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Stabilization and optimization of a power system with sensitivity considerations. Wedman, Leonard Nickolaus 1968

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S T A B I L I Z A T I O N AND O P T I M I Z A T I O N OF A POWER WITH S E N S I T I V I T Y  SYSTEM  CONSIDERATIONS  by LEONARD N I C K O L A U S B.Sc,  University  A THESIS  SUBMITTED  of IN  MASTER  in  Alberta, PARTIAL  REQUIREMENTS  FOR  Ve  accept  this  of  Head  the  the  of  of  as  conforming to  standard,  Department of  the  Electrical  . THE UNIVERSITY  Department Engineering  OF B R I T I S H  November,  \  SCIENCE  Committee  Members  THE  OF  Supervisor  Members  of  OF  Engineering  thesis  required  Research  FULFILMENT  Department  Electrical  1964  THE DEGREE  OF A P P L I E D  the  WEDMAN  1968  COLUMBIA  the  In p r e s e n t i n g  t h i s t h e s i s in p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s  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 B r i t i s h C o l u m b i a , the  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  I further agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s .  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 Date  Columbia  thesis or  publication  g a i n s h a l l n o t be a l l o w e d w i t h o u t  of  that  Study.  Department  It i s u n d e r s t o o d t h a t c o p y i n g o r  permission.  Department  and  copying of this  be g r a n t e d by t h e Head o f my  of this thesis for f i n a n c i a l written  for extensive  I agree  for  my  ABSTRACT  An i n v e s t i g a t i o n and  design  of  high  stabilization, controller solving  is  made  order  parameter  problems  some  systems.  and p a r a m e t e r  these  into  The  aspects  of  problems  the  analysis  treated  are  optimization,  computation  of  sensitivity.  The  developed  are  applied  to  a  methods  9^  an  system  optimal  order  linearized  shift  technique  f,or power  system. To used. both  stabilize Eigensystem  the  change  new  eigensystem parameter  used a  has  in  cost  an  method for  that  for  change  used  analysis  taken.  system  for  form.  and G r u v e r ' s with  a  approximation  of  the  is  by  the  initially  sensitivity  study  is  sensitivity  function  made  using  in  For  is  order  the  is  This  to  large  setting  to  computation  shifting a  steps  minimize of  method The  method  time  developed method  the  an  approximation  matrix.  Finally,  after  subsequently  recursive  Ricatti  is  determine  applied  that  successive fast  to  eigensystem  parameter  a method  determination.  new  method  eigenvalue  stable.  applied  method  This  conjunction  initiated  the  accuracy  quadratic Puri  and  is  A correction  may be  of  in  each  c a n be  the  eigenvalue  required  improving  controller, is  an  optimization procedure  solving  lation  change  been made.  functional  optimal  system,  sensitivity  parameter  the  in  the  developed calcuto  ensure  response  for  simultaneous  reduces  computation  (9) time the of  significantly investigation  parameters.  plied  to  the  over of  The  design  the  time  response  results of  conventional  of  the  a suboptimal  ii  method  thus  sensitivity  to  sensitivity  study  controller.  a  enabling  large are  number then  ap-  TABLE  OF  CONTENTS Page  ABSTRACT  (ii)  TABLE  (iii)  OF CONTENTS  LIST  OF T A B L E S  (v)  LIST  OF  (vi)  ILLUSTRATIONS  ACKNOWLEDGEMENT  . .  NOMENCLATURE  (viii)  1.  INTRODUCTION  2.  EIGENSYSTEM S E N S I T I V I T Y SYSTEM 2.1 2.2 2.3 2.4 2.5  3.  4.  4.3 4.4  Problem Formulation Eigensystem S e n s i t i v i t y Analysis An E i g e n v a l u e S h i f t i n g M e t h o d An E x t e n d e d C o l l a r a n d J a h n C o r r e c t i o n M e t h o d . A N u m e r i c a l Example A P P L I E D TO  20  FOR  20 21 22 23  A 26  Problem Formulation A R e c u r s i v e Method f o r O b t a i n i n g Successive Approximations A c c u r a c y of the R e c u r s i v e Method A Numerical Example  MULTIVARIABLE  5.2 5.3  CONTROLLER  SYSTEM  ANALYSIS  6 7 10 12 14  PARAMETER  Introduction . ... An E i g e n s y s t e m Form o f t h e C o s t F u n c t i o n a l . . . Performance Function M i n i m i z a t i o n A N u m e r i c a l Example .  SENSITIVITY  5.1  A P P L I E D TO 6  COMPUTATION OF AN O P T I M A L  4.1 4.2  6.  ANALYSIS  STABILIZATION  H I G H ORDER  5.  1  EIGENSYSTEM ANALYSIS OPTIMIZATION 3.1 3.2 3.3 3.4  (vii)  OF T H E  26 the  TIME RESPONSE  28 30 31 OF  SYSTEMS  39  S i m u l t a n e o u s C o m p u t a t i o n of Time Response Sensitivities t o a L a r g e Number o f P a r a m e t e r s . D e r i v a t i o n of the Computation A l g o r i t h m A Numerical Example  CONCLUSIONS  40 44 46 56  iii  Page APPENDIX A A.l A.2 A.3 A.4 APPENDIX B  FORMATION OF THE COEFFICIENT MATRIX A OF EQUATION ( 2 . 1 ) Third O r d e r M a c h i n e and T i e L i n e Equations Governor and H y d r a u l i c O p e r a t o r Equations Voltage Regulator-Exciter Equations Summary o f S y s t e m E q u a t i o n s INITIAL  REFERENCES  59 .....  62 63 64  OPERATING CONDITIONS OF A  POWER SYSTEM APPENDIX C  59  ,66  FLETCHER AND POWELL'S DESCENT METHOD .  68 69  iv  LIST  OF  TABLES  Table  Page  2.1  System Parameters  2.2  Eigensystem Shift  4.1  15 Through  Parameter  Adjustment  18  Error  32  of  K Approximation  v  LIST  OF  ILLUSTRATIONS  Figure 4.1  Page  Governor  Transfer  Operator  .  Function  and  Hydraulic 33  4.2  Voltage  4.3  Original  4.4  System  Responses  for  Q = d i a g [l  4.5  System  Responses  for  Q = d i a g [lO  5.1  Sensitivity of S to System Operating Conditions  5.2  Sensitivity  5.3  S-Response  Without  S-Response  with  5.4  of 5.5  Regulator-Exciter System  of  & to  l] . . .  36  10 1 1 1 1 1 1 l ] .  37  Parameters  Control,  and  Initial  48  Gains  Over  Control,  49  Variation Over  of  P  q  ..  Suboptimal  Control,  Over  Variation  P  o Control,  Optimal  Over  Control,  Variation Over  of  Q  ..  q  52 53  *o  S-Response  with Suboptimal  Control.  Over  Variation  Q *o  5.9  S-Response  5.10  S-Response w i t h  Optimal  5.11  S-Response w i t h  Suboptimal  to  52  Variation  Q  v.  51  Variation  o  S-Response w i t h  of  1 1 1 1 1 1 1  35  51  5.7  of  i  Controller  Optimal  S-Response W i t h o u t  5.8  33  Responses  5.6  of  Function  P  S-Response w i t h  of  Transfer  Without  . . . . . .  A.l  Governor  A.2  Voltage  Transfer  Control,  Over  Control,  Function  Regulator-Exciter  Variation Over  Control,  vi  v  ^ «« 0  54  Variation  Over  and H y d r a u l i c Transfer  of  53  Variation  Operator..  Function  55  63 64  ACKNOWLEDGEMENT I wish for  his  the  research  continued  many h e l p f u l Dr.  draft  encouragement  a n d M r . G.  valuable  the British  Also,  the material i n this  project, during  I have had thesis  with  Dawson.  suggestions.  b y my c o l l e a g u e s  of this  and guidance  thesis.  a r e d u e t o D r . M. S. D a v i e s  In a d d i t i o n , the support and  of this  d i s c u s s i o n s about  for offering  final  interest,  work and w r i t i n g  K. V o n g s u r i y a Thanks  and  t o t h a n k D r . Y. N. Y u , s u p e r v i s o r  i s duly from  Columbia Telephone  f o r reading The p r o o f  the  reading  manuscript ofthe  appreciated.  the National Research  Council  Company i s g r a t e f u l l y a c -  knowledged. I  owe a l a r g e d e b i t  patience  o f g r a t i t u d e t o my w i f e  and c o n t i n u i n g support  Doris  for her  t h r o u g h o u t my p o s t g r a d u a t e  vii  work.  NOMENCLATURE General A  nxn system  matrix  B  nxm  u  m-dimensional  control  b  n-dimensional  input  u  scalar  \^  eigenvalue  control  matrix vector  vector  input  T _  _  i '  X  X,  T i  V  V  X , w  y y  o f A and A  Vu\  eigenvectors  respectively  e i g e n v e c t o r m a t r i c e s composed o f columns v ^ , i = 1, . . . n , r e s p e c t i v e l y  initial  o f x. a n d  o f system states  state  variables  of y  q  parameter  A,  =  J  cost  Q  a positive  d e f i n i t e nxn state  V  a positive  d e f i n i t e mxm  K  nxn R i c a t t i  R  inverse  g(s)  c h a r a c t e r i s t i c polynomial  h^  coefficients  R(s)  adjoint  vector  functional variable  control  signal  weighting  matrix  weighting  matrix  matrix  of discrete  time  i n t e r v a l £ct of matrix  A  of g(s)  matrix  polynomial  of A  2 p^  c o e f f i c i e n t s of g(s)  z(t)  n-vector  w(t)  (2n-l)-vector  l\  1  X and V  approximate n-vector  Q  o f A and A  prefix  of state  variables  of state  denoting  of  variables  a linearized viii  (5.33) of  (5.34)  variable  subscript p  d/dt,  System H  field  x  d-axis x^  operator  mutual  d and q - a x i s  ;  reactance  transient  tie  G+jB  shunt  T^k  d-axis  damping c i r c u i t  I^JQ  d-axis  transient  line  Tqo  ^  -r^j  1  d-axis ^  a n <  a n c  ^  ^  q.-  (  i  a x  transient l -  a  £g  hydraulic  ^  dashpot  Y  water  x  i  at  time  generator  short  gate  actuator  Y <-ex  exciter  time  voltage  regulator  time  control  signal  H  generator  D  damping  time  and  time  constant time  constant  circuit  time  short  time  constants  constant  circuit  time  constants  constant  constant  constant  inertia  constant actuator  time  constant  loop  gain  constant  coefficient  voltage  regulator  gain  ^2  voltage  regulator  speed  or  governor  permanent  droop  ^  governor  temporary  droop  feedback  ix  bus  terminal  open c i r c u i t  time  infinite  constant  gate  governor  i  rotor  constant  -rj  •^•^  generator  open c i r c u i t  turbine  time  between  subtransient  s  and  reactance  subtransient  s  stator  reactance  impedance  admittance  tdo  between  synchronous  R+jX  a  condition  resistance  d-axis  Tq  operating  resistance  x\ a  'Cd'  initial  Parameters  R^ , ad  an  t i me d e r i v a t i v e  armature  a  x^,  denoting  System P,  Variables  Q  real  and r e a c t i v e  energy  conversion  mechanical generator  torque  CO  generator  angular  rated  *d' ^q ' d ' q ' ^ d V  V  '  angular  H[  linkages  to  ^ * anc  1~ 'ax:  terminal  V  infinite  bus  g  proportional  field  exciter  s  radians/second  radians/second  currents,  voltages  and  flux  respectively  armature  v  respectively  radians  v e l o c i t y , 377  v^  f d ' """f d ' ^ f d  in  velocity in  voltage  V  of generator  generator  angle  V p j j  Q  power  torque  input  ^  ,1  output  to f i e l d  current  voltage  voltage voltage,  control  ^/p  field  flux  Vp  field  voltage  g  p. u. gate  h  p. u. h y d r a u l i c  a  gate  a„  governor  current  voltage  linkage  proportional  proportional  movement  actuator  and f l u x  head  signal  feedback  signal  x  to  t o LfJ^^  v ^  linkages  respectively  1 INTRODUCTION  1.  In  the  problems the  of  to  steady  high  power  order  be  system  overcome.  state  systems.  The  systems  techniques  but  they  systems  the  power  ous  manipulation  calculation are  thus  costs system  size  attempt to  and  is  and  optimal  by  derived  These  in  apply  equations  reduce  approximati-ons the  the  or  to  problem. a  and  apply  to  In  number  the  of  method  thesis,  analysis  and  of  large a  laboriof Methods  equations  by  computation  accuracy,  methods  variable  system  Since  this  analysis  costs.  the  in  of  process  economize the  analysing  require  computation  efficient the  the  study  usually in  of  single  the  which  compromise  systematic  to  methods  difficult  control  for  extending  prohibitive  either  them  to  many  problem  are  algebraic  solve  the  techniques  economization f i r s t .  theory  these  still  established  essentially  some  are  sensitivity  field.  to  to  is  the  develop  trol  some  of  there  difficult  result  required  required  are  of  often  introducing  One  stability,  multivariable  in  design,  it  of  one is  reducing should  intended  from modern  design  of  con-  power  systems. The  first  stability. theory  for  to  large  is  usually  requirement  Although  described  The  method  by  best  system  and m u l t i v a r i a b l e  form  eigenvalue  for  achieving  Van  Ness  et.  by y  of  a =  methods  system  systems. set  of  In  system  operation  available  these  are  in  not  suited  modern a n a l y s i s ,  differential  For  is  control  the  equations  in  system  state  A y  (l.l)  an  stability and  are  stability,  determining  analysis.  al.  satisfactory  classical  determining  variable  for  stability unstable preferably  Laughton^"^  with  this  system by  suggest  formulation  a method  parameter a  is  is  required  adjustments.  stabilization  2• method which  involves  the  eigenvalues  with  the  eigenvalues  and  , are  used  value a  to  positive  determine  system  eigenvalues  was  were  stability  The  are  difficulty  by  the  parameter  parts.  and  successfully  found  the  real  eigenvectors  sensitivity.  large  calculating  In  their  methods  required  since  the  the of  latter  finding  handled  method  sensitivities  by  of  to  Van  both  former  determine  the  of  eigen-  eigensystem  Ness  et.  transforming  of  al..  A to  The  an  upper  (2) Hessenberg  form  eigenvectors values  and  were  using  then  into  the  QR t r a n s f o r m  f o u n d by  A X  =  XX  A V  =  XV  X  =  back  of  Francis.  substitution  of  The  the  eigen-  (1.2)  m  and  A  where Once the  the  eigensystem  real  part  into  account  than  one  a  new  of  an  more  eigensystem that  eigensystem  an  tivity  is  ously  calculated  algebraic These  q.  in  new  techniques  c o u l d be  eigenvalue can  each  change  be  that  found.  and to  shifted  To  allow  parameter can  However,  change.  be  found  take  more  generalized.  Laughton from  the  (1.3)  Aq  and ^lE/iq  Since  is  the  and Fadeev  terms  parameter  eigensystem  eigenvalue  determining  sensitivity  form  after  • • • X^~J  by  VE .  sensitivity  Laughtonf"^  eigenvector  for  the  method  eigensystem  a parameter  eigenvector Both  the  •• •  2  negatively  this  approximate  =  X  unstable  required  sensitivity  denotes to  one  change,  A E where E  obtained,  eigenvalue  than  parameter  suggests  was  diagj^  only  are  The  eigensystem  adapted  in  and  Chapter  the  were  previ-  the  calculation  derived  results  sensi-  was  adjustment,  additional  and F a d e e v a ^ " ^  the  eigensystem  sensitivity  a parameter  equations. of  the  required.  required  given  in  ^A/^q.  2 to  the  stabilization  3 of  a power  are  system.  computed  mate  the  new  sensitivity is  not  an  extended  often system  i s used.  the  and  problem making  settling  parameter  available  approximate  shift,  is  costly  response time.  structural is highly  This  they  mainly  this  engineer  response  the  i s faced  of  This  oscillatory  Although  are  For  changes.  requires  technique.  literature,  dynamic  eigensystem  developed.  a l r e a d y i n oj>eration, the the  approxi-  sensitivity  is required.  method  To  the  eigensystem  correction  improving  sensitivities  formulation.  eigenvalue  correction  optimization  i n the  an  Jahn  any  slow  eigenvector  a matrix  a  of  system  and  then  system  i f the  extremely  after  I f the  enough  Collar  without  required  a  eigensystem  a power  with  eigenvalue  s i m u l t a n e o u s l y from  accurate  For  Both  or  a may  be  has  an  application  of  there are applicable  many to  methods  low  (6) order  systems.  minimize  a q u a d r a t i c form  Normally, its  or  solution  to  solve  similarity  matrix,  form.  The  dependent details The initial  after  cost  are  given  methods design  calculating  or  type  cost  inversion  of  an  an  n (n+l)/2  as  well  x  n(n+l)/2  The of  to  systems.  functional  inversion  t r a n s f o r m a t i o n can to  form  the  be  cost  an  nxn  as  functional system  can  function,  optimal  then  used  to best  advantage  transformation matrix.  transformation, will be  without  have  the  Jordan  e v a l u a t e d , even any  matrix  The  canonical  with  a  inversion.  time The  3. control  improvement  optimal  for large  method  similarity transformation.  i n Chapter of  of  computational  simultaneous.equations.  functional  weighting  a  functional  e v a l u a t e d by  e i g e n v e c t o r s a r e .used  system  cost  the  n(n+l)/2 was  developed  this  obtained using a  The  of  of  involves  i t s derivative  matrix  if  a  derivative  matrix  Vongsuriya  of  a  controllers  t h e o r y may power  be  system.  for large  applied Recently  systems  have  to  the  methods appeared  in  the  l i t e r a t u r e .  These  methods  solve  determination  of  equation  involves  equation  leading  The  problem  is  the  R i c a t t i  equation  the  control  the to  to  solve  signals.  integration  enormous  for  the  which  in  Normally  of  a  turn  the  nonlinear  computation  R i c a t t i  effort  matrix.  enables  s o l u t i o n  a of  this  d i f f e r e n t i a l that  sometimes  (4) f a i l s  due  to  c a l c u l a t i o n  i n s t a b i l i t y .  Macfarlane  and  v  Freested  (7) et.  a l .  solved  matrix  by  an  for  the  R i c a t t i  eigensystem  But  their  method  not  completely  formulation  requires  r e l i a b l e  a  i f t  tioned.  Puri  method and  which  of  the  4  to  R i c a t t i  the  obtain  is  stable,  a It  the  as  w i l l  must  analysis the of  to  simple be  w i l l  be  as  is  an  that  a  is  are  great  order improvement.  space is  and  i l l -  is  condi-  The and  the  the  f i r s t  the  and  for  found  by  i n  c o n t r o l l e r  is  the  a  design,  of  operating  point  response,  time  is  used.  integral  it  very should  c o n t r o l l e r  s e n s i t i v i t y  i n  v a r i a t i o n a  2  system  dependent  chosen  any  the  usually  reasons  is  system.  an  design,  course,  Since  that  suboptimal  c o n t r o l l e r  stable.  Chapter  Chapter  4.  reason,  is  power  so  Chapter  i n  the  approximation  i n  evaluating  engineering  this  a  methods  of  system  applied  developed  for  time  the  that  previous  solution  approximation  optimal  system.  approximation  the  the  that  so  developed an  to  is  method  simplify  i n  a  matrix  c o n t r o l l e r  matrix  For  non-linear  v a r i a t i o n  double  storage  than  refined  optimal  although  possible.  parameters  space  chosen  method  applicable.  l i n e a r i z e d cause  be  technique  To  of  a  successive  convergence  R i c a t t i  implement  is  order  a  condition  must  noted  double  storage  approximations  designed.  system a  only  of  which  amount  present  v  s t a b i l i z a t i o n  recursive  d i f f i c u l t be  f i r s t  successive  using  The  Gruver's  the  the  from  o\  less  matrix  and  large  monotonic  c a l c u l a t i o n  To  The  fast  matrix.  Puri  Gruver  requires  exhibits  R i c a t t i  and  matrix  on  the in  case these  response  sensitivity tivity the time  two  study  is  required.  of  an  n^*  integration  of  an  for In  time  study  this,  of  Chapter  response equation  to  1  order  is  a method  variations  integrations  strated  in  a numerical  optimal  controller.  system  n«n]^ order  course,  5,  Kerlin for  stated  that  this  m parameters  system.  The  sensi-  requires  digital  computation  enormous. is  developed  in  m parameters  are  example  required. to  reduce  for  computing  system  simultaneously.  This the  the  method  is  structure  Only  demonof  the  EIGENSYSTEM SENSITIVITY  2.  ANALYSIS  A P P L I E D TO S Y S T E M  STABILIZATION The  methods  eigensystem  presented  sensitivity  stabilization  problem.  applied  eigenvalue  tend  move  to  author by a  This  method,  it  c a n be  routine  or  found by  The  inaccurate  unless  is  large  made  for  importance method  is  Van  et.  in  was  never  would  be  either  used  step  sizes.  with  of  the  high  how  the  Although  the  stabilizing  a  after  each  parameter  an  eigensystem from  order  application  or  an  in  this  used,  calculation  latter  a  time  systems, of  system  sensitivity  used  computation  latter  change.  c o n s u m i n g and t h e are  have  eigenvalues  method were  sizes  Since  system  this  time  step  a  of  Laughton^^  show  eigensystem  is  small  analysis  here  new  to  If  reentering  the  to  extension  applied  actually  required  by  be  an  changed.  applied.  method  very  is  of  are  a l . ^ ^ a n d  analysis  possibility  former  the  Ness  parameter  estimating  relations.  chapter  and w i l l  sensitivity  the  eigensystem  this  analysis  a system  mentioned  this new  if  in  correction is  the  of  utmost  sensitivity  extended  Collar  (12) and J a h n the  correction  correction  fewer  method  The  allows  point  differential  equations and  are  equations y  Matrix  to  larger  step  sizes  chapter.  and  Applying  consequently  Formulation  system  operating  state  developed  steps. Problem  2.1  method  A is  generally  variables.  move  in  a  small  a  are  assumed to  described in  the  =  A y  by  set  linearized of  first  about  order  an  linear  form (2.1)  nonsymmetric  A perturbation region  a  be  about  is the  matrix  and y  assumed which operating  is  a vector  causes  point.  The  the  of system  response  after  this  disturbance y(t)  where  \^  are  the  =  £  1=1  e^  eigenvalues  -  written 1  x  of  i  v  A,  i  as y  x^  >  o  = l ...n  1  (2.2)  f  eigenvectors  of  A,  and  T  v^  eigenvectors  of  .  For  A x\  =  \.  A v  =  T  As  is  in  the  well  known,  left  half  parameters  for  techniques  do  to  apply  sented a  c a n be  to  in  2.2  i  a  which  the for  order  matrix  form.  stable  is  (2.3)  the  stable.  selecting  if  they the  sensitivity  parameters  is  are  to  Although  problem,  expecially  Eigensystem  eigenvalues  problem then  this  systems,  method of  if  The  solving  determine  classical  are  difficult  system  is  analysis  such  located  that  repre-  provides  a high  order  stabilized.  errors  tivity  eigenvalues,  (2.4)  system  Calculation Fadeev  sensitivity  distinct "  is  Sensitivity  by  with  x.  Eigensystem  lyzed  system  complex p l a n e .  exist  c a n be  a  v.  system  high  convenient  system  A  of  Analysis  eigenvalues  and F a d e e v a ^ ^  analysis.  equations  For  similar  by  and  applying,  completeness  to  Fadeev  eigenvectors in  essence,  a derivation  and F a d e e v a ' s  is  of  were  ana-  eigensystem the  given  in  sensithis  section. The derived q,  that  sensitivity by  taking  of  the  an  partial  to  derivative  a parameter of  (2.3)  q  with  of  A  is  respect  to  is A ,  q  x  i  +  A x . ,  q  where  A,  and  *i'q  For  eigenvalue  convenience,  throughout  this  this thesis  =  \  =  W  =  partial unless  ^  i  X  f  q  x.  + X.  x.,  (2.5)  q  )>q  (2.6)  i ^ ^  derivative otherwise  (2.7) notation stated.  will  be  used  Premultiplying  —T  (2.5)  b y v\  vT1  Since,  from  one  obtains  + vT1  A, x. 'q 1 equation  for  =  \., " l ' q  v  x. 1 1  + A- v 1 1  T  T  x., l'q  v  (2.8) '  (2.4),  vT and  A x., l'q  A  \^ vT  =  a normalized eigenvector  (2.9)  product 1  -T v.  x.  J  1  equation  (2.8)  0  the  The by  i=j  (2.10)  for  becomes \ . , ^ 1' q  giving  . -<*1 f o r  =  sensitivity  eigenvector  premultiplying  of  =  vT A, x. 1 'q 1  \^  to  a parameter  sensitivity  this  v  is  equation  (2.11) '  q.  derived  from  equation  (2.5)  -T  by  v.  J A,  v. where  i  ^ j .  + v.  x.  Knowing  that A =  J  and b e c a u s e  of  the  x.  J  (2.12)  x\ ,  1  . A N,  may b e  x 1. x.  coefficients  equation  (2.16)  °ji Equation  q  written  where  c..  J  (2.12)  x.,  (2.13)  J  of  the  two  sets  of  eigenvectors  0  =  (2.14)  ,  c a n be  — x.. l'q  =  X  =  X  as  n.  q  J^l'  x.  X i  are  j = 1,  ,  i  f o u n d by  ...n  (2.16)  substituting  " V  '  f o r  1  ^  j  ( 2  *  1 7 )  as  ... "'*  c -  (2.15)  q  (2.15)  written  I  •  x.  vectors  fc,.,  ±  \  equation A  v \ -T - X.) vj 3c.,  /\  ( \  =  of  into  =  (2.16)  X.v.  +  X - vT  = =  x. ,  The  v.x.  X.,  becomes -T v. *j J  where  =  orthogonality vT  equation  A x.,  c - , 1 1  X  i '  **"  ...c  X  n]  .1  n3  J  T  (2.18) (2.19)  and  c..  =  0  xi Extending  (2.18)  sensitivities  to  include  c a n be w r i t t e n X,  By  assuming a s i m i l a r  the e i g e n v e c t o r quiring that  (2.21)  is  a unit  into  = 1,  in matrix  =  q  . . . n , the  eigenvector  f o r m as (2.20)  V K  T  q  matrix.  (2.21)  T  of matrix  + Aq.V, ) (X  equation (I  i  X C  sensitivity (V  I  =  q  i,  f o r m f o r "V,  V,  where  all  A  + Aq.X, )  results  + Aq.K )V X(l T  =  q  Substitution  (2.22)  may b e d e r i v e d I  (2.22)  of equation  (2.20)  and  in  + aq.C)  T  by r e -  =  I  (2.23)  T Since  the matrix  (2.10)  product V X equals  and a s s u m i n g t h a t  that  is  Aq K  then  the s o l u t i o n  Hence C is  given  Neglecting eigenvalues where  by  =  the second  are reasonably  eigenvalue,  sensitivity  order  of  (iq  large  analysis  is  terms  of  according to (2.23)  are  0  (2.23)  small,  (2.24)  is  -C  (2.25)  -V C  (2.26)  T  order  ±  for close  is  separated.  accuracy  poor  is  _T .  =  ii—.v. V 1  1  A, 1  valid  Near  merge  (2.23) m K Aq C) .  terms  eigenvalues  v, -  eigenvalues.  A,  t o be u s e d o n l y  r  points,  x. ^  second  i ^ j ^ k (2.27)  V  eigensystem  f o r the adjustment  real  the n e g l e c t e d  k  Since  l o n g as t h e  a multiple  —T .  x,  k  into since  —  as  breakaway  £j< v ^ x x  3  become  ^  matrix  (2.17).  complex conjugate  terms  £>q C  T  =  T  V,  where  the second order  o f K from K  a unit  k = 1  ...  n  sensitivity  of parameters,  the  close  eigenvalue  parameter An system the  problem  adjustment  important  available  by  step  required  performing  eigenvalue  diagonal  the  elements  of  eigenvector  A computational to  where  A,  column.  It  T  =  A ,  can  As  an  only  a  increasing  new  eigensystem.  formulation of  the  of  computation  sensitivities  the  eigen-  since  products  | (2.28)  equation  (2.11)  are  given  by  the  as  i j  (2.29)  matrix /  V  (  also  be  X  C from i  made  illustration element  verified  all  becomes  d..  one  J  be  D  by  X  q  from  d  can  ease  determining  .=  ij  saving  contains  the  sensitivity  advantage. q  V  i > q  proposed  matrix  matrix  C  used  =  recalculating  is  for the  circumvented  the  sensitivities  X and  of  equations  D The  be  and  feature  sensitivity  information  can  from  '  }  if of  a - , ij'q  (2.28)  (2.17) ^ "  1  the  J  of  consider  the  that  (  sparsity  this, in  as  i-th  D is  A,  the  row  given  2  *  3  0  )  is  q  case  and  J  i-th  simply  by  fjl, ... *•* jn] X  =  D  a. ., ij  'q  v. .  (2.31)  vni For  more  than  one  element  in  A,  ,  D becomes  a  sum o f  matrices  where  q 3 each are  matrix avoided  is  formed by  with  following  section  utilizing  the  2.3  k  elements for  The  parameters  in  solving  eigensystem  An E i g e n v a l u e  v e c t o r form For obvious  (2.31).  Thus  A, . q  the  - kn  An a p p r o a c h  problem  sensitivity  Shifting  (2n  2  of  ) is  system  multiplications taken  in  the  instability  equation.  Method  t h a t are a l l o w e d to change are w r i t t e n i n — T as q = jTj^, . . . q ^ , . . . q£j (2.32) r e a s o n s , the e i g e n v a l u e w i t h the l a r g e s t p o s i t i v e real  11 part,  \  ,  ments. are In  is  From  given  by parameter  (2.11),  to  the s e n s i t i v i t i e s \ X » m' q .  by  0  order  c h o s e n t o he c h a n g e d n e g a t i v e l y  J  to minimize  more  favorably  real  1  part  of  the e f f e c t  located  sensitivity  b  q., j  = 1,...Q, x  of parameter  eigenvalues,  eigenvalue  each  —T — v A, x m 'q . m  =  N  of X  adjust-  J  adjustments  the parameter Re \ \ I  , I <l t  m  k  1  S  .  (2.33) .  on o t h e r  q^ with  the  largest  , , , c h o s e n t o be  adjusted. Any  suitable  f r o m A^,  gradient  technique  but f o r the purpose  method s i m i l a r  of c o n t r o l l i n g  t o the Newton-Raphson * q  and d A p i s t h e s h i f t The  increment  in R e ^ X ^  A-1 i ' q  lating  k  i n the other • =  allowed  a  » i = l,...n  a l l AA-^  eigenvalues  for  '  to  l  increases change.  adjustment.  satisfactory^\^  i>(1  Once the  increment  is  calculated  then  k  is  Aq  k  has been  i n the eigenvectors  calculated  repeated  plane.  until  = l,...n  (2.29).  part  sensitive  then  If  (2.35)  after  o f one o f  obviously  q  parameter  the process  all  found,  is  X C. - k q ,  =  -V C . & q  from  a new A i s  c a n be c a l c u l a t e d  =  K  C  from  the r e a l  most  fails  i  as  calcu-  these k  cannot  be  may be c h o s e n  repeated  until  found.  AX  where  where  | X i  m  this  an a l l o w e d  used  a  m  c a n be c a l c u l a t e d i  b e y o n d Re ^ ^ ^ > t h e n  is  is  i n X>  (2.34)  ' Aq  I  X  The next  If  technique  eigenvalues  found that  s  the change  k  desired.  k  A  c a l c u l a t e £,q  = A X ^ R e ^ , ^  k  *Xi where  c a n be u s e d t o  calculated as  (2.36)  k  T  k  (2.30).  eigenvalues  and  k  The a d j u s t m e n t lie  process  i n the l e f t - h a l f  is  complex  It  should  parameter (2.35)  and  be  (2.36)  and  justment.  an  large  rection after  by  two  movement  size  each  to  argued of  used  in  This  adjustment.  The  small  change  region  this  time  case is  allows  a  correction  calculated  original  parameter  this  argument.  ad-  each  ad-  One,  unpredictable  time be  combined  if  with  eigensystem is  and  consuming.  method but  method  by  process  A after  can  fast  the  from  itself  sensitivity  small  consuming  may b e c o m e  in  to  about  larger  countering  b o t h methods  the  restricted  eigensystem  this  of  a  that  reasons  best  is  selecting  a new  in  method  eigensystem  calculation  the is  the  a method  are  that  this  applies  method a p p l i e d .  following 2.4  only  eigensystem  step  since  c o u l d be  There  believed  that  calculating  eigenvalue  two, is  It  avoided  justments  the  noted  adjustments  eigensystem. could  be  It a  a  cor-  calculation  detailed  in  the  by  first  section.  An E x t e n d e d  Collar  and J a h n  Correction  Method  (12) The forming  Collar the  and J a h n  matrix  correction  the  M and  R are  product  apjjroximate the  X^  diagonal  - 1  A  X E  (2.37)  be  and  (2.39)  into  (I  + E)  products  M- + R  the  (2.37)  X^  off  small  X  =  =  assumed to  Neglecting  applied  and  the  system  diagonal  elements  parts  since  matrix of  the  the  A. matrix  corrected  that X  with  and  R has  requires  Let  =  eigenvectors  respectively.  eigensystem  is  product XZ}A  using  method^  small  of  Xu, and  (2.38) _ 1  (M  M (I off  + E)  (2.39)  diagonal,  results  + R) (I  small  (2.38)  substitution  of  in  + E ) . •=  matrices  then  M.  R and E,  (2.40) (2.40)  can  be  13 written By  as  E M  insj^ection  the  -  M E  solution  of  =  E  R  (2.41)  is  e.. i j  =  r. . / ( i j  e..  =  0  •• -  m  JJ  m..) "  for  i.j  =  (2.42)  l....n  n The  corrected  matrix  X  is  then  determined  by  equation  (2.39).  In  T the in  thesis  -1  calculations,  (2.37)  instead  Collar  of  is  made  inverting  and J a h n ' s  the  method has  available  and  complex m a t r i x been  extended  is  used  for  X^  X^.  to  include  the  T correction This  is  one  First the  V  of a  so  that  the  N is  solution  to  V^ nearly  corrected  the  matrix  requirements  product  where The  of  a  an X  T  unit  of  be  normalized.  2.2. is  found by  forming (2.43)  and Z  requires =  T  V X will  N + Z  matrix  V X  Section  unnormalized V =  eigensystem  product  is  small  and  off-diagonal.  that  N  (2.44)  T where  V  is  assumed  to V  with  K assumed s m a l l  (2.45)  into  (2.44) (I  Neglecting written  be  given  =  T  and  (I off  by  + K)  V^  (2.45)  T  diagonal.  Substituting  (2.43)  and  gives + K)(N  products  as  of  small  K N  Hence  + Z)  =  =  N  (2.46)  matrices  K and  Z,  (2.46)  can  -Z  K  =  -Z  be  found  be (2.47)  N"  (2.48)  1  T A normalized V  can  (normalized and  from  (2.44)  V ) T  X  the =  requirement  that  I  (2.49)  as normalized V  Substituting  from  (2.45)  and  T  (2.47)  =  N  _  into  1  V  (2.50)  T  (2.50),  the  corrected  and  14 normalized  V  becomes V  Note  In the V  T  the  not  the  the  the  (2.35)  values.  these  to  At  repeated  the  The  the  correction  and  may  most  the  A matrix  application  eigensystem  is  the of  of  c a n be  by  the  be  high  written  in  in  the  need  for  to  the  method  change,  product T V X  product  is  corresponding  procedure  calculate  case  of in  method  is  first  iteration  moves  such  new eigen-  a discrete may  as  order  a  close  not  particularly  systems  the  the  eigenvectors  correction  order  a parameter  comparing  complete  eigensystem  the  by  checked.  the  except  diagonal.  eigenvalues  applied  sensitivity  stabilization  equations  to  eliminates  points,  N is  checked  need  (2.51)  1  inaccurate  method  it  the  be  N" )^  since  However,  eigenvalues  for  The  shifting  matrix.  Only  eigensystem  and  of  - Z  l  computed  accuracy  unit  closest  (2.34)  easily  process  required.  If of  is  eigensystem  X with  to  -1 N~  that  N (I  =  T  converge.  well  power  state  manner  suited  systems.  variable  (13) form the  and  the  eigensystem  eigenvalues  stabilizing of  after  the  procedure  A matrix.  stabilization  The  initiated position  indicates  the  of  by  calculating  the  degree  dominant of  stability  obtained. 2.5  A Numerical To  an  illustrate  example  will  also The  variable where  Example  of  an  the  application  unstable  demonstrate  the  power  eigensystem  power  system  form  y  A y  A is  derived  the  system  are  calculated  in  parameters for  =  a given  A.  initial P  is  stabilization presented.  correction  equations  are  method,  This  example  method.  written  in  state (2.1)  Appendix and  the  system  linearized as  of  ,  Q  This  matrix  operating and v,  as  is  calculated  conditions. derived  in  The  from latter  Appendix  B.  The  system  2.1  and  parameters  the  initial  =  chosen  operating  Qo  .753,  for  =  this  example  conditions  .030,  v  are  given  calculated  in  from (2.52)  1.05  to  Table  are i,  do  So All  .295, '  =  =  *  values  8  4  5  are  i  qo  '  ¥YO  in  per  Table  2.1  = .654, ' =  9  '  4  8  '  v, do V  = .393, '  k  k  l 2  o  System  Parameters Machine  Tie  Line  =  .003  .480  H  =  4.90  R = .110  =  .050  Tv = 1.90  D  =  2.00  X = .730  =  40.0  ?r  =  1.00  G = .100  =  1.00  =  .600  B  =  .270  =  9.00  = 7.60  d  X  cr = . 0 2 0  x' x  g  =  .500  (A.35)  A  the  coefficient  matrix  d  x q  .020  -.0204  (2.53)  unit.  r  From  .974  qo  .963  Voltage Regulator- Governor-Hydraulic Exciter Operator  t  v  L  do  becomes (2.54)  = •  -30.07  -4.300  -0.514  -0.187  = .130  38.47  57.70  1.000 1.000 -333.3 -2.080  42.06  -63.10  333.3 -19.61 -2.000 4.000  2.000 -1.053  -4.000  -0.133  -1.000  -50.00  -0.064  -0.480  -24.13  To  initiato  the  A computation  shifting  routine  process,  similar  to  an  eigensystem  that  used  of  by V a n  Ness  \ was  used  vectors the  to x^  claculate a n d v^  tialization ing  x^  by  In the  step,  the  the  the  scalar  v  shifting  ranges  A and X . f r o m  of  .0005  required.  et.  a l . ^ ^  T  found  correspondence  A is  in  x^.  this  to  v\  for  x  is  not  > T-,  >  from  require  A  .  The  eigen-  rearranging  established.  products  each  process,  and V  manner  eigenvector  J i  A,  are  As  a  since  final  n o r m a l i z e d by  The  > t  >  r  stabilizing  correction al  the  parameters  .1000,  part,  m  parameter 2. M  method  by  a  from  q^ from  specified  from  .0000  calculated  If  most  this  are  > £>,  adjusted >  within  1.000,  is  with  eigenvalue  the  largest  and. the  >  2  (2.55) 3.000  of'the  with  restrictions  with  k  application  summarized below  eigensystem  and  >  -3.000  of  addition(2.55).  positive  most  real  sensitive  (2.31). (2.34)  which  will  cause  a  in  other  eigenvalues  shift  the  change  the  that  q^. Aq^ exceeds is  such  the  change  that  parameter  must  after  searching  all  2 and  3 repeated  be the  until  allowed  Re  ">  found  and  parameters an  in  q^ or  Re ^ X , ^ 2 and  3  then  Calculate  the  new  eigenvectors  from  the  must  parameter-  (2.30),  Correct  the  approximate  eigensystem  by  the  the  repeated.  t h e n A.\j)  adjustable  if  (2.31)  method  be  can  (2.36). 6.  of  will  found. 5.  and  2.3  parameter  a Aq^ from  in  sensitive  and  2.4  the  600.0,  ^  a m o u n t AXJJ»  either  change  fails  decreased be  (2.29)  y  section  initial  changing  If  to  the  Calculate  4.  next  the  of  section  given  Calculate  3. result  of  Determine  A »  -600.0  process  consideration 1.  X  10.00,  divid-  i.  X  1.000  ini-  of  17 section  2.4.  7. new  Rer>eat  eigensystem  small  as For  was  the  reaches  determined the  procedure  by  specific  unstable  and  a  desired  at  negative  1 until value  Re  o r  or AAp  "  each  becomes  4. example  had  starting  of  this  section,  the  initial  system  eigenvalues  -333.53  + jO.  -2.1109 +  jO.  -25.091  + jO.  -1.0179 +  jO. (2.56)  The in  -15.087  + jO.  -0.0052  -4.7632  + jO.  +0.0465 +  eigenvalue Table  2.2  eigenvalues "sens" after  and the  shift  where found  those  and  only from  parameter the  the  after  stabilization  correction,  procedure  jO. J5.4124 results  eigenvalues  sensitivity  equations  are  designated  "corr".  The  eigenvalues  -19.368  + J41.389  -0.8611  -1.5.69  + jl.3869  -0.5316 +  values  are  jO.5316  + JO.  (2.57)  jO.  j5.5464 summarized below  with  the  final  underlined.  TJL  k  adjustments  The  are -1.3869 +  parameter  summarized  shown.  + jO.  +  are  are  -340.46  -0.5308 The  adjustment  dominant  each  +  2  $.  =  0.050  =  40.00 — -  =  — 0 . 0 0 7 2  327.40  - 453.30  1.000  —^-2.620  -2.940  =  0.480 0.002  P-0.1800 *- 0 . 0 0 1 0  =  7.600  *- 2 . 3 6 0 0  t  r  — 0 . 0 1 4 9  *  0.0143  »- 5 7 8 . 1 0 (2  0 . 0 8 3 0 — > ~ 0 . 0 3 4 0 — « - 0.012  58) —  1.1600  *-r Table  2.2  directly  indicates from  the  that  system  only  five  matrix  are  eigensystem required  but  reclaculations these  are  not  all  18  2.2  • Shift  Dominant  Eigenvalue  Roots  Sens.  Shift  1 and  2  Corr.  Through Parameter  Dominant 3 Sens.  Adjustment  Root  Parameter  Table  Corr.  New Parameter Value  -.0052  0  +.0465+J5.412  1  -.0035+J5.541  2  -.0285+J5.555  -.0239+J5.550  -.0052  -.0052  .0250  tl  .0149  3  -.0166+J5.513  -.0222+J5.464  -.0084  -.0130  .0031  St  .1797  4  -.0433+J5.404  -.0946+J5.383  -.0193  **  .0063  h  .0825  5  -.1630+J5.333  *  -.0376  *  .0125  k  O  6  -.1630+J5.335  -!"2000+j5.356  -.0626  **  .0250  t  7  -.2501+J5.357  *  -.1977  *  .0500  St  .0120  8  -.2807+J5.367  -.2885+J5.381  -.2477  -.3485  .0500  St  .0021  9  -.2947+J5.381  -.2946+J5.380  -.3485  -.3485  .0063  tl  .0074  *  -.3485  *  .0063  -.3258+J6.503  -.4941+J5.549  -.3507  -.3504  .0250  12 - . 4 9 4 1 + J 5 . 5 4 9  -.4941+J5.549  -.4487  -.4590  .1000  t  13 - . 4 9 8 9 + J 5 . 5 5 3  *  -.5090  .0500  St  .0001  l  453.3  10 - . 3 0 1 0 + J 5 . 3 9 2 11  ft  -.0052  14 - . 5 2 3 9 + J 5 . 5 5 4 15 - . 5 2 3 9 + J 5 . 5 4 7 16 - . 5 2 9 1 + J 5 . 5 4 3 17 - . 5 3 2 2 + J 5 . 5 4 3  Eigensystem  *•* -.5166+J5.547  *  -X- *  -.5269+J5.545  Recalculation  No C o r r e c t i o n  Required  •x-  -.0251  -.1477  .0500  -.5091  *  .0250  -.5341  **  .0250  -.5316 -.5316  From  -.5316  ** -.5316  System  .0125 .0031  Matrix  k  k  2  r  2  l  k  k  t  r  r  -2.621  0304  2 . 362  -2.943 327.4 1.505  1.164  l  578.1  tl  .0143  k  19 caused six,  by  close  eleven  change.  The  eigensystem  and  eigenvalues.  The  seventeen  caused  correction is  too  far  are  method from  the  recalculations  does  by not  correct  a too  required  large  converge  if  eigensystem.  by  shifts  parameter the  approximate  20 3.  EIGENSYSTEM  3.1  this  applied  minimize in  APPLIED  TO PARAMETER  OPTIMIZATION  Introduction In  is  ANALYSIS  chapter,  the parameter  to a parameter  a performance  quadratic  adjustment  optimization  criterion  method  problem.  specified  of Chapter  It i s desired  as a c o s t  functional  =  y Q  (3.1)  y dt  T  o  where  to  to  form J y  Q i s chosen  2"  =  A y  t o emphasize  emphasize  (2.1)  selected- state  i n t e g r a t i o n time.  variables  a n d of,  F o r an a s y m p t o t i c a l l y  ot>0,  stable  system  (3) with  initial  functional  conditions  B^ ^ +  i s solved  A with  =  initial  T  (3.3)  B_ . ,  i s solved  be h a n d l e d  These by  the largest minimizing  is  applied  does of  0  that  of a symmetric  this  matrix  cost B  « i h A  <-> =  =  relation  -B^  Q  *  B^  i n the normal  be i n v e r t e d .  Thus,  restricted  reduced  inverted  the computation  to A  i n this  solutions  i s found  (3.3).  of the eigensystem  was  then  was  As a f u r t h e r  the eigenvector  inversion. i n a single  n(n-»l)/2 that  requirements.  i n a method  transformation  nxn.  an  o f a system  by c o m p u t a t i o n a l  so t h a t  With  manner  the size  considerably  effort,  chapter  any matrix  of  2  (3.3)  were  matrix  ^ as 3  +  B_.,  f o r each  the desired  terms  B  whereby a s i m i l a r i t y  not require  cessive  +  o  must  requirements  to  i n terms  the recurrence  i s severely  Vongsuriya^^  and  h a s shown  condition  x n(n+l)/2 m a t r i x can  ?  from  B If  , Laughton  c a n be e v a l u a t e d J  where  y  developed applied  contribution transformation  the s o l u t i o n  o f B^  In a d d i t i o n ,  the solution  step  the s o l u t i o n  o f A, t h e s e n s i t i v i t y  thus of B  avoiding c<+  from  (3.3)  suc-  ^ expressed i n  equations  and c o r -  21 rection  method  of  Chapter  cost  functional  3.2  An E i g e n s y s t e m The  through  solution  generalized  later  substituted  AX  V Assuming  that  B  T  the  of  the  of  B-^ f r o m  to  arrive  into  =  +  B  at  minimize  is  investigated Let  the  first  eigensystem  is  form  of  A  (3.4) this  eigenvectors  the  and  T  XV  X  to  Functional  B^^.  XV  applied  adjustments.  Cost  then  X  easily  (3.3)  X  (3.3),  X  be  parameter  Form  A be  2 can  equation =  T  are  for  becomes  -Q  (3.5)  a normalized set  then  premultiply-  T ing  and  postmultiplying  Xx % X T  At  this  where  point,  BJ  is  a  it  is  this  B|  T  B  X  T  B  Q'  =  X  T  Q  n(n+l)/2  x] +  symmetric  elements  of  =  T  =  X  X  -X  X  (3.7b)  B  X  becomes (3.9)  diagonal,  be  obtained  B  |B'J  B-^ i s  then  found  from B  Similarly,  by  r--,  =  =  1  the by  solution  of  the  inspection  of  (3.9)  W  A  L J l J  ,  JB-JJ  (3.7a)  (3.8)  Q'  [ i].. = • - [Q'l.. and  since  X  =  can  (3.6)  (3.7a)  (3.6)  X is  Q X  T  in  X  matrix T  results  define  BJ X and  B^  and X r e s p e c t i v e l y  to  =  X  X  X  X  1  BJ  substitution,  is  B  T  convenient  XBSince  X  by  complex symmetric  [x With  +  1  (3.5)  i,  j = 1,  as  (3.10)  ...n  as  V.BJ V  substituting  (3.11)  T  (3.4)  into  (3.3)  and  premultiplying  T and  postmultiplying  analagous  to  (3.9)  by can  X  and X r e s p e c t i v e l y ,  be  written  as  a  general  equation  22 and by  inspection  Successive found  by  [B  1  +1  substitution  (3.10)  results  /( x  -  = ij  —  of  B j , BJ!,, ...B^  in  the  general  ij where to  Q'  B|,  is  given  (3.11),  by  can  be  V l Thus  the  value  of  the  J  =  y  Performance The  written  parameters  J.  method,  of  the  (3.16)  J, To  =  ' q:  evaluate  •••  (14)  to  B ^ ^,  is  B-^  transformation  applied  that  < ' 3  function -  J  from  (3.2)  1 5 )  becomes (3.16)  y.  minimization procedure  q^j  method  Taking a  y |v, B' .V oL'q.,. «+l  the  ke  n  (3.17) is  used  partial  found  the  for  requires  + VB' , , , V oc+l'q^  equation,  are  T  Appendix C,  c  this  qjQ  •••  descent  q^,  in  where  ^  the  T  J  same  ]_> s u c h  +  i1  v  for  function.  respect  c  (3.13)  Jij  the  +  summarized i n  performance with  B  d+l  jq.i>  and P o w e l l ' s  V  B J  into  Minimization  chosen  Fletcher  to  performance T V B' ,  Function  as  This  =  O  J  3.3  Also,  applied  (3.13)  solution  l_  (3.7b).  \.)  i +  —  minimizing  the  derivative  derivative  of  as T  + VB' a+1  partial  >q  ~ly J  J  derivative  o  (3.19) v  of  ^i equation  (3.14)  for  The  Q . 1  is  taken  elements  B  =  of ( - D  with B' oc+1  ,, T X,  + where  P.. ij  Substituting  = the  - fi  1  U*  +  1  ,~|  Jij  respect  (3.20a),  the  can  t h u s be T QX + X Q X ,  (3.7b)  written  /<x  substituted  as  >.)  i+  (3.20a)  (<* + l ) ( V , 1  eigenvector  elements  q^ w i t h  F.  of  q  + X-, J Ii  i  sensitivity X  into  to  .  +  J  X-)  (3.20b) .  (2.20)  l  +  1  equation  C. B ^ ^ ,  )/(/\  become  (-D  B  T  K + 1  Q'c, -JIJ  ij  + F  oc+l'q.  Thus  with  B^^,  evaluated  ^1  V, into  (3.19),  the  =  final  i ,  form  in  this  -V  Ct  In  the  • I C  parameter  system  incremental  tended  Collar  calculation that if,  a  of  J,  pose  of  the  are  «  +  i  "  B  and J a h n a new  *  +  of  of  in J,  computed f i r s t .  'q.  This  and  given  +  B  «i i' +  process Chapter  eigensystem  calculation  (3.21a)  (3.21b)  = 1, . . . n by  substituting  by  i c  correction  saving  j  manner  is  i  adjustment  procedure  considerable  in  '  B  /(V  (2.26)  q  -T  V  CK+l  ij  Vfl»q.U i J  B  23  '  avoids  (3.22)  y.  minimize  2 together is  used  each  computation and J ,  q i  to  method after  Tv  J,  with  to  the  eigen-  the  ex-  avoid  adjustment.  effort  can  the  vector  TV y o  the  computation  Note  be  J  repetitive also  realized  and of  its  trans-  matrix  products. 3.4  A Numerical To  system for k  2  If  this  study  system  study.  2 is  are  They  be  the  Chapter  and  would  are  illustrate of  the  Example  found  not  as 2  other  as  an  its  use  The  2.1  with  that  the  initial  stable,  invalid study  =  0.0149  =  -2.6207  adjustable  example.  the  Table  initially  f t  the  so  stabilization  k  and  from  chosen  unbounded and From the  chosen  taken  are  were  minimization procedure,  the  for  of  the  9 ^  initial  system  ,  stable. (3.1)  optimization  these  parameters  (3.23)  0.1797 parameters  of  functional  a parameter 2,  is  power  parameters  exception  cost  Chapter  order  from  Table  2.1  are  k,  =  40.00  =  7.600  (3.24)  1  T The  parameter  straints is  of  adjustment  (2.55).  not applicable  the  methods  ranges  Since  are given  the F l e t c h e r  to a problem with  available  for solving  by the i n e q u a l i t y  and P o w e l l  inequality  this  type  descent  method  constraints,  of problem  con-  and  usually (15)  require used.  large  a  number  H i s method  constrained  transforms  variable  Pi  =  of i t e r a t i o n s ,  the approach  a constrained  p^ b y t h e  arc s i n y ( q  )/(q  i  this  since  transformation,  now a n u n c o n s t r a i n e d  derivative  p  where  J, q  is  i  of q  q  given (3.25)  i'p *i  equal y  o  and  Q chosen  The  process  1  with ( i  P  after  =  and q . ,  as takes  l  q  i  l Q  nine  l =  is  2  l  l  l  diagQ.  is  i  p  s  i  then  chosen  Q  iterations  optimization  method  is  sought.  found  from  n  p  i  c  o  applicable The  the  partial  (3.27)  s  carried  out with  a value  of  as l 1  l  l ] 1  1  to f i n d  1  (3.28)  T  1  1  1  t h e minimum.  l]  (3.29)  The e i g e n -  are  + jO.  -1.0516 + JO.  -51.970 + j O .  -0.2062 + j O .  -9.9075 + j O .  -0.1257 +  -2.0319  -0.0192  the parameters  is  t o p^ a s  ^ min  process  to one, y [  min  (3.26)  i  respect  ~ max  q  =  -333.34  and  i  (3.25)  becomes  • q.,  by (3.22)  optimization  (3.1)  values  J,  max  i n p-space  1  derivative  The  i  =  q^ t o a n u n -  5"  ±  and P o w e l l ' s  optimum  of J i n p-space J,  a in  Fletcher  - q  i  min With  variable  is  transformation - q  i  o f Box  + jO. found are  J5.4459  + jO.  (3.30)  25  X  =  0.100  k,  =  1.032  l  £ f k  Note  that  This  is  usually other  quite used  is  This  very  for  at  indicates  constraints  compared to  in  view  good t r a n s i e n t are  can  be  =  1.000  =  1.000  3.000  2  small  surprising  parameters  (2.55). the  k^  t  their that  of  the  there  is  limits  fact'that  stability.  limits  relaxed.  its  Also  imposed by still  of  -600.  a high note  the  room f o r  and +600.  gain  that  is  the  constraints improvement  of if  26 4.  C O M P U T A T I O N  In by a  this  Puri high  chapter,  order  time  each  for  for  not  of  Problem  each  a  is  desired  linear  to  which 2  0  where  is  the  system  However  a  developed  of  e-  matrix.  computation  is  a  fast  limitation recursive  here.  This  the  products  only  method  -  A y  +  B  system  described  by  u"  control  (4.1)  vector  u(t)  time, 2 1  )  this  the K  of  [j? Q  +  u]dt  H ¥ T  subject  to  u(t)  = to  +  K  S  =  A  that  minimizes  the  K  S  +  with  =  [A - S  in  (4.4)  -A K  calculation  is  given  Q  equation  =  0  (4.4) (4.5)  T  the  substitution  of  (4.3)  becomes  K)y can  (4.6) be  f o u n d by  Ricatti-type -  T  K  B  is  (4.3)  -  1  (4.1)  The  y(t) Ricatti  W"  matrices.  constraint  matrix  K  (4.2)  definite  the  B  (4.1)  the  - V-VK  nonlinear =  y  T  J  equation,  solution  J o  positive  y  equation^  and  Ricatti  the  this  systems,  involves  the  solution  and  reverse  rapid,  inversion  developed  Ricatti  Since  order  symmetric  minimizes  T  The  lengthy.  the  S Y S T E M  controller  successive  to is  optimal  is  =  constant  A K  The  method  an  approximation  =  choose  )  K  O R D E R  p  Q and V are  (  converge  time-invariant  J  b y  for  requires  high  functional  u(t)  H I G H  (nxn).  y" It  is  to  Formulation  Consider  solve  this  A  F O R  a p p r o x i m a t i o n method  metiiod  of  matrix  order  to  finally  application  require  matrices  4.1  This  approximation  solving  C O N T R O L L E R  used  convergence  efficient  of  is  solutions.which the  does  O P T I M A L  a successive  system.  Although  method  A N  and G r u v e r  quation  for  O F  K A often  -  Q  matrix +  unstable.  K S  integrating,  in  differential K  Eigensystem  (4.7) methods  27 ( 4 . 4 ) are a v a i l a b l e ,  of  solving  of  a m a t r i x double  Furthermore, increasing eigensystem the in  accuracy usually  order.  developed  Although  i n Chapter  eigenvalues are too c l o s e l y Chapters  any  2 and 3 by f o r c i n g  neighboring Although  Ricatti order  but they r e q u i r e  the o r d e r of the o r i g i n a l  eigensystem  system  '  system  the  matrix  deteriorates  the c o r r e c t i o n  2 i s applicable, located.  eigensystem A.  with  procedure i t may  T h i s problem  f o r the  fail i f  was. a v o i d e d  t h e e i g e n v a l u e s t o pass  through  positions.  t h e r e a r e o t h e r methods a v a i l a b l e  e q u a t i o n , t h e most s u i t a b l e  systems was  found  f o r s o l v i n g the  one f o r a p p l i c a t i o n  to high  t o be t h e method o f s u c c e s s i v e a p p r o x i m a t i o n s (8)  developed  by P u r i  approximation generate  and G r u v e r .  of t h e R i c a t t i  A  where  +  A ^  and Puri  and G r u v e r  from  these  A  be  from  +  =  A  -  S K^  =  Q  +  K^  converge  matrix  m a t r i x must be c h o s e n left-half  K  have shown t h a t  equations  Ricatti  initial  matrix  matrix R i c a t t i equation (J) (J) [ (J)]T (J) . K  unique  T h i s method c a l c u l a t e s  7  (J)  the f o l l o w i n g  = 0  ^  ^S  de-  (4.8) (4.9)  K^" ^  (4.10)  1  the s u c c e s s i v e e v a l u a t i o n s of  m o n o t o n i c a l l y and r a p i d l y  i f  i s p r o p e r l y chosen.  so t h a t  the e i g e n v a l u e s of A ^ ^  complex p l a n e .  guess f o r  - 1  J _ 1  Q  the j - t h  If A^^  i s found  This  ^  to the initial  l i e i n the  t o be u n s t a b l e w i t h an  , then the e i g e n v a l u e s h i f t i n g  technique can  used. Solving  an n ( n + l ) / 2  for x n(n+l)/2  Thus, a l i m i t a t i o n analyzed  i n (4.8) u s u a l l y  involves  m a t r i x , where n i s t h e o r d e r o f m a t r i x  i s p l a c e d on t h e s i z e  practically.  the i n v e r s i o n of  o f t h e system  that  A.  can be  28 Instead,  a  recursive  here  which  with  nxn  4.2  A Recursive In  result  eliminates  lengthy  what is  Method  follows,  applicable (4.8)  can  is  (4.6)  to  now w r i t t e n  A and K are  result,  an  Thus  y  (  ?  y)  K  matrix  The  written where  as  N is  as  matrix  the  Ricatti  (4.12) Premultiplying  and  postmulti-  and  substituting  (4.12)  ( y Q y)  -  system,  a  A  chosen  integration  of  (4.13) w i t h  in y)dt  T  solution  (4.13)  T  J(y Q  =  o  =  e A t  to  (4.14)  _ y  be  expressed  in  terms  of  the  state  At  may  be  for  as  e an  t  A  over  these N  ^  integer  =  Q [e  T  evaluated  constant  =  (4.15)  0  y[e ]  =  t  y  (4.14) a n d c o m p a r i n g t e r m s , K b e c o m e s  K  matrix e  degenerate  since  as  integral  transition  omitted  (4.11)  respectively  =  (4.15) i n t o  each  be  Approximations  -0  results  y  This  will  the  matrices.  stable  y  homogeneous  Substituting  j  Successive  A y  and y  conditions  transition  tegrands  only  as  o the  operates  —  y„K y Let  and  developed  as  =  asymptotically  initial  is  (4.11) b e c o m e s  at For  j .  =  constant  (4.11) b y y  the  the  superscript  any  —T  into  for  inversions  Obtaining  K A  y  plying  solving  matrix  written  +  T  for  the  be  A K  where  of  matrices.  equation  and  method  N/R  a  by  A t  a  time  discrete  ]dt series interval time  = [^(lAO]* and  time  (4.16) of of  successive size At  intervals  may  =  inl/R.  be ( 4 > 1 ? )  as  (4.18)  29 M a t r i x e ^ ^ ' ^ i s c o n s t a n t f o r c o n s t a n t A and l / l t , designated  as  A^  =  w h i c h c a n be c a l c u l a t e d A For  1  a stable  =  (l/R)  (4.19)  f r o m t h e power  I + A/R + A / 2 ! R 2  2  series  + A /3!R 3  + ...  3  property  0  A t  Likewise, f o r a stable  as  t  »  (4.21)  s y s t e m , f r o m (4.17) and ( 4 . 1 9 ) , A^ has  (A^)^  f o r R chosen p o s i t i v e .  f 0  as  N  Substituting  oo  (4.22)  ( 4 . 1 9 ) and (4.17) i n t o  K c a n be a p p r o x i m a t e d by a s u m m a t i o n o f d i s c r e t e K  =  (Q + A?  Q A,)/2R  1  =  (Q/2  +  (A, Q A,  .  1  +  +(A ) Q 2  i  T  x  i n t e g r a l s as A )/2R 2  recursive  (4.24)  (A ) S A T  +  (A ) 2  +  2  t  4 T 4 (A,) S^A, + • • ( A ) S.AJL +  where  N  and  =  2  (4.26) i n t o t  integration  =  Q A  S  x  S,  =  S  2  S  =  S  2  i  2  3  • • S.  {  2  5  )  (4.26) =  S  (4.27)  1  (4.18), i n t e g r a t i o n  t i m e c a n be w r i t t e n as  2 /R  (4.28)  1  time i n c r e a s e s r a p i d l y w i t h i n c r e a s i n g i ,  and as a r e s u l t S ^ ^ c o n v e r g e s r a p i d l y t o S s i n c e +  property of equation  4  + 1  1  l i m S. i—&co  =  T  « • S  n  Note t h a t  reduced i f the  formula i s used.  2 Aj  Substituting  (4.23)  ^(AflViJ  S  2  +...  i x  e f f o r t o f c a l c u l a t i n g S f r o m (4.24) i s g r e a t l y  following  (4.16),  S)/R  wh e r e The  (4.20)  s y s t e m , t h e t r a n s i t i o n m a t r i x has t h e p r o p e r t y e  the  A e  and w i l l be  (4.22).  A-, has t h e  30 4.3  Accuracy  of  the  Theoretically recursive to  the  into a  formula  larger  large  the  number  series  creases. matrix  of  But,  with powers  of  of  number  the  since  by A  terms  by  large, and  A^  the  choosing  calculate  is  respect  divide  Also,  rapidly  the  with  will  areas. to  and  large  integral  decreases  elements  (4.23)  made  required  extremely  V  =  R is  accurate  (A/R)^  repeated  i  if  discrete  more  of  R chosen  K evaluated  improve  off-diagonal  A,  of  smaller  (4.20)  small  will  A since  the  with  Method  accuracy  (4.25)  R reduces  power  large  the  eigenvalues  a  Recursive  as  A^  N  almost  in  ina  calculation  unit of  multiplication A  i (4.29)  as  required  of  errors  by  in  the  recursive  (A^)^.  Thus  provement  obtainable  be  large  chosen  ation  of  the  round-off As with of  an  enough  integral  errors  3.  R,  to  The  9 ^  -70.33  + jO.  -42.26  + jO.  -2.070  +  is  considered  of  consider  indication  but  limit  large  errors not  causes the  R.  As  caused  too  large  an  accumulation  accuracy a result,  by  imR  discrete  since  in  should evalu-  this  case,  increase.  +  an  (4.25),  errors  reduce  (4.16)  -334.2  As  these  choosing a  illustration  various  Chapter  by  formula  the  an  order  error  example  from  has  -1.032  0.112 -0.005 approximation  in  approximating K  the  numerical  results  eigenvalues  + jO.  -0.301 +  jO. the  taken  A-matrix  jO.  of  involved  JO.  +  J5.450  + jO.  error  of  K,  the  matrix  E  where E  =  A K  +  K A  +  Q  (4.30)  31 From (4.11) i t i s known that E becomes a n u l l matrix i f K i s known exactly.  Table 4.1 i s a c o m p i l a t i o n of a..I i j I max  and  n V  Z-J  ij  E.. I ij I  (4.31)  f o r v a r i o u s R using s i n g l e and double p r e c i s i o n Three d i f f e r e n t approximations to  For  1.  A  2.  k  3.  A  x  Y  1  calculations.  are used, namely  =  I  +  A/R  +  A /2!R  =  I  +  A/R  +  A /2!R  2  +  A /3!R  =  I  +  A/R  +  A /2!R  2  +  A /3iR  2  2  2  2  3  3  3  small R, the e r r o r caused by d i s c r e t e e v a l u a t i o n  i s dominant  since  (4.32)  3  +  A /4!R 4  of the i n t e g r a l  double p r e c i s i o n c a l c u l a t i o n does not reduce the  error s i g n i f i c a n t l y .  As R i s chosen l a r g e r , the e r r o r  decreases and then increases  first  as round o f f e r r o r s accumulate  single precision calculations.  Although double p r e c i s i o n  l a t i o n s e x h i b i t d e c r e a s i n g e r r o r with i n c r e a s i n g  i n the  calcu-  R, the c a l c u l a t i o n  time i n each case i s double the time f o r s i n g l e p r e c i s i o n lation.  4  In t h i s example, the e r r o r caused by t r u n c a t i o n  calcuof the  power s e r i e s r e j i r e s e n t a t i o n of A-^ i s found minimal i f A^ i s approximated by 4.4  A  =  x  I  +  A/R  +  A /2!R 2  +  2  A /3!R 3  3  (4-33)  A Numerical Example The methods of s e c t i o n s  4.1 and 4.2 are applied  here to solve  f o r an optimal c o n t r o l l e r of the proposed power system of Appendix A.  To incorporate  the  following 1.  changes  I?)  and the s i g n a l  The t r a n s f e r  variable  function  by an actuator  'a' i s replaced  by a c o n t r o l  i s shown i n Figure 4.1.  signal  The s t a t e  equation f o r 'a^' becomes p a  where  are made.  The governor dashpot i s replaced  k ,/('l + f u-^.  the optimal c o n t r o l l e r s i n t o the power system,  u^  f  a /r;  =  -  =  (k  f  c l  /t  +  cl  c l  )  u  tlj i  (4.34) (4.35)  32 Table  4.1  - Error  of K  APPROXIIt  MATION NUMBER  500  900  1300  4300  7300  13300  Approximation  SINGLE PRECISION E. -  max  E. .  DOUBLE PRECISION  1  E. . i j Imax  1  .2619  .7350  .2619  .6280  2  .1259  .6091  .1259  .4710  3  .1464  .6363  .146.4  .4735  1  .0749  .3880  .0749  .1753  2  .0425  .3670  .0425  .1488  3  .0561  .4420  .0455  .1502  1  .0869  .5470  .0349  .0812  2  . 1040  . 5970.  .0210  .0719  3  .1025  .6190  .0219  .0724  i  .2120  1. 150  .0031  .0072  2  .2650  1. 420  .0019  .0066  3  .2640  1. 480  .0020  .0067  1  .3506  1. 618  .0011  .0025  2  .4511  2. 184  .0007  .0023  3  .4511  2. 271  .0007  .0023  1  .4420  3. 202  .0003  .0007  2  .59 10  4. 055  .0002  .0007  3  .5910  4. 204  .0002  .0007  33  r (i Figure 2.  4.1  a control  4.2.  P  v  t  =  2  Transfer  Function  t  *  has t h e  speed  equation  '  =  (1  and H y d r a u l i c  function  V  t  u  for  "  52&  a  k  •P1  >.  g  The t r a n s f e r  (kj^'l^/tj)  +  v  )  u,,.  v  u  P  1  ~(W  =  where  — AV  C  variable  s  t p)  +  regulator-exciter  signal  The s t a t e  and  P )  -t P)  >  cl  - Governor  The v o l t a g e  by  +T  (i.  (1  4g  1  '  s  s ^ l  V  +  v  Operator  signal  is  replaced  shown  in  becomes (4.36)  "2  +  (4.37)  5 3 ^F  a  Figure  (4.38)  2  l  s  1  .  + t p) x  u. Figure  4.2 - Voltage  The u^  and u  state 2  Regulator-Exciter  variable  equation  c a n be w r i t t e n y  where  y  and A i s  the  elements Since, y^  same  from  B  A y  J ^ , V,  as  that  (4.34),  respectively,  =  a  0  the  system  +  u  with  controllers  V  (4.39)  B u F»  » S»  v s  h  >  a  »  = a^  1  (4.36)  0  = 0 = 0  ,  & a  and (4.40)  n 9 n9  fJ  a  of Appendix A except  f o r the  = - -1 /l V/ - i t  u^ a n d u  2  c  l  appear  (4.40) following (4.41) i n y^ and  becomes  0  0  0  0  0  0  1  0  0  0  0  B  and  Function  as  =  a,-,  of  Transfer  0 U  2j  0 T  0  0  0  0  1  ^  T (4.42)  (4.43)  34 The  response  includes  a speed  shown  Figure  are  if  given For  as  ^ = .01 The be  stable  original  feedback The  =  Q  j0  with  without with  in  .2 of  control  eigenvalues  is 0  0  0  optimal  tested  given  in  Chapter  regulator  unstable.  parameter is  given  voltage  0 an  the  system  the  system  computation  together  system  the  4.3. y  the  of  The  0  0  for  a  0 j  of  is  conditions (4.44)  T  a value  Table  stability  which  dashpot,  initial  controller values  and  2,  of  2.1  and  are  is  used.  found  to  as  -333.5  + jO.  -1.566 +  jl.381  -100.0  + jO.  -.5344  jO.  +  (4.45) -15.56  + jO.  -.2189 +  -4.568 In  this  case  c a n be are  chosen  eigenvalue chosen  as  unit  matrix  is  method  together  solution 24.53  then  is  The  on an  .184,  is  and  Ii  IBM  nine  the  u  =  u  =  .028,  1  B  system For  response  a different Q  the an  =  of  this  improved  of  result, (4.2)  The  Ricatti  approximation section  single  4.2.  The  in  precision  K y  T  calcu-  from (4.46)  .050,-.367,-.362,  10  13.3,  example  Q chosen  c o r r e s p o n d i n g system  a  (4.47) .001,  diagjlO  as  approximations  using  calculated  - W"  1000.  successive  successive  and,  Q and W o f  as  technique  .196,-5.97,-.795,-.002,-.048, The  (4.19)  7044 c o m p u t e r  vector  necessary  Matrices  of  recursive  after  not  matrix.  applying  the  control  is -.019,  a null  found by with  JO.  shifting  matrices  obtained  seconds  lations.  as  +  J5.267  is  .102,-.048  10.6,-6.61,  shown  in  y  3.26_  Figure  4.4.  as 1  1  response  damping c h a r a c t e r i s t i c .  1 is  1  1  1  shown  in  However  l]  (4.48)  Figure this  has  4.5  and  been  has  obtained  Figure  4.4  - System  Responses  for  Q = diag ll  1 1 1 1 1 1 1 1  at  the  expense  signal  a^  to  of  increasing  three  A comparison double use  As  precision  the  For  made  and  Q given  by  a  of  in  Figure  the  small  calculations  versus  a  R = 1000  case  2  Single  precision  and  R = 1000  case  3  Single  precision  and R =  used  for  section  -.053,  4.3  the  calculation  comparison  1,  E  =  E  =  .436,  best  using  a  - W"  2xl0  - 6  ,  .064,  l  0.,  -5xlO~ ,  -  B  1  E  cases  the  200  accuracy  is  large  case  R;  obtained 1.  by  The  (4.49)  1 and  .238,-.Ill  34.6,  33.2,-14.8,  7.15_  2 and  cases  3  1 and  are (4.51)  0.,  0.,  2x10" ,  2xlO  5  2xlO" ,  0.,  7  0.,  - 5  ,  0.,  -lxlO" ,  0.  5  0.,  -lxlO"  =•  3  5  3xlO" -2xlO~^-4xlO~ -2xlO"^-2xlO" 4xlO" 5xlO" -lxlO~ lxlO"  3  4  3  4  seconds, 1000  to  case  be  cision of  times  3 -  4  time  concluded  the by  are:case  versus  single  this  with  1 -  seconds.  13$ b u t  from  calculation  accuracy  4  21.25  200 u s i n g  calculation can  3  4  calculation  a  4  (4.52)  -5xlO"^-lxlO"^lxlO~ -2xlO" -3xlO~ 9xlO~^lxlO~ -4xlO~^-2xlO~  The  the  matrix  2  0., - E  1  of  j  .052,-.830,-.893,  cases  6  E  versus  K  T  .002,  -3xl0" ,  0.,  5  All  single  (4.50)  between E  control  where  .709,-17.5,-2.16,-.001,-.111, differences  the  (4.48). and  precision  vising  R.  precision  case  The  large  Double  in  of  4.4.  1  concluded  is  is  that  maximum e x c u r s i o n  case  double E  times  the  3  Hote  accuracy  that  calculation  is  time.  case  2 -  a decrease  calculation  worsens  comparison that R  3  57.97 seconds,  precision  large  3  in  only  24.52 R  from  decreases  disproportionately.  the  definitely  use the  of  single  best  It  pre-  compromise  39 5.  SENSITIVITY SYSTEMS  ANALYSIS OF THE TIME RESPONSE OP MULTIVARIABLE  In t h i s c h a p t e r , the s e n s i t i v i t y of system response to parameter variations i s investigated.  Eigensystem s e n s i t i v i t y  analysis  was used i n Chapters 2 and 3 f o r system s t a b i l i s a t i o n and parameter optimization  r e s p e c t i v e l y , but i n i t s most d i r e c t a p p l i c a t i o n ,  eigensystem s e n s i t i v i t y a n a l y s i s could a l s o be used f o r e i g e n v a l u e movement s t u d i e s .  However, i n the f i n a l a n a l y s i s of a system  d e s i g n , a time response s e n s i t i v i t y study i s u s u a l l y r e q u i r e d  since  t h i s g i v e s the d e s i g n e r the d e s i r e d d i r e c t i n d i c a t i o n of changes of system response to parameter v a r i a t i o n s . For the a n a l y s i s of time response s e n s i t i v i t y , i t would be d e s i r a b l e to use an analog computer because of i t s f a s t i n t e g r a t i o n time.  But the s i z e of a system t h a t may be s t u d i e d  computer  on an analog  i s u s u a l l y l i m i t e d by the number of components a v a i l a b l e .  T h e r e f o r e , f o r l a r g e order systems, the d i g i t a l computer must be used.  However, i t s slow i n t e g r a t i o n time i s a p r a c t i c a l l i m i t a t i o n  on the number of parameters t h a t may be chosen f o r a s e n s i t i v i t y th s t u d y , s i n c e , as i s p o i n t e d out i n the f o l l o w i n g s e c t i o n , an n order e q u a t i o n must be i n t e g r a t e d  f o r each parameter  sensitivity  calculation. Kokotovic^^  developed a method of computing  sensitivities  f o r a l l parameters s i m u l t a n e o u s l y from two n^* o r d e r e q u a t i o n 1  integrations.  T h i s method, however,  i s r e s t r i c t e d t o a n a l y s i s of  s i n g l e v a r i a b l e systems where the parameters s t u d i e d must be chosen as the e q u a t i o n c o e f f i c i e n t s .  In t h i s c h a p t e r a simultaneous  m u l t i v a r i a b l e s e n s i t i v i t y method i s developed i n which any paramet e r may be chosen f o r i n v e s t i g a t i o n and o n l y one n^* and one 2n^^ 1  order e q u a t i o n need be i n t e g r a t e d parameter  sensitivities.  i n o r d e r to compute a l l the  40 5.1  S i m u l t a n e o u s C o m p u t a t i o n of Time a L a r g e Number o f P a r a m e t e r s . Consider  function  the  multivariable  example  Taking q^  the  E • = partial  results  in  N Thus,  system  to  0  [2  A  for  integration  following gration  ?1  »k  the  time  Let  of  0  0 of  is  on t h e  the  0  the  Laplace  initial  nomial  as is  in  the  the  (5.1)  k  terms  0  with  ~  y +  cT| ,  = 1  u  T  respect  to  a  (5.2)  parameter  u  y,  ,  <5  k = 1,...$,  -  3)  requires  k  integrations remove  the  parameters  of  (5.3).  dependance  investigated  =  R^  is  given  in  section  I  s  In of  the  inte-  and  to  compute  a matrix.  1  -h^s  + R  The  and  a  inverse  of  (sI-A)  characteristic  poly-  R(s)/g(s)  matrix -  (5.4)  zero.  adjoint  (5.5) polynomial  . . .-hus  1 1  polynomial l  S  n  "  The method  (5.5)  y(s) R(s)b  n  (5.1)  + ...  2  of  1 1 -  of R s i  *-  of  (sI-A),  . ..-h  namely (5.6)  ( s I - A ) , .'namely n  "  1  "  calculating  1  + ... R^  and  R h^  A  (5.7)  is  5.2.  Substituting becomes  an  to  b u(s)  1  characteristic s  n  applied  c h o s e n as  =  1  be A ] "  -  of  - AJ"' ' scalar  =  where  term,  ' to  of  \ll  adjoint  R(s)  Combining  derived  =  [jsl  is  '  transform  g(s) and R(s)  and  conditions  expressed  g(s)  0  simultaneously.  c a n be  single  forcing  (5.1)  0  sensitivity  number  y(s)  where  single  equation  N  +  (5.1)  a method  sensitivities  with  a  to  E u  +  sensitivity  =  solve  A y  derivative  the  k  =  q  one  with  Sensitivities  u y  For  Response  =  into  (5.8)  into  the  input-output  relation  R(s)•b•u(s)/g(s)  a vector  can  (5.4),  be  polynomial  written  (5.8) and u ( s ) / g ( s )  into  ci  41 y(s) where  =  z(s)  =  u(s)/g(s)  f(s)  =  7 s  '  v  f •  partial  sensitivity  (5.11)  1  1  ,-b"  n  _  (5.12)  1  of  (5.9)  with  respect  ?(s),  .z(s)  k  q  +.  k  derivative  z(s), q  with  (5.10),  = q  =  u ( s ) / \g(s)] of  ?(s). q  (5.16)  c a n be  tational  purposes  by  k  the  z(s),  -  breaking  written  k  (5.15) (5.13)  becomes  f(s).g(s).  .w(s)  a more  each  convenient  vector  (5.16)  k form  polynomial,  for  compu-  f(s),  and q  ?(s).g(s), q  define  F  k  ,  from  into the  the  the  first  q  where  S  second vector  as  ?(s).g(s), II, q  k  is  an  k  of  [}  =  .w(s)  nx(2n-l)  (5.12)  (5.16) =  F, q  1  times  s  k S  n  c a n be n  ^  (5.16)  =  - F.H,  _  written  1  "**  s  can  q  k  First  (5.17)  J  ,.z(s)  .1  of  constant  a vector.  k  as  ?  of  .z(s)  polynomial  q  where  k n  a matrix  ^  k  polynomial f(s),  The  of  column v e c t o r s F  Then,  product  as (5.14)  q  into  be  -w(s)  k put  k  can  2  (5.14),  .z(s)  k  Equation  - g(s). q  substitution  y(s),  q^,  (5.13) q  of  =  k  w(s)  Thus,  to  ?(s).z(s),  q  where  •  becomes  =  partial  1  derivative  y(s), From t h e  i.-.f-s "  1  H  . . £ b  n  equation  q  _  I  -  1  f the  n  (5.9) (5.10)  n  and •  Taking  f(s).z(s)  ***  l  also  .s"  n-diagonal  s "" 11  _ j ^  be w r i t t e n  (5.19) similarly  (5.20)  ~.v(s) matrix  (5.18)  given  by  42 17,  0  0 H, q  ,0  k  q  o  q.k.  k  (5.21)  o 0 q  and q  n'q  k 1 Q  '2n-2 Combining be  (5.18)  written  in  y(s), q  The  matrix  rived  from  written  in  c(s)  the  k form the  s  of  i  of  (s)  +  matrix  form  =  . ..  scalar  , +..a  l  F  E  matrix  the  times  b  response a  form come  where  s of  ^ is  S  2n-2*  equation  V  (  s  )  (  (5.20),  5  is  '  2  4  )  de-  \/ m , , +..b.s  l  m-i+, . . b, \;  (5.25)  m  1  n  o  0  b  o 0  . . . b .  . . .  y(s)  • •  s  l  (5.26)  2m  1  1  from  s  *. 0 .O'b  . . . b ,  i  1  (5.9)  (5.27)  J can  also  be  written  as  a  vector: y(s)  where  can  as  |_m Evidently  (5.16)  polynomials  m) ( s  ..a-^  =  (5.23)  equation  • H,  0. where  0  as  two  m , +..a.s  |a  sensitivity  second p o l y n o m i a l ,  product (s  the  n-l  the  (5.22)  q  2n-21T  form  k  !' k  k  s'  (5.20),  final  q  :  C(S)  and  i ' q  k  k  =  given  F  by  • i _ n  • z(s)  1  (5.19).  Taking  (5.28) the  inverse  Laplace  ( 5 . 2 4 ) and ( 5 . 2 8 ) , the s o l u t i o n s of y ( t ) and y ( t ) , y(t),. . = F. • z(t) + F . II, . w(t) k k k q  q  y(t)  =  z(t)  •' =  F  q  •  J "  z(t) 1  ^ . ! '  7  ^  3  ^  '=  -.Zi  - ^ n ]  1  transbeq  k  (5.29)  (5.30)  V  Since  the  -1  (t)  z(t)  2 n  _ .v(s)  Laplace  p z(t)-  . .-h.p " z'(t)-  can  be 0.1 0  ?2  i  from  0  the  of  1  0  v  (5.10)  i  =  canonical  form  n  •• 2n-l] v  can  .rh z(t)  0  0  "  V  transform  n  solved  [ ]  2  inverse n  then  i  u  be  (  written  h.  n  Similarly,  w|t)  w.  '  3  1  )  a;  (t)  (5.32)  0  . (5.33)  +  n  5  ...  h,  l can  be  solved  0  1  0  ....  0  0  1  0  u(t)  n  1  from 0  . . .  w,  0  w.  0 (5.34)  0 w 2n where be  P  p.^  are  ...  2 n  .  2n  the  expressed  rp  .  .  ***  . p  . i  .'1 ***  p  coefficients  l  w 2n  of  the  u(t)  [g( f] >  polynomial  s  2  which  can  as P  .  ...  P  rj  l  h  0  0  h  0 0  . . . (5.35)  0 where Thus, one  fh to  find  y(t)  integration  section  is  of  n  h n-  . h.  and y ( t ) , z(t)  a derivation  and w(t) of  the  l for is  .  . h, any  il  number  required.  algorithm  for  (5.36) of  parameters,  The  only  following  c o m p u t i n g F,  F, q  a n d II,  .  k  44 5.2  Derivation To  g(s) is  begin  of  the  and R(s)  in  is  rithm  computation,  h^  and two,  and  (5.17).  used  coefficients  Computation  (5.6)  computed from  cation^^  the  for  and the  the  (5.7)  coefficients  respectively  Leverrier's  this  R^  Algorithm  are  for  two  coefficients  can  algorithm of  be  for  easily  calculating  hu  from  the  a n d R^  Fadcev's  by  given  simple ,  k  q  the  algoas  a  k  equations.  by  the  -  1^1  F  modifi-  One,  and F,  algorithm  is  a  and  This  following  set  equations  A  = A  ±  A  h  = AR  2  = AR. , . i-l •  '  •  A The  h  X  •  A. l  = AR  n  •  These digital  h can  sparsity  of  A is  *  A  = tr  A /2  = tr .  R  = A  x  = A  2  x  -  2  •  A./i i  R. = A . l . i •  A /n n  checked =  R  for  2  (5.37)  .  = A  n  h I  - h.I l  .  •  n  - h  n  accuracy  by  n  I  the  last  identity  0  (5.38)  4  require  used  • R  1  2  = tr  n  but  (5.37)  = tr  •  be  equations computer  2  .  n  calculation  1  h. l  •  R  a  derived  calculated  II, q  result,  polynomials  reasons.  computed d i r e c t l y  sensitivity  the  method w i t h  calculation  c a n be  of  this to  A R  i  n  multiplications  can  be  reduced  advantage. ,  i  = 1  In  ...  when p r o g r a m m e d  considerably  the  matrix  if  on  the  products  of  n  (5.39) 2  if  only  p non-zero  elements  instead  of  all  n  elements  of  A  are 2  multiplied, A method record row With for  of  column numbers elements  element  multiplications  single  these.elements  non-zero each  number  proposed to  all  and  the  in  out a  the  may be  non-zero  reduced  elements  one-dimensional  array  and  in  another  two  one-dimensional  of  A taken  one  at  becomes  a matrix  with  a  only  time, one  the row  of  to  p-n  A is  .  to  their  arrays. product  filled.  A«R^ Thus,  45 the  complete  one-row can  be  matrix  product,  matrices.  Once  calculated  To  compute  R^  from  the  (5.39),  a n d h^  (5.12)  are  and  elements  of  can  be  found  determined  as  a  from  sum o f  these  (5.37),  F  (5.17).  II,  ,  Morgan's  method  is  used  k f o u n d by q  where  the  elements  h., 1  q  are  k  h, , 1  k  1  q  n'q where(*)  indicates  I * A,  = q  that  :  k  q  R  k =  i-1  R  ,  *  A  » q  *  A,  n-l  k  the  »q  inner n  (5.40)  k  k  product  is  to  be  taken,  namely (5.41)  w.. lk  ik The  solution  of  F,  i s d e r i v e d f r o m ( 5 . 3 7 ) as f o l l o w s . k derivative o f ( 5 . 1 2 ) w i t h r e s p e c t t o q , one q  Taking  the  partial  1  From  the  q  partial R  =  k  R  -  n  1  ,  derivative  x  n-i'q  of  A. '» q  F, q  k  (5.43)  +  E  k  q  R n  R  n-i  n-i'q  k  = Substituting  has  k  .  and A -  ~-i n  • E) q  h  (5.42)  k  . with n-i  respect  to  . I  into  (5.42)  and  using  f r-\  •• • ^ • • • • ^  A.R "'"n-i-l» the  q,  K  (5.43)  k  + • R "n-i-1 '  k  .  X  q  h  n-i'q  k  equation  .I  • , k  (5.12)  becomes F,  A, q  k  + e. l  where  e  , n-l  c . l  and As  in  the  =  A.e. =  . l+l  A, 'i  b L  R  n-i  C i  -  ••  C •  1 n-i  q  h-, ,  k  1  e . l  1  L  k  q  • •  1  .  c  n  computation  of  (5.44)  (5.45) (5.46)  k  R^  olJ  . n-l .  b  k  . • E, L  e  and  h^  where  the  sparsity  of  A  46 used  was  to  advantage,  the  computation  of  F,  and H ,  k extremely  economized  more  so  since  A,  k  q  5.3  A Numerical To  design  equation  is  the  of  A is  written  A  4 is  sensitivity  chosen  as  an  method,  example.  the  The  optimal  system  as  =  found  simultaneous  Chapter  y where  usually  q  Example  illustrate  control  is  be  Can  k sparse.  q  A y  from  E  +  the  =  u  (5.1)  numerical  results  of  Chapter  4  as  '  -.2041 -30.07  (5.47)  -4.300  38.47  57.70  .8303  .8934  1.000 -.5142  .0534  1.000  -.1865  -64.43  42.47  -333.3  333.3  -.0016  -20.05  -2.000  2.000  4.000  -1.053  -.1326 -.7091  u  and  to  17.52  E are  be  2.157  given  dependent  by  (5.2)  on t h e  =  ±  b It  angle  is  response,&  twenty  system  introduced (2.52). and  B  desired  in  Note  given  by  ,  1  >  H  2 =  eighteen  (4.42)  given  in  -1.000  -50.00  -33.23  14.80  -107.2  element  of  the  E  is  chosen  manner  the  (5.48) =  2  by  controller  Table  gains the  gains  2.1  initial  become  -l/H  (5.49)  sensitivity  controller  4 and t h r e e  simply  first  -19.6/2H  to  the  the  -4.000  19.6/2H  =  investigate  Chapter  -34.58  constant  to  parameters  that  where  inertia  b Hence  .1106  .0065  .1106  -.2381  in  of  the  given  by  addition  operating  for  tf  chosen  5^^  and  9 ^  torque (4.50), toTJ -^ = .01 c  conditions as rows  a unit of  the  of matrix  47 Ricatti  matrix.  The  matrices  F,  F,  and H, c a n be c a l c u l a t e d f o r all k k a l g o r i t h m o f s e c t i o n 5.2 b u t o n l y t h e s e c o n d q  parameters of  F,  by  the  q  and ( F * H , ) i n (5.29) are r e q u i r e d k k example to i n v e s t i g a t e the s e n s i t i v i t y  q  this  since  it  is  rows  chosen  in  q  variable  h only.  calculated  from  The the  time  responses  canonical  order  Runge-Kutta  integration  time,  the  second  desired  and % , k calculation S,  are  then  are  shown  forms  of  elements  evaluated  of  by  z(t)  (5.33)  routine  of  is  the  second  and w(t) and  y(t),  are  (5.34).  used.  For  The  then  A  each  and y ( t ) ,  (5.29).  state  4-th value  that  results  of  of  is this  q  shows  the  in  sensitivity  Figures  5.1  response  for q  parameters most  and  initial  sensitive  figures given  are  operating  controller  the  by  to  0  R  the  first  figure  sensitive  and the  Sensitivity percentage  =  k  The  most  conditions  the  ^u/( q .100./q )  5.2.  system  k  gains.  sensitivity  and  .  second for  the  responses  in  change  parameters  in  these  q /l00.  (5.50)  k  ^k The  most  sensitive  parameters  are  and K ( 9 , 8 ) .  initial  X and x^,  The  others  operating  and are  the  condition  gains  found  to  be  are  is  ,  K(9,2),  at  least  system  K(9,6),  ten  times  K(9 ?) ;  less  sens i t i v e . For  convenience,  the  calculation  for  that  Because of  to of  of  ration time takes  routine.  requires three  parameters  exciter  .00225 seconds  computer  sensitivity  system an  two  is  time  sixteen  minutes  to  thousand  and  and  twenty  in  are  controller initial  constant  required  Therefore  to  programs  the  of  integrate  and  order  of  seconds  one  the  a  Runge-Kutta seconds  z(t)  and w ( t ) .  an  other  time  nine  for  for  conditions.  seconds,  up t o  evaluations eight  gains  operating  .003  4-th  written,  IBM  interval integ of  real It  7044 com-  50 puter  to  troller to  one  compute gains.  gain,  seconds  is  is  more  than  to  eighteen  rameters and  of  a  tive  But  a  time it  the  time  parameters saving  in  results  new  suboptimal  of  and  the  four  variables,  state  in  to  eighteen  determining  five  minutes  new  the  time  Thus  is  con-  sensitivity  thirteen by  calculation  method.  the  and  method d e s c r i b e d  for  in  initial  to  -.6,  sensitivity By  (5.2).  of  for  This  sensitivity  eighteen  approximately  namely three  g,  S,  case  2 - System  with  an  case  3 - System  with  a  operating  v^  =  o  in  .9  control.  optimal  controller  pa-  one  the  1.2.  Also are  the  and  from  an  engineering  costly  to  implement.  of  hour  control,  good over only  control  viewpoint  5.11 .3  both  the be  since  to  an  the  a wide  can  only  is  Chapter  for  a  .9,  Q  illustrate  without  at  from  made  4  control  =  of  be  control  figures that  can  namely  through q  signal  design insensi-  A comparison  cases,  P  responses  suboptimal  a.  a  the  synthesized  without  5.3  over  differ  be  optimal  These  very  control  h and  suggest  neglecting  suboptimal  Figures  response  conditions Thus  system  conditions to  system  optimal  in  can  different  Original  shown  simply  control  1 -  are  calculation  regulator-exciter  case  results  less  the  governor  the S - r e s p o n s e f o r  design  old  controller.  eliminated  ranges.  of  calculation  this  voltage  operating  by  that  time the  sensitivity  minutes.  The  provement  found  required  the  The  is  using  ga.ins  of  response  calculation  required  the  forty  the  the  of  regarded  .6  imor  suboptimal  extreme  is  =  Q  optimal  range  it  variation  suband  initial  edges  of  these  as  the  best  easier  and  possibly  TIME-SEC. Figure  5.3  -  8-Response W i t h o u t  1.0  Control,  Over  Variation  of  P  Q  2.  TIME-SEC. Figure  5.4 -  S - R e s p o n s e -with O p t i m a l  Control,  Over  Variation  of  P  Q  ure  5.5  - ^-Response  with  Suboptimal  Control,  Over  Variation  0°  TJME-5EC. ure  5.6  -  ^-Response  Without  Control,  Over  Variation  of  Q  Q  of  P  TIME-SEC, Figure  5.8 - S-ftesponse w i t h S u b o p t i m a l  Control,  Over V a r i a t i o n  of Q  TIME-SEC. Figure  5.9  -  S-Response W i t h o u t  Control,  Over V a r i a t i o n  of  v  TIME-SEC. Figure  5.10  -  S-Rcsjionse w i t h  Optimal  Control,  Over  Variation  of v  TIME-SEC., Figure  5.11 -  S-Response  with  Suboptimal  Control,  Over  Variation  of  v  56 CONCLUSIONS  6.  Systematic optimization this  of  thesis.  system.  A  Chapter This  eigenvalue  from  method  system  is  procedure  shift  the  since  t i v i t y  and  is  the  method  system  also  justments.  takes When  system, the  i n  made  negative. of  of  voltage  It  of  be  power  system  performance  in  good.  the  of  the  the  system  of  found  the  that  obtained the  With  voltage  3  way the  as  to  new for  the  every  s e n s i -  case  of  might  d i r e c t l y  s t a b i l i z a t i o n on  be  parameter  ad-  order  9 ^  s t a b i l i z e d  i f  the  regulator-exciter improvements  decreasing droop  adjusts  the  the  unstable  can  transient  Chapter a  an  eigen-  eigensystems  c o r r e c t i o n  This  further  by  of  i n  an  correction  c a l c u l a t i o n ,  to  the  of  from  However,  the  After  necessary  r e s t r i c t i o n s  applied  such  not  in  from  Jahn's  found  required.  loop  index.  is  power  analysis.  using  accuracy  eigensystem  be  that  and  the  eigensystem  is  in  computed  c a l c u l a t i o n  a p p l i c a t i o n  and.  developed  derived  eigenvalues.  C o l l a r when  quite  w i l l  also  is  c o r r e c t i o n  the  new  regulator,  method  close  repetitive  account  can  of  applied  method  is  case  in  order  X  c a l c u l a t i o n  the  found  is  9^  eigensystem  eigensystem  feedback  s t a b i l i t y  l i n e a r i z e d quadratic  is  speed  degree  The  the  i t  gain  into  a  one  usually  matrix,  to  and  developed,  s e n s i t i v i t y  is  a  applied  been  eigensystem  This  then  have  and  repeated  and  s t a b i l i z a t i o n  s t a b i l i z a t i o n  avoided.  accuracy  the  systems  is  suspect.  for  been  for  the  new  matrix  relations  converge,  the  for  Thus  eigenvalues,  power  only  the  extended is  from  have  except  eigensystem  not  methods  s e n s i t i v i t y .  the  close  l i n e a r  eigenvalue  shift,  methods  order  requires  matrix  system  high  general from  2  e f f i c i e n t  These  method  system  and  the  the of  time the  a  the  constant governor.  parameters  minimize  eigensystem  in  is  time  of  a  weighted  formulation,  57 the  need  method, matrix  for only  is  extended  Collar  obtain  system  with  found  gain  of  initial  the  the  for  a  the  successive  zation  if  although a of  large the  number  example  into  optimal  the  the  is  is  variables  increased  the  be  be  discrete  made  time  of  the the  governor  number power  method  of  calculated  weighting  the of  The the  for  cost speed  is  to  is  is  is  for  as  intervals  controller solve  The  the  optimization  desired  integral time  In  different  functional. torque  required  to  is  stable The  weightings It  has  angle  been  choosing solution  is  relathe  In  matrix.  method,  by  regulator-exciter.  found  for  stabili-  are  system  It  response.  signals  not  to  regulator  to  chosen.  control  used  recursive  the  a  form.  matrix.  The  to  approximation  used  initiate  if  used  method  optimal  calculation  Ricatti  and  this  The  applied  procedure  an  accurate  system,  and v o l t a g e  calculation.  in  of  system  a good system  Ricatti  The  a  may be  successive  intervals  approximations. of  to  unstable. as  stabilization  a high voltage  method  used  initially  can  approximation then  of  from  unconstrained  calculate  recursive  can  2  stabilization  control  first  troller state  of  independant  example,  null  system  approximate,  numerical duced  the  successive  tively  A fast  the  transformation  that  to  4  in  eigenvalues.  also  apply  and G r u v e r ' s  Chapter  Chapter  to  detrimental  approximations of  To  Box's  example  Puri  in  system.  method  process  using  close  optimization  parameters  is  of  method  minimum.  order  As  calculation  case  constraints,  system  developed  the  correction  function  9^  power  large  for  constrained  from  eliminated.  and P o w e l l ' s  parameter  the  is  except  cost  is  eigensystem  and J a h n  A technique method  inversion  Fletcher the  transform is  one  required  necessary. to  matrix  introthis  initiate  the  with  optimal  a con-  applied  to  found  that  resulted  in  an  the  im-  an  proved done  for  damping, c h a r a c t e r i s t i c  at  the  expense  In  Chapter  time  response  a  5,  multivariable  of  of  the  increasing  the  simultaneous  systems.  sensitivity  This to  response. magnitude  sensitivity method  all  couse,  of  control.  method  allows  parameters  Of  the  from  is  this  is  developed  calculation  only  two  of  differ-  th ential order the  equation optimal  system  reactance machine  the  power  and  but  is  not  to  most  only  gains  for  the  sensitive  and  this  response  t h a t w i t h the Q and v, . o to  the  this  to  suboptimal  control  of  be  of  found  that  line  synchronous  is  is  control  over  9  neglecting  it  omitted  the  voltage,  By  Also,  4  is  controller  controller  Chapter  to  the  gains.  variables.  can  It  4.  reactance  suboptimal  signal  applied  terminal  regulator-exciter  control  optimal  a  is  Chapter  controller  state  voltage  with  of  sensitive  gains,  four  method  transient  of  controller  The  design  most  direct-axis  requires  system  system  response  insensitive which  integrations.  found  found  are  that  very  in-  altogether. is  very  a wide  the  The  close range  to of  P •,  K  It be  is  hoped t h a t  a p p l i e d , and the  application  of  these  results  these also  methods tested  methods  systems.  There  is  nonlinear  control  of  nonlinear  control  with  can  presented  on a c t u a l be  readily  much r e s e a r c h  a power the  system  to  and  linearized  or  be the  in  this  power  systems.  extended done  thesis  in  to the  coordination  steady  state  will The  multimachine area of  of the  control.  59 APPENDIX A Formation A.1  of  Third  tho  Coefficient  Order  Machine  Matrix  and T i e  A of  Line  Equation  2.1  Equations  (18) Park's V  d  =  V  q  =  CJ(^  equations P  =  <*^q  a  k  "  d  where  ^d "  "  X  q  (  d - ^ q  aV  R  G(p)  =  4  may b e  v  p  d  V  as  W  (  W  i  d  U  fad  ^  1 + T  kd ^  d  " -  and For ly  a power  ,  l  d  )  d  CJ  +  ( A a f )  +  (i+r^p)  (A.ig)  +  Q  system  (A.lh) study,  The  induced voltages  s p e e d d e v i a t i o n s dtiiy^  small  compared to 2.  pared  Armature  to  the  3. is  =  l b )  the  following  assumptions  are  usual-  made. 1.  the  1  q  CO  '  )  ^ ^ d o P H i ^ P ) ( q P >  x  q  a  u+x ;p)(i+r ;p)  f  d  (p)  l  (A.le)  p  x,(p) = x , (l+t p)(l tgp) x  >  (  d  \  R  A  ( A  - x (p)  f d  )  G(p) =  written  very  the  With  the V  d  v  q  =  are  is  ^  effects  voltages since  they  due  to  are  a n d ^> t ^ . o  q  since  to  the  neglected overall Park's  "Vq^o  v  the  neglected  neglected  are  assumptions,  ^ u )  and  it  is  small  com-  reactance.  relation  above =  a n d AtJifi  resistance  Subtransient in  a n d Pty  d  speed v o l t a g e s  armature  short  pi^  since  their  duration  response. equations  reduce  to (  A  -  2  a  )  ( A . 2b)  0  -  (1+t'p)  X  i  60 r.\  =  (./  ~x  rq Equation  (A.2c) ^ o  where  which  in  where the  may  Vd  v™,  turn  may  Plfip  field  flux  F  V  ^ d o  +  d (  *d x  =  v  -  =  t  p  d  v  ( v  o  F  p  as  d "  (  X  d  }  p  i  d  A  '  3  )  (A.4)  as 5)  .(A.  R  -  R  (x -x')i ) d  (A.6)  d  -L-VF  < -> A  7  ad  current i  the  x  written  X  and  ~  be  =  fd  field  written  FR  V  ( A . 2d) q  linkage  fr/ the  be  =  =  ^  i q  f  field  v  f  v III  =  d  x  (A.8)  ad  voltage v  =  d  (A.9)  p  ad  Equation  ( A . 5 ) together p(ku>)  with  the  (l/2H)(0)T.  =  general - CO T  O X pS  describe  a  =  torque  equation  D(^OJ))  -  (A.10a)  0 6 (A.10b)  A C O  third  order  e  %  machine  model  where  the  electrical  torque  i s g i v e n b y T  with  the  torque  base  is  speed  =  t  represent v  be  eliminated.  ~ K  chosen  the a  n  as  The  l  dU  (A-  1 radian/second.  The  prime  1  1  )  mover  (A.12)  h  third c  A  by  1.5  g •+  Au)> lfJ-p> p > . g  as to  \  approximated T  To  =  n  >  tie  order  model  with  variables line  equation,  state ^/ ,  from  i  the  variables d  and  i^  terminal  chosen need con-  61 ditions,  may be i„  written  =  as  (C+jB)v.  +  T  or  in  d-q  coordinates  R  -X  X  R  1 Uqj  Collecting  Vd  (A.2d),  -B  X  R  B  G  and  0  v  0  /x  and  X  equation  H d" d -^ d ( x  x  V  _q_  ^o  - l  l  0 two  (A.13b),  tie  line  equation  X  -1/x  R  X  R  1/x  ~1 _ 3 K  ~ 2 K  \  vd  q  G  -B  B  G  upon r e a r r a n g i n g K  U).  -X  rows  0  d  0 R  where,  0  Kl  V q  Vn  /x  X  (A.15)  ^ d  ^Vq  (A.16)  Vq  last  -X  o  ^d  the  R  d  (A.2b)  Substituting  the  ) / x  o  0 =  (A.14)  - q^o  X  d  L_q.  o d  0  (A.2a)  (A.13b)  v cos^ _ o _  v  0  From  sind  X  d  / dTdo  i  o  -t o< d-XcP  this  _q_  v  (A.6)  0  x  F R  (A.13a)  )  TR+IXT  G  w  inverting  o  v  - X  =.  v  t "  R  0 and  v  as  (A.3)  tdo  (  +  and  of  (A.15)  and  all  '  do d  of  (A.16)  into  becomes  +  R  -X  X  R  0  1  0  0  -1  0  1  1  0  Vd  Vi  (A.17)  vosinS vocos o  linearizing  vo cosS„ -v o sin§  R  0  -do 0  d X  (A.18)  K  where  =  x  (R/x  XG)  RB  d  = (1 + RG - XB + X / x ) K  =  3  d  K. = ( R / x - RB 4 q The  solutions * V  where  =  l  c  c  Y  o 4 ( K  V  = (  R  X  K ]  are  Aty  ^o  C O S  K  +  -  o  2  1  ( J  1  1  2 >/ do d l 4 r  X  ( K  3R)/t ;x (  K  +  4  K sinS )/(K K  ~  X  (A.20)  F  K sinS )/(K K  "  ^o  C O S  ( K  K  XG)  A>  - o 3  = < 4  2  and  d  -  = l/<CO, =  2  b  of A £ /  d  l  b  (A.19)  (1 + RG - XB +. X / x )  d  d  K  K  K  l  +  2  K  •  4  K  K ^ ) +  4  3  K ^ )  (A.21)  )  K K ) 2  3  The f i r s t third order model equation can now be written in state variable form by substituting first equation of p(M^,)  = Av  and the result into the linearized  (A.15) +  p  1  into the linearized  (A.20)  A (c (x -x )-x A: )^ d  1  d  d  d  d o  F  + (x -x )b d  d  (A.10b) P  with aubstitution of  (AUJ)  <£o eg  =  +  2H  P(A£>) = where  e  &T  e  qo i \  A.2  do  Governor The  Ai o  =-(i  Ar  o^ e T  -  2H  (A.22b)  A U>  (A.22c)  "do  q  r  qo  ' b , + i , 'b„)AS  qo  =  1  do  ( do V  -  Ai , + A(J,i  - U  s u b s t i t u t i o n of  where  W  and  (A.lOa)  H  =G)  after  -  . 7 5 Wo A h  into  (A.lOa)  A. T  which  (A.12)  Ab  d  X  other equations can be obtained by linearizing  (A.22a)  1  • The  (A.5)  2'  i  d  r  (A.15) -  (i  d  and  - A(J i ,  qo  ^q d o '  (A.20)  ' c, + i ' c qo 1 do  0  2  )  (A.23)  becomes - v, / , w , /,->.< i d o / x 7 J )/y ( A . 24) d  d o  p  ) (A.25)  = (lao  +  i  d 0  )  x  and H y d r a u l i c  governor  Equations  and h y d r a u l i c  transfer  function  is  given  by  (19)  63  A T, A  (l+t p)(l-T p) r  M  co (o-+r p +  {$ +o)v T?  diagram  given  Q  The  block  t  a  is  w  +  r  in  r  a  Figure  A a  1  2  g  A.l.  v  From  the (1  1 (1  (A.26)  t r p )(i+r p)(i+.5r p)  +  T p)  (1  g  transfer - T  p)  + .51  p)  A  T  t  S t V (1  Figure function state  +  A.l  r  Governor  (A.26)  variable  T p)  and the form  of  becomes  linearized the  _  a  (7  f  ^o^a  p(Mi)  = _ 2(ha.)  &a  (A.28b)  can  model  (Aa)  <- r  2(A^) _ 2(Ah) Tg Tv Equations  regulator-exciter 1  (i+r )(i+r l P  k  ~  diagram  crSt  -  f  Regulator-Exciter  to,  block  operator  the  (A.27)  fg  ^2  The  (A.12),  Ag  =  &co  hydraulic  + ~)(^ )  k  Ca  p(&g)  The v o l t a g e  and  equation  Operator  a  p(Aa ) =  Voltage  torque  and H y d r a u l i c  cr(Aa)  r a  A.3  Function  governor  ^ f  _ ACO  p(*a)  Transfer  l  k  e x  transfer  functions  are  given  (A.28a)  p)  2  (A.28b)  (i+r p)(n-r p) 1  is  shown  be w r i t t e n  in  e x  in  Figure  state  by  A.2.  variable  Equations form  as  (A.28a)  and  64  k  \ /  Figure  (l  A.2  s  l + T  Voltage  l  P  A.v  1  p  >-  )  (  1  +  Regulator-Exciter  tex*>  Transfer  k,k.  Function  Av  T p(*v )  =  p  r  where  =  which  when  d  linearized  A v t.  Av,  V  Summary  to  (A.22a,b,c), equations  (A.27)  equations  (A.29),  written  the  =  p»  A is  given  ' ^' by  (A.31)  . V ,  order  c  to  machine  order  =  governor  second  ninth  variable y  into  6  (A.31)  q'  l  v.  C  to  2  ; A (  (A.32)  ^F  Equations  and t h e  the n o t a t i o n  matrix  a  fourth  state  — y  where with  in  v, _ ' to  V j .  v , 2>^ to  the t h i r d  the  (A.30)  d'  of System  Combining  q  2  and (A.20)  l  b  v  becomes  x  (A.16)  r  +  v v, _ * to  Substituting  A.4  2  v,  (A.29)  order  order  form  and t i e  and h y d r a u l i c voltage  power  p  omitted  operator  equations  c a n be  as  v , p  equations  regulator-exciter  system  A y  lfJ ,  line  v , g  g , h,  a,  for simplicity.  T a ~j f  The  (A.33) ( A . 34) coefficient  A  (A.35)  =  _D 2H  '12  "13  32  33  L  l  k  2  2II  "ex _1_  a 53  '52  o  2H  'ex k  1.5 GO  <£o  ?1  -g  •w  g  tr  o a L  St ^  r  a  wh ere  a  12  l  "13 a 32 '33  U  + b i ')/2H ' "2 do' 0  (c i ' + c i ' 1 qo 2 do  o  n  (x -x )b /x d  d  1  -  I'l^iT*** l  (  v  q o  b  l  "  "53  "  k  l  (  v  q o  C  l  -  i  by  (A.25).  b^,  b , 2  c^,  c  l/r  •x•)/2H do d  d  k  constants  -  0  -  The o  o  co ( 1b , i l^qo  52  L  d  _1  2  (A.36)  *k^do /*k )  V  V  do 2 b  d o are  C  2  ) / v  )  /  v  t ri o  t o  given  r  i  by  (A.21a,b,c,d)  and  i  q 0  >  66 APPENDIX Initial  Operating  Vith  the  e>u)<^>  t^jJlfJ^)  Conditions  assumptions  armature  steady  state  (A.30)  may be 2 , to  form  of  of  a Power  System  appendix A neglecting  resistance  of  B  and  subtransient  (A.2a,b,c,d),  (A.6),  P^j  P^q»  effects,  (A.11),  the  (A.13b),  and  (6) V  v do  written =  V ,  "  ~Vqo  + V, qo 2  do  qo T  as 2  ^o  *ao =  eo  H>  V  i.)  u  d  0  "^o ^ q o  =  P  v  =  ^Fo  v  o  =  o  Fo  v i do do  i q qo  d  Fo  v i qo qo  Vdo  ~  -x  +  (  ^do  d  d  v o sinS o = (1 - XB + R G ) v do , v o cos& o and the  =  *o  V  Fo'  power  Hlo'  P  , o' Vqo'  v  i,  -  v,  q  0 and v, ^o to ^Fo'  °  V ,  do  do  - (RB + XG)v qo - R i ,do + X i qo  / ( P •j  allows  \lo'  =  x ) o q  X 1  q qo  i  do  &  2  +  (B.2) qo  the  \ o > P  l  )  flow  q o do  Choosing °  a  = (XG + R B ) v do , + (1 - XB + RG)v qo - X i ,do - R qo i  reactive  Q  B  o  n  d  8  determination o  f  r  o  m  {  B  '  l  v. to (v. to  2  + x p ) q^o'  2  ]  a  m  of l  (  v, , v , do' qo' B  '  2  )  a  S  v  o',  i do ,  -  (Qo • + x i • ) / vqo q qo  v„ Fo  =  v  V  +  qo  x . i ,  d do (B.3)  ^do^o  = q 0  Hlo  V  Wo  2  K  qo  /  w  o  - di d" d> do  =  r  ( x  X  i  t  +  d  i v  p o  and from the last two equations of ( B . l ) v  /-  o  '  V  2  ' -  do ,  +  qo  V  2 (B.4)  = arctan ( vdo' , / Vqo')  £°o  s  where Y  d o  vqo  -  =  (1 - XB  =  (XG  +  .RG)v  + RB)v  do  d o  +  -  ( R B  +  X G )  V  q  (1 - XB + R G ) v  o  qo  -  Ri  d  o  Xi,  do  +  X i  -  Ri  q  o  qo  APPENDIX C Fletcher  the  (14) 'Descent  and P o w e l l ' s  'Die  algorithm  1.  Compute t h e  parameter  of  cost  vector - l  T  g  Fletcher  p,  and P o w e l l ' s  functional  that  -  method, i s  gradient  with  as  respect  ri i ,  -l  p  x  ...  JT, iP j  ...  J,p  T i  i  is  2.  the  iteration  Adjust  the  i p .=p .  a minimum J  is  by  given  i and H  is  1  matrix  a  for  3.  =  1  1  =  k  is  k*  minimizes 4.  J  =  found 1  such  along  Evaluate H  where  1  D E  +  the  the  i^*  iteration  1  (C.2) matrix  usually  c h o s e n as  a  unit  5.  Repeat  component  of  then  s  1  (C.3)  1  ,  p  -i  + cr  tha.t  p  =  is  ^  H  1  +  D  and +  1  compute H**^ E  i  i  = the is  i  -  procedure less  than  W]  computed from p  1  T  from  <'> ¥1 <°- > C 5  f with  prescribed of  6  (C.7)  starting a  which  (C.4)  1  - -H -[y j-[? j - HVIFF  1  vector  1  = [ ^ { ^ ^ [ - J  1  th;* p a r a m e t e r  s .  (Cl)  +  1  1 +  gradient  g* ^ from  f  and  s*  A  -i  p  and  is  for  direction  iteration.  -i+1  J  s^  a gradient  Set 1  of  along  1  definite  first  5-  where  vector  f o u n d where  - H . g  positive the  is  1  (Cl)  number.  parameter  s""" u n t i l  to  T x  P=P where  follows  is  i -  = J>P  Method  the  last  step  2 until  accuracy. iteration.  The  every minimum  69 REFERENCES 1.  L a u g h t o n , M. A . , " T h e U s e o f S e n s i t i v i t y A n a l y s i s i n the Design of Generator E x c i t a t i o n C o n t r o l " , Proceedings o f the S e c o n d Power Systems C o m p u t a t i o n C o n f e r e n c e , V o l . 1, P a p e r 3 . 4 , S t o c k h o l m S w e d e n , 1 9 6 6 .  2.  F r a n c i s , J . G. F., " T h e QR T r a n s f o r m a t i o n " , P a r t I, The C o m p u t e r J o u r n a l , V o l . 4 , O c t o b e r 1 9 6 1 ; P a r t II ibid., J a n u a r y 1962.  3.  L a u g h t o n , M. A . , " S e n s i t i v i t y i n Dynamical System J o u r n a l o f E l e c t r o n i c s and C o n t r o l , V o l . 17, November 1964.  4.  Macfarlane, Linear 1963.  5.  Fadeev, of  D.  A. G. J . , Regulator  K.,  and  Linear  V.  Analysis", N o . 5,  "An E i g e n v e c t o r S o l u t i o n of the Optimal P r o b l e m " , i b i d . , V o l . 14, No. 6, June  N.  Fadeeva,  "Computational  Methods  Algebra",  Freeman,  San  1963.  Francisco,  6.  V o n g s u r i y a , K., "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 " , PhD. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, February 1968.  7.  F r e e s t e d , V. C . , R. F . W e b b e r , a n d R. V . B a s s , "The "GASP" C o m p u t e r P r o g r a m - An I n t e g r a t e d T o o l f o r O p t i m a l C o n t r o l and F i l t e r D e s i g n " , P r e p r i n t s o f t h e 1968 J o i n t Automatic C o n t r o l C o n f e r e n c e , pp. 198-202, University of M i c h i g a n .  8.  Puri,  9.  Kerlin,  N. N . , a n d W. A . G r u v e r , " O p t i m a l C o n t r o l D e s i g n v i a Successive Approximations", Preprints o f the 1967 J o i n t Automatic C o n t r o l C o n f e r e n c e , pp. 335-343, University of Pennsylvania. T.  W.,  "Sensitivities  Simulation, 10.  Van  11.  Morgan J r . ,  Vol.  B.  Householder,  A.  Analysis", 13.  Yu,  No.  by 6,  the  State  June  1967.  Variable  Approach",  Ness, J . E., J . M. B o y l e a n d F . P. Imad, "Sensitivities of L a r g e , M u l t i p l e - L o o p C o n t r o l S y s t e m s " , IEEE T r a n s a c t i o n s on A u t o m a t i c C o n t r o l , V o l . A C - 1 0 , J u l y 1 9 6 5 . S.,  Multivariable 12.  8,  "Sensitivity Systems",  S.,  "The  New  York,  Analysis  ibid.,  Theory  of  Vol.  and  Matrices  Blaisdell,  Synthesis  AC-11, in  July  of  1966.  Numerical  1964.  Y . N . , K. V o n g s u r i y a , a n d L . N. Wedman, "Application 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 S y s t e m " , Accepted f o r p u b l i c a t i o n i n I E E E T r a n s a c t i o n s on Power A p p a r a t u s and S y s t e m s .  70 14.  Fletcher, R. , and M. J . D. P o w e l l , "A R a p i d l y Convergent D e s c e n t M e t h o d f o r M i n i m i z a t i o n " , The C o m p u t e r Journal, V o l . 6, p p . 163-168, 1963.  15.  Box,  16.  Tomovic,  M. J . , "A Comparison of S e v e r a l C u r r e n t O p t i m i z a t i o n M e t h o d s a n d t h e Use o f T r a n s f o r m a t i o n s i n C o n s t r a i n e d P r o b l e m s " , i b i d . V o l . 9, pp. 67-77, 1966. R.,  "Sensitivity  McGraw-Hill, 17.  Gantmacher,  F.  Chelsea,  R., New  Analysis  of  New Y o r k ,  1963.  "Matrix  Theory",  York,  Dynamic  Vols.  I  Systems",  and  II,  I960.  18.  Park,  R. II., "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 Machines: I-Generalized Method of A n a l y s i s " , AIEE T r a n s a c t i o n s . V o l . 48, J u l y 1929, pp. 716-730.  19.  S i d d a l l , R. G . , " A P r i m e M o v e r - G o v e r n o r T e s t M o d e l f o r L a r g e Power S y s t e m s " , MASc. D i s s e r t a t i o n , University of B r i t i s h C o l u m b i a , J a n u a r y , 1968.  20.  Athans,  M.  Hill, 21.  Noton,  A.  a n d P. New R.  Control 22.  L.  York,  M.,  Falb,  "Optimal  Control",  McGraw  1966.  "Introduction  Engineering",  to  Variational  Pergamon  Press,  Methods  Oxford,  in  1965.  Wedman, L . N . , Y a o - N a n Y u , " C o m p u t a t i o n T e c h n i q u e s f o r t h e S t a b i l i z a t i o n and O p t i m i z a t i o n of H i g h Order Power S y s t e m s " , S u b m i t t e d f o r p u b l i c a t i o n t o t h e 1969 P I C A conference.  


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