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The effect of local motor loads on power system stability. Prior, Bruce George 1971

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THE EFFECT OF LOCAL MOTOR LOADS ON POWER SYSTEM STABILITY by BRUCE GEORGE PRIOR B.Eng. Nova S c o t i a T e c h n i c a l C o l l e g e , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t t h i s t h e s i s as c o n f o r m i n g to the r e q u i r e d s t a n d a r d R e s e a r c h S u p e r v i s o r , Members o f t he Committee Head o f t he Department Members o f t h e Department o f E l e c t r i c a l E n g i n e e r i n g THE UNIVERSITY OF ERITISH COLUMBIA September , 1971 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t 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 r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p urposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f £lECTr?<C*L- J L * W & t j } & r The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The e f f e c t o f l o c a l motor l o a d s on power s y s tem s t a b i l i t y i s i n v e s t i g a t e d . The power s y s tem c o n s i s t s o f a s y n c h r o n o u s g e n e r a t o r s u p p l y i n g a l a r g e s y s tem t h r o u g h a l o n g t r a n s m i s s i o n l i n e . The l o a d s s t u d i e d a re an i n d u c t i o n moto r , a s y n c h r o n o u s m o t o r , and the c o m b i n a t i o n o f the two, a l t h o u g h a g e n e r a l ca se o f any number o f l o c a l i n d u c t i o n and s ynchronous motor leads can be e a s i l y f o r m u l a t e d . S t a b i l i t y i s d e t e r m i n e d by o b s e r v i n g the r e s p o n s e o f the g e n e r a t o r and the motors o f t he s y s tem w i t h a f a u l t a t the t r a n s m i s s i o n l i n e . The r e s p o n s e i s c a l c u l a t e d f r om the m a t h e m a t i c a l model and i s - a l s o o b s e r v e d f rom t e s t s on a dynamic power s y s tem model i n t he l a b o r a t o r y . I t i s f ound f rom the s t u d i e s t h a t a l l the l o c a l motor l o a d s improve the s t a b i l i t y o f a power s y s t e m . TABLE OF CONTENTS ABSTRACT TABLE OF CONTENTS L I ST OF TABLES L I ST OF ILLUSTRATIONS ACKNOWLEDGMENT NOMENCLATURE 1. INTRODUCTION 2. SYNCHRONOUS GENERATOR WITHOUT LOCAL MOTOR LOADS 2.1 Synchronous G e n e r a t o r E q u a t i o n s 2.2 T r a n s m i s s i o n L i n e E q u a t i o n s 2.3 R e g u l a t o r and E x c i t e r E q u a t i o n s 2.4 Computa t i on and L a b o r a t o r y T e s t R e s u l t s 3. SYNCHRONOUS GENERATOR WITH A LOCAL INDUCTION MOTOR LOAD 3.1 I n d u c t i o n Motor E q u a t i o n s 3.2 T r a n s m i s s i o n L i n e E q u a t i o n s 3.3 Computa t i on and L a b o r a t o r y T e s t R e s u l t s 4. SYNCHRONOUS GENERATOR WITH A LOCAL SYNCHRONOUS MOTOR LOAD 4.1 Synchronous Motor E q u a t i o n s 4.2 T r a n s m i s s i o n L i n e E q u a t i o n s 4.3 Computa t i on and L a b o r a t o r y T e s t R e s u l t s 4.4 Computa t ion R e s u l t s U s i n g L a r g e - S y s t e m P a r a m e t e r s 5. SYNCHRONOUS GENERATOR WITH MULTIPLE LOCAL SYNCHRONOUS AND INDUCTION MOTOR LOADS 5.1 Mach ine E q u a t i o n s 5.2 T r a n s m i s s i o n L i n e E q u a t i o n s 5.3 Computa t i on and L a b o r a t o r y T e s t P.esults 6. CONCLUSION REFERENCES " i l l L IST OF TABLES T a b l e Page 2.1 Synchronous G e n e r a t o r , V o l t a g e R e g u l a t o r and T r a n s m i s s i o n 13 L i n e Pa ramete r s 2.2 Synchronous G e n e r a t o r System Wi thou t L o c a l Motor L o a d s . O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s 14 3.1 I n d u c t i o n Motor Pa ramete r s 27 3.2 Synchronous G e n e r a t o r System W i t h L o c a l I n d u c t i o n Motor L o a d . O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s 28 4.1 Synchronous Motor Pa ramete r s 39 4.2 Synchronous G e n e r a t o r System W i t h L o c a l Synchronous Motor L o a d . O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s 39 4.3 P a ramete r s o f L a r g e Synchronous Motor 48 4.4 P a ramete r s o f a L a r g e Power System 49 4.5 L a r g e Power System S t a b i l i t y S tudy . O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s 50 5.1 Synchronous G e n e r a t o r System W i t h L o c a l I n d u c t i o n and Synchronous Motor L o a d s . O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s 61 i v LIST OF ILLUSTRATIONS Figure Page 2.1 C i r c u i t Diagram of Ideal Synchronous Machine 5 2.2 One-line Diagram of One Machine-Infinite Bus System 6 2.3 One Machine System With Equivalent-ir Transmission Network 6 2.4 Block Diagram of Voltage Regulator and Ex c i t e r 12 2.5 Generator Torque Angle - No Motor Loads 16 2.6 Generator Voltage - No Motor Loads 17 2.7 Generator Current - No Motor Loads 18 3.1 Synchronous Generator System With Local Induction Motor Load 19 3.2 C i r c u i t Diagram of Ideal Induction Motor 20 3.3 Synchronous Generator And Induction Motor dq Terminal Voltages 22 3.4 Generator Torque Angle - Local Induction Motor Load 29 3.5 Generator Voltage - Local Induction Motor Load 30 3.6 Generator Current - Local Induction Motor Load 31 3.7 Induction Motor S l i p - Local Induction Motor Load 32 3.8 Induction Motor Current - Local Induction Motor Load 33 4.1 Synchronous Generator System With Local Synchronous Motor Load 35 4.2 Generator Torque Angle - Local Synchronous Motor Load 41 4.3 Synchronous Motor Torque Angle - Local Synchronous Motor Load 42 4.4' Generator Voltage - Local Synchronous Motor Load 43 4.5 Generator Current - Local Synchronous Motor Load 44 4.6 Synchronous Motor Current - Local Synchronous Motor Load 45 4.7 Generator Torque Angle - Local Synchronous Motor With Sinusoidal Shaft Load 46 4.8 Synchronous Motor Torque Angle - Local Synchronous Motor With Sinusoidal Shaft Load 47 4.9 Large Generator Torque Angle - No Motor Loads 51 L IST OF ILLUSTRATIONS F i g u r e Page 4.10 L a r g e G e n e r a t o r Torque A n g l e - L o c a l Synchronous Motor Load 52 5.1 Synchronous G e n e r a t o r System W i t h M u l t i p l e L o c a l Motor Loads 53 5.2 G e n e r a t o r Torque A n g l e - L o c a l I n d u c t i o n and Synchronous Motor Loads 62 5.-3 S ynchronous -Moto r Torque A n g l e — L o c a l I n d u c t i o n and Synchronous Motor Loads 63 5.4 I n d u c t i o n Motor S l i p - L o c a l I n d u c t i o n and Synchronous Motor Loads 64 5.5 G e n e r a t o r V o l t a g e - L o c a l I n d u c t i o n and Synchronous Motor Loads 65 5.6 G e n e r a t o r C u r r e n t - L o c a l I n d u c t i o n and Synchronous Motor Loads 66 5.7 Synchronous Motor C u r r e n t - L o c a l I n d u c t i o n and Synchronous Motor Loads 67 5.8 I n d u c t i o n Motor C u r r e n t - L o c a l I n d u c t i o n and Synchronous Motor Loads 68 v i ACKNOWLEDGEMENT I w i s h to e x p r e s s my s i n c e r e s t g r a t i t u d e to my s u p e r v i s o r , i D r . Y . N. Y u , f o r a l l h i s h e l p and u n d e r s t a n d i n g d u r i n g the p r e p a r a t i o n o f t h i s t h e s i s . The many h o u r s o f c o n s u l t a t i o n w i t h h i m w i l l be o f l a s t i n g b e n e f i t t o me. Thanks a r e a l s o due to D r . H. R. Ch inn f o r h i s c a r e f u l r e a d i n g o f the t h e s i s and f o r h i s a d v i c e c o n c e r n i n g programming methods. I a l s o w i s h t o thank my f e l l o w g radua te s t u d e n t s i n the power group f o r the many v a l u a b l e d i s c u s s i o n s we h a d . I am p a r t i c u l a r l y g r a t e f u l t o Dr . H. A. M. Moussa f o r the encouragement he gave me and a l s o f o r h i s a s s i s t a n c e i n some o f the l a b o r a t o r y t e s t s . The f i n a n c i a l a s s i s t a n c e o f the N a t i o n a l R e s e a r c h C o u n c i l o f Canada and of the B r i t i s h Co lumb ia Hydro and Power A u t h o r i t y i s g r a t e f u l l y a cknow ledged . F i n a l l y , I w i s h to e x p r e s s my deepes t a p p r e c i a t i o n to my w i f e , J e a n , f o r h e r p a t i e n c e and u n d e r s t a n d i n g 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 . NOMENCLATURE G e n e r a l t t i m e , s j complex o p e r a t o r , OJQ s ynchronous a n g u l a r v e l o c i t y , 377 r a d / s £ base o f n a t u r a l l o g a r i t h m s , 2.71828 T x t u r n s r a t i o o f i d e a l t r a n s f o r m e r between motor and g e n e r a t o r k s u p e r s c r i p t d e n o t i n g number o f p a r t i c u l a r motor i n m u l t i - m o t o r l o a d p d e r i v a t i v e o p e r a t o r , d / d t T r a n s m i s s i o n Network r + j x s e r i e s impedance, fi G + j B shunt a d m i t t a n c e , mho V Q i n f i n i t e bus v o l t a g e , V / p h [ V ] , [ I ] v o l t a g e and c u r r e n t v e c t o r s , V , A [Y] a d m i t t a n c e m a t r i x , mho i g t r a n s m i s s i o n l i n e i n s t a n t a n e o u s c u r r e n t , A Ig t r a n s m i s s i o n l i n e pha so r c u r r e n t , A 8g c u r r e n t phase angle., r a d k^ ,k2 ,C^ ,C2 t r a n s m i s s i o n l i n e c o n s t a n t s Synchronous Mach ines 3 s u b s c r i p t d e n o t i n g s ynchronous motor q u a n t i t y X ^ J X ^ . X Q s t a t o r dqO f l u x l i n k a g e s , Wb-T Xp f l u x l i n k a g e s t a t e v a r i a b l e , Wb-T e ^ . e q j e Q t e r m i n a l dqO v o l t a g e s , V v i i j v t t e r m i n a l phase v o l t a g e , V V T t e r m i n a l p h a s o r v o l t a g e , V a phase a n g l e o f t e r m i n a l v o l t a g e , r a d v r , v m r e a l and i m a g i n a r y t e r m i n a l v o l t a g e s , V e f f i e l d v o l t a g e , V v p 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 , V V p ^ a d e f i n e d v o l t a g e , e q u a t i o n ( 2 . 1 8 ) , V i j j j i q j i Q s t a t o r dqO c u r r e n t s , A i g , i a i n s t a n t a n e o u s a r m a t u r e c u r r e n t , A Ig a rma tu re pha so r c u r r e n t , A $ a rma tu re c u r r e n t phase a n g l e , r a d "^"Gr'^Gm r e a l and i m a g i n a r y a rmatu re c u r r e n t s , A i f f i e l d c u r r e n t , A a a n g l e f rom s t a t o r a -pha se a x i s t o r o t o r d i r e c t a x i s , e l e c t r i c a l r a d i a n s w a n g u l a r v e l o c i t y o f r o t o r , r a d / s 6 a n g l e f rom i n f i n i t e bus r e f e r e n c e t o d - a x i s , e l e c t r i c a l r a d i a n s S Q r o t o r s l i p P Q , Q Q r e a l and r e a c t i v e power o u t p u t , W and V A R TQ e l e c t r i c a l d e v e l o p e d t o r q u e , N -m R„ a rmatu re r e s i s t a n c e , ft ci Rf f i e l d r e s i s t a n c e , ft L j j j L q j L Q s t a t o r dqO i n d u c t a n c e s , H L a f ^ peak v a l u e o f m u t u a l i n d u c t a n c e between s t a t o r a - p h a s e and f i e l d w i n d i n g , H L f f f i e l d s e l f - i n d u c t a n c e , H x ^ , x q s t a t o r dq r e a c t a n c e s , ft ^ d ' » x d ' t r a n s i e n t d - a x i s i n d u c t a n c e and r e a c t a n c e , H and Q, ^df ' x d f m u t u a l i n d u c t a n c e and r e a c t a n c e , H and Q, T ^ ' t r a n s i e n t s h o r t c i r c u i t d - a x i s t ime c o n s t a n t , s T ^ Q ' t r a n s i e n t open c i r c u i t d - a x i s t ime c o n s t a n t , s J Q moment o f i n e r t i a , J - s / r a d Dg damping c o e f f i c i e n t , J - s / r a d T t p r i m e mover t o r q u e , N-m f g f r i c t i o n torque., N-m P O L E S Q number o f f i e l d p o l e s V o l t a g e R e g u l a t o r and E x c i t e r V R r e g u l a t o r o u t p u t v o l t a g e , V A^,A2 c o n s t a n t s d e f i n i n g r e g u l a t o r o u t p u t l i m i t s f ( v R ) f u n c t i o n d e f i n i n g r e g u l a t o r o u t p u t l i m i t s Tg e x c i t e r t ime c o n s t a n t , s T ^ £ r e g u l a t o r t ime c o n s t a n t , s r e g u l a t o r g a i n v r e f r e g u l a t o r r e f e r e n c e v o l t a g e , V I n d u c t i o n Motor 6 a n g l e used i n g e n e r a l dqO t r a n s f o r m a t i o n , r a d $2 a n g l e between s t a t o r a - p h a s e a x i s and r o t o r A - p h a s e a x i s , r a d 8 g a n g l e between r o t o r A - p h a s e a x i s and i n d u c t i o n motor d - a x i s , r a d e l d , e l q s t a t o r dq v o l t a g e s , V e 2 d » e 2 q r o t o r dq v o l t a g e s , V i-j^^ji^q s t a t o r dq c u r r e n t s , A ^ 2 d » ^ 2 q r o t o r dq c u r r e n t s , A i j ^ i n s t a n t a n e o u s s t a t o r c u r r e n t , A Ij4 p h a s o r s t a t o r c u r r e n t , A Y phase a n g l e o f s t a t o r c u r r e n t , r a d i-Mr'^Mm r e a l and i m a g i n a r y s t a t o r c u r r e n t s , A r o t o r s l i p T^j e l e c t r i c a l d e v e l o p e d t o r q u e , N-m ^ l d ' ^ l q s t a t o r dq f l u x l i n k a g e s , Wb-T ^ 2 d ' ^ 2 q r o t o r dq f l u x l i n k a g e s , Wb-T r-^ s t a t o r r e s i s t a n c e , ft X2 r o t o r r e s i s t a n c e , ft L - Q s t a t o r s e l f - i n d u c t a n c e , H r o t o r s e l f - i n d u c t a n c e , H peak v a l u e of. mutua l i n d u c t a n c e •^12» x 12 m u t u a l i n d u c t a n c e and r e a c t a n c e , H and ft L ' , x ' t r a n s i e n t i n d u c t a n c e and r e a c t a n c e , H and ft C-j,C^ i n d u c t i o n motor c o n s t a n t s D-^,D2,U3,D^ f u n c t i o n s o f 6 A a d e t e r m i n a n t , e q u a t i o n (3.40) Jj^ moment o f i n e r t i a , J - s / r a d D M damping c o e f f i c i e n t , J - s / r a d 1. INTRODUCTION The s t a b i l i t y o f e l e c t r i c power sys tems has been an a r e a o f i n t e r e s t i n r e s e a r c h f o r many y e a r s . As power sys tems i n c r e a s e i n s i z e and c o m p l e x i t y the need f o r m a i n t a i n i n g sy s tem s t a b i l i t y becomes more i m p o r t a n t . A l t h o u g h many p rob lems have been s o l v e d u s i n g e x c i t a t i o n c o n t r o l and o p t i m a l c o n t r o l o f modern t h e o r y o t h e r p rob lems r e m a i n . One o f them i s a more tho rough s tudy o f t he e f f e c t o f s y s tem l o a d s on power sy s tem s t a b i l i t y . In s t e a d y s t a t e s t a b i l i t y s t u d i e s l o a d s a r e u s u a l l y r e p r e s e n t e d by c o n s t a n t power, c o n s t a n t c u r r e n t , c o n s t a n t impedance, o r a c o m b i n a t i o n o f t he t h r e e [ 2 2 ] . I t i s known t h a t f o r s t e a d y s t a t e s t u d i e s t h e c o n s t a n t impedance r e p r e s e n t a t i o n g i v e s o p t i m i s t i c r e s u l t s , t he c o n s t a n t power r e p r e s e n t a t i o n g i v e s p e s s i m i s t i c r e s u l t s and the c o n s t a n t c u r r e n t r e p r e s e n t a t i o n g i v e s g e n e r a l l y s a t i s f a c t o r y r e s u l t s , somewhere i n between the f i r s t two. Not much work has been done on l o a d e f f e c t s i n t r a n s i e n t s t a b i l i t y s t u d i e s . D a h l [1] s u g g e s t e d a r e p r e s e n t a t i o n e i t h e r by c o n s t a n t impedance o r by t r a n s i e n t l o a d c h a r a c t e r i s t i c s but d i d n o t say how t h e l a t t e r were to be o b t a i n e d . C r a r y [2] p r o p o s e d a r e p r e s e n t a t i o n by c o n s t a n t impedance o r , a l t e r n a t i v e l y , c o n s t a n t impedance d u r i n g t h e f a u l t and c o n s t a n t power a f t e r t he f a u l t c l e a r a n c e . He f u r t h e r recommended t h a t i f t he r e s u l t s were s i g n i f i c a n t l y d i f f e r e n t u s i n g h i s two s u g g e s t i o n s an a t tempt s h o u l d be made to o b t a i n more a c c u r a t e l o a d d a t a . Weinbach [3] n e g l e c t e d the r e a l power a l t o g e t h e r and r e p r e s e n t e d t h e r e a c t i v e power as a c o n s t a n t r e a c t a n c e . K imbark [4] was i n f a v o u r w i t h the c o n s t a n t impedance r e p r e s e n t a t i o n . Hore [5] a g r e e d w i t h C r a r y but showed q u a l i t a t i v e l y t h a t a f t e r t he e l i m i n a t i o n o f the f a u l t i t would be more a d v i s a b l e to s i m u l a t e the l o a d as c o n s t a n t c u r r e n t . He f u r t h e r s t a t e d t h a t h i s recommendat ions were d i r e c t e d p r i m a r i l y toward sys tem p l a n n i n g and t h a t they were no t a d e q u a t e f o r t h e a n a l y s i s o f an e x i s t i n g sy s tem where s t a b i l i t y was a p r o b l e m . F o r t h a t , he c o n c l u d e d , t he n o n l i n e a r l o a d c h a r a c t e r i s t i c s r e p r e s e n t a t i o n was r e q u i r e d . Bauman, e t . a l . [6] c o n d u c t e d t e s t s to d e t e r m i n e the r e s p o n s e o f p a r t s o f a power sy s tem to s m a l l changes i n l o a d , v o l t a g e and f r e q u e n c y . They found t h a t an i n c r e a s e i n v o l t a g e r e s u l t e d i n an i n c r e a s e i n b o t h r e a l and r e a c t i v e l o a d . They a l s o e s t a b l i s h e d t h a t changes i n f r e q u e n c y w i t h i n r e a s o n a b l e l i m i t s would no t r e s u l t i n a p p r e c i a b l e changes i n l o a d p r o v i d e d a l l t he bus v o l t a g e s were m a i n t a i n e d c o n s t a n t . B r e r e t o n , e t . a l . [7] d i s c u s s e d the v a r i o u s methods o f r e p r e s e n t i n g i n d u c t i o n motor l o a d s d u r i n g power sys tem s t a b i l i t y s t u d i e s . They recommended t h a t b o t h m e c h a n i c a l and r o t o r e l e c t r i c a l t r a n s i e n t s o f an i n d u c t i o n motor s h o u l d be i n c l u d e d i n the r e p r e s e n t a t i o n i f a d i g i t a l computer i s emp loyed . Gevay and S c h i p p e l [8] i n v e s t i g a t e d the t r a n s i e n t s t a b i l i t y o f an i s o l a t e d r a d i a l power sys tem where the t o t a l l o a d was kep t c o n s t a n t bu t i t s d i v i s i o n , s ynchronous m o t o r s , i n d u c t i o n motor s and s t a t i c l o a d , was v a r i e d . They found t h a t t he p r e s e n c e o f s ynchronous motor l o a d was the most e f f e c t i v e i n m a i n t a i n i n g s t a b i l i t y . The sys tem was neve r s t a b l e i f t he s ynchronous motor l o a d was l e s s than 10 pe r c e n t o f t he t o t a l a c c o r d i n g to t h e i r s t u d y . They a l s o found t h a t b o t h s ynchronous and i n d u c t i o n motor s would swing w i t h r e l a t i v e l y l a r g e a m p l i t u d e s even i n s t a b l e c a s e s . K e n t , e t . a l . [9] emphas ized the need f o r a c o r r e c t 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 l o a d s i n s t a b i l i t y s t u d i e s and c o n d u c t e d l o a d v o l t a g e t e s t s on a power s y s tem. They gave methods f o r p r o p o r t i o n i n g bus l o a d s between c o n s t a n t impedance and c o n s t a n t power. The c l a s s i f i c a t i o n o f l o a d s was ba sed upon kWh s a l e s r e c o r d s . Rober t and Rob i chaud [10] b u i l t a l o a d s i m u l a t o r i n c o n j u n c t i o n w i t h a m i c r o - r e s e a u x and made c o m p a r a t i v e s t a b i l i t y t e s t s . They r e c o g n i z e d t h a t t he g e n e r a l l o a d p rob lem has two p a r t s wh i ch may be s t u d i e d s e p a r a t e l y . The f i r s t i s t o d e t e r m i n e the i n f l u e n c e o f t he c h a r a c t e r i s t i c s o f l o a d s on the s t a b i l i t y o f a power sy s tem and the second p a r t i s t he p r o b l e m o f l o a d r e p r e s e n t a t i o n . In t h i s t h e s i s t he e f f e c t o f l o c a l motor l o a d s on power sy s tem s t a b i l i t y i s i n v e s t i g a t e d . L i k e R o b e r t and R o b i c h a u d , a power s y s tem t e s t model i s u s e d . B u t , i n s t e a d o f s i m u l a t i n g the l o a d s by a r t i f i c i a l means, a c t u a l l o a d s , t h a t i s , i n d u c t i o n motor s and synchronous motor s a r e emp loyed . The motor s a r e c o n n e c t e d d i r e c t l y to t he g e n e r a t o r t e r m i n a l s t h r o u g h a t r a n s f o r m e r . Fo r a n a l y s i s t he s t a t e v a r i a b l e e q u a t i o n s a r e employed w i t h m e c h a n i c a l and e l e c t r i c a l r e l a t i o n s d e s c r i b e d i n d e t a i l . The g e n e r a t o r and t r a n s m i s s i o n l i n e e q u a t i o n s a r e d e v e l o p e d i n Chap te r 2. They a r e n o n l i n e a r e q u a t i o n s and many o f t he v a r i a b l e s commonly n e g l e c t e d i n s t a b i l i t y s t u d i e s a r e r e t a i n e d . The sy s tem w i t h o u t l o c a l motor l o a d s w i t h a t r a n s m i s s i o n l i n e f a u l t i s s o l v e d to p r o v i d e a b a s i s f o r compar i s on w i t h l a t e r s t u d i e s . In C h a p t e r 3 the i n d u c t i o n motor l o a d i s i n t r o d u c e d and s i m i l a r e q u a t i o n s d e r i v e d . F u r t h e r s t a b i l i t y s t u d i e s o f a t r a n s m i s s i o n l i n e f a u l t a r e c a r r i e d o u t . S i m i l a r s t u d i e s a r e c a r r i e d out i n Chap te r 4 but w i t h a l o c a l s ynch ronous m o t o r . In C h a p t e r 5 the most g e n e r a l i z e d f o r m u l a t i o n o f l o c a l motor l o a d s w i t h any number o f s ynchronous and i n d u c t i o n motor s i s p r e s e n t e d but o n l y one i n d u c t i o n motor and one s y n c h r o n o u s motor a r e used f o r t h e s tudy m a i n l y because o f t he r e s t r i c t i o n o f equ ipment . A dynamic t e s t model d e v e l o p e d a t t h e U n i v e r s i t y o f B r i t i s h Co lumb ia [ 1 1 , 1 2 , 1 3 , 1 4 ] i s u sed to v e r i f y the a n a l y t i c a l r e s u l t s o f t he t h e s i s . S i n c e t h e motor s used a r e s m a l l ones w i t h c o m p a r a t i v e l y h i g h l o s s e s f u r t h e r computed r e s u l t s o f s t u d i e s u s i n g l a r g e mach ine p a r a m e t e r s a r e i n c l u d e d i n C h a p t e r 4 to v e r i f y the u s e f u l n e s s o f t he r e s u l t s f r om s m a l l mach ine s t u d i e s . 2, SYNCHRONOUS GENERATOR WITHOUT LOCAL KOTOR LOAD The s t a t e e q u a t i o n s f o r a v o l t a g e r e g u l a t e d s y n c h r o n o u s g e n e r a t o r sy s tem w i t h o u t l o c a l motor l o a d a r e d e r i v e d . The m a t h e m a t i c a l mode l i s v e r i f i e d by a t r a n s i e n t s t a b i l i t y s t u d y . The computed r e s u l t s a r e compared w i t h the r e s u l t s o b t a i n e d d i r e c t l y f rom l a b o r a t o r y t e s t s on a power s y s tem t e s t m o d e l . 2.1 Synchronous G e n e r a t o r E q u a t i o n s The s ynchronous g e n e r a t o r e q u a t i o n s were o r i g i n a l l y d e r i v e d by Pa rk [15] i n p e r u n i t . Lewi s [16] r e w r o t e them i n MKS u n i t s as u sed h e r e . The f o r m u l a t i o n i s based upon the f o l l o w i n g a s s u m p t i o n s : 1) A l l i n d u c t a n c e s a r e i n d e p e n d e n t o f c u r r e n t ( s a t u r a t i o n n e g l e c t e d ) . 2) On ly t he second ha rmon i c o f t he permeance i n t he a i r gap i n a d d i t i o n t o the ave rage v a l u e i s c o n s i d e r e d . 3) The e l e c t r i c t r a n s i e n t o f t h e damper w i n d i n g i s n e g l e c t e d . The p o s i t i v e p o l a r i t i e s o f t he c u r r e n t s and v o l t a g e s and t h e d i r e c t i o n o f r o t a t i o n o f the r o t o r a r e shown i n F i g u r e 2 . 1 . The e q u a t i o n s o f t h e mach ine i n dqO c o o r d i n a t e s a r e e d = - R a i d " P A d " x q P C T (2 .1 ) " p X q + A d p o (2 .2 ) e 0 = ~ R a i 0 " P X 0 (2 .3 ) e f = R f i f + P A f (2 .4 ) F i g u r e 2 . 1 C i r c u i t D iagram o f I d e a l Synchronous Mach ine i Synchronous Generator Circui t Breaker I -Load Buses Fault Location i -A Circuit Breaker Infinite Bus F i g u r e 2.2 O n e - l i n e Diagram o f One M a c h i n e - I n f i n i t e Bus System Synchronous Generator B Infinite Bus F i g u r e 2.3 One Mach ine System w i t h E q u i v a l e n t - u T r a n s m i s s i o n Network where V V d + f W f ( 2 - 5 ) \ = V q ( 2 - 6 ) X Q = L 0 i Q (2 .7 ) X f = L f f ^ + jf W d ( 2 ' 8 ) F o r n o t a t i o n s , see n o m e n c l a t u r e . The f i e l d c u r r e n t i ^ i s u s u a l l y e l i m i n a t e d i n a n a l y s i s t o g i v e e d = -R^d ~ p A d - V 0 ( 2 - 9 ) eq " " V q " P \ + V ( 2 ' 1 0 ) A , = + i d (2 .11) " 0 R f U + T d 0 ' P ) w 0 ( 1 + ' T d O * P ) Xq X q = _ i i q (2 .12) In t h e e l i m i n a t i o n p r o c e s s t he f o l l o w i n g c o n s t a n t s a r e i n t r o d u c e d as d U d q 0 q „ t M T ' T ' - T a f M x d ~ "oLd L d - L d — — ^ L f f 3 L d f ~ 2 L a f M x d f " u 0 L d f Ad0 x d ^dO R f x d o (2 .13) 3L (2 .14) f Note t h a t a l l o f t he pa ramete r s o f (2 .9 ) t h r o u g h (2.12) a r e m e a s u r a b l e . The z e r o sequence c u r r e n t v a n i s h e s f o r b a l a n c e d o p e r a t i o n . E q u a t i o n s (2.9) t h r o u g h (2.11) a r e r e a r r a n g e d i n s t a t e v a r i a b l e f o rm as where [17] p X d = - e d - R a i d - o ) 0 ( l - s G ) A q (2 .15) PXq = " e q " V q + V 1 " SG> Xd <2-16> P X F = V F + V F R ( 2 ' 1 7 ) < ° 0 ( x d _ x d ' ) A d x d X F v p R = (2 .18) x d x d T d 0 x d f e f v p = (2 .19) * f X F u 0 A d i , = + (2 .20) x d x d 0 x d V q i = (2 .21) q x q a = a>Qt + 6 (2 .22) oi = pa = O)Q + p5 (2 .23) s G (2 .24) C 0 Q The e l e c t r o m e c h a n i c a l e q u a t i o n s o f t he s y n c h r o n o u s mach ine a r e p6 = - w 0 s G (2 .25) p s G = — J G POLES, 2u , 1(T G + f . - T t ) + D f l " 8.) (2.26) 2.2 T r a n s m i s s i o n L i n e E q u a t i o n s The t r a n s m i s s i o n l i n e i s m o d e l l e d w i t h lumped pa ramete r s wh i ch a r e v a l i d f o r f u n d a m e n t a l f r e q u e n c y v o l t a g e s and c u r r e n t s [ 1 8 J . F i g u r e 2.2 i s a o n e - l i n e d i a g r a m o f t h e sy s tem under c o n s i d e r a t i o n . The t r a n s m i s s i o n n e t w o r k , i n c l u d i n g l o c a l l o a d s r e p r e s e n t e d as impedances , i s r e d u c e d to an e q u i v a l e n t - u ne twork as shown i n F i g u r e 2 . 3 . The i n f i n i t e bus v o l t a g e i s t a k e n as t h e r e f e r e n c e f o r a l l pha so r q u a n t i t i e s . The c u r r e n t s f l o w i n g i n the s y s tem a r e assumed to be as f o l l o w s : From the g e n e r a t o r : ±G = f2 I G cos (u) Q t + 8) (2.27) From the i n f i n i t e b u s : i g = ^2 Ig cos(u)Qt + Gg) (2.28) The g e n e r a t o r t e r m i n a l v o l t a g e (phase " a " t o n e u t r a l ) i s v f c = Jl V t cos(u>Qt + a) (2.29) When the g e n e r a t o r c u r r e n t i s t r a n s f o r m e d i n t o i t s dqO c o o r d i n a t e s t he r e s u l t i n g dq c u r r e n t s a r e c o n s t a n t : i d = /3 I G c o s ( g - 6 ) i = - / 3 I. s i n ( 3 - 6 ) q U (2.30) (2 .31) The i n v e r s e t r a n s f o r m a t i o n f o r the g e n e r a t o r c u r r e n t i s i a = 3 [ i d c o s ( w 0 t + 6 ) + i q s i n ( u ) Q t + 6 ) ] (2 .32) w h i c h may be w r i t t e n i n p h a s o r f o rm as 10 o r j e _ d /3 /3 (2.34) The " i " t a k e s a n e g a t i v e s i g n becau se o f t h e c o o r d i n a t e s c h o s e n . L e t t h e l e f t s i d e o f (2.34) be w r i t t e n as i „ + j i „ and s e p a r a t e the r e a l and Gr Gm i m a g i n a r y p a r t s t o o b t a i n Ig cos B = x Gr d — cos 6 + — s i n 6 /3 /3 (2.35) I s i n 3 1 Gm d • * q — s i n o - — cos o /3 /3 (2.36) A s i m i l a r p r o c e d u r e f o r t he v o l t a g e y i e l d s ej e d q v_ = — cos 6 H s i n 6 7 3 r /3 (2.37) d • x q X v = — s i n o cos o m /3 /3 (2.38) The s t e a d y s t a t e p h a s o r e q u a t i o n o f t h e t r a n s m i s s i o n s y s tem i s [Y] [V] = [I] (2.39) where [Y] i s t he a d m i t t a n c e m a t r i x o f t h e n e t w o r k , [V] i s t h e bus v o l t a g e v e c t o r and [I] i s t h e c u r r e n t v e c t o r . Expanded , t h i s becomes [19] G + j B + r + j x r + j x r + j x 1 r + j x v r + J v m V + J i G m i B r + j ± B m (2.40) f where V Q i s t he i n f i n i t e bus v o l t a g e p h a s o r and i ^ + j i i s t h e pha so r c u r r e n t f rom the i n f i n i t e bus i n t o t h e ne twork . Expand ing the f i r s t e q u a t i o n o f (2 .40) and e q u a t i n g r e a l and i m a g i n a r y p a r t s y i e l d r G - xB + 1 - rB - xG v r _ v 0 + r - x ^•Gr rB + xG rG - xB + 1 v m 0 X r xGm _ _ (2 .41) E q u a t i o n s (2 .35) t h r o u g h (2.38) a r e s u b s t i t u t e d i n t o (2 .41) and t h e r e s u l t i n g e q u a t i o n s o l v e d f o r e^ and e q . The r e s u l t i s e d k l - k 2 k 2 k x V Q C O S <5 V Q s i n 6 C l C 2 - c 2 c 1 (2 .42) where r G - xB + 1 k „ = ( rG - xB + l ) 2 + ( rB + x G ) 2 rB + xG (rG - xB + l ) 2 + ( rB + x G ) 2 = k-^r + k 2 x ^2 ~~ ~ ^2^" (2 .43) (2 .44) (2 .45) (2 .46) 2*3 R e g u l a t o r and E x c i t e r E q u a t i o n s The b l o c k d i a g r am o f t he r e g u l a t o r and e x c i t e r sy s tem i s shown i n F i g u r e 2 .4 . The r e g u l a t o r o u t p u t l i m i t i s a c c o u n t e d f o r u s i n g a h y p e r b o l i c t a n g e n t f u n c t i o n [13] . . . f ( v R ) = t a n h ( A 2 v R ) (2 .47) where A-^  and A 2 a r e c o n s t a n t s d e t e r m i n e d f r om a l e a s t - s q u a r e s c r i t e r i o n . 7ref + \ K A V R f f ( v R ) 1 e f 1 1 + T R E P 1 + T E p F i g u r e 2.4 B l o c k D iagram o f 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 s t a t e e q u a t i o n s o f t he b l o c k d i a g ram a r e p e f = • [ f ( v R ) - e f ] T E P V R RE [ K A < v r e f " v t ) " V R1 The t e r m i n a l v o l t a g e i n terms o f dq v o l t a g e s i s v t = (2.48) (2.49) (2.50) In summary, the s t a t e e q u a t i o n s o f t h e comp le te sy s tem a r e (2.15) t o ( 2 . 1 7 ) , ( 2 . 2 5 ) , ( 2 . 2 6 ) , (2.48) and ( 2 . 4 9 ) . 2.4 Computa t i on and L a b o r a t o r y T e s t R e s u l t s The sy s tem e q u a t i o n s a r e v e r i f i e d w i t h l a b o r a t o r y t e s t s . The sys tem d i s t u r b a n c e f o r t he t r a n s i e n t s t a b i l i t y s t udy i s as f o l l o w s : a t h r e e -phase f a u l t o c c u r s a t one o f t he sys tem buses and the f a u l t e d l i n e i s i s o l a t e d a t 5 c y c l e s f o l l o w e d by a sy s tem r e s t o r a t i o n a t 30 c y c l e s . F i g u r e 2.2 shows the l o c a t i o n o f the f a u l t . The p a r a m e t e r s o f t he sy s tem a r e d e t e r m i n e d by d i r e c t measurement and a r e l i s t e d t o g e t h e r w i t h t h e T a b l e 2.1 Synchronous G e n e r a t o r , V o l t a g e R e g u l a t o r and T r a n s m i s s i o n L i n e Pa ramete r s Synchronous G e n e r a t o r Pa ramete r s 0.66 ft _ 16.2 ft 2.74 ft 4 .80 ft 4 0.00267 J - s / r a d R„ x d R £ P0LES ( x T d 0 * J a f M 9.71 ft 5.00 s 0.575 H 0.165 J - s 2 / r a d 1.585 N-m V o l t a g e R e g u l a t o r Pa ramete r s K A RE 0.152 0.035 s 0.050 s " 2 v r e £ 12.8 0.0943 72.6 V T r a n s m i s s i o n L i n e Pa ramete r s r X G B (ohm) (ohm) (mho) (mho) 1.42 21.0 0.0097 - 0 . 1 1 7 1.42 21.0 0 .0 0.0114 0.710 10.5 0.000033 0.0227 14 T a b l e 2.2 O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s O p e r a t i n g C o n d i t i o n s 100 W/ph Q G 85 .0 VAR/ph V t 57.6 V / p h V Q 31 .6 V / p h T X 1.5 T 3.73 N-m t I n i t i a l V a l u e s o f S t a t e V a r i a b l e s A d 0.2628 Wb-T v D 2.279 V X q 0.0617 Wb-T 6 133.1 e l e c t r i c a l d e g r e e s h 538 Wb-T S G ° e f 2.71 V I p r e f a u l t o p e r a t i n g c o n d i t i o n s i n T a b l e 2 . 1 . The computed and t e s t r e s u l t s a r e shown i n F i g u r e s 2.5 t h r o u g h 2 . 7 . F i g u r e 2.5 shows the g e n e r a t o r t o r q u e a n g l e . The maximum swing f rom the o p e r a t i n g p o i n t i s 52 e l e c t r i c a l d e g r e e s f rom c o m p u t a t i o n and 50 d e g r e e s f rom the l a b o r a t o r y t e s t w h i c h shows good ag reement . The t e s t r e s p o n s e i s somewhat s l o w e r than the computed r e s u l t . T h i s i n d i c a t e s t h a t t h e r e i s a l a r g e r damping e f f e c t i n t he mach ine than i n t he m a t h e m a t i c a l •model. The t e r m i n a l v o l t a g e i s shown i n F i g u r e 2 .6 . The s w i t c h i n g i n s t a n t s a r e e a s i l y s e e n . The c a l c u l a t e d c u r v e p r e d i c t s a v o l t a g e d i p to 45 v o l t s and a r i s e to 60 v o l t s . The a c t u a l t e s t shows a d i p to 43 v o l t s and a r i s e to 63 v o l t s . V o l t a g e s p i k e s a r e p r e d i c t e d i n c o m p u t a t i o n a t t h e s w i t c h i n g i n s t a n t s . They a r e a l s o seen i n t h e t e s t r e s u l t s i n t he l a b o r a t o r y bu t do no t have the same a m p l i t u d e s . T h i s i s because i t i s d i f f i c u l t t o r e a l i z e a f a u l t a t an e x a c t i n s t a n t . S m a l l o s c i l l a t i o n s o f t h e v o l t a g e waveform e n v e l o p e a r e e v i d e n t i n b o t h the f i g u r e s . The g e n e r a t o r c u r r e n t i s shown i n F i g u r e 2 . 7 . The c u r r e n t r o s e to 5.6 amperes i n t he t e s t . The p r e d i c t e d v a l u e was 6.2 amperes . S i n c e the t o r q u e a n g l e swing i s most i m p o r t a n t i t i s c o n c l u d e d t h a t t he m a t h e m a t i c a l model i s s u i t a b l e f o r t r a n s i e n t s t a b i l i t y s t u d i e s . 100.On § 80.0 LU y 60.0-LU _ J CD § 40.0 LU ZD C3 CtZ O a : o r— CC cr: LU z : LU CD 20.0-0 -/ / \ • / • • • • • • • • 1 • 1 * • 0 1 • • • — V • • • t I • • • • • | • * • % \ to****! f * i/ i: . t ' • * • * • * t : \ t • • \ / V / • V V Y I I s -20.0 » r T " 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME (SEC> F i g , 2.5 G e n e r a t o r T o r q u e A n g l e 17 I ' ' i i I ' i i i i i i i i i i i i i [ i i i i i i i i i i i i i r | i i » i [ 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T I M E ( S E C ) F i g . 2.7 G e n e r a t o r C u r r e n t 3. SYNCHRONOUS GENERATOR WITH A LOCAL INDUCTION MOTOR LOAD In t h i s chapter the e f f e c t of a l o c a l i n d u c t i o n motor load on power system s t a b i l i t y i s i n v e s t i g a t e d . A o n e - l i n e diagram of the system i s shown i n Figure 3.1. Transformer Induction Motor W W . — ^ M T 1 Infinite Bus-Figure 3.1 Synchronous Generator System With L o c a l Induction Motor Load 3.1 Induction Motor Equations The i n d u c t i o n motor, i s a round-rotor machine and the s a l i e n c y e f f e c t i s not present. Stanley [20] gave an a n a l y s i s of the i n d u c t i o n motor i n 1938. In h i s equations the machine c u r r e n t s , v o l t a g e s and f l u x l i n k a g e s were r e f e r r e d to axes f i x e d on the s t a t o r . S a t u r a t i o n was neglected. The equations used here f o l l o w c l o s e l y those of F i t z g e r a l d and K i n g s l e y [21] wi t h axes r o t a t i n g at the synchronous speed. Figure 3.2 shows the p o l a r i t y of c urrents and vo l t a g e s and the d i r e c t i o n of r o t a t i o n of the machine. The dqO trans f o r m a t i o n matrix i s chosen as F i g u r e 3.2 C i r c u i t D iagram o f I d e a l I n d u c t i o n Motor cos 8 2 . - s i n 3 1 3 2 cos (8 - s i n (6 1 3 2TT 2TT — ) 3 - cos (( l - s i n (8 1 3 4ir — ) 3 4TT — ) 3 where 8 = o^t f o r t he s t a t o r t r a n s f o r m a t i o n and 8 = u^t - 6 2 f o r the r o t o r . Note t h a t t h e n u m e r i c a l c o e f f i c i e n t s a r e d i f f e r e n t f rom F i t z g e r a l d and K i n g s l e y ' s . The e q u a t i o n s i n dq c o o r d i n a t e s a r e -Id = r l i l d + P X l d + w 0 X l q e l q = r l 1 l q + P A l q " V l d e 2 d = r 2 i 2 d + p X 2 d + X 2 q P 6 s 6 2 q = ^ Z q + p X 2 q " X 2 d P 6 s (3 .1 ) (3.2) (3.3) (3.4) where ^ld ~ L l l i l d + 2 L a A M i 2 d l l q ~ L l l i l q + 2 L a A M 1 2 q ^2d " L 2 2 i 2 d + 2 L a A M i l d ^2q ~ L 2 2 i 2 q + 2 L a A M i l q (3 .5) (3 .6 ) (3.7) (3 .8 ) The z e r o sequence c u r r e n t s v a n i s h f o r b a l a n c e d o p e r a t i o n . The r e l a t i o n s h i p between the g e n e r a t o r , and i n d u c t i o n motor t e r m i n a l v o l t a g e s i s shown i n F i g u r e 3 . 3 . The e q u a t i o n s a r e e,, = T (e, cos 6 + e n s i n 6) • (3.9) e-, = T (e cos <5 - e, s i n 6) (3.10) l q x q a where T v i s the turns r a t i o of an i d e a l transformer between the motor and the generator or motor t e r m i n a l v o l t a g e T x generator t e r m i n a l v o l t a g e Figure 3.3 Synchronous Generator and Indu c t i o n Motor dq Terminal Voltages F o l l o w i n g the suggestion of Brereton, e t . a l . [8] and Gabbard's d i s c u s s i o n the s t a t o r f l u x l i n k a g e s of the i n d u c t i o n motor are assumed to remain constant g i v i n g P X l d = P X l q = ° Since the r o t o r windings are s h o r t - c i r c u i t e d i n normal o p e r a t i o n , ^2d ~ e 2 q The i n d u c t i o n motor e q u a t i o n s now can be w r i t t e n as T x ( e d cos 6 + e q s i n 6) = r ^ ^ + u ^ I ^ i . ^ + w 0 L 1 2 1 2 q <3.11) T x ( e q cos 6 - e d s i n 6) = r i i l q - u 0 L l l i l d ~ u 0 L 1 2 i 2 d (3 .12) 0 = r 2 i 2 d + p X 2 d + A 2 q p 6 s (3 .13) 0 = r 2 i 2 q + P X 2 q " X 2 d P 6 s ' ( 3 - 1 4 ) X 2 d = L 2 2 i 2 d + L 1 2 i l d . (3 .15) X 2 q = L 2 2 1 2 q + L 1 2 i l q ( 3 ' 1 6 ) where 3 L 1 2 = 2 L a A M _ (3 .17) E l i m i n a t i n g i 2 d and i 2 q f rom ( 3 . 1 1 ) , ( 3 . 1 2 ) , (3 .15) and (3 .16) g i v e s hd = T x < D l e d + D 2 e q > + C 4 X 2 d " C 3 X 2 q < 3 - 1 8 ) ^ q = T x ( D l 6 q ~ D 2 e d > + C 3 X 2 d + C 4 X 2 q ( 3 ' 1 9 ) where x ' = C O Q L ' X 1 2 = W Q L 1 2 (3 .20) L 2 1 2 ' L ' = L 1 ± - (3 .21) L 2 2 1 D, = ( r , cos 6 + x 1 s i n 6) (3 .22) r ; L2 + ( x ' ) 2 1 D„ = • ( r , s i n 6 - x 1 cos 6) (3 .23) 2 r± 2 + ( x ' ) 2 1 r l x 1 2 C 3 = — - - (3 .24) L 2 2 [ r i 2 + ( x ' ) 2 ] C 4 = -x12x (3 .25 ) E q u a t i o n s (3 .13) and (3.14) may now be w r i t t e n i n s t a t e v a r i a b l e f o rm as P X 2 d = ~ r2hd ~ %sHX2q  (3- 26) p X 2 q = ' ^ Z q + %SKX2d  (3' 27) where 1^^ and a r e s o l v e d .from (3.15) and ( 3 . 1 6 ) . The e l e c t r o m e c h a n i c a l e q u a t i o n o f t he i n d u c t i o n m o t o r , a n a l o g o u s to ( 2 . 2 6 ) , i s P S M = — JM POLES M 2w r ( T L - T M) + % ( ! - s M ) (3 .28) 3.2 T r a n s m i s s i o n L i n e E q u a t i o n s The dq v o l t a g e e q u a t i o n (2 .46) i s no t v a l i d beca u se o f t h e a d d i t i o n o f l o c a l motor l o a d i n g . I f t he s t e a d y s t a t e c u r r e n t f l o w i n g i n t o the motor i s = /2 I M cos - ( u ) Q t + Y ) (3 .29 ) then i t s dq components a r e hd  = ^ T M C O S Y hq = _ / 3 ~ hi s i n Y (3 .30) (3 .31 ) F o l l o w i n g the p r o c e d u r e o u t l i n e d i n C h a p t e r 2 f o r d e f i n i n g g e n e r a t o r v o l t a g e and c u r r e n t , r e a l and i m a g i n a r y motor c u r r e n t s a r e d e f i n e d as x r = 7i x i d (3 .32 ) (3.33) These e q u a t i o n s a r e ana l ogous to (2 .35) and (2 .36) but do n o t c o n t a i n t h e t o r q u e a n g l e 6 because o f t he dq axes cho sen f o r t he i n d u c t i o n m o t o r . The t r a n s m i s s i o n l i n e e q u a t i o n , ana l ogous to ( 2 . 4 1 ) , i s rG - xB + 1 - r B - xG rB + xG r G - xB + 1 — v r — v 0 + r - x v m 0 X r Gr x M r (3.34) LGm _ ^ m U s i n g the c o n s t a n t s d e f i n e d i n (2 .42) t o (2 .45) and a l s o d e f i n i n g (3 .35) (3 .36) an e q u a t i o n f o r t he dq v o l t a g e s o f t h e s ynch ronous g e n e r a t o r i s o b t a i n e d as D 3 = C j cos S + C 2 s i n 6 ^4 = ^1 ^ - ^1 c o s ^ k 1 - k 2 k 2 k x V Q C O S 6 V Q s i n 6 C l C 2 - c 2 c 1 - T. x D3 " D4 D 4 D 3 S u b s t i t u t i n g i l d and i l q f r om (3.18) and (3.19) i n t o (3 .37) (3 .37) e d = /3 e q k l " k 2 k 2 k x V Q C O S 6 V Q s i n 6 + C l C 2 - c 2 c x D3 " D4 D4 D3 ••x D l D 2 - D 2 D1 C4 - C 3 C3 C4 ; 2d ^2q (3 .38) S o l v i n g once a g a i n f o r e^ and e , 1 A where Tx <D1D3 + D2 D4> + 1 Tx ( ¥ 4 ~ D2 D3> T X 2 ( D 2 D 3 - D l D 4 ) Tx ( D1 D3 + D 2 D 4 ) + 1 •3 k l " k 2 k 2 k x VQ COS 6 VQ s i n 6 + C l C 2 - C 2 Cj_ -T. x D 3 - D 4 D4 D3 C4 " C 3 C 3 C 4 ^2d l 2 q (3.39) A = [ ^ ( D ^ + D 2 D 4 ) + 1 ] 2 + T X 4 ( D 2 D 3 - D ^ ) 2 (3.40) In summary, t he s t a t e e q u a t i o n s o f the s ynchronous g e n e r a t o r s y s tem w i t h l o c a l i n d u c t i o n motor l o a d a r e (2 .15) t o ( 2 . 1 7 ) , ( 2 . 2 5 ) , ( 2 . 2 6 ) , ( 2 . 4 8 ) , ( 2 . 4 9 ) , ( 3 . 2 6 ) , (3 .27) and ( 3 . 2 8 ) . 3.3 Computa t i on and L a b o r a t o r y T e s t R e s u l t s The sy s tem p a r a m e t e r s and d i s t u r b a n c e f o r the t r a n s i e n t s t a b i l i t y s t u d y a r e the same as i n C h a p t e r 2. The p a r a m e t e r s o f the- i n d u c t i o n - m o t o r a r e d e t e r m i n e d f rom t e s t s and a r e l i s t e d i n T a b l e 3 . 1 . The p r e f a u l t o p e r a t i n g c o n d i t i o n s o f the e n t i r e sys tem a r e g i v e n i n T a b l e 3 .2 . The computed and t e s t r e s u l t s a r e shown i n F i g u r e s 3.4 t h r o u g h 3 .8 . A c o m p a r i s o n o f F i g u r e s 2.4 and 3.4 i n d i c a t e s t h a t t he power s y s tem i s more s t a b l e when s u p p l y i n g a l o c a l i n d u c t i o n motor l o a d . The v o l t a g e waveform of the g e n e r a t o r w i t h the i n d u c t i o n motor l o a d , F i g u r e 3 . 5 , i s e s s e n t i a l l y the same as t h a t i n F i g u r e 2.5 w i t h o u t t he l o a d . The T a b l e 3.1 I n d u c t i o n Motor Pa ramete r s I n d u c t i o n Motor Pa ramete r s r l 9 .10 Q. L l l 0 .426 H r 2 9.14 0, L 2 2 0.426 H L 1 2 0.409 H L a A M 0.273 H J M 0.0130 J - s2 / r a d P O L E S M 4 % 0.00133 J - s / r a d g e n e r a t o r c u r r e n t waveform, F i g u r e 3 .6 , i s t he same as i n F i g u r e 2.7 e x c e p t t h a t the a m p l i t u d e o f t h e o s c i l l a t i o n s a f t e r sy s tem r e s t o r a t i o n i s s m a l l e r when t h e r e i s a l o c a l i n d u c t i o n motor l o a d . F i g u r e 3.7 shows the computed i n d u c t i o n motor s l i p i n c r e a s e d u r i n g the f a u l t and o s c i l l a t i o n s a f t e r sy s tem r e s t o r a t i o n . T h i s r e s u l t was a l s o o b s e r v e d i n the l a b o r a t o r y but no t measured due to l a c k o f a s u i t a b l e m e a s u r i n g d e v i c e . The l a b o r a t o r y t e s t r e s u l t o f t h e g e n e r a t o r t o r q u e a n g l e swing i s shown i n F i g u r e 3 .4 (a ) wh i ch i n g e n e r a l a g r e e s w i t h the computed r e s u l t shown i n F i g u r e 3 .4 (b ) e x c e p t f o r t he i n i t i a l d e c r e a s e . The d e c r e a s e i n t o r q u e a n g l e i n d i c a t e s a sudden i n c r e a s e i n power o u t p u t o f t he g e n e r a t o r . U s u a l l y t he power o u t p u t o f t h e g e n e r a t o r d e c r e a s e s when a t r a n s m i s s i o n l i n e f a u l t o c c u r s , r e s u l t i n g i n a t o r q u e a n g l e i n c r e a s e . T h i s i s t r u e whether o r not the g e n e r a t o r has a l o c a l motor l o a d . Because o f t h e sudden l a r g e change i n a rmatu re c u r r e n t t h e r e must be a l a r g e c u r r e n t i n d u c e d i n t h e damper w i n d i n g s wh i ch r e s u l t s i n a l a r g e damping t o r q u e f o r t he g e n e r a t o r . Now i f t h e damping i s l a r g e r t han the a c c e l e r a t i n g e f f e c t due to t h e power T a b l e 3.2 O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s O p e r a t i n g C o n d i t i o n s P G 100 W/ph Q G 85 VAR/ph S M 0.033 v f c 57 .6 V / p h v 0 37.6 V / p h T x 1.5 T t 3.73 N-m T L 0.371 N-m I n i t i a l V a l u e s o f S t a t e V a r i a b l e s *d 0.263 Wb-T 6 120 e l e c t r i c a l d e g r e e s \ 0.0617 Wb-T s G 0 538 Wb- 0.121 Wb-T e f 2.71 V X„ 2q 0.348 Wb-T V R 2.28 V S M 0.033 ; / \ V 6 \ J . . . V I s 1 0 0 . 0 -i § 8 0 . 0 LU ib! 6 0 . 0 H LU _ J CD ^ 4 0 . 0 LU C3 CC O O I— CC ct: LU LU CD 2 0 . 0 H 0 ^ - 2 0 . 0 1 1 1 1 j ' ' ' ' | ' ' r-i | I I i i | t • - r - i i | i i i i | i i i i | i i j , | 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 d T I M E ( S E C ) . F i g . 3.4 G e n e r a t o r To rque A n g l e •imiilllll w h||||l I i i i i i i i i i l i l i l l i i ' l i i ii iiilliiipillll iJIIIHH! llilltilil IHMIIIIIII ' i i l l l i i l l l , i:n;!i;::;: Mlllllllll ' l inn 'mimm iiuiiiii; 0.4 s 6 5 n 07 §60 H 07 z : cr: Wc-e-co 55 cc > LU £50 H X Q_ cn o § 4 5 LU -z. LU CD 40 1 ' 1 1 i 1 • i ' ' ' ' | i ' i i | i i i i | i i i i | i \ i i | i i i i | 0 0.5 1.0 1.5 2 . 0 2 . 5 3.0 3.5 4.0 T I N E ( S E C ) F i g . 3.5 Generator Voltage F i g . 3.6 G e n e r a t o r C u r r e n t F i g . 3.7 I n d u c t i o n Motor S l i p c c CO 2: on 1.5 H L U cn Cr: o r — o £ 0 . 5 CJ ZD Q 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 | 1 1 1 1 I 1 i 1 1 ) 1 1 1 1 I 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T I M E ( S E C ) F i g . 3.8 I n d u c t i o n Moto r C u r r e n t o u t p u t d e c r e a s e , t he g e n e r a t o r t o r q u e a n g l e d e c r e a s e s i n s t e a d o f i n c r e a s i n g as p r e d i c t e d . T h i s phenomenon w i l l be more a p p a r e n t when t h e r e i s a l a r g e l o c a l l o a d . In t h e a n a l y s i s , however , t he e f f e c t o f damper w i n d i n g s has been n e g l e c t e d . Two c o n c l u s i o n s may be drawn f r om t h i s s t u d y . F i r s t , t h e e f f e c t o f damper w i n d i n g s must be i n c l u d e d i n s t a b i l i t y s t u d i e s i f e x a c t p r e d i c t i o n o f t o r q u e a n g l e swings i s r e q u i r e d . Second , a l o c a l i n d u c t i o n motor l o a d i n c r e a s e s t he s t a b i l i t y o f a power s y s t e m . 4. SYNCHRONOUS GENERATOR WITH A LOCAL SYNCHRONOUS MOTOR LOAD In t h i s chapter the l o c a l load e f f e c t of a synchronous motor on system s t a b i l i t y i s investigated. A one l i n e diagram of the system i s shown i n Figure 4 . 1 . Transformer Synchronous Motor J Infinite Bus^ Figure 4.1 Synchronous Generator System with Local Synchronous Motor Load 4.1 Synchronous Motor E q u a t i o n s The s y n c h r o n o u s motor has the same e q u a t i o n s as the synchronous g e n e r a t o r e x c e p t t h a t t h e p o s i t i v e d i r e c t i o n f o r s t a t o r c u r r e n t s i s now d e f i n e d t o be i n t o t h e m a c h i n e . The s u b s c r i p t " 3 " i s u sed to denote s ynch ronous motor q u a n t i t i e s . Ana logous t o s t a t e e q u a t i o n s (2 .15) t o ( 2 . 1 7 ) , (2 .25) and (2 .26) o f t he synchronous g e n e r a t o r , t he synchronous motor e q u a t i o n s a r e P A 3 d = ~ 6 3 d + R 3 a i 3 d " " S 3 ) X 3 < 1 pA„ = - e + R i + M ( i - s , ) X~ , 3q 3q 3a 3q 0 V V 3d p A 3 F = v 3 F + v 3 F R (4.1) (4.2) (4.3) 36 P 6 3 = - w 0 s 3 (4.4) p s 3 POLES. 2ov < T3L " T3M> + D 3 d " 8 3 ) (4.5) The a u x i l i a r y e q u a t i o n s , ana l o gou s t o (2.18) t o ( 2 . 2 4 ) , a r e i o 0 ( x 3 d - x 3 d ' ) X 3 D * 3 F v 3FR '3d ' 3F x 3d i3d0 x 3 d f e 3 f R 3f X 3 F u 0 A 3 d L 3d x 3 d ' T 3 d o ' x 3 d ' 0 3q L 3q x 3q a 3 = V + 6 3 to3 = p a 3 = W Q + p6 U)Q - u)3 0 -(4.6) (4 .7 ) (4 .8 ) (4 .9 ) (4 .10) (4 .11) (4 .12) S i n c e t h e t e r m i n a l v o l t a g e s o f t he g e n e r a t o r and t h e motor a r e common, t he dq v o l t a g e s o f t h e two mach ines a r e r e l a t e d by t ! - cos a 3 - cos (a 3 - cos (a 2ir 3 ATT 2 - s i n a 3 - s i n (a 3 - s i n (a 2TT 3 4ir -) - cos a 0 3 3 [ 3 co s ( a 3 -) 2TT 3 A 7T s i n a - s i n (0-3 - co s ( a 0 3 3 -) -) 2TT 3 4 7T - s i n (a^ ) e 3 d '3q '30 (4 .13) S o l v i n g f o r 63^ and 63^ w i t h e n = e ^ n = 0, '0 _ 30 e 3 d = T x [ e d cos (6 - 63) + e q s i n (6 - 63) ] e 3 q = T x ^ e q c o s ^ 6 " 6 3 ^ - e d s i n ^ ~ 6 3 ^ ^ (4 .14) (4 .15) 4.2 T r a n s m i s s i o n L i n e E q u a t i o n s The t r a n s m i s s i o n l i n e e q u a t i o n , ana l o gou s t o ( 3 . 3 4 ) , i s rG - xB + 1 - r B - xG r B + xG r G - xB + 1 m v 0 0 r - x ^-Gr x 3Mr 1Gm 13Mm (4 .16 ) where 3Mr 3Mm _x_ Tf cos 6 0 s i n 6. s i n <5- - c o s 6'. A3q (4 .17) E q u a t i o n (4.17) i s o b t a i n e d u s i n g the same arguments as t h o s e t h a t l e d to the w r i t i n g o f e q u a t i o n s (2.35) and ( 2 . 3 6 ) . S u b s t i t u t i n g ( 2 . 3 5 ) , ( 2 . 3 6 ) , ( 2 . 3 7 ) , (2 .38) and (4.17) i n t o (4 .16) and s o l v i n g f o r e and e , d q e d k l " k 2 k 2 k 2 V Q C O S 6 V Q s i n 5 D 5 - D 6 C l C 2 - c 2 C± x d L 3d ±3q (4 .18) where k-^, k 2 , and C 2 have been d e f i n e d i n e q u a t i o n s (2 .42) t o (2 .45 ) and D 5 = C x cos (<S - 63) + C 2 s i n (6 - 63) (4 .19) D, = C x s i n (6 - 63) - C 2 cos (6 - 63) (4 .20) I f e q u a t i o n (4.18) i s compared w i t h e q u a t i o n (3 .39) i t i s s een t h a t t he r e p r e s e n t a t i o n o f the s ynch ronous g e n e r a t o r sy s tem w i t h l o c a l s ynchronous motor l o a d i s much l e s s complex . The r e p r e s e n t a t i o n o f t h e s ynchronous g e n e r a t o r sy s tem w i t h l o c a l i n d u c t i o n motor l o a d i s more complex because the g e n e r a t o r c o o r d i n a t e s a r e f i x e d on the r o t o r whereas t h e i n d u c t i o n motor c o o r d i n a t e s a r e r o t a t i n g a t t he s y n c h r o n o u s s p e e d . In summary, t he s t a t e e q u a t i o n s o f t h e s ynchronous g e n e r a t o r sy s tem w i t h l o c a l s ynchronous motor l o a d a r e (2 .15) to ( 2 . 1 7 ) , ( 2 . 2 5 ) , ( 2 . 2 6 ) , ( 2 . 4 8 ) , (2.49) and (4.1) t o ( 4 . 5 ) . 4 .3 Computa t i on and L a b o r a t o r y T e s t R e s u l t s The sy s tem p a r a m e t e r s and d i s t u r b a n c e f o r t he t r a n s i e n t s t a b i l i t y s t u d y a r e t h e same as i n C h a p t e r s 2 and 3. The p a r a m e t e r s o f t h e s ynchronous motor a r e d e t e r m i n e d f rom t e s t s and l i s t e d i n T a b l e 4 . 1 . The r e a l power i n p u t t o t h e s ynchronous motor i s t h e same as t h a t o f t he 3 9 T a b l e 4.1 Synchronous Motor Pa ramete r s Synchronous Motor Pa ramete r s R 3 a 4.26 ft X 3 q 124.8 ft x 3 d 131.7 ft T 3 d 0 ' 0.0619 s x 3 d ' 40 .4 ft L 3 a f M 0.405 H R 3 f 10.8 ft J 3 0.0121 J - s2 / r a d POLES 3 4 D 3 0.000995 J - s / r a d T a b l e 4.2 O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s O p e r a t i n g C o n d i t i o n s P G 100 W/ph % 85 VAR/ph P 3 40 .0 W/ph Q3 19.3 VAR/ph v t 57.6 V /ph v 0 32.5 V / p h T x 1.5 T t 3.73 N-m T 3 L 0.431 N-m e 3 f 8.17 V I n i t i a l V a l u e s of S t a t e V a r i a b l e s A d 0.263 Wb-T S G 0 \ 0.0617 Wb - T A 3 d 0.270 Wb-T X F 538 Wb-T V - 0 . 2 7 8 Wb-T e f 2.71 V X 3 F 7.05 Wb-T V R 2.28 V 6 3 61.3 e l e c t r i c a l d e g r e e s 6 118 e l e c t r i c a l d e g r e e s s 3 0 i n d u c t i o n motor o f C h a p t e r 3. T y p i c a l i n d u c t i o n motor s l i p and synchronous motor power f a c t o r v a l u e s a r e s e t p r i o r to t he f a u l t o c c u r r e n c e . The o p e r a t i n g c o n d i t i o n s o f t h e e n t i r e sy s tem a r e l i s t e d i n T a b l e 4 . 2 . The computed and t e s t r e s u l t s a r e shown i n F i g u r e s 4.2 to 4 . 6 . A c o m p a r i s o n o f F i g u r e s 2.4 and 4.2 i n d i c a t e s t h a t t he power sys tem i s more s t a b l e when s u p p l y i n g a l o c a l s ynch ronous motor l o a d . The g e n e r a t o r c u r r e n t and v o l t a g e waveforms a r e e s s e n t i a l l y t h e same as i n Chap te r 3. A l l l a b o r a t o r y measurements a g r e e f a v o u r a b l y x^ith computed r e s u l t s e x c e p t t he t o r q u e a n g l e sw ing . The t o r q u e a n g l e swing o b s e r v e d i n the l a b o r a t o r y d e c r e a s e s i n i t i a l l y as i n the ca se o f i n d u c t i o n motor l o a d i n g and f o r the same r e a s o n . The s ynchronous motor o s c i l l a t e s d u r i n g the t e s t f o r a s h o r t p e r i o d wh ich i s a l s o p r e d i c t e d by c o m p u t a t i o n , F i g u r e 4 . 3 . To f u r t h e r e x p l o r e the n a t u r e o f a s ynch ronous motor l o a d , c o m p u t a t i o n i s ex tended to i n c l u d e the ca se o f s t e a d y s t a t e o p e r a t i o n o f t h e motor but w i t h a s i n u s o i d a l s h a f t l o a d . The o p e r a t i n g c o n d i t i o n s o f t he sy s tem f o r t h i s i n v e s t i g a t i o n a r e the same as i n T a b l e 4.2 e x c e p t t h a t t he s h a f t l o a d i s no l o n g e r c o n s t a n t a t 0.431 N-m bu t v a r i e s s i n u s o i d a l l y as T 3 L = ° - 4 3 1 + 0.1 cos 2nt Tha t i s , the peak l o a d i s 0.531 N-m and the f r e q u e n c y o f o s c i l l a t i o n i s 1.0 c y c l e pe r s e c o n d . The r e s u l t s a r e shown i n F i g u r e s 4 .7 and 4 . 8 . A l t h o u g h the g e n e r a t o r has s u s t a i n e d o s c i l l a t i o n s o f i t s t o r q u e a n g l e the sys tem rema in s s t a b l e f o r t he c a s e i n v e s t i g a t e d . 100 h •"^  / \ r 4f J -V > I 5 CD y 80 H o LU 60 H L u _ l CD cc 40 L U ZD C3 o o <x en LU L U CD 20 H OH - 2 0 0 0 ^ 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4 T IME (SEC) F i g . 4 .2 Generator To rque Angle i I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 T IME (SEC) F i g . 4 .3 Synchronous Motor To rque A n g l e 80 - i i — • -.—J o -> i n '—' -GE <x • -o60 -> UJ i n -cr X -Q_ CtZ 6D V-X Cr: UJ UJ CD 0.4 s n ^ l . O 1.5 2.0 2 . 5 3.0 3 . 5 4.0 T IME ISEC) F i g . 4.4 Generator Voltage - L o c a l Synchronous Motor Load 1 0 . O n a. 8 . 0 2: CE CO z: Cr: 6 . 0 L U err ZD CJ 4 . 0 H Cr: o 1— CE Cr: L U I 2 . 0 0 tut IH«!!<! I^tU.ll..!...'".'-'"""1 •>.imm rtv.'iV,i'.„i,. mem !!» < 0 . 4 S I ' ' ' I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4.1 TIME (SEC) F i g . 4.5 G e n e r a t o r C u r r e n t 2n CL CC oo CC U J Cr: cr: ZD CD Cr: CD r — OO ZD 6D 21 CD cn X C J 00 -0 ' i i . . ...tiiiii!! jj iiliiii IIIIIM!'!' iiilililti 0-4 S ' 1 ' 1 I ' 1 ' 1 I ' 1 1 1 I ' ' ' ' | I I I I I • I I I I I I I I I I 1 I I I 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 5 3 . 0 3 . 5 4 . 0 TIME (SEC) F i g . 4.6 Synchronous Motor C u r r e n t 1 4( 100 CD S 80 C J UJ ^ 60 UJ _ J CD cr 40 UJ ZD C3 cn 20 cn 63 az £ o UJ CD -20 ' ' ' ' | 1 1 1 1 I""1'1 1 1 P 1 1 1 i r-r-r-r-pi i i > J • t r t r 1 j-"t i r >~] 0 0.5 1.0 ' 1.5 2.0 2.5 3.0 3.5 4.0 TIME (SEC) F i g . 4.7 G e n e r a t o r Torque A n g l e 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 T IME (SEC) F i g . 4 .8 Synchronous Motor T o r q u e A n g l e 4 4.4 Computa t i on R e s u l t s U s i n g L a r g e - S y s t e m P a r a m e t e r s A l a r g e g e n e r a t o r w i t h a l o c a l s ynch ronous motor l o a d i s i n v e s t i g a t e d . The t r a n s m i s s i o n l i n e f a u l t i s a t t he same l o c a t i o n and o f t h e same d u r a t i o n as i n t h e s m a l l mach ine s t u d i e s . The sy s tem p a r a m e t e r s and o p e r a t i n g c o n d i t i o n s a r e g i v e n i n T a b l e s 4 . 3 , 4.4 and 4 . 5 . The t o r q u e a n g l e swings o f t he s ynchronous g e n e r a t o r w i t h o u t and w i t h s ynch ronous motor l o a d a r e shown i n F i g u r e s 4 .9 and 4.10 r e s p e c t i v e l y . The maximum swing i s 25.2 e l e c t r i c a l d e g r e e s w i t h o u t t he motor l o a d and 20.9 e l e c t r i c a l d e g r e e s when the motor i s i n c l u d e d . From t h e r e s u l t s o b t a i n e d i n t h i s c h a p t e r i t i s c o n c l u d e d t h a t a l o c a l s ynchronous motor l o a d w i l l a l s o i n c r e a s e the s t a b i l i t y o f a power s y s tem. T a b l e 4.3 Pa ramete r s o f L a r g e Synchronous Motor Pa ramete r s o f L a r g e Synchronous Motor R 3 a 0.00136 ft x 3 q 0.218 ft x 3 d 0.327 ft T 3 d 0 * 5.3 s x 3 d ' 0.0819 ft L 3 a f M 0.024 H R 3 f 0.25 ft J 3 11100 J - s 2 / r a d P 0 L E S 3 6 D 3 0 J - s / r a d I T a b l e 4.4 Pa ramete r s o f L a r g e Power System Synchronous G e n e r a t o r P a r a m e t e r s R 0.0 ft x n 1.04 ft a x d 1.85 ft T d Q ' 7.76 s x d ' 0.361 ft L a f M 0 , 0 6 1 H R f 0.182 ft J G 3 7 6 x l 04 J - s 2 / r a d P O L E S G 48 D Q 0 .0 J - s / r a d f G 0.0 N-m V o l t a g e R e g u l a t o r Pa ramete r s K A 0.925 v R ( m a x ) 1310 V T E 0.003 s v R ( m i n ) -1030 V T R E 0.05 s v r e f 8129 V T r a n s m i s s i o n L i n e P a r a m e t e r s r x G B (ohm) (ohm) (mho) (mho ) -0.00145 0.105 2.22 - 2 . 6 3 -0.00244 0.0815 2.86 3.98 -0.00286 0.0838 2.96 3.12 T a b l e 4.5 L a r g e Power System S t a b i l i t y S tudy O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s O p e r a t i n g C o n d i t i o n s P G 30 MW/ph 10 MVAR/ph P 3 26.7 MW/ph Q 3 14.3 MVAR/ph v t 7970 V / p h v 0 6157 V / p h T 0.239 X T t 5 . 7 3 x l 0 6 N-m T 3 L 6 . 3 3 x l 0 5 N-m e 3 f 88.6 V I n i t i a l V a l u e s o f S t a t e V a r i a b l e s X d 33.7 Wb-T S G 0.0 14.2 Wb-T X 3 d 5.1 Wb-T l . l l x l O 5 Wb-T X 3 q -7 .12 Wb-T e f 136 V X 3 F 1 . 2 7 x l 04 Wb-T V R 147 V 6 3 17.6 e l e c t r i c a l d e g r e e s 6 95.2 e l e c t r i c a l d e g r e e s s 3 0.0 F i g . 4.9 L a r ge G e n e r a t o r To rque A n g l e No Moto r Loads 5 5. SYNCHRONOUS GENERATOR WITH MULTIPLE LOCAL SYNCHRONOUS AND INDUCTION MOTOR LOADS In t h i s chapter a more general l o c a l load and i t s e f f e c t on power system s t a b i l i t y i s i n v e s t i g a t e d . The l o c a l load may c o n s i s t of any number of synchronous and i n d u c t i o n motors as w e l l as any other loads which may be represented by equivalent c i r c u i t s . A o n e - l i n e diagram of the system i s shown i n Figure 5.1. Note that loads represented by equ i v a l e n t c i r c u i t s are absorbed i n t o the equivalent-ir r e p r e s e n t a t i o n of the tr a n s m i s s i o n l i n e i n the network r e d u c t i o n process. Synchronous Generator / v y \ A /vV\A AAM /V<\A B I n d u c t i o n Motors Synchronous Mofor~s-ir Figure 5.1 Synchronous Generator System With M u l t i p l e L o c a l Motor Loads 5.1 Machine Equations The equations required to describe the synchronous generator system w i t h i t s equivalent-ir t r a n s m i s s i o n network, the i n d u c t i o n motors and the synchronous motors have already been developed i n Chapters 2 through 4. Synchronous G e n e r a t o r W i th V o l t a g e R e g u l a t o r P X d = ~ e d " R a i d - w 0 ( 1 " s G ) X c p X p = v F + v p R (5 .1 ) (5 .2 ) (5 .3 ) p e f = [ f ( v R ) - e f ) ] (5 .4 ) I n d u c t i o n M o t o r s : P V R RE [ K A < v r e f - v t > " V R ] POLES. 2uv p6 = - u 0 s G <TG + f G " Tt> + V 1 " SG> „-v k _ „ k. k .„ c k, k P X 2 d ~ " r 2 1 2 d " W 0 S M A 2 q P X 2 q k = " r 2 k i 2 q k + w 0 s M k x 2 d k k J M POLES. ( T L k - % k ) + D M k ( l - s M k ) 2u 0 (5 .5 ) (5 .6 ) (5 .7 ) (5 .8 ) (5 .9 ) (5 .10) :where f o r the m u l t i - m o t o r s y s tem the s u p e r s c r i p t k d e n o t e s t h e motor number, k = 1, 2, . . . , m Synchronous M o t o r s : P * 3 d k " " e 3 d k + R 3 a k i 3 d k " - O d " " 3 k > * 3 q k (5 .11) 55 where P A 3 q k = " e 3 q k + R 3 a k i 3 q k " V 1 - V 3 q ' p X 3 F k = v 3 F k + v 3 F R k P 6 3 k = -o) 0s 3 k ps 3 J3' POLES 2o)r < T3L " T 3 M K ) + D 3 (1 ~ s 3 K ) k = m+1, m+2, ..., n (5.12) (5.13) (5.14) (5.15) The o r d e r o f t he s y s tem i s 7 + 3m + 5(n-m) where m i s the number o f i n d u c t i o n motor s and (n-m) i s the number o f s ynchronous m o t o r s . Motor s o f t he same s i z e and w i t h s i m i l a r s h a f t l o a d c h a r a c t e r i s t i c s s h o u l d be lumped t o g e t h e r as one e q u i v a l e n t mach ine i n o r d e r to d e c r e a s e the number o f e q u a t i o n s . 5.2 T r a n s m i s s i o n L i n e E q u a t i o n s An e q u a t i o n s i m i l a r t o ( 2 . 4 0 ) , (3.34) and (4.16) i s w r i t t e n f o r the g e n e r a l c a s e as rG - xB + 1 - r B - xG r B + xG rG - xB + 1 v 0 r - x + O x r v, m i G r " ^ m ^ M r ^ ~ z n ( ± 3 r k ) iGm " *m<iMmk> " ^3m^ (5.16) where Z m and E n a r e o p e r a t o r s d e f i n e d by and m k= l n *n = >: k=m+l i k ^ r i Mra k ° /3 i k x 3 r £ n i k " /3 cos 5 3 K s i n 6 - j k s i n 6 0 - c o s 6-L 3d (5 .17) (5 .18 ) S u b s t i t u t i n g ( 2 . 3 7 ) , ( 2 . 3 8 ) , ( 2 . 3 5 ) , ( 2 . 3 6 ) , (5 .17) and (5.18) i n t o (5 .16) and s o l v i n g f o r the dq v o l t a g e s , = /3 cos <5 s i n 6 s i n 6 - c o s 6 cos <5 s i n 6 s i n 6 - c o s 6 cos 6 s i n 6 s i n 6 - c o s 6 k l k 2 - k 2 k x k l k 2 - k 2 k± k l k 2 - k 2 k± v 0 0 r - x x r r - x x r co s 6 s i n 6 s i n 6 - c o s 6 k V T x k L l d ) 57 cos 6 s i n 6 s i n 6 - c o s 6 k l k 2 " k 2 k l r - x x r £ n ( T x k cos 6 0 k s i n 6-s i n 6 o - c o s 6 ' i 3 d wh i ch r e d u c e s to (5 .19) e d = /3 k l k 2 - k 2 k x V Q COS 6 V Q s i n 6 C l C 2 - c 2 D 3 - D 4 D 4 D 3 ^ m ( T x k L l d V - ^ n ( T x k D 5 k - D 6 k Jit L 3d 1 3 q (5.20) where C-^  and C 2 a r e d e f i n e d by (2 .44) and ( 2 . 4 5 ) , D 3 and D 4 a r e d e f i n e d by (3.35) and (3.36) and D 5 k = C 1 cos ( 6 - 6 3 k ) + C 2 s i n ( 6 - 6 3 k ) D 6 k = s i n ( 6 - 6 3 k ) - C 2 cos ( 6 - < $ 3 k ) (5 .21) (5 .22) The i n d u c t i o n motor s t a t o r . c u r r e n t s a r e e l i m i n a t e d u s i n g (3 .18) and (3.19) i n the m a t r i x f o r m , i l d = x D x k D 2 k - D 2 k D-^ e d k r k c 4 - c 3 A 2 d V 2q (5 .23) r where D r - i k cos 6 + x'k s i n 6 k _ _± ( r k ) 2 + (x' k) 2 (5 .24) D, r- , k s i n 6 - x'k cos 6 k _ _t ' " ( r k ) 2 + (x' k) 2 (5 .25) r k k r l x 1 2 ' 3 " L 2 2 k [ ( r i k ) 2 + ( x ' k ) 2 ] (5 .26) x 1 2 x k ,k J 4 T kr , k. 2 , , ,k. 2, L 2 2 t ( r ! ) + (x' ) ] (5 .27) E q u a t i o n (5 .23) i s now s u b s t i t u t e d i n t o (5 .20) and the r e s u l t i n g e q u a t i o n s o l v e d f o r t he dq v o l t a g e s to o b t a i n 1 0 0 1 + D 3 - D 4 D 4 D 3 U ( T x k ) 2 D k h k  u l u 2 - D 2 k D l k -1 /3 k l ~ k 2 k2 kx V Q C O S 6 V Q s i n 6 + C l C 2 " C 2 C l x d D o - D / , D 4 D 3 ^m<T x k r k r k C 4 _ C 3 c 3 k c 4 k l 2 d V 2q M?x k " D 6 k D 6 k D 5 k x 3 d *3q (5 .28) The m a t r i x i n v e r s i o n o f (5 .28) i s 4x4 r e g a r d l e s s o f the number o f motor s i n v o l v e d and p r e s e n t s no c o m p u t a t i o n p r o b l e m s . In t he c a s e o f no i n d u c t i o n m o t o r s , ' D^ k = D 2 k = 0 f o r k = l , 2 , . . . , m , and no m a t r i x i n v e r s i o n i s n e c e s s a r y . The dq t e r m i n a l v o l t a g e s o f any i n d u c t i o n motor a r e c a l c u l a t e d f rom (3.9) and (3.10) and t h o s e o f any s ynch ronous motor f rom (4 .14) and ( 4 . 1 5 ) : e l d k = T x k ( e d cos 6 + e q s i n 6) (5 .29) e l q k = T x k ( e q cos 6 - e d s i n 6) (5 .30) e 3 d U = T x k f e d c o s <6 " 6 3 k ) + e q s i n ( S ~ 6 3 k > ] (5.31) e 3 q k = T x k t e q c o s ( 6 " 6 3 k ) " e d s i n ( 6 .~ 6 3 k ) 3 (5 .32) In summary, the s t a t e e q u a t i o n s o f a s ynch ronous g e n e r a t o r sy s tem s u p p l y i n g m u l t i p l e l o c a l i n d u c t i o n motor and s ynch ronous motor l o a d s a r e g i v e n by (5 .1) t h r o u g h ( 5 . 1 5 ) . A g e n e r a l t r a n s m i s s i o n l i n e e q u a t i o n r e l a t i n g a l l mach ine t e r m i n a l c o n d i t i o n s t o t he t r a n s m i s s i o n ne twork i s g i v e n by ( 5 . 2 8 ) . As f o r l o c a l l o a d s wh i ch may be r e p r e s e n t e d as e q u i v a l e n t c i r c u i t s , t h e y can be r e a d i l y i n c o r p o r a t e d i n t o the t r a n s m i s s i o n network r e p r e s e n t a t i o n . 5.3 Computa t i on and L a b o r a t o r y T e s t R e s u l t s A l t h o u g h a g e n e r a l f o r m u l a t i o n o f a s ynch ronous g e n e r a t o r sy s tem w i t h m u l t i p l e s ynchronous and i n d u c t i o n motor l o a d s was g i v e n i n t he l a s t s e c t i o n i t i s s u f f i c i e n t t o i n v e s t i g a t e the s t a b i l i t y o f a s ynch ronous g e n e r a t o r sy s tem s u p p l y i n g one l o c a l i n d u c t i o n motor and one l o c a l s ynchronous motor [ 8 ] . The sy s tem d i s t u r b a n c e and g e n e r a t o r and motor p a r a m e t e r s a r e g i v e n i n T a b l e s 2 . 1 , 3.1 and 4 . 1 . The p r e f a u l t o p e r a t i n g c o n d i t i o n s a r e g i v e n i n T a b l e 5 . 1 . The computed and t e s t r e s u l t s a r e shown i n F i g u r e s 5.2 t h r o u g h 5 .8 . A c o m p a r i s o n i s made o f t h e g e n e r a t o r t o r q u e a n g l e o f F i g u r e 5,2 w i t h t h a t o f F i g u r e s 3.4 and 4.2 where o n l y s i n g l e motor l o a d s were p r e s e n t . I t r e v e a l s t h a t the s t a b i l i t y o f t h e sy s tem i s i n c r e a s e d s t i l l f u r t h e r w i t h b o t h s ynchronous and i n d u c t i o n motor l o a d s . A d e c r e a s e i n g e n e r a t o r t o r q u e a n g l e i s seen i m m e d i a t e l y a f t e r t h e f a u l t o c c u r s i n t h e l a b o r a t o r y t e s t because o f t h e o m i s s i o n o f t he damper w i n d i n g s i n t h e a n a l y s i s as i n C h a p t e r s 3 and 4. The s ynchronous motor t o r q u e a n g l e i s shown i n F i g u r e 5 . 3 . The swing i s no t so s e v e r e as i n t he c a s e o f a s i n g l e s ynch ronous motor l o a d a l o n e , F i g u r e 4 . 3 . The i n d u c t i o n motor s l i p , F i g u r e 5 .4 , i s a lmos t t he same as i n C h a p t e r 3, t h e c a s e o f t h e g e n e r a t o r w i t h t h e i n d u c t i o n motor l o a d a l o n e . The g e n e r a t o r t e r m i n a l v o l t a g e , F i g u r e 5 . 5 , and the mach ine c u r r e n t s , F i g u r e s 5.6 to 5 . 8 , a r e much the same as i n t he p r e v i o u s t e s t s o f C h a p t e r s 2 to 4 e x c e p t t h a t o s c i l l a t i o n s v a n i s h more q u i c k l y a f t e r t h e t r a n s m i s s i o n l i n e i s r e s t o r e d . I t i s c o n c l u d e d f rom the s t u d i e s so f a r t h a t a l o c a l motor l o a d , e i t h e r i n d u c t i o n o r s ynchronous o r b o t h , w i l l improve t h e s t a b i l i t y o f a power s y s t e m . T a b l e 5.1 Synchronous G e n e r a t o r System Wi th L o c a l I n d u c t i o n And Synchronous Motor Loads O p e r a t i n g C o n d i t i o n s and I n i t i a l V a l u e s o f S t a t e V a r i a b l e s O p e r a t i n g C o n d i t i o n s P G 100 W/ph Q G 85 VAR/ph P 3 40 .0 W/ph Q 3 19.3 VAR/ph s M 0.033 v t 57 .6 V / p h V Q 39 .9 V /ph 1.5 e 3 f 8.17 V T t 3.73 N-m T 3 L 0.431 N-m T L 0.131 N-m I n i t i a l V a l u e s o f S t a t e V a r i a b l e s X d 0.263 Wb-T X 3 q --0.278 Wb-T X q 0.0617 Wb-T X 3 F 7.05 Wb-T X F 538 Wb-T ^3 51.8 e l e c t r i c a l d e g r e e s e f 2.71 V s 3 0.0 V R 2.28 V X 2 d 0.0486 Wb-T 6 109 e l e c t r i c a l d e g r e e s X 2 q 0.366 Wb-T X 3 d 0.270 Wb-T s M 0.033 S G 0.0 1 0 0 . On CD y 8 0 . 0 o LU gj 6 0 . 0 -LU — J CD cn 4 0 . 0 LU ZD O cn o cn o i— cc et: LU LU CD 2 0 . 0 -0 -- 2 0 . 0 V I S i i i i | i i i i | i i i i | I T i i ( r i i i | i i i i || i i r r | i r r f ] 0 0 . 5 1.0 1.5 2 . 0 2 . 5 3 . 0 3 . 5 4,0 TIME (SEC) F i g . 5.2 G e n e r a t o r Torque A n g l e 6: -25 i TIME (SEC) F i g . 5 . 3 S y n c h r o n o u s M o t o r T o r q u e A n g l e 65 60.0n CO co 55.0 i cr: UJ CD CE I— g 50.0 LU CO CE X OL | 45.0 x cr: LU LU CD 40.0 0.4 5 0 i i i i [ i i i i I r-0.5 1.0 i i I I r j i i i i I i i i i I i i i i i i i i i I 1.5 2.0 2.5 3.0 3.5 4.0 TIME (SEC) F i g . 5.5 G e n e r a t o r V o l t a g e § : 0 . 8 H cc CO z: or H-0.6 UJ cc ce ZD C J org 4 -C D U ' ^ (--O C J ?0.2H oo 0 r i i i | i i i t | i ; i i | i i r i | i i i ; j i i i i | i i i i | i i i i j 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 TIME (SEC) F i g . 5.7 Synchronous Motor Current 2 n CL 21 CE CO 2 1 cn UJ or ct: Si cn o o 0 ZD Q 0 TTfri'till!|'ll!'"?,'M!!!l ItUi ll^illlliillliiiiliii lilllln I 0.4 s r 1 1 1 1 1 1 1 1 I 1 1 ' 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 1 1 1 1 I 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 TIME (SEC) F i g . 5.8 I n d u c t i o n Motor C u r r e n t 6. CONCLUSION The e f f e c t o f l o c a l motor l o a d s on power sys tem s t a b i l i t y has been i n v e s t i g a t e d . From a n a l y t i c a l r e s u l t s o f b o t h l a r g e and s m a l l s ynchronous and i n d u c t i o n motor l o a d s and t e s t s on a power sys tem model u s i n g s m a l l mach ines the f o l l o w i n g c o n c l u s i o n s .can be drawn: 1. A l o c a l motor l o a d , e i t h e r i n d u c t i o n o r s ynchronous o r b o t h , w i l l improve the s t a b i l i t y o f a power s y s tem. 2. Bo th s ynchronous and i n d u c t i o n motor s o s c i l l a t e w i t h c o m p a r a t i v e l y l a r g e a m p l i t u d e s d u r i n g sy s tem d i s t u r b a n c e s even i f s t a b i l i t y i s m a i n t a i n e d . 3. The damper w i n d i n g s o f b o t h synchronous g e n e r a t o r s and s ynchronous moto r s have n o t i c e a b l e e f f e c t on the f i r s t swing a f t e r a f a u l t , as r e v e a l e d f rom the t e s t s . 4. Moto r s w i t h l a r g e p e r i o d i c a l m e c h a n i c a l l o a d s can cause o s c i l l a t i o n s to a power s y s tem. 5. The i n f l u e n c e o f l o c a l l o a d s on power sy s tem s t a b i l i t y can be e f f e c t i v e l y d e t e r m i n e d w i t h a power sy s tem t e s t m o d e l . C o n c l u s i o n s 1 and 2 were a l s o r e a c h e d by Gevay and S c h i p p e l [8] f o r the c a s e where the motor l o a d s were a t t he r e c e i v i n g end o f t he t r a n s m i s s i o n l i n e . C o n c l u s i o n 5 was a l s o r e a c h e d by Rober t and Rob i chaud [10] u s i n g s i m u l a t e d c o n s t a n t impedance , c o n s t a n t power and c o n s t a n t c u r r e n t l o a d s . The f o l l o w i n g recommendat ions a r e made f o r f u r t h e r s t u d i e s . F i r s t , t h e m a t h e m a t i c a l model o f s ynchronous mach ines s h o u l d be ex tended to i n c l u d e the damper w i n d i n g s i f e x a c t p r e d i c t i o n o f t o r q u e a n g l e swings i s r e q u i r e d . Second, more e x t e n s i v e t e s t s on the power sys tem model f o r d i f f e r e n t power and v o l t a g e l e v e l s and d i f f e r e n t power f a c t o r s o f t he g e n e r a t o r and the motor s s h o u l d be done i n o r d e r to r e a c h more g e n e r a l c o n c l u s i o n s . O ther t y p e s o f l o a d s such as l i g h t i n g and a r c f u r n a c e s s h o u l d a l s o be added i n t he a n a l y s i s . T h i r d , t he v a r i a t i o n o f bus v o l t a g e f r e q u e n c y and magn i tude s h o u l d a l s o be i n c l u d e d . The f o r m u l a t i o n p r e s e n t e d i n t h i s t h e s i s c an be r e a d i l y i n c o r p o r a t e d i n t o o p t i m a l s t a b i l i z a t i o n and c o n t r o l i n v e s t i g a t i o n s o f power sys tems s i n c e the s y s tem m o d e l l i n g i s a v e r y i m p o r t a n t p a r t o f t h e s t u d y . REFERENCES 1. O.G.C. Dahl, E l e c t r i c Power C i r c u i t s I I , New York: McGraw-Hill, f938.. 2. S.B. Crary, Power System S t a b i l i t y I I , New York: John Wiley, 1947. 3. M.P. Weinbach, E l e c t r i c Power Transmission, New York: MacMillan, 1948. 4 . E.W. Kimbark, Power System S t a b i l i t y I, New York: John Wiley, 1948. 5. R.A. Hore, Advanced Studies In E l e c t r i c a l Power System Design, London: Chapman and H a l l , 1966. 6. H.A. Bauman, O.W. Manz, J.E. McCormack, H.B. Seely, "System load swings," AIEE Trans., Vol. 60, pp. 541-547, 1941. 7. D.S. Brereton, D.G. Lewis, C C . Young, "Representation of induction-motor loads during power system s t a b i l i t y s t u d i e s , " AIEE Trans., Vol. 76, Pt. I l l , pp. 451-461, August 1957. 8. J . Gevay, W.H. Schippel, "Transient s t a b i l i t y of an i s o l a t e d r a d i a l power network with varied load d i v i s i o n , " IEEE Trans. Pwr. App. and Syst., Vol. 83, No. 9, pp. 964-970, September 1964. 9. M.H. Kent, W.R. Schmus, F.A. McCrackin, L.M. Wheeler, "Dynamic modelling of loads i n s t a b i l i t y s t u dies," IEEE Trans. Pwr. App. and Syst., Vol. PAS-88, No. 5, pp. 756-763, May 1969. 10. J . Robert, Y. Robichaud, "Load behaviour and transient s t a b i l i t y , " Conference Paper, IEEE Winter Power Meeting, New York, January 1970. 11. J.A. Bond, "A s o l i d state voltage regulator and e x c i t e r for a large power system model," U.B.C. M.A.Sc. Thesis, July 1967. 12. R.G. Siddal, "A prime mover-governor test model for large power systems," U.B.C. M.A.Sc. Thesis, January 1968. 13. G.E. Dawson, "A dynamic test model for power system s t a b i l i t y and control studies," U.B.C. Ph.D. Thesis, December 1969. 14. M.S. Metcalfe, "Resynchronization of a s l i p p i n g synchronous machine," U.B.C. M.A.Sc. Thesis, December 1969. 15. R.H. Park, "Two-reaction theory of synchronous machines I," AIEE Trans., Vol. 48, No. 2, pp. 716-730, July 1929. 16. W.A. Lewis, The P r i n c i p l e s of Synchronous Machines, 3rd edn., Chicago: I l l i n o i s I n s t i t u t e of Technology, 1959. 17. Y. Yu, K. Vongsuriya, L.N. Wedman, "Application of an optimal c o n t r o l theory to a power system," IEEE Trans. Pwr. App. and Syst., V o l . PAS-49, No. 1, pp. 55-60, January 1970. 72 18. D.W. Olive, " D i g i t a l simulation of synchronous machine t r a n s i e n t s , " IEEE Trans. Pwr. App. and Syst., Vol. PAS-87, No. 8, pp. 1669-1675, August 1968. 19. K. Vongsuriya, "The a p p l i c a t i o n of Lyapunov function to power system s t a b i l i t y analysis and c o n t r o l , " U.B.C. Ph.D. Thesis, February 1968. 20. H.C. Stanley, "An analysis of the induction machine," AIEE Trans., Vol. 57, pp. 751-757, 1938. 21. A.E. F i t z g e r a l d , C. Kingsley, E l e c t r i c Machinery, 2nd edn., New York: McGraw-Hill, 1961. 

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