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

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APPLICATION OF MODERN CONTROL TECHNIQUES TO POWER SYSTEMS by MOHAMMED SAMIR MOHAMMED MINIESY B . S c , A i n Shams U n i v e r s i t y , C a i r o , 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 , Ottawa, Canada, 1967 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n t h e Department of E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o the r e q u i r e d s t a n d a r d R e s e a r c h S u p e r v i s o r . Members o f Committee Head o f Department Members o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF BRITISH COLUMBIA O c t o b e r , 1971 In presenting this thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission f o r extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of <??/^c/^/ca f ecr-r/ sTf The University of B r i t i s h Columbia Vancouver 8, Canada Date A/oO- 2*£f //// ABSTRACT A power sy s t e m may be s u b j e c t e d t o d i f f e r e n t t y p e s o f d i s t u r b a n c e s . The c o n t r o l s t r a t e g y t o be t a k e n i n o r d e r t o p r e s e r v e s y s t e m s t a b i l i t y depends on t h e s e v e r i t y o f t h e d i s t u r b a n c e . F o r v e r y s e v e r e d i s t u r b a n c e s , power s y s t e m s t a b i l i t y can be i m p r o v e d by sudden changes i n t h e e l e c t r i c power n e t w o r k such as t h e i n s e r t i o n of b r a k i n g r e s i s t o r s , g e n e r a t o r d r o p p i n g o r l o a d s h e d d i n g . A u n i f i e d t r e a t m e n t o f 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 t h e s w i t c h i n g i n s t a n t s t o be e l e m e n t s o f a g e n e r a l i z e d 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 a p p l i e d 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 . L e s s s e v e r e d i s t u r b a n c e s can be overcome by e m p l o y i n g 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 c o n t r o l s . The g o v e r n o r c o n t r o l p r o b l e m f o r 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 i n v e s t i g a t e d v i a t h e m u l t i - l e v e l c o n c e p t . A t w o - l e v e l c o n t r o l l e r f o r i n t e r c o n n e c t e d power p l a n t s i s d i s c u s s e d . Each p l a n t has a f i r s t - l e v e l l o c a l o p t i m a l o r s u b o p t i m a l c o n t r o l l e r . The second l e v e l o f c o n t r o l i s an 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 c e n t r a l c o - o r d i n a t o r . I f a sudden s y s t e m d i s t u r b a n c e causes t h e s y s t e m a n g u l a r a c c e l e r a t i o n t o e xceed p r e s e t t o l e r a n c e s , a p r i o r i t y i n t e r r u p t t o t h e c e n t r a l c o - o r d i n a t o r i n i t i a t e s i n t e r v e n t i o n c o n t r o l . A n g u l a r v e l o c i t y d e v i a t i o n s of a l l p l a n t s a r e t r a n s m i t t e d t o t h e c o - o r d i n a t o r . T h i s d a t a i s used t o g e n e r a t e c o e f f i c i e n t d a t a f o r each p l a n t . On r e c e i v i n g i t s c o e f f i c i e n t d a t a , each p l a n t g e n e r a t e s a l o c a l s e c o n d -l e v e l 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 l o c a l c o n t r o l . The L oad-Frequency C o n t r o l p r o b l e m , due t o m i n o r o r r o u t i n e d i s t u r -bances caused by l o a d c h a n g e s , i s i n v e s t i g a t e d . S i n c e t h e i n c r e m e n t a l power demand i n a power s y s t e m i s n o t 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 t h e i i 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 use o f an i n t e g r a l -t y p e c o n t r o l o p e r a t i o n t o meet t h e s y s t e m o p e r a t i n g s p e c i f i c a t i o n s . T h i s . r e q u i r e m e n t i s i n t r o d u c e d i n t o 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 C o n t r o l p r o b l e m p r e s e n t e d i n t h i s t h e s i s . Two methods a r e s u g g e s t e d f o r demand i d e n t i f i c a t i o n . The f i r s t method makes use -of 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 . The second method makes use o f a L u e n b e r g e r o b s e r v e r t o i d e n t i f y unmeasured s t a t e s . The optimum c o n t r o l i s a l i n e a r f u n c t i o n of measured 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 i d e n t i f i e d i n c r e m e n t a l power demand. A method i s g i v e n f o r s o l v i n g , s u b o p t i m a l l y , t h e p r o b l e m of optimum-l o a d f r e q u e n c y s a m p l e d - d a t a c o n t r o l w i t h e i t h e r unknown d e t e r m i n i s t i c power demand or randomly 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 . I t i s shown how t o m o d i f y an optimum c o n t i n u o u s 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 of d i s c r e t e -d a t a t r a n s m i s s i o n and unknown d e t e r m i n i s t i c demand. The cas e o f random power demand and random d i s t u r b a n c e s i s t r e a t e d by i n t r o d u c i n g an a d a p t i v e o b s e r v e r . A t h r e e s t a g e s y s t e m a t i c d e s i g n p r o c e d u r e i s g i v e n . 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 a d a p t i v e o b s e r v e r i s i l l u s t r a t e d by an example. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS . i v LIST OF TABLES.. v i LIST OF FIGURES......; v i i ACKNOWLEDGEMENT .. i x 1. INTRODUCTION 1.1 System Decomposition 1 1.1.1 S p a t i a l Decomposition 1 1.1.2 Sequential Decomposition . . 2 1.2 Background and Thesis Layout 4 2. OPTIMUM NETWORK SWITCHING IN POWER SYSTEMS 2.1 Introduction 8 2.2. Problem Formulation for Optimum Switching Times 9 2.3 Algorithm. 13 2.4 Applications 13 2.4.1 C r i t i c a l Switching Time 14 2.4.2 Optimum Switching Time of a Braking Resistor 20 3. TWO-LEVEL CONTROL OF INTERCONNECTED POWER PLANTS 3.1 Introduction .- 27 3.2 Large Signal Model for Interconnected Power Plants 29 3.3 System Dynamics i n State Variable Form 32 3.4 Two-Level Structure of The Control Problem 35 3.5 On-Line Control Implementation........ 39 3.6 Off-Line Control Design 40 i v Page 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 42 3.6.2 D e s i g n o f The I n t e r v e n t i o n C o n t r o l . . . . 42 3.6.3 D e s i g n o f The A p p r o x i m a t e I n t e r v e n t i o n C o n t r o l . 43 3.7 A p p l i c a t i o n 44 4. OPTIMUM LOAD-FREQUENCY CONTINUOUS CONTROL WITH UNKNOWN DETERMINISTIC POWER DEMAND 4.1 I n t r o d u c t i o n .. .. 49 4.2 P r o b l e m F o r m u l a t i o n 50 4.3 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 54 4.4 A p p l i c a t i o n s 56 4.5 Demand I d e n t i f i e r - L u e n b e r g e r O b s e r v e r 63 5. OPTIMUM LOAD-FREQUENCY SAMPLED-DATA CONTROL WITH RANDOMLY VARYING SYSTEM DISTURBANCES 5.1 I n t r o d u c t i o n • .. 69 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 an Suboptimum 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 6. CONCLUSIONS 89 APPENDIX I 93 APPENDIX I I ....... 96 APPENDIX I I I • . . 98 APPENDIX IV 100 REFERENCES ............ 101 V L I S T OF TABLES _, Page T a b l e " (2.1) 18 T a b l e (2.2) ( P r o b . 1 ( a ) ) 18 T a b l e (2.3) ( P r o b . 1 ( b ) ) 18 T a b l e (2.4) D a t a and I n i t i a l C o n d i t i o n s 21 T a b l e (2.5) D u r i n g f a u l t a d m i t t a n c e s 22 T a b l e (2.6) A f t e r f a u l t a d m i t t a n c e s 22 T a b l e (2.7) ( P r o b . 1 ( c ) ) 22 T a b l e ( 2 . 8 ) N ( P r o b . 1 1 ( a ) ) . . : 24 T a b l e (2.9) ( P r o b . 1 1 ( b ) ) 24 T a b l e ( 2 . 1 0 ( ( P r o b . 1 1 1 ( c ) ) 26 T a b l e (3.1) 47 T a b l e (3.2) 48 T a b l e (4.1) 59 LIST OF FIGURES F i g u r e Page (2.1) One machine i n f i n i t e bus s y s t e m 17 (2.2) Power a n g l e d i a g r a m 17 (2.3) One l i n e d i a g r a m o f a f o u r - m a c h i n e s y s t e m 21 (2.4) Reduced sy s t e m of F i g . (2.3) 21 (2.5) ( P r o b . 1 1 ( a ) ) . . . . . . . . '. 24 (2.6) ( P r o b . 1 1 ( 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 d i a g r a m 30 •' (3.3) O n - l i n e c o n t r o l i m p l e m e n t a t i o n 41 (3.4) 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 change of a n g u l a r a c c e l e r a t i o n 47 (3.5) A n g u l a r and t i e - l i n e d e v i a t i o n f o r = 5, s W^Q = 0 w i t h (a) l o c a l c o n t r o l o n l y , (b) l o c a l p l u s i n t e r v e n t i o n c o n t r o l s 48 (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 and h y d r o a r e a s . . . . . . . 51 (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 a l o a d - f r e q u e n c y c o n t r o l l e r . 57 (4.3) System r e s p o n s e f o r (a) T q = 0.5, (b) T Q = 1 61 (4.4) Response c o m p a r i s o n (a) c o n v e n t i o n a l and (b) p r o p o s e d c o n t r o l s . . 62 (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 meas-urements i 67 (4.6) System r e s p o n s e - Example (4.4) '. .. 68 (5.1) Response c o m p a r i s o n f o r (a) c o n t i n u o u s c o n t r o l , (b) s a m p l e d - d a t a c o n t r o l T = 1 and (c) sampled d a t a c o n t r o l T = 2 74 (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 measurements sampled w i t h T=l • 83 F i g u r e Page (5.3) A verage c o s t as a f u n c t i o n o f t h r e s h o l d l e v e l g 85 (5.4) E f f e c t o f 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 o b s e r v e r (b) a d a p t i v e o b s e r v e r 86 (5.5) T r a c k i n g c a p a b i l i t y o f t h e a d a p t i v e o b s e r v e r 88 v i i i ACKNOWLEDGEMENT I am i n d e b t e d t o my S u p e r v i s o r , Dr. E.V. Bohn, f o r h i s g u i d a n c e and a s s i s t a n c e t h r o u g h o u t t h e p e r i o d o f t h i s r e s e a r c h . Thanks a re due t o Dr. Y.N. Yu f o r h i s i n t e r e s t and h i s v a l u a b l e comments on the d r a f t . R e a d i n g t h e D r a f t by Dr. M.S..Davies and Dr. B . J . K a b r i e l i s d u e l y a p p r e c i a t e d . 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 C u r r a n and E m i l e S t r u y k f o r p r o o f r e a d i n g t h e m a n u s c r i p t , I a l s o w i s h t o than k M i s s L i n d a M o r r i s f o r h e r p a t i e n c e i n t y p i n g the m a n u s c r i p t . The f i n a n c i a l s u p p o r t by the 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, g r a n t 67-3134 (1968-1971), and t h e S c h o l a r s h i p by B.C. Telephone Company (1970) a r e a p p r e c i a t e d . F i n a l l y , i n l i e u o f t h e t i m e I m i g h t have o t h e r w i s e s p e n t w i t h them, I w o u l d l i k e t o thank my w i f e N a d i a and my son Ehab f o r t h e i r p a t i e n c e and u n d e r s t a n d i n g . i x 1. INTRODUCTION I n t e r c o n n e c t i o n between a d j o i n i n g power systems i s an i n e v i t a b l e development i n t h e u t i l i t y i n d u s t r y b e c a u s e i t o f f e r s t h e m u t u a l b e n e f i t s o f i n h e r e n t economy, r e l i a b i l i t y o f o p e r a t i o n and impro v e d s t a b i l i t y . The p r o s -p e c t o f t r a n s f e r r i n g l a r g e b l o c k s o f power o v e r l o n g t r a n s m i s s i o n d i s t a n c e s 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 t i m e z o n e s , t o a s s i s t each i n t u r n t h r o u g h 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 v e r y a t t r a c t i v e and a g r e a t 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 . I n i t s r e a l i z a t i o n , many new problems and d i f f i c u l t i e s a r e c o n f r o n t i n g t h e power u t i l i t i e s . W i t h e v e r i n c r e a s i n g demands f o r e l e c t r i c power and t h e d e s i r e t o improve s t i l l f u r t h e r t h e q u a l i t y o f s e r v i c e , u t i l i t i e s must meet t h e c h a l l e n g e o f s e e k i n g i m p r o v e d methods o f r e g u l a t i n g power g e n e r a t i o n . 1.1 System D e c o m p o s i t i o n An i n t e r c o n n e c t e d , power s y s t e m i s .a complex s y s t e m whose p r i m a r y g o a l i s t o f u r n i s h e l e c t r i c a l e n ergy as r e q u i r e d by customers and as l o n g as. r e q u i r e d . R e q u i s i t e t o t h i s 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 s t a b l e e l e c t r i c a l f r e q u e n c y and v o l t a g e and by c o n t i n u i t y i n t i m e . A g e n e r a l s o l u t i o n t o t h e p r o b l e m o f c o n t r o l l i n g t h e w h o l e s y s t e m such t h a t a l l t h e o p e r a t i n g o b j e c t i v e s a r e s a t i s f i e d a t a l l t i m e s 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 s u c h a c o m p l i c a t e d p r o b l e m . 1.1.1 S p a t i a l D e c o m p o s i t i o n We can t a k e t h e whole power s y s t e m and decompose i t i n t o sub-systems o r a r e a s . The need 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 depend on: (1) The c o m p l e x i t y and c o m p u t a t i o n a l d i f f i c u l t y a s s o c i a t e d w i t h 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 o f g e n e r a t i n g s o u r c e s and t h e a s s o c i a t e d l o a d a l l o c a t i o n and p o l i t i c a l b o u n d a r i e s . (3) O v e r a l l c o n s i d e r a t i o n o f r e l i a b i l i t y . D e c o m p o s i t i o n i s g e n e r a l l y 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 network. F o r c o n t r o l r e l i a b i l i t y i t i s u s u a l l y r e q u i r e d t o have each •area as s e l f - s u f f i c i e n t as p o s s i b l e i n g e n e r a t i n g c a p a c i t y and i n i n t e r -c o n n e c t i o n s u p p o r t . 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 a r e a s c o n n e c t e d by weak t i e s t h e f o l l o w i n g p o l i c i e s a r e u s u a l l y recommended: a) each a r e a has t h e r e s p o n s i b i l i t y o f s e e i n g t h a t i t s i n t e r n a l d i s t u r b a n c e s do n o t d i s r u p t t h e i n t e r t i e s o r i m p a i r s e r v i c e i n o t h e r a r e a s ; b) b o t h t e r m i n a l a r e a s a r e r e s p o n s i b l e f o r c o u n t e r a c t i n g d i s t u r -bances on the i n t e r t i e s betx^een t h o s e a r e a s . 1.1.2 S e q u e n t i a l D e c o m p o s i t i o n Even f o r each area_, 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 a l l o p e r a t i n g s t a t e s . C h a r a c t e r i z a t i o n o f power sy s t e m o p e r a t i o n by o p e r a t i n g s t a t e s has been i n v e s t i g a t e d i n [1], a c c o r d i n g t o " c o n d i t i o n s f o r o p e r a t i o n " . I t i s ^ i n e f f e c t , decomposing t h e t o t a l o p e r a t i n g p r o b l e m i n t o a s e t o f s e q u e n t i a l p r o b l e m s , c o r r e s p o n d i n g e s s e n t i a l l y t o " b e f o r e , d u r i n g and a f t e r " a s e v e r e s y s t e m d i s t u r b a n c e . A c c o r d i n g t o [1], t h e power s y s t e m can be c o n s i d e r e d i n one o f t h e f o l l o w i n g s t a t e s ; (a) P r e v e n t i v e o p e r a t i n g s t a t e ( n o r m a l o p e r a t i o n ) : I n t h i s s t a t e t h e power s y s t e m i s b e i n g o p e r a t e d so t h a t t h e demands o f a l l customers a r e s a t i s f i e d a t s t a n d a r d f r e q u e n c y and v o l t a g e . The c o n t r o l p r o b l e m i n t h i s s t a t e i s t o c o n t i n u e i n d e f i n i t e l y t h e s a t i s f a c t i o n o f customer demand w i t h o u t i n t e r r u p t i o n and w i t h minimum c o s t . (b) Emergency o p e r a t i n g s t a t e : A s i d e f r o m t h e causes t h a t b r i n g s t h e system' t o t h a t , s t a t e , a n emergency o p e r a t i n g s t a t e comes about when t h e r a t i n g o f some components a r e e x c e e d e d , o r when t h e v o l t a g e cannot be m a i n t a i n e d a t a s a f e minimum, o r when t h e sys t e m f r e q u e n c y s t a r t s t o d e c r e a s e t o w a r d a v a l u e a t w h i c h motors w i l l s t a l l , o r when t h e e l e c t r i c a l s y s t e m i s i n t h e p r o c e s s o f l o o s i n g s y n c h r o n i s m . The c o n t r o l o b j e c t i v e i n t h i s s t a t e i s t o r e l i e v e 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 f u r t h e r d e g r a d a t i o n w h i l e s a t i s f y i n g a s p e c i f i e d customer demand. Economic c o n s i d e r a t i o n s become s e c o n d a r y i n t h i s s t a t e . (c) R e s t o r a t i v e s t a t e : T h i s i s the s t a t e where t h e power system w i l l be i n p a r t i a l l o a d o p e r a t i o n , when some customer l o a d s has been l o s t . U s u a l l y t h i s i s the a f t e r m a t h . o f an emergency. 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 s t a t e i s the s a f e t r a n s i t i o n f rom p a r t i a l t o 100 p e r c e n t s a t i s f a c t i o n o f a l l customer demand i n minimum t i m e . I t i s h a r d t o d e f i n e g e n e r a l b o u n d a r i e s between t h e d i f f e r e n t s t a t e s o f o p e r a t i o n . T h i s i s because o f t h e d i f f e r e n t d e s i g n and n a t u r e f o r 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 t h i s p o i n t , c o n s i d e r f o r example two a r e a s o f t h e same c a p a c i t y . Two e q u a l d i s t u r b a n c e s a t b o t h a r e a s may cause i n s t a b i l i t y t o one a r e a w h i l e t h e o t h e r may be a b l e t o s u r v i v e t h e d i s t u r b a n c e . The f i r s t a r e a w o u l d be i n t h e emergency s t a t e and emergency c o n t r o l a c t i o n s h o u l d be t a k e n a c c o r d i n g t o t h e n a t u r e o f d i s t u r b a n c e . 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 be n e c e s s a r y t o d e c e l e r a t e t h e a c c e l e r a t i n g g e n e r a t o r s . F o r the se c o n d a r e a , b e c a u s e o f t h e d e s i g n 4 o f i t s g e n e r a t o r s , s t a b i l i t y c o u l d be m a i n t a i n e d by u s i n g e x i s t i n g c o n t r o l s such, as g o v e r n o r s and/or y o L t a g e r e g u l a t o r s . There i s a l i m i t , however, on t h e amount o f d i s t u r b a n c e a f t e r which, the second a r e a must t a k e emergency c o n t r o l measures. T h i s t h e s i s i s c o n c e r n e d w i t h t h e p r o b l e m o f c o n t r o l i n t h e emergency and p r e v e n t i v e s t a t e . I n t h e n e x t S e c t i o n , we a r e g o i n g t o examine some c o n t r o l a s p e c t s r e l a t e d t o t h e s e s t a t e s from the energy b a l a n c e and dynamic b e h a v i o u r v i e x ^ p o i r t t , a l o n g w i t h the c u r r e n t xrork and t h e s i s l a y o u t . 1.2 Background arid T h e s i s L a y o u t D u r i n g s t e a d y s t a t e o p e r a t i o n o f pox^er s y s t e m t h e r e i s e q u i l i b r i u m between th e m e c h a n i c a l power i n p u t o f g e n e r a t o r s and t h e sum o f l o s s e s and e l e c t r i c a l power o u t p u t . N o n - e q u i l i b r i u m can o c c u r as a r e s u l t o f 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 o r a change i n e l e c t r i c a l o u t p u t . However, from a p r a c t i c a l v i e w p o i n t , i t i s o b v i o u s t h a t changes i n e l e c t r i c a l o u t p u t can o c c u r a l m o s t . i n s t a n t a n e o u s l y b e c a u s e o f change i n the n e t w o r k , whereas the m e c h a n i c a l i n p u t c a n n o t change n e a r l y so f a s t . The d i f f e r e n c e between i n p u t and o u t p u t power i s known as the a c c e l e r a t i n g o r d e c e l e r a t i n g power. The i mmediate s i g n o f n o n - e q u i l i b r i u m i s s y s t e m a c c e l e r a t i o n ( t h e a c c e l e r a t i n g o r d e c e l e r a t i n g power d i v i d e d by t h e i n e r t i a c o n s t a n t ) . Depending on t h e s e v e r i t y o f t h e d i s t u r b a n c e and c o n s e q u e n t l y the a c c e l e r a t i o n , a pox^er s y s t e m can be i n any o f t h e o p e r a t i n g s t a t e s m e n tioned b e f o r e . The c o n t r o l s f o r each s t a t e depend on t h e s e v e r i t y o f d i s t u r b a n c e . The f o l l o x ^ i n g p o i n t s b r i e f l y o u t l i n e t h e c o n t e n t s o f t h i s t h e s i s . 1. For a very severe disturbance caused by change i n the network, a discontinuous form of c o n t r o l i s required to prevent excessive " 2 3 4 5 6 7 system upset. This control problem has been i n v e s t i g a t e d ' ' ' ' ' , but no u n i f i e d technique has been given to determine the optimum switching times. In Chapter 2, the problem of optimum network switching i s 8 investigated . A u n i f i e d treatment of optimum switching i s presented by considering the switching instants to be elements of a generalized c o n t r o l vector. Dynamic optimization i s then applied to determine optimum switching i n s t a n t s . 2. Moderate disturbances can be overcome by employing: (a) faster c o n t r o l of prime movers ; (b) high speed e x c i t a t i o n system with supplementary signals for providing strong damping of swings. The above continuous type of c o n t r o l action can be augmented by discontinuous co n t r o l a c t i o n . The s o l u t i o n to the l i n e a r regulator problem with a quadratic cost index i s w e l l known. Its 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. These are the modelling of high order non-linear systems by a l i n e a r system, the computation, and transmission of large quantities of data between d i f f e r e n t plants. Some form of suboptimal control i s therefore e s s e n t i a l . Presently known suboptimal controllers,such as s p e c i f i c optimum controllers>reduce somex^hat the severity of the problem. However, i t i s d i f f i c u l t to 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 c o n t r o l l e r s without r e q u i r i n g continuous communication of large amounts of data between the plants. S i g n i f i c a n t improvement i n design technique, and system response over the conventional methods has been achieved by 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 to power systems w i t h some degrees, of sub-. , . 10,11,12 o p t i m a l i t y In Chapter 3, the p o s s i b i l i t y of implementing a simple c o n t r o l f o r a l a r g e s i g n a l model of interconnected 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 sec o n d - l e v e l c o n t r o l l e r co-ordinates the sub-13 systems by an i n t e r v e n t i o n open-loop c o n t r o l 3. Minor or r o u t i n e disturbances causes small d e v i a t i o n s from the f i x e d references and are correct e d by governor and/or v o l t a g e r e g u l a t o r c o n t r o l s . 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*. This i s the problem of r e g u l a t i n g the power output of e l e c t r i c generators w i t h i n a p r e s c r i b e d area such that each area s a t i s f i e s i t s own demand. For improved dynamic performance, the l i n e a r r e g u l a t o r s o l u t i o n 14 was adopted by Elgerd . In reference 14, the s t a t e d e v i a t i o n s are expressed i n terms of the f i n a l s t a t e s , the s t a t e s the system i s supposed to 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 unless the demand i s known a p r i o r i which i s not the s i t u a t i o n i n p r a c t i c e . 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 are suggested 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 approximation and i s s u i t a b l e f o r s l o w l y changing demands. In the second method the system s t a t e s are augmented by a demand equation, and a minimal order Luenberger observer i s u t i l i z e d . In the second method, i t i s assumed that the t i e - l i n e and frequency 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 d e f i n i t i o n . are the^ only measurements a v a i l a b l e . A modified form of the regulator problem s o l u t i o n ^ i s considered, which gives, i n conjunction with the demand i d e n t i f i e r , a f e a s i b l e suboptimum control for the LFC problem. 4. A p r a c t i c a l s i t u a t i o n which must be considered i s that the t i e - l i n e deviation may not be 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 the c o n t r o l l e r . In addition, for p r a c t i c a l systems, both plant and measure-ment devices are disturbed by noise. In the l a s t part of the t h e s i s , Chapte 5, these two points are investigated. The sampled-data r e g u l a t o r ^ , which i s s u i t a b l e for continuous systems that have a communication l i n k i n the feedback loop, i s considered. Because of the noise present i n the system, the system states are estimated by a suboptimal f i l t e r . The f i l t e r does not require d e t a i l e d a p r i o r i knowledge of noise s t a t i s t i c s . The f i l t e r i s e s s e n t i a l l y an adaptive observer and i s based on adaptively changing •a s c a l a r gain. Updating the gain .is based ,on-minimizing an instantaneous cost index. The cost index r e a l i z e s a trade o f f between a d e t e r m i n i s t i c performance index and an estimation e r r o r . The proposed scheme i s simple and does not require excessive computer memory or computation time. 2. OPTIMUM NETWORK SWITCHING IN POWER SYSTEMS 2.1 I n t r o d u c t i o n Power system s t a b i l i t y can be improved by discontinuous changes i n 2 the e l e c t r i c power network . The ac t i o n s to be taken i n order to b r i n g the system to e q u i l i b r i u m a f t e r a severe disturbance depend on the nature of the disturbance, b r i e f or prolonged. Sometimes there are d i f f e r e n t a c t i o n s to choose from. The choice of a c t i o n not only depends on the type of d i s t u r -bance, but a l s o on economical and p r a c t i c a l c o n s i d e r a t i o n s . Consider, f o r example, a b r i e f disturbance, 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 of a second, such as a short c i r c u i t c l e a r e d i n normal time. Such 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. This disturbance can be counteracted by a short 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 l o c a t e d at the generating p l a n t . Other di s t u r b a n c e s , f o r example the l o s s of a la r g e l o a d , produce a sustained n o n - e q u i l i b r i u m c o n d i t i o n . The c o n t r o l a c t i o n should l i k e w i s e be sustained,and i n t h i s event i t i s l o g i c a l to disconnect an amount of generating capacity equal to the l o s t load. In the l a t t e r case, prolonged a p p l i c a t i o n of a br a k i n g r e s i s t o r would be e f f e c t i v e , but i t would be uneconomical to provide r e s i s t o r s of the req u i r e d r a t i n g . In the former case, dropping of generation would be inappro-p r i a t e unless, i t could be r e s t o r e d to s e r v i c e r a p i d l y and w i t h accurate t i m i n g . Kimbark has discussed the p o s s i b i l i t i e s of improving power system 2 s t a b i l i t y by c o n t r o l of discontinuous changes i n the e l e c t r i c a l network . In h i s paper he does not present a systematic method f o r e v a l u a t i n g the swi t c h i n g times. A p r a c t i c a l implementation has been reported by the B r i t i s h 9 3 C o l u m b i a Hydro and Power A u t h o r i t y . A b r a k i n g r e s i s t o r i s s w i t c h e d on a t time o f i s o l a t i n g t h e f a u l t e d l i n e and s w i t c h e d o f f a t a l a t e r t i m e . T h i s time was d e t e r m i n e d by d i g i t a l s i m u l a t i o n and n u m e r i c a l e x p e r i m e n t a t i o n . T r a n s i e n t c o n t r o l by u s i n g n e t w o r k p a r a m e t e r s and by s e r i e s A- 5 6 7 c a p a c i t o r s w i t c h i n g has r e c e n t l y r e c e i v e d c o n s i d e r a b l e a t t e n t i o n ' ' ' . The methods d i s c u s s e d i n r e f e r e n c e s 4 and 5 a r e b a s e d on d e r i v a t i o n o f s w i t c h i n g f u n c t i o n s from energy f l o w c o n s i d e r a t i o n s . These methods do n o t appear s u i t a b l e f o r t h e case o f t h i r d o r h i g h e r o r d e r machine models o r f o r m u l t i - m a c h i n e s y s t e m s . R e f e r e n c e s 6 and 7 t r e a t s e r i e s c a p a c i t o r s w i t c h i n g by o p t i m a l c o n t r o l t h e o r y . The c o s t i n d e x used i s minimum t i m e . No s y s t e m a t i c i t e r a t i v e n u m e r i c a l method i s g i v e n f o r o b t a i n i n g t h e s o l u t i o n . A 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 a n a l o g computer x j i t h s u b s e q u e n t d i g i t a l computer r e f i n e m e n t i s used. F u r t h e r m o r e , t h e case o f g e n e r a l n e t w o r k s w i t c h i n g w i t h c o s t i n d i c e s o t h e r t h a n minimum t i m e i s n o t d i s c u s s e d . I t i s t h e p u r p o s e o f t h i s C h a p t e r t o 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 optimum s w i t c h i n g t i m e s . The method i s g e n e r a l and i s a p p l i -c a b l e t o any o f t h e p o s s i b i l i t i e s d i s c u s s e d by Kimbark. I n d e t e r m i n i n g the c r i t i c a l s w i t c h i n g t i m e , t h e method appears t o o f f e r b o t h c o m p u t a t i o n a l as w e l l as p r a c t i c a l a dvantages o v e r s t a b i l i t y approaches b a s e d on t h e c o n s t r u c t i o n o f L i a p u n o v f u n c t i o n s 2.2 P r o b l e m F o r m u l a t i o n f o r Optimum S w i t c h i n g Times A s t e e p e s t d e s c e n t 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 been p r o p o s e d by V a c h i n o . T h i s i s b a s e d on 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 f o r t h e bang-bang c o n t r o l as a p a r a m e t r i c c o n t r o l and augmenting i t w i t h a c o n t i n u o u s c o n t r o l t o f o r m a g e n e r a l i z e d c o n t r o l v e c t o r . A s i m p l i f i e d v e r s i o n o f t h i s method seems t o be a p p l i c a b l e t o the p r o b l e m o f optimum s w i t c h i n g t i m e s i n power s y s t e m s , as f o r m u l a t e d by Kimbark. T h i s i s a consequence of t h e s t r u c t u r a l form o f the d i f f e r e n t i 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 as w e l l as t h e r e l a t i v e l y s m a l l and known number o f s x ^ i t c h i n g i n s t a n t s . The d i f f e r e n t i a l e q u a t i o n s d e s c r i b i n g t h e s t a t e o f t h e s y s t e m have t h e f o r m x = f ( x , a k ) (2.1) where x i s an n s t a t e v e c t o r and i s a s y s t e m p a r a m e t e r v e c t o r w h i c h i s c o n s t a n t f o r t ^ ^ <: T < t ^ and w h i c h can change a t any o f N s w i t c h i n g i n s t a n t s t ^ , t^, ... t ^ . The p r o b l e m t o be c o n s i d e r e d i s t o choose t h 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 i n d e x <j> = <j>(xf), x f = x ( t f ) (2.2) i s m i n i m i z e d at a f i n a l t i m e t ^ = t ^ + ^ 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 . The i n i t i a l s t a t e , x ( t Q ) = x^, i s c o n s i d e r e d known. I n f o r m u l a t i n g a s o l u t i o n t o t h e o p t i m i z a t i o n p r o b l e m i t i s con-v e n i e n t t o c o n s i d e r t h e s e t o f 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 one d i f f e r e n t i a l e q u a t i o n o f t h e f o r m x = F ( x , a., v ) , x ( t ) = x (2.3) o o where A N + 1 F ( x , a, v) = E f ( x , a k ) [ h ( t - t k _ 1 ) - h ( t - t k ) ] (2.4) k = l I n ( 2 . 4 ) , h ( t ) i s t h e u n i t s t e p f u n c t i o n , a i s a c o m p o s i t e p a r a m e t e r v e c t o r formed from t h e a, and k v' = (tv t 2 , ... t N ) (2.5) i s c o n s i d e r e d a g e n e r a l i z e d c o n t r o l v e c t o r ( p r i m e denotes t r a n s p o s i t i o n ) . 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 by ( 2 . 2 ) , ( 2 . 3 ) , (2.4) and (2.5) can be 23 f o r m u l a t e d as an o p t i m a l c o n t r o l p r o b l e m . The s i m p l e s t n u m e r i c a l method o f s o l u t i o n o f o p t i m a l c o n t r o l p roblems i s s t e e p e s t d e s c e n t i n c o n t r o l f u n c t i o n s pace. However, t h e p a r a m e t r i c f o r m of t h e c o n t r o l v e c t o r (2.5) and t h e d i s c o n t i n u i t i e s i n (2.4) make i t n e c e s s a r y t o m o d i f y 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 by 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 change i n s t a t e fix r e s u l t i n g f r o m an i n c r e m e n t a l change 6v i n t h e c o n t r o l v e c t o r ( s ee ( 2 . 5 ) ) . 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 6x + F 6v, fix x V 0 (2.6) where F k x Hi 8x_ Hi 3x 3F n 3F n 8x, 3x (2.7) To a p p l y a s t e e p e s t d e s c e n t method,the i n c r e m e n t a l c o s t ficj> must be e x p r e s s e d i n terms o f t h e i n c r e m e n t a l c o n t r o l 5v. The d e s i r e d e x p r e s s i o n is where fi<p '= (Jr'fix^ = 4>'5v x f v x V _____ L 9 x l 9cj> 3x n J _____ }h ' " ' ' 8 t N (2.8) (2.9) I 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 t o x 5 a n d cp__ i s t h e g r a d i e n t x 12 of <j) with respect to v. In order to f i n d (j> an a l t e r n a t i v e expression must be found for 6<j>. This expression i s obtained by use of the i d e n t i t y (see (2.6)), (2.10) •_j(p'6x) = 6x ' [ p+ rp ] + p'F v6v where p i s the costate vector which i s defined by equating the c o e f f i c i e n t of 6x i n (2.10) to zero: p = -F xp ' (2.11) Integrating (2.10) with the i n i t i a l condition 6X Q = 0 and choosing Pf = -*_>_> (2.12) for the terminal condition, i t i s seen that 6(J) = <j>'6x_ = -p'6x_ = -J. F H*<5v dt x f f f J t v o where H = p'F i s the Hamilt.onian. Since 6v ..is' .constant ,it .follows .by comparison of (2.8) and (2.13) that (2.13) that <j) = - | H dt (2.14) v JX v o The evaluation of 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 f ~ = P'|f- = P'[f(x, a k ) - f ( x , a k + 1 ) ] A ( t - t k ) (2.15) since i - h ( t - t k ) = ~ A ( t - t k ) (2.16) where A(t) i s the unit impulse function. S u b s t i t u t i n g (2.15) into (2.14) y i e l d s v P ' C V I f ^ t , ) - f 2( t l>] P , ( t N ) [ f N ( V " fN+l (V ] (2.17) 13 where, f o r n o t a t l o n a l c o n v e n i e n c e , f ^ ( t ^ ) = f ( x ( t ^ ) , a^) , ^ 4^.^ ^^ ) = f ( x ( t ^ ) , c t k + ^ ) . I t f o l l o w s f r o m (2.8) t h a t t h e s t e e p e s t d e s c e n t a d j u s t -ment i s g i v e n by: where i ' > 0 i s a s t e p s i z e . <5v = -£<j>v (2.18) 2.3 A l g o r i t h m 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 a l g o r i t h m . 1. 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 s t a t e X q i n t e g r a t e (2.3) f o r w a r d i n t i m e from t Q u n t i l t h e s t o p p i n g c o n d i t i o n i s s a t i s f i e d , w h i c h d e f i n e s t ^ . 2. I n t e g r a t e (2.11) backward i n t i m e u s i n g (2.12) t o i n i t i a l i z e t h e c o s t a t e v e c t o r . 3. Use (2.18) t o update 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 s m a l l . 2.4 A p p l i c a t i o n s To i l l u s t r a t e i t s e f f e c t i v e n e s s , t h e g e n e r a l t e c h n i q u e d e v e l o p e d i n t h e p r e v i o u s s e c t i o n i s a p p l i e d t o t h e f o l l o w i n g p r o b l e m s : I . To f i n d t h e c r i t i c a l s w i t c h i n g t i m e f o r : (a) One machine i n f i n i t e bus s y s t e m ( s e c o n d o r d e r model; c o n s t a n t v o l t a g e b e h i n d t r a n s i e n t r e a c t a n c e ) . (b) One machine i n f i n i t e bus s y s t e m ( t h i r d o r d e r model; t a k i n g a c c o u n t o f f i e l d f l u x l i n k a g e c h a n g e s ) . (c) M u l t i - m a c h i n e s y s t e m ( s e c o n d o r d e r model f o r each machine) I I . To f i n d t h e optimum t i m e o f s w i t c h i n g o f f a b r a k i n g r e s i s t o r f o r ( a ) , (b) and ( c ) . 2.4.1 C r i t i c a l Switching Time 20 The system equations f or the multi-machine case are given by: 6 . = a), l l <*>. = ^ T - [ P .-D.w.-P . ] , i =1,2, N (2.19) l M. mi i i e i ' m l where N i s the number of machines. M., P . and D. are the i n e r t i a m I mi l constant, the mechanical input power and the damping c o e f f i c i e n t f o r the i t h machine, r e s p e c t i v e l y . For a s i n g l e machine the e l e c t r i c a l output power P and the damping c o e f f i c i e n t i n (2.19) are defined by, ^F 2 ^F P = C. (—) + (C. cos6 +C 0 s i n 6 ) ( — ) e 1 x 2 3 x o o + C. sin2<S +CC s i n 2 6 +C, cos 26. (2.20) 4 5 6 -C 1 1 + C 1 2 C 1 1 _ C 1 ? D = X 1 2 1 + • 2 cos26 (2.21) where = a flu x linkage proportional to f i e l d f l u x linkage, and x i s r O the d i r e c t - a x i s transient open c i r c u i t time constant. given by: where: In the case of a t h i r d order model the rate of change of i s * p = C 7+C g ,sin<S+C9 cos6+C 1 ( )^ F (2.22) C7 * "CloV0)"C9 c o s 6 ( ° > - C 8 sin6(0) (2.23) In the case of a second order model, ty^, i n (2.20) i s replaced by ii) = ^ ( O ) = constant. The parameters C , C 0, C. _ are the elements of the parameter vector a(see (2.4)) and t h e i r values depend on machine parameters and network impedances (see Appendix I I ) . 20 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 by: N A 2 m P . =E7G..+ E E.E.B.. sin(<5.-6.) (2.24) e i l i i l j 13 l j j r i D. = d. = c o n s t a n t i = 1, 2, . .., N (2.25) i i m where: E. /&. = i n t e r n a l p h a s o r v o l t a g e o f t h e i t h machine, G..+JB.. = i _____ — i j i j s h o r t c i r c u i t t r a n s f e r a d m i t t a n c e 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 at t h e i t h machine bus ( G i i i s n e g l e c t e d i n i i — ( 2 . 2 4 ) ) . The r o t o r a n g l e o f each machine i s f i x e d w i t h r e s p e c t t o t h e e l e c -t r i c a l phase a n g l e o f t h e v o l t a g e b e h i n d t h e t r a n s i e n t r e a c t a n c e . The a n g l e s a r e measured w i t h r e s p e c t 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 speed ( t h e i n f i n i t e bus i n t h e one machine c a s e ) . I n t h e m u l t i - m a c h i n e case t h e p a r a m e t e r s G.. and B.. a r e t h e el e m e n t s o f t h e a v e c t o r . i i 13 F i g u r e (2.1) i l l u s t r a t e s s i t u a t i o n s (a) and ( b ) . A s a l i e n t p o l e s y n c h r o n o u s g e n e r a t o r i s c o n n e c t e d t o an i n f i n i t e bus by two t r a n s m i s s i o n l i n e s . The machine s u p p l i e s a complex power P + jQ a t a t e r m i n a l v o l t a g e V . The i n f i n i t e bus has a f i x e d v o l t a g e V . A sudden t h r e e phase symme-t . 0 t r i c a l s h o r t c i r c u i t t o ground i s c o n s i d e r e d t o o c c u r a t p o s i t i o n ( x) a t t = 0. The f a u l t e d l i n e 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 a f t e r 8 c y c l e s at ti m e t . The f a u l t e d l i n e i s r e c o n n e c t e d a f t e r m c y c l e s w i t h t h e o f a u l t c l e a r e d . F i g u r e (2.2) i l l u s t r a t e s t h e power a n g l e d i a g r a m f o r t h e t h r e e s t a g e s : (a) f a u l t on, (b) f a u l t e d l i n e d i s c o n n e c t e d , (c) l i n e r e s t o r e d 16 w i t h f a u l t c l e a r e d . The c r i t i c a l s w i t c h i n g t i m e ( i n c y c l e s ) i s t h e maximum number o f c y c l e s , m, f o r w h i c h t h e machine s t a y s i n s y n c h r o n i s m w i t h t h e i n f i n i t e bus. An e q u i v a l e n t d e f i n i t i o n can be g i v e n by the e q u a l a r e a c r i t e r i o n (A.. = A.) w h i c h d e f i n e s the c r i t i c a l s w i t c h i n g t i m e t a t <S = <5(t ). I t i s e v i d e n t from F i g . ( 2 . 2 ) , t h a t t h e s w i t c h i n g t i m e e q u a l s t h e c r i t i c a l s w i t c h i n g t i m e , t = t , xjhen t h e c o n d i t i o n s s c u ( t f ) = 0 (2.26) w - w - ° a r e s a t i s f i e d , where t ^ i s t h e t i m e o f t h e f i r s t s w i n g . A p o s s i b l e c o s t i n d e x i s t ^ . The s w i t c h i n g t i m e becomes c r i t i c a l when t ^ i s a maximum s u b j e c t t o t h e c o n d i t i o n s ( 2 . 2 6 ) . However, t h e t e r m i n a l c o n d i t i o n s on the c o s t a t e a r e t h e n unknown and a more i n v o l v e d i t e r a t i v e p r o c e d u r e i s r e q u i r e d . T h i s c o m p l i c a t i o n can be a v o i d e d by c h o o s i n g a p e n a l t y f u n c t i o n c o s t i n d e x 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 o f t h e form 4 = 0 . 5 [ W l W 2 ( t f ) + W 2 ( P m ( t f ) - P e ( t f ) ) 2 ] (2.27) can be chosen, where and W2 a r e p o s i t i v e w e i g h t i n g f a c t o r s . The c h o i c e o f (2.27) a l l o w s t h e s i m p l e a l g o r i t h m g i v e n i n t h e p r e v i o u s S e c t i o n t o be used. The s t o p p i n g c o n d i t i o n , w h i c h d e f i n e s t ^ , i s t a k e n t o be t h e i n s t a n t o f t i m e when one of 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 f ) < 0 , ( 2 2 8 ) P ( t f ) - P ( t f ) > 0 • m r e r Pr 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 ) ; t h e v a l u e o f a f o r v a r i o u s s t a g e s a r e shown i n T a b l e ( 2 . 1 ) . The p r e - and p o s t - f a u l t v a l u e i s a„, d u r i n g t h e f a u l t i t i s a_ and w i t h t h e f a u l t e d l i n e d i s c o n n e c t e d i t Z r GENERATOR TRANSFORMER A 3 xt TRANSMISSION LINES " W f F i g . (2.1) One machine i n f i n i t e bus s y s t e m (ic: defined by Al = A2) F i g . (2.2) Power a n g l e d i a g r a m Table (2.1) bl(a)(b) tt (a)(b) << 1 U2 «3 <*/ *-2 C l 0. 226 0-244 0.359 0.345 . 38* 10~3 C2 -.258 -: 298 -.412 -.424 0 0. 519 0.90 7 0-528 0.921 0 C4 - .046 0.113 0. 059 0-124 0 C5 -.028 -.043 -.045 - .061 0 CS 0.034 0.056 0057 0-082 0 C7 1.258 1.268 • 1.268 1.268 1.268 C8 0-138 0.134 0.215 0-188 0. cg 0.395 0-678 0-428 0- 704 0 C,0 -.156 -. 187 -.159 -.190 -.397 C„ /M, 0. 272 0. 556 • 388 0- 626 0 0-036 -.090 • 056 0.106 0 Table (2.2) (Prob. 1(a) ) I ITERATION ts SECONDS OJ(tf) 0.1157 -.29 -38 0. 200 . 18 -.25 23 9.5 0.35 0-48 0.2167 • .06 -.21 4.8 0-52 0.233 .09 .15 3.4 0 58 0. 250 .03 .08 .69 0.71 *PA=Pm-Pe(<F) I ITERATION Table (2.3) (Prob. 1(b)) ts .1167 .133 • 15 • 1567 OJ (tf) • 087 • 045 .07 .02 .27 .24 .19 -.14 8.2 5.9 4.3 1.9 0-<0 04 3 0 48 19 i s a^. Equation (2.19) i s i n i t i a l i z e d at t = 0 with M = .0212 (pu power sec ), P = .735 (pu), i|v = 9.48 (pu), 6(0) = .9414 (rad), w(0) = 0(rad/sec) and m r a = a_,. (The subscript '1' i s omitted for the one machine case), r Problem 1(a) i s solved by integrating (2.19) with = constant = ili-(0) from t = 0 to t = t . . This y i e l d s x n (t ) = <$(t ) = 1.183 (rad), F o l o o x„(t ) = w(t ) = 4.14 (rad/sec) (a fourth-order Runge Kutta Subroutine i s Z o o used with a step size of 1 cycle = .01667 s e c ) . In (2.27) the choice W^  = W2 = 200 i s made. The results for the proposed method are given i n Table (2.2). 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 after fi v e i t e r a t i o n s . Problem 1(b) i s solved by using (2.22) with the same values at t = 0, defined i n 1(a). Integration during the fa u l t stage yields x^(t_) = 1.183 rad, x.(t ) = 4.14 rad/sec, and x.(t ) = i L ( t ) = 9.195 pu. The 2 O j O r O results -are shown i n Table (2.3). The c r i t i c a l •switching •time-is 1-0 cycles. Problem 1(c) i s the multi-machine problem. When a fau l t occursj the machine having the greater r a t i o of i n i t i a l accelerating power to momentum constant would be expected to accelerate faster than the other machines. In reference 20, the c r i t i c a l switching time i s defined with respect to the fastest machine and i t has been shown that this d e f i n i t i o n i s a useful one. Consequently, i t i s possible to use (2.27) for the m u l t i -machine problem, choosing the variables to be those of the fastest machine. The fault 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. The f a u l t i s assumed permanent. The c i r c u i t breakers open t seconds from fau l t occurence to i s o l a t e the faulted section. A four-machine system with data taken from 20 reference 20 i s considered. The relevant data are shown i n Tables ( 2 . 4 ) , (2.5) and (2.6). The one l i n e diagram of the system i s shown i n F i g . (2.3) and the reduced system i n F i g . (2.4). obtained from the proposed method are shown i n Table (2.7). The c r i t i c a l switching time i s 155 radians (time unit used, T , i n radians for comparison with reference 20, T = 2 r r f t . ) . This i s i n agreement with the numerical r e s u l t using a Liapunov function given i n reference 20. of the two approaches i s not p o s s i b l e . The advantage of the proposed approach i s that i t i s not a f f e c t e d by the order of the model. Governor and voltage regulator e f f e c t s can be included, i f desired. The approach based on constructing a Liapunov function becomes im p r a c t i c a l for system models greater than second order. 2.4.2 Optimum Switching Time of a Braking Resistor The braking r e s i s t o r i s connected to the generator bus at the instant t at which the c i r c u i t breaker opens. The r e s i s t o r i s discon-nected a f t e r a period t ^ . The s i z e of the r e s i s t o r i s usually determined by economic considerations and the power demand under normal operating 2 3 conditions ' . For a given r e s i s t o r and cost index there i s an optimum time of switching t ^ . For the multi-machine system, the r e s i s t o r i s applied to the f a s t e s t machine. A s u i t a b l e cost index f o r d e f i n i n g an optimal t, i s given by: The f a s t e s t machine i n t h i s case i s machine number 3. The r e s u l t s Since the methods are completely d i f f e r e n t , a d i r e c t comparison (2.29) o o T a b l e (2.4) (DATA AND INITIAL CONDITION) GEN. MVA 7APACITY "pu D p u 6 (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 T a b l e (2.5) (DURING FAULT ADMITTANCES) 1 2 3 4 1 -3.582 0-54 6 . 0-0 0.303 1 1.456 2 0.546 -.871 0.0 0-062 2 3 0-027 3 00 0.0 -2.0 0-0 0-0 4 0-303 0.062 0-0 - 1.216 4 0-22 T a b l e (2.6) \ 1 (AFTER 2 FAULT 3 ADMITTANCES) 4 7 -2.310 •664 -656 .751 1 •864 2 •664 •880 • 121 • 062 ' 3 • 029 3 • 656 • 121 -.868 • 062 .104 4 • 751 • 062 • 062 -•984 4 .225 T a b l e (2.7) (Prob 1(c)) ITER. Is (rad) 100 GJ PA 0 1 130 -.84 70-8 2 135 -.046 -.81 64.8 3 140 - . 11 - .76 58.0 4 145 -.12 -.68 47.0 5 155 -•045 -.39 15.4 The c o s t i n d e x (2.29) i s a measure o f t h e mean s q u a r e f r e q u e n c y and r o t o r a n g l e d e v i a t i o n . Augmenting (2.1) by x ^ + ^ = f n + 1_» x n + i ^ t 0 - ' = ^» ^ t seen t h a t t h e c o s t i n d e x i s d> = J = x , . ( t r ) . The c o s t a t e p has t o be aug-n+1 f mented by p ., and t h e H a m i l t o n i a n becomes H = p ' F + p M f ... I t f o l l o w s J r n + l n+1 n+1 from (2.11) and (2.12) t h a t N P - " H x " " S f ^ PthCt-t^-hCt-t^] + f p(t f) - 0 k = l (2.30) n+1 3x L. r n + l v f 3x ,. n+1 n+1 The a l g o r i t h m r e m a i n s 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. C o n s i d e r p r o b l e m TI(a). The b r a k i n g r e s i s t o r has a v a l u e o f 5.55 ( p u ) , e q u a l t o t h e l o c a l l o a d r e s i s t a n c e ( d o t t e d l i n e s i n F i g . ( 2 . 1 ) ) . The b r a k i n g r e s i s t o r i s a p p l i e d a t t = t and t h e c i r c u i t b r e a k e r i s assumed t o r e c l o s e a f t e r 12 c y c l e s . 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 s t a g e s a r e g i v e n i n T a b l e ( 2 . 1 ) . The r e s u l t s o b t a i n e d by t h e a l g o r i t h m f o r t ^ = 90 c y c l e s ( f r o m t ) a r e shown i n T a b l e ( 2 . 8 ) . The swing c u r v e s f o r t h e c a s e s (a) no b r a k i n g r e s i s t o r , (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 t i m e i n t e r v a l o f 18 c y c l e s , a r e shown i n F i g . ( 2 . 5 ) . I n p r o b l e m 11(b) t h e c i r c u i t b r e a k e r i s assumed t o r e c l o s e a f t e r 9 c y c l e s . The r e s u l t s a r e g i v e n i n T a b l e ( 2 . 9 ) . The optimum t i m e i n t e r v a l i s 18 c y c l e s . I n p r o b l e m 1 1 ( c ) t h e b r a k i n g r e s i s t o r has a v a l u e of 0.2 (pu) and i s a p p l i e d t o machine no. 3 a t t h e i n s t a n t t when t h e c i r c u i t b r e a k e r opens (155 r a d i a n s from t ) . The r e s u l t s o b t a i n e d f r o m t h e p r o p o s e d a l g o r i t h m -o 24 - T a b l e (2.8) (Prob E(a) ) ITER. 0 1 .0657 248.7 2 •0883 24 7.9 3 • 1 24 7.6 0 (no BR- ) =373.3 V 3 2 / / 1 • \ \ \ • . / \ N , s ^ -< 30 <^oy 50 720 r/Mf (cycles) -1 • -2 • F i g . (2 .5) - (ProbH(a) ) T a b l e (2.9) ITER. tb(sec) 0 1 • 0667 233-7 2 - •0833 231-5 3 • 1 229-8 4 .1167 228-6 5 • 133 227.9 6 .15 227.7 <f) (no BR.) =354-1 a r e g i v e n i n T a b l e (2.10). The optimum t i m e i n t e r v a l i s 80 r a d i a n s . The s w i n g c u r v e s f o r the c a s e s (a) no b r a k i n g r e s i s t o r , (b) b r a k i n g r e s i s t o r s w i t c h e d o f f a f t e r 200 r a d i a n s , a r e shown i n F i g . (2.6). Note t h a t t h e use o f b r a k i n g r e s i s t o r damps out t h e f i r s t s w i n g d u r i n g t h e f i r s t h a l f second (30 c y c l e s F i g . (2.5)). I t i s p o s s i b l e t o r e a p p l y t h e b r a k i n g r e s i s t o r and d i s c o n n e c t i t a g a i n f o r f u r t h e r damping i n t h e subsequent s w i n g s . T h i s , however w o u l d be u n e c o n o m i c a l s i n c e a b r a k i n g r e s i s t o r o f h i g h e r r a t i n g w o u l d be r e q u i r e d . The subsequent swings can be damped o u t by u s i n g 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 c o n t r o l s . I n t h e p r e v i o u s examples, t h o s e c o n t r o l s were t o t a l l y n e g l e c t e d , and t h e r e f o r e t h e r e i s no damping e f f e c t a f t e r t h e f i r s t s w i n g . I t i s t h e p u r p o s e o f n e x t C h a p t e r t o i n v e s t i g a t e t h e g o v e r n o r c o n t r o l f o r 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 . 26 Table (2.10) (Prob. 11(c)) ITER tb (rad) W~20 1 30 462 2 50 430 3 80 427 10 0(no B.R)= 643.2 • 3. TWO-LEVEL CONTROL OF INTERCONNECTED POWER PLANTS 3.1 Introduction Severe disturbances, caused by sudden changes i n the e l e c t r i c a l network, can be counteracted by discontinuous c o n t r o l as explained i n Chapter 2. However, for optimum system performance, discontinuous controls should be augmented by continuous or modulated controls namely (a) voltage regulator control and (b) governor c o n t r o l . By representing the dynamics of a power system by a l i n e a r model and choosing a quadratic cost index, the above con t r o l problems (a) and (b) can be formulated as the well known i n f i n i t e - t i m e l i n e a r state regulator 24 problem of optimal control theory . The state regulator c o n t r o l problem objective i s to control the system so that the system'states are kept small i n some sense. The s o l u t i o n of t h i s problem leads to an optimal c o n t r o l l e r which i s a l i n e a r function of the states of the 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 state regulator problem to the optimum control 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 bus system i s given i n reference 11. The optimum control was derived from a l i n e a r low-order model and was tested on a non-linear high order model representation for one machine and multimachine systems. Even though t h i s control was 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 attempt was made to f i n d the optimal control for a large s i g n a l model of interconnected machines. The problem of governor control of the prime mover for a one-machine i n f i n i t e bus system can be treated i n a s i m i l a r manner^. The regulator problem s o l u t i o n has also been suggested for the Load-Frequency 14 . * Control problem , and was applied to two interconnected s i m i l a r power plants . Interconnected areas were considered i n reference 14. Here, i t i s assumed that each area has only one plant. 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 t h e p l a n t s , t h e optimum c o n t r o l f o r each p l a n t , i s a l i n e a r c o m b i n a t i o n of i t s ox^n s t a t e s and t h e s t a t e s o f t h e o t h e r p l a n t s . I n r e f e r e n c e 14, f o r t h e example u s e d , t h e g a i n a s s o c i a t e d w i t h t h e s t a t e s o f t h e o t h e r p l a n t Was v e r y s m a l l and was n e g l e c t e d . That i s , each p l a n t uses o n l y l o c a l s t a t e i n f o r m a t i o n and c o n s e q u e n t l y i s c o n t r o l l e d i n a s u b o p t i m a l f a s h i o n . T h i s s u b o p t i m a l s o l u t i o n may be 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 cann o t be ad o p t e d as a g e n e r a l p o l i c y . T h i s 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 i n t e r c o n n e c t e d steam and h y d r o p l a n t s ' . I t was f o u n d t h a t a l o c a l s u b o p t i m a l 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 can r e s u l t i n s y s t e m i n s t a b i l i t y . C o n s e q u e n t l y , c o u p l i n g between p l a n t s and s y s t e m non-l i n e a r i t i e s cannot always be n e g l e c t e d and a good suboptimum c o n t r o l r e q u i r e s t h a t each p l a n t have i n f o r m a t i o n a v a i l a b l e about t h e s t a t e s o f o t h e r p l a n t s . I m p l e m e n t a t i o n o f such c o n t r o l w o u l d be e x p e n s i v e due t o t h e h i g h c o s t o f c o n t i n u o u s communication r e q u i r e d between t h e p l a n t s . The p u r p o s e o f t h i s C h a p t e r i s t o examine t h e p o s s i b i l i t y D f i m p l e m e n t i n g a s i m p l e r 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 i n t e r c o n n e c t e d poxjer p l a n t s . I t w i l l be assumed t h a t the g e n e r a t o r v o l t a g e s a r e h e l d c o n s t a n t by the v o l t a g e r e g u l a t o r c o n t r o l s . E x c i t a t i o n c o n t r o l , however, can be c o n s i d e r e d Xtfithout much d i f f i c u l t y . E o r l a r g e d e v i a t i o n s , t h e dynamics o f t h e s y s t e m can be r e p r e s e n t e d by a s e t o f f i r s t o r d e r n o n - l i n e a r d i f f e r e n t i a l e q u a t i o n s . By i n s p e c t i n g the 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 systems f o r w h i c h 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 d e v e l o p e d . The c o n c e p t 27 o f 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 s i n c e i t appears a p p l i c a b l e t o many complex systems such as power syst e m s . 29 A t w o - l e v e l govenor c o n t r o l o f a power s y s t e m i s p r o p o s e d . The f i r s t l e v e l c o n s i s t s o f i n d e p e n d e n t l i n e a r subsystems ( p l a n t s ) , w h i c h have l o c a l f e e d b a c k 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 t h e subsystems by an i n t e r v e n t i o n o p e n - l o o p c o n t r o l . The i n t e r v e n t i o n c o n t r o l i s used t o compensate f o r a d e c r e a s e i n s y s t e m p e r f o r m a n c e due t o n e g l e c t i n g 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 a t t h e l o c a l l e v e l . F o r ease i n r e a l - t i m e o n - l i n e i m p l e m e n t a t i o n , a s u b o p t i m a l 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 i n i t i a l c o n d i t i o n s and r e a l t i m e . The scheme i s s i m p l e , f a s t and r e q u i r e s a c o m p a r a t i v e l y s m a l l amount o f computer memory. 3.2 L a r g e S i g n a l Model f o r I n t e r c o n n e c t e d Power P l a n t s 1A 25 26 A b l o c k d i a g r a m r e p r e s e n t a t i o n f o r a steam and a h y d r o p l a n t ' i s shown i n F i g . (3.1) and F i g . (3.2) r e s p e c t i v e l y . The s t a t e v a r i a b l e s and c o n t r o l s f o r t h e i t h p l a n t a r e d e f i n e d as f o l l o x ^ s : ( t h e r e a r e 4 s t a t e s f o r t h e steam p l a n t and 5 s t a t e s f o r t h e h y d r o p l a n t ) x1 • l l = A6. l X 2 i A Aw, ! I X 3 i A AP . i n pu power (assuming - t h a t t h e t i m e c o n s t a n t o f t h e g e n e r a t o r 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 t i m e c o n s t a n t s , a l s o 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 ) . X 4 i ^ ^ ^ g i d e v i a t i o n i n g o v e r n o r p o s i t i o n i n pu power (steam p l a n t ) x ^ = 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 (hydro p l a n t ) X 5 i ^ ^ ^ g i d e v i a t i o n i n g o v e r n o r p o s i t i o n i n pu power ( h y d r o p l a n t ) u^ = A P c ^ speed changer p o s i t i o n i n pu power F i g . (3.1) Steam plant block diagram 7 1 -DfS + l •5D/S+7 X5i STiitl x4i GOVERNOR TURBINE GENERATOR *2i Tjj sin(<fj. •<fj) 1 Tm sin (tT/ -<(n) F i g . (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 . f l o w s i n pu power P t g^_, = s c h e d u l e d t i e - l i n e power between t h e i t h and j t h p l a n t . The dynamics o f t h e h y d r o p l a n t are d i f f e r e n t f r o m t h a t o f t h e steam p l a n t because o f Water i n e r t i a . I n t h e h y d r o p l a n t , when t h e g a t e s open, t h e t u r b i n 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 t h e n i n c r e a s e s . The e x t r a b l o c k i n F i g . (3.2) r e p r e s e n t s t h i s s i t u a t i o n . The t i e - l i n e f l o w s a r e g i v e n by P . = E P. . . (3.1) t i , t i j where where p«... = Iv- I |v.||Y..|sin ( 6 . - 6 . ) = T.. s i n 6 . . (3.2) t i j i J i j i j IJ IJ . V ' M ' j I I V (3-3) i s assumed c o n s t a n t , where 6.. = (6.-6.) (3.4) i j 1 J and where v = | | e J ± i s t h e v o l t a g e a t t h e i t h p l a n t bus b a r i n pu. Y.. i s t h e l i n e a d m i t t a n c e between t h e i t h and t h e i t h p l a n t i j — — ( l i n e r e s i s t a n c e i s n e g l e c t e d ) . The e x p r e s s i o n f o r t h e t i e l i n e d e v i a t i o n AP . i s d e r i v e d by c o n s i d e r i n g a d e v i a t i o n A6.. f r o m t h e n o m i n a l 6? . : t x J 5 i j x j P°. + AP„. = 2 T.. s i n ( 6 ? . + A 6 . . ) (3.5) t i " t i j i j i j i j ' s i n c e P°. = t J> . . = £ T, , s i n 6?, • (3.6) t i . t s i i , i n 11 3 3 j ^ i $H 32 I t follows that AP, , = I T. ,.[sin6°, CcosA<S. ,~l)+cos<$?,sinA6. ..] (3.7). t i . xj i j 13 i j 13 3 ^ i Equation (3.7) can be decomposed into 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. T?. ]A6. + .1 [T?, (cosAfi, .-1) t i L3 13 c 1 3 L 13 s 13 + T?, (sinA6.. - A6,)] (3.8) 13 c 13 1 where I.. - 1.. cos 0 . . , X 3 C 13 13' T? 4 = T.. s i n 6°., and X 3 S X3 X3' A6. . = A6 - A6.. X 3 i 3 No unique method exists for decomposing the system into l i n e a r subsystems for which a feedback control i s used and then accounting for subsystem interaction by an open-loop intervention control. ^ o r t' i e i n t e r connected plants discussed here, the decomposition (3.8) proved successful. This decomposition results i n the l o c a l voltage angle deviation being used i n a l o c a l feedback control. 3.3 System Dynamics In State Variable Form The dynamics of each plant can be represented i n the form: X-. = A.X. + B.U. + T.(X, ,X_, ... O (3.9) x x x x x x 1 2 N where X. i s a vector of state Variables of dimension n. (for the model x x used,n^=4 and 5 for steam and hydro plants, respectively )• U. 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 of dimension m.(m.=l f o r i . 1 1 both steam and hydro p l a n t s ) * F, i s a v e c t o r f u n c t i o n of dimension n,, which contains the 1 l coupling and n o n - l i n e a r terms. A^jB^ are time i n v a r i a n t matrices of a p p r o p r i a t e dimensions. The matrices A., B; and the v e c t o r F. are defined f o r each steam and hydro i l l p l a n t as f o l l o w s : Steam P l a n t A. k l 0 [ Y. T?. ]/M. -G_/M. l i e I i l J 3H 0 0 -E./T . i g i B! = [0 0 0 1/T .] i g i 0 1/M -1/T t i 0 0 1/Tt. " 1 / T g i 0 - £ E [T°. (cosA<S. .-1)+T°. (sinA6..-A6.)] M i l j s IJ i j c ' l j i ' J 0 Hydro P l a n t A. 4 i 0 [, LT°, ]/M. 3 ±3 c i 0 0 0 -G./M. 1/M. i i l 0 0 0 0 -2/D ( 2 / D I + 2 / T t ± ) -1/T t i -E./T . 0 0 0 -2/T t. 1/T. . t i -1/T . gi B^ = [0 0 0 0 1/T ] F.= i 1 } C T O i j s ( c o s A 6 i r l ) - r T 0 l j c ( s i n A 6 i j - A 6 . ) i 3 3*± 0 0 0 For N interconnected p l a n t s the dynamics of the system can be w r i t t e n as a composite s t a t e v e c t o r equation of the form (3.10) where the matrices A and B are knox-m b l o c k diagonal time i n v a r i a n t matrices so t h a t , f o r example X - AX + BU 4- F(X),- X ( t ) = X O 0 h o -... o A = A 2 (3.11) i i 0 The composite s t a t e v e c t o r X and the composite c o n t r o l v e c t o r U are composed of the s t a t e vectors X_^  and the 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 that X' = (X^,X 2,...,X^) and U' =(U^, U 2,...,U^). 35 (the prime i s used to denote transpose of a v e c t o r or m a t r i x ) . . S i m i l a r l y P« =CF' F' F') I t should be pointed out here that the c o n t r o l v e c t o r could 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 Would have a two-dimensional 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 can s t i l l be formulated i n the general form (3.10). E x c i t a t i o n c o n t r o l has been 12 shown to give damping e f f e c t on system o s c i l l a t i o n . However, to 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 , only governor c o n t r o l i s considered. 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 r e q u i r e d to f i n d a c o n t r o l v e c t o r U such that 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 disturbance i s minimized without excessive c o n t r o l e f f o r t . The problem can be formulated by i n t r o d u c i n g a cost index i r f c f J = -| j (X'QX + U'RU)dt (3.12) where Q and R are b l o c k diagonal weighting matrices (Q i s a p o s i t i v e or 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 choosing U so that J i s a minimum. The s t a t e X i s subject to the dynamical 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 that 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. (3.13) i i i f o r the l o c a l c o n t r o l l e r s , and that the second-level c o n t r o l l e r co-ordinates 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. The r e s u l t a n t composite c o n t r o l i s r u » -cx+ y C3.i4> where C i s a composite b l o c k diagonal m a t r i x of the form C 3.ll) composed from the C., i = l , 2, ... N. 1 The. f i r s t l e v e l of the c o n t r o l i s obtained by n e g l e c t i n g the coupling f u n c t i o n , F ( X ) , i n (3.10). The problem then reduces to the 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 cost index . The optimum c o n t r o l can be obtained-as a feedback c o n t r o l given by U = -R~ 1B ,KX = -CX (3.15) where K i s the s o l u t i o n of the matri x R i c a t t i equation K = -KA - A'K + KSK - G), K ( t ) = 0 (3.16) where S = BR- 1B f (3.17) Since A, B , Q and R are block diagonal m a t r i c e s , the s o l u t i o n f o r K i s a block diagonal matrix (from •(3.16) and (3.17)). The s o l u t i o n f o r each bl o c k R\ i s obtained by s o l v i n g K. = -K.A. - A!K.•+ K.S.K. - Q. , K.(t_) = 0, l i i i i I I l l i f ^ _3) S. = B.RTV i = 1,2,...N 1 1 1 1 For A, B , Oand R time i n v a r i a n t , l e t K be the steady s t a t e s s o l u t i o n of (3.16) 0 = -K A - A'K + K SK - Q. (3.19) s s s s 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 _ 1B'K (3.20) s i s used as a suboptimal l o c a l gain matrix. The choice (3.20) i s optimum i f t ^ = 0 0 i n (3.12) and i t of t e n gives an e x c e l l e n t suboptimal c o n t r o l . There i s an increase in- the cost index a s s o c i a t e d w i t h the 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 which haye been, neglected i n the suboptimal choice (3.20). To account f o r these 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 of the form V = t f V h '(3.21) i s introduced. To determine the optimum h, the o r i g i n a l problem must be reformulated. S u b s t i t u t i n g (3.14),(3.20), and (3.21) i n t o (3.10) gives (3.22) (3.23) X = (A-SK )X + F + Sh, X(0) = "X U = (-K X+h) s The problem 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 that the cost index (3.12) i s a minimum subject to the dynamical c o n s t r a i n t (3.22). The Hamiltonian f o r t h i s problem i s H = p* [ ( A - S K j X + F + Sh] - - i X'QX ~ (-K X+h)'S(-K X+h) 2 s S (3.24) (3.24) Applying the necessary c o n d i t i o n s of optimal c o n t r o l theory y i e l d s p = -H. = - (A-SK +F ) ' p + QX-K S(-K X+h) , X s X s s p ( t f ) = 0 (3.25) 0 = H h = S(p+K gX-h) X\fhere F £ 9^ 3 X N 9 X l " ' 9 X N H h ~ 9H_ 3h-9H Sh. N (3.26) (3.27) 38 Equations (3.22), (3.25) and (3.26) define a two-point boundary Value problem Xtfhose ^solution gives the optimum intervention control h. Since the equations are non-linear, i t e r a t i v e methods are required to obtain the solution. Consequently, since on-line implementation i s desired, optimal control i s not feasible and a good suboptimal control policy must be determined. I f h = h = 0, then l o c a l control only i s applied. Let X, p be the solution of (3.22) and (3.25) for this control. A fundamental result of optimal control theory i s that an incremental control 6h results i n an incremental cost index,given by fcf 6J = - | H h 6h dt '(3.28) t o where H, t= s(p + K X) . (3.29) h s The intervention control i s to be chosen so that system performance i s improved. That is,so that <SJ, the incremental cost index, i s decreased. Since S i s positive 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 size parameter, accomplishes this objective. In control theory terminology (3.30) i s a steepest descent increment i n function space. For computational reasons i t i s convenient to derive an equation for' A - -q = p + K X • . •" (3.31) s With the a i d of (3.19), (3.22), (.3.25) and (3.31) i t i s -seen that q = -(A-SK + F )'q+F'K X+K F , 4 C O = K X ( t f ) C3.32) (the terms with an overbar: are evaluated for the nominal X, which i s the s o l u t i o n of (3.22) for h - h = 0). The algorithm for fi n d i n g h i s simple one. Equation (3.22) i s integrated i n the forward d i r e c t i o n , taking h = 0. Equation (3.32) i s then integrated i n the backward d i r e c t i o n to f i n d q. The i n t e r v e n t i o n c o n t r o l i s then h = h + Sh = J£q. The optimum value i of ifwhich minimizes J can be found by a simple d i r e c t search procedure. The d e t a i l s of t h i s algorithm, i s given i n Section (3.6.2). However, on-line implementation of t h i s algorithm for computing h(t) i s not p r a c t i c a l . The next section considers t h i s problem. 3.5 On-line Control Implementation The intervention control h(t) i s a function of time and i n i t i a l conditions of a l l the subsystems (see (3.22) and (3.32)). Suppose that t h i s function i s knox<rn. To generate h ( t ) , i n i t i a l conditions must be transmitted to the second l e v e l . After generating h ( t ) , the second-level coordinator continuously transmits the components of h(t) back to the subsystems. A more f e a s i b l e way of implementing t h i s control i s to transmit a minimum amount of information between the l o c a l c o n t r o l l e r s and the c e n t r a l co-ordinator. In order for t h i s to be accomplishedj,h(t) must be approximated i n the form h.(t) = \ = g.(a.(X o),t) (3.33) where g. i s a non-linear function of time with unknown c o e f f i c i e n t s . Each 4C c o e f f i c i e n t i s non-linear function of i n i t i a l conditions. The c o e f f i c i e n t s are chosen such that | |h. (t) - ^ | | 2 i s minimum. A s u i t a b l e a l g e b r a i c form f o r g^ and i s given i n Section 0.6.3) and Appendix I I I , r e s p e c t i v e l y . On-line implementation would be as follows. Each subsystem i s l o c a l l y c o n t r o l l e d by a feedback control U. = -C.X.. At the time t = t of J J 1 i i o a system disturbance the i n i t i a l conditions of each subsystem are transmitted to the cen t r a l co-ordinator which generates the c o e f f i c i e n t s for (3.33). The intervention control h^(t) i s generated l o c a l l y by a function generator 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 (3.33), preliminary o f f - l i n e computations are ^required to determine the unknown c o e f f i c i e n t s which are stored by the c e n t r a l co-ordinator. On-line control requires r e l a t i v e l y few m u l t i p l i c a t i o n s and additions at the cen t r a l co-ordinator l e v e l ( a f t e r r e c e i v i n g X q from the •subsystems) to generate 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 , each i s a vector. For the system under i n v e s t i g a t i o n a dimension of four f o r each vector was found adequate for generating the h^s. 3.6 Off-Line Control Design Implementing (3.23) and (3.33), the control design follows three steps; a) design of the feedback co n t r o l u_^  = ~C/K.,b) design of the i n t e r -vention c o n t r o l h ( t ) , and c) design of the approximate in t e r v e n t i o n c o n t r o l h. The design d e t a i l s f o r each of the three steps follow. LOCAL CONTROLL NO.I ER Xl SUBSYSTEM N0.1 LOCAL LOCAL CONTROLLER CONTROLLER NO. 2 NO. N I X2 U2 XN UN — _ _ SUBSYSTEM SUBSYSTEM NO.2 NO.N F i g . (3.3) On-Line c o n t r o l implementation 42 3 . 6 . 1 D e s i g n o f t h e Feedback. C o n t r o l (IT ~ '-C^X ) S y n t h e s i z i n g t h i s c o n t r o l r e q u i r e s s o l u t i o n s o f (3.19) t o o b t a i n K « S i n c e c o u p l i n g i s n e g l e c t e d a t t h i s l e v e l , t h e s o l u t i o n r e d u c e s t o s o l v i n g (3.19) f o r each p l a n t s e p a r a t e l y . One method o f 28 s o l v i n g (3.19) i s by the Newton-Raphson t e c h n i q u e . F o r f a s t c o n v e r g e n c e , '? t h e i n i t i a l guess f o r K- ^  i s o b t a i n e d by i n t e g r a t i n g (3.18) f o r a s h o r t p e r i o d ( e . g . 10 s e c o n d s ) . C. i s g i v e n by C. = R.^BlK .. .f l ft • ' • 1 1 - 1 -Si 3 . 6 . 2 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 h ( t ) / The open l o o p 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 i n i t i a l c o n d i t i o n s . G i v e n X^, t h e optimum v a l u e h * ( t ) i s o b t a i n e d as f o l l o w s : (1) I n t e g r a t e (3.22) f o r w a r d f r o m t=t t o t = t f w i t h h=h= 0*The t r a j e c t o r y o b t a i n e d i s t h e n o m i n a l t r a j e c t o r y X ( t ) . E v a l u a t e t h e f i n a l c o n d i t i o n o f ( 3 . 3 2 ) , q ( t r ) = K X ( t ^ ) . r s f (2) I n t e g r a t e (3.32) backward from t = t - t o t=t ( w i t h X ( t ) = X ( t ) ) t o and s t o r e q ( t ) . (3) Choose a s t e p s i z e £> 0,and w i t h h ( t ) = £q(t) e v a l u a t e t h e c o s t i n d e x J . (4) F i n d t h e optimum s t e p H = ^ a t w h i c h J i s minimum. ( T h i s i s e a s i l y done by i n c r e m e n t i n g &); ' (5) 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 * = £ Qq(t). Note t h a t o n l y some components o f h ( t ) (and c o n s e q u e n t l y q ( t ) ) a r e t o be s t o r e d . F o r one c o n t r o l component i n each p l a n t , f o r example, o n l y one component o f h ( t ) wou l d be r e q u i r e d t o be s t o r e d s i n c e t h e o t h e r components a r e m u l t i p l e d by z e r o s ( s ee (3.17) f o r S and (3 .22) f o r S h ) . T h e r e f o r e , f o r the case of two interconnected power plants having the model of Section (3.2), only two components f o r h. (±) ( i a 1,2) are required. 3.6.3' Design of the "Approximate Intervention Control h As pointed out before.; the previous algorithm f o r h(t) cannot be implemented f o r on-line 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 function of known algebraic form, on-line implementation i s f e a s i b l e . Suitable functions are poly-nomials or s p l i n e s . By p l o t t i n g the int e r v e n t i o n controls as a function of time for each set of i n i t i a l conditions, k, i t was noticed that a cubic polynomial i n t could be f i t t e d to each h^(t) (the superscript k denotes that h^(t) i s evaluated f o r the set of i n i t i a l conditions k). Consider the case of approximating one int e r v e n t i o n c o n t r o l , f o r example h^(t) of the f i r s t plant. The same procedure i s followed f o r the int e r v e n t i o n controls of the other plants. (1) Let h^(t) be approximated by a cubic polynomial i n t: *u(t) = E A™"1 (3.34) i . m m=l A c u r v e - f i t t i n g routine based on a least-square3approach can be employed to f i n d the c o e f f i c i e n t s a^. This i s done f or d i f f e r e n t sets of i n i t i a l m conditions k=l,2,...M. (2) In general, the c o e f f i c i e n t s are functions of the i n i t i a l " ' m k conditions. Consider the mth c o e f f i c i e n t a . Let a be approximated by — m m J al^ajlh (3.35) m m o 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 . I t i s m o o r e q u i r e d to f i n d the c o e f f i c i e n t s i n a (X ). such t h a t • m o I|a k - a k | I 2 , m=l,2,3,4 , k=l,2,...M (3.36) m m ' i s minimum f o r a l l k. The choice of the f u n c t i o n (3.35) i s a r b i t r a r y . A choice of (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 given i n Appendix I I I . For o n - l i n e implementation the c o e f f i c i e n t s (3.35) are s t o r e d by 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 from each subsystem, the c e n t r a l co-ordinator computes four parameters a^, a^, and f o r each p l a n t and transmits them to the d i f f e r e n t subsystems. Each subsystem then generates i t s own i n t e r v e n t i o n c o n t r o l I K . 3.7 A p p l i c a t i o n 25 An example of _an interconnected steam and hydro p l a n t i s considered. Let the s u b s c r i p t s 1 and 2 denote the steam and hydro p l a n t s , r e s p e c t i v e l y . The f i r s t s i g n of impending t r o u b l e i n a power system d i s t u r b e d by l o s s of l o a d i s a c c e l e r a t i o n . The speed d e v i a t i o n (time i n t e g r a l of a c c e l e r a t i o n ) appears l a t e r , and the angular change ( i n t e g r a l of speed) s t i l l l a t e r (see F i g . ( 3 . 9 ) ) . Because of the r e l a t i v e l y long time constants a s s o c i a t e d w i t h the governor and t u r b i n e , t h e i r outputs do not change in s t a n t a n e o u s l y and can be assumed constant during the very short p e r i o d f o l l o w i n g a disturbance. Consequently, d e t e c t i o n of angular 2 a c c e l e r a t i o n i s the most promising 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 disturbance i s assumed to be a speed d e v i a t i o n from nominal at t=t. , a l l other s t a t e s are assumed zero. o The cost index i s chosen as 20 J a I j ^ A 6 1 + 4 + A 6 2 + A w 2 + U l + U 2 ^ d t That i s « 2 " , R^=l, R 2=l. The choice t ^ = 20 seconds i s made s i n c e the 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 represented by P i g . (3.1) 25 and F i g . (3.2) are as f o l l o w s M 1 = .04, G^.01, T f c l = 0.5, T = 0.5, E ] [ = .03 _ = 0.5. 1 g M 2 = .03, G 2 = .008, D 2 = 0.5, T f c 2 = 0.5, T « 1.2, E 2 = .013 T 1 2 = 0.05 , « J 2 = -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 obtained as expla i n e d i n Sec t i o n (3.6.1) and are given by: IL = -0.336 h6,-.607 Aw..-.416 AP ,-1.6 AX , 1 1 1 g l g l U 2 = ,515A6 2-l.l6Aw 2-11.5 AP g 2~41.6Ag 2~9.22 A X g 2 For the design of the i n t e r v e n t i o n c o n t r o l , 9 pets of speed d e v i a t i o n s are considered. Three d i f f e r e n t sets f o r each of the f o l l o w i n g cases are taken: (a) Disturbance at p l a n t (1) only. (b) Disturbance at p l a n t (2) only. (c) Disturbance at both p l a n t s simultaneously. Following the design procedure of Se c t i o n (3.6.3), i t x^as n o t i c e d t h a t h * ( t ) i s a l m o s t z e r o a f t e r 10 s e c o n d s . C o n s e q u e n t l y , the f i t t i n g r o u t i n e f o r (3.39) was g i v e n d a t a up t o 10 seconds o n l y . T h e r e f o r e , t h e g e n e r a t i o n o f h a t t h e l o c a l l e v e l i s done f o r 10 seconds a f t e r w h i c h i n t e r v e n t i o n c o n t r o l i s removed. The c o e f f i c i e n t s t o be s t o r e d a t the c e n t r a l c o - o r d i n a t o r w h i c h a r e used f o r g e n e r a t i n g h (see A p p e n d i x I I I ) a r e g i v e n i n T a b l e ( 3 . 1 ) . F o r the 9 s e t s o f i n i t i a l c o n d i t i o n s , a c o m p a r i s o n i s made between the c o s t f u n c t i o n J f o r the c ases (a) u s i n g l o c a l c o n t r o l o n l y ( J ); (b) u s i n g l o c a l p l u s i n t e r v e n t i o n c o n t r o l h * ( t ) , ( J , );and ( c ) u s i n g l o c a l b p l u s i n t e r v e n t i o n c o n t r o l h, w ) • The r e s u l t s a r e g i v e n i n T a b l e ( 3 . 2 ) . To show th e e f f e c t i v e n e s s o f o n - l i n e g e n e r a t i o n o f h ( t ) , a d i f f e r e n t s e t ( ^ o ^ ' °J2o=^^ W a S t e s t e ( 3 ( T e s t :> T a b l e ( 3 . 1 ) ) . W i t h l o c a l c o n t r o l o n l y the s y s t e m was u n s t a b l e . By i n t r o d u c i n g t h e i n t e r v e n t i o n c o n t r o l h t h e s y s t e m was s t a b i l i z e d . F i g u r e (3.5) shows t h e s e r e s u l t s . 47 SPEED I DETECTION • DELAYS F i g . (3.4) 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 change o f a n g u l a r a c c e l e r a t i o n T a b l e (3.1) SET NO-INITIAL CONDITIONS kJ/O, 0020 Ja h 1 o .446 .360 0.364 2 2,0 1.957 1.470 I 487 3 6.233 3-427 3.611 4 0, - 1.33 1. 625 1. 435 1. 453 5 0, -2.667 6.191 5-610 . 5.619 6 0,-4 13-95 12.738 12. 785 7 1, - 1.33 1. 5 1.45 1. 459 e 2, -2.657 5.9 5. 690 5.720 9 3,-4 13.73 13.3 13.35 TEST 5,0 10-4 xlO4 14.84 F i g . (3.5) A n g u l a r and t i e - l i n e d e v i a t i o n s f o r = 5, a>2Q = 0 w i t h (a) l o c a l c o n t r o l o n l y , (b) l o c a l p l u s i n t e r v e n t i o n c o n t r o l s . 4. OPTIMUM LOAD-FREQUENCY CONTINUOUS CONTROL WITH UNKNOWN DETERMINISTIC POWER DEMAND 4.1 I n t r o d u c t i o n Power s y s t e m d i s t u r b a n c e s c a u s e d by l o a d - f l u c t u a t i o n s r e s u l t i n changes i n t i e - l i n e 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 some form o f l o a d - f r e q u e n c y c o n t r o l . The form o f Lo a d - F r e q u e n c y C o n t r o l (LFC) p r e s e n t l y i n use i s b a s e d on an e r r o r s i g n a l x^hich i s a l i n e a r c o m b i n a t i o n of t h e n e t i n t e r c h a n g e and f r e q u e n c y e r r o r s . A s i m p l e i n t e g r a l - t y p e c o n t r o l a c t i o n d r i v e s t h e e r r o r s i g n a l s t o z e r o . Modern o p t i m a l c o n t r o l t h e o r y 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 h i g h - o r d e r s y s t e m s . The a p p l i c a t i o n s o f t h e s e t e c h n i q u e s t o improve LFC i s -i . . . 14,17,25,32 , , _ , 14 c u r r e n t l y r e c e i v i n g i n c r e a s i n g a t t e n t i o n . E l g e r d and Fosh a a p p l i e d t h e s o l u t i o n o f t h e s t a t e - r e g u l a t o r p r o b l e m t o t h e LFC p r o b l e m . T h i s a p p r o a c h , hoxtfever, r e q u i r e s knowledge o f t h e new s t e a d y - s t a t e o p e r a t i n g p o i n t . C o n s e q u e n t l y , t h e c o n t r o l 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 i n f o r m a t i o n r e q u i r e d f o r i t s i m p l e m e n t a t i o n i s n o t a v a i l a b l e . F e a s i b l 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 t h e i d e n t i f i c a t i o n o f t h e i n c r e m e n t a l power demand i n o r d e r t o o p t i m a l l y compensate f o r l o a d -f r e q u e n c y d e v i a t i o n s . 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 where a m o d i f i e d Kalman f i l t e r x^as i n t r o d u c e d t o p e r f o r m t h e i d e n t i f i c a t i o n . The approach s u g g e s t e d i n 17 h a s , hox^ever, s e v e r a l s h o r t c o m i n g s . The a s s u m p t i o n i s made t h a t t h e i n c r e m e n t a l t i e - l i n e power AP i s a known ( t h r o u g h measure-ments) f u n c t i o n o f t i m e . A c t u a l l y , AP depends on t h e s y s t e m s t a t e and t h i s must be a c c o u n t e d f o r i n t h e o p t i m a l c o n t r o l f o r m u l a t i o n o f t h e LFC p r o b l e m . A r a t h e r s e r i o u s s h o r t c o m i n g i n 17 i s t h e manner i n w h i c h t h e Kalman f i l t e r i s u sed. The Kalman f i l t e r o u t p u t g i v e s an e s t i m a t e x o f a s t a t e x. N o r m a l l y , i t i s t h e f i l t e r o u t p u t x w h i c h i s used f o r i m p l e m e n t i n g c o n t r o l a c t i o n . I n 17 however, t h e f i l t e r i n p u t x, i s used i n s t e a d o f t h e f i l t e r o u t p u t . I n v a r i a b l y , t h e f i l t e r i s r e a l i z e d i n a d i s c r e t e f o r m on a d i g i t a l computer. The n u m e r i c a l g e n e r a t i o n o f x from sampled d a t a c o u l d i n t r o d u c e h i g h l y u n d e s i r a b l e n o i s e p r o b l e m s . The u n c o n v e n t i o n a l r e q u i r e m e n t f o r x seems t o a r i s e out o f the manner i n w h i c h i n t e g r a l c o n t r o l a c t i o n i s i n t r o d u c e d i n t h e p r o b l e m f o r m u l a t i o n . I n t h e method p r e s e n t e d h e r e , x i s used t o implement c o n t r o l a c t i o n so t h a t the Kalman f i l t e r can be u s e d i n i t s c o n v e n t i o n a l form. 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 a s s o c i a t e d w i t h i m p l e m e n t i n g a Kalman f i l t e r . D e t a i l e d s t a t i s t i c a l d a t a about p l a n t and measurement n o i s e i s r e q u i r e d and such d a t a i s g e n e r a l l y n o t a v a i l a b l e f o r a power system. C o n s e q u e n t l y , i n s t e a d o f a Kalman f i l t e r , two a l t e r n a t i v e methods a r e s u g g e s t e d f o r demand i d e n t i f i c a t i o n . The f i r s t method i s e x t r e m e l y s i m p l e and i d e n t i f i e s i n c r e m e n t a l demand by t h e 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 t e c h n i -que. The second 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 b s e r v e r t o p e r f o r m t h e i d e n t i f i c a t i o n o f t h e demand and t o e s t i m a t e t h e unmeasured s t a t e s . The advantage o f t h e s e a l t e r n a t i v e methods i s t h a t t h e y do n o t r e q u i r e d a t a about n o i s e s t a t i s t i c s . 4.2 P r o b l e m F o r m u l a t i o n A t y p i c a l model of two i n t e r c o n n e c t e d power a r e a s i s shown i n F i g . (4. l ) 1 ^ ' ^ ^ ' T h e c o n t r o l l i n g s t a t i o n i n t h e f i r s t and second 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 and a h y d r o - p l a n t , r e s p e c t i v e l y . STEAM PLANT E. U, 1 • 1 sr9r STtrl A dl GOVERNOR TURBINE API AX, HYDRO PLANT A /___ STt2*1 -D? S+1 5 D2 S;'-l 12 APg2 + MJS+GJ 12c GOVERNOR TURBINE M2 S + G 2 AP d2 'Gj2rr AS-, F i g . (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 and h y d r o a r e a s . 52 In 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 the form x . = A.x . + B.u. + T.APj. + n..(u.) p i 1 p i 1 1 l d i i j j (4.1) •j $ i , j = 1, 2, . . . , N where f o r a steam-plant 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 Aw AP AX ] • P t g g (4.2) and f o r hydro-plant the s t a t e v e c t o r i s (4.3) The matrices 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 given by: (The matrices 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 ) . x* = [AP Aw AP Ag AX ] P t g 6 g A. = l 0 E T°. i j c j / i -1/M. -G./M. l i i 0 0 -E./T . 0 1/M. -1/T t i 0 1/Tt. •1/T . g i (4.4) B' = [0 0 0 1/T .] i . g i = [0 -1/M 0 0] 0!. = [-T°. 0 0 0] (4.5) (4.6) (4.7) where the terms i n (4.2)-(4.7) are as defined i n S e c t i o n (3.2). To avoid unnecessary co m p l i c a t i o n s i n n o t a t i o n , t h e coupling terms between the areas are set equal to zero i n the i n i t i a l problem f o r m u l a t i o n 53 and t h e s u b s c r i p t i i s dropped ( n o n - z e r o c o u p l i n g i s c o n s i d e r e d i n t h e ex a m p l e ) . E q u a t i o n (4.1) t h e n t a k e s t h e form x = A x + Bu + TAP, (4.8) p p d 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 LFC p r o b l e m must be c o n s i d e r e d t o be composed o f two s e p a r a t e p r o b l e m s : ( l ) P r o b l e m o f i d e n t i f y i n g t h e unknown power demand AP^. (2) P r o b l e m o f o p t i m a l l y c o n t r o l l i n g t h e dynamic r e s p o n s e so t h a t t h e g e n e r a t i o n becomes e q u a l t o t h e demand a t t h e s p e c i f i e d f r e q u e n c y . C o n s i d e r t h e second o f t h e above pr o b l e m s and assume f o r t h e moment t h a t AP, i s a known c o n s t a n t . The t e r m i n a l c o n d i t i o n s t o be s a t i s f i e d d a r e * AP ( o o ) = 0, Aw(<*>) = 0, AP (°°) - AP, = 0 and AX (~) - AP, = 0 (4.9) t • g ci g tt To f o r m u l a t e an o p t i m a l c o n t r o l p r o b l e m , 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 - pAP, (4.10) p d where p' = (0 0 1 1). (4.11) S u b s t i t u t i n g (4.10) i n t o (4.8) (and s e t t i n g AP^ t o z e r o ) y i e l d s ; x = Ax + Bu + (A p + P) A P , (4.12) d The t e r m i n a l c o n d i t i o n (4.9) r e q u i r e s t h a t i ( c o ) = o = X ( o o ) (4.13) I t i s seen from (4.11) and 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 t h a t 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 c o n t r o l i s t h e 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 on t h e e r r o r s i g n a l . To i n t r o d u c e t h i s c o n t r o l r e q u i r e m e n t i n t o t h e f o r m u l a t i o n o f an o p t i m a l c o n t r o l p r o b l e m n e c e s s i t a t e s augmenting (4.12) by See A p p e n d i x I f o r LFC c r i t e r i a so that where x n + ] _ ( t ) - u(t) - AP d n+1 ~ A • u = u The augmented system i s therefore (see (4.14)) where X = AX + B u X A *A B " ~ A ' 0" A A and B = X , , 0 0 1 - n+l_ X ^ The cost index i s taken to have the quadratic form 1 (4.15) (4.16) (4.17) (4.18) (4.19) J = (X'QX + u'Ru)dt (4.20) 0 where Q and R are p o s i t i v e d e f i n i t e matrices. The optimal control f o r the problem defined by (4.18) and (4.20) i s given by where u = c'X C ' = C S Sn+1 ] (4.21) (4.22) i s a constant vector which can be found by so l v i n g a steady-state R i c c a t i equation (see Section 3.6.1). In terms of the o r i g i n a l state variables the co n t r o l i s given by (see (4.10), (4.15) and (4.19)) u = s'x + s ,.u - (s'p + s ^ J A P , (4.23) p n+1 n+1 d where u(0) i s a r b i t r a r i l y taken to be zero. The co n t r o l (4.23) i s s i m i l a r to the conventional proportional plus i n t e g r a l c o n t r o l which i s presently used for load-frequency control. It i s seen, however, that the co n t r o l (4.23) i s f e a s i b l e only i f AP^ can be i d e n t i f i e d . The next Section discusses the demand i d e n t i f i e r , 4.3 Demand I d e n t i f i e r - D i f f e r e n t i a l Approximation To implement the control given by (4.23) requires the i d e n t i f i c a t i o n of the parameter A P ^ . A simple i d e n t i f i e r f o r A P ^ can be constructed by 23 using the method of d i f f e r e n t i a l approximation Let T = t. - t , be a f i x e d i d e n t i f i c a t i o n period which s t a r t s at o k k - i r time t, and terminates at time t, . The sequence (t , t - , t„, ...) defines k — i k o i z a set of i d e n t i f i c a t i o n i n t e r v a l s . In actual power systems, AP^ i s constant or slowly varying. Consequently, a reasonable approximation for AP^ i n (4.23) i s to use the demand i d e n t i f i e d during the previous period. That i s , by taking AP, = AP,(k-l) (4.24) d d for the i n t e r v a l tj < T £ t i c + i » where AP^(k-l) i s determined by i n t e g r a t i n g the power equilibrium .equation (see F i g . (4.1)). AP - AP, = MAoi + GAOJ + AP, (4.25) g d t over the previous i n t e r v a l t ^ _ ^ < r $ t^. This y i e l d s AP d(k-l) = i [-M(Aa)(tk) - A c j ( t k _ 1 ) ) o + J (-GAco(t)+AP (t)-AP ( t ) ) d t ] g V i •J (4.26) The quantities on the right-hand side of (4.26) are determined by measure-ments on the system. -The estimate given by (4.26) could be improved by averaging over several i d e n t i f i c a t i o n i n t e r v a l s . The type of averaging and the number of i n t e r v a l s used would depend on the kind of load disturbance. The structure of the composite plant and c o n t r o l l e r i s i l l u s t r a t e d i n F i g . (4.2). In load-frequency control there i s the problem of the c o n t r o l l e r following r a p i d l y changing random-load disturbances. This i s i n e f f i c i e n t and contributes to unncessary wear i n the c o n t r o l l e r mechanism. 19 ' Ross treated t h i s problem and suggested the use of an E r r o r Adaptive Control Computer (EACC). The EACC monitors the error signals and computes the p r o b a b i l i t y that load-frequency control action i s required. Control action i s i n i t i a t e d only when the computed p r o b a b i l i t y exceeds a preset threshold. As indicated i n F i g . (4.2), an EACC can be used to aug-ment proposed load-frequency control given by (4.23) and (4.26). 4.4 Applications The proposed load-frequency c o n t r o l l e r i s tested on two i n t e r -connected steam and hydro-plants (Fig. (4.1)). The parameter values used are as given i n Section (3.7) Due to the coupling between the plants, the optimum feedback con t r o l i s -a function of - a l l the states. -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 various forms of suboptimum c o n t r o l l e r s . Two d i f f e r e n t suboptimum c o n t r o l l e r s are considered, and they are compared to the optimum control. The optimum feedback, control has the form G0 * [ V L ~n02]> C 4 ' 2 7 ) "01 " S011X1 + S012X2> ( 4 - 2 8 ) "02 = S021X1 + S022V ( 4 - 2 9 ) where and are the state vectors ( i n the form (4.19)) f o r the f i r s t and second plant, r e s p e c t i v e l y . The gain vectors s Q J ^ (.1 > k=l, 2) are 24 solutions of a steady-state matrix R i c c a t i equation By neglecting the coupling between the plants, an optimum control 57 j DEMAND DECISION 1 ''EACC DEMAND IDENTIFIER dk UNIT DELAY ^VECTOR _ SCALAR (A.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 a l o a d - f r e q u e n c y c o n t r o l l e r of the form coupled system. r r l r 2 J ' (4.30) U r l " S r l l X l s (4.31) U r 2 = S r 2 2 X 2 J (4.32) The c o n t r o l (4.30) i s suboptimum f o r the : coupling terms i n the optimum c o n t r o l (4. 27), : form K " [ " s l (4.33) U s l = S 0 1 1 X 1 (4.34) U s 2 = S022 X2 (4.35) i s obtained. The type of suboptimal c o n t r o l given by (4.34) and (4.35) i s discussed i n reference 33. Example (4.1) The Q and R matrices i n (4.20) are chosen to be the u n i t matrices and the assumed demands are taken to be s l o w l y time-v a r y i n g and given by p _,0.1 sin(frt/20) 0 < t ? 10 A d l _ i 0 . 1 t > 10 A P d 2 =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 the f i n a l time tr are chosen to be 0.5 o f and 30 seconds, r e s p e c t i v e l y (at t ^ = 30 the system has e s s e n t i a l l y reached the s t e a d y - s t a t e ) . The i n i t i a l c o n d i t i o n s on the c o n t r o l l e r s are a r b i t r a r i l y set equal to zero. Table (4.1) gives the numerical values of the gain v e c t o r s ( i n terms of the o r i g i n a l s t a t e s (see (4.23)) and the performance cost J . Table (4.1) STATE FED BACK GAINS FOR THE CONTROLS % 'o °02 i °n \ "si . 0S2 STEAM PLANT A P. tl2 7.5 .. 15.62 7.5 STEAM PLANT A LO . .62 - .48 _ .5 _ .52 STEAM PLANT _ 8. -3.3 -7.2 -8. STEAM PLANT . 5.7 -1.4 -5.4 -5.7 STEAM PLANT u, -.4.76 - .4 7 -4 .66 -4.7-5. STEAM PLANT 18 .46 5.17 17.26 18.45 HYDRO PLANT A P 21. 27.5 21. HYDRO PLANT -.22 • 19 .22 .19 HYDRO PLANT .1.71 - .09 - .09 -.09 HYDRO PLANT * 9 2 -5.7 -10-7 -11.3 -10.7 HYDRO PLANT *X92 .2. 02 .12.9 -13.4 -12.9 HYDRO PLANT U> _ .47 -4.6 3 .4.7 -4.63 HYDRO PLANT ^ 2 9.9 28.32 29.49 28.32 J 0-5 3 0.7 8 0.65 Example (A.2) Table (4.1) shows that the suboptimal control u s gives a lower cost than u_. Consequently, u g is used in this Example which illustrates the effect of T on identification and control. The following o demands are assumed: r0.15 sin (irt/20) 0 < t jc 10 ' (4.37) AP = I fi dl l0.15 t > 10 , _ ,-0.1 sin (irt/20) 0 < t s? 10 M d 2 ~ l-0.1 t > 10 The system responses for (a) T_ = 0.5, and (b) T q = 1, are shown in Fig. (4.3). It is evident from Fig. (4.3) that the load-frequency control (4.34) and (4.35), with power demand identification given by (4.24) and (4.26) results in a satisfactory system response. Example (4.3). In this example the control u is compared with a s 25 conventional load-frequency controller given in transfer function form by u= (f ) (biir ) (cAu + V ( 4 - 3 8 ) The optimum parameter V a l u e s for the c o n t r o l l e r , as given i n reference 25, are: a) Steam plant; a^ = .09, b^ = 0.3, c^ = .02 b) Hydro plant; a^ = .4, b^ = 0.3, = .02 Since power demand i s not i d e n t i f i e d , the optimization i s performed a f t e r averaging over a s p e c i f i e d set of power demand p r o f i l e s . Consequently, for a given power demand, (4.38) i s suboptimum. Figure (4.4) i l l u s t r a t e s the comparison of the system responses f o r : (a) the conventional control u given by (4.38), and (b) the proposed 25 control u . The demands were chosen to be , s A P d l = " * ° 0 5 A P d 2 = , 0 ° 5 o.ooe 0.006 1 '0.004 30.0D2 £0.000 -0.002 H -0.004 10 15 20 TIME (SECONDS! o.ieo • 10 15 20 TIKE (SECONDS! 10 IS 20 TIKE (SECONDS) F i g . ( 4 . 4 ) Response Comparison (a) C o n v e n t i o n a l and (b) p r o p o s e d c o n t r o 63 I t i s e v i d e n t f r o m F i g . (4.4) t h a t t h e i d e n t i f i c a t i o n o f t h e power demand and t h e use o f t h e l o a d - f r e q u e n c y c o n t r o l u r e s u l t s i n a s i g n i f i c a n t s improvement i n s y s t e m r e s p o n s e . 4.5 Demand I d e n t i f i e r - L u e n b e r g e r O b s e r v e r T i e - l i n e power and f r e q u e n c y d e v i a t i o n s ( t h e f i r s t two s t a t e s i n t h e model g i v e n by (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 l o a d - f r e q u e n c y c o n t r o l l e r s . The optimum c o n t r o l g i v e n by (4.23) r e q u i r e s t h a t a l l t h e s t a t e s be measured and t h a t AP^ be i d e n t i f i e d . F o r an o b s e r v a b l e s y s t e m , measurements o f some o f t h e s t a t e s can be used t o 34 r e c o n s t r u c t t h e c o m p l e t e s t a t e by use of a L u e n b e r g e r o b s e r v e r C o n s i d e r a s y s t e m model and a measurement s y s t e m o f t h e form x = Ax + Bu (4.39) z = Hx where x i s an n - s t a t e v e c t o r and z i s an m measurement v e c t o r ( i n g e n e r a l m<n). I t i s assumed t h a t (4.39) i s o b s e r v a b l e . L u e n b e r g e r has shown t h a t a c l a s s o f (n-m) d i m e n s i o n a l o b s e r v e r s can be s t r u c t u r e d from (4.39). The o b s e r v e r o u t p u t s 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 un-measured s t a t e s . I n t h e o r y , a r b i t r a r i l y s m a l l s e t t l i n g t i m e o f t h e o b s e r v e r can be a c h i e v e d . To i l l u s t r a t e t h e use o f o b s e r v e r s t o i d e n t i f y unmeasured power sy s t e m s t a t e s and a c o n s t a n t AP^, 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 . The s y s t e m s t a t e e q u a t i o n s a r e augmented by AP^ = 0. T h i s y i e l d s x = A x + B u (4.40) a a where x' = [x AP, ], and where ( s e e ( 4 . 4 ) ) . 64 A = a 0 0 0 0 12c I -GjM. 1 -1/M. 0 ! - 1 / T t i 1 / T n -w i ° 0 1 0 -1/T gl 0 •1/M, 0 0 0 A ' A A l l | A12 A I A A21 i A22 (4.41) B' = [0 0 l o f 0] = [0 0 j" B ] gl Assuming that only the f i r s t two states are measured, the measurement matrix in (4.39) takes the form (4.42) H = 1 0 0 0 0 0 1 0 0 0 (4.43) The partitioning indicated in (4.41) and (4.42) is used to decompose (4.40) into the form *1 = A l l h + A12 ?2 . h = A21 h + A22 h + B 2 U where E,^ is the measured m-vector and E,^ 1 S a n n _ m vector which i s to (4.44) (4.45) be reconstructed by an observer. Consider the observer defined by 36 C = N n C + N 1 2 h + B 2 u M l l h> where N l l = M11 A12 + A22 (4.46) (4.47) (4.48) ^ ^ i i ^ n - ^ V + ^ i - ^ V -6: A In (4.48) M^^ i s an a r b i t r a r y (n-m)xm matrix. Let a = £ - £ be the A error between the unmeasured vector £ 2 and the observer output £ 2 , as given by (4.46) and (4.47). It i s seen that a - ( A 2 2 + M n A 1 2 ) a (4.49) Consequently, i f can be chosen so that (4.49) i s asymptotically stable, i t follows that £ 2(t) -> £ 2(t) as t •+ ». 35 To determine such a matrix, consider the a u x i l i a r y system » = ( A22 + M 1 1 A 1 2 } ' » ( 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 constant matrix. It follows from (4.50) and (4.51) that V = y'Ky (4.52) where By taking ( A22 + M 1 1 A 1 2 ) K + K ( A 2 2 + \lA12y + C = ° ( 4 - 5 3 ) M n = - | K A 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 matrix, (4.53) takes the form A 2 2K + KA 2 2 - KA^SA 1 2K + C = 0 (4.55) Equation (4.55) i s the algebraic matrix R i c c a t i equation which can be solved f o r a p o s i t i v e d e f i n i t e symmetric matrix. With t h i s choice of i t i s seen from (4.51) and (4.52) that the a u x i l i a r y system (4.50) i s asymptotically stable. Since (4.49) and (4.50) have the same eigenvalues, i t f o l l o w s t h a t (4.-49) i s a s y m p t o t i c a l l y s t a b l e . F i g u r e (4.5) i l l u s t r a t e s a 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 f t h e s t a t e and an o b s e r v e r t o r e c o n s t r u c t t h e r e m a i n i n g s t a t e components. N o t i c e t h a t t h e demand AP^ i s i n c l u d e d i n ^ w h i c h i s r e c o n s t r u c t e d by t h e o b s e r v e r f r o m (4.46) and ( 4 . 4 7 ) . Example (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 c o n s i d e r e d i n t h i s example i s t h e same as used i n Example ( 4 . 1 ) . O n l y t h e f i r s t two s t a t e s ( 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 ) a r e measured. D i g i t a l s i m u l a t i o n i n d i c a t e d t h a t 1 0 0 100 C = 1000 1 0 0 0 1 0 0 0 1 (4.56) .was a r e a s o n a b l e c h o i c e f o r t h i s example. -The.observer . m a t r i c e s r e s u l t i n g from t h i s c h o i c e a r e N 11 M. 11 -4.97 2 2.97 -1.33 -2 1.33 .79 0 -7.9 0 -.12 0 -.05 0 0.32 N 12 2.97 -1.4 1.33 -.73 -7.9 3.36 (4.57) (4.57) A c o n s t a n t demand o f AP^ = 0.1 i s assumed. F i g u r e (4.6) i l l u s t r a t e s t h e sy s t e m responses f o r Load-Frequency C o n t r o l u s i n g : (a) measurement of a l l the s t a t e s assuming AP_^  i s known; (b) measurement o f two o f the state 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 com-p o n e n t s . The s l i g h t d i f f e r e n c e i n r e s p o n s e s i n d i c a t e s t h a t t h e o b s e r v e r r e c o n s t r u c t s t h e unmeasured s t a t e s w i t h adequate ( f o r c o n t r o l p u r p o s e s ) a c c u r a c y . PLANT F i g . ( 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 Q >- ° -C_) UJ <3 L I J ID u- o-i o I fa) L - ^ » V - « l . - „ ' — J - | .i-...^., | ,,|,, ,[ I, | „,.^ i „ „ , ( 1-—; 1 H (b) in - i — i — i — i — | — i — i — I — i — | — i — i — i — i — | — i — i — i — i — | — i — i — i — i — | — i — r -10 20 30 40 50 TIME" (SECONDS) 60 ? 0 20 30 40 TI HE (SECONDS) 60 F i g . (4.6) System response - Example (4.4) 5. OPTIMUM LOAD-FREQUENCY SAMPLED-DATA CONTROL WITH RANDOMLY VARYING SYSTEM DISTURBANCES 5.1 I n t r o d u c t i o n A power s y s t e m a r e a 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 w h i c h a r e p h y s i c a l l y remote f r o m t h e c o n t r o l l i n g s t a t i o n o r d i s p a t c h i n g c e n t e r . Feedback r e g u l a t o r y c o n t r o l o f t h e s y s t e m r e q u i r e s t h e measurement o f t i e -l i n e f l o w s a t i n t e r c o n n e c t i o n s and t h e t r a n s m i s s i o n o f measured d a t a o v e r d a t a l i n k s t o t h e c o n t r o l l i n g p l a n t o r d i s p a t c h i n g c e n t e r . I t i s e s s e n t i a l , 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 t i m e on a c o n t r o l s t r a t e g y b a s e d 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 ^ . I n r e f e r e n c e 17 t h e e f f e c t o f t i e -l i n e measurement d e l a y i s t e s t e d on t h e c o n t i n u o u s system. No a t t e m p t , how-e v e r , i s made t o t a k e t h i s e f f e c t i n t o a c c o u n t i n t h e o p t i m a l c o n t r o l d e s i g n . T h i s C h a p t e r d e a l s , e s s e n t i a l l y . , w i t h t h e s a m p l e d - d a t a o r d i s c r e t e -t i m e 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. There i s , however, a n o t h e r i m p o r t a n t 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 i s c o n s i d e r e d . Many l o a d d i s t u r b a n c e s a r e random i n n a t u r e and measurements o f t h e s y s t e m s t a t e a r e o f t e n p e r t u r b e d by n o i s e . The p r o b l e m o f o p t i m a l c o n t r o l i n t h e p r e s e n c e o f p l a n t and measurement n o i s e i s known as t h e s t o c h a s t i c o p t i m a l c o n t r o l 31 p r o b l e m . The p r o b l e m has been 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 c o s t i n -dex. 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 ca 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 w i t h t h e s t a n d a r d optimum c o n t r o l l e r f o r a l i n e a r d e t e r m i n i s t i c s y s tem. The d e t a i l e d s t a t i s t i c a l d a t a about p l a n t and measurement n o i s e r e q u i r e d t o implement t h e Kalman f i l t e r 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. A s u b o p t i m a l 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 e x t e n s i v e s t a t i s t i c a l d a t a f o r i t s i m p l e m e n t a t i o n . The s m a l l number of p a r a m e t e r s i n t h e c o n t r o l l e r makes o n - l i n e t u n i n g f e a s i b l e . 5.2 O p t i m a l Sampled-Data R e g u l a t o r The p r o b l e m f o r m u l a t i o n f o r c o n t i n u o u s o p t i m a l l o a d - f r e q u e n c y c o n t r o l i s g i v e n by ( s e e C h a p t e r 4, (4.18) and (4.20)). X = A X + B u (5.1) (X' Q X + u' R u ) d t (5.2) t The i n t r o d u c t i o n o f a d a t a - l i n k i n t h e r e g u l a t o r y l o o p o f a power s y s t e m r e s u l t s i n a s a m p l e d - d a t a s y s t e m and a c o n t i n u o u s o p t i m a l c o n t r o l i s t h e n no l o n g e r r e a l i z a b l e . C o n s i d e r a s e t o f s a m p l i n g i n s t a n t s (t Q» t ^ , t^, ...) and l e t T = t j _ - t^ _^  be a c o n s t a n t s a m p l i n g i n t e r v a l ; I n a s a m p l e d - d a t a s y s t e m , t h e c o n t r o l i s c o n s t r a i n e d t o be c o n s t a n t between s a m p l i n g i n s t a n t s : u ( x ) - ; f c , t k = T < t k + 1 (5.3) The f o r m u l a t i o n o f an o p t i m a l c o n t r o l p r o b l e m w i t h a s a m p l e d - d a t a c o n t r o l o f t h e form (5.3) r e q u i r e s t h a t (5.1) and (5.2) be t r a n s f o r m e d i n t o a d i s c r e t e -t i m e e q u i v a l e n t s e t of e q u a t i o n s . S i n c e A and B a r e t i m e - i n v a r i a n t , t h e s o l u t i o n o f (5.1) f o r t h e i n t e r v a l ' t ^ = T < t f c + 1 i s g i v e n by \ + 1 = ~ X ( t k + 1 ) = ; ( t k + 1 , t k ) X k + t k + V t K + 1 , T ) B v k dx (5.4) = * \ + °\ where $ = c5(T,0). i s the s t a t e - t r a n s i t i o n m a t r i x o f (5.1): A *, $(0,0) = I (5.5) ( I i s t h e u n i t m a t r i x ) , and where J4I -'O T > $ ( T , t ) B d t . (5.6) The c o s t i n d e x (5.2) can be e x p r e s s e d as t h e sum o f N i n t e g r a l s , each i n t e g r a l b e i n g e v a l u a t e d o v e r a s a m p l i n g i n t e r v a l . U s i n g (5.4) i t i s see n t h a t J = f A ^ + 2 Xk\\ + ; k\V (5-7) k=0 where T „ Q = I * ' ( T , t ) Q $ ( T , t ) d t (5.8) •i, I M = $ ' ( T , t ) Q D ( T , t ) d t (5.9) 'o T R = | [R + D ' ( T , t ) Q D ( T , t ) ] d t (5.10) '0 The optimum f e e d b a c k 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 p r o b l e m g i v e n by (5.4) and (5.7) i s 1 6 \ = ~ C \ ( 5 ' n ) where C = R _ 1M' + (R + DIKD)~1DK0 (5.12) 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 d i f f e r e n c e e q u a t i o n \ - e' [ V i - Vi»(* + °'Vi^'W 0 + A (5-13) where K = 0 i s t h e boundary c o n d i t i o n and where 0 0 0 = I - D R _ 1 M' (5.14) A = Q - M R" 1 M' (5.15) Expressing (5.11) i n terms of the o r i g i n a l system s t a t e s (see Chapter 4, (4.8), (4.23)) i t i s seen that the f o l l o w i n g equations d e s c r i b e the optimum sampled-data load-frequency c o n t r o l : x (T) = A x (T) + B U ( T ) + r AP, x e l t - . t . , . ] (5.16) p p a K. krl u(x) = u k + t v k te[0,T] (5.17) V r " U k + T ; k ( 5 ' 1 8 ) ; k + l " S' X P k + S n + 1 \ " ( S'P + S n + l ) A P d ( 5 ' 1 9 ) C A ( s , s n + 1 ) (5.20) I t i s seen from (5.11), (5.12) and (5.17) that the 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 the sampling time T. I t i s shown i n reference 16 t h a t , as T -*• 0, the continuous optimum c o n t r o l i s the l i m i t i n g case of the sampled-data c o n t r o l . j Example (5.1) Consider the s i n g l e steam p l a n t discussed i n Chapter 4, Example (4.4). The f o l l o w i n g three c o n t r o l p o l i c i e s are considered f o r a demand AP, = 0.1: a (a) A continuous c o n t r o l u 1 which uses continuous s t a t e i n f o r m a t i o n (see Chapter 4, Table ( 4 . 1 ) ) . (b) A sampled-data c o n t r o l U2 which uses s t a t e i n f o r m a t i o n at a sampling ra t e of 1 second (T=l). (c) C o n t r o l 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 (5.20) f o r the above c o n t r o l s are: C' .= [15.62 -.5 -7.2 -5.4 -4.66] (5.21) CJ = [2.04 0.03 -0.18 -0.48 -1.58] (5.22) C^ = [.172 0.02 0.17 0.11 -0.51] (5.23) The c o s t i n d i c i e s f o r t h e t h r e e above c a s e s a r e : . C o n t r o l u^ u 2 Cost Index 0.754 1.465 2.319 The c o s t i n d e x has p r a c t i c a l u s e f u l n e s s o n l y i f t h e a s s o c i a t e d c o n t r o l s t r a t e g y r e s u l t s i n a c c e p t a b l e s y s t e m r e s p o n s e s . These a r e shown i n F i g . (5.1). I t i s se e n from F i g . (5.1) t h a t u^ g i v e s t h e f a s t e s t r e s p o n s e . Hox^ever, t h e r e i s no s i g n i f i c a n t d e t e r i o r a t i o n i n dynamic p e r f o r m a n c e by u s i n g a s a m p l i n g r a t e o f one second. 5.3 S t o c h a s t i c Optimum and Suboptimum C o n t r o l The s t o c h a s t i c o p t i m a l c o n t r o l p r o b l e m f o r a d i s c r e t e - t i m e l i n e a r s y s t e m w i t h l i n e a r measurements i s d e f i n e d by \ + i " W + \ \ + w k ( 5 - 2 4 )  z k + l = Hk+1 x k + l + v k + l . ( 5 ' 2 5 ) 1 N -J. = •=• £ x'.Q.x. + u,R.u_ (5.26) k 2 ^ = 1 j j j j j j where Wj_ i s an n - d i m e n s i o n a l p l a n t n o i s e v e c t o r , i s an m - d i m e n s i o n a l measurement v e c t o r and v, i s an m - d i m e n s i o n a l measurement n o i s e v e c t o r . k I n (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 m a t r i x sequence and 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 m a t r i x . J The p r o b l e m i s t o d e t e r m i n e a r e a l i z a b l e c o n t r o l sequence w h i c h m i n i m i z e s E(J^), t h e e x p e c t e d v a l u e o f J q , g i v e n t h e measurement sequence and t h e f o l l o w i n g s t a t i s t i c a l d a t a (E(«) i s t h e e x p e c t a t i o n o p e r a t o r ) : E ( x ) = x , E ( x x') = P , E(w. ) = E(v. ) = 0 o o ' o o o k k ^ 27) E( x ^ w k ) =0, E ( V j V k ) = 0 , E < W J V P - 0 ( 3 ^ ) E(w.w') = Q. E ( v _ v ! ) = R. J j J j J J 74 § " 1 I— tx UJ a UJ 1 o 1 I in ' 0 T — I — I — r 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1—I 1 1 1 1—1 1 ' 1 1 10 20 30 40 TIME (SECONDS! 50 60 UJ OJ • 1 ! 1 i 1 1—1—1—T—r—T—r 1 1 1 1 i ' •<-• r '• • 1 1 r ~ 10 20 30 40 50 TIME (SECONDS) 60 F i g . (5.D Response comparison f o r (a) continuous c o n t r o l , (b) sampled-data con t r o l T = 1 and (c) sampled data c o n t r o l T - 2. If the s t a t i s t i c s are Gaussian, i t i s known that the optimal c o n t r o l sequence i s given by u k = ~C k x k , (k = 0, 1, ... N-1) (5.28) where x k i s the c o n d i t i o n a l mean of x k and where C k i s the c o n t r o l gain 31 * for the optimum dete r m i n i s t i c (noise-free) c o n t r o l . The estimate x k can be generated r e c u r s i v e l y on-line by use of the Kalman f i l t e r : \ - ( \ - r D k - i c k - i ) ; k - i + V k ( 5 - 2 9 ) \ A = V \ ( V f V i c H ) ; H ( 5 - 3 0 ) The vector y k i s the d i f f e r e n c e between the measurement vector and the predicted measurement vector, and i s the optimum f i l t e r gain. I t i s seen from (5.24) - (5.30) that the error, e k = x k - x k > and the error covariance, P , are given r e s p e c t i v e l y by K. e k = <x " K k V ( V i e k - i + Wk-P - w ( 5 - 3 1 ) P k = E ( e k e k ) = d - W ^ A + V l ) ( I - W ( 5 > 3 2 ) The optimum f i l t e r gain, i ^ , minimizes T r ( P k ) , the trace of the error covariance matrix. From a mathematical point of view the s o l u t i o n i s s u r p r i s i n g l y simple. The control gain matrix-in (5.28) can be determined by s o l v i n g the noise-free optimal control problem and x k i s determined by s o l v i n g , separately, an optimal f i l t e r problem. From a p r a c t i c a l point of view, however, the on-line implementation of (5.28) r a i s e s severe problems, p a r t i c u l a r l y i n power system a p p l i c a t i o n s . Even i n the n o i s e - f r e e case, due to the complexity of the optimum c o n t r o l l e r , optimum c o n t r o l of a power system i s both impractical 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 to s i m p l i f y the system model. Consequently, the noise sequences i n (5.24) and (5.25) may i n part a r i s e from modelling e r r o r s , thus i n v a l i d a t i n g the Gaussian white noise assumption used i n d e r i v i n g (5.28). Furthermore, data 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 t i s reasonable, however, to r e t a i n the c o n t r o l s t r u c t u r e defined by (5.28), (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 . This f o l l o w s 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 observer f o r the system (5.24). The usefulness of an observer has been shown i n Chapter 4. The observer (5.29) gives improved estimates i f the observer g a i n i s chosen to minimize an e s t i m a t i o n e r r o r cost index (such as Tr(Pj_)). Let L be a constant observer m a t r i x g a i n which r e s u l t s i n a s t a b l e observer and l e t \ = g k L ( 5 ' 3 3 ) g where g k i s a s c a l a r gain. A c l a s s 0 of s t a b l e observers i s d e f i n e d by the s t a b i l i t y l i m i t s , S m < \ < % ( 5 ' 3 A ) on the s c a l a r gain. A l l subsequent observers are considered to belong to 0 . An optimum gain could be defined i n 0 by a s s o c i a t i n g a cost index w i t h (see (5.32)). Consider the choice Tr(07kPk) - a k g 2 - 2 3 k g k + y k (5.35) where . = *k = T r t \ L H k ( $ k - l P k - l $ k - l + Q k - 1 ) ] ( 5 3 6 ) \ - T r ( \ L ( V V i p k - i * k - i + V i ) H k + V L , ) and where Q k i s a p o s i t i v e s e m i - d e f i n i t e weighting m a t r i x . The optimum gain which minimizes the cost index (5.35) i s given by 77 8 k " hJak' (5.37) F i l t e r i n g and minimization of e s t i m a t i o n error 1 do not, however, represent the complete problem,which i s to determine a suboptimal c o n t r o l . Furthermore, even though there are only two s t a t i s t i c a l parameters i n (5.37), i t i s d e s i r a b l e to reduce f u r t h e r the need f o r s t a t i s t i c a l data. The dependency of the optimum s t o c h a s t i c c o n t r o l (5.28) on the data (5.27) a r i s e s out of the g l o b a l minimiziation of the cost index E ( J ). A suboptimal v o 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 (stage-wise) m i n i m i z a t i o n of an instantaneous cost 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 at stage k. The cost must be r e l a t e d i n some meaningful manner to (5.26). In the n o i s e - f r e e case, the system dynamics are x. = $. x. + D. u, , x = x. , j > k (5.38) J+l 3 3 J j k k ' J -the optimal c o n t r o l sequence i s given-by -u - a '-Gj-x^ , and -(5 .-26) -can -be expressed i n the closed form \ = i ; k V k ( 5 - 3 9 ) 31 where i s the s o l u t i o n of a d i s c r e t e m a t r i x - R i c c a t i equation . Let x^ be defined by (5.27) where x^_^ i s known from the previous stage and l e t e, = x, - x-, (5.40) 3 - 3 3 A simple estimate of the e f f e c t of the e r r o r e^ = e^ = x^ - x^ at stage k on the cost can be obtained by t a k i n g the p r e d i c t e d value of f u t u r e noise at any stage j > k to be equal to the mean va l u e , which i s zero, as given by (5.27). Consequently x.,, = ($, - D,C,)x, = k)x. (5.41) e j + 1 = (^ - D j C ; i ) e j = * <" j + 1 , k ) e k » ( j = k ) 78 a r e p r e d i c t e d f u t u r e v a l u e s o f and e^, r e s p e c t i v e l y . I n ( 5 . 4 1 ) , y ( j , k ) i s the sys t e m s t a t e t r a n s i t i o n m a t r i x . I n t r o d u c i n g (5.40) and (5.41) i n t o (5.26) and making use o f (5.39) y i e l d s J k = S k + e k \ X k + f e k \ \ ( 5 ' 4 2 ) where = A N Q, = Z ^ ( j , k ) ( Q +C'R C . ) y ( j , k ) (5.43) R j=k The e x p e c t a t i o n o f (5.42) i s E ( J k ) = E ( J k ) + T r C ^ E ^ e ^ ) ) + | T r ( Q k P k ) (5.44) E q u a t i o n (5.44) r e p r e s e n t s a d e c o m p o s i t i o n o f an e s t i m a t e o f t h e a v e r a g e c o s t i n t o a d e t e r m i n i s t i c c o n t r o l c o s t ( f i r s t term) and a c o s t o f e s t i -m a t i o n e r r o r ( t h i r d t e r m ) . The s e c o n d term c o u p l e s t h e two c o s t s . I t i s seen from (5.35) and (5.37) t h a t g k = g k m i n i m i z e s t h e t h i r d terra. C o n s i d e r t h e e f f e c t t h a t t h i s c h o i c e has on t h e second term. S i n c e x, .. i s k-1 assumed g i v e n , and E ( x k ^ ) - x____> l t : f o l l o w s t h a t E ( x k - 1 e k - l > " ° ( 5 ' 4 5 ) I t can be shown f r o m ( 5 . 2 9 ) , (5.31) and (5.45) t h a t ( s e e ( 5 . 3 5 ) , ( 5 . 3 6 ) ) . •k From (5.37) i t i s seen t h a t (5.46) v a n i s h e s when g k = g k , c o n s e q u e n t l y , i f \ - K+ i ( g k - + v a k ( 5 - 4 7 ) where a k = T_./ak> i t f o l l o w s t h a t • E ( J k ) = E ( J k ) (5.48) when g k = g k > E q u a t i o n (5.48) j u s t i f i e s c o n s i d e r i n g t h e c o n t r o l d e f i n e d by ( 5 . 2 8 ) , ( 5 . 2 9 ) , (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 . The c o s t i n d e x (5.47) has two terms. The f i r s t t erm i s t h e c o s t 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 t e r m i s a c o s t a s s o c i a t e d w i t h e s t i m a t i o n e r r o r . B o t h terms depend on and c o n s e q u e n t l y t h e r e i s a t r a d e - o f f between t h e two c o s t s , W i t h 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 w e i g h t i n g f a c t o r . T h i s s u g g e s t s t h e p o s s i b i l i t y o f c h o o s i n g g^ a d a p t i v e l y w i t h i n t h e o b s e r v e r c l a s s 0 so t h a t t h e r e i s a d e c r e a s e i n J, . To g k f o r m u l a t e s u c h 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 i n (5.47) be 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. From (5.30) and (5.36) i t i s seen t h a t a - Tr(Q L E ( y ^ L ' ) (5.49) " E(K L ' \ L V The random v a r i a b l e i n (5.49) i s always p o s i t i v e and v a n i s h e s o n l y when y k = 0. S i n c e e n t e r s (5.47) as a w e i g h t i n g f a c t o r , i t i s r e a s o n a b l e t o r e p l a c e (5.49) by an i n s t a n t a n e o u s e s t i m a t e a k £ y k W k y k (5.50) where Wk 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 m a t r i x . C o n s e q u e n t l y , (5.47) 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 i n d e x • J k -\K \ K+ i ( g k - 2 g g k + \K \ *k (5-51) where g i s a t h r e s h o l d l e v e l d e t e r m i n e d by o f f - l i n e computer s i m u l a t i o n ( o r by o n - l i n e t u n i n g ) . To p r e v e n t e r r a t i c g a i n changes due t o t h e e s t i m a t i o n (5.50), a s t e p s i z e c o n s t r a i n t ( g k " g k - l ) 2 = 6 j l ( 8 k - l ) 2 ( 5 - 5 2 ) where 6£ i s f i x e d , i s imposed. The Optimum a d a p t i v e g a i n i s d e f i n e d t o be the gain that minimizes (5.51) subject to the constraints (5.52) and (5.34). If (5.34) i s s a t i s f i e d , the optimum gain i s determined by the method of steepest descent,which y i e l d s g k = g k - l [ 1 ~ U s g n ( G k - l ) ] ( 5 ' 5 3 ) G ^ Gk-1 ~ 3 g k g j = g k - i \ 4 K-rViVi^k-i + g k - i L * k ( 5 - 5 5 ) I f (5.53) v i o l a t e s (5.34), then g^ i s replaced by the appropriate upper or lower bound. The adaptive nature of g, can be seen from (5.51). I f estimation er r o r becomes excessive, (5.50) increases and more x<reight i s given to choosing gj_ to minimize estimation error. As estimation error increases, i t i s desirable to place more weight on the use of measurements. This weighting i s done optimally i f (5.37) i s used. The threshold can be set so that g* approaches g^_ as estimation error increases. On the other hand, i f the estimation error i s acceptable, then more weight i s given to choosing g^ to minimize the cost of c o n t r o l . This means that, as long as the estimate i s acceptable, a small control e f f o r t should be used. The choice of i n (5.54) i s governed l a r g e l y by computational convenience. A simple and reasonable choice i s = w _^_> where w i s a p o s i t i v e number. Another simple p o s s i b i l i t y i s to choose w so that . G k - i = y{ \ c \ + b s g n ( S k - f i V (5-56) where b i s a p o s i t i v e number. The advantage of the adaptive approach i s that e x p l i c i t evaluation of s t a t i s t i c a l data i s not required. The c o n t r o l l e r i s "tuned" by o f f - l i n e computer simulations. The small number of tuning parameters (two) makes on-line tuning feasible. 5.4 Application - Single Steam Plant The model of a single steam plant given i n Example (5.1), i s used to evaluate the load-frequency control c a p a b i l i t i e s of the proposed adaptive controller. The augmented state model has the form (5.24). It i s assumed that frequency and t i e - l i n e deviations are the only measure-ments available so that the measurement matrix i n (5.25) has the form The discrete control sequence represented by u^ i n (5.24) must be replaced by U k = (u k , v k ) ' , where (see (5.17) - (5.19)) \ " V l +T V l . (5-58) V k = S' Xpk + Sn+1 \ - ( s' p + S n + l ) A P d ' ( 5 * 5 9 ) The $ and D matrices for the augmented model are given by (see Chapter 4, (4.40), and (5.4)) * = A * $(0,0) = I, (5.60) a D A [D 1 D 2] (5.61) where T * • J, 0 T *(T,t) B f t (5.62) a. (T,t) B • t • d t (5.63) 0 a For s i m u l a t i o n p u r p o s e s , t h e n o i s e V e c t o r s Wk and v k a r e t a k e n i n t h e form w. = a I , x. , , v, = a. I , 2, (5.64) k w wk k+1 k v vk k 8: where (see (4.40)), x' = [x; . AP „ ]. I , and I , are diagonal matrices k pk dk J wk vk whose elements are pseudo-random numbers with a uniform d i s t r i b u t i o n between -1 and +1. The scalars a and a are used to set noise l e v e l . w v . ( I t should be noticed that the noise (5.64) i s state-dependent. This occurs, for example, when the system parameters undergo random d i s -turbances. The control (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 (5.27), i s suboptimum for the type of noise given by (5.64). Tuning of the time-varying matrix gain to improve system performance i s impractical. The proposed suboptimum adaptive control, however, i s ea s i l y tuned to a variety of noise s t a t i s t i c s , including those defined by (5.64). The design of the adaptive control proceeds i n three stages. The f i r s t stage i s to determine the control gain matrix i n (5^28) for the deterministic system. This has been done i n Section (5.2) (Example (5.1)). The second stage i s the design of a deterministic observer with a constant gain L (see (5.33)). The design d e t a i l s are given i n Appendix IV where the result \ (.(.-x nn<; nai _ TOO TA/. " (5.65) T , = f. 663 .005 .091 -.128 .364 .007 .935 -.006. -.014 -.053 i s obtained. The system response for an incremental pother demand AP d = 0.1 using a suboptimum controller with the observer gain (5.65) i s shown i n Fig. (5.2). I t i s seen that the response meets the specified conditions i n that, as t -»- + o o , the frequency deviation and incremental generation approach zero and 0.1, respectively. In the absence of system noise, i t i s seen that the control (5.28), where x^ i s the deterministic observer output, gives acceptable dynamic performance. F i g . (5.2) Steam p l a n t r e s p o n s e . Frequency and t i e - l i n e measurements sampled w i t h T = 1. 84 The t h i r d stage i s the design, of the adaptive c o n t r o l l e r which i s based on changing the s c a l a r gain g^ according to the s t r a t e g y given by (5.53) and (5.56). The s t a b i l i t y l i m i t s (5.34) f o r the system under.con-s i d e r a t i o n are e a s i l y shown to be gffi = 0 and g^ = 2, r e s p e c t i v e l y . The" step-s i z e parameter 6£ i s set equal to 0.2 (see (5.52)). This choice i s a commonly used compromise between using a small step to meet l i n e a r i t y and numerical s t a b i l i t y requirements and using a l a r g e step to reduce the number of steps. The only parameters which 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 are b and g. For the system considered the choice b = 0.2 appeared reasonable a f t e r 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 . To i n v e s t i g a t e the e f f e c t of d i f f e r e n t Values f o r the t h r e s h o l d l e v e l g, the cost 60 _ - -J = E (X^ ^  + u£) (5.66) k=0 i s i n v e s t i g a t e d and averaged over ten runs. The i n i t i a l frequency d e v i a t i o n i s taken to be Aco(0) = 2 and a l l other i n i t i a l s t a t e s are set equal to zero. Figure (5.3) i l l u s t r a t e s the average cost as a f u n c t i o n of g f o r d i f f e r e n t noise l e v e l s . From F i g . (5.3) the best average value i s taken to be g = 0.5. The adaptive c o n t r o l l e r i s now "tuned" 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 evaluated. Figure (5.4) i l l u s t r a t e s the r e s u l t s f o r : (a) The d e t e r m i n i s t i c observer (g^ = 1). (b) The adaptive observer. I t i s seen that the adaptive observer r e s u l t s i n a lower cost. The three values chosen f o r a correspond to random changes i n w the elements of $ and D of 10%, 20% and 30%, r e s p e c t i v e l y . Heavy measurement noise ( a v = 1) could be considered to a r i s e when there are f a u l t y measurements or f a u l t y data t r a n s m i s s i o n . Fig. (5.3) Average cost as,a function of threshold l e v e l (5.4) E f f e c t of adaptive gain (a) de t e r m i n i s t i c observer (b) adaptive observer. The adaptive observer i s a f i l t e r whose output i s an- estimate of the s t a t e of the system. 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 tr a c k i n g ) c a p b i l i t i e s of the adaptive observer i n the presence of system nois e . For the e v a l u a t i o n , an incremental power demand of AP^ = 0.1 i s assumed and the noise l e v e l s are chosen to be a =0.2 and a =0.1. Figure v w 6 ,.(5.5) i l l u s t r a t e s the system frequency Aw, the estimated frequency Aw, the estimated demand AP^, and the 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) that the f i l t e r output gives a good estimate Of the average behaviour of the s t a t e s . /MO / A 1 1 ' ' 1 ' 1 1 1 20 40 60 TIME CSECONDSJ ( 5 . 5 ) T r a c k i n g c a p a b i l i t y o f t h e a d a p t i v e o b s e r v e r . 6. CONCLUSIONS F o r 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 power s y s t e m , an a l g o r i t h m i s p r e s e n t e d , i n C h a p t e r 2, f o r t h e e v a l u a t i o n o f optimum s w i t c h i n g i n s t a n t s f o r p a r a m e t e r changes i n t h e n e t w o r k so as t o improve s y s t e m s t a b i l i t y . The method appears t o o f f e r p r a c t i c a l as w e l l 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 v e r th e L i a p u n o v f u n c t i o n a p p r o a c h , i n f i n d i n g t h e c r i t i c a l s w i t c h i n g t i m e . 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 . P r e l i m i n a r y o f f - l i n e compu-t a t i o n c o u l d o b t a i n t h e r e l a t i o n s h i p s between optimum s w i t c h i n g i n s t a n t s and i n i t i a l s y s t e m s t a t e s . E f f i c i e n t n u m e r i c a l c u r v e - f i t t i n g methods a r e becoming 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 t o s t o r e t h e s e r e l a t i o n s h i p s i n p a r a m e t r i c form w h i c h r e q u i r e a l i m i t e d computer memory. T h i s w o u l d e l i m i n a t e t h e n e c e s s i t y f o r f a s t o n - l i n e s o l u t i o n o f s e t s o f d i f f e r e n t i a l e q u a t i o n s . F o r l e s s s e v e r e d i s t u r b a n c e s , d i g i t a l s i m u l a t i o n r e s u l t s show t h a t 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 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 . Because of t h e l a r g e number o f s t a t e v a r i a b l e s , o p t i m a l c o n t r o l o f an i n t e r c o n n e c t e d power sy s t e m i s n o t f e a s i b l e and some form o f s u b o p t i m a l c o n t r o l i s e s s e n t i a l . A suboptimum l o c a l c o n t r o l b a s e d on a l i n e a r i z e d model w h i c h n e g l e c t s p l a n t i n t e r a c t i o n i s p h y s i c a l l y f e a s i b l e b u t can r e s u l t i n s y s t e m i n s t a b i l i t y . By i n t r o d u c i n g t h e c o n c e p t o f t w o - l e v e l c o n t r o l a s a t i s f a c t o r y s u b o p t i m a l f e a s i b l e c o n t r o l i s o b t a i n a b l e . The l o c a l c o n t r o l i s augmented 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 c o n t r o l . A f e a s i b l e o n - l i n e method f o r g e n e r -a t i n g t h i s c o n t r o l i s g i v e n . O f f - l i n e c o m p u t a t i o n s a r e used t o d e t e r m i n e t h e i n t e r v e n t i o n c o n t r o l i n a parametric f o r m as a f u n c t i o n o f t i m e and i n i t i a l s t a t e s . 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 can g e n e r a t e t h e s e p a r a m e t e r s o n - l i n e and t r a n s m i t them back t o t h e subsystems w h i c h g e n e r a t e t h e l o c a l i n t e r v e n t i o n c o n t r o l s i g n a l s . Because most 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 , o t h e r 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 o n t r o l , and a s s o c i a t e d s y s t e m n o n - l i n e a r i t i e s can be. a c c o u n t e d f o r w i t h o u t much d i f f i c u l t y . The m u l t i - l e v e l c o n t r o l scheme w i t h 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 t h e p a r a m e t e r i z e d f o r m . s u g g e s t e d h e r e appears t o be a p r o m i s i n g f e a s i b l e a p p r o a c h t o t h e c o n t r o l o f i n t e r c o n n e c t e d power systems. The i n t e r v e n t i o n c o n t r o l i s an open-loop c o n t r o l w h i c h augments th e c l o s e d - l o o p l o c a l c o n t r o l l e r s . The c o m p o s i t e c o n t r o l r e s u l t s i n i m p r o v e d s y s t e m p e r f o r m a n c e . I n t e r v e n t i o n c o n t r o l Xtfould o n l y be a p p l i e d i f t h e s y s -tem d i s t u r b a n c e s a r e s i g n i f i c a n t . The 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 t o r o u t i n e s m a l l d i s t u r -bances e x p e r i e n c e d i n e v e r y d a y o p e r a t i o n o f power s y s t e m s , i s d i s c u s s e d i n C h a p t e r 4.. Because . 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 p r o b l e m o f o p t i m a l l o a d - f r e q u e n c y c o n t r o l cannot be s o l v e d by d i r e c t a p p l i -c a t i o n o f t h e o p t i m a l l i n e a r - s t a t e r e g u l a t o r c o n t r o l . A f e a s i b l e o p t i m a l 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 i d e n t i f i c a t i o n o f t h e i n c r e m e n t a l power demand. Two methods have been shown s u i t a b l e f o r power demand i d e n t i f i c a t i o n . One method i s bas e d on 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 v e r y s i m p l e . Improved i d e n t i f i c a t i o n a c c u r a c y can be a c h i e v e d by 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 . A f u r t h e r advantage o f t h e second method i s t h a t i t can cope w i t h t h e s i t u a t i o n where n o t a l l t h e s t a t e s a r e measured. The o b s e r v e r i s d r i v e n by measurements 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 o f t h e unmeasured s t a t e s and t h e i n c r e m e n t a l power demand. In. Chapter 5, a suboptimum s o l u t i o n to the problem of sampled data optimum load-frequency 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 incremental power demand i s given. Trade-off between system response and sampling r a t e can be e a s i l y s t u d i e d . The case of random system disturbance i s c o n s i -dered. 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 to 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 accurate models r e q u i r e d to achieve optimum performance are g e n e r a l l y not a v a i l a b l e . I t i s e s s e n t i a l , t h e r e f o r e , to study suboptimum c o n t r o l l e r s . A three stage procedure i s given f o r the design of a suboptimum s t o c h a s t i c c o n t r o l l e r . The f i r s t stage c o n s i s t s i n determining the c o n t r o l gain 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 stage c o n s i s t s i n the design of a c l a s s of d e t e r m i n i s t i c observers, the t h i r d and f i n a l stage c o n s i s t s i n a d a p t i v e l y choosing a s c a l a r observer gain so as to minimize an instantaneous cost index. An example i s used to i l l u s t r a t e the design .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 values show the effectiveness of the design procedure and the ease w i t h which " t u n i n g " can be accomplished. The proposed load-frequency c o n t r o l discussed i n Chapters 4 and 5 i s compatible w i t h an EACC-type c o n t r o l . The load-frequency c o n t r o l l e r i s a c t i v a t e d only a f t e r the EACC has decided that c o n t r o l a c t i o n i s r e q u i r e d . This prevents the load-frequency c o n t r o l l e r from attempting to c o r r e c t f o r r a p i d l y changing load f l u c t u a t i o n . An EACC c o n t r o l l e r could be programmed to make d e c i s i o n s concerning the type of c o n t r o l to be used. One mode of load-frequency c o n t r o l xrould be f o r unknown but d e t e r m i n i s t i c d i sturbances. The c o n t r o l l e r f o r t h i s mode could be of the form discussed i n Chapter 4 (or i n a sampled-data form as given i n S e c t i o n (5.2)). A second mode of c o n t r o l would occur i f the disturbances are random. The change from one mode of c o n t r o l to another i s b a s i c a l l y very simple. I t amounts to 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 disturbances and making g^ adaptive i n the case of random disturbances. APPENDIX I ,18,19 ( D e f i n i t i o n s ; 18 1. The term a r e a i d e n t i f i e s t h a t p a r t o f an i n t e r c o n n e c t e d power sy s t e m w h i c h i s t o absorb i t s own l o a d changes. I t may be a s i n g l e company, r e s p o n d i n g t o i t s own l o a d changes; i t may be p a r t o f a company o p e r a t i n g t o r e s p o n d t o l o a d changes t h a t o c c u r i n o n l y a g i v e n p a r t o f t h e company's n e t w o r k ; i t may "be a whole group o f companies p o o l e d t o g e t h e r t o a b s o r b t h e l o a d changes t h a t o c c u r anywhere w i t h i n t h e i r c o l l e c t i v e b o u n d a r i e s . 2. A s i n g l e a r e a i n t e r c o n n e c t e d s y s t e m i s one i n w h i c h l o a d changes a r e a b s o r b e d by t h e s y s t e m as a w h o l e , r e g a r d l e s s o f where on t h e s y s t e m t h e y o c c u r . No one p a r t o f t h e s y s t e m i s e x p e c t e d t o a d j u s t i t s own g e n e r a t i o n t o a b s o r b i t s own l o a d changes. Load changes t h a t o c c u r i n any p a r t o f t h e system may be a b s o r b e d e l s e w h e r e w i t h i n t h e s y s t e m , i n a c c o r d a n c e w i t h a l l o c a t i o n p r a c t i c e s p r e v a i l i n g a t t h a t p a r t i c u l a r t i m e . T i e - l i n e power f l o w s a r e , t h e r e f o r e , n e i t h e r s c h e d u l e d n o r c o n t r o l l e d . 3. A m u l t i p l e a r e a i n t e r c o n n e c t e d s y s t e m i s one t h a t c o n s i s t s o f a number o f o p e r a t i n g a r e a s , each o f w h i c h i s e x p e c t e d t o a d j u s t i t s own g e n e r a t i o n t o a bsorb i t s own l o a d changes. T i e - l i n e power f l o w s between a r e a s a r e s c h e d u l e d and m a i n t a i n e d . D u r i n g o u r s t u d y o f t h e 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 (as d e f i n e d l a t e r ) , t h e t e r m " p l a n t " s h a l l r e f e r t o t h e c o n t r o l l i n g g e n e r a t i n g s t a t i o n i n t h e a r e a . Two common t y p e s o f p l a n t s a r e c o n s i d e r e d , namely, steam and h y d r o p l a n t s . 94 4. Megavar-voltage (Q-V) c o n t r o l problem i s the problem of c o n t r o l l i n g the r e a c t i v e power i n the system. In t h i s problem, the main concern i s the vo l t a g e l e v e l at the d i f f e r e n t buses throughout the system. Due to the r e l a t i v e l y f a s t a c t i o n of voltage r e g u l a t o r s , i t i s common p r a c t i c e to assume that the bus voltages are maintained at t h e i r nominal v a l u e s . This assumption i s adopted i n t h i s t h e s i s . 5. Megawatt-Frequency (P-f) c o n t r o l problem i s the problem of c o n t r o l l i n g the r e a l power. Load-Frequency 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 p a r t i -c u l a r c o n t r o l job. The f o l l o w i n g d e f i n i t i o n of the LFC problem i s accepted by the IEEE (AIEE 94, Proposed D e f i n i t i o n s , December 1962): "Load-Frequency Control i s the r e g u l a t i o n of the power output of e l e c t r i c generators w i t h i n a p r e s c r i b e d area i n response to changes i n system frequency, t i e - l i n e l o a d i n g , or the r e l a t i o n of these to each other, so as to maintain the scheduled system frequency and/or the e s t a b l i s h e d interchanges w i t h other areas w i t h i n predetermined l i m i t s . " 6. LFC c r i t e r i a : Some of the c r i t e r i a of LFC as g e n e r a l l y defined are given below. N e g l e c t i n g the c o n t r a i n t s on measurements, c o n t r o l , and system 19 dynamics, then the 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 : Minimize ( a ) , (b) or (c) where (a) Area 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 + frequency b i a s x frequency d e v i a t i o n [ACE = AP + BAf -> 0 ] , ( I . l ) (b) Inadvertent Interchange ( I I ) = I n t e g r a l of the l i n e d e v i a t i o n [ I I = /AP dt 0] , • (1.2) 9 5 (c) Time D e v i a t i o n (TD) - -' ^jj x I n t e g r a l of frequency d e v i a t i o n [TD = ^ /Af dt ^ 0], (1.3) and a l s o mimimize (d) Area Supplementary C o n t r o l (ASC) = f u n c t i o n of ACE, I I , TD. [ASC = 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 of F i g . (2.1) can be reduced by Thevenin's theorem to a s e r i e s impedance r + j x and an equivalent i n f i n i t e bus voltage V'. The e l e c t r i c a l power, damping co-e f f i c i e n t and f l u x l i n k a g e equations can be reduced to the form given by (2.20), (2.21) and (2.22) r e s p e c t i v e l y ^ . The c o e f f i c i e n t s are defined by the folloxtfing r e l a t i o n s : C l 4 B l C2 ~ B2 C o S ^ + B 3 s i n Y C 3 = B^ c o s Y - B 2 s i n .$ C'4 = 0.5 B 4 cos(y-g) - -B^ cos y s i n 3 Cg - B^ s i n ycos g where: = A 0(A c+A./x ) B *2V"5 "W B2 - - A 1 V ; ( V A 4 / X q ) B 3 " A 2 A 3 V ; ( x d - X q ) / x d X q B 4 " A l A 3 V o ^ X q - X d ) / x d X q where: C 7 = constant (defined by (2.23)) Cg - Ag cosy C9 " A 6 s i n Y  C 1 0 " V T o °11 D l °12 D 2 V'(x - x " ) T " /(x +x )' o v q q' qo q e' B 2 4 ^ i - ^ y ^ v 2 g = Arct a n [ ( x d + * e ) / r e l v - Arctan [(x +x )/ r ] 1 q e e A, £ (/r 2+(x'+x )2)/A1x« 1 e d e i d A 2 - r e / A l x J A, A ( / r 2 + ( x x ) 2 ) / A , x 3 e q e 1 q A. = ( r +x (x +x ))/A,x x' 4 e e q e i q d A 5 " ( x q + X e ) / A l X d X q A 6 4 V X d ) A 3 V o / x d A ? 4 x d ( l - A A ) / x d + A 4 A. & r 2 + ( x +x )(x'+x )/x'x 1 e q • e d e d q In the above d e f i n i t i o n s : x d > x = synchronous reactance i n d and q axes, r e s p e c t i v e l y . 97 x' x' •= transient reactance in d and q axes, respectively, d q ' X d , X q ~ S u b s t r a r L S i e n < : reactance in d and q axes, respectively. T" ,T" = subtransient open-circuit time constant in d and q axes, respectively, do qo The values (in pu) of•the different parameters are: x, = 1.0, x = 0.6, x' = .27, x" = 0.22, x" = 0.29.x = 9.0, d * q ' d . ' d ' q "* o x" =.04, T " = .07, x, = .013, R = .15, x, « .7488, B = .067, do qo t X ' ' G = .18, V = 1.05, P+JQ = .753 + J.03. APPENDIX I I I As ex p l a i n e d i n S e c t i o n (3.7), the i n i t i a l c o n d i t i o n s f o r each p l a n t are taken to be zero w i t h the exception of the frequency d e v i a t i o n . Consequently a (X ) , f o r each of the two p l a n t s considered . i s assumed i n the J m o ' - ' form 2 2 a (X ) = B +B ^Au, +B ^AWA +B .J AO)- +B /AOU +B HAW- AW,, m o mo Mml l o m2 2o mi l o m4 2o tn5 l o 2o ( I I I . l ) where the c o e f f i c i e n t s B . >i~0, 1. ...5 are chosen so that (3.36) i s mi minimum. The algor 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 the optimum h*(.t) f o r M d i f f e r e n t sets of i n i t i a l c o n d i t i o n s (Section (3.6.2) ). (.2) Find the c o e f f i c i e n t s akj»m=l,2,3,4 f o r each set of i n i t i a l m c o n d i t i o n s k-1,2, ...M. (3) For m=l and k=l, ... M form I = -wr in m where M t m r = mo ml 5m5 and W 2 [1 Aw l 0 Aco 2 Q A u J o h J l Q Aw^.A^p]" 2 2 [1 Atow Ato0 Aw, Aw~ Aw„ • ••Aw -iM l o 2o l o Zo l o 20-" ( I I I . 2 ) 30 The o n l y unknown i n ( I I I - 2 ) i s T^. F o r M > 6, i s g i v e n by rw = (wwrV '^ (in. 3) (4) Repeat (3) f o r m=2,3,4. 100 APPENDIX IV 23 The optimum f i l t e r gain. can. be determined by s o l y i n g the equations K k + 1 = \ + 1 H ' \ l l V*'1** \ + 1 - \ + 1 - \ + i H ' ( H \ + i H'+WH V i .(lv-2> = + \ < I V - 3 ) r e c u r s i v e l y . The i n i t i a l covariance m a t r i x , P o > and the matrix sequences Q ^ , R^. (k = 0, 1, ...) are assumed knox^n. In ge n e r a l , however, the data P Q ^ , R ^ i s not a v a i l a b l e . However, equations (IV. 1), (IV. 2) and (IV. 3) s t i l l prove u s e f u l i n determining a constant observer m a t r i x gain L (see (5.33)). S i m u l a t i o n s t u d i e s or system operating experience u s u a l l y allow some i n i t i a l guess to be made f o r the unknown parameters. A reasonable choice i s to take constant p o s i t i v e - d e f i n i t e diagonal matrices P Q , Q q 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 stead y - s t a t e convariance matrix P . Equation (IV.1) ^ C O then y i e l d s the stead y - s t a t e gain m a t r i x and a reasonable choice f o r L i s to take L = K H' R " 1 (IV.4) co o For the example given i n Se c t i o n (5.4), the choice R = b. I . Q = b„ I" P = I (IV.5) o 1 ' o 2 ' o where b^ = = 10 \ i s made. E v a l u a t i o n of (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 to apply a d i s c r e t e v e r s i o n of the method presented i n reference 35. This a l t e r n a t i v e approach has been used i n Se c t i o n (4.5) to determine an observer gain. Equation (IV.4) i s to be p r e f e r r e d i f a l i m i t e d amount of s t a t i s t i c a l data i s a v a i l a b l e . 101 REFERENCES 1. T.E. Dy L i a c c o , "The A d a p t i v e R e l i a b i l i t y C o n t r o l System," IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-85, pp. 517-531, May 1967. 2. E.W. Kimbark, "Improvement o f Power System S t a b i l i t y by Changes i n t h e Network", IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-88, pp. 773-781, May 1969. 3. H.M. E l l i s , e t a l , "Dynamic S t a b i l i t y o f Peace R i v e r T r a n s m i s s i o n System", IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-85, pp. 586-600, June 1966. 4. O.J.M. S m i t h , "Power System T r a n s i e n t C o n t r o l by C a p a c i t o r S w i t c h i n g , " IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-88, pp. 28-35, J a n u a r y 1969. 5. W.A. M i t t l e s t a d t and J . L . Saugen, "A Method o f I m p r o v i n g Power System T r a n s i e n t S t a b i l i t y U s i n g C o n t r o l l a b l e P a r a m e t e r s " , IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-89, pp. 23-27, J a n u a r y 1970. 6. D.K. R e i t a n and N. Rama Rao, " O p t i m a l C o n t r o l o f T r a n s i e n t s i n a Power System," P r o c . IEEE. pp. 1448, August 1969. 7. N. Rama Rao and D.K. R e i t a n , "Improvement o f Power System T r a n s i e n t S t a b i l i t y U s i n g O p t i m a l C o n t r o l : Bang-Bang C o n t r o l o f R e a c t a n c e " , IEEE T r a n s . (Power A p p a r a t u s and S y s t e m s ) , V o l . PAS-89, pp. 975-984, May-June 1970. 8. S.M. M i n i e s y and E.V. Bohn, "Optimum Network S w i t c h i n g i n Power Systems," p r e s e n t e d a t t h e IEEE Power E n g i n e e r i n g S o c i e t y W i n t e r M e e t i n g , N.Y., p a p e r No. 71 TP 101-PWR. 9. R.H. P a r k , "Improved R e l i a b i l i t y o f fculk Power S u p p l y by F a s t Load C o n t r o l , " P r o c e e d i n g s o f t h e A m e r i c a n Power C o n f e r e n c e , C h i c a g o , 111., 102 A p r i l 1968, pp. 1128-1141. 10. Y.N. Yu, K. Vongsuriya and L. Wedman, "Application of an Optimal Control Theory to a Power System", IEEE Trans. (Power Apparatus and Systems), Vol. PAS-89, pp. 55-63, January, 1970. 11. Y.N. Yu and C. Siggers, " S t a b i l i z a t i o n and Optimal Control Signals f o r a Power System", presented at the IEEE Summer Power Meeting and EHV Conference, 1970, paper No. 70 TP531-PWR. 12. H.A. Moussa, and Y.N. Yu, "Optimal Power S t a b i l i z a t i o n Through E x c i t a -t i o n and/or Governor Control", presented at the 1971 IEEE Power Engineering Society Summer Meeting, Portland, Oregon, Paper No. 71TP 581-PWR. 13. S.M. Miniesy and E.V. Bohn, "Two-level Control of Interconnected Power Plants", presented at the 1971 IEEE. Power Engineering Society Summer Meeting, Portland, Oregon, paper No. 71 TP624-PWR. 14. C E . Fosha J r . , and 0.1. Elgerd, "The Megawatt Frequency Control Problem: A New Approach Via Optimal Control Theory", IEEE Trans. (Power Apparatus and Systems), Vol. PAS-89, pp. 563-578, A p r i l 1970. 15. C.D. Johnson, "Optimal Control of Linear Regulator with Constant D i s -turbances", IEEE Trans. (Automatic Control), V o l. AC-13, pp. 416-420, August 1968. 16. A.H. Levis, R.A. Schlueter and M. Athans, "On the Behaviour of Optimal Linear Sampled Data Regulators", I n t e r n a t i o n a l Journal on Control, 1971, Vol. 13, No. 2, pp. 343-361. 17. R.K. Cavin I I I , M.C. Budge J r . , and P. Rasmussen, "An Optimal Linear System Approach to Load-Frequency Control", presented at the IEEE 103 Power Engineering Society Winter Meeting, 1971, New York, paper No. 71 TP 52-PWR. 18. N. Cohn, " C o n t r o l of Interconnected Power Systems", Chapter 17, i n "The Handbook of Automation, Computation and C o n t r o l " j V o l . 3., J . Wiley: New York 1961. 19. C.W. Ross, " E r r o r Adaptive C o n t r o l Computer f o r Interconnected Power Systems", IEEE Trans. (Power Apparatus and Systems) V o l . PAS-85, pp. 742-749, J u l y 1966. 20. A.H. El-Abiad and K. Ngappan, "Transient S t a b i l i t y Regions of M u l t i -Machine Power Systems1.' , IEEE Trans. (Power Apparatus and Systems), V o l . PAS-85, pp. 169-179, February 1966. 21. Y.N. Yu and K. Vongsuriya, "Non-Linear Power System S t a b i l i t y Study by Liapunov Function and Zubov's Method",IEEE Trans. (Power Apparatus and Systems), V o l . PAS-85, pp. 1480-1485, December 1967. 22. R.F. Vachino, "Steepest Descent w i t h I n e q u a l i t y C o n s t r a i n t s on the C o n t r o l V a r i a b l e s " , J . SIAM, C o n t r o l , V o l . 4, pp. 245-261, 1966. 23. A.P. Sage, "Optimum Systems C o n t r o l " > (Book) by P r e n t i c e H a l l Inc., N.J., 1968. 24. M. Athans, and P.L. Falb, "Optimal C o n t r o l " , (Book), McGraw-Hill Inc., New York, N.Y., 1966, Chapter 9. 25. R.P. Aggarwal, F.R. Bergseth, "Large S i g n a l Dynamics of LFC C o n t r o l System and t h e i r O p t i m i z a t i o n Using Non-Linear Programming I " , IEEE Trans. (Power Apparatus and Systems), V o l . PAS-87, pp. 527-532, February, 1968. 26. L.K. Kirchmayer, "Economic C o n t r o l of Interconnected Power Systems", 104 (Book), J. Wiley, New York, 1959, Chapter 1. 27. M.D. Mesarovic, " M u l t i l e v e l Systems and Concepts i n Process Control", IEEE Proceedings Vol. 58, pp. 111-125, January, 1970. 28. T.'R. Blackburn, "Solution of the Algebraic Matrix R i c c a t i Equation Via Newton-Raphson I t e r a t i o n " , Joint Automatic Control Conference (JACC), The University of Michigan, June, 1968, pp. 940-945. 29. K. Vongsuriya, "The Application of Liapunov Function to Power System S t a b i l i t y , " Ph.D. thesis, University of B r i t i s h Columbia, Vancouver, February 1968. 30. D. Luenberger, "Optimization by vector space Methods", (Book), J. Wiley, New York, N.Y.,1969, Chapter 4. 31. J.S. Meditch, "Stochastic Optimal Linear Estimation and Control", (Book), McGraw-Hill Inc., New York, 1969. 32. 0.1. Elgerd, and CE. Fosha, "Optimum Megawatt-Frequency Control of Multi-Area E l e c t r i c Energy System", IEEE Trans. (Power Apparatus and Systems), Vol. PAS-89, pp. 556-563, A p r i l 1970. 33. R.L. Kosut, "Suboptimal Control of Linear Time-Invariant Systems Subject to Control Constraints", Proceedings of the Joint Automatic Control Conference, (JACC), 1970, pp. 820-828. 34. D.G. Luenberger, "Observers for Multivariable Systems", IEEE Trans. (Automatic Control), Vol. AC-11, pp. 190-197, A p r i l 1966. 35. G.W. Johnson, "A Deterministic Theory of Estimation and Control", IEEE Trans. (Automatic Control), Vol. AC-14, pp. 380-384, August 1969. 36. CD. Johnson and R.E. Skelton, "Optimal Desaturation of Momentum Exchange Control System", Proceedings of the Joint Automatic Control Conference, (JACC), 1970, pp. 683-694. 

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