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A dynamic test model for power system stability and control studies 1969

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A DYNAMIC TEST MODEL FOR POWER SYSTEM STABILITY AND CONTROL STUDIES by GRAHAM ELLIOTT DAWSON B.A. Sc., University of B r i t i s h Columbia, 1963 M.A. Sc., University of B r i t i s h Columbia, 1966 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Acting Head of the Department . . Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA December, 1969 In presenting t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree tha permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Depa rtment of Ele&f)rica,( En The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT A new model f o r power system t r a n s i e n t s t a b i l i t y t e s t s lia s been d e v e l o p e d . I t i n c l u d e s a dc motor s i m u l a t e d prime mover w i t h a governor c o n t r o l s y n t h e s i z e d by dc b o o s t e r g e n e r a t o r f i e l d c o n t r o l , a s o l i d s t a t e v o l t a g e r e g u l a t o r and e x c i t e r , a synchronous machine w i t h a l a r g e f i e l d t i m e con- s t a n t r e a l i z e d by n e g a t i v e r e s i s t a n c e i n t h e f i e l d c i r c u i t , a t r a n s m i s s i o n system w i t h time s e t t i n g SCR c o n t r o l l e d f a u l t and c l e a r sequence s w i t c h i n g s , an a c c u r a t e t o r q u e a n g l e de- v i a t i o n t r a n s d u c e r ( C h a p t e r 2 ) , and a n a l o g s to r e a l i z e con- v e n t i o n a l s t a b i l i z a t i o n and n o n l i n e a r o p t i m a l c o n t r o l ( C h a p t e r 5 ) . Three s t a t e v a r i a b l e m a t h e m a t i c a l models o f t h e t e s t model w i t h v a r i o u s degrees o f d e t a i l a r e derived, i n C h a p t e r 3 Comparisons o f r e s u l t s o f d i g i t a l c o m p u t a t i o n and r e a l model t e s t s o f a t j ' p i c a l power system d i s t u r b e d by a s h o r t c i r c u i t a r e g i v e n a l s o i n Chapter 3. A parameter s e n s i t i v i t y s t u d y i s c a r r i e d out i n C h a p t e r 4. Comparisons o f d i g i t a l computa- t i o n o f t r a n s i e n t s t a b i l i t y w i t h a n o n l i n e a r o p t i m a l c o n t r o l d e r i v e d i n t h i s t h e s i s and power and speed s t a b i l i z a t i o n d e r i v e d by a n o t h e r c o l l e a g u e o f t h e power group at U.B.C., w i t h the t r a n s i e n t s t a b i l i t y t e s t s on t h e t e s t model a r e g i v e n i n C h a p t e r 5. i i i TABLE OF CONTENTS Page ABSTRACT i i TABLE OF CONTENTS i i i L I S T OF TABLES v L I S T OF ILLUSTRATIONS v i ACKNOWLEDGEMENT . . v i i i NOMENCLATURE i x 1 . INTRODUCTION ' 1 2 . REALIZATION OF THE DYNAMIC TEST MODEL 4 2 . 1 . M o d e l i n g P r o c e d u r e 4 2 . 2 . A T y p i c a l Power S y s t e m 5 2 . 3 . S e t t i n g a T y p i c a l P o w e r S y s t e m on t h e T e s t M o d e l 6 2 . 3 . 1 D e t e r m i n i n g B a s e I m p e d e n c e , P o w e r and V o l t a g e 6 2 . 3 . 2 N u m e r i c a l V a l u e s o f S y s t e m a nd M o d e l G e n e r a t o r P a r a m e t e r s 8 2 . 3 . 3 N u m e r i c a l V a l u e s o f M o d e l B a s e Q u a n t i t i e s 9 2 . 3 . 4 M e a s u r e m e n t o f M o d e l G e n e r a t o r P a r a m e t e r s 10 2 . 4 . D e v e l o p m e n t o f Dynamic T e s t M o d e l f o r Power S y s t e m S i m u l a t i o n I I 2 . 4 . 1 G o v e r n o r - P r i m e M o v e r . . . . . . . n 2 . 4 . 2 R e g u l a t o r - E x c i t e r 14 2 . 4 . 3 T r a n s m i s s i o n L i n e and C i r c u i t B r e a k e r s . 2.8 2 . 5 . A u x i l i a r y M e a s u r i n g a n d C o n t r o l D e v i c e s . . 21 2 . 5 . 1 T i m e - S e q u e n c e C o n t r o l 21 2 . 5 . 2 T o r q u e A n g l e D e v i a t i o n M e a s u r e m e n t . . 23 2 . 5 . 3 F o r c e d E x c i t a t i o n C o n t r o l 2 5 i v Page 3. STATE EQUATION MODELS OF THE DYNAMIC MODEL • AND TEST RESULTS 28 3.1. S e v e n t h O r d e r S y n c h r o n o u s M a c h i n e S t a t e E q u a t i o n s 30 3.2. S t a t e E q u a t i o n s o f C o n t r o l l e r s and DC M o t o r - B o o s t e r 32 3.3. I n i t i a l S t a t e o f a Power S y s t e m 3^ 3.4. S t a t e E q u a t i o n s o f F i f t h O r d e r - S y n c h r o n o u s M a c h i n e and C o n t r o l l e r s 39 3.5. S t a t e E q u a t i o n s o f T h i r d O r d e r - S y n c h r o n o u s M a c h i n e and C o n t r o l l e r s 42 3.6. S t a t e E q u a t i o n s o f T h i r d O r d e r - S y n c h r o n o u s M a c h i n e and R e g u l a t o r - E x c i t e r bub W i t h o u t G o v e r n o r 45 3.7. C o m p u t a t i o n and T e s t R e s u l t s o f t h e Dynamic T e s t M o d e l . . 46 4. PARAMETER S E N S I T I V I T Y OF THE TEST MODEL . . . . . 58 4.1. S e n s i t i v i t y E q u a t i o n s 55 4.2. P a r a m e t e r S e n s i t i v i t y o f S y s t e m R e s p o n s e . . 55 5. NONLINEAR OPTIMAL STABILIZATION OF A POWER SYSTEM AND DYNAMIC MODEL TESTS . . . . . . . g 0 5.1. Power S y s t e m S t a b i l i z i n g S i g n a l . gC 5.2. Dynamic O p t i m i z a t i o n and C o m p u t a t i o n a l Method. go 5.3*. C o m p u t a t i o n and T e s t R e s u l t s g^ 6. CONCLUSION 10 7 APPENDIX 3A 109 APPENDIX 3B 114 REFERENCES 116 LIST OF TABLES 2.1 Model and A c t u a l S y s t e m G e n e r a t o r P a r a m e t e r V a l u e s g 5.1 P e r f o r m a n c e F u n c t i o n s Used D u r i n g T r a n s i e n t S t e p s 84 v i LIST OF ILLUSTRATIONS F i g u r e Page 2 . 1 Power System t o be Modeled 4 2 . 2 DC M o t o r - B o o s t e r U n i t 11 2 . 3 Model Governor-Prime Mover System 15 2 . 4 G o v e r n o r - H y d r a u l i c System T r a n s f e r F u n c t i o n . 15 2 . 5 B o o s t e r F i e l d Compensation and C u r r e n t A m p l i f i e r C i r c u i t • • 1.6 2 . 6 Model R e g u l a t o r - E x c i t e r System 16 2 . 7 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 • • • • 16 2 . 8 Scheme t o R e a l i z e N e g a t i v e R e s i s t a n c e . • • 17 2 . 9 One Phase Lumped Parameter E q u i v a l e n t o f a S i n g l e C i r c u i t 19 2 . 1 0 Model JT U n i t o f T r a n s m i s s i o n L i n e . . . . 20 2 . 1 1 One Phase o f t h e Model T r a n s m i s s i o n System . . 20 2 . 1 2 C i r c u i t B r e a k e r and F a u l t Time-Sequence C o n t r o l 22 6 2.13 Torque A n g l e D e v i a t i o n M e a s u r i n g D e v i c e . . . 24 2 .14 F o r c e d E x c i t a t i o n T i m e - S w i t c h i n g C o n t r o l . . 26 3 . 1 Dynamic T e s t Model o f One M a c h i n e - I n f i n i t e Bus System 2 9 3 . 2 Torque Angle T r a n s i e n t Responses 48 3 . 3 Governor A c t u a t o r P o s i t i o n T r a n s i e n t Responses. 4 9 3 . 4 Governor A c t u a t o r Feedback T r a n s i e n t Responses. 50 3 . 5 Governor Gate P o s i t i o n T r a n s i e n t Responses . . 5 1 3 . 6 T u r b i n e Torque T r a n s i e n t Responses . . . . 52 3 . 7 B o o s t e r Armature V o l t a g e T r a n s i e n t Responses . 53 3 . 8 T e r m i n a l V o l t a g e T r a n s i e n t Responses . . . . 54 3 . 9 R e g u l a t o r V o l t a g e T r a n s i e n t Responses . . . 5 5 v i i LIST OF ILLUSTRATIONS ( C o n t . ) F i g u r e Page 3.10 F i e l d V o l t a g e T r a n s i e n t R e s p o n s e s 56 3.11 F i e l d C u r r e n t T r a n s i e n t R e s p o n s e s 57 4.1 P a r a m e t e r S e n s i t i v i t } ' o f A i r Gap F l u x . . . 68 4.2 P a r a m e t e r S e n s i t i v i t y o f D - A x i s F l u x . . . . 69 4.3 P a r a m e t e r S e n s i t i v i t y o f Q - A x i s F l u x . . . . 70 4.4 P a r a m e t e r S e n s i t i v i t y o f A c t u a t o r P o s i t i o n . . 7 1 4.5 P a r a m e t e r S e n s i t i v i t y o f A c t u a t o r F e e d b a c k . • • 72 4.6 P a r a m e t e r S e n s i t i v i t y o f G a t e P o s i t i o n . . . 73 4.7 P a r a m e t e r S e n s i t i v i t y o f T u r b i n e T o r q u e . . . 74 4.8 P a r a m e t e r S e n s i t i v i t y o f B o o s t e r A r m a t u r e V o l t a g e 75 4.9 P a r a m e t e r S e n s i t i v i t y o f M e c h a n i c a l Speed . . 76 4.10 P a r a m e t e r S e n s i t i v i t y o f R e g u l a t o r V o l t a g e . . 77 4 . 1 1 P a r a m e t e r S e n s i t i v i t y o f F i e l d V o l t a g e . . . 78 4.12 P a r a m e t e r S e n s i t i v i t y o f D e l t a A n g l e . . . . 79 5.1 C a l c u l a t e d T r a n s i e n t R e s p o n s e s o f V a r i o u s P e r f o r m a n c e F u n c t i o n s 87 5.2 C a l c u l a t e d T r a n s i e n t R e s p o n s e s o f V a r i o u s S t a b i l i z i n g S i g n a l s 90 5.3 T r a n s i e n t R e s p o n s e s w i t h A c c e l e r a t i n g Power S t a b i l i z a t i o n 9 3 5.4 T r a n s i e n t R e s p o n s e s w i t h Speed D e v i a t i o n S t a b i l i z a t i o n 96 5.5 T r a n s i e n t R e s p o n s e s w i t h O p t i m a l C o n t r o l S t a b i l i z a t i o n 99 5.6 T r a n s i e n t R e s p o n s e s w i t h C o m p o s i t e S t a b i l i z a t i o n 102 5.7 T r a n s i e n t R e s p o n s e s w i t h A c c e l e r a t i n g Power S t a b i l i z a t i o n and C o m p o s i t e S t a b i l i z a t i o n . . 10.5 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 g u i d a n c e , i n t e r e s t and encouragement d u r - i n g t h e c o u r s e o f 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 . The d e v e l o p m e n t o f t h e s o l i d s t a t e v o l t a g e r e g u - l a t o r and e x c i t e r by Mr. J . Bond and t h e g o v e r n o r - p r i m e mover s y s t e m by Mr. R. S i d d a l l i s a c k n o w l e d g e d and t h e i r e f f o r t s g r e a t l y a p p r e c i a t e d . T h a n k s a r e due t o my c o l l e a g u e s f o r h e l p f u l d i s - c u s s i o n s and s u g g e s t i o n s , p a r t i c u l a r l y w i t h Mr. N. Thompson c o n c e r n i n g o p t i m a l c o n t r o l t h e o r y and w i t h Mr. C. S i g g e r s c o n c e r n i n g power s y s t e m s t a b i l i z i n g s i g n a l s . Mr. B. B l a c k b a l l ' s i n t e r e s t and e f f o r t s i n t h e c o n s t r u c t i o n and s u b s e q u e n t t e s t i n g o f t h e t e s t model were most h e l p f u l . S u p p o r t f r o m 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 Com- pany and t h e N a t i o n a l R e s e a r c h C o u n c i l o f Canada t h r o u g h a S t u d e n t s h i p award f o r 1967-1969 and g r a n t A3626 i s g r a t e f u l l y a c k n o w l e d g e d . T h a n k s a r e a l s o due t o Mrs. W. G r e i g f o r t y p i n g t h i s t h e s i s and Mr. A. H c K e n z i e f o r d r a u g h t i n g t h e f i g u r e s . I am g r a t e f u l t o my w i f e B e v e r l e y f o r h e r p a t i e n t u n d e r s t a n d i n g and encoura g e m e n t t h r o u g h o u t my g r a d u a t e p r o g r a m . NOMENCLATURE Genera1 H Hamiltonian J c o s t f u n c t i o n a l J, augmented cost f u n c t i o n a], a p d/dt, time d e r i v a t i v e o perator r e l a t i v e value of parameter q parameter q D i n t i a l v a lue of parameter x i a s e n s i t i v i t y c o e f f i c i e n t f o r s t a t e v a r i a b l e i with ' 1 r e s p e c t to parameter q r A 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 A c o s t a t e v a r i a b l e DC Machines Parameters L a F a coef f i c i e n t ;6J mL ap being the speed v o l t a g e c o e f f i c i e n t f o r the dc motor R dc motor-booster armature r e s i s t a n c e i n c l u d i n g R3 Rg m l o a d s e t t i n g r e s i s t a n c e i n the dc motor-booster armature c i r c u i t R̂ JJ b o o s t e r f i e l d r e s i s t a n c e T'fB booster f i e l d time constant V a r i a b l e s i ^ dc motor-booster armature c u r r e n t Ip dc motor f i e l d c u r r e n t T^ load torque v f B b o oster f i e l d v o l t a g e Vy b o o s t e r armature v o l t a g e v dc motor armature v o l t a g e Hi * ' v t d c c' c i n o ^ o r - b o o s t e r armature t e r m i n a l v o l t a g e (j mechanical angular speed of booster generator U)]U mechanical angular speed of dc motor k\ub mechanical angular base speed of dc motor, 188.5 rad/sec H y d r a u l i c Turbine and Governor Parameters G(p) governor t r a n s f e r f u n c t i o n H(p) h y d r a u l i c o perator t r a n s f e r f u n c t i o n H i n e r t i a . c o n s t a n t a c t u a t o r servomotor time constant Tj^ dashpot r e l a x a t i o n time T Q gate servomotor time constant T W water s t a r t i n g time TN.. mechanical s t a r t i n g time of the u n i t °̂ \\ c o e f f i c i e n t of net r e g u l a t i o n g~ permanent speed droop S temporary speed droop V a r i a b l e s a a c t u a t o r servomotor p o s i t i o n a.p a c t u a t o r feedback p o s i t i o n g gate servomotor p o s i t i o n n per u n i t r e l a t i v e angular speed change t t u r b i n e torque output Prime-Mover Governor Model Pa rameters J moment of i n e r t i a of t e s t model f KjW | -I- K 2 : f r i c t i o n K L a F ] F W g L a f , , . ^ — —h-.-,.'„ : model c o e f f i c i e n t ^ K K ~ — (l*FJr\ K K + \ / : c o e f f i c i e n t f o r the dc motor 4 1 — R torque c a n c e l l a t i o n K Bm load s e t t i n g r e s i s t a n c e i n dc motor-booster armature c i r c u i t r/ 3 ; s i m u l a t i o n c o e f f i c i e n t R e g u l a t o r - E x c i t e r Parameters a - | , a 2 constants -used to o b t a i n c h a r a c t e r i s t i c s of the f i e l d v o l t a g e l i m i t e r r e g u l a t o r gain Tj^p r e g u l a t o r time constant Tp e x c i t e r time constant V a r i a b l e s v^ r e g u l a t o r v o l t a g e v r e j r e g u l a t o r - e x c i t e r r e f e r e n c e v o l t a g e Synchronous Machine Parameters D damping c o e f f i c i e n t l- armature 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 R^ transformed f i e l d r e s i s t a n c e ' s l o p e 1 - r a t i o of synchronous machine steady s t a t e armature s h o r t c i r c u i t c u r r e n t and f i e l d c u r r e n t Tj^-| d - a x i s damper leakage time constant T' d-axis t r a n s i e n t s h o r t c i r c u i t time constant d T^ o d - a x i s t r a n s i e n t open c i r c u i t time constant 'T'a'T'q d- and q - 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 time constant r^do,''"qo C'~ a n C ' (i~ax^-s s u b t r a n s i e n t open c i r c u i t time constant x m u t u a l r e a c t a n c e b e t w e e n s t a t o r and r o t o r i n d - a x i s 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 s d q x' d - a x i s t r a n s i e n t r e a c t a n c e d x ,x d- and q - a x i s s u b t r a n s i e n t r e a c t a n c e s d q V a r i a b l e s i f j , i d — and q - a x i s c u r r e n t "*'fd f i e l d c u r r e n t P r e a l power o u t p u t o f t h e m a c h i n e Q r e a c t i v e power o u t p u t o f t h e m a c h i n e T 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 t o r q u e o u t p u t t o t h e r o t o r v ^ , v ^ d- and q - a x i s v o l t a g e s a r m a t u r e t e r m i n a l v o l t a g e f i e l d v o l t a g e v r a d v f | ; a 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 v o l t a g e R I V F R 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 VDR v o l t a g e p r o p o r t i o n a l t o d - a x i s damper w i n d i n g c u r r e n t V0R v o l t a g e p r o p o r t i o n a l t o q - a x i s damper w i n d i n g c u r r e n t ^d'fq ^~ a n c * c l " a x ^ s f l u x l i n k a g e s y f i e l d f l u x l i n k a g e V fLsjdV'c. , i f " l , l x p r o p o r t i o n a l t o f i e l d f l u x l i n k a g e F B I f a f ^D'^Q ^~ a n d Q - s x i s damper w i n d i n g f l u x l i n k a g e s (j^ e l e c t r i c a l a n g u l a r s p e e d CJeo b a s e e l e c t r i c a l a n g u l a r s p e e d , 377 r a d / s e c 8 t o r q u e a n g l e x i :i. i T r a n sni i s s i o n Sy s t em P a r a m e t e r s 13 s h u n t s u s c e p t a n c e G s h u n t c o n d u c t a n c e r s e r i e s r e s i s t a n c e x s e r i e s r e a c t a n c e V a r i a b l e s v i n f i n i t e bus v o l t a g e o INTRODUCTION 1 Modern power s y s t e m s have become so complex i n s t r u c - t u r e and so l a r g e i n s i z e t h a t o n - l i n e t e s t s o f some c o n t r o l schemes t o i m p r o v e t h e s t e a d y s t a t e and t r a n s i e n t s t a b i l i t y a r e e n t i r e l y p r o h i b i t i v e . T h i s i s b e c a u s e t h e e x p e r i m e n t s a r e n o t o n l y c o s t l y b ut a l s o p o s s i b l y d e s t r u c t i v e i n n a t u r e . T h e r e f o r e , i t i s d e s i r a b l e t o p e r f o r m t e s t s on s m a l l t e s t m o d e l s w h i c h h a ve s i m i l a r c h a r a c t e r i s t i c s t o t h e a c t u a l s y s t e m . The d e v e l o p m e n t o f t h e t e s t m o d e l s f o r l a r g e power s y s t e m s i s n o t new. Robert"*" *"*" i n 1950 c o n s t r u c t e d a m i c r o - m a c h i n e and r n i c r o r e s e a u x s y s t e m . He i n v e s t i g a t e d t h e e l e c t r o - m a g n e t i c and m e c h a n i c a l s i m i l a r i t i e s o f t h e model and r e a l s y s t e m . The same p e r u n i t r e a c t a n c e s , e q u a l t i m e c o n s t a n t s , s i m i l a r i n e r t i a c o n s t a n t s , and t o r q u e s p e e d c h a r a c t e r i s t i c s a r e u s e d i n t h e m o d e l i n g . A r o t a r y m a c h i n e was u s e d t o o b t a i n a n e g a t i v e r e s i s t a n c e t o i n c r e a s e t h e f i e l d t i m e c o n s t a n t . V e n i k o v d e s i g n e d a n o t h e r m i c r o m a c h i n e i n 1952. A commutator m a c h i n e was u s e d t o r e a l i z e t h e n e g a t i v e r e s i s t a n c e f o r t h e f i e l d t i m e c o n s t a n t . T h r e e r o t o r s w i t h d i f f e r e n t s a l i e n c y , x^ = 0.85? 0.55? 0.40 were u s e d . F l y w h e e l e f f e c t was a p p l i e d t o v a r y t h e i n e r t i a o f t h e m i c r o m a c h i n e . A d k i n s ' " m i c r o m a c h i n e was r e p o r t e d i n I960 and was u s e d t o i n v e s t i g a t e s h o r t c i r c u i t s , s y n c h r o n i z i n g and damping t o r q u e s , s w i n g c u r v e s , a s y n c h r o n o u s o p e r a t i o n and r e s y n c h r o n - i z a t i o n . The n e g a t i v e r e s i s t a n c e o f t h e f i e l d c i r c u i t was 2 r e a l i z e d by e l e c t r o n i c c i r c u i t s . The m i c r o t u r b i n e was r e a l - i z e d by a s e p a r a t e l y e x c i t e d dc m a c h i n e w i t h t h y r a t r o n c o n t r o l . More t e s t s a r e r e p o r t e d r e c e n t l y i n Canada by Roy^*^ u s i n g a m i c r o m a c h i n e and m i c r o r e s e a u x and i n t h e U.S.A. by 1 .5 D o u g h e r t y w i t h a t h y r i s t o r c o n t r o l l e d dc motor as t h e p r i m e mover f o r dc t r a n s m i s s i o n t e s t s . A new model f o r power s y s t e m dynamic s t u d i e s has been d e v e l o p e d a t U.B.C. P r e l i m i n a r y work has been done by J . Bond on t h e d e s i g n o f a s o l i d s t a t e v o l t a g e r e g u l a t o r and e x c i t e r and by R. S i d d a l l on t h e s i m u l a t i o n o f a g o v e r n o r - p r i m e mover. F u r t h e r d e v e l o p m e n t work has been c o m p l e t e d by t h i s t h e s i s i n c l u d i n g a d d i t i o n a l f e a t u r e s and i m p o r t a n t t e s t s o f t h e com- p l e t e s y s t e m . The model h a s t h e f o l l o w i n g f e a t u r e s . 1) I t i s v e r s a t i l e i n t h a t most power s y s t e m s w i t h c o n - v e n t i o n a l c o n t r o l l e r s c a n be s i m u l a t e d on t h e model on a p e r u n i t b a s i s . 2) I t has s o l i d s t a t e components and a n a l o g s i m u l a t e d r e g u l a t o r and e x c i t e r w i t h b o t h l i n e a r and f o r c e d e x c i t a t i o n c h a r a c t e r i s t i c s . 3) I t has an a n a l o g s i m u l a t e d 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 i n c o n j u n c t i o n w i t h a dc m o t o r s i m u l a t e d p r i m e mover. 4) I t has a s y n c h r o n o u s m a c h i n e and e x c i t e r w i t h a n e g a - t i v e r e s i s t a n c e i n t h e f i e l d c i r c u i t . The n e g a t i v e r e s i s t a n c e i s r e a l i z e d by e l e c t r o n i c c i r c u i t s . 5) I t has an ac t r a n s m i s s i o n l i n e w h i c h a t t h e moment i s d e s i g n e d f o r a s p e c i a l p r o j e c t . 3 6) I t has e l e c t r o n i c c o n t r o l l e d s w i t c h i n g ' to r e a l i z e f a u l t , f a u l t c l e a r e d and s u c c e s s f u l o r u n s u c c e s s f u l l i n e r e c l o s u r e a t p r e s e t t i m e s . 7) I t h a s an a c c u r a t e t o r q u e a n g l e d e v i a t i o n - t r a n s d u c e r u t i l i z i n g t h e z e r o c r o s s i n g s o f t h e t e r m i n a l v o l t a g e and r e f e r e n c e v o l t a g e waves. 8) I t h a s a s t a b i l i z i n g s i g n a l g e n e r a t o r . The t h e s i s a l s o i n c l u d e s t h e f o l l o w i n g f e a t u r e s . 1) D e r i v a t i o n o f s t a t e v a r i a b l e e q u a t i o n s f o r t h e t e s t model w i t h d i f f e r e n t d e g r e e s o f d e t a i l s , C h a p t e r 3. 2) C o m p r e h e n s i v e c o m p a r i s o n o f d i g i t a l c o m p u t a t i o n and m o del t e s t r e s u l t s o f a t r a n s i e n t s h o r t c i r c u i t on t h e s y s t e m , C h a p t e r 3. 3) P a r a m e t e r s e n s i t i v i t y s t u d y , C h a p t e r 4. 4) C o m p r e h e n s i v e c o m p a r i s o n o f d i g i t a l c o m p u t a t i o n and m odel t e s t s o f a "bang-bang" t y p e n o n l i n e a r o p t i m a l c o n t r o l o f t h e s y s t e m . I t i s hoped t h e t e s t model d e v e l o p e d w i l l be u s e f u l t o t h e t r a n s i e n t s t a b i l i t y s t u d y o f p r a c t i c a l power s y s t e m s . 2. REALIZATION OF THE DYNAMIC TEST MODEL 4 2.1 M o d e l i n g P r o c e d u r e A t y p i c a l one m a c h i n e - i n f i n i t e b u s s y s t e m , F i g . 2.1, i s c h o s e n t o be m o d e l e d f o r p o w e r s y s t e m d y n a m i c s s t u d i e s . The m e t h o d s and p r o c e d u r e o f m o d e l i n g a r e k e p t g e n e r a l so t h a t t h e y c a n be e x t e n d e d t o any m u l t i m a c h i n e power s y s t e m . II I Ll.H L II •-• IIIT' GOVERNOR - _ REGULATOR- EXCITER F i g . 2.1. Power S y s t e m t o be M o d e l e d The f i r s t s t e p o f m o d e l i n g i s t o o b t a i n a d e t a i l e d m a t h e m a t i c a l d e s c r i p t i o n o f a t y p i c a l p ower s y s t e m . F o r e x - a m p l e , t h e s y n c h r o n o u s m a c h i n e i s d e s c r i b e d by s e v e n t h o r d e r s t a t e e q u a t i o n s , f i f t h o r d e r f o r t h e e l e c t r i c a l and s e c o n d o r d e r f o r t h e m e c h a n i c a l , b u t w i l l be a p p r o x i m a t e d • by l o w e r o r d e r m o d e l s a f t e r c h e c k i n g w i t h t h e t e s t r e s u l t s . The s e c o n d s t e p i s t o d e c i d e what r a n g e s o f p a r a m e t e r v a l u e s o f t h e s y n c h r o n o u s m a c h i n e , t h e c o n t r o l l e r s , and t h e t r a n s m i s s i o n l i n e o f a c t u a l power s y s t e m s a r e t o be s i m u l a t e d . 5 The t h i r d s t e p i s t o f i n d t h e means, c i r c u i t s and machines, t o r e a l i z e t h e m a t h e m a t i c a l m o d e l . F o r example, how i s t h e g o v e r n o r - p r i m e mover s y s t e m t o be s i m u l a t e d by c o n v e n - t i o n a l d--c m a c h i n e s w i t h s o l i d s t a t e e l e c t r o n i c c i r c u i t s . T h e f o u r t h s t e p i s t h e c i r c u i t d e s i g n and c o n s t r u c t i o n d e t a i l s . E v i d e n t l y , i t i s i m p o r t a n t t o c a r r y t h r o u g h t h e t e s t s o f a l l t h e components, s u b s y s t e m s and t h e c o m p l e t e m o d e l . One o f t h e b a s i c p r o b l e m s i s how t o d e t e r m i n e t h e model p a r a - m e t e r s w i t h a c c u r a c y . The l a s t s t e p i s t o s e l e c t a b a s e impedance and a b a s e v o l t a g e o f t h e t e s t model f o r a p a r t i c u l a r s y s t e m u n d e r i n - v e s t i g a t i o n . 2.2 A T y p i c a l Power S y s t e m The t y p i c a l power s y s t e m t o be m o d e l e d by t h e dynamic t e s t model i s shown s c h e m a t i c a l l y i n F i g . 2.1. The f o u r main components a r e t h e g o v e r n o r - p r i m e mover w h i c h s u p p l i e s power t o and m a i n t a i n s a s y n c h r o n o u s s p e e d o f t h e s y s t e m , t h e s y n - c h r o n o u s m a c h i n e f o r t h e e l e c t r o m e c h a n i c a l e n e r g y c o n v e r s i o n , t h e r e g u l a t o r - e x c i t e r f o r t h e m a c h i n e t e r m i n a l v o l t a g e c o n t r o l , and t h e t r a n s m i s s i o n s y s t e m w h i c h c o n n e c t s t h e m a c h i n e t e r m i n a l t o t h e i n f i n i t e bus.. F o r t h e m o d e l i n g o f a one m a c h i n e - i n f i n i t e bus s y s t e m , t h e i n f i n i t e bus i s assumed t o be a m a c h i n e o f i n f i n i t e i n e r t i a and z e r o i n t e r n a l i mpedance. 2.3 S e t t i n g Up a T y p i c a l Power S y s t e m on t h e T e s t Model 2.3.1 D e t e r m i n i n g Base Impedance, Power and V o l t a g e The main o b j e c t i v e o f t h e m o d e l i n g i s t o a c h i e v e t h e same p e r u n i t v a l u e o f p a r a m e t e r s o f t h e a c t u a l s y s t e m on t h e d y n a m i c t e s t m o d e l . I n o t h e r words, t o make t h e t e s t model p e r u n i t i m p e d a n c e e q u a l t o t h e s y s t e m p e r u n i t i mpedance. Zm,pu = z p u P e r u n i t ( 2 * X ) where ^ Zm,pu - p e r u n i t Zmb and z m ) p U ~ P e r u n i t impedance v a l u e o f t e s t m o d e l, z p u = P e r u n i t i m pedance v a l u e o f a c t u a l s y s t e m , Z m = an impedance o f t h e model, Z ^ = model b a s e ohms. E q u a t i o n (2.1) a p p l i e s t o a l l s y s t e m components s u c h as gen- e r a t o r s , t r a n s f o r m e r s , t r a n s m i s s i o n l i n e s and l o a d s . T h e r e - f o r e , t h e model b a s e ohm (lmb) must be s e l e c t e d s u c h t h a t Zmb ~ £m_ ohm (2.2) ^pu T h i s c o n d i t i o n must be met i n o r d e r t o o b t a i n a good c o m p a r i s o n on t h e p e r u n i t b a s i s between t h e main p a r a m e t e r s o f t h e a c t u a l s y s t e m and t h a t o f t h e t e s t m o d e l . On t h e o t h e r hand, t o r e a l - i z e a n o n - e x i s t i n g i m p e d a n c e ( Z j n ) on t h e model, one may a p p l y t h e w e l l known f o r m u l a = Z m b Z p u - Z / k V m b \ 2 / M V A b \ ohm (2.3) 1<V, / I MVA , , o / \ mb/ 7 N e x t , t h e b a s e power o f t h e t e s t m o d e l i s e s t a b l i s h e d f r o m a c o m p a r i s o n on t h e p e r u n i t b a s i s o f t h e a c c e l e r a t i n g t o r q u e e q u a t i o n o f t h e t e s t m o d e l and t h a t o f t h e a c t u a l s y s - tem. As w i l l be r e v e a l e d i n S e c t i o n '2 . 4 . 1 * a s i m u l a t i o n p a r a m e t e r CX i s r e l a t e d t o t h e i n e r t i a c o n s t a n t I I ( s e c ) o f t h e a c t u a l p o w e r s y s t e m and t h e moment o f i n e r t i a J ( j o u l e - s e c 2 / r a d 2 ) o f t h e d y n a m i c t e s t m o d e l by t h e e q u a t i o n CX = JL j o u l e - s e c / r a d 2 ( 2 . 4 ) 2H a n d t h e b a s e t o r q u e o f t h e m o d e l i s g i v e n by Tmb = t * W m b j o u l e / r a d ( 2 . 5 ) w h e r e CO ^ ( 1 8 8 . 5 r a d / s e c ) i s t h e m o d e l b a s e m e c h a n i c a l s p e e d . S i n c e t h e b a s e power i s g i v e n by Pmb = ( J m b Tmb watt ( 2 . 6 ) s u b s t i t u t i o n o f ( 2 . 4 ) and ( 2 . 5 ) i n t o ( 2 . 6 ) g i v e s Pmb = J^mb w a t t ( 2 - 7 ) 211 E q u a t i o n ( 2 . 7 ) i s t h e c o n d i t i o n t o be met f o r s e t t i n g t h e b a s e power o f t h e m o d e l . F i n a l l y , t h e m o d e l b a s e v o l t a g e i s g i v e n by Vmb = Zmb Pmb < 2' 8) and e q u a t i o n ( 2 . 7 ) and ( 2 . 8 ) must be s a t i s f i e d s i m u l t a n e o u s l y . Of c o u r s e , t h e m o d e l v o l t a g e and power b a s e s must be w i t h i n 8 t h e r a t i n g o f t h e model g e n e r a t o r . I t i s d e s i r a b l e t o e s t a b - l i s h a b a s e v o l t a g e l e s s t h a n t h e model v o l t a g e r a t i n g f o r two r e a s o n s ; f i r s t t h e machine s a t u r a t i o n e f f e c t s c a n be n e g l e c t e d and s e c o n d , t h e s u s t a i n e d f a u l t c u r r e n t s i n t h e s t u d y w i l l n o t damage t h e g e n e r a t o r . 2.3.2 N u m e r i c a l V a l u e s o f S y s t e m and Model G e n e r a t o r P a r a m e t e r s N u m e r i c a l v a l u e s o f g e n e r a t o r p a r a m e t e r s o f an a c t u a l power s y s t e m and t h o s e o f t h e model a r e l i s t e d i n T a b l e 2.1 s i d e by s i d e on t h e p e r u n i t b a s i s f o r c o m p a r i s o n . The p a r a - P a r a m e t e r Mode! - S y s t e m A c t u a l S y s t e m M e a s u r e d S y n c h r o n o u s M a c h i n e P a r a m e t e r s r a .664-n. .0 42 pu .00247 pu x d 16. 2 1.025 pu .973 pu x q 9.71-n. .614 pu . 55 pu i x d 2.74.n. .173 pu .190 pu .247 s e c Ad j u s t a b l e * 5.00 s e c Base Impedance 15.8 .797 M e c h a n i c a l P a r a m e t e r s I n e r t i a f J - .165 j o u l e - s e c ^ / r a d ^ 2.67 x 10-3o> + 1 > 5 8 5 / 2 j o u l e - s e c / r a d H = 4.63 s e c II = 4.63 s e c •"-Section 2.4.2 T a b l e 2.1 Model and A c t u a l S y s t e m G e n e r a t o r P a r a m e t e r V a l u e s 9 m e t r i c v a l u e s o f t h e model, i n MKS u n i t s a r e d e t e r m i n e d d i r e c t l y f r o m t e s t s . The b a s e kV o f t h e s y s t e m g e n e r a t o r i s 13.9 and t h e b a s e MVA i s 242. To a r r i v e a t t h e p e r u n i t v a l u e s o f t h e m o d e l g e n e r a - t o r f o r a good s i m u l a t i o n o f t h e s y s t e m , e m p h a s i s must be p l a c e d on t h e most ' i m p o r t a n t ' p a r a m e t e r s s u c h a s x<^, x^, x^, II e t c . The g e n e r a t o r h a s so many p a r a m e t e r s t h a t a l l o f them c a n n o t be s i m u l a t e d c l o s e l y s i m u l t a n e o u s l y b e c a u s e o f so many c o n s t r a i n t s i n v o l v e d . F o r t h e p a r t i c u l a r p e r u n i t s e t up o f T a b l e 2.1 an e x a c t v a l u e o f H i s r e t a i n e d . 2.3.3 N u m e r i c a l V a l u e s o f M o d e l B a s e Q u a n t i t i e s F r om (2.7) a n d d a t a o f T a b l e 2.1 t h e m o d e l p o w e r b a s e e q u a l s Pmb = J to mb = -( .165)(188.5) 2 = 633 w a t t s • (2.9) 2H (2) (4.63) N e x t i f one w o u l d l i k e t o h a v e an e q u a l p e r u n i t v a l u e o f t d x , i n b o t h s y s t e m s , one s h a l l h a v e f r o m (2.1) zm,pu = x d •= 2.74 = 14.4" ohm (2.10) H o w e v e r , i n o r d e r t o g e t x.^, x ^ and x ^ s i m u l a t e d c l o s e l y s i m u l t a n e o u s l y , t h e b a s e ohm i s c h o s e n a s 15.8 ohms. The r e m a i n i n g b a s e v a l u e s a r e v , = 100 v l i n e t o l i n e v o l t (2.11) mb and I m b = 3.65 A amp (2.12) The n o m i n a l r a t i n g s o f t h e dynamic t e s t model g e n e r a t o r a r e P = 1600 w a t t s V = 208 v o l t s I •= 5 .5 amps 2 . 3 . 4 Measurement o f M o d e l G e n e r a t o r P a r a m e t e r s The s y n c h r o n o u s m a c h i n e r e a c t a n c e s and t i m e c o n s t a n t s a r e d e t e r m i n e d by t h e s t a n d a r d p r o c e d u r e s o f t h e I E E E T e s t 2 1 Code No. 1 1 5 . ' F o r example, t h e v a l u e o f t h e d i r e c t - a x i s f t r a n s i e n t r e a c t a n c e x^ i s o b t a i n e d f r o m t h e a r m a t u r e c u r r e n t e n v e l o p e o f a t h r e e - p h a s e sudden s h o r t c i r c u i t , and t h e d i r e c t - a x i s o p e n - c i r c u i t t r a n s i e n t t i m e c o n s t a n t T ^ Q i s o b t a i n e d f r o m t h e a r m a t u r e v o l t a g e e n v e l o p e when t h e f i e l d w i n d i n g w i t h e x c i t a t i o n i s s h o r t c i r c u i t e d . The v a l u e o f t h e d i r e c t - a x i s s y n c h r o n o u s r e a c t a n c e f o u n d by t h e s l i p t e s t compared f a v o u r a b l y w i t h t h e v a l u e o b t a i n e d f r o m t h e s t e a d y s t a t e open c i r c u i t and s h o r t c i r c u i t t e s t s i n t h e l i n e a r r e g i o n . The moment o f i n e r t i a J o f t h e d-c m o t o r - s y n c h r o n o u s m a c h i n e s e t i s d e t e r m i n e d from a r e t a r d a t i o n t e s t and t h e f r i c t i o n f i s o b t a i n e d f r o m a s t e a d y s t a t e t e s t u t i l i z i n g 2 2 t h e e n e r g y c o n v e r s i o n t o r q u e o f t h e d-c motor. The p a r a m e t e r v a l u e s t h u s o b t a i n e d a r e p r e s e n t e d i n T a b l e 2 . 1 . F i g . 2.2. DC M o t o r - B o o s t e r U n i t 2.4. D e v e l o p m e n t o f t h e D y n a m ic T e s t M o d e l f o r Pox^er S y s t e m S i m u l a t i o n 9 O 2.4.1 G o v e r n o r - P r i m e M o v e r 0 A d-c m o t o r i n s e r i e s w i t h a b o o s t e r , F i g . 2.2, i s u s e d t o s i m u l a t e t h e p r i m e m o v e r . Th e d~c m o t o r i s g i v e n a c o n s t a n t e x c i t a t i o n I f and t h e i n i t i a l a r m a t u r e c u r r e n t i ^ a n d e n e r g y c o n v e r s i o n t o r q u e a r e s e t by a l o a d s e t t i n g r e s i s t o r R g m . To o b t a i n an i n c r e m e n t a l t o r q u e a s a f u n c t i o n o f s p e e d d e v i a t i o n o f t h e d-c m o t o r s y n c h r o n o u s m a c h i n e s e t , t h e b o o s t e r r e c e i v e s an e x c i t a t i o n w h i c h w i l l c a u s e t h e m o d e l t o r e s p o n d s i m i l a r l y t o a p o w e r s y s t e m w i t h g o v e r n o r c o n t r o l . N e g l e c t i n g t h e t i m e v a r i a t i o n a l b ut n o t t h e s p e e d v o l t a g e e f f e c t o f t h e d-c m o t o r - b o o s t e r a r m a t u r e c i r c u i t , t h e a r m a t u r e c u r r e n t becomes A A = i | v t d c - W m L a P I P ' w g L a f v f J 3 \ R I amp {2.13) V R f B 1 + T f p P / T h e s e c o n d t e r m i n t h e b r a c k e t i s due t o c o n s t a n t e x c i t a t i o n and t h e t h i r d t e r m i s t h e s p e e d v o l t a g e o f t h e b o o s t e r . The v o l t a g e v t d c i s t h e t o t a l d-c v o l t a g e a p p l i e d t o t h e d-c motor- b o o s t e r a r m a t u r e c i r c u i t . The r e s i s t a n c e R i n c l u d e s two arma- t u r e r e s i s t a n c e s and t h e l o a d s e t t i n g r e s i s t o r R]3 m. The a c c e l e r a t i n g t o r q u e o f t h e d-c motor i s Jd(J m = L a F I F i A - ( K i « m + K 2 ) - T L j o u l e / r a d (2.14) dt The f i r s t t e r m o f t h e r i g h t hand s i d e o f t h e e q u a t i o n i s t h e e n e r g y c o n v e r s i o n t o r q u e , t h e s e c o n d t e r m t h e f r i c t i o n t o r q u e e x p e r i m e n t a l l y d e t e r m i n e d , and t h e l a s t t e r m t h e l o a d toi^que. F o r a l o a d d e v i a t i o n , (2.14) becomes J d A G J m = L f l F I F A i A ~ K]AoJm - A T L j o u l e / r a d (2.15) d t w h i c h c o r r e s p o n d s t o JdAU)m = G ( p ) l ! ( p ) ( - A W m ) - A T L (2.16) S u b s t i t u t i n g (2.13) i n t o (2.14) and (2.15) and c o m p a r i n g t h e r e s u l t s w i t h (2.16) y i e l d s A v f J = ~ l + T f B p ( « G ( p ) H ( p ) - K 4 ) ( - A « m ) v o l t (2.17)- K 3 where K 3 = L R p I p ( J ^ L a p amp-sec R Rpg r a d and = + ( L a p I p ) watt s e c ^ R r a d I t becomes e v i d e n t t h a t i f t h e v o l t a g e A V ^ c a n be r e a l i z e d a c c o r d i n g t o ( 2 . 1 7 ) , t h e n t h e d-c motor w i l l h a v e e x a c t l y t h e same t o r q u e - s p e e d c h a r a c t e r i s t i c s a s t h a t o f a l a r g e p r i m e mover w i t h a g o v e r n o r . D i v i d i n g t h r o u g h ( 2 . 1 6 ) by w t j ^ g i v e s J_ dn = G(p) 11 (p) (-n) - A T j p e r u n i t ( 2 . 1 8 ) w h i c h compares w i t h H o v e y ' s ^ ' ^ e q u a t i o n T m dn - G(p) l l ( p ) (-n) - A M p e r u n i t ( 2 . 1 9 ) d t H e nce = J_ } = 2H s e c ( 2 . 2 0 ) and T m b = « W m b . j o u l e / r a d ( 2 . 2 1 ) F o r t h e p a r t i c u l a r model p r i m e mover d e v e l o p e d a t U.B.C., t h e p a r a m e t e r s a r e L a p l p - 0 . 8 5 v o l t - s e c / r a d GJlTLaf = 73 ohm R^g - 5 1 . 6 ohm T f B - 0 . 5 s e c K X = 2 . 6 7 x 10""3 K 2 - 1 . 5 8 5 j o u l e / s e c j o u l e / r a d T he t o t a l r e s i s t a n c e R o f t h e a r m a t u r e c i r c u i t i n c l u d i n g t h e l o a d s e t t i n g r e s i s t a n c e R g m a t no l o a d i s R = 2 8 . 1 ohm ; v t d = 229 V The K 0 and K. c o n s t a n t s a r e 3 4 1 = 2 3 . 2 9 r a d = 0 . 0 2 5 8 w a t t s e c 2 K 3 amp-sec " r a d The m e c h a n i c a l p a r a m e t e r s a r e p r e s e n t e d i n S e c t i o n 2 . 3 . 2 . T h e g e n e r a l l a y o u t o f t h e m o d e l g o v e r n o r - p r i m e mover s y s t e m i s shown i n F i g . 2 . 3 . T h e g o v e r n o r - 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 G ( p ) l i ( p ) i s s e t up on an a n a l o g s p e c i a l l y b u i l t f o r t h i s p u r p o s e . The m o d e l i s c a p a b l e o f r e p r e s e n t i n g d i f f e r e n t g o v e r n o r - p r i m e m over c o n f i g u r a t i o n s . F o r t h e g o v e r n o r - 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 shown i n F i g . 2 . 4 * t h e p a r a m e t r i c v a l u e s a r e T A = 0 . 0 2 s e c CT= 0 . 0 6 $ = 0 . 5 T R = 5 . 0 0 s e c T G = 0 . 5 s e c T w = 1 . 6 s e c The b o o s t e r f i e l d c o m p e n s a t o r c i r c u i t ( l + T^^P) i s a l s o b u i l t w i t h a n a l o g c o m p o n e n t s . The c i r c u i t i s g i v e n i n F i g . 2 . 5 . A c u r r e n t a m p l i f i e r i s i n c l u d e d t o m a t c h t h e c u r r e n t l e v e l b e t w e e n t h e o p e r a t i o n a l a m p l i f i e r and t h e b o o s t e r f i e l d . No d i f f i c u l t i e s a r e e x p e r i e n c e d w i t h t h e d i f f e r e n t i a t i o n s i n c e t h e i n p u t s i g n a l f r e q u e n c y i s l e s s t h a n 1 Hz.. A s m a l l v a l u e o f c a p a c i t a n c e (C = l O O p f ) i s u s e d t o s t a b i l i z e t h e c i r c u i t . 2 . 4 . 2 R e g u l a t o r - E x c i t e r 2 , 5 The l a y o u t o f t h e model v o l t a g e r e g u l a t o r - e x c i t e r s y s t e m i s shown i n F i g . 2 . 6 . F o r t h e s t u d i e s i n t h i s t h e s i s , t h e t r a n s f e r f u n c t i o n R E ( p ) , F i g . 2 . 7 , o f t h e r e g u l a t o r - PERMANENT DROOP STrP 1+ Tr P GATE SERVO TEMPORARY DROOP AND DASHPOT 1-TWP t 1+.5TWP HYDRAULIC SYSTEM F i g . 2 . 4 . Governor-Hydraulic System T r a n s f e r F u n c t i o n 16 F i g . 2 . 5 . B o o s t e r F i e l d C o m p e n s a t i o n and C u r r e n t A m p l i f i e r C i r c u i t VOLTAGE SIGNAL FIELD VOLTAGE A ~U STABILIZING SIGNAL ANALOG COMPUTER r~ CURRENT AMPLIFIER NEGATIVE RESISTOR SYNCHRONOUS MACHINE FIELD F i g . 2 . 6 . M o d e l R e g u l a t o r - E x c i t e r S y s t e m VOLTAGE SIGNAL KA UTAP 1+T£P ———o FIELD "* VOLTAGE F i g . 2 . 7 . 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 e x c i t e r i s p a t c h e d o n t o a s p e c i a l l y b u i l t a n a l o g c o m p u t e r . The p a r a m e t e r v a l u e s o f an e x a m p l e a r e The v o l t a g e s i g n a l f r o m t h e a n a l o g c o m p u t e r i s t h e n a m p l i f i e d t o m a t c h t h e v o l t a g e and c u r r e n t l e v e l s o f t h e s y n c h r o n o u s m a c h i n e f i e l d . The m o d e l s y n c h r o n o u s m a c h i n e h a s a r e l a t i v e l y l a r g e r e s i s t a n c e i n t h e f i e l d c i r c u i t . The r e s i s t a n c e v a l u e i s r e d u c e d by a n e g a t i v e r e s i s t o r ^ • 4 t o o b t a i n t h e d e s i r e d o pen c i r c u i t f i e l d time, c o n s t a n t ( T ^ Q ) • T h e n e g a t i v e r e s i s t o r i s r e a l i z e d by a scheme shown i n F i g . 2.8. The i d e a i s t o o b t a i n (W100 T A = 0 . 0 5 s e c T £ = 0 . 0 3 5 s e c REGULATOR - EXCITER GROUND F i g . 2 . 8 . Scheme t o R e a l i z e N e g a t i v e R e s i s t a n c e a v o l t a g e v ^ w h i c h i s p r o p o r t i o n a l t o c u r r e n t v i = &L H g3 Rsi=1 v o l t ( 2 . 2 2 ) so c o n n e c t e d t h a t , V f d + V i = ( R f + R s + L f P ) i V O l t ( 2 - 2 3 ) T h u s one h a s v = ( R f + R e - g ) i + L P p i v o l t ( 2 . 2 4 ) f d S • i F o r e x a m p l e , t o o b t a i n an open f i e l d t i m e c o n s t a n t o f 5 s e c t h e o r i g i n a l r e s i s t a n c e , R f - 6 9 . 6 ohms, o f t h e m o d e l i s r e d u c e d t o 3 . 4 4 ohms. 2 . 4 . 3 T r a n s m i s s i o n L i n e and C i r c u i t B r e a k e r s A t r a n s m i s s i o n l i n e m o d e l i s b u i l t t o s i m u l a t e a 576 m i l e d o u b l e - c i r c u i t t h r e e - p h a s e t h r e e - s e c t i o n h i g h v o l t a g e t r a n s m i s s i o n l i n e . The d i s t r i b u t e d p a r a m e t e r s o f t h e h i g h v o l t a g e t r a n s m i s s i o n l i n e a r e z = 0 . 0 4 1 + j O . 5 3 0 9 o h m / m i l e y = J 7 . 8 8 x 1 0 " 6 m h o / m i l e E a c h s e c t i o n o f e a c h p h a s e o f e a c h c i r c u i t o f t h e l i n e i s s i m u l a t e d by a 3T s e c t i o n w i t h l u m p e d p a r a m e t e r s . The m o d e l s e c t i o n g i v e s t h e same p e r u n i t v o l t a g e and p e r u n i t c u r r e n t a t t h e ends a s t h a t o f t h e r e a l l i n e w i t h d i s t r i b u t e d p a r a - m e t e r s . T h i s i n c l u d e s t h e e f f e c t o f t h e s h u n t r e a c t o r s (13 5 MVAR a t 525 kV) o f t h e r e a l l i n e a t ea c h end o f t h e s e c t i o n . One c i r c u i t o f one p h a s e o f t h e lu m p e d p a r a m e t e r e q u i v a l e n t o f F i g . 2 . 9 . One P h a s e Lumped P a r a m e t e r E q u i v a l e n t o f a S i n g l e C i r c u i t 2 0 .67/ OHM — A A A 7.98juF F i g . 2.10. M o d e l TT U n i t o f T r a n s m i s s i o n L i n e MACHINE TERMINAL TT TT INFINITE BUS TT TRANSMISSION SYSTEM L 77" • 77 - 77 BRAKING RESISTOR FAULT F i g . 2.11. One P h a s e o f t h e M o d e l T r a n s m i s s i o n L i n e t h e t r a n s m i s s i o n s y s t e m and s h u n t r e a c t o r s i s shown i n F i g . 2.9. The p a r a m e t r i c v a l u e s o f t h e e q u i v a l e n t model J T u n i t a r e g i v e n i n F i g . 2.10. T h e r e a r e a l t o g e t h e r 18 s u c h u n i t s . They a r e b u i l t w i t h r e s i s t o r s , i r o n c o r e r e a c t a n c e s and c a p a c i t o r s w h i c h a r e c o m m e r c i a l l y a v a i l a b l e . A t h r e e - p h a s e b r a k i n g r e s i s t o r i s c o n n e c t e d t o t h e m a c h i n e t e r m i n a l f o r s p e c i a l s t u d i e s . T h r e e - p h a s e r e l a y s a r e u s e d as c i r c u i t b r e a k e r s f o r l i n e f a u l t , c l e a r i n g and b r a k i n g r e s i s t a n c e s w i t c h i n g . One p h a s e o f t h e model t r a n s - m i s s i o n s y s t e m i s shown i n F i g . 2.11. 2.5. A u x i l i a r y M e a s u r i n g and C o n t r o l D e v i c e s 2.5.1 T ime S e q u e n c e C o n t r o l The t i m e and s w i t c h i n g s e q u e n c e c o n t r o l o f t h e t r a n s - m i s s i o n l i n e and b r a k i n g r e s i s t o r c i r c u i t b r e a k e r s , and t h e f a u l t s i m u l a t i o n i s a c h i e v e d w i t h i n t e g r a t e d c i r c u i t l o g i c e l e m e n t s . The c o n t r o l i s s c h e m a t i c a l l y shown i n F i g . 2.12. The c r y s t a l c l o c k p r o d u c e s s q u a r e waves a t 16 kHz. The p u s h b u t t o n s e t s t h e f l i p - f l o p w h i c h a l l o w s t h e s q u a r e waves t o p a s s t h r o u g h t h e 2 i n p u t NOR g a t e t o t h e f r e q u e n c y d i v i d e r w h i c h h a s an o u t p u t f r e q u e n c y o f 1kHz. The 0 and 1 o u t p u t s o f t h e 12 b i t c o u n t e r a r e c o n n e c t e d t o u n i t s c o n t a i n - i n g 12 s i n g l e - p o l e d o u b l e - t h r o w t i m e s e l e c t i o n s w i t c h e s w h i c h i n t u r n a r e c o n n e c t e d t o 12 i n p u t NOR g a t e s w h i c h o u t p u t a p u l s e when a l l t h e i n p u t s a r e G. Two u n i t s a r e n e c e s s a r y , one p u l s e t o t u r n on and a s e c o n d p u l s e t o t u r n o f f t h e SCR s w i t c h , f o r t h e f a u l t s i m u l a t i o n and b r a k i n g c i r c u i t b r e a k e r o p e r a t i o n where a CLOSE-OPEN r e l a y a c t i o n i s 9 3.6 V* PUSH BUTTON SET 16 kHz SR FLIP FLOP \R 12 INPUT NOR GATE CRYSTAL CLOCK I 1kHz 2 INPU T NOR GATE H FREQUENCY! DIVIDER IrjFREQUENCY DIVIDER 1kHz \—H2) 12 BIT COUNTER IN VERTER J TIME SEL EC T ION SW/ TCHES 12 INPUT NOR GATE CLOSE V_ r JF OPEN SCR SWITCH I RELAY COIL BRAKING RESISTOR CIRCUIT BREAKER * FAULT SIMULATION J , J I 9 L _ ^ - I - - OPE A/j _C/.OSEj \OPEN i .TRANSMISSION i 5 Z./A/E CIRCUIT a 5 BREAKERS F i g . 2.12. C i r c u i t Breaker and F a u l t Time-Sequence C o n t r o l to so d e s i r e d . F o r t h e t r a n s m i s s i o n l i n e c i r c u i t b r e a k e r s i m u l a t i o n an a d d i t i o n a l u n i t i s i n c o r p o r a t e d f o r an OPEN-CLOSE-OPEN r e l a y a c t i o n t o r e p r e s e n t an u n s u c c e s s f u l r e c l o s u r e . The t i m e - s e q u e n c e c o n t r o l i s t e r m i n a t e d w i t h a f e e d b a c k p u l s e t o t h e s e t - r e s e t f l i p - f l o p 4 . 0 9 6 s e c o n d s a f t e r i n i t i a t i n g t h e a c t i o n w i t h t h e p u s h b u t t o n . T h i s t i m e i n t e r v a l a l l o w s f o r n o r m a l a n d a b n o r m a l t i m e - s e q u e n c e o p e r a t i o n o f t h e f a u l t s i m u l a t i o n , t r a n s m i s s i o n l i n e c i r c u i t b r e a k e r s , and b r a k i n g r e s i s t o r c i r - c u i t b r e a k e r s . 2 . 5 . 2 T o r q u e A n g l e D e v i a t i o n M e a s u r e m e n t F o r power s y s t e m d y n a m i c s t u d i e s , i t i s e v i d e n t l y v e r y i m p o r t a n t t o h a v e an a c c u r a t e t o r q u e a n g l e d e v i a t i o n s i g n a l w h i c h c a n be c o n s t a n t l y m o n i t o r e d a n d u s e d f o r c o n t r o l p u r - p o s e s . To t h i s e n d , a c o n t i n u o u s v o l t a g e s i g n a l p r o p o r t i o n a l t o t o r q u e a n g l e d e v i a t i o n i s o b t a i n e d by t h e scheme shown i n F i g . 2 . 1 3 . Two a-c v o l t a g e s i g n a l s ? o ne f r o m t h e i n f i n i t e b u s and one f r o m an a-c t a c h o m e t e r c o u p l e d t o t h e g e n e r a t o r s h a f t , a r e f e d i n t o s e p a r a t e c o m p a r a t o r s . The o u t p u t s q u a r e wave s i g n a l s a r e c o n n e c t e d t o m o n o s t a b l e s . T h i s e l i m i n a t e s i n t e r - m i t t e n t s w i t c h i n g c a u s e d by r i n g i n g a t t h e c o m p a r a t o r o u t p u t s . T h e t i m e d e l a y b e t w e e n t h e p u l s e s f r o m t h e m o n o s t a b l e s i s p r o - p o r t i o n a l t o t h e p h a s e s h i f t o f t h e two s i g n a l s . The a-c t a c h o m e t e r m o n o s t a b l e p u l s e s e t s t h e s e t - r e s e t f l i p f l o p t o a l l o w p u l s e s f r o m c l o c k 1 t h r o u g h t h e NOR g a t e t o t h e 9 b i t c o u n t e r 1 . The f r e q u e n c y o f c l o c k 1 i s 3 0 . 7 2 kHz so t h a t 512 p u l s e s w i l l be c o u n t e d i f t h e p h a s e s h i f t i s 360 d e g r e e s a t AC TACHOME TER COMPARATOR COMPARATOR 9 BIT COUNTER 2 9 SWITCHES 9 INPUT NOR GATE RESET OUN TER | 7 9BIT C JKFLIP FLOPS J L OUTPUT BUFFER JK FLIP FLOPS] TOGGLE INFINITE BUS 60 Hz DIGITAL TO ANALOG CONVERTER F i g . 2.13. T o r q u e A n g l e D e v i a t i o n M e a s u r i n g D e v i c e .to •fa. 60 Hz. The i n f i n i t e bus m o n o s t a b l e p u l s e l o w e r s t h e o u t p u t l e v e l o f t h e s e t - r e s e t f l i p - f l o p t o s t o p p u l s e s f r o m c l o c k 1. A t t h e same t i m e , t h e f a l l i n g edge o f t h e s e t - r e s e t f l i p - f l o p t o g g l e s a f l i p - f l o p t o a l l o w p u l s e s f r o m c l o c k 2 i n t o b o t h 9~ b i t c o u n t e r 1 and 2. T h i s scheme a l l o w s t h e s t e a d y s t a t e o u t - p u t a n a l o g s i g n a l t o be r e d u c e d t o z e r o by s e t t i n g t h e 9 d o u b l e - t h r o w s i n g l e - p o l e s w i t c h e s . The complement o f t h e b i n a r y v a l u e s e t i s added t o c o u n t e r 1 a t a p p r o x i m a t e l y 2.5 x 1 0 6 . b i t s p e r s e c o n d by c l o c k 2. The end o f t h e a d d i t i o n i s d e t e c t e d by an o u t p u t p u l s e f r o m a 9 - i n p u t NOR g a t e w h i c h i n i t i a t e s t h e f o l l o w i n g e v e n t s ; f i r s t t h e f l i p - f l o p i s t o g g l e d t o s h u t o f f t h e p u l s e s f r o m c l o c k 2, s e c o n d a p u l s e f r o m t h e NOR g a t e t o g g l e s t h e o u t p u t b u f f e r t o r e c e i v e t h e c o n t e n t s o f c o u n t e r 1, and t h i r d a m o n o s t a b l e i s t r i g g e r e d . T he m o n o s t a b l p r o v i d e s a d e l a y e d p u l s e so t h e c o n t e n t s o f c o u n t e r 1, b e f o r e b e i n g r e s e t , c a n be t r a n s f e r e d t o t h e o u t p u t b u f f e r . T he o u t - p u t b u f f e r i s c o n v e r t e d t o a c o n t i n u o u s v o l t a g e s i g n a l by a d i g i t a l t o a n a l o g c o n v e r t e r w i t h a r a n g e 0 t o -10 v o l t s . The s a m p l i n g r a t e i s 60 t i m e s a s e c o n d . 2.5.3 F o r c e d E x c i t a t i o n C o n t r o l P r o v i s i o n i s made f o r t h e f o r c e d e x c i t a t i o n c o n t r o l t o p r o v i d e a s t a b i l i z i n g s i g n a l f o r t h e power s y s t e m d u r i n g and a f t e r a s y s t e m d i s t u r b a n c e ; C h a p t e r 5. The s t a b i l i z i n g s i g n a l i s an o p e n - l o o p b a n g - b a n g v o l t a g e s i g n a l summed w i t h t h e t e r - m i n a l and r e f e r e n c e v o l t a g e t o p r o v i d e a t o t a l v o l t a g e e r r o r s i g n a l , F i g . 2.6. The b l o c k d i a g r a m o f F i g . 2.14 shows a ©-> SR FLIP R FLOP [ INVERTER NOR GATE JL I 1 DECADE COUNTER I 10 100 1000 I L THUMBWHEEL SV/ITCHES INVERTER r~ NOR GATE EMERGENCY 6 i STOP o 3.5 V INVERTER LONG PULSE MONO SHORT PULSE MONO r REPEATED 8 TIMES I. A A A THUMBWHEEL SWITCHES INVERTER 1 X NOR GATE J F i g . 2.14. Forced E x c i t a t i o n Time-Switching C o n t r o l NEGATIVE VOLTAGE I TRANSISTOR SWITCH TRANSISTOR SWITCH POSITIVE VOLTAGE scheme t o r e a l i z e t h e s w i t c h i n g t i m e s a n d t h e number o f s w i t c h i n g s o f t h e s t a b i l i z i n g s i g n a l . The e v e n t s o f t h e f o r c e d e x c i t a t i o n c o n t r o l a r e s t a r t e d a t t h e i n s t a n t o f f a u l t by a s i g n a l f r o m p u s h b u t t o n s e t o f t h e t i m e - s e q u e n c e c o n t r o l , F i g . 2.12. T h i s s i g n a l r e s e t s a s e t - r e s e t f l i p - f l o p a l l o w i n g l k l i z s q u a r e waves f r o m t h e f r e q u e n c y d i v i d e r , F i g . 2.12, t o be c o u n t e d by t h e d e c a d e c o u n t e r . Ten u n i t s c o n s i s t i n g o f t h u m b w h e e l s w i t c h e s , i n - v e r t e r s and NOR g a t e s s e t t h e t i m e s o f s w i t c h i n g . A t t h e i n s t a n t t h e e v e n t s a r e i n i t i a t e d , t h e c o m p l e m e n t a r y l e v e l f r o m t h e s e t - r e s e t f l i p - f l o p i s f e d i n t o an i n v e r t e r whose o u t p u t p e r f o r m s two f u n c t i o n s . F i r s t , two m o n o s t a b l e s a r e t r i g g e r e d t h r o u g h an i n v e r t e r . One m o n o s t a b l e w i t h a l o n g p u l s e e s t a b - l i s h e s t h e i n i t i a l p o l a r i t y o f t h e f o r c e d e x c i t a t i o n and t h e s e c o n d s h o r t e r p u l s e t o g g l e s t h e i n i t i a l c o n d i t i o n i n t o f l i p - f l o p 2. S e c o n d , f l i p - f l o p 1 i s t o g g l e d w h i c h a l l o w s t h e NOR g a t e s f o r n e g a t i v e and p o s i t i v e v o l t a g e s t o be e n a b l e d . The t i m e s e t t i n g u n i t s t o g g l e f l i p - f l o p 2 t o p r o d u c e a b a n g - b a n g t y p e o f f o r c e d e x c i t a t i o n v o l t a g e s i g n a l . The l a s t t i m i n g u n i t o u t p u t f e e d s b a c k t o t h e s e t - r e s e t f l i p - f l o p t o s t o p t h e t i m i n g s e q u e n c e . A t t h e same t i m e , t h e h i g h l e v e l o u t p u t o f f l i p - f l o p 1 c l a m p s t h e n e g a t i v e and p o s i t i v e v o l t a g e s t o a z e r o v a l u e . 3. STATE EQUATION MODELS OF THE DYNAMIC MODEL AND TEST RESULTS One o f t h e most i m p o r t a n t d e c i s i o n s t o be made i n p o w e r s y s t e m d y n a m i c s s t u d i e s i s how much o f t h e d e t a i l o f t h e s y n c h r o n o u s m a c h i n e s h o u l d be d e s c r i b e d by e q u a t i o n s . The r e s u l t s o f a m a t h e m a t i c a l m o d e l i s a c o m p r o m i s e b e t w e e n c o m p u t a t i o n t i m e and a c c u r a c y and i s t o be d e c i d e d f r o m d i r e c t c o m p a r i s o n o f c o m p u t a t i o n and t e s t , n o t by a r b i t r a r y c h o i c e . I t i s t h e o b j e c t i v e o f t h i s c h a p t e r t o i n v e s t i g a t e a n d t o c o m p a r e t h e c o m p u t a t i o n r e s u l t s o f s y n c h r o n o u s m a c h i n e m o d e l s o f d i f f e r e n t d e g r e e s o f d e t a i l s w i t h t h o s e f r o m m a c h i n e t e s t s . A n o t h e r i m p o r t a n t p o i n t i n m o d e l i n g i s t h a t t h e m a c h i n e must be d e s c r i b e d by e q u a t i o n s w i t h m e a s u r e a b l e p a r a m e t e r s . T h e r e i s much more f r e e d o m i n t h e m a t h e m a t i c a l m a n i p u l a t i o n o f e q u a t i o n s t h a n t h e m e t h o d s a v a i l a b l e i n ob- t a i n i n g r e l i a b l e p a r a m e t e r v a l u e s d i r e c t l y f r o m t e s t s . F o r e x a m p l e , t h e s y n c h r o n o u s m a c h i n e r e a c t a n c e s c a n be d e t e r m i n e d w i t h much b e t t e r a c c u r a c y t h a n t h e l e a k a g e r e a c t a n c e s . A t h i r d p o i n t i n m o d e l i n g i s t h a t s i n c e most o p t i m a l c o n t r o l t h e o r y , c o m p u t a t i o n and n o n l i n e a r s t a b i l i t y a n a l y s i s t e c h n i q u e s a r e d e v e l o p e d f r o m s y s t e m e q u a t i o n s i n t h e s t a t e v a r i a b l e f o r m , i t i s d e s i r a b l e t o m o d e l t h e s y s t e m e q u a t i o n s a s s u c h t o a l l o w t h e a p p l i c a t i o n o f t h e t h e o r y and o b t a i n t h e s o l u t i o n by known c o m p u t a t i o n a l m e t h o d s . I n t h i s c h a p t e r a one m a c h i n e - i n f i n i t e b u s s y s t e m m o d e l s c h e m a t i c a l l y shown i n F i g . 3.1 w i l l be d e v e l o p e d . The s t a t e e q u a t i o n r e p r e s e n t a t i o n o f t h e r e g u l a t o r - e x c i t e r i s Rp 'A + Bm SUMMING AMPLIFIER MACHINE TERMINAL AAA - W g INFINITE BUS G —— —- B F i g . 3.1. Dynamic T e s t M o d e l o f One M a c h i n e - I n f i n i t e Bus S y s t e m t o b a s e d on F i g . 2.1, and t h a t o f t h e g o v e r n o r - h y d r a u l i c s y s t e m o n F i g . 2 . 4 i n c o r p o r a t i n g t h e s t a t e e q u a t i o n o f t h e d-c m o t o r - b o o s t e r o f F i g . 2 . 2 . 3 . 1 . S e v e n t h O r d e r S y n c h r o n o u s M a c h i n e S t a t e E q u a t i o n s P a r k ' s e q u a t i o n s f o r a s y n c h r o n o u s m a c h i n e i n d-q 3 1 c o o r d i n a t e s ' a r e : v d = P^ d ~ W e t F q ~ V - d ( 3- 1) v q - PH>q + W e ^ " r a i q ( 3 . 2 ) Vd = 1+ Tmp xad_ v f d _ ( 1 + T d p ) (1+Tdp) _ x d i d ( l + T d o p ) ( l + T d ' o P ) W E O R F ( l + T d o p ) ( l + T d o p ) 4 0eo ( 3 . 3 ) H'q = - 1 + Tq-£ _JEfl i q ( 3 . 4 ) 1 + Tq 0P C J e o which can be r e a r r a n g e d i n t o t h e f o l l o w i n g s t a t e v a r i a b l e form P<Pd = v d + WeH>q + r a i d ( 3 ' 5 ) P^q = v q ~ W e ^ d + Vq (3'6) p V F = R v F - v F R ( 3 . 7 ) P*D = ~ V D R ( 3 ' 8 ) P V Q ^ - V Q R ( 3 ' 9 ) where Vp^, i d and v ^ a r e s o l v e d from i d V DR x d ( T d o + T D ) - x d ( T d o - T c ^ e o x d ( ? it tt , x d T d o T d o x d T d o x d T D d o xdo x d T x d ] d o - CJ, eo tt x d T d o do x!i T" d J d o ' d o x d ldo x , T d c v " T" T ' x d T d o J d o ^ d 9 D ( 3 . 1 0 ) a n d i q a n d V Q R f r o m e o • W e o ( x q - x q ) L q x q I q o it " x q T q o Q ( 3 . 1 1 ) T h e d e r i v a t i o n o f (3.5) t h r o u g h ( 3 . 1 1 ) i s p r e s e n t e d i n A p p e n - d i x 3A. F o r t h e s t u d y o f t h e m a c h i n e - i n f i n i t e b u s s y s t e m a s s h o w n i n F i g . 3 . 1 , t h e d - a n d q - a x i s m a c h i n e t e r m i n a l v o l t a g e c a n b e e x p r e s s e d i n t e r m s o f t h e i n f i n i t e b u s v o l t a g e a n d t h e t o r q u e a n g l e b e t w e e n s y n c h r o n o u s m a c h i n e q - a x i s a n d i n f i n i t e b u s v o l t a g e a n d t h e m a c h i n e d - a n d q - a x i s c u r r e n t a s f o l l o w s . k-^r+k^x - ( k n x - k 2 r ) v d q k ^ x - k r k ^ r + k 0 x + k l k 2 • k 2 k x v Q sinS v c o s S o ( 3 . 1 3 ) where k± = (1-xB+rG)/((1-xB+rG) 2 + (xG+rB) 2) (3.12) k 2 = (xG +rB) /((1-xB+rG) 2 + (xG+rB) 2) To complete the d e s c r i p t i o n of the synchronous machine dynamics, two a d d i t i o n a l equations are obtained as f o l l o w s . From the e q u i l i b r i u m equation f o r torque P wm = J < T i ~ DcJm " T e ) (3.13) where the t h r e e phase e l e c t r i c a l torque i s T e - 3 P ° * e s (<P d i q - * q i d ) (3.14) From the r e l a t i o n between e l e c t r i c a l torque angle and mechanical speed PS = poles u - to (3.15) 2 m GO Equations (3.5) through (3.9), (3.13) and (3.15) are the seventh order synchronous machine equations i n the s t a t e v a r i a b l e form and (3.10), (3.11), (3.12) and (3.14) are the a u x i l i a r y e q u a t i o n s . 3.2. S t a t e Equations of C o n t r o l l e r s and DC Motor-Booster The t r a n s f e r f u n c t i o n o f the r e g u l a t o r - e x c i t e r , F i g . 2.7, i n s t a t e equation form i s " - ~ V R + | * > r e f - v t ) IRE !RE f o r t h e r e g u l a t o r and P v f d , X. f ( v R ) _ i Y f d (3.17) f o r t h e s y n c h r o n o u s machine f i e l d v o l t a g e . The c h a r a c t e r - i s t i c s o f t h e f i e l d v o l t a g e l i m i t e r a r e a p p r o x i m a t e d by f ( v R ) = ai t a n h . ( a 2 v R ) (3.18) where t h e v a l u e s o f a^ and a 2 a r e d e t e r m i n e d f r o m a l e a s t s q u a r e s c r i t e r i o n . The t e r m i n a l v o l t a g e i s V t = / v d + V q <3.19) The t r a n s f e r f u n c t i o n o f t h e g o v e r n o r - h y d r a u l i c s y s t e m , F i g . 2.4* i n t h e s t a t e e q u a t i o n a r e as f o l l o w s . F o r t h e a c t u a t o r p o s i t i o n pa = JT a -._! a f - , (3.20) . " T A T A T A f o r t h e a c t u a t o r f e e d b a c k p o s i t i o n p a f = -UJL a - STV+ T A a f ~ 6_&">m (3.21) T A T r T A • T A f o r t h e g a t e p o s i t i o n oo- = JL a - _ i _ S (3.22) P C > T G T G and f o r the t u r b i n e torque output pt - - a + Tg + TW g - , ,1 .„ t (3.23) . 5T G . 5 T G T w . 5T W The booster armature v o l t a g e s t a t e equation from (2.13) i s p v B = _ ̂ g L a f v f B - 1 v B (3.24) R f B T f B T f B where the boo s t e r f i e l d v o l t a g e s i g n a l of F i g . 2.2 i s computed by v f B = X (t_) _ fe^R ^ + ^ £ f i pk) K 3 \ 21 \ K 3 K3/ 2 K 3 V2' _ T f B f 2 ^ - P ( ^ ) (3.25) The s t a t e equation of the mechanical speed of the t e s t model of (3.13) becomes Nm\ = L a F j F Lk _ *1 (j^m) ^2 _ £e (3.26) P \ 2 / 2J J \ 2 / 2J 2J where the d-c motor torque of (3.13) i s computed by T ± = L a F i F i A (3.27) the d-c motor and booster armature c u r r e n t by i A = K d c " vm - V B ) / R (3.28) and t h e motor a r m a t u r e v o l t a g e by vm 2 L a F i p p (3.29) The s t a t e v a r i a b l e tJ m i n s t e a d o f U) i s c h o s e n b e c a u s e t h e ~r m t a c h o m e t e r o u t p u t v o l t a g e i s 94.25 v o l t s a t s y n c h r o n o u s s p e e d (188.5 r a d / s e c ) . The damping c o e f f i c i e n t d e t e r m i n e d e x p e r i m e n t a l l y i s D = KJL + K 2 W~ (3.30) m F i n a l l y , t h e e l e c t r i c a l t o r q u e a n g l e s t a t e e q u a t i o n f r o m (3.15) becomes pS = 4 _ W E O (3.31) 3.3. I n i t i a l S t a t e o f a Power S y s t e m The i n i t i a l s t a t e s o f a power s y s t e m , v d , Vq, i ^ , i q , v Q and S, a r e d e t e r m i n e d f r o m t h e o p e r a t i n g c o n d i t i o n s , i . e . , t h e r e a l power P, t h e r e a c t i v e power Q and t h e v o l t a g e a t t h e m a c h i n e t e r m i n a l f r o m t h e f o l l o w i n g n o n l i n e a r a l g e b r a i c e q u a t i o n s . P = v d + v q ̂  (3.32) 2 ^ v q i d - v d i q (3.33) v t = V d + v q (3.34) v d = " V d + x q i q ( 3 ' 3 5 > v d = k - ^ v ^ s i n S + r i d - x i ) + k 2 ( v 0 c o s S + x i d + r i ) (3.36) v q - k 1 ( v Q c o s S + x i d + r i ) - k 2 ( v Q s i n S + r i d ~ x i q ) (3-37) F o r t h e p a r t i c u l a r c a s e s s t u d i e d i n t h i s t h e s i s , t h e s e e q u a - t i o n s a r e s o l v e d by t h e me t h o d o f F l e t c h e r a n d P o w e l l . An i n i t i a l e s t i m a t e o f t h e s o l u t i o n o f i q , v d , V q and i d i s o b t a i n e d f r o m t h e c l o s e d f o r m s o l u t i o n s 3 • 4 w h i c h n e g l e c t s a r m a t u r e r e s i s t a n c e and t h a t o f v Q and 8 f r o m t h e t r a n s m i s s i o n c o n f i g u r a t i o n o f F i g . 3.1. 1 q = • — (3.38) 4 ( P x q ) 2 + ( v | + Q)2 v d = x q i q (3.39) v q = H " V d (3.40) . 2 q Q + X n i r i , i d = v 9 q ( 3 . 4 D 'b = \ / [ > d - r ( i d - v d G+VqB) + x ( i q - v d B - v q G ) ] 2 2 7 + l > q - r ( i q - v d B ~ v q G ) " x ( i d ~ v d G + v q B ) ] (3.42) S = a r c t a n v d - r ( i d - v d G + v q B ) + x ( i q - v d B - v q G ) V q - r ( i q - V d B-v 6 ) - x ( i d - v d G + v q B ) (3.43) The i n i t i a l v a l u e o f t h e f i e l d v o l t a g e , r e q u i r e d f o r t h e i n t e g r a t i o n o f ( 3 . 1 7 ) , i s d e t e r m i n e d f r o m (3.1) a nd (3.3), ( v q + X d A d + Vaiq) Ri x ad (3.44) where W = & e eo, and R f ^ad R f \ / r a + X / ( r a + x d x ) 'slope' (3.45) The 'slope' i s determined by the slope of the steady s t a t e s h o r t c i r c u i t c h a r a c t e r i s t i c of the synchronous machine, r e l a t i n g RMS armature phase curren t and d-c f i e l d c u r r e n t . The d e r i v a t i o n of the expression f o r i s presented i n Appendix 3B. x a d The i n i t i a l values of ^p, tp d and are determined from (3A. Tdo _ T d o 1 - x d U)eo 0 0 0 0 -x. eo (3.46) The f o l l o w i n g c o n d i t i o n s are a p p l i e d DI T 0 T d o = ° V 0 T T d TD 0 VDR = 0 VQR ~ 0 (3.47) Since V p R = Vp d u r i n g steady s t a t e , the i n i t i a l r e g u l a t o r v o l t a g e i s VR arctanh a 2 \ a x (3,48) and the r e f e r e n c e v o l t a g e i s e s t a b l i s h e d from V r e f = ( V R + KA V / \ ( 3 ' 4 9 ) The i n i t i a l v a l u e s of the g o v e r n o r - h y d r a u l i c prime mover are a = 0 (3.50) a f = 0 (3.5D g - 0 (3.52) t - 0 (3.53) The i n i t i a l value of the booster v o l t a g e of (3.24) i s v B = 0 (3.54) s i n c e v^g = 0, (3.25). The i n i t i a l r e a l power of the dynamic t e s t model i s s e t by the l o a d s e t t i n g r e s i s t o r i n t h e d-c motor-booster armature c i r c u i t . The value of r e s i s t - ance i s e s t a b l i s h e d from (3.28), (3.20) and (3.54) R = K d c " u m L a F i f ) / lA ( 3 . 5 5 ) where i A i s o b t a i n e d from the steady s t a t e torque equation and i s equal to LA = ( K l + K2 + Vd ^ ~ «Pq id) / L a F % ( 3'56) The f i r s t two terms on the r i g h t hand s i d e of (3.56) are torque terms due to f r i c t i o n and the l a s t two terms are the torque of the synchronous machine. The mechanical synchron- ous speed of the t e s t model i s tO m - 188.5 rad/sec (3.57) 39 3.4. S t a t e Equations of F i f t h Order-Synchronous Machine and C o n t r o l l e r s The f i f t h order-synchronous machine s t a t e equations are obtained by n e g l e c t i n g the damper winding e f f e c t s of the seventh order model. By removing (3.8) and (3.9) and s e t t i n g DI 0 , T do 0 , T" = 0 , T % 0 , T" = 0 qo (3.58) A = 1, B = 0, (3.59) and T = T, X d T d o T D = 0 (3.60) i n (3A.2) , Park's equations reduce to P^d = v d + (Jeij> + r a i p<uq = V q - w e y d + r a i q P^F = V F ~ VFR (3.61) (3.62) (3.63) where (3.10) i s r e p l a c e d by FR d̂ x d "^eo ( x d ~ x d ) x d ido x d J d o eo x d V, Yd (3.64) and (3.11) becomes eo ^ q (3.65) F u r t h e r e l i m i n a t i o n o f v d , Vq, i d , i q a n d Vp-^ i n (3.6l) t h r o u g h (3.63) u s i n g (3.12), (3.64) and (3.65) r e s u l t s i n t h e f o l l o w i n g s t a t e e q u a t i o n s . PVF = - x d V F + ^ e o K l - X d ) Yd x d T d o x d (3.66) + ( r a + X d X q ) t s l ° p e ' V f d P ^ c l = k l r + k 2 x + r a ^ F " k l r + k 2 x + r a ̂ e o ^ d x d ldo x d + k l X " k 2 r Weo«Pq + 4 * ° ^ 4>q + k l v o s i n 8 + k 2 V o c p s S x 2 q (3.67) P < f q = k l X " k 2 r V F " k l X " k 2 r W e o " 4 - ° ̂ m ^ d x d T d o x d 2 ~ k l r + k 2 x + r a ^ e o ^ q " k l v 0 s i n 8 + k l v o c o s S X q (3.68) The s t a t e e q u a t i o n s (3.26) and (3.31) f o r m a c h i n e d y n a m i c s , (3.16) and (3.17) f o r r e g u l a t o r - e x c i t e r , (3.20) t h r o u g h (3.23) f o r g o v e r n o r - h y d r a u l i c s y s t e m , (3.24) f o r b o o s t e r a r m a t u r e v o l t - a g e , and h e n c e t h e a u x i l i a r y e q u a t i o n s o f them r e m a i n u n c h a n g e d e x c e p t t h e e l e c t r i c a l t o r q u e T o f (3.26) and (3.14) now e q u a l t o T e - ecoeo U - J L \ - ~ - V W F (3.69) \ x d x q / x d T d o t h e d- and q- a x i s v o l t a g e s o f ( 3.19) f r o m 3 . 6 1 ) , ( 3 . 6 2 ) , ( 3 . 6 4 ) and ( 3 . 6 5 ) now e q u a l t o v d = P^d ~ we4>q " • i r a i ~ VP + £iL_^eo Vd ( 3 . 7 0 ) x d i d o x d v q = PVq + w e V d + r a »eo Vq ( 3 . 7 1 ) x q T h e s e e q u a t i o n s c o n s i s t o f t h e c o m p l e t e s e t o f s t a t e and a u x i l i a r y e q u a t i o n s f o r t h e power s y s t e m w i t h a f i f t h o r d e r - s y n c h r o n o u s m a c h i n e and c o n t r o l l e r s . The i n i t i a l c o n d i t i o n s f o r t h e s t a t e e q u a t i o n s o f t h e f i f t h o r d e i — s y n c h r o n o u s and c o n t r o l l e r s a r e t h e same as p r e - s e n t e d i n S e c t i o n 3 . 3 . The v a r i a b l e s i d * i q , v d , v q , v^. and i p d a r e c a l c u l a t e d f r o m t h e s o l u t i o n o f t h e s t a t e v a r i a b l e s a t e a c h i n t e g r a t i o n s t e p . 1 VF - 0 0 eo y d ( 3 . 7 2 ) a. d x d T d o 1 q ~ ^ Vq ( 3 . 7 3 ) v d ~ ( k i r + k 2 x ) i d ~ ( k ^ x - k 2 r ) i . q + ^ l v o s * - n ^ + k o v c o s S * o ( 3 . 7 4 ) v q = ( k 1 x - k 2 r ) i d + ( k 1 r + k 2 x ) i q + k, v Q c o s 8 - k 2 v 0 s i n S ( 3 . 7 5 ) ( 3 . 1 9 ) 4- - f d r 2 4- x 2 1 a x q x d V F ( x d " X d ) ^eoVd^ ( r 2 + x d x q ) ' s l o p e ' V Tdo xd t x d (3.76) 3-5. S t a t e E q u a t i o n s o f T h i r d O r d e r - S y n c h r o n o u s M a c h i n e and C o n t r o l l e r s The t h i r d o r d e r s y n c h r o n o u s m a c h i n e s t a t e e q u a t i o n s a r e o b t a i n e d f r o m t h e f i f t h o r d e r model w i t h two more assump- t i o n s . F i r s t , t h e s p e e d v o l t a g e e f f e c t s due t o s p e e d v a r i a t i o n i n (3.61) and (3.62) a r e n e g l i g i b l e ; w e<pd ~ weo(|;d , wecpq ~ u>eoH>q (3.77) S e c o n d , t h e i n d u c e d v o l t a g e e f f e c t s due t o t h e change o f f l u x l i n k a g e s a r e much s m a l l e r t h a n t h e s p e e d v o l t a g e s ; p<rd < < "eoYq ., P^ q << Vd (3.78) As a r e s u l t , t h e two s t a t e e q u a t i o n s (3.61) and (3.62) r e d u c e t o a l g e b r a i c e q u a t i o n s and may be w r i t t e n 1 0 0 1 0 OJ •U) 0 e ^ d + r 0 a 0 r . (3.79) where v d and v c a n be e l i m i n a t e d u s i n g (3.12) and i d and i . q u s i n g (3.64) and (3.65). F i n a l l y , ty^ and <f a r e e x p r e s s e d a s 4.3 Vq 0) eo o + x _ ( x n + x n ) o x o J ,1 x q x d ^do • r 0 ( x 0 + x d ) + r Q x 0 1 2 m i x,, T do V, Vt r o k l - k 2 ( x o + X q ) r k +k ( x +x ) o 2 1 o q l r Q k 2 4 k 1 ( x Q + x d ) r o k l - K 2 ( x o + x q ) t x d x. v Q s i n 8 v Q c o s S (3.80) where A = C 0 2 O r 2 + (x Q+x q) (xQ+x-d) x d x and ^o = k l r + k 2 x + r a c Q = k l X - k 2 r (3.81) A f t e r e l i m i n a t i n g V̂ , (3.63) r e d u c e s t o 1 / x d - ( X d - X d ) ( rg+ X o( x 0+x q ))\ifF + ( r!+x dx q) ' s l o p e ' x d T d o ^ o + ( x o + X q ) ( x o + x d ) \ / r a + xq R . f d + ( x d ~ x d ) ( r o k l - K 2 ( x o + x q ^ v o s i n S + ( x d - x d ) ( r Q k 2+l< 1 ( x 0 + x q ) ) v Q c o s 8 r^+Uo+x ) ( x D + x d ) ^ o + ( x o + x q ) ( x o + x d ) (3.82) The o t h e r two s y n c h r o n o u s m a c h i n e d y n a m i c s t a t e e q u a t i o n s (3.26) and (3.31) and t h e a u x i l i a r y e q u a t i o n (3.69) r e m a i n u n c h a n g e d . 44 The e x c i t e r s t a t e equations (3.16) and (3.17) and the t e r m i n a l equation (3.19) remain the same. However, w i t h the c o n d i t i o n s (3.77) and (3.78), (3.70) reduces to "d " -"e<A " _ J ^ _ f" *d ( 3 _ 8 3 ) x d T d o x d and (3.71) to v q = "eo^q 4 r a ^eo V q X q (3.84) The s t a t e equations d e s c r i b i n g the governor-hydraulic prime mover (3.20) to (3.23) and the d-c motor-booster s t a t e equation (3.24) remain unchanged. In summary (3.82), (3. 26), (3.31), (3.16), (3.17), (3.20) to (3.23) and (3.24) form the s t a t e equations of the t h i r d order-synchronous machine wi t h c o n t r o l l e r s , and the a u x i l i a r y equations are (3.28), (3.29), (3.69), (3.85), (3.86), (3.18), (3.19), (3.83), (3.84) and (3.25). The method of e v a l u a t i n g the i n i t i a l c o n d i t i o n s i s described i n S e c t i o n 3.3. The v a r i - ables .^d, <p , , 1^, v^, V q , v̂ . and i ^ are c a l c u l a t e d from the s o l u t i o n of s t a t e v a r i a b l e s at each step as f o l l o w s 2 ' m = . r p + x 0 (x 0+x q) ( j ; p + x d ( r p k j - k 2 (x 0+x q+x q) ) s±n S w e o T d o ( r ^ + C xo + xq)C^ + xd) w e o + X d ( r p k z - t - ^ C x o + X q ) ) cos S (3.85) 4 5 = " q ^ o ^ ^ o ^ o ^ ^ V F + x q ( r o k 2 + k 2 ( x o + x d > > s i n 5 w e o x d T d o " e o + x q ( r 0 k 1 - k 2 ( x 0 + x d ) ) C Q s S ( 3 . 8 6 ) ^ e o i H - - ^ e o <Pd ( 3 . 8 7 ) * d?do x d i q = - ^ e o <pq ( 3 . 8 8 ) . x q v d = " Weo ^ q " r a i d ( 3 . 8 9 ) v q = "eo <Pd " r a i q ( 3 . 9 0 ) v t - 7 Vd + V q ( 3 ' 9 1 ) ± f c i = \ / r a + x q / x d ~ ( x d - x d ) "eoVd^j ( 3 . 9 2 ) ( r | + x d x q ) s l o p e \ x d T d o x d / 3 . 6 . S t a t e E q u a t i o n s of T h i r d O r d e r - S y n c h r o n o u s M a c h i n e and R e g u l a t o r - E x c i t e r b u t w i t h o u t G o v e r n o r F o r t h e s t u d y of e l e c t r i c a l t r a n s i e n t s of t h e s y s t e m , t h e g o v e r n o r - h y d r a u l i c p r i m e mover d y n a m i c s a r e r e p l a c e d w i t h a c o n s t a n t t o r q u e i n p u t . The e q u a t i o n s ( 3 . 8 2) and ( 3 . 3 1 ) d e - s c r i b i n g t h e s y n c h r o n o u s m a c h i n e a n d ( 3 . 1 6 ) and ( 3 . 1 7 ) d e s c r i b - i n g t h e r e g u l a t o r - e x c i t e r r e m a i n u n c h a n g e d . The s t a t e e q u a t i o n f o r m e c h a n i c a l s p e e d ( 3 . 2 6 ) becomes 4 6 i 2 ./ 2Jw m b J 2 J 2 (3.93) J \ x d x q / ^ d o The a u x i l i a r y e q u a t i o n s a r e (3.85), (3.86), (3.18), (3.19), (3.83) and (3.84). The i n i t i a l c o n d i t i o n s a r e e v a l u a t e d by t h e t e c h n i q u e s d e s c r i b e d i n S e c t i o n 3.3 and t h e v a r i a b l e s Yd, Yq, i d } l q > v q > v t a n d ±fd (3.85) t h r o u g h (3.92). 3.7. C o m p u t a t i o n and T e s t R e s u l t s o f t h e Dynamic T e s t M o d e l I n t h i s S e c t i o n , c o m p u t a t i o n r e s u l t s o f v a r i o u s s t a t e v a r i a b l e m o d e l s o f S e c t i o n 3.4, 3.5 and 3.6 and r e s u l t s f r o m a c t u a l t e s t s a r e summarized and a c o m p a r i s o n i s made i n o r d e r t o v e r i f y t h e m a t h e m a t i c a l m o d e l s . A t r a n s i e n t t e s t i s c a r r i e d o u t on t h e power s y s t e m d e s c r i b e d i n C h a p t e r 2 s i m u l a t e d on t h e d y n a mic t e s t m o d e l . The s y s t e m h a s a t h r e e - p h a s e f a u l t b e f o r e t h e f a u l t l i n e o f a d o u b l e c i r c u i t t r a n s m i s s i o n s y s t e m i s i s o l a t e d a t 5 c y c l e s . The f a u l t i s c l e a r e d and t h e s y s t e m r e s t o r e d a f t e r 30 c y c l e s . The t e s t r e s p o n s e s o f t h e t o r q u e a n g l e 8, g o v e r n o r a c t u a t o r p o s i t i o n a, g o v e r n o r a c t u a t o r f e e d b a c k a f , g a t e t r, t u r b i n e t o r q u e t , b o o s t e r a r m a t u r e v o l t a g e vg, t e r m i n a l v o l t a g e v t , r e g u l a t o r v o l t a g e v ^ and f i e l d c u r r e n t i P { j a r e r e c o r d e d on a V i s i c o r d e r . The r e s u l t s a r e p l o t t e d a l o n g w i t h t h e computa- t i o n r e s u l t s o f t h e t h r e e d i f f e r e n t s t a t e v a r i a b l e m o d e l s . T h e c u r v e s i n F i g . 3.2 t h r o u g h 3.11 a r e i d e n t i f i e d a s f o l l o w s : 1. t h i r d o r d e r m a c h i n e w i t h e x c i t e r , 2. t h i r d o r d e r m a c h i n e v / i t h e x c i t e r and g o v e r n o r , 3. f i f t h o r d e r m a c h i n e w i t h e x c i t e r and g o v e r n o r , 4. d y n a m i c m o d e l t e s t r e s u l t s . Hamming's n u m e r i c a l i n t e g r a t i o n m e t h o d 3 " ^ i s u s e d f o r c o m p u t a t i o n w i t h an i n t e g r a t i o n s t e p s i z e o f 0.000 25 s e c o n d s f o r t h e f i f t h o r d e r m a c h i n e w h i c h i n c l u d e s p f ^ and pV q and 0.005 s e c o n d s f o r t h e o t h e r two m o d e l s . The c o m p u t a t i o n r e s u l t s o f a l l t h r e e m o d e l s a r e v e r y c l o s e e x c e p t t h e t e r m i n a l v o l t a g e r e s p o n s e o f t h e f i f t h o r d e r m a c h i n e m o d e l , c u r v e 3, F i g . 3.8. The f i f t h o r d e r m o d e l p r e d i c t s t h e v o l t a g e s p i k e s due t o l i n e s w i t c h i n g a t f a u l t c l e a z ^ e d and s y s t e m r e s t o r e d w h i c h i s s u b s t a n t i a t e d by a c t u a l t e s t r e s u l t s . I t i s a l s o o b s e r v e d t h a t b o t h t h e t h i r d and f i f t h o r d e r m o d e l s w i t h g o v - e r n o r , c u r v e s 2 and 3, F i g . 3.2, a r e s l i g h t l y more u n s t a b l e t h a n t h e t h i r d o r d e r i v d t h o u t g o v e r n o r , c u r v e 1, F i g . 3.2. T h i s phenomena i s o b s e r v e d a l s o f r o m d i r e c t m o d e l t e s t s . A c l o s e c o r r e l a t i o n b e t w e e n c o m p u t a t i o n a l and t e s t m o d e l r e s u l t s i s o b s e r v e d e x c e p t f o r t h e r e g u l a t o r v o l t a g e , F i g . 3.9, and t h e f i e l d v o l t a g e , F i g . 3.10, a t t h e i n s t a n t t h e s y s t e m i s r e s t o r e d . T h i s may be a t t r i b u t e d t o t h e i m p e r - f e c t m a t h e m a t i c a l m o d e l . F o r e x a m p l e , i t d o e s n o t i n c l u d e t r a n s m i s s i o n l i n e s w i t c h i n g , s i n c e i t i s d e s c r i b e d by s t e a d y s t a t e e q u a t i o n s . H o w e v e r , t h e p r e v a i l i n g f r e q u e n c y o f o s c i l - l a t i o n i s t h e same.  F i g . 3.3. Governor Actuator P o s i t i o n Transient Responses o  1.5n T I M E ( S E C ) F i g . 3.5. G o v e r n o r Gate P o s i t i o n T r a n s i e n t R e s p o n s e s 1—1 F i g . 3.6. T u r b i n e T o r q u e T r a n s i e n t R e s p o n s e s 3 n TIME (SEC) F i g . 3.7. B o o s t e r A r m a t u r e V o l t a g e T r a n s i e n t R e s p o n s e s O J 100 - i cc 3 0 S 2 0 H 10 0 fau/f cleared \system restored 0 I — I — I — I — I — I — I — I — 1 — I — I — I I I I 0^5 l!o 1.5 2.0 T I M E ( S E C ) i — i — i — j — i — i — i — i — ] — i — i — i — i — i — i — i ' ' i 2.5 3.0 3.5 4 . 0 F i g . 3 . 8 . Terminal Voltage Transient Responses F i g . 3.9. R e g u l a t o r V o l t a g e T r a n s i e n t R e s p o n s e s  F i g . 3.11. F i e l d C u r r e n t T r a n s i e n t R e s p o n s e s 4. PARAMETER SE N S I T I V I T Y OF THE TEST MODEL S e n s i t i v i t y a n a l y s i s ^ " J ^.2 i s a p p l i e d i n t h i s c h a p - t e r t o i n v e s t i g a t e t h e e f f e c t o f model p a r a m e t e r s on s y s t e m r e s p o n s e . As shown i n S e c t i o n 2.3.3, o n l y i m p o r t a n t model p a r a m e t e r s , n o t a l l o f them, can be m a tched s i m u l t a n e o u s l y w i t h a c t u a l s y s t e m v a l u e s ( T a b l e 2.1). The model b a s e imped- a n c e c h o s e n i s a compromise. 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 s e e w h e t h e r t h i s a p p r o a c h i s j u s t i f i e d . The c o n t r o l l e r s a r e i n c l u d e d i n t h e i n v e s t i g a t i o n . The s t a t e e q u a t i o n s f o r t h e s y n c h r o n o u s m a c h i n e and c o n t r o l l e r s a r e g i v e n i n S e c t i o n 3.4. 4.1. S e n s i t i v i t y E q u a t i o n T h e s t a t e v a r i a b l e e q u a t i o n i s w r i t t e n as x i ~ ' % ( x > q r ) > q r = q / q 0 ( 4 . i ) where q r i s t h e r e l a t i v e v a l u e o f t h e p a r a m e t e r , q t h e t r u e v a l u e and q Q a c o n s t a n t e q u a l t o i n i t i a l v a l u e o f t h e p a r a - m e t e r . The s e n s i t i v i t y e q u a t i o n f o r s m a l l p a r a m e t e r p e r t u r - b a t i o n s becomes n S * f i . x , _ + q Q b £ i x i > a „ = f £ i . x L, + q Q i i i (4.2) K J- ox, d q where x , ^ = cbq, ( 4 > 3 ) d q r 59 The r i g h t hand s i d e o f t h e s e n s i t i v i t y e q u a t i o n ( 4 . 2 ) c o n s i s t s o f two t e r m s ; t h e f i r s t t e r m i n c l u d e s t h e s t a t e v a r i a b l e s e n s i t i v i t y c o e f f i c i e n t s o n l y and t h e s e c o n d t e r m d e pends upon p a r a m e t e r s e x p l i c i t l y . The two t e r m s w i l l be i d e n t i f i e d as f l . ( x , x q ) and f 2 . (x) r e s p e c t i v e l y , where n f l i a < x> xi, „ ) = S M i x. n (4.4) i > ^ r k , q r k = l T k , q r and f 2 i > q r ( x ) = q Q M i U > 5 ) q When t h e r e s u l t s a r e a p p l i e d t o s y s t e m e q u a t i o n s o f S e c t i o n 3.4, t h e t e r m s i n d e p e n d e n t o f p a r a m e t e r s a r e fH X d v + W e o ( xd~ xd) x. „ f l v F , q r = " r ^ y ^ r ^ r X l t f d > q r (4.6) + ( r a 4 x H x a ) ' s l o p e ' + 2 , SL_3 v f d ^ r 7 r 2 + x2, R a q r f 1 , r o ro "eox + x o w e o x _ V d ^ r x ^ X ^ F ^ r ^ V d ^ r + 7Z + 4 em x + 4 ( j ) q x ^ ^ q r + k 1 v o c o s 8 s i n U ^ . ) 2 Q ^ I** n q> *r "2" k 0 v s i n S c o s (x ) 2 ° S ' q - (4.7) 60 f l _ x o d do x o u e o x x d 4 w m x - 4 ip, x - r o u e o x 2 r d ^ r d «m,<Ir x V q r 2 q k-,v cosSsin (x ) - k,v sinScos (x ) x ° 6>q n 1 ° s>q„ (4.8) f l 2 ( L aF F JR Ha> qr 2 - L a F x F x v 2JR - i ^eo / ^ _ j W « p d x + V x J ^5 x q j \ ^q> qr q Y d , q r / Jxd^d^ I F r V q r q ^ F ^ r j (4.9) = 4 x 6J _21> 1 x K A TRE R' r T R E v t _ 1 _ (vd( r 0 - r a) + v x Q )x x d T d o 4 v F , q r (0. f ( v d ( r 0 - r a ) + v q X Q ) + ̂  C v d x 0 - v q ( r 0 - r a ) ) r V q r + C v d^ k i v 0 c o s^~' s'2 V o s i n ^ ) + vq( ~' c2 V o C O S ^ ~ k ] V o S " ' " n ^ ^ 'S,q, (4.11) 61 2, pi = _ _ ! x + a l A 2 s e c h ( a , v R ) x /, -, ,-> \ f S d , q r T E ^ f d ^ r " T T 2 V R ' q r ( 4 " 1 2 ) f l a , q - ~ Z. *a,q " J L x a - _2_ x ( 4.1 3) T A M r T A f q r T A - f ' q r f l = -<rj x - & T r -I- T A x - 2 _ x (4.14) a f > q r T A a ' q r T r T A a f q r T A ^ ^ r f l f f n = x a a " _1_ x q (4.15) g ' q r T G 3 , q r T Q g , q r t , q = ~ . 1 x a q + T G + T W x - 1 , x. (4.16) • 5T G . 5 T G T W S , q r > 5 T w t , q r f l v a = - 2* <*>gLaf 1 x + ( 2 * « R _ 2 K 4 _ \ O g L a f x B > q r K 3 R f f i T f B 2 ' q r V K3 K3 / R f f i T f B S n , q r 2 « T f B " g L a f 1 f l , a + T f p f 2 o ( c t R - 2K 4\<J g L a f f l K3 R f B T f B 2 ' V \ K3 K 3 / R f B T f B % i , 4 r 2 1 X T f B V B > q r (4.17) The p a r a m e t e r s i n v o l v e d i n t h e s e n s i t i v i t y i n v e s t i g a - ' t t x o n a r e r a , x d , X q , x d , T d o and J o f t h e s y n c h r o n o u s m a c h i n e , K A , T̂ JJ- and T^ o f t h e r e g u l a t o r - e x c i t e r and S, T, T^, T R , T G and T^ o f t h e 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 . F o r t h e a r m a t u r e r e s i s t a n c e r . (4.5) y i e l d s f2... = r a T s l o p e ' ( 2 x/r^+x2, - V ^ + X d X 2 ) \ v (4.18) T ^ T V a q ~l^r~~ td 62 f2 = r a w - ra ( l >eo o> (4.19) UJ . r- —-r;:rr— T P T > D rVi r n ~"n;rr- ' F d icio x d f 2 = - r a u e o <tf (4.20) iu r — T q T q, 1 a x r f o r the d-axis synchronous reactance f2 x - - VF • + ^ M 2 V d + y ^ s l o p e t V ^ X d x ^ T ^ R f r 2 + x 2 f o r the q-axis synchronous reactance x 'slope' / / 2. 2 x ( r + X |x„)\ /. f2„ := - X o u e o ID ' (4.23) V x q ~ x ^ ~ ^ q f2 = r o w e o <p (4.24) u; x T'q Tq, q X q f2 = A Xq C v . f ? + v _ ( f 2 ~ a' e o y _)) (4.25) f 2 = - 3w e o v d ^ q (4.26) (j x ~7 T i ) q J x q 2 f 2 v x = Vffi/^ ~ --4 !̂ £al.V2» x (4.27) V B , X q q 1 K3 K3 , . R f B T f J % , x q 7 2 63 f o r the d-axis t r a n s i e n t reactance f 2 v v t = X d ; VF - x d f t o HJd (4.28) F' xd x d T d o . x d f , « _ r o + r o w e o ip , (4.29) t 2 y , v » — i — — F — i — d Y d , x d x d T d o x d f 2 , = - x o Vr- + V'eo V H (4.30) f2 , = - x o V p + x o w e o V H " " V X d Wd~o x d ~ r a S ^ ' " ^ , x d x ^ T d o * x j : f 2 „ k . - - ̂ d _ ( v d ( f 2 ^ + * F - - W d q " V V (4.31) f2 , = 3 U e o Vd V q -__J? "V F V q (4.32) *2* f 2 y x t = x d T f B - Hi ]«3iaf " (4.33) VB> xd \ Ko K 3 ; R f B T f B -f>*d f o r the d-axis o p e n - c i r c u i t t r a n s i e n t time constant F 2 V F ' T i o = ^ (4.34) *d> do x d l d Q (4.36) f 2 y ry, f R» do K A T d o ( v , ( f 2 , + V d ( 1 ) , 1 ' • rp • Y a > do x d l d o " T R E v t 5T" V  + W ao (4.37) f 2 u f ""TV- ( 4 , 3 8 ) m r P » J x ,X , 1 7 ' M o d d o T . = T d o T f B f ^ " £ ^ L \ ^ A F _ ^2 f (4.39) 1 3 , 1 do y K 3 K 3 / R f B T f B — ° f 2 v m a n d f o r t h e moment o f i n e r t i a J ±A = I ( v t - 2 L a p i F f£m - v B ) (4.40) f2 = f l n - L a F i F i A + S + 3 C J e o / l - l V <p ^m^ J 2~ 2J 2J J l x d x q ] q 2 _ _ J _ _ S _ V F cu (4.41) J x d T d o q 'm = - 94.25 (4.42) 2 f2 , = - _ _ J _ " g L a f t + £ V V a f ^ m V B , J K 3 T m R f f iT f B K 3 T m R f B T f B 2 + J T f B f f ^ . ^ \ ^ f . f 2 f i ) j (4.43) V K 3 K 3 / f B f B -f > N e x t f o r t h e g a i n o f t h e r e g u l a t o r - e x c i t e r , (4.5) y i e l d s f2 " Kef ~ V <4-44) R A LRE 6 5 f o r the r e g u l a t o r time constant T ^ f \ TPE = ^ " ( V " f " V t ' U - 4 5 ) ' K L X R E I R E and f o r the e x c i t e r time constant T^ f 2 v T = J L v f , - f l tanh ( a 2 v R ) ( 4 . 4 6 ) vfd> E T E 1 Q To F i n a l l y , f o r the t r a n s i e n t droop S of the governor-hydraulic operator, ( 4 . 5 ) y i e l d s ••= <im » 9 4 . 2 5 ( 4 . 4 7 2 2 f 2 ' = - ^ g _ a - a p - 2% ^m ( 4 . 4 8 ) f ' T A T A T A 2 f o r the permanent droop f 2 a j V r = - «L a ( 4 . 4 9 ) TA = - ^ i a ( 4 . 5 0 ) a f » v T A f o r the actuator servomotor time constant T A f 2 B 1 a + 1_ a f + 2_„ *_Sn ( 4 - 5 1 ) A T A T A T A 2 f 2 T =^S a + 1_ a p + 2S ( 4 . 5 2 ) af> A T A T A T A 2 66 f o r t h e d a s h p o t damping t i m e c o n s t a n t T R f 2 - _ = _1 a p (4.53) f o r t h e g a t e s e r v o m o t o r t i m e c o n s t a n t T g f2 m - - _1_ ( a - g ) (4.54) G T G f2. r„ = _2_ ( a - g ) (4.55) ' G T G and f o r t h e w a t e r s t a r t i n g t i m e c o n s t a n t T w f 2 t T = - _2_ ( g - t ) (4.56) ^ W T W The e q u a t i o n s , (4.6) t h r o u g h (4.56), a r e u s e d f o r t h e c o m p u t a t i o n . 4.2. P a r a m e t e r S e n s i t i v i t y o f S y s t e m R e s p o n s e P a r a m e t e r s e n s i t i v i t y o f s y s t e m r e s p o n s e i s i n v e s t i - g a t e d i n t h i s s e c t i o n . T h e r e a r e f i f t e e n p a r a m e t e r s and t w e l v e s t a t e v a r i a b l e s i n v o l v e d . From e a c h s e n s i t i v i t y c u r v e t h e max- imum and minimum a r e f o u n d . A c o n v e n i e n t c r i t e r i a i s s e t t h a t any p a r a m e t e r whose s e n s i t i v i t y c u r v e maximum and minimum f o r a s t a t e v a r i a b l e a r e l e s s t h a n o n e - s e v e n t h o f t h o s e o f a l l t h e s e n s i t i v i t y c u r v e s w i l l be c o n s i d e r e d i n s e n s i t i v e . T he r e s u l t s c a n be summarized i n t h r e e c a t e g o r i e s . One, p a r a m e t e r s a r e v e r y s e n s i t i v e i n t h e b e g i n n i n g and r e m a i n s e n s i - 67 t i v e f o r the r e s t of the time p e r i o d . Two, parameters are very- s e n s i t i v e i n the beginning but not towards the end. Three, parameters are i n s e n s i t i v e i n the beginning but become s e n s i - t i v e towards the end. In the f i r s t category, i t i s noted that the s e n s i t i v i t y c u r v e s H ' p ^ ( F i g . 4.1), V d Y ( F i g . 4.2), v R j X ( F i g . 4.10), q '"q q and v.p d x ( F i g . 4.11), a l l with respect to x„, have t h e i r ' q maximum du r i n g the f a u l t and f a u l t c l e a r e d p e r i o d and remain s e n s i t i v e a f t e r the system has been r e s t o r e d . A s i m i l a r r e s u l t i s observed f o r the parameter T d o except f o r typ <j>» ( F i g . 4.1), 5 do which i n c r e a s e s d u r i n g the f i r s t 0.2 seconds and remains l a r g e throughout the remaining p e r i o d . In the second category, i t i s observed t h a t Vd x ' ' d ( F i g . 4.2), ipn x i ( F i g . 4.3) and v R K ( F i g . 4.10) are l a r g e H , D K , I V A only d u r i n g the f a u l t p e r i o d ("Co.08 seconds). In the t h i r d category, i t i s n o t i c e d that the moment of i n e r t i a J i s i n s e n s i t i v e d u r i n g the f a u l t and f a u l t c l e a r e d p e r i o d but becomes very s e n s i t i v e towards the end. t I t i s found, i n general, from t h i s study that x q , T d o and J are the most s e n s i t i v e parameters and that to a c e r t a i n extent x d , K^, and T R E but not r&, x r f, T^ and the governor- h y d r a u l i c operator* parameters. RIR GRP FLUX IWEBERS) 0 50 100 150 200 250 300 350 I i • i ' 1 I I I 1 I 1 I I I I 1 I— I 1—I— I !— I— I—I 1—!—1 1—I— I 1— I — J—I PRRRMETER SENSITIVITY -400 -200 0 200 400 89 D-RXIS FLUX (WEBERS) -0.00 0.05 0.10 0.15 0.20 i i i • i i i i i i — i — i — i — i — i — i — i — i — i — i — I PRRRMETER SENSITIVITY -0.2 -0.1 -0.0 0.1 0.2 69 Q-RXIS FLUX (V/EBERS) -0.10 -0 .08 -0 .06 -0 .04 -0 .02 -0 .00 0.02     BOOSTER RRMRTURE VOLTRGE (VOLTS) - 3 - 2 - 1 0 1 2 3 I i i i i 1 i i i i I i i i i I i i i i !—i—i—i—i—I—i—i—i—i—I PRRRMETER SENSITIVITY -30 -20 -10 0 10 20 30 MECH SPEED / 2 (RRD/SEC) 0 20 40 60 80 100 11111 • 11111 • • i • i • • • 11 • 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 PRRRMETER SENSITIVITY -10 -5 0 5 10 15 91 REGULATOR VOLTRGE (VOLTS) -20 0 - 20 40 60 i i • i i i i i • • i i i i i i i i i i 1.1.. i i i 1.1 i i i * i i i i 1.1 PRRRMETER SENSITIVITY w Lt FIELD VOLTRGE tVOLTS) -15 -10 -5 0 5 10 15 DELTR RNGLE IRRD) 61 5 . NONLINEAR OPTIMAL S T A B I L I Z A T I O N OF A POWER SYSTEM AND DYNAMIC MODEL TESTS 5 . 1 . Power Sysbem S t a b i l i z i n g S i g n a l The m a i n p u r p o s e s o f a s t a b i l i z i n g s i g n a l a r e t o p r o v i d e a d d i t i o n a l s y n c h r o n i z i n g t o r q u e d u r i n g t h e f i r s t t o r q u e a n g l e s w i n g a f t e r a t r a n s i e n t d i s t u r b a n c e and t o p r o - v i d e d a m p i n g t o r q u e f o r s u b s e q u e n t o s c i l l a t i o n s . E l l i s and o t h e r s ^ • 1 > 5 . 2 , 5 . 3 were a b l e t o o b t a i n e f f e c t i v e s t a b i l i z a t i o n u s i n g s p e e d , a c c e l e r a t i n g power and f r e q u e n c y d e v i a t i o n s i g - n a l s . The t h e o r e t i c a l b a s i s i s f o u n d f r o m a l i n e a r m o d e l by d e M e l l o and C o n c o r d i a . " ' * 4 T h e s e a r e t h e c o n v e n t i o n a l s t a b i l - i z a t i o n t e c h n i q u e s . J o n e s ^ ' ^ d i d a b a n g - b a n g c o n t r o l t e s t on a m o d e l power s y s t e m . I n t h i s C h a p t e r , o p t i m a l c o n t r o l t h e o r y i s a p p l i e d t o a power s y s t e m d e s c r i b e d by n o n l i n e a r s t a t e e q u a t i o n s i n c l u d i n g f i e l d v o l t a g e and s t a b i l i z i n g s i g n a l l i m i t s . The power s y s t e m i s shown i n F i g . 3 . 1 . 5 . 2 . D y n a m i c O p t i m i z a t i o n and C o m p u t a t i o n a l M e t h o d The p r o b l e m b e i n g c o n s i d e r e d i s t o f i n d an o p t i m u m c o n t r o l u J , ;'(t) w h i c h m i n i m i z e s t h e p e r f o r m a n c e f u n c t i o n a l J - 0 ( x ( t f ) ) +j F ( x ) d t ( 5 . D t o s u b j e c t t o t h e d y n a m i c a l c o n s t r a i n t s o r s y s t e m e q u a t i o n s x = f ( x , u ) ( 5 . 2 ) The v a r i a t i o n a l c a l c u l u s method o f L a g r a n g e M u l t i p l i e r s i s u s e d t o f o r m an augmented f u n c t i o n a l 81 5 . 6 J a = # ( x ( t f ) ) + J ( - H ( x , A , u) + A 1 x ) d t ( 5 . 3 ) c o where H i s t h e H a m i l t o n i a n H ( x , A , u) - - F ( x ) + A T f ( x , u ) ( 5 . 4 ) and A. i s a t i m e - v a r y i n g c o s t a t e v a r i a b l e v e c t o r . The c o n d i - 5.7 t i o n s w h i c h must be s a t i s f i e d a t t h e optimum a r e x = f ( x , u) x ( t Q ) = X q s t a t e e q u a t i o n s ( 5 . 5 ) A = - H x ( x , A , u) c o s t a t e e q u a t i o n s ( 5 . 6 ) 0 = H ( x , A , u) g r a d i e n t c o n d i t i o n (5.7) A ( t ^ ) = ( x ( t ^ ) ) t r a n s v e r s a l i t y c o n d i t i o n ( 5 . 8 ) E q u a t i o n s ( 5 . 5 ) t h r o u g h ( 5 . 8 ) r e p r e s e n t a n o n l i n e a r t w o - p o i n t b o u n d a r y - v a l u e p r o b l e m ( T P B V P ) . F o r t h e s o l u t i o n o f t h e TPBVP, t h e g r a d i e n t ^ * ^ and r 7 Newton-Raphson w i t h R i c a t t i T r a n s f o r m a t i o n J ' ' methods a r e t e s t e d f i r s t on a s i m p l i f i e d t h i r d - o r d e r power s y s t e m . T h e r e i s no d i f f i c u l t y w i t h t h e c o m p u t a t i o n . The method o f Newton- Raphson w i t h R i c a t t i t r a n s f o r m a t i o n i s t h e n a p p l i e d t o t h e f i f t h - o r d e r s y s t e m u s i n g t h e g r a d i e n t method t o s t a r t . Num- e r i c a l i n s t a b i l i t y i n t h e s o l u t i o n o f t h e R i c a t t i e q u a t i o n i s e x p e r i e n c e d . The s o l u t i o n o f t h e TPBVP i s f i n a l l y o b t a i n e d w i t h t h e g r a d i e n t method. I t t a k e s f i f t y i t e r a t i o n s o r l e s s . T h r e e d i f f e r e n t p e r f o r m a n c e f u n c t i o n s a r e c h o s e n and t e s t e d f o r e a c h s t e p o f t h e l i n e f a u l t , f a u l t c l e a r e d (one c i r c u i t o f a d o u b l e c i r c u i t r e m o ved), and l i n e r e s t o r e d ( s u c c e s s f u l r e c l o s u r e o f c i r c u i t ) . The g e n e r a l f o r m o f t h e p e r f o r m a n c e f u n c t i o n f r o m (5.1) i s 0 ( x ( t f ) ) =  W i ^ J + W2 <aS)2 ( 5 ' 9 } F(x) = w 3 {&QJmj2 + w 4 ( & S ) 2 + w5̂ &wmj2 (5.10) The f i r s t two s t e p s a r e o f f i x e d t i m e e s t a b l i s h e d by t h e c i r c u i t b r e a k e r s e t t i n g s . I n t h e s e two s t e p s , (f) ( x ( t ^ ) ) i s s e t t o z e r o . S t u d i e s a r e made o f a l t e r n a t i v e l y c o n s i d e r - i n g t h e w e i g h t i n g f a c t o r s , o r a l o n e . The same p r o - c e d u r e s a r e r e p e a t e d f o r t h e f i n a l s t e p but i n c l u d i n g W-̂  and ^2' ^ ° e m P n a s i z e s m a l l d e v i a t i o n s o f s p e e d and t o r q u e a n g l e a t t h e f i n a l t i m e and V/̂  a r e s e t e q u a l t o t e n , W-̂ = ^ * The f i n a l t i m e o f t h e s y s t e m r e s t o r e d s t e p i s e s t i m a t e d f r o m a s t a b i l i z i n g s i g n a l s t u d y (~1.7 s e c o n d s ) a n t i c i p a t i n g a s h o r t e r s e t t l i n g t i m e (1.25 s e c o n d s ) . The H a m i l t o n i a n i s H = - F ( x ) + A-^pYp) + A 2 ( p ̂ Dl) + \3 (pS) 2 + ( p v R + u j + A 5 ( p v f d ) (5.11) * TRF/ ' 83 where pSJp, P V R a r e S ' i v e n by (3.82) and (3.16) r e s p e c t i v e l } ' The H a m i l t o n i a n i s m a x i m i z e d o r t h e p e r f o r m a n c e f u n c t i o n a l i s m i n i m i z e d w i t h r e s p e c t t o u i f t h e c o n d i t i o n u = umax s S n * 4 i s s a t i s f i e d . T h i s i s a "bang-bang" t y p e o f o p t i m a l c o n t r o l . The s o l u t i o n o f u i s o b t a i n e d i n d i r e c t l y by o p t i m i z i n g t h e 1 r g u n c o n s t r a i n e d v a l u e u r e l a t e d by Box's t r a n s f o r m a t i o n as f o l l o w s u - U m a x s i n (u) (5.12) I n t h e g r a d i e n t method, t h e c o r r e c t i o n a p p l i e d t o u a t e a c h p o i n t i s A u - -k Hu (5.13) I t i s r a t h e r d i f f i c u l t t o c h o o s e k"*'^. I n t h i s c o m p u t a t i o n t h e f o l l o w i n g v a l u e o f k~*'^ i s c h o s e n k = S l (5.14) H T H d t ii u where S l i s a s t e p s i z e c o n s t r a i n t . 5.3. C o m p u t a t i o n and T e s t R e s u l t s C o m p u t a t i o n r e s u l t s from an IBM 36O M o d e l 67 and t e s t r e s u l t s f r o m t h e dynamic t e s t model a r e summarized i n t h i s S e c t i o n as f o l l o w s . 8 4 F o r t h e c o m p u t a t i o n , i n v e s t i g a t i o n o f t h e v a r i o u s p e r f o r m a n c e f u n c t i o n s o u t l i n e d i n S e c t i o n 5.2 r e v e a l s t h a t f o r t h e f a u l t s t e p , t h e t h r e e p e r f o r m a n c e f u n c t i o n s y i e l d t h e same o p t i m a l c o n t r o l and f o r t h e f a u l t c l e a r e d and s y s t e m r e s t o r e d s t e p s , t h e o p t i m a l c o n t r o l i s t h e same f o r F = t&^nxj and F = ( A S ) 2 but d i f f e r e n t f o r F = T h u s , a s i n g l e t r a j e c t o r y f o r t h e s y s t e m v a r i a b l e s i s o b t a i n e d d u r i n g t h e f a u l t s t e p , two t r a j e c t o r i e s d u r i n g t h e f a u l t c l e a r e d s t e p , and two t r a j e c t o r i e s f r o m e a c h o f t h e p r e v i o u s two d u r i n g t h e s y s t e m r e s t o r e d s t e p . T h e s e r e s u l t s a r e p r e - s e n t e d i n F i g . 5.1 w i t h t h e p e r f o r m a n c e f u n c t i o n s summarized i n T a b l e 5.1. C u r v e F i g . 5 . 1 . F a u l t F a u l t C l e a r e d S v s t e m R e s t o r e d 1 1 0 ^ « m j 2 + 10 (AS ) 2 + J^ wmj 2 d t 2 i o ^ w a j 2 + i o (&S) 2 + jpm)2dt 3 1 0 ^ u m j 2 + 10 (AS) 2 + j ^ ^ j 2 J L 4 1 0 ^ « m j 2 + 10 (AS) 2 + j ^ j 2 d t T a b l e 5.1. P e r f o r m a n c e F u n c t i o n s Used D u r i n g T r a n s i e n t S t e p s 85 C o m p a r i s o n s a r e t h e n made between t h e s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l s i g n a l s and t h o s e w i t h o u t . I t i s o b s e r v e d t h a t t h e s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l s i g n a l s a r e much more damped t h a n t h o s e w i t h o u t . Once the. c o n t r o l i s removed a f t e r 1 . 2 5 s e c o n d s t h e s y s t e m o s c i l l a t e s w i t h a r e d u c e d mag- n i t u d e . I t i s o b s e r v e d a l s o t h a t a l l t h e s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l s i g n a l s s t a y v e r y c l o s e . The s p e e d de- v i a t i o n , t o r q u e a n g l e and f i e l d v o l t a g e r e s p o n s e s a r e shown i n F i g . 5 . 1 a , b, and c r e s p e c t i v e l y . C o m p a r i s o n i s t h e n made between c o n v e n t i o n a l s p e e d d e v i a t i o n and a c c e l e r a t i n g power s t a b i l i z i n g s i g n a l r e s p o n s e s and one o f t h e o p t i m a l c o n t r o l s ( c u r v e 2> F i g . 5 . 1 ) r e s p o n s e s . I n g e n e r a l , i t seems t h a t t h e o p t i m a l c o n t r o l y i e l d s a b e t t e r d amping e f f e c t , s p e e d d e v i a t i o n i n F i g . 5 . 2 a and t o r q u e a n g l e F i g . 5 . 2 b , t h a n t h e s t a b i l i z i n g s i g n a l s e x c e p t f o r t h e f i e l d v o l t a g e i n F i g . 5 . 2 c b e c a u s e o f t h e n a t u r e o f t h e f o r c e d e x c i t a t i o n . Power s t a b i l i z a t i o n t e s t r e s u l t s o b t a i n e d d i r e c t l y f r o m t h e dynamic t e s t model a r e p l o t t e d a l o n g w i t h c o m p u t a t i o n r e s u l t s i n F i g . 5 . 3 a , 5 . 3 b , and 5 . 3 c f o r c o m p a r i s o n . A l l s y s t e m r e s p o n s e s a r e c l o s e . S i m i l a r c o m p a r i s o n s o f c o m p u t a t i o n and t e s t r e s u l t s a r e made i n F i g . 5 . 4 a , b and c f o r s y s t e m r e s p o n s e s w i t h a s p e e d s t a b i l i z i n g s i g n a l . I n b o t h c a s e s s t e a d y s t a t e o s c i l - l a t i o n s a r e o b s e r v e d as e x p e r i e n c e d i n p r a c t i c e . However, b e c a u s e o f t h e v e r y n a t u r e o f t h e s t e a d y s t a t e o s c i l l a t i o n s 86 i t i s d i f f i c u l t t o r e a l i z e t h e same i n i t i a l c o n d i t i o n f o r c o m p u t a t i o n on t h e t e s t m o d e l . C o m p a r i s o n i s t h e n made between t e s t and c o m p u t a t i o n o f s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l . They a g r e e w i t h e a c h o t h e r very w e l l , F i g . 5 . 5 a and 5 . 5 b , e x c e p t t h e s w i t c h i n g t r a n s i e n t d i s t u r b a n c e i n t h e f i e l d v o l t a g e F i g . 5 . 5 c w h i c h i s o b s e r v e d a l s o i n F i g . 3 . 8 . The l a s t c o m p a r i s o n s o f t e s t and c o m p u t a t i o n r e s u l t s a r e c a r r i e d o u t on s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l f o r t h e f i r s t 1 . 2 5 s e c o n d s and power s t a b i l i z i n g s i g n a l f o r t h e r e m a i n i n g p e r i o d , F i g . 5 . 6 a , b, and c . N o t e t h a t t h e c o m p a r i - s o n o f t h e f i r s t 1 . 2 5 s e c o n d s i s made i n F i g . 5 . 5 a , b and c . I t i s o b s e r v e d t h a t t h e o v e r a l l r e s p o n s e s o f t h e t e s t and c o m p u t a t i o n a r e v e r y c l o s e . I n F i g . 5 . 7 , s y s t e m r e s p o n s e s w i t h o p t i m a l c o n t r o l and power s t a b i l i z i n g s i g n a l a r e p l o t t e d a l o n g w i t h s y s t e m r e s p o n s e s w i t h power s t a b i l i z i n g s i g n a l a l o n e . I t i s i n t e r e s t - i n g t o n o t e t h a t t h e c o m p o s i t e s i g n a l y i e l d s t h e b e s t o v e r a l l s y s t e m r e s p o n s e and t h i s i s r e a l i z e d t o a l e s s e r d e g r e e on t h e t e s t m o d e l . From t h e c o m p a r i s o n s made above i t i s c o n c l u d e d t h e dynamic t e s t model c a n be us e d t o p e r f o r m c o m p l i c a t e d power- s y s t e m t e s t s , s u c h as s t a b i l i z i n g s i g n a l c o n t r o l , and u s e d t o c h e c k c o m p u t a t i o n a l p r e d i c t i o n s . MECHRNICRL SPEED DEVIRTION (RRD/SEC) ^8 DELTA ANGLE (RAD) -0.25 0 0.25 0.50 0.75 1.00 1.25 1.50 38 69 MECHRNICRL SPEED DEVIATION (RRD/SEC) i i no * o »—* rv) F i g . 5.2 (b) Torque Angle C a l c u l a t e d T r a n s i e n t Responses of V a r i o u s S t a b i l i z i n g S i g n a l s C O I— _ J Q LU CE > a _i L U speed deviation F i g . -j 1 1 1 1 ^ 1 1 1 1 1 1 1 1 1 j 1 r 0.25 0.50 0.75 1.00 T I M E ( S E C ) (c) F i e l d Voltage 5.2 C a l c u l a t e d Transient Responses of Various S t a b i l i z i n g S i g n a l s 1.25 to 2 n TIME (SEC) (a) Mechanical Speed D e v i a t i o n F i g . 5.3 Transient Responses with A c c e l e r a t i n g Power S t a b i l i z a t i o n 1.50 -i calculated .test model sr. i—|— i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i i | 1 <~ 1.5 2.0 2.5 3.0 3.5 TIME (SEC) i—]—i—i—i—i—|—r 0.5 1.0 (c) F i e l d Voltage g. 5.3 Transient Responses wi t h A c c e l e r a t i n g Power S t a b i l i z a t i o n MECHRNICRL SPEED DEVIATION 1RRD/SEC) 1.50 -i T I M E ( S E C ) (b) T o r q u e A n g l e F i g . 5.4 T r a n s i e n t R e s p o n s e s w i t h Speed D e v i a t i o n S t a b i l i z a t i o n ^ 15-i CO f— _ J LU CD cn > o I LU 4.0 ( c ) F i e l d V o l t a g e F i g . 5.4 T r a n s i e n t R e s p o n s e s w i t h Speed D e v i a t i o n S t a b i l i z a t i o n o co 2 CJ LU CO \ CD cr cn 1 rz: o CE UJ Q Q LU LU CL-IO 0 ^ 1 CJ - 1 CE CJ LU -2 n o stabilization fault cleared system restore — i -T 1 1 j 1 1 1 1 j 1 1 1 1 1 1 1 1 1 1 r 0 0.25 0.50 0.75 1 .00 T I M E ( S E C ) (a) Mechanical Speed D e v i a t i o n F i g . 5-5 Transient Responses wi t h Optimal C o n t r o l S t a b i l i z a t i o n T 1 r 1.25 vO o c r L U I LD rz. CE cr 1.50 - i 1.25H 1 .00 H 0.75 H 0.50 H 0 .25- •0.25 J^.calcula ted 1.25 T I M E ( S E C ) (b) Torque Angle F i g . 5.5 Transient Responses wi t h Optimal C o n t r o l S t a b i l i z a t i o n o o TIME (SEC) • ( c ) F i e l d V o l t a g e F i g . 5.5 T r a n s i e n t R e s p o n s e s w i t h O p t i m a l C o n t r o l S t a b i l i z a t i o n g MECHRMICRL SPEED D E V I A T I O N (RRD/5EC) i i f\j >-* O f\j ZOT  no stabilization TIMF ( S E C ) (c) F i e l d Voltage F i g . 5.6. Transient Responses w i t h Composite S t a b i l i z a t i o n 2n TIME (SEC) (a) Mechanical Speed D e v i a t i o n F i g . 5.. 7 Transient Responses w i t h A c c e l e r a t i n g Power S t a b i l i z a t i o n M and Composite S t a b i l i z a t i o n p o c r or LU I CD CC cr i L U Q 1 . 5 0 1 . 2 5 - 1 . 0 0 - 0 . 7 5 0 . 5 0 H 0 . 2 5 - 0 . 2 5 0 0 . 5 1 . 0 1.5 2.0. 2 . 5 T I M E ( S E C ) 3 . 0 3 . 5 (b) Torque Angle F i g . 5.7 Transient Responses wi t h A c c e l e r a t i n g Power S t a b i l i z a t i o n r'* ' and Composite S t a b i l i z a t i o n o O N 1 07 • CONCLUSION A new model.for power system t r a n s i e n t s t a b i l i t y and con- t r o l t e s t s has been developed capable of -representing any conventional one machine i n f i n i t e bus system i n d e t a i l . The model system has a synchronous machine w i t h an adjustable f i e l d time constant, a s o l i d s t a t e r e g u l a t o r - e x c i t e r system and an a d a p t e d dc motor prime mover with t y p i c a l speed governor c h a r a c t e r i s t i c s . The model can be used to i n v e s t i g a t e power system dynamics over a wide range of operating c o n d i t i o n s . A general f i f t h order s t a t e v a r i a b l e model f o r a synchronous machine has been derived, from Park's equations. I t has been shown that parameters i n t h i s s t a t e v a r i a b l e model can be determined d i r e c t l y , from experimental machine- t e s t s . This i s an improvement over e x i s t i n g models which are based on parameters not d i r e c t l y measurable. Three v a r i a t i o n s on t h i s general model have been shown to be u s e f u l i n p r e d i c t i n g the dynamic behaviour of a synchronous machine and interconnected systems. From the comparison of computation and model t e s t r e s u l t s , i t has been found that the f i r s t order synchronous machine (plPcO and the second order dynamics ( pS,pCJj are s u f f i c i e n t f o r most studies except subtransient and switching phenomenon which has not been included i n t h i s study. A parameter s e n s i t i v i t y study has been-carried out. I t has been found that x , T, and J are the most s e n s i t i v e narameters q' do but not r , x-, , T̂ , and the governor h y d r a u l i c operator parameters. a Q i i The usefulness of the model has been demonstrated with the study of the s t a b i l i z i n g s i g n a l . A "bang-bang" type nonlinear optimal c o n t r o l s i g n a l f o r a f a u l t - f a u l t c l e a r e d - s y s t e m r e s t o r e d power s y s t e m has been treated as a t w o - p o i n t - b o u n d a r y - v a l u e problem i n t h i s t h e s i s . A gradient method has been applied to obtain the s w i t c h i n g times of the c o n t r o l . 'The c o n t r o l s i g n a l thus obtained has been implemented on the t e s t model. The model test r e s u l t s have been shown to agree favourably with those obtained from computation. Thus, i t has proven that the test model provides a convenient means to check the design. The t e s t model has a l s o been used to t e s t conventional speed and a c c e l e r a t i n g p o w e r ' s t a b i l i s i n g s i g n a l s . The experimental r e s u l t s have compared favourably with computed values. I t has been demonstrated experimentally t h a t the nonlinear optimal c o n t r o l provides b e t t e r system damping than conventional s i g n a l s . The p r i n c i p a l c o n t r i - bution of the t h e s i s i s the development of a compatible t e s t model and higher order state v a r i a b l e synchronous machine models f o r power system dynamic s t u d i e s . For future studies i t i s f e l t that 1) the mathematical model could be improved, i f necessary, to include subtransient and switching phenomenon, 2) comparison of a l t e r n a t i v e schemes of prime mover s i m u l a t i o n should be made, 3) the prototype t e s t model should be developed to include a u n i v e r s a l transmission system since the present one i s of s p e c i a l purpose, 4) and the model d e v e l o p e d could be m u l t i p l i e d f o r m u l t i - ma chine s t ud i e s. 109 APPENDIX 3A Park's Equations i n the State V a r i a b l e Form Equation (3.3) i s expanded to o b t a i n d R ^ e o ( l + T d o P ) W e o (1+T^p)^ 1+TcP + T D p \ Vweo ( l + T d o p ) "eo ( l + T d o ^ > / X d x d where vp R x a d v f d t RF t T d o - T *D1 T d o - T" xdo it T d o " TD1 T^o " T d o T d o (Td + T d o ) t II T d T d t " T d o ~ T d o _ it , II t it t it TD - ~ Tdo ( T d + T d - T d o ) + T d TH T d o ~ T d o Terms i n (3A.1) are combined to o b t a i n Yd = J _ ( ] l l F (1+T cp) x d i d + x d i d 1 + T d o p - x d ! d eo + _ L _ |SvF - Tnpxdid eo 1 + T d > (3A.1) (3A.2) 110 w h i c h c a n be w r i t t e n Y d = - x d j - d + V F R + ^DR ( 3 A . 3 ) W e o U e o ^eo wher e v p R Rvp + ( T d o - T c ) x d p i P 1 d ( 3 A . 4 ) 1 + T d o P V ° R = S v F - T D p x d i d (3A . 5 ) 1 + T d o P From (3A.4) one h a s V F = - i - (VFR + P ( T d o v F R - x d ( T d o - T c ) i d ) ) ( 3 A . 6 ) a n d when c o m p a r e d w i t h Rvp = v F R + p Y p (3A.7) t h e r e s u l t s y i e l d ^F - T d o V F R ~ x d < Tdo - T c ) ^ <3A.8) From (-3A.5) one h a s 0 = v D R - S v F + p ( T d o v D R + T D x d i d ) (3A.9) and when c o m p a r e d w i t h 0 - v D R + P^D (3A.10) t h e r e s u l t s y i e l d = T d o v D R + T D x d i d (3A.11) a n d v — v — Sv DR DR F I l l E q u a t i o n s ( 3 A . 3 ) , (3A . 8 ) and ( 3 A . l l ) c an be w r i t t e n i n m a t r i x f o r m ^ d T do 1 - x d ( T d o - T c ) 0 -x d to eo 0 eo eo xd^D T do V F R (3.A.12) .» i d and VDR a r e V F R x d ( T d o + T D ) - x d ( T d o - T c ) W e o x d ( T d o - T c ) xd T d o T d o x d T d o x d T d o T d o i d 1 1 ^ d x d T d o *3 " I I x d T d o VDR " *d TD w e o xd T n xdTo. _ x3 I d o T d o x d T d o xJl T d' 0 T d o _ (3.10) A s i m i l a r a p p r o a c h can be u s e d t o o b t a i n s t a t e equa- t i o n s i n t h e q - a x i s . The f l u x l i n k a g e s due t o t h e a r m a t u r e w i n d i n g and t h e damper w i n d i n g a r e o b t a i n e d by r e w r i t i n g (3.4) a s f o l l o w s Vq = - (T+TqP) Xgiq + x q i q _ x q i ( (3A.13) ( l + T q 0 p ) W e o u e o W e eo 1 1 2 When ( 3 A . 1 3 ) i s compared with ( 3 A . 1 4 ) eo eo the r e s u l t y i e l d s P x q ( T q o T n ) i f (3A.15) 1+T qo Equation ( 3 A . 1 5 ) can be f u r t h e r w r i t t e n 0 V Q R + P ( T a o v0P- - x n ( T n o - T q ) i n ) qo VQR ~xq<- x qo_J q-'xq^ ( 3 A . 1 6 ) and when i t i s compared to 0 - v QR + PVQ (3A.17) the r e s u l t y i e l d s p. : = T q o (VQR - ( x q - x q ) *q) (3A.18) Equations ( 3 A . 1 5 ) and ( 3 A . 1 8 ) can be w r i t t e n i n matrix form a q eo eo - ( x q x q ) T q' 0 T " q o RQR (3A.19) and the s o l u t i o n s of i q and V Q R are 113 tt), eo 1 x q i q o - W e o ( x q - x q ) Q (3.11) Therefore Park's equations can be v s ' r i t t e n i n the s t a t e v a r i a b l e form as f o l l o w s P^q = v q - + r a ± q (3.5) (3.6) Rv, FR (3.7) P^D = ~ VDR (3.8) (3.9) 114 APPENDIX 3B Determination of E f f e c t i v e V o l t a g e R a t i o R f / x a d The v a l u e of Rf i n (3.3) i s i n f a c t an e f f e c t i v e xad v o l t a g e r a t i o of the synchronous machine and can be determined from a s h o r t c i r c u i t t e s t and machine parameters. For a t h r e e phase steady s t a t e s h o r t c i r c u i t o f the synchronous machine Park's equations become "eoY'q - - r a i d <3B.l) "eoTd = r a i q (3B.2) ^eo^d = f a d v f d ~ xdld (3B.3) Rf w e o V q = - x q i q ( 3 B . 4 ) Combining ( 3 B . 1 ) and(3B.3) y i e l d s 1q ~ £a i d (3B.5) X q S u b s t i t u t i n g (3B.2) and (3B.5) i n t o ( 3 B.3) g i v e s i d = *q x a d v f d ( 3 B . 6 ) r a + x d x q R f S u b s t i t u t i n g i d i n t o (3B.5) y i e l d s 115 i = . r a f a d v f d (3B.7) r a + x d x q R f Thus the a-phase short c i r c u i t c urrent equals i = /i 2 + = / r f + x^ x_, v 2 2 a + x q x a d v f d (3B .8 ) r a + x d x q R f When ( 3 B . 8 ) i s compared to the t e s t r e s u l t s obtained from the steady s t a t e short c i r c u i t t e s t , one has 'slope' = i / / v ^ j (3B.9) Thus RF = R f yjrl + x2 ( 3 > 4 5 ) cad ( r a + x d x q ) 'slope' 116 REFERENCES R. R o b e r t , " M i c r o m a c h i n e and m i c r o r e s e a u x : s t u d y o f t h e p r o b l e m s o f t r a n s i e n t s t a b i l i t y by t h e u s e o f m o d e l s s i m i l a r e l e c t r o m e c h a n i c a l l y t o e x i s t i n g m a c h i n e s and s y s t e m s , " C.I.G.R.E., v o l . I l l , 1950. V. A. V e n i k o v , " R e p r e s e n t a t i o n o f e l e c t r i c a l p h e n - omena on p h y s i c a l m o d e l s a s a p p l i e d t o power s y s t e m , C.I.G.R.E., v o l . I l l , 1952. B. A d k i n s , " M i c r o m a c h i n e s t u d i e s a t I m p e r i a l C o l l e g e E l e c t r i c a l T i m e s , J u l y I960. J . Roy, " E f f e c t s o f s y n c h r o n o u s m a c h i n e p a r a m e t e r s o n d y n a m i c and t r a n s i e n t s t a b i l i t y , " P a p e r p r e s e n t e d t o C.E.A. W i n n i p e g M e e t i n g , M a r c h 1967." J . J . D o u g h e r t y and V. C a l e c a , "The E E I a c / d c t r a n s - m i s s i o n m o d e l , " I E E E T r a n s a c t i o n s , v o l . PAS-87, pp . 504-512, F e b r u a r y 1968. " T e s t p r o c e d u r e s f o r s y n c h r o n o u s m a c h i n e s , " I E E E p u b l i c a t i o n No. 115, M a r c h 1965. Y. N, Yu and G. E. Dawson, " M o d e l i n g a four- e l e c t r i c m a c h i n e s y s t e m on a n a l o g u s i n g p a r a m e t e r s d i r e c t l y d e t e r m i n e d f r o m t e s t s , " I E E E T r a n s a c t i o n s , v o l . PAS-87, p p . 632-641, M a r c h 1968. R. G. S i d d a l l , "A p r i m e m o v e r - g o v e r n o r t e s t m o d e l f o r l a r g e p o w e r s y s t e m s , " U.B.C. MASc. T h e s i s , J a n u a r y 1968. L. M. Hovey, "Optimum A d j u s t m e n t o f G o v e r n o r s i n H y d r o G e n e r a t i n g S t a t i o n s , " E n g i n e e r i n g I n s t i t u t e o f Canada J o u r n a l , p p. 64-71, Nove. I960. J . A. Bond, "A s o l i d s t a t e v o l t a g e r e g u l a t o r a nd e x c i t e r f o r a l a r g e po\vrer s y s t e m t e s t m o d e l , " U.B.C. MASc. T h e s i s , J u l y 1967. 117 C h a p t e r 3 3.1 R. H. P a r k , "Two r e a c t i o n t h e o r y o f s y n c h r o n o u s m a c h i n e s , g e n e r a l i z e d method o f a n a l y s i s , " A I E E T r a n s a c t i o n s , v o l . 4 8 , pp. 716-730, J u l y 1929. 3.2 K. V o n g s u r i y a , "The a p p l i c a t i o n o f L y a p u n o v f u n c - t i o n t o power s y s t e m s t a b i l i t y a n a l y s i s and c o n t r o l , " U.B.C. PhD T h e s i s , F e b r u a r y 1968. 3.3 R. F l e t c h e r and M. J . D. P o w e l l , "A r a p i d l y c o n v e r - g e n t d e s c e n t m e t h o d f o r m i n i m i z a t i o n , " The C o m p u t e r J o u r n a l , v o l . 6, p p . 163-168, 1963. 3.4 Y. N. Y u and K. V o n g s u r i y a , " S t e a d y s t a t e s t a b i l i t y l i m i t s o f a r e g u l a t e d s y n c h r o n o u s m a c h i n e c o n n e c t e d t o an i n f i n i t e s y s t e m , " I E E E T r a n s a c t i o n s , v o l . P A S - 8 5 , PP. 759-767, J u l y 1966. C h a p t e r 4 4.1 4 . 2 R. T o m o v i c , S e n s i t i v i t y a n a l y s i s o f d y n a m i c s y s t e m s . New Y o r k : M c G r a w - H i l l , 1963. A. P. S a g e , Optimum S y s t e m s C o n t r o l , E n g l e w o o d C l i f f s , N . J . : P r e n t i c e H a l l , 1968, C h a p t e r 12. C h a p t e r 5 5.1 5 .2 5 .3 5 .4 P. L. Dandenc e t a l . , " E f f e c t o f h i g h - s p e e d r e c t i f i e r e x c i t a t i o n s y s t e m on g e n e r a t o r s t a b i l i t y l i m i t s , " I E E E T r a n s a c t i o n s , v o l . P A S - 8 7 , p p . 1 9 0 - 2 0 1 , J a n u a r y 1 9 6 8 . R. H. S c h i e r and A. L. B l y t h e , " F i e l d t e s t s o f d y n a m i c s t a b i l i t y u s i n g a s t a b i l i z i n g s i g n a l a n d c o m p u t e r p r o g r a m v e r i f i c a t i o n , " I E E E T r a n s a c t i o n s , v o l . P A S - 8 7 , pp. 3 1 5 - 3 2 2 , F e b r u a r y 1 9 6 8 . F. R. S c h l i e f e t a l . , " C o n t r o l o f r o t a t i n g e x c i t e r s f o r p ower s y s t e m d a m p i n g - p i l o t a p p l i c a t i o n s a n d e x - p e r i e n c e , " I E E E T r a n s a c t i o n P a p e r 69 TP 155-PWR. F. P. d e M e l l o a n d C. C o n c o r d i a , " C o n c e p t s o f s y n - c h r o n o u s m a c h i n e s t a b i l i t y a s a f f e c t e d by e x c i t a t i o n c o n t r o l , " I E E E T r a n s a c t i o n s , v o l . P A S - 8 8 , p p . 3 1 6 - 3 2 9 , A p r i l 1 9 6 9 . 118 G. A. J o n e s , " T r a n s i e n t s t a b i l i t y o f a s y n c h r o n o u s g e n e r a t o r u n d e r c o n d i t i o n s o f b a n g - b a n g e x c i t a t i o n s c h e d u l i n g , " I E E E E T r a n s a c t i o n s , v o l . PAS-84, pp. 114-121, F e b r u a r y 1965. A. P. S a g e , Optimum s y s t e m s c o n t r o l , E n g l e w o o d C l i f f s , N . J . : P r e n t i c e H a l l , 1968. C. II. S c h l e y , J r . and I . L e e , " O p t i m a l C o n t r o l c o m p u t a t i o n by t h e Ne w t o n - R a p h s o n method and t h e R i c c a t i t r a n s f o r m a t i o n , " I E E E T r a n s a c t i o n s , v o l . AC-12, p p . 139-144, A p r i l 1967. M. J . Box, "A c o m p a r i s o n o f s e v e r a l c u r r e n t o p t i m i - z a t i o n m e t h o d s , a nd t h e u s e o f t r a n s f o r m a t i o n s i n c o n s t r a i n e d p r o b l e m s , " The C o m p u t e r J o u r n a l , v o l . 9, pp. 67, 1966. J . W. S u t h e r l a n d , "The s y n t h e s i s o f o p t i m a l c o n t r o l - l e r s f o r a c l a s s o f a e r o d y n a m i c a l s y s t e m s , a n d t h e n u m e r i c a l s o l u t i o n o f n o n l i n e a r o p t i m a l c o n t r o l p r o b l e m s , " U.B.C. PhD T h e s i s , May 1967. Y. N. Yu and C. S i g g e r s , " S t a b i l i z a t i o n and o p t i m a l c o n t r o l s i g n a l s f o r a power s y s t e m . " A c c e p t e d a s t r a n s a c t i o n p a p e r f o r I E E E 1970 Summer Power M e e t i n g .

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