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The effect of local motor loads on power system stability. Prior, Bruce George 1971

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THE E F F E C T OF LOCAL MOTOR LOADS  ON POWER  SYSTEM  STABILITY  by  BRUCE GEORGE B.Eng.  Nova  A THESIS  Scotia  PRIOR  Technical College,  SUBMITTED IN  PARTIAL FULFILMENT  THE REQUIREMENTS FOR THE DEGREE  MASTER  in  OF A P P L I E D  We a c c e p t  this  thesis  as  of  Head o f  the  conforming  to  ,  the Committee  Department  Members of  of  standard  Supervisor  Members  OF  Engineering  required  Research  of  OF  SCIENCE  the Department  Electrical  1969  the  Electrical  Department Engineering  THE U N I V E R S I T Y OF E R I T I S H COLUMBIA  September,  1971  the  In p r e s e n t i n g an the  this  thesis i n partial  advanced degree a t t h e U n i v e r s i t y Library  s h a l l make i t f r e e l y  f u l f i l m e n t of the requirements f o r o f B r i t i s h Columbia, I agree  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  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 extensive for  that  copying of t h i s  thesis  s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by h i s r e p r e s e n t a t i v e s .  I t i s understood that  of t h i s  thesis f o rfinancial  written  permission.  Department o f  £lECTr?<C*L-  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, C a n a d a  gain  s h a l l n o t be a l l o w e d w i t h o u t my  J L * W & t j  Columbia  copying or p u b l i c a t i o n  }  & r  ABSTRACT The e f f e c t investigated. supplying studied of  the  two,  synchronous  system  although  through  a general  loads  consists  an i n d u c t i o n m o t o r ,  a long  the  with  at  response the  case  system model i n  the  the  loads  of  of  and i s - a l s o  laboratory.  any n u m b e r o f  improve the  and  is  found  stability  line. and  local  is  from t e s t s from the of  The  the  the motors  a power  is  generator loads  combination  induction  Stability  The r e s p o n s e  observed  It  motor,  formulated.  line.  stability  a synchronous  transmission  the g e n e r a t o r  transmission  the m a t h e m a t i c a l model  l o c a l motor  of  on power s y s t e m  a synchronous  motor leads can be e a s i l y  by o b s e r v i n g a fault  l o c a l motor  The power s y s t e m  a large  are  of  is  of  determined  the  system  calculated on a dynamic  studies system.  and  that  from power all  T A B L E OF CONTENTS  ABSTRACT T A B L E OF CONTENTS L I S T OF TABLES L I S T OF ILLUSTRATIONS ACKNOWLEDGMENT NOMENCLATURE 1.  INTRODUCTION  2.  SYNCHRONOUS  3.  Synchronous  2.2 2.3 2.4  Transmission Line Equations R e g u l a t o r and E x c i t e r E q u a t i o n s C o m p u t a t i o n and L a b o r a t o r y T e s t  SYNCHRONOUS  5.  Equations  I n d u c t i o n Motor Equations Transmission Line Equations C o m p u t a t i o n and L a b o r a t o r y T e s t  SYNCHRONOUS  Synchronous  4.2 4.3  Transmission Line Equations C o m p u t a t i o n and L a b o r a t o r y T e s t  4.4  Computation R e s u l t s  Motor  Using  MOTOR  LOAD  Results  Large-System  GENERATOR WITH M U L T I P L E  5.2  Transmission Line  5.3  C o m p u t a t i o n and L a b o r a t o r y T e s t  Parameters  LOCAL SYNCHRONOUS  LOADS  Machine  REFERENCES  Results  Equations  5.1  CONCLUSION  Results  GENERATOR WITH A LOCAL SYNCHRONOUS  4.1  SYNCHRONOUS  LOADS  GENERATOR WITH A LOCAL INDUCTION MOTOR LOAD  INDUCTION MOTOR  6.  Generator  LOCAL MOTOR  2.1  3.1 3.2 3.3 4.  GENERATOR WITHOUT  Equations Equations P.esults  AND  "ill  LIST  OF TABLES  Table  2.1  Page  Synchronous Line  2.2  3.1 3.2  Generator, Voltage  Synchronous  Generator  Conditions  Induction  Motor  Synchronous Operating  13  Generator  Conditions  4.2  Synchronous  Generator  Motor  Loads.  State  Variables  14 27  Initial  Values  Induction of  State  Motor  Load.  Variables  28  Parameters  of  Large  4.4  Parameters  of  a Large  4.5  Large  39  System W i t h L o c a l and  4.3  Synchronous  of  Parameters  Conditions  Power  Local  Values  System W i t h L o c a l and  Motor  Initial  Initial  Parameters  Synchronous  Operating  System Without and  4.1  5.1  and T r a n s m i s s i o n  Parameters  Operating  and  Regulator  Initial  Synchronous Power  System  Stability  Values  of  Generator  State  Values  of  State  Motor  Variables  Motor  System Study.  Load. 39 48 49  Operating  Conditions  Variables  System W i t h L o c a l  Synchronous Motor Loads. Values of State V a r i a b l e s  Synchronous  50 Induction  Operating Conditions  and  and Initial 61  iv  LIST OF ILLUSTRATIONS  Figure  Page  2.1  C i r c u i t Diagram of I d e a l Synchronous Machine  5  2.2  O n e - l i n e Diagram of One M a c h i n e - I n f i n i t e Bus System  6  2.3  One Machine System With E q u i v a l e n t - i r T r a n s m i s s i o n Network  6  2.4  B l o c k Diagram of V o l t a g e R e g u l a t o r and E x c i t e r  12  2.5  Generator Torque Angle - No Motor Loads  16  2.6  G e n e r a t o r V o l t a g e - No Motor Loads  17  2.7  Generator C u r r e n t - No Motor Loads  18  3.1  Synchronous G e n e r a t o r System With L o c a l I n d u c t i o n Motor Load  19  3.2  C i r c u i t Diagram o f I d e a l I n d u c t i o n Motor  20  3.3  Synchronous G e n e r a t o r And I n d u c t i o n Motor dq T e r m i n a l V o l t a g e s  22  3.4  G e n e r a t o r Torque Angle - L o c a l I n d u c t i o n Motor Load  29  3.5  G e n e r a t o r V o l t a g e - L o c a l I n d u c t i o n Motor Load  30  3.6  G e n e r a t o r C u r r e n t - L o c a l I n d u c t i o n Motor Load  31  3.7  I n d u c t i o n Motor S l i p - L o c a l I n d u c t i o n Motor Load  32  3.8  I n d u c t i o n Motor C u r r e n t - L o c a l I n d u c t i o n Motor Load  33  4.1  Synchronous Generator System With L o c a l Synchronous Motor Load  35  4.2  G e n e r a t o r Torque Angle - L o c a l Synchronous Motor Load  41  4.3  Synchronous Motor Torque Angle - L o c a l Synchronous Motor Load  42  4.4'  Generator V o l t a g e - L o c a l Synchronous Motor Load  43  4.5  Generator C u r r e n t - L o c a l Synchronous Motor Load  44  4.6  Synchronous Motor C u r r e n t - L o c a l Synchronous Motor Load  45  4.7  Generator Torque Angle - L o c a l Synchronous Motor With S i n u s o i d a l Shaft Load  46  Synchronous Motor Torque Angle - L o c a l Synchronous Motor With S i n u s o i d a l Shaft Load  47  Large Generator Torque Angle - No Motor Loads  51  4.8  4.9  L I S T OF I L L U S T R A T I O N S  Figure  Page  4.10  Large Generator Torque Angle  5.1  Synchronous  5.2  5.-3  5.4  -  Local  Induction  and  Load  52  Loads  53  Synchronous  Loads  62  Synchronous-Motor  Torque Angle — L o c a l I n d u c t i o n  Synchronous Motor  Loads  I n d u c t i o n Motor Motor  5.5  L o c a l Synchronous Motor  G e n e r a t o r System W i t h M u l t i p l e L o c a l Motor  Generator Torque Angle Motor  -  Slip  -  and 63  Local Induction  and  Synchronous  Loads  64  Generator Voltage  -  Local  Induction  and S y n c h r o n o u s  Motor  Loads 5.6  65  Generator Current -  Local Induction  and  Synchronous  Motor  Loads 5.7  5.8  66  Synchronous Motor Motor Loads I n d u c t i o n Motor Motor Loads  Current -  Current -  Local Induction  and  Synchronous 67  Local Induction  and  Synchronous 68  vi  ACKNOWLEDGEMENT  I i Dr. of  Y.  N.  wish  Yu,  to express  for  this  thesis.  lasting  benefit  the  thesis I  group  for  and  also  for his wish  to  due  and u n d e r s t a n d i n g of  he  to  advice thank  valuable  am p a r t i c u l a r l y  consultation  t o my  during with  supervisor, the  preparation  him w i l l  be  of  Dr.  H.  R.  Chinn  for  his  c o n c e r n i n g programming  my f e l l o w  discussions grateful  g a v e me a n d a l s o  graduate  careful  reading  methods.  students  in  the  power  we h a d .  to Dr.  for his  H.  A.  M.  assistance  Moussa in  some  for  the  of  the  tests. The  Canada  are  and  also  encouragement laboratory  help  gratitude  t o me.  t h e many I  his  T h e many h o u r s  Thanks of  all  my s i n c e r e s t  of  financial  assistance  the B r i t i s h  of  the N a t i o n a l  Columbia Hydro  Research  and Power A u t h o r i t y  Council is  of  gratefully  acknowledged. Finally, Jean,  I  wish  for her patience  to e x p r e s s  my d e e p e s t  and u n d e r s t a n d i n g  appreciation  throughout  t o my w i f e ,  my u n i v e r s i t y  career.  NOMENCLATURE  General  t  time,  j  complex  OJQ  synchronous  £  base  T  and k  p  operator,  of  turns  x  + jx  ratio  logarithms, ideal  superscript  denoting  multi-motor  load  377  series  2.71828  transformer  number o f  operator,  impedance,  VQ  infinite  between  motor  particular  motor  d/dt  admittance,  voltage  bus  fi mho  voltage,  and c u r r e n t matrix,  V/ph  vectors,  V,A  [Y]  admittance  ig  transmission  line  instantaneous  Ig  transmission  line  phasor  8g  c u r r e n t phase  k^,k2,C^,C2  transmission  Synchronous  rad/s  Network  shunt  [I]  of  velocity,  generator  G + jB  [V],  angular  natural  derivative  Transmission r  s  mho  angle., line  current,  current,  A  A  rad  constants  Machines  3  subscript  X^JX^.XQ  stator  Xp  flux  e^.eqjeQ  terminal  denoting  dqO f l u x  linkage  synchronous  linkages,  state  Wb-T  variable,  dqO v o l t a g e s ,  V  motor  Wb-T  quantity  in  viij  v  t  t e r m i n a l phase  V  T  t e r m i n a l phasor  a  phase  v ,v r  m  real  angle and  voltage,  of  voltage,  V  terminal voltage,  imaginary  ef  f i e l d voltage,  vp  voltage  Vp^  a defined voltage,  ijjjiqjiQ  stator  ig,i  instantaneous  a  V  rad  terminal voltages,  V  V  p r o p o r t i o n a l to  field voltage,  equation  (2.18),  V  V  dqO c u r r e n t s , A armature  Ig  armature phasor  $  armature  c u r r e n t phase  "^"Gr'^Gm  real  imaginary  if  field  current, A  a  angle  from s t a t o r  and  electrical w  angular  6  angle  current, A  current, A angle,  armature  a-phase  rad  currents,  axis  to  A  rotor  direct  axis,  radians  velocity  of  rotor,  from i n f i n i t e bus  rad/s  r e f e r e n c e to  d-axis,  electrical  radians SQ  rotor  slip  PQ,QQ  r e a l and r e a c t i v e power o u t p u t ,  TQ  e l e c t r i c a l d e v e l o p e d t o r q u e , N-m  R„  armature  W and V A R  r e s i s t a n c e , ft  ci  Rf  field  LjjjLqjLQ  stator  L f^  peak v a l u e  a  r e s i s t a n c e , ft dqO i n d u c t a n c e s , of  mutual  and f i e l d w i n d i n g , Lff x^,x  field  q  stator  H  i n d u c t a n c e between s t a t o r  H  self-inductance, d q r e a c t a n c e s , ft  H  a-phase  ^d'» d'  transient  ^df ' d f  mutual  T^'  transient  short  T^Q'  transient  open c i r c u i t  d-axis  JQ  moment  inertia,  /rad  Dg  damping  T  p r i m e mover  x  x  t  d-axis  inductance  of  number  Regulator  and  of  field  constants  defining  f(v )  function defining  Tg  exciter  T^£  regulator  time  regulator  gain  regulator  reference voltage,  f  Induction  6 $2  8  g  e  ld  ,  e  lq  2d» 2q e  constant,  output  limits  s  time  voltage,  V  regulator regulator  constant,  output  limits  s  constant,  s  V  Motor  angle  used  in  general  angle  between s t a t o r  axis,  rad  angle  between r o t o r  d-axis, e  time  s  Exciter  A^,A2  e  constant,  poles  output  r  time  N-m  regulator  v  H a n d Q,  N-m  VR  R  H a n d Q,  J-s/rad  torque,  POLESQ  d-axis  J-s  coefficient,  f r i c t i o n torque.,  and r e a c t a n c e ,  and r e a c t a n c e ,  circuit  fg  Voltage  inductance  stator rotor  i-j^^ji^q  stator  ^2d»^2q  rotor  a-phase  A-phase  rad dq v o l t a g e s ,  dq v o l t a g e s ,  V V  dq c u r r e n t s , dq c u r r e n t s ,  dqO t r a n s f o r m a t i o n ,  A A  axis  axis  and  and  rad  rotor  A-phase  induction  motor  ij^  instantaneous  stator  Ij4  phasor  current, A  Y  phase  i-Mr'^Mm  real  stator angle  of stator  and i m a g i n a r y  rotor  current, A  current, rad  stator  currents,  slip  T^j  e l e c t r i c a l developed torque,  ^ld'^lq  stator  ^2d'^2q  rotor  r-^  stator  X2  rotor  L-Q  stator rotor  dq f l u x dq f l u x  linkages, linkages,  Wb-T Wb-T  r e s i s t a n c e , ft self-inductance, self-inductance, of. m u t u a l  •^12» 12  mutual  L',x'  transient  C-j,C^  i n d u c t i o n motor  D-^,D2,U3,D^  functions  A  a determinant,  Jj^  moment  D  damping  M  N-m  r e s i s t a n c e , ft  peak v a l u e x  A  inductance  of  H H  inductance  and r e a c t a n c e ,  inductance  a n d r e a c t a n c e , H a n d ft  constants  of 6 equation  (3.40)  inertia, J-s /rad  coefficient,  H a n d ft  J-s/rad  1.  The s t a b i l i t y interest  of  INTRODUCTION  electric  power  i n r e s e a r c h f o r many y e a r s .  complexity  the need f o r  important.  maintaining  A l t h o u g h many p r o b l e m s  control  and o p t i m a l  c o n t r o l of  them i s  a more t h o r o u g h  study  As  systems power  system  have been s o l v e d  the e f f e c t  been an a r e a  systems  stability  modern t h e o r y of  has  increase  becomes using  system  in  size  and  more  excitation  other problems of  of  loads  remain.  One  on power  of  system  stability. In by c o n s t a n t of  the  steady power,  three  state  stability  constant  [22].  It  is  current,  known  impedance r e p r e s e n t a t i o n g i v e s  pessimistic  representation  gives  generally  first  two.  stability  Dahl  impedance o r by t r a n s i e n t latter  impedance o r ,  and c o n s t a n t if  attempt  should  neglected  the r e a l  a constant  to  as  [2]  the f a u l t  Hore  [5]  constant  and  [4]  current.  fault  a  studies  the constant results,  combination the  constant  power  current  somewhere effects  in  in  between  transient  a r e p r e s e n t a t i o n e i t h e r by but  proposed constant  clearance.  d i d not  say  constant  how  the  a r e p r e s e n t a t i o n by impedance d u r i n g  the  fault  He f u r t h e r r e c o m m e n d e d  d i f f e r e n t using  his  two  data.  that  suggestions Weinbach  an  [3]  and r e p r e s e n t e d t h e r e a c t i v e power was  agreed with the  or  represented  the constant  o b t a i n more a c c u r a t e l o a d  Kimbark  the e l i m i n a t i o n of  load  Crary  power a l t o g e t h e r  reactance.  representation.  the  b e made  results,  characteristics  were s i g n i f i c a n t l y  state  b e e n done on l o a d  suggested  load  are u s u a l l y  impedance,  steady  satisfactory  alternatively,  power a f t e r  the r e s u l t s  after  [1]  were t o be o b t a i n e d .  constant  results  N o t much w o r k h a s  studies.  for  optimistic  gives  loads  constant  that  representation  the  studies  it  in  favour with  C r a r y but would  He f u r t h e r  showed  the constant  stated  that  impedance  qualitatively  b e more a d v i s a b l e his  to  as  that  simulate  recommendations  were d i r e c t e d p r i m a r i l y adequate  for  problem.  the a n a l y s i s  For  that,  r e p r e s e n t a t i o n was determine voltage in  an  changes  in  constant.  in  i n d u c t i o n motor  stability constant load, was  but  was  should  its  if  the  according  to  cases. of  Kent,  electrical  o n a power constant based  [7]  be  loads  Gevay  and  radial  power  synchronous study.  would al.  loads  system.  emphasized  stability  impedance and c o n s t a n t  simulator  in  stability  tests.  l o a d was  found  studies  records.  power. Robert  methods  of  stability  studies.  transients a the  total  for  t h a n 10 p e r c e n t  amplitudes  bus  The c l a s s i f i c a t i o n and R o b i c h a u d  [10]  and made  the general  load  kept  and  static  motor  load  never  of  both synchronous  proportioning  an  transient  l o a d was  synchronous  a correct  of  digital  the  total  and  even i n  stable  representation  and c o n d u c t e d l o a d v o l t a g e  conjunction with a micro-reseaux They r e c o g n i z e d t h a t  maintained  T h e s y s t e m was  the need f o r  that  in  i n d u c t i o n motors  that  load,  resulted  result  [8] i n v e s t i g a t e d  less  in  established  electrical  stability.  They gave methods  u p o n kWh s a l e s  system  the presence of  They a l s o  in voltage  the v a r i o u s  motors,  a  to  changes  v o l t a g e s were  swing w i t h r e l a t i v e l y l a r g e [9]  in  motor  small  system where t h e  synchronous that  to  the r e p r e s e n t a t i o n i f  Schippel  was  conducted t e s t s  would not  power  not  characteristics  They a l s o  the bus  during  included in  They found  et.  limits  discussed  effective in maintaining  i n d u c t i o n motors  [6]  an i n c r e a s e  provided a l l  al.  division,  their  al.  load  a power s y s t e m  reasonable  load  isolated  varied.  t h e most  stable  an  et.  that  t h e y were  system where s t a b i l i t y  b o t h m e c h a n i c a l and r o t o r  employed.  of  and t h a t  and r e a c t i v e l o a d .  i n d u c t i o n motor  They recommended t h a t  of  They found  Brereton, et.  is  Bauman,  parts  i n both r e a l  changes  representing  computer  of  frequency within  appreciable  an e x i s t i n g  required.  frequency.  increase  of  system p l a n n i n g  he c o n c l u d e d , t h e n o n l i n e a r  the response  and  toward  loads of  built  between  loads a  tests  was  load  comparative p r o b l e m has  two  parts  w h i c h may b e s t u d i e d  influence system  of  and  the c h a r a c t e r i s t i c s  the In  second p a r t  this  system  stability  system  test  thesis is  model  is  means,  motors  employed.  terminals are  stability  with a with and  But,  loads,  of  on t h e  is,  of  equations  later  fault with  In  Chapter  studies. equations are  carried out.  a local  synchronous  l o c a l motor  i n d u c t i o n motors synchronous  is  motor  line  is  and  state  In  Chapter  the  only  load  of  c a r r i e d out 5 t h e most  loads  comparison  introduced a  transmission  in  Chapter 4  generalized  synchronous  because of  2.  neglected  for is  detail.  Chapter  l o c a l motor  one i n d u c t i o n motor  study mainly  equations  in  commonly  studies  w i t h a n y number o f  p r e s e n t e d but  variable  described  i n d u c t i o n motor  motor.  generator  are developed i n  are  by  synchronous  the  provide a basis  studies  a power  the loads  the v a r i a b l e s  Similar  for  the  relations  to  power  on p o w e r  i n d u c t i o n motors  Further s t a b i l i t y  loads  a r e used  loads  The s y s t e m w i t h o u t solved  a  the  representation.  simulating  equations  3 the  derived.  of  and R o b i c h a u d ,  For a n a l y s i s  are retained. fault  determine  are connected d i r e c t l y to  a n d many o f  line  to  stability  load  Like Robert  that  is  l o c a l motor  instead  The motors  studies  f o r m u l a t i o n of  and  and  one  the  restriction  equipment. A dynamic  test  Columbia  [11,12,13,14]  thesis.  Since  losses are  effect  and t r a n s m i s s i o n  transmission  similar  line  of  loads  the problem of  a transformer.  They a r e n o n l i n e a r  but  used.  The f i r s t  employed w i t h m e c h a n i c a l and e l e c t r i c a l  The g e n e r a t o r  in  the  actual  through  is  of  investigated.  artificial are  separately.  is  the motors  model developed at used  used are  f u r t h e r computed r e s u l t s  included in  from small  Chapter4  machine  to  of  to v e r i f y  studies.  verify small  the U n i v e r s i t y of  the a n a l y t i c a l ones  studies  with  using  the u s e f u l n e s s  British  results  of  comparatively  large of  machine  the  the  high  parameters  results  2,  SYNCHRONOUS GENERATOR WITHOUT LOCAL KOTOR  The s t a t e system without verified  by  equations  l o c a l motor  a transient  with  the r e s u l t s  test  model.  2.1  Synchronous  [15]  in  Generator  1)  is  study.  All  synchronous  generator  The m a t h e m a t i c a l m o d e l  The computed r e s u l t s  from l a b o r a t o r y  tests  are  is  compared  on a power  system  Equations  generator  Lewis  based  regulated  are d e r i v e d .  obtained d i r e c t l y  per u n i t .  The f o r m u l a t i o n  load  a voltage  stability  The s y n c h r o n o u s Park  for  LOAD  [16]  were o r i g i n a l l y  r e w r o t e t h e m i n MKS  upon t h e  inductances  equations  following  are  units  d e r i v e d by as  used  here.  assumptions:  independent of  current  (saturation  neglected). 2)  Only  the  second harmonic  addition 3) The p o s i t i v e rotation machine  of in  to  the  The e l e c t r i c polarities the r o t o r  the  dqO c o o r d i n a t e s  of  currents  shown i n  the permeance i n  value  transient  of  are  average  of  is  gap  in  considered.  t h e damper w i n d i n g and v o l t a g e s  Figure  the a i r  2.1.  and  is  neglected.  the d i r e c t i o n  The e q u a t i o n s  of  of  the  are  e  d  - aid  =  R  e  e  "  P d  "  pX  "  A  x  +  q  0 = ~ a 0 " P 0 R  f  =  R  f  i  i  f  X  +  P f A  qP  C T  A po d  (2.1)  (2.2)  (2.3)  (2.4)  Figure  2.1  C i r c u i t Diagram  of  Ideal  Synchronous  Machine  i  -Load Buses  Synchronous Generator  Circuit Breaker  Circuit Breaker  I ion Fault Location  Figure  2.2  O n e - l i n e Diagram of  A -  One M a c h i n e - I n f i n i t e  Infinite Bus  Bus  System  Synchronous Generator Infinite Bus  B  Figure  2.3  One M a c h i n e  System w i t h  Equivalent-u Transmission  Network  where  V V d + f W f  \ =Vq X  X  For n o t a t i o n s ,  see  f  =  Q  f f^  L  = L  +  current  i ^ is  d  e  e  5 )  ( 2  -  6 )  (2.7)  Q  jf W  usually  - R^d  =  ~  p  d  ( 2  V  -  q " "Vq " P \  in  analysis  U  R  +  T  '  8 )  V  +  X  X  process d  q  w  0  (  ( 2  +' dO*P  1  T  x  1 0 )  (2.11)  d  q  = _ i i  (2.12)  q  the f o l l o w i n g U d  constants are q 0 q  introduced  as (2.13)  3L  t  d  '  )  o  „  give  ( 2 - 9 )  i  d0'P)  to  0  + "0 f  elimination  eliminated  d  A  A, =  the  -  nomenclature.  The f i e l d  In  i  0  ( 2  M  ~  T  "o Ld  '  T L  '  d  -  -  T L  a  d  f  M  — — ^ ff L  L  df  ~  3 2  L  afM  x  df  "  u  0 df L  (2.14) A  d0  x  R  f  d  ^dO x  f  d  Note The  that  all  of  the parameters  of  (2.9)  zero sequence c u r r e n t v a n i s h e s Equations  form  (2.9)  for  through  balanced  are  measurable.  operation.  through  (2.11)  are rearranged  = -e  R i  o) (l  in  state  variable  as  pX  d  -  d  a  -  d  P F  =  V  F  <°0 d  _  x  d') d  X  -  0  PXq = " e q " V q + V  where  (2.12)  +  V  s )A G  (2.15)  q  " G> Xd  <2-16>  S  1  FR  (  2  '  1  7  )  [17]  ( x  v  p  R  =  x  A  d  =  p  X  d  x  x  v  d F  x  T  (2.18)  d0  df f e  (2.19)  *f X  i , = x  i  F  d  q  x  =  0 d  u  A  + d0  x  (2.20)  d  Vq  (2.21)  x  q a = a> t +  6  Q  (2.22)  oi = p a = O)Q + p5  s  (2.24)  G  C0  The  (2.23)  Q  electromechanical equations  p6 = - w s 0  of  G  the synchronous  machine  are  (2.25)  POLES, ps  =  G  — J  2.2  Transmission  The valid  for  network,  diagram  including  the  The  as  the  system  line  of  the  resulting  local as  loads shown  T ) + D f l " 8.)  assumed  modelled with  to  under  and  in  be as  2.3.  quantities.  infinite  ig  = ^2 I g  dq  currents  are  transformation  i  =  3  w h i c h may b e w r i t t e n  in  a  [ i  is  is  reduced to  The c u r r e n t s  voltage  an is  flowing  in  bus:  (phase  = Jl  V  "a"  transformed  cos(u) t +  8)  (2.27)  cos(u)Qt  Gg)  (2.28)  Q  to n e u t r a l )  cos(u>Qt +  t  G  into  +  is  (2.29)  a)  its  dqO c o o r d i n a t e s  the  constant:  i  inverse  G  is  2.2  are  follows:  From the  current  which  transmission  The i n f i n i t e bus  = f2 I  f c  Figure  The  impedances,  ±  terminal voltage  [18J.  consideration.  Figure  phasor  lumped p a r a m e t e r s  currents  generator:  generator  (2.26)  t  r e p r e s e n t e d as  all  i  The  -  From t h e  generator  the  is  system  v  When  f.  +  frequency voltages  reference for  are  G  Equations  transmission  e q u i v a l e n t - u network taken  G  Line  fundamental  a one-line  1(T 2u,  q  = /3  d  G  = - / 3 I.  for  d  I  the  cos(w t  phasor  0  form  U  cos(g  -  6)  (2.30)  sin(3  -  6)  (2.31)  generator  +  as  6) +  i  current  q  sin(u) t Q  is  + 6)]  (2.32)  10 or  je  _  d  (2.34)  /3  The  "i  left  "  takes  side  imaginary  a negative  (2.34)  of  parts  to  /3  s i g n because  be w r i t t e n  as  of  the  coordinates  i„ + ji„ Gr Gm  and  I  —  cos  + —  6  sin  3  • *  d 1  Gm  procedure for  —  the v o l t a g e  ejd  sin  r  v  steady  state  voltage  vector  and  • sin  /3  phasor  G + jB  is  the  H  q  q  x  /3  [V]  =  jx  sin  6  cos  o  (2.37)  of  the  +  jx  transmission  system  the network,  r  is  (2.39)  current vector.  r  (2.38)  X  [I]  v  +  o  (2.36)  o - —  o  + r  cos  (2.35)  q  equation of  admittance matrix [I]  6  73  [Y]  the  6  /3  = — m  sin  yields  cos  d  is  and  /3  e  v_ = —  [Y]  the  /3  /3  where  the r e a l  d  B = x Gr  cos  /3  The  separate  Let  obtain  Ig  A similar  chosen.  +  [V]  Expanded,  J m v  is  the  this  V + Ji  bus becomes  [19]  G m  (2.40) 1 i  r  +  jx  r  +  jx  f  Br  +  j ±  Bm  where V Q i s current  the  i n f i n i t e bus  from the  equation of  infinite  (2.40)  rG -  voltage  bus  into  and e q u a t i n g  xB + 1  -  r B + xG  rB  rG -  -  phasor  i ^  the network.  real  and  xG  v  xB +  and  1  v  v  _  0  the  the  parts  yield  r  -x  (2.35)  resulting  equation  e  through  (2.38)  solved  d  k  for  are  r  X  substituted  e^ and e .  l  -  k  VQ  2  into  (2.41)  The r e s u l t  q  C O S <5  C  first  (2.41) x  _  Equations  phasor  ^•Gr  +  0  m  is  Expanding  imaginary  r  + j i  Gm  and  _  the  is  l  C  2 (2.42)  k  k  2  VQ  x  sin  -c  6  c  2  1  where  rG -  xB +  1 (2.43)  (rG  -  xB + l ) rB +  k„  +  2  Figure tangent  2.4.  and E x c i t e r  xB +  l)  +  2  of  the  output  xG)  2  k x  (2.45)  ~ ^2^"  (2.46)  2  regulator limit  is  and e x c i t e r  accounted  for  system using  are  is a  constants  =  tanh  (A v ) 2  R  determined from a l e a s t - s q u a r e s  shown  in  hyperbolic .  R  2  +  [13]  f(v )  w h e r e A-^ a n d A  (rB  Equations  diagram  The r e g u l a t o r  function  2  (2.44) -  ^2 ~~  The b l o c k  xG)  xG  = k-^r +  Regulator  +  = (rG  2*3  (rB  .  .  (2.47)  criterion.  7  ref  +  \  K  A  V  f  R  1 1 +  Figure  The s t a t e  2.4  Block  equations  of  T  R E  P  Diagram  of  the b l o c k  pe  = •  f  T  PV  e  1 +  Regulator  and  f  T p E  Exciter  are  [f(v )  -  R  e ]  (2.48)  f  E  [ A< ref K  R  R  Voltage  diagram  1  f(v )  v  "  v  t)  "  V  (2.49)  R1  RE  The  terminal voltage  in  terms  of  v  In to  (2.17),  2.4  summary,  (2.25),  the  (2.26),  t  state  system phase  disturbance fault  isolated Figure  at  2.2  occurs  for at  5 cycles shows  the  (2.50)  equations  (2.48)  equations  and  Test  of  the  the  f o l l o w e d by  the l o c a t i o n of  complete system are  Results  stability  system buses  laboratory  study and  is  the  a r e d e t e r m i n e d by d i r e c t measurement  as  tests.  30  fault.  The p a r a m e t e r s  and a r e  listed  The  follows:  faulted  a system r e s t o r a t i o n at the  (2.15)  (2.49).  are v e r i f i e d with  transient  one o f  is  =  C o m p u t a t i o n and L a b o r a t o r y  The s y s t e m  dq v o l t a g e s  together  line  a threeis  cycles. of  the  with  system the  Table Synchronous  Generator, Voltage  Regulator  Synchronous R„  0.66  xd  1 6 . 2 ft  ft  2.1 and T r a n s m i s s i o n  Generator  £  P0LES  (  Parameters  Parameters  _  9 . 7 1 ft  x d0*  T  2 . 7 4 ft  R  Line  s  5.00 0.575  H  4 . 8 0 ft  0.165  J-s /rad  4  1.585  N-m  0.00267  J  A  RE  2  J-s/rad Voltage  K  afM  Regulator  Parameters  0.152  12.8  0.035  s  0.050  s  Transmission  0.0943  "2 v  Line  re£  72.6  V  Parameters  r  X  G  (ohm)  (ohm)  (mho)  1.42  21.0  0.0097  1.42  21.0  0.0  0.0114  0.710  10.5  0.000033  0.0227  B (mho) -0.117  14 Table Operating  Conditions  and  Initial  Operating  V T  of  State  Variables  Conditions Q  G  57.6  V  Q  V/ph  85.0  VAR/ph  31.6  V/ph  3.73  N-m  t X  d  X  T  1.5  t  Values of  State  0.2628  Wb-T  v  0.0617  Wb-T  6  D  Variables 2.279  V  133.1  electrical  q  h e  Values  100 W/ph  Initial  A  2.2  f  538  Wb-T S  2.71  V  I  G  °  degrees  prefault are  operating  shown i n  conditions  Figures  Figure  2.5  from the o p e r a t i n g degrees  from the is  there  a larger  •model. are  easily  have  fault  at  envelope in  point  somewhat  seen. to  63 v o l t s .  instants. not  shows is  slower  damping  Figure  test  which  than  the  effect  Voltage  an e x a c t  spikes  also  concluded  that  torque angle. degrees  in  the machine Figure  the  The c u r r e n t r o s e amperes.  test  Since  the  This  the  from computation  computed r e s u l t .  shown i n  Small  i n both  results  T h e maximum  shows g o o d a g r e e m e n t .  test is  than  This in  2.6.  the  The  to  dip  the  difficult  oscillations  of  the voltage  amperes  torque angle  the mathematical model  is  in  the  swing  suitable  for  most  that  instants volts a  rise  switching  to  but  do  realize  a  waveform  current  test.  is  and  laboratory  is  5.6  test  45  the  in  it  to  to  43 v o l t s  The g e n e r a t o r  50  mathematical  because  figures.  and  indicates  computation at  results  swing  The s w i t c h i n g  a voltage  shows a d i p  are p r e d i c t e d i n  seen  instant.  are evident  6.2  in  The a c t u a l  t h e same a m p l i t u d e s .  v a l u e was  studies.  60 v o l t s .  test  2.7.  52 e l e c t r i c a l  is  The computed and  The c a l c u l a t e d c u r v e p r e d i c t s  They a r e  2.7.  2.1.  the generator  The t e r m i n a l v o l t a g e  and a r i s e to  through  laboratory  response is  2.5  in Table  is  shown  The p r e d i c t e d important  transient  it  stability  is  •  / t  I  f to****!  • • • • •  /\  •  • • • • • • •  • • • | •  1  • • •  * • *  1  • •  %  •  i/  *  /  0 —V  1  \  *  i: .  *  •  *t '  •  t  •  :  \ •  t V  Y  V  \ /  •  V /  I  I s  100.On  §  80.0  LU  y  60.0-  LU _J  CD  §  40.0  LU ZD C3  CtZ  O  20.0-  a: o r—  CC cr: LU  z:  0-  LU CD »  -20.0 0  0.5  1.0  Fig,  2.5  1.5 2.0 2.5 TIME (SEC> Generator Torque  3.0  Angle  r  T  "  3.5  17  I ' 0  ' i i  I '  0.5  i i i  i  i i i i  1 .0  i  i i i i [ i i i i  1.5  2.0  TIME Fig.  2.7  i  2.5  i i i i  3.0  (SEC)  Generator  i  Current  i i i  r | 3.5  i i » i [  4.0  3.  SYNCHRONOUS GENERATOR WITH A LOCAL INDUCTION MOTOR LOAD  In t h i s chapter  t h e e f f e c t o f a l o c a l i n d u c t i o n motor l o a d on  power system s t a b i l i t y i s i n v e s t i g a t e d .  A o n e - l i n e diagram o f t h e system i s  shown i n F i g u r e 3.1.  W W . — ^ M T  1  Infinite BusTransformer Induction Motor  F i g u r e 3.1 Synchronous G e n e r a t o r System W i t h L o c a l I n d u c t i o n Motor Load  3.1  I n d u c t i o n Motor  Equations  The i n d u c t i o n motor, i s a r o u n d - r o t o r machine and t h e s a l i e n c y e f f e c t i s not present. i n 1938.  Stanley  I n h i s equations  [20] gave an a n a l y s i s o f t h e i n d u c t i o n motor  t h e machine c u r r e n t s , v o l t a g e s and f l u x l i n k a g e s  were r e f e r r e d t o axes f i x e d on t h e s t a t o r . equations  S a t u r a t i o n was n e g l e c t e d .  The  used here f o l l o w c l o s e l y those o f F i t z g e r a l d and K i n g s l e y [21]  w i t h axes r o t a t i n g a t t h e synchronous speed.  F i g u r e 3.2 shows t h e p o l a r i t y  of c u r r e n t s and v o l t a g e s and t h e d i r e c t i o n o f r o t a t i o n o f t h e machine. dqO t r a n s f o r m a t i o n m a t r i x i s chosen as  The  Figure  3.2  Circuit  Diagram  of  Ideal  Induction  Motor  2TT cos  8  2 cos  4ir  (8  -  cos  ((  —)  3  where  8 = o^t  rotor.  Note  -  sin  and K i n g s l e y ' s .  (6  l-  —)  sin  (8  —)  3  3  1  1  1  3  3  3  f o r the stator  that  4TT  2TT  2 . - sin 3  transformation  and  8 = u^t -  6  2  the n u m e r i c a l c o e f f i c i e n t s are d i f f e r e n t from The e q u a t i o n s  -Id  e  lq  e  2d  6  2q  =  =  =  Fitzgerald  i n dq c o o r d i n a t e s a r e  l  r  =  f o r the  r  r  l d  i  l  1  P ld  +  l q  2 2d i  X  +  +  ^ Z q  +  +  P lq  p X  p X  X  "  A  2d  +  2q  (3.2)  V l d  X  "  (3.1)  0 lq  w  X  (3.3)  2qP s 6  2dP  (3.4)  6 s  where  ^ld ~  l  The  lq  ~  L  l l  i  l d  +  2  L  aAM 2d  L  l l  i  l q  +  2  L  aAM 2q  ^2d "  L  22 2d i  +  2  L  ^2q ~  L  22 2q  +  2  L  zero sequence c u r r e n t s v a n i s h The r e l a t i o n s h i p  terminal  voltages  is  shown  i  (3.5)  i  (3.6)  1  (3.7)  aAM ld i  (3.8)  aAM lq i  f o r balanced  operation.  between t h e g e n e r a t o r , and i n d u c t i o n i n Figure  3.3.  The e q u a t i o n s  are  motor  e,, = T  ( e , cos 6 + e  n  s i n 6)  •  e-, = T (e cos <5 - e, s i n 6) lq x q a where T  v  i s the t u r n s r a t i o of an i d e a l t r a n s f o r m e r  the g e n e r a t o r  (3.9)  (3.10)  between t h e motor and  or motor t e r m i n a l v o l t a g e T x  generator  terminal voltage  Figure  3.3  Synchronous G e n e r a t o r and I n d u c t i o n Motor dq T e r m i n a l V o l t a g e s F o l l o w i n g t h e s u g g e s t i o n o f B r e r e t o n , e t . a l . [8] and Gabbard's d i s c u s s i o n t h e s t a t o r f l u x l i n k a g e s o f t h e i n d u c t i o n motor a r e assumed t o remain c o n s t a n t g i v i n g  P ld X  =  P lq X  =  °  Since the r o t o r windings are s h o r t - c i r c u i t e d i n normal o p e r a t i o n ,  ^2d ~ 2 q e  The  i n d u c t i o n motor  equations  now c a n b e w r i t t e n  as  T  x  (e  d  cos  6 + e  q  sin  6)  = r ^ ^  + u^I^i.^ +  T  x  (e  q  cos  6 -  d  sin  6)  =  -  2  + pX  e  0 = r  0  =  r  i  2  d  2 2q i  r  i  i  l  q  +  2 d  P 2q "  +  u  X  A  L 0  2 q  ~  i  p6  <3.11)  0 12 2d  (3.12)  L 0  u  1  L  i  (3.13)  s  2dP s  X  '  6  X  2d  =  L  22 2d i  +  L  12 ld  X  2q  =  L  22 2q  +  L  12 lq  1  l l l d  12 2q  w  (  -  3  1  4  )  .(3.15)  i  i  (  3  '  1  6  )  where 3  L  Eliminating  i  2  d  and  i  hd  ^ q  2  from  q  =  T  x  =  T  x  D  D  2  =  (3.11),  < l d  (  12  e  l  6  q  L  aAM  _  (3.12),  (3.15)  +  D  2 q>  +  C  4 2d  ~  D  2 d>  +  C  3 2d  e  e  "  X  X  C  +  C  and  3 2q  (3.16)  (3.17)  gives  < -  X  3  4 2q X  (  3  '  1  1 8  9  )  )  where  x'  =  COQL'  X  1  W Q L  =  2  1  (3.20)  2  2  L  12  '  L ' = L  1  -  ±  (3.21) L  22  1 D,  =  (r, 2 r ; L  +  (x')  cos  6 + x  1  sin  6)  (3.22)  sin  6 -  1  cos  6)  (3.23)  2  1 D„ 2  =  r±  • 2  +  (r,  (x')  2  r  C  3  = L  x  1  2  2  [  l 12 — + x  2  r  i  (x') ] 2  (3.24)  -x x 12  C  Equations  (3.13)  and (3.14)  4  may now b e w r i t t e n  P 2d X  p X  where  1^^ a n d  ~ r2hd  =  2q  ' ^ Z q  =  a r e s o l v e d .from  (2.26),  PS  M  =  2w  M  Transmission  Line  o f l o c a l motor  M  (T  '  (3 27)  S  and  (3.16).  of the i n d u c t i o n motor,  analogous  - T ) + % ( !  L  M  (3.28)  - s ) M  r  equation  (2.46)  loading.  If  i s not v a l i d  the steady  state  because  of the  current  flowing  into  is  = /2 I  then  -  (3 26)  s  Equations  The dq v o l t a g e  the motor  form as  % K X2d  +  (3.15)  — J  addition  variable  is  POLES  3.2  in state  % H X2q  ~  The e l e c t r o m e c h a n i c a l e q u a t i o n to  (3.25)  =  cos- (u) t +  M  Q  Y  )  (3.29)  i t s dq components a r e  hd hq  Following voltage  =  the procedure o u t l i n e d  and c u r r e n t ,  real  ^  =  _  /  3  T  M  ~ hi  C  O  S  Y  s  i  n  Y  i n Chapter  and i m a g i n a r y  r =  7i  x  id  (3.31)  2 for defining  motor  x  (3.30)  currents  generator  are defined  as  (3.32)  (3.33)  These e q u a t i o n s are analogous torque  angle  6 because of  transmission  line  to  (2.35)  t h e dq a x e s  equation,  and  (2.36)  b u t do n o t  contain the  chosen f o r  the i n d u c t i o n motor.  to  is  analogous  (2.41),  The  —  rG  -  xB + 1  -rB  -  xG  r  v  v  r  0  Using  + xG  rG -  xB +  the constants defined  in  ^4  equation for  k  (2.42)  =  -k  VQ  2  ^  of  X  (2.45)  and a l s o  ^1  -  sin  2  c  o  s  r  k  2  sin  VQ  x  Gm  _  C  l  defining  (3.35)  (3.36)  ^  2  C  -c  6  c  2  ^m  6  k  L  the synchronous generator i s  6  COS  to  Mr  (3.34)  0  S + C  ^1  t h e dq v o l t a g e s  1  m  v  = C j cos  D3  an  1  x  Gr  +  —  rB  -x  D  3  D  4  obtained  as  "4 D  T. x  1  D  3  (3.37) Substituting  e  d  k  = e  q  i  l  l  d  and i  l  from (3.18)  q  " 2  VQ  COS  6  k  VQ  sin  6  k  /3  and  C  (3.19)  l  C  into  (3.37)  2  + k  2  D  x  3  "4 D  -c  D  l  c  2  D  2  x  C  4  - C  3  ;  2d (3.38)  ••x D  4  D  3  -D  2  D  1  C  3  C  4  ^2q  Solving  once a g a i n f o r  T  1  x  A  <13 D  T  2 X  D  ( D  +  D  D  2  e^ and e  2 4> + D  3  -  1  T  D )  D l  x ( ¥ 4 ~ 2 3> D  T  4  l  k  ,  x ( 1 3 D  k  +  D  2 4 D  6  COS  VQ  " 2  D  D  1  l  C  •3  ) +  C  2  + k  k  2  D  s in  VQ  x  -D  3  C  4  6  -C  4  " 3  Cj_  2  ^2d  C  (3.39)  -T.  x D  4  D  3  C  3  C  4  l  2q  where  A =  In system w i t h (2.48),  3.3  summary, local  (2.49),  are  the  voltage  is  the  same a s  in  conditions test  of  waveform of the  are  2  are  generator as  that  D ^ )  synchronous  (2.15)  to  for  The p a r a m e t e r s  listed  shown  Figures  -  3  (2.17),  (3.40)  2  generator (2.25),  (2.26),  Results  2.  and a r e  (D D  the  and d i s t u r b a n c e  in  2.4  when s u p p l y i n g  same  of  4 X  in  Figures and 3.4  a local  with  the t r a n s i e n t of  3.4  The  in  2.5  prefault  Table 3.2.  through  indicates  i n d u c t i o n motor  without  the  The  3.8.  that  the i n d u c t i o n motor  in Figure  stability  the- i n d u c t i o n - m o t o r  Table 3.1.  the e n t i r e system are g i v e n  of  the  + T  2  (3.28).  Test  Chapter  results  more s t a b l e  + 1 ]  load  and  system parameters  essentially  4  equations  and L a b o r a t o r y  A comparison  is  (3.27)  The  computed and  system  2  state  determined from t e s t s  operating  + D D )  i n d u c t i o n motor  (3.26),  Computation  study are  [ ^ ( D ^  the  power  load.  load, load.  The  Figure The  3.5,  Table  r  l  r  2  L  12  L  aAM  POLES  generator  M  the amplitude  when  there  is  fault  observed  Induction  Motor Parameters  3.7  the  L  l l  9.14  0,  L  22  J  M  0.409  H  0.273  H  of  Figure  the o s c i l l a t i o n s i n d u c t i o n motor  shows  the  laboratory  computed  after but  test  3.4(a)  which  shown i n  Figure  3.4(b)  except  not  angle  indicates  t h e power  occurs,  in  a  output  resulting  the g e n e r a t o r  change  armature  has  in  H  J-s /rad 2  same a s  system  J-s/rad  in  Figure  restoration  2.7 is  except  smaller  load. i n d u c t i o n motor restoration.  measured  of in  t h e damping  the  due  to  slip  increase  This  result  lack  the generator agrees  initial  increase  in  with  of  a  during  was  also  suitable  power  a torque angle  increase.  t h e r e must  larger  in than  Because  be a l a r g e  a large  swing  the  result  computed  damping  The d e c r e a s e  output  decreases  load.  torque angle  decrease.  the generator  a l o c a l motor  current  is  of  general  for  sudden  damper w i n d i n g s w h i c h r e s u l t s Now i f  not  0.426  0.00133  the  after  system  result  Figure  fault  is  H  device.  shown i n  Usually  3.6,  0.426  0.0130  %  4  The l a b o r a t o r y  torque  Parameters  Q.  and o s c i l l a t i o n s in  measuring  Motor  9.10  a local  Figure the  Induction  c u r r e n t waveform,  that  3.1  of  in  generator.  when a t r a n s m i s s i o n This of  the  current torque  the a c c e l e r a t i n g  the  is  is  t r u e whether  sudden induced  for  effect  line  the  due  to  large in  the  generator. the  power  or  Table Operating  C o n d i t i o n s and I n i t i a l  Operating  P  G  S  M  v  0  T  t  \  0.033  v  f c  f  V  R  2.71  V  2.28  V  G  37.6  V/ph  T  x  3.73  N-m  T  L  Values of  State  Variables  85  VAR/ph  57.6  V/ph  1.5 0.371  N-m  State Variables 6  Wb-T  s  120 e l e c t r i c a l  G  538 Wb-  e  of  Conditions Q  0 . 2 6 3 Wb-T 0.0617  Values  100 W/ph  Initial  *d  3.2  0 0.121  Wb-T  X„ 2q  0.348  Wb-T  M  0.033  S  degrees  ; /  V  \  6 \J...  I  s  i |  i  V  1 0 0 . 0 -i  § 8 0 . 0 LU  ib! 6 0 . 0 H LU _J CD  ^  40.0  LU C3 CC O O I— CC ct: LU LU CD  20.0  H  0 ^  -20.0  1  0  1  1  1  j ' ' ' ' | ' '  0.5  r-i  1.0  |  II  1.5  i  i  2.0  TIME Fig.  3.4  | t• -r-i  2.5  (SEC)  Generator  i i i | i i  3.0  .  Torque  Angle  i  i | i  3.5  i  j , |  d  IHMIIIIIII  w •imiilllll  iiuiiiii;  i i i i i i i i i l i l i l l i i ' l i i ii ' i i l l l i i l l l ,  h||||l I  iiilliiipillll  iJIIIHH! i:n;!i;::;: llilltilil Mlllllllll'  linn  'mimm  0.4 s  65n 07  §60 H 07  z: cr:  Wc-e-  55  co cc  > LU  £50 H X Q_ cn  o  §45  LU -z. LU CD  40  1  0  '  1  1  i  1  0.5  • i  ' ' ' ' | i ' i i |i i i i |i i i i | i \ i i |i i i i |  1.0  1.5  2.0  TINE Fig.  3.5  2.5  (SEC)  Generator  Voltage  3.0  3.5 4.0  Fig.  3.6  Generator  Current  Fig.  3.7  I n d u c t i o n Motor  Slip  cc  CO  2: 1.5  on  H  LU  cn Cr: o r—  o  £ 0 . 5  CJ ZD Q  1 1 1 1 I 1 1 1 1 I  0  0.5  1 1 1 1 I 1 1 1 1 I  1.0  1.5  2.0  TIME  Fig.  3.8  1 1 1 1 | 1 1 1 1 I  2.5  (SEC)  Induction Motor  Current  3.0  1 i  1 1 ) 1 1 1 1 I  3.5  4.0  output as  decrease,  predicted.  local been  load.  the generator  This In  torque  phenomenon w i l l  the a n a l y s i s ,  decreases  be more a p p a r e n t  however,  the  effect  of  instead  of  when t h e r e  increasing is  a  damper w i n d i n g s  large has  neglected. Two c o n c l u s i o n s  may b e d r a w n  damper w i n d i n g s m u s t b e i n c l u d e d of  angle  torque angle  increases  the  swings  stability  is of  in  required. a power  from t h i s stability Second,  system.  study. studies  a local  First, if  exact  the  effect  prediction  i n d u c t i o n motor  load  of  4.  SYNCHRONOUS GENERATOR WITH A LOCAL SYNCHRONOUS MOTOR LOAD  In t h i s  chapter the l o c a l  on system s t a b i l i t y  load  i s investigated.  effect  o f a synchronous motor  A one l i n e  diagram o f t h e system i s  shown i n F i g u r e 4 . 1 .  J  Infinite B u s ^ Transformer Synchronous Motor  4 . 1 Synchronous G e n e r a t o r System w i t h L o c a l Synchronous Motor Load  Figure  4.1  Synchronous  The generator defined  except that  t o be i n t o  motor  (2.25)  Equations  synchronous  synchronous motor (2.17),  Motor  m o t o r h a s t h e same e q u a t i o n s  the positive  the machine. quantities.  and (2.26)  direction for stator The s u b s c r i p t  Analogous  currents  " 3 " i s used  to state  of the synchronous  as the synchronous  equations  generator,  i s now  to denote (2.15)  to  the synchronous  equations are  (4.1) P A  3d  pA„ 3q  =  ~ 3d 6  = -e  +  3q  R  3  a  i  3  "  d  "  + R i + M (i 3 a 3q 0  pA  V  3 F  = v  3  F  +  v  3  F  R  S  3  ) X 3 <  1  s,)X~, V 3d  (4.2)  (4.3)  36  6  P  =  3  -w s 0  (4.4)  3  POLES. ps  < 3L " T  3  2ov The a u x i l i a r y  equations,  analogous  io (x 0  to  3M> +  (2.18)  - x  3 d  T  3 d  3 d " 83)  D  to  ')  X  (2.24),  3  D  (4.5)  are  * 3 F  -(4.6)  v 3FR  '3d  x  x  3df 3f e  '3F  R  X  L  3 d 3d0 i  (4.7)  3f  3F  u  0 3d A  (4.8)  3d x  3d' 3do' T  x  3d'  0 3q L  a  to  3  3q  3  (4.9)  x 3q  =  = pa  +  V  6  =  WQ +  U)Q  - u)  3  (4.10)  3  (4.11)  p6  3  (4.12) 0  Since common,  the terminal voltages  t h e dq v o l t a g e s  of the generator  o f t h e two m a c h i n e s  t  and t h e motor a r e  a r e r e l a t e d by  ! - cos 3  2 - sin 3  a  a  2ir - cos 3  (a  -  (a  2TT  3  - sin 3  (a  -  (a  3  4ir  ATT cos  sin  - cos 3  a  -)  sin  0  3  a  2TT cos  [3  (a  2TT -  -)  3  s i n (0-3  -)  3  Solving  for  a n d 63^  63^  e  e  4.2  3  = T  d  3q  =  T  with  [e  x  x  (a  4 7T  ^ q e  Transmission Line  xB + 1  n  c  o  s  =  n  _  6  "  6  3^  (a^  )  '30  0,  63) + e  (6 -  ^  sin  -  e  d  sin  q  s  i  (6 -  ^  n  ~  6  (4.14)  63) ]  (4.15)  3^ ^  Equations  The t r a n s m i s s i o n  rG -  e = e^ '0 30  cos  d  -  -)  0 3  (4.13)  '3q  3  A 7T - cos 3  3d  e  -rB  l i n e equation, analogous  -  xG  v  r  0  to  (3.34),  -x  ^-Gr  is  x  3Mr (4.16)  r B + xG  rG -  xB +  1  0  m  1  Gm  3Mm  1  where  3Mr  cos  Tf 3Mm  Equation  (4.17)  is  6  0  sin  6.  _x_  (4.17) sin  obtained using  <5-  -cos  6'.  A  3q  t h e same a r g u m e n t s  as  those  that  led  to  the w r i t i n g (2.37),  of  equations  (2.38)  e  and  (2.35)  (4.17)  d  k  into  l  k  and  (2.36).  Substituting  (4.16)  and  solving  " 2  VQ  COS  6  k  VQ  sin  5  k  2  2  D  -D  5  for  C  -c  L  6  e  l  (2.35), and  d  C  2  e  x  q  (2.36),  ,  d  C  2  ±  3d (4.18)  ±  w h e r e k-^,  k ,  and C  2  D  5  D,  If that  have been d e f i n e d i n  2  = C  x  cos  (<S -  = C  x  sin  (6 - 63)  equation  (4.18)  the r e p r e s e n t a t i o n of  the  synchronous  generator  system with  local  the generator  coordinates  are  In  summary,  system w i t h l o c a l (2.26),  4.3  (2.48),  synchronous and  (4.1)  C o m p u t a t i o n and L a b o r a t o r y  are  the  synchronous real  power  same a s  motor input  are to  in  sin  (6  -  2  cos  (6 -  C  to  Test  (2.45)  at  (4.20)  (3.39)  it  is  system w i t h  load  of are  seen  local  The r e p r e s e n t a t i o n o f  the the  load  synchronous  is  more  to  the complex  the  speed.  synchronous  (2.15)  and  63)  f i x e d on t h e r o t o r whereas  generator  (2.17),  (2.25),  (4.5).  Results  2 and  synchronous  to  (4.19)  i n d u c t i o n motor  3.  motor  for  the t r a n s i e n t  The p a r a m e t e r s  determined from t e s t s  the  (2.42)  63)  equation  and d i s t u r b a n c e  Chapters  -  generator  complex.  equations  motor  The s y s t e m p a r a m e t e r s study  2  are r o t a t i n g  the s t a t e  (2.49)  + C  synchronous  much l e s s  coordinates  equations  compared w i t h  motor  i n d u c t i o n motor  is  is  63)  synchronous  because  load  3q  is  and l i s t e d the  of  stability  the  in Table 4.1.  same a s  that  of  the  The  39  Table  R  3a  x  3d  x  3d'  R  3f  POLES  Synchronous  Motor  Parameters  Synchronous  Motor  Parameters  4 . 2 6 ft  X  3q  1 3 1 . 7 ft  T  3d0'  4 0 . 4 ft  L  3afM  1 0 . 8 ft  J  3  D  3  4  3  Table Operating  Conditions  and  100  P  G  P  3  v  t  T  x  T  3L  W/ph  57.6  V/ph  d  \ X  F  e  f  V  R  6  0.263  V a l u e s of  0 . 0 6 1 7 Wb - T 538  of  Wb-T  J-s /rad 2  State  85  Q3  Wb-T  H  0.000995  %  N-m  Initial  0.405  s  J-s/rad  Variables  Conditions  1.5 0.431  0.0619  0.0121  Values  W/ph  40.0  1 2 4 . 8 ft  4.2  Initial  Operating  A  4.1  State  S  G  A  3d  V  v  0  T  t  e  3f  VAR/ph  19.3  VAR/ph  32.5  V/ph  3.73  N-m  8.17  V  Variables 0 0.270  Wb-T  -0.278 7.05  Wb-T  Wb-T  2.71  V  X  2.28  V  63  61.3 e l e c t r i c a l  s  0  118 e l e c t r i c a l  degrees  3F  3  degrees  i n d u c t i o n motor synchronous  motor  occurrence. Table  of  Chapter  more  generator 3.  results  except  the  All  Figure  for  a  2.4  the  same r e a s o n .  short  as  are  shown i n  and  the  fault  that  synchronous  are e s s e n t i a l l y agree  favourably  the  case  also  of  p r e d i c t e d by  in  4.2  4.6.  to  power  load.  same a s  The in  x^ith c o m p u t e d swing observed  induction  motor  listed  the  motor the  The t o r q u e a n g l e in  are  Figures  indicates  The s y n c h r o n o u s  p e r i o d which i s  to  slip  e n t i r e system  a local  swing.  initially  the  prior  and 4 . 2  waveforms  torque angle  set  results  when s u p p l y i n g  decreases  and f o r  test  Figures  of  l a b o r a t o r y measurements the  laboratory  loading  test  c u r r e n t and v o l t a g e  Chapter  the  stable  are  conditions  The computed and A comparison of  is  T y p i c a l i n d u c t i o n motor  power f a c t o r v a l u e s  The o p e r a t i n g  4.2.  system  3.  in  motor  oscillates  during  computation,  4.3. To f u r t h e r  computation i s the motor  but  the  system  the  shaft  for load  explore the nature of  extended to with  a sinusoidal  this is  include  shaft  investigation  no  longer  the  case  of  load.  are  constant  a synchronous  the at  steady  state  The o p e r a t i n g  same a s  0.431  motor  operation  of  conditions  of  i n T a b l e 4.2  N-m b u t  load,  varies  except  that  sinusoidally  as  T  That 1.0  is,  the peak  c y c l e per  Although  load  second.  the generator  system remains  stable  3L  is  =  ° - 4 3 1 + 0.1  0.531  N-m a n d  The r e s u l t s has for  are  sustained the case  cos  2nt  the frequency of shown i n  Figures  oscillations  investigated.  of  its  oscillation 4.7  and  is  4.8.  torque angle  the  h /  \  r  J  •"^ >  4f  -V  I 5  100  CD  y  80  H  60  H  o LU  Lu _l CD  cc  40  LU  ZD C3  o o <x en  LU  20  H  OH  LU CD  -20 0  0^5  1.0  Fig.  4.2  1.5 2.0 2.5 TIME (SEC) Generator  Torque  Angle  3.0  3.5  4  i  0  I I I 0.5  I I I  I  I I I I  1.0  Fig.  4.3  I  I I I I  I  I I I I  I  Motor  Torque  I I I I  1.5 2.0 2.5 TIME (SEC)  Synchronous  I  I I I I  3.0  Angle  I  I I I I  3.5  I  4.0  0.4 s  80 i—  -i  -  .—J•  -  o >  in '—'  GE  -  <x o60 •  >  UJ in  cr X Q_  -  -  CtZ  6D V-  X Cr: UJ UJ CD  n^l.O F i g . 4.4  Generator  1.5  2.0  TIME  2.5 ISEC)  3.0  3.5  V o l t a g e - L o c a l Synchronous Motor Load  4.0  IH«!!<!  tut  •>.imm rtv.'iV,i'.„i,. 1  mem  I  I  ^ItU.ll..!...'".'-'"""  10.On  a. 2:  0.4  !!» <  S  8.0  CE CO  z: Cr:  6.0 LU  err ZD CJ  Cr: 4 . 0 o 1—  H  CE  Cr:  I LU  2.0  0  I  0  ' ' ' I  I  0.5  I I  I  I  I I  1.0  I I  I  I  1.5  I  I  I  I  I I  2.0  I I  I  2.5  TIME (SEC) Fig.  4.5  Generator  I  Current  I I  I I  3.0  I I  I  I  3.5  I I  I  I  4.1  ' i i . . ...tiiiii!!  IIIIIM!'!'  jj  iiliiii  iiilililti  2n  0-4 S  CL  CC oo CC  U J  Cr: cr: ZD CD Cr: CD  r—  OO ZD 6D 21 CD cn  X CJ  00  0  '  0  1  '  1  I '  1  '  1  0.5  I '  1  1.0  1  1  I  '  '  1.5  '  '  |  I  I  2.0  I  I  I  •  I  2.5  I  I  I  I  3.0  TIME (SEC) Fig.  4.6  Synchronous  1  Motor  I  Current  I  I  I  I  1  3.5  I  I I  4.0  4(  100  CD  80  S CJ UJ  ^  60  UJ _J  CD  cr 40 UJ  ZD C3 cn  20 cn 63  az £  o  UJ CD  -20  ' ' ' ' |  0  1  0.5  1  1  1  I""'  11 1  1  1.0' Fig.  4.7  P  1  1  1  i r-r-r-r-pi i i > J • t r t r j-"t i r >~] 1  2.0 2.5 TIME (SEC)  1.5  Generator  Torque  3.0  Angle  3.5  4.0  0  0.5  1.0  Fig.  4.8  1.5 2.0 2.5 TIME (SEC) Synchronous  Motor  Torque  3.0  Angle  3.5  4  4.4  Computation Results  A large investigated. the  The t r a n s m i s s i o n  and o p e r a t i n g  in  the  conditions  angle  swings of  motor  load are  swing  is  25.2  in Figures  electrical  when t h e m o t o r  synchronous  synchronous  fault  is  at  i n Tables  4.9  4.3,  and 4 . 1 0  without  motor  the  generator without  degrees  is  motor  line  Parameters  small machine s t u d i e s .  the synchronous shown  a local  are given  From t h e r e s u l t s local  Large-System  generator with  same d u r a t i o n a s  degrees  Using  same  load  l o c a t i o n and  The s y s t e m 4.4  The t o r q u e  synchronous  respectively. load  T h e maximum  and 2 0 . 9  obtained  load w i l l  in  also  Table  this  chapter  increase  it  the  is  concluded that  stability  of  4.3  Parameters  of  Large  Synchronous  Motor  Parameters  of  Large Synchronous  Motor  R  3a  0.00136 ft  x  3d  x  R  0 . 2 1 8 ft  x  3q  0 . 3 2 7 ft  T  3d0*  3d'  0 . 0 8 1 9 ft  L  3afM  3f  0 . 2 5 ft  J  3  11100  6  D  3  0  3  electrical  included.  system.  P0LES  of  parameters  and 4 . 5 .  and w i t h  the motor  is  I  5.3  s  0.024  H J-s /rad  J-s/rad  2  a power  a  Table Parameters  of  Synchronous R  a  4.4  Large  Power  Generator  Parameters  0 . 0 ft  x  n  T  d  x  d  1.85  x  d'  0 . 3 6 1 ft  L  R  f  0 . 1 8 2 ft  J  48  DQ  POLES  f  G  0.0  G  ft  T  T  A E  R E  Q  1.04  ft  7.76  s  '  afM  0  ,  0  6  1  H  376xl0  G  0.0  4  J-s /rad 2  J-s/rad  N-m Voltage  K  System  Regulator Parameters  0.925  1310 V  v (min)  -1030  v  8129 V  R  0.003 0.05  v (max) s s  Transmission  R  r  e  f  V  Line Parameters r (ohm)  x (ohm)  G (mho)  (mho )  -0.00145  0.105  2.22  -2.63  -0.00244  0.0815  2.86  3.98  -0.00286  0.0838  2.96  3.12  B  Table Large Operating  Power  Conditions  and  System  G  30 MW/ph  P  3  2 6 . 7 MW/ph  v  t  7970  5  N-m  d  f  V  R  6  of  State  33.7  Wb-T  S  G  14.2  Wb-T  X  3d  X  3q  136 V  X  3F  147  6  l.llxlO  e  Values  95.2  5  Wb-T  V electrical  degrees  s  3  3  3  0  v  Initial  X  Q  V/ph  6.33xl0  L  Study of  State  Variables  Conditions  0.239  X  3  Values  10  P  T  Stability  Initial  Operating  T  4.5  T  t  e  3f  MVAR/ph  14.3  MVAR/ph  6157  V/ph  5.73xl0  6  N-m  88.6 V  Variables 0.0 5.1 -7.12  Wb-T Wb-T  1.27xl0 17.6 0.0  4  Wb-T  electrical  degrees  Fig.  4.9  Large  Generator Torque Angle  No M o t o r  Loads  5  5.  SYNCHRONOUS GENERATOR WITH MULTIPLE LOCAL SYNCHRONOUS AND INDUCTION MOTOR LOADS  In t h i s c h a p t e r a more g e n e r a l l o c a l l o a d and i t s e f f e c t on power system s t a b i l i t y i s i n v e s t i g a t e d .  The l o c a l l o a d may c o n s i s t of any number  of synchronous and i n d u c t i o n motors as w e l l as any o t h e r l o a d s w h i c h may be by e q u i v a l e n t c i r c u i t s .  represented  A o n e - l i n e diagram o f t h e system i s  shown i n F i g u r e 5.1. Note t h a t l o a d s r e p r e s e n t e d  by e q u i v a l e n t  circuits  a r e absorbed i n t o t h e e q u i v a l e n t - i r r e p r e s e n t a t i o n o f t h e t r a n s m i s s i o n i n t h e network r e d u c t i o n  Synchronous Generator  / v  y\A  line  process.  /vV\A  AAM  /V<\A B  Induction Motors  F i g u r e 5.1  5.1  Machine  Synchronous Mofor~s-ir  Synchronous G e n e r a t o r System W i t h M u l t i p l e L o c a l Motor  Loads  Equations  The e q u a t i o n s  r e q u i r e d t o d e s c r i b e t h e synchronous  generator  system w i t h i t s e q u i v a l e n t - i r t r a n s m i s s i o n network, t h e i n d u c t i o n motors and t h e synchronous motors have a l r e a d y been d e v e l o p e d i n C h a p t e r s 2 t h r o u g h 4.  Synchronous  Generator With Voltage  P d X  ~ d "  =  e  Regulator  aid  R  -  0  w  (  "  1  G  s  (5.1)  c  ) X  (5.2)  pX  pe  = v  p  =  f  +  F  v  p  [f(v )  -  R  P R  [ A< ref  V  K  -  v  (5.3)  R  v  (5.4)  e )] f  t>  "  V  (5.5)  R]  RE  p6 =  -u s 0  (5.6)  G  POLES. <G T  2uv  Induction  +  f  G  t>  T  "  V  +  1  "  S  (5.7)  G>  Motors:  „-v k _ „ k. k P 2d ~ " 2 2d " X  r  P 2q X  = " 2  k  r  k  i  2 q  (T  :where f o r  2u  M  the m u l t i - m o t o r  W  S  +  k  (5.8)  w  0  A  s  k  x  M  2  d  (5.9)  k  k  POLES.  J  .„ k, k 0 M 2q c  1  -  k L  k %  )  + D  k M  (l  -  s  k M  (5.10)  )  0  system  the  superscript  k denotes  the  motor  number, k = 1,  Synchronous  2,  .. ., m  Motors:  P*3d  k  "  " 3d e  k  +  R  3a i3d k  k  "  - O d  "  "  k 3  >*3q  k  (5.11)  55  P 3q A  = " 3q  k  e  k  +  pX  3a  R  i  = v  k 3 F  P  k  6  3q  k  k 3 F  "  V  + v  = -o) s  k 3  0  -  1  V  (5.12)  3q'  (5.13)  k 3 F R  (5.14)  k 3  POLES  ps  < 3L T  3  2o)  J3'  "  T  3M ) K  +  D  3  (1 ~  s  K 3  (5.15)  )  r  where  k  The  order of  motors  and  and w i t h  the  (n-m)  similar  system is  is  = m+1,  7 + 3m + 5 ( n - m ) w h e r e m i s  t h e number o f  shaft  load in  5.2  Equations  Line  order  An e q u a t i o n s i m i l a r the  rG  general  -  case  xB + 1  r B + xG  synchronous  characteristics  one e q u i v a l e n t m a c h i n e  Transmission  ..., n  m+2,  to  to  decrease  (2.40),  t h e number o f  motors.  should  Motors  t h e same  be lumped t o g e t h e r  t h e number o f  (3.34)  of  and  induction size  as  equations.  (4.16)  is  written  for  as  -rB  rG -  -  xG  xB +  v  1  v, m  r  0  -x  iGr  "  ^m^Mr^  ~  z  n( 3r ) ±  k  (5.16)  + O  x  r  iGm "  *m<iMm > k  "  ^3m^  where Z  m  and E  n  are operators defined  by  m  k=l  n  >:  *n =  k=m+l  and  i ^ r  k  ° i  i x  and  solving  =  (2.37),  3r  t h e dq  cos  <5  sin  6  cos  53  sin 6-j  K  k L  £n "  3d (5.18)  /3 sin  k  (2.38),  for  k  cos  k  i  Substituting  Mra  (5.17)  /3  -cos  60  (2.35),  (2.36),  6-  (5.17)  and  (5.18)  into  voltages,  sin  6  -cos  6  <5  sin  6  sin  6  -cos  6  cos  6  sin  6  k  l  k  2  v  0  /3 -k  k  -k  k  k  2  l  k  2  k  ±  2  l  x  k  2  0  r  -x  cos  6  sin  6  x  r  sin  6  -cos  6  r  -x  k L  V T sin  6  -cos  6  -k  2  k  ±  x  r  x  k  ld  )  (5.16)  57  cos  sin  6  6  l  k  r  2  k  -x £ ( x T  cos  6  0  sin  6-  sin  6o  -cos  6'  k  i3d  k  n  sin  6  -cos  6  " 2 k  x  l  k  r  (5.19) which reduces to  e  d  k  l  2  VQ  COS  6  k  VQ  sin  6  k  C  l  2  C  = /3 -k  2  D  x  -D  3  L  4  ^ D  D  4  D -  ^n(T  -c  m  (T  (3.35)  and  2  are defined  (3.36)  k  V  3  -D  k 5  k L  6  3d (5.20)  k x  by  1  (2.44)  and  3q  (2.45),  D 3 and D  are defined  4  k 5  k 6  =  C  1  =  cos  (6  -  6  3  sin  (6  -  6  3  k  k  )  + C  )  -  C  2  2  sin  (6  -  6  cos  (6  -  <$  k  )  (5.21)  k  )  (5.22)  3  3  The i n d u c t i o n motor s t a t o r . c u r r e n t s a r e e l i m i n a t e d u s i n g and  (3.19)  in  i  by  and  D  D  ld  x  Jit  w h e r e C-^ a n d C  2  the m a t r i x  form,  D  ld  (3.18)  D  k x  k e  2  d  c  k 4  r  -c  k  A  2d  3  = x  (5.23) -D  k 2  D-^  V  r  2q  where  k _ r-i _± k  D  + x'  k  6  cos  (r ) + (x' ) k  2  6  sin  k  (5.24)  2  r - , s i n 6 - x' c o s k _ _t D, ' " (r ) + (x' ) k  k  k  2  r  "  3  L  k 2  2  [ (  k r  i  Equation solved  (5.23)  for  is  4  T  to  )  (5.26)  +  2  (x' ) ] k  2  12  x  (5.27)  r  r  L  now s u b s t i t u t e d  t h e dq v o l t a g e s  12  x  k ,k k , k. 2 , , ,k. 2, 2 2 t ( ! ) + (x' ) ] x  J  (5.25)  2  k k  l  r  '  k  6  into  (5.20)  and t h e r e s u l t i n g  equation  obtain  -1 1  0  D  - D  3  D u  4  + 0  U(T  1  l  k  D  4  D  VQ  ~ 2 k  )  k x  -D  6  l  C  /3  l  u  k  2  2  3  COS  h  k  k  k  2  D  C  2  C  l  x  l  d  +  k Do  2  k  VQ  x  4  D  6  r  -D/,  C  ^m<T D  sin  r  k  _ C  3  l  2d  V  2q  k x  c  k 3  "D  M?x  C  k  4  c  3  " 2  x  k 6  k 4  3d  k  (5.28) D  k 6  D  k 5  *3q  The m a t r i x motors  involved  and p r e s e n t s  induction motors, inversion  is  inversion  ' D^  = D  k  of  no  (5.28)  for  (3.9)  regardless In  k = l,2,...,m,  of the  and no  t h e number case  of  of  no  matrix  necessary.  The dq t e r m i n a l v o l t a g e s from  4x4  computation problems.  = 0  k 2  is  and  (3.10)  and  those  of  of  any  i n d u c t i o n motor  any  synchronous  motor  are from  calculated (4.14)  and  (4.15):  In system loads  as  is  l  q  k  = T  x  = T  x  k  k  (e  d  cos  6 +  e  q  sin  6)  (5.29)  (e  q  cos  6 -  e  d  sin  6)  (5.30)  =  T  x f d  c  o  s  <  e  3q  k  =  T  x t q  c  o  s  (  k  e  k  summary,  given  transmission  e  k  U  given  equivalent  d  3d  e  6  6  "  6  3 )  "  6  3 )  k  +  "  k  q  e  e  d  i  s  n  i  equations  multiple  local  i n d u c t i o n motor  all by  machine  (5.28).  circuits, network  through  they  (5.15).  terminal As  for  ( .~ 6  6  3 >]  (5.31)  3 )3  (5.32)  k  6  k  a synchronous and  conditions  local  ~  S  A general  loads  can be r e a d i l y  representation.  (  n  state  (5.1)  of  s  the  by  equation r e l a t i n g network  l  e  supplying are  e  to  generator  synchronous  motor  transmission the  transmission  w h i c h may b e  incorporated  line  into  represented the  5.3  C o m p u t a t i o n and L a b o r a t o r y  Although  synchronous  section  sufficient  generator  is  system  synchronous  and to  supplying  motor  [8].  investigate one l o c a l  in  Tables  conditions  are given  in  Table 5.1.  in Figures  5.2  through  It  reveals  Figures  that  the  both synchronous angle  is  because  seen of  Chapters  is  3.4  and 4.2  alone, as  in  is  Figure Chapter  3.1  given  of  a  and 4 . 1 .  in  and o n e  test  the  last  local and  The p r e f a u l t  The c o m p u t e d a n d  system  synchronous  and g e n e r a t o r  the generator  motor  operating  results  torque angle  the  system  is  i n d u c t i o n motor  loads.  the omission  not  i n d u c t i o n motor  single  of  immediately a f t e r  3 and  stability  where o n l y  of  are  shown  the  fault  Figure  motor  loads  were  increased  still  further  A decrease  occurs  t h e damper w i n d i n g s  of  in  in  in  the  present.  generator  laboratory  the a n a l y s i s  as  5,2  with  torque  test in  4.  The s y n c h r o n o u s swing  made o f  stability  and  2.1,  was  generator  5.8.  A comparison of  the  loads  The s y s t e m d i s t u r b a n c e  are given  that  a synchronous  i n d u c t i o n motor  parameters  with  Results  a g e n e r a l f o r m u l a t i o n of  with multiple it  Test  so  s e v e r e as  4.3. 3,  motor in  torque angle  the case  of  The i n d u c t i o n m o t o r  the case  of  is  shown i n  a single slip,  the generator  synchronous  Figure  with  Figure  the  5.4,  is  5.3.  motor  The load  almost  the  i n d u c t i o n motor  same  load  alone. The g e n e r a t o r currents, Chapters  Figures 2 to  transmission It either power  5.6  is  is  that  Figure  a r e much t h e  oscillations  5.5,  same a s  in  and  the  machine  the previous  v a n i s h more q u i c k l y  after  tests  of  the  restored.  concluded from the  i n d u c t i o n or system.  to 5.8,  4 except line  terminal voltage,  synchronous  studies  so  or both, w i l l  far  that  a l o c a l motor  improve the  stability  load, of  a  Table Synchronous  5.1  G e n e r a t o r System With L o c a l And S y n c h r o n o u s  Operating  Conditions  and  G  100  P  3  40.0  s  V Q  W/ph  d  V/ph V  e  3  f  8.17  T  3  L  0.431  X  0.263  Values  Wb-T  of  State  F  e  f  V  R  6 X  S  3d G  538  Wb-T  t  57.6  V/ph  T  t  T  L  3.73  N-m  0.131  N-m  Variables  7.05  Wb-T  51.8  electrical  3  2.28  V  X  2d  X  2q  0.0  VAR/ph  3F  s  Wb-T  19.3 3  X  V  0.270  VAR/ph  -0.278 -  ^3  degrees  85  3q  2.71  109 e l e c t r i c a l  Variables  X  q X  State  1.5  N-m  Wb-T  0.0617  Q  v  Initial  X  QG  W/ph  39.9  of  Conditions  0.033  M  Induction  Loads  I n i t i a l Values  Operating P  Motor  s  M  Wb-T  0.0 0.0486 0.366 0.033  Wb-T Wb-T  degrees  V  1 0 0 . On I  S  CD  y  80.0  o LU  gj 6 0 . 0 LU —J CD  cn  40.0  LU ZD O  cn o  cn  o i— cc et: LU  20.0-  0 -  LU CD  -20.0  i i i i | i i i i | i i i i | I T i i ( r i i i | i i i i || i i  0  0.5  1.0 Fig.  1.5 2.0 2.5 TIME (SEC) 5.2  Generator  Torque  3.0  Angle  r r | i r  3.5  r f ]  4,0  6:  -25  i  TIME Fig.  5.3  (SEC)  Synchronous Motor Torque  Angle  65  0.4  5  60.0n CO  co 55.0 i cr: UJ CD CE  — I  g  50.0  LU CO CE X OL  | 45.0 x cr: LU  LU CD  40.0  i i  0  i i  [ i  0.5  i i  i I  r-  i i  1.0 Fig.  II  r  j i i i i I i i i i  1.5 2.0 2.5 TIME (SEC) 5.5  Generator  I  i i i i  3.0  Voltage  i  i  3.5  i i  i  I  4.0  §:0.8H cc CO  z: or H-0.6 UJ cc ce  ZD CJ  org CD  (--  U  '  4-  ^  O  CJ  ?0.2H oo  0  r i  0  i  i  |  0.5  i  i  i  t | i  ;  i i  1.0  | i  1.5  i  i  r  2.0  TIME F i g . 5.7  | i i  i ;  j  2.5  i  i i i | i  3.0  (SEC)  Synchronous Motor C u r r e n t  i  i  i  | i  3.5  i i  i  j  4.0  TTfri'till!|'ll!'"?,'M!!!l  ItUi ll^illlliillliiiiliii  lilllln  I  0.4 s  2n  CL 21  CE CO 21  cn UJ  or  ct:  Si cn o o  0  ZD Q  0  r 1 1 1 1 1 1 1 1 I  0  0.5  1 1 '  1.0 Fig.  5.8  1 I  1  1  1  1  I  1  1  1  1  I  1  1  1  1.5 2.0 2.5 TIME (SEC) Induction Motor  1  I  1  1  1  3.0  Current  1  I  1  1  1  1  3.5  I  6.  The e f f e c t o f investigated. and  machines  loads  the f o l l o w i n g 1.  l o c a l motor  From a n a l y t i c a l  i n d u c t i o n motor  load,  improve  the  Both  synchronous  comparatively  3.  results  and  large  motors  a fault,  Motors  to  i n d u c t i o n or of  a power  1 and 2 were a l s o  c a s e where t h e motor line.  Conclusion  simulated  constant  loads  5 was  during  system  include  is  required.  loads  constant  recommendations  different  power a n d v o l t a g e  generator  and  the motors  o n power  system  can  swing  cause  and  Schippel the  and R o b i c h a u d  a r e made f o r machines  be done i n  should  order  to  [8]  can  model. for  the  transmission [10]  using loads.  studies. be  extended  torque angle  o n t h e power  a n d d i f f e r e n t power  test  current  further  exact p r e d i c t i o n of tests  stability  system  power and c o n s t a n t  synchronous  levels  should  on t h e f i r s t  tests.  t h e r e c e i v i n g end o f  S e c o n d , more e x t e n s i v e  and  system.  r e a c h e d by R o b e r t  if  both,  disturbances  p e r i o d i c a l mechanical loads  local  t h e damper w i n d i n g s  or  with  generators  have n o t i c e a b l e e f f e c t  the m a t h e m a t i c a l model of  to  synchronous  both synchronous  r e a c h e d by G e v a y  impedance,  The f o l l o w i n g First,  also  small  maintained. of  were a t  been  synchronous  oscillate  b e e f f e c t i v e l y d e t e r m i n e d w i t h a power Conclusions  has  system.  i n d u c t i o n motors  a power  The i n f l u e n c e o f  small  system model u s i n g  r e v e a l e d from the  with large  oscillations 5.  as  and  stability  drawn:  amplitudes  is  system  both large  on a power  stability  stability  synchronous  4.  of  either  T h e damper w i n d i n g s  after  o n power  c o n c l u s i o n s .can b e  will  even i f  loads  and t e s t s  A l o c a l motor  2.  CONCLUSION  swings  system model factors  r e a c h more  of  for  the  general  conclusions. should  also  Other be added  types in  frequency  and m a g n i t u d e  presented  in  stabilization modelling  is  this and  the  loads  control  s u c h as  analysis.  should  thesis  a very  of  also  Third,  be  part  of  The  incorporated  investigations the  of  and a r c  the v a r i a t i o n  included.  c a n be r e a d i l y  important  lighting  power  study.  furnaces of  bus  voltage  formulation into  systems  optimal since  the  system  REFERENCES  1.  O.G.C. D a h l , E l e c t r i c Power C i r c u i t s I I , New  2.  S.B.  C r a r y , Power System S t a b i l i t y  3.  M.P.  Weinbach,  4.  E.W.  Kimbark,  5.  R.A. Hore, Advanced S t u d i e s In E l e c t r i c a l Power System D e s i g n , London: Chapman and H a l l , 1966.  6.  H.A. Bauman, O.W. Manz, J . E . McCormack, H.B. S e e l y , "System swings," AIEE T r a n s . , V o l . 60, pp. 541-547, 1941.  7.  D.S. B r e r e t o n , D.G. Lewis, C C . Young, " R e p r e s e n t a t i o n of i n d u c t i o n motor l o a d s d u r i n g power system s t a b i l i t y s t u d i e s , " AIEE T r a n s . , V o l . 76, P t . I l l , pp. 451-461, August 1957.  8.  J . Gevay, W.H. S c h i p p e l , " T r a n s i e n t s t a b i l i t y of an i s o l a t e d r a d i a l power network w i t h v a r i e d l o a d d i v i s i o n , " IEEE T r a n s . Pwr. App. and S y s t . , V o l . 83, No. 9, pp. 964-970, September 1964.  9.  M.H. Kent, W.R. Schmus, F.A. M c C r a c k i n , L.M. Wheeler, "Dynamic m o d e l l i n g of l o a d s i n s t a b i l i t y s t u d i e s , " IEEE T r a n s . Pwr. App. S y s t . , V o l . PAS-88, No. 5, pp. 756-763, May 1969.  I I , New  York: M c G r a w - H i l l , f938..  York: John W i l e y , 1947.  E l e c t r i c Power T r a n s m i s s i o n , New Power System S t a b i l i t y  I, New  York: M a c M i l l a n , 1948.  York: John W i l e y , 1948.  load  and  10.  J . 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Lewis, The P r i n c i p l e s of Synchronous Machines, 3rd edn., C h i c a g o : I l l i n o i s I n s t i t u t e of Technology, 1959.  17.  Y. Yu, K. V o n g s u r i y a , L.N. Wedman, " A p p l i c a t i o n o f an o p t i m a l c o n t r o l t h e o r y to a power system," IEEE T r a n s . Pwr. App. and S y s t . , V o l . PAS-49, No. 1, pp. 55-60, January 1970.  and  machine,"  I , " AIEE  72  18.  D.W. O l i v e , " D i g i t a l s i m u l a t i o n of synchronous machine t r a n s i e n t s , " IEEE T r a n s . Pwr. App. and S y s t . , V o l . PAS-87, No. 8, pp. 1669-1675, August 1968.  19.  K. V o n g s u r i y a , "The a p p l i c a t i o n of Lyapunov f u n c t i o n to power system s t a b i l i t y a n a l y s i s and c o n t r o l , " U.B.C. Ph.D. T h e s i s , February 1968.  20.  H.C. S t a n l e y , "An a n a l y s i s o f the i n d u c t i o n V o l . 57, pp. 751-757, 1938.  21.  A.E. F i t z g e r a l d , C. 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