f f e r e n t i a l high are  theoretical  order  a  analysis  standard  complete  the  control  description  equations.This  d i f f e r e n t i a l  of  is  theory of  usually  equations  dynamic  the  can  s t a b i l i t y be  system  considered  with  of  a  a p p l i e d . ^ * ^ * ^ with  tedious  complicated  because  coefficients  involved. Simplification  only  one  reactance  of  and  the  one  problem  voltage  is  usually  source  for  made  the  by  using  synchronous  (17) machine.  The  accurate,  because  transients,  and  included  the  results many  obtained,  important  voltage  nevertheless,  changes  regulator  and  like  are  not  voltage  and  governor  actions  current  are  not  (4) i n  equations.  (26) As be  for  used  parameters  for are  Work  has  C o n c o r d i a ^ ^ of  voltage  synchronous accurate  been  operating  at  investigations,  done  regulation were  Park's  equations  since  a l l  can  important  always machine  included.  discussed  equations  machines,  used  unity  Concordia's  i n  this  direction  and  pointed  out  with  respect  to  and power work  the  stability  factor was  was  the  since  1944  important  s t a b i l i t y limit  of  when effects  l i m i t . a  Park's  round-rotor  machine  studied.  f o l l o w e d by  several  others.  Among  (3) them on  were  Park's  Heffron  and  equations,  P h i l l i p s for  the  who  gave  investigation  a of  method, a  also  based  round-rotor  generator operating stability limits  at any  i n the  e s t a b l i s h e d with the Analogue  power f a c t o r .  The  dynamic  l e a d i n g power f a c t o r r e g i o n were  a i d of an analogue computer. computer m e t h o d s ^ ' ^ ' ^ w e r e  developed by Messerle and Bruck, ^ " " ^  and  Aldred  also  and  (7)  Shackshaft  i n t h e i r dynamic s t a b i l i t y  and Bruck had  gone a step f u r t h e r i n t a k i n g the prime mover  control into consideration to determine the  •  studies.  Nyquist c r i t e r i o n  stability limit  Messerle  was  also  employed  of a round-rotor generator. (8)  Messerle's work was  f u r t h e r extended by Goodwin  a network analyser,  determined the  initial  conditions  f o r the  external  stability  ' who,  v  using  impedance and  study  the  of a water wheel  generator. Reports from the C.I.G.R.E. ( ' 1 0  1 : L  » ^  indicated  12  that  s i m i l a r approaches to the problem of dynamic s t a b i l i t y studies been done i n the  U.S.S.R. i n the past two  r e a c t i o n theory f o r the  decades.  synchronous machine has  The  hav  two-  a l s o been  employed. The  previous work of H e f f r o n and  Phillips,  Messerle have been extended i n t h i s t h e s i s . the  A new  study of a s a l i e n t - p o l e machine operating  f a c t o r has  been e s t a b l i s h e d .  The  round-rotor machine, and  which were normally n e g l e c t e d , are Voltage r e g u l a t o r region  are  and  i s used to determine the  The  method f o r  at any  The  power  e f f e c t s of  the t i e - l i n e  taken i n t o  governor e f f e c t s i n the  investigated.  of  method, includes ,.  the round-rotor machine as a s p e c i a l case. s a l i e n c y i n the  and, also t h a t  consideration. under-excited  w e l l known Routh-Hurwitz  stability limits.  resistance^  The  criterion  D-partition  4 (24)  method i s used f o r the i n v e s t i g a t i o n of the e f f e c t of two p a r t i c u l a r parameters on s t a b i l i t y . A d i g i t a l computer i s used f o r the computations.  5  2.  2.1  REVIEW OP GENERAL THEORIES  C l a s s i c a l Synchronous Machine Theory(22.23,26,27) When the e q u a t i o n s d e s c r i b i n g  the performance of a  synchronous machine are w r i t t e n i n phase c o - o r d i n a t e s , t h e y become l i n e a r e q u a t i o n s w i t h t i m e - v a r y i n g c o e f f i c i e n t s because the mutual i n d u c t a n c e s between the s t a t o r and r o t o r are p e r i o d i c  functions  circuits  of r o t o r angular p o s i t i o n .  I n the case of a s a l i e n t - p o l e machine the mutual i n d u c t a n c e s between s t a t o r phases are a l s o p e r i o d i c of r o t o r a n g u l a r p o s i t i o n .  functions  This causes d i f f i c u l t i e s i n  analysis. (26) Park  i n t r o d u c e d the w e l l known d—q t r a n s f o r m a t i o n  w i t h the f o l l o w i n g 1.  The s t a t o r w i n d i n g s are s i n u s o i d a l l y d i s t r i b u t e d  around - 2.  assumptions:  the a i r gap.  The e f f e c t o f the s l o t s on the a i r gap f l u x  i s not a p p r e c i a b l e w i t h v a r i a t i o n i n the r o t o r 3.  Saturation  angle.  e f f e c t s are n e g l e c t e d . (27)  P a r k ' s t r a n s f o r m a t i o n m a t r i x , j n o d i f i e d by Yu following K a K  b  K c  =  forms 1 /2  Y3  cos ©  - sin 9  , has the K  d  1  cos  (0 - 120) - s i n (© - 120)  K  1  cos  (© + 120) - s i n (0 + 120)  K  (2-1) 0  where K  , K, , K £L  flux the  D  r e p r e s e n t v a r i a b l e s s u c h as  l i n k a g e s or" v o l t a g e s corresponding  axis  currents,  i n phase c o - o r d i n a t e s  v a r i a b l e s i n the  and  ,  ,  direct-, quadrature-and  zero-  coordinates. When t h e  s y n c h r o n o u s machine e q u a t i o n s  by  the  transformation matrix  ( 2 — l ) f r o m the  to  the  d-q-o  time-varying  inductances The  c o o r d i n a t e s , the are  transformation,  in effect,  machine i n t o  direct—  quadrature—axis  and  a zero  the  are  transformed  phase  coordinates  coefficients  of  the  e l i m i n a t e d . "»  w i n d i n g s of the  that  armature  C  r e s o l v e s the  a two-phase model w i t h windings,  a x i s w i n d i n g w h i c h can be  and  analysed  three  phase  the  i n a d d i t i o n to separately.  d-axis  d—axis  q-axis q—axxs  (a)  S y n c h r o n o u s Machine i n Phase C o o r d i n a t e s Fig.  2-1  (b)  S y n c h r o n o u s Machine i n Park's Coordinates  S y n c h r o n o u s Machine M o d e l s  7 equations i n the o p e r a t o r  Park's  v  d  V  V  = Hi "  \&  px  c a n be  written:  a d -  q 0  form  r  (2-2)  i a q  = pw •^o  - r  _ SLEI  i  a o  _  V  tt  v  X  fd  tt  d(p) • .d  a>  (2-3)  q  X  T  0  = - -2- i tt  where  x (p) =  0  x ( l + - C > ) ( 1 + T* p) d  d  d  (1 + < p ) ( l  +r  o  (l +  X X  n^) q  = " (1 9  d o  p)  -£V)  (2-4)  7 ~ ~  + T* P qo^  x  G(p) Hpd  Por d e t a i l s Additional conditions  +f  d  0  P)(l  o f symbols  +^ p) o  see A p p e n d i x  equations r e l a t i n g  are t  2 m  2  d q  ( 2  + T e  .4, 'ar -'v\-.J^v;^»k'-- ^fe..;=: •';  terminal  q  = J p 0 +^pO r  the machine  2^2  d  q * T  I.  :  _  5 )  8  where  i s the t e r m i n a l v o l t a g e  energy c o n v e r s i o n  2.2  and T , e  the  electromechanical  torque.  S t a b i l i t y of a Dynamical System 2.2.1  S t a b i l i t y i n the s m a l l The system e q u a t i o n s d e s c r i b i n g the performance  of a r e g u l a t i n g p r o c e s s are u s u a l l y n o n l i n e a r equations.  differential  However, i n the s t u d y of a synchronous machine i n  a power system, a s m a l l d e v i a t i o n from an i n i t i a l e q u i l i b r i u m point i s usually  considered.  L i a p o u n o f f has s h o w n ^ ^ * " ^ t h a t the s t a b i l i t y of a non2  l i n e a r system under s m a l l d e v i a t i o n s from e q u i l i b r i u m ^ i§ > 0  c o m p l e t e l y determined by the l i n e a r l y p e r t u r b a t e d  system  e q u a t i o n s which are the f i r s t v a r i a t i o n of the d i f f e r e n t i a l equations.  There are a l s o c r i t i c a l cases when the method  cannot be a p p l i e d .  Then e i t h e r a l l l i n e a r p e r t u r b a t i o n  terms  v a n i s h or the c h a r a c t e r i s t i c r o o t s of the l i n e a r i z e d system e q u a t i o n s possess a zero r o o t or a p a i r (or p a i r s ) of p u r e l y imaginary r o o t s . I n power system dynamic s t a b i l i t y s t u d i e s , however, i t has been assumed t h a t the r e g u l a t i n g system i s s t i l l  a t the  s t a b i l i t y boundary when these cases a r i s e . 2.2.2  S t a b i l i t y of a system w i t h l i n e a r i z e d equations The s t a b i l i t y of a l i n e a r d i f f e r e n t i a l  i s determined by the homogeneous d i f f e r e n t i a l  differential equation  equation th  d e s c r i b i n g the f r e e motion of the system.  For a system of n  o r d e r the e q u a t i o n of t h e f o l l o w i n g form w i l l be o b t a i n e d : a  .n o tt  S  (  t  )  +  a  ,-n-l l ^dt" S  (  t  )  + —  —  • a S(t) = 0 n  (2-5) The s o l u t i o n o f which i s w e l l known as n S(t)  =  A.e  1  i = 1 where  are the r o o t s o f t h e c h a r a c t e r i s t i c e q u a t i o n of the  form a P Q  n , n-1 . + a^p + •>.••.  h =  a  0  (2-6)  The s t a b i l i t y of t h e system i s then determined by the r e a l p a r t of Zc whose n a t u r e can be i n v e s t i g a t e d by a number of c r i t e r i a , w h i c h make i t p o s s i b l e t o judge the s t a b i l i t y w i t h o u t r e c o u r s e t o the computation o f Z^,. 2.2.3  The Routh-Hurwitz S t a b i l i t y  Criterion  I n o r d e r t o ensure t h a t a l l r o o t s o f the e q u a t i o n (2—6) have n e g a t i v e r e a l p a r t s , i t i s n e c e s s a r y and s u f f i c i e n t that, with a  Q  g r e a t e r t h a n z e r o , a l l the d i a g o n a l d e t e r m i n a n t s  of the a r r a y c o n s t r u c t e d from t h e c o e f f i c i e n t s of ( 2 — 6 ) , a  a  l o  a  3 a 2 h  a_ « • • 0 j a. 4 (2-7)  *  6  0  . • • a n  10 be g r e a t e r t h a n z e r o ;  i,e« a  l  3  a  a  a  0  ^> 0, ....  4  a^ • • . 0 a. A  n  a, (2-8)  =  0  0  . . . a n (24)  2.2.4  D - . p a r t i t i o n method f o r two r e a l  parameters  Vhen the e f f e c t of .tvo r e a l parameters on ,th.e4 s t a b i l i t y of a r e g u l a t i n g system i s t o be c o n s i d e r e d and i f the two parameters can be e x p r e s s e d e x p l i c i t l y  i n the c h a r a c t e r i s t i c  e q u a t i o n , the s t a b i l i t y b o u n d a r i e s w i t h r e s p e c t t o the two parameters can be d e t e r m i n e d . Let  the c h a r a c t e r i s t i c e q u a t i o n of the system be e x p r e s s e d  i n the form *"P(p) + l f i ( p )  + B(p) = 0  where f and X. are the two r e a l parameters  (2-9)  under c o n s i d e r a t i o n  and Q ( p ) , P ( p ) and R(p) are p o l y n o m i a l s w i t h c o e f f i c i e n t s t h a t are  independent of C and Z « Por a s t a b l e system a l l t h e zeroes of the c h a r a c t e r i s t i c  e q u a t i o n must be i n the l e f t h a l f z - p l a n e .  I f the v a l u e s of  if and z are a l l o w e d t o v a r y c o n t i n u o u s l y so t h a t a t l e a s t one of t h e zero l o c a t i o n s moves towards and f i n a l l y reaches the  11 i m a g i n a r y a x i s , t h e system i s on the s t a b i l i t y l i m i t and becomes o s c i l l a t o r y .  The l o c u s o f p o i n t s i n t h e  {ti-z)  plane  r e p r e s e n t i n g such a s i t u a t i o n i s c a l l e d the s t a b i l i t y boundary. One  s i d e o f t h e boundary i s t h e s t a b l e r e g i o n which  corresponds  to t h e case o f a l l t h e zeroes o f t h e c h a r a c t e r i t i c e q u a t i o n l y i n g i n the l e f t h a l f  z-plane.  The o t h e r s i d e of t h e boundary i s  the u n s t a b l e r e g i o n c o r r e s p o n d i n g t o the case where a t l e a s t one o f t h e zeroes o f t h e c h a r a c t e r i s t i c e q u a t i o n i s i n t h e right half  z-plane.  T h e r e f o r e , i f a l l t h e p o s s i b l e l o c i which  correspond  to t h e j — a x i s zeroes of t h e c h a r a c t e r i s t i c e q u a t i o n are found i n the (if — z) p l a n e , t h e s t a b i l i t y boundary can be determined from t h e l o c i .  The l o c i would d i y i d e the (tf-z) plane  into  s t a b l e and u n s t a b l e r e g i o n s p r o v i d e d t h a t s t a b l e r e g i o n s e x i s t . To f i n d t h e l o c i of t h e s t a b i l i t y boundary i n t h e  (ti-z)  p l a n e , l e t p = jw i n t h e c h a r a c t e r i s t i c e q u a t i o n , rB(j ) w  + ^Q(j ) + B(3 ) = 0 t t  W  (2-10)  L e t P, Q and R f u r t h e r be s e p a r a t e d i n t o r e a l and i m a g i n a r y parts: P(j«)  P^w)  + jP («0  Q(j»)  Q (»)  + 3'Q («)  R(j«)  R (a>) + jR (a>)  n  2  2  x  2  or tfP^a)  +ZQ (<a) 1  + R-^tt) = 0  sP (a>) + tQ (a>) + R ( « ) = 0 2  2  2  (2-12)  12 wiieuce  Z=  -R^tt) P  1  ( « )  -R (<o)  P  T 2  Q (o>)  2  ( » )  -R^tt)  1  (2-13)  Q (a>)  -R (o>)  2  where  Q («) 1  2  P (w) 1  (2-14)  A = Q (») 2  P (») 2  N o n t r i v i a l s o l u t i o n s ofif andZ e x i s t when A £ 0 .  I fA= 0  the n o n t r i v i a l s o l u t i o n s do n o t e x i s t which means t h a t t h e r e i s no p o s s i b l e p o i n t i n t h e (jT—T) plane which g i v e s an i m a g i n a r y axis root. By v a r y i n g a> from — o o t o ©o a l l t h e l o c i o f t h e j - a x i s c h a r a c t e r i s t i c r o o t can be o b t a i n e d .  I n the computation  it is  s u f f i c i e n t t o v a r y w from 0 t o co s i n c e b o t h i f a n d r are the r a t i o s of odd p o l y n o m i a l s . Q^ &2  a  r  e  f  Note t h a t P ^ *  and R^ a r e even, and P , 2  p o l y n o m i a l s i n tt*  S t r a i g h t l i n e l o c i a r e o b t a i n e d by s u b s t i t u t i n g tt = 0 and to  =oointo  equation  (2-12)  because they s a t i s f y i t .  These  l i n e s i n t h e (f-z) plane represent the c h a r a c t e r i s t i c roots j o andjoo and can, t h e r e f o r e , be p o s s i b l e s t a b i l i t y  boundaries.  A f t e r t h e l o c i a r e o b t a i n e d t h e s t a b l e and u n s t a b l e r e g i o n s can be i d e n t i f i e d by some other  s t a b i l i t y Criterion*  3.  DERIVATION OP SYSTEM EQUATIONS FOR THE DYNAMIC STABILITY  3*1  STUDIES  The System Under Study A power system n o r m a l l y c o n s i s t s o f a number o f  synchronous machines s u p p l y i n g power t o the system a t v a r i o u s points*  F o r the dynamic s t a b i l i t y study o f one p a r t i c u l a r  synchronous machine o f the system, t h e r e m a i n i n g p a r t s of t h e  system are n o r m a l l y c o n s i d e r e d as an i n f i n i t e bus t o  which the machine i s connected through an e x t e r n a l  impedance*  The synchronous machine o f the system under study is  c o n t r o l l e d by a v o l t a g e r e g u l a t o r and a speed g o v e r n o r ,  b o t h o f the c o n t i n u o u s l y — a c t i n g t y p e . A schematic diagram o f the system i s shown i n F i g . 3-1. oil relay system  ~<SH  •turbine  fawrrtar  "speed  rekttnee  3-1  ^ injmite  sensing element.  vaWe  Fig.  t i e . Iine —TnSOT^—wvv.  exciter ffBTHT)  generator ^  —TjtTOfX—VWfV-  MOltagt,  regulator  Schematic o f System Under Study E  Fig,  3-2  V o l t a g e Phasor Diagram o f the I n f i n i t e a Synchronous Generator  Bus and  14  3*2  System E q u a t i o n s  3.2.1  The s y n c h r o n o u s  machine  Park's equations w i l l stability  study.  In order to f i n d  s y n c h r o n o u s machine on s t a b i l i t y additional 1.  assumptions  will  be a p p l i e d t h e major  be made:  c a n be  neglected  compared t o t h e speed v o l t a g e s © U ^  and ©ty, due t o t h e c r o s s a  because  of the  The i n d u c e d v o l t a g e s due t o t h e change o f f l u x  they are small  2.  effects  of the system, the f o l l o w i n g  l i n k a g e s pvj/^ and p\|/^ i n t h e d and q axes because  t o t h e dynamic  The a r m a t u r e  excitations, resistance voltage  d r o p c a n be  neglected  i t i s small. 3.  c o n s t a n t s T . ' , • r f * 7">  The t i m e  d relatively  small, w i l l  W i t h t h e above  v  Y  d  a l s o be  7  do  ^q  T* ^qo  and T _ , b e i n g Dl  neglected.  assumptions,  = -v e P  <o  d  v  <o  f d  d  x (p) V  T  = -  i  q  w  q  e  = V d  " V^q  x-,(p)= x,  ~  t  " (1 + *d>> x  (p) = x  G(p)  q  ad  = Hp(l  +t  d 0  P)  ^  15  The  other  relevant  equations are T = J p © + oc p© + T m r r g  v  3.2.2  - v  2 t  + v  2 4  2  The t i e l i n e  An a s s u m p t i o n w i l l namely t h a t  (3-2)  4  i t sresistance  be made f o r t h e e x t e r n a l impedance,  and r e a c t a n c e  Two more e q u a t i o n s  are  elements are  obtained  constant.  from the steady  s t a t e p h a s o r d i a g r a m , F i g . 3-2,  v., = v  d  v  3.2.3  q  The v o l t a g e Since  causes a small  = v  sini5 + f i , - x i  o o  : d  cosS  n  regulator A V  g(p)  and t h e g o v e r n o r  change i n machine t e r m i n a l  change i n t h e f i e l d  f u n c t i o n of a voltage  -P  c a n be w r i t t e n  small  details  as  A  (3-4)  =  f o r a speed  governor,  =1$  (3-5)  change i n s p e e d o r f r e q u e n c y c a u s e s a  change i n p r i m e mover The  generally  AT  f(p) A  voltage  e x c i t a t i o n , the t r a n s f e r  A s i m i l a r t r a n s f e r f u n c t i o n c a n be w r i t t e n ,  (3-3)  + x i +r i d q  regulator  a small  q  small  input.  of the t r a n s f e r f u n c t i o n s  are g i v e n i n  Appendix I I , 3.3  Linearization  of t h e System  F o r the dynamic vai'iaus v a r i a b l e s taken into from  Equations  stability  around  an i n i t i a l  consideration*  a set of l i n e a r i z e d  s t u d y o n l y s m a l l changes of operating point w i l l  Hence s t a b i l i t y  equations  of the  c a n be original  be  determined system  equations. The The  frequency  instantaneous  acceleration  of the  infinite  bus  i s considered  a n g u l a r displacement,., v e l o c i t y  fixed  and  of the- m a c h i n e , r e s p e c t i v e l y , can be w r i t t e n as  © = .©  + &  o  p© = to + p £  , where p ©  Q  = «  and p © = p 8 Then t h e dynamic whether or not  stability  of the  6" goes t o i n f i n i t y  system  can be  w i t h time  d e f i n e d by  after  a small l o  disturbance. From e q u a t i o n s linearized  system A  V j  Av Av AT  d q id  (3—l)  through  equations =  -V  qo  pA<5  are +  x  q  obtained? Ai  q  =. V , A<5 + x, ( p ) A i , do d d r  =  g(p)Av  x  = f'(p)pA£  (3-5),.the  + G(p)Av„, fd  following  17 Av, = v o cos rSoA6 + r A i d, - x A i q = -v  Av  i  6>AT  o  do 3  s i n S A6 + x A i , + r A i o d q Av,, + i Av + v , A i , + v A i - T pA6 d qo q do d qo q eo  J p A6 + ocpAd + AT Av^ v  By  ^ to  Av, + d  v  Av *  to  = v' Av, + v' Av do"'d ' qo q  defining (3-8)  h(p) £ g ( p ) G ( p ) J(p) A j  2 p  + (  a  - -ja)p - f ( p ) ,  where f ( p ) = p f ( p ) , and e l i m i n a t i n g A T , AT * Av^ from the system e q u a t i o n s , the m  g  f o l l o w i n g f i v e homogeneous equations are o b t a i n e d : 1  0  -r  -v cos 6  X  0  0  1  1  0  -h( )v; P  ido  3«4  o  -X  0  -h(p)v^ l +  i  v s i n <5  ~\  V  d  do  0  A v  d  Av 0  q •= 0 Ai  0  x (p)  V  qo  .—r  0  q a>j(p)  V  qo  A<§ (3-9)  The C h a r a c t e r i s t i c E q u a t i o n o f the System T h i s i s o b t a i n a b l e from the c h a r a c t e r i s t i c d e t e r m i n a n t  of the above m a t r i x e q u a t i o n .  The c h a r a c t e r i s t i c determinant  18 A = J(p)[h(p)A _+ x ( p ) A ]  h(p) [ A P  d  +  4  A  + A,  2  + x (p) A p  +  6  d  5  A  + JA p  y  g  + Aql  (3-10)  where A. = -tt v , r x 1 do q A  n  = « fx + x  A  0  =  2  I  ttIr +  - v ( r + x qo  x x  2  I  3  x x ) q  q  x +  2  +  q  A. = Y -v' ( v , x + v r + i ( r +x 4 qo doj do qo qo < A  5  c  = v  J ^ qoj^do" v  r  v + i,„(r + ' q o " ' "do  x ).  2  x  ) + ') 2  s i n S v., l r x — v ' -(v, (x+x ) + r ( v + i , x )) qo \ do q qo do q'J do qo q o  o  + v cos S  o  o  v ' x (v, + i x) - v ' do q do qo qo\  do  r + x(v  qo  +i )l do / J A  A, = Y (v - i, x + i r) 6 qo qo do qo A „ =-v sinS 7 o A A  8  =  Y do  + v  qo  A  9  Q  {  _ - v ^ r + v x - x, ( r do qo do  2 2 (r +x )+ xx  + i, r x do q  + x )  r + i., r x s i n 8 v , (x+x ) + v do q qo do q o o  = v  "V c o s  o  Since  i (x+x ) + v cos S (v - i , x + i r) o qo q o o qo do q qo  5  o  -v, r + v x + do qo  the t r a n s f e r f u n c t i o n  i x x do q n  of the v o l t a g e  -H (1+7P) B  g (p) = {  1+ ( T  e  +  V s  )  p  +  regulator (3-11)  V s ?  2  ]  19 and  the t r a n s f e r  function —u  f(p)  "  =  (1+  o f the speed  governor  m Tl  TjpHl+^p)  (3-12)  or f(p)  (i+rp)(n-Tp)  J ( p ) and h ( p ) f u n c t i o n s  become  o J ( p ) = J p + (a - - § £ )  u p  T  2  P  +  ( l + ^ p ) ( l + r p) 2  x h(p)  = -  ad  A (1 + < P ) |  +  r  0  e ^  )  2  , / A ad where u, = u, — ^ — , *e ^e Rp Therefore, the c h a r a c t e r i s t i c  A =  J  p  2  +  (  a  „ ^P_  determinant a p  ) p  -M-e ( 1 + tP ) A X  +  (i+r )(i+r p) lP  x + —  (1+? )A —  (  2  1+  r  d>H  1+  ( r  e  +  ^ >  +  e V '  r  P  a  (  1  +  ^o  p  + A.  (i ^ p){i (r  )  +  A p+ A 6  d  <  i + r  From e q u a t i o n  ?  o  +  A p+ g  A  q  e +  r/ )p T rp' s  +  A P+ 4  (3-14)  d>> (3—14), the c h a r a c t e r i s t i c  A  e  equation i s obtained?  5  20 7 a  o  P  6  +  a  l  P  +  a  2  P  5 3  +  a  P  +  a  4 4  3  P  +  a  5  P  2 +  6  a  p  +  a  7  =  (3-15)  0  where a  o  = T J B. a 1  a, = T B + T, J B, 1 a 4 d 1 a,  = T B_+ a 5  a-  =  T  a V  T J B + T,B . - jx c 1 d 4 e r  T  b  J  B  l  +  T  c 4 B  - ^ ( J A . V a.  =  T  b 4 B  +  c V  T  T  6  d 3 B  =  T  b V  T  c 3 B  a,  T  a. For 3.5  B  b V 3  "  %  (  B  > W e V  + 5  T )  l T 2  s  ^ { V V V s *  +  +  -,;{JA  B  n  +  V I W V E )  E  r  d  T  B r  V (J W a  +  A, J ^ 1 1 2 s  B  A T  1 +  5  E +  2 " ^'eV  2  (  W s >  B T ) 6  +  F  - ^ <  A 5  J B  2 e*s T  I  ~ B  +  B  A  i s) r  4  V  B  6  )  +  B  5  ^e 5 A  d e t a i l s of the d e r i v a t i o n ^  see A p p e n d i x I I I .  The R o u t h - H u r w i t z D e t e r m i n a n t s o f t h e C h a r a c t e r i s t i c Equation R o u t h - H u r w i t z D e t e r m i n a n t s c a n be w r i t t e n a  ]  *< 0 0 0 0 0  a,  a„  a„  0  0  0  a,  a  a.  0  0  0  ]  a.  a,  a,7  0  0  a,  a  a,  0  0  0  a-,  a.  a,-  a  0  0  a0  a.  a,  a,  0  0  0  a.  a-  a  a.  a  a  0  4  P  as  (3-16)  21 Separately w r i t t e n , they  A  A  l  =  2  =  a  i a ] L  a  -  2  A  4  =  a  4 3  A  5  =  a  5 4 ~  A  +  A  a a Q  ~~  SL^&2  =  are  2  A  a  3  i^o ' ^  a  i  (  a  w  A  e r e  o  l 6 " 0 7 " a  a  3 6 3 a  A  +  A  a  6  + a a 2  2  a  The  D-partition  To f i n d  the  a  on  used.  procedure,  characteristic explicitly  then  +  a  )  +  0 5  a  a  a  0 5 0 a  A  l 6 5 " a  a  a  l 4 ?)~ 0 3 7 0 a  a  a  ?  a  a  A  2  0  7  1  a^^)  6  a  O 6 5^ 7 a  a  a  '  2  (3-17)  6  Equations  stability,  of r e l a t i v e  v a l u e s of two  real  the R o u t h — H u r w i t z C r i t e r i o n  however, i s r a t h e r t e d i o u s .  e q u a t i o n of the  system  c a n be  real  can a l s o  be  When the  expressed  parameters,  the  D-partition  applied.  characteristic  e q u a t i o n o f the  t h e D — p a r t i t i o n method can be and  a  as t h e f u n c t i o n o f two  method can be The  2 5  a  ~  a  a a -(a a -  = a A  ?  effect  parameters The  jA + Q  i a  A 3.6  a  l 4  a^ayA^+ a^a^ayA^*^ A ^ ^ a ^ a ^ a ^ — a 2^7  a^A^—  =  a  2^ 2 3 7  -a-^ (a^ a^-a^a^.)  A  =  separated into  real  first and  system  under study  w r i t t e n as e q u a t i o n  i m a g i n a r y p a r t s as  by  (2—9)  equation  (2-12). In interest,  terms o f t h e \i and \i , w h i c h a r e o f e s e q u a t i o n (2-12) becomes  particular  (/ P (tt) - j/ Q (tt) + Rn (tt)  =0 , (x P.(tt) - (/ Q„(tt) + R.(tt) = 0 s  1  e  1  1  (3-18)  22 The this  stability  e q u a t i o n and Detailed  Appendix I I I .  boundaries  c a n be  by v a r y i n g tt f r o m  derivation  determined  zero  of e q u a t i o n  to  by  solving  infinity.  (3-18) i s g i v e n i n  23 4.  SYSTEM PARAMETERS AND I N I T I A L CONDITIONS  OF SYNCHRONOUS  MACHINES  4.1  System The  Parameters system under study has parameters  w i t h the f o l l o w i n g  values. Synchronous Machines. will  behave i n  values  Synchronous machines o f v a r i o u s r a t i n g s  e x a c t l y t h e same way as l o n g as t h e p e r u n i t  o f t h e machine p a r a m e t e r s a r e t h e same.  analysis  the d i r e c t -  and q u a d r a t u r e — a x i s  s y n c h r o n o u s machine and t h e t i e l i n e f°  r  reactances of the  impedance w i l l  the f o l l o w i n g parameters which w i l l J  be k e p t  be v a r i e d  except  constant:  = 120 TIH= 4000 = 2000  p.u.  = 600  p.u.  a = 3.0  p.u.  r  d0  T  d  Voltage settings  I n the  Regulator  Settings.  The v o l t a g e r e g u l a t o r  a r e assumed, u n l e s s o t h e r w i s e U  = 800  p.u.  = 800  p.u.  specified,  t o be n o r m a l l y  e r  s 30 *e r  s  Governor S e t t i n g s . functions For time  Two d i f f e r e n t  governor  are considered, unless otherwise a governor  constant  w i t h one t i m e  r e s p e c t i v e l y are  transfer  specified.  constant,  t h e g a i n and t h e ,  :  24  T ^ = 377  p.u.  t  p.u.  = 0  For a governor  w i t h two  b  T^=  4.2  T = 2  300  Initial  p.u.  are  time  constants u = 5 m r  and  assumed.  C o n d i t i o n s o f S a l i e n t - f r O t e S y n c h r o n o u s Machine  A new method f o r f i n d i n g i n i t i a l c o n d i t i o n s , i ^ > "^qo' •V * M*-, , "V » v „ , and 8 f o r a s a l i e n t - p o l e synchronous qo do' qo ido o 0  machine w i l l  be  (3—3) we  do'  developed.  Consider to  V  the  steady  state.  From P a r k ' s  equations  (3-l)  have v„ do  = x  v  = ^  qo  i  q R  p  P = v, do  (4-1  qo v „ , - x, i , fdo d do  (4-2  i , + v do qo  (4-3  Q = v i , - v, * qo do do v, do  = v  v  = v qo  i i  qo  (4-4  qo  o  sin S + r i ,- x i o do qo  o  cos o  v , = v to do 2  2  6 + x  + v  i , + ,do  r  (4-5  i  (4—6 qo (4-7  2  qo  equations v, . v , i , , i , ^ do* qo' do qo'  In these  unknown w h i l e ^. > P V  Q  w i t h 7 unknowns.  a  n  d  From  Q are g i v e n . (4-3)  and  P = i , i x + v do qo q qo  There  v o, A ~o ,  are 7  u  v  fdo  are  equations  (4-1) i  qo  (4-8)  25 From  (4-4)  and  (4-1)  .2  ,— x Q = v qo l do q From  (4-7)  and  L2 ~  qo  ~  \l  x  2  2  q  (4-10)  qo  2  (4-9) ±  _  = d  i  a—ap_  0  (4-n)  qo  V  (4-11) and (4-8) ( Q P  From  ±  to  Q + x  From  (4-9)  qo  (4-1)  V  From  1  /  * q X  =  .2 q°  1  Y^  x  \ i  + v  q qo  o  i  (4-12)  qo qo  (4-12) and (4-10) (Q + x i ) q n> / ~2 _ x 2 9 ± 2.o F to q qo 2  p  =  x  i + /v q. q.° /  -  2  t  o  2 .2  x i q  qo  i  qo  X  (4-13) Hence P I  v, to  (4-14)  qo  V  =  X  do  (4-15)  1  q qo  2  2 y  qo  to  (4-16)  do  •2  + x i „ v * a—a°.  ^•do  Q  V  .  Q  °  (4-17) R  (v x , i , J -x2 " q o - "d~do' ad  fdo  (4-18)  consequently  V  do  V  qo  = -v = v  do  qo  (4-19) (4-20)  26 Next f r o m  (4-5) and  vo =  \  V  (  (4-6)  r  A  do  ~  r  1* do +  x  1  ) + (v qo 2  - x i qo  — r i do  ) qo 2  (4-21) S  All  = arc t a n (^ ~ qo do  Q  the i n i t i a l  I  of  *  have b e e n  The e x t e r n a l  included.  (4-22)  The  t o a s a l i e n t - p o l e machine  as a r o u n d - r o t o r m a c h i n e , t h e l a t t e r  the former.  fe) qo  c o n d i t i o n s have t h u s b e e n f o u n d .  method i s g e n e r a l and a p p l i c a b l e well  fet do  t i e line  being a special  reactance,and  as case  resistance  27 5.  5.1  DYNAMIC S T A B I L I T Y STUDIES USING ROUTH-HURWITZ CRITERION  E f f e c t of Saliency The  on t h e Dynamic  synchronous s a l i e n c y  Stability  i n a r o u n d - r o t o r machine has (14-15)  usually show  b e e n assumed t o be z e r o , b u t t e s t s on a c t u a l  that  saliency  some s a l i e n c y  f o r round—rotor  i s present.  machines  Thus t h e a s s u m p t i o n  machines i s n o t v a l i d .  o f non—  I n the  f o l l o w i n g , t h r e e c a s e s w i l l be s t u d i e d . The f i r s t c a s e , x = x , c o r r e s p o n d s t o an i d e a l r o u n d - r o t o r machine; t h e s e c o n d d q c a s e , x = 0.85 x , a r o u n d — r o t o r machine w i t h s a l i e n c y e f f e c t d q considered; typical  salient-pole Fig.  dynamic factor  and t h e t h i r d  stability  normal  to a  machine*  5-1 shows t h e e f f e c t o f r o t o r  saliency  o f t h e system u n d e r u n i t y  operation.  counted  c a s e , x,= 0.6 x , c o r r e s p o n d s ' d. q  The a c t i o n  o f t h e speed  b y s e t t i n g fj, = 0, w h i l e  on t h e  and l e a d i n g governor  power  i s dis-  the voltage regulator  has t h e  settings.  Curve 1 i n d i c a t e s t h e dynamic s t a b i l i t y l i m i t o f t h e i d e a l r o u n d - r o t o r machine w i t h x ='x = 1.0, p.u., c u r v e 2 t h e d q s t a b i l i t y l i m i t o f t h e r o u n d — r o t o r machine w i t h a s a l i e n c y effect limit seen  considered  (x = 1, x ..= 0 . 8 5 ) , and c u r v e 3 t h e s t a b i l i t y d q o f t h e s a l i e n t - p o l e machine (x,= 1, x = 0 . 6 ) . I t c a n be d q  that  the r o t o r  saliency  increases  l i m i t , e s p e c i a l l y under l e a d i n g family without  of curves t i e line  corresponds included.  t h e dynamic  power f a c t o r o p e r a t i o n .  shown i n d o t t e d l i n e s c o r r e s p o n d s resistance  stability The  t o t h e system  ( r = 0 ) , and t h o s e shown i n s o l i d  t o t h e system w i t h t h e t i e l i n e  resistance  lines  28  0.5  Power  0.5  u  y)  P 1.0  Limits  2.0  without  1.5  P-U.  Governor  2.0  PU.  with a one - time - constant Governor  Similar effect and  computations  included,  and  The  results  constant,  the  are  with  summarized  while  same c o n c l u s i o n  s p e e d g o v e r n o r of the  i n F i g , 5—3 can  be  Effect The  the  o f the  Short  Circuit  short  circuit  ratio  modern d e s i g n  i t has  drawn t h a t  s y n c h r o n o u s machine i n c r e a s e s  5.2  repeated  governor  in Pig.  5-2  5—3. I n P i g . 5-2  time  the  are  two  the  of  time  one  constants.  saliency effect  dynamic  Ratio  s y s t e m has  stability  a  limit.  o f a S y n c h r o n o u s Machine  a s y n c h r o n o u s machine to — . d  i s approximately equal  of  A high  of  short  x  circuit  ratio  circuit  machine  c o s t s more t h a n one  limited  stability  voltage  r e g u l a t o r , but  considerably 5—4  set  by  limit  using  shows t h e  stability indicate  limit the  a system w i t h  the  as  of the  Z^=  of the  short  1.35  limit  short  the  short  a voltage  regulator  circuit  regulator.  ratio The  on the ;  dotted  voltage  regulator  lines  are  circuit  ratio  regulator  can  When can  i n the  i s equal  hardly  the be  a  those  regulator.  same p . f .  lines  a system w i t h o u t  solid  limit  have  increased  voltage  i s u s e d the  a t the  be  voltage  machine.  power a t u n i t y p . f .  p.u.  will  machine w i t h  r e g u l a t o r , and  ratios  w i t h o u t a modern  3000 t o a p p r o x i m a t e  example, when the  acting voltage as h i g h  stability  a continuously-acting  p.u.  circuit  a continuously-acting  machine w i t h o u t  b e y o n d 0.9  short  f o r an u n d e r — e x c i t e d  and  e  For  the  power l i m i t  a t \i' = 0.01  low  i f i t i s operated  effect  continuously-acting  •—^  a low  ratio. Machines d e s i g n e d w i t h  Fig.  with  be  for  to loaded  continuouslyincreased  dynamic  to  stability  0.5  = 1  X  r  =0.1  Soliency  Ejfect  P.U.  - 30,Ms  7e  =Zs =800  P.U.  Zi  =Z  P.U.  2  =300  1. Xot - 1, X ,  Short  Circuit  on. Power  Power  Ratio Limits  =1  2. Xrf * 1, Xq = 0-85  U  =  Xq = 0-6  Limits'-  with a Two- time - constant Governor.  Ro\ 5 -.4  = 5,Mm = 5  Me  3  Fi^5-3  30  2.o p.u.  15  1.0  Effect on  31 region.  5.3  E f f e c t of the T i e Line 5.3.1  E f f e c t of the l i n e It  tie  line  limit.  Larger  value  limits  Closer  a r e e x t e n d e d by i n c r e a s i n g t h e r e s i s t a n c e the reactance  i s kept  regulator  constant  and g o v e r n o r a r e  The r e s u l t s a r e shown i n F i g . 5-5 and 5-6.  e x a m i n a t i o n o f F i g . 5-6 f u r t h e r r e v e a l s  resistance  on s t a b i l i t y  limit  The e f f e c t o f t h e l i n e Fig.  studies  by v a r y i n g  difference  a t u n i t y power f a c t o r  reactance  5-7 and 5-8 show t h e r e s u l t s o f the value  of t i e l i n e  between the s t a b i l i t y  reactance.  l i m i t with line  stability The  resistance  included  and t h a t w i t h o u t is. i n s i g n i f i c a n t , when t h e . ,  tie  reactance  line  i s comparatively  f i c a n t when t h e r e a c t a n c e  that  the e f f e c t  w i t h d i f f e r e n t degrees of s a l i e n c y . 5.3.2  "the  stability  a r e o b s e r v e d when t h e t i e l i n e  I n a l l cases the v o l t a g e  settings.  the l i n e  e f f e c t on t h e dynamic  i n the computations.  f r o m 0 t o 0.3 p . u . w h i l e  a t normal  and  stability  studies  a t 1.0 p . u .  of  has a d i s t i n c t  i s included  The  resistance  i s o b s e r v e d f r o m F i g . 5-1, 5-2 and 5-3 t h a t t h e  resistance  resistance  Impedance  r e s u l t s of studies the smaller  stability  limit.  i s large.  small.  B u t i t becomes  F i g . 5-8 e s p e c i a l l y shows  f o r u n i t y power f a c t o r .  the t i e l i n e  signi-  reactance,  I t also  reveals  the l a r g e r i s the  32  \  i.o  1  '2/  2.0  r.u  Xi  3  ^  /  = 1, X q = 0 . 8 5  *30,Ms-  S,Mm*S  ?e.*-7* = 8 0 0  P.U  /  = Z  r  o.o  2.  f  =  0.1  3.  f  •  0.3  X--1.0  Tie  Line on.  Resistance  Power  Effect  Limits.  1.  X i  = 1, Xq. = 1  2.  X i = 1 , Xq  = 0-8  3.  X i  «= 0.6  - 1,  P.f.  0.25  0.5  = 300  2  r  (i-  rip. 5-5  0-75  Xq  =  1,0  10  r Fi£L 5-6 Power  Tie Limits  Line at  Resistance Unity  P.U.  Effect  Power Factor.  on  P.U.  33  34 5,4  Effect  of E x c i t e r  The from  about  exciter  of c o n v e n t i o n a l e x c i t e r  and g o v e r n o r e  time  r  effect  time  on s t a b i l i t y  E f f e c t of the Governor Stability Limit Governor  stability  limit  shows t h e s t a b i l i t y power f a c t o r  tends  to increase  By  and k e e p i n g t h e g a i n  &  However, t h e maximum shows t h a t  for a  occurs i n the  gain.  w i t h one time  constant  governor  of a synchronous  limit  limit  Time C o n s t a n t s on t h e Dynamic  c o n s i d e r a b l y by making t h e time  has a s i n g l e  time  g e n e r a t o r c a n be  constant small.  o f t h e system under  constant,  improved  F i g . 5-11  unity  and l e a d i n g  o p e r a t i o n as a f f e c t e d by t h e v a l u e o f t h e g o v e r n o r  constant.  The s o l i d  and t h e d o t t e d l i n e s  lines  a r e f o r t h e r o u n d - r o t o r machine  are f o r the s a l i e n t - p o l e  s e t s , o f c u r v e s show t h a t the  Z  Z^ o f 800 p . u . t h e maximum power l i m i t  When t h e s p e e d  time  limit  o f two o t h e r s t u d i e s .  limit.  r e g i o n o f low v o l t a g e - r e g u l a t o r  the  I t c a n be seen t h a t f o r  F i g . 5-10 a l s o  Q  5.5.1  of the  i n t h e v a l u e o f Z seems t o have a e  depends on b o t h (/ and Z »  5.5  of varying  constant.  5-10 shows t h e r e s u l t s  \i = 30, t h e i n c r e a s e e  fixed  constant.  g a i n l/ = 30 t h e s t a b i l i t y  the v o l t a g e - r e g u l a t o r  favourable  constants v a r i e s  P i g . 5-9 shows t h e e f f e c t  are kept  with increasing exciter Fig.  time  c o n s t a n t w h i l e the other parameters  particular  varying  Constant  0.5 t o 3 s e c o n d s .  time  regulator the  range  Time  fast  improvement o f t h e system  acting dynamic  governors  machine.  Both  are d e s i r a b l e f o r  stability.  35  5-10  Exciter Effect  on  Time  Constant  Power  Limits.  and  &ain  36  05  1.0  2.0  1.5  P.U.  X « 1, f - 0,1  \  \\ Mi = 3 0 , Ms \ \ Mm = 25  =5  = Zs = 800 P.U  ~frZe '/"/  P.U.  i. ?i = too , z  2  =0  l.  2  =0  7, = 3 7 7 ,  3  3. Z i = 2 0 0 0 , Z  -0  t  = -I , X<j =• 0 6  Xi = 1 ,  Ro\ 5-11  Effect Governor  5-12  Effect  =  1  of tKe One - time-constant on. Power  of the Gain  Crovernor  *q  on  of a Power  Limits.  One-time-constant Limits.  37 Fig. stability larger  5-12 shows t h e e f f e c t  limit.  I t c a n be s e e n t h a t  i s the s t a b i l i t y  constant  i twill  limit.  have l i t t l e  can be s e e n by c o m p a r i n g Fig.  of governor  g a i n s on t h e  the l a r g e r  the g a i n the  F o r a governor with a large  effect  on t h e s t a b i l i t y  limit.  c u r v e 1 o f F i g . 5-1 w i t h c u r v e 1 o f  Governor  w i t h two t i m e  constants  When t h e s p e e d g o v e r n o r has two e q u a l time effect  on t h e s y s t e m  ment i n t h e s t a b i l i t y time  limit  stability  Curve  stability governor  limit, time  limit  an e x p e c t e d governor from  g i v e s the s m a l l e s t  machine.  the s o l i d  of the r o u n d - r o t o r  i n t h e same f i g u r e  Very  This  lines  little  indicate  improvement i n time  constant  1000 p . u . t o - z e r o .  5—14 shows t h e s l i g h t  gain.  While  limit  , the d o t t e d l i n e s  machine f r o m  increase  stability  occurs f o r a  i s o b t a i n e d by r e d u c i n g t h e g o v e r n o r  a salient-pole Fig.  c o n s t a n t , between z e r o  computations,  the s t a b i l i t y  those of the s a l i e n t - p o l e stability  time  c o n s t a n t v a l u e o f 200 p . u .  machine  improve-  i n F i g . 5-13 shows t h a t t h e l o w e s t  among t h e f o u r  i n F i g . 5-13 i n d i c a t e synchronous  line)  The  1000 t o as low as 100 p . u .  i s a value of governor  3 (solid  changes.  constants  i s v e r y s m a l l even when t h e g o v e r n o r  the h i g h e s t p o s s i b l e v a l u e , t h a t  limit.  of  dynamic  c o n s t a n t s are decreased from  Moreover,there and  This  5-11. 5.5.2  its  time  of s t a b i l i t y  decrease obtained instead of limit  w i t h an i n c r e a s e i n  i s c o n t r a d i c t o r y to the r e s u l t s  obtained  t h e s t u d y o f t h e system w i t h a g o v e r n o r w i t h one time  constant. The  above c o n c l u s i o n s s u g g e s t t h a t  on t h e dynamic  stability  limit  the governor  must be i n c l u d e d  effect  i f the governor  F t ^ . 5713  Effect  of the Two - time-constant  G-overnor on. rower  i.o  0.5  1.5  2.0  1  2  Limits.  |  \  v  II  -f.O  jf  p. u.  x = i,r=o.i p.u. Ml ^ JO , Ms * 5 7e = 7s - 800 P U . Z i = Z s 300 P U . a  Jl.^mHO X e l = 1 , * « ' 0 . 6 | 2 . JMn* 5  X<M,X,=1 2.0  Fip^.  5-14  Effect  constant  of  the S a i n Governor  o} a  Two-time  on Power Limits.  is the be  represented effect  by  a single  time  o f the  governor  i s r a t h e r s m a l l and  neglected  constants.  i f the  governor  constant.  i s represented  On  by  the  other  hand,  could therefore two  equal  time  40 6.  DYNAMIC STABILITY STUDIES USING THE  D-PARTITION METHOD  Por a g i v e n o p e r a t i n g l o a d c o n d i t i o n , the boundary-of the dynamic s t a b i l i t y of a system can be found i n the  (|x -u. )  plane by u s i n g the D - p a r t i t i o n Method. An <c v a l u e r a n g i n g  from 0 to 0.02  i s found to be  f o r f i n d i n g a l l the r e s u l t s of p r a c t i c a l i n t e r e s t . (0 = 0 t o 0.001 \i s  f  sufficient  Usually  y i e l d s the boundary of maximum p e r m i s s i b l e  and <o = 0.001  to 0.011  y i e l d s the boundary of the maximum  permissible The  s t u d i e s are c a r r i e d out on the system by  varying  system o p e r a t i n g c o n d i t i o n s and a l s o some of the system parameters. I n g e n e r a l , the s t a b i l i t y r e g i o n of i n t e r e s t i n each study i s shown w i t h two boundaries i n the  (\x - j / ) p l a n e . 6  The  S  u p p e r — l e f t boundary w i l l be r e f e r r e d to h e r e a f t e r as the high-p.' s bound and the l o w e r - r i g h t boundary as the h i g h - ^ bound. 6.1  The E f f e c t of S a l i e n c y on the S t a b i l i t y Boundaries Fig.  6-1  b o u n d a r i e s i n the  shows the e f f e c t of s a l i e n c y on the {\i - |/ ) p l a n e .  The  stability  governor a c t i o n i s  e x c l u d e d by s e t t i n g \i = 0. Curves 1, 2 and 3 correspond to x = ^ m q 0.6, 0.85, and 1.0 r e s p e c t i v e l y w h i l e x i s k e p t c o n s t a n t at d unity. I t i s observed t h a t f o r the p o r t i o n of the curves shown, the presence of s a l i e n c y i n c r e a s e s the s t a b i l i t y r e g i o n at the high-p/ bound, but decreases i t i n the h i g h - j / bound.  However,  i n the v e r y low \i r e g i o n the presence of s a l i e n c y decreases the  41  42 s t a b i l i t y r e g i o n a t the h i g h - u ^ bound. Curves 3 and 2 i n P i g . 6—1,  which correspond r e s p e c t i v e l y  to the s t a b i l i t y r e g i o n of a r o u n d - r o t o r machine w i t h and rotor saliency, clearly  without  i n d i c a t e t h a t the s a l i e n c y y i e l d s  a r a t h e r p e s s i m i s t i c r e s u l t f o r the s t a b i l i t y r e g i o n a t the high-|/  e  bound and a r a t h e r o p t i m i s t i c r e s u l t f o r the  stability  r e g i o n a t the h i g h - [ / bound. s F i g . 6-2  shows t h a t f o r the same power l i m i t a s a l i e n t -  p o l e machine must be operated w i t h a h i g h e r [/ g a i n t h a n f o r the r o u n d - r o t o r 6.2  machine.  The E f f e c t of the T i e L i n e R e s i s t a n c e and Reactance F i g . 6-3  shows the e f f e c t of the t i e l i n e r e s i s t a n c e  on the s t a b i l i t y r e g i o n of the system. s e t a t 1.0  p.u.,  The  The  l i n e reactance  is  and the — r a t i o i s v a r i e d .  s t a b i l i t y r e g i o n i s i n c r e a s e d c o n s i d e r a b l y by  i n c r e a s e i n the ^ r a t i o from z e r o .  an  By comparing curves 1 and  2  i n F i g . 6-3 , i t can be seen t h a t a v e r y p e s s i m i s t i c r e s u l t i n the s t a b i l i t y r e g i o n would be o b t a i n e d i f the t i e l i n e r e s i s t a n c e i s completely  ignored.  T h i s i s r a t h e r m i s l e a d i n g and might  r e s u l t i n an i n a p p r o p r i a t e c o o r d i n a t i o n of the v o l t a g e r e g u l a t o r parameters. Curves 1 and 2 of F i g . 6—4 reactance  on the s t a b i l i t y r e g i o n .  decrease i n the t i e l i n e r e a c t a n c e region.  show the e f f e c t of t i e l i n e I t can be seen t h a t the gives a larger s t a b i l i t y  44  6.3  N e g l e c t i n g Both the T i e L i n e R e s i s t a n c e and the S a l i e n c y i n a Round-jiotor Machine Fig.  6-5 shows the e f f e c t of b o t h t i e l i n e r e s i s t a n c e  and s a l i e n c y on the s t a b i l i t y r e g i o n of a r o u n d - r o t o r  machine  o p e r a t i n g a t u n i t y power f a c t o r f o r v a r i o u s power v a l u e s . I t can be seen t h a t the n e g l e c t of b o t h l i n e r e s i s t a n c e and s a l i e n c y g i v e s a s m a l l e r s t a b i l i t y r e g i o n a t the h i g h - ^ bound*, For the s t a b i l i t y r e g i o n a t the h i g h - [ / bound, the e f f e c t of n e g l e c t i n g both l i n e  r e s i s t a n c e and s a l i e n c y becomes v e r y  when x, i s l a r g e or the s h o r t c i r c u i t r a t i o d be seen from F i g . 6-6. 6*4  The E f f e c t of Short C i r c u i t Fig.  i s small.  T h i s can  Ratio  6-6 a l s o shows t h a t the s t a b i l i t y r e g i o n a t the  h i g h — b o u n d decreases w i t h s h o r t c i r c u i t r a t i o . s t a b i l i t y r e g i o n a t the high—u' circuit ratio  great  As f o r the  bound, the decrease i n s h o r t  s u s u a l l y i n c r e a s e s the s t a b i l i t y r e g i o n .  The  solid  l i n e s show the s t a b i l i t y r e g i o n of the r o u n d - r o t o r machine w i t h x a s a l i e n c y —^- = 0.85, and the d o t t e d l i n e s are those of the d x  r o u n d — r o t o r machine w i t h s a l i e n c y and t i e l i n e r e s i s t a n c e neglected. 6.5  The E f f e c t of the E x c i t e r Time Constant Fig.  6-7 shows the s t a b i l i t y r e g i o n of a  round-rotor  machine, x,= x , o p e r a t i n g a t u n i t y power f a c t o r s u b j e c t e d t o d q the v a r i a t i o n of £ .. N o t i c e a b l e e f f e c t s are observed o n l y i n the s t a b i l i t y r e g i o n a t the high-p,^ bound, e s p e c i a l l y f o r low \i  values.  45  1.0  j Xi L 1  I  r  p . 1 p - 1 ?-1  F i o ^6 - 5  Effect  and  Saliency  o f N e o l e c t i n o Tts o nthe  P.U. - 1 ,X, » r - 01 =  0.0  .45 P . U . . 5 5 p.u. .65 P . U .  Line Resistance  o r a b i l iity tv  Re^on.  S.C.R »1 S . C . R • l/l.l S.C.R - 1/i.s . 0 , j £ i . 1.0 ' 'X=1-0 0 . 1 , ^ 1 = 0.819  .  . 1.45 P.U. . - 1  -3 • 3 0 0 Zj-eoo P.u.  .5, t,  Fjo\ 6-6  Effect the  of Short  Stability  Circuit Region .  Ratio  on  2  P.U.  rip 6-a  Exciter Reoijn  Time of  a  Constant Salient-pole  Effect  on  Machine.  trie  Stability  47 S i m i l a r computations are r e p e a t e d f o r a s a l i e n t - p o l e machine i n s t e a d of the r o u n d — r o t o r machine. i n P i g . 6-8  The  r e s u l t s are  which shows t h a t the same c o n c l u s i o n can be drawn f o r low \x v a l u e s s  f o r b o t h types of machine, namely *  the  s m a l l e r the e x c i t e r time c o n s t a n t , the s m a l l e r the s t a b i l i t y a t the high-|/ 6.6  given  bound/  region  ,  The E f f e c t of the S t a b i l i z e r Time Constant The decrease i n T  the s t a b i l i t y r e g i o n .  u s u a l l y r e s u l t s i n an i n c r e a s e i n s The s t a b i l i t y r e g i o n a t the h i g h - ^  bound and the s t a b i l i t y r e g i o n a t the high-p,'  bound i n c r e a s e s decreases which can be seen from F i g . 6-9 and 6-10. How—  as T  s e v e r , f o r a s a l i e n t - p o l e machine, x^= in z \i  1 and x^=  0.6,  a decrease  a y cause a decrease i n the s t a b i l i t y r e g i o n a t the h i g h — s bound f o r low \i v a l u e s , as i s seen i n F i g . 6-10.  6.7  m  The E f f e c t of the Governor Time Constants Fig.  6-11  shows the e f f e c t of a governor w i t h two  c o n s t a n t s on the s t a b i l i t y r e g i o n of a r o u n d - r o t o r x^= q= x  1»0.  time  machine,  I t can be seen t h a t the two-time-constant governor  a f f e c t s the s t a b i l i t y r e g i o n a t the h i g h - u ^ bound but not  the  h i g h — \i bound. s For h i g h \x v a l u e s , the s m a l l e r the governor time c o n s t a n t s , the l a r g e r would be the s t a b i l i t y r e g i o n a t the high—(/  bound.  However, t h e r e e x i s t s a case, T, = T = 0  t h a t y i e l d s the s m a l l e s t s t a b i l i t y The  200  p.u.,  region.  above c o n c l u s i o n a p p l i e s a l s o to the case of a s a l i e n t -  p o l e machine, x,=  1, x = 0.6,  as seen i n F i g .  6-12.  48  49  100  »  T O  00 SO  /  fo  f\f  10  i/ i/ if ii  a 7  <  X « « 1 , X q ' 1 P.U.  x  .  Te  - Z i • 800  .  5  7, = 7 ,  .  ^2 1.  I  ^  4  4  1  2 . 7, -  Cj t,  J.  tl =  4.  7, = 7  5.  7  t  Fig. 6 - 1 1 on  the  «  7 S • 10  P.U  70  o TUU  200 600  -  P.U.  30  -  t  1.45  P.U.  1  P.f..  hi  -01  = ?» -  P  2  5  i,r  MC  Effect  of  Stability  the  Two - time - constant  Report  of  a Round - rotor  G-overnor Machine .  too 90  70 to  40 30  10  <  ft  10 f  x . 1 , r - 01 Zt - ? » . eoo p.u.  e 7  *  Mm  J, 2.3.4  3>  9  <?  4  iii  t  .A  _  • 7, - Z  3.7,  -Z  -  200  4.7,  =7,  =  600  Ft^ on  6 • 12 the  07  •  rl/ I'  0 10  Effect .Stability  Mi. of the  Two - time - constant  flexion  of  a Salient-pole  Governor Machine.  - 30 - 70  z  t  P • P.f -  /» f  5  1. ? , 2.7,  ij  t  .  1.45  1  P.U  50 The e f f e c t o f a g o v e r n o r w i t h one t i m e c o n s t a n t s t a b i l i t y r e g i o n i s shown i n F i g . 6-13. the  s t a b i l i t y region considerably.  The t i m e c o n s t a n t  yf- where a v e r y f a s t a c t i n g g o v e r n o r may s  s t a b i l i t y r e g i o n a t t h e high—\x  changes  A f a s t e r governor u s u a l l y  g i v e s a l a r g e r s t a b i l i t y r e g i o n a t t h e high-p,' low  on t h e  bound.  bound, except f o r cause a s m a l l e r  00 »  60 7o  U  Wft 40 30  20  2  ~V  / /  10  8  >^  7  7  6 S  Xd • X< • 1 P.U. x - 1 , r - o . i p.u. Ze »Z« -flOO P U .  ^/m = 25 1. Zi - o = e 2. T = 100, Z , = 0 3. 7 i = 3 7 7 , Z - 0 2  4  A  3  /  2  *  J  i  » 6 7 a e ID  .,  1  F"Lo\  6-13  Effect on  of the  17  2  ,  /  P = 1.45 P.f =1  •  10  o  One-time - constant  tke Stability  Region .  4. Ti = 2000,Z,- 0 PU.  looo Governor  52 7.  CONCLUSIONS  Prom the r e s u l t s o f t h e s t u d i e s i n t h i s t h e s i s , the p r i n c i p a l c o n c l u s i o n s t h a t can be drawn are as f o l l o w s : 1.  I t has been found from Routh-Hurwitz C r i t e r i o n  s t u d y t h a t f o r f i x e d s e t t i n g s o f c o n t r o l parameters the s a l i e n c y o f a synchronous machine c o n s i d e r a b l y i n c r e a s e s the dynamic  s t a b i l i t y l i m i t o f a power system and from the D-  p a r t i t i o n s t u d y t h a t f o r a f i x e d power l i m i t the s a l i e n c y of a synchronous machine i n c r e a s e s the s t a b i l i t y r e g i o n a t the high—bound  except f o r low [i v a l u e s , b u t d e c r e a s e s i t a t t h e  6  S  high—ji' bound c o n s i d e r a b l y . s 2. The s h o r t c i r c u i t r a t i o of a synchronous machine 1_  has e f f e c t s s i m i l a r t o t h a t o f , • d x  r a t i o , the l a r g e r i s the dynamic  The l a r g e r the s h o r t  stability limit.  i n c r e a s e s the s t a b i l i t y r e g i o n a t the high-|/  e  circuit  I t also  bound and  d e c r e a s e s i t a t the high-[/ bound. s 3.  The t i e l i n e impedance has a s i g n i f i c a n t e f f e c t  upon t h e dynamic  stability.  B o t h t h e s t a b i l i t y l i m i t i n the  Routh—Hurwitz C r i t e r i o n s t u d y and the s t a b i l i t y r e g i o n i n the D — p a r t i t i o n study increase w i t h the t i e l i n e r e s i s t a n c e , but decrease w i t h an i n c r e a s e i n t h e l i n e  reactance. For a f i x e d  r a t i o o f —, t h e e f f e c t o f t i e l i n e r e s i s t a n c e on s t a b i l i t y  limit  i s r a t h e r s m a l l f o r a s m a l l t i e l i n e r e a c t a n c e b u t becomes larger f o r a larger t i e line reactance. 4.  B o t h the s t a b i l i t y l i m i t and the s t a b i l i t y  are g r e a t l y reduced i f the s a l i e n c y and the t i e l i n e are n e g l e c t e d .  region  resistance  A c t u a l l y a l a r g e r ^ r a t h e r t h a n a s m a l l e r , \i  ,  53  g a i n i s p e r m i s s i b l e f o r t h e m a c h i n e when t h e s a l i e n c y and t h e t i e l i n e resistance 5.  are considered*  I t has been found t h a t a l a r g e r e x c i t e r time  gives a l a r g e r s t a b i l i t y  limit  fora particular  parameters, or a l a r g e r s t a b i l i t y  constant  s e t t i n g of  region f o r a particular  However, i n o r d e r t o o b t a i n a maximum s t a b i l i t y  limit,  power.  the time  c o n s t a n t s a n d t h e g a i n o f a n e x c i t e r must be a p p r o p r i a t e l y  coordinated. 6. the a  s  A decrease i n the s t a b i l i z e r time constant  s t a b i l i t y r e g i o n a t the high-|/ bound.  e  increases  bound and a l s o a t t h e h i g h —  However, f o r l o w \L v a l u e s , d e c r e a s i n g t h e s  s t a b i l i z e r time c o n s t a n t may r e s u l t i n a d e c r e a s e o f t h e s t a b i l i t y r e g i o n a t t h e high—u.^ 7.  The g o v e r n o r h a s c o m p a r a t i v e l y  dynamic s t a b i l i t y . its  bound.  F o r a g o v e r n o r w i t h two e q u a l  e f f e c t on t h e dynamic s t a b i l i t y  limit  cases i t c a n e v e n d e c r e a s e t h e s t a b i l i t y w i t h one t i m e c o n s t a n t , large  •  l e s s e f f e c t on t h e constants,  i s s m a l l and i n some limit.  i t s e f f e c t on s t a b i l i t y  F o r high-jx' v a l u e s  time  a t the high-[/  For a governor i s comparatively  bound, t h e s m a l l e r  the governor time c o n s t a n t , the l a r g e r i s the s t a b i l i t y F o r t h e l o w \i p a r t o f t h e h i g h - | / time—constant  region.  bound, however, a s m a l l  g o v e r n o r may d e c r e a s e t h e s t a b i l i t y  the  other hand, the governor time constant  the  stability  region.  has l i t t l e  On  e f f e c t on  r e g i o n a t t h e high-|/ bound. s  The s t u d y  i n t h i s t h e s i s has been c o n f i n e d t o t h e dynamic  s t a b i l i t y i n the small without dynamic s t a b i l i t y  considering nonlinearities.  The  i n t h e l a r g e o f a power s y s t e m i n c l u d i n g n o n -  linearities i s l e f t f o rfuture studies.  54 APPENDIX I . SYMBOLS AND UNITS  S u b s c r i p t o denotes an i n i t i a l  condition  P r e f i x A denotes a small change about the i n i t i a l  operating  value p = ^ £ = time d e r i v a t i v e i ^ # i ^ = armature c u r r e n t s i n d— and q-axes  respectively  v,,v  respectively  = armature v o l t a g e s i n d— and q-axes = armature terminal v o l t a g e  v^  = a p p l i e d v o l t a g e i n f i e l d winding  V  = i n f i n i t e bus-bar v o l t a g e  q  ^d'^q E  =  "k flux l i n k a g e s i n d- and q-axes r e s p e c t i v e l y x , —§— v_, = armature open c i r c u i t v o l t a g e  a r m a  u r e  x^»x^ = synchronous reactance i n d- and q-axes  respectively  x , = mutual reactance between the s t a t o r and r o t o r i n aa d—axis x = t i e - l i n e reactance' between the generator and the bus—bar Rp = f i e l d winding r e s i s t a n c e r  = armature winding r e s i s t a n c e i n d- or q-axis  circuit  r = t i e - l i n e r e s i s t a n c e between the generator and the bus—bar 8 = power angle T  m  = mechanical input torque to the r o t o r  T = energy conversion torque e P = r e a l power output of the machine Q = r e a c t i v e power output of the machine H = inertia  constant  J = moment of i n e r t i a  55  a = damping c o e f f i c i e n t © = i n s t a n t a n e o u s a n g u l a r p o s i t i o n of r o t o r *p0 = a n g u l a r v e l o c i t y o f machine # =. r a t e d a n g u l a r v e l o c i t y f = r a t e d system f r e q u e n c y 7/ = d i r e c t - a x i s t r a n s i e n t do  o p e n - c i r c u i t time c o n s t a n t  X^ = d i r e c t - a x i s t r a n s i e n t s h o r t - c i r c u i t time c o n s t a n t X? - d i r e c t - a x i s s u b t r a n s i e n t s h o r t - c i r c u i t time c o n s t a n t Mo r ^ = q u a d r a t u r e - a x i s s u b t r a n s i e n t o p e n - c i r c u i t time c o n s t a n t o  q u a d r a t u r e - a x i s s u b t r a n s i e n t s h o r t - c i r c u i t time constant = d i r e c t - a x i s damper leakage time c o n s t a n t Te = e x c i t e r time c o n s t a n t = s t a b i l i z e r time c o n s t a n t ^1*^2  =  H  g°  time c o n s t a n t s  v e r n o r  = exciter gain 6 X  fi  r  H  = converter gain = amplifier  gain  £t •  p, , = s t a b i l i z e r g a i n He = H e x ^ r ii  e  = —§— \i  / , H_ = 1  *p +  i  r  e  g  u  l  a  t  o  r  g  f t i n  = over a l l r e g u l a t o r g a i n  e  r  =  o  v  r  l  c  +  A = over a l l s t a b i l i z e r g a i n  Throughout t h e t h e s i s a l l c a l c u l a t i o n s p e r - u n i t system based on M.K.S. u n i t s .  a r e made u s i n g t h e  56  The u n i t of time i s 1 r a d i a n ; a t f = 60 Hz, 1 second = 2nf = 377 r a d i a n s .  Moment of i n e r t i a J = 4n;fH, and p.u. power and  p.u. torque are n u m e r i c a l l y i d e n t i c a l ,  i . e . P = T^  Q  p.u.  57  APPENDIX I I . VOLTAGE REGULATOR AND SPEED GOVERNOR TRANSFER FUNCTIONS  a.  Voltage Regulator Transfer  F i g . I . Voltage  Function^'^  Regulator  The v o l t a g e r e g u l a t i n g system under c o n s i d e r a t i o n i s shown i n F i g . I . I t has t h e f o l l o w i n g p a r t s : 1.  A means o f measuring t h e t e r m i n a l v o l t a g e e r r o r .  2.  A device to convert the voltage e r r o r to a s u i t a b l e  signal (Av ). r  The c o n v e r t e r may be d e f i n e d by an a m p l i f i c a t i o n  f a c t o r (|i ) and zero time 3.  constant.  A s t a t i c or r o t a t i n g a m p l i f i e r with a large gain  (ji ) and a s m a l l time c o n s t a n t 4.  ("C ) .  An e x c i t e r s u p p l y i n g t h e f i e l d c u r r e n t t o a  synchronous machine. 5.  A s t a b i l i z i n g t r a n s f o r m e r f e e d i n g back i n t o the  r e g u l a t o r c i r c u i t a s i g n a l p r o p o r t i o n a l t o the r a t e o f change o f the f i e l d c u r r e n t .  58  Since Av  Av  =-u Av,, converter r t  r  r  F  (i Ay = —-—-$ a m p l i f i e r  a  *a  1+  Av, =  P  Av„,, s t a b i l i z e r  Si S |  and |X  Av., = , r a  6  X  l+rp e  (Av - Av ), f i e l d a  D  the v o l t a g e r e g u l a t o r t r a n s f e r f u n c t i o n  g( ) £  A v  P  Q i + y y  fd A V  where  t  u; = l l  <e b.  r  d+VfHVV^^^eV]  LX [X ,  ex a r r  r  IX = u, , u r  s  "svex  Speed Governor T r a n s f e r F u n c t i o n ^ * * The  speed governor system, i n c l u d i n g the time lags  due t o steam or water and some intermediate a c t u a t i n g member can be s p e c i f i e d by two time constants, and one g a i n . Thus the t r a n s f e r f u n c t i o n of the governor // v A f(p) =  ~ %  (r +i)(r p+i) lP  2  I f one j, of the time constants can be neglected the t r a n s f e r f u n c t i o n becomes (i+^p)  59  APPENDIX I I I .  DERIVATION OP THE CHARACTERISTIC EQUATION  Prom e q u a t i o n  the following characteristic  ( 3 - 1 4 ) ,  equation i s obtained  \P)(l+^p){l+-(VVs>P  '(l+^ P)(l + 0  = p() P  +  ( < 1  l P  p ) ( l + 7  l  p ) ( l +  eV J A 2i  (i+ ^ p ) ( i + r ) (I+T P){i+(z + r n ) p +  ( i + r ) (i+ ^p) 2  T +  i P  e  2  s  jh( ) P  VH  e y +  1 + ( r  e sP ) V 2  T  ) p n  s  (  9  + A  )  = 6  To expand, let VdV 3 do  V W B  3  d =  +  3  A  V d  +  B  4  = V d V  B  5  =  ( A  ^  V  T  .= T  a  r  A  6  + A  T T 2  A  7 d  * 1  9  A  r  ^ T  es  T  8 do  ) K  r  d  +  +  8  A  +  B  A  +  4  l A  (  a  " TT>  9 do T  +  B  2  +  (  a  2  B  "  J  ^  60  Nov  J(p)<l+y> V (l+  |= p T J B  + p  7  a  +P +P  T  a 2 B  1  6 T  -  ( a  ^  ) ( 1+  a { l B  (  - ¥ )  a  +  J  B  2 ;  VV diV - ¥ ^ T  T.JB.+T {B. (a b 1 cl 1 - a l e s ^ T T. (B. (a - -££) M l tt  a  2 K V i V s T T So.) + J B J + T,B„(a - -££•) a 2) d 2 w ' T A (jT_+(a ££•) 7 r j . \ e l ( E » ' 1 2 s) T + JB-1 + T B_(a - -££) + H {B. ( r + r u' )+B r r ) 2J c 2 « ml 1 e s s 2 e sj eo tt•) j + JB,1 B  J  7  1  r  o  y  T  +P  Next c o n s i d e r h(p)(i+r p)(i+r p)(i T p)|i+(r +r ^ ) + r r p f y  d  2  o  - ^ ( P V I ¥ S  1  +  +  P ^ 4 E 3  A  T  2  +  e  A  5 l 2 s T  T  7  s  ) +  P  s  (A P+A )  P (A VA T ) 2  4  5  E  4  + P  (A  5  4 +  A T ) 5  P  61 (l+r^)(i+r p)(i+r p){l+(T +r 1  4 + P  |T A a  3  2 +  P  T  T (A  +  6 d  D  6 +  T  |T (A b  e  6T  4  A  T  7  5  A  T  A  T  Wdo  + P {T A  r  +  C  + {T A P  b  9 +  9+  T (A b  8 +  P  6  P + A  d  9  9  D Q  Hence t h e f i n a l w r i t t e n as e q u a t i o n  )  +  T  dVdo)  (  A  +  r  'V? )  A r ; ) + AgT^J  A+ A r J 8  d  d V 9 do)}  T  +  2  2  Vr)  +  T  P { b 8 do . V V V i o * 3  s  D  6  +  +  +r T p | x ( ) ( A  A^) +T^*A r j  6 +  T  A  ^ J p  A T )}  P a 8 do P { a<V 9 do  + P { a 9 +  c  P ( bW-. c<VVd>  +  =  T A  ? +  2  d  + A  9  form o f the c h a r a c t e r i s t i c e q u a t i o n i s  (3-15)•  ?  )  APPENDIX I V . DERIVATION OF THE D-PARTITION EQUATIONS  Let  T  rz +  r« +  =" T  c  1 2  e s  1  2  e  e q u a t i o n (3-15) becomes P  +  .6J B  _  ... 9  i i V s *r{ l.«V 2 s 4 l 2 s] M l 3 4 l 2 s r  Vl 2 sj T  T  P |B (r r )T:  3  B  3  P  1 +  2  ViV.  5 j  +P {T A  +  B r  E  5+  p T JB  P  7  A  {T B  4 + P  +  A  P { 3  2 + P  t /  5  B c  (T B C  +p(T B P  3  3  6 1 +  T  JA  6 +  A  C  4 +  + B  5  %  5  %  1  B  B  4  +  %  G  3  P  3  r ] + P {T B A  e  T JB P  1  B ) • + B, 2  1  ( T  + T  5 +  B  +  V W l V s ) + A  6 +  T ^ *  ?  T ^ t ]  »r*f * )  1 +  e  s  % l 2 s e} (B  J +  (B  T  + B  5  + B  T  ^mVs)  P { E 6  +  J B  3  5  B l  E +  W +  P  +  W  + B T  S  A X j + p(T A  E  5  2  T  T  W  +  VlVs)  T J  4 +  E  B  P  1 +  T B T  E 3V  + T B  %  T  + B  + B T )  g  W  {T B  T B  3 +  +  P  r  r  5  +  T B  2  s  r  )  Vs  +  P { 3 i 2 s  +  2  +  T  JB  +  T  +B  )r  B T )) 2  E  = 0  The c h a r a c t e r i s t i c e q u a t i o n can be e x p r e s s e d as j/  P ( p ) - |/Q(p) + R(p) = 0  L e t p = j«, we have U { P («) + j P ( « ) J - n | ^ ( « ) + /  ,  S  where  1  2  e  dQ (»)J 2  +{B (»)' + 1  jE («o)J= 2  0  ) r  63  T (») 1  A W . -« ( 3 V 2 2  P (») 2  (  B  °V lV 4 i 2 s ' V l V ^  + <  B  s  ) X  r  5 s r  + B  • {jB (r +r )T .+ +  B  1  2  5 V 2 (  ) T  r  k +  " ^ i V s " ^  6  {  {E6  TT3  T  B  T  B  T 7 { 3 lVs B  + •  *  2  {  + t t 4  B3T3  E V  T  { a 3 T  + B,r • ~4^s  T  T  F 6 B  +  J A  V W l V . ]  +  a V W e )  T  % 2 s)  +  ^  +  VlVsj -  « {V i 4  8  a  ) r  + r  B ^ r ^ ]  5  1  ( r  B  B  V V  +  1  %Vs) T_A_+ F 5  *  +  V  +  5  V  V  +H B z r 1 - « ( T ' B - + T B + B + jx (B + B T ) ' 2  r  m  VT JB A  3  T  B  A  B  +  B  5  0  W e  + 5  5  +  »m  F5  + T' JB  5  " ( P 3 T  «. (T B  1 +  " » { 'o +  I c 3  1 e s ]  +  B  2  V  + 4  4  1+  J  m i 2 e  T BC ] E  V  4  E  "» V (  Vs  ) r  ,  +  A  5  B + p, A , 6 m 1 r  B  1  64 REFERENCES  1.  Concordia, C , "Steady S t a t e S t a b i l i t y of Synchronous Machines as A f f e c t e d by V o l t a g e - R e g u l a t o r C h a r a c t e r i s t i c s " , T r a n s . A . I . E . E . , V o l . 63, 1944, p . 215.  2.  Concordia, C , " E f f e c t of Buck-Boost V o l t a g e Regulator Steady S t a t e S t a b i l i t y L i m i t " , Trans. A.I.E.E. V o l . P a r t I , 1950, p . 380.  3.  H e f f r o n , V..G... and R.A. P h i l l i p s , " E f f e c t o f a Modern Amplidyne V o l t a g e R e g u l a t o r on Under E x c i t e d O p e r a t i o n o f L a r g e T u r b i n e G e n e r a t o r s " , T r a n s * A . I . E . E . V o l . 71, P a r t I I I , 1952, p . 692.  4.  M e s s e r l e , H.F., and R.W. B r u c k , "Steady S y n c h r o n o u s G e n e r a t o r as A f f e c t e d by G o v e r n o r s " , P r o c * I . E . E . , V o l . 102C,  5.  M e s s e r l e , H.K., " R e l a t i v e Dynamic S t a b i l i t y o f L a r g e S y n c h r o n o u s G e n e r a t o r s " , P r o c . I . E . E . , V o l . 103C, p . 234.  State S t a b i l i t y R e g u l a t o r s and 1955, p . 24.  on 69,  of  1956,  6.  M e s s e r l e , H.K., "Dynamic S t a b i l i t y of A l t e r n a t o r s as A f f e c t e d by M a c h i n e R e a c t a n c e and T r a n s m i s s i o n L i n k s " , C.I.G.R.E., V o l . I l l , 1958, No. 315.  7*  A l d r e d , A.S. and G. S h a c k s h a f t , "The E f f e c t o f V o l t a g e R e g u l a t o r s on S t e a d y S t a t e and T r a n s i e n t S t a b i l i t y o f a S y n c h r o n o u s G e n e r a t o r " , P r o c . I . E . E . , V o l . 105A, 1958, p . 420.  8.  Goodwin, C . J . , " V o l t a g e R e g u l a t o r R e q u i r e m e n t s f o r S t e a d y S t a t e S t a b i l i t y o f ¥ a t e r Wheel G e n e r a t o r s C o n n e c t e d t o a System t h r o u g h Long T r a n s m i s s i o n L i n e s " , C.I.G.R.E., V o l . I l l , 1962, No. 301.  9.  Farnham, S*B., and R.W. Swarthout, " F i e l d E x c i t a t i o n R e l a t i n g t o M a c h i n e and System O p e r a t i o n " , T r a n s . V o l . 72, P a r t I I I , 1953, p . 1215.  in A.I.E.E.,  10.  L e b e d e v , S.A., " A r t i f i c i a l S t a b i l i t y of Synchronous C.I.G.R.E., V o l . I l l , 1948, No. 334.  Machine",  11.  Z u c k e r n i c , L.V., "The Compounding of f o r I n c r e a s i n g System S t a b i l i t y " , 1952, No. 340.  12.  V e n i k o v , V.A. and I.V. L i t k e n s , " E x p e r i m e n t a l and A n a l y t i c a l I n v e s t i g a t i o n o f Power System S t a b i l i t y w i t h G e n e r a t o r A u t o m a t i c V o l t a g e R e g u l a t o r s " , C.I.G.R.E., V o l . I l l , 1956, No. 324.  Synchronous Generator C.I.G.R.E., V o l . I l l ,  65 13.  P h i l l i p s R.A. and A.S. R u b e n s t e i n , " O p e r a t i o n o f L a r g e S y n c h r o n o u s G e n e r a t o r s i n t h e Dynamic S t a b i l i t y R e g i o n w i t h Modern Amplidyne V o l t a g e R e g u l a t o r s " , T r a n s . A . I . E . E . , V o l . 95, P a r t I I I , 1956, p . 762.  14.  S h a c k s h a f t , G., "A G e n e r a l - p u r p o s e T u r b o - a l t e r n a t o r M o d e l " , P r o c . I . E . E . , V o l . 110, No. 4, A p r i l 1963, p . 703.  15.  Busemann, F., " R e s u l t s o f F u l l S c a l e S t a b i l i t y T e s t on the B r i t i s h 132 k y . G r i d - S y s t e m " , P r o c . I . E . E . , V o l . 105A, 1958, p . 347. .  16.  B o t h w e l l , F.E., " S t a b i l i t y of V o l t a g e R e g u l s t o r s " , T r a n s . A . I . E . E . , V o l . 69, P a r t I I , 1950, p . 1430.  17.  A d k i n s , B., " A n a l y s i s o f H u n t i n g by Means o f V e c t o r D i a g r a m " , J o u r n . I . E . E . * V o l . 93, P a r t I I I , 1946,  18.  p.  541.  C o n c o r d i a , C., C r a r y , S.B. and E . E . P a r k e r , " E f f e c t o f P r i m e Mover Speed. G o v e r n o r C h a r a c t e r i s t i c s on Power-System F r e q u e n c y V a r i a t i o n s , and T i e - L i n e Power Swings", T r a n s . A. I . E . E . V o l . 60, 1941, p . 559. f  19.  C o n c o r d i a , C. and L.K. K i r c h m a y e r , " T i e L i n e Power and F r e q u e n c y C o n t r o l o f E l e c t r i c Power S y s t e m s " , T r a n s . A * I . E . E . V o l . 72, P a r t I I I , p . 562<  20.  R u e d e n b e r g , R., "The F r e q u e n c y o f N a t u r a l Power O s c i l l a t i o n s i n I n t e r — c o n n e c t e d G e n e r a t i n g and D i s t r i b u t i o n S y s t e m s " , T r a n s . A . I . E . E . , V o l . 62, 1943, p . 791.  21.  C r a r y , S.B., "Power System S t a b i l i t y , V o l s . Wi l e y and. Sons, I n c . j New Y o r k , 1945.  22.  C o n c o r d i a , C., "Synchronous Machine T h e o r y and J o h n W i l e y and S o n s , I n c . * New Y o r k , 1957.  23.  A d k i n s , B., "The G e n e r a l T h e o r y o f E l e c t r i c a l W i l e y and Sons, I n c . , New Y o r k , 1957.  24.  Meerov, M.V., Regulating  I and I I , J o h n Performance, Machines,  John  " I n t r o d u c t i o n t o t h e Dynamics o f A u t o m a t i c of E l e c t r i c a l Machines", B u t t e r w o r t h s , London,  25.  L i a p o u n o f f , M.A., "The G e n e r a l P r o b l e m o f S t a b i l i t y P r i n c e t o n U n i v e r s i t y , P r i n c e t o n , N.J., 1947.  26.  P a r k , R.H., "Two R e a c t i o n T h e o r y o f S y n c h r o n o u s M a c h i n e s , G e n e r a l i z e d Method o f A n a l y s i s " , T r a n s . A . I . E . E . , V o l . 48, J u l y , 1929, p . 716.  27.  Yu, Y.N., "The Torque T e n s o r o f the G e n e r a l M a c h i n e " , A . I . E . E . , V o l . 81, P a r t I I I , 1963, p . 623.  28.  B r o a d b e n t , D« and C h i h n , H.R. , "The A p p l i c a t i o n o f S i g n a l Flow Diagrams t o I n t e r c o n n e c t e d E l e c t r i c Power S y s t e m s " , A u s t . J . o f A p p l . S c i . , V o l . 11, No. 3, 1960, p . 353.  1961.  of Motion",  Trans.  

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