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UBC Theses and Dissertations

Application of modern control techniques to power systems Miniesy, Mohammed Samir Mohammed 1971

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A P P L I C A T I O N OF MODERN CONTROL TECHNIQUES TO POWER SYSTEMS  by  MOHAMMED SAMIR MOHAMMED M I N I E S Y B.Sc,  A i n Shams U n i v e r s i t y ,  Cairo,  E g y p t , 1962  M.E., C a r l e t o n U n i v e r s i t y , O t t a w a , C a n a d a , 1 9 6 7  A THESIS SUBMITTED I N P A R T I A L FULFILLMENT  OF  THE REQUIREMENTS FOR THE DEGREE OF  DOCTOR OF PHILOSOPHY  in  t h e Department of  Electrical  We a c c e p t  this  thesis  required  Engineering  as c o n f o r m i n g  to the  standard  Research Supervisor  .  Members o f C o m m i t t e e  Head o f D e p a r t m e n t Members o f t h e D e p a r t m e n t o f Electrical THE UNIVERSITY  Engineering  OF B R I T I S H COLUMBIA  O c t o b e r , 1971  In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree  at the U n i v e r s i t y o f B r i t i s h Columbia,  the L i b r a r y s h a l l make i t f r e e l y  I agree  a v a i l a b l e f o r r e f e r e n c e and  study.  I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying o f t h i s for  s c h o l a r l y purposes  may  by h i s r e p r e s e n t a t i v e s .  be granted by the Head o f my  I t i s understood  of t h i s t h e s i s f o r f i n a n c i a l  <??/^c/^/ca  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  A/oO-  2*£f  ////  Department or  g a i n s h a l l not be allowed without  f  Columbia  thesis  t h a t copying or p u b l i c a t i o n  written permission.  Department o f  that  ecr-r/ sTf  my  ABSTRACT A p o w e r s y s t e m may The  control strategy  d e p e n d s on  the  generator  optimum s w i t c h i n g elements of  electric  to preserve  system  disturbances.  stability  i s presented  power s y s t e m s t a b i l i t y  p o w e r n e t w o r k s u c h as  dropping or load  a generalized  to d i f f e r e n t types of  disturbance.  severe disturbances,  sudden changes i n the  resistors,  subjected  taken i n order  s e v e r i t y of the  For very by  t o be  be  by  shedding.  considering  the  A unified the  can  be  improved  i n s e r t i o n of treatment  switching  braking  of  i n s t a n t s to  c o n t r o l v e c t o r . Dynamic o p t i m i z a t i o n i s t h e n  be  applied  t o d e t e r m i n e optimum s w i t c h i n g i n s t a n t s . Less severe disturbances voltage  regulator controls.  model of  interconnected  A two-level p l a n t has l e v e l of If  The  a first-level c o n t r o l i s an  local  tolerances,  plant. level  co-ordinator. On  optimal  power p l a n t s  or suboptimal  causes the i n t e r r u p t to  Angular v e l o c i t y This  data  controller.  system angular the  data,  a central  second co-ordinator.  a c c e l e r a t i o n to  data  transmitted for  each second-  local control.  to minor or r o u t i n e  Since  exceed  initiates  each p l a n t generates a l o c a l  changes,is i n v e s t i g a t e d .  demand i n a p o w e r s y s t e m i s n o t  The  d e v i a t i o n s of a l l p l a n t s are  L o a d - F r e q u e n c y C o n t r o l p r o b l e m , due load  and/or  Each  central co-ordinator  i n t e r v e n t i o n c o n t r o l w h i c h augments f i r s t - l e v e l  b a n c e s c a u s e d by  i s discussed.  i s used to generate c o e f f i c i e n t  receiving i t s coefficient  The  employing governor  i n t e r v e n t i o n c o n t r o l p e r f o r m e d by  a priority  control.  overcome by  i s i n v e s t i g a t e d v i a the m u l t i - l e v e l concept.  c o n t r o l l e r for interconnected  intervention to the  be  governor c o n t r o l problem f o r a l a r g e s i g n a l  power p l a n t s  a sudden system d i s t u r b a n c e  preset  can  the  incremental  disturpower  a l w a y s known a p r i o r i ^ d i r e c t a p p l i c a t i o n o f  ii  the  optimum l i n e a r - s t a t e r e g u l a t o r t o L o a d - F r e q u e n c y C o n t r o l i s n o t p o s s i b l e . F u r t h e r m o r e , L o a d - F r e q u e n c y C o n t r o l g e n e r a l l y r e q u i r e s t h e u s e o f an  integral-  type  This .  control operation  requirement  t o meet  i s introduced into  Control problem presented Two  the system operating s p e c i f i c a t i o n s .  t h e f o r m u l a t i o n o f t h e optimum L o a d - F r e q u e n c y  i n this  methods a r e s u g g e s t e d  m e t h o d makes u s e -of d i f f e r e n t i a l of a Luenberger observer a linear  f o r demand i d e n t i f i c a t i o n .  approximation.  to identify  first  The s e c o n d m e t h o d makes u s e  unmeasured s t a t e s .  The o p t i m u m c o n t r o l i s  i n c r e m e n t a l p o w e r demand. A method i s g i v e n  frequency  for solving,  s u b o p t i m a l l y , t h e p r o b l e m o f optimum-  sampled-data c o n t r o l w i t h e i t h e r  demand o r r a n d o m l y v a r y i n g s y s t e m d i s t u r b a n c e s . optimum c o n t i n u o u s data  The  f u n c t i o n o f m e a s u r e d s t a t e s , i d e n t i f i e d unmeasured s t a t e s , and t h e  identified  load  thesis.  unknown  d e t e r m i n i s t i c power  I t i s shown how  to modify  an  c o n t r o l t o o b t a i n optimum c o n t r o l i n t h e case o f d i s c r e t e -  t r a n s m i s s i o n a n d unknown d e t e r m i n i s t i c demand. The c a s e o f random p o w e r demand a n d r a n d o m d i s t u r b a n c e s  by  i n t r o d u c i n g an a d a p t i v e  is  given.  observer  observer.  A three stage  systematic design  The e f f e c t i v e n e s s o f L o a d - F r e q u e n c y C o n t r o l u s i n g an i s i l l u s t r a t e d b y an e x a m p l e .  i i i  i s treated procedure  adaptive  TABLE OF CONTENTS  Page  ABSTRACT  i i  TABLE OF CONTENTS  .  LIST OF TABLES..  v i  LIST OF FIGURES......; ACKNOWLEDGEMENT 1.  1.2  3.  v i i ..  ix  INTRODUCTION 1.1  2.  iv  System Decomposition  1  1.1.1  S p a t i a l Decomposition  1  1.1.2  S e q u e n t i a l Decomposition  Background  . .  and T h e s i s Layout  2 4  OPTIMUM NETWORK SWITCHING IN POWER SYSTEMS 2.1  Introduction  8  2.2.  Problem F o r m u l a t i o n  2.3  Algorithm.  13  2.4  Applications  13  f o r Optimum S w i t c h i n g  2.4.1  C r i t i c a l Switching  2.4.2  Optimum S w i t c h i n g  Times  Time  9  14  Time o f a B r a k i n g  Resistor  20  TWO-LEVEL CONTROL OF INTERCONNECTED POWER PLANTS 3.1  Introduction  .-  27  3.2  Large S i g n a l Model f o r I n t e r c o n n e c t e d  3.3  System Dynamics i n S t a t e V a r i a b l e Form  32  3.4  Two-Level  35  3.5  On-Line  3.6  O f f - L i n e C o n t r o l Design  Power P l a n t s  S t r u c t u r e of The C o n t r o l Problem  C o n t r o l Implementation........  29  39 40  iv  Page  3.7 4.  3.6.1  D e s i g n o f The L o c a l  Feedback C o n t r o l  3.6.2  D e s i g n o f The I n t e r v e n t i o n  3.6.3  D e s i g n o f The A p p r o x i m a t e  42  Control....  42  Intervention  Control.  43  Application  44  OPTIMUM LOAD-FREQUENCY CONTINUOUS CONTROL WITH UNKNOWN D E T E R M I N I S T I C POWER DEMAND  5.  4.1  Introduction  .. ..  4.2  Problem  4.3  Demand I d e n t i f i e r  4.4  Applications  4.5  Demand I d e n t i f i e r  Formulation  49 50  - D i f f e r e n t i a l Approximation  54 56  - Luenberger  Observer  63  OPTIMUM LOAD-FREQUENCY SAMPLED-DATA CONTROL WITH RANDOMLY VARYING SYSTEM DISTURBANCES  6.  5.1  Introduction  • ..  5.2  O p t i m a l Sampled-Data R e g u l a t o r  70  5.3  S t o c h a s t i c Optimum a n S u b o p t i m u m C o n t r o l  73  5.4  A p p l i c a t i o n - S i n g l e Steam P l a n t . . . . . . . . . . .  81  CONCLUSIONS  89  APPENDIX I  93  APPENDIX I I APPENDIX I I I  ....... •  96 ..  98  APPENDIX I V REFERENCES  69  100 ............  101 V  L I S T OF  TABLES  T a b l e " (2.1)  _, Page 18  Table  (2.2)  (Prob. 1(a))  18  Table  (2.3)  (Prob. 1(b))  18  Table  ( 2 . 4 ) D a t a and  Table  (2.5) D u r i n g f a u l t  Table  (2.6) A f t e r  Table  (2.7)  Table  (2.8) (Prob. 11(a))..:  24  Table  (2.9)  24  Table  (2.10((Prob. 111(c))  26  Table  (3.1)  47  Table  (3.2)  48  Table  (4.1)  59  Initial  fault  Conditions  admittances admittances  (Prob. 1 ( c ) ) N  (Prob. 11(b))  21 22 22 22  L I S T OF  FIGURES  Figure  Page  (2.1)  One  (2.2)  Power a n g l e d i a g r a m  17  (2.3)  One  21  (2.4)  Reduced system of F i g . (2.3)  (2.5)  (Prob. 11(a))........  (2.6)  (Prob. 11(c))  26  (3.1)  Steam p l a n t b l o c k d i a g r a m  30  (3.2)  Hydro p l a n t b l o c k diagram  30  On-line control implementation  41  •' ( 3 . 3 ) (3.4)  line  system  17  diagram of a four-machine system  21 '.  24  A n g l e and a n g u l a r speed as a f u n c t i o n o f t i m e f o r a s t e p of  (3.5)  machine i n f i n i t e bus  change  angular acceleration  A n g u l a r and local  47  tie-line deviation  control  only,  for  = 5,  s  W^Q  (b) l o c a l p l u s i n t e r v e n t i o n i n t e r c o n n e c t e d steam  =  0 with  (a)  controls  (4.1)  B l o c k d i a g r a m o f two  (4.2)  B l o c k d i a g r a m o f a power p l a n t w i t h  (4.3)  System  (4.4)  Response comparison  (4.5)  B l o c k diagram o f optimum l o a d - f r e q u e n c y c o n t r o l w i t h p a r t i a l  r e s p o n s e f o r (a) T  q  = 0.5,  System  Q  (a) c o n v e n t i o n a l and  = 1  (5.1)  Response comparison  (b) p r o p o s e d  controls..  c o n t r o l T = 1 and  w i t h T=l  (c) sampled  response.  67  (b)  and t i e - l i n e measurements •  68  sampled-data  data control T = 2  Frequency  62  meas-  '. ..  f o r (a) c o n t i n u o u s c o n t r o l ,  57 61  response - Example (4.4)  Steam p l a n t  51  i  (4.6)  (5.2)  areas.......  a load-frequency controller.  (b) T  urements  and h y d r o  48  74 sampled 83  Figure  Page  (5.3)  A v e r a g e c o s t as  (5.4)  Effect  (5.5)  a function of threshold l e v e l  of adaptive  gain  (a) d e t e r m i n i s t i c  g  observer  85 (b)  adaptive  observer  86  Tracking c a p a b i l i t y of the adaptive observer  88  viii  ACKNOWLEDGEMENT I and  am i n d e b t e d  t o my S u p e r v i s o r , D r . E.V. B o h n , f o r h i s  a s s i s t a n c e throughout the p e r i o d o f t h i s  research.  T h a n k s a r e due t o D r . Y.N. Y u f o r h i s i n t e r e s t comments on t h e d r a f t . is  duely  Struyk  appreciated.  for proof I  the  valuable  I am g r a t e f u l  t o my c o l l e a g u e s T e r r y  Kabriel  Curran  and Emile  for her patience  i n typing  the manuscript,  t o thank Miss  Linda Morris  manuscript. The  grant  and h i s  Reading t h e D r a f t by Dr. M.S..Davies and Dr. B.J.  reading  also wish  guidance  financial  support  by t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada,  6 7 - 3 1 3 4 ( 1 9 6 8 - 1 9 7 1 ) , a n d t h e S c h o l a r s h i p b y B.C. T e l e p h o n e Company  (1970) a r e a p p r e c i a t e d . Finally, them, I w o u l d l i k e and  i n l i e u o f the time  I might have o t h e r w i s e  spent  t o t h a n k my w i f e N a d i a a n d my s o n Ehab f o r t h e i r  understanding.  ix  with patience  1. Interconnection development i n t h e u t i l i t y inherent  INTRODUCTION  b e t w e e n a d j o i n i n g p o w e r s y s t e m s i s an i n e v i t a b l e  i n d u s t r y because i t o f f e r s the mutual b e n e f i t s o f  economy, r e l i a b i l i t y  of operation  pect of t r a n s f e r r i n g large blocks  and i m p r o v e d s t a b i l i t y .  o f power o v e r l o n g  transmission  The  pros-  distances  between n e i g h b o u r i n g systems i n d i f f e r e n t time zones, t o a s s i s t each i n t u r n through i t s r e s p e c t i v e  peak l o a d p e r i o d  i s very  encouragement t o l a r g e s c a l e i n t e r c o n n e c t i o n . problems and d i f f i c u l t i e s increasing the  are confronting  demands f o r e l e c t r i c  1.1  System  electrical  With ever  to improve s t i l l  must meet t h e c h a l l e n g e  of seeking  further improved  generation.  interconnected,  to furnish electrical  Requisite  t h e power u t i l i t i e s .  Decomposition An  is  great  I n i t s r e a l i z a t i o n , many new  power and t h e d e s i r e  q u a l i t y of service, u t i l i t i e s  methods o f r e g u l a t i n g power  a t t r a c t i v e and a  to this  p o w e r s y s t e m i s .a c o m p l e x s y s t e m w h o s e p r i m a r y  e n e r g y as r e q u i r e d by c u s t o m e r s and as l o n g  o b j e c t i v e i s a q u a l i t y o f s e r v i c e c h a r a c t e r i z e d by  f r e q u e n c y and v o l t a g e A general  as. r e q u i r e d . stable  and by c o n t i n u i t y i n time.  s o l u t i o n t o the problem of c o n t r o l l i n g the whole  such that a l l the o p e r a t i n g  objectives are  satisfied  system  at a l l times i s i m p r a c t i c a l .  D e c o m p o s i t i o n o f t h e s y s t e m " s p a c e - w i s e " and " t i m e - w i s e " i s n e c e s s a r y t o s o l v e a complicated  1.1.1  goal  such  problem.  S p a t i a l Decomposition We  c a n t a k e t h e w h o l e power s y s t e m and decompose i t i n t o  systems o r areas. complexity  The n e e d f o r s u b d i v i s i o n i n t o a r e a s w i l l  and c o m p u t a t i o n a l d i f f i c u l t y  associated  with  depend on:  sub(1)  The  a s i n g l e c o n t r o l model.  2  (2)  The g e o g r a p h i c a l  l o c a t i o n of generating  load  a l l o c a t i o n and p o l i t i c a l  boundaries.  s o u r c e s and t h e (3)  associated  Overall consideration  of  reliability. Decomposition i s generally network.  For c o n t r o l r e l i a b i l i t y  •area a s s e l f - s u f f i c i e n t connection  support.  by weak t i e s a) disturbances b)  g u i d e d by t h e s p e c i f i c n a t u r e o f t h e  i t i s usually required  as p o s s i b l e i n g e n e r a t i n g  are u s u a l l y  each area has t h e r e s p o n s i b i l i t y do n o t d i s r u p t terminal  the i n t e r t i e s  Sequential  and i n i n t e r areas  connected  recommended: of seeing  or impair  areas are r e s p o n s i b l e  b a n c e s on t h e i n t e r t i e s betx^een t h o s e  1.1.2  capacity  I n a power s y s t e m c o n s i s t i n g o f s t r o n g  the f o l l o w i n g p o l i c i e s  both  to have each  that  i t s internal  service i n other  f o r counteracting  areas ; distur-  areas.  Decomposition  E v e n f o r e a c h a r e a _ , s i n g l e c o n t r o l p o l i c y w o u l d be i n a d e q u a t e f o r all  operating  operating  Characterization  It is^in  e f f e c t , decomposing the t o t a l  and a f t e r " a s e v e r e s y s t e m According  following  by  to "conditions f o r  operating  problem  a set of sequential problems, corresponding e s s e n t i a l l y to  during  "before,  disturbance.  t o [1], t h e p o w e r s y s t e m c a n b e c o n s i d e r e d  i n one o f t h e  states; (a)  Preventive  In t h i s  operating  state  (normal  operation):  s t a t e t h e power s y s t e m i s b e i n g  demands o f a l l c u s t o m e r s a r e s a t i s f i e d The c o n t r o l p r o b l e m i n t h i s of  o f power s y s t e m o p e r a t i o n  s t a t e s h a s b e e n i n v e s t i g a t e d i n [1], a c c o r d i n g  operation". into  states.  o p e r a t e d so t h a t  at standard  state i s to continue  c u s t o m e r demand w i t h o u t i n t e r r u p t i o n a n d w i t h  f r e q u e n c y and  indefinitely minimum  cost.  the  voltage.  the s a t i s f a c t i o n  (b)  Emergency o p e r a t i n g  Aside  from the causes that b r i n g s  emergency o p e r a t i n g  o r when t h e s y s t e m f r e q u e n c y stall,  starts  in  this  c u s t o m e r demand.  a t a s a f e minimum,  t o decrease toward a v a l u e  further degradation  while  E c o n o m i c c o n s i d e r a t i o n s become s e c o n d a r y  (c)  Restorative state:  This  i s t h e s t a t e where t h e power system w i l l  an emergency.  state i s the safe t r a n s i t i o n  be i n p a r t i a l  It  i s hard  states of operation.  from p a r t i a l  t o 100  percent  satisfaction  time.  t o d e f i n e g e n e r a l boundaries between the d i f f e r e n t This  i s because of the d i f f e r e n t  d i f f e r e n t p a r t s o f t h e system.  To c l a r i f y  e x a m p l e two a r e a s o f t h e same c a p a c i t y . a r e a s may c a u s e i n s t a b i l i t y survive the disturbance.  The f i r s t  design  disturbances  at both  t h e o t h e r may b e a b l e t o  a r e a w o u l d be i n t h e emergency s t a t e be taken  according  to the nature  F o r e x a m p l e , b r a k i n g r e s i s t o r s may b e n e c e s s a r y  a c c e l e r a t i n g generators.  and n a t u r e  this point,consider f o r  Two e q u a l  t o one a r e a w h i l e  emergency c o n t r o l a c t i o n s h o u l d  disturbance.  Usually this i s  The c o n t r o l o b j e c t i v e i n t h e r e s t o r a t i v e  o f a l l c u s t o m e r demand i n m i n i m u m  the  satisfying  state.  the a f t e r m a t h . o f  and  of  state i s to relieve  l o a d o p e r a t i o n , when some c u s t o m e r l o a d s h a s b e e n l o s t .  for  a t which  system i s i n the process  The c o n t r o l o b j e c t i v e i n t h i s  the s y s t e m d i s t r e s s and f o r e s t a l l  o f some c o m p o n e n t s  cannot be m a i n t a i n e d  o r when t h e e l e c t r i c a l  l o o s i n g synchronism.  a specified  t h e system' t o that, s t a t e , a n  s t a t e comes a b o u t when t h e r a t i n g  a r e e x c e e d e d , o r when t h e v o l t a g e  motors w i l l  state:  of  to decelerate  F o r the second area, because o f t h e design  4  of  i t s generators, s t a b i l i t y  controls  such, as  h o w e v e r , on  the  could  be  m a i n t a i n e d by  using  governors and/or yoLtage r e g u l a t o r s . amount o f d i s t u r b a n c e  existing  There i s a  a f t e r which, the  limit,  second a r e a must  t a k e emergency c o n t r o l measures. This e m e r g e n c y and  t h e s i s i s concerned with preventive  state.  In the  e x a m i n e some c o n t r o l a s p e c t s r e l a t e d t o b a l a n c e and and  1.2  thesis  these states  we  are  going  from the the  the to  energy  current  xrork  Layout  between the m e c h a n i c a l power i n p u t e l e c t r i c a l power o u t p u t .  of  o u t p u t can  n e t w o r k , whereas the  The  d i f f e r e n c e between i n p u t  acceleration  power.  (the  can  mechanical input  The  and  the  sum  o c c u r as  i t i s obvious that  occur almost.instantaneously  the  or d e c e l e r a t i n g  g e n e r a t o r s and  is of  because of  the  immediate s i g n of n o n - e q u i l i b r i u m  of  in  change i n  cannot change n e a r l y  a c c e l e r a t i n g or d e c e l e r a t i n g  losses  output.  changes  o u t p u t p o w e r i s known as  equilibrium  a result  or a change i n e l e c t r i c a l  However, from a p r a c t i c a l v i e w p o i n t , electrical  o f pox^er s y s t e m t h e r e  Non-equilibrium  e i t h e r a change i n m e c h a n i c a l i n p u t  so  fast.  accelerating i s system  p o w e r d i v i d e d by  the  constant). D e p e n d i n g on  the  next Section,  control in  layout.  B a c k g r o u n d arid T h e s i s  inertia  problem of  dynamic b e h a v i o u r viex^poirtt, along w i t h  During steady s t a t e operation  and  the  the  s e v e r i t y of  a c c e l e r a t i o n , a pox^er s y s t e m c a n  mentioned before. disturbance.  The  The  controls  be  the  disturbance  i n any  f o r each s t a t e  follox^ing points  briefly  of  the  and  operating  d e p e n d on  o u t l i n e the  consequently  the  states  severity  contents of  of  this  thesis.  1.  For a very severe  network, a d i s c o n t i n u o u s  d i s t u r b a n c e caused by  change i n the  form o f c o n t r o l i s r e q u i r e d to p r e v e n t  excessive  " 2 3 4 5 6 7 system upset.  T h i s c o n t r o l problem has  no u n i f i e d t e c h n i q u e times.  been i n v e s t i g a t e d ' ' ' ' ' , but  has been g i v e n to determine the optimum s w i t c h i n g  In Chapter 2, the problem of optimum network s w i t c h i n g i s 8  investigated  .  A u n i f i e d treatment  of optimum s w i t c h i n g i s p r e s e n t e d  by  c o n s i d e r i n g the s w i t c h i n g i n s t a n t s to be elements of a g e n e r a l i z e d c o n t r o l vector.  Dynamic o p t i m i z a t i o n i s then a p p l i e d to determine optimum  switching i n s t a n t s . 2.  Moderate d i s t u r b a n c e s  can be overcome by employing:  (a)  f a s t e r c o n t r o l of prime movers ; (b) h i g h speed e x c i t a t i o n system w i t h supplementary s i g n a l s f o r p r o v i d i n g s t r o n g damping of swings. continuous  type o f c o n t r o l a c t i o n can be  augmented by  The  above  discontinuous  control action. The  s o l u t i o n to the l i n e a r r e g u l a t o r problem w i t h  cost index i s w e l l known.  I t s p r a c t i c a l a p p l i c a t i o n to the c o n t r o l of  power systems, however, poses severe problems. o f h i g h o r d e r n o n - l i n e a r systems by and  a quadratic  These  are the  a l i n e a r system, the  modelling  computation,  t r a n s m i s s i o n of l a r g e q u a n t i t i e s of data between d i f f e r e n t p l a n t s .  Some form of suboptimal suboptimal somex^hat  control i s  therefore  c o n t r o l l e r s , s u c h as s p e c i f i c optimum the s e v e r i t y of the problem.  essential.  controllers>reduce  However, i t i s d i f f i c u l t  account f o r system i n t e r a c t i o n and n o n - l i n e a r i t i e s i n such without  r e q u i r i n g continuous  the p l a n t s . over  Significant  P r e s e n t l y known  to  controllers  communication o f l a r g e amounts of data between  improvement i n d e s i g n technique, and  the c o n v e n t i o n a l methods  has been a c h i e v e d by  system response  the a p p l i c a t i o n of  6'  the  l i n e a r r e g u l a t o r problem t o power systems w i t h some degrees, of s u b -  . , . 10,11,12 optimality In  Chapter 3, the p o s s i b i l i t y o f i m p l e m e n t i n g a s i m p l e c o n t r o l  for  a l a r g e s i g n a l model o f i n t e r c o n n e c t e d power p l a n t s i s examined v i a  the  concept of m u l t i - l e v e l c o n t r o l .  A t w o - l e v e l c o n t r o l i s proposed.  The f i r s t l e v e l c o n s i s t s of independent l i n e a r subsystems, Xtfhich have l o c a l feedback c o n t r o l l e r s .  The s e c o n d - l e v e l c o n t r o l l e r c o - o r d i n a t e s the sub13  systems by an i n t e r v e n t i o n open-loop 3.  control  M i n o r o r r o u t i n e d i s t u r b a n c e s causes s m a l l d e v i a t i o n s from the  f i x e d r e f e r e n c e s and a r e c o r r e c t e d by g o v e r n o r and/or v o l t a g e r e g u l a t o r controls.  The main problem i n t h i s s t a t e i s Load-Frequency  C o n t r o l (LFC)  problem*.  T h i s i s the problem o f r e g u l a t i n g the power output o f e l e c t r i c  g e n e r a t o r s w i t h i n a p r e s c r i b e d a r e a such t h a t each a r e a s a t i s f i e s i t s own demand. For  improved dynamic p e r f o r m a n c e , t h e l i n e a r r e g u l a t o r  solution  14 was adopted by E l g e r d  .  I n r e f e r e n c e 14, the s t a t e d e v i a t i o n s a r e  e x p r e s s e d i n terms o f t h e f i n a l s t a t e s , the s t a t e s the system i s supposed t o reach a f t e r a c e r t a i n demand i s a p p l i e d .  The f i n a l s t a t e s cannot be  known u n l e s s the demand i s known a p r i o r i w h i c h i s n o t the s i t u a t i o n i n practice.  A f e a s i b l e optimum c o n t r o l must i d e n t i f y the unknown demand. In  Chapter 4, two methods a r e s u g g e s t e d f o r i d e n t i f y i n g the demand.  The f i r s t method uses d i f f e r e n t i a l a p p r o x i m a t i o n and i s s u i t a b l e f o r s l o w l y changing demands.  I n the second method the system s t a t e s a r e augmented by  a demand e q u a t i o n , and a m i n i m a l o r d e r Luenberger o b s e r v e r i s u t i l i z e d . the  In  second method, i t i s assumed t h a t t h e t i e - l i n e and f r e q u e n c y d e v i a t i o n s  •k  See Appendix I f o r d e t a i l e d  definition.  are the^ only measurements a v a i l a b l e .  A modified  form of the r e g u l a t o r  problem s o l u t i o n ^  i s c o n s i d e r e d , which g i v e s , i n c o n j u n c t i o n w i t h  demand i d e n t i f i e r ,  a f e a s i b l e suboptimum c o n t r o l f o r the LFC  4. tie-line  A p r a c t i c a l s i t u a t i o n which must be  d e v i a t i o n may  the c o n t r o l l e r .  not be  these  is  s u i t a b l e f o r continuous  The  sampled-data r e g u l a t o r ^ , which  Because of the n o i s e p r e s e n t a suboptimal  filter.  i n the system,  The  filter  r e q u i r e d e t a i l e d a p r i o r i knowledge of n o i s e s t a t i s t i c s . an a d a p t i v e  •a s c a l a r g a i n . cost index.  Updating  The  performance index and does not  Chapte  systems t h a t have a communication l i n k i n the  the system s t a t e s are e s t i m a t e d by  essentially  the  In the l a s t p a r t of the t h e s i s ,  p o i n t s are i n v e s t i g a t e d .  feedback l o o p , i s c o n s i d e r e d .  is  considered i s that  In a d d i t i o n , f o r p r a c t i c a l systems, both p l a n t and measure-  5,  not  problem.  instantaneously a v a i l a b l e f o r u t i l i z a t i o n i n  ment d e v i c e s are d i s t u r b e d by n o i s e . two  the  observer  and  The  does filter  i s based on a d a p t i v e l y changing  t h e g a i n .is based ,on-minimizing  an  instantaneous  cost index r e a l i z e s a t r a d e o f f between a d e t e r m i n i s t i c and  an e s t i m a t i o n e r r o r .  The  proposed scheme i s  r e q u i r e e x c e s s i v e computer memory or computation  time.  simple  2. 2.1  OPTIMUM NETWORK SWITCHING IN POWER SYSTEMS  Introduction Power system s t a b i l i t y can be improved by d i s c o n t i n u o u s changes i n 2  the e l e c t r i c power network .  The a c t i o n s t o be t a k e n  system t o e q u i l i b r i u m a f t e r a s e v e r e d i s t u r b a n c e disturbance, b r i e f o r prolonged. choose from.  Sometimes  i n order t o b r i n g the  depend on t h e n a t u r e o f t h e  there are d i f f e r e n t actions to  The c h o i c e o f a c t i o n n o t o n l y depends on t h e type o f d i s t u r -  bance, b u t a l s o on e c o n o m i c a l  and p r a c t i c a l c o n s i d e r a t i o n s .  C o n s i d e r , f o r example, a b r i e f d i s t u r b a n c e , l a s t i n g t y p i c a l l y a f r a c t i o n o f a second, such as a s h o r t c i r c u i t c l e a r e d i n n o r m a l t i m e . a f a u l t near a generating p l a n t a c c e l e r a t e s the generators.  Such  This disturbance  can be c o u n t e r a c t e d by a s h o r t a p p l i c a t i o n of a shunt b r a k i n g r e s i s t o r 3  located a t the generating plant . Other d i s t u r b a n c e s , f o r example t h e l o s s o f a l a r g e l o a d , produce a sustained non-equilibrium condition.  The c o n t r o l a c t i o n s h o u l d  likewise  be s u s t a i n e d , a n d i n t h i s event i t i s l o g i c a l t o d i s c o n n e c t an amount o f generating capacity equal t o the l o s t load. I n t h e l a t t e r case, p r o l o n g e d  application of a braking r e s i s t o r  would be e f f e c t i v e , b u t i t would be u n e c o n o m i c a l t o p r o v i d e r e s i s t o r s o f t h e required rating.  I n t h e former  case, d r o p p i n g o f g e n e r a t i o n would be i n a p p r o -  p r i a t e u n l e s s , i t c o u l d be r e s t o r e d t o s e r v i c e r a p i d l y and w i t h a c c u r a t e t i m i n g . Kimbark has d i s c u s s e d t h e p o s s i b i l i t i e s o f i m p r o v i n g power system 2 s t a b i l i t y by c o n t r o l o f d i s c o n t i n u o u s changes i n t h e e l e c t r i c a l network . I n h i s paper he does n o t p r e s e n t a s y s t e m a t i c method f o r e v a l u a t i n g t h e switching times.  A p r a c t i c a l implementation  has been r e p o r t e d by t h e B r i t i s h  9  3 Power A u t h o r i t y .  C o l u m b i a H y d r o and time  of i s o l a t i n g  t i m e was  the  faulted  d e t e r m i n e d by Transient  line  digital  A braking resistor  and  switched  s i m u l a t i o n and  c o n t r o l by  i s switched  o f f at a l a t e r  numerical  on  time.  at  This  experimentation.  u s i n g n e t w o r k p a r a m e t e r s and  by  series  A- 5 6 7 c a p a c i t o r s w i t c h i n g has The  methods d i s c u s s e d  switching  i n references  f u n c t i o n s from energy  appear s u i t a b l e multi-machine by  recently received considerable attention ' ' ' .  optimal  f o r the  flow  systems.  References  c o n t r o l theory.  The  switching with It  is used.  analog  t o any  critical well  as p r a c t i c a l  of Liapunov 2.2  o f the p o s s i b i l i t i e s  s w i t c h i n g time,  f o r o b t a i n i n g the  than  for  solution.  case of g e n e r a l  minimum t i m e  i s not  Chapter to present The  stability  discussed.  method i s g e n e r a l  d i s c u s s e d by  Kimbark.  digital  network  a systematic and  method is  appli-  In determining  the method appears t o o f f e r b o t h  advantages over  not  No  computer x j i t h subsequent  Furthermore, the  f o r e v a l u a t i n g optimum s w i t c h i n g t i m e s . cable  u s e d i s minimum t i m e .  method i s g i v e n  i s the purpose of t h i s  T h e s e m e t h o d s do  7 treat series capacitor switching  cost index  cost i n d i c e s other  d e r i v a t i o n of  or h i g h e r o r d e r machine models or  6 and  t r i a l - a n d - e r r o r a p p r o a c h u s i n g an  computer refinement  5 a r e b a s e d on  considerations.  case of t h i r d  systematic iterative numerical A  4 and  the  computational  a p p r o a c h e s b a s e d on  the  as  construction  functions  Problem Formulation A steepest  f o r Optimum S w i t c h i n g  descent  Times  method f o r s o l v i n g a combined c o n t i n u o u s  and  22 bang-bang optimum c o n t r o l p r o b l e m has b a s e d on  c o n s i d e r i n g the  parametric  c o n t r o l and  been p r o p o s e d by  Vachino  .  This  is  s w i t c h i n g i n s t a n t s f o r t h e b a n g - b a n g c o n t r o l as  augmenting i t w i t h  a continuous  c o n t r o l to form  a  a  generalized control vector. be a p p l i c a b l e  A simplified  t o t h e p r o b l e m o f optimum s w i t c h i n g  as f o r m u l a t e d b y K i m b a r k . the  differential  relatively  version of this  T h i s i s a consequence  m e t h o d seems t o  t i m e s i n power  of the s t r u c t u r a l  e q u a t i o n s d e s c r i b i n g power s y s t e m dynamics  s m a l l a n d known number The d i f f e r e n t i a l  of sx^itching  systems, form o f  as w e l l  as t h e  instants.  equations describing  the s t a t e of the system  have the form  x = f(x, a )  (2.1)  k  w h e r e x i s an n s t a t e v e c t o r a n d  i s a system parameter  vector which i s  constant  f o r t ^ ^ <: T < t ^ a n d w h i c h c a n c h a n g e a t a n y o f N  instants  t ^ , t^,  ... t ^ . The p r o b l e m t o b e c o n s i d e r e d i s t o c h o o s e  s w i t c h i n g i n s t a n t s so t h a t a c o s t  x  f  i s minimized at a f i n a l  In  the  index  <j> = <j>(x ),  The i n i t i a l  switching  time t ^ = t ^  +  f  = x(t )  (2.2)  f  ^ d e f i n e d by a g i v e n s t o p p i n g c o n d i t i o n .  s t a t e , x ( t ) = x ^ , i s c o n s i d e r e d known. Q  formulating a solution  to the o p t i m i z a t i o n problem i t i s con-  venient to consider the s e t of d i f f e r e n t i a l one d i f f e r e n t i a l  e q u a t i o n s g i v e n by  (2.1) as  equation of the form x = F ( x , a., v ) ,  x(t ) = x o o  (2.3)  where A  F(x, In  a, v) =  N  +  E k=l  1  f ( x ,a ) [ h ( t - t _ ) k  k  1  - h(t-t )] k  (2.4), h ( t ) i s the u n i t step f u n c t i o n , a i s a composite parameter  formed  (2.4) vector  f r o m t h e a, and k v ' = (t  v  t  2  , ... t ) N  (2.5)  is  considered  a generalized control vector  The o p t i m i z a t i o n p r o b l e m d e f i n e d b y  (2.2),  (prime  denotes t r a n s p o s i t i o n ) .  (2.3),  (2.4) and (2.5) can be  23 formulated  as an o p t i m a l  of s o l u t i o n of o p t i m a l f u n c t i o n space. and  c o n t r o l problem  The s i m p l e s t n u m e r i c a l  c o n t r o l problems i s steepest  However, t h e p a r a m e t r i c  the d i s c o n t i n u i t i e s  .  method  descent i n c o n t r o l  form of t h e c o n t r o l v e c t o r (2.5)  i n ( 2 . 4 ) make i t n e c e s s a r y  to modify t h i s  method.  The r e q u i r e d m o d i f i c a t i o n i s o b t a i n e d b y c o n s i d e r i n g t h e i n c r e m e n t a l c h a n g e i n s t a t e fix r e s u l t i n g control vector  (see (2.5)).  f r o m an i n c r e m e n t a l  change 6v i n t h e  L i n e a r i z a t i o n o f (2.3) y i e l d s :  fix = F 6 x + F 6 v , V x  0  fix  (2.6)  where  Hi 3x  Hi 8x_  F  x  (2.7)  k 3F  3F  n 8x,  To a p p l y be e x p r e s s e d  a steepest  descent method,the i n c r e m e n t a l  i n terms of the i n c r e m e n t a l  is  n  3x  c o n t r o l 5v.  c o s t ficj> m u s t  The d e s i r e d  expression  (2.8)  fi<p '= (Jr'fix^ = 4>'5v x f v  where _____  x  L  9  x  3x  l  9cj> V  In  }h  n J  (2.9)  _____  '"  ' '  8 t  N  ( 2 . 8 ) <j>__ i s t h e g r a d i e n t o f <j) w i t h r e s p e c t x  t o x a n d cp__ i s t h e g r a d i e n t 5  12  of <j) w i t h r e s p e c t  to v.  must be found f o r 6<j>. (see  In order  to f i n d (j> an  This expression  alternative  i s obtained  expression  by use o f t h e i d e n t i t y  (2.6)),  •_j(p'6x) = 6 x ' [ p + r p ] + p'F 6v  (2.10)  v  where p i s the c o s t a t e v e c t o r which i s d e f i n e d by e q u a t i n g  the c o e f f i c i e n t  of 6x i n (2.10) to z e r o : p = -F p  '  x  Integrating  (2.10) w i t h the i n i t i a l  (2.11)  c o n d i t i o n 6 X = 0 and choosing Q  Pf = -*_>_> for the terminal  (2.12)  c o n d i t i o n , i t i s seen t h a t  (2.13)  6(J) = <j>'6x_ = -p'6x_ = - J . H*<5v d t x f f f J t v o F  where H = p'F i s t h e Hamilt.onian.  Since  6v ..is' .constant , i t .follows .by  comparison o f (2.8) and (2.13) t h a t <j) = - | v JX  H dt v  o  (2.14)  The e v a l u a t i o n o f H__ i s s t r a i g h t f o r a a r d .  From  (2.4) i t i s seen  that f ~  = P'|f- = P'[f(x,  a )-f(x, a  -h(t-t )  = ~A(t-t )  k  k + 1  )]A(t-t ) k  (2.15)  since i  k  k  where A ( t ) i s the u n i t impulse f u n c t i o n . yields  S u b s t i t u t i n g (2.15) i n t o  (2.16) (2.14)  P ' C V I f ^ t , ) - f ( >] 2  tl  (2.17)  v  P  , ( t  N  ) [ f  N V (  "  f  N+l V (  ]  13  where, f o r n o t a t l o n a l convenience,  f^(t^)  = f(x(t^),  (2.8) t h a t t h e s t e e p e s t  ct ^) . k +  I t f o l l o w s from  = f(x(t^),  a^) ,  ^^4.^^^) descent  adjust-  ment i s g i v e n b y : <5v = -£<j>  (2.18)  v  where i ' > 0 i s a s t e p  2.3  size.  Algorithm A s y s t e m a t i c method f o r e v a l u a t i n g v i s g i v e n by t h e f o l l o w i n g  algorithm. 1. X  q  is  i n t e g r a t e (2.3) forward  2.4  from t  Q  until  the stopping  condition  Integrate  (2.11) backward i n time  using  (2.12) t o i n i t i a l i z e  costate vector. 3.  Use (2.18) t o u p d a t e t h e c o n t r o l ( v = v + 6 v ) .  4.  T e r m i n a t e t h e c o m p u t a t i o n when  ||<j> || i s s u f f i c i e n t l y  illustrate  the previous I.  constant  i t s effectiveness, the general  technique  developed  s e c t i o n i s a p p l i e d t o the f o l l o w i n g problems:  To f i n d  the c r i t i c a l s w i t c h i n g time  (a) One m a c h i n e i n f i n i t e b u s s y s t e m voltage behind t r a n s i e n t reactance).  for: (second  order  (b) One m a c h i n e i n f i n i t e b u s s y s t e m ( t h i r d o r d e r t a k i n g account of f i e l d f l u x l i n k a g e changes). (c)  for  small.  Applications To  in  i n time  state  s a t i s f i e d , which defines t ^ . 2.  the  S t a r t i n g w i t h a n o m i n a l c o n t r o l v and a g i v e n i n i t i a l  Multi-machine  system  I I . To f i n d t h e o p t i m u m t i m e ( a ) , (b) and ( c ) .  (second  model;  model;  o r d e r model f o r each machine)  of switching o f f a braking  resistor  2.4.1  C r i t i c a l S w i t c h i n g Time 20 The system e q u a t i o n s f o r the m u l t i - m a c h i n e case a r e g i v e n by: 6 . = a), l l <*>. = ^ T - [ .-D.w.-P . ] , l M. mi i i e i l  i =1,2,  P  where N i s the number o f machines. m  N ' m  M., P . and D. a r e t h e i n e r t i a I mi l  c o n s t a n t , the m e c h a n i c a l i n p u t power and the damping i t h machine,  (2.19)  coefficient  respectively.  For a s i n g l e machine the e l e c t r i c a l output power P damping c o e f f i c i e n t  ^F = C. ( — ) e 1 x o  ^F + (C. cos6 +C s i n 6 ) ( — ) 2 3 x o 0  + C. sin2<S +C  4 C  D =  r  X  11 1  + C  1  s i n 6 +C, c o s 6 . 2  C  5 12  2  C  +  11  _ C  •  transient  (2.20)  2  6  -  1?  cos26  2  (2.21)  = a f l u x linkage p r o p o r t i o n a l to f i e l d  the d i r e c t - a x i s  and t h e  i n (2.19) a r e d e f i n e d by,  2  P  where  f o r the  open c i r c u i t  f l u x l i n k a g e , and x i s O  time c o n s t a n t .  In t h e case of a t h i r d o r d e r model the r a t e o f change o f  is  g i v e n by: *  p  = C + C ,sin<S+C 7  g  9  cos6+C  1()  ^  (2.22)  F  where: C  7  *  " loV " 9 C  0)  C  c o s 6  (°>-  C 8  sin6(0)  (2.23)  In the case o f a second o r d e r model, ty^, i n (2.20) i s r e p l a c e d by ii)  = ^ ( O ) = constant.  The parameters C , C ,  of the parameter v e c t o r a ( s e e  0  C. _ a r e t h e elements  (2.4)) and t h e i r v a l u e s depend on machine  parameters and network impedances (see Appendix I I ) .  I n t h e m u l t i - m a c h i n e case t h e q u a n t i t i e s i n (2.19) a r e g i v e n b y :  20  N A 2 P . =E7G..+ ei l i i  m  E  E.E.B.. s i n ( < 5 . - 6 . ) l j 13 l j  (2.24)  jri  D. = d. = c o n s t a n t i i where: short  E. /&. = i n t e r n a l p h a s o r v o l t a g e i _____ circuit  i = 1, 2, . ..,  N  (2.25)  m  o f t h e i t h m a c h i n e , G..+JB.. = — i j i j  t r a n s f e r admittance between t h e i t h and t h e j t h machine, and  G.. i s t h e l o a d c o n d u c t a n c e a t t h e i t h m a c h i n e b u s ( G i i i s n e g l e c t e d i n ii — (2.24)). The  r o t o r angle  trical  phase angle  angles  a r e measured w i t h  speed  o f each machine i s f i x e d w i t h  of the voltage behind respect  respect  to the e l e c -  the transient reactance.  The  t o a common a x i s r o t a t i n g a t s y n c h r o n o u s  ( t h e i n f i n i t e b u s i n t h e one m a c h i n e c a s e ) .  I n the multi-machine  c a s e t h e p a r a m e t e r s G.. a n d B.. a r e t h e e l e m e n t s o f t h e a v e c t o r . ii 13 Figure  (2.1) i l l u s t r a t e s  synchronous generator  s i t u a t i o n s (a) and ( b ) .  A salient  pole  i s c o n n e c t e d t o a n i n f i n i t e b u s b y two t r a n s m i s s i o n  lines. The m a c h i n e s u p p l i e s a c o m p l e x p o w e r P + j Q a t a t e r m i n a l v o l t a g e V . T h e i n f i n i t e b u s h a s a f i x e d v o l t a g e V . A s u d d e n t h r e e p h a s e symmet . 0 trical  short  t = 0.  t o ground i s c o n s i d e r e d  The f a u l t e d l i n e  cycles fault  circuit  at time t . o  to occur  at p o s i t i o n (x) at  s e c t i o n between A and B i s d i s c o n n e c t e d  The f a u l t e d l i n e  i s reconnected  after  8  after m cycles with the  cleared. Figure  stages:  (2.2) i l l u s t r a t e s  (a) f a u l t  t h e power a n g l e  on, (b) f a u l t e d l i n e  diagram f o r the three  disconnected,  (c) l i n e  restored  16  with  fault  cleared.  number o f c y c l e s , m, infinite  bus.  criterion  s w i t c h i n g time  An e q u i v a l e n t  i s evident  definition  can be g i v e n by t h e e q u a l  the c r i t i c a l  t u  (  f  t  )  =  -°  where t ^ i s t h e time o f t h e f i r s t  swing.  i s t ^ . The s w i t c h i n g t i m e b e c o m e s c r i t i c a l when to the conditions  the  (2.26)  0  w - w  subject  a t <S = <5(t ) .  = t , xjhen t h e c o n d i t i o n s c  s  index  switching time t  area  from F i g . ( 2 . 2 ) , t h a t the s w i t c h i n g time equals  s w i t c h i n g time,  are s a t i s f i e d ,  ( i n c y c l e s ) i s t h e maximum  f o r which the machine s t a y s i n synchronism w i t h t h e  (A.. = A.) w h i c h d e f i n e s It  critical  The c r i t i c a l  (2.26).  A possible  cost  t ^ i s a maximum  However, t h e t e r m i n a l c o n d i t i o n s on t h e  c o s t a t e a r e t h e n unknown a n d a more i n v o l v e d i t e r a t i v e  procedure i s required.  This  function cost  complication  can be a v o i d e d  by choosing  a penalty  b a s e d on (2.26) as a t a r g e t s e t . That i s , a c o s t i n d e x 4 = 0.5[W can be c h o s e n , where of  (2.27) a l l o w s  used.  and W  the simple  The s t o p p i n g  2 l W  of t h e form  (t )+W (P (t )-P (t )) ]  (2.27)  2  f  2  index  2  m  f  e  f  are p o s i t i v e weighting  algorithm given  factors.  i n the previous  c o n d i t i o n , which defines  t ^ , i s taken  The  choice  S e c t i o n t o be  t o be t h e i n s t a n t  o f t i m e when one o f t h e f o l l o w i n g c o n d i t i o n s i s s a t i s f i e d ; u)(t ) < 0  ,  P  f  f  m  ( 2  (t )-P ( t )> 0 r e r f  is  f o r various a„, d u r i n g Z  stages  a r e shown i n T a b l e  the fault  8  )  •  P r o b l e m s 1 ( a ) and (b) a r e d e f i n e d by ( 2 . 1 9 ) - ( 2 . 2 3 ) ; a  2  the value  of  ( 2 . 1 ) . The p r e - and p o s t - f a u l t v a l u e  i t i s a_ and w i t h r  the faulted l i n e  disconnected i t  GENERATOR  TRANSFORMER  t  x  Fig.  (2.1)  A  TRANSMISSION  LINES  3  W  One m a c h i n e i n f i n i t e b u s s y s t e m  (ic: defined by Al = A2)  Fig.  "  (2.2)  Power a n g l e  diagram  f  T a b l e (2.1) bl(a)(b)  << 1  tt (a)(b) 0. 226  l  C  -.258  2  C  4  C  5  C  S  C  7  C  8  C  c  g  ,0  C  C„  /M,  U2 «3  <*/  *-2  0-244  0.359  0.345  -: 298  . 38* 10~  -.412  -.424  0  0. 519  0.90 7  0-528  0.921  0  -  0.113  0. 059  0-124  0  -.043  -.045  - .061  0  0.056  0057  0-082  0  1.258  1.268  • 1.268  1.268  1.268  0-138  0.134  0.215  0-188  0.  0.395  0-678  0-428  0- 704  0  -. 187  -.159  -.190  .046  -.028 0.034  -.156 0. 272 0-036  0. 556 -.090  -.397  • 388  0- 626  0  • 056  0.106  0  T a b l e (2.2) (Prob. 1(a) )  I  ITERATION  ts SECONDS 0.1157  OJ(tf) -.29  -38  23  0.35  -.25  9.5  0-48  0. 200  . 18  0.2167  • .06  -.21  4.8  0-52  0.233  .09  .15  3.4  0 58  0. 250  .03  .69  0.71  .08  *PA=Pm-Pe(< ) F  T a b l e (2.3) (Prob.  I  ITERATION  1(b))  ts  OJ (t )  .1167  • 087  .27  8.2  0-<0  .133  • 045  .24  5.9  04 3  • 15  .07  .19  4.3  0 48  • 1567  .02  f  -.14  1.9  3  19  i s a^. P  m  Equation  = .735  a = a_,.  (2.19) i s i n i t i a l i z e d  (pu), i|v = 9.48 r  at t = 0 w i t h M = .0212  (pu), 6(0) = .9414  (rad), w(0)  = 0(rad/sec) and  (The s u b s c r i p t '1' i s omitted for the one machine case),  r  Problem 1(a) i s solved by i n t e g r a t i n g (2.19) with = ili-(0) from t = 0 to t = t . . F o x„(t ) = w(t ) = 4.14 Z o o  2  = 200 i s made.  (2.2).  =  constant  This y i e l d s x (t ) = <$(t ) = 1.183 l o o  (rad/sec)  n  (rad),  (a fourth-order Runge Kutta Subroutine i s  used w i t h a step s i z e of 1 cycle = .01667 s e c ) . W  (pu power sec ),  In (2.27) the choice W^  =  The r e s u l t s f o r the proposed method are given i n Table  Note the steady decrease i n the cost index.  A c r i t i c a l switching  time of 15 cycles i s found a f t e r f i v e i t e r a t i o n s . Problem 1(b) i s solved by using (2.22) w i t h the same values at t = 0, defined i n 1(a).  I n t e g r a t i o n during the f a u l t stage y i e l d s x ^ ( t _ ) =  1.183  r a d / s e c , and x . ( t ) = i L ( t ) = 9.195  rad, x . ( t ) = 4.14 2  O  j  r e s u l t s -are shown i n Table (2.3).  O  r  pu.  The  O  The c r i t i c a l •switching •time-is 1-0 c y c l e s .  Problem 1(c) i s the multi-machine problem.  When a f a u l t occursj  the machine having the greater r a t i o of i n i t i a l a c c e l e r a t i n g power to momentum constant would be expected to accelerate f a s t e r than the other machines.  In reference 20, the c r i t i c a l switching time i s defined w i t h  respect to the f a s t e s t machine and i t has been shown that t h i s d e f i n i t i o n i s a u s e f u l one.  Consequently, i t i s p o s s i b l e to use (2.27) f o r the m u l t i -  machine problem, choosing the v a r i a b l e s to be those of the f a s t e s t machine. The f a u l t i s taken to be a sudden three phase symmetric short c i r c u i t to ground at any one of the t i e l i n e s between two machines. assumed permanent.  The c i r c u i t breakers open t  to i s o l a t e the f a u l t e d s e c t i o n .  The f a u l t i s  seconds from f a u l t occurence  A four-machine system w i t h data taken from  20  r e f e r e n c e 20 i s c o n s i d e r e d . (2.5) and and  (2.6).  the reduced The  obtained  The  one  The  r e l e v a n t d a t a are shown i n T a b l e s  l i n e diagram of the system i s shown i n F i g .  f a s t e s t machine i n t h i s  case i s machine number 3.  from the proposed method are shown i n T a b l e radians  r e s u l t u s i n g a Liapunov  (2.7).  The  The  results  critical  (time u n i t used, T , i n r a d i a n s f o r comparison  w i t h r e f e r e n c e 20, T = 2 r r f t . ) .  T h i s i s i n agreement w i t h the  function given i n reference  S i n c e the methods are c o m p l e t e l y the two  (2.3)  system i n F i g . ( 2 . 4 ) .  s w i t c h i n g time i s 155  of  (2.4),  approaches i s not p o s s i b l e .  The  numerical  20.  different, a direct  comparison  advantage of the proposed  approach i s t h a t i t i s not a f f e c t e d by the o r d e r of the model.  Governor  and v o l t a g e r e g u l a t o r e f f e c t s  approach  can be  based on c o n s t r u c t i n g a Liapunov  included, i f desired.  The  f u n c t i o n becomes i m p r a c t i c a l f o r system  models g r e a t e r than second o r d e r .  2.4.2  Optimum S w i t c h i n g Time of a B r a k i n g R e s i s t o r The b r a k i n g r e s i s t o r i s connected  instant nected  t  at which the c i r c u i t b r e a k e r  after a period t^.  The  by economic c o n s i d e r a t i o n s and  to the g e n e r a t o r bus  opens.  The  at  the  resistor i s discon-  s i z e of the r e s i s t o r i s u s u a l l y  determined  the power demand under normal o p e r a t i n g  2 3 conditions  ' .  For a g i v e n r e s i s t o r and  c o s t index t h e r e i s an optimum  time of s w i t c h i n g t ^ . For the multi-machine system, the r e s i s t o r i s a p p l i e d to the machine.  A s u i t a b l e c o s t index f o r d e f i n i n g an o p t i m a l t, i s g i v e n  fastest  by:  (2.29) o  o  Table (2.4) (DATA GEN.  MVA 7APACITY  AND  INITIAL  CONDITION)  "pu  D  6 p u  (to) rorf  ™  P  pu  1  100  75350  1.0  1.0004  • 0013  .332  2  15  1130  12.0  1.0410  .1030  • 100  3  40  2260  2.5  1.1900  .1970  • 300  4  30  1508  6.0  1.0710  .0772  • 200  22  Table (2.5) (DURING  1 1  FAULT  2  -3.582  0-54 6  2  0.546  3  00  4  0-303  ADMITTANCES)  3 .  -.871 0.0  4  0-0  0.303  1  1.456  0.0  0-062  2  0-027  0-0  3  0-0  1.216  4  0-22  1  •864  -2.0  0.062  -  0-0  Table (2.6)  \  FAULT  (AFTER 1  2  ADMITTANCES)  3  4  7 -2.310  •664  -656  .751  2  •664  •880  • 121  • 062  3  • 656  • 121  4  • 751  • 062  -.868  • 062  • 062  • 029  '  -•984  3  .104  4  .225  Table (2.7) (Prob ITER.  Is  (rad)  1(c)) 100 GJ  A  0  -.84  70-8  -.81  64.8  P  1  130  2  135  3  140  - . 11  - .76  58.0  4  145  -.12  -.68  47.0  5  155  -•045  -.39  15.4  -.046  The  cost index  ( 2 . 2 9 ) i s a m e a s u r e o f t h e mean s q u a r e  angle deviation. seen  Augmenting  (2.1) by x ^ ^ = f +  n+1  t h a t t h e c o s t i n d e x i s d> = J = x , . ( t ) . n+1 f  _»  x n  frequency  +i^  t 0  -'  ^» ^  =  m e n t e d b y p ., a n d t h e H a m i l t o n i a n b e c o m e s H = p ' F + p f ... n+l n+1 n+1 from  t  T h e c o s t a t e p h a s t o b e aug-  r  J  and r o t o r  M  r  I t follows  (2.11) and (2.12) t h a t N  P - " x " " S f ^PthCt-t^-hCt-t^] + f H  k=l  p(t ) - 0 f  (2.30) n+1 The  3x . n+1  r  L  a l g o r i t h m remains  n+l  v  f  3x ,. n+1  o t h e r w i s e unchanged.  I n (2.29) t h e c h o i c e  = 100,  = 10 i s made. Consider problem  TI(a).  The b r a k i n g r e s i s t o r h a s a v a l u e o f 5.55  (pu), equal t o the l o c a l load r e s i s t a n c e braking resistor i s applied at t = t t o r e c l o s e a f t e r 12 c y c l e s . are given i n Table cycles  (2.1).  i n F i g . (2.1)).  The  and t h e c i r c u i t b r e a k e r i s assumed  The v a l u e s o f t h e a v e c t o r f o r v a r i o u s  stages  The r e s u l t s o b t a i n e d b y t h e a l g o r i t h m f o r t ^ = 90  ( f r o m t ) a r e shown i n T a b l e  (a) no b r a k i n g r e s i s t o r ,  (dotted lines  (2.8).  The s w i n g  curves f o r t h e cases  (b) a b r a k i n g r e s i s t o r a p p l i e d f o r t h e optimum  time  i n t e r v a l o f 1 8 c y c l e s , a r e shown i n F i g . ( 2 . 5 ) . In problem 9 cycles. is  The r e s u l t s  the c i r c u i t breaker  are given i n Table  i s assumed t o r e c l o s e a f t e r  ( 2 . 9 ) . The optimum t i m e  interval  18 c y c l e s . In problem  is  11(b)  11(c)  t h e b r a k i n g r e s i s t o r h a s a v a l u e o f 0.2 ( p u ) a n d  a p p l i e d t o machine no. 3 a t t h e i n s t a n t  (155 r a d i a n s f r o m t ) . o  t  when t h e c i r c u i t b r e a k e r  The r e s u l t s o b t a i n e d from t h e p r o p o s e d  opens  algorithm-  24  -  Table (2.8) (Prob  E(a)  )  0  ITER. 1  .0657  248.7  2  •0883  24 7.9  •1  3  0 (no BR- )  V  24 7.6 =373.3  3  2  1•  \  \  /  \  • .  /  -2  \ s  •  •  Fig.  N  , ^ 50  <^oy  30 -1  /  (2  .5)  - (ProbH(a)  )  Table (2.9) ITER.  t (sec)  0  1  • 0667  233-7  2 -  •0833  231-5  b  3  •1  229-8  4  .1167  228-6  5  • 133  227.9  6  .15  227.7  <f) (no BR.)  =354-1  r/Mf  -<720 (cycles)  are  given  (2.10).  i n Table  swing curves f o r the cases switched  200  off after  The o p t i m u m t i m e i n t e r v a l ( a ) no b r a k i n g  r a d i a n s , a r e shown i n F i g .  Note t h a t the use of b r a k i n g during  the f i r s t  reapply in  t h e subsequent  braking  h a l f second  the braking  resistor  swings.  the  first  governor  neglected,  swing.  cycles Fig.  and d i s c o n n e c t  r a t i n g would  swings  regulator controls.  were t o t a l l y  (30  resistor  be  80  resistor  damps o u t t h e f i r s t  i t again  swing  I t i s p o s s i b l e to f o r f u r t h e r damping  be u n e c o n o m i c a l  since a  required.  In the previous  I t i s the purpose  The  (2.6).  c a n b e damped o u t b y u s i n g  and t h e r e f o r e t h e r e  radians.  (b) b r a k i n g  (2.5)).  T h i s , however would  r e s i s t o r of higher The s u b s e q u e n t  voltage  resistor,  is  examples,  governor  those c o n t r o l s  i s no d a m p i n g e f f e c t  of next Chapter  and/or  after  to investigate the  c o n t r o l f o r l a r g e s i g n a l model of i n t e r c o n n e c t e d  power p l a n t s .  26  Table (2.10) (Prob. ITER  t  b  (rad)  11(c))  W~ 0 2  1  30  462  2  50  430  3  80  427  10 0(no B.R)= 643.2  •  3.  3.1  TWO-LEVEL CONTROL OF  Introduction Severe d i s t u r b a n c e s ,  network, can be Chapter 2. should voltage  be  counteracted  augmented by  by  sudden changes i n the  discontinuous  regulator  continuous or modulated  c o n t r o l and  representing  choosing a quadratic  can be  caused by  electrical  c o n t r o l as e x p l a i n e d  However, f o r optimum system performance, d i s c o n t i n u o u s  By and  INTERCONNECTED POWER PLANTS  formulated  (b)  c o n t r o l s namely  in controls  (a)  governor c o n t r o l .  the dynamics of a power system by  a l i n e a r model  c o s t index, the above c o n t r o l problems  as the w e l l known i n f i n i t e - t i m e l i n e a r s t a t e  (a) and  (b)  regulator  24 problem of o p t i m a l  c o n t r o l theory  .  The  state regulator  c o n t r o l problem  o b j e c t i v e i s to c o n t r o l the system so t h a t the s y s t e m ' s t a t e s are kept i n some sense.  The  s o l u t i o n of t h i s problem l e a d s  which i s a l i n e a r f u n c t i o n of the  s t a t e s of the  small  to an o p t i m a l c o n t r o l l e r  system.  A p p l i c a t i o n of the s o l u t i o n of the l i n e a r s t a t e r e g u l a t o r  problem  to the optimum c o n t r o l of machine e x c i t a t i o n i n a one-machine i n f i n i t e system i s g i v e n  i n reference  11.  l i n e a r low-order model and was representation c o n t r o l was  f o r one  The  optimum c o n t r o l was  t e s t e d on a n o n - l i n e a r  machine and  high  m u l t i m a c h i n e systems.  found e f f e c t i v e i n damping o s c i l l a t i o n s , no  f i n d the o p t i m a l The  derived order  from a model  Even though  attempt was  c o n t r o l f o r a l a r g e s i g n a l model of i n t e r c o n n e c t e d  problem of governor c o n t r o l of the prime mover f o r a  machine i n f i n i t e bus  system can be  r e g u l a t o r problem s o l u t i o n has C o n t r o l problem  14  , and was  treated i n a s i m i l a r manner^.  bus  this  made to machines. oneThe  a l s o been suggested f o r the Load-Frequency  a p p l i e d to two  I n t e r c o n n e c t e d areas were c o n s i d e r e d t h a t each area has only one p l a n t .  .  interconnected  i n reference  14.  *  s i m i l a r power p l a n t s . Here, i t i s assumed  Because o f t h e c o u p l i n g  i n t h e model r e p r e s e n t i n g  c o n t r o l f o r each plant, i s a l i n e a r states  of the other plants.  associated  with  the states  That i s , each p l a n t  the plants,  c o m b i n a t i o n o f i t s ox^n s t a t e s  In reference  14,  local  controlled i n a suboptimal fashion.  very small  state information This  and t h e  f o r t h e example used, t h e g a i n  o f t h e o t h e r p l a n t Was  uses o n l y  t h e optimum  a n d was  neglected.  and c o n s e q u e n t l y i s  s u b o p t i m a l s o l u t i o n may b e  adequate f o r s m a l l - s i g n a l model as i n t h e example used, b u t i t cannot be a d o p t e d as a g e n e r a l  policy.  This  f a c t i s shown i n t h e p r e s e n t  s t u d y o f two  25 26 t y p i c a l interconnected l o c a l suboptimal  that  '  .  I t was f o u n d t h a t  Consequently, coupling  cannot always be n e g l e c t e d  each p l a n t have i n f o r m a t i o n  between p l a n t s  and s y s t e m n o n -  and a good suboptimum c o n t r o l  a v a i l a b l e about t h e s t a t e s  continuous communication r e q u i r e d purpose of t h i s  poxjer p l a n t s .  be assumed t h a t  can  be c o n s i d e r e d Eor  by the  a s e t of f i r s t  regulator  Xtfithout much  large  cost o f  Chapter i s t o examine t h e p o s s i b i l i t y  c o n t r o l f o r a l a r g e - s i g n a l model o f  c o n s t a n t by t h e v o l t a g e  plants.  between t h e p l a n t s .  implementing a simpler Itwill  requires  of other  I m p l e m e n t a t i o n o f s u c h c o n t r o l w o u l d b e e x p e n s i v e due t o t h e h i g h  The  a  c o n t r o l b a s e d on a l i n e a r i z e d s y s t e m model c a n r e s u l t i n  system i n s t a b i l i t y . linearities  steam and h y d r o p l a n t s  f  interconnected  the generator voltages  controls.  D  are held  E x c i t a t i o n c o n t r o l , however,  difficulty.  deviations,  t h e dynamics o f t h e s y s t e m can be  order non-linear  differential  equations.  By  inspecting  s y s t e m dynamics i t i s f o u n d t o b e l o n g t o a c l a s s o f n o n - l i n e a r  f o r which a m u l t i - l e v e l s u b o p t i m a l c o n t r o l can be developed.  represented  systems  The c o n c e p t  27 of m u l t i - l e v e l c o n t r o l  i s a t t r a c t i v e since  c o m p l e x s y s t e m s s u c h as p o w e r  systems.  i t appears a p p l i c a b l e  t o many  29  A two-level  govenor  first  level  local  feedback c o n t r o l l e r s .  subsystems is  c o n t r o l o f a power system i s p r o p o s e d .  c o n s i s t s of independent l i n e a r  subsystems  The s e c o n d - l e v e l  b y an i n t e r v e n t i o n o p e n - l o o p  ( p l a n t s ) , which  c o n t r o l l e r co-ordinates  control.  and n o n - l i n e a r i t i e s a t t h e l o c a l For  ease i n r e a l - t i m e  on-line  have the  The i n t e r v e n t i o n c o n t r o l  used t o compensate f o r a d e c r e a s e i n system performance  interaction  The  due t o n e g l e c t i n g  level.  implementation,a suboptimal s o l u t i o n  t o t h e optimum i n t e r v e n t i o n c o n t r o l i s d e t e r m i n e d as a f u n c t i o n o f  initial  conditions  a  and r e a l  comparatively  3.2  small  time.  The s c h e m e i s s i m p l e ,  f a s t and r e q u i r e s  amount o f c o m p u t e r memory.  Large S i g n a l Model f o r Interconnected  Power  Plants 1A 25 26  A block is  f o r the i t h plant  f o r t h e steam  1• =  x  X  A  2i A  X  3i  f o r a steam  shown i n F i g . ( 3 . 1 ) a n d F i g . ( 3 . 2 ) r e s p e c t i v e l y .  and c o n t r o l s  l l  diagram r e p r e s e n t a t i o n  A6. Aw,l I  plant  and 5 s t a t e s  are defined  and a h y d r o p l a n t The s t a t e  as f o l l o x ^ s :  f o r the hydro  (there  variables are 4  plant)  AP . (assuming - t h a t  the time constant  of the generator  i s n e g l i g i b l e i n c o m p a r i s o n t o g o v e r n o r and t u r b i n e also 4i  x ^ X  5i  states  !  i n pu power  X  '  t h e two p l a n t s  a r e assumed o f e q u a l c a p a c i t y ) .  ^ ^^gi  deviation  = Ag^  d e v i a t i o n i n g a t e p o s i t i o n i n pu power  ^ ^^gi  deviation  u^ = A P ^ c  speed  i  i  n  n  time  g o v e r n o r p o s i t i o n i n pu power  (hydro  g o v e r n o r p o s i t i o n i n pu power  changer p o s i t i o n i n pu power  (steam  plant)  plant)  (hydro  plant)  constants,  Fig.  (3.1) Steam  7  1 X  5i  STiitl  GOVERNOR  x  4i  plant  block  diagram  -DfS + l •5D/S+7  *2i GENERATOR  TURBINE  Tjj sin(<fj.•<fj) 1  T sin (tT/ -<(n) m  Fig.  (3.2)  Hydro  plant  block  diagram  31  A.P ^  = d e v i a t i o n i n t i e - l i n e .flows  P ^_, = s c h e d u l e d t i e - l i n e dynamics of  the  hydro p l a n t  steam p l a n t because of Water i n e r t i a . open, the The  turbine  extra block The  power  power b e t w e e n the  tg  The  i n pu  i t h and  j t h plant.  are d i f f e r e n t  from that of  I n t h e h y d r o p l a n t , when t h e  t o r q u e t e n d s t o d e c r e a s e m o m e n t a r i l y and  in Fig.  (3.2)  represents  tie-line  flows  are  P  ti  . =  this  given ,  E  then  the gates  increases.  situation.  by  P. . . t i j  (3.1)  where  «... p  tij  = Iv- I | v . | | Y . . | s i n ( 6 . - 6 . ) = T.. i  J  i j  i  j  s i n 6..  IJ  .  IJ  (3.2) V  '  where  M'jIIV is  assumed c o n s t a n t ,  where  6.. ij and  where v  = Y.. ij  (line AP  | |  e  i s the  ±  J  J  =  i s the  line  5  +  P°. ti  =  voltage  (3.4) at the  The  expression  a d e v i a t i o n A6.. i j  considering  P°. ti  (6.-6.) 1 J i t h p l a n t bus  admittance between the  resistance i s neglected).  . i s d e r i v e d by tx  (3-3)  AP„. = 2 " t i j  i t h and —  f o r the  the  bar  pu.  i t h plant —  t i e line  from the nominal  T.. s i n ( 6 ? . + A 6 . . ) i j i j i j '  in  deviation 6?.: xj  (3.5)  since .  t  3  j^i  J>  .. = tsii  ,  £  3  $H  T, , s i n in  6?,  11  •  (3.6)  32  I t follows that AP, , = I ti .  T. ,.[sin6°, CcosA<S. ,~l)+cos<$?,sinA6. ..] xj ij 13 i j 13  (3.7).  3^i Equation (3.7) can be decomposed i n t o a l i n e a r l o c a l term which i s a'function of A6^ only, and non-linear coupling terms: AP.. = [.T. ti 3 L  T?. ]A6. + .1 [T?, (cosAfi, .-1) 13 c 1 3 13 s 13 L  + T?, (sinA6.. - A6,)] 13 c 13 1  (3.8)  where I..  - 1.. cos  T?  4  X3S  0 . . ,  13'  13  X3C  = T.. s i n 6°., and X3  A6. . = A6 X3  X3'  - A6.. i  3  No unique method e x i s t s f o r decomposing  the system i n t o l i n e a r  subsystems f o r which a feedback c o n t r o l i s used and then accounting f o r subsystem i n t e r a c t i o n by an open-loop i n t e r v e n t i o n c o n t r o l .  ^  o r  t  '  ie  i n t e r  connected plants discussed here, the decomposition (3.8) proved s u c c e s s f u l . This decomposition r e s u l t s i n the l o c a l voltage angle d e v i a t i o n being used i n a l o c a l feedback c o n t r o l . 3.3  System Dynamics In State V a r i a b l e Form The dynamics of each plant can be represented  i n the form:  X-. = A.X. + B.U. + T.(X, ,X_, ... O x x x x x x 1 2 N  (3.9)  where X. x  i s a vector of s t a t e V a r i a b l e s of dimension n. ( f o r the model x used,n^=4 and 5 f o r steam and hydro p l a n t s , r e s p e c t i v e l y )•  U. i  i s a v e c t o r of c o n t r o l v a r i a b l e s o f d i m e n s i o n m.(m.=l f o r . 1 1 b o t h steam and hydro p l a n t s ) *  F, 1  i s a v e c t o r f u n c t i o n of d i m e n s i o n n,, w h i c h c o n t a i n s l  the  c o u p l i n g and n o n - l i n e a r terms. A^jB^  a r e time i n v a r i a n t m a t r i c e s of a p p r o p r i a t e  The m a t r i c e s A., i plant  dimensions.  B; and the v e c t o r F. a r e d e f i n e d f o r each steam and l l  as f o l l o w s :  Steam P l a n t 0 [ Y. T?. ]/M. lie I J  A. l  i  1/M  0  -1/T  0  =  0  3H  k  -E./T . i gi  0  B!  -G_/M. i l  0  [0  0  0  1/T  1/T . t  ti "  1 / T  gi  .] gi  0 - £ M  E [T°. (cosA<S. .-1)+T°. ( s i n A 6 . . - A 6 . ) ] ljs IJ ijc' l j i ' J  i  0  hydro  Hydro P l a n t 0 [, T°, L  3  A. i  ±3  4  B^ = [0  -G./M. i i  ]/M. i  0  0  0  0  0  -E./T  0  0  1 F.= i  c  0  } i  CT  1/T  O i j s  1/M. l  -2/D  0  0  0  0  (2/D +2/T ) I  -1/T .  t ±  ti  -2/T . t  1/T. . ti -1/T  0  .  gi  ]  (cosA6 l)-rT  0  i r  l j c  (sinA6 -A6.) i j  3 3*±  0 0 0  F o r N i n t e r c o n n e c t e d p l a n t s t h e dynamics o f t h e system can be w r i t t e n as a c o m p o s i t e s t a t e v e c t o r e q u a t i o n o f t h e form X - AX + BU 4- F(X),- X ( t ) = X O  0  (3.10)  where t h e m a t r i c e s A and B a r e knox-m b l o c k d i a g o n a l t i m e i n v a r i a n t m a t r i c e s so t h a t , f o r example h  A =  o -... o A  (3.11)  2  i  i  0 The composite s t a t e v e c t o r X and t h e composite c o n t r o l v e c t o r U are composed o f t h e s t a t e v e c t o r s X_^ and t h e c o n t r o l v e c t o r u\, i = l , 2,...,N, r e s p e c t i v e l y so t h a t X' = (X^,X ,...,X^) and U' =(U^, 2  U ,...,U^). 2  35  ( t h e prime i s used t o denote t r a n s p o s e P«  =CF'  F'  of a v e c t o r or m a t r i x ) . . S i m i l a r l y  F')  I t s h o u l d be p o i n t e d out h e r e t h a t the c o n t r o l v e c t o r c o u l d be augmented to i n c l u d e e x c i t a t i o n c o n t r o l and i n t h i s case x-je W o u l d have a t w o - d i m e n s i o n a l c o n t r o l v e c t o r f o r each p l a n t .  The system dynamics  s t i l l be f o r m u l a t e d i n the g e n e r a l form (3.10).  can  E x c i t a t i o n c o n t r o l has been 12  shown t o g i v e damping e f f e c t on system o s c i l l a t i o n  .  However, t o c l e a r l y  i l l u s t r a t e the proposed c o n t r o l , o n l y governor c o n t r o l i s c o n s i d e r e d . The c o n t r o l problem i s the f o l l o w i n g .  I t i s required to f i n d a  c o n t r o l v e c t o r U such t h a t the d e v i a t i o n s i n the s t a t e s r e s u l t i n g from a system d i s t u r b a n c e i s m i n i m i z e d w i t h o u t e x c e s s i v e c o n t r o l e f f o r t . problem can be f o r m u l a t e d by i n t r o d u c i n g a c o s t  i J = -|  The  index  r f f c  j  (X'QX + U'RU)dt  where Q and R a r e b l o c k d i a g o n a l w e i g h t i n g m a t r i c e s  (3.12) (Q i s a p o s i t i v e o r  s e m i p o s i t i v e d e f i n i t e m a t r i x , R i s a p o s i t i v e d e f i n i t e m a t r i x ) , and U so t h a t J i s a minimum.  choosing  The s t a t e X i s s u b j e c t t o t h e d y n a m i c a l c o n s t r a i n t  (3.10). 3.4  Two-Level S t r u c t u r e Of the C o n t r o l Problem A t w o - l e v e l s t r u c t u r e i s chosen.  I t i s assumed t h a t the s t r u c t u r e  s p e c i f i e s a feedback c o n t r o l of the form U. = -C.X. i i i  (3.13)  f o r t h e l o c a l c o n t r o l l e r s , and t h a t the s e c o n d - l e v e l c o n t r o l l e r the subsystems  ( p l a n t s ) by an i n t e r v e n t i o n c o n t r o l V.  co-ordinates  The r e s u l t a n t c o m p o s i t e  control i s  r  u » -cx+  y  C3.i4>  where C i s a c o m p o s i t e b l o c k d i a g o n a l m a t r i x o f t h e form C 3 . l l ) composed from t h e C., i = l , 2, ... N. 1 The. f i r s t l e v e l of t h e c o n t r o l i s o b t a i n e d by n e g l e c t i n g t h e c o u p l i n g f u n c t i o n , F ( X ) , i n (3.10).  The problem then r e d u c e s t o t h e w e l l  24 known l i n e a r c o n t r o l problem w i t h q u a d r a t i c c o s t i n d e x can be o b t a i n e d - a s  .  The optimum c o n t r o l  a feedback c o n t r o l g i v e n by U = - R ~ B K X = -CX 1  where K i s t h e s o l u t i o n o f t h e m a t r i x R i c a t t i K = -KA - A'K + KSK - G),  where  (3.15)  ,  S = BR- B 1  equation  K(t ) = 0  (3.16) (3.17)  f  S i n c e A, B , Q and R a r e b l o c k d i a g o n a l m a t r i c e s , t h e s o l u t i o n f o r K i s a block diagonal matrix  (from •(3.16) and ( 3 . 1 7 ) ) .  The s o l u t i o n f o r each  b l o c k R\ i s o b t a i n e d by s o l v i n g K. = -K.A. - A!K.•+ K.S.K. - Q. , l i i i i I I l l S. = B . R T V 1 1 1 1  K . ( t _ ) = 0, i f  ^  _3)  i = 1,2,...N  F o r A, B , Oand R time i n v a r i a n t , l e t K s s o l u t i o n o f (3.16)  be t h e s t e a d y  state  0 = -K A - A'K + K SK - Q. s s s s  (3.19)  An e a s i l y implemented c o n t r o l r e s u l t s i f C = R B'K  (3.20)  _1  i s used as a s u b o p t i m a l if t^ =  00  s  l o c a l gain matrix.  The c h o i c e  (3.20) i s optimum  i n (3.12) and i t o f t e n g i v e s an e x c e l l e n t s u b o p t i m a l  control.  There i s an i n c r e a s e i n - t h e c o s t i n d e x a s s o c i a t e d w i t h t h e subsystem  37  i n t e r a c t i o n and system n o n - l i n e a r i t i e s w h i c h haye been, n e g l e c t e d i n t h e suboptimal  c h o i c e (3.20).  To a c c o u n t f o r t h e s e e f f e c t s a s e c o n d - l e v e l  c o n t r o l o f the form V = i s introduced.  tfVh  '(3.21)  To d e t e r m i n e the optimum h, the o r i g i n a l p r o b l e m must be  reformulated.  S u b s t i t u t i n g ( 3 . 1 4 ) , ( 3 . 2 0 ) , and (3.21) i n t o X = (A-SK )X + F + Sh, U =  (-K  s  (3.10) g i v e s (3.22)  X(0) = "X X+h)  (3.23)  The p r o b l e m i s to choose the i n t e r v e n t i o n c o n t r o l h so t h a t the c o s t  index  (3.12) i s a minimum s u b j e c t to t h e d y n a m i c a l c o n s t r a i n t ( 3 . 2 2 ) .  Hamiltonian  The  f o r t h i s problem i s H = p* [ ( A - S K j X + F + Sh] - - i  (3.24)  X'QX  (3.24) ~ (-K X+h)'S(-K X+h) 2 s S A p p l y i n g the n e c e s s a r y  c o n d i t i o n s of optimal c o n t r o l theory  p = -H.  X  = - (A-SK +F ) ' p + QX-K s  X  s  S(-K X+h) s  yields  , (3.25)  p(t ) = 0 f  0 = H  h  = S(p+K X-h)  (3.26)  g  X\fhere 9^ F  £  H 3 X  N  h ~  9H_ 3h(3.27)  9  X  l"'  9  X  N  9H Sh. N  38  Equations (3.22), (3.25) and (3.26) define a two-point boundary Value problem Xtfhose ^solution gives the optimum i n t e r v e n t i o n c o n t r o l h. Since the equations are n o n - l i n e a r , i t e r a t i v e methods are required to obtain the s o l u t i o n .  Consequently,  since on-line implementation i s d e s i r e d ,  optimal c o n t r o l i s not f e a s i b l e and a good suboptimal c o n t r o l p o l i c y must be determined. I f h = h = 0, then l o c a l c o n t r o l only i s applied. the s o l u t i o n of (3.22) and (3.25) f o r t h i s c o n t r o l .  Let X, p be  A fundamental r e s u l t  of optimal c o n t r o l theory i s that an incremental c o n t r o l 6h r e s u l t s i n an incremental cost index,given by f  fc  6J = -  | t  H  6h dt  h  '(3.28)  o  where H, t= s(p + K X) .  (3.29)  s  h  The i n t e r v e n t i o n c o n t r o l i s to be chosen so that system performance i s improved.  That i s , s o that <SJ, the incremental cost index, i s decreased.  Since S i s p o s i t i v e d e f i n i t e , i t i s seen from (3.28) and (3.29) that 6h = Up  + K X),  (3.30)  s where £> 0 i s a step s i z e parameter, accomplishes t h i s o b j e c t i v e . In c o n t r o l theory terminology (3.30) i s a steepest descent increment i n f u n c t i o n space. For computational reasons i t i s convenient to derive an equation for'  A -  -  q = p + K X • s  .  •" (3.31)  With the a i d o f (3.19), q = -(A-SK (the  (3.22),  (.3.25) and  + F )'q+F'K X+K  (3.31) i t i s -seen t h a t  F , 4CO  = K X(t )  C3.32)  f  terms w i t h an overbar: are e v a l u a t e d f o r the nominal  X, which i s the  s o l u t i o n of (3.22) f o r h - h = 0 ) . The  a l g o r i t h m f o r f i n d i n g h i s simple one.  i n t e g r a t e d i n the forward d i r e c t i o n ,  t a k i n g h = 0.  i n t e g r a t e d i n the backward d i r e c t i o n to f i n d q. then h = h + Sh = J£q.  The  optimum v a l u e i  found by a simple d i r e c t s e a r c h procedure. is  g i v e n i n S e c t i o n (3.6.2).  Equation Equation  The  of i which f  The  (3.32) i s then  intervention control i s minimizes  d e t a i l s of t h i s  However, o n - l i n e implementation  a l g o r i t h m f o r computing h ( t ) i s not p r a c t i c a l .  (3.22) i s  J can  be  algorithm,  of  this  The next s e c t i o n c o n s i d e r s  t h i s problem.  3.5  On-line Control The  Implementation  i n t e r v e n t i o n c o n t r o l h ( t ) i s a f u n c t i o n of time and  c o n d i t i o n s of a l l the subsystems (see (3.22) and f u n c t i o n i s knox<rn. to  the second  To generate h ( t ) , i n i t i a l  level.  (3.32)).  initial  Suppose t h a t t h i s  c o n d i t i o n s must be t r a n s m i t t e d  A f t e r g e n e r a t i n g h ( t ) , the s e c o n d - l e v e l c o o r d i n a t o r  c o n t i n u o u s l y t r a n s m i t s the components o f h ( t ) back to the subsystems. A more f e a s i b l e way  o f implementing  t h i s c o n t r o l i s to t r a n s m i t  a minimum amount of i n f o r m a t i o n between the l o c a l c o n t r o l l e r s central co-ordinator. approximated  i n the  and  the  In o r d e r f o r t h i s to be accomplishedj,h(t) must be  form h.(t)  = \  = g.(a.(X ),t) o  where g. i s a n o n - l i n e a r f u n c t i o n o f time w i t h unknown c o e f f i c i e n t s .  (3.33) Each  4C  coefficient are  i s non-linear  chosen s u c h t h a t  form f o r g^ and  | |h. ( t ) - ^ | |  i s given  On-line l o c a l l yJ  f u n c t i o n of i n i t i a l  conditions.  i s minimum.  2  A suitable algebraic  0.6.3) and Appendix  i n Section  implementation would be as f o l l o w s .  the i n i t i a l  t o the c e n t r a l c o - o r d i n a t o r  conditions  III, respectively.  Each subsystem i s  c o n t r o l l e d by a feedback c o n t r o l U. = -C.X.. J 1 i i  a system d i s t u r b a n c e  The c o e f f i c i e n t s  A t the time t = t o  of  o f each subsystem a r e t r a n s m i t t e d  which g e n e r a t e s the c o e f f i c i e n t s f o r (3.33).  The i n t e r v e n t i o n c o n t r o l h ^ ( t ) i s g e n e r a t e d l o c a l l y by a f u n c t i o n g e n e r a t o r a f t e r r e c e i v i n g the c o e f f i c i e n t s from the c e n t r a l co-ordinator  (see F i g .  (3.3)). In implementing ^required  (3.33), p r e l i m i n a r y  additions  On-line  control requires  a t the c e n t r a l c o - o r d i n a t o r  •subsystems)  relatively  level  a r e s t o r e d by the c e n t r a l  few m u l t i p l i c a t i o n s and  (after receiving X  q  from the  t o g e n e r a t e the c o e f f i c i e n t s .  -As w i l l be shown l a t e r , i n v e s t i g a t i o n a dimension o f f o u r for  generating  3.6  Off-Line  each  Control  s t e p s ; a) d e s i g n  f o r each v e c t o r  F o r the system  was found  under  adequate  Design (3.23) and (3.33), the c o n t r o l d e s i g n  o f the feedback c o n t r o l u_^ = ~C/K.,b)  c o n t r o l h ( t ) , and c) d e s i g n  The d e s i g n  i s a vector.  the h ^ s .  Implementing  h.  computations a r e  t o determine the unknown c o e f f i c i e n t s which  co-ordinator.  vention  off-line  design  follows  three  o f the i n t e r -  o f t h e approximate i n t e r v e n t i o n c o n t r o l  d e t a i l s f o r each o f the t h r e e  steps follow.  LOCAL CONTROLL NO.I  LOCAL CONTROLLER NO. 2  ER  l  X  X  2  LOCAL CONTROLLER NO. N  I U  2  X  N  U  — _ _ SUBSYSTEM  SUBSYSTEM NO.2  N0.1  Fig.  (3.3)  On-Line c o n t r o l  SUBSYSTEM NO.N  implementation  N  42  3.6.1  D e s i g n o f t h e Feedback. C o n t r o l Synthesizing  obtain K  «  Since  this  control requires  coupling  r e d u c e s t o s o l v i n g (3.19)  (IT ~ '-C^X )  i s neglected  s o l u t i o n s o f (3.19) t o  at this  level,  f o r each p l a n t s e p a r a t e l y .  the s o l u t i o n  One m e t h o d o f  28 solving the  (3.19)  g u e s s f o r K- ^ i s o b t a i n e d  ( e . g . 10 s e c o n d s ) .  l  b y i n t e g r a t i n g (3.18)  convergence,  f o r a short  • ' • 1 1 - 1 -Si  ft  Design o f t h e I n t e r v e n t i o n Control h ( t ) The o p e n l o o p  (1) trajectory  q ( t ) = K X(t^). r s f I n t e g r a t e (3.32) backward  initial  h * ( t ) i s obtained  as f o l l o w s :  from t=t  with  to t=t  i s the nominal t r a j e c t o r y X ( t ) .  c o n d i t i o n o f (3.32), (2)  (3.22) f o r w a r d  Integrate  obtained  /  i n t e r v e n t i o n c o n t r o l h ( t ) depends on t h e  c o n d i t i o n s . G i v e n X^, t h e o p t i m u m v a l u e  and  Forfast  C. i s g i v e n b y C. = R . ^ B l K ..  .f 3.6.2  .  '?  initial  period  i s by the Newton-Raphson t e c h n i q u e  f  Evaluate  h=h=0*The the final  r  from t = t - t o t = t ( w i t h t o  X(t)=X(t))  store q(t). (3)  Choose a s t e p  s i z e £> 0 , a n d w i t h h ( t ) = £q(t)  evaluate the  cost index J . (4) is  Find  t h e optimum s t e p  e a s i l y done b y i n c r e m e n t i n g (5)  a t w h i c h J i s minimum.  (This  '  The optimum i n t e r v e n t i o n c o n t r o l i s chosen t o be h * = £ q ( t ) . Q  Note that only to be stored.  &);  H = ^  some c o m p o n e n t s o f h ( t ) ( a n d c o n s e q u e n t l y q ( t ) ) a r e  F o r one c o n t r o l component i n each p l a n t , f o r e x a m p l e ,  one c o m p o n e n t o f h ( t ) w o u l d b e r e q u i r e d are m u l t i p l e d by zeros  ( s e e (3.17)  t o be s t o r e d s i n c e the other  f o r S and (3.22) f o r S h ) .  only components  Therefore,  for  the  case of two i n t e r c o n n e c t e d  (3.2), o n l y  power p l a n t s h a v i n g  two components f o r h. (±)  (i  1,2) a r e r e q u i r e d .  a  3.6.3' Design o f t h e "Approximate I n t e r v e n t i o n As p o i n t e d be  out before.;  the model o f S e c t i o n  Control h  the p r e v i o u s  algorithm  f o r h(t)  cannot  implemented f o r o n - l i n e c o n t r o l of the power system under i n v e s t i g a t i o n .  However, by approximating h ( t ) by a s u i t a b l e f u n c t i o n o f known a l g e b r a i c form, o n - l i n e implementation i s f e a s i b l e .  Suitable functions are poly-  nomials o r s p l i n e s . By p l o t t i n g for  the i n t e r v e n t i o n c o n t r o l s as a f u n c t i o n o f time  each s e t o f i n i t i a l  polynomial  c o n d i t i o n s , k, i t was n o t i c e d t h a t a c u b i c  i n t c o u l d be f i t t e d  that h ^ ( t ) i s e v a l u a t e d  t o each h ^ ( t )  (the s u p e r s c r i p t k denotes  f o r the s e t o f i n i t i a l  conditions k ) .  Consider the  case o f a p p r o x i m a t i n g one i n t e r v e n t i o n c o n t r o l , f o r example h ^ ( t ) o f t h e first  plant.  The same procedure i s f o l l o w e d  c o n t r o l s of the o t h e r (1)  f o r the i n t e r v e n t i o n  plants.  L e t h ^ ( t ) be approximated by a c u b i c p o l y n o m i a l  *u(t) = i  int:  E A™" . m m=l  (3.34)  1  A c u r v e - f i t t i n g r o u t i n e based on a l e a s t - s q u a r e 3 a p p r o a c h can be employed to f i n d the c o e f f i c i e n t s a^. m  T h i s i s done f o r d i f f e r e n t s e t s o f i n i t i a l  c o n d i t i o n s k=l,2,...M. (2) conditions.  I n g e n e r a l , the c o e f f i c i e n t s " ' Consider  the mth c o e f f i c i e n t —  al^ajlh m  m  o  m a . m  a r e f u n c t i o n s o f the i n i t i a l  k  L e t a be approximated by m J  (3.35)  where a (X ) i s a s p e c i f i e d f u n c t i o n of X \<rith unknown c o e f f i c i e n t s . mo o required  It is  to f i n d the c o e f f i c i e n t s i n a (X ). s u c h t h a t • mo  I|a  k  i s minimum f o r a l l k.  - a | I , m=l,2,3,4 , k=l,2,...M m '  The  (3.36)  2  k  m  c h o i c e of the f u n c t i o n  (3.35) i s a r b i t r a r y .  A c h o i c e o f (3.35) s u i t a b l e f o r the power system s t u d i e d i s g i v e n i n Appendix I I I . For o n - l i n e i m p l e m e n t a t i o n the c o e f f i c i e n t s (3.35) are s t o r e d the c e n t r a l c o - o r d i n a t o r .  On  r e c e i v i n g the i n i t i a l c o n d i t i o n s  subsystem, the c e n t r a l c o - o r d i n a t o r f o r each p l a n t and  from each  computes f o u r parameters a^, a^,  transmits them to the d i f f e r e n t subsystems.  and  Each  intervention control I K .  subsystem then generates i t s own 3.7  by  Application 25 An example of  considered.  _an  interconnected  Let the s u b s c r i p t s 1 and  steam and hydro p l a n t  is  2 denote the steam and h y d r o p l a n t s ,  respectively. The  f i r s t s i g n of impending t r o u b l e i n a power system  by l o s s o f l o a d i s a c c e l e r a t i o n . acceleration)  appears l a t e r , and  s t i l l l a t e r (see F i g . ( 3 . 9 ) ) .  The  speed d e v i a t i o n  period  and  (time i n t e g r a l o f  the a n g u l a r change ( i n t e g r a l of speed)  Because of the r e l a t i v e l y l o n g  c o n s t a n t s a s s o c i a t e d w i t h the governor and change i n s t a n t a n e o u s l y  time  t u r b i n e , t h e i r o u t p u t s do  can be assumed c o n s t a n t d u r i n g  following a disturbance.  disturbed  the v e r y  not  short  C o n s e q u e n t l y , d e t e c t i o n of a n g u l a r 2  a c c e l e r a t i o n i s the most p r o m i s i n g way  of q u i c k l y i n i t i a t i n g c o n t r o l a c t i o n .  In t h i s a p p l i c a t i o n ^ t h e d i s t u r b a n c e  i s assumed to be a speed  d e v i a t i o n from n o m i n a l a t t=t. , a l l o t h e r s t a t e s a r e assumed z e r o . o i n d e x i s chosen as 20 J  a  j  I  ^  A 6  1  +  4  + A 6  2  + A w  2  +  l  U  +  U  2^  The c o s t  d t  That i s ,  «2"  R^=l, R = l . 2  The c h o i c e t ^ = 20 seconds i s made s i n c e t h e system s e t t l i n g time i s around 30 seconds. The d i f f e r e n t parameters f o r each p l a n t r e p r e s e n t e d  by P i g . (3.1)  25 and F i g . (3.2) a r e as f o l l o w s M M T  1 2  1 2  = .04, G ^ . 0 1 , T = 0.5, T = 0.5, E = .03 = .03, G = .008, D = 0.5, T = 0.5, T « 1.2, E fcl  2  = 0.05  ][  2  , « J  2  fc2  2  = .013  _ = 0.5. 1 g  = -1  The l o c a l feedback c o n t r o l s f o r each p l a n t i s o b t a i n e d  as e x p l a i n e d i n  S e c t i o n (3.6.1) and a r e g i v e n b y : IL = -0.336 h6,-.607 1 1 U  2  Aw..-.416 AP ,-1.6 AX , 1 gl gl  = ,515A6 -l.l6Aw -11.5 AP ~41.6Ag ~9.22 2  2  g2  2  AX  g 2  F o r t h e d e s i g n o f t h e i n t e r v e n t i o n c o n t r o l , 9 p e t s o f speed deviations are considered.  Three d i f f e r e n t s e t s f o r each o f t h e f o l l o w i n g  cases a r e t a k e n : (a) (b) (c)  Disturbance Disturbance Disturbance  a t p l a n t (1) o n l y . a t p l a n t (2) o n l y . at both plants simultaneously.  Following the design procedure of Section  (3.6.3), i t x^as n o t i c e d  that  h*(t)  i s almost zero  ( 3 . 3 9 ) was  given  of h at the  data  cost  (b) u s i n g  10  seconds.  t o 10  the  seconds only.  The  coefficients h  f u n c t i o n J f o r the  Therefore,  fitting the  routine  generation  t o be  s t o r e d at  the  central co-ordinator  given  i n Table  (3.1).  c o n d i t i o n s , a c o m p a r i s o n i s made b e t w e e n  cases  (a) u s i n g  local  intervention control h*(t),  c o n t r o l only ( J , );and  ( J );  (c) u s i n g  local  b plus  i n t e r v e n t i o n c o n t r o l h, To  different  )•  The  show t h e e f f e c t i v e n e s s o f  set  c o n t r o l only control h  w  the  (^o^' the  ° 2o ^^ J  =  s y s t e m was  s y s t e m was  W a S  t  e  for  seconds a f t e r which i n t e r v e n t i o n  (see Appendix I I I ) are  9 sets of i n i t i a l  local plus  Consequently, the  i s done f o r 10  used f o r g e n e r a t i n g For  the  up  local level  c o n t r o l i s removed. which are  after  s  t  e  (  unstable.  stabilized.  results  are  given  on-line generation 3  ( By  T e s t :  > Table  i n Table  (3.2).  of h ( t ) , a  (3.1)).  With  local  i n t r o d u c i n g the i n t e r v e n t i o n  Figure  (3.5)  shows t h e s e  results.  47  SPEED  I DETECTION DELAYS Fig.  •  (3.4) Angle and a n g u l a r speed as a f u n c t i o n o f time s t e p change o f a n g u l a r a c c e l e r a t i o n  Table (3.1) NO-  SET  INITIAL CONDITIONS kJ/O, 20 o  a  1 2  2,0  3  h  J  00  .446  .360  0.364  1.957  1.470  I 487  6.233  3-427  3.611  4  0, - 1.33  1. 625  1. 435 1. 453  5  0, -2.667  6.191  5-610 .  13-95  12.738 12. 785  6  0,-4 1.33  7  1, -  e  2, -2.657  9  TEST  3,-4 5,0  5.619  1. 5  1.45  5.9  5. 690 5.720  13.73 10-4  xlO  4  13.3  1. 459  13.35 14.84  f o ra  Fig.  (3.5)  A n g u l a r and (a)  local  t i e - l i n e deviations  control only,  (b)  for  l o c a l plus  = 5,  a>2 = 0 Q  intervention  with controls.  4.  OPTIMUM LOAD-FREQUENCY CONTINUOUS CONTROL WITH UNKNOWN D E T E R M I N I S T I C POWER DEMAND  4.1  Introduction Power s y s t e m d i s t u r b a n c e s  changes i n t i e - l i n e  of  i n use i s b a s e d on an e r r o r  drives  the error  signals  Modern o p t i m a l  some  The f o r m o f L o a d - F r e q u e n c y C o n t r o l ( L F C ) s i g n a l x^hich  t h e n e t i n t e r c h a n g e and frequency e r r o r s .  action  result i n  r e a l power and s y s t e m f r e q u e n c y , n e c e s s i t a t i n g  form o f l o a d - f r e q u e n c y c o n t r o l . presently  caused by l o a d - f l u c t u a t i o n s  i s a l i n e a r combination  A simple integral-type  control  t o zero.  c o n t r o l theory has l e d t o t h e development o f d e s i g n  t e c h n i q u e s w h i c h can r e s u l t i n s i g n i f i c a n t improvement i n t h e c o n t r o l o f high-order systems.  -i  currently applied  The a p p l i c a t i o n s  . . .  receiving  increasing  attention  14,17,25,32  the solution of the state-regulator  a p p r o a c h , hoxtfever,  requires  Consequently, the c o n t r o l information  required  Feasible of  o f t h e s e t e c h n i q u e s t o improve LFC i s .  k n o w l e d g e o f t h e new s t e a d y - s t a t e  operating  This point.  the identification  compensate f o r l o a d -  T h i s f a c t was r e c o g n i z e d i n r e f e r e n c e 17  o f time.  t i e - l i n e p o w e r AP  A c t u a l l y , AP  must b e a c c o u n t e d f o r i n t h e o p t i m a l  where a  to perform the i d e n t i f i c a t i o n .  i n 17 h a s , hox^ever, s e v e r a l  the incremental  ments) f u n c t i o n  , 14  f o ri t s implementation i s not a v a i l a b l e .  m o d i f i e d K a l m a n f i l t e r x^as i n t r o d u c e d  i s made t h a t  ,_  and Fosha  i s n o t a f e a s i b l e optimum c o n t r o l , s i n c e t h e  optimum l o a d - f r e q u e n c y c o n t r o l r e q u i r e s  frequency deviations.  ,  problem t o t h e LFC problem.  t h e i n c r e m e n t a l p o w e r demand i n o r d e r t o o p t i m a l l y  approach suggested  Elgerd  shortcomings.  The  The a s s u m p t i o n  i s a known ( t h r o u g h m e a s u r e -  depends on t h e s y s t e m s t a t e and t h i s  control  formulation  o f t h e LFC problem.  A rather serious shortcoming is  i n 17 i s t h e m a n n e r  Normally,  i t i s the f i l t e r  output output  action.  I n 17 h o w e v e r , t h e f i l t e r  output.  Invariably,  computer. highly  the f i l t e r  The n u m e r i c a l  x which  i s used f o r implementing  i s realized  In  a c t i o n so t h a t the Kalman f i l t e r  Consequently, suggested and que.  can  The s e c o n d  states.  could introduce  requirement  control  identification.  action i s  f o r a power  a  system.  The f i r s t m e t h o d i s e x t r e m e l y approximation  method i s b a s e d on u s i n g a L u e n b e r g e r - t y p e o f t h e demand  noise  two a l t e r n a t i v e m e t h o d s a r e simple techni-  observer to  and t o e s t i m a t e t h e unmeasured  The a d v a n t a g e o f t h e s e a l t e r n a t i v e m e t h o d s i s t h a t t h e y  do n o t  statistics.  Formulation  A t y p i c a l m o d e l o f two i n t e r c o n n e c t e d p o w e r a r e a s (4. l )  form.  d a t a about p l a n t a n d m e a s u r e m e n t  i n c r e m e n t a l demand b y t h e d i f f e r e n t i a l  the i d e n t i f i c a t i o n  Problem  for x  a s s o c i a t e d w i t h implementing  data i s g e n e r a l l y not a v a i l a b l e  r e q u i r e d a t a about n o i s e  digital  be used i n i t s c o n v e n t i o n a l  i n s t e a d o f a Kalman f i l t e r ,  identifies  perform  integral  on a  h e r e , x i s used t o implement c o n t r o l  Detailed statistical  f o r demand  sampled d a t a  The u n c o n v e n t i o n a l  However, t h e r e i s a p r a c t i c a l d i f f i c u l t y  r e q u i r e d and s u c h  form  filter  formulation.  t h e method p r e s e n t e d  Kalman f i l t e r .  i n a discrete  g e n e r a t i o n of x from  i n the problem  control  i n p u t x, i s used i n s t e a d o f the  u n d e s i r a b l e n o i s e problems.  introduced  Fig.  filter  g i v e s an e s t i m a t e x o f a s t a t e x.  seems t o a r i s e o u t o f t h e m a n n e r i n w h i c h  4.2  t h e Kalman  used. The K a l m a n f i l t e r  is  i n which  1  ^ ' ^ ^ ' T h e  controlling  a r e a i s t a k e n t o be a s t e a m - p l a n t  i s shown i n  s t a t i o n i n the f i r s t  and a h y d r o - p l a n t ,  and  respectively.  second  STEAM  PLANT  E.  dl sr  U,  1 •  1  r  A  ST l tr  9  GOVERNOR  TURBINE  API  HYDRO  AX,  GOVERNOR  Fig.  12  12c  PLANT  /___ ST  MJS+GJ  A  -D S+1  g2 +  AP  ?  'Gj rr  5 D S;'-l  t2*  1  M  2  TURBINE  AS-,  2  2  S  +  G  2  AP d2  ( 4 . 1 ) B l o c k d i a g r a m o f two i n t e r c o n n e c t e d steam a n d h y d r o  areas.  52  I n s t a t e v a r i a b l e form the i - t h c o n t r o l l i n g p l a n t dynamics i n an N - i n t e r connected system has x  the form + T.APj. + n . . ( u . ) l di i j j  . = A.x . + B.u. pi 1 pi 11  •j $ i , j = 1,  (4.1) 2,  . . . , N  where f o r a s t e a m - p l a n t the s t a t e v e c t o r i s (prime denotes t r a n s p o s i t i o n ) x* = [AP • P t and  f o r hydro-plant  Aw  matrices  AX  g  g  (4.2)  ]  the s t a t e v e c t o r i s x* = [AP P t  The m a t r i c e s  AP  Aw  AP  Ag  g  AX  6  g  (4.3)  ]  i n ( 4 . 1 ) , f o r a s t e a m - c o n t r o l l i n g p l a n t are g i v e n by:  (The  f o r a hydro p l a n t have a s i m i l a r s t r u c t u r e ) . 0  E  T°. ij c  0  0  j/i A. l  =  -1/M.  -G./M. i i  l  1/M. -1/T  0 -E./T  0  B' = [0 i = [0 0!.  = [-T°.  (4.4) ti  .  0  1/T . t  •1/T  0  1/T  . -1/M  0 0  0  .] gi  (4.5)  0]  (4.6)  0]  (4.7)  where the terms i n ( 4 . 2 ) - ( 4 . 7 ) a r e as d e f i n e d i n S e c t i o n To a v o i d u n n e c e s s a r y c o m p l i c a t i o n s  . gi  (3.2).  i n notation,the  coupling  terms between the areas are s e t e q u a l to z e r o i n the i n i t i a l p r o b l e m  formulation  53  and  t h e s u b s c r i p t i i s dropped  example).  Equation  (4.1)  (non-zero coupling  then takes x  A x  =  p  i s considered  t h e form  (4.8)  + Bu + TAP, d  p  To o b t a i n a f e a s i b l e c o n t r o l , t h e L F C p r o b l e m m u s t b e t o b e c o m p o s e d o f two s e p a r a t e unknown p o w e r demand AP^. response so t h a t  problems:  (2)  considered  ( l ) Problem of i d e n t i f y i n g the  Problem of o p t i m a l l y  the generation  i nthe  becomes e q u a l  controlling  t h e dynamic  t o t h e demand a t t h e s p e c i f i e d  frequency. Consider  t h e s e c o n d o f t h e above p r o b l e m s and assume f o r t h e  moment t h a t A P , i s a known c o n s t a n t . d  The t e r m i n a l  conditions  t o be  satisfied  are* AP ( o o ) = 0, Aw(<*>) = 0, AP (°°) - A P , = 0 a n d AX (~) - A P , = 0 t • g ci g tt To f o r m u l a t e  an o p t i m a l  (4.9)  c o n t r o l problem, a change o f v a r i a b l e s i s i n t r o d u c e d :  .' x = x  p  - pAP,  (4.10)  d  where  p' = (0 Substituting  0  1  1).  (4.11)  (4.10) i n t o (4.8) ( a n d s e t t i n g AP^ t o z e r o ) y i e l d s ; x = A x + B u + (A p + P ) A P ,  (4.12)  d T h e t e r m i n a l c o n d i t i o n (4.9)  requires  that  i ( c o ) = o = X(oo) It  i s seen from  (4.13)  (4.11) a n d t h e d e f i n i t i o n o f t h e s y s t e m m a t r i c e s  that  A p + r = -B  (4.14)  An e s s e n t i a l c h a r a c t e r i s t i c o f t h e l o a d - f r e q u e n c y r e q u i r e m e n t f o r an i n t e g r a l - t y p e o p e r a t i o n this  See  on t h e e r r o r s i g n a l .  c o n t r o l requirement i n t o the formulation  necessitates  augmenting  A p p e n d i x I f o r LFC  (4.12) b y criteria  control i s the  o f an o p t i m a l  To  introduce  c o n t r o l problem  x  n + ]  _(t) - u(t) -  AP  (4.15)  d  so t h a t (4.16)  n+1  where  ~ A •  (4.17)  u = u The augmented  system i s t h e r e f o r e (see (4.14)) X = AX  +  (4.18)  B u  where X X ^  A  *A  A A  B"  ~  and 0  X  ,, - n+l_  A  ' 0" (4.19)  B = 1  0  The c o s t i n d e x i s taken to have the q u a d r a t i c form  J =  1  (4.20)  (X'QX + u'Ru)dt 0  where Q and R a r e p o s i t i v e d e f i n i t e The o p t i m a l  matrices.  c o n t r o l f o r the problem d e f i n e d by (4.18) and  i s g i v e n by  (4.20) (4.21)  u = c'X  where C  i s a constant equation  '  =  C S  S  n+1  (4.22)  ]  v e c t o r which can be found by s o l v i n g a s t e a d y - s t a t e  Riccati  (see S e c t i o n 3.6.1). In terms o f the o r i g i n a l s t a t e v a r i a b l e s the c o n t r o l i s g i v e n by  (see  (4.10),  (4.15) and (4.19)) u = s'x  where u(0) i s a r b i t r a r i l y  p  + s ,.u - (s'p + s ^ J A P , n+1 n+1 d  taken to be z e r o .  (4.23)  The c o n t r o l (4.23) i s s i m i l a r  to the c o n v e n t i o n a l p r o p o r t i o n a l p l u s i n t e g r a l c o n t r o l which i s p r e s e n t l y used f o r l o a d - f r e q u e n c y is  I t i s seen, however,  f e a s i b l e o n l y i f AP^ can be i d e n t i f i e d .  demand 4.3  control.  t h a t the c o n t r o l (4.23)  The next S e c t i o n d i s c u s s e s the  identifier,  Demand I d e n t i f i e r - D i f f e r e n t i a l A p p r o x i m a t i o n To implement the c o n t r o l g i v e n by  (4.23) r e q u i r e s the i d e n t i f i c a t i o n  of the parameter A P ^ .  A simple  identifier  u s i n g the method of d i f f e r e n t i a l Let T time t, and k—i  approximation  constructed  by  23  = t. - t , be a f i x e d i d e n t i f i c a t i o n p e r i o d which s t a r t s k k-i  o  r  terminates  at time t, . k  a s e t of i d e n t i f i c a t i o n i n t e r v a l s . or s l o w l y v a r y i n g . (4.23) i s to use by  f o r A P ^ can be  The  sequence ( t , t - , t„, o i z  ...)  In a c t u a l power systems, AP^  Consequently, a r e a s o n a b l e  approximation  the demand i d e n t i f i e d d u r i n g the p r e v i o u s  at  defines  i s constant  f o r AP^  period.  in That i s ,  taking AP, = A P , ( k - l ) d d  f o r the i n t e r v a l tj <  T  £  t  where A P ^ ( k - l )  i +i» c  the power e q u i l i b r i u m .equation AP over the p r e v i o u s  g  i s determined by i n t e g r a t i n g  (see F i g . ( 4 . 1 ) ) .  - AP, = MAoi + GAOJ + AP, d t  interval  t ^ _ ^ < r $ t^.  AP (k-l)  = i  d  (4.24)  This  (4.25)  yields  [-M(Aa)(t ) - A c j ( t _ ) ) k  k  1  o (4.26) + J  (-GAco(t)+AP (t)-AP g  •J  (t))dt]  Vi  The  q u a n t i t i e s on  the r i g h t - h a n d s i d e of (4.26) a r e determined by measure-  ments on the system. The  estimate  g i v e n by  (4.26) c o u l d be improved by  several identification intervals.  The  type of a v e r a g i n g  and  averaging  over  the number of  i n t e r v a l s used would depend on the k i n d of l o a d d i s t u r b a n c e . The i n F i g . (4.2).  s t r u c t u r e of the composite p l a n t and In l o a d - f r e q u e n c y  controller following rapidly  controller i s illustrated  c o n t r o l t h e r e i s the problem of  changing random-load d i s t u r b a n c e s .  the This i s  inefficient  and  contributes  19  to unncessary wear i n the  '  Ross  treated  Control  t h i s problem and  Computer (EACC).  The  Control preset  action i s i n i t i a t e d threshold.  As  suggested the  only when the  indicated  in Fig.  and required.  computed p r o b a b i l i t y exceeds a  (4.2),  an  EACC can  (4.23) and  be  used to  aug-  (4.26).  proposed l o a d - f r e q u e n c y c o n t r o l l e r i s t e s t e d  connected steam and  hydro-plants  as g i v e n i n S e c t i o n Due  to the  ( F i g . (4.1)).  between the p l a n t s ,  of - a l l the  states.  two  inter-  parameter v a l u e s used  d i f f e r e n t suboptimum c o n t r o l l e r s are  the  optimum feedback  -The'complexity-of .such .a c o n t r o l l e r  makes i t e s s e n t i a l to i n v e s t i g a t e v a r i o u s  t o the  The  on  (3.7)  coupling  c o n t r o l i s -a f u n c t i o n  forms of suboptimum c o n t r o l l e r s . c o n s i d e r e d , and  they are  compared  optimum c o n t r o l . The  optimum feedback, c o n t r o l has  0  *  "01  "  G  "02 where and  error signals  Adaptive  Applications The  Two  of an E r r o r  load-frequency control action i s  ment proposed l o a d - f r e q u e n c y c o n t r o l g i v e n by  are  use  EACC m o n i t o r s the  computes the p r o b a b i l i t y that  4.4  c o n t r o l l e r mechanism.  and  are  second p l a n t ,  solutions  the  =  [  S  S  the  form  VL  ~ 02 > n  011 1 X  021 1 X  +  +  state vectors  respectively.  The  S  S  C  012 2> X  ( 4  022V  ( 4  ( i n the form gain vectors  s  (4.19)) f o r the QJ^  of a s t e a d y - s t a t e  matrix R i c c a t i equation  By n e g l e c t i n g  coupling  the  ]  (.1 > k = l ,  2)  -  -  4  '  2  2 8 )  2 9 )  first are  24  between the p l a n t s ,  an optimum  control  7  )  57  j  DEMAND DECISION  1 ''EACC  dk DEMAND  UNIT DELAY  IDENTIFIER  ^VECTOR _  (A.2)  SCALAR  B l o c k d i a g r a m o f a power p l a n t w i t h a l o a d - f r e q u e n c y  controller  o f t h e form r U  rl  U  r2  r l "  S  =  S  J  rll l X  r22 2 X  (4.30)  r2 '  (4.31)  s  (4.32)  J  The c o n t r o l (4.30) i s suboptimum f o r the coupled  system. : c o u p l i n g terms i n t h e optimum c o n t r o l (4. 2 7 ) , : form  K i s obtained.  (4.33)  " " s l [  U  sl  =  S  011 1  U  s2  =  S  022 2  (4.34)  X  The type o f s u b o p t i m a l  (4.35)  X  c o n t r o l g i v e n by (4.34) and (4.35)  i s d i s c u s s e d i n r e f e r e n c e 33. Example (4.1)  The Q and R m a t r i c e s i n (4.20) a r e chosen t o be  the u n i t m a t r i c e s and t h e assumed demands a r e t a k e n t o be s l o w l y t i m e v a r y i n g and g i v e n by p A  _,0.1 s i n ( f r t / 2 0 ) d l 0.1  AP  _ i  d 2  0 < t ? 10 t > 10  =0.0  The i d e n t i f i c a t i o n i n t e r v a l T  and t h e f i n a l time t  r  o  a r e chosen t o be 0.5  f  and 30 seconds, r e s p e c t i v e l y ( a t t ^ = 30 t h e system has e s s e n t i a l l y the s t e a d y - s t a t e ) .  The i n i t i a l  reached  c o n d i t i o n s on t h e c o n t r o l l e r s a r e  a r b i t r a r i l y set equal to zero. Table  (4.1) g i v e s t h e n u m e r i c a l v a l u e s of t h e g a i n v e c t o r s ( i n  terms o f t h e o r i g i n a l s t a t e s ( s e e (4.23)) and t h e p e r f o r m a n c e c o s t J .  Table (4.1)  GAINS  STATE FED BACK A P. A  LO  STEAM  PLANT  tl2  u  ,  FOR THE i  'o  %  °02  7.5 ..  . .62  - .48  PLANT  U  >  ^ 2  J  HYDRO  7.5  _ .5  _ .52 -8.  . 5.7  -1.4  -5.4  -5.7  - .4 7 -4 .66  -.4.76  5.17  0  18.45 21.  27.5 • 19  .22  - .09  S2  -4.7-5.  17.26  21.  .1.71  X  15.62  -7.2  -.22  * 92  "si .  °n  -3.3  A P  9  \  _ 8.  18 .46  * 2  CONTROLS  .19  - .09  -.09  -10-7  -11.3  -10.7  .2. 02 .12.9  -13.4  -12.9  .4.7  -4.63  -5.7  _ .47 -4.6 9.9  3  28.32  0-5  3  29.49  0.7  8  28.32 0.65  Example (A.2)  Table (4.1) shows that the suboptimal control u s i s used i n this Example which  gives a lower cost than u_.  Consequently, u  i l l u s t r a t e s the effect of T  on i d e n t i f i c a t i o n and control.  g  The following  o demands are assumed:  = I  AP fi  dl  , d2  l  0.15 s i n (irt/20) 0.15  0 < t jc 10 ' t > 10  r  _ ,-0.1 s i n (irt/20) ~ -0.1  M  0 < t s? 10 t > 10  l  The system responses f o r (a) T_ = 0.5, and (b) T (4.3).  (4.37)  q  = 1, are shown i n F i g .  I t i s evident from F i g . (4.3) that the load-frequency  control (4.34) and (4.35), with power demand i d e n t i f i c a t i o n given by (4.24) and (4.26) results i n a s a t i s f a c t o r y system response. Example (4.3).  In this example the control u i s compared with a s 25 conventional load-frequency c o n t r o l l e r given i n transfer function form by  = f biir  u  The optimum parameter  (  )(  )(cAu +  V  (4  -  38)  V a l u e s f o r t h e c o n t r o l l e r , as g i v e n i n r e f e r e n c e 25,  are: a)  Steam p l a n t ; a^ = .09, b ^ = 0.3, c^ = .02  b)  Hydro p l a n t ; a^ = .4, b ^ = 0.3,  S i n c e power demand i s n o t i d e n t i f i e d ,  = .02  the o p t i m i z a t i o n i s performed  a v e r a g i n g over a s p e c i f i e d s e t o f power demand p r o f i l e s .  after  Consequently,  f o r a g i v e n power demand, (4.38) i s suboptimum. Figure for:  (4.4) i l l u s t r a t e s  the comparison  o f t h e system  (a) t h e c o n v e n t i o n a l c o n t r o l u g i v e n by (4.38), and (b)  control u . s  25 The demands were chosen t o be ,  A P  dl  =  "*°  0 5  A P  d2  =  ,  0  °  5  responses t h e proposed  o.ooe 0.006 1  '0.004  3  0.0D2  £0.000  -0.002 H  -0.004  10  15 20 TIME (SECONDS!  o.ieo •  10  Fig.  10  15 20 TIKE (SECONDS!  ( 4 . 4 ) Response Comparison  (a)  Conventional  and  IS 20 TIKE (SECONDS)  (b)  proposed  contro  63  It  i s e v i d e n t f r o m F i g . (4.4)  and  t h e use  of the load-frequency  improvement i n system  4.5  t h a t the i d e n t i f i c a t i o n control u  T i e - l i n e power and  in a  significant  Observer  frequency  deviations (the f i r s t  two  states i n  (4.1)) a r e t h e o n l y measurements r e q u i r e d i n p r e s e n t l y  used load-frequency requires  results  response.  Demand I d e n t i f i e r - L u e n b e r g e r  t h e model g i v e n by  s  o f t h e p o w e r demand  controllers.  The  optimum c o n t r o l g i v e n by  t h a t a l l t h e s t a t e s b e m e a s u r e d and  an o b s e r v a b l e s y s t e m ,  t h a t AP^  be  (4.23)  identified.  m e a s u r e m e n t s o f some o f t h e s t a t e s c a n be  For  used  to  34 reconstruct  the complete  Consider  s t a t e by  a system  use  of a Luenberger  m o d e l and  observer  a measurement system  x = Ax +  of the  form  Bu  (4.39)  z = Hx w h e r e x i s an n - s t a t e v e c t o r a n d m<n).  I t i s a s s u m e d t h a t (4.39) i s o b s e r v a b l e .  class The  z i s an m m e a s u r e m e n t v e c t o r  of  (n-m)  dimensional observers  observer outputs  measured s t a t e s . observer  can be To  In theory, a r b i t r a r i l y  illustrate  system  state equations [x  shown t h a t a  (4.39).  small settling  time of  un-  the  achieved.  s t a t e s and  =  s t r u c t u r e d from  has  g i v e an a s y m p t o t i c a l l y c o r r e c t e s t i m a t e o f t h e  system  w h e r e x'  c a n be  Luenberger  ( i n general  the use  a constant  AP^,  of observers  unmeasured power  a s i n g l e steam p l a n t i s c o n s i d e r e d .  a r e a u g m e n t e d b y AP^  x = A x + B u  AP, ], and w h e r e  to i d e n t i f y  a (see  a (4.4)).  = 0.  This  yields  The  (4.40)  64  0  12c -GjM.  a  -1/M.  1  0 A  0  I  ! -  =  1 / T  ti  1 / T  -w i °  0  0  0  0  0  n  -1/T  A  A  ll  '  A  | 12 A  (4.41) 0  gl  0  1  B' = [0  •1/M,  0  A  A  I  A  2 1 i 22 A  0  l o f 0]  = [0  0 j" B ]  (4.42)  gl Assuming that only the f i r s t two states are measured, the measurement matrix i n (4.39) takes the form 1 0  H =  0 0 1 0  0 0  0 0  (4.43)  The p a r t i t i o n i n g indicated i n (4.41) and (4.42) i s used to decompose (4.40) into the form *1 .  h  =  A  =  A  l l h  A  +  21 h  +  A  12  ?  (4.44)  2  22 h  + B  2  where E,^ i s the measured m-vector and E,^  1 S  (4.45)  U  a n  n  _  m  vector which i s to  be  reconstructed by an observer. Consider the observer defined by C = N  C + N  n  M  ll  1 2  h  36  + B  2  u  h>  (4.46) (4.47)  where N  ll  =  M  11 12 A  +  A  22  ^^ii^n-^V + ^i-^V-  (4.48)  6:  A In (4.48) M^^ i s an a r b i t r a r y  (n-m)xm m a t r i x .  Let a = £  - £  be t h e A  e r r o r between the unmeasured v e c t o r g i v e n by (4.46) and (4.47).  it  follows  and the o b s e r v e r output £ , as  2  2  I t i s seen that  a - (A Consequently, i f  £  2 2  + M A n  )a  1 2  can be chosen so t h a t  (4.49)  (4.49) i s a s y m p t o t i c a l l y  stable,  that £ ( t ) -> £ ( t ) 2  t •+ ».  as  2  To determine such a m a t r i x , c o n s i d e r t h e a u x i l i a r y »  =  ( A  22  +  M  system  35  11 12 '» A  }  ( 4  '  5 0 )  and l e t • V = -y'Cu  (4.51)  where C i s an a r b i t r a r y p o s i t i v e d e f i n i t e c o n s t a n t m a t r i x . from (4.50) and (4.51)  I t follows  that V = y'Ky  (4.52)  where ( A  By  22  +  M  11 12 A  ) K  +  K ( A  22  +  \l 12 A  y  +  C  =  °  ( 4  -  5 3 )  taking M  = -|KA  n  i 2  S  (4.54)  where S i s an a r b i t r a r y p o s i t i v e - d e f i n i t e m a t r i x , (4.53) takes t h e form A Equation  2 2  K + KA  2 2  - KA^SA  1 2  K + C = 0  (4.55) i s the a l g e b r a i c m a t r i x R i c c a t i e q u a t i o n which can be  s o l v e d f o r a p o s i t i v e d e f i n i t e symmetric m a t r i x . it  (4.55)  i s seen from (4.51) and (4.52) t h a t  asymptotically  stable.  With t h i s  choice of  the a u x i l i a r y system  (4.50) i s  S i n c e (4.49) and (4.50) have the same e i g e n v a l u e s ,  it  follows  that  (4.-49) i s a s y m p t o t i c a l l y  Figure frequency  control with  to reconstruct is  (4.5) i l l u s t r a t e s  included  partial  stable.  a block  d i a g r a m o f optimum  measurements o f t h e s t a t e and an o b s e r v e r  t h e r e m a i n i n g s t a t e components.  in ^  load-  which i s reconstructed  Notice  that  t h e demand  by t h e o b s e r v e r from  AP^  (4.46) and  (4.47). Example this  ( 4 . 4 ) The d a t a f o r t h e s i n g l e s t e a m - p l a n t  e x a m p l e i s t h e same a s u s e d i n E x a m p l e ( 4 . 1 ) .  states  ( t i e - l i n e and f r e q u e n c y d e v i a t i o n s )  indicated  Only the f i r s t  a r e measured.  0 100  .was a r e a s o n a b l e c h o i c e  11  M. 11 A constant illustrates  C = 1000  for this  example.  simulation  (4.56)  -The.observer .matrices  -4.97 -1.33 .79 0 0 0  2 -2 0  2.97 1.33 -7.9  N  2.97 1.33 -7.9  12  resulting  (4.57)  demand o f AP^  (4.57)  = 0.1 i s a s s u m e d .  Figure  Control (b)  (4.6)  using:  d i f f e r e n c e i n responses i n d i c a t e s that  t h e unmeasured s t a t e s w i t h  adequate  (a)  measurement o f  components and o b s e r v e r r e c o n s t r u c t i o n o f r e m a i n i n g  The s l i g h t  reconstructs  -1.4 -.73 3.36  -.12 -.05 0.32  t h e s y s t e m responses f o r L o a d - F r e q u e n c y  o f t h e state  accuracy.  Digital  1 0 0 0 1 0 0 0 1  m e a s u r e m e n t o f a l l t h e s t a t e s a s s u m i n g AP_^ i s k n o w n ;  ponents.  two  choice are  N  two  in  that 1 0  from t h i s  considered  com-  the observer  (for c o n t r o l purposes)  PLANT  Fig.  ( 4 . 5 ) B l o c k d i a g r a m o f optimum l o a d - f r e q u e n c y c o n t r o l w i t h p a r t i a l measurements  o  QC  i—i  > UJ  fa)  Q  >C_)  L-^»V-«l.-„'—J-  ° -  |  .i-...^., | ,,|,,  ,[  I, | „,.^  i„„,  (  1-—;  1 H  (b)  UJ <3 LIJ  ID  u- oi  o I  in - i — i — i — i — | — i — i — I — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — r -  10  ? 0  20  30 40 TIME" (SECONDS)  20  30 40 (SECONDS)  TI HE Fig.  50  (4.6) System response - Example (4.4)  60  60  5.  OPTIMUM LOAD-FREQUENCY SAMPLED-DATA CONTROL WITH RANDOMLY VARYING SYSTEM DISTURBANCES  5.1  Introduction A power s y s t e m a r e a  physically  g e n e r a l l y has i n t e r c o n n e c t i o n s which a r e  remote from t h e c o n t r o l l i n g  Feedback r e g u l a t o r y  station or dispatching  c o n t r o l of the system r e q u i r e s  line  flows  at interconnections  and t h e t r a n s m i s s i o n  data  links  to the controlling plant  or dispatching  center.  t h e measurement o f t i e o f measured data center.  I t i s essential,  t h e r e f o r e , t o i n v e s t i g a t e t h e e f f e c t o f s a m p l i n g time on a c o n t r o l based e n t i r e l y  on c o n t i n u o u s s i g n a l s ^ .  l i n e measurement d e l a y ever,  i s made t o t a k e This  In reference  time v e r s i o n o f t h e c o n t r o l l e r d i s c u s s e d  i n C h a p t e r 4.  a s p e c t o f power s y s t e m c o n t r o l t h a t  load disturbances  a r e random i n n a t u r e  often perturbed  of p l a n t  by n o i s e .  and measurement n o i s e  control  design.  the sampled-data o r d i s c r e t e -  another important  are  No a t t e m p t , how-  e f f e c t i n t o account i n the optimal  Chapter deals, essentially., w i t h  strategy  17 t h e e f f e c t o f t i e -  i s t e s t e d on t h e c o n t i n u o u s s y s t e m . this  over  There i s , however, i s considered.  Many  and measurements o f t h e system  The p r o b l e m o f o p t i m a l  state  c o n t r o l i n the presence  i s known a s t h e s t o c h a s t i c o p t i m a l  control  31 problem dex.  .  The p r o b l e m h a s b e e n s o l v e d  f o r t h e case o f a q u a d r a t i c  cost i n -  The r e s u l t i n g c o n t r o l l e r c o n s i s t s o f a c a s c a d e c o m b i n a t i o n o f a Kalman  f i l t e r with  the standard  The d e t a i l e d s t a t i s t i c a l  optimum c o n t r o l l e r f o r a l i n e a r data  implement t h e Kalman f i l t e r A suboptimal extensive  about p l a n t  d e t e r m i n i s t i c system.  and measurement n o i s e  required to  i s g e n e r a l l y n o t a v a i l a b l e i n a power  system.  s t o c h a s t i c c o n t r o l l e r i s i n v e s t i g a t e d w h i c h does n o t r e q u i r e  statistical  data  f o ri t s implementation.  The s m a l l number o f  parameters i n the  5.2  c o n t r o l l e r makes o n - l i n e t u n i n g  O p t i m a l Sampled-Data The  control  Regulator  problem formulation  i s g i v e n by  (see  feasible.  f o r continuous  4, (4.18) a n d  Chapter  optimal  load-frequency  (4.20)).  X = A X + B u  (5.1)  (X' Q X + u'  (5.2)  R u)dt  t The  i n t r o d u c t i o n of a d a t a - l i n k i n the r e g u l a t o r y loop  results no  i n a s a m p l e d - d a t a s y s t e m and  longer  T = t j _ - t^_^  be  a set of sampling  a constant  control i s constrained  f o r m u l a t i o n o f an  of the  form  (5.3)  time e q u i v a l e n t s o l u t i o n of  t o be  \  +  1  =~X(t  - ;  optimal  ,  fc  k + 1  ) = ;  (  interval;  t  c o n t r o l i s then  k  < t  T  (5.1)  and  k  +  1  ,  k  +  (5.2)  let system,  instants:  (5.3)  1  t )X k  be  S i n c e A and  interval 't^ = T < t  t  and  In a sampled-data  between sampling  =  ...)  c o n t r o l problem w i t h a sampled-data c o n t r o l  s e t of e q u a t i o n s . f o r the  t ^ , t^,  Q  constant  requires that  (5.1)  i n s t a n t s (t »  sampling  u(x) The  optimal  system  realizable. Consider  the  a continuous  of a power  k  +  t k +  f  Vt  transformed  into  a discrete-  B are t i m e - i n v a r i a n t , the c  +  1  K + 1  i s given  ,T)Bv  k  by  dx  (5.4) =  *\  +  °\  w h e r e $ = c5(T,0). i s t h e s t a t e - t r a n s i t i o n m a t r i x o f  A *,  $(0,0) = I  (5.1):  (5.5)  (I i s t h e u n i t m a t r i x ) , and where T > $(T,t)B dt.  I  J4 The c o s t i n d e x  -'O (5.2) c a n b e e x p r e s s e d  (5.6)  a s t h e sum o f N i n t e g r a l s ,  i n t e g r a l b e i n g evaluated over a sampling  interval.  Using  each  (5.4) i t i s s e e n  that J =  f A ^ k=0  where  + 2X  k\\ k \ V + ;  (5  -  7)  T „  Q = I  *'(T,t)Q  •i, I  M =  $(T,t)dt  (5.8)  $'(T,t)QD(T,t)dt  (5.9)  'o  R=  T  |  [R + D ' ( T , t ) Q D ( T , t ) ] d t  (5.10)  '0  The o p t i m u m f e e d b a c k by  (5.4) a n d (5.7) i s  1  c o n t r o l f o r t h e d i s c r e t e - t i m e problem  given  6  \  =  ~  C  \  (  5  '  n  )  where  C = R M ' + ( R + D KD)~ DK0 _ 1  I  (5.12)  1  The nxm c o n s t a n t m a t r i x K i s t h e s t e a d y - s t a t e s o l u t i o n o f t h e m a t r i x - R i c c a t i difference  equation  \ -' V i - Vi» * ° ' V i ^ ' W e  where K  [  (  +  = 0 i s t h e boundary c o n d i t i o n  0  +A  (5  -  13)  and where  0 0  0 = I - D R  M'  (5.14)  A = Q - M R " M'  (5.15)  _ 1  1  Expressing Chapter 4, ( 4 . 8 ) ,  (5.11) i n terms o f t h e o r i g i n a l system s t a t e s ( s e e  (4.23)) i t i s seen t h a t t h e f o l l o w i n g e q u a t i o n s d e s c r i b e  the optimum sampled-data l o a d - f r e q u e n c y c o n t r o l : x ( T ) = A x ( T ) + B U ( T ) + r AP, p  p  u(x) = u V r " ;  U  k l  " ' S  +  C  k  k  a  + t v +  T  P  k  X  (s, s  A  xelt-.t.,.]  ;  +  K.  (5.16)  krl  te[0,T]  k  (5.17)  k S  ( 5  n 1 \ +  n + 1  "  ( S  'P  +  S  n l  )  A  P  +  d  ( 5  '  '  1 9 )  )  I t i s seen from ( 5 . 1 1 ) ,  1 8 )  (5.20) (5.12) and (5.17) t h a t t h e optimum  c o n t r o l f o r a sampled-data system depends p a r a m e t r i c a l l y on t h e s a m p l i n g time T.  I t i s shown i n r e f e r e n c e  16 t h a t , as T -*• 0, t h e c o n t i n u o u s optimum  c o n t r o l i s t h e l i m i t i n g case o f t h e sampled-data c o n t r o l . j  Example (5.1) C o n s i d e r t h e s i n g l e steam p l a n t d i s c u s s e d i n Chapter 4, Example ( 4 . 4 ) .  The f o l l o w i n g t h r e e  control p o l i c i e s are  considered  f o r a demand AP, = 0.1: a (a) A c o n t i n u o u s c o n t r o l u w h i c h uses c o n t i n u o u s s t a t e i n f o r m a t i o n ( s e e Chapter 4, T a b l e ( 4 . 1 ) ) . (b) A sampled-data c o n t r o l U2 w h i c h uses s t a t e i n f o r m a t i o n s a m p l i n g r a t e o f 1 second ( T = l ) . 1  (c)  Control u^  s  same as (b) above w i t h T=2.  The feedback c o e f f i c i e n t s C' .= [15.62  at a  -.5  (5.20) f o r t h e above c o n t r o l s a r e :  -7.2  -5.4  -4.66]  (5.21)  CJ = [2.04 0.03  -0.18  -0.48  -1.58]  (5.22)  C^ = [.172  0.17  0.11  -0.51]  (5.23)  0.02  The  cost  indicies  f o r t h e t h r e e above cases a r e : . Control  u^  Cost Index The  cost  (5.1).  I t i s seen from F i g .  Hox^ever, t h e r e using  5.3  0.754  index has p r a c t i c a l usefulness  strategy r e s u l t s i n acceptable  1.465  only  2.319  i fthe associated  (5.1) t h a t u ^ g i v e s  control  T h e s e a r e shown i n F i g . the f a s t e s t response.  d e t e r i o r a t i o n i n dynamic performance by  r a t e o f one second.  S t o c h a s t i c Optimum a n d S u b o p t i m u m The  2  system responses.  i s no s i g n i f i c a n t  a sampling  u  stochastic optimal  Control  c o n t r o l problem f o r a d i s c r e t e - t i m e  linear  system w i t h l i n e a r measurements i s d e f i n e d by \ z  i "W  +  k+l  =  H  1 k 2 ^  J.  +  k+1  N  = •=• £ =  w h e r e Wj_ i s a n n - d i m e n s i o n a l  1  x  k+l  \ \ +  v  +  k  w  k+l  ( 5  .  ( 5  x'.Q.x. + u,R.u_ j j j j j j  plant noise vector,  The  and  matrix  the expected value  the following s t a t i s t i c a l E(x  ) = x o  , o '  E ( x ^ w ) =0, k  data  vector.  sequence and  matrix.  problem i s t o determine a r e a l i z a b l e  m i n i m i z e s E(J^),  2 5 )  i s an m - d i m e n s i o n a l  (5.26), Q_. i s a p o s i t i v e s e m i - d e f i n i t e w e i g h t i n g  R. i s a - p o s i t i v e d e f i n i t e w e i g h t i n g J  2 4 )  (5.26)  m e a s u r e m e n t v e c t o r a n d v, i s a n m - d i m e n s i o n a l m e a s u r e m e n t n o i s e k In  '  -  of J  q  , given  c o n t r o l sequence which  t h e measurement  (E(«) i s t h e e x p e c t a t i o n E(w. ) = E ( v . ) = 0 k k  E ( x x') = P , o o o E(VjV ) = 0 k  E(w.w') = Q. J j J  ,  E  < J PW  V  0  E ( v _ v ! ) = R. j J J  (3^)  sequence  operator):  ^  27)  74  §"1 I—  tx UJ  a  UJ  1  o 1 I  in  1  T—I—I—r  10  ' 0  1  1  1  1  1  1  1  1  20 TIME  1  1  1  1  1  1—I  1  1  1  1—1  1  50  30 40 (SECONDS!  '  1  1  60  UJ OJ  •1  !  1 i  1 1—1—1—T—r—T—r  10  1 1 1  20 TIME  Fig. (5.D  1 i  ' •<-• r  30 40 (SECONDS)  '• •  1  1  r~  50  f o r (a) c o n t i n u o u s c o n t r o l , (b) Response comparison t r o l T = 1 and ( c ) sampled d a t a c o n t r o l T - 2. con  60  sampled-data  If  the  statistics  are  sequence i s g i v e n  for be  the  i s the  k  = ~C  k  x  k  ,  k  (k = 0,  c o n d i t i o n a l mean of x  optimum d e t e r m i n i s t i c  -  \ vector  predicted  y  A  =  i s the  k  (  (noise-free)  control  V \  c  control  use ) ;  31  i s the  control  .  * estimate x  The  of the Kalman  k-i  and the  )  error, e  can  k  filter: ( 5  H  ;  i s the  given r e s p e c t i v e l y  gain  Vk  +  H  c  k  (5.28)  optimum f i l t e r = x  k  -  ( 5  d i f f e r e n c e between the measurement v e c t o r  seen from (5.24) - (5.30) t h a t , are  k  V f V i  (  measurement v e c t o r ,  covariance, P  optimal  ... N-1)  where C  \-r k-i k-i D  1,  and  generated r e c u r s i v e l y o n - l i n e by  \  The  the  by u  where x  Gaussian, i t i s known that  k  - x  and  k >  and  gain. the  2 9 )  -  3 0 )  the It i s  error  by  K. e  P  The  k  < " kV Vi k-i  =  x  K  = E(e e ) =  k  k  optimum f i l t e r  k  gain,  (  e  +  W  - w  k-P  d - W ^ A  ( 5  + V l  i ^ , minimizes T r ( P ) ,  the  k  )  (  I  -  - W  t r a c e of the  3 1 )  ( 5 > 3 2 )  error  covariance matrix. From a m a t h e m a t i c a l p o i n t simple.  The  c o n t r o l gain m a t r i x - i n  the n o i s e - f r e e separately,  however, the o n - l i n e  due  to the  problem.  implementation of  x  system i s b o t h i m p r a c t i c a l  determined by  solving  i s determined by  solving,  From a p r a c t i c a l p o i n t  of view,  k  (5.28) r a i s e s s e v e r e problems,  i n power system a p p l i c a t i o n s .  complexity of the  solution is surprisingly  (5.28) can be  o p t i m a l c o n t r o l problem and  an o p t i m a l f i l t e r  particularly  of view the  Even i n the n o i s e - f r e e  case,  optimum c o n t r o l l e r , optimum c o n t r o l o f a power  and  uneconomical.  I t i s e s s e n t i a l to i n v e s t i g a t e suboptimum c o n t r o l l e r s and s i m p l i f y the system model. and  (5.25) may  Consequently,  to  t h e n o i s e sequences i n (5.24)  i n p a r t a r i s e from m o d e l l i n g e r r o r s , thus i n v a l i d a t i n g  G a u s s i a n w h i t e n o i s e a s s u m p t i o n used i n d e r i v i n g  (5.28).  the  Furthermore,  d a t a i n the form (5.27) i s g e n e r a l l y not a v a i l a b l e f o r a power system. i s r e a s o n a b l e , however, to r e t a i n t h e c o n t r o l s t r u c t u r e d e f i n e d by (5.29) and  (5.30) i n a suboptimum s t o c h a s t i c c o n t r o l l e r .  (5.28),  This follows  from the f a c t t h a t , f o r a r b i t r a r y Kj_, (5.29) i s an o b s e r v e r  f o r t h e system  (5.24).  The  observer  (5.29) g i v e s improved e s t i m a t e s i f t h e o b s e r v e r g a i n  to m i n i m i z e  u s e f u l n e s s o f an o b s e r v e r has been shown i n C h a p t e r 4.  an e s t i m a t i o n e r r o r c o s t i n d e x  It  The  i s chosen  (such as T r ( P j _ ) ) .  L e t L be a c o n s t a n t o b s e r v e r m a t r i x g a i n w h i c h r e s u l t s i n a s t a b l e o b s e r v e r and l e t \ where g the  k  i s a scalar gain.  =  k  g  L  A class 0  ( 5  of s t a b l e observers  g  '  3 3 )  '  3 A )  i s d e f i n e d by  stability limits,  S on the s c a l a r g a i n . 0 . with  \  < m  %  <  ( 5  A l l subsequent o b s e r v e r s  An optimum g a i n c o u l d be d e f i n e d i n 0 (see ( 5 . 3 2 ) ) .  Consider  the  Tr(07 P ) - a k  where  and where Q  k  .  k  by a s s o c i a t i n g a c o s t  index  choice k  g  - 23  2  k  g  + y  k  (5.35)  k  *k = \  T  t\  r  r  (  \  L  H  L  k (  (  $  k-l k-l k-l  V V i  P  p  $  k - i * k - i  +  Q  +  k-1  ) ]  V i  ( 5  )  H  k  i s a p o s i t i v e semi-definite weighting matrix.  gain which minimizes  to  = T  -  are c o n s i d e r e d t o b e l o n g  the c o s t i n d e x (5.35) i s g i v e n by  V  +  The  L  ,  )  optimum  3 6 )  77  8  k " hJ k'  (5.37)  a  F i l t e r i n g and minimization o f e s t i m a t i o n e r r o r r e p r e s e n t t h e complete problem,which Furthermore,  1  do n o t , however,  i s t o determine a suboptimal  control.  even though t h e r e a r e o n l y two s t a t i s t i c a l parameters i n  ( 5 . 3 7 ) , i t i s d e s i r a b l e t o reduce f u r t h e r t h e need f o r s t a t i s t i c a l  data.  The dependency o f t h e optimum s t o c h a s t i c c o n t r o l (5.28) on t h e d a t a (5.27) a r i s e s out o f t h e g l o b a l m i n i m i z i a t i o n o f t h e c o s t i n d e x E ( J ) . A s u b o p t i m a l o v  s t o c h a s t i c c o n t r o l can be determined by a l o c a l of an i n s t a n t a n e o u s c o s t The c o s t  (stage-wise) m i n i m i z a t i o n  a s s o c i a t e d w i t h a c o n t r o l d e c i s i o n a t s t a g e k.  must be r e l a t e d i n some m e a n i n g f u l manner t o ( 5 . 2 6 ) . In t h e n o i s e - f r e e case, t h e system dynamics a r e x. = $. x. + D. u, , x = x. , j > k J+l 3 3 J j k k ' -  (5.38)  J  the o p t i m a l c o n t r o l sequence i s g i v e n - b y -u - a '-Gj-x^ , and -(5 .-26) -can -be e x p r e s s e d i n t h e c l o s e d form \  = i  ;  k V k  ( 5  -  3 9 )  31 where  i s the s o l u t i o n of a d i s c r e t e m a t r i x - R i c c a t i equation  . Let  x^ be d e f i n e d by (5.27) where ^_^ i s known from t h e p r e v i o u s s t a g e and l e t x  e, = x , - x-, 3  -  3  (5.40)  3  A s i m p l e e s t i m a t e o f t h e e f f e c t o f t h e e r r o r e^ = e^ = x ^ - x ^ a t s t a g e k on the c o s t  can be o b t a i n e d by t a k i n g t h e p r e d i c t e d v a l u e o f f u t u r e  n o i s e a t any s t a g e j > k t o be e q u a l t o t h e mean v a l u e , which i s z e r o , as g i v e n by (5.27).  Consequently (5.41) x.,, = ( $ , - D,C,)x, = e = (^ - j ; i j * " D  j + 1  C  ) e  =  <  j + 1 ,  k  k)x. k » )  e  ( j  =  k  )  78  are p r e d i c t e d f u t u r e values y(j,k)  of  a n d e^, r e s p e c t i v e l y .  i s the system s t a t e t r a n s i t i o n matrix.  (5.41) i n t o  (5.26) and making  J  k =  S  k  u s e o f (5.39)  +  k \  e  k  X  +  f  I n (5.41),  Introducing  (5.40) and  yields  k  e  \  \  ( 5  '  4 2 )  where =  A  Q, = R  The e x p e c t a t i o n  k  cost  = E(J ) + TrC^E^e^))  into  a d e t e r m i n i s t i c c o n t r o l cost  theeffect  assumed g i v e n ,  term).  ^) - ____> E ( x  It  c a n b e shown f r o m ( 5 . 2 9 ) ,  From (5.37) i t i s s e e n t h a t \ where a  - K  +  = T_./ > i t f o l l o w s k  • when g by  k  = g  (5.28),  k >  The  k-1  l t :  k  term) and a c o s t o f e s t i t h e two c o s t s .  minimizes the third  e  follows  k-l>  i  k  ( g  I t  terra.  Since  x, .. k-1  " °  -  i s  that ( 5  (5.31) and (5.45) t h a t  when g +  k  (see (5.35),  '  (5.36)).  •k = g , consequently, i f k  v k a  ( 5  E(J )= E(J ) k  -  4 7 )  (5.48)  k  considering  the control defined  (5.33) and (5.37) as a suboptimum s t o c h a s t i c c o n t r o l .  cost index  4 5 )  that  E q u a t i o n (5.48) j u s t i f i e s  (5.29),  = g  (5.46) v a n i s h e s  a  k  k  o f t h e average  c h o i c e h a s on t h e s e c o n d term.  x  k  (first  (5.44)  k  o f an estimate  The s e c o n d t e r m c o u p l e s  that t h i s  and E ( x  k  a decomposition  seen from (5.35) and (5.37) t h a t g  Consider  (5.43)  +|Tr(Q P )  k  (5.44) r e p r e s e n t s  mation e r r o r ( t h i r d is  Z ^ ( j , k ) ( Q +C'R C . ) y ( j , k ) j=k  o f (5.42) i s  E(J ) Equation  N  (5.47) h a s two t e r m s .  The f i r s t  term i s t h e cost  79  o f d e t e r m i n i s t i c c o n t r o l and t h e s e c o n d term estimation error.  Both  terms depend on  t r a d e - o f f b e t w e e n t h e two c o s t s , W i t h weighting within  factor.  This suggests  the observer  class 0  i s a cost associated with  and c o n s e q u e n t l y  ot^ r e p r e s e n t i n g t h e t r a d e - o f f o r  the p o s s i b i l i t y  of choosing  so t h a t t h e r e i s a d e c r e a s e  in  g^ a d a p t i v e l y  i n J, . k  g formulate such  there i s a  To  an a d a p t i v e c o n t r o l s t r a t e g y r e q u i r e s t h a t t h e p a r a m e t e r s  (5.47) b e e s t i m a t e d ,  as f a r as t h i s p o s s i b l e , from t h e a v a i l a b l e  measurements. F r o m (5.30) a n d (5.36) i t i s s e e n  a  - Tr(Q L E  that  (y^L') (5.49)  " K E(  The random v a r i a b l e y  k  = 0.  Since  L  '  \  i n (5.49) i s a l w a y s p o s i t i v e  enters  (5.47) a s a w e i g h t i n g  t o r e p l a c e (5.49) b y a n i n s t a n t a n e o u s a where W  i s a positive  k  V  L  k  £ y  k  J  k -\K  factor,  W  y  k  i k -  +  ( g  where g i s a t h r e s h o l d l e v e l  when  i t i s reasonable  (5.50)  k  definite weighting matrix.  \ K  only  estimate  can be r e p l a c e d by t h e i n s t a n t a n e o u s . c o s t  •  and v a n i s h e s  2 g  determined  g  Consequently,  (5.47)  index  k  +  \K  by o f f - l i n e  \ *  -  (5 51) k  computer s i m u l a t i o n  (or by o n - l i n e t u n i n g ) . To p r e v e n t  erratic  g a i n c h a n g e s due t o t h e e s t i m a t i o n  (5.50),  a step s i z e c o n s t r a i n t ( g  w h e r e 6£ i s f i x e d ,  k  "  g  k - l  i s imposed.  )  2  =  6 j l (  8k-l  ) 2  ( 5  -  T h e Optimum a d a p t i v e g a i n i s d e f i n e d t o b e  5 2 )  the g a i n t h a t m i n i m i z e s (5.34).  (5.51) s u b j e c t to the c o n s t r a i n t s (5.52) and  I f (5.34) i s s a t i s f i e d ,  the optimum g a i n i s determined by  the  method of s t e e p e s t descent,which y i e l d s g  G k-1 G  ^ ~  k  3 g  =  g  k-l  If  U  s  g  n  (  k-l  G  j  =  g  )  ]  '  ( 5  5 3 )  k-i  K-rViVi^k-i  4  (5.53) v i o l a t e s  ~  1  k g  \  [  (5.34),  +  k - i  g  *  L  then g^ i s r e p l a c e d by  (  5  k  -  5  5  )  the a p p r o p r i a t e upper o r  lower bound. The  adaptive nature  e r r o r becomes e x c e s s i v e , choosing is  of g,  can be  (5.50) i n c r e a s e s  gj_ to m i n i m i z e e s t i m a t i o n e r r o r .  seen from (5.51).  If estimation  and more x<reight i s g i v e n As  estimation error increases, i t  d e s i r a b l e to p l a c e more weight on the use of measurements.  weighting  to  i s done o p t i m a l l y i f (5.37) i s used.  The  This  t h r e s h o l d can be  so t h a t g* approaches g^_ as e s t i m a t i o n e r r o r i n c r e a s e s .  On  the  set  other  hand, i f the e s t i m a t i o n e r r o r i s a c c e p t a b l e , then more weight i s g i v e n to  choosing  g^ to m i n i m i z e the c o s t of c o n t r o l .  l o n g as the e s t i m a t e The convenience.  i s acceptable, a small c o n t r o l e f f o r t  choice of A simple  p o s i t i v e number.  G  and  reasonable  k - i = y{ \  where b i s a p o s i t i v e number. that e x p l i c i t  s h o u l d be  i n (5.54) i s governed l a r g e l y by choice i s  Another simple p o s s i b i l i t y .  T h i s means t h a t , as  c  \ The  +  b  s g n (  = ^__> w  used.  computational  where w i s a  i s to choose w so t h a t  Sk-f  i  V  -  (5  56)  advantage of the a d a p t i v e approach i s  e v a l u a t i o n of s t a t i s t i c a l d a t a i s not r e q u i r e d .  The  controller  i s "tuned" by o f f - l i n e computer s i m u l a t i o n s .  The small number of tuning  parameters (two) makes o n - l i n e tuning f e a s i b l e . 5.4  A p p l i c a t i o n - Single Steam P l a n t The model of a s i n g l e steam p l a n t given i n Example (5.1),  i s used to evaluate the load-frequency adaptive c o n t r o l l e r .  c o n t r o l c a p a b i l i t i e s of the proposed  The augmented s t a t e model has the form (5.24).  I t i s assumed that frequency and t i e - l i n e d e v i a t i o n s are the only measurements a v a i l a b l e so that the measurement matrix i n (5.25) has the form  The d i s c r e t e c o n t r o l sequence represented by u^ i n (5.24) must be replaced by U  k  = ( u , v ) ' , where (see (5.17) - (5.19)) k  k  \ " Vl Vl . +T  V  k  =  S  ' p k n+1 \ X  +  S  ( s  '  p + S  n+l  (5  ) A P  d'  ( 5  *  -  58)  5 9 )  The $ and D matrices f o r the augmented model are given by (see Chapter 4, (4.40), and  (5.4)) * = A D  A  *  a  [D  $(0,0) = I ,  (5.60) (5.61)  D ]  1  2  where T  (5.62)  *(T,t) B f t a.  0  * • J, T  (T,t)  0  For in  s i m u l a t i o n purposes,  B  (5.63)  • t • dt a  the n o i s e Vectors W  k  and v  k  are  taken  t h e form w. = a I , x. , k w wk k+1  ,  v, = a. I , 2, k v vk k  (5.64)  8:  where (see (4.40)), x' = [x; . AP „ ]. I , and I , are diagonal matrices k pk dk wk vk whose elements are pseudo-random numbers w i t h a uniform d i s t r i b u t i o n between J  -1 and +1.  The s c a l a r s a and a are used to set noise l e v e l . w v . (  I t should be n o t i c e d that the noise (5.64) i s state-dependent. This occurs, f o r example, when the system parameters undergo random d i s turbances.  The c o n t r o l (5.28), with x^ given by the Kalman f i l t e r , whose  gain i s based on the Gaussian s t a t i s t i c s type of noise given by (5.64).  (5.27), i s suboptimum f o r the  Tuning of the time-varying matrix gain to  improve system performance i s i m p r a c t i c a l .  The proposed suboptimum adaptive  c o n t r o l , however, i s e a s i l y tuned to a v a r i e t y of noise s t a t i s t i c s , i n c l u d i n g those defined by (5.64). The design of the adaptive c o n t r o l proceeds i n three stages. The f i r s t stage i s to determine the c o n t r o l gain matrix i n (5^28) f o r the d e t e r m i n i s t i c system.  This has been done i n Section (5.2) (Example (5.1)).  The second stage i s the design of a d e t e r m i n i s t i c observer with a constant gain L (see (5.33)). where the r e s u l t T  i s obtained.  ,  =  The design d e t a i l s are given i n Appendix IV  f.\ 663 (.(.-x.005 nn<; .007  .935  .091 nai -.006.  -.128 _ TOO -.014  .364 TA/. " -.053  The system response f o r an incremental pother demand AP  (5.65) d  = 0.1  using a suboptimum c o n t r o l l e r w i t h the observer gain (5.65) i s shown i n Fig.  (5.2).  I t i s seen that the response meets the s p e c i f i e d conditions i n  that, as t -»- + o o , the frequency d e v i a t i o n and incremental generation approach zero and 0.1, r e s p e c t i v e l y .  In the absence of system noise, i t i s seen that  the c o n t r o l (5.28), where x^ i s the d e t e r m i n i s t i c observer output, gives acceptable dynamic performance.  Fig.  (5.2)  Steam p l a n t  r e s p o n s e . F r e q u e n c y and t i e - l i n e s a m p l e d w i t h T = 1.  measurements  84  The t h i r d s t a g e i s t h e design, o f the a d a p t i v e c o n t r o l l e r w h i c h i s based on changing (5.53) and (5.56).  t h e s c a l a r g a i n g^ a c c o r d i n g t o the s t r a t e g y g i v e n by The s t a b i l i t y l i m i t s  s i d e r a t i o n a r e e a s i l y shown t o be g  ffi  s i z e parameter 6£ i s s e t e q u a l t o 0.2  (5.34) f o r t h e system under.con-  = 0 and g^ = 2, r e s p e c t i v e l y . (see (5.52)).  The" s t e p -  This choice i s a  commonly used compromise between u s i n g a s m a l l s t e p t o meet l i n e a r i t y and numerical s t a b i l i t y requirements of s t e p s . g.  and u s i n g a l a r g e s t e p t o reduce t h e number  The o n l y parameters w h i c h r e q u i r e d e t a i l e d i n v e s t i g a t i o n a r e b and  F o r t h e system c o n s i d e r e d t h e c h o i c e b = 0.2  some p r e l i m i n a r y s i m u l a t i o n s t u d i e s .  appeared r e a s o n a b l e  after  To i n v e s t i g a t e t h e e f f e c t o f d i f f e r e n t  Values f o r t h e t h r e s h o l d l e v e l g, t h e c o s t  J =  60 _ E (X^ ^ k=0  + u£)  i s i n v e s t i g a t e d and averaged over t e n r u n s .  The i n i t i a l  (5.66) frequency d e v i a t i o n  i s taken t o be Aco(0) = 2 and a l l o t h e r i n i t i a l s t a t e s a r e s e t e q u a l t o z e r o . F i g u r e (5.3) i l l u s t r a t e s the average c o s t as a f u n c t i o n o f g f o r d i f f e r e n t noise levels.  From F i g . (5.3) the b e s t average v a l u e i s t a k e n t o be g =  0.5.  The a d a p t i v e c o n t r o l l e r i s now " t u n e d " and i t s e f f e c t on system performance w i t h d i f f e r e n t n o i s e l e v e l s can be e v a l u a t e d . i l l u s t r a t e s the r e s u l t s f o r : (b)  The a d a p t i v e o b s e r v e r .  a lower c o s t .  (a)  Figure  The d e t e r m i n i s t i c o b s e r v e r  (5.4)  ( g ^ = 1).  I t i s seen t h a t t h e a d a p t i v e o b s e r v e r r e s u l t s i n  The t h r e e v a l u e s chosen f o r a c o r r e s p o n d w  t o random changes i n  the elements o f $ and D o f 10%, 20% and 30%, r e s p e c t i v e l y .  Heavy measurement  n o i s e ( a = 1) c o u l d be c o n s i d e r e d t o a r i s e when there a r e f a u l t y measurements v  or f a u l t y data t r a n s m i s s i o n .  F i g . (5.3)  Average cost as,a function of threshold l e v e l  (5.4)  E f f e c t of a d a p t i v e g a i n (a) d e t e r m i n i s t i c (b) a d a p t i v e o b s e r v e r .  observer  The a d a p t i v e o b s e r v e r i s a f i l t e r whose o u t p u t i s an- e s t i m a t e o f the s t a t e o f t h e system. tracking) c a p b i l i t i e s noise.  I t i s of i n t e r e s t to evaluate the f i l t e r i n g (or  of the adaptive observer i n the presence  o f system  F o r t h e e v a l u a t i o n , an i n c r e m e n t a l power demand o f AP^ = 0.1 i s  assumed and t h e n o i s e l e v e l s a r e chosen t o be a = 0 . 2 v  and a = 0 . 1 . w  Figure 6  ,.(5.5) i l l u s t r a t e s t h e system f r e q u e n c y Aw, t h e e s t i m a t e d f r e q u e n c y Aw, the e s t i m a t e d demand AP^, and t h e c o n t r o l s i g n a l u.  I t i s seen from F i g . (5.5)  t h a t t h e f i l t e r output g i v e s a good e s t i m a t e Of t h e average b e h a v i o u r states.  of the  /MO /  A  1 1 20 TIME  (5.5)  '  1 40 CSECONDSJ  Tracking capability  '  '  1  1  of the adaptive  1 60  observer.  6.  For  V e r y s e v e r e d i s t u r b a n c e s i n a p o w e r s y s t e m , an a l g o r i t h m  p r e s e n t e d , i n C h a p t e r 2, f o r parameter  f o r the evaluation  changes i n t h e network  method a p p e a r s the  CONCLUSIONS  s o as t o i m p r o v e  t o o f f e r p r a c t i c a l as w e l l  Liapunov function  o f optimum s w i t c h i n g  could  initial  obtain  approach, i n f i n d i n g  the r e l a t i o n s h i p s  system s t a t e s .  the c r i t i c a l  eliminate  the necessity  switching  Preliminary  to store  for fast on-line  s o l u t i o n of sets  time. compuand  are  relationships  This  of  over  instants  these  a l i m i t e d c o m p u t e r memory.  The  off-line  c u r v e - f i t t i n g methods  b e c o m i n g a v a i l a b l e x^hich w o u l d make i t p o s s i b l e require  system s t a b i l i t y .  between optimum s w i t c h i n g  E f f i c i e n t numerical  i n p a r a m e t r i c form which  instants  as c o m p u t a t i o n a l a d v a n t a g e s  O n - l i n e i m p l e m e n t a t i o n seems p o s s i b l e . tation  is  would  differential  equations. For  less severe disturbances,  s y s t e m n o n - l i n e a r i t i e s and p l a n t of  the large  number o f s t a t e  digital  simulation  r e s u l t s show  i n t e r a c t i o n must be a c c o u n t e d f o r .  variables,  optimal control  o f an  plant  c o n t r o l based  interaction i s physically By  introducing  on a l i n e a r i z e d m o d e l w h i c h  suboptimal f e a s i b l e control  i s obtainable.  The  control  local  control  control.  ating  O f f - l i n e computations are used  the  control  intervention  states.  i s given.  control  instability.  a satisfactory  by a s e c o n d - l e v e l i n t e r v e n t i o n this  i s essential.  neglects  f e a s i b l e b u t can r e s u l t i n system  the concept of t w o - l e v e l  A feasible on-line  i n a p a r a m e t r i c f o r m as a f u n c t i o n  i s augmented  method f o r generto determine  o f t i m e and  When a s y s t e m d i s t u r b a n c e o c c u r s , a s e c o n d - l e v e l c o o r d i n a t o r  generate these parameters  on-line  and  transmit  them b a c k  Because  interconnected  p o w e r s y s t e m i s n o t f e a s i b l e a n d some f o r m o f s u b o p t i m a l c o n t r o l A suboptimum l o c a l  that  to the  initial can  subsystems  which  generate the l o c a l  intervention control signals.  B e c a u s e m o s t o f t h e c o m p u t a t i o n s a r e done o f f - l i n e , c o n t r o l s s u c h as e x c i t e r v o l t a g e c a n be. a c c o u n t e d f o r w i t h o u t  control,  and a s s o c i a t e d s y s t e m  much d i f f i c u l t y .  with h i g h - l e v e l i n t e r v e n t i o n c o n t r o l i n the h e r e a p p e a r s t o be a p r o m i s i n g power  feasible  The m u l t i - l e v e l parameterized  approach  closed-loop  local  tem d i s t u r b a n c e s  controllers.  form.suggested  control results  i n improved  c o n t r o l problem,  i n everyday  due t o r o u t i n e s m a l l  o p e r a t i o n o f power s y s t e m s ,  distur-  i s discussed  B e c a u s e . i n c r e m e n t a l p.ow.er ^demand i s n o t .known a . . p r i o r i , , .the  problem of optimal  load-frequency  c a t i o n of the optimal l i n e a r - s t a t e  c o n t r o l cannot be s o l v e d by d i r e c t regulator control.  A feasible  c o n t r o l i s o b t a i n a b l e by a s t a t e v a r i a b l e t r a n s f o r m a t i o n and by o f t h e i n c r e m e n t a l p o w e r demand. p o w e r demand i d e n t i f i c a t i o n . m a t i o n and i s v e r y  simple.  One m e t h o d i s b a s e d  The o b s e r v e r  approxi-  a c c u r a c y can be A  further  achieved advantage  the s i t u a t i o n where n o t a l l  i s d r i v e n b y m e a s u r e m e n t s o f some  o f t h e s t a t e s and i t s o u t p u t i s an e s t i m a t e p o w e r demand.  optimal  identification  on d i f f e r e n t i a l  Improved i d e n t i f i c a t i o n  o f t h e second method i s t h a t i t can cope w i t h the s t a t e s are measured.  appli-  Two m e t h o d s h a v e b e e n shown s u i t a b l e f o r  t h e second method w h i c h uses a L u e n b e r g e r o b s e r v e r .  incremental  augments  are s i g n i f i c a n t .  bances experienced C h a p t e r 4..  The c o m p o s i t e  c o n t r o l which  I n t e r v e n t i o n c o n t r o l Xtfould o n l y b e a p p l i e d i f t h e s y s -  The l o a d - f r e q u e n c y  by  c o n t r o l scheme  systems.  system performance.  in  non-linearities  to the c o n t r o l of interconnected  The i n t e r v e n t i o n c o n t r o l i s an o p e n - l o o p the  other  o f t h e unmeasured s t a t e s and t h e  In. Chapter 5, a suboptimum s o l u t i o n t o t h e problem  of sampled  d a t a optimum l o a d - f r e q u e n c y c o n t r o l w i t h unknown d e t e r m i n i s t i c i n c r e m e n t a l power demand i s g i v e n .  T r a d e - o f f between system response  r a t e can be e a s i l y s t u d i e d . dered.  The  and  sampling  case o f random system d i s t u r b a n c e i s c o n s i -  The optimum s t o c h a s t i c c o n t r o l l e r i s e x c e s s i v e l y complex t o be  used i n c o n t r o l l i n g a power system.  Furthermore,  the s t a t i s t i c a l  data  and  a c c u r a t e models r e q u i r e d to a c h i e v e optimum performance a r e g e n e r a l l y n o t available.  I t i s e s s e n t i a l , t h e r e f o r e , t o s t u d y suboptimum c o n t r o l l e r s .  A three stage procedure controller.  The  i s g i v e n f o r t h e d e s i g n of a suboptimum s t o c h a s t i c  f i r s t s t a g e c o n s i s t s i n d e t e r m i n i n g the c o n t r o l g a i n f o r a  d e t e r m i n i s t i c optimum c o n t r o l , the second  s t a g e c o n s i s t s i n the d e s i g n o f a  c l a s s of d e t e r m i n i s t i c o b s e r v e r s , t h e t h i r d and f i n a l s t a g e c o n s i s t s i n a d a p t i v e l y c h o o s i n g a s c a l a r o b s e r v e r g a i n so as t o m i n i m i z e an cost index.  An example i s used t o i l l u s t r a t e  instantaneous  the d e s i g n .procedure.  Compara-  t i v e s t u d i e s of system performance f o r d i f f e r e n t parameter v a l u e s show the e f f e c t i v e n e s s o f the d e s i g n p r o c e d u r e be  and t h e ease w i t h w h i c h " t u n i n g " can  accomplished. The proposed  l o a d - f r e q u e n c y c o n t r o l d i s c u s s e d i n Chapters 4 and 5  i s c o m p a t i b l e w i t h an EACC-type c o n t r o l .  The l o a d - f r e q u e n c y c o n t r o l l e r i s  a c t i v a t e d o n l y a f t e r the EACC has d e c i d e d t h a t c o n t r o l a c t i o n i s r e q u i r e d . T h i s p r e v e n t s t h e l o a d - f r e q u e n c y c o n t r o l l e r from a t t e m p t i n g t o c o r r e c t f o r r a p i d l y changing l o a d f l u c t u a t i o n .  An EACC c o n t r o l l e r c o u l d be programmed  to make d e c i s i o n s c o n c e r n i n g the t y p e of c o n t r o l t o be used. l o a d - f r e q u e n c y c o n t r o l xrould be f o r  One mode o f  unknown but d e t e r m i n i s t i c d i s t u r b a n c e s .  The  c o n t r o l l e r f o r t h i s mode c o u l d be o f t h e form d i s c u s s e d i n Chapter  (or  i n a sampled-data form as g i v e n i n S e c t i o n ( 5 . 2 ) ) . A second mode o f  4  c o n t r o l would o c c u r i f the d i s t u r b a n c e s a r e random. mode o f c o n t r o l t o another i s b a s i c a l l y v e r y s i m p l e .  The  change from  one  I t amounts t o s e t t i n g  g^ = 1 i n the case of d e t e r m i n i s t i c d i s t u r b a n c e s and making g^ a d a p t i v e i n the case o f random d i s t u r b a n c e s .  APPENDIX I ,18,19 (Definitions; 1.  The  term area  18  identifies  w h i c h i s t o a b s o r b i t s own responding  t o i t s own  t h a t p a r t o f an  l o a d changes.  n e t w o r k ; i t may "be  A s i n g l e area the  s y s t e m as  No  one  p a r t of the  be  own  Load  areas,  a b s o r b i t s own and  the  boundaries.  the  system they  i t s own i n any  part of  T i e - l i n e power f l o w s  absorbed  occur.  generation  system, i n accordance w i t h  time.  to absorb  the  to  absorb  system  may  allocation practices  are,  therefore,  nor c o n t r o l l e d .  A m u l t i p l e area  operating  together  to  company's  i n w h i c h l o a d changes are  changes t h a t o c c u r  at that p a r t i c u l a r  n e i t h e r scheduled  a s i n g l e company,  collective  system i s expected to a d j u s t  absorbed elsewhere w i t h i n the  prevailing  3.  anywhere w i t h i n t h e i r  system  p a r t o f a company o p e r a t i n g  a w h o l e , r e g a r d l e s s o f w h e r e on  l o a d changes.  power  i n only a given part of the  i n t e r c o n n e c t e d s y s t e m i s one  by  its  be  be  a whole group of companies p o o l e d  l o a d changes t h a t o c c u r  2.  I t may  l o a d c h a n g e s ; i t may  respond t o l o a d changes t h a t o c c u r  interconnected  interconnected  s y s t e m i s one  that  each of which i s expected to a d j u s t  l o a d changes.  c o n s i s t s o f a number i t s own  generation  T i e - l i n e power f l o w s between a r e a s a r e  of  to scheduled  maintained. During  later), area. plants.  the Two  our  study  of the Load-Frequency C o n t r o l problem  term " p l a n t " s h a l l common t y p e s  r e f e r to the  of p l a n t s are  controlling  considered,  (as  generating  n a m e l y , s t e a m and  defined  station in hydro  the  94  4.  M e g a v a r - v o l t a g e (Q-V) c o n t r o l problem i s t h e problem o f c o n t r o l l i n g t h e  r e a c t i v e power i n t h e system.  I n t h i s problem, t h e main c o n c e r n i s t h e v o l t a g e  l e v e l a t t h e d i f f e r e n t buses t h r o u g h o u t t h e system.  Due t o t h e r e l a t i v e l y  f a s t a c t i o n o f v o l t a g e r e g u l a t o r s , i t i s common p r a c t i c e t o assume t h a t t h e bus v o l t a g e s a r e m a i n t a i n e d a t t h e i r n o m i n a l v a l u e s . adopted i n t h i s  5.  thesis.  Megawatt-Frequency  r e a l power.  T h i s assumption i s  ( P - f ) c o n t r o l problem i s t h e problem o f c o n t r o l l i n g t h e  Load-Frequency  cular control job.  C o n t r o l (LFC) i s an a l t e r n a t e term f o r t h i s  parti-  The f o l l o w i n g d e f i n i t i o n o f t h e LFC problem i s a c c e p t e d  by t h e IEEE (AIEE 94, Proposed D e f i n i t i o n s , December 1962): "Load-Frequency  C o n t r o l i s t h e r e g u l a t i o n o f t h e power o u t p u t o f  e l e c t r i c g e n e r a t o r s w i t h i n a p r e s c r i b e d a r e a i n r e s p o n s e t o changes i n system f r e q u e n c y , t i e - l i n e l o a d i n g , o r t h e r e l a t i o n o f t h e s e t o each o t h e r , so as t o m a i n t a i n t h e s c h e d u l e d system f r e q u e n c y and/or t h e e s t a b l i s h e d i n t e r c h a n g e s w i t h o t h e r areas w i t h i n predetermined l i m i t s . "  6.  LFC c r i t e r i a :  g i v e n below.  Some o f t h e c r i t e r i a o f LFC as g e n e r a l l y d e f i n e d a r e  N e g l e c t i n g t h e c o n t r a i n t s on measurements, c o n t r o l , and system 19  dynamics,  then t h e i d e a l " S t a t i c " c o n t r o l c r i t e r i a may be s t a t e d as f o l l o w s  :  M i n i m i z e ( a ) , (b) o r (c) where (a)  A r e a C o n t r o l E r r o r (ACE) = t i e - l i n e d e v i a t i o n + f r e q u e n c y b i a s x frequency d e v i a t i o n [ACE = AP  (b)  + BAf -> 0 ] ,  Inadvertent Interchange ( I I ) = I n t e g r a l of the l i n e [ I I = /AP  dt  0] , •  (I.l) deviation (1.2)  95  (c)  Time D e v i a t i o n  (TD) - -' ^jj x I n t e g r a l o f f r e q u e n c y d e v i a t i o n  [TD = ^ and a l s o  /Af d t ^ 0],  (1.3)  mimimize (d)  A r e a Supplementary C o n t r o l [ASC = f (ACE, I I , TD)  (ASC) = f u n c t i o n o f ACE, I I , TD.  minimum]  (1.4)  APPENDIX I I The t r a n s m i s s i o n system and l o c a l impedance o f F i g . (2.1) can be reduced by Thevenin's theorem t o a s e r i e s impedance r e q u i v a l e n t i n f i n i t e bus v o l t a g e V'.  + j x  and an  The e l e c t r i c a l power, damping c o -  e f f i c i e n t and f l u x l i n k a g e e q u a t i o n s can be reduced t o t h e form g i v e n by ( 2 . 2 0 ) , (2.21) and (2.22) The l  C  C  C  2 ~ 2 B  C  c o e f f i c i e n t s a r e d e f i n e d by t h e folloxtfing r e l a t i o n s :  l  4 B  C  o  ^  S  + B  3  = B^ c o s - B  3  respectively^.  Y  C' = 0.5 B 4  s  i  n  C  cos(y-g)  4  = constant ( d e f i n e d by (2.23))  Cg - Ag cosy  Y  s i n .$  2  7  9  °11  Cg - B^ s i n y c o s g  A  s  i  1 0 " V  C  - -B^ cos y s i n 3  " 6  Y  o  l  D  °12  T  n  D  2  where: = A (A +A./x ) *2 "5 "W  B  0  V'(x - x " ) " / ( x +x )' o q q' qo q e'  c  B  2 - - 1 ; V 4  B  3 " 2 3 ;  B  4 "  A  A  V  A  A  V  (  ( x  A  / X  d- q X  A  V  X  X  2 ^i-^y^v  B  q  ) / x  l 3 o^ q- d  T  v  V  4  )  g = Arctan [ ( x + * ) / r l  d q  ) / x  2  d  X  e  e  v - A r c t a n [ ( x +x ) / r ] q e e  d q X  1  where: A, £ ( / r + ( x ' + x ) )/A x« 1 e d e i d  A  5  A  A  6  2  2  1  - r /  2  e  A,  A l  xJ  (/r +(x x) )/A,x e q e 1 q A. = ( r +x (x +x ))/A,x x' 4 e e q e i q d A  2  2  A  ?  " 4  ( x  q  + X  V d X  e  ) / A  ) A  l d q X  3 o V  X  / x  d  4 x (l-A )/x +A d  A  d  4  3  A. & r + ( x +x )(x'+x ) / x ' x 1 e q •e d e dq 2  In t h e above d e f i n i t i o n s : x  d >  x  = synchronous  r e a c t a n c e i n d and q axes,  respectively.  97  x' x' •= transient reactance i n d and q axes, respectively, d q ' X  d  , X  q ~  S u  b  reactance i n d and q axes, respectively.  s t r a r L S i e n < :  T" ,T" = subtransient do qo The values ( i n pu)  open-circuit time constant i n d and q axes, respectively,  of•the d i f f e r e n t parameters are:  x , = 1.0, x = 0.6, x' = .27, x" = 0.22, x" = 0.29.x = d * q ' d . ' d ' q "* o x" =.04, do G  T " = .07, qo  = .18, V  = 1.05,  x, = .013, t  R = .15,  P+JQ = .753 +  J.03.  x, « .7488, B = X '  9.0, .067, '  APPENDIX I I I As e x p l a i n e d i n S e c t i o n ( 3 . 7 ) , t h e i n i t i a l p l a n t a r e t a k e n t o be z e r o w i t h t h e e x c e p t i o n  c o n d i t i o n s f o r each  o f the frequency d e v i a t i o n .  C o n s e q u e n t l y a (X ) , f o r each o f t h e two p l a n t s c o n s i d e r e d . i s assumed i n t h e m o ' ' J  form  am (Xo ) = Bmo +B ^Au, +B ^AWA m l l o m2 2o M  2 l o m4  2 2o  +B .J AO)- +B /AOU +B HAW-  mi  (III.l)  AW,,  tn5 l o 2o  where t h e c o e f f i c i e n t s B . >i~0, 1. ...5 a r e chosen so t h a t (3.36) i s mi  minimum.  The a l g o r i t h m f o r f i n d i n g B . f o r one p l a n t i s as f o l l o w s (1)  F i n d t h e optimum h*(.t) f o r M d i f f e r e n t s e t s o f i n i t i a l  c o n d i t i o n s ( S e c t i o n (3.6.2) ). F i n d t h e c o e f f i c i e n t s a j»m=l,2,3,4 f o r each s e t o f i n i t i a l m c o n d i t i o n s k-1,2, ...M. (.2)  k  (3)  F o r m=l and k = l , ... M form I  = -wr  in  (III.2)  m  where mo  r  M t m  =  ml  m5  5  [1  Aw  [1  Atow  l0  Aco  2Q  AuJ  o  hJ  l Q  Aw^.A^p]"  W 2 lo  Ato  0  2o  2 2 Aw, Aw~ lo Zo  Aw„ •••Aw -iM l o 20-"  and  The o n l y unknown i n ( I I I - 2 ) i s T^.  F o r M > 6,  r = (wwrV^' w  (4)  Repeat  (3)  f o r m=2,3,4.  i s given by  30  (in. 3)  100  APPENDIX I V The  optimum f i l t e r  K  k 1  \  =  +  \  + 1  +  -\  gain. can. be determined  1  H  1  ( H  +  \ i' W V i H  +  . ->  H  (lv  +  <  \  +  23  V*' **  -\ i '  =  recursively.  \ll  '  H  + 1  by s o l y i n g t h e e q u a t i o n s  The i n i t i a l c o v a r i a n c e m a t r i x , P  Q^,  R^. (k = 0, 1, ...) a r e assumed knox^n.  P  Q ^ , R ^ i s not a v a i l a b l e .  o >  I V  -  3 )  and t h e m a t r i x sequences  I n g e n e r a l , however, t h e d a t a  However, e q u a t i o n s  (IV. 1 ) , (IV. 2) and (IV. 3)  s t i l l prove u s e f u l i n d e t e r m i n i n g a c o n s t a n t o b s e r v e r m a t r i x g a i n L ( s e e (5.33)).  S i m u l a t i o n s t u d i e s o r system o p e r a t i n g e x p e r i e n c e  usually  a l l o w some i n i t i a l guess to be made f o r t h e unknown parameters. reasonable P , Q  Q  q  c h o i c e i s t o take c o n s t a n t p o s i t i v e - d e f i n i t e d i a g o n a l m a t r i c e s  and R . q  Based on t h i s c h o i c e , (IV.2) and (IV.3) can be r e c u r s i v e l y  solved f o r the steady-state convariance matrix P . ^  Equation  (IV.1)  C O  then y i e l d s t h e s t e a d y - s t a t e g a i n m a t r i x and a r e a s o n a b l e to  A  choice f o r L i s  take  H' R " co o F o r t h e example g i v e n i n S e c t i o n ( 5 . 4 ) , t h e c h o i c e L = K  R where b^ =  o  = b. I 1 = 10 \  . '  Q = b„ I" o 2 ' i s made.  (IV.4)  1  P  o  = I  (IV.5)  E v a l u a t i o n o f (IV.4) y i e l d s  (5.65).  An a l t e r n a t i v e approach i s t o a p p l y a d i s c r e t e v e r s i o n o f t h e method p r e s e n t e d i n r e f e r e n c e 35.  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