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Analysis of waterhammer effect on speed change and governing stability of hydraulic turbines El-Fitiany, Farouk Abdalla 1978

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ANALYSIS OF WATERHAMMER EFFECT ON SPEED CHANGE AND GOVERNING STABILITY OF HYDRAULIC TURBINES by FAROUK ABDALLA EL-FITIANY B.Sc. (Hons.) ( C i v i l Engineering), University of Alexandria, Egypt, 1970 M.A.Sc. ( C i v i l Engineering), University of Alexandria, Egypt, 1973-M.Eng. ( C i v i l Engineering), University of Br i t i s h Columbia, Canada, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY " in • . THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA November, 19 78 ©Farouk Abdalla El-Fitiany, 1978 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 l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of 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 of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head of my department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l 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 . Department of C i v i 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 Vancouver, Canada V6T 1W5 Date CQ&z /0 . ABSTRACT Research S u p e r v i s o r : P r o f e s s o r E. Ruus A l i n e a r i z e d m a t h e m a t i c a l model f o r the a n a l y s i s o f t u r b i n e speed g o v e r n i n g problem i s p r e s e n t e d . T h i s model i n c l u d e s t h e e l a s t i c i t y e f f e c t s of b o t h water and p e n s t o c k w a l l s , the e f f e c t s o f t u r b i n e c h a r a c t e r i s t i c s l o p e s , and t h e l i n e a r i z e d e q u a t i o n o f a p r o p o r t i o n a l - i n t e g r a l h y d r a u l i c governor. The p r o c e d u r e can be extended t o a governor w i t h a d e r i v a t i v e term as w e l l . E x a c t , t r a v e l l i n g - w a v e s o l u t i o n s f o r t h e d i f f e r e n t p o s s i b l e c a ses a r e d e r i v e d and p r e s e n t e d i n s i m p l e forms. T a y l o r ' s e x p a n s i o n f o r m u l a s which o f f e r a more e f f i c i e n t a l t e r n a t i v e f o r n u m e r i c a l e v a l u a t i o n o f the speed a r e a l s o d e r i v e d . S o l u t i o n s i n b o t h cases take i n t o c o n s i d e r a t i o n p e n s t o c k f r i c t i o n l o s s e s i n a l i n e a r i z e d form, as w e l l as permanent speed droop ( o r speed r e g u l a t i o n ) and s e l f - r e g u l a t i o n c o e f f i c i e n t . These t r a v e l l i n g - w a v e s o l u t i o n s a r e then compared to n u m e r i c a l s o l u t i o n by t h e method o f c h a r a c t e r i s t i c s w i t h c l o s e agreement i n the r e s u l t s . T h i s new a n a l y t i c a l method i s then used t o stu d y the i n f l u e n c e o f e l a s t i c i t y on maximum speed d e v i a t i o n and on the s t a b i l i t y o f o s c i l l a t i o n s and to d e f i n e the l i m i t a t i o n of a r i g i d - c o l u m n a n a l y s i s . F o r high-head p l a n t s , e l a s t i c i t y of t h e water and p e n s t o c k w a l l s i n c r e a s e s maximum speed i f governor s e t t i n g s a r e kept the same. The major i n f l u e n c e o f t h e e l a s t i c waves, however, i s the r e d u c t i o n o f the s t a b i l i t y m a rgin f o r the speed v a r i a t i o n . Optimum governor s e t t i n g s w i l l be, t h e r e f o r e , d i f f e r e n t from those o b t a i n e d by n e g l e c t i n g e l a s t i c i t y of the water and pens t o c k w a l l s . i i TABLE OF CONTENTS ABSTRACT i i -LIST OF FIGURES v NOTATION v i ACKNOWLEDGEMENTS x i Chapter I . INTRODUCTION 1 I I . SYSTEM COMPONENTS AND EQUATIONS GOVERNING THEIR RESPONSES. 18 2.1 I n t r o d u c t i o n 18 2.2 System Components 18 2.3 T u r b i n e D i s c h a r g e and Torque E q u a t i o n s 22 2.4 E q u a t i o n o f M o t i o n f o r R o t a t i n g P a r t s 26 2.5 Governor E q u a t i o n . . 28 2.6 P e n s t o c k E q u a t i o n 35 I I I . SPEED DIFFERENTIAL EQUATION AND ITS SOLUTION 45 3.1 Gove r n i n g E q u a t i o n s 45 3.2 S o l u t i o n o f Speed E q u a t i o n 49 3.3 E v a l u a t i o n o f h . ( t ) , q . ( t ) and z ± ( t ) 55 3.4 Summary 57 3.5 S o l u t i o n w i t h l i n e a r i z e d f r i c t i o n 58 3.6 Approximate S o l u t i o n 60 3.7 Summary 64 IV. VERIFICATION OF THE TRAVELLING-WAVE SOLUTION AND APPLICATIONS 65 4.1 Comparison w i t h the Method of C h a r a c t e r i s t i c s ' S o l u t i o n 65 4.2 Comparison w i t h L a p l a c e - T r a n s f o r m S o l u t i o n 70 4.3 E f f e c t of F r i c t i o n L o s s e s i n the Pens t o c k 72 4.4 The E f f e c t s o f Permanent Droop a and S e l f -R e g u l a t i o n a 76 4.5 E l a s t i c i t y E f f e c t s on Maximum S p e e d - D e v i a t i o n and S t a b i l i t y - L i m i t 76 V.' SUMMARY AND CONCLUSIONS 83 i i i BIBLIOGRAPHY 85 APPENDICES A. E x a c t S o l u t i o n 89 B. Approximate S o l u t i o n 98 C. C o n s t a n t s 1 ° 2 i v LIST OF FIGURES No. 1. Two-Port R e p r e s e n t a t i o n of a F l u i d T r a n s m i s s i o n L i n e 10 2. G e n e r a l i z e d Two-Port R e p r e s e n t a t i o n o f a F l u i d T r a n s m i s s i o n L i n e 14 3. A D i s c r e t e Time Model f o r a D i s t r i b u t e d - P a r a m e t e r System . . . 16 4. B a s i c Elements of a Hydropower P l a n t 19 5. F u n c t i o n a l B l o c k Diagram of a Hydro P l a n t 23 6. C h a r a c t e r i s t i c s ' S l o p e s f o r a T u r b i n e O p e r a t i n g a t P o i n t of Best E f f i c i e n c y 27 7-a. A Schematic Diagram of a M e c h a n i c a l - H y d r a u l i c Governor . . . . 30 7- b. A Schematic Diagram of an E l e c t r i c - H y d r a u l i c Governor . . . . 31 8- a. M e c h a n i c a l Governor R e p r e s e n t a t i o n . 32 8-b. E l e c t r i c Governor R e p r e s e n t a t i o n 33 9. R i g i d Water-Column Model 36 10. G r i d P o i n t s f o r Method of C h a r a c t e r i s t i c s ' S o l u t i o n 41 11. A p p r o a c h i n g and R e f l e c t e d Waves f o r T r a v e l l i n g - W a v e S o l u t i o n 41 12. Speed V a r i a t i o n : A n a l y t i c a l and N u m e r i c a l S o l u t i o n s 68 13. Speed V a r i a t i o n : E f f e c t o f T r u n c a t i o n E r r o r i n T a y l o r ' s E x p a n s i o n 68 14. Head, D i s c h a r g e , and Gate V a r i a t i o n s 71 15. Speed V a r i a t i o n : Comparison w i t h o t h e r A n a l y t i c a l S o l u t i o n s 71 16. Speed V a r i a t i o n f o r Po = 0.7 > P o * 73 17. Head, D i s c h a r g e and Gate V a r i a t i o n s f o r pg = 0.7 > p0* . . . . 74 18. E f f e c t of Pens t o c k F r i c t i o n on Speed V a r i a t i o n 75 19. E f f e c t of Pens t o c k F r i c t i o n on Head V a r i a t i o n 75 20. E f f e c t o f Permanent Droop a and S e l f - R e g u l a t i o n a on Speed V a r i a t i o n 77 21-a,b E l a s t i c i t y E f f e c t on Maximum S p e e d - D e v i a t i o n 78 21-c,d E l a s t i c i t y E f f e c t on Maximum S p e e d - D e v i a t i o n . 79 22. E l a s t i c i t y E f f e c t on t h e S t a b i l i t y - L i m i t 81' 23-a. U n s t a b l e S p e e d - V a r i a t i o n 82 23-b. S t a b l e S p e e d - V a r i a t i o n 82 v NOTATION The f o l l o w i n g symbols a r e used i n t h i s t h e s i s : A = c r p s s - s e e t i o n a l a r e a o f t h e pe n s t o c k , a l s o a f r i c t i o n term, A = e x p ( - 2 R f / p 0 ) a = c e l e r i t y o f water'hammer wave a\ to a-jz = c o n s t a n t s i n terms of the system parameters B ± ( e ) , B^(0) = p o l y n o m i a l s o f 0 BN_^  = a c o n s t a n t i n t h e s o l u t i o n of the speed f o r i - 1 <_ t < i b. ., b! . = c o e f f i c i e n t s o f t h e p o l y n o m i a l s B . ( e ) and B ' . ( e ) , r e s p e c t i v e l y C ± ( e ) . , C^(6) = p o l y n o m i a l s o f 6 CN^, CH^, CQ^ = c o n s t a n t s i n t h e s o l u t i o n s o f t h e speed, the p i e z o m e t e r i c head, and the d i s c h a r g e , r e s p e c t i v e l y , f o r i — 1 <_ t <_ i c = a c o n s t a n t i n Eq. 28 c. ., e'. . = c o e f f i c i e n t s o f the p o l y n o m i a l s C . ( e ) and C ! ( e ) , 1,3 i , J i x r e s p e c t i v e l y D = p e n s t o c k d i a m e t e r ; a l s o a d i f f e r e n t i a l o p e r a t o r , D' = - j — , a d i f f e r e n t i a l o p e r a t o r . do d = • , a d i f f e r e n t i a l o p e r a t o r e = t h i c k n e s s of t h e w a l l s of t h e p e n s t o c k E = Young's modulus o f e l a s t i c i t y f o r the p e n s t o c k w a l l s F " = a p r e s s u r e wave t r a v e l l i n g a t the upstream d i r e c t i o n t o F 6 = f i r s t o r d e r d i f f e r e n t i a l o p e r a t o r s o f D F C = l i n e a r f r i c t i o n c o e f f i c i e n t , F ^ = g H ^ Q / ( L VQ) f = a p r e s s u r e wave t r a v e l l i n g a t the downstream d i r e c t i o n G l l > G12» G 2 i , G22 = f u n c t i o n s o f Z ( s ) and r ( s ) GH. = a c o n s t a n t i n t h e s o l u t i o n of the h e a d - d e v i a t i o n x f o r h ± ( t ) . g = a c c e l e r a t i o n due t o g r a v i t y v i g l t o gg = c o n s t a n t s i n terms of the system parameters H ( x , t ) = a b s o l u t e p i e z o m e t e r i c head a l o n g the p e n s t o c k H ( t ) = a b s o l u t e net p r e s s u r e head a t the t u r b i n e n = a b s o l u t e p r e s s u r e head l o s s due to f r i c t i o n ? , \ H(x,t) , . . , , h ( x , t ) = —7% , r e l a t i v e p x e z o m e t r i c head a l o n g the Hn0 p e n s t o c k \ H ( x , t ) - H(x,0) . . . t . . j J • 4- • h ( x , t ) = — 2 , r e l a t i v e p i e z o m e t n c head d e v i a t i o n Hn0 a l o n g the p e n s t o c k h ( t ) = h'X0,t) , r r e l a t i v e net p r e s s u r e head d e v i a t i o n a t the t u r b i n e , w i t h the t a i l water l e v e l as a datum I Q ( y ) , I 2 ( y ) = m o d i f i e d B e s s e l f u n c t i o n s j = r ^ i . K = b u l k modulus o f e l a s t i c i t y o f t h e water K,, K., K = d e r i v a t i v e , i n t e g r a l and p r o p o r t i o n a l g a i n s , a 1 p r e s p e c t i v e l y K! = K..T , 1 1 ch L = p e n s t o c k l e n g t h ; a l s o a second o r d e r d i f f e r e n t i a l o p e r a t o r of D I , , a , l = p a r t i a l d e r i v a t i v e s of t h e t u r b i n e output t o r q u e w i t h mh mn mz r e s p e c t to head, speed, and gate opening, r e s p e c t i v e l y I , , Z , H = p a r t i a l d e r i v a t i v e s of the t u r b i n e d i s c h a r g e w i t h qh qn qz r e s p e c t to head, speed, and gate opening, r e s p e c t i v e l y M(t) = a b s o l u t e output t o r q u e of t h e t u r b i n e M = h i g h e s t power i n the t r u n c a t e d T a y l o r ' s expansions M l 5 M 2, M 3 = c o n s t a n t s , M x = M + 1, M 2 = M + 2, M 3 = M + 3 M(t) - M Q m ( t ) = zz , r e l a t i v e d e v i a t i o n of the t o r q u e output 0 N ( t ) = a b s o l u t e speed of r o t a t i o n o f the t u r b i n e N ( t ) = TT1?- , u n i t speed u H-5 n n ( t ) = ^ ^ \ T — , r e l a t i v e d e v i a t i o n o f the speed NQ v i i P ( t ) = a b s o l u t e power output of t h e t u r b i n e P_ D 2 H" p P ( t ) = -r-pr , u n i t power n P ( t ) = " " n , r e l a t i v e d e v i a t i o n o f the power output ^0 Q ( t ) = a b s o l u t e t u r b i n e d i s c h a r g e Q D2. H 2 Q u ( t ) = — - — v , u n i t d i s c h a r g e n q ( t ) = — , r e l a t i v e d e v i a t i o n o f the t u r b i n e d i s c h a r g e Qo R = r a d i u s of g y r a t i o n of the combined mass of the g e n e r a t o r and t h e t u r b i n e ; a l s o speed r e g u l a t i o n c o n s t a n t f o r t h e e l e c t r i c governor H f 0 R = — — , r e l a t i v e head l o s s due to f r i c t i o n a t the f H n 0 i n i t i a l s t e a d y s t a t e R ± ( e ) , RH ± ( 8 ) , RQ^Ce) = p o l y n o m i a l s o f 6 r . ., r h . ., r q . . = c o e f f i c i e n t s o f the p o l y n o m i a l s R . ( e ) , RH . ( e ) , and i , j 1,3 i,3 x^'> l , . ( 8 ) . l . 1 RQ.(0), r e s p e c t i v e l y S , ( Q ) , SH . ( e ) , SQ ^ ( 9 ) = p o l y n o m i a l s o f 6 1-1 SM. = Z h. .(x.) 3=1 X ' 3 2 s. ., sh. ., sq. . = c o e f f i c i e n t s o f * S . ( 6 ) , S H . ( 6 ) and S O . ( 6 ) , r e s p e c t i v e l y 1,3 i , J i»3 i i l s = L a p l a c e ' s o p e r a t o r T = time, a b s o l u t e 2L . . . T , = — , c h a r a c t e r i s t i c time c o n s t a n t ch a T = governor response time T m 0 = m e c h a n i c a l s t a r t i n g time (at i n i t i a l s t e a d y s t a t e ) T mO T' = — — , r e l a t i v e m e c h a n i c a l s t a r t i n g time ml) T . ch v i i i T = p i l o t v a l v e l a g time P T = dashpot r e s e t time r T T^ , = ~— , r e l a t i v e dashpot r e s e t time ch L V 0 . . T „ = — 7 ; — , water s t a r t i n g time c o n s t a n t wO g H n Q T t = — — , r e l a t i v e time ch I L ( 8 ) = p o l y n o m i a l of 6 u. . = c o e f f i c i e n t s o f U.(8) i , J 1 V ( x , t ) = a b s o l u t e v e l o c i t y o f water i n the p e n s t o c k V\(9) = p o l y n o m i a l of 0 v ( x , t ) = ^ ^ ' ^ , r e l a t i v e v e l o c i t y of water i n t h e p e n s t o c k 0 V ( x , t ) - V 0 v ' ( x , t ) = =z , r e l a t i v e d e v i a t i o n o f the v e l o c i t y V 0 v ( t ) = v ' ( 0 , t ) = q ( t ) , r e l a t i v e d e v i a t i o n o f the v e l o c i t y a t t h e t u r b i n e v . = c o e f f i c i e n t s o f V.(0) 1 , 1 1 W = combined weight of t h e t u r b i n e and g e n e r a t o r r o t a t i n g masses w = u n i t weight of the water X = a b s o l u t e d i s t a n c e from the t u r b i n e i n l e t a l o n g the p e n s t o c k , p o s i t i v e i n the upstream d i r e c t i o n X , . x = — , r e l a t i v e d i s t a n c e YLD(s) = l o a d a d m i t t a n c e i n L a p l a c e ' s domain Z ( t ) = t u r b i n e gate opening r e l a t i v e t o t h e f u l l g ate opening Z ( s ) = c h a r a c t e r i s t i c impedance of the p e n s t o c k i n L a p l a c e ' s domain Z Ct ") — Z z ( t ) = - k , r e l a t i v e d e v i a t i o n o f t h e gate opening z 0 i x Greek l e t t e r s : = speed s e l f - r e g u l a t i o n c o e f f i c i e n t , a = a. - I I mn a = l o a d damping c o e f f i c i e n t alt c t 2 , a3, a[:, •' = i n t e g r a t i o n c o n s t a n t s r(s) = the p r o p a g a t i o n o p e r a t o r i n L a p l a c e ' s domain <5 . =• r e l a t i v e t r a n s i e n t speed droop A-i, A.2 = g o v e r n i n g parameters f o r an i d e a l impulse t u r b i n e , T T wO , . wO ^ 1 = "fj. - a n d , A 2 = -j,— mO r Al_. , X2_., X3j = c o e f f i c i e n t s i n the t r u n c a t e d T a y l o r ' s expansions v = k i n e m a t i c v i s c o s i t y of the water a Vo . - , . t ' PO •- o — 5 — ' p i p e l i n e o r A l l i e v i s c o n s t a n t 2 g H n 0 p = d e n s i t y of t h e water w J o = r e l a t i v e permanent speed droop 6 = C O T , a f u n c t i o n of the time t T = t - - i + 1, f o r i - 1 < t < i = / a 2 - 2a2. , a c o n s t a n t S u b s c r i p t s i = v a l u e s a t the time i n t e r v a l i - 1 <_ t < i 0 = i n i t i a l s t e a d y s t a t e v a l u e s x = p a r t i a l d e r i v a t i v e w i t h r e s p e c t to the d i s t a n c e x t = p a r t i a l d e r i v a t i v e w i t h r e s p e c t to the time t ACKNOWLEDGEMENTS The a u t h o r wishes t o ex p r e s s h i s s i n c e r e g r a t i t u d e t o P r o f e s s o r E. Ruus, t h e s i s s u p e r v i s o r , f o r gui d a n c e , s u g g e s t i o n s and c o n t i n u o u s encouragement. To the o t h e r members of the committee, P r o f e s s o r s Y. Yu, S.O. R u s s e l l , and W.F. C a s e l t o n , the au t h o r e x p r e s s e s h i s thanks f o r t h e i r v a l u e d a d v i c e and i n t e r e s t . S p e c i a l thanks go to B.C. Hydro, H y d r o e l e c t r i c D e s i g n D i v i s i o n f o r s u p p l y i n g some t e s t dat The a u t h o r i s a l s o i n d e b t e d t o the N a t i o n a l Research C o u n c i l of Canad and t o the U n i v e r s i t y o f B r i t i s h Columbia f o r f i n a n c i a l a s s i s t a n c e . x i CHAPTER I INTRODUCTION The problem of h y d r o - t u r b i n e g o v e r n i n g i s as o l d as the water wheel i n v e n t i o n i t s e l f . F o l l o w i n g a l o a d change, an unbalanced t o r q u e w i l l e x i s t due to the d i f f e r e n c e between t u r b i n e output and e l e c t r i c l o a d demand. The r o t a t i n g mass of g e n e r a t o r and t u r b i n e s t a r t s then to a c c e l e r a t e or d e c e l -e r a t e r e s u l t i n g i n a speed e r r o r . S e n s i n g such a d e v i a t i o n , the t u r b i n e governor comes i n t o a c t i o n and responds by i n c r e a s i n g or d e c r e a s i n g the gate opening t o r e g a i n t o r q u e b a l a n c e . F o r r u n - o f - r i v e r p l a n t s and' p l a n t s w i t h s h o r t p e n s t o c k s governor a c t i o n a c h i e v e s the r e q u i r e d change i n a v e r y s h o r t time w i t h no p a r t i c u l a r d i f f i c u l t y . F o r hydro p l a n t s w i t h r e l a t i v e l y l o n g p e n s t o c k s , speed r e g u l a t i o n i s c o m p l i c a t e d because i n e r t i a and the e l a s t i c i t y e f f e c t s o f the water column tend t o produce i n i t i a l output change o p p o s i t e t o t h a t o f the governor. Thus, t h e s e e f f e c t s always i n t r o d u c e a d e s t a b i l i z i n g i n f l u e n c e and t h e r e f o r e a d d i t i o n a l measures to i n c r e a s e the s t a b i l i t y margin must be c o n s i d e r e d . Moreover, p e n s t o c k t r a n s i e n t s s h o u l d be p r o p e r l y a n a l y z e d i f optimum speed c o n t r o l i s t o be rea c h e d . E l a s t i c i t y of water and penstock w a l l s g r e a t l y a f f e c t s b o t h phase and amplitude o f p r e s s u r e and d i s c h a r g e o s c i l l a t i o n s . T u r b i n e o u t p u t , hence speed d e v i a t i o n w i l l be a l s o a f f e c t e d . The main g o a l of t h i s t h e s i s i s t o i n c o r p o r a t e these e l a s t i c i t y e f f e c t s i n a l i n e a r model t h a t d e s c r i b e s t r a n s i e n t s of the speed, head, d i s c h a r g e , and gate r e s u l t i n g from a s t e p - l o a d change. For v e r y s m a l l l o a d changes, n o n - l i n e a r i t i e s caused by f r i c t i o n , dead band and l a g i n d i f f e r e n t governor and t u r b i n e components have to be c o n s i d e r e d whereas o t h e r k i n d s o f n o n - l i n e a r i t i e s , such as s a t u r a t i o n l i m i t s 1 2 and t u r b i n e c h a r a c t e r i s t i c s , must be i n c l u d e d f o r l a r g e changes e.g. t o t a l l o a d r e j e c t i o n . L i n e a r r e p r e s e n t a t i o n , commonly used i n p r e l i m i n a r y d e s i g n s to check governor s t a b i l i t y and e f f e c t i v e n e s s , i s l i m i t e d to l o a d changes between 2.0 and 10 p e r c e n t of i n i t i a l s t e a d y s t a t e l o a d , a p p r o x i m a t e l y . G o v e r n i n g of h y d r a u l i c t u r b i n e s i n g e n e r a l has been a f i e l d of i n t e n -s i v e s t u d i e s by e a r l y i n v e s t i g a t o r s [1 to 6 ] . The-'''most comprehensive study u t i l i z i n g a l i n e a r model i s perhaps t h a t made by P a y n t e r [7 to 9 ] . In f o r m u l a t i n g e q u a t i o n s of t h a t model, P a y n t e r and o t h e r s b e f o r e him assumed t h a t : 1) water and p e n s t o c k w a l l s a r e i n e l a s t i c ( r i g i d water column) 2) t u r b i n e e f f i c i e n c y s t a y s c o n s t a n t f o r s m a l l v a r i a t i o n s i n speed, head, and gate opening 3) e l e c t r i c l o a d i s p u r e l y r e s i s t i v e w i t h i n s t a n t a n e o u s v o l t a g e r e g u l a t i o n 4) a s i n g l e hydro u n i t s u p p l i e s power to an i s o l a t e d l o a d A l l r e s p o n s e s were assumed to be l i n e a r w i t h no l a g or dead band i n the governor, so t h a t a f i r s t o r d e r governor e q u a t i o n c o u l d be used. Curves of l i m i t i n g s t a b i l i t y and optimum governor s e t t i n g s were o b t a i n e d d i r e c t l y T T - , :'".w0 . , wO , ._ by u s i n g an a n a l o g computer, i n terms of = — and = — — - •where.-TWQ 1 S m r water s t a r t i n g time, 6 i s temporary speed droop, T .Q i s m e c h a n i c a l s t a r t i n g time and T r i s dashpot r e s e t time. Hovey [10,11] used the same model to get s t a b i l i t y l i m i t s a n a l y t i c a l l y . He made f u r t h e r s t u d i e s f o r optimum governor s e t t i n g s . Hovey and Bateman [12] c a r r i e d out f i e l d t e s t s f o r s m a l l l o a d changes a t the K e l s e y P l a n t , which sup-p l i e d an i s o l a t e d l o a d . The measured speed d e v i a t i o n s were always l e s s than 3 those computed. The i n t e r e s t i n g d i s c u s s i o n which f o l l o w e d t h i s paper sug-g e s t e d t h a t b o t h l o a d damping and I n i t i a l s l o p e s of t u r b i n e c h a r a c t e r i s t i c s were p o s s i b l e causes of these d i f f e r e n c e s . By i n c l u d i n g s e l f - r e g u l a t i o n and permanent speed droop, Chaudhry [13] showed t h a t the s t a b l e r e g i o n w i l l i n c r e a s e s u b s t a n t i a l l y . In a subsequent work by Thorne and H i l l [14] s t a t e - v a r i a b l e s t e c h n i q u e r a t h e r than Routh-Hurwitz c r i t e r i o n was used' to a n a l y z e s t a b i l i t y b o u n d a r i e s . They extended the l i n e a r model to c o n s i d e r the e f f e c t o f o p e r a t i o n p o i n t on s l o p e s o f t u r b i n e c h a r a c t e r i s t i c s f o r a K a p l a n t u r b i n e of The Mactaquac P l a n t i n New Brunswick. I t was found t h a t the s t a b l e r e g i o n w i l l be g r e a t l y reduced as the o p e r a t i o n p o i n t moves from 75 to 100 p e r c e n t of r a t e d l o a d . Comparison between f i e l d t e s t r e s u l t s and those o b t a i n e d by a c o n t i n u o u s system model-l i n g program (C.S.M.P.) was made f o r the same p l a n t [15]. The more g e n e r a l l i n e a r model used f o r t h i s purpose i n c l u d e d a b e t t e r r e p r e s e n t a t i o n of the l o a d by an e q u i v a l e n t synchronous machine, a p r o p o r t i o n a l - i n t e g r a l - d e r i v a t i v e governor as w e l l as t u r b i n e s l o p e s . E l a s t i c i t y e f f e c t s i n the p e n s t o c k were not c o n s i d e r e d , however. F i e l d r e c o r d s f o r a l o a d i n c r e a s e of about 28 p e r -cent of r a t e d v a l u e compared f a v o u r a b l y w i t h computed r e s u l t s . Though no i n f o r m a t i o n was g i v e n about p e n s t o c k l e n g t h , head, or d i s c h a r g e v a l u e s , i t i s w e l l known t h a t K a p l a n t u r b i n e s a r e used f o r low head, l a r g e d i s c h a r g e i n s t a l l a t i o n s f o r which e l a s t i c i t y e f f e c t s a r e l e s s pronounced. N u m e r i c a l s o l u t i o n s u s i n g d i g i t a l - c o m p u t e r s f o r g l o b a l n o n - l i n e a r models were develo p e d by s e v e r a l r e s e a r c h e r s . In h i s Ph.D. t h e s i s , Vaughan [16] d e v e l o p e d an a l g e b r a i c model f o r s t e a d y - s t a t e t u r b i n e c h a r a c t e r i s t i c s i n which b o t h head H and t o r q u e M were approximated by p o l y n o m i a l s of speed N, d i s c h a r g e Q and gate Z. V a l i d i t y of the model f o r g e n e r a l a p p l i c a t i o n was 4 not v e r i f i e d . Governor n o n - l i n e a r i t i e s were i n c l u d e d i n the computer program w h i l e assuming a r i g i d water column. Vaughan s t u d i e d a l s o e f f e c t s of changing o p e r a t i o n p o i n t u s i n g a l i n e a r m o d e l ' s i m i l a r t o P a y n t e r ' s . Optimum s e t t i n g s were o b t a i n e d i n terms of e q u i v a l e n t impulse t u r b i n e parameters X* = T / (6 x ), X* = x /T and H v * • l w e m 2 w r 1 3m (5m h y d r a u l i c d i f f i c u l t y , d e f i n e d by Vaughan as r = - ^ ' ^ z ^ ^ 6z^ ' where „_ //3h. r. o. ,/Snu , _ ,-. , n c Tw.6m 9m. N „ 3m X = 2T J (TT- ) , o = a/(-=—), and x = T (1 + 0.5 — (-j T r - ) ) - Here, -5— w wO 3q e oz m m T 6z 3z 3z m i s the p a r t i a l d e r i v a t i v e of t o r q u e w i t h r e s p e c t t o gate w i t h d i s c h a r g e "q" 5m as an i n p u t v a r i a b l e and head "h" as an output one. For — the head i s an i n p u t w h i l e d i s c h a r g e becomes an output v a r i a b l e . Optimum v a l u e s were g i v e n as: x i = i^h- and x*= ^ Vr +1 f <to ~ • He concluded that §ov-2 m e r n o r r e s p o n s e degrades w i t h i n c r e a s i n g gate or d e c r e a s i n g head even i f r e o p t i m i z a t i o n i s made. Though he i n d i c a t e d t h a t water hammer e f f e c t s may be d i s r e g a r d e d f o r i n s t a l l a t i o n s of l a r g e pg ( p i p e - l i n e or A l l i e v i ' s con-s t a n t ) v a l u e s , no thorough i n v e s t i g a t i o n was made. Wozniak [17] c o n s i d e r e d e l a s t i c i t y e f f e c t s i n h i s computer program by u s i n g an approximate i n v e r s e t r a n s f o r m of the impedance f u n c t i o n f o r a s i n g l e p i p e l i n e : * ^ S ^ = _ 2p tanh ( 4 s + F ) , where h and q a r e head and d i s c h a r g e q ( s ; u d a t t u r b i n e end o f the p i p e , F i s a l i n e a r f r i c t i o n f a c t o r , and s i s L a p l a c e ' s o p e r a t o r . T h i s approximate i n v e r s e t r a n s f o r m which was o b t a i n e d by T a y l o r ' s e x p a n s i o n o f s i n h y / c o s h y, where y = (^ s + F ) , does n o t , however, y i e l d s a t i s f a c t o r y r e p r e s e n t a t i o n of p e n s t o c k t r a n s i e n t s , as w i l l be e x p l a i n e d below. V a l i d i t y o f a l i n e a r model f o r s m a l l d i s t u r b a n c e s was checked by Wozniak and B l a i r [18] by comparing analogue computer r e s u l t s ( l i n e a r ) w i t h those of the g l o b a l d i g i t a l model. 5 A v a l u a b l e a n a l y s i s of h y d r o - p l a n t t r a n s i e n t s was made by a group of r e s e a r c h e r s i n The M a s s a c h u s e t t s I n s t i t u t e of T e c h n o l o g y . [ 1 9 ] . A d e t a i l e d computer program was develop e d i n two s t a g e s . The f i r s t s t a g e d i d not i n -c l u d e governor a c t i o n ; i n s t e a d the gate c l o s u r e curve f o l l o w i n g a t o t a l l o a d r e j e c t i o n was assumed. An i m p l i c i t f i n i t e - d i f f e r e n c e scheme was u t i l i z e d to s o l v e momentum and c o n t i n u i t y e q u a t i o n s of the e l a s t i c water column. R e s u l t s were i n v e r y good agreement w i t h b o t h the c h a r a c t e r i s t i c s method s o l u t i o n and f i e l d t e s t measurements a t G a r r i s o n and Oahe power p l a n t s . I n the second s t a g e , the governor e q u a t i o n was added to the program. I t was not p o s s i b l e , however, to c a r r y out f i e l d t e s t s f o r s m a l l l o a d changes because p l a n t s were s t r o n g l y connected to a l a r g e system whereas computations were based on the i s o l a t e d l o a d assumption. Comparisons w i t h f i e l d t e s t s f o r t o t a l l o a d r e j e c t i o n s were a l s o made by Ghaudhry [ 2 0 ] . C l o s e agreement w i t h computed r e s u l t s was o b t a i n e d f o r speed d e v i a t i o n and gate opening. S l i g h t d i f f e r e n c e s i n phase and damping a f t e r peak v a l u e , observed f o r head and d i s c h a r g e , were s u s p e c t e d to r e s u l t from i n a c c u r a c i e s o f t u r b i n e d a t a f o r s m a l l gate o p e n i n g s . Equa-t i o n s of the e l a s t i c water column were s o l v e d by an e x p l i c i t f i n i t e - d i f f e r e n c e scheme based on the c h a r a c t e r i s t i c s method and almost a l l governor non l i n e a r -i t i e s were c o n s i d e r e d . No comparison f o r s m a l l l o a d changes was made, howerer. Sm a l l l o a d d i s t u r b a n c e / f i e l d t e s t s f o r s i m u l a t e d i s o l a t i o n were performed f o r Grand Coulee T h i r d Power P l a n t w i t h an e l e c t r o - h y d r a u l i c p r o p o r t i o n a l - i n t e g r a l - d e r i v a t i v e governor [21]. The a c t u a l r e sponse com-pared w e l l w i t h computed speed d e v i a t i o n i n b o t h d u r a t i o n and shape and was w i t h i n 4.0 p e r c e n t f o r the magnitude. I d e a l impulse t u r b i n e s l o p e s ( P a y n t e r ' s model) were used w h i l e a r i g i d water column was assumed. C o r r e s p o n d i n g water-hammer c o n s t a n t p n was a p p r o x i m a t e l y 3.0. 6 I n s t a b i l i t y caused by e l a s t i c i t y of water and p e n s t o c k w a l l s was e x p e r i e n c e d i n Upper and Lower M o l i n a Power P l a n t s [22] which have v e r y l o n g p e n s t o c k s w i t h 2L/a = 20 and 12.5 seconds, and = 0.233 and 0.374, r e s p e c -t i v e l y . None o f the normal adjustments i n governor s e t t i n g s would e l i m i n a t e the i n s t a b i l i t y . S t a b l e o p e r a t i o n s was o b t a i n e d o n l y when the s l o p e o f the cam a t the speed-no-load p o s i t i o n was reduced to p r o v i d e s m a l l e r n e e d l e move-ment f o r a g i v e n d e f l e c t o r movement and a "dead time" was i n t r o d u c e d i n the ne e d l e r e l a y v a l v e . These changes e f f e c t i v e l y reduced the r a t i o o f s t e a d y -s t a t e n e e d l e to d e f l e c t o r movements by a f a c t o r o f 2, a c o n c l u s i o n t h a t was c o n f i r m e d by a f r e q u e n c y response a n a l y s i s (Bode diagram) as shown by Leum i n the d i s c u s s i o n o f the paper. D i r e c t comparison between e l a s t i c and r i g i d water column t h e o r i e s was made by Wood [23].. He showed t h a t f o r p a r t i a l and t o t a l u n i f o r m gate c l o s u r e s p r e s s u r e r i s e s r e s u l t i n g from water hammer may be 40 p e r c e n t h i g h e r than c a l -c u l a t e d by r i g i d column e x p r e s s i o n s , f o r s m a l l v a l u e s ( P Q < 1.5) and r e l a -3L t i v e l y r a p i d c l o s u r e s (T < — ) . The d i f f e r e n c e d i m i n i s h e s f o r slow c l o s u r e s C 9 . (T > ^ u ^ ) . i t a l s o d e c r e a s e s as p„ i n c r e a s e s and may become n e g a t i v e . For C 9 . v 2L Pg = 3 and T c = a r i g i d water column assumption y i e l d s p r e s s u r e r i s e 65 p e r c e n t h i g h e r than t h e a c t u a l one ( e l a s t i c t h e o r y ) . From t h e above r e v i e w one comes t o the c o n c l u s i o n t h a t a l t h o u g h the l i n e a r i z e d model used by P a y n t e r , Hovey and o t h e r s to study h y d r o - p l a n t s t r a n s i e n t s i s g e n e r a l l y v a l i d f o r s m a l l l o a d changes, e l a s t i c i t y e f f e c t s on t r a n s i e n t s i n the penstock, p a r t i c u l a r l y f o r h i g h head p l a n t s ( s m a l l P Q ) , have not been a d e q u a t e l y i n c l u d e d i n such a model. I n v e s t i g a t i o n s of the problem o f m o d e l l i n g f l u i d - l i n e t r a n s i e n t s s t a r t e d more than a c e n t u r y ago. Three b a s i c l i n e a r models which a r e 7 s u i t a b l e f o r m a t h e m a t i c a l treatment have e v o l v e d : a) l o s s l e s s f l u i d l i n e w i t h no energy d i s s i p a t i o n , b) l i n e a r f r i c t i o n model w i t h l o s s e s p r o p o r t i o n a l to mean v e l o c i t y , and c) a d i s s i p a t i v e model t h a t i n c l u d e s v i s c o s i t y e f f e c t s . In a d d i t i o n t h e r e a r e q u a s i - l i n e a r models t h a t c o n s i d e r a c o n v e c t i v e a c c e l e r a -t i o n term (V r—) and n o n - l i n e a r but c o n s t a n t s t e a d y s t a t e f r i c t i o n f a c t o r have dx a l s o been w i d e l y used f o r water l i n e s [ 2 4 ] . F o r a l o s s l e s s h o r i z o n t a l p i p e l i n e , momentum and c o n t i n u i t y e q u a t i o n s a r e reduced t o : gH + V = 0 ° x t 2 and H + — V = 0 t g x where H( x , t ) and V ( x , t ) are p i e z o m e t r i c head and v e l o c i t y p e r t u r b a t i o n s , " a " i s wave v e l o c i t y , and s u b s c r i p t s x, and t denote p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t to d i s t a n c e ( p o s i t i v e downstream) and time, r e s p e c t i v e l y . The g e n e r a l s o l u t i o n of these e q u a t i o n s takes the form: H ( x , t ) = F ( t - f ) + f ( t + f) V ( x , t ) = f [ F ( t - f) - f ( t + f)] w i t h F and f b e i n g unknown f u n c t i o n s determined by boundary c o n d i t i o n s . A l l i e v i [25] u t i l i z e d the boundary c o n d i t i o n of an upstream, c o n s t a n t - h e a d r e s e r v o i r to get the w e l l known r e l a t i o n : H ( L , t ) - H ( L , t - ^) = f [ V ( L , t ) - V ( L , t - ^ ) ] a g a A n a l y s i s was made f o r a v a r i e t y of v a l v e movements as a downstream boundary c o n d i t i o n . G r a p h i c a l s o l u t i o n s based on t h i s f o r m u l a were extended 8 to o t h e r boundary c o n d i t i o n s by l a t e r i n v e s t i g a t o r s [26,27], but were not a p p l i e d to t u r b i n e g o v e r n i n g problem. Wood [28] and R i c h [29,30] a p p l i e d H e a v i s i d e ' s o p e r a t i o n a l t h e o r y to waterhammer a n l a y s i s f o r f r i c t i o n l e s s p i p e l i n e s and p i p e l i n e s w i t h l i n e a r f r i c t i o n "F c"> Model 2, f o r which momentum e q u a t i o n becomes: gH + V. + F «V = 0 ° x t c The s o l u t i o n i n terms of the o p e r a t o r p = t o g e t h e r w i t h boundary c o n d i -t i o n s l e a d t o h y p e r b o l i c e x p r e s s i o n s s i m i l a r to those o b t a i n e d by a p p l y i n g L a p l a c e ' s t r a n s f o r m . These can be expanded i n terms of e x p o n e n t i a l f u n c t i o n s w hich may be i n t e r p r e t e d as u n i t s t e p s w i t h d i f f e r e n t l a g t i m e s . O p e r a t i o n a l t e c h n i q u e s a r e l i m i t e d to problems where e i t h e r H or V a t the downstream end of p e n s t o c k i s a known f u n c t i o n of time. The s o l u t i o n s become unduly com-p l i c a t e d , f o r a l l but the s i m p l e s t boundary c o n d i t i o n s . Analogy between f r i c t i o n l e s s p i p e l i n e s and d i s s i p a t i o n l e s s e l e c t r i c a l t r a n s m i s s i o n l i n e s was u t i l i z e d by P a y n t e r to study the resonance phenomena a s s o c i a t e d w i t h a r h y t h m i c gate-motion [ 3 1 ] . R e l a t i v e head and v e l o c i t y changes at the gate were r e l a t e d by: — = - 2p„ t a n — v K o a where "oo" i s f r e q u e n c y of gate o s c i l l a t i o n . The head was always l a g g i n g the v e l o c i t y by 90°. Resonance s t u d i e s were f u r t h e r extended to more complex systems by u s i n g b o t h the impedance method [32] and the t r a n s f e r m a t r i x method [33]. Though the complete s o l u t i o n of momentum and c o n t i n u i t y e q u a t i o n s de-pends on b o t h i n i t i a l and boundary c o n d i t i o n s , any terms i n t r o d u c e d by i n i t i a l c o n d i t i o n s a r e t r a n s i e n t i n n a t u r e and they e v e n t u a l l y w i l l d i s a p p e a r i n a 9 p h y s i c a l s i t u a t i o n . The r e m a i n i n g p a r t o f the s o l u t i o n i s known as the s t e a d y -o s c i l l a t o r y r e s p o n s e o f the system. S t e a d y - o s c i l l a t o r y r e s p o n s e , r a t h e r than a complete s o l u t i o n , i s the main concer n o f such resonance s t u d i e s and a s i n u -s o i d a l head', d i s c h a r g e , and gate v a r i a t i o n i s always assumed. In the l a s t two decades i n t e n s i v e work has been made to r e p r e s e n t a t t e n u a t i o n and d i s p e r s i o n e f f e c t s of a f l u i d l i n e , Model 3 [34 to 38]. A two-dime n s i o n a l model has to be used i n t h i s case w i t h v e l o c i t y and p r e s s u r e v a r i a t i o n s i n r a d i a l d i r e c t i o n " r " i n t r o d u c e d to momentum and c o n t i n u i t y e q u a t i o n s : where V q i s k i n e m a t i c v i s c o s i t y a t ste a d y s t a t e . I t has been shown t h a t heat t r a n s f e r has n e g l i g i b l e e f f e c t s on w a t e r - l i n e t r a n s i e n t s . Due to v i s -c o s i t y , f r i c t i o n becomes a frequency-dependent f u n c t i o n . Laplace-domain e x p r e s s i o n s were g i v e n f o r head andflow a t the downstream end f o r v e r y s i m p l e boundary c o n d i t i o n s such as sudden v a l v e c l o s u r e , a p p l i c a t i o n o f a r e c t a n g u l a r p r e s s u r e p u l s e , s t e p p r e s s u r e o r f l o w change a t one end. Even f o r such s i m p l e c a s e s , e x p r e s s i o n s o b t a i n e d i n v o l v e d B e s s e l f u n c t i o n s , and the i n v e r s e t r a n s -form t o time domain was too c o m p l i c a t e d f o r p r a c t i c a l use. t i o n of a f l u i d p i p e l i n e r e sponse has e v o l v e d ( F i g . 1). I t i s summarized by two f u n c t i o n s , namely the c h a r a c t e r i s t i c - i m p e d a n c e , Z ( s ) and the p r o p a g a t i o n o p e r a t o r T ( s ) , s b e i n g L a p l a c e ' s o p e r a t o r . The head a t a p a r t i c u l a r c r o s s s e c t i o n i s r e l a t e d to the f l o w a t t h a t s e c t i o n by: gH + V = \> (V + - V & x t • o r r r r a 2 V H t + T ^ x + V r + 7> - ° and Analogous t o the two-port ( 4 - t e r m i n a l ) networks, a u s e f u l r e p r e s e n t a -10 H, F i g . 1 T w o - P o r t - R e p r e s e n t a t i o n o f a F l u i d T r a n s m i s s i o n L i n e 11 Q(x,s) A { S ) whereas head a t a downstream l o c a t i o n x 2 °f a s e m i - i n f i n i t e p i p e i s r e l a t e d to the upstream head a t x^ by: X ~x H(x 2,s) - r ( s ) ( - ^ - ^ ) H(x 1,s) = £ £ i s a r e f e r e n c e l e n g t h . F o r the t h r e e models of a water p i p e l i n e , c h a r a c t e r i s t i c impedances and p r o p a g a t i o n o p e r a t o r s a r e : L o s s l e s s l i n e : Z ( s ) = -^7- , A i s c r o s s - s e c t i o n a r e a gA r ( s ) = ^ L i n e a r - f r i c t i o n l i n e : Z ( s ) = gA F a . c T ( s ) = — V I + a 1 F c D i s s i p a t i v e l i n e : Z ( s ) = (k) a , / o ' gA V l 2 ( k ) Ls A 0 0 , D Is r ( s ) = i f Vi too' k = f v v :2 o I , and I 2 a r e m o d i f i e d B e s s e l f u n c t i o n s , and D i s the p i p e d iameter. Leonard [39] i n v e s t i g a t e d d i f f e r e n t p o s s i b l e c o mbinations o f i n p u t s to a two-port r e p r e s e n t a t i o n o f a l o s s l e s s l i n e and d e r i v e d r e l a t i o n s between i n p u t and output v a r i a b l e s i n a m a t r i x form: 12 G12 G • G.V 21 • 22 where G ^ , G^* ^21' a n c * ^22 a r e f u n c t : i - o n s °f Z ( s ) and T ( s ) . For a c o n s t a n t head r e s e r v o i r -at)the upstream end o f ^ p e n s t o c k (h = 0 ) , r e l a t i v e head and f l o w changes a t the downstream end ar e r e l a t e d by: h(s-)> q( s ) - 2 p 0 / t a n h (rj-) Many attempts have been made to s i m p l i f y or approximate the complex e x p r e s s i o n s f o r Z ( s ) and F ( s ) of a d i s s i p a t i v e l i n e . I t has been found t h a t a power s e r i e s e x p a n s i o n f o r the h y p e r b o l i c f u n c t i o n s which appeared f r e q u e n t l y i n Laplace-domain e x p r e s s i o n s may i n d i c a t e an i n s t a b i l i t y which does not e x i s t i n t h e a c t u a l p h y s i c a l system. An i n f i n i t e p r o d u c t e x p a n s i o n i n terms of the z e r o s of the f u n c t i o n has been suggested to overcome t h i s problem [40]. Though such a p r o c e d u r e g i v e s s a t i s f a c t o r y a p p r o x i m a t i o n s of the l i n e f r e -quency response f o r s i m p l e boundary c o n d i t i o n s , the w r i t e r has found t h a t t r a n s i e n t r e sponse (time domain) o b t a i n e d by u s i n g such an e x p a n s i o n f o r t u r b i n e g o v e r n i n g i s not adequate p a r t i c u l a r l y f o r the f i r s t few time i n t e r v a l s . Leonard [39] assumed a d i s p e r s i o n i n the form of a s i m p l e f i r s t o r d e r l a g and t r e a t e d a t t e n u a t i o n as a lumped r e s i s t a n c e t o g e t h e r w i t h the l o s s l e s s model. Time c o n s t a n t of the l a g and v a l u e of the r e s i s t a n c e were determined by a b e s t - f i t p r o c e d u r e to approximate to the exact s o l u t i o n o b t a i n e d by a f i n i t e d i f f e r e n c e t e c h n i q u e . 13 Karam [41,42] g e n e r a l i z e d the two-port r e p r e s e n t a t i o n to i n c l u d e two f i l t e r s of the impulse response h ( t ) of a s e m i - i n f i n i t e l i n e . By c o n v o l u t i o n w i t h the a r b i t r a r y i n p u t X ( t ) , l i n e r esponse becomes: t y ( t ) = | X(A) h ( t - X) dX 0 The o r i g i n a l b l o c k diagram s t r u c t u r e was m o d i f i e d to overcome the s e v e r e n u m e r i c a l i n s t a b i l i t y problem encountered w i t h d i s c r e t e c a l c u l a t i o n by d i g i t a l computer. The s t a n d a r d p r o c e d u r e (see F i g . 2) f o r e v a l u a t i n g t r a n s i e n t response f o r an a r b i t r a r y i n p u t i n v o l v e d : Q 2 ( s ) a) d e r i v a t i o n of l o a d a d m i t t a n c e , YLD(s) = —— j—r-, H 2 ^ S ; b) s u b s t i t u t i o n of YLD(s) i n the t r a n s f e r f u n c t i o n r e l a t i n g i n c i d e n t f i l t e r t o v a r i a b l e of i n t e r e s t , say H 2, c) i n v e r s e L a p l a c e •' t r a n s f o r m of the r e s u l t i n g f u n c -t i o n , and d) c o n v o l u t i o n of the r e s u l t w i t h the o u t p u t of the i n c i d e n t f i l t e r , y ( t ) Such a p r o c e d u r e i s too c o m p l i c a t e d to be u t i l i z e d f o r the a n a l y s i s of the t u r b i n e g o v e r n i n g problem. In a d d i t i o n to d i f f i c u l t i e s e ncountered to get the i n v e r s e t r a n s f o r m , n u m e r i c a l e v a l u a t i o n f o r the c o n v o l u t i o n i n t e g r a l i s n e c e s s a r y . A p r o c e d u r e f o r i n c l u d i n g frequency-dependent f r i c t i o n i n s t e a d of s t e a d y - s t a t e f r i c t i o n i n the s o l u t i o n by method of c h a r a c t e r i s t i c s was de-v e l o p e d by Z i e l k e [37] and l a t e r improved by T r i k h a [43] to reduce the e x c e s -s i v e computer s t o r a g e and e x c u t i o n time r e q u i r e d . Though a f a i r l y a c c u r a t e + i n c i d e n t f i l t e r h ( T ) Y L D ' Z ( s ) - l YLD«Z(s)+l r e f l e c t e d f i l t e r 1 YLE>'Z(s)+l YLD Y L D ' Z ( s ) + l G e n e r a l i z e d Two-Port Repre s e n t a t i o n of a F l u i d Transmission L i n e , [4.1] 15 r e p r e s e n t a t i o n o f a l i q u i d l i n e has been o b t a i n e d , i t i s by n a t u r e a n u m e r i c a l s o l u t i o n . The use of sampled ( d i s c r e t e ) d a t a models to a n a l y s e d i s t r i b u t e d -parameter systems was a l s o suggested [44,45]. A s o l u t i o n u s i n g z - t r a n s f o r m a p p l i e d to components o f the response f u n c t i o n i n L a p l a c e ' s domain i n c l u d e d i -Ts 1 — e a sample-and-hold c i r c u i t o f the form H (s) = , w i t h T b e i n g the 0 s sampling p e r i o d ( F i g . 3 ) . I n v e r s e z - t r a n s f o r m i s then a p p l i e d to the o v e r a l l r e s u l t i n g f u n c t i o n t o get the time-domain o u t p u t . T h i s p r o c e d u r e was b a s i c a l l y i n t r o d u c e d to overcome d i f f i c u l t i e s i n h e r e n t i n d e t e r m i n i n g the i n v e r s e L a p l a c e t r a n s f o r m . However, a d i g i t a l computer has to be used f o r i n v e r s e z - t r a n s f o r m . T h i s complex p r o c e d u r e l i m i t s the u s e f u l n e s s o f such a t e c h n i q u e f o r the ma t h e m a t i c a l a n a l y s i s o f speed t r a n s i e n t s i n hydro p l a n t s . S e v e r a l f r e q u e n c y response s t u d i e s have been c a r r i e d out f o r the t u r b i n e g o v e r n i n g problem u s i n g t r a n s f e r f u n c t i o n s o f f r i c t i o n l e s s or l i n e a r f r i c t i o n models, e.g. [46 to 50]. A v a r i e t y of f r e q u e n c y domain models o f d i f f e r e n t degrees of a c c u r a c y a r e a v a i l a b l e but v e r y few attempts have been made to extend them t o the time domain. R a n s f o r d and R o t t n e r [51] o b t a i n e d t r a n s i e n t speed s o l u t i o n by u s i n g a s i x t h - o r d e r T a y l o r ' s e x p a n s i o n o f the h y p e r b o l i c f u n c t i o n s i n L a p l a c e ' s domain. Compared to the a c c u r a t e n u m e r i c a l s o l u t i o n by the c h a r a c t e r i s t i c s method, the w r i t e r has found R a n s f o r d ' s s o l u t i o n u n s a t i s f a c t o r y , as w i l l be shown i n ch a p t e r IV of t h i s t h e s i s . Moreover, power s e r i e s e x p a n s i o n may i n t r o d u c e an a r t i f i c i a l i n s t a b i l i t y as mentioned b e f o r e . An approximate a n a l y s i s o f water hammer (non-wave approach) was i n v e s -t i g a t e d by M. Macagno and E. Macagno [52]. They rewrote momentum and con-t i n u i t y e q u a t i o n s as o r d i n a r y d i f f e r e n t i a l e q u a t i o n s i n terms of i n l e t and o u t l e t v a r i a b l e s . T h i s y i e l d e d a second.order d i f f e r e n t i a l e q u a t i o n o f the form: 16 F ( t ) . F * ( t ) a) The Z-Transform: Z [ f ( t ) ] = L [ F * ( t ) ] , F * ( t ) = I F ( n T ) * S ( t - n T ) n=0 Lumped Parameters Components input_ G L ( s ) -Tv s ) D i s t r i b u t e d Parameters Components 1 J I n t r o d u c e d Sample-and-H o l d f u n c t i o n VsTl Output b) B l o c k Diagram F i g . 3 A D i s c r e t e T i m e M o d e l f o r a D i s t r i b u t e d - P a r a m e t e r S y s t e m 17 Jl + ( T > 2 h + < T > \ o £ - o a t where h and v a r e r e l a t i v e changes a t the o u t l e t . T h i s , i n f a c t , i s the same as r i g i d - c o l u m n e q u a t i o n w i t h the a d d i t i o n o f a second d e r i v a t i v e of the head. When a p p l i e d to a s h o r t p i p e (shock a b s o r b e r ) , good r e s u l t s have been o b t a i n e d f o r average v a r i a t i o n . The w r i t e r has found, however, t h a t such an approxima-t i o n i s n o t adequate f o r speed t r a n s i e n t s a n a l y s i s where the pe n s t o c k has a c o n s i d e r a b l e l e n g t h . In t h i s t h e s i s , exact t r a n s i e n t r e sponse i s d e r i v e d f o r a l i n e a r i z e d model of a hydro p l a n t i n which e l a s t i c i t y e f f e c t s a r e i n c l u d e d . A f r i c t i o n -l e s s p e n s t o c k i s c o n s i d e r e d f i r s t , and l a t e r a pen s t o c k w i t h l i n e a r f r i c t i o n l o s s e s i s a n a l y z e d . The g e n e r a l s o l u t i o n of a wave e q u a t i o n i n terms o f two t r a v e l l i n g waves F and f has been found to be the most u s e f u l one f o r such a ma t h e m a t i c a l trea t m e n t . In c h a p t e r I I , b a s i c g o v e r n i n g e q u a t i o n s a r e f i r s t p r e s e n t e d . D e r i v a -t i o n of t h e speed d i f f e r e n t i a l e q u a t i o n f o l l o w s i n c h a p t e r I I I , where an exact s o l u t i o n i s o b t a i n e d w i t h o u t and w i t h the f r i c t i o n e f f e c t . A l t e r n a t i v e forms of the s o l u t i o n u s i n g T a y l o r ' s expansions o f the f i n a l r e s u l t s a r e approximate but s i m p l e r and more e f f i c i e n t f o r n u m e r i c a l a p p l i c a t i o n . I n c h a p t e r IV, these s o l u t i o n s a r e v e r i f i e d by comparison w i t h the w e l l e s t a b l i s h e d method of c h a r a c t e r i s t i c s and o t h e r methods. A d i s c u s s i o n f o l l o w e d by summary and c o n c l u s i o n s completes the body of the t h e s i s . CHAPTER I I SYSTEM COMPONENTS AND EQUATIONS GOVERNING THEIR RESPONSES 2.1 I n t r o d u c t i o n The c h a p t e r b e g i n s w i t h a g e n e r a l d e s c r i p t i o n of d i f f e r e n t components of an i s o l a t e d hydro-power p l a n t t h a t r e p r e s e n t the p h y s i c a l model to be a n a l y s e d . D i f f e r e n t i a l e q u a t i o n s which govern the b e h a v i o r of t h e s e components a r e then p r e s e n t e d . Two main approaches f o r s o l v i n g the p e n s t o c k e q u a t i o n , w i t h e l a s t i c i t y e f f e c t s i n c l u d e d , a r e of p a r t i c u l a r i n t e r e s t : the method of c h a r a c t e r i s t i c s combined w i t h t h e use of a d i g i t a l computer, and the g e n e r a l s o l u t i o n o f l i n -e a r homogeneous wave e q u a t i o n s i n terms of t r a v e l l i n g and r e f l e c t e d waves. The second approach i s t h e one adopted l a t e r t o r e p r e s e n t p e n s t o c k e f f e c t s i n t h e m a t h e m a t i c a l model. The assumptions i n v o l v e d i n t h e d e r i v a t i o n s of t u r b i n e , governor and p e n s t o c k e q u a t i o n s a r e d i s c u s s e d i n c o n j u n c t i o n w i t h t h e s e d e r i v a t i o n s . 2.2 System Components . For speed t r a n s i e n t s study of a hydro-power p l a n t under i s o l a t e d - l o a d c o n d i t i o n s , t h e most important elements (see F i g . 4) a r e : 2.2.1 R e s e r v o i r a t the up-stream (U/S) end of t h e p e n s t o c k . A c o n s t a n t head r e s e r v o i r w i t h an u n l i m i t e d water s u r f a c e a r e a i s assumed. T h i s assump-t i o n i s v a l i d even i f a surge tank of a l i m i t e d c r o s s - s e c t i o n a r e a e x i s t s i n p l a c e o f t h e r e s e r v o i r s i n c e water l e v e l o s c i l l a t i o n s i n t h e surge tank a r e 18 19 R e s e r v o i r ,3> Hv G e n e r a t o r N T r a n s m i s s i o n L i n e s ~V7 P e n s t o c k F i g . 4 Basic Elements of a Hydropower P l a n t 20 v e r y slow compared t o the t r a n s i e n t s of e l a s t i c pressure-waves (waterhammer). V a r i a t i o n s of water l e v e l i n a surge tank s h o u l d be c o n s i d e r e d o n l y i f l a r g e changes of e l e c t r i c power (l o a d ) a r e s t u d i e d . 2.2.2 P e nstock . The p e n s t o c k i s t r e a t e d as a s i n g l e h o r i z o n t a l p i p e h a v i n g c o n s t a n t d i a m e t e r , w a l l t h i c k n e s s and w a l l m a t e r i a l . T h i s p i p e ex-tends from the r e s e r v o i r to the t u r b i n e r u n n e r . P r e s s u r e wave speed " a " w i l l be, t h e r e f o r e , c o n s t a n t over the e n t i r e p e n s t o c k l e n g t h . E l a s t i c i t y of p e n s t o c k w a l l s i s c o n s i d e r e d i n c a l c u l a t i n g t h e wave speed but p r o p a g a t i o n of e l a s t i c waves through w a l l s themselves i s d i s r e g a r d e d . Head l o s s e s due to w a l l f r i c t i o n and o t h e r s o u r c e s a r e u s u a l l y l e s s than 5% of the t o t a l head. The use of a f r i c t i o n l e s s f l o w model w i l l n o t , t h e r e f o r e , i n t r o d u c e a l a r g e e r r o r . L a t e r , an approximate s o l u t i o n which i n c l u d e s t h e s e l o s s e s i n a l i n e a r i z e d form i s i n v e s t i g a t e d . Compared to p e n s t o c k l e n g t h , d r a f t tube i s v e r y s h o r t and i t s t r a n s i e n t s can be n e g l e c t e d i n most c a s e s . '2.2.3 T u r b i n e a t P e n s t o c k D/S end. F o r t h e s t e a d y s t a t e o p e r a t i o n , t u r b i n e performance i s c o m p l e t e l y d e f i n e d by t u r b i n e " c h a r a c t e r i s t i c s c u r v e s " o b t a i n e d from model t e s t s . U s u a l l y , d i s c h a r g e or e f f i c i e n c y and power output a r e g i v e n i n terms of head, gate and speed. A c c u r a t e r e p r e s e n t a t i o n of t h e s e c u r v e s by a g e n e r a l m a t h e m a t i c a l model i s v e r y d i f f i c u l t [16] and [ 1 7 ] . For d e t a i l e d computer s t u d i e s i t i s more c o n v e n i e n t t o use s t o r e d d a t a p o i n t s . In t h i s t h e s i s o n l y s l o p e s o f t h e s e c h a r a c t e r i s t i c c u r v e s a t the p o i n t of i n i t i a l s t e a d y s t a t e o p e r a t i o n a r e used to d e f i n e r e l a t i o n s between d i s c h a r g e , head, g a t e , speed and power. T h i s i s a r e a s o n a b l e a p p r o x i m a t i o n f o r a study of t r a n s i e n t s due to s m a l l l o a d changes [18]. 2.2.4 Speed governor. C o n t r o l of t u r b i n e - g e n e r a t o r s h a f t speed may be a t t a i n e d by e i t h e r a m e c h a n i c a l o r an e l e c t r i c governor. In f a c t , a p r o p o r -t i o n a l - i n t e g r a l e l e c t r i c governor has a r e s p o n s e f u n c t i o n s i m i l a r t o a 21 m e c h a n i c a l one of dashpot type. A d d i t i o n of a d e r i v a t i v e term ( p r o p o r t i o n a l -i n t e g r a l - d e r i v a t i v e governor) r a i s e s t h e o r d e r of t h i s f u n c t i o n . In o r d e r t o a v o i d a d d i n g c o m p l i c a t i o n s to the a n a l y s i s , o n l y dashpot or p r o p o r t i o n a l -i n t e g r a l g o v ernors a r e c o n s i d e r e d . The i n p u t to a speed governor i s the speed d e v i a t i o n from a r e f e r e n c e v a l u e . The output i s a c o r r e s p o n d i n g gate change to c o r r e c t t h a t d e v i a t i o n . D e t a i l s of governor components may be found i n l i t e r a t u r e e.g., [9, 53, 54]. These d e t a i l s a r e dependent on m a n u f a c t u r e r s ' d e s i g n s , but t h e g e n e r a l f e a t u r e s a r e the same. Advanced t y p e s of g o vernors such as d i g i t a l c o n t r o l -l e r s and o n - l i n e computer c o n t r o l s which a r e s t i l l b e i n g d e v e l o p e d w i l l p r o -v i d e much g r e a t e r f l e x i b i l i t y i n o p t i m i z i n g governor s e t t i n g s and w i l l r e s u l t i n more a c c u r a t e c o n t r o l . 2.2.5 Load. Governor s t a b i l i t y and • c o n t r o l e f f i c i e n c y a r e examined by s t u d y i n g speed t r a n s i e n t s r e s u l t i n g from s m a l l l o a d changes. For t h i s purpose a s t e p l o a d - o f f or l o a d - o n change, m^ = 2 - 10% of the r a t e d power i s assumed. N e i t h e r t i e l i n e power surges nor v o l t a g e t r a n s i e n t s a r e c o n s i d -e r e d h e r e . Load damping e f f e c t which depends on t y p e of g r i d l o a d * i s non-l i n e a r and i s d i f f i c u l t to e s t i m a t e . ..If l i n e a r i z e d about the i n i t i a l p o i n t of o p e r a t i o n , the l o a d damping e f f e c t becomes p r o p o r t i o n a l to speed. The t o t a l r e l a t i v e l o a d change may then be w r i t t e n as m^ + a ^ n ( t ) . The most adve r s e c case i n p r a c t i c e o c c u r s when the l o a d i s p u r e l y r e s i s t i v e and v o l t a g e i s t i g h t l y r e g u l a t e d , f o r which ( l o a d damping c o n s t a n t ) = -1 [16]. S i n c e windage and f r i c t i o n l o s s e s a t t h e t u r b i n e - g e n e r a t o r s h a f t a r e v e r y s m a l l compared to r a t e d power o u t p u t , they a r e n e g l e c t e d . * F o r p u r e l y r e s i s t i v e l o a d s e.g., l i g h t i n g or e l e c t r i c f u r n a c e s , system l o a d i s independent of speed whereas f o r o t h e r l o a d s t h e system l o a d i s p r o p o r t i o n a l to the speed. For t u r b o machines the l o a d v a r i e s w i t h the cube of the speed. 22 F i g . 5 shows how t h e s e components d e s c r i b e d above r e l a t e d t o each o t h e r . 2.3 T u r b i n e D i s c h a r g e and Torque E q u a t i o n s 2.3.1 The g e n e r a l model. At any time ( t ) , t u r b i n e o p e r a t i o n i s com-p l e t e l y d e f i n e d by f i v e v a r i a b l e s , namely net head a c t i n g on the t u r b i n e H^, d i s c h a r g e Q, gate opening Z, speed of r o t a t i o n N, and t u r b i n e output t o r q u e M. Only two of them, u s u a l l y Q and M, can be e x p r e s s e d i n terms of the t h r e e r e m a i n i n g v a r i a b l e s i n the form of m o d e l - t e s t s ' c u r v e s . Q(t) = Q(H,Z,N) (1-a) and M(t) = M(H,Z,N) (1-b) U s i n g s i m i l i t u d e r e l a t i o n s , i t i s p o s s i b l e t o r e p r e s e n t r e s u l t s o f t h e s e t e s t s i n a more comprehensive form. I t i s well-known t h a t f o r any two homol-ogous t u r b i n e s , t h a t i s , t u r b i n e s h a v i n g g e o m e t r i c and k i n e m a t i c s i m i l a r i t y t h e f o l l o w i n g r e l a t i o n s " a r e s a t i s f i e d : c o n s t a n t (2-a) D 2 N 2 = c o n s t a n t , or — ^ - y - » c o n s t a n t (2-b) D 3N D 2H ^ n P P and =* c o n s t a n t , or -r-pr =* c o n s t a n t (2-c) D % 3 D 2H 3 / 2 n where D i s t h e diameter of t u r b i n e r u n n e r . U n i t speed, u n i t d i s c h a r g e and u n i t power a r e d e f i n e d a s : M DN , o \ N - — (3-a) u H ^ n ( 3 " b ) D^H. 2 and P = -r-pr (3-c) U D 2H 3 / 2 Thus, a group of homologous t u r b i n e s w i l l have the same u n i t v a l u e s . By 23 R e s e r v o i r Load M' L H. R Pe n s t o c k H T u r b i n e M Machine I n e r t i a MT T r a n s m i s -s i o n L i n e Governor F i g . 5 Functional Block Diagram of a Hydro Plant 24 u s i n g e x p r e s s i o n s 3-a, b, and c u n i t d i s c h a r g e and u n i t power become f u n c -t i o n s of two v a r i a b l e s o n l y ( t u r b i n e c h a r a c t e r i s t i c c u r v e s ) : Q ( t ) = Q. (N ,Z) (4-a) u u u and P ( t ) - P (N ,Z) (4-b) u u u The t o t a l i n c r e m e n t a l change may be th e n e x p r e s s e d a s : 9Q 8Q A ^ u - jf A N u + i f A Z ( 5 ' a ) u a n d A P u ~w" A N u + A Z ( 5 " b ) u F o r s m a l l d e v i a t i o n s from i n i t i a l s t e a d y s t a t e , s l o p e s o f c h a r a c t e r -i s t i c c u r v e s can be approximated by t h e i r i n i t i a l s t e a d y s t a t e v a l u e s and a t any time " t " i n t h e t r a n s i e n t s t a t e : t 3Q t 9Q t T, AQ - (-~r) n E AN + ( —^-) n E AZ (6-a) 0 u 9 N u 0 Q u 9Z 0 Q t 3 J? t 9 P t and E AP - ( - 7 % E AN + (-f-) E AZ (6-b) Q u 9N u 0 Q u 3Z 0 Q I n t r o d u c i n g r e l a t i v e speed d e v i a t i o n n, d i s c h a r g e d e v i a t i o n q, head d e v i a t i o n h, gate d e v i a t i o n z and power d e v i a t i o n p where: N( t ) - N Q ( t ) - Q H ( t ) - H n ( t ) = , q ( t ) = , h ( t ) -N 0 Q 0 H n 0 z ( t ) - Z Q P ( t ) - P Q z ( t ) = , and p ( t ) = - , ^0 ^0 e q u a t i o n (3) becomes a f t e r l i n e a r i z a t i o n : N =* N (1 + n - %h) (7-a) U U 0 0 = Q (1 + q - hh) (7-b) 0 and P - P (1 + p - | h ) (7-c) u U Q 2 25 M(t) - M Q R e l a t i v e t o r q u e d e v i a t i o n , m(t) =•= — , r a t h e r than r e l a t i v e power 0 d e v i a t i o n p ( t ) i s t o be used i n dynamic e q u a t i o n of r o t a t i n g p a r t s . Eq. 7-c i s t h e r e f o r e r e w r i t t e n u t i l i z i n g t h e l i n e a r i z e d power-torque r e l a t i o n -s h i p : P = P (1 + m + n - y h ) (7-d) u u Q 2 S u b s t i t u t i o n of r e l a t i o n s g i v e n by 7-a, 7-b and 7-d i n t o Eqns. 6-a and 6-b l e a d s f i n a l l y t o t h e f o l l o w i n g e q u a t i o n s which d e s c r i b e t u r b i n e d i s -charge and t o r q u e l i n e a r i z e d about i n i t i a l s t e a d y s t a t e o p e r a t i o n v a l u e s : q ( t ) = £ h ( t ) + £ n ( t ) + £ z ( t ) (8-a) qn qn qz and m(t) .=* £ . h ( t ) + £ n ( t ) + £ z ( t ) (8-b) mn mn mz where ,N n 3Q V J S I 1 - - Q T ' ( w L ) O 1 V ( 9 " a ) uO u N 9Q t , _u0 ^ u k ( 9 _ b ) qn Q . ^3N }0 ^ b ) uO u £ A ( 9" C ) qz Q u Q 8 Z >0 N 3P N 3P mn P . ^3N ; 0 1 ^ e ; uO u Z 3P A N D ^ " P T ^ O - ( 9 " F ) uu I f the e f f i c i e n c y "e" c u r v e s a r e g i v e n i n s t e a d of u n i t d i s c h a r g e t h e n : z - i + A - ^ ( ^ qn mn e n V3N ; 0 ' V g ; u u £• - Js(l - I ) (9-h) qh qn 26 Z and £ = £ - -f - (ff)„ ( 9 - i ) qz mz eQ 32. 0 U n i t v a l u e s r e p r e s e n t a t i o n reduces the number o f s l o p e s needed f o r c a l c u l a t -i n g t u r b i n e d i s c h a r g e and t o r q u e output from s i x t o f o u r . 2.3.2 I d e a l Impulse t u r b i n e model. For an impulse t u r b i n e o p e r a t i n g a t p o i n t of b e s t e f f i c i e n c y , s l o p e s of c h a r a c t e r i s t i c c u r v e s a r e w e l l approximated by: 3Q <#><>- ° ( 1 ° - a ) u 3Q Q n u 3P P a n d <ir>o • - i f ( 1 0- d> S u b s t i t u t i n g i n Eq. 8, d i s c h a r g e and t o r q u e r e l a t i o n s f o r an i d e a l i m p ulse t u r b i n e become: q ( t ) - 0.5h(t) + z ( t ) (11-a) and m(t) = 1.5h(t) - n ( t ) + z ( t ) (11-b) In g e n e r a l , f o r any t u r b i n e o p e r a t i n g a t p o i n t of b e s t e f f i c i e n c y b o t h £ and £ a r e a p p r o x i m a t e l y e q u a l t o u n i t y a n d £ s £ - 1. qz mz r r J J mn qn Va l u e s o f £ . and £ depend on t h e s p e c i f i c speed N of the t u r b i n e [ 9 ] , qn mn r r s as shown i n F i g . 6. 2.4 E q u a t i o n of M o t i o n f o r R o t a t i n g P a r t s In a steady s t a t e c o n d i t i o n the t u r b i n e - g e n e r a t o r common s h a f t r o t a t e s a t " c o n s t a n t speed "oo" w i t h two e q u a l and o p p o s i t e t o r q u e s a c t i n g on i t , h y d r a u l i c t o r q u e M, and e l e c t r i c t o r q u e M^. As e l e c t r i c t o r q u e v a r i e s due 27 Impulse .Re a c t i o n P r o p e l l e r —H h * 1 1 1 1 1 I 1 1 J 1 h • 1 i j j 1 i • i il i i 1 I i 0 .  40 80 120 160 200 S p e c i f i c Speed, N i Fig. 6 Characteristics Slopes for a Turbine Operating at Point of Best E f f i c i e n c y [9] 28 to a l o a d change, the r o t a t i n g mass of g e n e r a t o r , s h a f t , and t u r b i n e s t a r t s to a c c e l e r a t e or d e c e l e r a t e a c c o r d i n g to r o t a t i o n a l motion e q u a t i o n : WR2 'du. M - M. (12) g dT ' £ WR2 The t o t a l moment of i n e r t i a i n c l u d e s the i n e r t i a o f g e n e r a t o r , s h a f t , t u r b i n e and mass of water i n s i d e the t u r b i n e . U s u a l l y the t u r b i n e ' s i n e r t i a i s q u i t e s m a l l compared t o the g e n e r a t o r ' s i n e r t i a . In terms of r e l a t i v e , d e v i a t i o n s , Eq. 12 y i e l d s Tm0 oY - m " m * ( 1 3 ) The m e c h a n i c a l s t a r t i n g time T „ i s e s s e n t i a l l y the r a t i o of i n i t i a l mO k i n e t i c energy of r o t a t i n g mass to i n i t i a l power P^, 2 2TTN 2 T m Q ( s e c o n d s ) - ^ - (-J>) /P Q (14) The speed i s e x p r e s s e d i n r.p.m. A s t e p l o a d change i s u s u a l l y assumed f o r g o v e r n i n g s t u d i e s , hence m^ becomes a c o n s t a n t , p o s i t i v e f o r l o a d - o n ease. The t o t a l l o a d change, however, i n c l u d e s a l s o a l o a d damping e f f e c t p r o p o r t i o n a l to n ( t ) and Eq. 13 i s more a c c u r a t e l y w r i t t e n : Tm0 I f ' m - ( m £ + a £ n ) ( 1 5 ) T h i s r e p r e s e n t s t h e r o t a t i n g p a r t s e q u a t i o n of m o t i o n t o be s o l v e d s i m u l -t a n e o u s l y w i t h t u r b i n e , governor and p e n s t o c k e q u a t i o n s . 2.5 Governor E q u a t i o n Speed d e v i a t i o n i s d e t e c t e d by a speed s e n s i n g d e v i c e , u s u a l l y a f l y b a l l head, p r o d u c i n g a p r o p o r t i o n a l s i g n a l which i s t h e n t r a n s m i t t e d to the g a t e servomotor. T r a n s m i s s i o n i s c a r r i e d out e i t h e r m e c h a n i c a l l y through a system of l e v e r s or e l e c t r i c a l l y u s i n g permanent magnet g e n e r a t o r s and e l e c t r i c a m p l i f i e r s . In b o t h cases a h y d r a u l i c system c o n s i s t i n g m a i n l y o f a p i l o t v a l v e , p i l o t servomotor, and a d i s t r i b u t i o n v a l v e has to be used 29 to m a g n i f y ' t h i s s i g n a l and to c o n v e r t i t i n t o a m e c h a n i c a l f o r c e s t r o n g enough t o move the gate servomotor p i s t o n i n the p r o p e r d i r e c t i o n . A feedback system a c t u a t e d by gate movement checks any o v e r - t r a v e l i n the same d i r e c t i o n . The feedback c o n s i s t s o f a s m a l l p a r t of permanent n a t u r e (permanent speed droop, a) and a r e l a t i v e l y l a r g e one o f a t r a n s i e n t n a t u r e ( t r a n s i e n t speed droop, S ) . In m e c h a n i c a l governors t r a n s i e n t droop i s u s u a l l y o b t a i n e d by u s i n g a h y d r a u l i c dashpot w i t h an a d j u s t a b l e r e s e t time T , t o g e t h e r w i t h a r e s t o r i n g s p r i n g system. In e l e c t r i c g o v ernors a t h r e e - t e r m c o n t r o l l e r ( a m p l i f i e r ) i s used i n s t e a d . F i g . 7 shows sc h e m a t i c diagrams of a m e c h a n i c a l - h y d r a u l i c and an e l e c t r i c h y d r a u l i c g o v e r n o r s . P o s i t i o n and r a t e s a t u r a t i o n l i m i t s f o r p i l o t v a l v e , d i s t r i b u t i o n v a l v e , gate servomotor, and dashpot, dead band and dead time due t o f r i c t i o n between d i f f e r e n t p a r t s r e s u l t i n a n o n l i n e a r f u n c t i o n i n g o f the governor. For s m a l l speed d e v i a t i o n s , however, s a t u r a t i o n l i m i t s a r e seldom reached and modern governor d e s i g n s have kept dead band and dead time v e r y s m a l l . A l i n e a r governor r e s p o n s e i s t h e r e f o r e w e l l j u s t i f i e d f o r s m a l l l o a d changes. A f u n c t i o n a l b l o c k diagram as w e l l as t h e c o r r e s p o n d i n g l i n e a r mathe-m a t i c a l r e p r e s e n t a t i o n f o r a m e c h a n i c a l - h y d r a u l i c governor a r e shown i n F i g . 8-a. F o l l o w i n g Leum [54] an e l e c t r i c - h y d r a u l i c governor may be r e p r e s e n t e d by a b l o c k diagram as shown i n F i g . 8-b. R e f e r r i n g t o F i g . 8-a, m e c h a n i c a l governor d i f f e r e n t i a l e q u a t i o n becomes: [T T T d 3 + (T +T )T d 2 + (T +T (a+6))d + a] z ( t ) - -(1+T d) n ( t ) (16) p r g p r' g g r r i n which T •* p i l o t v a l v e time c o n s t a n t , s e c , P T = dashpot time c o n s t a n t , s e c , T = "L =* governor r e s p o n s e time c o n s t a n t , s e c , a =* permanent speed droop r e l a t i v e t o r a t e d speed, 30 F i g . 7-a A S c h e m a t i c D i a g r a m o f a M e c h a n i c a l - H y d r a u l i c G o v e r n o r speed s i g n a l p u l s e g e n e r a t o r g e n e r a t o r p u l s e (1) p p u l s e (2) speed sense to speed s w i t c h e s /- summing power / bus a m p l i f i e r . g e n e r a t o r s h a f t p u l s e ( 3 ) , to the tachometer l o a d computer from c u r r e n t -t r a n s f o r m e r s - speed r e g u l a t i o n from p o t e n t i a l t r a n s f o r m e r s 1 >\"Y""''''""-iA^iVr i m p i l o t , -^j^i l s e r v omotor a \». gate l i m i t hrl -d i s t r i b u f tit t i o n v a l v e 0 gate servomotor E l e c t r i c P a r t H y d r a u l i c P a r t F i g . 7-b A Schematic Diagram of An E l e c t r i c - H y d r a u l i c Governor (54] LO 32 speed d e v i a t i o n L o v _ - ^ p i l o t v a l v e and s e r v o d i s t r i b u t i o n v a l v e and gate s e r v o gate J d e v i a t i o n permanent droop t r a n s i e n t droop (dashpot)  (i) F u n c t i o n a l B l o c k D i a g r a m a 6 T r s 1+T s r ' ( i i ) L i n e a r M a t h e m a t i c a l R e p r e s e n t a t i o n k l k 2 s 1+T s F i g . 8-a M e c h a n i c a l G o v e r n o r speed r e g u l a t i o n (or speed droop) speed d e v i a t i o n p u l s e g e n e r a t o r & power a m p l i f i e r s t r a n s d u c e r p i l o t v a l v e & s e r v o d i s t r i b u t i o n v a l v e & gate s e r v o ) gate d e v i a t i o n (i) F u n c t i o n a l Block Diagram R'm^ (or az) k ( 1 + T i s ) ^ ( l + X 2 s ) ( i i ) L i n e a r Mathematical R e p r e s e n t a t i o n k : a m p l i f i e r s g a i n T-^ : p u l s e g e n e r a t o r and power am-p l i f i e r s time c o n s t a n t ^2 '• t r a n s d u c e r time c o n s t a n t F i g . 8-b E l e c t r i c Governor [ 5 4 ] 34 <5 * t r a n s i e n t speed droop r e l a t i v e t o r a t e d speed, and d s t a n d s f o r , where T i s t h e time i n seconds, a i S i n c e b o t h T and T a r e much s m a l l e r than T , Eq. 16 may be s i m p l i -P g r f i e d t o : ( T r ( a + 6 ) d + a ) z = - ( 1 + T r d ) n (17) A f t e r some rearrangement, d i f f e r e n t i a l e q u a t i o n f o r e l e c t r i c governor p r e s e n t e d i n F i g . 8-b can be e x p r e s s e d a s : K. z = -(K + T + R , d ) ( n + R m „ ) p a d l or dz = - ( K , d 2 + K d + K.)n - K. R m (18) d p l l % where K^, K^, and a r e p r o p o r t i o n a l , i n t e g r a l and d e r i v a t i v e g a i n s . Speed r e g u l a t i o n R ( s l o p e o f l o a d - s p e e d r e l a t i o n s h i p ) r a t h e r t h a n speed droop a ( s l o p e of gate-speed r e l a t i o n s h i p ) i s commonly used i n e l e c t r i c g o v e r n o r s . I f permanent feedback i s o b t a i n e d from gate p o s i t i o n i n s t e a d o f l o a d , Eq. 18 changes t o : K. z - -(K + - i + K, d) (n + az) p d d or ( aK, d 2 + 41-hjK ) d + aK.) z •» - (K, d 2 + K d + K.) n (19) d -- — p I d p l For a p r o p o r t i o n a l - i n t e g r a l g overnor, Eq. 19 reduces t o : ((l+o-K ) d + CTK.)z = -(K. + K d) n (20) p i l p I n f a c t , Eq. 20 i s analogous t o Eq. 17 f o r a m e c h a n i c a l governor w i t h ^ p r o p o r t i o n a l g a i n and (-HJT o oT^ governor, t h e r e f o r e , e x h i b i t s t h e same b e h a v i o r as t h e c l a s s i c a l m e c h a n i c a l governor w i t h a dashpot. Though o n l y a f i r s t o r d e r r e s p o n s e g i v e n by e i t h e r Eq. 17 or 20 w i l l be c o n s i d e r e d i n the f o l l o w i n g a n a l y s i s , s i m i l a r p r o c e d u r e can be used as w e l l f o r response e q u a t i o n s of h i g h e r o r d e r , i . e . , f o r a p r o p o r t i o n a l - i n t e g r a l — d e r i v a t i v e e l e c t r i c governor o r a m e c h a n i c a l governor w i t h a p p r e c i a b l e r e s p o n s e time T . The a n a l y s i s , however, becomes more c o m p l i c a t e d . -(j) as (T^~) as i n t e g r a l one. A (PI) e l e c t r i c 35 2.6 P e n s t o c k E q u a t i o n As the governor s t a r t s t o change the gate opening f o l l o w i n g a l o a d change, p r e s s u r e waves a r e c r e a t e d a t t h e upstream f a c e o f t h e t u r b i n e g a t e . These waves t r a v e l a l o n g the p e n s t o c k and a r e r e f l e c t e d w i t h an o p p o s i t e s i g n a t the r e s e r v o i r end. These r e f l e c t e d waves t r a v e l back towards the t u r b i n e gate and a r e superimposed on new p r e s s u r e waves t r a v e l l i n g upstream. A t any time T a f t e r the f i r s t g ate movement, p r e s s u r e head and d i s c h a r g e v a l u e s a t the t u r b i n e , and hence t u r b i n e o u t p u t , depend upon p r e v i o u s r e f l e c -t i o n s as w e l l as upon newly g e n e r a t e d waves. Thus, a c o n t i n u o u s i n t e r a c t i o n e x i s t s between governor o p e r a t i o n , wave p r o p a g a t i o n and t u r b i n e o u t p u t . A r e v i e w of the " r i g i d water column" assumption which has been e x t e n -s i v e l y used i n s i m i l a r l i n e a r a n a l y s e s o f speed g o v e r n i n g i s g i v e n below. T h i s i s f o l l o w e d by the more a c c u r a t e " e l a s t i c water column" t h e o r y . Among s e v e r a l n u m e r i c a l methods o f s o l u t i o n based on t h i s t h e o r y , " c h a r a c t e r i s t i c s method" i s chosen f o r comparison w i t h the a n a l y t i c a l s o l u t i o n o b t a i n e d i n t h i s t h e s i s , s i n c e t h e former i s w e l l e s t a b l i s h e d , has been c o n f i r m e d by e x p e r i -ments [20,24] and p r o v i d e s g r e a t e r f l e x i b i l i t y i n m o d e l l i n g d i f f e r e n t boundary c o n d i t i o n s . The g e n e r a l s o l u t i o n o f the s i m p l e wave e q u a t i o n i s , however, the o n l y one a d a p t a b l e to the r e q u i r e d m a t h e m a t i c a l a n a l y s i s of speed t r a n s i e n t s t h a t i n c l u d e s e l a s t i c i t y e f f e c t s . 2.6.1 R i g i d - w a t e r column assumption. I f e l a s t i c i t y e f f e c t s o f b o t h water i n the p e n s t o c k and of p e n s t o c k w a l l s a r e i g n o r e d , then water can be t r e a t e d as a r i g i d column e x t e n d i n g from the r e s e r v o i r t o the downstream end o f t h e p e n s t o c k . Such a column w i l l o s c i l l a t e w i t h a v e l o c i t y which, a t any g i v e n time, i s c o n s t a n t a l o n g the p e n s t o c k . A p p l y i n g Newton's law of motion (see F i g . 9 ) , we g e t wA(H n + H - H - H.) = - L 4? (21) nO fO n £' g dT ' 36 R e s e r v o i r ~ T H f o 1 9 1 S t - S t . HGL I" A H n ( t ) T a i l Water wAH f(t) wA(H +H f ) HI wAH n(t) F i g . 9 Rigid Water-Column Model 37 I f t h e head l o s s due t o f r i c t i o n i s assumed t o be p r o p o r t i o n a l t o t h e square o f t h e v e l o c i t y , t h e n : H f ( T ) - H f Q ( ^ p - ) 2 (22) Then e q u a t i o n 21 can be reduced t o dq =" — 7~ [h + R q(2+q)] (23) wO L Vo i n which T _ = — - — , i s known as "water s t a r t i n g t i m e " , and R f  W ° g H n O „ f H f 0 i s t h e i n i t i a l s t e a d y s t a t e f r i c t i o n r a t i o , — — • H n 0 With a l i n e a r i z e d f r i c t i o n , p e n s t o c k e q u a t i o n (Eq. 23) becomes: dq - - (h + 2 R q) (24) wO F o r a f r i c t i o n l e s s f l o w , H^Q = 0 and Eq. 24 i s f u r t h e r s i m p l i f i e d t o : dq = - h (25) wO Most i n v e s t i g a t o r s have used t h i s s i m p l e r e l a t i o n s h i p t o r e p r e s e n t p e n s t o c k i n f l u e n c e on p r e s s u r e head and d i s c h a r g e a t the t u r b i n e end. A l t h o u g h i t y i e l d s r e a s o n a b l y good r e s u l t s f o r r e l a t i v e l y slow gate changes, p r o p e r m o d e l l i n g t h a t t a k e s e l a s t i c i t y e f f e c t s i n t o account i s n e c e s s a r y when r a p i d changes a r e c o n s i d e r e d and t h e s e a r e u s u a l i n speed g o v e r n i n g problems. 2.6.2 E l a s t i c water-column (waterhammer) t h e o r y . F o r n o n - v i s c o u s o r l o w - v i s c o s i t y l i q u i d s , s u c h as water, f l o w i n g through a c o n s t a n t diameter p i p e -l i n e , a o n e - d i m e n s i o n a l approach can be used f o r s t u d y i n g the dynamics o f motion. Thus, p r e s s u r e and v e l o c i t y d i s t r i b u t i o n s a t any c r o s s s e c t i o n a r e assumed u n i f o r m . I t i s a l s o assumed t h a t b o t h water and p e n s t o c k w a l l s a r e p e r f e c t l y e l a s t i c and obey Hook's law of l i n e a r e l a s t i c i t y . Use o f a steady s t a t e f o r m u l a f o r f r i c t i o n l o s s e s e v a l u a t i o n i s extended t o t r a n s i e n t s t a t e . More a c c u r a t e r e p r e s e n t a t i o n o f t r a n s i e n t f r i c t i o n l o s s e s i s f r e q u e n c y -38 dependent [34] and adds c o n s i d e r a b l e c o m p l i c a t i o n t o the a n a l y s i s . S i n c e f r i c t i o n has a secondary i n f l u e n c e on t u r b i n e speed d e v i a t i o n s , i t i s not n e c e s s a r y t o use such an approach. C o n s i d e r i n g d i f f e r e n t f o r c e s a c t i n g on a mass element of t h e f l o w , t h e e q u a t i o n of motion t a k e s t h e form: - g H x - VV X + V T + ^ V| V| - 0 (26) S u b s c r i p t s X and T denote p a r t i a l d i f f e r e n t i a t i o n w i t h r e s p e c t t o d i s t a n c e ( p o s i t i v e i n upstream d i r e c t i o n ) and time, r e s p e c t i v e l y , and f i s the Darcy-Weisbach f r i c t i o n f a c t o r . C o n t i n u i t y e q u a t i o n i s develop e d by c o n s i d e r i n g changes w i t h i n a c o n t r o l volume embodying t h i s mass element. The net r a t e o f mass i n f l o w t o the c o n t r o l volume i s e q u a l to time r a t e o f mass i n c r e a s e w i t h i n i t . F o r a h o r i z o n t a l p i p e t h i s may be f i n a l l y e x p r e s s e d a s : " V H X + H T " T V X = ° ( 2 7 ) i n which a 2 « (JL)/[1 +c(f)<£).1 ( 2 g ) where a i s the p r e s s u r e wave v e l o c i t y , K i s the b u l k modules of e l a s t i c i t y f o r water, E i s t h e w a l l ' s modules of e l a s t i c i t y , p i s water d e n s i t y , D i s pe n s t o c k d i a m e t e r , e i s the w a l l t h i c k n e s s , and c i s a c o n s t a n t dependent on t h e l o n g i t u d i n a l r e s t r a i n t of the pe n s t o c k . For t h e d e r i v a t i o n d e t a i l s of Eqns. 26 and 27 see r e f s . [20,24,27]. U s i n g n o r m a l i z e d v a r i a b l e s : X . T , 2L C , . H ( x , t ) . V ( x , t ) x - - , t =- - — where T = — , h ( x , t ) = — — 2 — and v ( x , t ) - — r - 2 — , L T , c h a H n V ch nO 0 Eqns. 26 and 27 become: V 2 aV - h - ~ - v v + „ I T U v + R £ v l v l = 0 (29). X g H n 0 X 2 g H n O fc f ' where R f = H f ( J / H . 39 2V n 2aV. and h - v h — - v =» 0 (30) t a x g H n 0 x An o r d e r of magnitude st u d y of d i f f e r e n t c o e f f i c i e n t s r e v e a l s t h a t c o n t r i b u t i o n s by c o n v e c t i v e a c c e l e r a t i o n term (v v ) and f r i c t i o n term a r e x s m a l l compared to o t h e r terms i n Eq. 29. S i m i l a r l y f o r t h e term (v h ) when compared t o r e m a i n i n g terms of Eq. 30. With t h e s e terms dropped, equ a t i o n s r o f motion a n d ( c o n t i n u i t y e q u a t i o n become: - n x + p Q v t = 0 (31) and n t - ^P.Q = 0 (32) The c o n s t a n t i s known as " p i p e l i n e o r A l l i e v i ' s c o n s t a n t " , a V o P ° " 2 g H n O ' I f t h e f r i c t i o n term i s p r e s e r v e d , e q u a t i o n o f ~ m o t i o n t a k e s the form: - n + p r . v + R v 2 = i 0 f o r v > 0 (33) x K 0 f , R e l a t i v e d e v i a t i o n s from i n i t i a l s t e a d y s t a t e h'(x,t) and v ' ( x , t ) a r e r e l a t e d t o n o r m a l i z e d v a l u e s : n ( x , t ) - 1 + R x + h ' ( x , t ) (34) and v ( x , t ) » 1 + v ' ( x , t ) (35) S u b s t i t u t i n g i n Eqns. 32 and 33 we f i n a l l y g e t : h ^ ( x , t ) - 4 p Q v ^ ( x , t ) =* 0 (36) and h x ( x , t ) - p Q y^.(x,t) = R f ( 1 + v ' ) 2 - R f (37-a) o r , w i t h l i n e a r i z e d f r i c t i o n : h x ( x , t ) - p Q v£(x,t) = 2R fv* (37-b) For f r i c t i o n l e s s f l o w R^ = 0, Eq. 37 y i e l d s : h x ( x , t ) - p Q v£(x,t) - 0 (38) S e v e r a l a n a l y t i c a l , n u m e r i c a l and g r a p h i c a l methods have been used to s o l v e the p a i r o f f i r s t o r d e r h y p e r b o l i c p a r t i a l d i f f e r e n t i a l Eqns. 36 and 37. C l o s e d form a n a l y t i c a l s o l u t i o n s a r e not a v a i l a b l e except f o r a few problems 40 of v e r y s i m p l e boundary c o n d i t i o n s . 2.6.2a C h a r a c t e r i s t i c s method. Though a s o l u t i o n based on t h i s method i s a v a i l a b l e f o r bo t h e q u a t i o n of motion and c o n t i n u i t y e q u a t i o n i n t h e i r o r i g i n a l forms (29) and (30), d i s c u s s i o n w i l l be r e s t r i c t e d h e r e t o l i n e a r -i z e d - f r i c t i o n Eqns. 37 and 36 which a r e used i n d e r i v a t i o n o f an a n a l y t i c a l s o l u t i o n f o r speed t r a n s i e n t s . R e f e r r i n g t o F i g . 10, t o t a l d e r i v a t i v e s a l o n g t h e p a t h c + t r a v e r s e d by a p r e s s u r e wave o r i g i n a t i n g a t t u r b i n e end t a k e the p a r t i c u l a r v a l u e s : dh' , h . dx + , dt x dt t » 2 h' + h' (39-a) and dv' , dx-:. , ——- =c v ' h v dt x dt t = 2 v 1 + v' (39-b) x t U s i n g t h e s e r e l a t i o n s , a l i n e a r c o m b i n a t i o n o f Eqns. 36 and 37 can be w r i t t e n as a s i n g l e o r d i n a r y d i f f e r e n t i a l e q u a t i o n : f - * o ^ - - « f T ' " ° < 4 ° > -dx which i s v a l i d o n l y a l o n g l i n e s of s l o p e = +2 i n x - t p l a n e , r e f e r r e d : ; t 6 as c + c h a r a c t e r i s t i c s . dx S i m i l a r l y , a l o n g any l i n e of s l o p e - j j / =•= -2 which r e p r e s e n t s t h e p a t h o f a r e f l e c t e d wave from the r e s e r v o i r end, r e f e r r e d t o as c c h a r a c t e r i s t i c : f + 2 P o - ^ + 4 R f - • '<*«• Together w i t h the boundary c o n d i t i o n s , f i n i t e - d i f f e r e n c e forms of Eqns. 40 and 41 may be s o l v e d n u m e r i c a l l y t o determine v a l u e s o f h ' ( x , t ) and Ax v ' ( x , t ) a t each p o i n t o f a r e c t a n g u l a r g r i d o f s i d e s r a t i o — = 2, F i g . 10. 5 t - A t 1.0 x+Ax x x-Ax 0.0 R e s e r v o i r t u r b i n e F i g . 10 G r i d P o i n t s f o r Method of C h a r a c t e r i s t i c ' s S o l u t i o n F i g . 11 Approaching and R e f l e c t e d Waves f o r T r a v e l l i n g - w a v e S o l u t i o n 42 2.6.2b T r a v e l l i n g - w a v e s o l u t i o n . The two f i r s t o r d e r homogeneous p a r t i a l d i f f e r e n t i a l Eqns. 36 and 38 may be combined i n t o one second o r d e r p a r t i a l d i f f e r e n t i a l e q u a t i o n i n e i t h e r h ' ( x , t ) or v ' ( x , t ) : h' - 4h'~ " 0 (42) t t xx and v 1 - 4 v , _ - 0 (43) t t XX These r e p r e s e n t a c l a s s i c a l o n e - d i m e n s i o n a l wave e q u a t i o n f o r which the g e n e r a l s o l u t i o n i s : h ' ( x , t ) » F ( t - 0.5x) + f ( t + 0.5x) (44) and v ' ( x , t ) > TT^- f F ( t - 0.5x) - f ( t + 0.5x] (45) Shapes of the unknown f u n c t i o n s F and f depend upon i n i t i a l and boundary c o n d i t i o n s . They r e p r e s e n t p r e s s u r e waves t r a v e l l i n g w i t h v e l o c i t y a i n p o s i t i v e and n e g a t i v e x d i r e c t i o n s , r e s p e c t i v e l y (see F i g . 11). A p p l y i n g boundary c o n d i t i o n a t r e s e r v o i r end ( h 1 ( l , t ) = 0.0), we get from Eq. 44: f ( t + 0.5) = - F ( t - 0.5) or f ( t ) =» - F ( t - 1) = - F ( T I ) where T . = t - i (46) L e t h ^ ( t ) and q ^ ( t ) be r e l a t i v e head and d i s c h a r g e d e v i a t i o n s a t x * 0 ( t u r b i n e end) a t a time t where i - 1 _< t < i , i = 1, 2, 3, . . . From Eqns. 44 and 45 we have: h . ( t ) = F . ( t ) + f . _ 1 ( t ) (47) and q ( t ) ~ ( F . ( t ) - F. At)) (48) x 2 p Q x x-1 S u b s t i t u t i n g f o r r £ _ ^ ( t : ) from Eq. 46, t h e n : h ± ( t ) = F ± ( t ) - F ± _ 1 ( T 1 ) -1 2p, and q . ( t ) = (F. ( t ) + F. - . ( T ^ ) x 2 p Q x x-1 1 (49) (50) S i m i l a r l y , r e l a t i v e d e v i a t i o n s a t time ( t - 1) a r e : 43 h i _ i ( T l ) ' * F i - i ( T l ) ' " F i - 2 ( T 2 ) ( 5 1 ) and q - . ^ x i ) = ^ ( F ^ ^ i ) + F ± _ 2 ( t 2 ) ) ( 5 2 ) The unknown f u n c t i o n F can be e l i m i n a t e d and a d i r e c t r e l a t i o n between r e l a t i v e d e v i a t i o n s a t two subsequent time i n t e r v a l s i s o b t a i n e d : h . ( t ) + h ^ C x ! ) = - 2 p 0 ( q i ( t ) - q ^ O i ) ) f o r i ^ l <_ t < x (53) T h i s r e l a t i o n i s known as " A l l i e v i ' s i n t e r l o c k e d s e r i e s " . In t h i s t h e s i s , a new d i f f e r e n t form o f t h a t r e l a t i o n w i l l . b e u t i l i z e d t o r e p r e s e n t p e n s t o c k e q u a t i o n . A s u c c e s s i v e a p p l i c a t i o n of Eq. 53 r e s u l t s xn: F o r 0 £ t < 1: h x ( t ) - - 2 p Q q x ( t ) F o r 1 <_ t < 2: h 2 ( t ) = - 2p- q 2 ( t ) + 2 Q q l ( x l ) - I h ^ ) . = - 2 p Q q 2 ( t ) - 2 h x ( T l ) For 2 < t < 3: h 3 ( t ) - - 2 p Q q 3 ( t ) + 2 p Q q 2 ( T l ) - h 2 ( T l ) - - 2 p Q q 3 ( t ) - 2 ( h 2 ( T l ) + h j C x a ) ) e t c . G e n e r a l l y , f o r i - 1 <_ t < i , " t h e pe n s t o c k e q u a t i o n " which summarizes p e n s t o c k i n e r t i a l and e l a s t i c e f f e c t s on p r e s s u r e head and d i s c h a r g e a t t h e t u r b i n e end, becomes: i - 1 h . ( t ) = - 2 p Q q ( t ) - 2 E h ( T . ) 1 1 3=1 1 J 3 -1 1 - 1 or q . ( t ) - ( h . ( t ) + 2 E h. . ( x . ) ) i - 1 < t < i (54) 2 P q x j = 1 x-j 3 S i n c e i t i s a p i e c e - w i s e c o n t i n u o u s r e l a t i o n s h i p , a n a l y t i c s o l u t i o n to f o l l o w w i l l n e c e s s a r i l y have same c h a r a c t e r . Use of a computer i s not n e c e s s a r y i f o n l y t r a n s i e n t s d u r i n g f i r s t few i n t e r v a l s , 4 t o 5 of (—) SL seconds, a r e needed. Such a time i s l o n g enough i n most cases to a c h i e v e maximum speed d e v i a t i o n . Now, w i t h a l l g o v e r n i n g e q u a t i o n s f o r d i f f e r e n t system components ready, we p r o c e e d to a g e n e r a l a n a l y t i c a l s o l u t i o n f o r v a r i a b l e s i n v o l v e d and i n p a r t i c u l a r f o r speed. »-.•••;• CHAPTER I I I SPEED DIFFERENTIAL EQUATION AND ITS SOLUTION In t h i s c h a p t e r , a n a l y t i c a l s o l u t i o n s t a r t s w i t h the d e r i v a t i o n of the speed d i f f e r e n t i a l e q u a t i o n f o r the f r i c t i o n l e s s f l o w c a s e . The complete s o l u t i o n i s the n p r e s e n t e d . I n analogy t o a d i s t o r t i o n l e s s t r a n s m i s s i o n l i n e , t h e s o l u t i o n of t h e wave e q u a t i o n w i t h a l i n e a r i z e d f r i c t i o n i s o b t a i n e d . E x p r e s s i o n s f o r speed d e v i a t i o n s t a y b a s i c a l l y t h e same, however. U s i n g T a y l o r ' s e x p a n s i o n s , e x a c t s o l u t i o n s a r e r e a r r a n g e d and e x p r e s s e d by approximate but more e f f i c i e n t f o r m u l a s . S o l u t i o n s f o r head and d i s c h a r g e d e v i a t i o n s a r e a l s o p r e s e n t e d . 3.1 Gov e r n i n g E q u a t i o n s S u b s t i t u t i n g f o r h y d r a u l i c t o r q u e as g i v e n by Eqn. ( 8 - b ) , the e q u a t i o n of motion f o r t h e r o t a t i n g mass becomes: T' Dn = £ , h + £ n + £ z - (m + a n ) ( 5 5 ) mO mh mn mz £ £ T i n which T' - and D = mO T , dt ch Load damping may be combined w i t h t u r b i n e - s p e e d r e g u l a t i o n t o g i v e t o t a l s e l f - r e g u l a t i o n c o e f f i c i e n t , a =» a„ - £ . With t h i s we g e t : £ mn F i [ n ( t ) ] =•= £ , h + £ z - mn ( 5 6 ) 1 mh mz £ where F-, = T' D + a ( 5 7 ) 1 mU For low s p e c i f i c speed t u r b i n e s £ - - 1, F i g . 6 , whereas a v a r i e s mn £ from -1 f o r ohmic l o a d s w i t h v o l t a g e r e g u l a t i o n t o a p p r o x i m a t e l y 4 i n t h e 45 46 case of n o - v o l t a g e r e g u l a t i o n [ 4 ] . Hence, the t o t a l s e l f - r e g u l a t i o n ranges between 0 to 5. I t f o l l o w s from Eq. 55 t h a t a p o s i t i v e s e l f -r e g u l a t i o n produces a c o u n t e r e f f e c t t o l o a d c c h a n g e s , thus i n c r e a s i n g system s t a b i l i t y . Dependent on governor type and feedback d e s i g n , governor response may be r e p r e s e n t e d by one of the Eqns. 17, 18 (with & d =•= 0) or 20, which have the g e n e r a l form: F 2 [z"(t)J = - F 3 [ n ( t ) ] + gk (58) F 2 and F 3 a r e f i r s t o r d e r d i f f e r e n t i a l o p e r a t o r s : F 2 = g l D + g 2 (59) and F 3 = g 3 D + 1 (60) The c o n s t a n t s g^ through g^ a r e g i v e n i n the f o l l o w i n g t a b l e . Governor's C o n s t a n t s i l §2 83 84 M e c h a n i c a l governor, gate f e e d back (6+a)T^_ a T^ _ 0 (PI) E l e c t r i c g overnor, l o a d feedback 1/KV 0 K p / K i " R m & (PI) E l e c t r i c governor, gate feedback (1+oK )'/K'. a K /K| 0 p x p i where T* = T /T , , and K'. - K. T ^ r r ch x x ch Thus, f o r a p a r t i c u l a r time i n t e r v a l i - 1 _< t < i , i = 1, 2, 3, ... , system g o v e r n i n g e q u a t i o n s a r e : R o t a t i n g p a r t s : F n [ n . ( t ) ] - I , h . ( t ) + I z . ( t ) - m (61) 1 x mh x mz x £ Governor: F 2 [ z j L ( t ) ] =» - F 3 [ n ± ( t ) ] + gk (62) T u r b i n e d i s c h a r g e : q . ( t ) = % . h . ( t ) + & z . ( t ) + £ n . ( t ) (63) 6 H i v *qh x v *-qz i ^qn i Penstock: q . ( t ) = -^=- h. ( t ) - — SM. (64) 1 2 po 1 p 0 1 i - 1 where SM. - I h. . ( T.) (65) x . -, x - i j 3=1 E l i m i n a t i n g h., q. and z_^, we get speed d i f f e r e n t i a l e q u a t i o n : a 3 L [ n . ( t ) ] - - — n - F 2[SM.]• + ah (66) 1 H mO 1 where: L = D 2 + 2& 1 D + 2a2 (67-a) a l " 2gTT^ [ g 2 TmO + a g l + g 3 ( V z " P0 & 3 £ q z ) + § 1 P0 ^ ( 6 7 _ b ) 3 2 - 2 1 ^ [ a g 2 + *mz ~ p0 3 3 V + S 2 P0 a 3 V ( 6 7 _ c ) a 3 " l + 2 p m £ a n d ( 6 7 _ d ) P0 qh 'k l g 2 ^ ~ § t t ( £mz " p 0 a 3 % z ) ] (67-e) g l m 0 £ q The r i g h t - h a n d s i d e o f Eq. 66 s t i l l , however, i n c l u d e s head v a r i a t i o n s f o r p r e v i o u s time i n t e r v a l s , 0 < t < i - 1 . By e l m i n a t i n g q. and z. from Eqns. 61, 63 and 64, a r e l a t i o n between h ^ ( t ) and n ^ ( t ) i s o b t a i n e d : h . ( t ) ' p. a 5 ( F K [ n . ( t ) ] + — £ S M . + £ m ) x M 0 5 x p Q mz x qz £ where: (68) F 4 E £ T' D + (a £ + £ £ ) (69-a) H qz mO qz mz qn a 5 " " 2 / U m z ( 1 " p 0 / p 0 ) ] ( 6 9 _ b ) and p * = 0.5 £ /(£ , £ - £ £ , ) (69-c) ^0 mz mh qz mz qh For the s p e c i a l case p ^ =• p * , Eq. 68 i s not v a l i d and a s e p a r a t e t r e a t m e n t i s n e c e s s a r y . T h i s s i n g u l a r i t y c o r r e s p o n d s to =•= 0.5 f o r a t u r b i n e o p e r a t i n g a t t h e p o i n t of b e s t e f f i c i e n c y . S u c c e s s i v e a p p l i c a t i o n s of Eqns. 65, 66 and 68 y i e l d : F o r i =- 1, 0 <_ t < 1: SMl =•= 0 L [ n x ( t ) ] =•= a.h h l ( t ) =" P 0 a 5 ( ^ [ ^ ( t ) ] + £ q z mA) 48 For i =•= 2, 1 <_ t < 2: SM 2 - h i ( T x ) p 0 a s C F i t E n v d ! ) ] + £ q z m^) L [ n 2 ( t ) ] - - - = 1 - M 5 ^ ] + a 4 g l T m O a 6 — F 2 F i J n l ( ^ l ) ] + a 4 + a 7 g i T ; 0 where: a 6 = - p 0 a 3 a 5 (70-a) §2 a 6 and a 7 =>= —=77— I m (70-b) g l T ; 0 £ h 2 ( t ) - p 0 a 5 { F 4 [ n 2 ( t ) + a 5 £ m z n ^ ) ] + (1 + a 5 A ^ ) m£} For i = 3 , 2 £ t < 3 : SM 3 - h 2 ( T l ) + h x ( x 2 ) - P 0 a 5 { F t + [ n 2 ( T 1 ) + a 8 n 1 ( r 2 ) ] + (1 + a 8 ) m^} where: a R = 1 + a^Jl (70-c) 0 3 mz " a 3 L [ n 3 ( t ) ] - -—r- F 2 [ S M 3 ] + ah  g l mO a 6 — F 2 F I t [ n 2 ( T l ) + a 8 n 1 ( x 2 ) ] + a 4 + a y (1 + a 8 ) g l T m O h 3 ( t ) - p 0 a 5 { F [ + [ n 3 ( t ) + a 5 A m z ( n 2 ( T l ) + n 1 ( x 2 ) ' ) ] + (1 + ^ S ^ 1 + a 8 ) H q z F o r i = * 4 , 3 < _ t < 4 : SM 4 = h 3 ( T l ) + h 2 ( x 2 ) + h 1 ( x 3 ) > P 0 a 5 { F 4 [ n 3 ( T l ) + a 8 n 2 ( x 2 ) + a 2 n i ( x 3 ) ] + (1 + a 8 + a 2 ) m^} " a 3 LEiUfCt)] = „ T , F 2 [ S M 2 ] + a 4 S l m 0 a 6 g l T m O — F 2 F l t [ n 3 ( x 1 ) + a 8 n 2 ( x 2 ) + a 2 n 1 ( x 3 ) ] + &h + a70- + aQ + a 2 ) 49 hi+(t) = P 0 a 5 { F i t [ n i f ( t ) + a 5 I ( ^ ( x ^ + a 8 n 2 ( x 2 ) + a 2 n 1 ( x 3 ) ] + TJ1Z 8 ( l ~ f a s £ (1 + a f t + a 2 ) ) £ m } 5 qz B 8 qz £ e t c . Hence, f o r i - 1 <_ t < i , d i f f e r e n t i a l e q u a t i o n o f r e l a t i v e speed d e v i a t i o n becomes: a 6 1-1 _. . i - 1 L [ n . ( t ) ] ^ T - F 2 F 4 [ ^ 4~J~ n . ( x . _ . ) ] + a^ + a 7 E a^ ,1=1,2,3, ... (71) 1 g l im0 j = l J 1 2 j = l A l s o , r e l a t i v e head d e v i a t i o n a t t u r b i n e end i s g i v e n by: i - 1 1 - 1 1---1 h . ( t ) = p 0 a 5 { F 4 [ n i ( t ) + a 5 _E_^ a j J V ^ . J ] + (1 + a 5 £ E aj! 1 ) £ m > 1=1,2,3, ... (72) 0 mz . , 0 qz £••' In b o t h e q u a t i o n s , a l l summations reduce t o z e r o f o r i = 1. 3.2 S o l u t i o n of Speed E q u a t i o n L e t n ! ( t ) = n . ( t ) - a 8 n. ( T . - . ) l l 0 l - l l - l S i n c e L(D) i s a l i n e a r o p e r a t o r , we get from Eq. 71: L [ n ! ( t ) I - a 6 £• (D 2 + a 9 D + a 1 0 ) [ n . - ( t ] ) ] + a 7 + ak(l - a 8 ) (73) where g? £ £ a . z . mz qn /-?/ \ a 9 = + - + ^ r — 3 - (74-a) mO 1 mO qz and g 2 £ £ r T " '! mO -qz S o l u t i o n o f Eq. 73 w i l l be t h e sum of a complementary f u n c t i o n n ^ ( t ) and a p a r t i c u l a r i n t e g r a l n * * ( t ) . The c o r r e s p o n d i n g a u x i l i a r y e q u a t i o n w i l l have two r e a l and d i f f e r e n t , e q u a l or complex c o n j u g a t e r o o t s i f a 2 i s g r e a t e r than, e q u a l to or l e s s than 2 a 2 , r e s p e c t i v e l y . P r o c e d u r e s a r e the same f o r any of t h e s e c a s e s , so we 50 2 w i l l c o n s i d e r o n l y the c a s e a^ < 2 a 2 i n d e t a i l . S o l u t i o n s f o r the o t h e r two cases a r e p r e s e n t e d i n Appendix A a l o n g w i t h those f o r = p^. L e t (- a 1 ± jco) be t h e r o o t s of t h e a u x i l i a r y e q u a t i o n , w i t h j = / - I and co 3 / 2 a 9 - a 2 , where a 2 < 2 a 2 . The complementary f u n c t i o n t h e n becomes: n * ( t ) = expC-aj^t) [cqsincot + a 2 c o s u j t ] (75) The c o n s t a n t s a\ and a 2 a r e determined from i n i t i a l c o n d i t i o n s a t t =•= i - 1. I n terms of 0. , , - t h i s may be r e w r i t t e n a s : -l - l n i ^ 9 i - l ^ ' e x P ( a l l e i _ i ) [ a l s i n 9 i _ i + a 2 c o s 6 1 _ 1 ] (76) where 9 i - l ' ^ i - l ~ ^(.t - ( i - 1 ) ) (77-a) a l and a-i -i =» (77-b) to In f a c t , t h e parameters x and 9 w i l l always v a r y between 0.0 t o 1.0 and 0.0 t o co, r e s p e c t i v e l y , d u r i n g each time i n t e r v a l c o n s i d e r e d . T h e i r sub-s c r i p t s , t h e r e f o r e , w i l l be dropped out from h e r e on w i t h the u n d e r s t a n d i n g t h a t f o r ± - l < L t < ± : j = ' t - ( i - 1 ) and 0 = cox. 3.2.1 P a r t i c u l a r i n t e g r a l . R e s u l t s f o r the f i r s t few i n t e r v a l s show t h a t t h e complete s o l u t i o n f o r i - 1 <_ t < i takes the form: n ± ( 0 ) =- e x p ( a n 6 ) [ R ± ( 0 ) s i n e + S ± ( 0 ) cos0] + CKL (78) R^ and S^ a r e p o l y n o m i a l s of degree ( i - 1 ) : i R.(0) - Z ' r . . 0 i _ J (79-a) 1 j - 1 1 , J i ._. and S . (6) - Z s. . 0 1 J (79 - b ) 1 • -i i»3 3-1 J C1SL i s a c o n s t a n t f o r t h i s p a r t i c u l a r i n t e r v a l . Then, on s u b s t i t u t i o n o f ri^_^(8), Eq. 73 becomes: 51 L [ n ! ( 6 ) ] = a6l e x p ( a , i 0 ) [ B . ( 6 ) s i n 0 + G.(0) cos0] X 1z X X + a 7 + a u ( l - a 8 ) + a 6 a 1 Q H q z C N i _ 1 (80) The p o l y n o m i a l s B^ and C^ a r e d e f i n e d a s : i - 1 i - j - 1 B.(0) = E b. . 0 (81-a) 3=1 i - 1 i - j - 1 and C.(6) - E c. . 0 (81-b) 1 • i 1>3 3=1 3.2.2 R e l a t i o n s between B., C , and R. , , S. ,.• Comparing the r i g h t -x x' x-1 x-1 f b hand s i d e s of Eqns. 73 and 80, we get B ± ( 6 ) » w 2 [ ( a 1 2 + a 1 3 D' + D' 2) R^ C e ) - ( a x 3 + 2D') S ^ ( 6 ) ] (82-a) and C ± ( 0 ) =• c o 2 [ ( a 1 2 + a 1 3 D' + D' 2) S ± _ 1 ( 0 ) + ( a 1 3 + 2D') R ± _ 1 ( 0 ) ] (82-b) where D' -= £ (83-a) a10 a 9 a l l a 1 ? » + + a 2 - 1 (83-b) 1 2 2 co 11 ' a 9 and a 7 q = h 2ai i (83-c) 13 ^ i 1 A p p l y i n g t h e d i f f e r e n t i a l o p e r a t o r s t o R^_^ and ^, c o e f f i c i e n t s of B^ and C. a r e o b t a i n e d : x b i , j ' " 2 [ a l 2 r i - l , j " a 1 3 ^ i - l , j + <^^±„iti-i ~ 2 s i - l , j - l + ( i - j + 1) r •)] (84-a) x x, j z and c ± ) . = a ) 2 [ a 1 2 ^ _ 1 } j + a ^ r . ^ . + ( i - j ) ( a ^ s . ^ ._ ± + 2 r i - l , j - l + ( i " j + 1 ) S i - l , j - 2 ) ] ' j - * 1 5 2 ' 1 - 1 ( 8 4 _ b ) where r . n i = s. = 0 f o r k < 1 (84-c) 52 S i n c e the f i r s t r e f l e c t e d wave a r r i v e s a t t u r b i n e end a t t = 1, t h e s e r e l a t i o n s a r e not used f o r t < 1. The r i g h t - h a n d s i d e of Eq. 73 w i l l be mere l y a c o n s t a n t f o r the f i r s t i n t e r v a l , o _< t < 1. A l s o , as a g e n e r a l r u l e , h e n c e f o r t h , any s u b s c r i p t e d v a r i a b l e y IC , Jo s h o u l d be a u t o m a t i c a l l y s e t t o z e r o i f k < 1 or I < 1. The arguments of the e x p o n e n t i a l , s i n e and c o s i n e terms a p p e a r i n g on the r i g h t - h a n d s i d e of Eq. 80 a r e t h e r e a l and im a g i n a r y p a r t s o f (- a^ ± jw) which a r e a t t h e same time c o n j u g a t e r o o t s o f m u l t i p l i c i t y 1 f o r t h e a u x i l i a r y e q u a t i o n . T h i s i s m a t h e m a t i c a l l y c a l l e d a " r e s o n a n t " case of d i f f e r e n t i a l e q u a t i o n s , but i t does not n e c e s s a r i l y imply resonance c o n d i t i o n s i n the p h y s i c a l model. The p a r t i c u l a r i n t e g r a l f o r such a case can be w r i t t e n a s : n**(0) = e x p ( a n e ) [B!! (e) s i n e + C^(e) cose] + BN ± (85) Degrees of p o l y n o m i a l s B^ and C^,are one s t e p h i g h e r than B_^  and CL: B !(e) = E b! . e i _ j (86-a) j - l . 1 > 3 i . . • c!(e) = z c l . e 1 " 3 (86-b) 1 • i I,J j = i w i t h b! . - c! . = 0 (86-c) 1 , 1 1 , 1 BN. i s a c o n s t a n t f o r i - 1 < t < i . x — S i n c e n * A ( 6 ) s h o u l d s a t i s f y Eq. 80, we g e t : x a 6 D J- 2B! - 2D'C! - - 9 - A B. (87-a) x x o)z qz x a 6 D' 2C| + 2D'B! - - 5 - £, C. (87-b) i . i ^ qz l and BN. = T T — [ a 7 + a^ (1 - a 8 ) + a 6 a 1 0 i > CN. ] (87-c) 1 ^clV> C[Z X ~ J . 1 The p a r t i c u l a r i n t e g r a l w i l l be s l i g h t l y d i f f e r e n t i f a 2 = 0, as shown i n Appendix A. Eqns. 87-a and 87-b can be reduced t o : 53 1 3 6 b! . = _,T .v ( — I c. . - ( i - j ) ( i - j + l ) c! . ,) (88-a) x,j 2 (1-3) co qz 1,3 i , J - l -1 3 6 c! . = 9 r . X . , ( — A b. . - ( i - j ) ( i - j + l ) b! . J (88-b) 1,3 2 (1-3) " i z I,J 1,3-1 and b! . = c! . = 0 (88-c) 1 , 1 1 , 1 A d i r e c t r e l a t i o n s h i p between B^, C!^  and R j ^ ' "*"s °btained by e l i m i n a t i n g B. and C. from Eqns. 84 and 88: 1 1 b i,3 " T \ z {TI=iy ( a l 2 S i - l , J + " ^ i - l , ^ + I " S i - 1 , 3 - 1 + ( 2 + — } r i - l , 3 + IsCl-j+l) U^.-^.,'+ ( i - 1 + 2 ) ( r . _ l j . _ 3 - b ^ ) ] } (89-qz a 6 . 1 • ' a 1 3 c'. . I {-7-.—rr ( a 1 ? r , . . - a n s , . .) + — — r . ., . - -1,3 2 qz L ( i - 3 ) 1 Z i - l , 3 l i i - l , 3 2 1-1,3"! a 1 2 (2 + — ) s . _ 1 ) j _ 1 - % ( i - 3 + D [ a 1 3 s . _ 1 J _ 2 + (i-i+2) -( s , , . . " -j— c' . ? ) ] } , j - 1,2,3, i - 1 (89-b) 1 - 1 , 3~J aZ%„.y 1>3~ Z and b! . =- c\ . - 0 (89-c) 1 , 1 1 , 1 3.2.3 G e n e r a l s o l u t i o n . I f we d e f i n e b! . = a i , and c! .=• a , , the 1 , 1 x ' 1 , 1 1 g e n e r a l s o l u t i o n f o r Eq. 73 becomes: n^(e) - e x p ( a n 6 ) [B^(6) s i n 6 + C^(e) cose] + BN ± (90) S i n c e n^(e) =* n^(e) + a 8 n_^_^(0), we f i n a l l y g e t : n ± ( e ) =- e x p ( a u e ) [R ± ( e ) s i n e + S (e) cose] + CN (91) which has the same form as p r e d i c t e d b e f o r e i n Eq. 78. C o e f f i c i e n t s of R. and S. a r e determined by: 1 1 J 1 a 1 5 r . A =" ~ — r ( a i u s . .. . + a v s r - 1 •) H T~ s • . 1 + a 1 R r . . . . + 1,3 1-3 1 4 i - l , 3 l b 1-1,3 2 1-1,3-1 1 6 1-1,3-1 JgCl-j+l) [ a 1 : 5 r i _ 1 > j _ 2 + 3s(i-j+2) ( - r ^ j - 2 + a l 7 r i _ l , j - 3 ) ] ( 9 2 _ a ) 54 _1 a 1 5 and s. . = -—r (aii+r. , - - a 15 s - n .)' T~ r - i • i + a 16 s - i • i + i,3 i - 3 i - l , 3 i - l , 3 2 i - l , j - l 1 D l - l , j - 1 ^ ( i - j + D E a i s s ^ ^ ^ + hi±-±+2)(-su^2 + a i 7 s i _ 1 ) j _ 3 ) ] (92-b) 3 =* I , 2 , • • • , i - 1 where =•= h3-s a12^^z (93-a) a 1 5 - J s a 6 a 1 3 £ q z (93-b) a16 ' a 6 \ z U + + a 8 ( 9 3 " c ) a17 ' a 6 * q z + a8 ( 9 3"d) The c o n s t a n t CN. i s g i v e n by: a4 CN], - 7777/ (94-a) and CN ± = a 1 8 CN + a 1 9 , i > 1 (94-b) where a6 a10 a18 - * 8 +-2aVJ-*qz ( 9 4 " C ) and a 1 9 = ( a 7 + a^Cl-ag)) (94-d) Both r . . and s. . a r e s t i l l t o be determined from i n i t i a l c o n d i t i o n s 1 , 1 1 , 1 f o r t h e i n t e r v a l i - 1 _< t < i . 3.2.4 I n i t i a l c o n d i t i o n s . I t can be e a s i l y shown t h a t b o t h q ( t ) and h ( t ) as w e l l as n ( t ) and z ( t ) a r e c o n t i n u o u s a t the b e g i n n i n g of each time i n t e r v a l . I t f o l l o w s from Eq. 55 t h a t t h e f i r s t d e r i v a t i v e of n ( t ) i s a l s o c o n t i n u o u s a t t h e s e i n s t a n t s , except a t t = 0. Hence, a t t = i - 1 we have: n.(0) - n. . (OJ) (95-a) l l - l dn. dn a n d * T U ' ^ f 1 U ( 9 5" b ) 55 A p p l i c a t i o n of t h e s e c o n d i t i o n s y i e l d s : i - 1 J i , i = " i - l " ^1 T * 2 0 ^ Ka2lL±-l,j " a i - l , j - ' 1 ^ 1 s. . = CN - CN. + a 2 0 S ( a ^ r , 1 ^ + s_. 1 ) OJ j (96-a) i - 1 and r . r . ^ - a n C C N . - CN.^) - + a 2 0 ^ [ ( r . ^ . - a ^ s . ^ . ) + ( a 2 1 r . . . + s. . .)] w 1^" 1 (96-b) OJ Z i i - l , 3 i - l , 3 — m_ F o r i = 1, ^ 1 = + a n CN X (96-c) ' mO where a 2 0 = exp(-a^) C O S O J (97-a) and a 2 i = tanu (97-b) The exact s o l u t i o n f o r n^(e) i s now complete and w e l l d e f i n e d p r o v i d e d t h a t s o l u t i o n of n. „(e) has been a l r e a d y determined. l - l E x p r e s s i o n s f o r o t h e r v a r i a b l e s can be d i r e c t l y d e r i v e d from t h i s s o l u t i o n as f o l l o w s . 3.3 E v a l u a t i o n o f . h . ( t ) , q ± ( t ) , and z ± ( t ) R e l a t i o n s between r e l a t i v e head d e v i a t i o n s f o r two c o n s e c u t i v e i n t e r -v a l s c o u l d be o b t a i n e d from Eq. 72: h (9) - a 8 h (9) .« p o a g F j n ^ e ) - n ^ C e ) ] (98) L e t t h e g e n e r a l form f o r h_^(e) be: h (e) - e x p ( a 1 1 9 ) [RH i(6) sinO + SH ± ( e ) cose] + CH.. (99) RH^ and SH^ a r e a l s o p o l y n o m i a l s of degree ( i - 1 ) : i . . RH.(6) - E r h . . 0 1 3 (100-a) j - l i . . and SH . ( e ) = E sh. . QX~3 (100-b) 3=1 56 A p p l y i n g Eq. 98, we g e t : r h i , j ' a 8 r h i - l , j - l + a ^ ( a 2 3 r i 5 . - s ^ . ) - a ^ C a^r.,^.^ - s . ^ . ^ ) + a 2 2 ( i - j + l ) ( r - r ) (101-a) 1 , J X -L J. , J and sh i , j • a « 8 h l - l , j - l + a " < a « s l f j + r ± > j ) - a 2 2 ( a 2 3 s . _ l 5 . _ 1 + r . ^ . ^ ) + a 2 2 ( i - j + l ) ( s - s. ) , j - 1,2,3, ... , i (101-b) X j J X X X , J £ . The c o n s t a n t CH. i s a l s o o b t a i n e d : x CHi = Po a5& m 0 + a 2 4 C N ! (102-a) and CH ± =* a 8 C H i _ 1 + a 2 i + ( C N - CN ) , i > 1 (102-b) where a 2 2 = P 0 a 5 w r o £ q z (103-a) a 2 3 =* a n + (103-b) a 2 2 and aot, = p n a c ( a J l + Z I ) (103-c) ^ u 3 qz • mz qn R e l a t i v e d i s c h a r g e d e v i a t i o n q ^ ( t ) may be o b t a i n e d i n a s i m i l a r way. U s i n g t h e h e a d - d i s c h a r g e r e l a t i o n as g i v e n by Eq. 64, we g e t : q.(6) - q, .(9) - - ~ - (h.(9) + h. . .(e)) (104) x x—x zp o x x—x Hence, w i t h d i s c h a r g e f o r m u l a o f the form: q ± ( 6 ) - e x p ( a n 9 ) [RQ ± ( e ) s i n e + SQ ± ( e ) cose] + CQ ± (105) the c o e f f i c i e n t s o f t h e p o l y n o m i a l s RQ^ and SQ_^  become: r q . . = r q . 1 . - ~ - ( r h . . + r h . 1 . ,) (106-a) M i , 3 M x - l , j - l 2p 0 1,3 x - l , j - l y s q . . = sq - ~ - (sh. . + sh. ., . -.) (106-b) i . J M x - l , j - l 2p 0 i , 3 i - l , 3 - l . -j = 1,2,3, ..., i and t h e c o n s t a n t CO. = CO. . - - ~ (CH. + CH. ,) (106-c) x x ^ i - l 2p 0 x x-1 S o l u t i o n f o r r e l a t i v e gate d e v i a t i o n s w i l l have a s i m i l a r form which can be e a s i l y d e r i v e d from e i t h e r governor Eq. 62 or t u r b i n e d i s c h a r g e Eq. 63. I t i s s i m p l e r , however, t o use the l a t t e r i f h^, q_^  and a r e a l r e a d y determined. 3.4 Summary I f a{ and a2, as d e f i n e d by Eq. 67, a r e such t h a t a 2 < 2 a 2 , and Po ^ Po*' e x a c t s o l u t i o n f o r the speed n ( t ) , f o r a p a r t i c u l a r time i n t e r v a l i - 1 < t < i i s : n ± ( 0 ) =•= e x p ( a n 6 ) [R (0) s i n e + S ± ( 9 ) cose ] + CN ± where 0 = w(t - ( i - 1 ) ) R^ and S^ a r e p o l y n o m i a l s i n 6 of degree ( i - 1 ) . T h e i r c o e f f i c i e n t s a r e d etermined from Eqns. 92 and 96. CIsL i s a c o n s t a n t f o r t h i s p a r t i c u l a r i n t e r v a l and i s g i v e n by Eq. 94. The head and t h e d i s c h a r g e a r e s i m i l a r l y e x p r e s s e d a s : h ± ( 0 ) = e x p ( a n 6 ) [RH i ( 6 ) s i n e + SH ± ( 6 ) cose] + CH ± q ± ( t ) =- e x p ( a i : e ) [RQ ±(0) s i n e + SQ ^ e ) cos0 ] + CQ ± A g a i n , RH , SH_, RQ^ and SQ^ a r e p o l y n o m i a l s of degree ( i - 1 ) , CH^ and CQ^ a r e c o n s t a n t s f o r i - 1 _< t < i . Eqns. 101 through 106 a r e used t o determin e the d i f f e r e n t c o e f f i c i e n t s and c o n s t a n t s of t h e s e f o r m u l a s . R e l a t i v e gate d e v i a t i o n a t any time t , i - 1 _< t < i j can be o b t a i n e d d i r e c t l y from t u r b i n e d i s c h a r g e Eq. 63. I f a 2 > 2 a 2 ^ 0 and p 0 ^ P o * > a v e r y s i m i l a r s o l u t i o n but i n terms of h y p e r b o l i c f u n c t i o n s i s o b t a i n e d , as shown i n Appendix A. The s p e c i a l case of a 2 = 2 a 2 and p 0 ^ P Q * r e s u l t s i n a s o l u t i o n of t h e 58 form: n^(x) = e x p ( - a ^ R (x) + C1SL, where x = t - ( i - 1 ) , but t h e p o l y - . n o m i a l R^ w i l l be o f a degree ( 2 i - l ) i n s t e a d of ( i - 1 ) . D e t a i l s of t h i s c ase and of the case a 2 > 2 a 2 = 0, p 0 ^ p 0 * a r e a l s o g i v e n i n Appendix A. S i n c e t h e head-speed r e l a t i o n , Eq. 68 i s not v a l i d f o r P n : x P n * = , : 0 . 5 £ / ( £ , I -I £ , ) , a s e p a r a t e treatment i s n e c e s s a r y . u u mz mh qz mz qh T h i s l e a d s to f o u r d i f f e r e n t cases as f o r the g e n e r a l case p 0 ^ P o * # R e s u l t s a r e p r e s e n t e d i n Appendix A. 3.5 S o l u t i o n w i t h L i n e a r i z e d F r i c t i o n C o n t i n u i t y e q u a t i o n 36, and e q u a t i o n of motion 37-b, may be w r i t t e n a s : h ' ( x , t ) - 4 p 0 v ' ( x , t ) = - E h ' ( x , t ) , w i t h £ + 0 (107-a) L X and h ' ( x , t ) - p 0 v ' ( x , t ) = 2 R v ' ( x , t ) (107-b) A s o l u t i o n e x i s t s f o r t h e s e two non-homogeneous e q u a t i o n s o n l y i f the p e n s t o c k i s d i s t o r t i o n l e s s , i . e . , f r i c t i o n causes a t t e n u a t i o n of the t r a v e l -l i n g waves w i t h o u t d i s t o r t i n g t h e i r shapes. T h i s c o n d i t i o n i s s a t i s f i e d i f [ 5 5 ] : 2R e - — (108) PO S i n c e the a c t u a l v a l u e of E i s z e r o , the p r e s s u r e wave w i l l always e x p e r i e n c e some d i s t o r t i o n u n l e s s the p e n s t o c k i s t o t a l l y f r i c t i o n l e s s . L a t e r a p p l i c a t i o n s , however, show t h a t th e assumption of a d i s t o r t i o n l e s s p e n s t o c k y i e l d s f a i r l y a c c u r a t e r e s u l t s f o r speed t r a n s i e n t s over the time range of p r a c t i c a l i n t e r e s t . Assuming a d i s t o r t i o n l e s s l i n e , g e n e r a l . s o l u t i o n f o r Eqn. 107 i s : R f R h ' ( x , t ) =* exp( x) F ( t - 0.5x) + exp ( — x) f ( t + 0.5x) (109-a) PO PO -1 R f R f and v ' ( x , t ) = - — [exp( x) F ( t - 0.5x) - exp ( — x) f (t + 0.5x] (109-b) 2 P 0 PO PO 59 F o l l o w i n g s i m i l a r p r o c e d u r e (see d e r i v a t i o n s t a r t i n g from Eq. 46), p e n s t o c k e q u a t i o n becomes: q . ( t ) - - ~ [ h . ( t ) + 2 . E A J h. . ( T . ) ] (110) i 2p 0 i i-2 J where - 2 R f A - exp( -) (111) PO A l s o the r e l a t i o n between .h.(t) and n . ( t ) f o r p n ^ p n* becomes: x x u u i - 1 h . ( t ) - p 0 a 5 { F i J n.Ct) + A a 5 £ E a i 1 j 1 n.(x. .)] + x U J H - L ° mz . .. ° 1 l - i 3-1 i - 1 . , (1 + A a 5 £ E a J J X ) £ m l (112) ° mz . , ° qz I 3=1 where a 8 = Aag (113-a) which i s e q u i v a l e n t t o Eq. 72 f o r f r i c t i o n l e s s f l o w case. The s o l u t i o n f o r n ^ ( t ) w i l l be the same as b e f o r e w i t h a 6 and a 8 b e i n g r e p l a c e d by ag and ag, where ag = Aag. (113-b) H e n c e f o r t h , t h e primes on b o t h ag and ag w i l l be dropped w i t h the u n d e r s t a n d i n g t h a t the c o e f f i c i e n t A i s i n c l u d e d i n a l l c o n s t a n t s i n v o l v i n g ag and ag. S o l u t i o n s f o r h ( t ) and q ^ ( t ) change s l i g h t l y from the f r i c t i o n -l e s s c a s e . F o r i - 1 £ t < i , and f o r a 2 < 2 a 2 : h . (8 ) - a 8 h . re) = p 0 a 5 { F i t - [ n . ( e ) - An. , ( 6 ) ] + (1-A):.£ m (114) x ° x-1 u o t x q Z £ and w i t h h (9) =» e x p ( a n 9 ) [RH ± ( e ) s i n e + SR\(9) cos9 ] + CH ± (115) we get r h i , j ' a 8 r V l , j - l + a 2 2 ( a 2 3 r i 5 J " s i 5 J ) - A a 2 2 ( a 2 3 ' 1 . 1 , j . 1 " 8 i - l , J - l > + a 2 2 ( i - j + l ) ( r - Ar ) , j - 1,2, ... , i (116-a) x > 3 -1- x x , j z S h i , j " a 8 S V l , j - l + a 2 2 ( a 2 3 S i s j + r . 5.) - A a 2 2 ( a 2 3 s . _ l j . _ 1 + r . ^ . ^ ) + a 2 2 ( i - j + l ) ( s . - As ) , j - 1,2, ... , i (116-b) x , j x X X , J z 60 CH = asCH. , + a 2u(CN.- A CN. J + p n a 5 ( l - A ) £ m , i > 1 (116-c) i i - 1 i i - 1 u 3 qz £ CH-^  i s g i v e n by Eq. 102-a. A l s o , we have q . O ) - A q. , (6) - (h.(0) + A h. (0)) (117) x x—x zp o x x—x t o g e t h e r w i t h q (0) = e x p ( a n 0 ) [RQ ±(0) s i n 0 + SQ ±(0) cos0] + CQ ± (118) From which, we g e t : r q . . =A-rq. , . , ( r h . . + A r h . . . n ) (119-a) M r , j n x - l , j - l 2p 0 x,j 1-1,1-1 sq . . - A sq. .. . 1 --^-(sh. . + A s h . . . ,) (119-b) i , J H x - l , j - l 2 p 0 x,j i - l , J - 1 j =•= 1, 2, .. . , i and CQ. - A CQ. 1 - (CH. + A CH. .) (119-c) x ^ i - l 2pQ l i - l S o l u t i o n s f o r o t h e r cases a r e m o d i f i e d i n the same way t o i n c l u d e l i n e a r i z e d f r i c t i o n e f f e c t (see Appendix A ) . 3.6 Approximate S o l u t i o n 3.6.1 Speed v a r i a t i o n . A l t h o u g h the exact s o l u t i o n i s s i m p l e and s t r a i g h t f o r w a r d , a p p l i c a t i o n s have shown some i n s t a b i l i t y problems p a r t i c u -l a r l y i n the neighbourhood of Po*. The c o e f f i c i e n t s o f the d i f f e r e n t p o l y -n o m i a l s grow so r a p i d l y w i t h every new r e f l e c t i o n t h a t a c c u r a c y i s l o s t a f t e r about seven r e f l e c t i o n s . S e v e r a l approaches were t r i e d t o overcome the i n s t a b i l i t y problem. S u b s t a n t i a l improvement i s a c h i e v e d by u s i n g T a y l o r ' s e x p a n sions f o r b o t h e x p o n e n t i a l and t r i g o n o m e t r i c or h y p e r b o l i c f u n c t i o n s . L e t * Mi M x-j e x p ( a n Q ) sin© =•= E XI. 6 3=1 J and e x p ( a n Q ) cos6 = E A2. 0 3=1 2 where Mi = M + 1, M i s the h i g h e s t power used i n , t h e e x p a n s i o n . i - 1 _< t < i , s o l u t i o n f o r t h e speed takes the form: . Mi M j - j n..(6) E r . . 0 = R. (0) S u b s t i t u t i n g i n Eqn. 51, we g e t : Mi L [ n ' ( 0 ) ] = a 6 l E [ a i 0 r + u a 9 ( M 2 - j ) r + X q^. -1- -L»J -1- X » J M l "3 w 2 ( M 3 - j ) ( M 2 - j ) r . _ . ] 0 + ay + 3 ^ ( 1 - 3 8 ) where M 2 = M+-2, and M 3 - M + 3., . •• ,. -:. A p a r t i c u l a r s o l u t i o n of Eq. 122 can be w r i t t e n a s : Mi M i - j n**(0) = E . b. . 0 ' = B. (0) 1 • i 1,3 1 3"! *A1 and X2 may be s y s t e m a t i c a l l y e v a l u a t e d as f o l l o w s : 3 3 . L e t Bi = 1, and Y i = 0 For j =* 2, 3, ... , Mi : & = a i l B j - l " Y j - 1 ' a n d Y. = a n V l + 3.., Then, XI = YM2-j / ( M2 - :? ) ! a n d * 2 j ' 3 M 2 - j 1 ( M2~ j ) V, ' ™ 2 F o r a 2 > 2 a 2 : 3. = a n 3 . . + y. 3 3-1 3-1 62 The p o l y n o m i a l s B^Ce) and R^-^Ce) a r e r e l a t e d by: M r L [ B . ( 9 ) ] - a 6 £ q z ^ [ ^ l o V l J + V l J - l + M i - j u2 ( M 3 - j ) ( M 2 - j ) r ± _ 1 ] e + a y + a ^ l - a g ) (124) which, f o r j =* 1, 2, M 1, y i e l d s b i , j = 2a7 [ a6 a10V r i _ l , j + ^ ( M 2 - j ) ( a 6 a 9 \ z r ± - l , j - l " ^ i . J - l ' + u2 ( M 3 - j ) ( M 2 - j ) ( a 6 £ q z r . _ l 5 . _ 2 - b . ^ ) ] (125-a) a n d b i , M l " b i , M l + a 1 9 < 1 2 5" b> The approximate s o l u t i o n f o r the case o f a 2 1 0 i s t r e a t e d s e p a r a t e l y i n Appendix B. The g e n e r a l s o l u t i o n of the speed d i f f e r e n t i a l e q u a t i o n becomes: n i ( e ) = e x p ( a : x e ) ( a i s i n e + a 2 c o s e ) + C ± (e) ( 1 2 6 i C . (e) i s a l s o a p o l y n o m i a l of degree M: (127) Ml Mj - j c ( e ) = B . (e) + a 8R. ,(e) = z c. . e 1 1 ° l - l . v 1>J I t s c o e f f i c i e n t s a r e g i v e n by: ooa^ ' a i 8 r i - u + ( M 2 " j V [ a 2 5 r i - i , j - i " I T f c i f j - l + ( M 3 . - j ) ( a 2 6 r . _ l j . _ 2 - -g- c . 5 . _ 2 ) ] , j - 1, 2, ... , M l (128-a) and c. „, =» c. „ + a i q (128-b) I . M J i,M x where a 2 5 - ( a 6 a 9 £ q z + 2 a i a 8 ) (129-a) U , 2 and a 2 6 = ^ - ( a 6 £ ^ + a 8 ) (129-b) i 2 - qz 63 The c o n s t a n t s ai and a 2 a r e determined from the i n i t i a l c o n d i t i o n s (Eq. 95): M l M - i 3 (130) a i = a n c i , M 1 - C i , M + .l± ( a l + M l " j ) r i - l , j Mi M - j and a 2 = - c + E r . ., . w (131) F i n a l l y , u s i n g T a y l o r ' s expansions (120), the g e n e r a l s o l u t i o n which i s g i v e n by Eq. 121 i s o b t a i n e d . In t h i s e q u a t i o n : r - a i A l . + a 2A2 + c. . , j = 1, 2, ... , Mi (132) I n i t i a l c o n d i t i o n s f o r the f i r s t i n t e r v a l a r e a f f e c t e d by the s t e p - l o a d change which i n t r o d u c e s an a d d i t i o n a l term t o 0 4 : m ~ + a n c n (133-a) c, . - 0 , j • 1, 2, ... , M (133-b) i > 3 a4 and c. =» -Tr— (133-c) l, M i 2 a 2 3.6.2 E v a l u a t i o n of h ^ ( t ) and q ^ ( t ) . S u b s t i t u t i n g n^(g) and n^_^(e) i n Eq. 114, a s i m i l a r e x p r e s s i o n f o r h ^ ( t ) i s o b t a i n e d : Mi M - j h.(e) - i u. . e' ' - u.(e) (134) 3=1 , 3 1 The c o e f f i c i e n t s u. . a r e g i v e n by: U i , j " a 8 u i - l , j + ^ ( r i , j - A r i - l , j ) + a 2 2 ( M 2 - J ) ( r i , j - l - A r i - l , j - l ) ' j = 1, 2, ... , M (135-a) and u. M = h. (u) (135-b) i,M x l - l For i - 1, u. =•= 0 (135-c) 1 ,Mi 1 64 The d i s c h a r g e q . ( t ) can a l s o be e x p r e s s e d a s : 1 i n which: Mi M - j q . ( t ) - Z V. . 0 - V.(0) (136) 3=1 ^ v. . = A v_. n - - ~ - (u_. J + A u_. n ,) , j = 1,2, . . . , Mi i , j i-1,3 2p 0 1,3 x-1,3 (137) 3.7 Summary Approximate s o l u t i o n s f o r n ^ t ) , h_^(t) and q ^ ( t ) which u t i l i z e T a y l o r ' s expansions t r u n c a t e d to the power M w i l l be p o l y n o m i a l s of the same degree M. Formulas o b t a i n e d above t o determine t h e i r c o e f f i c i e n t s i n terms of p r e v i o u s time i n t e r v a l s o l u t i o n a r e v a l i d f o r b o t h a 2 < 2 a 2 and a 2 > 2 a 2 ^ 0, f o r a l l v a l u e s of Po except PQ ^ Po*- F o r a i > 2 a 2 , h y p e r b o l i c f u n c t i o n s a r e used t o d e t e r m i n e XI and X2 i n Eq. 120. The s p e c i a l case of a 2 = 2 a 2 i s s e p a r a t e l y i n v e s t i g a t e d . A p a r a l l e l a n a l y s i s has been c a r r i e d out f o r pg " P Q * - R e s u l t s of a l l t h e s e cases a r e p r e s e n t e d i n Appendix B. A l l c o n s t a n t s a p p e a r i n g i n the d i f f e r e n t e x a c t and approximate s o l u t i o n s a r e p r e s e n t e d i n Appendix C. CHAPTER IV VERIFICATION OF THE TRAVELLING-WAVE SOLUTION AND APPLICATIONS 4.1 Comparison w i t h t h e Method of C h a r a c t e r i s t i c s ' S o l u t i o n To check the r e s u l t s o b t a i n e d by the a n a l y t i c a l , t r a v e l l i n g - w a v e s o l u t i o n , d a t a f o r the high-head power p l a n t w i t h a d o u b l e - r u n n e r , d o u b l e - j e t i mpulse t u r b i n e p r e s e n t e d i n [51] a r e used. I n i t i a l s t e a d y s t a t e output i s 10800.0 m e t r i c horsepower, s u p p l i e d under a head o f 347.0 meters and an a v e r -age p e n s t o c k v e l o c i t y o f 3.118 m/sec. Pen s t o c k l e n g t h i s 632.7 m. and d i a m e t e r 1.031mwith n e g l i g i b l e f r i c t i o n l o s s e s . Wave v e l o c i t y i s e s t i m a t e d to be lOOO.Om/sec. The t o t a l i n e r t i a of the t u r b i n e and g e n e r a t o r (WR2) i s 17.5 m e t r i c ton-m 2 and t u r b i n e speed i s 500.0 r.p.m. With i s o l a t e d l o a d c o n d i t i o n assumed, t o t a l s e l f r e g u l a t i o n a i s taken e q u a l t o 1.5. oo o Based on t h e c r i t e r i o n : I-Q/ n z d t =•= min., optimum s e t t i n g s suggested f o r a m e c h a n i c a l governor a r e : temporary droop, 6 = 0.243, and dashpot t i m e , T^ =•= 2.64 s e c . [ 5 1 ] . Permanent speed droop i s n e g l e c t e d . P i p e l i n e c o n s t a n t c o r r e s p o n d i n g t o i n i t i a l s t e a d y s t a t e i s p 0 = 0.458, and m e c h a n i c a l s t a r t - u p time, T^ =•= 6.0 s e c . S i n c e the t u r b i n e i s o p e r a t i n g i n the v i c i n i t y o f t h e p o i n t of b e s t e f f i c i e n c y , i d e a l impulse t u r b i n e s l o p e s , Eq. 11, can be adopted h e r e . From Eq. 69-c we get p 0 * = 0 . 5 which i s d i f f e r -ent from p 0 . Furthermore, we f i n d from Eq. 67 t h a t a 2 > 2 a 2 ^ 0. T h e r e f o r e , the e x a c t s o l u t i o n f o r speed d e v i a t i o n t a k e s t h e form: n.(e) = exp (a-i n 0 )[R. ( 0 ) s i n h e + S.(e) coshe ] , i - 1 < t < i 65 66 S u b s t i t u t i o n of t h e p l a n t d a t a i n t h e d i f f e r e n t e x p r e s s i o n s g i v e n i n Appendix A ( p a r t I, case 2) y i e l d s : F o r 0 x t < 1 : h (6)/m = exp(-1.890176) [- 2.176629 sinhG] 1 £ t < 2 : n 2 ( 8 ) / m = exp(-l.890170)[(12.585759 - 40.85876) s i n h e + (39.035609 - 0.17484) cosh9] 2 £ t < 3 : n 3 ( 6 ) / m A ' - exp(-1.890176)[(-386.41856 2 + 234.71149 - 1032.119) sinhG + (-225.71256 2 + 1032.9056 - 0.1860) coshe] 3 < t < 4 : XH+ ( 6 ) / m ^ - exp(-l.890176) [(0.20940936 3 - 1.260946 2 + 0.5909226 -2.952214) si n h 6 + (0.2745056 3 - 0.59137706 2+ 2.9523026-0.5998 E-5) cosh6] «,.E4 and so on, w i t h 6 = 0.09620 ( t - i + 1) and E(k) - 1 0 k . F i g . 12 shows a comparison between t h i s s o l u t i o n and r e s u l t s o b t a i n e d by u s i n g the method of c h a r a c t e r i s t i c s f o r waterhammer a n a l y s i s w i t h p e n s t o c k l e n g t h d i v i d e d i n t o 10 r e a c h e s . The agreement i s v e r y c l o s e as b o t h s o l u t i o n s p r a c t i c a l l y ^ y i e l d the same r e s u l t s . I t i s n o t i c e d , however, t h a t s t a b i l i t y o f the a n a l y t i c a l s o l u t i o n becomes more c r i t i c a l i f t h e v a l u e of PQ i s v e r y c l o s e t o p$* or i f ( a 2 - 2 a 2 ) approaches z e r o . In such cases the p o l y n o m i a l ' s c o e f f i c i e n t s i n c r e a s e so r a p i d l y w i t h each time i n t e r v a l t h a t a c c u r a c y i s l o s t a f t e r o n l y a few i n t e r v a l s . By u s i n g T a y l o r ' s e x p a n s i o n up to a power M , s t a b i l i t y i s improved and t h e s o l u t i o n t a k e s the form (Appendix B, case 2 o f p a r t I ) : Though t r u n c a t i o n e r r o r w i l l i n c r e a s e w i t h time, e x p a n s i o n up t o t h e se v e n t h power y i e l d s a c c u r a t e r e s u l t s f o r time range o f p r a c t i c a l i n t e r e s t , M+1 n. (6) = Z r . 6 M-j+1 i - 1 < t < i 67 F i e . 13. V a l u e s o f r . . t o r . _ f o r f i r s t f o u r i n t e r v a l s a r e as f o l l o w s : & i , 3 i , 8 \ i 3 \ 1 2 3 4 1 -0. 345761 E - l 0. 357643 0. 162129 E l 0.595426 E l 2 0. 841937 E - l -0. 536452 -0. 283146 E l -0.810118 E l 3 -0. 175505 0. 449132 0. 224687 E l -0.139908 E l 4 0. 303607 0. 258289 0. 207159 E l 0.151027 E2 . 5 -0. 414356 -0. 137427 E l -0. 556561 E l -0.122546 E l 6 0. 404036 0. 153438 E l 0. 690198 -0.635202 7 -0. 215701 -0. 147521 0. 114238 0.982289 E - l 8 0. 0 -0. 175874 E - l -0. 186010 E - l -0.582921 'E-2 E x p r e s s i o n s f o r t h e head h, the d i s c h a r g e q and t h e gate z w i l l t a k e s i m i l a r forms: For i - 1 < t < i : h.(6) l 8 E u. 3=1 1,3 q.(9) =» E v . . j-1 ^ and z-'(8) = E w. . 1=1 x>* i~3 P o l y n o m i a l s c o e f f i c i e n t s u. ., v. . and w. . f o r t h e same d a t a and f o r 1,3 i,3 i,3 i =* 1 t o 4 a r e g i v e n below. A comparison w i t h t h e r e s u l t s o f n u m e r i c a l s o l u -t i o n by the c h a r a c t e r i s t i c s method i s shown i n F i g . 14. 68 0.04 0.00 -0.04 F i g . 12 Speed V a r i a t i o n : A n a l y t i c a l and Numerical S o l u t i o n s F i g . 13 Speed V a r i a t i o n : E f f e c t of T r u n c a t i o n E r r o r i n T a y l o r ' s Expansion J \ 1 2 3 4 \ 1 0. 565566 -0. 193159 E2 0.419916 E3 -0. 964894 E4 2 -0. 153572 -0. 225184 -0.204249 E l -0. 205519 E2 3 0. 316914 0. 138014 E l 0.872856 E l 0. 203897. E2 4 -0. 529800 -0. 296356 E l -0.750554 E l 0. 502060 E2 5 0. 638122 0. 206262 E l -0.151576 E2 0. 112294 E2 6 -0. 324525 0. 347167 E l -0.181076 E l -0. 284266 E l 7 -0. 556967 0. 160025 0.603776 -0. 108814 8 0. 0 -0. 570041 E - l -0.650833 E-2 0. 204747 E - l 1 -0. 617430 0. 198524 E2 -0.417484 E3 0. 965787 E4 2 0. 167655 0. 581144 0.305677 E l 0. 277231 E2 3 -0. 345976 -0. 219866 E l -0.132344 E2 -0. 450229 E2 4 0. 578384 0. 439210 E l 0.158212 E2 -0. 307950 E2 5 -0. 696640 -0. 364505 E l 0.106508 E2 0. 149392 E2 6 0. 354285 -0. 308146 E l -0.489468 E l 0. 185469 7 0. 608043 0. 104139 E l 0.207542 -0. 332810 8 0. 0 0. 622321 E - l 01131569 0. 116322 1 -0. 900213 0. 295103 E2 -0.627442 E3 0. 144823 E5 2 0. 244441 0. 693736 0.407802 E l 0. 379991 E2 3 -0. 504433' -0. 28887 . ..El -0.175986 E2 -0. 552177 E2 4 0. 843284 0. 587388 E l 0.195740 E2 -0. 558980 E2 5 -0. 101570 -0. 467636 E l 0.182296 E2 -0. 367549 E l 6 0. 516548 -0. 481730 E l -0.398930 E l 0. 160680 E l 7 0. 886527 0. 961374 -0.943460 E - l -0. 278403 8 0. 0 0. 907341 E - l 0.134823 0.106085 70 With u . v a l u e s known, v. . and w . a r e determined from the s i m p l i -1,3 i-3 i,3 f i e d r e l a t i o n s : v. . =•= v . - (u. . + u. .) i, 3 i - l , 3 2 p 0 i , j i - l , 3 and w . . = v . . - 0 . 5 u . . 1,3 i>3 i,3 A l t e r n a t i v e l y , z ( t ) can be d i r e c t l y o b t a i n e d from: r ( t ) = q ( t ) - 0.5 h ( t ) 4.2 Comparison w i t h L a p l a c e - T r a n s f o r m S o l u t i o n A p p l y i n g L a p l a c e ' s t r a n s f o r m t o t h e b a s i c system Eqns. 31, 32, 35, and 37, r e l a t i v e speed d e v i a t i o n i n L a p l a c e ' s domain n ( s ) can be f i n a l l y e x p r e s s e d a s : n ( s ) / m 0 - - 6T'/[6T' T' s 2 + ( a 6 + f ( s ) ) T's + f ( s ) ] a r r mU r where f ( s ) = (1 - 2p 0 t g h ( s / 2 ) / ( l + p 0 t g h ( s / 2 ) ) There i s no d i r e c t i n v e r s e t r a n s f o r m a t i o n a v a i l a b l e f o r such a f u n c t i o n , however. U s i n g s i x t h o r d e r expansions of s i n h ( s / 2 ) and c o s h ( s / 2 ) , R a n s f o r d and R o t t n e r [51] o b t a i n e d an e x p r e s s i o n f o r the speed f o r the same p l a n t d a t a d i s c u s s e d above: n ( t ) / m ^ =• 0.10188 exp(-3.1530t) - 0.25606 exp(-0.57303t) + 0.09826 exp(-0.4077t) s i n ( 5 . 5 2 5 2 t + 0.4782) + 0.11212 exp(-0.25855t) cos(1.8726t + 0.2391) F i g . 15 i l l u s t r a t e s t h i s r e l a t i o n s h i p t o g e t h e r w i t h t h e t r a v e l l i n g -wave s o l u t i o n d e v e l o p e d i n t h i s t h e s i s . I t i s c l e a r t h a t t h e L a p l a c e -t r a n s f o r m s o l u t i o n does not y i e l d s a t i s f a c t o r y r e s u l t s i n the time domain. Even w i t h the t e n t h o r d e r expansions of t h e h y p e r b o l i c f u n c t i o n s l i t t l e improvement i s g a i n e d . F i g . 15 a l s o shows t h e r e s u l t s o f r i g i d water-column a n a l y s i s f o r t h e same d a t a . Though i t compares more f a v o u r a b l y w i t h the exact s o l u t i o n , 0.04 a* 0.00 -0.04 -0.08 -0.12 -0.16 -0.20 -0.24 Same d a t a as i n F i g . 12 1 1 0 ? / / t V / / / / • I —i j—j \ / A n a l y t i c a l ( T r a v e l l i n g - w a v e ) S o l u t i o n \\ \\ \\ / ; ; / / i / \\ ; '. / ' '••/: L a p l a c e T r a n s f o r m S o l u t i o n R i g i d Water-column S o l u t i o n \j F i g . 15 Speed V a r i a t i o n : Comparison with Other A n a l y t i c a l S o l u t i o n s a p p r e c i a b l e d i f f e r e n c e s i n damping c h a r a c t e r i s t i c s : : a s w e l l as the f i r s t peak s t i l l e x i s t . F u r t h e r check f o r the a n a l y t i c a l s o l u t i o n i s c a r r i e d out f o r the case PQ = 0.7 which might be a t t a i n e d by i n c r e a s i n g the i n i t i a l v e l o c i t y VQ to 4.75 m/sec, F i g s . 16 and 17. S i n c e a 2 > 2 a 2 , s i m i l a r e x p r e s s i o n s a r e o b t a i n e d f o r t h e speed and o t h e r v a r i a b l e s , as b e f o r e . S o l u t i o n b e h a v i o r shows c o n s i d e r a b l e improvement as p j gets f a r t h e r from t h e s i n g u l a r i t y pQ* = 0.5. Governor s e t t i n g s , however, s h o u l d be m o d i f i e d to r e g a i n optimum response c o r r e s p o n d i n g to t h i s i n i t i a l c o n d i t i o n . From F i g . 16 i t appears t h a t the d e s t a b i l i z a t i o n i n f l u e n c e i n t r o d u c e d by water hammer i n the p e n s t o c k i n c r e a s e s as governor s e t t i n g s d e p a r t f u r t h e r from t h e optimum. 4.3 E f f e c t o f F r i c t i o n L o s s e s i n the P e n s t o c k U s u a l l y f r i c t i o n l o s s e s a r e kept below 5% of the t o t a l head a v a i l -a b l e . Assuming R =» H /H = 0.05, o t h e r parameters i n the o r i g i n a l t u r U n(J d a t a b e i n g t h e same, a s i m i l a r e x p r e s s i o n f o r speed d e v i a t i o n i s o b t a i n e d w i t h s l i g h t m o d i f i c a t i o n s i n the c o n s t a n t s a 6 and a 8 as g i v e n i n Appendix C. Comparing t h e r e s u l t s w i t h t h o s e f o r f r i c t i o n l e s s c a s e , F i g . 18, i t can be seen t h a t maximum d e v i a t i o n i s s l i g h t l y i n c r e a s e d , whereas the damping c h a r a c t e r i s t i c s have been s i g n i f i c a n t l y improved by the e x i s t e n c e of f r i c t i o n . S i n c e maximum speed d e v i a t i o n , f o r t h i s example, i s a t t a i n e d a f t e r o n l y one complete t r i p o f t h e p r e s s u r e wave ( t =• 1.65), i t i s h a r d l y a f f e c t e d by p e n s t o c k f r i c t i o n . I t can be a l s o seen t h a t the assumption -of a d i s t o r t i o n l e s s p e n s t o c k does not i n t r o d u c e any s i g n i f i c a n t e r r o r t o the speed v a r i a t i o n . The d i f f e r e n c e i s more n o t i c e a b l e f o r t h e head d e v i a t i o n h, F i g . 19. F r i c t i o n has a s t r o n g e r damping e f f e c t h e r e compared to the speed. 73 F i g . 16 Speed V a r i a t i o n f o r p n = 0.7 > p„ E i g . 17 Head, Discharge and Gate V a r i a t i o n s f o r p n = 0.7 > p* 0.04 0.00'. ^—s e 5 -0.04 -0.08 -0.12 -0.16/ -0.20 -0.24 R,. = r—°- = 0.05, o t h e r d a t a f ^  H _ 0 n Q a r e the same as i n F i g . 12 6 8 10 12 L -/ t -/ / 1 / An r\ l y t i c a l ( d i s t o r t i o n l e s s l i n e ^ r a c t e r i s t i c s method, d i s t o r -/ / / ' / © oo Cha / / / / / / t i o n e f f e c t i n c l u d e d F r i c t i o n l e s s Case 1 V / / / / / V F i g . 18 Ef f e c t of Penstock F r i c t i o n on Speed Variation Same d a t a as i n F i g . 18 10 h - f (h /m4) 12 A n a l y t i c a l ( d i s t o r t i o n l e s s l i n e ) C h a r a c t e r i s t i c s , d i s t o r t i o n e f f e c t i n c l u d e d F r i c t i o n l e s s Case F i g . 19 Ef f e c t of Penstock F r i c t i o n on Head Variation 76 4.4 The E f f e c t s of Permanent Droop a and S e l f - R e g u l a t i o n a F i g . 20 shows speed d e v i a t i o n f o r t h e o r i g i n a l d a t a (a » 1.5 and o = 0.0).as w e l l as two o t h e r c a s e s : a) a ' 0.0, cr =• o.O and b) a - 0.0, cr - 0.03 I t becomes apparent t h a t a p o s i t i v e s e l f - r e g u l a t i o n c o e f f i c i e n t has a f a r g r e a t e r s t a b i l i z i n g i n f l u e n c e t h a n permanent speed droop. The assumption of z e r o s e l f - r e g u l a t i o n w i l l be, t h e r e f o r e , on the c o n s e r v a t i v e s i d e f o r most d e s i g n c a s e s . While a permanent droop cr > 0.0 s l i g h t l y i n c r e a s e s the maximum d e v i a t i o n , i t a l s o i n t r o d u c e s an a p p r e c i a b l e damping. S i m i l a r c o n c l u s i o n s d e r i v e d from r i g i d water-column a n a l y s i s have been r e a c h e d by Chaudhry[13], and o t h e r s . 4.5 E l a s t i c i t y E f f e c t s on Maximum S p e e d - D e v i a t i o n and S t a b i l i t y - L i m i t F o r an i d e a l impulse t u r b i n e w i t h ct =* a - R = 0.0, maximum speed-d e v i a t i o n can be e x p r e s s e d i n terms o f =« T _/ST ~, 2^ * T ~/T , wO mO wO r K =•= 10 T „.m 0/T _ and P n . F i g s . 21-a through 21-d i l l u s t r a t e v a r i a t i o n wO * mO of (n /K) w i t h A l l i e v i ' s c o n s t a n t p, f o r s e v e r a l c o mbinations o f max 0 Ai and A 2 where: * i =•= T „/(<5T .) wO mO * 2 " T w 0 / T r and,K = 1 0 ( T w 0 . m £ / T m 0 ) E l a s t i c i t y o f the water and p e n s t o c k w a l l s g e n e r a l l y i n c r e a s e s ( n m a x / K ) • The d i f f e r e n c e may r e a c h 15% f o r high-head p l a n t s ( s m a l l Po) but i t d i m i n i s h e s r a p i d l y f o r l a r g e r Po v a l u e s and becomes l e s s than 5% f o r P 0 > 1.0. The major i n f l u e n c e o f e l a s t i c i t y i s , however, on t h e damping c h a r a c t e r i s t i c s , hence the s t a b i l i t y of speed o s c i l l a t i o n s . T h i s i s F i g . 2 0 E f f e c t of Permanent Droop " a " and Self-Regulation on Speed Variation 30 29 28 27 26 25 24 23 I d e a l Impulse T u r b i n e w i t h a = G = R f = 0 . 0 t 0 A2 = 0 .15 N X l = 0, 40 Zst : a b i l i t > N t 0 45 J) 1>— L] .mit 0. 50 . J j M 0 p ft 0. 55 =_ o >, rn . CL 60 _ 0.25 0 .5 1 .0 2 . 0 4 . 0 8 .0 1 6 . 0 F i g . 2 1-a E l a s t i c i t y E f f e c t on Maximum • T w Q «m £ Speed-Deviation (k = 1 0 -"• m " .) T. mO I d e a l Impulse T u r b i n e w i t h a = a = R f = 0 . 0 ro A2 = 0 .20 T A. = 0 .40 £ S t a Lim b i l i t . i t —_ \ 0 .45 cn (U y 0 .50 hptot n _____ cn •— rt - ' I w — 30 29 28 27 26 25 24 23 0 .25 0 .5 1 .0 4 . 0 8 .0 1 6 . 0 F i g . 21-b E l a s t i c i t y E f f e c t on Maximum Speed-Deviation 79 .28 ,27 e .26 .25 .24 .23 I d e a l Impulse T u r b i n e w i t h a = o = R f Q = 0 . 0 X 2 = 0.25 V. \ = 0. 40 / > _ l _ 0. 45 - — — • c a 4-i i St£ L I T b i l i l i t 0. 50 y uijj LU n y uijj LU w • w <J 0.25 0.5 1.0 2.0 4.0 8.0 16.0 p o — F i g . 21-c E l a s t i c i t y E f f e c t on Maximum S p e e d - D e v i a t i o n .29 .28 .27 .26 .25 .24 .23 .22 I d e a l Impulse T u r b i n e w i t h a = a = Rf = 0 . 0 r 0 x. -v. 2 X, = 0.40 / L f n .45 c c 3 D / ty 50 1 1 nptot ^ S t a b i l i 0 .55' 1 1 nptot L i i n i t e i < 0 0.25 0.5 1.0 2.0 4.0 8.0 16.0 p o — ^ F i g . 21-d E l a s t i c i t y E f f e c t on Maximum S p e e d - D e v i a t i o n 80 c l e a r l y shown by the narrower s t a b i l i t y range f o r s m a l l e r v a l u e s of PQ, F i g . 22. In comparison t o governor s e t t i n g s o b t a i n e d by u s i n g the r i g i d water-column approach, a l a r g e r temporary droop <5 and dashpot time T (or p r o p o r t i o n a l and i n t e g r a l g a i n s ) s h o u l d be used t o o b t a i n optimum governor r e s p o n s e . Examples of speed-time p l o t s f o r two p o i n t s on e i t h e r s i d e o f t h e s t a b i l i t y — l i m i t f o r pn = 0.8 and A 2 =» 0.1 a r e shown i n F i g s . 23-a and 23-b. I t i s n o t i c e d t h a t a d e c r e a s i n g d e v i a t i o n f o r t h e f i r s t few i n t e r v a l s does not n e c e s s a r i l y i n d i c a t e s t a b l e o s c i l l a t i o n . 81 F i g . 22 E l a s t i c i t y E f f e c t on the S t a b i l i t y - L i m i t 82 F i g . 2 3-b S t a b l e S p e e d - V a r i a t i o n 83 CHAPTER V SUMMARY AND CONCLUSIONS The p r i m a r y o b j e c t i v e of t h i s t h e s i s has been t o i n c l u d e e l a s t i c i t y e f f e c t s of b o t h water and pe n s t o c k w a l l s i n the m a t h e m a t i c a l a n a l y s i s o f speed t r a n s i e n t f o r a l i n e a r i z e d model. The downstream boundary c o n d i -t i o n imposed by governor and t u r b i n e c h a r a c t e r i s t i c s w i l l be an o r d i n a r y d i f f e r e n t i a l e q u a t i o n of second or h i g h e r o r d e r i n v o l v i n g the head and the d i s c h a r g e . An a n a l y t i c a l s o l u t i o n has been o b t a i n e d h e r e by u t i l i z i n g t h e t r a v e l l i n g - w a v e s o l u t i o n o f t h e wave e q u a t i o n i n the form o f A l l i e v i ' s i n t e r c o n n e c t e d s e r i e s . F i n a l r e s u l t s a r e so s i m p l e and s t r a i g h t f o r w a r d i n a p p l i c a t i o n t h a t a hand c a l c u l a t o r may be used for. t h e f i r s t few time i n t e r v a l s . U s i n g T a y l o r ' s e x p a n s i o n s , even s i m p l e r forms of the s o l u t i o n a r e d e r i v e d which are.more e f f i c i e n t n u m e r i c a l l y i n s p i t e o f t r u n c a t i o n e r r o r s i n t r o d u c e d . These s o l u t i o n s , however, a r e p i e c e - w i s e c o n t i n u o u s , 2L i . e . , v a l i d o n l y f o r a p a r t i c u l a r time i n t e r v a l ( — ) , which i s t h e time 3. f o r a complete t r i p of the p r e s s u r e wave. The a n a l y t i c a l s o l u t i o n was v e r i f i e d by comparing t h e r e s u l t s with those o b t a i n e d by t h e method o f c h a r a c t e r i s t i c s . Though no f i e l d d a t a were a v a i l a b l e f o r a d i r e c t comparison, the v a l i d i t y o f the l i n e a r i z e d model and of t h e use of c h a r a c t e r i s t i c s method i n hydro p l a n t s t r a n s i e n t s have been checked by s e v e r a l i n v e s t i g a t o r s [15, 18, 19, 20, 2 1 ] . S i n g u l a r i t i e s e x i s t a t p g = P o * and/or a 2 = 2 a 2 which a r e t r e a t e d s e p a r a t e l y but some n u m e r i c a l i n s t a b i l i t y i s encountered i n the v i c i n i t y of 84 t h e s e v a l u e s . The main c o n c l u s i o n s of the a n a l y s i s made he r e a r e : 1. Compared t o r i g i d water-column a n a l y s i s , e l a s t i c i t y of the water and p e n s t o c k w a l l s i n c r e a s e s maximum speed d e v i a t i o n i f governor s e t t i n g s a r e kept t h e same. T h i s i n c r e a s e i s more n o t i c e a b l e f o r high-head p l a n t s w i t h Po < 0.6 (at r a t e d c o n d i t i o n s ) but i t becomes l e s s t h a n 5% f o r P O > 1.0, i . e . , low-head p l a n t s . 2. The major i n f l u e n c e of t h e e l a s t i c waves i s on the s t a b i l i t y o f speed o s c i l l a t i o n s . Optimum governor s e t t i n g s d e r i v e d from a r i g i d w ater-column a n a l y s i s may r e s u l t i n a t o t a l l y u n s t a b l e o p e r a t i o n f o r high-head p l a n t s because of e l a s t i c i t y e f f e c t s . The d e s t a b i l i z a t i o n produced by waterhammer may'be n e g l e c t e d f o r low head p l a n t s o f p 0 > 2.0 ( a t r a t e d v a l u e s ) . 3. P e n s t o c k f r i c t i o n may s l i g h t l y i n c r e a s e maximum s p e e d - d e v i a t i o n but i t i n t r o d u c e s a s i g n i f i c a n t damping i n f l u e n c e . 4. The assumption of a d i s t o r t i o n l e s s p e n s t o c k w i t h f r i c t i o n f o r t h e m a t h e m a t i c a l treatment does not r e s u l t i n any a p p r e c i a b l e e r r o r . 5. Permanent speed droop and p o s i t i v e s e l f r e g u l a t i o n c o e f f i c i e n t have a s t r o n g s t a b i l i z i n g e f f e c t on speed o s c i l l a t i o n s . T h i s i s s i m i l a r t o r e s u l t s o b t a i n e d by a r i g i d column a n a l y s i s [13] but t h e amount of damping h e r e w i l l be a f u n c t i o n of P Q , as w e l l as of o t h e r parameters. 85; BIBLIOGRAPHY 1. Gaden, D. , " C o n t r i b u t i o n A L'Etude Des R e g u l a t e u r s De V i t e s s e . C o n s i d e r -a t i o n s Sur Le Probleme De L a S t a b i l i t e " . 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E. , "Step Responses o f L i q u i d L i n e s w i t h Frequency-Dependent E f f e c t s o f V i s c o s i t y " , J . o f B a s i c Engg., T r a n s . ASME, S e r i e s D, V o l . 89, no. 2, June 1965. 36. Holmboe, E. L., and Rouleau, W. T. ,"The E f f e c t of V i s c o u s Shear on T r a n s i e n t s i n L i q u i d L i n e s " , J . of B a s i c Engg., T r a n s . ASME, S e r i e s D, V o l . 89, no. 1, March 1967. 37. Z i e l k e , W. ,"Frequency-Dependent F r i c t i o n i n T r a n s i e n t P i p e Flow", J . o f B a s i c Engg., T r a n s . ASME, S e r i e s D, V o l . 90, No. 1, March 1968. 38. K a n t o l a , R. , " T r a n s i e n t Response o f F l u i d L i n e s I n c l u d i n g Frequency Modulated I n p u t s " , J . o f B a s i c Engg., T r a n s . ASME, S e r i e s D, June 1971. 39. Leonard, R. G. ?"A S i m p l i f i e d Model f o r a F l u i d T r a n s m i s s i o n L i n e " , Ph.D. T h e s i s , The P e n n s y l v a n i a S t a t e U n i v . , June 1970. 40. Oldenburger, R., and Goodson, R. E. , " S i m p l i f i c a t i o n o f H y d r a u l i c L i n e Dynamics by Use of I n f i n i t e P r o d u c t s " , J . o f B a s i c Engg., T r a n s . ASME, V o l . 86, no. 1, March 1964. 41. Karam, J . T., J r . , "A Simple Time Domain Model f o r F l u i d T r a n s m i s s i o n L i n e Systems", Ph.D. T h e s i s , Purdue U n i v . , 1972. 42. , and Leonard, R. G. , "A Simple Yet T h e o r e t i c a l l y Based Time Domain Model f o r F l u i d T r a n s m i s s i o n L i n e Systems", J . of F l u i d s Engg., T r a n s . ASME, S e r i e s I, V o l . 95, no. 4, 1973. 43. T r i k h a , A. K. , "An E f f i c i e n t Method of S i m u l a t i n g Frequency-Dependent F r i c t i o n i n T r a n s i e n t L i q u i d Flow", J . o f F l u i d s Engg., T r a n s . ASME, March 1975. 8 a 44. P i e r r e , D. A., and H i g g i n s , T. J . , "Sampled Data R e p r e s e n t a t i v e System: An E f f e c t i v e Concept f o r Use i n t h e A n a l y s i s o f D i s t r i b u t e d Parameter Systems", Instrument Soc. of America (ISA) T r a n s . , V o l . 3, no. 3, J u l y 1964. 45. G a r r e y , D. C , and D'Souza, A. F. , " D i s c r e t e - T i m e Models and S t a b i l i t y of D i s t r i b u t e d - P a r a m e t e r Systems", J . of B a s i c Engg., T r a n s . ASME, S e r i e s D, June 1967. 46. Raabe,- J . , " S t a b i l i t ' a t s b e t r a c h t u n g e n an W a s s e r t u r b i n e n r e g l e r n " , Der Maschinenmarket, (13a), Wurzburg, Nr. 12, 10 Feb., 1961. 47. S t r e e t e r , V. L., and W y l i e , E. B. , "Resonance i n Governed Hydro P i p i n g Systems", P r o c . of I n t . Symposium on Water Hammer i n Pumped Sto r a g e P r o j e c t s , ASME, Chicago , Nov., 1965. 48. Agnew, P. W. ,"The G o v e r n i n g of F r a n c i s T u r b i n e s " , Water Power, London, A p r i l 1974. 49. Brekke, H. / ' S t a b i l i t y S t u d i e s f o r a Governed T u r b i n e O p e r a t i n g Under I s o l a t e d Load C o n d i t i o n s " , Water Power, London, Sept., 1974. 50. , "A Study of th e I n f l u e n c e of T u r b i n e C h a r a c t e r i s t i c s on T u r b i n e G o v e r n i n g " , Second I n t . Conf. on P r e s s u r e Surges, London, Sept. 1976, Paper J2. 51. R a n s f o r d , G., and R o t t n e r , J . , "The O p t i m i z a t i o n of H y d r a u l i c Governor Performance T a k i n g Account of th e G r i d I n h e r e n t S t a b i l i t y F a c t o r and E l a s t i c Water Hammer E f f e c t s " , La H o u i l l e B l a n c h e , no. 1, Janv.-Feb., 1959. 52. Macagno, M., and Macagno, E. 0. > "Non-Wave, Approximate A n a l y s i s of P r e s s u r e Surges", F i r s t I n t . Conf. on P r e s s u r e Surges, C a n t e r b u r y , England, Sept., 1972. 53. "Prime Mover C o n t r o l C o n f e r e n c e " , L e c t u r e Notes, Woodward Governor Company, R o c k f o r d , I l l i n o i s , 1975. 54. Leum, M. ,"The Development and F i e l d E x p e r i e n c e of a T r a n s i s t o r E l e c t r i c Governor f o r Hydro T u r b i n e s " , IEEE T r a n s . , Power Apparatus and Systems, V o l . PAS-85, No. 4, A p r i l 1966. 55. A d l e r , R. B., Chu, L. J . , and Fano, R. M. , " E l e c t r o m a g n e t i c Energy T r a n s m i s s i o n and R a d i a t i o n " , W i l e y & Son, I n c . , New York, 1960. APPENDIX A Ex a c t S o l u t i o n A l l c o n s t a n t s a p p e a r i n g i n t h e s o l u t i o n s a r e summarized i n Appendix C. A s i n g u l a r i t y o c c u r s i f p n =* Pn* =* 0.5 £ / (£ I -I £ , ) which c o r r e -b J u u mz mn qz mz qh sponds t o PQ = 0.5 f o r a t u r b i n e o p e r a t i n g a t the p o i n t of b e s t e f f i c i e n c y . S o l u t i o n s f o r d i s c h a r g e and gate w i l l have s i m i l a r forms as f o r t h e head. These can be o b t a i n e d by a p p l y i n g r e l a t i o n s (98) and (41). In the e x p r e s -s i o n s below, a s u b s c r i p t e d v a r i a b l e y, = 0 whenever k and/or £ < 1. I t k,£ i s a l s o u n d e r s t o o d t h a t f o r any time i n t e r v a l i - 1 <_ t < i , T = t - i + l and 8 = OJT . 1- P o ^ P o * ( T l i e G e n e r a l Case) A c c o r d i n g t o v a l u e s o f a^ and a 2 , f o u r d i f f e r e n t s o l u t i o n s may e x i s t c o r r e s p o n d i n g t o the c a s e s : 1. a 2 < 2 a 2 2. a 2 > 2 a 2 ^ 0 3. a 2 > 2 a 2 = 0 and 4. a 2 =» 2 a 2 The f i r s t two a r e t h e more g e n e r a l ones. S o l u t i o n f o r a 2 < 2 a 2 has been p r e s e n t e d i n Chapter I I I . 2. a 2 > 2 a 2 ^ 0 For i - 1 <_ t < i , s o l u t i o n s f o r speed and head t a k e the form. n ± ( e ) =* e x p ( a n 6 ) [1^(6) sinhe + S (6) coshe] + CN ± and h ± ( 6 ) = e x p ( a n 0 ) [RH ± ( e ) sinhe + SH (0) cosh0 ] + CH ± w i t h 0 =• COT , and to = / a 2 - 2 a 2 P o l y n o m i a l s R^, S_/, RH^ and SH^ a r e of degree i - 1 : i ._. i ._. R . ( e ) = E r . . e 1 3 , s.(0) = z s . . e 1 J J - l J - l J RH.(6) - E r h . . O 1 3 and SH.(6) = E sh. . CN and CH. a r e c o n s t a n t s f o r the time i n t e r v a l c o n s i d e r e d , l l For i = 1 ( f i r s t i n t e r v a l ) , CNi = a-t/2a2 r i , i - - v u T ; 0 + a i 1 C N i ' s i , i C H i P 0 a _ U ' A q z m £ + a 2 4 C N 1 r h l f l " a 2 2 ( a 2 3 r ^ + s ^ ) , s h ^ For i_>_ \v- CN. = a 1 8 CN + a 1 9 a 1 5 JsCi-J+DLais + (-r. § ._ 2 + a 1 7 r . _ 1 ) . _ 3 ) ] a15 • s ± j . - £ ...» • a 1 5 • — + - " «i-> i ( i - . + l ) [ a 1 5 3 . ^ ^ . 2 + W l - 3 + 2 ) ( - s . _ . _ 2 + a 1 7 s ^ - j - j * ) • i - 1 r. . - a n ( C N . - CN ) - s L + a 2 0 ;Z ^ - 1 , 3 + * 2 1 V ^ C a 2 1 r i . 1 > 1 + . ^ . j ) ! and s. . = n , (co) - CN -1 , 1 I - 1 1 A l s o , r h = a 8 r h . _ + a 2 2 ( a 2 3 r . ^ + 8 ± > j > - A a22 ^ a2 B i . l , ; ] - ^ + a 2 2 ( i - 3 + 1 ^ r i 5 J - l - A r i - l , j - 2 ) S h . 5 . - a 8 s h ^ ^ + a 2 2 ( a 2 3 S l 5 J + *_.,;,> ' A a22 ( a23 S i - 1 ^ ^ + 1 « S i ) j - l - A S H , i - 2 ) ' ' " 2 ' ' and CH. = a 8 C H . ^ + a 2 1 + ( C N . - A CN.^) + p 0 a 5 ( l - A) £ q z m 3. a 2 >: 2a2 - 0 In t h i s s p e c i a l c a s e, s o l u t i o n s become: n.(6) => e x p ( - 26) R.(6) + S.(6) x x x and h.(6) - e x p ( - 26) RH.(0) + SH.(9) X X X R. and RH_^  a r e p o l y n o m i a l s of degree i - 1 , whereas S^ and SH_^  a r e of degree For i - 1 : r i , i • - h ( s i , i + v ( a i t ; 0 ) > s i , i - a 4 / 2 a i . s i , 2 - - r i , i and r h 1 ± =* a 2 2 ( a 3 t t - 2) ^ ± , s t ^ ± = a 2 2 a 3 4 s . ^ , s h 1 > 2 " " r h l , l F o r i > 1: r i , 3 " i=j~ 3 3 0 r i - l , 3 + 3 3 1 r i " l , 3 - l + J S ( 1 " J + 1 ) ( r i , 3 " l " 3 1 7 r i - l , J - 2 ) ' j " 1, 2, ... , i - 1 s. . =* . "I", - a 3 2 s. . . + a 3 3 s. T . 1 - % ( i - j + 2 ) ( s . . .. - a 1 7 s. . 0 ) 1,3 l - 3 + l 1-1,3 1-1,3-1 1,3-1 1-1,3-2 j = 1 , 2, ... , i s. . =•= s. . + — ^ - ( a 7 + ai+(l - a s ) ) x , i 2 a 2 1 - 1 1 r . . - h(r. . .. + s. .) - h I [ — e x p ( - 2 a i ) . (-2a x + i - j - 1 ) r . . . + 1 , 1 i , i - l 1 , 1 a x i - l , 3 d - j ) s , i a r 1 1 " 1 x x, J s. . = n ( a i ) - r . . 1,1+1 i - l 1 , 1 A l s o , r h - a 8 r h + a 2 2 ( a 3 t + - 2) ( r - A r ..) + 1 , 3 x 1 , 3 x x , j x x , j x a 2 2 ( i - j + l ) ( r - A r ) , j = 1, 2, ... , i x , j x x x , j z sh - a 8 sh + a 2 2 a 3 l + ( s - A s ) + a 2 2 ( i - j + 2 ) -X , J X X , J X X , J X X , J X ( s - i " A s - ! _ i n - ? ) i " 1 » 2 » • • • » 1 + 1 X , J X X X , J <i sh. . ,. =•= sh. . + p 0 a 5 ( l - A) I m„ x,x+l i , i + l qz £ 4. a 2 = 2 a 2 S o l u t i o n i n t h i s case reduces t o : n ^ x ) = e x p ( - a i T) R^T) + CN ± and h . ( x ) = e x p ( - a i T) RH.(T) + CH. 1 1 1 i R. and RH^ a r e p o l y n o m i a l s of a degree ( 2 i - l ) , CN^ and CH^ a r e c o n s t a n t s f o r i - 1 <_ t < i . I f a 2 = 2 a 2 = 0, CN_^  v a n i s h e s p r o v i d e d t h a t a^ = 0 a l s o , which i s t r u e f o r a l l p r a c t i c a l s i t u a t i o n s . F o r i = 1: C^i = ai+/2a 2 r i , l - " a l C N 1 " V T m O > r l , 2 - " C N 1 and CHj = p 0 a 5 m^ + alh CN X r h l , l = a38 r i , l > r h l , 2 = " C H 1 For i > 1: CN ± = a 1 8 C N ^ + a 1 9 a 3 5 a36 + TT—r r . '. . + a 1 7 r . . . „ , j = 1, 2, ... , 2 i - 2 r i , j ( 2 i - j ) ( 2 i - j - l ) r i - l , j 2 i - j L i - l , j - l * 1 7 i - l , j - 2 ' 2 I—2 / o • • -i \ r . „. = aiCCN. . - CN.) + e x p ( - a : ) 1 r i - l , j - l x , 2 x - l 1 x-1 x x r . = n. (1) - CN. x,2x x-1 x and CH ± = a 8 C H ± _ 1 + azk(CN± - A C N ^ ) + p 0 a 5 ( l - A) £ q z m^ r h . . = a a r h . . . 0 + a 3 8 ( r . . - A r . . ) + a 3 7 ( 2 i - j + l ) ( r - A r „) i , 3 a x - l , j - 2 d 0 i , j x-1,3-2 x , 3 - l i - l , j - 3 j = 1, 2, ... , 2 i I I . S p e c i a l Case of p 0 = P o * Speed d i f f e r e n t i a l e q u a t i o n i n t h i s c ase becomes: L:[:n !U)] = akl F 6 [ n i _ 1 ( T l ) ] + a k z + a 4 (1 - a ^ ) where L = D 2 + 2a\ D + 2a2 , F 6 = gs D + g6 and n ^ ( t ) = n ± ( t ) - a 4 3 n i _ 1 ( T l ) For t h e head: F 2 [ h ^ ( t ) ] = P o a 4 0 ( F e t n ^ e ) - A n i _ 1 ( 0 ) ] + (1 - A) gk] where h | ( t ) - h ± ( t ) - a ^ h^C 1.) S i m i l a r t o t h e g e n e r a l case P o ^ P o * » f o u r d i f f e r e n t s o l u t i o n s may e x i s t depending on v a l u e s of t h e c o n s t a n t s a i and a 2 . 1. a\ < 2a2 , and 2. a 2 > 2a2 ^ 0 S o l u t i o n s f o r t h e s e two cases a r e p r e s e n t e d below t o g e t h e r w i t h s i n O , cos8 f o r the f i r s t and s i n h B , cosh9 f o r the second. I n some p l a c e s where two s i g n s e x i s t t h e lower one c o r r e s p o n d s t o the second c a s e . n.(6) - e x p ( a 1 i 6 ) [ R . ( 0 ) s i n G + S.(6) C O S B ] + CN. i r\ Li '•>. xi ' s i n h x x ' c o s h X and h.(6) - e x p ( a i i 6 ) [ R H . ( 6 ) S : m , e + SH. (6) ' c o s , 0.] + l r». ± i x sxnh x cosh g2 GH.exp( 6) + CH. , f o r g 2 f 0 or GH. + CH..0 , f o r g 2 - 0 x x R.' S., RH. and SH. a r e p o l y n o m i a l s o f degree i - 1, CN., CH. and GH. x x x x x x x a r e c o n s t a n t s f o r i - 1 <_ t < i . For i - 1 : Same fo r m u l a s of cases (1) and ( 2 ) , PQ f Po* can be used h e r e t o determine r , ,., s. and CN. . E x p r e s s i o n s below f o r r h . ., sh. ., CH. 1,1' 1,1' 1 x , j x , j x and GH. a r e v a l i d a l s o f o r i - 1 . x F o r i > 1: r i , j " T = J ( a ^ 5 V i . j + a ^ 6 r i - i , j ) + — 8 i - i f j - i + a " r i - i f j - i • ha-i+D K 6 r H ) . . 2 + J.(i-j+2)(-r l f j_ 2 + a , 3 r . ^ ^ ) ] 1 -s i , j " i=J ( T 3 4 5 r i - i>3 + a ^ 6 s i - i , J ) + TT r i - i , j - i + ^ 8 i - i , J - : i _ ( i - j + l ) [ a „ 6 V l , j - 2 + h^-3+2)(- sif._2 + a , 3 s ^ ^ ) ] j =1, 2, i-1 r i , i = a H ( C N i " C N i - l } " ^ i - l + ^ 0 [ ( r i - l , j + ^ 1 S i - l , j ) + ( a 2 1 r . - . + s. _ .)] a ) i - j - 1 o) 2 1 1-1,3 i - l , 3 s. . - n. . ( t o ) - CN 1 , 1 i - l i and CN ± = a k 8 C N ^ + a k 9 A l s o , r h ^ . = a, 3 r h . ^ ._± + b ± J , s h . ^ . = s h . ^ . ^ + c ± > J j - 1, 2, ... where b. . - a 5 4 r * ± a 5 5 s j + ( i - j + 1 ) [ a 5 6 r ' ± a 5 7 s ' _ ± -i,3 3 3' ^ a c o b. . + a S n c. . . ] b S 1,3-1 i , 3 - l and c = a 5 1 +- s^ - a 5 5 r_! + (i-j+1) [ a 5 6 s ^ - a 5 ? r ' ^ -a t r o C . . . + a,-q b. . - ] 5 8 1,3-1 b y i , 3 - l i n which r ! - r . . - A r and s' = s - A s 3 1,3 1-1,3-1 3 i,3 i - l , 3 i CH. = a 4 3 GH._ 1 + — j - [ g 6 ( C N . - A CN.^) + (1 - A) g l + j ^ ] g 2 , f o r g 2 ^ 0 where k - ^ wgl , f o r g-2 = 0 and GH. = h. . ( t o ) - (CH. + s h . .) , g 2 + 0 l i - l i 1 , 1 ^ = h. , ( t o ) - sh. . , g 2 = 0 i - l 1 , 1 z 3. a 2 > 2 a 2 = 0 R e l a t i v e speed d e v i a t i o n i s g i v e n by: n.(0) = e x p ( - 20) R.(0) + S.(0) l i i where R. and S. a r e p o l y n o m i a l s of degrees i - 1 and i , r e s p e c t i v e l y . E x p r e s s i o n s f o r r , s.. ^  and s . a r e the same as i n case (3) f o r 1 , 1 1 , 1 i»z PO ^ P o * -F o r i > 1: r i , j * i - j ( a 6 1 - a 6 0 ) r . ^ ^ . + ( a 4 3 - a 5 1 ) r . ^ ^ . ^ + % ( i - j + l ) ( r - a 4 3 r „) , j = 1, 2, ... , i - 1 i j j i i i> J ^ a60 s. . =» -—T7T s, -. . + ( a u 3 + a g i ) s. .. -i , J i-J+1 i - l , J • b i i - l , J - 1 J s ( i - j + 2 ) ( s ._, - a 4 3 s „) , j - 1, 2, ... , i i , J i i i , J ^ S i , i * S i , i + 2 a f ( a " 2 + 3 4 ( 1 " i - 1 1 r . . - % ( r . n + s .) -h E [ — e x p ( - 2 a i ) . ( - 2 & 1 + i - j - 1 ) r + 1 , 1 1 , 1 - 1 1 , 1 j = l S i - l , j ] a l " j " 1 and s. n ( a x ) - r . . i , i + l i - l 1 1 , 1 P r o v i d e d t h a t g 2 ^ 0, r e l a t i v e head d e v i a t i o n can be e x p r e s s e d a s : - g2 h.(0) =* e x p ( - 20) RH.(0) + S H . ( 0 ) + GH. exp( 0) i i l l a x g x w i t h RH_ and SH^ b e i n g p o l y n o m i a l s o f degrees i - 1 and i , r e s p e c t i v e l y , and GH. i s a c o n s t a n t f o r i - 1 < t < i . l — r h . . - auo r h . _ . n + ap? r ! + a f i 3 ( i - j + l ) v\ , - . i , j ^ i - l , J - 1 6 2 J J-1 a 6 u ( i - J + l ) ( r h . . , - r h - s _ i - i - ^ ' j - 1 , 2, ... , i ' i , J - 1 i i , J ^ S h i , j " a h S S h i - l , j - l + 3 6 5 S j + a 6 6 ( i " J + 2 ) s j _ l " a 6 7 ( i - j + 2 ) ( s h - a k 3 sh ) , j = 1, 2, ... , i + 1 i , J i i i , J ^ s h i , i - i 3 s h i , i + i + P o a ^ a - A ) V i i w h e r e r j • r i , 3 - A r i - 1 , 3 - 1 ' a n d s _ = s i , j - A s i - l 5 j - l GH. - h. . (a.) - r h . . - sh. ... . 1 i-1 1 1 , 1 i , i + l .. However, i f g 2 • 0, s o l u t i o n changes s l i g h t l y t o : h (0) = e x p ( - 20) RH (6) + SH ±(0) RH^ and SH^ a r e p o l y n o m i a l s o f degrees i - 1 and i + 1, r e s p e c t i v e l y . E x p r e s s i o n s f o r r h . . a r e the same. a 6 8 sh. . - a,,o s h . . " + .. ... s! + a 6 9 s' , j = * l , 2, ... , i + l 1,3 1-1,3-1 1-3+I 3 b y 3-1 and sh. . .. * h. . (a - i ) - r h . . i , i + l l - l 1 i > i 4. a 2 = 2 a 2 r S o l u t i o n f o r the speed becomes n (x) - e x p ( - &x x) R ± ( x ) + CN ± where R^ i s a p o l y n o m i a l of degree (2 i - 1) and CN_^  i s a c o n s t a n t . F o r i = 1: CN X - a i + / 2 a 2 , r » - a x CN X - n ^ / T ^ , and - - CN X A g a i n , f o r aj = a 2 =• 0, a^ w i l l a l s o be z e r o f o r most p r a c t i c a l cases and CN. - 0. l F or i > 1: at.l(g6 " a l g5> a t l §5 i , j " ( 2 i - j ) . ( 2 i - . j - l ) r i - l , j + 2 i - j r i - l , j - l + * k 3 r i - l , 3 - 2 ' j = - l , 2 , ... , 2 i - 2 2 i - 2 r i , 2 i - l " a l < C N i - l " C V + e x P ( " a'l> \ ( 2 i - J - 1 ) r i - l , j - ] j = l r . .. - n. (1) - CN. i , 2 i i - l v 1 . CN ± - ay 8 C N ^ + a i t 9 For g 2 # 0, s o l u t i o n f o r t h e head i s g i v e n by: g2 h ( T ) = e x p ( - a i x) R H . ( T ) + CH. + GH. . exp( x) i i i i g l where RH^ i s a p o l y n o m i a l of degree 2 i - 1, and CH.^ and GH^ a r e c o n s t a n t s . r h . s . - a ^ r h . ^ ._ 2 +-a 7 o(r ± fj - A r ± _ l f J _ 2 ) + ( 2 i - j + l ) [ a 7 l ( r i , j - l " A r i - l , j - 3 ) " a 7 2 ( r h i , j - l " 3 4 3 r h i - l , j - 3 ) ] ' j " 1 , 2, ... , 2 i PO ak0 CH, = 343 CH. + — [ g 6 ( C N . - A CN ) + (1 - A) gk I ] GH. - h. .(1) - r h . _. - CH. x i - l i , 2 i l I f g2 =* 0, s o l u t i o n becomes h.(x) - e x p ( - a i x) RH.(x) + CH. .x + GH. l i i i Formulas f o r r h . . a r e t h e same. PO a t+0 CH. =- 343 CH. - + [g6(CN. - A CN. ) + (1 - A) g 4 I ] l i - l g l l l - l qz GH. - h. (1) - r h . l i - l i , 2 i These e x p r e s s i o n s a r e a l s o v a l i d f o r i - l . 98 . APPENDIX B Approximate S o l u t i o n Remarks made i n the b e g i n n i n g of Appendix A a r e a p p l i c a b l e h e r e a l s o , but r e l a t i o n s (116) and (41) a r e used t o determine d i s c h a r g e and gate a f t e r s o l v i n g f i r s t f o r speed and head. I t has been shown t h a t s o l u t i o n s w i l l d i f f e r a c c o r d i n g t o v a l u e s of a^ and a 2 as w e l l as pQ. Approximate formulas u s i n g T a y l o r ' s expansions f o r the two cases a 2 < 2 a 2 and a 2 > 2 a 2 ^ 0 (PO ^ P o * ) were d i s c u s s e d i n Chapter I I I . S o l u t i o n s f o r the a d d i t i o n a l s p e c i a l c a ses a r e p r e s e n t e d below. The cases of a 2 > 2 a 2 - 0 and a 1 =* a 2 0 a r e d i s r e g a r d e d h e r e s i n c e e x p a n s i o n f o r m u l a s f o r them have no o r l i t t l e advantage over the ex a c t ones. I- PO ^ P O * 4. a 2 =- 2 a 2 i 0 S o l u t i o n f o r the speed y i e l d s , f o r i - 1 <_ t < i : n. (x) " E r . . x 1 • i i>3 where =* M + 1, and M i s t h e h i g h e s t power i n the ex p a n s i o n Mi Mj-j e x p ( - a x T ) = Z X. x 3=1 J r i , j = a l A j + 1 + a 2 A j + C i , j ' 3 = 1 , 2 , ... , M and r . „ =" n. . (1) x,Mx . 1 - 1 a l i n which c . 5 . - a 1 8 r . _ l 5 . + ( M 2 - j ) [ a 2 7 r . ^ . ^ - — c . ^ + ( M 3 - j ) ( a 2 8 r . _ l 5 . _ 2 - c ± J _ 2 ) ] , j - 1, 2 , . . . . Mj and c. ,. 1 c. ,. + a i q 1,1-! l , - ^ 1 9 where M 2 = M + 2 and M 3 = M + 3 The c o n s t a n t s a\ and a 2 a r e g i v e n by: Ml a l " " a l C i , M i " C i , M + .l± ( a l + M l " j ) r i - l , j and az - n . ^ C D - c . ^ F o r i = 1: C l , j " ° ' j * 1 5 2 ' ' M ^ °l,Mi * 2a a l s o a i - - m £/ T; 0 - a x c ^ , a 2 - -The head can be s i m i l a r l y e x p r e s s e d a s : 2 1 M i - j M h . ( r ) =- E u. . T j-1 ^ and u. M - h. (1) i , M i x-1 where r'. - r . . - A r . . 3 x,3 x-1,3 1 1 • P o " P o ' 1. a 2 < 2a2 and 2. a 2 > 2a 2 ^ 0 F o r i - 1 < t < i : M 1 M x-j n.(e) - E r . . j-1 l ' J 99 w i t h u. . - a„ u. - . + a 2 U r ! + a o 7 ( M 2 - j ) r ! , , j - 1, 2, ... , M 1,3 a x-1,3 / H 3 d / J - i : . . * a i A l . + a 2 X2. + c . x,3 x 3 z 3 i , : and r . - n. (a)) i , M i x-1 X l j , A.2., a-i and a 2 a r e c a l c u l a t e d i n the same way as i n cases (1) and (2) f o r P Q 4 p 0 * ( f o r b o t h i >_ 1) * ^ 8 r i - l , j + ( ^ > [ a 7 3 . r i - l , J - 1 " " a 7 u,2 ( M 3 - j ) ( a , 3 V ^ . ^ - c ± J _ 2 > ] , C - 0 , and c =- 3^/232 1,3 X j l ^ l A l s o , s o l u t i o n f o r the head i s : M / s x f i n h.(0) - E u Q 1 R M, - j . . 9 1 J w i t h u. , » u ; , . + aq 1,3 * 6 i - l , 3 ' and u. „ = h. (co) i,Mj i - l 8 2 M l « A3, i s g i v e n by: e x p ( - 9 ) - E A3. 6 1 J u g i ^ 3 . b, , - [Po a40 §6 ^ + W ( M 2 - J ) ( P 0 a«f0 §5 i>3 &2 J PO a40 §4 and b. „ - b. .. + (1 - A) Jl i . M i i . M i g 2 qz where r ! = r . . - A r . . 3 1,3 i - l , 3 aq = h. n (to) - b. - a ^ c i u. . 3 1-1 I J M J H 3 i - l , 4. a 2 - 2 a 2 ^ 0 For i - 1 <_ t < i : ---- Mi n.(x) - E r . . x j = l ^ :. . - cn A ." -. + a 2 X . + c i,3 . 1 3+1 2 3 a n d r i , M l = n i - l ( 1 ) X. i s d e f i n e d by: exp(-a^ T) - E X:. T 1 3 j = l 1-101 C i , j " ^ 8 r i - l , j + < M 2 - J ) [ a 7 3 r i - l , j - l ' ^ C i , j - 1 + 2 ^ ( M 3 " J ) ^ 3 r._lf._2 - c ± f j _ 2 ) ] , j - 1, 2, ... , M l a n d C i , M l •• C ± , K 1 + a ^ The c o n s t a n t s 0 4 and a2 a r e g i v e n by: Ml a l " " c 4 . - M - a i c ... , ., + E (ai + Mn-j) r . 1 . i,M 1 i , M i j - i 1 - 1 »J and a 2 > n . ^ U ) - c . ^ F o r i _ = _ l j c l , j . " " ° ' C l , M i • 3 4 / 2 3 2 ' A I = " V T m O ' 3 1 C l , M i ' a n d K 2 * " C l , M i A l s o , h . ( T ) - E u. . x 1 J j - 1 ^ The c o e f f i c i e n t s u. . a r e determined by u s i n g f o r m u l a s of c a s e s (1) and (2) f o r po = PQ*, w i t h co b e i n g r e p l a c e d by 1.0 APPENDIX C Con s t a n t s The c o n s t a n t s through gg and a± t h r o u g h a 7 3 a r e v a l i d f o r a l l time t . Only a few of them a r e used f o r a p a r t i c u l a r case of p-, a± and a 2 . D e f i n i t i o n s o f a b s o l u t e and r e l a t i v e system parameters a r e p r e s e n t e d i n the b e g i n n i n g o f t h i s t h e s i s t o g e t h e r w i t h o t h e r n o t a t i o n s . g! t o g 4 (see t a b l e on page 46) p n * = 0.5 I /(I , I - I I , ) K U mz mh qz mz qh L e t 7 " £ m z ( 1 " Po/PO*)/^ 1 + 2P0 A q h ) 3 1 - ( g 2 Tm0 + a g l + g 3 y + g l 3 3 PO V / ( 2 g l T m 0 } a 2 = ( a g 2 + Y + g 2 a 3 p 0 A q n)/'(2g- T^_ a 3 ' 2 ^ m h / ( 1 + 2 p 0 V a 4 =• ( g L f y - g 2 m £ ) / - ( g i T^ Q) a 5 - - 2/[l (1 - '-^-) ] 5 mz PO"* A = e x p ( - 2 R f / p 0 ) a 6 = - pQ A a 3 a 5  A 7 ' S2 a 6 J l q z m £/.(gi-T; 0) a 8 > A ( l •+ a 5 £ m 7 ) mz a 9 = a / r 0 + g 2 / g l + £ m z £ q n / ( T ; 0 , / q z ) a i o = 82 Ca + £mz V / £ q z > / ( S l ^ / ..a2 - 2 a 9 1 z a i i " - a i /w a 1 2 ^ a 1 0 / w 2 + a 9 a n / u . + a*. + 1 (+ f o r a 2 > 2 a 2 ) a i 3 * ag/co + 2a i : L a 1 4 - 0.5 a 6 a 1 2 £ q z a 1 5 - 0.5 a 6 a 1 3 £ q Z a 1 6 •a 6 £ (1 ± 0.25 a 1 2 ) + a 8  D qz a 1 7 a 6 * '+ a 8 q^. a 1 8 - a 8 + 0.5 a 6 a 1 0 J l / a 2 q^ a 1 9 0 . 5 ( a 7 + a^Cl - a 8 ) ) / a 2 a 2 0 e x p ( - a-^) GOSCO , a 2 < 2 a 2 o r . a 2 1 tanco , a 2 < 2 a 2 o r tanhco , a 2 2 rx p n a 5 T' w Si K U 0 mO qz a 2 3 =- a n + (a + I I IH T' 1 1 mz qn qz mO a 2 4 p n a 5 (a I + H J l ) u 0 qz mz qn a 2 5 = co(a 6 a g i l + 2a! a 8 ) / 2 a 2 qz a 2 6 W 2 ( a 6 A + a 8 ) / 2 a 2 a 2 7 ( a 6 a 9 J l + 2 a : a 8 ) / 2 a 2 q-4 a 2 8 ( a6 * + a 8 ) / 2 a 2 qz, a 3 0 a 6 £ q z ^ a 1 0 - 2 a l a 9 + 4 a 2 ) / 2 a 2 a 3 1 a 8 " a 6 ^ <a9 ~ 4 a i ) / 2 a i a 3 2 a 6 a 1 0  l q z / 2 a l a 3 3 a 8 + a 6 a 9 £ q z / 2 a i a3U a i l + J l J l qz mz qn a 3 5 =» a 6 £ q z ( a 1 0 " a l a 9 + a 2 ) a 3 6 = a 6 * q z ( a 9 " 2 a l > a 3 7 po a s T ; 0 i q z a38 - a2k ~ a l a 3 7 103. (- f o r a 2 > 2 a 2 ) ex p ( - a ^ c o shto , a 2 > 2 a 2 a 2 > 2a; 85 ' 81 * q n " 83 * q Z §6 ' 82 S- Q N " % Z a t t 0 - - 2/(1 + 2p 0 J l q n ) aitx — A PO a 3 a 4 o / ( 8 l : - : T m o ^ at+2 = ai+i gi+ I qz ai>3 ' A ( l + ak0) akk =* a l l §5 + §6/ w a^g = g 5 a^/203 347 - ± a^i 344 / 4 1 0 (- for a 2 > 2s 2) a^e = a^g + g 6 a 4 1 / 2 a 2 a^g = ( a t t 2 + 3^(1 - 3[ t3))/2s 2 a50 = a l l Si + 82/w a51 ' ' / K o ± 8?) (- for a 2 > 2a 2) (- for a 2 > 2a 2) 344 a 5 0 ± g l g 5 a53 = §1 akk ~ §5 a50 a54 * PO a40 a51 a52 a55 * PO a40 a51 a53 a56 = PO a40 a50 a51 §5 a57 - PO a40 a51 81 85 a58 " a50 a51 81 a59 ' a51 S 2  a6 0 = a41 8 6/2 a 2  a61 " a41 8 5 / 2 a l a62 " PO aU0(S 6 - 2 a l 85)/.(82 " 2 a l Si) a63 =* PO a^0 a l 85/(82 ~ 2 a l 81) a6k = a l 81/(g2 " 2 a l 81) a65 * PO a40 86/S2 a66 2P0 a40 85/82" a67 = a l 81/82 a68 " PO a40 8 5 / ( a l Si) 105 a 6 9 =* 2 p 0 aito gs/gl ayo =* Po ai+o(g6 " a l g5>/(g2 ~ a l gl) a 7 i = po ai+o gs/(g2 " a l gl) a72 = g l / ( g 2 " a l gl) a 7 3 =* toCai+i g 5 + 2 a x a i +3)/2a 2 

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