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UBC Theses and Dissertations

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

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STAB I L I ZAT ION AND OPTIMIZATION OF A POWER SYSTEM WITH SENS IT IV ITY CONSIDERATIONS by LEONARD NICKOLAUS WEDMAN B . S c , U n i v e r s i t y o f A l b e r t a , 1964 A THES IS SUBMITTED IN PART IAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPL IED SC IENCE i n t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g Ve 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 t h e 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 the Commi t t ee Head o f . THE \ t h e D e p a r t m e n t Members o f t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g UNIVERSITY OF BR IT ISH COLUMBIA November , 1968 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and S t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada D a t e ABSTRACT An i n v e s t i g a t i o n i s made i n t o some a s p e c t s o f t he a n a l y s i s and d e s i g n o f h i g h o r d e r s y s t e m s . The p r o b l e m s t r e a t e d a r e s y s t e m s t a b i l i z a t i o n , p a r a m e t e r o p t i m i z a t i o n , c o m p u t a t i o n o f an o p t i m a l c o n t r o l l e r and p a r a m e t e r s e n s i t i v i t y . The methods d e v e l o p e d f,or s o l v i n g t h e s e p r o b l e m s a r e a p p l i e d t o a 9 ^ o r d e r l i n e a r i z e d power s y s t e m . To s t a b i l i z e t h e s y s t e m , an e i g e n v a l u e s h i f t t e c h n i q u e i s u s e d . E i g e n s y s t e m s e n s i t i v i t y a n a l y s i s i s a p p l i e d t o d e t e r m i n e b o t h t he p a r a m e t e r change r e q u i r e d and t h e new e i g e n s y s t e m a f t e r t he change has b e e n made. A c o r r e c t i o n method i s a p p l i e d t o t he new e i g e n s y s t e m f o r i m p r o v i n g a c c u r a c y i n o r d e r t h a t l a r g e s t e p s i n p a r a m e t e r change may be t a k e n . T h i s method i s s u b s e q u e n t l y u s e d i n an o p t i m i z a t i o n p r o c e d u r e f o r p a r a m e t e r s e t t i n g t o m i n i m i z e a c o s t f u n c t i o n a l o f q u a d r a t i c f o r m . F o r t h e c o m p u t a t i o n o f an o p t i m a l c o n t r o l l e r , P u r i and G r u v e r ' s s u c c e s s i v e a p p r o x i m a t i o n method i s u s e d i n c o n j u n c t i o n w i t h a f a s t r e c u r s i v e method d e v e l o p e d f o r s o l v i n g e a c h a p p r o x i m a t i o n o f t he R i c a t t i m a t r i x . The c a l c u -l a t i o n can be i n i t i a t e d by t he e i g e n v a l u e s h i f t i n g method t o e n s u r e t h a t the s y s t e m i s i n i t i a l l y s t a b l e . F i n a l l y , a t i m e r e s p o n s e s e n s i t i v i t y s t u d y i s made u s i n g a method d e v e l o p e d f o r s i m u l t a n e o u s s e n s i t i v i t y f u n c t i o n d e t e r m i n a t i o n . T h i s method r e d u c e s c o m p u t a t i o n (9) t i m e s i g n i f i c a n t l y o v e r t he c o n v e n t i o n a l method t h u s e n a b l i n g the i n v e s t i g a t i o n o f t i m e r e s p o n s e s e n s i t i v i t y t o a l a r g e number o f p a r a m e t e r s . The r e s u l t s o f t he s e n s i t i v i t y s t u d y a r e t h e n a p -p l i e d to the d e s i g n o f a s u b o p t i m a l c o n t r o l l e r . i i TABLE OF CONTENTS Page ABSTRACT ( i i ) TABLE OF CONTENTS ( i i i ) L I S T OF TABLES (v ) L I ST OF ILLUSTRATIONS ( v i ) ACKNOWLEDGEMENT . . ( v i i ) NOMENCLATURE ( v i i i ) 1. INTRODUCTION 1 2. E IGENSYSTEM SENS IT IV ITY ANALYSIS APPL IED TO SYSTEM STAB I L I ZAT ION 6 2.1 P r o b l e m F o r m u l a t i o n 6 2.2 E i g e n s y s t e m S e n s i t i v i t y A n a l y s i s 7 2 . 3 An E i g e n v a l u e S h i f t i n g Me thod 10 2 . 4 An E x t e n d e d C o l l a r and J a h n C o r r e c t i o n M e t h o d . 12 2.5 A N u m e r i c a l Examp le 14 3. E IGENSYSTEM ANALYSIS APPL IED TO PARAMETER OPTIMIZATION 20 3.1 I n t r o d u c t i o n . . . . 20 3 . 2 An E i g e n s y s t e m Form o f t he C o s t F u n c t i o n a l . . . 21 3 . 3 P e r f o r m a n c e F u n c t i o n M i n i m i z a t i o n 22 3 . 4 A N u m e r i c a l Examp le . 23 4 . COMPUTATION OF AN OPTIMAL CONTROLLER FOR A HIGH ORDER SYSTEM 26 4 . 1 P r o b l e m F o r m u l a t i o n 26 4 . 2 A R e c u r s i v e Method f o r O b t a i n i n g the S u c c e s s i v e A p p r o x i m a t i o n s 28 4 . 3 A c c u r a c y o f t he R e c u r s i v e Me thod 30 4 . 4 A N u m e r i c a l Examp le 31 5. S ENS IT IV ITY ANALYSIS OF THE TIME RESPONSE OF MULTIVARIABLE SYSTEMS 39 5.1 S i m u l t a n e o u s C o m p u t a t i o n o f T ime Response S e n s i t i v i t i e s t o a L a r g e Number o f P a r a m e t e r s . 40 5.2 D e r i v a t i o n o f t he C o m p u t a t i o n A l g o r i t h m 44 5 . 3 A N u m e r i c a l Examp le 46 6. CONCLUSIONS 56 i i i Page APPENDIX A FORMATION OF THE COEFFICIENT MATRIX A OF EQUATION (2.1) 59 A . l T h i r d Order Machine and T i e L i n e E q u a t i o n s 59 A.2 Governor and H y d r a u l i c Operator E q u a t i o n s 62 A.3 V o l t a g e R e g u l a t o r - E x c i t e r E q u a t i o n s ..... 63 A.4 Summary of System E q u a t i o n s 64 APPENDIX B INITIAL OPERATING CONDITIONS OF A POWER SYSTEM ,66 APPENDIX C FLETCHER AND POWELL'S DESCENT METHOD 68 REFERENCES . 69 i v L I ST OF TABLES T a b l e Page 2.1 S y s t e m P a r a m e t e r s 15 2.2 E i g e n s y s t e m S h i f t T h r o u g h P a r a m e t e r A d j u s t m e n t 18 4.1 E r r o r o f K A p p r o x i m a t i o n 32 v L IST OF ILLUSTRATIONS F i g u r e Page 4 .1 G o v e r n o r T r a n s f e r F u n c t i o n and H y d r a u l i c O p e r a t o r . 33 4 .2 V o l t a g e R e g u l a t o r - E x c i t e r T r a n s f e r F u n c t i o n 33 4 . 3 O r i g i n a l S y s t em R e s p o n s e s i 35 4 . 4 S y s t em R e s p o n s e s f o r Q = d i a g [ l 1 1 1 1 1 1 1 l ] . . . 36 4 . 5 S y s t em R e s p o n s e s f o r Q = d i a g [lO 10 1 1 1 1 1 1 l ] . 37 5.1 S e n s i t i v i t y o f S t o S y s t e m P a r a m e t e r s and I n i t i a l O p e r a t i n g C o n d i t i o n s 48 5.2 S e n s i t i v i t y o f & t o C o n t r o l l e r G a i n s 49 5.3 S-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f P q . . 51 5.4 S-Response w i t h O p t i m a l C o n t r o l , Ove r V a r i a t i o n o f P 51 o 5.5 S-Response w i t h S u b o p t i m a l C o n t r o l , Ove r V a r i a t i o n o f P 52 o 5.6 S-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f Q q . . 52 5 .7 S-Response w i t h O p t i m a l C o n t r o l , Ove r V a r i a t i o n o f Q 53 *o 5.8 S-Response w i t h S u b o p t i m a l C o n t r o l . Ove r V a r i a t i o n o f Q 53 *o 5.9 S-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f v ^ 0 « « 54 5 .10 S-Response w i t h O p t i m a l C o n t r o l , Ove r V a r i a t i o n 5.11 S-Response w i t h S u b o p t i m a l C o n t r o l , Ove r V a r i a t i o n o f v . . . . . . . 55 t o A . l G o v e r n o r T r a n s f e r F u n c t i o n and H y d r a u l i c O p e r a t o r . . 63 A . 2 V o l t a g e R e g u l a t o r - E x c i t e r T r a n s f e r F u n c t i o n 64 v i ACKNOWLEDGEMENT I w i s h t o t h a n k D r . Y. N. Yu, s u p e r v i s o r o f t h i s p r o j e c t , f o r h i s c o n t i n u e d i n t e r e s t , e n c o u r a g e m e n t and g u i d a n c e d u r i n g t h e r e s e a r c h work and w r i t i n g o f t h i s t h e s i s . A l s o , I have had many h e l p f u l d i s c u s s i o n s a b o u t t h e m a t e r i a l i n t h i s t h e s i s w i t h D r . K. V o n g s u r i y a and Mr. G. Dawson. Thanks a r e due t o D r . M. S. D a v i e s f o r r e a d i n g t h e m a n u s c r i p t and f o r o f f e r i n g v a l u a b l e s u g g e s t i o n s . The p r o o f r e a d i n g o f t h e f i n a l d r a f t b y my c o l l e a g u e s i s d u l y a p p r e c i a t e d . I n a d d i t i o n , t h e s u p p o r t f r o m t h e N a t i o n a l R e s e a r c h C o u n c i l and t h e B r i t i s h C o l u m b i a T e l e p h o n e Company i s g r a t e f u l l y a c -k n o w l e d g e d . I owe a l a r g e d e b i t o f g r a t i t u d e t o my w i f e D o r i s f o r h e r p a t i e n c e and c o n t i n u i n g s u p p o r t t h r o u g h o u t my p o s t g r a d u a t e work. v i i NOMENCLATURE G e n e r a l A nxn s y s t e m m a t r i x B nxm c o n t r o l m a t r i x u m - d i m e n s i o n a l c o n t r o l v e c t o r b n - d i m e n s i o n a l i n p u t v e c t o r u s c a l a r i n p u t T \^ e i g e n v a l u e o f A and A _ _ T X i ' V i e i g e n v e c t o r s o f A and A r e s p e c t i v e l y X , V e i g e n v e c t o r m a t r i c e s composed o f columns o f x. and v ^ , i = 1, ... n, r e s p e c t i v e l y 1 X w , Vu\ a p p r o x i m a t e X and V y n - v e c t o r o f system s t a t e v a r i a b l e s y Q i n i t i a l s t a t e s o f y q p a r a m e t e r v e c t o r A, = J c o s t f u n c t i o n a l Q a p o s i t i v e d e f i n i t e nxn s t a t e v a r i a b l e w e i g h t i n g m a t r i x V a p o s i t i v e d e f i n i t e mxm c o n t r o l s i g n a l w e i g h t i n g m a t r i x K nxn R i c a t t i m a t r i x R i n v e r s e o f d i s c r e t e t i m e i n t e r v a l £ct g ( s ) c h a r a c t e r i s t i c p o l y n o m i a l of m a t r i x A h^ c o e f f i c i e n t s o f g ( s ) R ( s ) a d j o i n t m a t r i x p o l y n o m i a l o f A 2 p^ c o e f f i c i e n t s o f g ( s ) z ( t ) n - v e c t o r o f s t a t e v a r i a b l e s of (5.33) w ( t ) ( 2 n - l ) - v e c t o r o f s t a t e v a r i a b l e s o f (5.34) l\ p r e f i x d e n o t i n g a l i n e a r i z e d v a r i a b l e v i i i s u b s c r i p t d e n o t i n g an i n i t i a l o p e r a t i n g c o n d i t i o n p d / d t , t i me d e r i v a t i v e o p e r a t o r S y s t e m P a r a m e t e r s H a r m a t u r e r e s i s t a n c e a R^ f i e l d r e s i s t a n c e ; i x , d - a x i s m u t u a l r e a c t a n c e be tween s t a t o r and r o t o r ad x^ , x^ d and q - a x i s s y n c h r o n o u s r e a c t a n c e x\ d - a x i s t r a n s i e n t r e a c t a n c e a R+jX t i e l i n e impedance be tween g e n e r a t o r and i n f i n i t e bus G+jB s h u n t a d m i t t a n c e a t g e n e r a t o r t e r m i n a l T ^ k d - a x i s damp ing c i r c u i t t i m e c o n s t a n t I^JQ d - a x i s t r a n s i e n t open c i r c u i t t ime c o n s t a n t tdo Tqo ^ a n < ^ q . - a x i s s u b t r a n s i e n t open c i r c u i t t ime c o n s t a n t s -r^ j 1 d - a x i s t r a n s i e n t s h o r t c i r c u i t t i m e c o n s t a n t 'Cd' Tq ^ a n c ^ ( l - a x i s s u b t r a n s i e n t s h o r t c i r c u i t t i m e c o n s t a n t s £ g h y d r a u l i c t u r b i n e g a t e t ime c o n s t a n t ^ d a s h p o t t i m e c o n s t a n t Y w a t e r t i m e c o n s t a n t -rja g a t e a c t u a t o r t ime c o n s t a n t Y e x c i t e r t i m e c o n s t a n t <-ex v o l t a g e r e g u l a t o r t ime c o n s t a n t •^ •^  g o v e r n o r c o n t r o l s i g n a l a c t u a t o r t i m e c o n s t a n t H g e n e r a t o r i n e r t i a c o n s t a n t D damp ing c o e f f i c i e n t v o l t a g e r e g u l a t o r g a i n ^2 v o l t a g e r e g u l a t o r s p e e d f e e d b a c k l o o p g a i n or g o v e r n o r pe rmanen t d r o o p ^ g o v e r n o r t e m p o r a r y d r o o p i x System V a r i a b l e s P, Q r e a l and r e a c t i v e o u t p u t power o f g e n e r a t o r r e s p e c t i v e l y e n e r g y c o n v e r s i o n t o r q u e m e c h a n i c a l i n p u t t o g e n e r a t o r ^ g e n e r a t o r t o r q u e a n g l e i n r a d i a n s CO g e n e r a t o r a n g u l a r v e l o c i t y i n r a d i a n s / s e c o n d , 1 r a t e d a n g u l a r v e l o c i t y , 377 r a d i a n s / s e c o n d *d' ^ q ' V d ' V q ' ^ d ' H[ ^ a n c* 1~ax:'-s c u r r e n t s , v o l t a g e s and f l u x l i n k a g e s r e s p e c t i v e l y V p j j v o l t a g e p r o p o r t i o n a l t o f i e l d c u r r e n t v ^ a r m a t u r e t e r m i n a l v o l t a g e V Q i n f i n i t e bus v o l t a g e V f d' """f d' ^ f d f i e l d v o l t a g e , c u r r e n t and f l u x l i n k a g e s r e s p e c t i v e l y v g e x c i t e r c o n t r o l v o l t a g e ^ / p f i e l d f l u x l i n k a g e p r o p o r t i o n a l t o LfJ^^ V p f i e l d v o l t a g e p r o p o r t i o n a l t o v ^ g p. u. g a t e movement h p. u. h y d r a u l i c head a g a t e a c t u a t o r s i g n a l a„ g o v e r n o r f e e d b a c k s i g n a l x 1 1 . INTRODUCTION In t he power s y s t e m d e s i g n , t h e r e a r e s t i l l many d i f f i c u l t p r o b l e m s t o be o v e r c o m e . One o f t h e s e i s the p r o b l e m o f a n a l y s i n g t h e s t e a d y s t a t e s t a b i l i t y , s e n s i t i v i t y and o p t i m a l 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 e s t a b l i s h e d t e c h n i q u e s f o r t h e a n a l y s i s o f m u l t i v a r i a b l e s y s t e m s a re d e r i v e d by e x t e n d i n g s i n g l e v a r i a b l e t e c h n i q u e s b u t t h e y a r e d i f f i c u l t t o a p p l y t o the s t u d y o f l a r g e s y s t e m s i n t h e power f i e l d . T h e s e methods u s u a l l y r e q u i r e a l a b o r i -ous m a n i p u l a t i o n o f a l g e b r a i c e q u a t i o n s w h i c h i n the p r o c e s s o f c a l c u l a t i o n o f t e n r e s u l t i n p r o h i b i t i v e c o m p u t a t i o n c o s t s . Me thods a r e t h u s r e q u i r e d t o e i t h e r r e d u c e t h e number o f s y s t e m e q u a t i o n s by i n t r o d u c i n g some a p p r o x i m a t i - o n s or t o e c o n o m i z e t he c o m p u t a t i o n c o s t s r e q u i r e d t o s o l v e the p r o b l e m . S i n c e t h e method o f r e d u c i n g s y s t e m s i z e i s e s s e n t i a l l y a compromise i n a c c u r a c y , one s h o u l d a t t e m p t t h e e c o n o m i z a t i o n f i r s t . In t h i s t h e s i s , i t i s i n t e n d e d t o d e v e l o p some s y s t e m a t i c and e f f i c i e n t methods f r o m modern c o n -t r o l t h e o r y and a p p l y them t o t h e a n a l y s i s and d e s i g n o f power s y s t e m s . The f i r s t r e q u i r e m e n t f o r s a t i s f a c t o r y s y s t e m o p e r a t i o n i s s t a b i l i t y . A l t h o u g h c l a s s i c a l methods a r e a v a i l a b l e i n c o n t r o l t h e o r y f o r d e t e r m i n i n g s y s t e m s t a b i l i t y , t h e s e a r e n o t s u i t e d t o l a r g e and m u l t i v a r i a b l e s y s t e m s . In modern a n a l y s i s , t h e s y s t e m i s u s u a l l y d e s c r i b e d by a 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 i n s t a t e v a r i a b l e f o r m y = A y ( l . l ) The b e s t method o f d e t e r m i n i n g s t a b i l i t y w i t h t h i s f o r m u l a t i o n i s by e i g e n v a l u e a n a l y s i s . F o r an u n s t a b l e s y s t e m a method i s r e q u i r e d f o r a c h i e v i n g s y s t e m s t a b i l i t y p r e f e r a b l y by p a r a m e t e r a d j u s t m e n t s . Van Ness e t . a l . and L a u g h t o n ^ " ^ s u g g e s t a s t a b i l i z a t i o n 2 • method w h i c h i n v o l v e s c a l c u l a t i n g t he p a r a m e t e r s e n s i t i v i t i e s o f t he e i g e n v a l u e s w i t h p o s i t i v e r e a l p a r t s . In t h e i r methods b o t h t he e i g e n v a l u e s and e i g e n v e c t o r s a r e r e q u i r e d s i n c e the f o r m e r , a r e u s e d t o d e t e r m i n e s t a b i l i t y and the l a t t e r t o d e t e r m i n e e i g e n -v a l u e s e n s i t i v i t y . The d i f f i c u l t y o f f i n d i n g the e i g e n s y s t e m o f a l a r g e s y s t e m was s u c c e s s f u l l y h a n d l e d by Van Ness e t . a l . . The e i g e n v a l u e s were f o u n d by t he method o f t r a n s f o r m i n g A t o an u p p e r (2) H e s s e n b e r g f o rm and u s i n g the QR t r a n s f o r m o f F r a n c i s . The e i g e n v e c t o r s were t h e n f o u n d by b a c k s u b s t i t u t i o n o f t h e e i g e n -v a l u e s i n t o A X = X X m (1.2) and A A V = X V where X = d i a g j ^ X 2 • • • • • • X^~J Once t h e e i g e n s y s t e m was o b t a i n e d , the p a r a m e t e r change t h a t s h i f t e d t h e r e a l p a r t o f an e i g e n v a l u e n e g a t i v e l y c o u l d be f o u n d . To t a k e i n t o a c c o u n t more t h a n one u n s t a b l e e i g e n v a l u e and t o a l l o w more t h a n one p a r a m e t e r c h a n g e , t h i s method c a n be g e n e r a l i z e d . Howeve r , a new e i g e n s y s t e m i s r e q u i r e d a f t e r e a ch p a r a m e t e r c h a n g e . L a u g h t o n s u g g e s t s t h a t an a p p r o x i m a t e new e i g e n s y s t e m can be f o u n d f r o m t h e e i g e n s y s t e m s e n s i t i v i t y by A E = VE . A q ( 1 . 3 ) where E d e n o t e s t he e i g e n s y s t e m and ^ lE/ iq t he e i g e n s y s t e m s e n s i -t i v i t y t o a p a r a m e t e r q . S i n c e e i g e n v a l u e s e n s i t i v i t y was p r e v i -o u s l y c a l c u l a t e d f o r d e t e r m i n i n g a p a r a m e t e r a d j u s t m e n t , t he e i g e n v e c t o r s e n s i t i v i t y i s t he o n l y a d d i t i o n a l c a l c u l a t i o n r e q u i r e d . B o t h L a u g h t o n f " ^ and F a d e e v and F a d e e v a ^ " ^ d e r i v e d t h e r e q u i r e d e i g e n v e c t o r s e n s i t i v i t y e q u a t i o n s . The r e s u l t s were g i v e n i n a l g e b r a i c f o rm i n t e rms o f t h e e i g e n s y s t e m and ^ A / ^ q . T h e s e t e c h n i q u e s a r e a d a p t e d i n C h a p t e r 2 t o the s t a b i l i z a t i o n 3 o f a power s y s t e m . B o t h e i g e n v a l u e and e i g e n v e c t o r s e n s i t i v i t i e s a r e computed s i m u l t a n e o u s l y from a m a t r i x f o r m u l a t i o n . To a p p r o x i -mate the new e i g e n s y s t e m a f t e r an e i g e n v a l u e s h i f t , t h e e i g e n s y s t e m s e n s i t i v i t y i s u s e d . I f the a p p r o x i m a t e e i g e n s y s t e m s e n s i t i v i t y i s n o t a c c u r a t e enough t h e n a c o r r e c t i o n i s r e q u i r e d . F o r t h i s an e x t e n d e d C o l l a r and J a h n c o r r e c t i o n method i s d e v e l o p e d . F o r a power sy s t e m a l r e a d y i n o j > e r a t i o n , the e n g i n e e r i s f a c e d o f t e n w i t h the p r o b l e m o f i m p r o v i n g t h e dynamic r e s p o n s e o f a s y s t e m w i t h o u t making any c o s t l y s t r u c t u r a l c h a n g e s . T h i s may be r e q u i r e d i f the s y s t e m r e s p o n s e i s h i g h l y o s c i l l a t o r y o r has an e x t r e m e l y s l o w s e t t l i n g t i m e . T h i s r e q u i r e s t h e a p p l i c a t i o n o f a p a r a m e t e r o p t i m i z a t i o n t e c h n i q u e . A l t h o u g h t h e r e a r e many methods a v a i l a b l e i n the l i t e r a t u r e , t h e y a r e m a i n l y a p p l i c a b l e t o low (6) o r d e r s y s t e m s . V o n g s u r i y a d e v e l o p e d a c o m p u t a t i o n a l method t o m i n i m i z e a q u a d r a t i c form c o s t f u n c t i o n a l f o r l a r g e s y s t e m s . N o r m a l l y , a s o l u t i o n of t h i s t y p e o f c o s t f u n c t i o n a l as w e l l as i t s d e r i v a t i v e i n v o l v e s the i n v e r s i o n o f an n (n+l)/2 x n ( n + l ) / 2 m a t r i x t o s o l v e n ( n + l ) / 2 s i m u l t a n e o u s . e q u a t i o n s . The c o s t f u n c t i o n a l o r i t s d e r i v a t i v e was e v a l u a t e d by an i n v e r s i o n o f an nxn s y s t e m m a t r i x o b t a i n e d u s i n g a s i m i l a r i t y t r a n s f o r m a t i o n . The s i m i l a r i t y t r a n s f o r m a t i o n can be u s e d t o b e s t advantage i f e i g e n v e c t o r s a r e .used t o form t h e t r a n s f o r m a t i o n m a t r i x . The s y s t e m m a t r i x , a f t e r t r a n s f o r m a t i o n , w i l l have th e J o r d a n c a n o n i c a l f o r m . The c o s t f u n c t i o n a l can t h e n be e v a l u a t e d , even w i t h a time dependent w e i g h t i n g f u n c t i o n , w i t h o u t any m a t r i x i n v e r s i o n . The d e t a i l s a r e g i v e n i n C h a p t e r 3. The methods o f o p t i m a l c o n t r o l t h e o r y may be a p p l i e d t o the i n i t i a l d e s i g n or improvement of a power sy s t e m . R e c e n t l y methods o f c a l c u l a t i n g o p t i m a l c o n t r o l l e r s f o r l a r g e systems have a p p e a r e d i n t h e l i t e r a t u r e . T h e p r o b l e m i s t o s o l v e f o r t h e R i c a t t i m a t r i x . T h e s e m e t h o d s s o l v e t h e R i c a t t i e q u a t i o n w h i c h i n t u r n e n a b l e s a d e t e r m i n a t i o n o f t h e c o n t r o l s i g n a l s . N o r m a l l y t h e s o l u t i o n o f t h i s e q u a t i o n i n v o l v e s t h e i n t e g r a t i o n o f a 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 l e a d i n g t o e n o r m o u s c o m p u t a t i o n e f f o r t t h a t s o m e t i m e s (4) f a i l s d u e t o c a l c u l a t i o n i n s t a b i l i t y . M a c f a r l a n e v a n d F r e e s t e d (7) e t . a l . s o l v e d f o r t h e R i c a t t i m a t r i x f r o m a d o u b l e o r d e r m a t r i x b y a n e i g e n s y s t e m f o r m u l a t i o n w h i c h i s a g r e a t i m p r o v e m e n t . B u t t h e i r m e t h o d r e q u i r e s a l a r g e a m o u n t o f s t o r a g e s p a c e a n d i s n o t c o m p l e t e l y r e l i a b l e i f t h e d o u b l e o r d e r m a t r i x i s i l l - c o n d i -t o \ t i o n e d . P u r i a n d G r u v e r v p r e s e n t a s u c c e s s i v e a p p r o x i m a t i o n m e t h o d w h i c h r e q u i r e s l e s s s t o r a g e s p a c e t h a n t h e p r e v i o u s m e t h o d s a n d e x h i b i t s f a s t m o n o t o n i c c o n v e r g e n c e t o t h e s o l u t i o n o f t h e R i c a t t i m a t r i x . T h e o n l y c o n d i t i o n i s t h a t t h e f i r s t a p p r o x i m a t i o n o f t h e R i c a t t i m a t r i x m u s t b e c h o s e n s o t h a t t h e s y s t e m i s s t a b l e . P u r i a n d G r u v e r ' s m e t h o d i s r e f i n e d a n d a p p l i e d i n C h a p t e r 4 t o t h e c a l c u l a t i o n o f a n o p t i m a l c o n t r o l l e r f o r a p o w e r s y s t e m . T o o b t a i n t h e f i r s t R i c a t t i m a t r i x a p p r o x i m a t i o n s o t h a t t h e s y s t e m i s s t a b l e , t h e s t a b i l i z a t i o n m e t h o d d e v e l o p e d i n C h a p t e r 2 i s u s e d . T h e s u c c e s s i v e a p p r o x i m a t i o n s a r e f o u n d b y e v a l u a t i n g a n i n t e g r a l u s i n g a r e c u r s i v e t e c h n i q u e d e v e l o p e d i n C h a p t e r 4. I t m u s t b e n o t e d t h a t a n o p t i m a l c o n t r o l l e r i s u s u a l l y v e r y d i f f i c u l t t o i m p l e m e n t a l t h o u g h f o r e n g i n e e r i n g r e a s o n s i t s h o u l d b e a s s i m p l e a s p o s s i b l e . F o r t h i s r e a s o n , a s u b o p t i m a l c o n t r o l l e r w i l l b e d e s i g n e d . T o s i m p l i f y t h e c o n t r o l l e r d e s i g n , s e n s i t i v i t y a n a l y s i s i s a p p l i c a b l e . T h e d e s i g n , o f c o u r s e , i s d e p e n d e n t o n t h e s y s t e m p a r a m e t e r s a n d t h e o p e r a t i n g p o i n t c h o s e n i n t h e c a s e o f a l i n e a r i z e d n o n - l i n e a r s y s t e m . S i n c e a n y v a r i a t i o n i n t h e s e w i l l c a u s e a v a r i a t i o n i n t h e t i m e r e s p o n s e , a t i m e r e s p o n s e s e n s i t i v i t y s t u d y i s r e q u i r e d . K e r l i n s t a t e d t h a t t h i s s e n s i -t i v i t y s t u d y o f an n^*1 o r d e r s y s t e m f o r m p a r a m e t e r s r e q u i r e s t h e i n t e g r a t i o n o f an n « n ] ^ o r d e r s y s t e m . The d i g i t a l c o m p u t a t i o n t i m e f o r t h i s , o f c o u r s e , i s e n o r m o u s . In C h a p t e r 5, a method i s d e v e l o p e d f o r c o m p u t i n g the s y s t e m t ime r e s p o n s e to v a r i a t i o n s i n m p a r a m e t e r s s i m u l t a n e o u s l y . O n l y two e q u a t i o n i n t e g r a t i o n s a r e r e q u i r e d . T h i s method i s demon -s t r a t e d i n a n u m e r i c a l example t o r e d u c e t he s t r u c t u r e o f t he o p t i m a l c o n t r o l l e r . 2. E IGENSYSTEM SENS IT IV ITY ANALYSIS APPL IED TO SYSTEM STAB I L I ZAT ION The methods p r e s e n t e d i n t h i s c h a p t e r a r e an e x t e n s i o n o f e i g e n s y s t e m s e n s i t i v i t y a n a l y s i s and w i l l be a p p l i e d t o a s y s t e m s t a b i l i z a t i o n p r o b l e m . Van Ness e t . a l . ^ ^ a n d L a u g h t o n ^ ^ have a p p l i e d e i g e n v a l u e s e n s i t i v i t y a n a l y s i s t o show how t h e e i g e n v a l u e s t e n d t o move i f a s y s t e m p a r a m e t e r i s c h a n g e d . A l t h o u g h the l a t t e r a u t h o r m e n t i o n e d the p o s s i b i l i t y o f a c t u a l l y s t a b i l i z i n g a s y s t e m by t h i s m e t h o d , i t was n e v e r a p p l i e d . I f t h i s method were u s e d , a new e i g e n s y s t e m w o u l d be r e q u i r e d a f t e r each p a r a m e t e r c h a n g e . T h i s c a n be f o u n d e i t h e r b y r e e n t e r i n g an e i g e n s y s t e m c a l c u l a t i o n r o u t i n e or by e s t i m a t i n g the new e i g e n s y s t e m f r o m s e n s i t i v i t y r e l a t i o n s . The f o r m e r method i s t i m e c o n s u m i n g and t h e l a t t e r i n a c c u r a t e u n l e s s v e r y s m a l l s t e p s i z e s a r e u s e d o r a c o r r e c t i o n i s made f o r l a r g e s t e p s i z e s . S i n c e c o m p u t a t i o n t i m e i s o f u t m o s t i m p o r t a n c e i n t he a n a l y s i s o f h i g h o r d e r s y s t e m s , t he s e n s i t i v i t y method i s u s e d h e r e w i t h the a p p l i c a t i o n o f an e x t e n d e d C o l l a r (12) and J a h n c o r r e c t i o n method d e v e l o p e d i n t h i s c h a p t e r . A p p l y i n g t he c o r r e c t i o n method a l l o w s l a r g e r s t e p s i z e s and c o n s e q u e n t l y f e w e r s t e p s . 2.1 P r o b l e m F o r m u l a t i o n The s y s t e m e q u a t i o n s a r e assumed t o be l i n e a r i z e d a b o u t an o p e r a t i n g p o i n t and a r e d e s c r i b e d by a s e t o f f i r s t o r d e r 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 i n t he f o r m y = A y (2.1) M a t r i x A i s g e n e r a l l y a n o n s y m m e t r i c m a t r i x and y i s a v e c t o r o f s t a t e v a r i a b l e s . A p e r t u r b a t i o n i s assumed w h i c h c a u s e s the s y s t e m t o move i n a s m a l l r e g i o n a b o u t t h e o p e r a t i n g p o i n t . The r e s p o n s e a f t e r t h i s d i s t u r b a n c e c a n be w r i t t e n as y ( t ) = £ e ^ 1 x i v i y o > 1 = l f . . . n ( 2 . 2 ) 1=1 and where \^ a r e t he e i g e n v a l u e s o f A , x^ e i g e n v e c t o r s o f A , - T v^ e i g e n v e c t o r s o f A . F o r a s y s t e m w i t h d i s t i n c t e i g e n v a l u e s , A x\ = \ . x . " ( 2 . 3 ) A T v i = v . ( 2 . 4 ) As i s w e l l known, a s y s t e m i s s t a b l e i f t h e e i g e n v a l u e s a r e l o c a t e d i n t h e l e f t h a l f comp l ex p l a n e . The p r o b l e m t h e n i s t o d e t e r m i n e p a r a m e t e r s f o r w h i c h t he s y s t e m i s s t a b l e . A l t h o u g h c l a s s i c a l t e c h n i q u e s do e x i s t f o r s o l v i n g t h i s p r o b l e m , t h e y a r e d i f f i c u l t t o a p p l y t o h i g h o r d e r s y s t e m s , e x p e c i a l l y i f t he s y s t e m i s r e p r e -s e n t e d i n m a t r i x f o r m . E i g e n s y s t e m s e n s i t i v i t y a n a l y s i s p r o v i d e s a c o n v e n i e n t method o f s e l e c t i n g p a r a m e t e r s s u c h t h a t a h i g h o r d e r s y s t e m c a n be s t a b i l i z e d . 2 .2 E i g e n s y s t e m S e n s i t i v i t y A n a l y s i s C a l c u l a t i o n e r r o r s o f e i g e n v a l u e s and e i g e n v e c t o r s were a n a -l y z e d by F a d e e v and F a d e e v a ^ ^ by a p p l y i n g , i n e s s e n c e , e i g e n s y s t e m s e n s i t i v i t y a n a l y s i s . F o r c o m p l e t e n e s s a d e r i v a t i o n o f t he s e n s i -t i v i t y e q u a t i o n s s i m i l a r t o F a d e e v and F a d e e v a ' s i s g i v e n i n t h i s s e c t i o n . The s e n s i t i v i t y o f an e i g e n v a l u e t o a p a r a m e t e r q o f A i s d e r i v e d by t a k i n g the p a r t i a l d e r i v a t i v e o f ( 2 . 3 ) w i t h r e s p e c t t o q , t h a t i s A , q x i + A x . , q = \ i f q x . + X. x . , q ( 2 . 5 ) where A , = W )>q ( 2 . 6 ) and * i ' q = ^ X i ^ ^ ( 2 . 7 ) F o r c o n v e n i e n c e , t h i s p a r t i a l d e r i v a t i v e n o t a t i o n w i l l be u sed t h r o u g h o u t t h i s t h e s i s u n l e s s o t h e r w i s e s t a t e d . P r e m u l t i p l y i n g —T ( 2 . 5 ) by v\ one o b t a i n s vT A, x . + vT A x . , = \ . , v T x. + A- v T x . , ( 2 . 8 ) 1 ' q 1 1 l ' q " l ' q 1 1 1 1 l ' q v ' S i n c e , f r om e q u a t i o n ( 2 . 4 ) , vT A = \^ vT ( 2 . 9 ) and f o r a n o r m a l i z e d e i g e n v e c t o r p r o d u c t 1 *1 f o r i = j 0 f o r e q u a t i o n ( 2 . 8 ) becomes -T - . v . x . = -< 1 J ( 2 . 1 0 ) \ . , = vT A, x. ( 2 . 1 1 ) ^ 1 ' q 1 ' q 1 v ' g i v i n g t he s e n s i t i v i t y o f \^ t o a p a r a m e t e r q . The e i g e n v e c t o r s e n s i t i v i t y i s d e r i v e d f r o m e q u a t i o n ( 2 . 5 ) -T by p r e m u l t i p l y i n g t h i s e q u a t i o n by v . J v. A , x. + v . A x., = X., v . x . + X.v. x., ( 2 . 1 2 ) where i ^ j . Know ing t h a t A = X - vT ( 2 . 1 3 ) J J J and b e c a u s e o f t he o r t h o g o n a l i t y o f t he two s e t s o f e i g e n v e c t o r s vT x. = 0 ( 2 . 1 4 ) J 1 e q u a t i o n ( 2 . 1 2 ) becomes -T . - / \ v \ -T -v. A , x . = J q 1 where x\ , may be w r i t t e n as *j N x. = ( \ ± - X.) vj 3 c . , q (2 .15 ) n. • c - i , j = 1, . . . n ( 2 . 1 6 ) x . , = \ x . The c o e f f i c i e n t s c . . o f v e c t o r s x . a r e f o u n d by s u b s t i t u t i n g e q u a t i o n ( 2 . 1 6 ) i n t o e q u a t i o n ( 2 . 1 5 ) °ji = A , q X i " V ' f o r 1 ^ j ( 2 * 1 7 ) E q u a t i o n ( 2 . 1 6 ) c a n be w r i t t e n as — T x . . = X f c , . , . . . c - , . . . c .1 ( 2 . 1 8 ) l ' q I 1 1 n3J where X = J^l' " ' * X i ' * * " X n ] ( 2 . 1 9 ) and c . . = 0 xi E x t e n d i n g ( 2 . 1 8 ) t o i n c l u d e a l l i , i = 1, . . . n , the e i g e n v e c t o r s e n s i t i v i t i e s c an be w r i t t e n i n m a t r i x f o r m as X , q = X C ( 2 . 2 0 ) By a s s u m i n g a s i m i l a r f o r m f o r "V, V , q = V K ( 2 . 2 1 ) T t he e i g e n v e c t o r s e n s i t i v i t y o f m a t r i x A may be d e r i v e d by r e -q u i r i n g t h a t (V + A q . V , q ) T ( X + A q . X , q ) = I ( 2 . 2 2 ) where I i s a u n i t m a t r i x . S u b s t i t u t i o n o f e q u a t i o n ( 2 . 2 0 ) and ( 2 . 2 1 ) i n t o e q u a t i o n ( 2 . 2 2 ) r e s u l t s i n ( I + A q . K T ) V T X ( l + a q . C ) = I ( 2 . 2 3 ) T S i n c e t he m a t r i x p r o d u c t V X e q u a l s a u n i t m a t r i x a c c o r d i n g t o ( 2 . 1 0 ) and a s s u m i n g t h a t t h e s e c o n d o r d e r t e r m s o f ( 2 . 2 3 ) a r e s m a l l , t h a t i s A q K T £>q C ^ 0 ( 2 . 2 4 ) t h e n t he s o l u t i o n o f K f r o m ( 2 . 2 3 ) i s K T = -C ( 2 . 2 5 ) Hence V , = -V C T ( 2 . 2 6 ) where C i s g i v e n b y ( 2 . 1 7 ) . N e g l e c t i n g t he s e c o n d o r d e r t e rms i s v a l i d as l o n g as t he e i g e n v a l u e s a r e r e a s o n a b l y s e p a r a t e d . Near b r eakaway p o i n t s , where c o m p l e x c o n j u g a t e e i g e n v a l u e s merge i n t o a m u l t i p l e r e a l e i g e n v a l u e , s e n s i t i v i t y a c c u r a c y i s p o o r s i n c e the n e g l e c t e d s e c o n d o r d e r t e rms o f ( 2 . 2 3 ) _T . — —T . m ii—.v. A , x, v, A , x . ( i q K Aq C) ± . = V 1 1 1 k - k ^ i ^ j ^ k ( 2 . 2 7 ) 3 £j< v ^ x x r V k = 1 . . . n become l a r g e f o r c l o s e e i g e n v a l u e s . S i n c e e i g e n s y s t e m s e n s i t i v i t y a n a l y s i s i s t o be u s e d o n l y f o r the a d j u s t m e n t o f p a r a m e t e r s , t he c l o s e e i g e n v a l u e p r o b l e m can be c i r c u m v e n t e d by i n c r e a s i n g the p a r a m e t e r a d j u s t m e n t s t e p and r e c a l c u l a t i n g a new e i g e n s y s t e m . An i m p o r t a n t f e a t u r e o f t he p r o p o s e d f o r m u l a t i o n o f the e i g e n -s y s t e m s e n s i t i v i t y e q u a t i o n s i s the ease o f c o m p u t a t i o n s i n c e a l l 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 d e t e r m i n i n g s e n s i t i v i t i e s becomes a v a i l a b l e by p e r f o r m i n g the m a t r i x p r o d u c t s | D = V T A , q X ( 2 . 2 8 ) The e i g e n v a l u e s e n s i t i v i t i e s f r o m e q u a t i o n ( 2 . 1 1 ) a r e g i v e n by t he d i a g o n a l e l e m e n t s o f m a t r i x D as X i > q .= d.. ( 2 . 2 9 ) and the e i g e n v e c t o r s e n s i t i v i t y m a t r i x C f r o m ( 2 . 1 7 ) as C i j = d i j / ( V X i } ' 1 ^ J" ( 2 * 3 0 ) A c o m p u t a t i o n a l s a v i n g c a n a l s o be made i f t h e s p a r s i t y o f A , q i s u s e d t o a d v a n t a g e . As an i l l u s t r a t i o n o f t h i s , c o n s i d e r t he c a s e where A , c o n t a i n s o n l y one e l e m e n t a - , i n t h e i - t h row and i - t h q J i j ' q J c o l u m n . I t c an be v e r i f i e d f r o m ( 2 . 2 8 ) t h a t D i s g i v e n s i m p l y by D = a . . , i j ' q v . . v n i fjl, ... *•* Xjn] ( 2 . 3 1 ) F o r more t h a n one e l e m e n t i n A , , D becomes a sum o f m a t r i c e s where q 3 2 ea ch m a t r i x i s f o r m e d by ( 2 . 3 1 ) . Thus (2n - kn ) m u l t i p l i c a t i o n s a r e a v o i d e d w i t h k e l e m e n t s i n A , q . An a p p r o a c h i s t a k e n i n the f o l l o w i n g s e c t i o n f o r s o l v i n g t he p r o b l e m o f s y s t e m i n s t a b i l i t y u t i l i z i n g t he e i g e n s y s t e m s e n s i t i v i t y e q u a t i o n . 2 .3 An E i g e n v a l u e S h i f t i n g Me thod The p a r a m e t e r s t h a t a r e a l l o w e d t o change a r e w r i t t e n i n — T v e c t o r f o rm as q = jTj^, . . . q ^ , . . . q£j ( 2 . 3 2 ) F o r o b v i o u s r e a s o n s , t he e i g e n v a l u e w i t h t he l a r g e s t p o s i t i v e r e a l 11 p a r t , \ , i s c h o s e n t o he c h a n g e d n e g a t i v e l y by p a r a m e t e r a d j u s t -m e n t s . From ( 2 . 1 1 ) , t he s e n s i t i v i t i e s o f X t o each q., j = 1,...Q, \ —T — a r e g i v e n by X » = v A , x ( 2 . 3 3 ) 0 J N m ' q . m ' q . m x . In o r d e r t o m i n i m i z e the e f f e c t o f p a r a m e t e r a d j u s t m e n t s on o t h e r more f a v o r a b l y l o c a t e d e i g e n v a l u e s , the p a r a m e t e r q^ w i t h t he l a r g e s t r e a l p a r t o f e i g e n v a l u e s e n s i t i v i t y Re \ \ , I . , , , 1 b J I m <l k t 1 S c h o s e n t o be a d j u s t e d . Any s u i t a b l e g r a d i e n t t e c h n i q u e can be u s e d t o c a l c u l a t e £ ,q k f r o m A^, b u t f o r the p u r p o s e o f c o n t r o l l i n g the change i n Xm> a method s i m i l a r t o t h e Newton-Raphson t e c h n i q u e i s u s e d where * q k = A X ^ R e ^ , ^ ( 2 . 3 4 ) and d A p i s t he s h i f t i n R e ^ X ^ d e s i r e d . The i n c r e m e n t i n t he o t h e r e i g e n v a l u e s c a n be c a l c u l a t e d as *Xi •= Xi>(1 I ' A q i i = l , . . . n ( 2 . 3 5 ) k | X i where A-1 » i = l , . . . n i s c a l c u l a t e d f r o m ( 2 . 2 9 ) . I f a f t e r c a l c u -A i ' q k ' l a t i n g a l l AA-^ l s f o u n d t h a t t he r e a l p a r t o f one o f t h e s e e i g e n v a l u e s i n c r e a s e s beyond Re ^ ^ m ^ > t h e n o b v i o u s l y q k c a n n o t be a l l o w e d t o c h a n g e . The n e x t most s e n s i t i v e p a r a m e t e r may be c h o s e n f o r a d j u s t m e n t . I f t h i s f a i l s t h e n t he p r o c e s s i s r e p e a t e d u n t i l a s a t i s f a c t o r y ^ \ ^ i s f o u n d . Once an a l l o w e d A q k has been f o u n d , a new A i s c a l c u l a t e d and the i n c r e m e n t i n t he e i g e n v e c t o r s c a n be c a l c u l a t e d as A X = X C. -kq, KT k ( 2 . 3 6 ) = -V C k . & q k where C k i s c a l c u l a t e d f r o m ( 2 . 3 0 ) . The a d j u s t m e n t p r o c e s s i s t h e n r e p e a t e d u n t i l a l l e i g e n v a l u e s l i e i n the l e f t - h a l f comp l ex p l a n e . I t s h o u l d be n o t e d t h a t t h i s method i s r e s t r i c t e d to s m a l l p a r a m e t e r a d j u s t m e n t s s i n c e the e i g e n s y s t e m change c a l c u l a t e d by ( 2 . 3 5 ) and ( 2 . 3 6 ) o n l y a p p l i e s t o a s m a l l r e g i o n a b o u t t he o r i g i n a l e i g e n s y s t e m . I t c o u l d be a r g u e d t h a t t h i s t i m e c o n s u m i n g p r o c e s s c o u l d be a v o i d e d by a method o f s e l e c t i n g l a r g e r p a r a m e t e r a d -j u s t m e n t s and c a l c u l a t i n g a new e i g e n s y s t e m f r o m A a f t e r e a ch a d -j u s t m e n t . T h e r e a r e two r e a s o n s c o u n t e r i n g t h i s a r g u m e n t . One , t he e i g e n v a l u e movement i n t h i s c a se may become u n p r e d i c t a b l e and two , an e i g e n s y s t e m c a l c u l a t i o n i s i n i t s e l f t ime c o n s u m i n g . I t i s b e l i e v e d t h a t t h e b e s t o f b o t h methods can be c o m b i n e d i f a l a r g e s t e p s i z e i s u s e d i n t he s e n s i t i v i t y method b u t w i t h a c o r -r e c t i o n method a p p l i e d . T h i s a l l o w s a f a s t e i g e n s y s t e m c a l c u l a t i o n a f t e r e a c h a d j u s t m e n t . The c o r r e c t i o n method i s d e t a i l e d i n the f o l l o w i n g s e c t i o n . 2 .4 An E x t e n d e d C o l l a r and J a h n C o r r e c t i o n Method (12) The C o l l a r and J a h n c o r r e c t i o n method^ i s a p p l i e d by f i r s t f o r m i n g the m a t r i x p r o d u c t XZ}A X^ = M- + R ( 2 . 3 7 ) u s i n g t he ap j j r ox ima te e i g e n v e c t o r s X^ and the s y s t e m m a t r i x A . M and R a r e t h e d i a g o n a l and the o f f d i a g o n a l p a r t s o f the m a t r i x p r o d u c t r e s p e c t i v e l y . R has s m a l l e l e m e n t s s i n c e t he c o r r e c t e d e i g e n s y s t e m r e q u i r e s t h a t X - 1 A X = M ( 2 . 3 8 ) L e t X = Xu, ( I + E) ( 2 . 3 9 ) w i t h E assumed t o be s m a l l and o f f d i a g o n a l , t h e n s u b s t i t u t i o n o f ( 2 . 3 7 ) and ( 2 . 3 9 ) i n t o ( 2 . 3 8 ) r e s u l t s i n ( I + E ) _ 1 ( M + R) (I + E) . •= M. ( 2 . 4 0 ) N e g l e c t i n g p r o d u c t s o f s m a l l m a t r i c e s R and E, ( 2 . 4 0 ) can be 13 w r i t t e n as E M - M E = R ( 2 . 4 1 ) By i n s j ^ e c t i o n t he s o l u t i o n o f E i s e . . = r . . / ( m • • - m . . ) i j i j JJ " ( 2 . 4 2 ) e . . = 0 f o r i . j = l . . . . n n The c o r r e c t e d m a t r i x X i s t h e n d e t e r m i n e d by e q u a t i o n ( 2 . 3 9 ) . In T -1 t he t h e s i s c a l c u l a t i o n s , i s made a v a i l a b l e and i s u s e d f o r X^ i n ( 2 . 3 7 ) i n s t e a d o f i n v e r t i n g t he comp l ex m a t r i x X^. C o l l a r and J a h n ' s method has been e x t e n d e d t o i n c l u d e the T c o r r e c t i o n o f V so t h a t t he m a t r i x p r o d u c t V X w i l l be n o r m a l i z e d . T h i s i s one o f t he r e q u i r e m e n t s o f S e c t i o n 2 . 2 . F i r s t a s o l u t i o n t o an u n n o r m a l i z e d V i s f o u n d by f o r m i n g the p r o d u c t V ^ T X = N + Z ( 2 . 4 3 ) where N i s n e a r l y a u n i t m a t r i x and Z i s s m a l l and o f f - d i a g o n a l . The c o r r e c t e d e i g e n s y s t e m r e q u i r e s t h a t V T X = N ( 2 . 4 4 ) T where V i s assumed t o be g i v e n by V T = (I + K) V ^ T ( 2 . 4 5 ) w i t h K assumed s m a l l and o f f d i a g o n a l . S u b s t i t u t i n g ( 2 . 4 3 ) and ( 2 . 4 5 ) i n t o ( 2 . 4 4 ) g i v e s ( I + K ) ( N + Z) = N ( 2 . 4 6 ) N e g l e c t i n g p r o d u c t s o f s m a l l m a t r i c e s K and Z , ( 2 . 4 6 ) can be w r i t t e n as K N = -Z ( 2 . 4 7 ) Hence K = -Z N " 1 ( 2 . 4 8 ) T A n o r m a l i z e d V c an be f o u n d f r o m the r e q u i r e m e n t t h a t ( n o r m a l i z e d V T ) X = I ( 2 . 4 9 ) and f r o m ( 2 . 4 4 ) as n o r m a l i z e d V T = N _ 1 V T ( 2 . 5 0 ) S u b s t i t u t i n g ( 2 . 4 5 ) and ( 2 . 4 7 ) i n t o ( 2 . 5 0 ) , the c o r r e c t e d and 14 n o r m a l i z e d V becomes V T = N  l(I - Z N " 1 ) ^ ( 2 . 5 1 ) -1 Note t h a t N~ i s e a s i l y computed s i n c e N i s d i a g o n a l . In the p r o c e s s o f s h i f t i n g t he e i g e n v a l u e s by a p a r a m e t e r c h a n g e , t he e i g e n s y s t e m a c c u r a c y may be c h e c k e d by c o m p a r i n g the p r o d u c t T T V X w i t h t he u n i t m a t r i x . Howeve r , the c o m p l e t e p r o d u c t V X i s n o t r e q u i r e d . O n l y the most i n a c c u r a t e e i g e n v e c t o r s c o r r e s p o n d i n g t o t he c l o s e s t e i g e n v a l u e s need t o be c h e c k e d . I f the c o r r e c t i o n method i s a p p l i e d i n t h e i t e r a t i o n p r o c e d u r e o f ( 2 . 3 4 ) and ( 2 . 3 5 ) i t e l i m i n a t e s the need t o c a l c u l a t e a new e i g e n s y s t e m f o r t h e A m a t r i x e x c e p t f o r t h e c a s e o f c l o s e e i g e n -v a l u e s . A t t h e s e p o i n t s , the e i g e n s y s t e m moves i n a d i s c r e t e manner and r e p e a t e d a p p l i c a t i o n o f t h e c o r r e c t i o n method may n o t c o n v e r g e . The e i g e n s y s t e m s e n s i t i v i t y method i s p a r t i c u l a r l y w e l l s u i t e d t o t h e s t a b i l i z a t i o n o f h i g h o r d e r s y s t e m s s u c h as power s y s t e m s . The e q u a t i o n s c a n be w r i t t e n i n t he f i r s t o r d e r s t a t e v a r i a b l e (13) f o r m and the s t a b i l i z i n g p r o c e d u r e i n i t i a t e d by c a l c u l a t i n g t he e i g e n s y s t e m o f t he A m a t r i x . The p o s i t i o n o f t he d o m i n a n t e i g e n v a l u e s a f t e r s t a b i l i z a t i o n i n d i c a t e s t h e d e g r e e o f s t a b i l i t y o b t a i n e d . 2 .5 A N u m e r i c a l Examp le To i l l u s t r a t e the a p p l i c a t i o n o f t h e s t a b i l i z a t i o n m e t h o d , an example o f an u n s t a b l e power s y s t e m i s p r e s e n t e d . T h i s example w i l l a l s o d e m o n s t r a t e the e i g e n s y s t e m c o r r e c t i o n m e t h o d . The l i n e a r i z e d power s y s t e m e q u a t i o n s a r e w r i t t e n i n s t a t e v a r i a b l e f o r m as y = A y ( 2 . 1 ) where A i s d e r i v e d i n A p p e n d i x A . T h i s m a t r i x i s c a l c u l a t e d f rom the s y s t e m p a r a m e t e r s and i n i t i a l o p e r a t i n g c o n d i t i o n s . The l a t t e r a r e c a l c u l a t e d f o r a g i v e n P , Q and v , as d e r i v e d i n A p p e n d i x B. The s y s t e m p a r a m e t e r s c h o s e n f o r t h i s example a r e g i v e n i n T a b l e 2.1 and the i n i t i a l o p e r a t i n g c o n d i t i o n s c a l c u l a t e d f r o m = . 7 5 3 , Q = . 0 3 0 , v o t o 1.05 a r e i, = . 2 9 5 , i = . 6 5 4 , do ' qo ' v , = . 3 9 3 , v do ' qo . 9 7 4 S o = * 8 4 5 ' ¥YO = 9 ' 4 8 ' V o A l l v a l u e s a r e i n p e r u n i t . . 9 6 3 T a b l e 2.1 Sys tem P a r a m e t e r s ( 2 . 5 2 ) ( 2 . 5 3 ) V o l t a g e R e g u l a t o r -E x c i t e r G o v e r n o r - H y d r a u l i c O p e r a t o r Mach ine T i e L i n e t = . 0 0 3 . 4 8 0 H = 4 . 9 0 R = . 1 1 0 = . 0 5 0 Tv = 1.90 D = 2 .00 X = . 7 3 0 k l = 4 0 . 0 ? r = 7 . 6 0 X d = 1.00 G = . 1 0 0 k 2 = 1 .00 cr = r g = . 0 2 0 . 5 0 0 . 0 2 0 x ' x d x q L d o = . 6 0 0 = . 2 7 0 = 9 . 0 0 B = . 1 3 0 From ( A . 3 5 ) the c o e f f i c i e n t m a t r i x becomes A = • ( 2 . 5 4 ) - . 0 2 0 4 - 3 0 . 0 7 - 4 . 3 0 0 3 8 . 4 7 5 7 . 7 0 1 .000 - 0 . 5 1 4 - 0 . 1 8 7 1 .000 - 3 3 3 . 3 3 3 3 . 3 - 2 . 0 8 0 4 2 . 0 6 - 6 3 . 1 0 - 1 9 . 6 1 - 2 . 0 0 0 2 . 0 0 0 4 . 0 0 0 - 1 . 0 5 3 - 4 . 0 0 0 - 0 . 1 3 3 - 1 . 0 0 0 - 5 0 . 0 0 - 0 . 0 6 4 - 0 . 4 8 0 - 2 4 . 1 3 To i n i t i a t o the s h i f t i n g p r o c e s s , an e i g e n s y s t e m o f A i s r e q u i r e d . A c o m p u t a t i o n r o u t i n e s i m i l a r t o t h a t u s e d by Van Ness e t . a l . ^ ^ \ T was u s e d t o c l a c u l a t e A and X . f r o m A, and V f r o m A . The e i g e n -v e c t o r s x^ and v^ f o u n d i n t h i s manner r e q u i r e r e a r r a n g i n g s i n c e t he c o r r e s p o n d e n c e o f x^ . t o v\ i s n o t e s t a b l i s h e d . As a f i n a l i n i -t i a l i z a t i o n s t e p , t h e e i g e n v e c t o r p r o d u c t s a r e n o r m a l i z e d by d i v i d -i n g x^ by t he s c a l a r vJ x i f o r e a ch i . In t h e s h i f t i n g p r o c e s s , t h e p a r a m e t e r s a r e a d j u s t e d w i t h i n t he r a n g e s .0005 > T-, > . 1 0 0 0 , . 0000 > £>, > 1 . 0 0 0 , X ( 2 . 5 5 ) 1 .000 > t r > 1 0 . 0 0 , - 6 0 0 . 0 y ^ 6 0 0 . 0 , - 3 . 0 0 0 > k 2 > 3 .000 The s t a b i l i z i n g p r o c e s s o f s e c t i o n 2 .3 w i t h a p p l i c a t i o n o f ' t h e c o r r e c t i o n method o f s e c t i o n 2 .4 i s s u m m a r i z e d b e l o w w i t h a d d i t i o n -a l c o n s i d e r a t i o n g i v e n t o t he p a r a m e t e r r e s t r i c t i o n s o f ( 2 . 5 5 ) . 1. D e t e r m i n e t h e e i g e n v a l u e w i t h t he l a r g e s t p o s i t i v e r e a l p a r t , A m » f r o m t h e i n i t i a l e i g e n s y s t e m and. the most s e n s i t i v e p a r a m e t e r q^ f r o m ( 2 . 2 9 ) and ( 2 . 3 1 ) . 2 . C a l c u l a t e a A q ^ f r o m ( 2 . 3 4 ) w h i c h w i l l c a u s e a s h i f t o f X M by a s p e c i f i e d amount AXJJ» 3. C a l c u l a t e the change i n t h e o t h e r e i g e n v a l u e s t h a t w i l l r e s u l t f r o m c h a n g i n g q ^ . 4 . I f e i t h e r A q ^ e x c e e d s t he change a l l o w e d i n q^ o r i f t h e c a l c u l a t e d change i n i s s u c h t h a t Re "> Re ^ X , ^ t h e n the n e x t most s e n s i t i v e p a r a m e t e r must be f o u n d and 2 and 3 r e p e a t e d . I f t h i s f a i l s a f t e r s e a r c h i n g a l l t he p a r a m e t e r s t h e n A.\j) must be d e c r e a s e d and 2 and 3 r e p e a t e d u n t i l an a d j u s t a b l e parameter- can be f o u n d . 5. C a l c u l a t e t h e new e i g e n v e c t o r s f r o m ( 2 . 3 0 ) , ( 2 . 3 1 ) and ( 2 . 3 6 ) . 6. C o r r e c t the a p p r o x i m a t e e i g e n s y s t e m by t he method o f 17 s e c t i o n 2 . 4 . 7. Rer>eat t h e p r o c e d u r e s t a r t i n g a t 1 u n t i l Re o r " e a c h new e i g e n s y s t e m r e a c h e s a d e s i r e d n e g a t i v e v a l u e o r A A p becomes s m a l l as d e t e r m i n e d by 4 . F o r t he s p e c i f i c example o f t h i s s e c t i o n , the i n i t i a l s y s t e m was u n s t a b l e and had e i g e n v a l u e s - 3 3 3 . 5 3 + j O . - 2 . 1 1 0 9 + j O . - 2 5 . 0 9 1 + j O . - 1 . 0 1 7 9 + j O . ( 2 . 5 6 ) - 1 5 . 0 8 7 + j O . - 0 . 0 0 5 2 + j O . - 4 . 7 6 3 2 + j O . +0 .0465 + J 5 . 4 1 2 4 The e i g e n v a l u e s h i f t and p a r a m e t e r a d j u s t m e n t r e s u l t s a r e s u m m a r i z e d i n T a b l e 2.2 where o n l y t he d o m i n a n t e i g e n v a l u e s a r e shown. The e i g e n v a l u e s f o u n d f r o m t h e s e n s i t i v i t y e q u a t i o n s a r e d e s i g n a t e d " s e n s " and t h o s e a f t e r e a c h c o r r e c t i o n , " c o r r " . The e i g e n v a l u e s a f t e r t he s t a b i l i z a t i o n p r o c e d u r e a r e - 3 4 0 . 4 6 + j O . - 1 . 3 8 6 9 + j O . 5 3 1 6 - 1 9 . 3 6 8 + J 4 1 . 3 8 9 - 0 . 8 6 1 1 + JO . ( 2 . 5 7 ) - 1 . 5 . 6 9 + j l . 3 8 6 9 - 0 . 5 3 1 6 + j O . - 0 . 5 3 0 8 + j 5 . 5 4 6 4 The p a r a m e t e r a d j u s t m e n t s a r e s u m m a r i z e d b e l o w w i t h t h e f i n a l v a l u e s u n d e r l i n e d . TJL = 0 . 0 5 0 — 0 . 0 1 4 9 — 0 . 0 0 7 2 * 0 . 0 1 4 3 = 4 0 . 0 0 — - 3 2 7 . 4 0 - 4 5 3 . 3 0 »- 5 7 8 . 1 0 k 2 = 1 .000 — ^ - 2 . 6 2 0 - 2 . 9 4 0 (2 58) $. = 0 . 4 8 0 P - 0 . 1 8 0 0 0 . 0 8 3 0 — > ~ 0 . 0 3 4 0 — « - 0 . 0 1 2 — t 0 . 0 0 2 *- 0 . 0 0 1 0 r = 7 . 6 0 0 *- 2 . 3 6 0 0 1 .1600 *-r T a b l e 2 .2 i n d i c a t e s t h a t o n l y f i v e e i g e n s y s t e m r e c l a c u l a t i o n s d i r e c t l y f r o m the s y s t e m m a t r i x a r e r e q u i r e d b u t t h e s e a r e n o t a l l 18 T a b l e 2.2 E i g e n v a l u e S h i f t T h r o u g h P a r a m e t e r A d j u s t m e n t • Shift Dominan t R o o t s 1 and 2 Dominan t Roo t 3 Parameter New P a r a m e t e r V a l u e • Shift S e n s . C o r r . S e n s . C o r r . Parameter 0 + .0465+ J5 .412 - . 0 0 5 2 1 - . 0 0 3 5 + J 5 . 5 4 1 ft - . 0 0 5 2 •x-. 0 5 0 0 k 2 - 2 . 6 2 1 2 - . 0 2 8 5 + J 5 . 5 5 5 - . 0 2 3 9 + J 5 . 5 5 0 - . 0 0 5 2 - . 0 0 5 2 . 0 2 5 0 t l . 0149 3 - . 0 1 6 6 + J 5 . 5 1 3 - . 0 2 2 2 + J 5 . 4 6 4 - . 0 0 8 4 - . 0 1 3 0 .0031 St . 1797 4 - . 0 4 3 3 + J 5 . 4 0 4 - . 0 9 4 6 + J 5 . 3 8 3 - . 0 1 9 3 ** - . 0 2 5 1 .0063 h . 0825 5 - . 1 6 3 0 + J 5 . 3 3 3 * - . 0 3 7 6 * . 0125 k O 0 3 0 4 6 - . 1 6 3 0 + J 5 . 3 3 5 - ! " 2000+ j5 .356 - . 0 6 2 6 ** - . 1 4 7 7 . 0250 t r 2. 362 7 - . 2 5 0 1 + J 5 . 3 5 7 * - . 1 9 7 7 * . 0 5 0 0 St . 0 1 2 0 8 - . 2 8 0 7 + J 5 . 3 6 7 - . 2 8 8 5 + J 5 . 3 8 1 - . 2 4 7 7 - . 3 4 8 5 . 0 5 0 0 St .0021 9 - . 2 9 4 7 + J 5 . 3 8 1 - . 2 9 4 6 + J 5 . 3 8 0 - . 3 4 8 5 - . 3 4 8 5 .0063 t l . 0074 10 - . 3 0 1 0 + J 5 . 3 9 2 * - . 3 4 8 5 * . 0063 k 2 - 2 . 9 4 3 11 - . 3 2 5 8 + J 6 . 5 0 3 - . 4 9 4 1 + J 5 . 5 4 9 - . 3 5 0 7 - . 3 5 0 4 .0250 k l 3 2 7 . 4 12 - . 4 9 4 1 + J 5 . 5 4 9 - . 4 9 4 1 + J 5 . 5 4 9 - . 4 4 8 7 - . 4 5 9 0 . 1 0 0 0 t r 1.505 13 - . 4 9 8 9 + J 5 . 5 5 3 * - . 5 0 9 0 . 0 5 0 0 St .0001 14 - . 5 2 3 9 + J 5 . 5 5 4 - . 5 0 9 1 * . 0 2 5 0 k l 4 5 3 . 3 15 - . 5 2 3 9 + J 5 . 5 4 7 *•* - . 5 1 6 6 + J 5 . 5 4 7 - . 5341 ** - . 5 3 1 6 . 0 2 5 0 t r 1.164 16 - . 5 2 9 1 + J 5 . 5 4 3 * - . 5 3 1 6 .0125 k l 578 .1 17 - . 5 3 2 2 + J 5 . 5 4 3 -X- * - . 5 2 6 9 + J 5 . 5 4 5 - . 5 3 1 6 ** - . 5 3 1 6 .0031 t l . 0143 E i g e n s y s t e m R e c a l c u l a t i o n From Sys t em M a t r i x No C o r r e c t i o n R e q u i r e d 19 c a u s e d by c l o s e e i g e n v a l u e s . The r e c a l c u l a t i o n s r e q u i r e d by s h i f t s s i x , e l e v e n and s e v e n t e e n a r e c a u s e d by a t o o l a r g e p a r a m e t e r c h a n g e . The c o r r e c t i o n method does n o t c o n v e r g e i f t he a p p r o x i m a t e e i g e n s y s t e m i s t o o f a r f r om t h e c o r r e c t e i g e n s y s t e m . 20 3. EIGENSYSTEM ANALYSIS APPLIED TO PARAMETER OPTIMIZATION 3.1 I n t r o d u c t i o n In t h i s c h a p t e r , the p a r a m e t e r a d j u s t m e n t method o f C h a p t e r 2" i s a p p l i e d t o a p a r a m e t e r o p t i m i z a t i o n p r o b l e m . I t i s d e s i r e d t o m i n i m i z e a p e r f o r m a n c e c r i t e r i o n s p e c i f i e d as a c o s t f u n c t i o n a l i n q u a d r a t i c form J = y T Q y d t (3.1) o where y = A y (2.1) Q i s c h o s e n t o emphasize s e l e c t e d - s t a t e v a r i a b l e s and of, ot>0, t o emphasize i n t e g r a t i o n t i m e . F o r an a s y m p t o t i c a l l y s t a b l e system (3) w i t h i n i t i a l c o n d i t i o n s y , L a u g h t o n has shown t h a t t h i s c o s t f u n c t i o n a l c an be e v a l u a t e d i n terms o f a symmetric m a t r i x B ^ as J = ? 0 B « + i h <3-2> where B^+^ i s s o l v e d f r o m t h e r e c u r r e n c e r e l a t i o n T A B_ . , + B_. , A = -B^ w i t h i n i t i a l c o n d i t i o n (3.3) B = Q o * I f (3.3) i s s o l v e d f o r each B^ i n t h e normal manner t h e n an n(n-»l)/2 x n(n+l)/2 m a t r i x must be i n v e r t e d . Thus, t h e s i z e o f a sys t e m t h a t can be h a n d l e d i s s e v e r e l y r e s t r i c t e d by c o m p u t a t i o n a l r e q u i r e m e n t s . These r e q u i r e m e n t s were r e d u c e d c o n s i d e r a b l y i n a method d e v e l o p e d by V o n g s u r i y a ^ ^ whereby a s i m i l a r i t y t r a n s f o r m a t i o n was a p p l i e d and t h e l a r g e s t m a t r i x i n v e r t e d was nxn. As a f u r t h e r c o n t r i b u t i o n t o m i n i m i z i n g the c o m p u t a t i o n e f f o r t , the e i g e n v e c t o r t r a n s f o r m a t i o n i s a p p l i e d t o A i n t h i s c h a p t e r so t h a t t h e s o l u t i o n o f B^ from (3.3) does n o t r e q u i r e any m a t r i x i n v e r s i o n . In a d d i t i o n , t h e s o l u t i o n o f t h e d e s i r e d i s f o u n d i n a s i n g l e s t e p t h u s a v o i d i n g s u c -c e s s i v e s o l u t i o n s o f (3.3). With t h e s o l u t i o n o f B c < +^ e x p r e s s e d i n terms o f t h e e i g e n s y s t e m o f A, t h e s e n s i t i v i t y e q u a t i o n s and c o r -21 r e c t i o n method o f C h a p t e r 2 c a n be e a s i l y a p p l i e d t o m i n i m i z e the c o s t f u n c t i o n a l t h r o u g h p a r a m e t e r a d j u s t m e n t s . 3.2 An E i g e n s y s t e m Form o f t he C o s t F u n c t i o n a l The s o l u t i o n o f B-^  f r o m ( 3 . 3 ) i s i n v e s t i g a t e d f i r s t and i s g e n e r a l i z e d l a t e r t o a r r i v e a t B^^ . L e t the e i g e n s y s t e m fo rm o f A A = X X V T ( 3 . 4 ) be s u b s t i t u t e d i n t o ( 3 . 3 ) , t h e n t h i s e q u a t i o n f o r becomes V A X T B X + B X X X V T = -Q ( 3 . 5 ) A s s u m i n g t h a t t h e e i g e n v e c t o r s a r e a n o r m a l i z e d s e t t h e n p r e m u l t i p l y -T i n g and p o s t m u l t i p l y i n g ( 3 . 5 ) by X and X r e s p e c t i v e l y r e s u l t s i n Xx T %1 X + X T B 1 X X = - X T Q X ( 3 . 6 ) A t t h i s p o i n t , i t i s c o n v e n i e n t t o d e f i n e B J = X T B X X ( 3 . 7 a ) Q ' = X T Q X ( 3 . 7 b ) where B J i s a c o m p l e x s y m m e t r i c m a t r i x s i n c e [xT B X x ] T = X T B X X ( 3 . 8 ) W i t h t h i s s u b s t i t u t i o n , ( 3 . 6 ) becomes X B - + B J X = Q' ( 3 . 9 ) S i n c e B| i s s y m m e t r i c and X i s d i a g o n a l , the s o l u t i o n o f t he n ( n + l ) / 2 e l e m e n t s o f B^ c a n be o b t a i n e d by i n s p e c t i o n o f ( 3 . 9 ) as [Bi].. = • - [Q'l.. A W r - - , L J l J ( 3 . 1 0 ) and |B'J = JB-JJ , i , j = 1, . . . n B-^  i s t h e n f o u n d f r o m ( 3 . 7 a ) as B1 = V . B J V T ( 3 . 1 1 ) S i m i l a r l y , by s u b s t i t u t i n g ( 3 . 4 ) i n t o ( 3 . 3 ) and p r e m u l t i p l y i n g T and p o s t m u l t i p l y i n g by X and X r e s p e c t i v e l y , a g e n e r a l e q u a t i o n a n a l a g o u s to ( 3 . 9 ) c an be w r i t t e n as and by i n s p e c t i o n [ B 1 + 1 = -i j — — /( x i + \.) 22 ( 3 . 1 3 ) S u c c e s s i v e s u b s t i t u t i o n o f B j , BJ!,, ...B^ i n t o ( 3 . 1 3 ) where B-^  i s f o u n d by ( 3 . 1 0 ) r e s u l t s i n t he g e n e r a l s o l u t i o n i j l_ J i j where Q' i s g i v e n by ( 3 . 7 b ) . A l s o , t he same t r a n s f o r m a t i o n a p p l i e d to B|, ( 3 . 1 1 ) , c a n be a p p l i e d t o B c J + ]_> s u c h t h a t V l = V Bi +1 ^ < 3' 1 5 ) Thus t he v a l u e o f t he p e r f o r m a n c e f u n c t i o n J f r o m ( 3 . 2 ) becomes ( 3 . 1 6 ) J = y V B' , J O d+l T -v y. 3.3 P e r f o r m a n c e F u n c t i o n M i n i m i z a t i o n The p a r a m e t e r s c h o s e n f o r t h e m i n i m i z a t i o n p r o c e d u r e a r e T w r i t t e n as jq . i> • • • q ^ j ••• qjQ (14) ( 3 . 1 7 ) F l e t c h e r and P o w e l l ' s d e s c e n t method i s u s e d f o r m i n i m i z i n g J . T h i s m e t h o d , s u m m a r i z e d i n A p p e n d i x C , r e q u i r e s t h e d e r i v a t i v e o f t he p e r f o r m a n c e f u n c t i o n . T a k i n g t h e p a r t i a l d e r i v a t i v e o f ( 3 . 1 6 ) w i t h r e s p e c t t o q ^ , c a n ke f o u n d as J , = y T |v, B' . V + VB ' , , , V T + V B ' ~ly ( 3 . 1 9 ) ' q : J o L ' q . , . « + l oc+l 'q^ a+1 > q J J o v To e v a l u a t e B ^ ^ , i n t h i s e q u a t i o n , t h e p a r t i a l d e r i v a t i v e o f ^ i e q u a t i o n ( 3 . 1 4 ) i s t a k e n w i t h r e s p e c t t o q^ w i t h ( 3 . 7 b ) s u b s t i t u t e d f o r Q 1 . The e l e m e n t s o f B' , , c a n t h u s be w r i t t e n as B = ( - D oc+1 T T X , QX + X QX, /<x i + >.) + F. where P . . = - f i 1 ,~| (<* + l ) ( V , + X-, )/(/\ + X-) i j U * + 1 J i j 1 q i J I i 1 . J S u b s t i t u t i n g t he e i g e n v e c t o r s e n s i t i v i t y e q u a t i o n X C. l ( 3 . 2 0 a ) ( 3 . 2 0 b ) . ( 2 . 2 0 ) i n t o ( 3 . 2 0 a ) , t he e l e m e n t s o f B ^ + ^ , become B i j ( - D K + 1 + F T Q'c, -JIJ / ( V V CK+l i j Boc+l'q. ^1 Vfl»q. i , j = 1, . . . n 23 ( 3 . 2 1 a ) ( 3 . 2 1 b ) U i J Thus w i t h B^^ , e v a l u a t e d i n t h i s manner and by s u b s t i t u t i n g V , = -V Ct i n t o ( 3 . 1 9 ) , t he f i n a l f o r m o f J , i s g i v e n by q i -T • CI B « + i " B * + i c + B «i + i ' q i T-v y . ( 2 . 2 6 ) ( 3 . 2 2 ) In t he p a r a m e t e r a d j u s t m e n t p r o c e s s t o m i n i m i z e J , t he e i g e n -s y s t e m i n c r e m e n t a l p r o c e d u r e o f C h a p t e r 2 t o g e t h e r w i t h t he e x -t e n d e d C o l l a r and J a h n c o r r e c t i o n method i s u s e d to a v o i d r e p e t i t i v e c a l c u l a t i o n o f a new e i g e n s y s t e m a f t e r e a c h a d j u s t m e n t . Note a l s o t h a t a c o n s i d e r a b l e s a v i n g i n c o m p u t a t i o n e f f o r t c an be r e a l i z e d T -i f , i n t h e c a l c u l a t i o n o f J , and J , t he v e c t o r V y and i t s t r a n s -' ' q . ' J o pose a r e computed f i r s t . T h i s a v o i d s t he c o m p u t a t i o n o f m a t r i x p r o d u c t s . 3.4 A N u m e r i c a l Examp le To i l l u s t r a t e the m i n i m i z a t i o n p r o c e d u r e , t he 9 ^ o r d e r power s y s t e m o f C h a p t e r 2 i s c h o s e n as an e x a m p l e . The i n i t i a l p a r a m e t e r s f o r t h i s s t u d y a r e t a k e n f rom T a b l e 2.1 w i t h t he e x c e p t i o n o f , k 2 and They a r e c h o s e n so t h a t t h e i n i t i a l s y s t e m i s s t a b l e . I f t he s y s t e m were n o t i n i t i a l l y s t a b l e , t he c o s t f u n c t i o n a l ( 3 . 1 ) w o u l d be unbounded and i t s use i n v a l i d f o r a p a r a m e t e r o p t i m i z a t i o n s t u d y . F rom t h e s t a b i l i z a t i o n s t u d y o f C h a p t e r 2, t h e s e p a r a m e t e r s a r e f o u n d as = 0 . 0 1 4 9 k 2 = - 2 . 6 2 0 7 ( 3 . 2 3 ) f t 0 . 1 7 9 7 and the o t h e r a d j u s t a b l e p a r a m e t e r s f r om T a b l e 2.1 a r e k, = 4 0 . 0 0 1 ( 3 . 2 4 ) T = 7 . 6 0 0 The p a r a m e t e r a d j u s t m e n t r a n g e s a r e g i v e n by the i n e q u a l i t y c o n -s t r a i n t s o f ( 2 . 5 5 ) . S i n c e t he F l e t c h e r and P o w e l l d e s c e n t method i s n o t a p p l i c a b l e t o a p r o b l e m 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 , and the methods a v a i l a b l e f o r s o l v i n g t h i s t y p e o f p r o b l e m u s u a l l y (15) r e q u i r e a l a r g e number o f i t e r a t i o n s , t h e a p p r o a c h o f Box i s u s e d . H i s method t r a n s f o r m s a c o n s t r a i n e d v a r i a b l e q^ t o an u n -c o n s t r a i n e d v a r i a b l e p^ by t he t r a n s f o r m a t i o n P i = a r c s i n y ( q i - q i ) / ( q i - q± 5" ( 3 . 2 5 ) m in max min W i t h t h i s t r a n s f o r m a t i o n , F l e t c h e r and P o w e l l ' s method i s a p p l i c a b l e s i n c e now an u n c o n s t r a i n e d opt imum i n p-space i s s o u g h t . The d e r i v a t i v e o f J i n p-space becomes J , = J , • q . , ( 3 . 2 6 ) p i q i 1 P i where J , i s g i v e n by ( 3 . 2 2 ) and q . , i s f o u n d f r o m the p a r t i a l q i 1 p i d e r i v a t i v e o f ( 3 . 2 5 ) w i t h r e s p e c t t o p^ as q i ' p = ( q i ~ q i ^ 2 s i n p i c o s (3 . 2 7 ) * i max m in The o p t i m i z a t i o n p r o c e s s i s t h e n c a r r i e d ou t w i t h a v a l u e o f a i n ( 3 . 1 ) e q u a l t o o n e , y Q c h o s e n as y o = [ l l l l l l l l l ] T ( 3 . 2 8 ) and Q c h o s e n as Q = d i a g Q . 1 1 1 1 1 1 1 l ] ( 3 . 2 9 ) The p r o c e s s t a k e s n i n e i t e r a t i o n s t o f i n d t h e min imum. The e i g e n -v a l u e s a f t e r o p t i m i z a t i o n a r e - 3 3 3 . 3 4 + j O . - 1 . 0 5 1 6 + JO . - 5 1 . 9 7 0 + j O . - 0 . 2 0 6 2 + j O . - 9 . 9 0 7 5 + j O . - 0 . 1 2 5 7 + J 5 . 4 4 5 9 - 2 . 0 3 1 9 + j O . - 0 . 0 1 9 2 + j O . ( 3 . 3 0 ) and t h e p a r a m e t e r s f o u n d a r e 25 Xl = 0 . 1 0 0 k, = 1 .032 £ t = 1 .000 f = 1 .000 k 2 3 .000 Note t h a t k^ i s v e r y s m a l l compared t o i t s l i m i t s o f - 6 0 0 . and +600. T h i s i s q u i t e s u r p r i s i n g i n v i e w o f the f a c t ' t h a t a h i g h g a i n i s u s u a l l y u s e d f o r good t r a n s i e n t s t a b i l i t y . A l s o n o t e t h a t t he o t h e r p a r a m e t e r s a r e a t t h e i r l i m i t s i m p o s e d by t he c o n s t r a i n t s o f ( 2 . 5 5 ) . T h i s i n d i c a t e s t h a t t h e r e i s s t i l l room f o r improvemen t i f t he c o n s t r a i n t s c a n be r e l a x e d . 26 4 . C O M P U T A T I O N O F A N O P T I M A L C O N T R O L L E R F O R A H I G H O R D E R S Y S T E M In t h i s c h a p t e r , a s u c c e s s i v e a p p r o x i m a t i o n method d e v e l o p e d by P u r i and G r u v e r i s u s e d t o s o l v e f o r an o p t i m a l c o n t r o l l e r o f a h i g h o r d e r s y s t e m . T h i s meti iod r e q u i r e s s u c c e s s i v e R i c a t t i e-q u a t i o n s o l u t i o n s . w h i c h f i n a l l y c o n v e r g e t o t he R i c a t t i m a t r i x . A l t h o u g h the c o n v e r g e n c e o f t h i s method i s r a p i d , the c o m p u t a t i o n t i m e f o r e a c h a p p r o x i m a t i o n i s l e n g t h y . S i n c e t h i s i s a l i m i t a t i o n f o r e f f i c i e n t a p p l i c a t i o n to h i g h o r d e r s y s t e m s , a f a s t r e c u r s i v e method f o r s o l v i n g e a c h a p p r o x i m a t i o n i s d e v e l o p e d h e r e . T h i s method does n o t r e q u i r e m a t r i x i n v e r s i o n and i n v o l v e s o n l y t he p r o d u c t s o f m a t r i c e s o f o r d e r ( n x n ) . 4 .1 P r o b l e m F o r m u l a t i o n -C o n s i d e r a l i n e a r t i m e - i n v a r i a n t s y s t e m d e s c r i b e d by y" = A y + B u" ( 4 . 1 ) I t i s d e s i r e d t o c h o o s e t h e c o n t r o l v e c t o r u ( t ) t h a t m i n i m i z e s t h e f u n c t i o n a l p J = J [j?TQ y + H T¥ u ] d t ( 4 . 2 ) o Q and V a r e c o n s t a n t s y m m e t r i c p o s i t i v e d e f i n i t e m a t r i c e s . The u ( t ) w h i c h m i n i m i z e s J s u b j e c t t o the c o n s t r a i n t ( 4 . 1 ) i s g i v e n b y ( 2 0 ) u ( t ) = - V -VK y ( t ) ( 4 . 3 ) where K i s t he s o l u t i o n to t h e m a t r i x R i c a t t i e q u a t i o n A T K + K A - K S K + Q = 0 ( 4 . 4 ) and S = B W " 1 B T ( 4 . 5 ) The s y s t e m e q u a t i o n , ( 4 . 1 ) w i t h t he s u b s t i t u t i o n o f ( 4 . 3 ) becomes y = [A - S K ) y ( 4 . 6 ) The s o l u t i o n o f K i n ( 4 . 4 ) c an be f o u n d by i n t e g r a t i n g , i n r e v e r s e t i m e , t h e n o n l i n e a r R i c a t t i - t y p e m a t r i x d i f f e r e n t i a l e q u a t i o n ^ 2 1 ) K = - A T K - K A - Q + K S K ( 4 . 7 ) However t h i s c a l c u l a t i o n i s o f t e n u n s t a b l e . E i g e n s y s t e m methods 27 of s o l v i n g ( 4 . 4 ) are a v a i l a b l e , ' but they r e q u i r e the eigensystem of a matrix double the order of the o r i g i n a l system matrix A. Furthermore, eigensystem accuracy u s u a l l y d e t e r i o r a t e s with i n c r e a s i n g system order. Although the c o r r e c t i o n procedure f o r the eigensystem developed i n Chapter 2 i s a p p l i c a b l e , i t may f a i l i f the eigenvalues are too c l o s e l y l o c a t e d . This problem was. avoided i n Chapters 2 and 3 by f o r c i n g the eigenvalues to pass through any neighboring p o s i t i o n s . Although there are other methods a v a i l a b l e f o r s o l v i n g the R i c a t t i equation, the most s u i t a b l e one f o r a p p l i c a t i o n to high order systems was found to be the method of succ e s s i v e approximations (8) developed by P u r i and Gruver. 7 This method c a l c u l a t e s the j - t h approximation of the R i c a t t i matrix from the f o l l o w i n g de-generate matrix R i c a t t i equation K ( J ) A ( J ) + [ A ( J ) ] T K ( J ) +. Q ( J ) = 0 (4.8) where A ^ = A - S K^ J _ 1^ (4.9) and = Q + K ^ - 1 ^ S K^" 1^ (4.10) P u r i and Gruver have shown that the succ e s s i v e e v a l u a t i o n s of ^ from these equations converge monotonically and r a p i d l y to the unique R i c a t t i matrix i f i s p r o p e r l y chosen. This i n i t i a l matrix must be chosen so that the eigenvalues of A ^ ^ l i e i n the l e f t - h a l f complex plane. I f A ^ ^ i s found to be unstable with an i n i t i a l guess f o r , then the eigenvalue s h i f t i n g technique can be used. S o l v i n g f o r i n (4.8) u s u a l l y i n v o l v e s the i n v e r s i o n of an n(n+l)/2 x n(n+l)/2 matrix, where n i s the order of matrix A. Thus, a l i m i t a t i o n i s placed on the s i z e of the system that can be analyzed p r a c t i c a l l y . 28 I n s t e a d , a r e c u r s i v e method o f s o l v i n g f o r i s d e v e l o p e d h e r e w h i c h e l i m i n a t e s l e n g t h y m a t r i x i n v e r s i o n s and o p e r a t e s o n l y w i t h nxn m a t r i c e s . 4.2 A R e c u r s i v e Me thod f o r O b t a i n i n g t h e S u c c e s s i v e A p p r o x i m a t i o n s In what f o l l o w s , t he s u p e r s c r i p t j w i l l be o m i t t e d s i n c e the r e s u l t i s a p p l i c a b l e t o any j . Thus t he d e g e n e r a t e m a t r i x R i c a t t i e q u a t i o n (4.8) c a n be w r i t t e n as A T K + K A = -0 (4.11) and (4.6) i s now w r i t t e n as y = A y (4.12) where A and K a r e c o n s t a n t m a t r i c e s . P r e m u l t i p l y i n g and p o s t m u l t i -—T — p l y i n g (4.11) by y and y r e s p e c t i v e l y and s u b s t i t u t i n g (4.12) i n t o t h e r e s u l t , (4.11) becomes at ( y ? K y) = - (y T Q y) ( 4 . 1 3 ) F o r an a s y m p t o t i c a l l y s t a b l e s y s t e m , i n t e g r a t i o n o f (4.13) w i t h i n i t i a l c o n d i t i o n s y r e s u l t s i n y „ K y o = J(yTQ y ) d t (4.14) o _ L e t t he homogeneous s o l u t i o n t o y be e x p r e s s e d i n t e rms o f t he s t a t e t r a n s i t i o n m a t r i x as y = e A t y 0 ( 4 . 1 5 ) S u b s t i t u t i n g (4.15) i n t o (4.14) and c o m p a r i n g t e r m s , K becomes K = y[eAt]T Q [ e A t ] d t (4.16) T h i s i n t e g r a l may be e v a l u a t e d by a s e r i e s o f s u c c e s s i v e i n -t e g r a n d s each as a c o n s t a n t o v e r a t ime i n t e r v a l o f s i z e A t = l/R. The t r a n s i t i o n m a t r i x f o r t h e s e d i s c r e t e t ime i n t e r v a l s may be w r i t t e n as e A t = e A N ^ = [ ^ ( l A O ] * ( 4 > 1 ? ) where N i s c h o s e n as an i n t e g e r and t ime as t = N/R (4.18) 29 M a t r i x e ^ ^ ' ^ i s c o n s t a n t f o r c o n s t a n t A and l / l t , and w i l l be d e s i g n a t e d as A^ = e A ( l / R ) (4.19) which can be c a l c u l a t e d from the power s e r i e s A 1 = I + A/R + A 2/2!R 2 + A 3/3!R 3 + ... (4.20) For a s t a b l e system, the t r a n s i t i o n m a t r i x has the p r o p e r t y e A t 0 as t » (4.21) L i k e w i s e , f o r a s t a b l e system, from (4.17) and (4 . 1 9 ) , A^ has the p r o p e r t y ( A ^ ) ^ f 0 as N oo (4.22) f o r R chosen p o s i t i v e . S u b s t i t u t i n g (4.19) and (4.17) i n t o ( 4 .16), K can be approximated by a summation of d i s c r e t e i n t e g r a l s as K = (Q + A? Q A,)/2R + (A, Q A, + ( A 2 ) T Q A 2)/2R +... 1 1 . i x i x (4.23) = (Q/2 + S)/R wh e r e S ^ ( A f l V i J (4.24) The e f f o r t of c a l c u l a t i n g S from (4.24) i s g r e a t l y reduced i f the f o l l o w i n g r e c u r s i v e f o r m u l a i s used. 2 Aj + ( A 2 ) T Q A 2 = S x ( A 2 ) T S t A 2 + S, = S 2 4 T 4 (A,) S^A, + S 2 = S 3 { 4 2 5 ) • « • • • • ( A n ) S.AJL + S i S . + 1 where N = 2 1 (4.26) and l i m S. = S (4.27) i—& c o 1 S u b s t i t u t i n g (4.26) i n t o ( 4 . 1 8 ) , i n t e g r a t i o n time can be w r i t t e n as t = 21/R (4.28) Note t h a t i n t e g r a t i o n time i n c r e a s e s r a p i d l y w i t h i n c r e a s i n g i , and as a r e s u l t S^ +^ converges r a p i d l y t o S s i n c e A-, has the p r o p e r t y of e q u a t i o n (4.22). 30 4 . 3 A c c u r a c y o f t he R e c u r s i v e Me thod T h e o r e t i c a l l y the a c c u r a c y o f K e v a l u a t e d by ( 4 . 2 3 ) and the r e c u r s i v e f o r m u l a ( 4 . 2 5 ) w i l l i m p r o v e i f R i s made l a r g e w i t h r e s p e c t t o t h e e i g e n v a l u e s o f A s i n c e t he d i s c r e t e i n t e g r a l w i l l d i v i d e i n t o a l a r g e r number o f s m a l l e r more a c c u r a t e a r e a s . A l s o , c h o o s i n g a l a r g e R r e d u c e s t h e number o f t e rms r e q u i r e d t o c a l c u l a t e A^ i n t h e power s e r i e s ( 4 . 2 0 ) s i n c e (A/R )^ d e c r e a s e s r a p i d l y as N i n -c r e a s e s . B u t , w i t h R c h o s e n e x t r e m e l y l a r g e , A^ i s a l m o s t a u n i t m a t r i x w i t h s m a l l o f f - d i a g o n a l e l e m e n t s and the c a l c u l a t i o n o f l a r g e powers o f A , by r e p e a t e d m u l t i p l i c a t i o n as r e q u i r e d by t he r e c u r s i v e f o r m u l a ( 4 . 2 5 ) , c a u s e s an a c c u m u l a t i o n o f e r r o r s i n (A^ )^ . Thus t h e s e e r r o r s l i m i t the a c c u r a c y i m -p rovemen t o b t a i n a b l e by c h o o s i n g a l a r g e R. As a r e s u l t , R s h o u l d be c h o s e n l a r g e enough t o r e d u c e e r r o r s c a u s e d by d i s c r e t e e v a l u -a t i o n o f t he i n t e g r a l ( 4 . 1 6 ) b u t n o t t o o l a r g e s i n c e i n t h i s c a s e , r o u n d - o f f e r r o r s i n c r e a s e . As an i l l u s t r a t i o n o f t he e r r o r i n v o l v e d i n a p p r o x i m a t i n g K w i t h v a r i o u s R, c o n s i d e r an example t a k e n f r o m the n u m e r i c a l r e s u l t s o f C h a p t e r 3. The 9 ^ o r d e r A - m a t r i x has e i g e n v a l u e s A i = V A i ( 4 . 2 9 ) - 3 3 4 . 2 + j O . - 1 . 0 3 2 + j O . - 7 0 . 3 3 + j O . - 0 . 3 0 1 + JO . - 4 2 . 2 6 + j O . 0 . 112 + J 5 . 4 5 0 - 2 . 0 7 0 + j O . - 0 . 0 0 5 + j O . As an i n d i c a t i o n o f t he a p p r o x i m a t i o n e r r o r o f K, t he m a t r i x E i s c o n s i d e r e d where E = A K + K A + Q ( 4 . 3 0 ) 31 From (4.11) i t is known that E becomes a nu l l matrix i f K is known exactly. Table 4.1 is a compilation of n a..I and V E.. (4.31) i j I max Z-J I i j I i j for various R using single and double precision calculations. Three different approximations to are used, namely 1. A x = I + A/R + A 2/2!R 2 2. kY = I + A/R + A 2/2!R 2 + A 3/3!R 3 (4.32) 3. A1 = I + A/R + A 2/2!R 2 + A 3/3iR 3 + A 4/4!R 4 For small R, the error caused by discrete evaluation of the integral is dominant since double precision calculation does not reduce the error s i g n i f i c a n t l y . As R is chosen larger, the error f i r s t decreases and then increases as round off errors accumulate in the single precision calculations. Although double precision calcu-lations exhibit decreasing error with increasing R, the calculation time in each case is double the time for single precision calcu-l a t i o n . In this example, the error caused by truncation of the power series rejiresentation of A-^  is found minimal i f A^ i s ap-proximated by A x = I + A/R + A 2/2!R 2 + A 3/3!R 3 (4-33) 4.4 A Numerical Example The methods of sections 4.1 and 4.2 are applied here to solve for an optimal controller of the proposed power system of Appendix A. To incorporate the optimal controllers into the power system, the following changes are made. 1. The governor dashpot is replaced by an actuator k ,/(' l + f I ? ) and the signal 'a' is replaced by a control signal u-^ . The transfer function is shown in Figure 4.1. The state variable equation for 'a^' becomes p a f = - a f / r ; c l + tlj (4.34) where u^ = ( k c l / t c l ) u i (4.35) 32 T a b l e 4.1 - E r r o r o f K A p p r o x i m a t i o n It APPROXI- SINGLE PRECISION DOUBLE PRECISION MATION NUMBER E. - max E. . E . . 1 i j Imax 500 1 .2619 .7350 .2619 .6280 2 .1259 .6091 .1259 .4710 3 .1464 .6363 .146.4 .4735 900 1 .0749 .3880 .0749 .1753 2 .0425 .3670 .0425 .1488 3 .0561 .4420 .0455 .1502 1300 1 .0869 .5470 .0349 .0812 2 . 1040 . 5970. .0210 .0719 3 .1025 .6190 .0219 .0724 4300 i .2120 1. 150 .0031 .0072 2 .2650 1. 420 .0019 .0066 3 .2640 1. 480 .0020 .0067 7300 1 .3506 1. 618 .0011 .0025 2 .4511 2. 184 .0007 .0023 3 .4511 2. 271 .0007 .0023 13300 1 .4420 3. 202 .0003 .0007 2 .59 10 4. 055 .0002 .0007 3 .5910 4. 204 .0002 .0007 33 r P ) c l 1 4 g ( i . + t g p ) > (1 - t v P ) >. ( i + T C 1 P ) F i g u r e 4 .1 - G o v e r n o r T r a n s f e r F u n c t i o n and H y d r a u l i c O p e r a t o r 2. The v o l t a g e r e g u l a t o r - e x c i t e r has t he speed s i g n a l r e p l a c e d by a c o n t r o l s i g n a l u,,. The t r a n s f e r f u n c t i o n i s shown i n F i g u r e 4 . 2 . The s t a t e v a r i a b l e e q u a t i o n f o r ' v s ' becomes P v s = ~(W V t " V s ^ l + "2 where v t = a 5 2 & + a 5 3 ^F and u 2 = ( k j ^ ' l ^ / t j ) u 2 ( 4 . 3 6 ) ( 4 . 3 7 ) ( 4 . 3 8 ) — AV t + •P k l s 1 1 * (1 + t x p ) . u. F i g u r e 4 . 2 - V o l t a g e R e g u l a t o r - E x c i t e r T r a n s f e r F u n c t i o n The s t a t e v a r i a b l e e q u a t i o n o f t he s y s t e m w i t h c o n t r o l l e r s u^ and u 2 c a n be w r i t t e n as y = A y + B u ( 4 . 3 9 ) ( 4 . 4 0 ) where y = J ^ , V, V F » v s » S» h > a » a f J and A i s t he same as t h a t o f A p p e n d i x A e x c e p t f o r t he f o l l o w i n g = 0 , a 9 9 = - l / t c l e l e m e n t s a,-, a 0 1 = a ^ 0  & n n  -1/V-i ( 4 . 4 1 ) S i n c e , f r o m ( 4 . 3 4 ) , ( 4 . 3 6 ) and ( 4 . 4 0 ) u^ and u 2 a p p e a r i n y ^ and y ^ r e s p e c t i v e l y , B becomes 0 0 0 0 1 0 0 0 0 ^ and B u 0 0 0 0 0 0 0 0 1 T T U 2 j ( 4 . 4 2 ) ( 4 . 4 3 ) 34 The r e s p o n s e o f t he o r i g i n a l s y s t e m g i v e n i n C h a p t e r 2, wh i ch i n c l u d e s a s p e e d f e e d b a c k i n the v o l t a g e r e g u l a t o r and a d a s h p o t , i s shown i f F i g u r e 4 . 3 . The s y s t e m i s u n s t a b l e . The i n i t i a l c o n d i t i o n s a r e g i v e n as y Q = j 0 .2 0 0 0 0 0 0 0 j T ( 4 . 4 4 ) F o r t he c o m p u t a t i o n o f an o p t i m a l c o n t r o l l e r a v a l u e o f ^ = .01 t o g e t h e r w i t h t he p a r a m e t e r v a l u e s o f T a b l e 2.1 a r e u s e d . The s y s t e m w i t h o u t c o n t r o l i s t e s t e d f o r s t a b i l i t y and i s f o u n d t o be s t a b l e w i t h e i g e n v a l u e s g i v e n as - 3 3 3 . 5 + j O . - 1 . 5 6 6 + j l . 3 8 1 - 1 0 0 . 0 + j O . - . 5 3 4 4 + j O . ( 4 . 4 5 ) - 1 5 . 5 6 + j O . - . 2 1 8 9 + J 5 . 2 6 7 - 4 . 5 6 8 + JO . In t h i s c a se e i g e n v a l u e s h i f t i n g i s n o t n e c e s s a r y a n d , as a r e s u l t , c a n be c h o s e n as a n u l l m a t r i x . M a t r i c e s Q and W o f ( 4 . 2 ) a r e c h o s e n as u n i t m a t r i c e s and Ii o f ( 4 . 1 9 ) as 1000 . The R i c a t t i m a t r i x i s t h e n f o u n d by a p p l y i n g t he s u c c e s s i v e a p p r o x i m a t i o n method t o g e t h e r w i t h t h e r e c u r s i v e t e c h n i q u e o f s e c t i o n 4 . 2 . The s o l u t i o n i s o b t a i n e d a f t e r n i n e s u c c e s s i v e a p p r o x i m a t i o n s i n 2 4 . 5 3 s e c o n d s on an IBM 7044 c o m p u t e r u s i n g s i n g l e p r e c i s i o n c a l c u -l a t i o n s . The c o n t r o l v e c t o r c a l c u l a t e d f r o m u = - W" 1 B T K y ( 4 . 4 6 ) i s u = ( 4 . 4 7 ) - . 0 1 9 , . 1 8 4 , . 0 2 8 , . 0 0 1 , . 0 5 0 , - . 3 6 7 , - . 3 6 2 , . 1 0 2 , - . 0 4 8 . 1 9 6 , - 5 . 9 7 , - . 7 9 5 , - . 0 0 2 , - . 0 4 8 , 1 3 . 3 , 1 0 . 6 , - 6 . 6 1 , 3.26_ The s y s t e m r e s p o n s e o f t h i s example i s shown i n F i g u r e 4 . 4 . F o r a d i f f e r e n t Q c h o s e n as Q = d i a g j l O 10 1 1 1 1 1 1 l ] ( 4 . 4 8 ) t h e c o r r e s p o n d i n g s y s t e m r e s p o n s e i s shown i n F i g u r e 4 .5 and has an i m p r o v e d damp ing c h a r a c t e r i s t i c . However t h i s has been o b t a i n e d y F i g u r e 4 .4 - S y s t em R e s p o n s e s f o r Q = d i a g l l 1 1 1 1 1 1 1 1 a t the expense o f i n c r e a s i n g the maximum e x c u r s i o n o f the c o n t r o l s i g n a l a^ to t h r e e t i m e s t h a t i n F i g u r e 4.4. A c o m p a r i s o n i s made o f t he c a l c u l a t i o n s v i s i ng s i n g l e v e r s u s d o u b l e p r e c i s i o n and a s m a l l v e r s u s a l a r g e R. A l l o f the c a s e s use t h e Q g i v e n by ( 4 . 4 8 ) . c a s e 1 D o u b l e p r e c i s i o n and R = 1000 j c a s e 2 S i n g l e p r e c i s i o n and R = 1000 c a s e 3 S i n g l e p r e c i s i o n and R = 200 As c o n c l u d e d i n s e c t i o n 4 . 3 t he b e s t a c c u r a c y i s o b t a i n e d by the d o u b l e p r e c i s i o n c a l c u l a t i o n u s i n g a l a r g e R; c a se 1. The m a t r i x E i s u s e d f o r c o m p a r i s o n where E = - W " 1 B T K ( 4 . 4 9 ) F o r c a s e 1, E = ( 4 . 5 0 ) - . 0 5 3 , . 4 3 6 , . 0 6 4 , . 0 0 2 , . 0 5 2 , - . 8 3 0 , - . 8 9 3 , . 2 3 8 , - . I l l . 7 0 9 , - 1 7 . 5 , - 2 . 1 6 , - . 0 0 1 , - . 1 1 1 , 3 4 . 6 , 3 3 . 2 , - 1 4 . 8 , 7.15_ The d i f f e r e n c e s be tween c a s e s 1 and 2 and c a s e s 1 and 3 a r e E l - E 2 ( 4 . 5 1 ) 2 x l 0 - 6 , 0 . , - 3 x l 0 " 6 , 0 . , 0 . , 2 x 1 0 " 5 , 2 x l O - 5 , - l x l O " 5 , 0 . - 5 x l O ~ 5 , 0 . , 0 . , 2 x l O " 7 , 0 . , 0 . , 0 . , 0 . , - l x l O " 4 E1 - E 3 =• ( 4 . 5 2 ) - 5 x l O " ^ - l x l O " ^ l x l O ~ 4 - 2 x l O " 3 - 3 x l O ~ 3 9 x l O ~ ^ l x l O ~ 4 - 4 x l O ~ ^ - 2 x l O ~ 5 3 x l O " 4 - 2 x l O ~ ^ - 4 x l O ~ 4 - 2 x l O " ^ - 2 x l O " 4 4 x l O " 3 5 x l O " 3 - l x l O ~ 3 l x l O " 3 The c a l c u l a t i o n t i m e s a r e : c a s e 1 - 5 7 . 9 7 s e c o n d s , c a s e 2 - 2 4 . 5 2 s e c o n d s , c a s e 3 - 2 1 . 2 5 s e c o n d s . Ho te t h a t a d e c r e a s e i n R f r om 1000 t o 200 u s i n g the s i n g l e p r e c i s i o n c a l c u l a t i o n o n l y d e c r e a s e s c a l c u l a t i o n t i m e by 13$ b u t a c c u r a c y wo r sens d i s p r o p o r t i o n a t e l y . I t c a n be c o n c l u d e d f r o m t h i s c o m p a r i s o n t h a t the use o f s i n g l e p r e -c i s i o n c a l c u l a t i o n w i t h a l a r g e R i s d e f i n i t e l y t he b e s t compromise o f a c c u r a c y v e r s u s c a l c u l a t i o n t i m e . 39 5. SENSITIVITY ANALYSIS OF THE TIME RESPONSE OP MULTIVARIABLE SYSTEMS In t h i s chapter, the s e n s i t i v i t y of system response to parame-te r v a r i a t i o n s i s i n v e s t i g a t e d . Eigensystem s e n s i t i v i t y a n a l y s i s was used i n Chapters 2 and 3 for system s t a b i l i s a t i o n and parameter o p t i m i z a t i o n r e s p e c t i v e l y , but i n i t s most d i r e c t a p p l i c a t i o n , eigensystem s e n s i t i v i t y a n a l y s i s could also be used f o r eigenvalue movement s t u d i e s . However, i n the f i n a l a n a l y s i s of a system design, a time response s e n s i t i v i t y study i s u s u a l l y required since t h i s gives the designer the desired d i r e c t i n d i c a t i o n of changes of system response to parameter v a r i a t i o n s . For the a n a l y s i s of time response s e n s i t i v i t y , i t would be d e s i r a b l e to use an analog computer because of i t s f a s t i n t e g r a t i o n time. But the s i z e of a system that may be studied on an analog computer i s u s u a l l y l i m i t e d by the number of components a v a i l a b l e . Therefore, f o r large order systems, the d i g i t a l computer must be used. However, i t s slow i n t e g r a t i o n time i s a p r a c t i c a l l i m i t a t i o n on the number of parameters that may be chosen f o r a s e n s i t i v i t y t h study, s i n c e , as i s pointed out i n the f o l l o w i n g s e c t i o n , an n order equation must be integrated f o r each parameter s e n s i t i v i t y c a l c u l a t i o n . K o k o t o v i c ^ ^ developed a method of computing s e n s i t i v i t i e s f o r a l l parameters simultaneously from two n^*1 order equation i n t e g r a t i o n s . This method, however, i s r e s t r i c t e d to a n a l y s i s of s i n g l e v a r i a b l e systems where the parameters studied must be chosen as the equation c o e f f i c i e n t s . In t h i s chapter a simultaneous m u l t i v a r i a b l e s e n s i t i v i t y method i s developed i n which any parame-te r may be chosen f o r i n v e s t i g a t i o n and only one n^*1 and one 2n^^ order equation need be integrated i n order to compute a l l the parameter s e n s i t i v i t i e s . 40 5.1 S i m u l t a n e o u s C o m p u t a t i o n o f Time Response S e n s i t i v i t i e s t o  a L a r g e Number o f P a r a m e t e r s . C o n s i d e r t he m u l t i v a r i a b l e s y s t e m w i t h a s i n g l e f o r c i n g f u n c t i o n u y = A y + E u ( 5 . 1 ) F o r examp le E • = [2 0 0 0 0 0 0 0 cT|T, u = 1 ( 5 . 2 ) T a k i n g the p a r t i a l d e r i v a t i v e o f ( 5 . 1 ) w i t h r e s p e c t t o a p a r a m e t e r q^ r e s u l t s i n t he s e n s i t i v i t y e q u a t i o n N k = A ?1»k + N k ~y + u < 5 - 3 ) T h u s , t o s o l v e f o r t he s e n s i t i v i t y y , , k = 1 , . . . $ , r e q u i r e s q k one i n t e g r a t i o n o f ( 5 . 1 ) and ' ' i n t e g r a t i o n s o f ( 5 . 3 ) . In the f o l l o w i n g a method i s d e r i v e d t o remove the d e p e n d a n c e o f i n t e -g r a t i o n t i m e on the number o f p a r a m e t e r s i n v e s t i g a t e d and t o compute s e n s i t i v i t i e s s i m u l t a n e o u s l y . L e t t he L a p l a c e t r a n s f o r m be a p p l i e d t o ( 5 . 1 ) y ( s ) = \ll - A ] " 1 b u ( s ) ( 5 . 4 ) w i t h the i n i t i a l c o n d i t i o n s c h o s e n as z e r o . The i n v e r s e o f ( s I-A) c a n be e x p r e s s e d i n t e rms o f an a d j o i n t and a c h a r a c t e r i s t i c p o l y -n o m i a l as [jsl - AJ" ' 1 ' = R ( s ) / g ( s ) ( 5 . 5 ) where g ( s ) i s t he s c a l a r c h a r a c t e r i s t i c p o l y n o m i a l o f ( s I - A ) , n a m e l y g ( s ) = s n - h ^ s 1 1 . . . - h u s 1 1 - * - . . . - h ( 5 . 6 ) and R ( s ) i s the a d j o i n t m a t r i x p o l y n o m i a l o f ( s I-A ) , . 'namely R ( s ) = I s n - 1 + R l S n " 2 + . . . R i s n " 1 " 1 + . . . R A ( 5 . 7 ) where R^  i s a m a t r i x . The method o f c a l c u l a t i n g R^  and h^ i s g i v e n i n s e c t i o n 5 . 2 . S u b s t i t u t i n g ( 5 . 5 ) i n t o ( 5 . 4 ) , the i n p u t - o u t p u t r e l a t i o n becomes y ( s ) = R ( s ) • b • u ( s ) / g ( s ) ( 5 . 8 ) C o m b i n i n g R ( s ) b i n t o a v e c t o r p o l y n o m i a l and u ( s ) / g ( s ) i n t o ci s i n g l e t e r m , ( 5 . 8 ) c an be w r i t t e n 41 y ( s ) = f ( s ) . z ( s ) ( 5 . 9 ) where z ( s ) = u ( s ) / g ( s ) ( 5 . 1 0 ) f ( s ) = 7 s n _ 1 i . - . f - s 1 " 1 ( 5 . 1 1 ) v ' n I 1 • and • f • - H ,-b" 1 . . n _ 1 ( 5 . 1 2 ) f £ b n T a k i n g the p a r t i a l d e r i v a t i v e o f ( 5 . 9 ) w i t h r e s p e c t t o q ^ , the s e n s i t i v i t y e q u a t i o n becomes y ( s ) , = ? ( s ) , . z ( s ) +. ? ( s ) . z ( s ) , ( 5 . 1 3 ) q k q k q k From t h e p a r t i a l d e r i v a t i v e o f ( 5 . 1 0 ) , z ( s ) , c an be w r i t t e n as q k z ( s ) , = - g ( s ) . -w(s) ( 5 . 1 4 ) q k q k where w(s ) = u ( s ) / \g(s)]2 ( 5 . 1 5 ) T h u s , w i t h s u b s t i t u t i o n o f ( 5 . 1 4 ) , ( 5 . 1 3 ) becomes y ( s ) , = ? ( s ) . . z ( s ) - f ( s ) . g ( s ) . .w(s ) ( 5 . 1 6 ) q k q k q k E q u a t i o n ( 5 . 1 6 ) c a n be p u t i n t o a more c o n v e n i e n t f o r m f o r c o m p u -t a t i o n a l p u r p o s e s b y b r e a k i n g e a c h v e c t o r p o l y n o m i a l , f ( s ) , and q k ? ( s ) . g ( s ) , , i n t o t he p r o d u c t o f a m a t r i x t i m e s a v e c t o r . F i r s t q k d e f i n e F f r om the co l umn v e c t o r s o f ( 5 . 1 2 ) as F k ^ ? n J ( 5 . 1 7 ) T h e n , t he f i r s t p o l y n o m i a l o f ( 5 . 1 6 ) c a n be w r i t t e n f ( s ) , . z ( s ) = F, .1 , . z ( s ) ( 5 . 1 8 ) q k q k n _ 1 _ where S n 1 = [} s S ^ "** s l * * * s11"" j ^ ( 5 . 1 9 ) The s e c o n d v e c t o r p o l y n o m i a l o f ( 5 . 1 6 ) c an a l s o be w r i t t e n s i m i l a r l y as ? ( s ) . g ( s ) , .w (s ) = - F . H , .s" ~.v(s) ( 5 . 2 0 ) q k q k where II, i s an n x ( 2 n - l ) c o n s t a n t n - d i a g o n a l m a t r i x g i v e n by q k 42 H, q k 17, 0 q k ,0 0 0 q. o k. o q k ( 5 . 2 1 ) and q k n ' q k i ' q k ! ' q k '2n-2 Q 1 s s' 2n-21T 0 ( 5 . 2 2 ) ( 5 . 2 3 ) C o m b i n i n g ( 5 . 1 8 ) and ( 5 . 2 0 ) , t he s e n s i t i v i t y e q u a t i o n ( 5 . 1 6 ) c a n be w r i t t e n i n t he f i n a l f o rm as y ( s ) , q k q k n- l ( s ) + F • H , S 2 n - 2 * V ( s ) ( 5 ' 2 4 ) The m a t r i x f o rm o f the s e c o n d p o l y n o m i a l , e q u a t i o n ( 5 . 2 0 ) , i s d e -r i v e d f r o m the p r o d u c t o f two s c a l a r p o l y n o m i a l s C ( S ) : w r i t t e n i n m a t r i x f o r m as c ( s ) = |a . . . . .a-^ 1 i m , , \ / m , , m-i , , \ (s + . . a . s + . . a ) ( s + . . b . s + . . b ; l m l m ( 5 . 2 5 ) n o o 0 b 0 . . . 0. * . 0 . O ' b 1 s • • l s 2m ( 5 . 2 6 ) where ( 5 . 2 7 ) E = b . . . b . . . . b , 1 |_m i 1 J E v i d e n t l y t he r e s p o n s e y ( s ) f r o m ( 5 . 9 ) c a n a l s o be w r i t t e n as a m a t r i x t i m e s a v e c t o r : y ( s ) = F • i n _ 1 • z ( s ) ( 5 . 2 8 ) where s ^ i s g i v e n by ( 5 . 1 9 ) . T a k i n g the i n v e r s e L a p l a c e t r a n s -f o r m o f ( 5 . 2 4 ) and ( 5 . 2 8 ) , t he s o l u t i o n s o f y ( t ) and y ( t ) , b e -q k come where y ( t ) , . . = F. • z ( t ) + F . II, . w ( t ) q k q k q k ( 5 . 2 9 ) y ( t ) = F • z ( t ) z ( t ) •' = J " 1 ^ . ! ' 7 ^ 3 ^ '= - . Z i - ^ n ] 1 ( 5 . 3 0 ) V ( t ) -1 i 2 n _ 2 . v ( s ) [ V] " v i • • v 2 n - l ] ( 5 ' 3 1 ) S i n c e the i n v e r s e L a p l a c e t r a n s f o r m o f ( 5 . 1 0 ) can be w r i t t e n a; p n z ( t ) - . . - h . p n " i z ' ( t ) - . r h n z ( t ) = u ( t ) t h e n z ( t ) c an be s o l v e d f rom the c a n o n i c a l f o rm ( 5 . 3 2 ) ?2 n 0 . 1 0 0 0 0 1 0 . n h. . . . h , l 1 n + 0 S i m i l a r l y , w | t ) c an be s o l v e d f r o m 0 1 0 . . . . 0 0 0 1 0 . . . w. w 2n 0 . . . . . . '1 P 2 n * * * p i * * * p l w, w. w 2n u(t) 0 0 ( 5 . 3 3 ) u ( t ) ( 5 . 3 4 ) where p.^  a r e the c o e f f i c i e n t s o f t he p o l y n o m i a l [ g ( s f ] 2 > w h i c h c a n be e x p r e s s e d as r p 2 n . . . P . . . . P l rj h 0 0 0 h 0 . . . where fh h n n- . h. l 0 . . h, il ( 5 . 3 5 ) ( 5 . 3 6 ) T h u s , t o f i n d y ( t ) and y ( t ) , f o r any number o f p a r a m e t e r s , o n l y one i n t e g r a t i o n o f z ( t ) and w ( t ) i s r e q u i r e d . The f o l l o w i n g s e c t i o n i s a d e r i v a t i o n o f t he a l g o r i t h m f o r c o m p u t i n g F, F, q k and II, . 44 5.2 D e r i v a t i o n o f t h e C o m p u t a t i o n A l g o r i t h m To b e g i n the c o m p u t a t i o n , the c o e f f i c i e n t s o f the p o l y n o m i a l s g ( s ) and R ( s ) i n ( 5 . 6 ) and ( 5 . 7 ) r e s p e c t i v e l y a r e c a l c u l a t e d and F i s computed f rom ( 5 . 1 7 ) . L e v e r r i e r ' s method w i t h F a d c e v ' s m o d i f i -c a t i o n ^ ^ i s u sed f o r t h i s c a l c u l a t i o n f o r two r e a s o n s . One , the c o e f f i c i e n t s h^ and R^  c an be computed d i r e c t l y by a s i m p l e a l g o -r i t h m and two , t he s e n s i t i v i t y c o e f f i c i e n t s II, and F, , as a q k q k r e s u l t , c a n be e a s i l y d e r i v e d f r o m the a l g o r i t h m e q u a t i o n s . T h i s a l g o r i t h m f o r c a l c u l a t i n g hu and R^  i s g i v e n by the f o l l o w i n g s e t o f e q u a t i o n s A± = A h 1 = t r A 1 • R x = A x - 1^1 A 2 = A R X h 2 = t r A 2 / 2 R 2 = A 2 - h 2 I • • • A. = AR. , h . = t r A . / i R. = A . - h . I ( 5 . 3 7 ) l . i - l l . i l . i l • ' • • . * . • • • . A = AR h = t r A /n R = A - h I n n n n n n n The c a l c u l a t i o n c a n be c h e c k e d f o r a c c u r a c y by the l a s t i d e n t i t y R = 0 ( 5 . 3 8 ) n 4 These e q u a t i o n s r e q u i r e n m u l t i p l i c a t i o n s when programmed on a d i g i t a l c o m p u t e r b u t t h i s c a n be r e d u c e d c o n s i d e r a b l y i f t he s p a r s i t y o f A i s used t o a d v a n t a g e . In t he m a t r i x p r o d u c t s o f ( 5 . 3 7 ) A R i , i = 1 . . . n ( 5 . 3 9 ) 2 i f o n l y p n o n - z e r o e l e m e n t s i n s t e a d o f a l l n e l e m e n t s o f A a r e 2 m u l t i p l i e d , the number o f m u l t i p l i c a t i o n s may be r e d u c e d t o p-n . A method p r o p o s e d to s i n g l e ou t the n o n - z e r o e l e m e n t s o f A i s t o r e c o r d a l l t h e s e . e l e m e n t s i n a o n e - d i m e n s i o n a l a r r a y and t h e i r row and co lumn numbers i n a n o t h e r two o n e - d i m e n s i o n a l a r r a y s . W i th n o n - z e r o e l e m e n t s o f A t a k e n one a t a t i m e , the p r o d u c t A«R^ f o r e a c h e l e m e n t becomes a m a t r i x w i t h o n l y one row f i l l e d . T h u s , 45 the c o m p l e t e m a t r i x p r o d u c t , ( 5 . 3 9 ) , can be f o u n d as a sum o f t h e s e one-row m a t r i c e s . Once R^  and h^ a r e d e t e r m i n e d f rom ( 5 . 3 7 ) , F can be c a l c u l a t e d f r o m ( 5 . 1 2 ) and ( 5 . 1 7 ) . To compute t he e l e m e n t s o f II, , M o r g a n ' s method i s used q k where t he e l e m e n t s h . , a r e f o u n d by 1 q k h, , = I * A , 1 q k : q k 1 q k R i - 1 * A » q k ( 5 . 4 0 ) n ' q k = R , * A , n- l » q k w h e r e ( * ) i n d i c a t e s t h a t the i n n e r p r o d u c t i s t o be t a k e n , name l y n w.. l k ( 5 . 4 1 ) i k The s o l u t i o n o f F, i s d e r i v e d f r o m ( 5 . 3 7 ) as f o l l o w s . q k T a k i n g the p a r t i a l d e r i v a t i v e o f ( 5 . 1 2 ) w i t h r e s p e c t t o q k , one has = R E + R . • E) n-i 1 q k n - 1 , q k n ~ X q k From the p a r t i a l d e r i v a t i v e o f R . and A . w i t h r e s p e c t t o q, x n- i n- i K ( 5 . 4 2 ) R n - i ' q k n - i ' q k - h = A . • R q k + A .R . I ' » " n - i - 1 ' " ' " n - i - l » - h • , q k n - i ' q k ( 5 . 4 3 ) . I S u b s t i t u t i n g ( 5 . 4 3 ) i n t o ( 5 . 4 2 ) and u s i n g the e q u a t i o n ( 5 . 1 2 ) F, becomes q k F, where A , q k L f r-\ • • • ^ • • • • ^ + C i • • C • • • 1 1 1 c n e. = A . e . . l l + l n-i q k e , = A , n- l ' i Lk b - h-, , . b 1 q k e . l e . ol n-l . J ( 5 . 4 4 ) ( 5 . 4 5 ) and c . l R . • E , n-i L k ( 5 . 4 6 ) As i n the c o m p u t a t i o n o f R^  and h^ where the s p a r s i t y o f A 46 was u sed to a d v a n t a g e , t he c o m p u t a t i o n o f F, and H , Can be q k q k e c o n o m i z e d more so s i n c e A , i s u s u a l l y e x t r e m e l y s p a r s e . q k 5.3 A N u m e r i c a l Examp le To i l l u s t r a t e the s i m u l t a n e o u s s e n s i t i v i t y m e t h o d , t he o p t i m a l c o n t r o l d e s i g n o f C h a p t e r 4 i s c h o s e n as an e x a m p l e . The s y s t e m e q u a t i o n i s w r i t t e n as y = A y + E u ( 5 . 1 ) where A i s f o u n d f rom the n u m e r i c a l r e s u l t s o f C h a p t e r 4 as A = ' ( 5 . 4 7 ) - . 2 0 4 1 - 3 0 . 0 7 - 4 . 3 0 0 3 8 . 4 7 5 7 . 7 0 1 .000 - . 5 1 4 2 - . 1 8 6 5 1 .000 - 3 3 3 . 3 3 3 3 . 3 . 0534 4 2 . 4 7 - 6 4 . 4 3 - . 0 0 1 6 - 2 0 . 0 5 . 8303 . 8934 - . 2 3 8 1 .1106 - 2 . 0 0 0 2 . 0 0 0 4 . 0 0 0 - 1 . 0 5 3 - 4 . 0 0 0 - . 1 3 2 6 - 1 . 0 0 0 - 5 0 . 0 0 - . 7 0 9 1 1 7 . 5 2 2 . 1 5 7 .0065 . 1 1 0 6 - 3 4 . 5 8 - 3 3 . 2 3 1 4 . 8 0 - 1 0 7 . 2 u and E a r e g i v e n by ( 5 . 2 ) where t he f i r s t e l e m e n t o f E i s c h o s e n t o be d e p e n d e n t on the i n e r t i a c o n s t a n t i n t h e manner b± = 2 = 1 9 . 6 / 2 H ( 5 . 4 8 ) Hence b 1 > H = - 1 9 . 6 / 2 H 2 = - l / H ( 5 . 4 9 ) I t i s d e s i r e d t o i n v e s t i g a t e the s e n s i t i v i t y o f the t o r q u e a n g l e r e s p o n s e , & , t o e i g h t e e n c o n t r o l l e r g a i n s g i v e n by ( 4 . 5 0 ) , t w e n t y s y s t e m p a r a m e t e r s g i v e n by T a b l e 2.1 i n a d d i t i o n toTJ c -^ = .01 i n t r o d u c e d i n C h a p t e r 4 and t h r e e i n i t i a l o p e r a t i n g c o n d i t i o n s o f ( 2 . 5 2 ) . Note t h a t the c o n t r o l l e r g a i n s f o r tf c h o s e n as a u n i t m a t r i x and B g i v e n by ( 4 . 4 2 ) s i m p l y become the 5^^ and 9 ^ rows o f the 47 R i c a t t i m a t r i x . The m a t r i c e s F, F, and H, c a n be c a l c u l a t e d f o r a l l q k q k p a r a m e t e r s by t he a l g o r i t h m o f s e c t i o n 5.2 b u t o n l y t he s e c o n d rows o f F, and ( F * H , ) i n ( 5 . 2 9 ) a r e r e q u i r e d s i n c e i t i s c h o s e n i n q k q k t h i s example t o i n v e s t i g a t e t he s e n s i t i v i t y o f t he s e c o n d s t a t e v a r i a b l e h o n l y . The t ime r e s p o n s e s o f z ( t ) and w ( t ) a r e t h e n c a l c u l a t e d f r o m the c a n o n i c a l f o rms ( 5 . 3 3 ) and ( 5 . 3 4 ) . A 4- th o r d e r R u n g e - K u t t a i n t e g r a t i o n r o u t i n e i s u s e d . F o r each v a l u e o f t i m e , t he d e s i r e d s e c o n d e l e m e n t s o f y ( t ) , and y ( t ) , t h a t i s S, and % , a r e t h e n e v a l u a t e d by ( 5 . 2 9 ) . The r e s u l t s o f t h i s q k c a l c u l a t i o n a r e shown i n F i g u r e s 5.1 and 5 . 2 . The f i r s t f i g u r e shows the s e n s i t i v i t y r e s p o n s e f o r t he most s e n s i t i v e s y s t e m q k p a r a m e t e r s and i n i t i a l o p e r a t i n g c o n d i t i o n s and the s e c o n d f o r t he most s e n s i t i v e c o n t r o l l e r g a i n s . S e n s i t i v i t y r e s p o n s e s i n t h e s e f i g u r e s a r e t he s e n s i t i v i t y t o t he p e r c e n t a g e change i n p a r a m e t e r s g i v e n by ^ u / ( 0 q R . 1 0 0 . / q k ) = . q k / l 0 0 . ( 5 . 5 0 ) ^k The most s e n s i t i v e i n i t i a l o p e r a t i n g c o n d i t i o n i s , s y s t e m p a r a m e t e r s a r e X and x^, and t h e g a i n s a r e K ( 9 , 2 ) , K ( 9 , 6 ) , K ( 9 ; ? ) and K ( 9 , 8 ) . The o t h e r s a r e f o u n d t o be a t l e a s t t e n t i m e s l e s s sens i t i v e . F o r c o n v e n i e n c e , two c o m p u t e r p rog r ams a r e w r i t t e n , one f o r t he c a l c u l a t i o n o f s e n s i t i v i t y t o c o n t r o l l e r g a i n s and the o t h e r f o r t h a t t o s y s t e m p a r a m e t e r s and i n i t i a l o p e r a t i n g c o n d i t i o n s . B e c a u s e o f an e x c i t e r t ime c o n s t a n t o f . 003 s e c o n d s , a t i m e i n t e r v a l o f . 00225 s e c o n d s i s r e q u i r e d i n t he 4- th o r d e r R u n g e - K u t t a i n t e g r a t i o n r o u t i n e . T h e r e f o r e t o i n t e g r a t e up t o n i n e s e c o n d s o f r e a l t ime r e q u i r e s s i x t e e n t h o u s a n d e v a l u a t i o n s o f z ( t ) and w ( t ) . I t t a k e s t h r e e m i n u t e s and twen t y e i g h t s e c o n d s f o r an IBM 7044 com-50 p u t e r t o compute the t i m e r e s p o n s e s e n s i t i v i t y t o e i g h t e e n c o n -t r o l l e r g a i n s . Bu t i t i s f o u n d t h a t i n d e t e r m i n i n g the s e n s i t i v i t y t o one g a i n , a c a l c u l a t i o n t ime o f f i v e m i n u t e s and t h i r t e e n s e c o n d s i s r e q u i r e d u s i n g the o l d method d e s c r i b e d by ( 5 . 2 ) . T h i s i s more t h a n the t i m e r e q u i r e d f o r the c a l c u l a t i o n o f s e n s i t i v i t y t o e i g h t e e n p a r a m e t e r s by the new m e t h o d . Thus f o r e i g h t e e n p a -r a m e t e r s t he s a v i n g i n c a l c u l a t i o n t ime i s a p p r o x i m a t e l y one h o u r and f o r t y m i n u t e s . The r e s u l t s o f t h i s s e n s i t i v i t y c a l c u l a t i o n s u g g e s t a d e s i g n o f a new s u b o p t i m a l c o n t r o l l e r . By s i m p l y n e g l e c t i n g t he i n s e n s i -t i v e ga. ins the v o l t a g e r e g u l a t o r - e x c i t e r c o n t r o l s i g n a l c an be e l i m i n a t e d and the g o v e r n o r c o n t r o l can be s y n t h e s i z e d f r o m o n l y f o u r s t a t e v a r i a b l e s , name l y S , g , h and a . A c o m p a r i s o n i s made o f t h e S - r e s p o n s e f o r t h r e e d i f f e r e n t c a s e s , name l y c a s e 1 - O r i g i n a l s y s t e m w i t h o u t c o n t r o l c a s e 2 - S y s t em w i t h an o p t i m a l c o n t r o l , C h a p t e r 4 c a s e 3 - S y s t em w i t h a s u b o p t i m a l c o n t r o l The r e s u l t s a r e shown i n F i g u r e s 5.3 t h r o u g h 5.11 f o r a v a r i a t i o n i n i n i t i a l o p e r a t i n g c o n d i t i o n s o f P q = .3 t o . 9 , Q Q = .6 t o - . 6 , v ^ o = . 9 t o 1 . 2 . T h e s e f i g u r e s i l l u s t r a t e the i m -p r o v e m e n t i n s y s t e m r e s p o n s e o v e r t h a t w i t h o u t an 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 . A l s o the r e s p o n s e s o f b o t h t h e s u b o p t i m a l and o p t i m a l c o n t r o l l e r a r e v e r y good o v e r a w ide r ange o f i n i t i a l o p e r a t i n g c o n d i t i o n s and d i f f e r o n l y a t t he ex t reme edges o f t h e s e r a n g e s . Thus the s u b o p t i m a l c o n t r o l c an be r e g a r d e d as t he b e s t d e s i g n f rom an e n g i n e e r i n g v i e w p o i n t s i n c e i t i s e a s i e r and p o s s i b l y l e s s c o s t l y t o i m p l e m e n t . T I M E - S E C . F i g u r e 5.3 - 8-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f P Q 1.0 2. TIME-SEC. F i g u r e 5.4 - S-Response -with O p t i m a l C o n t r o l , Over V a r i a t i o n o f P Q ure 5.5 - ^-Response w i t h S u b o p t i m a l C o n t r o l , Ove r V a r i a t i o n o f P 0° TJME-5EC. u r e 5.6 - ^-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f QQ TIME-SEC, F i g u r e 5.8 - S-ftesponse w i t h S u b o p t i m a l C o n t r o l , Over V a r i a t i o n of Q TIME-SEC. F i g u r e 5.9 - S-Response W i t h o u t C o n t r o l , Ove r V a r i a t i o n o f v TIME-SEC. F i g u r e 5 .10 - S-Rcsjionse w i t h O p t i m a l C o n t r o l , Over V a r i a t i o n o f v TIME-SEC., F i g u r e 5.11 - S-Response w i t h S u b o p t i m a l C o n t r o l , Ove r V a r i a t i o n o f v 56 6. C O N C L U S I O N S S y s t e m a t i c a n d e f f i c i e n t m e t h o d s f o r t h e s t a b i l i z a t i o n a n d o p t i m i z a t i o n o f h i g h o r d e r l i n e a r s y s t e m s h a v e b e e n d e v e l o p e d , i n t h i s t h e s i s . T h e s e m e t h o d s h a v e b e e n a p p l i e d t o a 9^X o r d e r p o w e r s y s t e m . A g e n e r a l p r o c e d u r e f o r s t a b i l i z a t i o n i s d e r i v e d i n C h a p t e r 2 f r o m e i g e n v a l u e a n d e i g e n s y s t e m s e n s i t i v i t y a n a l y s i s . T h i s m e t h o d r e q u i r e s o n l y o n e e i g e n s y s t e m c a l c u l a t i o n f r o m t h e s y s t e m m a t r i x e x c e p t f o r t h e c a s e o f c l o s e e i g e n v a l u e s . A f t e r a n e i g e n v a l u e s h i f t , t h e n e w e i g e n s y s t e m i s c o m p u t e d u s i n g t h e e i g e n -s y s t e m s e n s i t i v i t y . T h u s t h e r e p e t i t i v e c a l c u l a t i o n o f e i g e n s y s t e m s f r o m t h e s y s t e m m a t r i x i s a v o i d e d . C o l l a r a n d J a h n ' s c o r r e c t i o n m e t h o d i s e x t e n d e d a n d i s a p p l i e d w h e n t h e a c c u r a c y o f t h e n e w e i g e n s y s t e m i s s u s p e c t . T h i s c o r r e c t i o n i s n o t n e c e s s a r y f o r e v e r y s h i f t s i n c e t h e a c c u r a c y o f t h e e i g e n s y s t e m f o u n d f r o m t h e s e n s i -t i v i t y r e l a t i o n s i s u s u a l l y q u i t e g o o d . H o w e v e r , i n t h e c a s e o f c l o s e e i g e n v a l u e s , r e p e a t e d a p p l i c a t i o n o f t h e c o r r e c t i o n m i g h t n o t c o n v e r g e , a n d t h e n a n e w e i g e n s y s t e m c a l c u l a t i o n , d i r e c t l y f r o m t h e s y s t e m m a t r i x , w i l l b e r e q u i r e d . T h i s s t a b i l i z a t i o n m e t h o d a l s o t a k e s i n t o a c c o u n t t h e r e s t r i c t i o n s o n p a r a m e t e r a d -j u s t m e n t s . W h e n t h e m e t h o d i s a p p l i e d t o a n u n s t a b l e 9 ^ o r d e r p o w e r s y s t e m , i t i s f o u n d t h a t t h e s y s t e m c a n b e s t a b i l i z e d i f t h e g a i n i n t h e s p e e d f e e d b a c k l o o p o f t h e v o l t a g e r e g u l a t o r - e x c i t e r i s m a d e n e g a t i v e . I t i s a l s o f o u n d t h a t f u r t h e r i m p r o v e m e n t s i n t h e d e g r e e o f s t a b i l i t y c a n b e o b t a i n e d b y d e c r e a s i n g t h e t i m e c o n s t a n t o f t h e v o l t a g e r e g u l a t o r , a n d . t h e t r a n s i e n t d r o o p o f t h e g o v e r n o r . T h e m e t h o d d e v e l o p e d i n C h a p t e r 3 a d j u s t s t h e p a r a m e t e r s o f a l i n e a r i z e d p o w e r s y s t e m i n s u c h a w a y a s t o m i n i m i z e a t i m e w e i g h t e d q u a d r a t i c p e r f o r m a n c e i n d e x . W i t h t h e e i g e n s y s t e m f o r m u l a t i o n , 57 t he need f o r m a t r i x i n v e r s i o n i s e l i m i n a t e d . As i n the s t a b i l i z a t i o n m e t h o d , o n l y one i n i t i a l e i g e n s y s t e m c a l c u l a t i o n f rom a s y s t e m m a t r i x i s r e q u i r e d e x c e p t f o r t he c a se o f c l o s e e i g e n v a l u e s . The e x t e n d e d C o l l a r and J a h n c o r r e c t i o n method a l s o may be a p p l i e d i f n e c e s s a r y . F l e t c h e r and P o w e l l ' s o p t i m i z a t i o n p r o c e d u r e i s u s e d t o o b t a i n the c o s t f u n c t i o n min imum. To a p p l y t h i s method to a s y s t e m w i t h p a r a m e t e r c o n s t r a i n t s , B o x ' s t r a n s f o r m a t i o n i s u sed t o t r a n s f o r m the c o n s t r a i n e d p a r a m e t e r s t o u n c o n s t r a i n e d f o r m . I t i s f o u n d f rom t h e 9 ^ o r d e r example t h a t a h i g h v o l t a g e r e g u l a t o r g a i n o f the power s y s t e m i s d e t r i m e n t a l t o a good s y s t e m r e s p o n s e . A t e c h n i q u e u s i n g P u r i and G r u v e r ' s s u c c e s s i v e a p p r o x i m a t i o n method i s d e v e l o p e d i n C h a p t e r 4 t o c a l c u l a t e an o p t i m a l c o n t r o l l e r f o r a l a r g e s y s t e m . A f a s t r e c u r s i v e method i s u sed t o s o l v e f o r t he s u c c e s s i v e a p p r o x i m a t i o n s o f t he R i c a t t i m a t r i x . The s t a b i l i -z a t i o n method o f C h a p t e r 2 c a n be u s e d to i n i t i a t e t he o p t i m i z a t i o n p r o c e s s i f t he s y s t e m i s i n i t i a l l y u n s t a b l e . The r e c u r s i v e m e t h o d , a l t h o u g h a p p r o x i m a t e , c a n be made as a c c u r a t e as d e s i r e d by c h o o s i n g a l a r g e number o f d i s c r e t e t i m e i n t e r v a l s f o r the i n t e g r a l s o l u t i o n o f t he s u c c e s s i v e a p p r o x i m a t i o n s . The c a l c u l a t i o n t i m e i s r e l a -t i v e l y i n d e p e n d a n t o f t he number o f i n t e r v a l s c h o s e n . In the n u m e r i c a l example o f t he power s y s t e m , c o n t r o l s i g n a l s a r e i n t r o -d u c e d i n t o the g o v e r n o r and v o l t a g e r e g u l a t o r - e x c i t e r . In t h i s e x a m p l e , the s t a b i l i z a t i o n method i s n o t r e q u i r e d to i n i t i a t e the o p t i m a l c o n t r o l c a l c u l a t i o n . The s y s t e m i s f o u n d s t a b l e w i t h a n u l l f i r s t a p p r o x i m a t i o n o f t he R i c a t t i m a t r i x . The o p t i m a l c o n -t r o l l e r i s t h e n c a l c u l a t e d f o r d i f f e r e n t w e i g h t i n g s a p p l i e d t o the s t a t e v a r i a b l e s i n the c o s t f u n c t i o n a l . I t has b e e n f o u n d t h a t an i n c r e a s e d w e i g h t i n g o f speed and t o r q u e a n g l e r e s u l t e d i n an im-p r o v e d damping , c h a r a c t e r i s t i c o f the r e s p o n s e . Of c o u s e , t h i s i s done a t the expense o f i n c r e a s i n g the m a g n i t u d e o f c o n t r o l . In C h a p t e r 5, a s i m u l t a n e o u s s e n s i t i v i t y method i s d e v e l o p e d f o r m u l t i v a r i a b l e s y s t e m s . T h i s method a l l o w s the c a l c u l a t i o n o f t ime r e s p o n s e s e n s i t i v i t y t o a l l p a r a m e t e r s f r o m o n l y two d i f f e r -t h e n t i a l e q u a t i o n i n t e g r a t i o n s . The method i s a p p l i e d to t he 9 o r d e r o p t i m a l power s y s t e m d e s i g n o f C h a p t e r 4. I t i s f o u n d t h a t t he s y s t e m r e s p o n s e i s most s e n s i t i v e t o t e r m i n a l v o l t a g e , l i n e r e a c t a n c e and d i r e c t - a x i s t r a n s i e n t r e a c t a n c e o f t he s y n c h r o n o u s machine b u t n o t t o most o f the c o n t r o l l e r g a i n s . By n e g l e c t i n g t he i n s e n s i t i v e c o n t r o l l e r g a i n s , a s u b o p t i m a l c o n t r o l l e r i s f o u n d w h i c h r e q u i r e s o n l y f o u r s t a t e v a r i a b l e s . A l s o , i t i s f o u n d t h a t the g a i n s f o r t he v o l t a g e r e g u l a t o r - e x c i t e r c o n t r o l a r e v e r y i n -s e n s i t i v e and t h i s c o n t r o l s i g n a l c an be o m i t t e d a l t o g e t h e r . The s y s t e m r e s p o n s e w i t h t h i s s u b o p t i m a l c o n t r o l l e r i s v e r y c l o s e t o t h a t w i t h the o p t i m a l c o n t r o l o f C h a p t e r 4 o v e r a wide r ange o f P •, Q and v , . K o t o I t i s hoped t h a t t h e s e methods p r e s e n t e d i n t h i s t h e s i s w i l l be a p p l i e d , and t h e r e s u l t s t e s t e d on a c t u a l power s y s t e m s . The a p p l i c a t i o n o f t h e s e methods c a n be r e a d i l y e x t e n d e d t o m u l t i m a c h i n e s y s t e m s . T h e r e a l s o i s much r e s e a r c h t o be done i n t he a r e a o f n o n l i n e a r c o n t r o l o f a power s y s t e m and the c o o r d i n a t i o n o f the n o n l i n e a r c o n t r o l w i t h t he l i n e a r i z e d o r s t e a d y s t a t e c o n t r o l . 59 APPENDIX A F o r m a t i o n o f tho C o e f f i c i e n t M a t r i x A o f E q u a t i o n 2.1 A .1 T h i r d O r d e r M a c h i n e and T i e L i n e E q u a t i o n s (18) P a r k ' s e q u a t i o n s may be w r i t t e n as V d = P ^ d " k a 4 d - ^ q W ( A > l a ) V q = " R a V V d W ( A ' l b ) CJ(^ d = G(p) v f d - x d ( p ) i d ( A . l c ) <*^q = " X q ( p ) \ U , l d ) where G(p) = f a d ^ 1 + T k d p ^ (A.le) Rf u+xd;p)(i+rd;p) x , ( p ) = x , (l+tdp)(l+tgp) ( A a f ) " ^ ^ d o P H i ^ P ) x (p) - x ( 1 + q P > q q (i+r^p) ( A . i g ) and CO = CJ Q + ( A . l h ) F o r a power s y s t e m s t u d y , t he f o l l o w i n g a s s u m p t i o n s a r e u s u a l -l y made . 1. The i n d u c e d v o l t a g e s p i ^ d and Pty and the v o l t a g e s due t o t he s p e e d d e v i a t i o n s dtiiy^ and AtJifi a r e n e g l e c t e d s i n c e t h e y a r e s m a l l compared t o t he s p e e d v o l t a g e s ^ q and ^>o t ^ . 2. A r m a t u r e r e s i s t a n c e i s n e g l e c t e d s i n c e i t i s s m a l l com-p a r e d t o t he a r m a t u r e r e a c t a n c e . 3. S u b t r a n s i e n t e f f e c t s a r e n e g l e c t e d s i n c e t h e i r d u r a t i o n i s v e r y s h o r t i n r e l a t i o n t o t he o v e r a l l r e s p o n s e . W i t h t he above a s s u m p t i o n s , P a r k ' s e q u a t i o n s r e d u c e t o V d = " V q ^ o ( A - 2 a ) v q = ^ u ) 0 (A . 2b) v - ( 1 + t ' p ) X i 6 0 r.\ (./ = ~x i (A . 2d) r q q q E q u a t i o n ( A . 2 c ) may be w r i t t e n as ^ o Vd = V F R ~ x d *d ( A ' 3 ) where v ™ , = V F + ^ d o ( x d " X d } p i d (A .4) w h i c h i n t u r n may be w r i t t e n as Plfip = v p - v p R . ( A . 5) where ^ = t d o ( v F R - ( x d - x ' ) i d ) ( A .6) t h e f i e l d f l u x l i n k a g e fr/fd = -L-VF <A-7> X a d t h e f i e l d c u r r e n t v x i f d = I I I ( A . 8 ) ad and t h e f i e l d v o l t a g e v f d = v p (A.9) a d E q u a t i o n ( A . 5 ) t o g e t h e r w i t h t he g e n e r a l t o r q u e e q u a t i o n p(ku>) = ( l / 2 H ) ( 0 ) T . - CO T - D ( ^O J ) ) ( A . 1 0 a ) O X 0 6 pS = A C O ( A . 1 0 b ) d e s c r i b e a t h i r d o r d e r mach ine mode l where t he e l e c t r i c a l t o r q u e i s g i v e n b y T e = % \ ~ K A d U ( A - 1 1 ) w i t h the base s p e e d c h o s e n as 1 r a d i a n / s e c o n d . The p r i m e mover t o r q u e i s a p p r o x i m a t e d by T t = g •+ 1.5 h ( A . 1 2 ) To r e p r e s e n t t he t h i r d o r d e r mode l w i t h s t a t e v a r i a b l e s c h o s e n as Au)> lfJ-p> v p > . g a n c l n > v a r i a b l e s ^/ , i d and i^ need t o be e l i m i n a t e d . The t i e l i n e e q u a t i o n , f r o m the t e r m i n a l c o n -d i t i o n s , may be w r i t t e n as i „ = (C+ jB ) v . + ( v t " v o ) T TR+IXT 61 ( A . 1 3 a ) o r i n d-q c o o r d i n a t e s as R - X X R 1 U q j R - X X R G -B B G 0 v L _ q . v s i n d o v c o s ^ _ o _ ( A . 1 3 b ) C o l l e c t i n g ( A . 2 d ) , ( A . 3 ) and (A.6) - t d o < X d - X c P tdo Vd =. 0 wo / xd 0 0 -Xq^o ( A . 1 4 ) and i n v e r t i n g t h i s e q u a t i o n x /XdTdo 0 From ( A . 2 a ) and ( A . 2 b ) v F R i _ q _ H ( x d " x d ) / x d -^0/xd o o 0 V X q Kl Vn ( A . 1 5 ) v d 0 - l ^d V _q_ = ^ o l 0 Vq ( A . 1 6 ) S u b s t i t u t i n g the l a s t two rows o f ( A . 1 5 ) and a l l o f ( A . 1 6 ) i n t o ( A . 1 3 b ) , t he t i e l i n e e q u a t i o n becomes R - X X R - 1 / x d 0 1/x 0 R - X X R q G -B B G vd + R - X X R + 1 0 0 1 ' do d 0 Vi ( A . 1 7 ) 0 -1 1 0 w h e r e , upon r e a r r a n g i n g and l i n e a r i z i n g U). ~K1 ~ K 2 ^ d v cosS„ o 0 -v sin§ o 0 _ K 3 \ V^q -do d Vd R X v sinS o v cos o o ( A . 1 8 ) where K x = ( R / x d RB XG) = (1 + RG - XB + X/x ) K 3 = (1 + RG - XB +. X / x d ) K. = (R/x - RB - XG) 4 q (A .19) The s o l u t i o n s o f A £ / d and Aty a r e * V d = l/< CO, A> F ( A . 2 0 ) where ( A . 2 1 ) b l = Y o ( K 4 C O S ^ o " K 2 s i n S o ) / ( K 1 K 4 + K ^ ) b 2 = - V o ( K 3 C O S ^ o ~ K 1 s i n S ( J ) / (K 1 K 4 + K ^ ) c l = < K 4 R + K 2 X > / r d o X d ( K l K 4 + K 2 K 3 ) c 2 = ( K ] X - K 3 R ) / t d ; x d ( K l K 4 • K 2 K 3 ) The f i rs t third order model equation can now be written in state variable form by substituting ( A . 2 0 ) into the linearized f irst equation of ( A . 1 5 ) and the result into the linearized ( A .5 ) p (M^ , ) = A v p + 1 A d ( c 1 ( x d - x d ) - x d A : d o ) ^ F + (x d -x d )b 1 ( A . 2 2 a ) • Ab X d The other equations can be obtained by linearizing ( A . l O a ) and ( A . 1 0 b ) with aubstitution of ( A . 1 2 ) into ( A . l O a ) P ( A U J ) = <£o eg + . 75 Wo A h - Wo^ Te - A U> 2H H 2H P(A£>) = A.co ) where A. T = G ) A i - U A i , + A ( J , i - A(J i , e o " do q r q o d r d qo ^q do ' w h i c h a f t e r s u b s t i t u t i o n o f ( A . 1 5 ) and ( A . 2 0 ) becomes ( A . 2 2 b ) ( A . 2 2 c ) ( A . 2 3 ) & T = - ( i ' b , + i , 'b„)AS - ( i ' c , + i ' c 0 - v , / , w , /,->.< i e qo 1 do 2' qo 1 do 2 d o / x d 7 J d o )/yp( A . 24) where qo do = ( V d o - i ) ( A . 2 5 ) iAr\ = (lao + i d 0 ) x A . 2 G o v e r n o r and H y d r a u l i c E q u a t i o n s The g o v e r n o r and h y d r a u l i c t r a n s f e r f u n c t i o n i s g i v e n by ( 1 9 ) A T, ( l + t r p ) ( l - T w p ) A M coQ(o-+rap + {$t+o)vrT? + t a r r p 2 ) ( i + r g p ) ( i + . 5 r v p ) The b l o c k d i a g r a m i s g i v e n i n F i g u r e A . l . From the t r a n s f e r 63 ( A . 2 6 ) 1 A a 1 (1 - T p) A T t (1 + T g p ) (1 + . 51 p) S t V (1 + T r p ) F i g u r e A . l G o v e r n o r T r a n s f e r F u n c t i o n and H y d r a u l i c O p e r a t o r f u n c t i o n ( A . 2 6 ) and the l i n e a r i z e d t o r q u e e q u a t i o n ( A . 1 2 ) , t he s t a t e v a r i a b l e f o r m o f t he g o v e r n o r and h y d r a u l i c o p e r a t o r mode l becomes p(*a) _ ACO ^ a f _ cr(Aa) r a a p ( A a f ) = ^o ^ a (7 k + ~ ) ( ^ f ) -C a <- r crSt (Aa) p ( & g ) = &a Ag ( A . 2 7 ) p ( M i ) = _ 2(ha.) 2 (A^) _ 2 (Ah) fg T g T v A . 3 V o l t a g e R e g u l a t o r - E x c i t e r E q u a t i o n s The v o l t a g e r e g u l a t o r - e x c i t e r t r a n s f e r f u n c t i o n s a r e g i v e n by ^2 to, 1 ( i + r l P ) ( i + r e x p ) k l k 2 ( A . 2 8 a ) ( A . 2 8 b ) &co ~ ( i + r 1 p ) ( n - r e x p ) The b l o c k d i a g r a m i s shown i n F i g u r e A . 2 . E q u a t i o n s ( A . 2 8 a ) and ( A . 2 8 b ) c an be w r i t t e n i n s t a t e v a r i a b l e f o r m as 64 \ k l s 1 A.v p / ( l + T l P ) ( 1 + tex*> >-F i g u r e A . 2 V o l t a g e R e g u l a t o r - E x c i t e r T r a n s f e r F u n c t i o n k,k. p ( * v p ) = A v T r r where 2 2 = v, + v d q w h i c h when l i n e a r i z e d becomes A v . v , d v , t o t o t x _ * ' V j . _ ' q ' S u b s t i t u t i n g ( A . 1 6 ) and ( A . 2 0 ) i n t o ( A . 3 1 ) ( A . 2 9 ) ( A . 3 0 ) ( A . 3 1 ) A v , V b l v , a2>^6 . V , c l v . C 2 ; A ( ^ F t o t o t o t o ( A . 3 2 ) A . 4 Summary o f S y s t em E q u a t i o n s C o m b i n i n g t h e t h i r d o r d e r mach ine and t i e l i n e e q u a t i o n s ( A . 2 2 a , b , c ) , t he f o u r t h o r d e r g o v e r n o r and h y d r a u l i c o p e r a t o r e q u a t i o n s ( A . 2 7 ) and the s e c o n d o r d e r v o l t a g e r e g u l a t o r - e x c i t e r e q u a t i o n s ( A . 2 9 ) , t he n i n t h o r d e r power s y s t e m e q u a t i o n s can be w r i t t e n i n s t a t e v a r i a b l e f o r m as y = A y ( A . 3 3 ) — T where y = p» lfJp, v p , v g , g , h , a , af~j (A . 34) w i t h t he n o t a t i o n ' ^ ' o m i t t e d f o r s i m p l i c i t y . The c o e f f i c i e n t m a t r i x A i s g i v e n by A = ( A . 3 5 ) _ D 2H k l k 2 '12 L 32 '52 "13 33 a 53 ' ex "ex _1_ ?1 <£o 2H 1.5 GO o 2II -g g •w t r o L a ^ r a a St _1 w h e r e l12 "13 co ( b , i o 1 + b 0 i ' ) / 2 H a 32 '33 L52 "53 l^qo ' "2 do ' U ( c n i ' + c 0 i ' - l / r •x•)/2H o 1 qo 2 do do d ( x d - x d ) b 1 / x d I'l^iT*** - *k^do )/*k - k l ( v q o b l " V d o b 2 ) / v t o r i " k l ( v q o C l - V d o C 2 ) / v t o r i ( A . 3 6 ) The c o n s t a n t s b^, b 2 , c^ , c 2 a r e g i v e n by ( A . 2 1 a , b , c , d ) and i q 0 > i d o by ( A . 2 5 ) . 66 APPENDIX B I n i t i a l O p e r a t i n g C o n d i t i o n s o f a Power S y s t em V i t h t he a s s u m p t i o n s o f a p p e n d i x A n e g l e c t i n g P ^ j P^q» t^jJlfJ^) e>u)<^ > a r m a t u r e r e s i s t a n c e and s u b t r a n s i e n t e f f e c t s , the s t e a d y s t a t e f o r m o f ( A . 2 a , b , c , d ) , ( A . 6 ) , ( A . 1 1 ) , ( A . 1 3 b ) , and (6) ( A . 3 0 ) may be w r i t t e n as 2 2 , 2 V , = V , + V t o do qo v do " ~Vqo ^ o qo * a o o T = P = v i + v i eo o do do qo qo H> V d 0 = v F o ~ V d o ( B a ) i.) u = -x i "^o ^qo q qo ^ F o v d Fo ^do d d do v sinS = (1 - XB + R G ) v , - (RB + XG)v - R i , + X i o o do qo do qo v cos& = (XG + R B ) v , + (1 - XB + RG)v - X i , - R i o o do qo do qo and t h e r e a c t i v e power f l o w Q = v i , - v , i ( B .2 ) *o qo do do qo C h o o s i n g P , 0 and v , a l l o w s the d e t e r m i n a t i o n o f v , , v , v , ° o ' ^o t o d o ' q o ' o' V F o ' Hlo' Vqo' ^ F o ' \ l o ' \ o > & n d 8 o f r o m { B ' l ] a m l ( B ' 2 ) a S P v . o t o l q ° / ( P x ) 2 + (v . 2 + x p ) •j o q t o q ^ o ' 2 V , = X 1 do q qo i , - (Q • + x i • 2 ) /v do Ko q qo qo v „ = v + x . i , Fo qo d do V q 0 = ^do^o Hlo V q o / w o Wo = - r d i ( x d " X d > i d o + t d i v p o and from the last two equations of ( B . l ) v / - 2 ' - 2 o ' V , + V do qo £ = arctan ( v , / V ) °o s do' qo' where (B.3) (B .4) Y d o = (1 - XB + . R G ) v d o - ( R B + X G ) V q o - R i d o + X i q o v - = (XG + RB)v + (1 - XB + RG)v - X i , - R i qo do qo do qo APPENDIX C (14) F l e t c h e r and P o w e l l ' s ' D e s c e n t Method ' D i e a l g o r i t h m o f F l e t c h e r and P o w e l l ' s method, i s as f o l l o w s 1. Compute t h e c o s t f u n c t i o n a l g r a d i e n t w i t h r e s p e c t t o t he p a r a m e t e r v e c t o r p, t h a t i s - l T i -g = J>P - - l P=P r i T i T i i , p x . . . J , P j . . . J , p x T i ( C l ) p .=p . where i i s t he i t e r a t i o n number . 2. A d j u s t t h e p a r a m e t e r v e c t o r a l o n g a g r a d i e n t d i r e c t i o n s""" u n t i l a minimum J 1 i s f o u n d where s^ f o r the i^*1 i t e r a t i o n i s g i v e n by i 1 = - H 1 . g 1 ( C . 2 ) and H 1 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 u s u a l l y c h o s e n as a u n i t m a t r i x f o r the f i r s t i t e r a t i o n . 3. S e t 5-1 = k A s*1 ( C . 3 ) - i+1 - i , - i and p = p + cr where k* i s f o u n d s u c h tha.t p 1 + ^ i s th;* p a r a m e t e r v e c t o r w h i c h m i n i m i z e s J 1 a l o n g the g r a d i e n t s 1 . 4 . E v a l u a t e g* + ^ f r o m ( C l ) and compute H**^ f r o m H 1 + 1 = H 1 + D 1 + E 1 ( C . 4 ) where D 1 = [ ^ { ^ ^ [ - J W] <C'5> E 1 - -H i-[y ij-[? ijT- HVIFF ¥1 <°-6> and f = - f (C .7) 5. R e p e a t t he p r o c e d u r e s t a r t i n g w i t h s t e p 2 u n t i l e v e r y component o f s 1 i s l e s s t h a n a p r e s c r i b e d a c c u r a c y . The minimum o f J i s t h e n computed f rom p 1 o f t he l a s t i t e r a t i o n . 69 REFERENCES 1. L a u g h t o n , M. A . , " The Use o f S e n s i t i v i t y A n a l y s i s i n the D e s i g n o f G e n e r a t o r E x c i t a t i o n C o n t r o l " , P r o c e e d i n g s  o f the S e c o n d Power Sys tems C o m p u t a t i o n C o n f e r e n c e , V o l . 1, P a p e r 3 . 4 , S t o c k h o l m Sweden , 1966 . 2 . F r a n c i s , J . G . F . , "The QR T r a n s f o r m a t i o n " , P a r t I, The Compute r J o u r n a l , V o l . 4 , O c t o b e r 1 9 6 1 ; P a r t I I i b i d . , J a n u a r y 1 9 6 2 . 3. L a u g h t o n , M. A . , " S e n s i t i v i t y i n D y n a m i c a l S ys tem A n a l y s i s " , J o u r n a l o f E l e c t r o n i c s and C o n t r o l , V o l . 1 7 , No . 5, November 1964 . 4 . M a c f a r l a n e , A . G . J . , " A n E i g e n v e c t o r S o l u t i o n o f t he O p t i m a l L i n e a r R e g u l a t o r P r o b l e m " , i b i d . , V o l . 14 , No . 6 , June 1 9 6 3 . 5. F a d e e v , D. K . , and V . N. 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