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Resynchronization of a slipping synchronous machine Metcalfe, Malcolm Stuart 1969

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R E S Y N C H R O N I Z A I I O N OP A S L I P P I N G SYNCHRONOUS MA C H I N E b y MALCOLM STUART M E T C A L F E B.A.Sc., 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 , 1967 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF THE R E Q U I R E M E N T S FOR THE DEGREE OF M ASTER OF A P P L I E D S C I E N C E i n t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g We a c c e p t . t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d R e s e a r c h S u p e r v i s o r M e m b e r s o f t h e C o m m i t t e e A c t i n g H e a d o f t h e D e p a r t m e n t M e m b e r s o f t h e D e p a r t m e n t o f E l e c t r i c a l E n g i n e e r i n g THE U N I V E R S I T Y OF B R I T I S H COLUMBIA D e c e m b e r , 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment 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 not 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 . Depa r tment The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada i i ABSTRACT I n t h i s t h e s i s , r e s y n c h r o n i z a t i o n o f a s y n c h r o n o u s m a c h i n e f r o m a s l i p p i n g s t a t e i s c o n s i d e r e d . A s e t o f e q u a t i o n s f o r t h e s l i p p i n g m a c h i n e a r e d e v e l o p e d . S l i p t e s t s o n a d y n a m i c t e s t m o d e l a r e d o n e t o c o m p a r e w i t h c o m p u t e d r e s u l t s . A c r i t i c a l d a m p i n g n e c e s s a r y f o r r e s y n c h r o n i z a t i o n i s i n v e s t i g a t e d , a n d a n e x p r e s s i o n i s d e r i v e d . T e s t s a r e a g a i n c a r r i e d o u t o n a p o w e r s y s t e m s e t u p o n t h e d y n a m i c t e s t m o d e l t o v e r i f y t h e r e s u l t . F i n a l l y , a n o n l i n e a r o p t i m a l c o n t r o l i s d e v e l o p e d a n d i m p l e m e n t e d o n t h e t e s t m o d e l . i i i TABLE OP CONTENTS 'ABSTRACT i i TABLE OP CONTENTS ' i i i LIST OP ILLUSTRATIONS i v '.ACKNOWLEDGEMENT • v NOMENCLATURE v i 1 . INTRODUCTION 1 1 .1 The Resynchronization Problem 1 1 .2 Thesis Approach 2 2 . STATE EQUATIONS OP A SLIPPING SYNCHRONOUS MACHINE' 3 2 . 1 Co-ordinate System 3 2 . 2 Stator Currents 5 2 . 3 Swing Equation 9 2 .4 State Equations of a Synchronous Machine f o r Resynchronization Studies. 12 3 . COMPARISON OP COMPUTATION AND DYNAMIC MODEL TESTS 13 3 .1 Torque Angle Transducer 13 3 . 2 S l i p Sensing 15 3 . 3 D i g i t a l Readout 18 3 .4 Dynamic Test Model 18 3 . 5 S l i p Tests 21 4 . CRITICAL DAMPING POR RESYNCHRONIZATION 26 4 . 1 C r i t i c a l Damping f o r Resynchronization 26 4 . 2 Numerical Examples 30 5 . APPLICATION OP OPTIMAL CONTROL 32 ' 5 - 1 Optimization Problem 32 5 .2 A p p l i c a t i o n of the Method to the Resynchronization Problem 34 6. CONCLUSION ' 39 APPENDIX - .40 REFERENCES 42 LIST OP ILLUSTRATIONS F igure Page 3.1 Zero Cross ing Detector 14 3.2 Angle Counter and Reference Angle Subt rac tor 15 3.3 B ina ry Adder 17 3 .4 S i x - C y c l e Counter 19 3.5 S l i p Sensor 19 3 . 6 S l i p and S ta to r Current Envelope f o r an Unexci ted S l i p p i n g Synchronous Machine 22 3-7 Comparison of Computed and Test Resu l t s for an Unexci ted S l i p p i n g Synchronous Machine 23 3.8 S l i p and S ta to r Current Envelope f o r an E x c i t e d S l i p p i n g Synchronous Machine 24 3.9 Comparison of Computed and Test Resu l t s f o r an E x c i t e d S l i p p i n g Synchronous Machine 25 4 .1 The R i g i d Pendulum 27 5.1 Phase-Plane T r a j e c t o r i e s f o r Constant F i e l d and f o r Optimal C o n t r o l E x c i t a t i o n 35 5.2 Comparison of Computed and Test Resul ts . f o r Constant Vol tage E x c i t a t i o n 36 5.3 Comparison of Computed and Test Resu l t s f o r Optimal E x c i t a t i o n 37 V ACKNOWLEDGMENT Sincere thanks are due to the many people who have a s s i s t e d i n producing the r e s u l t s given here. In p a r t i c u l a r , I would l i k e to thank the pr o j e c t supervisor, Dr. Y.N. Yu, f o r h i s continuous i n t e r e s t , patience and guidance. Several members of the power group are al s o acknowledged f o r t h e i r many h e l p f u l d i s c u s s i o n s , and t h e i r c a r e f u l proof-reading of the f i n a l d r a f t . I am g r a t e f u l f o r the e f f o r t s of Mr. W.R. B l a c k h a l l , who prepared the model t e s t s . I wish to thank Dr. E. V. Bonn and Dr. H. Chinn f o r reading the d r a f t , and f o r t h e i r valuable comments. The f i n a n c i a l assistance of the N a t i o n a l Research Council i s g r a t e f u l l y acknowledged. NOMENCLATURE General f n-dimensional s t a t e v a r i a b l e equation ve.ctor H system Hamiltonian k constant used f o r c o n s t r a i n t i n c o n t r o l problem i ' ^ . a s t a t e v a r i a b l e P n-dimensional costate vector p d/dt t time t~£ f i n a l time f o r c o n t r o l study u c o n t r o l v a r i a b l e x n-dimensional s t a t e v a r i a b l e Pendulum Analogy Parameters and Va r i a b l e s I moment of i n e r t i a .K damping c o e f f i c i e n t K c c r i t i c a l damping c o e f f i c i e n t L 'maximum r e s t o r i n g torque T ap p l i e d torque Synchronous Machine Parameters and. Va r i a b l e s A,E,C, machine swing equation c o e f f i c i e n t s a-^ , a^, b-^  , bp constants of s t a t o r e x c i t a t i o n component s t a t o r d-axis current C1' C2 , <^1 ,^2 constants of s t a t o r e x c i t a t i o n component s t a t o r q-axis current D machine damping c o e f f i c i e n t D c c r i t i c a l damping c o e f f i c i a n t i - , , i d and q-axis s t a t o r currents d' q 1 V l l i ., , i d and q-axis s t a t o r e x c i t a t i o n components of a a q a s t a t o r current i - , f , i ,, d and q-axis f i e l d e x c i t a t i o n components of s t a t o r current M r o t oi' momentum Pa s t a t o r excitation.component of e l e c t r i c a l " power output P^ t o t a l e l e c t r i c a l power output P f i e l d e x c i t a t i o n component of e l e c t r i c a l power output P. mechanical power input s per-unit s l i p V R.M.S. phase-phase te r m i n a l voltage V ,V ,V a,b and c-phase-neutral voltages a' b' c ' r D V J,,V o,d and q a x i s voltage at the machine terminals o' d' q ' ^ to V-p f i e l d voltage r e f l e c t e d to the s t a t o r V f i e l d voltage (3 power angle w i t h respect to the q-axis 6 power angle w i t h respect to the d-axis y ,, y d and q-axis f l u x l i n k a g e s a q ^ 9 angle between r o t o r d-axis and s t a t o r a-winding Synchronous Machine Parameters H machine i n e r t i a constant J moment of i n e r t i a R armature r e s i s t a n c e R-n f i e l d r e s i s t a n c e F d-axis short c i r c u i t time constant d-axis open c i r c u i t time constant T^'jT^" d-axis short c i r c u i t t r a n s i e n t and subtransient time constants ''"'"do" d-axis open c i r c u i t t r a n s i e n t and a 0 0 subtransient time constants T" q-axis short c i r c u i t subtransient time q qo constant T" q-axis open c i r c u i t subtransient time constant X,,X d and q-axis synchronous reactances d' q 1 . INTRODUCTION 1 1 .1 The? Resynchronization. Problem In the past,- l o s s of synchronism of a synchronous machine i n a power system has r e s u l t e d i n an i n t e r r u p t i o n of s e r v i c e because of the separation of that machine from the system. This could cause a serious problem i n a larg e system. With the advance of system technology, many studies have been d i r e c t e d at i n c r e a s i n g the s t a b i l i t y l i m i t s . Less of synchro-ism and r e s y n c h r o n i z a t i o n , however, s t i l l represent serious problems and have received renewed i n t e r e s t i n recent years. P a r k , i n 1929» derived current and torque equations f o r a synchronous machine during s t a r t i n g . Edgerton and Four-marier, i n 1931» i n v e s t i g a t e d the r e s y n c h r o n i z a t i o n process, and found that the response of a machine was dependent upon a r e l a t i v e (3) damping term. In 1947 , Concordia and Temoshok, using a d i f f e r e n t i a l analyzer, examined the system impedance e f f e c t on s e l f - s y n c h r o n i z a t i o n . More r e c e n t l y , Malik and C o r y ^ ^ 1966 , simulated a synchronous machine on an analog computer and compared the analog r e s u l t s with those from micrcmachine • (5) t e s t s . In 1969 , Lay* et. a l . developed equations f o r a s l i p -ping synchronous me.ch.ine from Park's equations, with s e v e r a l approximations, f o r the re s y n c h r o n i z a t i o n study of a cr o s s -compound t u r b o a l t e r n a t o r s e t . 2 1. 2 Thesis Approach In t h i s t h e s i s , a new approach to the problem i s taken. A new set of equations f o r a s l i p p i n g synchronous machine i s developed i n Chapter 2. The f o l l o w i n g assumptions are made. 1. The machine i s i d e a l as defined by Park. 2. The s l i p i s small and v a r i e s slowly. The armature r e s i s t a n c e i s a l s o considered to be s m a l l . As a r e s u l t , a l l second and higher order terms of armature r e s i s t a n c e and s l i p are neglected-. A dynamic t e s t model developed at U.B.C. ^ ^ ^ ^ ^ f o r power-system s t a b i l i t y and c o n t r o l studies i s used f o r r e a l machine t e s t s . In a d d i t i o n to the e x i s t i n g system, • d i g i t a l devices f o r measuring s l i p and torque angle are developed. Small s l i p t e s t s on the dynamic model are compared wi t h computation. The r e s u l t s are given i n Chapter 3« In Chapter 4 , the concept of c r i t i c a l damping i s defined. From the equation f o r the s l i p p i n g synchronous machine, the c r i t i c a l damping i s c a l c u l a t e d and implemented on the dynamic t e s t model to see i f the machine w i l l resynchronize or not i f over or underdamped. F i n a l l y , i n Chapter 5 , an optimal c o n t r o l f o r r e s y n c h r o n i z a t i o n i s c a l c u l a t e d and implemented. 2 . STATE EQUATIONS OP A SLIPPING STTNCHRONOUS MACHINE 3 2 . 1 C o- ord ina. te Sys t em (1) Applying Park's transformation, the s t a t o r phase voltages and cur-rents of a synchronous machine can be expressed i n d and q components. 'The d and q axes can be f i x e d to the s l i p p i n g r o t o r , t u r n i n g at an angular v e l o c i t y where co CO w e ( l - s ) e l e c t r i c a l angular v e l o c i t y ( 2 . 1 ) (radians/sec.) cog = e l e c t r i c a l synchronous speed (120 JC radians/sec.) s = s l i p , or per-unit r e l a t i v e speed of the r o t o r w i t h respect tc the s t a t o r r e v o l v i n g f i e l d . Let V be the R.'M.S. linve v o l t a g e , and V a V. b V = c 2 V cos(to t ) 3 6 V cos (co t - 2TC ) 6 3 " ( 2 . 2 ) cos(co t - Q% ) e 5 be the phase voltages. The transformation matrix of Park (9) (1) modified by Y u ^ 1 has the form a b I'd q cos 6-- s i n -Q-1 / / 2 cos (-0--sin(-0-2TC) 3 3 cos 3 is) 3 ( 2 . 3 ) 4 where O i s the angle between the r o t o r d - ax i s and the s t a t o r a -winding a x i s . A p p l y i n g the t r ans format ion (2.3) to the vo l t ages (2.2) y i e l d s V = 0. o V d = V c o s(e - w t ) (2.4) V = V s i n(e - w t ) q e The zero a x i s component e x i s t s only for the a s y m e t r i c a l case and can be sepa ra t e ly ana lyzed . Let 5 be the angle by which the r o t o r d - ax i s leads the s t a t o r r e v o l v i n g f i e l d . 6 = -e - w e t (2.5) Then we have V = 0 . o V d = V cos 5 ' ( 2 . 4 a ) V = - V s i n 5 We a l s o have 6 = - sw g (2.6) since •9- = co (1 - s) (2.7) e The angle 5 may be determined by i n t e g r a t i n g the r e l a t i o n ( 2 . 6 ) 2.2 S t a t o r C u r r e n t s P a r k ' s e q u a t i o n s f o r a synchronous machine are V, r= pH< - Ri., - cow d ^ • d d q q ^ q q a where and x d ( p ) G-(p) Y d = - u i d + — V P e e X a ( p ) q w e q v = _ _A i x . ( P ) = { 1 + T d P ) ( l + TdP>*c d ( 1 + T d o p ) ( l + T 3 o p ) G(p) (1 + T D l p )  (1 + T d o p ) ( l + T J Q p ) (2.8) (2.8a) / N (1 + T"p)x x q ( p ) = gJL_SL ( 2 > 8 b ) (1 + T ^ p ) XAT) V - —-Y V P - R p V f d S e v e r a l assumptions are made i n the f o l l o w i n g t o s i m p l i f y the a n a l y s i s . The f i r s t a ssumption i s t h a t the s l i p v a r i e s s l o w l y , w h i c h r e s u l t s i n s i n u s o i d a l d and q v o l t a g e s . The s t e a d y s t a t e s o l u t i o n can thus be found by s e t t i n g P' = - J s w e (2.9) 2 The second assumption i s that since the s l i p i s s m a l l , a l l s terms may be neglected. Equations (2.8b) become (1 + T,p) x,(p) = x, (1 + T 2p) a (1 + T"p) x (p) = ~ q — x (2.8c) * (1 + T" p)1 qo^ G(p) (1 + T J J J P ) (1 + ^ 2P) T, = T' + T" 1 d d (2.8d) TV, = T' + T" 2 do do The t h i r d assumption i s that the armature r e s i s t a n c e R i s small 2 and, as a r e s u l t , R and Rs terms w i l l be neglected. The s t a t o r current s o l u t i o n s of equations (2.8) are d i v i d e d i n t o two p a r t s : d da df i - i + i „ q qa. qf where i ^ and i ^ are the s t a t o r current components r e s u l t i n g from a. f i e l d e x c i t a t i o n , and the current components i ^ and i ^ a are f u n c t i o n s of the s t a t o r voltage and torque angle. The cur-rent expressions found i n the Appendix are 7 d a (Rco + px )V e 1 q d (1 + T 2p)V Q (1 - s ) 2 [ l + + ^ ' ) p ] w e x d x q (1 - s ) ( l + T l P) x d (1 + T" p)V ( R c oe + P x d } Vf, (2.10) ^ a (1 - s ) ( l + T"p)x q (1 - s ) 2 [ l + ( T 2 + T»)p]we x w x d q "df I (1 + T p ip) (1 + T l P ) x d " V F (2.11) RV F (1 - s) (1 + ( T 2 + + T 2 ) p : ) x d x ( i S u b s t i t u t i n g (2.4a) i n t o (2.10), d i f f e r e n t i a t i n g and s i m p l i f y i n g the r e s u l t y i e l d s i d a a V [ ( a i + a 2 s ^ s i r l |5 + ("b-^  + b 2s) cos 5J (2.12) x "qa ^[(c-^ + c 2 s ) s i n 6 + (d-^ + d 2s) cos 5] where a l - (1 - s): d^ a, 1 (1 - s)' vd b l = -R ( l - s ) 2 x , x d q R - ( T 2 - T l ) W e (1 - s) X d (T" - T")co qo q e (2.12a) 8 cL = — 1 d - 1 1 (1 - s ) x q 2 (1 - s ) 2 x q For d e t a i l s see the Appendix. As the voltage Vp i s u s u a l l y not s i n u s o i d a l , the approach used thus f a r cannot be a p p l i e d . The s t a t e equations f o r i ^ and i £ are derived as f o l l o w s . From (2.11) we have x d f x (1 + T D 1 p ) V p T 1 x d • (P + ^ l -R V F qf - x,x T (1 - s i l d A < i A o u " b J (p + 1/T Q) where T - T + T + T" o 1 + 2 + q which can be w r i t t e n i n the s t a t e v a r i a b l e form : , _ V F ^ f cif - T,x, T, I d 1 • _ 1 ( R V F _ . x (2.11a) (2.13) where x d f = ^ f + T l x d (2.13a) 2.3 Swing Equation The mechanical torque equation of a r o t a t i n g device may he w r i t t e n n2w (2.14) J ^ - T 2 - a dt' where Then 2 J = moment of i n e r t i a of the r o t o r (kg - m ) 0 = the mechanical angle of the r o t o r (radians) ? 2 -2-T &= the net a c c e l e r a t i n g torque (kg - m - sec co t + 6 0 = 6 (P/2) 1 at' d t 2 (2.15) where P/2 = the number of pole p a i r s 6 = the power angle ( e l e c t r i c a l radians) S u b s t i t u t i n g the r e s u l t s i n t o (2.14) and m u l t i p l y i n g both sides of the equation by com, the mechanical synchronous speed co m u e/(P/2) y i e l d s the swing equation M = P ' d t 2 a c where the angular momentum M = Jwm/(P/2) and the a c c e l e r a t i n g power Kr, = T CO ac a m (2.16) (2.17) ( j o u l e - s e c / e l e c t r i c a l radian) (2.17a) (watts) (2.17b) 10 Neglecting mechanical damping, the a c c e l e r a t i n g power may be w r i t t e n P = P. - P (2.18) ac m e where P^ n i s the power supplied to the machine by the prime mover and P g -is the e l e c t r i c a l power output. (1) The power output of a synchronous machine i s given by P e = w ( ? x I ) (2.19) Expanded i t becomes P e = ( V d + W + R ( ± c l 2 + \ 2 ) + (- p V d " ( 2 ' 1 9 a ) The terms on the r i g h t hand side of (2.19a) represent the te r m i n a l power, the i n t e r n a l r e s i s t i v e l o s s e s , and the change i n i n t e r n a l stored energy r e s p e c t i v e l y . The l a s t two terms are u s u a l l y neglected r e s u l t i n g i n P = V i + V i (2.19b) e d d q q Let P be separated i n t o two terms e x P e = P a + P f (2.20) where P a = V a a + V q a ( 2 ' 2 0 a ) E f = V d f + V q f ( 2 - 2 0 b ) S u b s t i t u t i n g (2.12) i n t o (2.20a) r e s u l t s i n v2 P_ .= | I - B a s i n 25 + (t>2 - c 2 ) s + (b + c 2 ) s cos 25 a 9 + ( b 1 - C l ) l (2.21) where 11 B = °-The value of P^ can be found a f t e r s o l v i n g f o r i ^ and ± ^  i n (2.13) and (2.13a). S u b s t i t u t i n g (2.20) i n t o (2.17) y i e l d s M d ! & = P ± n - ( P a + P f) (2.17o) d t ^ S u b s t i t u t i n g (2.20a) and (2.20b) i n t o (2.17c) y i e l d s the second order swing equation. M d2^. + Ad 0 + B = C (2.22) ,,2 dt dt where 9 ( b 0 + c„) A = f (c2 - ^ + T£—£ cos 26) 2 o J % 2 2 (2.22a) V B = - 2 B a s i n . 25 + V i d f c o s 5 ' - Virgin 6 C = P.. - V 2 (b t - c,) i n 2 i 1 When the machine i s i n a steady sta t e c o n d i t i o n a f t e r synchro-n i z a t i o n , o 6 = 0 6 = 0 S e t t i n g R = 0 P = 6 - it/2 and s u b s t i t u t i n g i n t o (2.22) y i e l d s P. = Y V P s i n " + Y W " _ f o l s i n 2(3 m r — a — ' x-, 2x-,x d d q a f a m i l i a r r e s u l t . 12 2.4 State Equations of a Synchronous Machine f o r Resynchronization Studies The s t a t e equations f o r i ^ , and i Q f . V P df Ldf - I v x , T, i d 1 qf - T, x,x ( i -I d q' - l qf (2.13) "df l d f + V a (2.13a) and the swing equation ,2 M~72 + i t + B = C dt • (2.22) are f i n a l l y w r i t t e n i n the st a t e v a r i a b l e form • kt' 0 0 0 " 0 ' 0- -1/T ' o 0 0 V + 0 6 0 0 0 1 6 0 6 . 0 0 0 -A/M • .6. - C-B . M J .1 T l x d R T x , x I l-s .o d q "0 "0 V F (2.23) 3 . COMPARISON OF COMPUTATION AND DYNAMIC MODEL TESTS In Chapter 2, a mathematical model f o r a s l i p p i n g machine was developed. ' To compare r e s u l t s from dynamic model t e s t s w i t h computed values, d i g i t a l devices to measure torque angle and s l i p have been designed and constructed. Comparisons are made between dynamic model t e s t r e s u l t s and computational r e s u l t s at a small s l i p , w i t h and without e x c i t a t i o n . 3.1 Torque Angle - Transducer A new torque angle transducer with a r e s o l u t i o n of 2 / 3 e l e c t r i c a l degrees i s designed and constructed. The torque angle 6, defined by the d i f f e r e n c e between the r o t o r angle and the s t a t o r voltage angle, i s found by d i g i t a l l y measuring the time between the zero crossings of the a-phase s t a t o r v o l t a g e , and that of a voltage generated i n an a u x i l i a r y machine on the t e s t model s h a f t . The two s i g n a l s are reduced, using a voltage d i v i d e r , to a l e v e l s u i t a b l e f o r the l o g i c . The r e s u l t i n g low l e v e l s i g n a l s d r i v e a pa.ir of comparators ( F a i r c h i l d uL710) , which convert the p o s i t i v e p o r t i o n of the s i n u s o i d a l waves to square s i g n a l s , ( (l) , (2) Figure 3«l) which i n tu r n d r i v e a p a i r of short duration monostable m u l t i v i b r a t o r s . The two r e s u l t i n g s i g n a l s ( (3) & (4) Figure 3 - i ) are pulses at each p o s i t i v e going zero c r o s s i n g . The a u x i l i a r y machine s i g n a l d r i v e s the set input of an R-S f l i p - f l o p , and the synchronous machine ter m i n a l voltage s i g n a l d r i v e s the reset input. The output of the R-S f l i p - f l o p i s a square s i g n a l ((5) Figure 3 - l ) , the dur a t i o n of which i s the time between the two zero c r o s s i n g s . R-S FLIP FLOP MONOSTABLE © LJ Figure 3«1- Zero Crossing Detector Since the time d u r a t i o n of the output s i g n a l , (5), i s propor-t i o n a l to the torque angle 5 , i t i s measured by gating the s i g n a l from a clock i n t o the angle counter. . The clock • 9 operates at 30.72 kHz r e s u l t i n g i n 512 or 2 pulses per cycle at 60 Hz. F i g . 3-2 Angle Counter- and Reference Angle Subtractor 16 As the a u x i l i a r y machine i s not coupled to the synchronous machine at a s p e c i f i c angle, i t i s necessary to add a constant reference angle to determine 5 • A device was designed to per-form t h i s by counting at 2.5 mHz. , a fixed, number of pulses i n t o the angle counter immediately a f t e r the angle count has been corn-completed (Figure 3 . 2 ) . The input s i g n a l (5) ga.tes the 30.72 kHz cl o c k pulses i n t o the 9- b i t angle counter. When the gating s i g n a l (5) becomes zero, i t stops the count, and toggles a J-K f l i p - f l o p , s t a r t i n g the reference angle count. This i s counted i n t o both the angle counter, and an a u x i l i a r y 9 b i t counter. V/hen the a u x i l i a r y counter readout equals the pre-set value on the reference angle swit c h , a pulse i s generated to toggle the J-K f l i p - f l o p again, stopping the count. The same pulse (?) toggles the output r e g i s -t e r , r e p l a c i n g the old values of (S) with the updated value (&). 3•2 S l i p Sensing The value of s l i p i s computed from the angle over a period of time. Since 6 = - sw e and • 6 (t) - 5(t - At) 6 = A(t) Then s = -~JT (6(t - At) - 6 ( t ) ) e The r e s o l u t i o n of the s l i p sensor i s determined from the number of pulses read i n a given time At. Since the value of At i s set at 6 c y c l e s , and each cycle i s resolved i n t o 512 p o r t i o n s , the 17 r e s o l u t i o n i s l . / ( 5 1 2 x 6 ) p.u.. The s u b t r a c t i o n i s c a r r i e d out using one's complement a r i t h m e t i c , by complementing the l a t t e r term and then adding. The c i r c u i t i s b u i l t around a binary adder (Figure 3 o ) which c a r r i e s out the necessary a d d i t i o n . X Y X-Y F i gure 3• 3 Binary Adder The sequence i s a c t i v a t e d by a cycle counter, which counts to s i x and resets. (Figure 3 - 4 ) . I t i s d r i v e n by the reset pulse (8) of the torque angle transducer c i r c u i t (Figure 3 . 2 ) . At the count of s i x , i t generates a pulse, which toggles the new value 18 of s l i p i n t o the s l i p r e g i s t e r , and then sets the storage r e g i s t e r equal to the l a s t torque angle r e g i s t e r output (Figure 3 . 5 ) . 3 . 3 D i g i t a l Readout A v i s u a l d i g i t a l readout f o r torque angle and f o r s l i p are constructed. Binary to decimal conversion i s done by means of binary to pulse converter, ( 2 . 5 mHz). A s e r i e s of decade con-v e r t e r s are used to d r i v e the readout devices. 3 . 4 Dynamic Test Model The synchronous ma.chine of a dynamic t e s t model developed f o r power system s t a b i l i t y and c o n t r o l studies i s used f o r a l l machine t e s t s of t h i s p r o j e c t . The machine i s set up to simulate a t y p i c a l one m a c h i n e - i n f i n i t e bus power system on the same per-(8) u n i t b a s i s . The base power P-^  i s found as Jco2 P b = = : 6 ^ watts f o r where H = 4 . 6 3 sec co = m 3 7 , / 2 = mechanical synchronous speed (rad/sec) H = the i n e r t i a constant (sec) 2 2 J =r . 1 6 5 = moment of i n e r t i a ( j o u l e sec /rad ) The base impedance i s chosen = 1 4 . 4 ohms Then, the base voltage found frcm b b b 19 * OUTPUT Figure 3-4 Six Cycle Ccuuter SIX CYCLE COUNTER MONO-STABLE SLIP REGISTER OUTPUT f i g u r e 3.5 S l i p Sensor 20 i s V. =.- 100 v o l t s b Since the ohmic value of x-, and x of the machine are d q x d = 14.5 X q - 9 - 5 the per-unit values become x^ p.u. = 1.01 x p.u. = 0.64 q * 3.5 S l i p Tests To check the v a l i d i t y and accuracy of the mathematical model, two small s l i p t e s t s are done at p o s i t i v e s l i p with the mechanical input set at P. = -392 watts m P ± n p.u. = - 392/633 = - 0.62 p.u. The t e s t r e s u l t s are shown i n Pigure 3.6. The s t a t o r current envelope i s approximately a s i n u s o i d of double s l i p frequency because of the s a l i e n c y , and the machine speed o s c i l l a t e s at the same frequency. The maximum current corresponds approximately to the smallest s l i p . The r e s u l t s are compared, w i t h those p r e d i c t e d using the stat e equations (2.23) i n Pigure 3-7, where ,. . f? (.2 .2x1/2 M =* h ( i d + v The•second t e s t i s done w i t h the same mechanical lo a d , but w i t h a f i e l d e x c i t a t i o n of ten v o l t s , which corresponds to V p = 34 v o l t s The machine does not synchronize, and the t e s t r e s u l t s are shown 21 i n Figure J.8. Again the double s l i p frequency component i s observed, but i t i s d i s t o r t e d by a s l i p frequency component due to the d.c. e x c i t a t i o n . This phenomenon r e s u l t s from the f a c t that the unexcited r o t o r tends to l i n e up any of i t s poles with the s t a t o r mmf, while the e x c i t e d r o t o r w i l l l i n e up only every other pole w i t h the s t a t o r mmf. Since i , and i are s i n u s o i d a l q.a qa at s l i p frequency f o r the unexcited case, the transformation r e s u l t s i n double frequency components. A d d i t i o n of constant i,_P and i „ r e s p e c t i v e l y to i - , and i r e s u l t s i n both f i r s t df qf ^ J da qa and second harmonic terms. The r e s u l t s a.re compared w i t h those p r e d i c t e d using the s t a t e equations (2.23) i n Figure 3«9-23 / COMPUTED VALUE 2 DYNAMIC TEST VALUE ~f 1 i 1 r .2 .4 .6 .8 1.0 1.2 TIME (SECONDS) TIME (SECONDS) Figure 3.7 Compa.rison of Computed and. Test Results f o r an Unexcited S l i p p i n g Synchronous Machine 25 TIME 7 SECONDS) ~\ 1 1 1 1 I I O .2 .4 ,6 .8 1.0 1.2 TIME (SECONDS) Figure 3.9 Comparison of Computed and Test Results f o r an Excit e d S l i p p i n g Synchronous Machine t 26 4. CRITICAL DAMPING FOR RESYNCHRONIZATION The mathematical model f o r a s l i p p i n g synchronous machine was developed i n Chapter 2. Computational r e s u l t s were compared with those d i r e c t l y obtained from dynamic model t e s t s i n Chapter 3. In t h i s chapter, attempts are made to p r e d i c t the c r i t i c a l damping and to optimize the e x c i t a t i o n c o n t r o l f o r r e s y n c h r o n i z a t i o n . 4•1 C r i t i c a l Damping f o r Resynchronization An accurate p r e d i c t i o n of conditions required f o r synchro-n i z a t i o n r e q u i r e s the s o l u t i o n of 2 M ^ + A | r 5 + B = C (2.22) dt^ a Z ( ? ) An exact s o l u t i o n i s d i f f i c u l t . Edgerton and Pourmarier used a s i m p l i f i e d equation and found that the response of a synchronous machine depended upon a r e l a t i v e damping constant. A c r i t i c a l damping which determines whether or not a machine w i l l synchronize by i t s e l f i s determined as f e l l o w s . Neglecting smaller terms, equation (2.22) i s w r i t t e n 2 M ^ + D ( l + r c o s 2 5 ) | | + V i d f c o s 5 = C (4.1) where D = | H c 2 . b 2 ) . . (4.1a) e ( b 2 + c 2 ) r = Tb 2 - c 2) 27 Since the torque angle i s u s u a l l y def ined as the angle between the t e r m i n a l vo l tage and a vo l tage i n the r o t o r q - a x i s , not tha t i n the d - ax i s as & was def ined i n Chapter 2, l e t (3 '= 6 - Jt/2 (4.2) and be s u b s t i t u t e d i n t o (4-1-). Cne has ,2, M - | + D ( l - rcos 2 p ) f f + V i H - p s i n P =: dt "df , C (4 which y i e l d s a w e l l known steady s t a t e equa t ion . V i d f s inp = C (Note tha t (3<0 f o r a synchronous motor) . The equat ion (4.3) bears a c lose resemblance to the r i g i d pendulum equat ion ( 4 . 4 ) . ,2, I d i g d t 2 r/- dB . „ + dt + ^ s i n P T (4.4) where 1 K L T moment of i n e r t i a , damping c o e f f i c i e n t . mgl = weight x l e n g t h to centre of mass, a p p l i e d to rque . K a ) R i g i d Pendulum b) S t a b l e Eq u i 1 j 1) r i. u:a c) U n s t a b l e E q u i 1 i b r i u m F i g u r e 4.1 The R i g i d Pendulum 28 Steady sta t e s o l u t i o n s to (4.3) and (4-4) are analogous. For a steady sta t e to e x i s t w i t h time d e r i v a t i v e s zero, the i n e q u a l i t y that the maximum r e s t o r i n g power V j i ^ l must be greater than the magnitude of the a p p l i e d power C , v l i d f l > ° or i n the case of the pendulum 1 > ITI must be obeyed. Consequently, two s o l u t i o n s e x i s t f o r each equa-t i o n . The f i r s t s o l u t i o n w i l l be s t a b l e while the second w i l l not. The s o l u t i o n s are fi = s i n " 1 ^ 0 - ) O f ? p < TC/2 o v i d f o and JC/2 < it or B Q = sin _ 1(£) 0 ^ (3 o< tt/2 and %/2 < 8^ < % and are shown i n Figure 4.1. Hence, should the pendulum,- or the synchronous machine, under t r a n s i e n t conditions a t t a i n an angle greater than 8^, i t w i l l continue i n c r e a s i n g to an angle greater than it. However, a synchronous machine may a t t a i n an angle greater than TC/2, but l e s s than 8^ without l o o s i n g s t a b i l i t y . ' The term c r i t i c a l damping i s defined as the damping by which a system, s t a r t i n g from with a given•input, w i l l t u r n and'reach the c r i t i c a l angle p^ at a v e l o c i t y P = 0 An approximate value of p f o r the pendulum i s determined from energy c o n s i d e r a t i o n s . With no damping p = M (1 + cos p ) l / 2 (4.5) I Applying the i n i t i a l c o n d i t i o n s , 0 = -rc P '= 0 (4.6) and since the energy l o s s due to damping equals work done by torque plus the change 'in p o t e n t i a l energy, at the c r i t i c a l angle p' P' ^~p' Ko j Ko K c p dp = / T dp + 1(1 + cos p^) (4.7) -it J -it the c r i t i c a l damping c o e f f i c i e n t becomes \h K i t + J3') + L ( i + cos p' ) K c = S . 2_. ( 4 < ? a ) 4(1 + s i n po ) 2 For the synchronous machine D = M ™-j C ( J C + p') + B ( l + cos p') x d f l ° ° (4.7b) c ~ _ rK f 0 (1 + cos p ) 1 ^ 2 ( l - rcos 2p)dp By approximating P' = it Ko i t i s found 30 M 2TCC , » n _ 1 V l X a i _ 2 T C C _ I M , v " = 8 [ r + r ( ^ 7 5 - 3/5)] = 8 ( 1 - " . 1 3 r ) y v j± d f| T -P i do) I f r equals z e r o D. = 2g-° . A n f i = 0.780 ( 4 . c = 6 \/ YM = ' v v | i a f , To apply c o n d i t i o n ( 4 . 8 ) to a synchronous machine f o r synchro-n i z a t i o n , an e x c i t a t i o n must be ap p l i e d to decrease the c r i t i c a l • damping l e v e l so that i t becomes l e s s than the a c t u a l damping. V 2 D = 2 ^ - ( c 2 - b 2) ( 4 . 1 a ) e 4 •2 Numerical Examples The t e s t done i n Chapter 3 f o r the s l i p p i n g synchronous machine w i t h e x c i t a t i o n i s examined. Prom the t e s t data Vj, = 34 v o l t s V = 1 0 0 v o l t s C = - 4 5 0 watts I t i s found from ( 4 . 1 a ) that the a c t u a l damping V 2 D = ~—(c„ - b^) = 6 5 . 5 newton-meters e i s smaller than the c r i t i c a l damping T> 2irC / M n „ r . , D c = 8 ( 1 - . 1 3 r ) V V j i ^ l = 1 0 5 n e w t o n m e t e r s H e n c e t h e m a c h i n e d i d n o t s y n c h r o n i z e . A n o t h e r t e s t i s c a r r i e d o u t , i n c r e a s i n g t h e e x c i t a t i o n g r a d u a l l y t o f i n d t h e c r i t i c a l d a m p i n g r e q u i r e d f o r s y n c h r o n -i z a t i o n . The d a t a a t t h e moment o f s y n c h r o n i z a t i o n are a s f o l l o w s . V-p = 61 v o l t s V = 1 1 0 v o l t s C = - 4 5 0 watts The a c t u a l damping found from equation ( 4 - l a ) i s D - 7 9 » 4 newton-meters which i s greater than the c r i t i c a l value frcm equation ( 4 . 8 ) . D = 78 newton-meters As p r e d i c t e d , the machine synchronized s u c c e s s f u l l y without c o n t r o l . 32 APPLICATION OP OPTIMAL CONTROL The concept of c r i t i c a l damping f o r r e s y n c h r o n i z a t i o n i s to f i n d a constant voltage a p p l i e d to the f i e l d winding a l l of the time, to decrease the c r i t i c a l damping s u f f i c i e n t l y f o r the machine to resynchronize i t s e l f . E v i d e n t l y , to apply a constant voltage to the f i e l d winding of a synchronous machine i s not optimal. Therefore, optimal c o n t r o l may be a p p l i e d to cause sy n c h r o n i z a t i o n to occur i n an optimal f a s h i o n . I t may even allow s y n c h r o n i z a t i o n when s e l f - s y n c h r o n i z a t i o n by c r i t i c a l damping would f a i l . 5.1 Optimization Problem The problem i s to obtain an optimal c o n t r o l to minimize the performance f u n c t i o n J = &2' dt (5.1) J 0 subject to the c o n s t r a i n t x = f (5.2) where the x corresponds to the state v a r i a b l e s i ^ f > > 6 > o and 5, and f corresponds to the r e s p e c t i v e ' s t a t e equations. • As the c o n t r o l i s constrained, the performance f u n c t i o n (5.1) i s modified as J = T t f ( ku 2) dt (5.1a) J 0 where u i s the c o n t r o l v a r i a b l e . The value of k i s made small i n i t i a l l y , and i s increased during the computation. The equation (5.1a) i s w r i t t e n as a. st a t e v a r i a b l e where x = f (5-3) G O f = &2 + k u 2 The performance f u n c t i o n i s augmented by Lagrange mu l t i p l i e r to include the s t a t e equations as •t J f f ( p t x - H ) d t (5.4) J n where the Hamiltonian a . / Q H - - f P f + P f 4- P f + P f i o 1 1 + 2 2 4 3 3 4 4 At the optimum the f o l l o w i n g conditions are met x = f ( s t a t e equations) (5.2) P = -H, (costate equations) (5.5) H^ = 0 (gradient condition) (5.6) P(t^) = 0 ( t r a n s v e r s a l i t y condition) (5-7) A gradient method i s ap p l i e d to optimize the performance f u n c t i o n i n f u n c t i o n space. The s t a t e equations (5.2) are in t e g r a t e d forward from t = 0 to t ^ , w i t h an i n i t i a l value of u. Applying the t r a n s v e r s a l i t y c o n d i t i o n ( 5 . 7 ) , the costate equations (5-5) are i n t e g r a t e d backwards, using the values of x stored during the forward i n t e g r a t i o n . Values of H are — D & u c a l c u l a t e d and stored at each point on the backward i n t e g r a t i o n . At the end of the i n t e g r a t i o n , the values of H are used to D ' u cor r e c t the values of u. Several i t e r a t i o n s are made u n t i l an optimal, i s a,pproached. 34 5.2 A p p l i c a t i o n of the Method to the Resynchronization Problem The s t a t e equations c o n s i s t of (2.23) and x = f 0 o The costate equations (5-5) found are 1/T1 0 0 V - jyj cos X3 P 2 0 1/T 0 V s i n M p., P3 0 0 0 A' P 5 . P4. 0 0 1 - ^ ( l + rcos 2x^ .P4 + 0 0 0 L 2 x 4 . (5.5a) where • A' 4Dxz V 2B V i r s m x^ + 3 cos 2x df M Vx 3 M s i n x. 2 ]V[ cos x^  The c o n t r o l v a r i a b l e i s V p and hence u V-A case whei"-e the machine would not s e l f - s y n c h r o n i z e . i s s e l e c t e d f o r the o p t i m i z a t i o n study. A previous example i n d i c a t e d that the machine w i t h V = 110 v o l t s V-p = 61 v o l t s C = « 450 watts was j u s t c r i t i c a l l y damped. By s e t t i n g V p = 42 v o l t s the machine would not s e l f - s y n c h r o n i z e . The optimal c o n t r o l 35 described above i s a p p l i e d w i t h a c o n s t r a i n t value on V p of + 42 v o l t s . Computed phase-plane t r a j e c t o r i e s are p l o t t e d i n Figure 5-1 f o r cases with and without optimal c o n t r o l . Results from dynamic model t e s t s are compared with the computed r e s u l t s f o r the underdamped case i n Figure 5.2, and f o r the case w i t h optimal c o n t r o l i n Figure 5-3. / OPTIMAL CONTROL TRAJECTORY. 2. CONSTANT Vc TRAJECTORY. r TORQUE ANGLE (6) (RADIANS) F i g . 5.1 Phase Plane T r a j e c t o r i e s f o r Constant F i e l d and f o r Optimal Control E x c i t a t i o n O 1.0 2.0 3.0 TIME (SECOND) F i g . 5.2 Comparison of Computed and Test Results f o r Constant Voltage E x c i t a t i o n VA 1 F i g . 5-3 Comparison of Computed and Test Results f o r Optimal E x c i t a t i o n co 6. CONCLUSIONS The fundamental equations f o r a s l i p p i n g synchronous machine have been derived from Park's equations by approximating the s t a t o r currents as phasors at the s l i p frequency, and s e t t i n g p = - jsw , then n e g l e c t i n g higher order terms. The equations are a p p l i e d to compute the s l i p response of a t y p i c a l power system, and the t e s t result's compared favourably w i t h those obtained d i r e c t l y from dynamic model t e s t s . •A c r i t i c a l damping c r i t e r i o n f o r r e s y n c h r o n i z a t i o n i s developed from an analogy between a synchronous machine and a r i g i d pendulum, and i s v e r i f i e d by c a l c u l a t i o n and machine t e s t s . A nonlinear optimal c o n t r o l i s computed by a gradient method, and i s implemented on the t e s t model. Both computation and machine t e s t s i n d i c a t e that the synchronous machine can synchronize i n an optimal f a s h i o n even w i t h a f i e l d voltage of only 2/3 that required f o r c r i t i c a l damping. However, as the optimal c o n t r o l i s pre-computed and hence a p p l i e s only to cases where r e p e t i t i v e c o n d i t i o n s occur, i t would be d e s i r a b l e to have an on l i n e optimal c o n t r o l l e r . A P P E N D I X 40 S o l u t i o n f o r the Stator Current Components i d & and i q & S u b s t i t u t i n g (2.1) and (2.8a) i n t o Park's equations (2.8) y i e l d s 1 -v d V q • - ( : px d(p) co . + R) (1 - s)x (p) q •(1 - s)x,(p) - ( : ,p* (p) CO R) i d P / U S + i (1 - s) q or OJ = W H + DO &(p)vp The s o l u t i o n f o r I i s as f o l l o w s Li] = [A] vj - [B]-O(P)VJ The expression (A.3) suggests that the current s o l u t i o n s c o n s i s t of two sepa.rate components. Let x d a i qa = { A ] _ 1 [ V ] = - [ A ] _ 1 [ B ] G ( P ) V I I n v e r s i o n of the A matrix y i e l d s (1 + [T 2 + T»]p) - A -1 ^ 1 L A 0 01 x '  (1 - s ) 2 ( l + [T± + T^]p)x dx q p* (p) & ( P ) V F (A.l) ( A . 2 ) ( A . 3 ) (A.4) ( A . 5 ) co . + R) - (1 - s)x (p) (1 - s ) x d ( p ) -( ,px-,(p) CO + (A 4 1 S u b s t i t u t i o n of (A.6 ) i n t o (A.4 ) and (A.5) y i e l d s -(px + Rio ) V, ( 1 + T o P) Vq Q e d 2 ^ ( * n \ i d a = - (A. 7a) ( l - s ) 2 ( l + [T : + T^J p)co ex dx ( 1-s) ( 1 + T 1p)x d ( 1 + T" p) V, (px, + Rco ) V qo^ d ^ d e ao i = . - * (A.7b) qa o — ( 1-s) ( 1 + T"p) x ( 1-s) ( 1 + [T-, + T"J p)co x x q q L I q J ^ e d q M u l t i p l y i n g both numerator and denominator of the f i r s t term by ( l - T + T" p) , those of the second, term by J. q 2 ( l - T^p), and om i t t i n g the p terms y i e l d s _ - ( p x q + Rcoe V d _ ( 1 + [T 2 - T j p) q da — ( l - s ) co x,x " (l-s)x-, e d q d (A.8a) S i m i l a r l y 1 + [ T n n - T n I ^ V d ( ? Xd + R C ° P } V n i = I q ° q J 1 - d f ^ (A. 8b) qa -p ( l - s ) x ( l - s ) co x,x q e d q S u b s t i t u t i n g the voltages (2.4a) and (2.9) i n t o (A.8a) and. (A.8b) y i e l d s (2.12) 42 REFERENCES 1. Park, R.H., Two Reaction Theory of Synchronous Machines Generalized Method of A n a l y s i s - Part I, AIEE Transactions, V o l . 48, J u l y 1929, pp. 716-730. 2. Edgerton, H.E., and Fourmarier, P., The P u l l i n g i n t o Step of a Synchronous Motor, AIEE'Transactions, V o l . 50, J u l y , 1931, pp. 769-777. 3. Concordia, C , and Temoshok, M. , Resynchronizing of Generators,-AIEE Transactions, V o l . 66, pp. 1512-1518. 4. Ma l i k , O.P., and Cory, B.J., Study of Asynchronous Operation and Resynchronization of Synchronous Machines by Mathematical Models, IEE Proceedings, V o l . 113. No. 12, December, 1966, pp. 1977-1990. 5. l a y , R.K., l e n f e s t , E.H., Temoshok, M. and Winchester, R.L., E l e c t r i c a l Resynchronization of a Large Cross-Ccmpaund Turbine-Generator Set Part 1, A n a l y t i c a l Study. ' IEEE Trans-a c t i o n s , V o l . PAS-88, No.7, J u l y , 1969, pp.1137-1145• 6. Bond, J.A., A S o l i d State Voltage Regulator and E x c i t e r For a Large Power System Test Model, MASc. Thesis U n i v e r s i t y of B r i t i s h Columbia, J u l y , 1967. 7. S i d d a l l , R.G., A Prime Mover-Governor Test Model For Large Power Systems, MASc. Thesis, U n i v e r s i t y of B r i t i s h Columbia, January, 1968. 8. Dawson, G.E., A Dynamic Test Model For Power System S t a b i l i t y and Control Studies Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, December, 1969. 9. Yu, Y.N., The Torque Tensor of the General Machine, AIEE Transactions, V o l . 81, Part IV, 1963, pp 623-628. 10. Smolinski, W.J., C r i t i c a l Damping Levels of a Hydraulic Turbine Generator, The Engineering J o u r n a l , V o l . 52, No. 9, Sept., 1969, pp. I-VI 11. Kimbark, E.W., Power System S t a b i l i t y . V o l . 1, Wiley, 1948. 12. M a l i k , O.P., and Cory, B„J., Automatic Resynchronization of Synchronous Machines, IEE Proceedings, V o l . 118, No. 12, December, 1966, pp. 1972-1976. 13. Levine, D.L., Lay, R.K. , and Temoshok, M. , E l e c t r i c a l Resynchron-i z a t i o n of a Large Cross-Compound Turbine-Generator Set, Part I I F i e l d Tests, IEEE Transactions, V o l . PAS-88, No. 7, J u l y , 1969 pp. 1146-1150. 

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