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Excitation control of torsional oscillations due to subsynchronous resonance Yan, Andrew 1979

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EXCITATION CONTROL OF TORSIONAL OSCILLATIONS DUE TO SUBSYNCHRONOUS RESONANCE by Andrew {^ <an B.S.E.E. Un i v e r s i t y of Texas at Arl i n g t o n , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES i n 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 THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1979 0 Andrew Yan, 1979 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. ( Andrew Yan ) Department of E l e c t r i c a l E n g i n e e r i n g The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 n.tP 9 A?r11 979 3E-6 B P 75-5 1 I E ABSTRACT S u b s y n c h r o n o u s r e s o n a n c e phenomena . i n a power s y s t e m w i t h s e r i e s - c a p a c i t o r - c o m p e n s a t e d t r a n s m i s s i o n l i n e s may c a u s e damaging t o r s i o n a l o s c i l l a t i o n s i n t h e s h a f t s o f t h e t u r b i n e g e n -e r a t o r . I n t h i s t h e s i s a h i g h - o r d e r power s y s t e m mode l f o r s u b s y n c h r o n o u s r e s o n a n c e s t u d i e s i s d e r i v e d . An e x c i t a t i o n p r o c e d u r e f o r c o n t r o l o f t o r s i o n a l o s c i l l a t i o n s i s p r e s e n t e d . The e x c i t a t i o n c o n t r o l i s o f t h e l i n e a r o p t i m a l t y p e s y n t h e s i z e d f rom t h e s y s t e m ' s o u t p u t s i g n a l s . Dynamic p e r f o r m a n c e t e s t s o f t h e e x c i t a t i o n c o n t r o l l e r on t h e n o n l i n e a r mode l show t h a t a l l mode o s c i l l a t i o n s can be s t a b i l i z e d s i m u l t a n e o u s l y w i t h i n t h e r ange o f d y n a m i c s t a b i l i t y f o r a w i d e r a n g e o f c a p a c i t o r c o m p e n s a t i o n , b u t n o t f o r a s e v e r e t r a n s i e n t f a u l t . 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 ACKNOWLEDGMENT v i i i NOMENCLATURE i x 1. INTRODUCTION 1 1.1 S u b s y n c h r o n o u s Resonance P r o b l e m s and C o r r e c t i v e M e a s u r e s . . . ± 1.2 R e c e n t Work on E x c i t a t i o n C o n t r o l o f SSR 2 1.3 Scope o f t h e T h e s i s 3 2 . MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE STUDIES . . . . 5 2 . 1 I n t r o d u c t i o n 5 2 . 2 M o d e l l i n g t h e M e c h a n i c a l S y s t e m 6 2 . 3 M o d e l l i n g t h e E l e c t r i c a l Sys t em 10 2 . 4 L i n e a r i z e d M o d e l 17 2 . 5 Reduced O r d e r M o d e l 18 2 . 6 E i g e n v a l u e A n a l y s i s 18 3 . L INEAR OPTIMAL EXCITATION CONTROL DESIGN 2 4 3 . 1 I n t r o d u c t i o n . . ; 24 3 . 2 L i n e a r S t a t e R e g u l a t o r P r o b l e m 24 3 .3 S o l u t i o n t o t h e M a t r i x R i c c a t i E q u a t i o n 2 6 3 .4 Wide Range SSR S t a b i l i z e r and t h e C h o i c e o f W e i g h t i n g M a t r i c e s . 27 4 . EFFECT OF EXCITATION CONTROLLER ON STEADY STATE AND TRANSIENT SSR. . 32 4 . 1 I n t r o d u c t i o n 32 4 . 2 C o n t r o l l e r f o r S t e a d y S t a t e SSR 32 4 . 3 C o n t r o l l e r f o r T r a n s i e n t SSR 44 5 . CONCLUSION 5 6 i i i Page REFERENCE 57 APPENDIX I 59 APPENDIX I I 60 i v LIST OF TABLES Table Page 2.1 Eigenvalues of various order SSR model at 30% capacitor compensation f or P=0.9 p.u. at 0.9 power factor lagging. . . . . 20 2.2 Eigenvalues of various order SSR model at 50% capacitor compensation f or P=0.9 p.u. at 0.9 power factor lagging. . . . . 21 2.3 Eigenvalues of various order SSR model at 70% capacitor compensation f or P=0.9 p.u. at 0.9 power factor lagging. . . . . 22 2.4 Eigenvalues of various order SSR model at 80% capacitor compensation f or P=0.9 p.u. at 0.9 power factor lagging. . . . . 23 3.1 Eigenvalues of the system . with l i n e a r optimal control at various degrees of capacitor compensation f or P=0.9 p.u. power factor = 0.9 lagging 29 3.2 Shaft modes- of the system with and. without l i n e a r optimal co n t r o l at various degrees of capacitor compensation for P= 1.25 p.u. at 0.9 power factor lagging 30 3.3 Shaft modes of the system with and without l i n e a r optimal control at various degrees of capacitor compensation for P=0.5 p.u. at 0.9 power factor leading • • • 31 v L I S T OF ILLUSTRATIONS F i g u r e - Page 2 . 1 F u n c t i o n a l b l o c k d i a g r a m o f a power s y s t e m f o r SSR s t u d y . . . 5 2 . 2 M o d e l o f t h e s t e am t u r b i n e s y s t e m 6 2 . 3 The g e n e r a t i n g u n i t m a s s - s p r i n g s y s t e m 8 2 . 4 M o d e l l i n g o f t h e m a s s - s p r i n g s y s t e m i n t h e v i c i n i t y o f t h e i r o t a t i o n a l mass 8 2 . 5 The speed g o v e r n o r mode l 10 2 . 6 The t r a n s m i s s i o n s y s t e m 11 2 . 7 V o l t a g e r e g u l a t o r and e x c i t e r mode l 12 2 . 8 A s i x - w i n d i n g g e n e r a t o r mode l 14 2 . 9 E q u i v a l e n t c i r c u i t o f a f i v e - w i n d i n g g e n e r a t o r mode l f rom a s i x - w i n d i n g m o d e l 14 2 . 1 0 A r m a t u r e c u r r e n t r e s p o n s e s o f t h e s i x - w i n d i n g and t h e f i v e -w i n d i n g g e n e r a t o r mode l 15 2 . 1 1 Q - a x i s damper c u r r e n t o f t h e f i v e - w i n d i n g g e n e r a t o r mode l . . 16 2 . 1 2 Q - a x i s damper c u r r e n t s o f t h e s i x - w i n d i n g g e n e r a t o r mode l . . 16 4 . 1 Dynamic r e s p o n s e s o f t h e s y s t e m w i t h o u t c o n t r o l when s u b j e c t e d t o 10% l o a d change 33 4 . 2 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t c o n t r o l when s u b j e c t e d t o a p u l s e t o r q u e d i s t u r b a n c e 35 4 . 3 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l when s u b j e c t e d t o a p u l s e t o r q u e d i s t u r b a n c e 39 4 . 4 T o r s i o n a l o s c i l l a t i o n o f t h e g e n e r a t o r - e x c i t e r s h a f t a t 80% c o m p e n s a t i o n ( w i t h o u t c o n t r o l ) 43 4 . 5 T o r s i o n a l o s c i l l a t i o n s o f t h e g e n e r a t o r - e x c i t e r s h a f t a t 80% c o m p e n s a t i o n ( w i t h c o n t r o l ) 43 4 . 6 E l e c t r i c a l n e t w o r k f o r t h e s i m u l a t i o n o f s u b s y n c h r o n o u s r e s o n a n c e 44 4 . 7 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t c o n t r o l when s u j e c t e d t o a t h r e e - p h a s e f a u l t a t t h e r emote end (X =0 .01 p . u . ) 46 i i 4 . 8 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l when s u b j e c t e d t o a t h r e e - p h a s e f a u l t a t t h e r emote end (Xg=0.01 p . u . ) 50 v i F i g u r e Page 4.9 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h o u t c o n t r o l when s u b j e c t e d t o a t h r e e - p h a s e f a u l t ( X =0.06 p . u . ) 54 4 . 1 0 Dynamic r e s p o n s e o f t h e power s y s t e m w i t h c o n t r o l when s u b j e c t e d t o a t h r e e phase f a u l t (X =0.06 p . u . ) 55 v i i ACKNOWLEDGEMENT I w o u l d l i k e t o e x p r e s s my most g r a t e f u l t h a n k s and d e e p e s t g r a t i t u d e t o D r . Y a o - n a n Yu and D r . M . D . Wvong, s u p e r v i s o r s o f t h i s p r o j e c t , f o r t h e i r c o n t i n u e d i n t e r e s t , encouragement and g u i d a n c e d u r i n g t h e r e s e a r c h work and w r i t i n g o f t h i s t h e s i s . I a l s o w i s h t o e x p r e s s my deep a p p r e c i a t i o n t o M r . E l - S h a r k a w i f o r h i s i n t e r e s t and t i m e l y a d v i c e . The f i n a n c i a l s u p p o r t o f t h e N a t u r a l S c i e n c e s and E n g i n e e r i n g R e s e a r c h C o u n c i l o f Canada and t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a i s g r a t e f u l l y a c k n o w l e d g e d . I am g r a t e f u l t o my p a r e n t s , members o f my f a m i l y , and M r . F r a n k Wong f o r t h e i r encouragment t h r o u g h o u t my u n i v e r s i t y c a r e e r . v i i i NOMENCLATURE General A system matrix B control matrix u control vector x state vector of the unmeasurable model y state vector of the measurable model H transformation matrix Q symmetric p o s i t i v e semi-definite weighting matrix R symmetric p o s i t i v e d e f i n i t e weighting matrix K R i c c a t i matrix G closed loop system matrix M composite matrix as defined i n (3.14) A,X eigenvalue matrix,eigenvector matrix of M o subscript-denoting i n i t i a l condition x time d e r i v a t i v e of x t superscript denoting transpose -1 superscript denoting inverse s d i f f e r e n t i a l operator A p r e f i x denoting a l i n e a r i z e d v a r i a b l e j complex operator, S-T Mass-spring system M i n e r t i a c o e f f i c i e n t K shaft s t i f f n e s s constant D damping 0 rotor angle to rotor speed CJ^ synchronous speed i x S y n c h r o n o u s M a c h i n e i i n s t a n t a n e o u s v a l u e o f c u r r e n t V i n s t a n t a n e o u s v a l u e o f v o l t a g e T f l u x - l i n k a g e R r e s i s t a n c e X r e a c t a n c e 6 t o r q u e a n g l e e l e c t r i c t o r q u e i t e r m i n a l c u r r e n t V t e r m i n a l v o l t a g e j Q ^ g e n e r a t o r o u t p u t power d , q s u b s c r i p t d e n o t i n g d i r e c t - and q u a d r a t u r e - a x i s s t a t o r q u a n t i t i e s f s u b s c r i p t d e n o t i n g f i e l d c i r c u i t q u a n t i t i e s D , Q , S s u b s c r i p t d e n o t i n g d i r e c t - and q u a d r a t u r e - a x i s damper q u a n t i t i e s &,L s u b s c r i p t d e n o t i n g l e a k a g e impedance i n damper and s t a t o r c s u b s c r i p t d e n o t i n g q u a n t i t i e s a s s o c i a t e w i t h c a p a c i t o r a s u b s c r i p t d e n o t i n g a r m a t u r e phase q u a n i t i e s T r a n s m i s s i o n L i n e X , R r e a c t a n c e and r e s i s t a n c e o f t h e t r a n s f o r m e r X , R r e a c t a n c e and r e s i s t a n c e o f t h e t r a n s m i s s i o n l i n e e ' e X r e a c t a n c e o f c a p a c i t o r c V i n f i n i t e bus v o l t a g e o E x c i t o r and V o l t a g e R e g u l a t o r r e g u l a t o r g a i n T. r e g u l a t o r t i m e c o n s t a n t T e x c i t e r t i m e c o n s t a n t L V r e f e r e n c e v o l t a g e r e f • x G o v e r n o r and Steam T u r b i n e Sys t em K a c t u a t o r g a i n g T ^ , T 2 a c t u a t o r t i m e c o n s t a n t T^ s e r v o m o t o r t i m e c o n s t a n t a a c t u a t o r s i g n a l P power a t g a t e o u t l e t VJV T s t eam c h e s t t i m e c o n s t a n t LH T r e h e a t e r t i m e c o n s t a n t Kri T c r o s s - o v e r t i m e c o n s t a n t F h i g h p r e s s u r e t u r b i n e power f r a c t i o n r l r F^.p i n t e r m e d i a t e p r e s s u r e t u r b i n e power f r a c t i o n F L P A l o w p r e s s u r e t u r b i n e A power f r a c t i o n F ^ p g l o w p r e s s u r e t u r b i n e B power f r a c t i o n T^p h i g h p r e s s u r e t u r b i n e t o r q u e T^p i n t e r m e d i a t e p r e s s u r e t u r b i n e t o r q u e T L P A ' T L P B ^ O W P r e s s u r e t u r b i n e t o r q u e x i 1 1. INTRODUCTION 1.1 S u b s y n c h r o n o u s Resonance P r o b l e m s and C o r r e c t i v e M e a s u r e s The i n c r e a s i n g e l e c t r i c power demand, t h e u n a v a i l a b i l i t y o f g e n e r a t i o n s i t e s a t heavy l o a d c e n t r e s , and t h e d i f f i c u l t i e s i n o b t a i n i n g t h e r i g h t o f way t o b u i l d new t r a n s m i s s i o n l i n e s due t o e n v i r o n m e n t a l c o n s i d e r a t i o n s have f r e q u e n t l y made l o n g s e r i e s - c a p a c i t o r - c o m p e n s a t e d t r a n s m i s s i o n l i n e s n e c e s s a r y f o r t r a n s f e r r i n g b u l k p o w e r . H o w e v e r , t h e use o f s e r i e s c a p a c i t o r s f o r t r a n s m i s s i o n l i n e c o m p e n s a t i o n may cause e l e c t r i c a l s y s t e m r e s o n a n c e and damage. These o s c i l l a t i o n s a r e t h e r e s u l t o f i n t e r a c t i o n be tween r e s o n a n c e phenomena i n t h e e l e c t r i c a l s y s t e m and t h e m e c h a n i c a l o s c i l l a t i o n o f t h e t u r b i n e - g e n e r a t o r m a s s - s p r i n g s y s t e m . The h a z a r d s o f t h e s e o s c i l l a t i o n s were n o t s e r i o u s l y c o n s i d e r e d u n t i l a f t e r t h e two s h a f t f a i l u r e s a t Mohave g e n e r a t i n g s t a t i o n i n 1970 and 1971 [ 1 ] . The t e r m " S u b s y n c h r o n o u s Resonance ( S S R ) " i s u sed t o d e s i g n a t e t h e g e n e r a l phenomemon e n c o m p a s s i n g t h e o s c i l l a t o r y a t t r i b u t e s o f e l e c t r i c a l and m e c h a n i c a l v a r i a b -l e s a s s o c i a t e d w i t h t h e t u r b i n e - g e n e r a t o r s when c o u p l e d t o a s e r i e s -c a p a c i t o r - c o m p e n s a t e d t r a n s m i s s i o n s y s t e m . D e s p i t e t h e p o t e n t i a l l y damaging SSR e f f e c t , u t i l i t i e s s t i l l f a v o r t h e use o f s e r i e s c a p a c i t o r s t o i n c r e a s e t h e power t r a n s f e r c a p a b i l i t y as an a l t e r n a t i v e t o a d d i t i o n a l t r a n s m i s s i o n l i n e s and more c a p i t a l i n v e s t -men t . H e n c e , i n o r d e r t o overcome t h e p r o b l e m s due t o SSR, e x t e n s i v e e f f o r t has been made i n a n a l y z i n g t h e two s h a f t f a i l u r e s . P r o b l e m s a r e i d e n t i f i e d as i n d u c t i o n g e n e r a t o r e f f e c t , t o r s i o n a l i n t e r a c t i o n , and t r a n s i e n t t o r q u e s [ 2 ] . SSR phenomenon may o c c u r i n two d i f f e r e n t f o r m s : s e l f - e x c i t e d o r s t e a d y s t a t e SSR and t r a n s i e n t SSR. S e l f - e x c i t e d SSR i n v o l v e s s p o n t a n e o u s o s c i l l a t i o n s t h a t a r e e i t h e r s u s t a i n e d o r i n c r e a s e d i n m a g n i t u d e w i t h t i m e . 2 Two mechanisms t h a t p r o d u c e t h e s e o s c i l l a t i o n s a r e t h e i n d u c t i o n g e n e r a t o r e f f e c t where t h e n a g a t i v e r e s i s t a n c e o f t h e g e n e r a t o r a t s u b s y n c h r o n o u s f r e q u e n c i e s exceeds t h e r e s i s t a n c e o f t h e t r a n s m i s s i o n s y s t e m ; and t h e b i l a t e r a l c o u p l i n g known as t o r s i o n a l i n t e r a c t i o n be tween t h e m e c h a n i c a l modes o f t h e m a s s - s p r i n g s y s t e m and t h e n a t u r a l o s c i l l a t i o n mode o f t h e e l e c t r i c a l n e t w o r k . T r a n s i e n t SSR i n v o l v e s t h e t r a n s i e n t t o r q u e s on segments o f t h e t u r b i n e - g e n e r a t o r s h a f t . These t o r q u e s r e s u l t f rom s u b s y n c h r o n o u s o s c i l l a t i o n c u r r e n t s i n t h e e l e c t r i c a l n e t w o r k c a u s e d by f a u l t s o r s w i t c h i n g o p e r a t i o n s . A f t e r a n a l y z i n g and i d e n t i f y i n g t h e SSR p r o b l e m s , c o r r e c t i v e measures have been p r o p o s e d [ 2 ] , e . g . , a d d i t i o n a l a m o r t i s s e u r w i n d i n g s t o r e d u c e t h e i n d u c t i o n g e n e r a t o r e f f e c t ; s t a t i c h i g h - Q f i l t e r s t o b l o c k t h e s u b s y n c h r o n o u s c u r r e n t s a t c r i t i c a l f r e q u e n c i e s ; s u p p l e m e n t a r y 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 d d i t i o n a l s y s t e m d a m p i n g ; a c a p a c i t o r d u a l gap f l a s h i n g scheme t o m i n i m i z e t h e m a g n i t u d e o f t h e t r a n s i e n t t o r q u e s i n t h e s h a f t ; and f i n a l l y , a s u b s y n c h r o n o u s o v e r c u r r e n t r e l a y t o p r o t e c t t h e g e n -e r a t i n g u n i t i n c a s e o f s u s t a i n e d s u b s y n c h r o n o u s o s c i l l a t i o n s . M o s t o f t h e s e p r o p o s a l s have a l r e a d y been pu t i n t o p r a c t i c e . 1.2 R e c e n t Work on E x c i t a t i o n C o n t r o l o f SSR A l t h o u g h a p e r f e c t l y t u n e d h i g h - Q b l o c k i n g f i l t e r c a n p r o v i d e a d e q u a t e damping t o t h e s y s t e m , when d e t u n e d , t h e h i g h - Q f i l t e r r e s i s t a n c e a t a mode f r e q u e n c y i s s h a r p l y r e d u c e d r e s u l t i n g i n r e d u c e d damping p e r f o r m a n c e . D e t u n i n g o c c u r s b e c a u s e o f a m b i e n t t e m p e r a t u r e v a r i a t i o n , c a p a c i t o r f a i l u r e , and changes i n s y s t e m f r e q u e n c i e s d u r i n g s w i n g c o n d i t i o n s . I n v i e w o f t h e s h o r t c o m i n g s o f s t a t i c f i l t e r , s u p p l e m e n t a l e x c i t a t i o n c o n t r o l may p r o v i d e a b e t t e r a l t e r n a t i v e t o s u p p r e s s t h e s t e a d y s t a t e SSR phenomon. 3 S a i t o e t . a l . [ 3 ] p r o p o s e d a N e g a t i v e Damping S t a b i l i z e r w h i c h i s an e x c i t a t i o n c o n t r o l l e r u s i n g r e a c t i v e power as t h e c o n t r o l s i g n a l ; Hamdam and Hughes [ 4 ] p r o p o s e d a s i m i l a r t y p e c o n t r o l l e r u s i n g r e a c t i v e o r r e a l power f e e d b a c k . I n e x a m i n i n g t h e i r a p p r o a c h i n d e s i g n i n g t h e s e c o n t r o l s , i t i s found t h a t t h e i r a i m i s t o s u p p r e s s t h e e l e c t r i c a l r e s o n a n c e phenomena whereas SSR i s much more c o m p l i c a t e d due t o t h e e l e c t r o m e c h a n i c a l i n t e r a c t i o n . To a c c o u n t f o r t h e e l e c t r o m e c h a n i c a l i n t e r a c t i o n , a u n i f i e d e l e c t r o m e c h a n i c a l s y s t e m mode l s h o u l d be u sed i n a n a l y s i s and d e s i g n . Fouad and Khu [ 5 ] p r o p o s e d a c o n t r o l l e r u s i n g speed s i g n a l where t h e p i c k - u p speed d e v i a t i o n w i l l be p r o d u c e d s o l e l y by t h e mode t o be c o n t r o l l e d . Y u , Wvong and Tse [ 6 ] p r o p o s e d a w i d e r a n g e l i n e a r o p t i m a l c o n t r o l l e r u s i n g m e a s u r a b l e s t a t e f e e d b a c k as t h e c o n t r o l s i g n a l . E l - S e r a f i and S h a l t o u t [ 7 ] s u g g e s t e d a m u l t i - l o o p e x c i t a t i o n c o n t r o l l e r by u s i n g f i l t e r e d and phase s h i f t e d t e r m i n a l v o l t a g e as s i g n a l s . I n t h e a n a l o g s t u d i e s by Fouad and K h u , e x c i t a t i o n v o l t a g e c e i l i n g l i m i t s a r e n e g l e c t e d . I n t h e l a s t two p a p e r s [ 6 , 7 ] , t h e r e s u l t s were n o t v e r i f i e d on t h e n o n - l i n e a r power s y s t e m . H e n c e , t h e l a c k o f n o n l i n e a r t e s t i n g and v o l t a g e c e i l i n g l i m i t s may have l e d to:. . o p t i m i s t i c c o n c l u s i o n s ' abou t e x c i t a t i o n " c o n t r o l o f " SSR.. 1 .3 Scope o f t h e T h e s i s I n t h i s t h e s i s , an e x c i t a t i o n c o n t r o l l e r o f t h e l i n e a r o p t i m a l t y p e u s i n g m e a s u r a b l e s t a t e v a r i a b l e s f o r t h e f e e d b a c k w i l l be d e v e l o p e d and t e s t e d on b o t h l i n e a r and n o n l i n e a r mode l s ,[21], I n C h a p t e r 2, a u n i f i e d n o n -l i n e a r power s y s t e m m o d e l i s d e v e l o p e d ; t h e mode l i s t h e n l i n e a r i z e d and i t s o r d e r i s r e d u c e d t o f a c i l i t a t e t h e c o n t r o l l e r d e s i g n ; and t h e e i g e n v a l u e s o f t h e s y s t e m a t v a r i o u s d e g r e e s o f c a p a c i t o r c o m p e n s a t i o n a r e s t u d i e d . I n C h a p t e r 3 , a w i d e - c a p a c i t o r - r a n g e l i n e a r o p t i m a l c o n t r o l l e r i s d e s i g n e d and 4 i t s e f f e c t i v e n e s s i s v e r i f i e d by e i g e n v a l u e a n a l y s i s . I n C h a p t e r 4 , t h e c o n t r o l l e r i s f u r t h e r t e s t e d on t h e o r i g i n a l n o n l i n e a r power s y s t e m mode l d e v e l o p e d i n C h a p t e r 2 . The d y n a m i c r e s p o n s e s o f t h e s y s t e m w i t h and w i t h -o u t c o n t r o l a r e o b t a i n e d . F i n a l l y , a summary o f i m p o r t a n t f i n d i n g s i s g i v e n i n C h a p t e r 5 . 5 2 . MODELLING A POWER SYSTEM FOR SUBSYNCHRONOUS RESONANCE STUDIES 2 . 1 I n t r o d u c t i o n The component r e p r e s e n t a t i o n o f a power s y s t e m f o r s u b s y n c h r o n o u s r e s o n a n c e s t u d i e s i s q u i t e d i f f e r e n t f rom t h a t o f a c o n v e n t i o n a l power s y s t e m s t a b i l i t y s t u d y . I n o r d e r t o a c c o u n t f o r t h e t o r s i o n a l r e s o n a n c e o f t h e m e c h a n i c a l m a s s - s p r i n g s y s t e m due t o SSR, f a c t o r s t h a t a r e u s u a l l y n e g l e c t e d i n c o n v e n t i o n a l s t a b i l i t y s t u d i e s a r e now i m p o r t a n t : name ly t h e g e n e r a t o r and t u r b i n e s h a f t t o r s i o n a l s t i f f n e s s e s [ 8 ] , g e n e r a t o r r o t o r a m o r t i s s e u r p a r a m e t e r s [9 . ] , a r m a t u r e t r a n s i e n t s and n e t w o r k t r a n s i e n t s . I n t h e f i r s t p a r t o f t h i s c h a p t e r , a d e t a i l e d h i g h o r d e r m a t h e m a t i c a l m o d e l i s d e r i v e d . The m o d e l c o n s i s t s o f t h e g e n e r a t o r t u r b i n e m a s s - s p r i n g s y s t e m , g o v e r n o r , s t e am t u r b i n e t o r q u e s , g e n e r a t o r , e x c i t e r , and c a p a c i t o r - c o m p e n s a t e d t r a n s m i s s i o n l i n e . A f u n c t i o n a l b l o c k d i a g r a m o f t he mode l i s shown i n F i g u r e 2 . 1 . I n t h e s e c o n d p a r t , t h e s y s t e m e q u a t i o n s d e s c r i b i n g t h e h i g h o r d e r m o d e l a r e l i n e a r i z e d a b o u t a n o m i n a l o p e r a t i n g p o i n t and t h e s y s t e m ' s e i g e n v a l u e s a r e e x a m i n e d . F i n a l l y a r e d u c e d o r d e r m o d e l f o r f u r t h e r i n v e s t i g a t i o n i s o b t a i n e d . U E x c i t e r and V o l t a g e R e g -u l a t o r T r a n s m i s s i o n S y s t e m P Q V e , e , t J f D G e n e r a t o r E l e c t r o m a g n -e t i c Dynamics G o v e r n o r " Steam T u r b i n e m M a s s - s p r i n g S y s t e m F i g u r e 2 . 1 F u n c t i o n a l b l o c k d i a g r a m o f a power s y s t e m f o r SSR s t u d i e s 6 2 . 2 M o d e l l i n g t h e M e c h a n i c a l S y s t e m Steam t u r b i n e : The s t eam t u r b i n e s y s t e m mode l i s shown i n F i g u r e 2 . 2 . I t i s a tandem compound, s i n g l e r e h e a t s y s t e m . The m o d e l i s m a i n l y b a s e d on t h e I E E E commi t t ee r e p o r t [ 10 ] . 1 + sT CH 1 + sT RH HP 1 + sT CO I P f T T HP I P F i g u r e 2 . 2 M o d e l o f t h e s t eam t u r b i n e s y s t e m The c o r r e s p o n d i n g e q u a t i o n s a r e LPA LPB T T L P A L P B HP T C H G V T C H H P ( 2 . 1 ) "IP I P F H P T R H H P T I P RH ( 2 . 2 ) L P A L P A F T I P I P CO Tco LPA ( 2 . 3 ) LPB L P B  F L P A L P A ( 2 . 4 ) 7 Generating Unit Mass-spring System : The generating unit t o r s i o n a l system consists of one high pressure turbine (HP), one intermediate pressure turbine (IP), two low pressure turbines (LPA,LPB), one generator (G), and one e x c i t e r (EX) which are a l l mechanically coupled together by shafts as shown i n Figure 2.3 . To make the analysis simple, the following assumptions are made : (a) Each mass-spring element has a lumped mass of i n e r t i a constant M. (b) The mass of the shaft between any two elements i s n e g l i g i b l e and behaves l i k e a l i n e a r t o r s i o n a l spring. (c) Only mechanical damping i s considered.' Figure 2.4 i l l u s t r a t e s the various t o r s i o n a l forces experienced by the i * " * 1 element i n the system - a p o s i t i v e t o r s i o n a l torque ±+±('®±+± 0. ) on the l e f t , a negative torque -K.,., .( 0. - 0. , ) on the r i g h t , 1 ' l + l , x x x - 1 ' an external torque T_^  ,a p o s i t i v e a c celerating torque M^io^ , and a negative damping torque ~®±<ii± . A general equation of motion of the i ^ rotor i s as follows M.u>. = T. - D.u. + K. ... ( 6... - 8. ) - K . ( 6 . - 0 ) (2.5) X X X X X 1,1+1 V 1+1 X 1,1 - 1 X x - 1 where M. : i n e r t i a constant of the i * " ^ rotor x th 6^  : r o t a t i o n displacement of the i rotor K. . ,.. : t o r s i o n a l s t i f f n e s s constant of the shaft between i * " * 1 and x .1+1 ...th l + l rotor F i g u r e 2 . 4 M o d e l l i n g o f t h e m a s s - s p r i n g s y s t e m i n t h e v i c i n i t y o f t h e i r o t a t i o n a l mass 9 A p p l y i n g e q u a t i o n ( 2 . 5 ) t o t h e s i x mass t o r s i o n a l s y s t e m , t w e l v e d i f f e r e n t i a l e q u a t i o n s a r e o b t a i n e d : H i g h p r e s s u r e t u r b i n e I n t e r m e d i a t e L = K 1 2 p r e s s u r e t u r b i n e 2 M 1 2 u)„ = K 1 2 M 2 1 K 1 2 a ° 1 ^ THP S 7 ~ e i " MT w i M 7 " < ^ * 2 + M, 23 D T 2 ^ IP ^ u 2 + s r ( 2 . 6 ) ( 2 . 7 ) ( 2 . 8 ) ( 2 . 9 ) Low p r e s s u r e t u r b i n e A ( D „ = 23 M 3 2 K 2 3 + K 34 K 3 4 D T 3 , LPA ( 2 . 1 0 ) ( 2 . 1 1 ) Low p r e s s u r e t u r b i n e B to. = 34 M. 3 4 rK 3 4 + K 4 5 . K 4 5 D T 4 LPB M, 4 M. 4 4 ( 2 . 1 2 ) G e n e r a t o r V w 4 " % } OJ . = 45 M 5 4 , K 4 5 + K 5 6 , . M K 5 6 ( ) 6 + D c T 5 _ _e M 5 " " M 5 ( 2 . 1 3 ) ( 2 . 1 4 ) E x c i t e r 6 = 0), ( 0) D O J , = 56 K 6 - 56 M, 6 M, 6 6 6 ( 2 . 1 5 ) ( 2 . 1 6 ) V W 6 " u o } ( 2 . 1 7 ) where 6 : e l e c t r i c a l a n g u l a r d i s p l a c e m e n t i n e l e c t r i c a l r a d i a n w h i c h i s equ a l , t o t h e m e c h a n i c a l r a d i a n f o r a two p o l e machine. t i l O K : speed o f t h e i r o t o r i n p e r u n i t . 03-q : s ynchronous speed w h i c h i s e q u a l t o 1 p e r u n i t . o), : base speed w h i c h i s e q u a l t o 377 r a d i a n / s e c o n d , b 6 : m e c h a n i c a l a n g u l a r d i s p l a c e m e n t i n r a d i a n . : e l e c t r i c t o r q u e a c r o s s t h e a i r gap i n p e r u n i t , 10 Speed G o v e r n o r : The speed g o v e r n o r m o d e l shown i n F i g u r e 2 . 5 i s b a s e d on an I E E E c o m m i t t e e r e p o r t [ 1 0 ] . The i n i t i a l power P q i s t h e l o a d r e f e r e n c e w h i c h combines w i t h t h e i n c r e m e n t s due t o speed d e v i a t i o n t o o b t a i n t h e t o t a l p o w e r , P , , , s u b j e c t t o t h e t i m e l a g , To, i n t r o d u c e d by t h e s e r v o m o t o r m e c h a n i s m . F i g u r e 2 . 5 The speed g o v e r n o r The c o r r e s p o n d i n g s t a t e e q u a t i o n s a r e P < P < P GV . - GV - GV m m max 2 . 3 M o d e l l i n g t h e E l e c t r i c a l S y s t e m T r a n s m i s s i o n S y s t e m : The t r a n s m i s s i o n s y s t e m i s r e p r e s e n t e d by a s i n g l e l i n e c o n n e c t e d t o an i n f i n i t e bus as shown i n F i g u r e 2 . 6 . The r e s i s t a n c e and r e a c t a n c e o f t h e t r a n s f o r m e r a r e r e p r e s e n t e d by R and X t r e s p e c t i v e l y . The r e s i s t a n c e and r e a c t a n c e o f t h e t r a n s m i s s i o n l i n e a r e d e n o t e d by R and X . V o l t a g e a c r o s s t h e c a p a c i t o r i s V c , whereas V r e p r e s e n t s t h e t e r m i n a l v o l t a g e a t 11 the capacitor. f" R t X t \ Gen } - 4 - ^ - A / " \ A ^ ~ - / y V V Figure 2.6 The transmission system ct The general voltage equation for the system shown i n Figure 2.6 i s : [ v ] , _ - r R 1 [ I 1 , + [ L ] [ I ] + f V ] , 1 t Jphase L J 1 t Jphase L J d t L t phase ' c phase o phase (2.20) A l l the quantities i n phase coordinate are transformed into Park's coordinate [ 11 ] by a transformation matrix [ T ] = cos© cos(9 - 120) cos (9 + 120) -sine -sin(9 - 120) -sin(6 + 120) 1 1 1 (2.21) Since balance operation i s assumed, the o-component of the d-q-o coordinate i s zero. Hence the equations of the transmission l i n e are : Terminal voltage X + X V, = ( . £ ) I, - ( X_ + X ) I + ( R + R ) I, + V d to d t e q t e d cd + V sinfi o (2.22) V = ( X t + X e ) I + ( X + X ) I j + ( R +R ) I + V q q t e d t e 7 q cq + V cos6 o (2.23) 12 C a p a c i t o r v o l t a g e V c d = V + a) X I , c q c d ( 2 . 2 4 ) -£SL_ = _ •V , + co X I c d c q ( 2 . 2 5 ) V o l t a g e R e g u l a t o r and E x c i t e r The e x c i t e r and v o l t a g e r e g u l a t o r mode l i s shown i n F i g u r e 2 . 7 . I t i s a c o n t i n u o u s a c t i n g t y p e w h i c h i s b a s e d on an I E E E c o m m i t t e e r e p o r t [12] w i t h some s i m p l i f i c a t i o n s - t h e r e g u l a t o r i n p u t f i l t e r t i m e c o n s t a n t , t h e s a t u r a t i o n , and t h e s t a b i l i z i n g f e e d b a c k l o o p a r e n e g l e c t e d . r e f max 1 + sT R m i n J f D F i g u r e 2 . 7 V o l t a g e r e g u l a t o r and e x c i t e r m o d e l where U i s t h e s u p p l e m e n t a r y c o n t r o l s i g n a l E K i s t h e v o l t a g e r e g u l a t o r g a i n T ^ i s t h e v o l t a g e r e g u l a t o r t i m e c o n s t a n t T„ i s t h e e x c i t e r t i m e c o n s t a n t E E ^ D i s t h e o u t p u t v o l t a g e o f t h e e x c i t e r V. R m i n and V. R max a r e t h e r e g u l a t o r c e i l i n g v o l t a g e The c o r r e s p o n d i n g e q u a t i o n s a r e : K A V = —— ( V - V + U ) -R I , 1 r e f t V A 7 K m i n — "R — "R max ( 2 . 2 6 ) 13 E f D = -Yl\ ' -T-hv ( 2 ' 2 7 ) V = / V 2 + V 2 • ( 2 . 2 8 ) t d q The s y n c h r o n o u s g e n e r a t o r I n p r e v i o u s w o r k [13] , a s i x - w i n d i n g s y n c h r o n o u s g e n e r a t o r m o d e l , as shown i n F i g u r e 2 . 8 , i s a s sumed . S i n c e t h e damper w i n d i n g s a r e p e r m a n e n t l y s h o r t c i r c u i t e d , t h e q u e s t i o n i s : s h o u l d t h e two damper w i n d i n g s on t h e Q - a x i s be r e p r e s e n t e d by one e q u i v a l e n t w i n d i n g ? By t a k i n g t h e p a r a l l e l e q u i v a l e n t o f t h e two Q-damper l e a k a g e i m p e d a n c e , as shown i n F i g u r e 2 . 9 , a f i v e w i n d i n g - mode l i s d e r i v e d . The dynamic r e s p o n s e s o f t h e s i x - w i n d i n g and f i v e - w i n d i n g mode l s when c o n n e c t e d t o . t h e r e s t o f t h e s y s t e m a r e shown i n F i g u r e 2 . 1 0 t o F i g u r e 2 .12 . I t i s f o u n d t h a t t h e a r m a t u r e c u r r e n t s o f t h e two mode l s a r e t h e same, and i o f t h e f i v e w i n d i n g mode l i s s i m p l y t h e sum o f i and i 0 Q b o f t h e s i x - w i n d i n g m o d e l . I n o t h e r w o r d s , t h e e l e c t r i c t o r q u e p r o d u c e d by t h e two mode l s a c r o s s t h e a i r gap i s t h e same. H e n c e , a f i v e - w i n d i n g mode l r e p r e s e n t a t i o n i s s u f f i c i e n t . The v o l t a g e e q u a t i o n s o f t h e r o t o r c i r c u i t a r e : V , = f . - u f - R i ( 2 . 2 9 ) d d q a d V ¥ + m l ' - R i ( 2 . 3 0 ) q q d a q V = ¥ - R i ( 2 . 3 1 ) 0 - % " ; R D S> ( 2 ' 3 2 ) o - *Q - V Q ( 2"3 3 ) The e l e c t r i c t o r q u e e q u a t i o n i n p e r u n i t i s T = i - y i , ( 2 . 3 4 ) e d q q d q - a x i s d - a x i s D ( T O d f F i g u r e 2 . 8 A s i x - w i n d i n g g e n e r a t o r m o d e l r i i i j Si V i i i X T • qL d - a x i s R • a -AA/V X mq q - a x i s F i g u r e 2 . 9 E q u i v a l e n t c i r c u i t o f a f i v e - w i n d i n g g e n e r a t o r mode l f r o m a s i x - w i n d i n g mode l ( s o l i d l i n e -damper l i n k a g e impedance o f t h e f i v e - w i n d i n g m o d e l ; d o t t e d l i n e - d a m p e r l e a k a g e impedances o f t h e s i x - w i n d i n g m o d e l ) 15 F i g u r e 2 . 1 0 A r m a t u r e c u r r e n t r e s p o n s e o f t h e s i x - w i n d i n g and f i v e - w i n d i n g g e n e r a t o r m o d e l 16 F i g u r e 2 . 1 2 Q - a x i s damper c u r r e n t s o f t h e s i x - w i n d i n g g e n e r a t o r mode l 17 The f l u x l i n k a g e e q u a t i o n i s : ' - x d X md X md ±A q - X q X mq i q = md X f X md ±f X md X md X D 1 Q J - X mq X Q . . V ( 2 . 3 5 ) 2 . 4 L i n e a r i z e d M o d e l The m o d e l d e r i v e d above i s n o n l i n e a r . E x c l u d i n g t h e c e i l i n g l i m i t s i m p o s e d on t h e v o l t a g e r e g u l a t o r and g o v e r n o r , t h e o t h e r n o n l i n e a r i t i e s a r e : t h e q u a d r a t i c t e rms o f t h e c u r r e n t s i n t he e l e c t r i c t o r q u e e q u a t i o n , t h e t r i g o n -m e t r i c f u n c t i o n s due t o P a r k ' s t r a n s f o r m a t i o n , and t h e speed v o l t a g e t e r m s , S t a b i l i t y a n a l y s i s , i s q u i t e complex f o r n o n l i n e a r s y s t e m s . I t i s , h o w e v e r , e x p e c t e d t h a t t h e s t a b i l i t y c r i t e r i a f o r l i n e a r s y s t e m s c o u l d be a p p l i e d t o n o n l i n e a r sy s t ems i f t h e d e v i a t i o n s f rom the : e q u i l i b r i u m s t a t e a r e s u f f i c i e n t l y s m a l l so t h a t t h e n o n l i n e a r i t y has o n l y a m i n o r e f f e c t [ T 4 ] . By n e g l e c t i n g t h e c e i l i n g l i m i t s , t h e o r i g i n a l s y s t e m o f e q u a t i o n s i s l i n e a r i z e d abou t a n o m i n a l o p e r a t i n g p o i n t as d e s c r i b e d i n [ 15 ] w h i c h r e s u l t s i n a s e t o f s t a n d a r d s t a t e v a r i a b l e e q u a t i o n s i n t h e f o r m o f [ x ] = [ A ] [ x ] ( 2 . 3 6 ) E q u a t i o n ( 2 . 3 6 ) c a n be p a r t i t i o n e d i n t o e l e c t r i c and m e c h a n i c a l s u b s y s t e m s as f o l l o w s f X I — X I I V i V I I A A I I . I I I , I I f x 1 I X I I I J ( 2 . 3 7 ) 18 where [ x^. ] a r e t h e s t a t e v a r i a b l e s o f t h e m e c h a n i c a l s y s t e m [ X-J-J] a r e t h e s t a t e v a r i a b l e s o f t h e e l e c t r i c a l s y s t e m 2 . 5 Reduced O r d e r M o d e l F o r s y s t e m d e s i g n p u r p o s e s , i t i s d e s i r a b l e t o have a l o w o r d e r mode l t o a p p r o x i m a t e t h e h i g h o r d e r mode l so as t o m i n i m i z e t h e c o m p u t a t i o n a l e f f o r t . D e c i d i n g on w h i c h component t o n e g l e c t m a i n l y depends on t h e t i m e s p a n o f s t u d y , component v a l u e s , and d e g r e e o f c o u p l i n g o f t h e component t o t h e r e s t o f t h e s y s t e m . H o w e v e r , one p r i n c i p l e i s t h a t t h e l o w e r o r d e r mode l s h o u l d r e t a i n t o a c e r t a i n e x t e n t a l l t h e dominan t p r o p e r t i e s o f t h e h i g h o r d e r m o d e l . S i n c e t h e e x c i t e r mass s p r i n g c o n s t a n t i s s m a l l i n c o m p a r i s o n to t h e r e s t o f t h e m a s s - s p r i n g s y s t e m , i t i s n e g l e c t e d . F u r t h e r m o r e , t h e g o v e r n o r and s t eam t u r b i n e s a r e n e g l e c t e d b e c a u s e o f i t s l a r g e t i m e c o n s t a n t . Thus t h e 26*"^ o r d e r t h m o d e l i s r e d u c e d t o a 19 o r d e r m o d e l . 2 . 6 E i g e n v a l u e A n a l y s i s I n modern a n a l y s i s , s y s t e m s a r e u s u a l l y d e s c r i b e d by a s e t o f d i f f e r e n t i a l e q u a t i o n s i n s t a t e s p a c e f o r m . H e n c e , t h e s t a b i l i t y o f a l i n e a r t i m e i n v a r i a n t s y s t e m can e a s i l y be d e t e r m i n e d by e x a m i n i n g t h e e i g e n v a l u e s o f t h e s y s t e m m a t r i x . The r e a l p a r t o f t h e e i g e n v a l u e s d i s c l o s e s t h e s t a b i l i t y o f t h e s y s t e m : s t a b l e i f a l l i t s e i g e n v a l u e s have n e g a t i v e r e a l p a r t s ; u n s t a b l e i f any o f i t s . . e i g e n v a l u e s have a p o s i t i v e r e a l p a r t . The i m a g i n a r y p a r t o f t h e e i g e n v a l u e s i n d i c a t e s t h e s y s t e m ' s n a t u r a l f r e q u e n c i e s . F o r t h e s y s t e m unde r c o n s i d e r a t i o n h e r e , most o f t h e e i g e n v a l u e s do n o t change w i t h d i f f e r e n t l o a d i n g s [ 15 ] , so t h e e i g e n v a l u e a n a l y s i s i s c o n f i n e d t o t h e f o l l o w i n g c o n d i t i o n s : P q = 0 . 9 p . u . , P . F . = 0 . 9 l a g g i n g , V F C = 1.0 p . u . The e i g e n v a l u e s o f t h e 27^ o r d e r mode l [ 15 ] , 26*"^ o r d e r m o d e l , t h and 19 o r d e r mode l f o r v a r i o u s d e g r e e s o f c o m p e n s a t i o n a r e l i s t e d i n 19 T a b l e 2 . 1 - T a b l e 2 . 4 . I t i s found t h a t t h e s y s t e m i s u n s t a b l e when t h e n a t u r a l f r e q u e n c y o f t h e e l e c t r i c a l mode i s c l o s e t o a m e c h a n i c a l mode, and as t h e d e g r e e o f c o m p e n s a t i o n i n c r e a s e s , two m e c h a n i c a l modes a r e e x c i t e d s i m u l t a n o u s l y . From t h e T a b l e s , we can see t h a t t h e c o r r e s p o n d i n g e i g e n v a l u e s o f t h e 2 7 t ' 1 o r d e r and 2 6 ^ mode l a r e c l o s e , so i t p r o v i d e s f u r t h e r e v i d e n c e t h a t t h 26 o r d e r mode l i s s u f f i c i e n t . F u r t h e r m o r e , t h e dominan t e i g e n v a l u e s o f t h e 2 6 t ' 1 o r d e r mode l a r e r e t a i n e d i n t h e 1 9 ^ o r d e r m o d e l . H e n c e , i t . i s d e c i d e d t h a t t h e 19 o r d e r mode l c a n be u s e d f o r s t a b i l i z e r d e s i g n . 20 2 7 t h o r d e r ut h , 26 o r d e r r e d u c e d 1 9 ^ mode l mode l o r d e r mode l - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± j 2 9 8 . 1 8 " +0 .1541 ± j 2 0 4 . 3 5 +0 .1544 ± j 2 0 4 . 3 6 +0 .0652 ± j 2 0 3 . 7 1 s h a f t modes - 0 . 2 4 9 6 ± J 1 6 0 . 7 2 - 0 . 2 4 9 7 ± .1160.72 - 0 . 2 3 9 7 ± j 1 6 0 . 1 2 - 0 . 6 7 0 6 ± J 1 2 7 . 0 3 - 0 . 6 7 0 6 ± J 1 2 7 . 0 3 - 0 . 2 8 7 7 ± j 9 9 . 2 1 0 - 0 . 2 8 7 7 ± j 9 9 : 2 1 0 - 0 . 2 4 7 7 ± j l O l . 1 3 - 0 . 0 4 7 9 ± j 8 . 4 8 0 1 - 0 . 0 5 2 8 ± j 8 . 4 8 0 0 - 0 . 0 8 4 9 ± J 8 . 5 4 0 0 - 0 . 1 4 1 7 - 0 . 1 4 1 7 - 4 . 6 1 6 0 - 4 . 5 9 7 6 T u r b i n e and - 3 . 0 3 3 6 - 3 . 1 0 4 9 G o v e r n o r - 4 . 6 7 3 2 ± j O . 6 2 6 9 - 4 . 6 6 4 2 ± J 0 . 5 9 7 8 S t a t o r - 7 . 0 2 2 4 ± j 5 4 2 . 8 0 - 7 . 0 2 2 7 ± j 5 4 2 . 8 0 - 7 . 0 2 2 8 ± J 5 4 2 . 8 0 and N e t w o r k - 6 . 1 9 8 4 ± j 2 0 9 . 2 0 - 6 . 1 9 8 6 ± j 2 0 9 . 1 9 - 6 . 1 0 5 7 ± J 2 0 9 . 3 2 - 8 . 4 4 0 4 - 8 . 4 7 0 2 - 8 . 5 2 0 6 M a c h i n e - 3 1 . 9 2 0 - 3 1 . 9 8 8 - 3 1 . 9 8 5 r o t o r - 2 5 . 4 0 4 - 1 . 9 8 3 0 - 2 . 0 2 1 8 - 2 . 2 5 9 1 E x c i t e r - 4 9 9 . 9 7 - 5 0 0 . 0 0 - 5 0 0 . 0 0 and V o l t a g e - 1 0 1 . 9 1 - 1 0 1 . 9 3 - 1 0 1 . 9 3 R e g u l a t o r T a b l e 2 . 1 E i g e n v a l u e s o f v a r i o u s o r d e r SSR mode l .at 30% c a p a c i t o r c o m p e n a t i o n f o r P = 0 . 9 p . u . a t 0 . 9 power f a c t o r l a g g i n g . 21 2 7 t h o r d e r 26 o r d e r r e d u c e d 19 ^ m o d e l mode l o r d e r m o d e l - 0 . 1 8 1 8 + J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 + J 2 9 8 . 1 8 +0 .1560 + J 2 0 2 . 6 8 +0 .1560 ± J 2 0 2 . 6 8 +0 .1513 + J 2 0 2 . 1 6 S h a f t +0 .9100 + J 1 6 1 . 4 2 +0.9106 ± J161.42 +0.7977 + J 1 6 0 . 8 6 modes - 0 . 6 7 9 9 + J 1 2 7 . 0 8 - 0 . 6 7 9 8 ± J 1 2 7 . 4 9 - 0 . 3 5 4 5 + J 9 9 . 7 9 0 - 0 . 3 5 4 6 ± J 9 9 . 4 9 0 - 0 . 3 2 5 1 + j l O l . 4 6 - 0 . 2 6 7 4 + J 9 . 5 4 5 9 - 0 . 2 7 4 9 ± J 9 . 5 5 0 - 0 . 2 9 5 8 + J 9 . 6 0 0 0 - 0 . 1 4 1 8 - 0 . 1 4 1 8 T u r b i n e and - 4 . 0 4 9 6 - 3 . 8 6 0 4 G o v e r n o r - 3 . 3 3 3 5 - 3 . 5 1 0 4 - 4 . 7 9 3 9 + j O . 3 1 9 8 - 4 . 8 1 4 7 ± J 0 . 2 9 9 2 S t a t o r - 7 . 0 8 0 0 + J 5 9 1 . 1 5 - 7 . 0 8 0 0 ± J 5 9 1 . 1 6 - 7 . 0 8 0 0 + J 5 9 1 . 1 6 and N e t w o r k - 6 . 8 3 8 7 + J 1 6 1 . 4 7 - 6 . 8 3 8 6 ± J 1 6 1 . 4 7 - 6 . 7 0 5 9 + J 1 6 1 . 4 1 - 8 . 1 2 7 7 - 8 . 1 7 4 6 - 8 . 2 2 9 3 M a c h i n e - 3 2 . 8 0 8 - 3 2 . 8 8 2 - 3 2 . 8 7 9 r o t o r - 2 5 . 4 2 3 - 1 . 9 0 7 0 - 1 . 9 4 4 7 - 2 . 1 7 0 0 E x c i t e r - 4 9 9 . 9 7 - 5 0 0 . 0 0 - 5 0 0 . 0 0 and V o l t a g e - 1 0 1 . 7 6 - 1 0 1 . 7 7 - 1 0 1 . 7 7 R e g u l a t o r T a b l e 2 . 2 E i g e n v a l u e s o f v a r i o u s o r d e r SSR m o d e l a t 50% c a p a c i t o r c o m p e n s a t i o n f o r P = 0 . 9 p . u . a t 0 . 9 power f a c t o r l a g g i n g . 22 2 7 t h o r d e r „ ^ t h 26 o r d e r r e d u c e d 1 9 ^ mode l mode l o r d e r mode l - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 +0 .0386 ± J 2 0 2 . 7 8 +0.0386 ± J 2 0 2 . 7 9 +0 .0371 ± J 2 0 2 . 2 6 S h a f t ^ 0 . 0 3 3 1 ± j 1 6 0 . 3 6 - 0 . 0 3 3 1 ± J 1 6 0 . 3 7 - 0 . 0 3 2 6 ± J 1 5 9 . 7 8 modes - 0 . 2 3 3 5 ± J 1 2 6 . 8 3 - 0 . 2 3 3 8 ± J 1 2 6 . 8 3 - 0 . 4 5 3 2 ± J 1 0 0 . 5 7 - 0 . 4 5 3 3 ± J 1 0 0 . 5 7 - 0 . 4 1 7 4 ± J 1 0 2 . 8 5 - 0 . 5 7 1 2 ± j l O . 9 0 5 — 0 . 5 8 1 9 ± j l O . 9 1 0 - 0 . 5 9 3 1 ± j l O . 9 8 0 - 0 . 1 4 1 9 - 0 . 1 4 1 9 T u r b i n e and - 3 . 5 3 4 0 ± j O . 4 5 9 5 - 3 . 5 3 3 4 ± j O . 5 2 6 6 G o v e r n o r - 4 . 9 3 2 5 ± j O . 1 8 6 1 - 4 . 9 4 3 6 ± j O . 1 6 7 2 S t a t o r and - 7 . 1 2 0 3 + J 6 3 0 . 4 5 - 7 . 1 2 0 2 ± J 6 3 0 . 4 5 - 7 . 1 2 0 3 ± J 6 3 0 . 4 5 N e t w o r k - 5 . 4 9 5 1 ± i l 2 2 . 2 2 - 5 . 4 9 4 3 ± J 1 2 2 . 2 2 - 5 . 0 3 9 0 ± J 1 2 1 . 6 3 - 7 . 7 1 5 4 - 7 . 7 7 9 1 - 7 . 8 3 9 7 M a c h i n e - 3 4 . 1 9 8 - 3 4 . 2 0 7 7 - 3 4 . 2 0 4 r o t o r - 2 5 . 4 5 5 - 1 . 8 1 1 6 - 1 . 8 4 6 3 - 2 . 0 3 9 7 E x c i t e r and - 4 9 9 . 9 7 - 5 0 0 . 0 0 - 5 0 0 . 0 0 V o l t a g e R e g u l a t o r - 1 0 1 . 5 6 - 1 0 1 . 5 6 - 1 0 1 . 5 6 F i g u r e 2 . 3 E i g e n v a l u e s o f v a r i o u s o r d e r S.SRimodel a t 70% c a p a c i t o r c o m p e n s a t i o n f o r P = 0 . 9 p . u . a t 0 . 9 power f a c t o r l a g g i n g . 23 27*"'1 o r d e r mode l 26 o r d e r mode l r e d u c e d 19^ o r d e r mode l - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± j 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 +0 .0173 i j 2 0 2 . 8 1 +0 .0173 ± J 2 0 2 . 8 1 +0 .0166 ± J 2 0 2 . 2 9 S h a f t modes - 0 . 0 8 1 2 ± - 0 . 5 9 3 1 ± j 1 6 0 . 4 3 j 1 2 6 . 8 9 - 0 . 0 8 8 2 1 - 0 . 5 9 3 1 ± J 1 6 0 . 4 3 j 1 2 6 . 8 9 - 0 . 0 8 6 2 ± J 1 5 9 . 8 4 +2 .9035 ± J 1 0 2 . 2 8 +2 .9061 ± J 1 0 2 . 7 7 +3 .8541 ± J 1 0 3 . 6 8 - 0 . 7 7 4 1 ± j 1 1 . 7 5 1 - 0 . 7 8 7 2 ± j l l . 7 6 0 - 0 . 7 9 4 2 ± j l l . 8 4 0 - 0 . 1 4 1 9 - 0 . 1 4 1 9 T u r b i n e and - 3 . 4 7 8 2 ± j O . 5 5 1 4 - 3 . 4 7 8 3 ± j O . 5 9 8 9 G o v e r n o r - 4 . 9 7 5 9 ± j O . 1 0 8 0 - 4 . 9 8 3 7 ± j O . 0 8 0 1 S t a t o r and N e t w o r k - 7 . 1 3 6 5 ± - 7 . 9 7 6 0 ± J 6 4 7 . 9 7 J 1 0 2 . 7 7 - 7 . 1 3 6 5 ± - 7 . 9 7 7 4 ± J 6 4 7 . 9 7 J 1 0 2 . 7 7 - 7 . 1 3 6 5 ± J 6 4 7 . 6 9 - 8 . 7 9 5 7 ± J 1 0 3 . 6 9 - 7 . 4 4 7 4 - 7 . 5 2 8 4 - 7 . 5 9 2 4 M a c h i n e r o t o r - 3 5 . 1 2 0 - 2 5 . 4 7 2 - 1 . 7 4 5 0 - 3 5 . 1 3 2 - 1 . 7 7 8 0 - 3 5 , 1 2 8 - 1 . 9 4 6 4 E x c i t e r and V o l t a g e R e g u l a t o r - 4 9 9 . 9 7 - 1 0 1 . 4 2 - 5 0 0 . 0 0 - 1 0 1 . 4 2 - 5 0 0 . 0 0 - 1 0 1 . 4 2 -F i g u r e 2 . 4 E i g e n v a l u e s o f v a r i o u s o r d e r SSR mode l a t 80% c a p a c i t o r c o m p e n s a t i o n f o r P = 0 . 9 p . u . a t 0 . 9 power f a c t o r l a g g i n g . 24 3 . L I N E A R OPTIMAL EXCITATION CONTROL DESIGN 3 . 1 I n t r o d u c t i o n W i t h t h e a d v e n t o f l a r g e d i g i t a l compu te r s and n u m e r i c a l a n a l y s i s t e c h n i q u e s , some modern c o n t r o l t h e o r i e s can r e a d i l y be a p p l i e d t o e l e c t r i c a l , e c o n o m i c , and o t h e r l a r g e s y s t e m s . A c e r t a i n c l a s s o f l i n e a r o p t i m a l c o n t r o l t h e o r y , known as t h e s t a t e r e g u l a t o r p r o b l e m , has b e e n a p p l i e d t o t h e s t a b i l i z e r d e s i g n t o i m p r o v e t h e d y n a m i c r e s p o n s e o f power s y s t e m s , e . g . , t h e l o w f r e q u e n c y s y s t e m o s c i l l a t i o n [16 -19 ] and t h e SSR [ 6 ] . I t was found t h a t t h e l i n e a r o p t i m a l c o n t r o l l e r n o t o n l y c a n p r o v i d e good damping t o t h e s y s t e m , b u t a l s o can s t a b i l i z e t h e s y s t e m o v e r a w i d e - p o w e r - r a n g e o p e r a t i o n [ 1 8 , 1 9 ] . t h e power s y s t e m due t o s u b s y n c h r o n o u s r e s o n a n c e . B e c a u s e o f t h e m e r i t s o f l i n e a r o p t i m a l c o n t r o l , a l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l l e r w i l l be d e s i g n e d t o s t a b i l i z e t h e s u b s y n c h r o n o u s r e s o n a n c e i n t h e s y s t e m . F o r t h e d e s i g n , a r e d u c e d 1 9 ^ o r d e r mode l w i t h a l l m e a s u r a b l e s t a t e v a r i a b l e s i s t h c h o s e n . Bu t t h e c o n t r o l i s a p p l i e d t o t h e 26 o r d e r f u l l mode l f o r a l i n e a r t e s t i n t h i s C h a p t e r , where t h e e i g e n v a l u e s o f t h e c l o s e d l o o p s y s t e m a t v a r i o u s o p e r a t i n g and c o m p e n s a t i o n c o n d i t i o n s w i l l be e x a m i n e d . 3 . 2 L i n e a r S t a t e R e g u l a t o r P r o b l e m The l i n e a r o p t i m a l r e g u l a t o r may be f o r m u l a t e d as f o l l o w s : C o n s i d e r t h e l i n e a r i z e d s y s t e m s t a t e e q u a t i o n s As f o u n d i n C h a p t e r 2 , m u l t i p l e u n s t a b l e e igenmodes may e x i s t i n x Ax + Bu ( 3 . 1 ) y Hx ( 3 . 2 ) F i n d t h e o p t i m a l p e r f o r m a n c e f u n c t i o n 1 oo . t t J = 9- / ( y Q y + u R u) d t ( 3 . 3 ) 0 s u b j e c t t o t h e s y s t e m d y n a m i c c o n t r i a n t ( 3 . 1 ) and ( 3 . 2 ) 25 s u b s t i t u t i n g ( 3 . 2 ) i n t o ( 3 . 3 ) g i v e s 1 °° t t t J = ^ / [ x ( H Q H) x + u R u ] d t ( 3 . 4 ) 2 0 A H a m i l t o n i a n was fo rmed b y a p p e n d i n g ( 3 . 1 ) t o ( 3 . 4 ) , H = | [ x t ( H t Q H) x + u f c R u] + p C (Ax + Bu) ( 3 . 5 ) where p f c i s t h e c o s t a t e v e c t o r o r L a g r a n g e m u l t i p l i e r s . The o p t i m a l c o n t r o l c a n be f o u n d f rom 8f//8u , r e s u l t i n g o r u = - R 1 B t K x ( 3 . 6 ) u = - R ^ K H _ 1 y ( 3 . 7 ) where i n t h e p a r t i c u l a r LOC d e s i g n i n t h i s t h e s i s , H i s a i n v e r t a b l e s q u a r e m a t r i x . K o f ( 3 . 6 ) and ( 3 . 7 ) i s t he R i c c a t i m a t r i x w h i c h s a t i f i e s t h e n o n l i n e a r m a t r i x a l g e b r a i c e q u a t i o n K A + A t K - K B R _ 1 B t K = - H f cQ H ( 3 . 8 ) I n ( 3 . 8 ) , Q i s a p o s i t i v e s e m i - d e f i n i t e m a t r i x and R i s a p o s i t i v e d e f i n i t e m a t r i x . W i t h u d e c i d e d , t h e c l o s e d l o o p s y s t e m e q u a t i o n becomes x = Gx ( 3 . 9 ) where G = ( A - B R _ 1 B t K ) ( 3 . 1 0 ) F o r t h e p a r t i c u l a r c a s e K A t B = [ 0 0 0 ] ( 3 . 1 1 ) T A x = [ A o ^ . A e ^ Au> 2, A 0 2 , Aw A 9 3 , Ao> 4, A 6 4 , Atii , A6 ; A i , , A i , A i . , A i , A i . , AV , , AV , A V n , AE d ' q ' f D ' Q ' c d ' c q ' R ' f D J ( 3 . 1 2 ) y = [ A w 1 , A e i , A w 2 , A 6 2 , Aw A 0 ^ , A u ^, AQ^, Aco , A6 ; A P e , A Q e , A V t , A i a , A i f , A V c , A V ^ . A V R , A E f l ) ] t ( 3 . 1 3 ) 26 The t r a n s f o r m a t i o n m a t r i x H i s shown i n t h e A p p e n d i x I. 3 .3 S o l u t i o n t o the M a t r i x R i c c a t i E q u a t i o n The n o n l i n e a r a l g e b r a i c m a t r i x e q u a t i o n ( 3 . 8 ) i s s o l v e d by t h e c o m p o s i t e m a t r i x method [ 2 0 ] . F o r t h e c o n t i n u o u s c o n t r o l s y s t e m o f e q u a t i o n ( 3 . 1 ) , ( 3 . 2 ) , and c o s t f u n c t i o n o f e q u a t i o n ( 3 . 4 ) , t h e 2n x 2n c o m p o s i t e m a t r i x [ M ] i s g i v e n by TM ] - H t Q H -B R - V -A ( 3 . 1 4 ) The 2n e i g e n v a l u e s o f m a t r i x [ M ] a r e s y m m e t r i c a l l y d i s t r i b u t e d on t h e r i g h t and t h e l e f t p a r t s o f t h e c o m p l e x p l a n e . L e t t h e e i g e n v a l u e m a t r i x be ( A . [ A ] = II ( 3 . 1 5 ) and t h e c o r r e s p o n d i n g e i g e n v e c t o r m a t r i x be [ X ] I I I I X I I x i v ( 3 . 1 6 ) where [ A ^ ] c o n s t i t u t e s n s t a b l e e i g e n v a l u e s o f [ M ] w h i c h a r e t h e e i g e n -v a l u e s o f t h e c l o s e d l o o p s y s t e m G i n ( 3 . 9 ) , and [ A ] = - [ ] . Then t h e R i c c a t i m a t r i x [ K ] may be computed f r o m [ K ] = [ X T I ] [ X I ]" ( 3 . 1 7 ) 27 3 . 4 Wide Range SSR S t a b i l i z e r and t h e C h o i c e o f W e i g h t i n g M a t r i c e s The p r o p e r t i e s o f t h e c o m p o s i t e m a t r i x [ M ] s u g g e s t t h a t f o r any c h o s e n w e i g h t i n g m a t r i c e s [Q] and [R] o f t h e s p e c i f i e d t y p e , t h e r e w i l l a l w a y s be an o p t i m a l c o n t r o l w h i c h w i l l s t a b i l i z e t h e s y s t e m . A l t h o u g h t h e above i n f o r m a t i o n s u g g e s t s t h a t c o n t r o l l e r s f o r v a r i o u s d e g r e e s o f c a p a c i t o r comp-s a t i o n can a l w a y s be d e s i g n e d , i t i s d e s i r a b l e t o have a s i n g l e c o n t r o l l e r t h a t w i l l s t a b i l i z e t h e s y s t e m f o r a w i d e r ange o f c a p a c i t o r c o m p e n s a t i o n and o p e r a t i n g c o n d i t i o n s . The s o l u t i o n [K] f o r e q u a t i o n ( 3 . 7 ) i s u n i q u e f o r a g i v e n s e t o f w e i g h t i n g m a t r i c e s [Q] and [ R ] , so t h e amount o f damping a c h i e v e d i n the c l o s e d l o o p s y s t e m and t h e g o a l o f w i d e r a n g e s t a b i l i z a t i o n o f t h e s y s t e m depend m a i n l y upon t h e c h o i c e o f t h e w e i g h t i n g m a t r i c e s . A l t h o u g h i t e r a t i v e scheme [18] c a n be u s e d t o o b t a i n [Q] f o r t h e p r e s c r i b e d c l o s e d l o o p e i g e n v a l u e s , c h o i c e o f e l e m e n t s o f [Q] a c c o r d i n g t o t h e r e s u l t s o f p r e v i o u s w o r k [ 1 6 - 1 8 ] may a l s o g i v e s u f f i c i e n t damping t o t h e s y s t e m . Common p r a c t i c e f o r s e r i e s c a p a c i t o r c o m p e n s a t i o n i n most u t i l i t i e s i s up t o 70%. W i t h i n t h a t c o m p e n s a t i o n r a n g e , r e s u l t s f rom C h a p t e r 2 d i s c l o s e t h a t s t a b i l i t y o f t h e s y s t e m u n d e r c o n s i d e r a t i o n i s t h e w o r s t a t 50% compen-s a t i o n ; t h e r e a r e two u n s t a b l e modes . H e n c e , t h e c o n t r o l l e r d e s i g n w i l l be b a s e d on 50% c a p a c i t o r c o m p e n s a t i o n . P r e v i o u s w o r k [ 1 6 - 1 8 ] has s u g g e s t e d t h a t a speed d e v i a t i o n be w e i g h t e d more t h a n an a n g l e d e v i a t i o n , and a m e c h a n i c a l v a r i a b l e d e v i a t i o n more t h a n an e l e c t r i c a l v a r i a b l e d e v i a t i o n . A c c o r d i n g l y , t h e f i n a l w e i g h t i n g m a t r i c e s a r e c h o s e n as f o l l o w s : [R] = 1 [Q] = d i a g [ 1 0 0 , 1 0 , 5 0 0 0 , 5 0 , 5 0 0 0 , 5 0 , 5 0 0 0 0 , 5 0 , 5 0 0 0 , 5 0 ; 1 5 0 , 1 5 0 , 1 , 1 , 0 , 1 , 1 , 0 , 0 ] ( 3 . 1 5 ) 28 I n t h i s c a s e , a s p e c i a l w e i g h t i s g i v e n t o t h e speed d e v i a t i o n o f t h e l o w p r e s s u r e t u r b i n e . W i t h t h i s c h o i c e , t h e u n d e s i r a b l e e i g e n v a l u e s can be e f f e c t i v e l y s h i f t e d to t h e l e f t w i t h o u t e x c e s s i v e i n d i v i d u a l g a i n s . W i t h t h e w e i g h t i n g m a t r i c e s c h o s e n a b o v e , a c o n t r o l l e r d e s i g n e d f o r 50% c o m p e n s a t i o n has t h e f o l l o w i n g g a i n : F o r [x ] mode l - [ R _ 1 B t K ] = [ - 2 . 4 0 7 2 , - 1 8 . 9 6 4 , 3 4 . 3 8 0 , 9 . 4 4 1 7 , 4 1 1 . 0 4 , 1 5 3 . 8 2 , 4 9 2 . 4 5 , 3 0 5 . 4 4 , 3 9 . 7 1 4 , 1 3 5 . 2 7 ; 8 0 . 7 0 2 , 3 4 . 3 1 2 , - 7 7 . 1 6 3 , - 7 6 . 5 2 8 , - 2 9 . 1 0 3 , 1 3 . 4 4 5 , - 8 . 2 6 8 1 , - 0 . 0 2 6 1 , - 0 . 0 0 8 6 ] ( 3 . 1 6 ) F o r [y ] m o d e l - [ R ~ 1 B t K H _ 1 ] = [ - 2 . 4 0 7 2 , - 1 8 . 9 6 4 , 3 4 . 3 8 0 , 9 . 4 4 1 7 , 4 1 1 . 0 4 , 1 5 3 . 8 2 , 4 9 2 . 4 5 , 3 0 5 . 4 4 , 3 9 . 7 1 4 , 1 7 3 . 7 4 ; - 0 . 8 7 3 8 , - 4 9 . 6 3 0 , 6 6 . 5 6 7 , - 0 . 2 7 2 9 , - 0 . 6 4 6 5 , 5 7 . 2 4 2 , - 5 2 . 0 0 2 , - 0 . 0 2 6 1 , - 0 . 0 0 8 6 ] ( 3 . 1 7 ) The c o n t r o l i s t h e n a p p l i e d t o t h e l i n e a r i z e d 261"*1 o r d e r m o d e l -L and t h e e i g e n v a l u e s o f t h e c o n t r o l l e d s y s t e m , a t a c o m p e n s a t i o n (X / x £ ) r a n g i n g f rom 10% t o 90% and f o r t h r e e d i f f e r e n t l o a d i n g c o n d i t i o n s (P = 0 . 9 , 1 . 2 5 , and 0 . 5 p . u . ) , a r e e x a m i n e d . T y p i c a l r e s u l t s a r e g i v e n i n T a b l e s 3 . 1 - 3 .3 . I t i s f o u n d t h a t f o r P £ = 0 . 9 p . u . and 1 .25 p . u . , t h e c o n t r o l l e d s y s t e m i s s t a b l e w i t h i n t h e p r e s c r i b e d r ange o f c o m p e n s a t i o n . H o w e v e r , f o r l i g h t l o a d (P = 0 . 5 p . u . ) t h e s y s t e m i s o n l y s t a b l e up t o 70% c o m p e n s a t i o n . Thus t h e l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l has p r o v e d t o be e f f e c t i v e i n s t a b i l i z i n g SSR w i t h i n t h e l i n e a r - o p e r a t i o n - r a n g e o f t h e s y s t e m . B u t , t h i s may n o t be v a l i d f o r t h e n o n l i n e a r r a n g e . So t h e l i n e a r o p t i m a l c o n t r o l l e r w i l l be f u r t h e r t e s t e d on t h e o r i g i n a l n o n l i n e a r s y s t e m as d e r i v e d i n C h a p t e r 2 , and p r e s e n t e d i n C h a p t e r 4 . 29 c a p a c i t o r 30% 50% 80% c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± .1298.18 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 0 6 4 ± J 2 0 1 . 6 5 - 0 . 6 0 5 2 ± J 2 0 2 . 6 9 - 0 . 4 7 3 6 ± J 2 0 2 . 6 2 S h a f t - 0 . 5 4 8 1 ± J 1 6 0 . 4 6 - 1 . 0 6 2 0 ± J 1 6 1 . 1 2 - 0 . 3 2 0 8 ± J 1 6 0 . 7 1 modes - 0 . 7 1 3 4 ± J 1 2 7 . 1 5 - 0 . 7 3 6 2 ± J 1 2 7 . 2 8 - 0 . 4 1 6 6 ± J 1 2 7 . 0 6 - 0 . 9 6 2 6 ± J 9 8 . 9 4 1 - 1 . 1 7 5 1 ± J 9 8 . 9 8 1 - 0 . 4 1 8 5 ± J 9 9 . 9 6 3 - 3 . 5 5 5 9 ± J 4 . 8 2 5 4 - 5 . 0 3 3 6 ± J 6 . 8 6 1 7 - 9 . 6 7 3 1 ± j l l . 2 7 5 - 0 . 1 4 0 8 - 0 . 1 4 1 5 - 0 . 1 4 2 2 T u r b i n e and - 2 . 9 2 9 6 - 3 . 2 8 7 7 - 3 . 7 4 1 7 ± j O . 8 1 7 8 G o v e r n o r - 4 . 7 5 6 8 - 4 . 7 4 2 6 - 5 . 1 7 0 7 ± j 2 i 2 1 1 7 - 4 . 6 4 8 9 ± j l . 5 9 1 5 - 4 . 8 1 2 2 ± j O . 2 2 6 2 S t a t o r - 7 . 0 1 9 9 ± j 5 4 2 . 7 7 - 7 . 0 8 8 4 ± J 5 9 1 . 1 6 - 7 . 1 5 2 3 ± J 6 4 8 . 0 1 and N e t w o r k - 7 . 9 1 1 2 ± J 2 1 3 . 1 4 - 1 1 . 0 5 7 ± j 1 6 3 . 7 0 - 4 4 . 1 7 8 ± j 1 1 1 . 4 0 - 7 0 . 1 9 3 ± j 7 6 . 1 1 0 - 6 3 . 5 5 2 ± J 7 1 . 6 4 8 - 2 2 . 3 9 1 ± J 6 3 . 2 2 7 M a c h i n e r o t o r - 2 . 0 4 1 7 - 2 . 1 1 6 5 - 2 . 2 0 0 8 E x c i t e r - 5 0 0 . 0 0 - 5 0 0 . 0 0 - 5 0 0 . 4 4 and V o l t a g e - 1 1 8 . 7 1 - 1 2 0 . 5 4 - 1 3 0 . 7 4 R e g u l a t o r T a b l e 3 . 1 E i g e n v a l u e s o f t h e s y s t e m w i t h l i n e a r o p t i m a l c o n t r o l a t v a r i o u s d e g r e e s o f c a p a c i t o r c o m p e n s a t i o n . O p e r a t i n g c o n d i t i o n : P=0.9 p . u . , P . F . = 0 . 9 l a g g i n g , V = 1.0 p . u . 30 30% 50% 70% 80% c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n -0 .1818 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 .1818 ± J 2 9 8 . 1 8 - 0 .2227 ± J 2 0 4 . 6 9 +0 .2282 ± J 2 0 2 . 6 9 +0.0719 ± j 2 0 2 . 7 9 +0 .0427 ± j 2 0 2 . 8 1 - 0 .2965 ± J 1 6 0 . 7 1 +0 .8909 ± j 1 6 1 . 1 6 +0 .0144 + ' j 1 6 0 . 3 8 - 0 .0926 ± J 1 6 0 . 4 6 - 0 . 6783 ± J 1 2 7 . 0 3 - 0 . 6 9 7 7 ± J 1 2 7 . 0 8 - 0 . 1 7 2 7 ± J 1 2 6 . 8 8 - 0 .6164 ± J 1 2 6 . 9 4 - 0 .3557 ± J 9 9 . 1 7 8 - 0 . 4 6 5 7 ± J 9 9 . 4 6 4 - 0 . 7 4 6 8 ± j 1 0 0 . 5 9 +2 .9013 ± J 1 0 2 . 1 5 +0 .3544 ± J 8 . 1 8 2 1 +0 .0760 ± J 9 . 3 1 3 2 - 0 . 2 9 7 5 ± j l O . 7 5 8 - 0 .5416 ± j 1 1 . 6 5 6 (a) 30% 50% 70% 80% c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n - 0 .1818 ± J 2 9 8 . 1 8 - 0 .1818 ± j 2 9 8 . 1 8 - 0 .1818 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 .5033 ± J 2 0 1 . 6 1 - 0 .5054 ± J 2 0 2 . 7 1 - 0 .4337 ± J 2 0 2 . 6 0 - 0 . 4 3 0 6 ± J 2 0 2 . 5 8 - 0 .6497 ± J 1 6 0 . 4 5 - 1 .1105 ± j l 6 3 . l l - 0 .2363 ± j l 6 0 . ' 7 7 - 0 . 2 9 2 7 ± J 1 6 0 . 6 9 - 0 . 7233 ± J 1 2 7 . 1 7 - 0 .7406 ± J 1 2 7 . 3 2 - 0 .2397 ± J 1 2 7 . 2 2 - 0 . 4 0 4 3 ± J 1 2 7 . 0 4 -1 .1814 ± J 9 8 . 8 0 3 - 1 .4867 ± J 9 9 . 0 3 0 - 1 .5509 ± j l O O . 4 3 - 0 . 1 4 9 1 ± j l O O . 0 3 -6 .9895 ± j 2 . 3612 - 6 .6118 ± J 3 . 7 6 5 5 -9 .1417 ± J 6 . 9 3 7 3 - 1 2 . 1 6 2 ± J 8 . 1 3 0 9 (b) T a b l e 3 .2 (a) S h a f t modes< o f t h e s y s t e m a t v a r i o u s d e g r e e s o f c o m p e n s a t i o n a t P=1.25 p . u . , P . F . = 0 . 9 l a g g i n g V =1.0 p . u . ( w i t h o u t c o n t r o l ) (b) S h a f t modes o f t h e s y s t e m w i t h l i n e a r o p t i m a l c o n t r o l a t v a r i o u s d e g r e e s o f c o m p e n s a t i o n a t P=1.25 p . u . , P . F . =0.9 l a g g i n g , V =1 .0 p . u . 31 30% 50% 70% 80% c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± j 2 9 8 .18 - 0 .1818 ± J 2 9 8 . 1 8 +0 .8503 ± J 2 0 4 . 2 9 +0 .0901 ± j 2 0 2 . 6 1 +0 .0064 ± j 2 0 2 .77 - 0 .0072 ± J 2 0 2 . 8 0 - 0 . 1 9 4 9 ± J 1 6 0 . 7 9 +1 .2050 ± J 1 6 1 . 6 7 - 0 . 0 7 5 9 ± J 1 6 0 .33 - 0 .1167 ± j 1 6 0 . 4 1 - 0 . 6 6 0 8 ± J 1 2 7 . 0 5 - 0 . 6 5 8 3 ± j l 2 7 . l l - 0 . 2 5 3 5 ± j l 2 6 .73 - 0 .6122 ± J 1 2 6 . 8 7 - 0 . 2 1 8 7 ± J 9 9 . 4 0 5 - 0 . 2 3 9 0 ± J 9 9 . 6 9 5 - 0 . 1 2 4 7 ± j l O O .84 +3 .3952 ± j 1 0 1 . 8 4 - 0 . 4 0 5 7 ± J 9 . 9 9 3 0 - 0 . 5 6 2 5 ± j l O . 7 5 6 - 0 . 7 8 8 3 ± j l l . 773 - 0 .9451 ± J 1 2 . 4 2 7 (a) 30% 50% 70% 80% c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n c o m p e n s a t i o n - 0 . 1 8 1 8 ± j29 .8 .18 - 0 .1818 ± J 2 9 8 . 1 8 - 0 .1818 ± J 2 9 8 . 1 8 - 0 . 1 8 1 8 ± J 2 9 8 . 1 8 - 0 . 2 2 5 1 ± j 2 0 1 . 8 6 - 0 .4357 ± j 2 0 2 . 6 1 - 0 .3604 ± J 2 0 2 . 5 5 - 0 . 3 5 2 5 ± J 2 0 2 . 5 2 - 0 . 5 7 3 8 ± J 1 6 0 . 5 6 - 0 .5739 ± J 1 6 1 . 2 2 - 0 .2447 ± J 1 6 0 . 6 3 - 0 . 3 0 0 4 ± J 1 6 0 . 5 8 - 0 . 6 9 7 1 ± .1127.19 - 0 .6559 ± J 1 2 7 . 3 3 - 0 .2837 ± J 1 2 7 . 0 8 - 0 . 4 6 8 3 ± J 1 2 7 . 0 0 - 1 . 0 1 5 1 ± J 9 9 . 0 2 5 -1 .1638 ± J 9 9 . 2 7 3 - 0 .7136 ± j 1 0 0 . 1 9 +0 .1498 ± J 9 9 . 6 5 7 - 4 . 6 6 9 6 ± J 7 . 4 4 6 3 - 6 .0982 ± J 8 . 9 2 5 4 -9 .1094 ± j l l . 2 3 3 - 1 2 . 5 5 3 ± J 1 2 . 5 6 2 (b) T a b l e 3 . 3 (a) S h a f t modes G f t h e s y s t e m a t v a r i o u s d e g r e e s o f c o m p e n s a t i o n a t P=0.5 p . u . , P . F = 0 . 9 l e a d i n g , . V =1.0 p . u . ( w i t h o u t c o n t r o l ) (b) S h a f t modes o f t h e s y s t e m w i t h l i n e a r o p t i m a l c o n t r o l a t v a r i o u s d e g r e e s o f c o m p e n s a t i o n a t P=0.5 p . u . , P . F . = 0 . 9 l e a d i n g , V =1.0 p . u . 32 4 . EFFECT OF EXCITATION CONTROLLER ON STEADY STATE AND TRANSIENT SSR 4 . 1 I n t r o d u c t i o n I t has been shown t h a t l i n e a r o p t i m a l c o n t r o l l e r d e s i g n e d i n C h a p t e r 3 can s u p p r e s s t h e SSR phenomenon. H o w e v e r , i t must be n o t e d t h a t a l l t e s t s were p e r f o r m e d on t h e l i n e a r i z e d s y s t e m w i t h o u t c o n s i d e r i n g e x c i t e r v o l t a g e l i m i t s and o t h e r n o n l i n e a r i t i e s . I n some c a s e s , t h e d i s -t u r b a n c e s have c a u s e d t h e e x c i t e r v o l t a g e t o s w i n g w e l l b e y o n d any r e a s o n -a b l e c e i l i n g v o l t a g e . I n t h i s C h a p t e r , t h e dynamic p e r f o r m a n c e o f t h e s y s t e m , w i t h and w i t h o u t s u p p l e m e n t a r y e x c i t a t i o n c o n t r o l , d e s i g n e d i n C h a p t e r 3 , w i l l be i n v e s t i g a t e d . o n t h e n o n l i n e a r s y s t e m m o d e l d e s c r i b e d by 26 f i r s t o r d e r d i f f e r e n t i a l e q u a t i o n s . A l l g o v e r n o r and e x c i t e r p h y s i c a l l i m i t s a r e i n c l u d e d . T r a n s i e n t t o r q u e s on the s h a f t s o f t h e g e n e r a t o r m a s s - s p r i n g s y s t e m and t h e dynamic r e s p o n s e s o f t h e g e n e r a t o r w i t h and w i t h o u t c o n t r o l w i l l be examined when the s y s t e m i s s u b j e c t e d t o v a r i o u s d i s t u r b a n c e s . 4 . 2 C o n t r o l l e r f o r S t e a d y S t a t e SSR I n d a i l y o p e r a t i o n , a power s y s t e m i s f r e q u e n t l y e x p o s e d t o d i s -t u r b a n c e s u c h as l o a d c h a n g e ; s u c h d i s t u r b a n c e may n o t be a p r o b l e m o f s t e a d y s t a t e s t a b i l i t y b u t s e l f - e x c i t e d SSR may a r i s e . F i g u r e 4 . 1 shows t h e d y n a m i c r e s p o n s e s o f t h e s y s t e m when i t i s s u b j e c t e d t o a 10% l o a d c h a n g e ; t h e s y s t e m i s i n i t i a l l y o p e r a t i n g a t P o f 0 . 9 p . u . and t h e 10% l o a d change i s r e p r e s e n t -ed as f o l l o w s T + O . l t 0 < t < 1 s e c . mo - -T = { T + 0 . 1 t < 1 s e c . mo • • - -I n F i g u r e 4 . 1 , g r o w i n g s h a f t t o r q u e s due t o SSR a r e o b s e r v e d . 33 1 1 1 1 1 1 1 1 1 1 2 0.0 1.0 2.0 3.0 4.0 5.0 TIME(SEC) XT F i g u r e 4 . 1 Dynamic r e s p o n s e s o f t h e s y s t e m w i t h o u t c o n t r o l when s u b j e c t e d t o 10% l o a d c h a n g e . 34 F o r f u r t h e r s t u d i e s , a more s e v e r e d i s t u r b a n c e , i . e . , a p u l s e t o r q u e o f 20% f o r 0 . 2 s e c o n d i s assumed f o r v a r i o u s o p e r a t i n g c o n d i t i o n s . T y p i c a l r e s u l t f o r 0 . 9 p e r u n i t g e n e r a t o r l o a d i n g and 50% c a p a c i t o r com-p e n s a t i o n a r e shown i n F i g u r e s 4 . 2 and 4 . 3 . W i t h c o n t r o l , t h e s y s t e m has no e x c e s s i v e r e s p o n s e and t h e d i s t u r b e d s y s t e m r e t u r n s t o i t s n o r m a l o p e r -a t i n g c o n d i t i o n i n a w e l l - d a m p e d manner as shown i n F i g u r e 4 . 3 . I n c o n t r a s t , F i g u r e 4 . 2 shows s u s t a i n e d o s c i l l a t i o n s and g r o w i n g t o r s i o n a l s h a f t t o r q u e s i n t h e s y s t e m w i t h o u t c o n t r o l . The same d i s t u r b a n c e i s a p p l i e d t o t h e s y s t e m w i t h 80% c a p a c i t o r c o m p e n s a t i o n and t y p i c a l r e s u l t s a r e shown i n F i g u r e 4 . 4 and F i g u r e 4 . 5 . W i t h o u t c o n t r o l , F i g u r e 4 . 4 , t h e t o r s i o n a l o s c i l l a t i o n s on t h e s h a f t s e x c e e d t h e i r o n c e - i n - a - l i f e t i m e l i m i t so t h a t t h e s h a f t s w o u l d be damaged. H o w e v e r , as shown i n F i g u r e 4 . 5 , t h e dange rous t o r s i o n a l o s c i l l a t i o n s due to SSR c a n be a l l e v i a t e d when t h e s y s t e m has t h e l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l . A l t h o u g h a p u l s e t o r q u e i n p u t t o t h e s y s t e m i s a pseudo d i s t u r b -a n c e , n o t n o r m a l l y e n c o u n t e r e d i n p r a c t i c e , i t can be t a k e n as an ex t r eme c a s e i n t h e d a i l y power s y s t e m o p e r a t i o n . H e n c e , as shown i n t h e above r e s u l t s , t h e l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l can e f f i c i e n t l y c o n t r o l t h e s t e a d y s t a t e SSR p r o b l e m f o r a w i d e r a n g e o f c a p a c i t o r c o m p e n s a t i o n . CO CO U J a in. in 0 . 0 — < o . X — 0 _ c n 03 0 . 0 1 .0 I I 1 2 . 0 3 . 0 T I M E ( S E C ) 4 . 0 I 5 . 0 I 1 .0 ~1 1 1— 2 . 0 3 . 0 T I M E ( S E C ) 4 . 0 5.0 2 . 0 3 . 0 T I M E ( S E C ) Figure 4.2 Dynamic responses of the power system at 0.9 p.u. generator loading, 0.9 power facter lagging, and 50% capacitor compensation when subjected to a pulse torque disturbance (without control) Co 36 in o 0 . 0 ) . 0 2 . 0 3 . 0 4 . 0 5 . 0 TIME(SEC) F i g u r e 4 . 2 ( c o n t i n u e d ) 37 F i g u r e 4 . 2 ( c o n t i n u e d ) 38 in 0 . 0 1.0 2 . 0 3 . 0 4 . 0 5 . 0 TIME(SEC) F i g u r e 4 . 2 ( c o n t i n u e d ) F i g u r e 4 . 3 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l a t 0 .9 p . u . g e n e r a t o r l o a d i n g , 0 . 9 power f a c t o r l a g g i n g , and 50% c a p a c i t o r c o m p e n s a t i o n w h e n , s u b j e c t e d t o a p u l s e t o r q u e d i s t u r b a n c e . CO TORQUE (LPB-GEN) P.U. 0.88 0.9S 1.04 1.12 1.2 1.28 1.35 o TORQUE CLPR-LPB) P.U. 0.64 0.72 0.8 0.88 0.9S 1.04 1.12 in 0.0 1 .0 n i I 2.0 3.0 TIME(SEC) 4.0 F i g u r e 4 . 3 ( c o n t i n u e d ) 43 F i g u r e 4 . 5 T o r s i o n a l o s c i l l a t i o n s o f t h e g e n e r a t o r - e x c i t e r s h a f t a t 80% c o m p e n s a t i o n ( w i t h c o n t r o l ) 44 4 . 3 C o n t r o l l e r f o r T r a n s i e n t SSR The t r a n s i e n t t o r q u e s e x p e r i e n c e d by t h e t u r b o g e n e r a t o r s h a f t s as a r e s u l t o f e l e c t r i c a l t r a n s i e n t s c a u s e d by t r a n s m i s s i o n f a u l t s a r e now c o n s i d e r e d . I n o r d e r t o e v a l u a t e t h e e f f e c t i v e n e s s o f t h e d e s i g n e d e x c i t a t i o n c o n t r o l l e r , a s i m u l t a n o u s t h r e e f a u l t was a p p l i e d a t bus B i n F i g u r e 4 . 6 a t t i m e t=0 and t h e n removed a f t e r 0 . 0 7 5 s e c o n d . The f a u l t impedance and s e r i e s c a p a c i t o r a r e assumed t o be 0 . 0 4 p . u . and 0 . 2 8 p . u . r e s p e c t i v e l y . The f a u l t l o c a t i o n , hence t h e impedance be tween bus B and t h e i n f i n i t e b u s , Xg , i s a l s o v a r i e d . F i g u r e 4 . 6 E l e c t r i c a l n e t w o r k f o r t h e s i m u l a t i o n o f s u b s y n c h r o n o u s r e s o n a n c e . 45 With the f a u l t l o c a t e d at the remote end o f the t r a n s m i s s i o n l i n e , i . e . , Xg equal to 0 . 0 1 p . u . , the responses of the system wi thou t and w i t h c o n t r o l are shown i n F igu res 4 . 7 and 4 . 8 r e s p e c t i v e l y . F igu re 4 . 7 shows c l e a r l y the t o r s i o n a l i n t e r a c t i o n between the e l e c t r i c a l and mechanica l system. The f o r c i n g frequency of the e l e c t r i c a l torque ( P = T^ i n p . u . ) i s c l o s e to a t o r s i o n a l mode frequency of the mass-spr ing system so tha t the mode i s e x c i t e d , and growing t o r s i o n a l shaft torques are observed. But no excess ive o s c i l l a t i o n s are observed i n F i g u r e 4 . 8 for the system w i t h e x c i t a t i o n c o n t r o l . As the f a u l t l o c a t i o n i s moved c l o s e r to the generator t e r m i n a l , the d i s tu rbance to the system becomes more severe . The system i s subjec ted to the s tandard t e s t as d i s c r i b e d i n the Benchmark Model [13] w i t h X equal to 0 . 0 6 p . u . and c a p a c i t o r equal to 0 . 2 8 p . u . . T y p i c a l r e s u l t s are shown i n F igures 4 . 9 and 4 . 1 0 . The f a u l t e f f e c t seems more severe than the p rev ious case that the shaf ts exper ience l a r g e r t o r s i o n a l s t r e s s and i t takes a longer time fo r o s c i l l a t i o n s to decay. Furthermore, when the c o n t r o l i s t e s t ed on the system at 80% c a p a c i t o r compensation and the system i s subjec ted to the same d i s t u r b a n c e , i t i s uns tab le and the shaf t torque inc reases as those shown i n F igu re 4 . 4 , because of the e x c i t a t i o n v o l t a g e c e i l i n g s and other l i m i t a t i o n s that are not cons idered i n a l i n e a r c o n t r o l l e r . To summarize, the l i n e a r o p t i m a l e x c i t a t i o n c o n t r o l can p rov ide s u f f i c i e n t damping to s t a b i l i z e the system i n most cases . But i t i s not recommended f o r t r a n s i e n t s t a b i l i t y c o n t r o l o f a power system w i t h a very severe f a u l t . Figure 4.7 Dynamic responses of the power system without control at 0.9 p.u. generator loading, 0.9 power factor lagging,and at 50% capacitor compensation when subjected to a three-phase fault at the remote end (X =0.01 p.u.) 47 F i g u r e 4.7 ( c o n t i n u e d ) 48 Figure 4.7 (continued ) 49 o • o 1 D_(D o o H 1 r 1 1 1 1 1 i i i 0.0 • l .O 2.0 3.0 4.0 5.0 TIMECSECJ F i g u r e 4.7 ( c o n t i n u e d ) O co 01 •cn rsj o " 0 . 0 " A f i r r y i iiw mi -i n 1 1— 2 . 0 3 . 0 T I M E ( S E C ) 4 . 0 1.0 ~T 1 1— 2 . 0 3 . 0 T I M E t S E C ) I — 4 . 0 -1 5 . 0 — ! 5 . 0 _ o " I X o " ^o 'o CL -, .o. LU ' o O o . LU 1 LUm C 0 0 " . . i ! ! ill i t 41 J, 0 . 0 ~ i — 1.0 i 1 1— 2 . 0 3 . 0 T I M E t S E C ) o -C L , CC LU O o ' CL. U3 0 . 0 n 1 I I I 1.0 2 . 0 3 . 0 T I M E t S E C ) 4 . 0 4 . 0 F i g u r e 4 . 8 Dynamic r e s p o n s e s o f t h e power s y s t e m w i t h c o n t r o l a t 0 . 9 p . u . g e n e r a t o r l o a d i n g , 0 . 9 power f a c t o r l a g g i n g , and 50% c a p a c i t o r c o m p e n s a t i o n when s u b j e c t e d t o a t h r e e - p h a s e f a u l t a t t he remote end (Xg=0.01 p . u . ) 5.0 ~1 5 . 0 o 51 - J t D " •—4 CL. CD o H i i i i i — — i 1 1 1 1 0.0 1.0 2.0 3.0 4.0 5.0 TIME(SECJ F i g u r e 4.8 ( c o n t i n u e d ) 52 i n o tM o l 1 1 1 1 1 1 1 1 1 1 0.0 1 .0 2.0 3.0 4.0 5.0 TIME(SEC) F i g u r e 4 . 8 ( c o n t i n u e d ) 53 F i g u r e 4 . 8 ( c o n t i n u e d ) 54 i n C M " 3 ° . • C V J D_ LU LD I a Q _ _ J ' i n L U ° " ZD O a cn -. 1 "'1 ti'feli ijliifliifjlliilj J;?'U'l'jjj,.Vd. " "I''! ! ! ' " ' : ' ! ! ' ! iiiiihi! lllii J K ! 31 r i n 0 . 0 1 .0 - i 1 r— 2 . 0 3 . 0 TIME(SEC) 4 . 0 5 . 0 C D • C M CL CL I I -< I D :!!:, i III 'fe I'lll III'1 I I/it 81 §11 ill JJ iijil i 0 . 0 1 .0 n I 1 2 . 0 3 . 0 TIME(SEC) 4 . 0 5 . 0 Figure 4.9 Dynamic responses of the power system without c o n t r o l for a three-phase f a u l t (Xg = 0.06 p.u.) a t 50% compensation. 55 Figure 4.10 Dynamic responses of the power system with c o n t r o l f o r a three-phase f a u l t (XB=0.06 p.u.) at 50% compensation. 56 5 . CONCLUSION A h i g h - o r d e r n o n l i n e a r power s y s t e m mode l f o r s t u d y i n g t h e t o r s i o n a l o s c i l l a t i o n s due t o s u b s y n c h r o n o u s r e s o n a n c e has been d e v e l o p e d . From e i g e n v a l u e a n a l y s i s o f t h e l i n e a r i z e d m o d e l , i t i s found t h a t more t h a n one m e c h a n i c a l mode can be e x c i t e d s i m u l t a n o u s l y f o r a h i g h d e g r e e o f t h c a p a c i t o r c o m p e n s a t i o n . E i g e n v a l u e a n a l y s i s o f t h e r e d u c e d 19 o r d e r mode l r e v e a l e d t h a t t h e s e u n s t a b l e m e c h a n i c a l modes r e m a i n even though t h e e x c i t e r mass i s n e g l e c t e d . H e n c e , t h e p o t e n t i a l d a n g e r s o f s u b s y n c h r o n o u s r e s o n a n c e c a n n o t be n e g l e c t e d i n a t u r b i n e g e n e r a t o r u n i t even w i t h a s t a t i c e x c i t e r . A l i n e a r o p t i m a l c o n t r o l l e r has been d e s i g n e d and i t s e f f e c t i v e -n e s s i s i n i t i a l l y t e s t e d by e i g e n v a l u e a n a l y s i s . I t n o t o n l y s t a b i l i z e s t h e s y s t e m f o r c a p a c i t o r c o m p e n s a t i o n r a n g i n g f rom 10% t o 90% a t n o r m a l o p e r a t i n g c o n d i t i o n s , b u t a l s o i m p r o v e s t h e s t a b i l i t y c o n s i d e r a b l y a t o t h e r o p e r a t i n g c o n d i t i o n s . Dynamic p e r f o r m a n c e t e s t s u s i n g a h i g h - o r d e r n o n l i n e a r mode l f u r t h e r show t h a t t h e s y s t e m w i t h o u t c o n t r o l e x h i b i t s g r o w i n g s h a f t t o r q u e s n o t o n l y i n t h e s e c t i o n be tween t h e g e n e r a t o r and the e x c i t e r , b u t a l s o i n o t h e r s e c t i o n s s u c h as t he s h a f t s on e i t h e r s i d e o f t h e i n t e r m e d i a t e p r e s s u r e t u r b i n e . W i t h mode ra t e d i s t u r b a n c e , t h e s y s t e m w i t h e x c i t a t i o n c o n t r o l shows damped r e s p o n s e s . H o w e v e r , w i t h more s e v e r e d i s t u r b a n c e , s u c h as a t h r e e - p h a s e f a u l t c l o s e t o t h e g e n e r a t o r t e r m i n a l , t h e l i n e a r o p t i m a l c o n t r o l l e r i s no l o n g e r e f f e c t i v e . I n summary, an e x c i t a t i o n c o n t r o l l e r o f t h e l i n e a r o p t i m a l t y p e can be d e s i g n e d t o e f f e c t i v e l y s t a b i l i z e t h e t o r s i o n a l o s c i l l a t i o n s w i t h i n t h e r a n g e o f d y n a m i c s t a b i l i t y , b u t i t i s n o t recommended f o r t o r s i o n a l o s c i l l a t i o n s o f t h e s e v e r e t r a n s i e n t t y p e . 57 REFERENCES [1] M.C. H a l l and D.A. Hodges, "Experience with 500kV Subsynchronous Resonance and Resulting Turbine Generator Shaft Damage at Mohave Generating Station", IEEE P u b l i c a t i o n 76CH1066-0-PWR, pp. 22-25, 1976. [2] R.G. Farmer, A.L. Schalb and E l i Katz, "Navajo Project Report on Subsynchronous Resonance Analysis and Solutions", IEEE P u b l i c a t -ion 76Chl066-0-PWR, pp. 55-58, 1976. [3] 0. Saito, H. Mukae and K. Murotani, "Suppression of Se l f Excited O s c i l l a t i o n s i n Series-compensated Transmission Lines By E x c i t a -t i o n Control of Synchronous Machine", IEEE Trans, on PAS, Vol. PAS 94, pp. 1777-1788, Sept/Oct. 1975. [4] H.M.A. Hamdan and F.M. Hughes, " E x c i t a t i o n Controller Design For The Damping of Self Excited O s c i l l a t i o n s i n Series Compensated Lines", Paper A78 565 -4 , IEEE PES Summer Meeting, Los Angeles, July 1978. [5] A.A. Fouad and K.T. Khu, "Damping of Torsional O s c i l l a t i o n s i n Power Systems With Series-compensated Lines",IEEE Trans, on PAS, Vol. PAS 97, pp. 744-751, May/June 1978. [6] Yao-nan Yu, M.D. Wvong and K.K. Tse, "Multi-Mode Wide-Range Sub-synchronous Resonance S t a b i l i z a t i o n " , Paper A78 554-8, IEEE PES Summer Meeting, Los Angeles, July 1978. [7] A.M. E l - S e r a f i and A.A. Shaltout, "Control of Subsynchronous Re-sonance O s c i l l a t i o n s By Multi-loop E x c i t a t i o n C o n t r o l l e r " Paper A79 076-1, IEEE PES Winter Meeting, New York C i t y , Feb. 1979. [8] R.A. Hedin, R.C. Dancy and K.B. Stump, "An Analysis of the Sub-synchronous Interaction of Synchronous Machine and Transmission Networks", Proceeding of the American Power Conference, Vol. 35, 1973, pp. 1112-1119. [9] C. Concordia and R.P. Schulz, "Appropriate Component Representa-t i o n f o r the Simulation of Power Sysyem Dynamics", IEEE Pu b l i c a -t i o n 75CH0970-4-PWR, pp. 16-23, 1975. [10]. IEEE Committee Report, "Dynamic Models for Steam and Hydro Turb-ines i n Power System Studies", IEEE Trans, on PAS, Vol. PAS 92, pp. 1904-1915, Nov/Dec. 1973. [11] E.W. Kimbark, " Power System S t a b i l i t y " , Vol. III. , Wiley, New York 1956. [12] IEEE Committee Report, "Computer Representation of E x c i t a t i o n Systems", IEEE Trans, on PAS, Vol PAS 87, pp. 1460-1470, June/July 1968. 58 [13] I E E E Commit tee R e p o r t , " F i r s t Benmark M o d e l f o r Computer S i m u l a t i o n o f S u b s y n c h r o n o u s R e n s o n a n c e " , I E E E T r a n s , on P A S , V o l . PAS 9 6 , p p . 1 5 6 5 - 1 5 7 2 , S e p t . / O c t . 1 9 7 7 . [14] J . L . W i l l e m s , " S t a b i l i t y T h e o r y o f D y n a m i c a l S y s t e m s " , Book P u b l i s ^ hed by J o h n W i l e y & S o n s , I n c . , New Y o r k , 1 9 7 0 . [15] K . K . T s e , " H i g h O r d e r S u b s y n c h r o n o u s Resonance M o d e l s and M u l t i -Mode S t a b i l i z a t i o n " , M a s t e r T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , V a n c o u v e r B . C . , June 1 9 7 7 . [16] Y a o - n a n Y u , K . V o n g s u r i y a and L . N . Wedman, " A p p l i c a t i o n o f an O p t i m a l C o n t r o l T h e o r y t o a Power S y s t e m " , I E E E T r a n s , on P A S , V o l . PAS 8 9 , p p . 5 5 - 6 2 , J a n . / F e b . 1 9 7 0 . [17] Y a o - n a 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 " , I E E E T r a n s , on P A S , V o l . PAS 8 9 , N o . 1 , J a n u a r y 1 9 7 0 . [18] H . A . M . Moussa and Y a o - n a n Y u , " O p t i m a l Power S y s t e m S t a b i l i z a t i o n T h -r o u g h E x c i t a t i o n a n d / o r G o v e r n o r C o n t r l " , I E E E T r a n s , on P A S , V o l . PAS 9 1 , p p . 1 1 6 6 - 1 1 7 4 , M a y / J u n e 1972 [19] B . H a b i b u l l a h and Y a o - n a n Y u , " P h y s i c a l l y R e a l i z a b l e Wide Power Range O p t i m a l C o n t r o l l e r s f o r Power S y s t e m s " , I E E E T r a n s , on P A S , V o l . PAS 9 3 , p p . 1 4 9 8 - 1 5 0 6 , S e p t . / O c t . 1 9 7 4 . [20] J . E . P o t t e r , " M a t r i x Q u a d r a t i c S o l u t i o n s " , S I A M , J o u r n a l o f A p p l i e d M a t h . , V o l . 1 4 , N o . 3 , p p . 4 9 6 - 5 0 6 , May 1 9 6 6 . [21] A . Y a n , M . D . Wvong, and Y a o - n a n Y u , " E x c i t a t i o n C o n t r o l o f T o r s i o n a l O s c i l l a t i o n s " , P a p e r s u b m i t t e d t o I E E E PES Summer M e e t i n g , V a n c o u v e r , J u l y 1 9 7 9 . APPENDIX I EQUATIONS FOR THE TRANSFORMATION MATRIX H AP = (X - X , ) i A i , + [(X - X , ) i , + E._ ]Ai + i X ,Ai_ + i X ,Ai_ e q d qo d q d do fD q qo md f — qo md D - i , X A i n do mq... Q AQ = ( E ^ - 2XAj ) A i , - 2X i A i + X , i , A i . + X , i , Ai + X i A i . x e fD d do d q qo q md do f md do D mq qo Q AV = [(V R + V X)Ai, +(-V, X + RV ) A i + V, AV , + V AV + t do qo d do qo q do cd qo cq V (V,, cos6 - V sin6 )A6 ]/ V^ o do o qo o to Ai = ( 1 , A i , + i A i )/ i a do d qo q ao AV = ( V „ AV + V ' AV ) / V c cdo cd cqo cq co AV = [ V , AV , + V 1AV + V ( V , cos6 - V sin6 )A6 ]/ V ct cdo cd cqo cq o cdo o cqo o co where X = X + X e t R = R + R e t 60 APPENDIX I I NUMERIAL VALUES OF MODEL I N P . U . SYSTEM M a s s - s p r i n g s y s t e m M l = 0 .185794 K 1 2 = 1 9 . 3 0 3 M 2 = 0 . 3 1 1 1 7 8 K 2 3 = 3 4 . 9 2 9 M 3 = 1 .717340 K „ , = 5 2 . 0 3 8 34 M 4 - 1 .768430 K . _ = 7 0 . 8 5 8 45 M 5 = 1 .736990 K _ , = 2 . 8 2 2 0 M 6 - 0 . 0 6 8 4 3 3 D . . = 0 . 1 i = 1 , 2 , . . . . , 6 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 X d = 1.79 X f = 1 .6999 R f = X md 1.66 X D = 1 .6657 % = X = q 1 .71 X Q = 1 .6845 R Q X = mq 1 .58 X g = 1 .8250 R s " R = a 0 . 0 0 1 5 E x c i t e r and V o l t a g e R e g u l a t o r K A = 50 T„ = 0 . 0 0 2 L T A = E x c i t e r v o l t a g e c e i l i n g l i m i t s = ± 7 . 0 T u r b i n e and G o v e r n i n g Sys t em K = g 25 T1 = 0 . 2 T 2 " T = 3 0 . 3 T CH= ° ' 3 T RH= T co= 0 . 2 F HP= ° ' 3 • F I P = F L P 1 = 0 . 2 2 F L P 2 = ° - 2 2 P GV = ± 0 . 1 0 . 0 0 1 0 5 0 . 0 0 3 7 1 0 . 0 0 5 2 6 0 . 0 1 8 2 0 0 . 0 7 .0 

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