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Modelling of hydraulic components for hydroelectric power generating units Thiessen, Peter Stewart 1973

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MODELLING OF HYDRAULIC COMPONENTS FOR HYDROELECTRIC POWER GENERATING UNITS by PETER STEWART THIESSEN B.A.Sc, University of B r i t i s h  Columbia, 1963  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE  REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE  i n the Department of Electrical  Engineering  We a c c e p t t h i s t h e s i s as conforming required standard  THE  t o the  UNIVERSITY OF BRITISH COLUMBIA  May 1973  In p r e s e n t i n g an  this  thesis i n partial  advanced degree a t t h e U n i v e r s i t y  the  Library  s h a l l make i t f r e e l y  f u l f i l m e n t of the requirements f o r o f B r i t i s h Columbia, I agree  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 that permission f o r extensive for  copying of t h i s  thesis  s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by h i s r e p r e s e n t a t i v e s .  I t i s understood that  of t h i s  thesis f o rfinancial  written  permission.  Department o f  ms)y  gain  / 7 , /9~72>  Columbia  copying or p u b l i c a t i o n  s h a l l n o t be a l l o w e d w i t h o u t my  £L,(TCy/Q/o/QL  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, C a n a d a  Date  that  &AJ<S-/A/67^/A>  ABSTRACT  The t h e s i s d e a l s w i t h a p r o c e d u r e f o r d e v e l o p i n g m a t h e m a t i c a l models o f h y d r a u l i c components f o r h y d r o e l e c t r i c power g e n e r a t i n g u n i t s . H y d r a u l i c system components m o d e l l e d a r e t h e p e n s t o c k , r e s e r v o i r , s u r g e t a n k , F r a n c i s t u r b i n e and t h e w i c k e t g a t e a c t u a t o r .  A modelling p h i l o -  sophy i s s u g g e s t e d . The t h e s i s proposes t h a t f o r a g e n e r a t o r e x p e r i e n c i n g a sharp t r a n s i e n t , such as l i n e b r e a k e r r e c l o s i n g , i t s c a p a c i t y t o m a i n t a i n s y n c h r o n i s m can be enhanced by t a k i n g advantage o f waterhammer t o s h a r p l y reduce  the t u r b i n e ' s h y d r a u l i c t o r q u e i n p u t t o the g e n e r a t o r d u r i n g  t h e f i r s t few swings of t h e t i e - l i n e . i n c l u d e d support t h i s  claim.  i  R e s u l t s of computer s i m u l a t i o n s  TABLE OF CONTENTS  ABSTRACT TABLE OF CONTENTS 1.  INTRODUCTION  2.  COMMENTS ON MODELLING  3.  THE PENSTOCK  4.  PENSTOCK TERMINAL CONSTRAINTS 4.1  The Reservoir  4.2  The Surge Tank  5.  THE ACTUATOR  6.  THE FRANCIS TURBINE 6.1  Introduction  6.2  Turbine-Penstock End Constraints  6.3  Turbine Torque  6.4 7.  8.  The Turbine By-Pass Valve  COMPUTER SIMULATIONS 7.1  Introduction  7.2  The System Model.  7.3  Discussion of Simulation Results  7.4  Comments  SUPPLEMENTARY CONSIDERATIONS 8.1  The Draft Tube  8.2  Actuator Design  8.3  The Unit's Structural  9.  CONCLUSIONS  10.  REFERENCES  Strength  ii  ACKNOWLEDGMENT  The author wishes to express h i s gratitude to Dr. E. Sigurdson, h i s thesis supervisor.  The suggestions, c r i t i c i s m s and help from the  members of the committee, Dr. M. S. Davies, Dr. E. Ruus and Dr. Y. N. Yu are also appreciated.  iii  MODELLING OF HYDRAULIC COMPONENTS FOR HYDROELECTRIC POWER GENERATING UNITS  1.  Introduction  This thesis deals with the development  of mathematical models  of selected hydraulic components .of .a hydroelectric power generating u n i t . The work was motivated by the p o s s i b i l i t y of enhancing a generator's capacity to maintain synchronism when experiencing a sharp system d i s t urbance by exercizing rapid control of the hydraulic torque during the transient condition.  Rapid control of torque i s not expected from gate  actuation alone, but by the combined e f f e c t of gate motion and waterhammer.  The objective of rapid torque control i s to reduce the hydraul-  i c torque applied to the unit's r o t a t i n g assembly during the f i r s t of the power system following l i n e breaker r e c l o s i n g .  swing  This w i l l reduce  the p o s s i b i l i t y of pole s l i p p i n g and enhance the unit's capacity to weather transient conditions without f a l l i n g out of synchronism. The fundamental quantity which determines whether a synchronous e l e c t r i c power generating unit i s i n synchronism when under the influence of a severe transient i s the e l e c t r i c torque angle.  1  Since a c t u a l f i e l d  2  p r o f i l e s are non-linear '  and r e q u i r e s e v e r a l F o u r i e r terms t o model  them a c c u r a t e l y , a p h y s i c a l d e f i n i t i o n f o r s y n c h r o n i s m i s t h a t t h e a i r gap t o r q u e a c t s i n a d i r e c t i o n o p p o s i t e t o t o r q u e a n g l e growth.  Once  t h e t o r q u e changes s i g n , t h e r o t o r i s s a i d t o have s l i p p e d a. p o l e . S l i p p i n g a p o l e once o r s e v e r a l t i m e s may n o t be d e t r i m e n t a l t o t h e u n i t ' s s t r u c t u r e and w i n d i n g s ;  however c o n t i n u e d s l i p p i n g r e p r e s e n t s  l o s s o f s y n c h r o n i s m and i n c u r r e n t W e s t e r n p r a c t i c e t h e u n i t i s t h e n 3 u s u a l l y removed from t h e e l e c t r i c g r i d by p r o t e c t i v e r e l a y s .  (Venekov  c l a i m s u n i t s i n t h e U.S.S.R. a r e a t t i m e s kept r u n n i n g a s y n c h r o n o u s l y and a t t e m p t s a r e made t o r e s y n c h r o n i z e w i t h o u t t r i p p i n g them.) u n i t may o f c o u r s e l a t e r be r e s y n c h r o n i z e d and r e l o a d e d ; t h e power system w i l l have o p e r a t e d m o m e n t a r i l y  The  nonetheless  a t a n e t power d e f i c i t  and customers i n p r o x i m i t y o f t h e u n i t w i l l have e x p e r i e n c e d v o l t a g e and f r e q u e n c y v a r i a t i o n s u n t i l o t h e r u n i t s have been a b l e t o i n c r e a s e their generation. The t y p e o f s e v e r e t r a n s i e n t s b e i n g c o n s i d e r e d a r e t h o s e due to  l i n e c i r c u i t b r e a k e r s r e c l o s i n g f o r l i n e f a u l t c l e a r i n g o r by o t h e r  nearby u n i t s i n t h e g r i d t r i p p i n g o f f - l i n e as a r e s u l t o f i n t e r n a l d e t e c t i o n o r a c c i d e n t a l o p e r a t i o n by o p e r a t o r s . c h a r a c t e r i z e d by t o t a l o r near t o t a l l o s s  fault  The t r a n s i e n t s a r e  of e l e c t r i c r e a c t i o n torque  i n the generators. In  t h e l i t e r a t u r e , t o r q u e a n g l e has been g i v e n d e f i n i t i o n s  r a n g i n g from t h e a n g l e between t h e fundamentals 4 and t h e r o t o r magnetomotive f o r c e phasor and t h e d i r e c t o r q u a d r a t u r e  o f t h e a i r gap f l u x  t o t h e a n g l e between the bus v o l t a g e axes.^'^  3  A d o p t i n g t h e l a t e r , t h e t o r q u e a n g l e speed i s g i v e n by  dG = n  co  dT  r  -  (1-1)  co s  where co i s t h e speed o f t h e r o t o r t o w h i c h t h e d i r e c t a x i s i s f i x e d r r  by d e f i n i t i o n , n i s t h e number of p o l e p a i r s and co^ i s t h e speed of t h e bus v o l t a g e phasor a t t h e u n i t ' s t e r m i n a l s o r a t some c o n v e n i e n t reference point.  I n e q u a t i o n (1-1) co^_ i s an a b s o l u t e v e l o c i t y w i t h  r e s p e c t t o an i n e r t i a l r e f e r e n c e . e l e c t r i c a l radians/second.  U n i t s o f speed f o r co^ and 8 a r e  A c c e l e r a t i o n of t h e r o t o r , assembly i s  g i v e n by  —  = J  . ( h y d r a u l i c t o r q u e i n p u t - magnetic torque - l o s s e s )  reaction (1-2)  From e q u a t i o n s (1-1) and (1-2) we s e e t h a t a c c e l e r a t i o n o f t h e t o r q u e a n g l e toward l o s s o f s y n c h r o n i s m i s d i r e c t l y i n f l u e n c e d by t h e h y d r a u l i c t o r q u e a p p l i e d t o t h e r o t a t i n g assembly by t h e t u r b i n e . The h y d r a u l i c and magnetic  a i r gap t o r q u e s a r e t h e two dom-  i n a n t f a c t o r s g o v e r n i n g t h e b e h a v i o r o f t h e t o r q u e a n g l e and t h e r e f o r e t h e u n i t ' s a b i l i t y t o m a i n t a i n s y n c h r o n i s m o r , a f t e r s l i p p i n g a few p o l e s , t o r e c a p t u r e synchronism.  C o n t r o l o f t h e s e t o r q u e s i s thus o f  v i t a l importance i n g u i d i n g a u n i t t h r o u g h a t r a n s i e n t . I n r e c e n t t i m e s much a t t e n t i o n has been d i r e c t e d toward t h e c o n t r o l o f e l e c t r i c t o r q u e w h i c h i s a c h i e v e d by t h e use o f h i g h  speed,  h i g h peak v o l t a g e s t a t i c e x c i t e r s w h i c h f i e l d t e s t s have shown t o be 7 8 s u c c e s s f u l and p r a c t i c a l . '  4  Of the two dominant torques, t h i s thesis d i r e c t s attention to the hydraulic torque applied to the r o t a t i n g assembly by the turbine and proposes that rapid control of hydraulic torque may be f e a s i b l e i n helping a unit maintain synchronism, p a r t i c u l a r l y f o r plants with long penstocks, by taking advantage of waterhammer i n the penstocks.  To  t h i s end, t h i s study emphasizes the hydraulic components of a hydroe l e c t r i c unit. The f i r s t step i n an i n v e s t i g a t i o n to study the f e a s i b i l i t y of hydraulic torque control i s to obtain a model of a l l the dominant and necessary dynamics of the physical hardware.  This study l i m i t s  i t s e l f to the presentation of an approach to be taken when f a b r i c a t i n g a (mathematical) model and points out precautions to be taken i n t h e i r preparation.  •Although some models are given, a complete model set i s  not offered, f o r a model set i s v a l i d only f o r one p a r t i c u l a r generating unit.  To obtain values of c o e f f i c i e n t s and model s i m p l i f i c a t i o n s  i t i s further necessary to run f i e l d tests on a p a r t i c u l a r unit to v e r i f y the v a l i d i t y of numerous assumptions, model adequacy and fication.  simpli-  F i e l d tests on e x i s t i n g units werenot part of t h i s study.  The work i n t h i s study however prepares the ground f o r f i e l d t e s t s to confirm the models, which would be the next step. The development of t h i s report begins with a modelling approach philosophy.. It then considers the penstock which leads water to the turbine from a nearby reservoir or from an intermediate surge tank, the wicket gate servomotors which control the flow of water to the turbine, and f i n a l l y the turbine i t s e l f .  5  Since turbine torque depends on the pressure head i n the s c r o l l case, the dynamics of the e n t i r e hydraulic system, p a r t i c u l a r l y the penstock, a f f e c t the turbine torque.  It i s therefore necessary to  precede the derivation of the turbine torque expressions with the modelling of the penstock and the surge tank i n order to solve for the pressure head. Turbine torque also depends on the volume of water flowing through the turbine.  This flow i s regulated by the wicket gates and  since the dynamics of the servomotors which d r i v e the gate or valve mechanism have an appreciable influence on the o v e r - a l l dynamics, the servomotors too must be  modelled.  A l l models w i l l be sets of f i r s t order d i f f e r e n t i a l or difference equations to f a c i l i t a t e t h e i r use i n dynamic optimization methods f o r generating optimal c o n t r o l sequences.  6  2.  COMMENTS ON MODELLING  B e f o r e embarking upon m o d e l l i n g o f h y d r a u l i c  components, i t i s  e s s e n t i a l t o e s t a b l i s h f i r s t an approach p h i l o s o p h y t o m o d e l l i n g .  Model-  l i n g can be c o n t r o v e r s i a l f o r t h e r e can be many v a r i a t i o n s , p a r t i c u l a r l y when d e a l i n g w i t h tics.  devices having nonlinear  The f o l l o w i n g d e s c r i b e s  veloping  and d i s c o n t i n u o u s  a s u g g e s t e d approach t o be t a k e n when de-  models and i t i s used as a guide i n t h i s  thesis.  The key word i n e n g i n e e r i n g i s p r e d i c t i o n . concerned w i t h  characteris-  Our e f f o r t s a r e  the p r e d i c t i o n o f e f f e c t s i n t h e f u t u r e .  In order to  p r e d i c t the e v o l u t i o n o f a p h y s i c a l p r o c e s s , however, i t i s n e c e s s a r y f i r s t t o o b t a i n a model o f the p h y s i c a l p r o c e s s and then r u n the model a t a time s c a l e f a s t e r than r e a l - t i m e .  While running at the f a s t e r time  r a t e , the e v o l u t i o n o f t h e model s t a t e i s the p r e d i c t i o n o f how the p h y s i c a l process w i l l evolve i n the r e a l - t i m e  future.  P r e d i c t i o n i s the  essence o f e n g i n e e r i n g and the model o f t h e p h y s i c a l p r o c e s s p r o v i d e s the means o f a c h i e v i n g  the p r e d i c t i o n .  P r e d i c t i o n can n e v e r be e x a c t . finite  There w i l l  always be some  e r r o r between t h e e v o l u t i o n o f t h e model's s t a t e t r a j e c t o r y and  that o f the p h y s i c a l process l a t e r i n r e a l - t i m e .  The e r r o r i n p r e d i c -  t i o n n o n e t h e l e s s can be reduced by the use o f an a c c u r a t e model o f t h e process. Processes contain  an i n f i n i t e number o f s t a t e v a r i a b l e s and  p a r a m e t e r s , and the l a r g e r the number o f q u a n t i t i e s i n c o r p o r a t e d the model, the b e t t e r t h e p r e d i c t i o n .  into  The s a l i e n t q u e s t i o n i s how t o  l i m i t t h e number o f e f f e c t s t o be p r e d i c t e d by t h e model, and how t o determine xtfhich parameters and s t a t e v a r i a b l e s s h o u l d be  included.  7  T h i s i s the dilemma of the model d e v e l o p e r . At an e a r l y s t a g e i n model development ponse must be e s t a b l i s h e d . speeds may  the time range o f r e s -  The dominant i n f l u e n c e s f o r p r o c e s s e s a t low  be q u i t e d i f f e r e n t from t h o s e d e s c r i b i n g the e v o l u t i o n o f the  s t a t e a t h i g h speeds.  F o r example,  i f a h y d r a u l i c system c o n s i s t s o f  a t e n m i l e t u n n e l , a surge tank and a 4000 f o o t p e n s t o c k and the o b j e c t i v e i s to c o n t r o l a u n i t ' s s t e a d y - s t a t e power g e n e r a t i o n , then t h e hyd r a u l i c components may systems.  be s a t i s f a c t o r i l y m o d e l l e d as lumped parameter  However, i f the o b j e c t i v e i s t o c o n t r o l the t u r b i n e ' s h y d r a u l i c  t o r q u e d u r i n g a one second system s w i n g , then the. p e n s t o c k must be led in  model-  as a d i s t r i b u t e d parameter system and the surge tank and t u n n e l may c e r t a i n p l a n t s be n e g l e c t e d .  The response time of i n t e r e s t has a  d e f i n i t e i n f l u e n c e on the v a r i a b l e s and parameters  to be  incorporated  i n t o the model and on the c o m p l e x i t y , o r s i m p l i c i t y , o f the model. In  our a p p l i c a t i o n , the time i n t e r v a l d u r i n g w h i c h  i s sought i s between z e r o and f o u r seconds. have e i t h e r m a i n t a i n e d or l o s t  prediction  After t h i s , a unit  will  synchronism.  The e v e n t u a l need t o compromise between model s i m p l i c i t y a c c u r a c y has been i m p l i e d e a r l i e r .  and  I n so d o i n g , c o g n i z a n c e must be  t a k e n o f the f a c t t h a t c e r t a i n parameters are more dominant t h a n o t h e r s . By s e n s i t i v i t y methods a sequence lished. are  of dominant parameters can be e s t a b -  R e l a t i v e dominance o f parameters w i l l s h i f t as s e n s i t i v i t y methods  applied along d i f f e r e n t state  trajectories.  W i t h the rank o f parameters i n hand, the c u t - o f f p o i n t must n e x t be e s t a b l i s h e d .  The p r o p e r p r o c e d u r e cannot be g i v e n a t t h i s  t i a l s t a g e of model development ment .  ini-  b u t one can s u r m i s e the f o l l o w i n g a r g u -  8  In the end, the o v e r a l l of d o l l a r s .  cost function must have the units  Each a d d i t i o n a l parameter included increases the model's  complexity and necessitates i t s measurement.  Inclusion of the parameter  w i l l also increase the accuracy of the p r e d i c t i o n and hence lead to an improved r e a l i s t i c minimum f o r the cost function and a truer optimal control.  The reduced cost function i s the b e n e f i t , which must be assigned  a d o l l a r value.  The model complexity cut-off point i s then determined  by the point i n the sequence of decreasing parameter dominance where the cost of including the parameter and a l l i t s associated costs equals the cost benefit i n a lower, truer minimum cost value of the cost tion.  In a sense  func-  this i s an embedded optimization problem.  The above philosophy following observations  of approach to modelling  leads to the  and considerations.  In the o v e r - a l l scope of a p r o j e c t , model development i s an i t e r a t i v e process.  From f i e l d experience with devices one f i r s t begins  with a best guess of the c h a r a c t e r i s t i c s , and writes t h e i r  interdepen-  dences i n shorthand form c a l l e d an i n i t i a l mathematical model. interdependencies equations. has captured  These  are expressed as sets of d i f f e r e n t i a l and algebraic  After this, f i e l d  tests must be run to v e r i f y that the model  the dominant c h a r a c t e r i s t i c s of the device and i s able to  p r e d i c t i t s dynamic behaviour to the s a t i s f a c t i o n of the user.  As a  r e s u l t of these tests modifications to the i n i t i a l model usually follow, some of which may simplify the model, others may complicate. i s true i f e a r l i e r neglected  The l a t t e r  d i s c o n t i n u i t y e f f e c t s of dead-time influence  the behaviour s u f f i c i e n t l y to require t h e i r i n c l u s i o n i n the model. Modelling of p h y s i c a l hardware i s therefore not a single process; i t i s iterative.  9  The scope of this thesis excludes f i e l d tests and hence the modelling w i l l be l i m i t e d to the i n i t i a l model l e v e l . Physical processes involving the transmission of energy mass are d i s t r i b u t e d parameter i n nature.  through  When beginning model develop-  ment f o r a device, i t i s thus expedient to f i r s t view the device as a d i s t r i b u t e d parameter system.  However, provided the e f f e c t s of energy  and mass transfer delays throughout the system's space have n e g l i g i b l e e f f e c t on the end use of the model, the modelling equations may  then be  approximated by the simpler lumped parameter equations and represented by ordinary d i f f e r e n t i a l equations. Model accuracy of components or subsystems having l e s s e r i n fluence need not be as exact as those e x h i b i t i n g the more dominant a f f e c t s . Thus the dynamics of the hydraulic preamplifier are less important than those of the power servomotor  appearing l a t e r i n the cascade.  Similarly,  the surge tank need not be modelled too p r e c i s e l y for i t s influence i s less dominant.  (Surge tank effects are important, however, f o r evaluation  of pseudo-steady-state conditions f o r the optimal trajectory.)  It is  thus evident that i n model development each device must be considered separately, f i r s t whether i t s e f f e c t s warrant i t being included i n the model, and secondly, the degree of exactness required of i t s model. The above dealt p r i n c i p a l l y with dynamic models for optimal trajectory determination and, as mentioned, dynamic accuracy may  not  be e s s e n t i a l f o r certain devices and hence omitted to simplify the dynamic model. conditions may  However, a precise s t a t i c model describing the boundary s t i l l be required.  For example, the f i n a l steady-state  pressure head i n the s c r o l l case may be required to e s t a b l i s h the boundary value for that v a r i a b l e .  (It i s assumed that r e l i e v i n g the dynamic  10  model of the r e s p o n s i b i l i t y f o r p r e c i s e end  conditions  algorithm,  s e l e c t p r o c e d u r e were used  as i t would i f a space f l o o d and  to f i n d the g l o b a l o p t i m a l  trajectory.)  A m o d e l l i n g p h i l o s o p h y must be guide before proceding with  s i m p l i f i e s the  established  to s e r v e as  the d e t a i l s o f development.  The  model development of h y d r a u l i c components o f a h y d r o e l e c t r i c u n i t w i l l p r o c e e d i n the l i g h t o f the above  discussion.  a  following generating  11  3.  THE PENSTOCK  Penstocks is  are f u l l  f l o w i n g p r e s s u r e c o n d u i t s whose f u n c t i o n  t o convey water from a r e s e r v o i r o r an i n t e r m e d i a t e surge tank t o a  t u r b i n e a t i t s downstream end. In h y d r o e l e c t r i c p l a n t l a y o u t d e s i g n , p e n s t o c k s a r e k e p t as s h o r t as p r a c t i c a l i n o r d e r n o t o n l y t o reduce  the c o s t o f m a t e r i a l ,  i n s t a l l a t i o n and e x c a v a t i o n f o r s u b s u r f a c e i n s t a l l a t i o n s b u t a l s o t o enhance the g o v e r n i n g a b i l i t y o f the t u r b i n e g o v e r n o r s . is  A l o n g penstock  d e t r i m e n t a l t o u n i t r e s p o n s e , p a r t i c u l a r l y i f the nominal p r e s s u r e  head i s low. Long penstocks e x h i b i t governing.  The d i f f i c u l t y a r i s e s because  s u r e waves t o t r a v e l it  c h a r a c t e r i s t i c s d e t r i m e n t a l to f a s t o f the time r e q u i r e d f o r p r e s -  from ,the t u r b i n e gates t o the r e s e r v o i r , i n f o r m i n g  t h a t a change i n flow i s r e q u i r e d a t the t u r b i n e end.  2000 f e e t l o n g , the round t r i p  For a penstock  d e l a y i s a p p r o x i m a t e l y one second.  Thus  when a torque i n c r e a s e i s r e q u i r e d the w i c k e t gates i n c r e a s e t h e i r t u r e s to i n c r e a s e the flow o f water. i n c r e a s e by a c e r t a i n percentage  Although  immediately,  the f l o w o f water does the p r e s s u r e head  reduces  by a g r e a t e r p e r c e n t a g e so t h a t the n e t torque i s i n d e e d reduced of i n c r e a s e d .  The reduced torque e x i s t s u n t i l  instead  the wave o f r e d u c e d p r e s -  s u r e t r a v e l s t o the r e s e r v o i r , where t h e p r e s s u r e d i f f e r e n t i a l more w a t e r t o flow i n t o the penstock and t h i s  aper-  causes  flow i n c r e a s e accompanied  by t h e p r e s s u r e i n c r e a s e o f the r e t u r n i n g wave i s f e l t a t the t u r b i n e e n d .  9 I n one extreme case t h i s r e q u i r e s 2.5 seconds Similarly,  .  f o r a r e q u i r e d torque d e c r e a s e , i n r e d u c i n g t h e i r  a p e r t u r e the gates cause  the p r e s s u r e head t o r i s e and thus i n c r e a s e the  12  torque. and  As the p r e s s u r e waves t r a v e l t o and f r o between t h e r e s e r v o i r  the t u r b i n e they e v e n t u a l l y damp out and the torque i s reduced  to i t s  new lower v a l u e . In  this  t h e s i s i t i s proposed  water hammer can be used t o advantage que  that  the d e t r i m e n t a l e f f e c t o f  i n controlling  the h y d r a u l i c  o f a u n i t w h i l e i t i s e x p e r i e n c i n g a sharp t r a n s i e n t .  torque b e f o r e the t r a n s i e n t and a f t e r w i l l be i d e n t i c a l power g e n e r a t i o n s e t p o i n t i s assumed to remain transient period),  S i n c e the  (the s t e a d y - s t a t e  c o n s t a n t d u r i n g the  the w i c k e t gate c o n t r o l l e r would i n i t i a t e  c o n t r o l sequences.  D u r i n g the f i r s t  tor-  p o r t i o n o f the system's  the f o l l o w i n g oscillation  when a r e d u c t i o n i n torque i s r e q u i r e d , the gates would be d r i v e n open to  reduce the t o r q u e .  D u r i n g the second p o r t i o n of' the c y c l e when an i n -  c r e a s e i n torque i s r e q u i r e d , the gates would be r e t u r n e d t o t h e i r  ini-  t i a l p o s i t i o n and i n the course o f t h e i r -motion i n c r e a s e the t o r q u e . T h i s c o n t r o l sequence i s the r e v e r s e o f t h a t g e n e r a t e d by conventional t u r b i n e speed-error governors.  C o n v e n t i o n a l governors a r e  d e s i g n e d t o reduce the gate a p e r t u r e on s e n s i n g a speed i n c r e a s e and t o i n c r e a s e the a p e r t u r e on s e n s i n g a speed d e c r e a s e . signifies  A speed i n c r e a s e  an i n c r e a s i n g torque a n g l e , which must be d e c r e a s e d , n o t  i n c r e a s e d , t o ensure synchronism.  C o n v e n t i o n a l governor a c t i o n i s t h e r e -  f o r e o p p o s i t e t o t h a t r e q u i r e d , and f o r t h i s r e a s o n p r e s e n t gate are  d e s i g n e d t o remain For  controls  inactive during transient periods.  an o p t i m a l c o n t r o l l e r t o be c a p a b l e o f d i s c e r n i n g when t o  i n i t i a t e r e v e r s e a c t i o n , i t must o f n e c e s s i t y p o s s e s s an a c c u r a t e model of  t h e dynamics o f both the a c t u a t o r and the p e n s t o c k .  effect i s critical.  That i s , the d i s t r i b u t e d parameter  penstock becomes a p p r e c i a b l e .  The time d e l a y nature o f the  13  The m o d e l l i n g approach  adopted  c o n s i d e r s the p e n s t o c k  and i t s c o n t a i n e d f l u i d t o be a d i s t r i b u t e d parameter system.  metal After  t h e i r d e r i v a t i o n , the s e t of 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 s a r e  reduced  to a s e t of 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 by the method o f c h a r a c t e r istics.  F i n a l l y , s i n c e the model w i l l i n the end be programmed on a  d i g i t a l computer, the 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 are c o n v e r t e d difference  e q u a t i o n s , n o t f o r g e t t i n g the c h a r a c t e r i s t i c curves  by the e a r l i e r c o n v e r s i o n . t i o n s resembles The p e n s t o c k  generated  Each e q u a t i o n i n the s e t of d i f f e r e n c e  a lumped parameter model of  a  equa-  reach of a penstock.  can be c o n s i d e r e d t o have been b r o k e n up i n t o reaches f o r  each o f w h i c h  the lumped parameter model i s adequate.  Application  of the method o f c h a r a c t e r i s t i c s t o p e n s t o c k  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 s i s w e l l documented i n r e f e r e n c e 10. convenience  the p h y s i c a l model ( F i g u r e 1 ) , the d e r i v e d p a r t i a l  e n t i a l e q u a t i o n s , the g e n e r a t e d 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 their  into  For  differand  a s s o c i a t e d c h a r a c t e r i s t i c curves w i l l be g i v e n . A p p l i c a t i o n o f the e q u a t i o n o f m o t i o n  and the e q u a t i o n of  c o n t i n u i t y t o an e l e m e n t a l f l u i d d i s c as d e f i n e d i n F i g u r e 1 y i e l d s a f t e r m o d i f i c a t i o n the f o l l o w i n g p a i r of n o n - l i n e a r p a r t i a l  differential  equations: 9H  at  9H 3x  +  v  +  V 9V  3x  (3-1)  HH  +  g  —  I  i  (3-2)  + FV | V I = 0  where 'a', the v e l o c i t y o f sound i n the w a t e r , i s g i v e n by a  2  K/p  (3-3)  14  PA  F j g u r e 1.  +  d  F r e e body diagram of e l e m e n t a l f l u i d i n a conduit.  w = u n i t weight of water p = p r e s s u r e head A = penstock area z = e l e v a t i o n above datum  disc  15  and where V = v e l o c i t y o f water i n f t / s e c , H = p r e s s u r e head i n f e e t  of water,  K = b u l k modulus f o r water (43.2  10^ l b / s q . f t ) ,  p = d e n s i t y o f water, 2 g = acceleration  of g r a v i t y ,  32.2  ft/sec  E = e l a s t i c modulus f o r the penstock w a l l m a t e r i a l ( a p p r o x i m a t e l y 4.32 F = f/2D = H g / £ V  10  lb/sq.ft for steel),  2  f  where  i s the l i n e f r i c t i o n head as measured by  controller  instrumentation, f = friction  f a c t o r i n the Darcy-Weisbach  JI = l e n g t h o f the  penstock  D = p i p e diameter, t  and  = pipe w a l l thickness.  1  A c c o r d i n g to the method of c h a r a c t e r i s t i c s , and  (3-2) w i t h an unknown m u l t i p l i e r ,  f o r the unknown m u l t i p l i e r i n the two  generate  unknowns, H and V,  e q u a t i o n s , and  these new  the former e q u a t i o n s Combining  and L = L  and s o l v i n g  formula,  1  any  and  r e a l and  (3-1)  d i s t i n c t values  o f the o r i g i n a l p a r t i a l d i f f e r e n t i a l a l l the c h a r a c t e r i s t i c s  of  (3-2).  as + X L  (3-4)  2  f o r the unknown m u l t i p l i e r  X yields  X = + a/g which are two  two  equations  two 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  equations possess  (3-1)  i n combining  r e a l , d i s t i n c t values giving  (3-5) the f o u r e q u a t i o n s below.  16  For A = + a/g, c a l l e d the p o s i t i v e c h a r a c t e r i s t i c curve, we obtain  +  dt  +  g dt  *  g  F V 1| V |  '  =  0  (3-6)  ~ = V + a  (3-7)  For A = - a/g, the negative c h a r a c t e r i s t i c curve, we have  £  (3-9)  - V - a  dt  The physical significance of these two characteristic: curves along which the solution of the ordinary d i f f e r e n t i a l equations i n V and H must l i e i s that they represent the d i r e c t i o n of wave propagation; downstream propagation along the p o s i t i v e .characteristic and upstream for the negative c h a r a c t e r i s t i c . In p r a c t i c e , V i s usually less than 2 0 ft/sec and the v e l o c i t y of sound greater than  ft/sec and so equations  3000  (3-7)  and  (3-9)  can  be s i m p l i f i e d to  | |  =  +  a  .  (3-10)  (3-11)  respectively. Conversion of the ordinary d i f f e r e n t i a l equations to form i s f a c i l i t a t e d by equations  (3-10)  and  (3-11).  difference  These equations  define the r e l a t i o n between the reach length and the time i n t e r v a l to be Ax  =  a  • At  (3-12)  17  I f our i n t e r c o n n e c t e d e l e c t r i c network c o n t a i n s o t h e r  penstocks  h a v i n g d i f f e r e n t wave v e l o c i t i e s , then the number o f reaches each pens t o c k can have i s p r e d e t e r m i n e d , ded i n t o e q u a l l e n g t h r e a c h e s .  and hence each p e n s t o c k cannot be C o n s i d e r a t i o n must t h e r e f o r e be  to the e n t i r e i n t e r c o n n e c t e d system and  penstock's  given  from i n s p e c t i o n o f w o r s t  j u d i c i o u s s e l e c t i o n of a s u i t a b l e time i n t e r v a l made.  divi-  cases,  T h e r e a f t e r each  l e n g t h i s a r b i t r a r i l y a d j u s t e d to the n e a r e s t m u l t i p l e o f i t s  r e a c h , as d e f i n e d by = a^ • A t ,  follows.  j = j*"'  penstock'  V a r i a b l e s H and V i n e q u a t i o n s  (3-6)  S i n c e the s o l u t i o n to e q u a t i o n  (3-6) must l i e on i t s a s s o c i a t e d  c h a r a c t e r i s t i c , namely e q u a t i o n equation  1  and  (3-8) are s o l v e d as  ( 3 - 1 0 ) , the d i f f e r e n t i a l s dH and dV i n  (3-6) at i n s t a n t s k and k + 1  are:  dH = H. . . . — H. .. . i,k+l x-l,k and i,k+l  l-l,k  S i m i l a r l y the s o l u t i o n f o r e q u a t i o n equation  (3-11).  The  i n s t a n t s k and k+1  (3-8) must l i e on i t s c h a r a c t e r i s t i c  d i f f e r e n t i a l s dH and dV i n e q u a t i o n  (3-8)  at  are:  dH H . =  ..-—H.._. i,k+l l+l,k  and dV = V. . ,. i,k+l Figure 2 i l l u s t r a t e s The  -V.. l+l,k the r e l a t i o n s h i p s as f u n c t i o n s o f i and  above s e t s of d i f f e r e n t i a l s are s u b s t i t u t e d i n t o  respective equations.  The  r e s u l t s are:  their  k.  18  x  o  x  i - l i x  x  i+l  x  n  DISTANCE FROM RESERVOIR Figure 2.  The relationship between i and k for velocity V and pressure H on the x-t plane.  19  H  -x , ki +a.1l - -i i i ,+ k~ g ( - i ,k+l i i n - V.l .- .l) ,+ k— g F A t V.l -.l ,.k |V. -l i-. l ,. k I= 0 H  v  1  1  (3-13)  and i,k+l  l+l,k  g  i,k+l  l+l,k  g  l+l,k  1  I+I,k  1  (3-14)  A d d i n g e q u a t i o n s ( 3 - 1 3 ) and ( 3 - 1 4 ) e l i m i n a t e s V, ,  H.  i,k+l  = ^"{H. 2  + H  l - l , k  +  f  F A t ( v  . + - ( V . . . - V. . . . )  i+I,k  g  -i , k, j.i +l  V  i+i,k! i+i,kl v  " V i , k l i - i , k i v  )  }  ( 3  "  1 5 )  from ( 3 - 1 4 ) e l i m i n a t e s H. , . t o g i v e  = T W . ., . + V. . . + ^ ( H . . . - H. ,. . ) 2 i+I,k l - l , k a i-l,k i+l,k  " where i =  i+I,k  l-l,k  s u b t r a c t i n g equations (3-13)  and  to give  F A t ( v  i+i,kl i+i,;J V  +  v  i-i.,kl i-i,kl> v  < " >  }  3  16  0,1,2,3,...,n-l,n  r e p r e s e n t s t h e end o f r e a c h s t a t i o n s f o r the n r e a c h e s , and  r e p r e s e n t i n s t a n t s o f t i m e , whose f i n a l element must be e s t a b l i s h e d l a t e r when t h e o p t i m i z a t i o n problem Equations (3-15)  i s being formulated.  and ( 3 - 1 6 )  d e s c r i b e i n d i s c r e t i z e d form t h e  dynamic b e h a v i o u r o f t h e w a t e r i n t h e p e n s t o c k . they may be d i r e c t l y programmed on a d i g i t a l  Being  computer.  discretized, S i n c e each new  V and H depends o n l y on i t s a d j a c e n t n e i g h b o u r , e q u a t i o n s ( 3 - 1 5 ) and ( 3 - 1 6 ) c o v e r t h e e n t i r e l e n g t h o f t h e p e n s t o c k e x c e p t t h e two ends, w h i c h terminal constraints f o r solution.  Equations  a r e the m a t h e m a t i c a l model o f t h e p e n s t o c k .  require  ( 3 - 1 5 ) and ( 3 - 1 6 ) t h e r e f o r e They account f o r t h e  20  e l a s t i c i t y o f w a t e r , the e l a s t i c i t y o f the c o n f i n i n g p e n s t o c k w a l l m a t e r i a l and f r i c t i o n o f the f l o w i n g w a t e r .  21  4.  PENSTOCK TERMINAL CONSTRAINTS Upstream t e r m i n a l  configurations  with  for  a l a r g e r e s e r v o i r o r a s i m p l e surge tank.  downstream t e r m i n a l be  c o n s t r a i n t s w i l l n e x t be d e r i v e d  condition  r e p r e s e n t e d by a h y d r a u l i c  plant  The  turbine  will  considered i n a l a t e r section.  4.1  The R e s e r v o i r The  r e s e r v o i r i s defined  as a body o f w a t e r o f s u f f i c i e n t  s u r f a c e a r e a t o ensure t h a t i t s w a t e r s u r f a c e e l e v a t i o n may be c o n s i d e r e d constant i n s p i t e of v a r i a t i o n s i n water flow i n t o the penstock. The  r e s e r v o i r o f f e r s one c o n s t r a i n t H  , . = H  o,k+l  o,k 1  namely,  = H°  (4-1)  where H ° i s the e l e v a t i o n o f the r e s e r v o i r w a t e r s u r f a c e . city  the v e l o c i t y head i s n o t i n c l u d e d  For s i m p l i -  i n e q u a t i o n ( 4 - 1 ) ; i t can be  c o n s i d e r e d t o be an e q u i v a l e n t f r i c t i o n l o s s i n c l u d e d w i t h friction loss.  the l i n e  Although t h i s w i l l r e s u l t i n a discrepancy i n the pres-  s u r e p r o f i l e n e a r the r e s e r v o i r end o f t h e p e n s t o c k , t h e e f f e c t not be a p p r e c i a b l e a t t h e t u r b i n e friction  end.  will  I n the model, t h e p e n s t o c k  f a c t o r m e r e l y appears l a r g e r . Derivation  of the upstream c o n d i t i o n s  begins with  rewriting  e q u a t i o n (3-14) as H. ., = - V . ... + C i,k+l g i,k+l o  (4-2)  where C = H - . - - V . . + - FAt V... , IV,,. . I o i+l,k g l+l,k g l+l,k i+ljk 1  1  (4-3)  E x p r e s s i o n (4-3) c o n t a i n s the d e l a y e d e f f e c t s which t r a v e l  22  .upstream through the f i r s t r e a c h i n time A t . , - i n (4-2)  Constraining H  t o remain c o n s t a n t at the r e s e r -  v o i r e l e v a t i o n H° and s o l v i n g f o r V. , . y i e l d s 1 j K."t~X  V  o,k+1  = f (H° - C ) a o  (4-3)  and H  o,k l " °  < "  H  4  +  Equations (4-3)  and ( 4 - 4 )  4)  p r o v i d e the m i s s i n g v a l u e s a t s t a t i o n z e r o  l o c a t e d a t the r e s e r v o i r end of the p e n s t o c k . 4.2  The Surge Tank I n s i t u a t i o n s where a r e s e r v o i r i s s e v e r a l m i l e s from the  powerhouse, i t i s n e c e s s a r y to b r e a k up the l o n g l e n g t h o f c o n d u i t i n t o two segments by p r o v i d i n g a f r e e w a t e r s u r f a c e and a'volume of  adequate  c a p a c i t y to e f f e c t i v e l y decouple the surge e f f e c t s between these two segments.  T h i s v e s s e l i s c a l l e d a surge tank. The main types o f surge tanks a r e the s i m p l e s u r g e t a n k , the  r e s t r i c t e d o r i f i c e surge t a n k , and the d i f f e r e n t i a l s u r g e t a n k .  The  downstream segment w h i c h connects the s u r g e tank to the p l a n t i s made as s h o r t as p r a c t i c a l .  I t s minimum l e n g t h i s l i m i t e d by the need t o  p r o v i d e a f r e e s u r f a c e a t the h y d r a u l i c grade l i n e l e v e l i n the surge tank. A f u n c t i o n o f a surge tank i s to decouple the two  conduits,  e f f e c t i v e l y p r e s e n t i n g the t u r b i n e w i t h a s h o r t p e n s t o c k and so e n h a n c i n g t h e governor's a b i l i t y t o r e g u l a t e Surge tank dynamics  speed.  need not be i n c l u d e d i n the dynamic model.  However the h i g h speed model must then be s u p p o r t e d by a s l o w e r model  23  w h i c h w i l l p r e d i c t the t e r m i n a l c o n s t r a i n t s on t h e s o l u t i o n The  trajectory.  s l o w e r model p r e d i c t s the new s t e a d y - s t a t e head a t the surge  w h i c h can be c o n s i d e r e d by the p e n s t o c k  tank,  as a r e s e r v o i r o f l i m i t e d  whose s u r f a c e e l e v a t i o n v a r i e s w i t h f l o w .  size  I f the d e c o u p l i n g i s adequate  the segment upstream o f the tank can be c o n s i d e r e d as a lumped p a r a m e t e r system and m o d e l l e d by 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 .  The e q u a t i o n s  are: t . = H + -r±- • (V. - V , ) A t o,k+l o,k A k o,k s A  H  Vk+i • i  V l - \  (4-5)  < „,k i - o> E  c  < 4  -  6 )  ( 4  -  7 )  +  +  ^  <  H  ° -  H  c , k -  F  \ l  R e f e r t o F i g u r e 3 f o r symbol d e f i n i t i o n s .  V  k l ' -  r e p r e s e n t s the v e l o c i t y  a t i n s t a n t k o f the mass o f w a t e r i n t h e t u n n e l w h i c h i s c o n s i d e r e d t o be a lumped parameter system. The  accuracy o f (4-5) and (4-7) can be improved by u s i n g  average f l o w s i n p l a c e of the v a l u e s a t t h e end o f the p r e v i o u s interval.  time  I n p r a c t i c e , however, t h e w a t e r s u r f a c e l e v e l does n o t a l t e r  s i g n i f i c a n t l y and t h e rougher  a p p r o x i m a t i o n may p r o v e t o be adequate.  For c e r t a i n i n s t a l l a t i o n s the two c o n d u i t s may e x h i b i t ant c h a r a c t e r i s t i c s .  reson-  F o r p e r i o d i c m o t i o n o f t h e w i c k e t gates a t s u i t -  a b l e f r e q u e n c i e s , a resonance node may appear n e a r the surge r e n d e r i n g i t i n e f f e c t i v e i n d e c o u p l i n g the two c o n d u i t s .  tank  Studies should  be made w i t h the t u n n e l m o d e l l e d by d i s t r i b u t e d parameter e q u a t i o n s t o v e r i f y t h a t t h e lumped parameter model o f e q u a t i o n s i s adequate. The  The t u n n e l would be m o d e l l e d  (4-5), (4-6) and (4-7)  i d e n t i c a l l y to the penstock.  r e s e r v o i r end o f t h e t u n n e l i n v o l v e s e q u a t i o n s  (4-3) and (4-4).  DATUM  Figure  3.  Schematic l a y o u t of a h y d r a u l i c s y s t e m showing r e s e r v o i r , t u n n e l , s u r g e t a n k , p e n s t o c k and t u r b i n e .  25  However, the surge tank junction now requires constraints between V ,. ,,, n,k+l n,k+l  o,k+l  o,k+l  H  n,k+1  As 4dt f"  =  H  A  0  H  H  o,k+l  * t in,k v  A  = g  o,k+l  w. n,k+l  J  p i o,k  (4-10) v  V . ,. + C o,k+l o  " fg  =  v  , a.-! + n,k+l  v  c  (4-11) ^ ' (4-12)  n  where Cn » H n-l,k . . + - gV n-l,k . - - g F A t Vn-l,k . V n-l,k 1  1  1  1  and i s obtained s i m i l a r l y to C . o Applying these four constraints and solving f o r the unknowns y i e l d s  H  i .n o,k+l J  H  V  =  H  i+T^(A V . - A V . J! o,k t n,k p o,k' A  v , i = Hn,k , + A n,k+l  v :  s  (A t Vn,k. - Ap  o , k l = \ < o,k l " H  +  V  +  C  Vo,k' ,)  o>  n , k 1 = " \ < , k l - n> H  +  v  (4-13) '  v  (4-14) '  s  C  n  +  (  4  "  1  5  )  <*" > 16  Equations (4-13) to (4-16) are the difference equations f o r the conditions of the tunnel and the penstock at the base of the surge tank. Other types of surge tanks can be modelled by f i r s t  obtaining  the required constraint equations and then following the above procedure to solve f o r the unknowns i n difference equation form.  26  The r e s t r i c t e d o r i f i c e surge tank requires an a d d i t i o n a l constraint since i t s surface elevation w i l l not be i d e n t i c a l to the pressure head at the junction. pressure  The r e s t r i c t i n g o r i f i c e w i l l e x h i b i t a  drop as water i s forced to flow through i t " ^ .  Further, the  o r i f i c e i s usually not r e c i p r o c a l i n that i t offers more resistance to inflow than to outflow. A d i f f e r e n t i a l surge tank can be viewed as a combination of a simple surge tank and a r e s t r i c t e d o r i f i c e surge tank and modelled accordingly. Surge tanks are often equipped with an over-flow spillway. However, i t i s used only on t o t a l load r e j e c t i o n , and the n o n - l i n e a r i t y does therefore not a f f e c t our modelling. In the foregoing, modelling  equations f o r several penstock  upstream terminal conditions were derived and items deserving s p e c i a l considerations were noted. The downstream terminal constraint i s the turbine.  Before  considering the turbine, i t s flow control actuation mechanism must f i r s t be analyzed.  This i s necessary because the turbine model i s a function  of the gate p o s i t i o n .  27  5.  THE ACTUATOR  W i c k e t gate s e r v o m o t o r s c o n t r o l t h e w i c k e t gates t o r e g u l a t e the f l o w o f w a t e r t h r o u g h the t u r b i n e . s i g n a l f e d i n t o an e l e c t r o m e c h a n i c a l  They r e s p o n d t o a weak e l e c t r i c transducer.  Between t h e t r a n s d u c e r  and  servomotors one f i n d s a number o f h y d r a u l i c d e v i c e s w h i c h  and  condition  amplify  t h e s i g n a l as i t p a s s e s from t h e t r a n s d u c e r t o t h e s e r v o -  12 motors  .  T h i s c o l l e c t i o n o f h y d r a u l i c d e v i c e s w i l l f o r c o n v e n i e n c e be  c a l l e d the actuator. Of a l l t h e h y d r a u l i c s t a g e s i n an a c t u a t o r  system, our a t t e n -  t i o n w i l l be d i r e c t e d toward t h e f i n a l h y d r a u l i c power s t a g e . inant  c h a r a c t e r i s t i c s o f the h y d r a u l i c  system a r e i n h e r e n t  The dom-  ini t .  The  f i n a l s t a g e can be e i t h e r an i n t e g r a t o r o r a power a m p l i f i e r .  For  smaller  units  units i t i s often a hydraulic integrator.  i n which l a r g e r a c t u a t i o n  forces  are required  For l a r g e r  t o overcome s t i c t i o n ,  f r i c t i o n and h y d r a u l i c r e a c t i o n t o r q u e s , t h e f i n a l s t a g e i s u s u a l l y a power a m p l i f i e r .  This a m p l i f i e r stage i s preceded  by an i n t e g r a t o r  of l e s s e r c a p a c i t y whose f u n c t i o n i s t o e n s u r e t h e gates m a i n t a i n a f i x e d p o s i t i o n i n the s t e a d y - s t a t e .  The e s s e n t i a l d i f f e r e n c e between  these two types o f f i n a l s t a g e a c t u a t o r s  i s the power a m p l i f i e r w h i c h i s an  i n t e g r a t o r w i t h p o s i t i o n feedback t o i t s a c t u a t i n g g r a t o r w i l l be c o n s i d e r e d  signal.  The i n t e -  first.  The common model f o r an i n t e g r a t o r i s £  - A ,  (5-1)  where y = p o s i t i o n of i n t e g r a t o r  piston,  28  x = p o s i t i o n of p i l o t v a l v e A = a The  and  constant.  above model i s v a l i d o n l y i f the d r i v i n g f o r c e o f  p i s t o n i s zero o r n e g l i g b l e i n r e l a t o n t o the p r e s s u r e ' f l u i d source. t h i s model to  For a l l but the f i n a l s t a g e , be  the  of the h y d r a u l i c  f i e l d measurements may  show  adequate.  For the f i n a l s t a g e to be r e p r e s e n t a b l e by  the above s i m p l i -  f i e d model, the a c t u a t o r would have t o be g r o s s l y o v e r d e s i g n e d , w h i c h economics p r o h i b i t . estimate  Indeed, i n d e s i g n i n g an a c t u a t o r , m a n u f a c t u r e r s  the magnitudes o f the v a r i o u s types o f r e s i s t i v e f o r c e s a s s o -  c i a t e d w i t h a gate mechanism.  They then d e s i g n  the a c t u a t o r w i t h j u s t  enough m o t i v e c a p a c i t y to overcome t h e s e f o r c e s and  a c c e l e r a t e the mech-  anism mass a t a speed t h a t w i l l meet the customer's response cations.  The  pressure  d i f f e r e n c e across  specifi-  the p i s t o n s w i l l t h e r e f o r e  not  be n e g l i g i b l e and i t w i l l be n e c e s s a r y to d e r i v e a model w h i c h a c c o u n t s f o r non-zero p i s t o n f o r c e s . Before proceeding  to d e r i v e the model f o r a h y d r a u l i c power  i n t e g r a t o r , c o n s i d e r the r e s i s t i v e f o r c e s to be overcome by The  dominant f o r c e s  the  actuator.  are:  - mechanism b e a r i n g  stiction,  - mechanism b e a r i n g  friction,  - unbalanced h y d r a u l i c t o r q u e on the g u i d e v a n e s ,  and  - i n e r t i a of the mechanism mass, w h i c h l i e s m o s t l y i n the r o t a t i o n a l i n e r t i a of the w i c k e t  gates about t h e i r  bearing  axes. For any p a r t i c u l a r u n i t , the above e f f e c t s must be measured to d e t e r m i n e whether some are e i t h e r n e g l i g i b l e or dependent on  other  29  factors.  For example, bearing s t i c t i o n and f r i c t i o n may be e f f e c t e d by  unsatisfactory operation of the l u b r i c a t i o n system. Figure A i l l u s t r a t e s a t y p i c a l actuator configuration showing the accumulator tank, the p i l o t valve and the integrator power p i s t o n . The accumulator tank i s p a r t l y f i l l e d with compressed a i r to provide pressure energy for the f l u i d . hydraulic f l u i d .  The lower portion contains  Since a i r dissolves i n the o i l which flows i n t o the  actuator, the accumulator tank must be replenished with both o i l (from the sump) and a i r .  The accumulator tank i s a closed v e s s e l and so i t s  pressure w i l l decrease when a large volume of o i l i s withdrawn during a heavy actuation control sequence.  During normal operation the source  pressure w i l l vary, decreasing by approximately pumps s t a r t on low o i l l e v e l .  20% before the governor  I t may sometimes be necessary  f o r the  c o n t r o l l e r model to take this v a r i a t i o n i n t o account i n order to p r e d i c t actuator response.  R e c a l l that our models are intended to be v a l i d f o r  a unit experiencing large s i g n a l actuation sequences. P i l o t valve spools can be of varying types, such as overlapped, zero-lapped and underlapped. characteristics.  Wear can also a l t e r the e f f e c t i v e spool  For the sake of generality t h i s derivation incorporates  a lap factor where X = the magnitude of the lapping, k = -1 for overlap, = 0 for zero-lap, and = +1 for underlap. One may then obtain the applicable model by s u b s t i t u t i n g the appropriate values defined above.  Figure  4.  S c h e m a t i c - d i a g r a m o f an i n t e g r a t i n g w i c k e t g a t e a c t u a t i o n s y s t e m c o m p l e t e w i t h p r e s s u r e s o u r c e , p i l o t v a l v e , s e r v o m o t o r s and g a t e l i n k a g e mechanism.  31  The equations r e l a t i n g flow through spool ports and the port pressure drops are;  q  x  = cjx+kx]  /j.P  q  2  = C |-x + kX| / | P - p | sgn (P  q  3  = C |-x + kX|  q  4  = C j x + k | / |p - P |  s  -  P ; L  | sgn ( P -  g  2  / |  3  P  l  2  - P | sgn (  2  where f o r (5-2) and (5-5) and f o r (5-3) and (5-4)  u(x + kX)  g  e  e  (5-2)  g  - p ) u(-x + kX)  (5-3)  P l  - P ) u(-x + kX)  (5-4  2  e  sgn ( p - P ) u(x + kX) 2  e  (5-5)  x $ D - kX -x >, D - kX .  The function u( • ) i s the unit step function. In equations  (5-2) to (5-5) above, flow was assumed to vary  as the -square -root of -the pressure drop. I f as a r e s u l t of f i e l d  This i s an approximation.  tests the r e l a t i o n i s found to be markedly d i f -  ferent, and i f this difference s i g n i f i c a n t l y affects our end r e s u l t , namely, the torque control, then the r e l a t i o n should be modelled by a t h i r d or f i f t h order polynomial.  By using odd powers only, the equations  simplify since absolute value functions and the sign functions do not appear.  The step functions and the t r a v e l l i m i t s , however, must be r e -  tained. P i l o t valve pistons often have t r a v e l l i m i t s , r e s t r i c t i n g t h e i r range to less than the spool port t r a v e l .  Travel l i m i t s are usually  not considered when modelling f o r small s i g n a l applications.  However,  i n large s i g n a l applications the p i l o t spools do t r a v e l t h e i r f u l l range and so t r a v e l l i m i t s must be included i n the model. A set of constraints r e l a t i n g the p i l o t valve flows and pressures  32  on both, s i d e s o f the p i s t o n s a r e c o n s i d e r e d n e x t . c o m p r e s s i b i l i t y o f the f l u i d  They i n c l u d e t h e  i n t h e p i s t o n chambers.  O t h e r w i s e , when  b o t h p o r t s a r e c l o s e d s i m u l t a n e o u s l y as they w i l l be f o r the o v e r - l a p p e d s p o o l and may be f o r t h e z e r o - l a p p e d s p o o l , the p r e s s u r e s become i n d e t e r m i n a t e and t h e model f a i l s .  T h i s can be v i s u a l i z e d i f one c o n s i d e r s  the gate to be i n m o t i o n when the o v e r l a p p e d s p o o l v a l v e reaches i t s c e n t e r p o s i t i o n where the p o r t s a r e a l l c l o s e d . n o t accounted  I f compressibility i s  f o r , then an i m p u l s e f o r c e must a c t on the p i s t o n s t o  a r r e s t the momentum o f the gate mechanism i n z e r o t i m e .  By i n c l u d i n g  c o m p r e s s i b i l i t y , the p r e s s u r e v a l u e s , a l t h o u g h b e i n g v e r y l a r g e , remain f i n i t e  will  and manageable i n the a l g o r i t h m .  D i f f e r e n t i a t i n g the b u l k modulus d e f i n i t i o n dP - K  ^ V  w i t h r e s p e c t t o time g i v e s dP dt  K dV " dt  <"> 5  6  where K = b u l k modulus o f the o i l , w h i c h i s a p p r o x i m a t e l y psi,  250,000  depending on t h e type o f o i l and t h e amount o f a i r  dissolved i n i t . V = combined volume o f t h e commoned chambers o f b o t h  cylinders  and a l l p i p i n g up t o t h e s p o o l v a l v e s , and P = the p r e s s u r e i n t h e chambers. From F i g u r e 4 the n e t i n f l o w i n t o c y l i n d e r chamber 1 i s dV I F  ^  1  =  q  and i n t o chamber 2  l ' 3 q  +  A  dt  "  K  l  ( P  1  _  P  2  }  •  ( 5  "  7 )  33  dV-  ,  - d T  =  q  2 -  4 -  q  a f  A  +  K  l  (  p  l -  p  2 )  <"> 5  8  where = p i s t o n l e a k a g e c o e f f i c i e n t w h i c h i t may be n e c e s s a r y t o i n c l u d e i n the model i f s t e a d y - s t a t e d r i f t i s to be p r e dicted, s = d i s p l a c e m e n t o f the p i s t o n , ds — = v e l o c i t y o f the p i s t o n ,  and  A = combined f a c e a r e a of b o t h p i s t o n s exposed t o each chamber, S u b s t i t u t i n g e q u a t i o n s (5-7) and (5-8) i n t o (5-6) g i v e s  d p  v  1  ~dt = V-TTS  i " 3 + A f " l 1 " 2> q  V  K  ( P  p  }  <5  + A(S-s)  {"q2 H  0  4 ~ A dt " q,. %+ H  K1 ^( 1p  )T } V  P  o  (5-10)  where V' = the sum o f the volume i n a c y l i n d e r n o t s t r o k e d by  the  p i s t o n s and the volume o f the e n t i r e p i p i n g between the c y l i n d e r and the s p o o l v a l v e p o r t s , and S = piston travel  range.  The f i n a l elements o f the a c t u a t o r t o be m o d e l l e d a r e the gates and the l i n k a g e s i n the gate mechanism.  For convenience, t h e i r  m o t i o n w i l l be t r a n s l a t e d t o t h e i r l i n e a r e q u i v a l e n t r e f e r e n c e d to t h e servomotor p i s t o n s h a f t .  9)  He;  K  ?  dp  dt  { q  The r o t a t i o n a l i n e r t i a o f a l l r o t a t i n g l i n k -  ages and gate vanes w i l l thus be e x p r e s s e d as an e q u i v a l e n t l i n e a r i n e r t i a f i x e d t o the servomotor s h a f t .  S i m i l a r l y , a l l forces including  34  the f r i c t i o n and hydraulic torque acting on the guide vanes w i l l also be translated to t h e i r equivalent forces acting on Applying Newton's Law  the servomotor shaft,  of Motion to the equivalent lumped  mass gives (5-11)  "5T '  M_1  ( (  P i " *2> " s . v V A  (£  +  £  +  (5  -12>  where f  eF s ";| | + e  s g n  ( 8  Bl  f  = v  F  v  S  1  )  ( 5  l  ~  1 3 )  <" > 5  14  M = equivalent mass on the shaft, s^ = v e l o c i t y of the p i s t o n , and f, h  represents the e f f e c t of the hydraulic torque acting on the guide vanes as the water flows through them.  Equation  (5-13) represents s t i c t i o n e f f e c t s present i n mech-  anism bearings when motion i s near s t a n d s t i l l . value of the s t a r t i n g f r i c t i o n .  .F  represents the peak  Parameter 'e' defines the gradient with  which the s t i c t i o n force decreases as v e l o c i t y increases.  The  combina-  t i o n of the absolute value and sign functions ensures that the s t i c t i o n expression w i l l be an odd function. Equation velocities.  (5-14) accounts  f o r viscous f r i c t i o n present at greater  For i n i t i a l model purposes, the viscous f r i c t i o n term i s  assumed to be l i n e a r i n v e l o c i t y .  I f f i e l d tests prove t h i s assumption  to be inadequate, higher power terms must then be appended to obtain satisfactory correlation.  F r i c t i o n must be presented by an odd function.  35  The approximate  shape of the f u n c t i o n r e l a t i n g the h y d r a u l i c  t o r q u e a c t i n g on the guide vanes t o servomotor  stroke i s i l l u s t r a t e d  25 i n Figure 5  .  A f o u r t h o r d e r p o l y n o m i a l f i t s h o u l d prove  adequate.  V a l u e s of torque must be o b t a i n e d from f i e l d measurements.  F i g u r e 5 i n d i c a t e s t h a t the t o r q u e a t z e r o gate a c t s i n the opening d i r e c t i o n and a t f u l l g a t e , a c t s t o c l o s e the vanes. ensures  This design  t h a t upon l o s s o f h y d r a u l i c f l u i d p r e s s u r e the g a t e s w i l l a u t o -  m a t i c a l l y d r i v e t o about 20% gate and so l i m i t t u r b i n e power o u t p u t . The a c t u a t o r model c o n s i s t s o f e q u a t i o n s and  (5-12) .  (5-9) and  Equations  (5-9) , ( 5 - 1 0 ) , (5-11)  ( 5 - 2 ) t o (5-5) must be s u b s t i t u t e d i n t o  (5-10) and e q u a t i o n s  (5-13) and  equations  (5-14) i n t o e q u a t i o n ( 5 - 1 2 ) .  To i l l u s t r a t e the r e d u c t i o n o f the above model t o the  linear  i n t e g r a t o r i n e q u a t i o n ( 5 - 1 ) , assume a l l r e s i s t a n c e f o r c e s and the of  the gate mechanism to be z e r o .  s p o o l to have p e r f e c t z e r o l a p .  inertia  I n a d d i t i o n , assume the p i l o t v a l v e Neglecting f l u i d compressibility  con-  s t r a i n t s , we have l "  q  q  and  2  p  1  =  q  q  =  4  3 p . 2  S o l v i n g f o r d s / d t , the servomotor ds dt  _v A *\  w h i c h i s e q u i v a l e n t to ( 5 - 1 ) .  P  piston velocity, yields  — r ~ i S  - P „ 6  x  (5-15)  F i g u r e 5. V a r i a t i o n of h y d r a u l i c t o r q u e a c t i n g on each w i c k e t gate w i t h servomotor s t r o k e .  37  The h y d r a u l i c i n t e g r a t o r i s c o n v e r t e d  i n t o a power a m p l i f i e r  by f e e d i n g back t h e servomotor s h a f t p o s i t i o n to t h e p i l o t v a l v e summing mechanism.  I n response to an a c t u a t i o n s i g n a l y, s p o o l d i s p l a c e m e n t  becomes  v £  where  and  a r e  t n e  x  - V 1  +  2  l  mechanism arm l e n g t h s .  The range o f x ( t ) i s l i m i t e d by t r a v e l l i m i t s and t h e r e f o r e i t may be n e c e s s a r y  t o impose s t a t e bounds on these v a r i a b l e s .  I n summary, a power a c t u a t o r i s a n o n - l i n e a r - d e v i c e w i t h continuous  functions.  F o r an e c o n o m i c a l l y  designed  dis-  a c t u a t o r the gate  mechanism d r i v i n g f o r c e i s a p p r e c i a b l e and so p i s t o n r e a c t i o n f o r c e s cannot be n e g l e c t e d .  F o r h y d r a u l i c p r e a m p l i f i e r s , however, p i s t o n  r e a c t i o n f o r c e s can be n e g l e c t e d , and the l i n e a r i n t e g r a t o r model may be adequate. Models o f power a c t u a t o r s w i t h o v e r l a p p e d  spools i n t h e i r  p i l o t v a l v e s must i n c l u d e the e l a s t i c e f f e c t s o f t h e f l u i d when t h e i n e r t i a o f t h e w i c k e t gate mechanism i s a p p r e c i a b l e .  O t h e r w i s e , i f the s p o o l  c l o s e s w h i l e the gate mechanism i s i n m o t i o n , the impulse instantaneously  required to  d e c e l e r a t e the mechanism t o s t a n d s t i l l causes the model  t o f a i l because the p r e s s u r e s  i n the c y l i n d e r s become  indeterminate.  38  6.  THE FRANCIS TURBINE  6.1 Introduction An hydraulic turbine i s an assembly of components consisting of a s c r o l l case, stay vanes, guide vanes or wicket gates, a runner and a draft tube.  When studying a turbine's performance,  these components  must be considered c o l l e c t i v e l y . There are two classes of turbines, impulse and reaction turbines.  Examples of these are the Pelton Wheel and the Francis turbine  respectively.  These classes d i f f e r i n the manner they use to extract  momentum from the pressurized water entering the confines of the turbines. Our attention w i l l be l i m i t e d to the Francis turbine.  Equations  sought are those describing the penstock end constraints and those r e l a t i n g pressure head, flow and gate stroke to the torque acting on the turbine. The Francis turbine, being of the reaction type, i s a f u l l flowing turbine i n which the s c r o l l case pressure head i s not t o t a l l y converted into v e l o c i t y head, as i s the case with impulse turbines. Only a portion i s converted as water flows through the wicket gates. The runner and the gates therefore are not decoupled and so they behave l i k e a compound o r i f i c e . A b r i e f physical description of water flowing through a turbine follows.  Water enters the turbine assembly at the entrance to the  s c r o l l case, a c i r c u l a r duct of decreasing cross-sectional area which guides the incoming water i n a c i r c u l a r path around the runner axis, converting the incoming water's l i n e a r momentum to angular momentum. As water flows inward toward the runner, i t encounters the fixed stay vanes  39  and a c t u a t o r c o n t r o l l e d guide vanes w h i c h f u r t h e r add a n g u l a r momentum to t h e w a t e r . minimizes  The guide vanes a l s o d i r e c t t h e f l o w i n a d i r e c t i o n t h a t  e n t r y shock t o t h e r u n n e r .  Between t h e s t a y vanes and guide  vanes and between t h e guide vanes and the r u n n e r , w a t e r f l o w s as a f r e e v o r t e x , i n c r e a s i n g i t s t a n g e n t i a l v e l o c i t y as i t f l o w s i n w a r d .  Finally,  h a v i n g g i v e n up i t s a n g u l a r momentum, i t l e a v e s t h e c o n f i n e s o f t h e .runner and d i s c h a r g e s i n t o t h e t a i l r a c e through  the d r a f t  tube.  Inf. s e e k i n g c o n s t r a i n t e q u a t i o n s , i n i t i a l e f f o r t s were d i r e c t e d toward  an a n a l y t i c d e r i v a t i o n from f i r s t p r i n c i p l e s .  pound o r i f i c e was found d i f f i c u l t  t o model.  However, t h e com-  The main " d i f f i c u l t y was  due t o f r e e v o r t e x m o t i o n t a k i n g p l a c e n o t o n l y i n t h e p l a n e o f r u n n e r r o t a t i o n b u t a l s o i n t h e v e r t i c a l p l a n e w h i c h sweeps around t h e r u n n e r periphery.  A p p l i c a t i o n o f Flow P o t e n t i a l Theory r e q u i r e s c o n f o r m a l map-  p i n g i n three dimensions.  Furthermore,  t o compute t o r q u e and o r i f i c e  back p r e s s u r e r e q u i r e s t h e t u r b i n e b l a d e s and gate b l a d e s t o be c o n f o r mallymapped into  s i m p l e r g e o m e t r i c shapes.  a l s o be t r a n s f o r m e d . may n o t j u s t i f y .  The v o r t e x m o t i o n must  The e x p r e s s i o n would be complex w h i c h t h e end use  Moreover, P o t e n t i a l Flow Theory does n o t account  for  shock and f r i c t i o n l o s s e s , f o r i n t h e t h e o r y f l u i d s a r e i n v i s c i d . A r e l a t i o n s h i p between p r e s s u r e , v e l o c i t y , t u r b i n e speed, and s t r o k e can, n o n e t h e l e s s , be o b t a i n e d by f i t t i n g a h y p e r s u r f a c e t o s e t s o f v a l u e s o b t a i n e d from f i e l d measurements.  A satisfactorily  f i t t e d s u r f a c e w i l l s i m u l a t e t h e t u r b i n e ' s . c h a r a c t e r i s t i c s as they a c t u a l l y are. Runner speed i s among the v a r i a b l e s t h a t determine characteristics.  a turbine's  S e t s o f d a t a r e l a t i n g p r e s s u r e , f l o w , s t r o k e and  speed c o u l d n o t be found and so t h e c o n s t r a i n i n g e q u a t i o n s  considered  40  assume c o n s t a n t speed.  In computer s i m u l a t i o n s p r e s e n t e d l a t e r , the  g r e a t e s t speed d e v i a t i o n was l e s s 6.2  Turbine-Penstock  than f i v e  percent.  End C o n s t r a i n t s  The v a r i a b l e s i n v o l v e d i n these end c o n s t r a i n t e q u a t i o n s a r e head, v e l o c i t y and s t r o k e . An e q u a t i o n r e l a t i n g V  these v a r i a b l e s  f o r the t u r b i n e i s  2 . . = a + a. s. + a_H , ., + a_ s H + a. s. n,k+l o 1 k+1 2 n,k+l 3 k+1 n,k+l 4 k+1  Equation fit.  (6-1)  n  (6-1) i s a p o l y n o m i a l i n head and s t r o k e and g i v e s a r e a s o n a b l e  The v a l u e s o f the c o e f f i c i e n t s depend on the u n i t ' s d e s i g n head,  d e s i g n flow and the c h a r a c t e r i s t i c s o f the t u r b i n e ' s compound  orifice.  V a l u e s w i l l be g i v e n i n the f o l l o w i n g c h a p t e r . An e q u a t i o n r e l a t i n g head and flow f o r the end o f the p e n s t o c k is  o b t a i n e d by r e w r i t i n g e q u a t i o n V  (3-13) w i t h i = n,  . - £ (C n,k+l a n  - H  ,..)•. n,k+l  <6-2)  where C = H . + — (V - . - FAt V , , | V , ,|) n n-l,k g n-l,k n-l,k n-l,k' 1  S o l v i n g f o r head and v e l i c i t y i n e q u a t i o n s  H • ..  n,k+l  C = a n —  g  Equations  (6-1) and  (6-2) y i e l d s  - (a + a.s, + a.s ,..) o 1 k+1 4 k+1 2  ,* (6-4)  + a_ + a s . 2 3 k+1  (6-2) and (6-4) g i v e the v a l u e s f o r p r e s s u r e and  velocity  f o r the p e n s t o c k - t u r b i n e end c o n d i t i o n .  equation  (6-4) must be computed  (6-2).  (6-3)  first,  In the computer  and i t s v a l u e s u b s t i t u t e d  program, into  41  6.3  Turbine  Torque  An e q u a t i o n e x p r e s s i n g torque as a f u n c t i o n of head s t r o k e can be variables. will  o b t a i n e d from a p o l y n o m i a l f i t to d a t a r e l a t i n g  Data c o n t a i n i n g runner speed was  t h e r e f o r e be  assumed to be  not  found.  and these  Runner  speed  c o n s t a n t and e q u a l t o the u n i t ' s  syn-  chronous speed. An e q u a t i o n f o r torque i s 2 T  k+1  =  b  o  +  b  l k+1 S  +  Again,  b  5  +  \  b  +  2  l  H  n,k+1  H  +  b  3  S  k+1  H  n,k+1  +  b  4  S  k+1  n,k 1  ( 6  +  the v a l u e s o f the c o e f f i c i e n t s b  d e s i g n head, d e s i g n flow and  Q  t o b,. depend on  the c h a r a c t e r i s t i c s o f the compound  E q u a t i o n "(6-5) g i v e s the h y d r a u l i c torque c o u p l e d from w a t e r t o the runner time k.  Together,  "  5 )  the  orifice. the  as a f u n c t i o n o f head and s t r o k e at i n s t a n t s o f t h i s h y d r a u l i c torque and the e l e c t r i c torque  on the r o t o r determine  the u n i t ' s speed  deviation during  acting  transient  conditions. In making the f i t ,  the r e g i o n near the r a t e d power contour  a head a g a i n s t d i s c h a r g e p l o t must be lower power zones.  f i t more p r e c i s e l y  U n i t s are u s u a l l y designed  T h i s t o g e t h e r w i t h the f a c t t h a t a i r gap  on  than t h a t a t  to accept 15%  overload.  torque i s g r e a t e s t f o r the  o v e r l o a d c o n d i t i o n r e q u i r e s the f i t t o be most p r e c i s e a l o n g the o v e r l o a d power contour, and d e c r e a s i n g toward lower powers. 40% power c o n t o u r , an e r r o r i n f i t of s e v e r a l p e r c e n t may l o a d i s a l s o of l e s s e r c o n c e r n .  15%  Below the  be a c c e p t a b l e .  The  r e g i o n beyond 15%  The p r e c i s i o n  fit  s h o u l d f o l l o w the power c o n t o u r s , b e i n g most p r e c i s e n e a r  the  of  42  overload 6.4  rating.  The T u r b i n e By-Pass V a l v e Our main concern i s the c o n t r o l of the h y d r a u l i c t o r q u e c o u p l e d  i n t o the t u r b i n e r u n n e r . v a l v e may  I t i s proposed t h a t a s m a l l f a s t - a c t i n g  e n a b l e a u n i t to m a i n t a i n s y n c h r o n i s m f o r any l o a d .  by-pass  The e f -  f e c t i v e a r e a o f a s m a l l v a l v e would be a p p r o x i m a t e l y between one  and  two p e r c e n t o f the p e n s t o c k ' s c r o s s - s e c t i o n a l a r e a . In  t h i s scheme the by-pass v a l v e i s i n s t a l l e d about f i v e pen-  s t o c k d i a m e t e r s upstream o f the s c r o l l case e n t r a n c e and i t by-passes w a t e r around the h y d r a u l i c t u r b i n e i n t o the t a i l r a c e .  I t thereby enables  w a t e r f l o w i n g down the p e n s t o c k t o be d i v e r t e d around the t u r b i n e w i t h o u t i t s energy b e i n g c o u p l e d i n t o the r u n n e r . I n i t s c o n s t r u c t i o n , the v a l v e assembly  c o n s i s t s of a needle  v a l v e h e l d i n p l a c e by p e n s t o c k p r e s s u r e a p p l i e d to a p i s t o n i n a d o u b l e ported c y l i n d e r .  Upon d e t e c t i n g a t r a n s i e n t c o n d i t i o n , the h o l d i n g  p r e s s u r e would be s w i t c h e d o f f and the c y l i n d e r v e n t e d t o  atmosphere,  a l l o w i n g the p e n s t o c k p r e s s u r e a c t i n g on the n e e d l e v a l v e ' s p o r t a r e a to r a p i d l y  (about 0.2  seconds) d r i v e the v a l v e f u l l y open.  Subsequently  p e n s t o c k p r e s s u r e would be r e a p p l i e d and, v i a a r e s t r i c t i n g o r i f i c e , n e e d l e v a l v e would  the  c l o s e g r a d u a l l y i n o r d e r n o t t o i n d u c e unwanted  positive pressures. The e q u a t i o n f o r t o r q u e i s i d e n t i c a l t o e q u a t i o n ( 6 - 5 ) , because the head i s common to b o t h the t u r b i n e and the v a l v e . The e q u a t i o n r e l a t i n g head and f l o w f o r the p e n s t o c k - t u r b i n e i n t e r f a c e i s s i m i l a r t o e q u a t i o n ( 6 - 4 ) ; the f l o w through the v a l v e must be added t o e q u a t i o n ( 6 - 1 ) .  Flow changes caused by the v a l v e w i l l have  a l a r g e e f f e c t on the head a t the t u r b i n e ' s  inlet.  43  The head and f l o w through  the v a l v e a r e r e l a t e d by  Q = ( C A) y ^ i ?  (6-6)  d  The c o r r e s p o n d i n g  p e n s t o c k v e l o c i t y due t o t h i s f l o w i s  —An i =A —£ A  V  =  P  p  p  ^  /2eH .  (6-7)  A =  v^gH • G P A = C, A, t h e e f f e c t i v e v a l v e a r e a , and A = effective f u l l g d ' g o v a l v e area G = v a l v e s t r o k e , 0 £ G <: 1. Thus A = A G g go  where  Equation  (6-7) i s n o t l i n e a r i n head.  In order to s i m p l i f y  the s o l u t i o n f o r head when i t i s added t o e q u a t i o n l i n e a r i z e d e q u i v a l e n t i s sought.  (6-1), a s u i t a b l e  S i n c e the head d u r i n g the t r a n s i e n t  c o n d i t i o n s w i l l be reduced as a r e s u l t o f v a l v e o p e r a t i o n by 20%, a r e a s o n a b l e  approximately  l i n e a r i z a t i o n i n head can be o b t a i n e d i n a s m a l l  r e g i o n below the u n i t ' s d e s i g n head v a l u e .  The l i n e a r i z a t i o n i s  A G = 2 f J  V P  fo  <° H  +  H  >  <"> 6  where H ° = d e s i g n head. Combining e q u a t i o n s  (6-1) and (6-8) y i e l d s 2  V  n,k+1  =  a  o  +  a  +  l k+1 S  a  5  (H  +  a  2 n,k+1 H  ° ,k i> + H  n  +  +  *  a  G  3 k+1 n,k+1 S  H  +  a  k r +  where A a  5  2A P  •  Is ZN|  The p e n s t o c k end c o n s t r a i n t i s e q u a t i o n S o l v i n g f o r head i n e q u a t i o n s  (6-2).  (6-9) and (6-2) y i e l d s  4 k+1 S  (6  "  9)  8  44  H  =  a  C  n  ! a  where  2  - [a +  + a.s. + a.s. ,, + a H°G, ] 1 k+1 4 k+1 5 k+1 2  o  , (6-10)  q  VW1  +  "5  G k +  l  +  n N  ?  i s g i v e n by e q u a t i o n ( 6 - 3 ) . Equations  (6-10) and  (6-2) g i v e the s o l u t i o n f o r head  and  v e l o c i t y at the j u n c t i o n of the p e n s t o c k , the t u r b i n e and the by-pass valve. Remarks B e f o r e a p p l y i n g s u r f a c e f i t s , the d e f i n i t i o n f o r head H on the f i e l d d a t a curves must be c o n f i r m e d ; i t may.present the head  dif-  f e r e n c e between the s c r o l l case i n l e t and t a i l ' w a t e r l e v e l o r t h a t between the s c r o l l case and runner e x i t a t shroud r i n g l e v e l .  Head H  i s t o t a l head and i n c l u d e s v e l o c i t y heads a t the e n t r a n c e t o the  scroll  19 case and a t the e x i t p o i n t ' . An advantage to o b t a i n i n g these r e l a t i o n s h i p s i n the form o f a p o l y n o m i a l f i t t e d t o f i e l d d a t a i s t h a t a l l the c h a r a c t e r i s t i c s r e l a t i n g these v a r i a b l e s are i n c l u d e d i n the e q u a t i o n .  In a n a l y t i c  deri-  v a t i o n s i t r e q u i r e s a c o n s c i o u s e f f o r t t o ensure the dominant e f f e c t s have been i n c l u d e d .  45  7.  7..1  COMPUTER SIMULATIONS  Introduction I t has  synchronism  been proposed t h a t the a b i l i t y of a u n i t to m a i n t a i n  following a l i n e clearing operation  c o u l d be enhanced by  moving the w i c k e t gates i n a r e v e r s e d i r e c t i o n to t h a t i n c u r r e n t t i c e f o r speed r e g u l a t i o n , and reduce the h y d r a u l i c peak has  thus take advantage of waterhammer t o  t o r q u e i n p u t t o the machine u n t i l the f i r s t  simulations  w h i c h , f o r the models used, s u g g e s t the t h e s i s p r o p o s i t i o n The steady-state  c r i t e r i o n f o r improvement i s the i n c r e a s e a c t i v e power w h i c h a u n i t and  t o the i n f i n i t e bus  transmission  to be  in prefault l i n e can  For  t h i s study l o s s of s y n c h r o n i s m i s d e f i n e d  wherever the t o r q u e a n g l e (between the r o t o r and  a  pro-  to o c c u r  the i n f i n i t e bus)  e l e c t r i c a l degrees w i t h i n f i v e seconds o f b r e a k e r  This allows  deliver  In p r a c t i c e  s l i p p o l e s s e v e r a l times b e f o r e the u n i t i s t r i p p e d by  tective relays.  valid.  and y e t r e t a i n s y n c h r o n i s m f o l l o w i n g l i n e c l e a r i n g .  L o s s of s y n c h r o n i s m i s d i f f i c u l t t o d e f i n e .  ceeds 180  swing  passed. T h i s c h a p t e r p r e s e n t s the r e s u l t s o f computer  r o t o r may  prac-  ex-  reclosure.  f o r a p p r o x i m a t e l y f i v e system s w i n g s .  During the t r a n s i e n t c o n d i t i o n s  f i e l d e x c i t a t i o n plays  an  i m p o r t a n t r o l e i n d e t e r m i n i n g the a b i l i t y o f a u n i t to m a i n t a i n s y n chronism.  A l t h o u g h the a c t u a t o r  reduces the h y d r a u l i c i n p u t  torque  t h r o u g h the f i r s t swing peak, i t i s the duty o f the e x c i t e r to b u i l d up e x c i t a t i o n i n the i n t e r i m so t h a t i t can keep the u n i t i n s y n c h r o n i s m f o r the f o l l o w i n g swings.  The  bounds on the e x c i t e r c o n t r o l space p a r -  t i c u l a r l y a f f e c t the a b i l i t y o f a u n i t to m a i n t a i n s y n c h r o n i s m .  46  It i s assumed that i f syncrhonism i s maintained  for f i v e  swings, optimal e x c i t e r control would be capable of returning the system to i t s o r i g i n a l , p r e f a u l t steady-state condition. The e l e c t r i c power transmission system shown i n Figure s i s t s of a generator-transformer  6 con-  connecting the unit to a switchyard  bus, and a p a i r of p a r a l l e l transmission lines connecting  the  switchyard  bus to an i n f i n i t e capacity power g r i d . The disturbance i s assumed to be a balanced applied to one ofthe p a r a l l e l l i n e s . breakers, not single-pole breakers.  3-phase l i n e f a u l t  Line c l e a r i n g i s effected by 3-pole This assumption obviates the need  to model negative and zero sequence c h a r a c t e r i s t i c s of the e l e c t r i c a l system. For the measure of improvement to be consistent, each test requires -a set of i d e n t i c a l i n i t i a l .conditions.  In .stipulating  constant  i n i t i a l conditions, a choice can be made from among the following: switchyard bus voltage, generator terminal voltage, reactive power passing the switchyard bus or the generator terminals, the power factors at  these two points or at the i n f i n i t e bus, the f i e l d e x c i t a t i o n .  In  p r a c t i c e , o v e r - a l l power network generation scheduling would d i c t a t e the various generation requirements. system i s r a d i a l l y  For our study however, since our  connected to the g r i d we s h a l l i n s i s t that each  r a d i a l system connected to the i n f i n i t e grid supply i t s own reactive power and deliver only active power to the g r i d .  The condition for  each test therefore i s unity power factor into the i n f i n i t e bus node. The other two i n i t i a l conditions are given by the active power flow which i s determined by the gate p o s i t i o n and by the i n f i n i t e bus voltage, defined to be 1 p.u.  47  0.13  Synchronous Generator F i g u r e 6.  p.u.  Generator Transformer  ^ S w i t chyard Bus  x = 1.20  p.,u.  x = 1,20  p..u.  Transmission Lines  C o n f i g u r a t i o n o f e l e c t r i c system s i m u l a t e d .  o-  Infinite Bus  ^-Line Breakers  48  7.2  The  System Model The h y d r o - e l e c t r i c power g e n e r a t i n g system s e l e c t e d f o r the  s i m u l a t i o n s t u d i e s i s the s i n g l e u n i t e q u i v a l e n t of the g e n e r a t o r s the W.A.C. Bennett  Dam,  Hudson Hope, Canada.  o b t a i n e d from the d e s i g n notes the p r o j e c t .  and  The  t e c h n i c a l data  at was  the c o n t r a c t documents p r e p a r e d  for  S e v e r a l q u a n t i t i e s were a l t e r e d to emphasize t h e i r e f -  fects. Models w i t h v a l u e s  f o r the h y d r a u l i c and  the e l e c t r i c systems  follow. 7.2.1  The  H y d r a u l i c System  The h y d r a u l i c system c o n s i s t s o f a 2000 f o o t penstock the r e s e r v o i r and 1200  Although  f e e t , the l o n g e r l e n g t h was  of l e n g t h on was  the t u r b i n e .  the a c t u a l l e n g t h i s o n l y  taken i n o r d e r to emphasize the  the system's b e h a v i o u r .  also simulated.  between  In the s t u d y ,  a 1000  effect  foot length  In s p i t e of the waterhammer wave t r a v e l  h a l v e d , the o v e r a l l e f f e c t on the u n i t ' s power l i m i t was  about  time  l e s s than  being one  percent. F r i c t i o n l o s s i n penstocks o f d e s i g n head.  Inorder  g e n e r a l l y l i e s below f i v e  to emphasize the e f f e c t s o f f r i c t i o n ,  o f ten p e r c e n t i s assumed.  The  system parameters a r e :  Design head,  500  Design  flow,  5850 cu.  Design  power ( t u r b i n e ) ,  310,000 brake horsepower,  Design  efficiency  Design  speed,  (turbine),  feet,  94.0  150  RPM,  Rated head,  455  feet,  Rated f l o w ,  6750 cu.  ft/sec,  %  ft/sec,  percent a loss  49 Penstock diameter,  18 feet,  F r i c t i o n loss at design flow,  50 feet W.C.  System conditions are: Reservoir elevation 550 feet above t a i l r a c e elevation, Servomotor stroke at 70%. The equations f o r the penstock are:  H  i,k 1 " 1  { H  +  +  i-l,k  F  A  +  H  i+l,k  2  where  -  F  A  i-l,k- i l,k V  t  ) / C  ( 7  +  v  v  )  /  c  -  X )  }  k  x+l,k  •  ( V  ( i i, - i-i,k  t  +  i,k+l  +  l-l,k  < i+i,k v  +  v  l-l,k  i+I,k  i-i,k»  i=l,2,...,9 C = g/a At = time increment, a = 4000 f t . / s e c , the v e l o c i t y of sound i n the penstock, F = 0.0402/V  2  n,o The absolute value function of v e l o c i t y i s not used i n each simulation since the flow i s always p o s i t i v e .  At the penstock's r e s e r v o i r end, the  equations are:  At  H  o,k+l -  V  o,k+l  =  5  5  C ( H  0  ( 7  o , k " l,k> H  +  V  l,k "  F  A  t  V  i ,  ( 7 k  "  '  3 )  4 )  the penstock's turbine end, the equations are: H  n,k+1 -  <C H _ n  {  I  +  a  2  1 ) k  +  a  + (V^-FAt  3 \+l  }  V _ 2  1 > k  ) - ( a ^  s ^ + a ^ )  (7-5)  50 v  n,k+l t  x  i  =  C  <  n -1 l1 , k  H  "  n,k+l i  H  n  )  1 n -1 l , k- FAt Vn - l . ,k  (7-6)  2  +  v  n  For  a t u r b i n e w i t h a by-pass v a l v e , the e q u a t i o n s f o r the t u r b i n e end o f  the  penstock are: H  ... = ( C H + V - FAt V n,k+l n~l>k n-l,k n-l,k 2  "  { a  V  n,k+1  S  k+1  =  ( a  2  =  o  +  +  a  k  3 k-fl S  +  +  a  +  T~  U  a  a  4 k+1 S  5 k+1 G  ^n-l.k " \  C  S  l V l  a  >  k  +  +  l  +  a  5 ° H  G  k+1  ) } /  -  C }  )  +  ( 7  Vl.k "  F A t  V  n-l,k  ~  7 )  ( 7  l,(k+1)  "  ( 7  8 )  '  9 )  V a l u e s f o r the c o e f f i c i e n t s a r e : n = 10> the number o f r e a c h e s i n the p e n s t o c k , a  o  = +44.605  a  a  2  = .01375  a  3  = .02365  a /  = -22.935  a  = .000704 A  4  A  = -12.952  5  8°  = by-pass v a l v e e f f e c t i v e  go T =2.5 a  area,  seconds, servomotor f u l l s t r o k e t r a v e l p e r i o d  u^(') = a c t u a t o r c o n t r o l s i g n a l , and H° = 500 f e e t W.C., 7.2.2  the d e s i g n head.  The E l e c t r i c System The e l e c t r i c system c o n s i s t s o f a g e n e r a t o r , a g e n e r a t o r  t r a n s f o r m e r and a p a r a l l e l s e t o f t r a n s m i s s i o n l i n e s .  The system p a r a -  meters e x p r e s s e d i n t h e p e r u n i t system a r e : D i r e c t - a x i s synchronous r e a c t a n c e , Q u a d r a t u r e - a x i s synchronous r e a c t a n c e  x^ = 0.90 p.u. x^ = 0.55 p.u.  51  Direct-axis  transient  reactance  x' = 0.18  p.u.  d Direct-axis  open-circuit  time c o n s t a n t ,  Machine i n e r t i a , Transmission  WR  l i n e reactance  equations  d$ "dT  f  (X  do  * F  d  - <o(t) -  ft -  =  g  10  6  lb. ft  x = 1.20  .13  second 9  p.u.  p.u.  f o r the t h i r d o r d e r synchronous machine a r e :  -X "  = 220  (each l i n e ) ,  Generator transformer reactance x The  9  = 7.76  ( t )  +  26  -X')  -^7~  d  C  °  S  6  (  t  )  +  U  2  ( t )  (  7  ~  1  0  )  (7-11)  %  " - c h  " elec>  <" >  T  7  12  where ^ T  elec  (t)  =  do and T  X' - X  d  -cs  1  d  ^ i s g i v e n by e q u a t i o n In e q u a t i o n  -YTY  +  6(t)}  sine(t)  (7-13), the m o d i f i e d r e a c t a n c e s  X, = x, + x + -r- x d d g 2 X = x + x + - ^ - x q q g 2 X' = x* + x + ^-'x d d q 2  iji  r  are:  = d i r e c t axis  flux  linkages,  6 = torque angle between d - a x i s and i n f i n i t e The  c o e f f i c i e n t s i n equation  (7-13)  (6-5).  (7-10) and  The e l e c t r i c a l v a r i a b l e s  "  q  (6-5)  are:  bus,  are:  52 t> = -.000061 2  b  = +.007530  3  b.  +.153  =  4 b  = -.00368  c  5  ^(t),  the g e n e r a t o r f i e l d e x c i t a t i o n c o n t r o l s i g n a l , i s g i v e n u„  =  0.3  (OJ-U) )  2  +  o  for  -2  <  u„  Fo  <  .5  2  = 10 f o r  i n e q u a t i o n (7-14)  .5,  = -2  i n e q u a t i o n (7-14) $  -2.  for  by:  and  where CJ Vp  Q  q  = 120TT r a d i a n s / s e c o n d , the e l e c t r i c a l speed,  = the p r e f a u l t s t e a d y - s t a t e e x c i t a t i o n as o b t a i n e d from initial The  by  trial  i n the  and  and  conditions.  above e x c i t a t i o n s i g n a l g e n e r a t i n g f u n c t i o n was error.  the  I t was  found to be  obtained  a simple yet e f f e c t i v e  function  simulations. For  sumed t o be  the synchronous g e n e r a t o r , the machine parameters are  constant.  E f f e c t s o f m a g n e t i c s a t u r a t i o n are  as-  neglected.  S i n c e , w h i l e on the v e r g e of a u n i t l o s i n g s y n c h r o n i s m , f l u x e s  and  currents  saturation  assume l a r g e v a l u e s , f i e l d t e s t s may  prove n e g l e c t of  to i n t r o d u c e s i g n i f i c a n t e r r o r . A p p r o p r i a t e d e l a y s are i n s e r t e d i n t o the e x c i t e r and tuator  f o r improved r e a l - l i f e s i m u l a t i o n .  For b o t h , a 50 ms.  allowed f o r c o n t r o l d e c i s i o n l o g i c to e s t a b l i s h that a l i n e operation i s required. for  For  the  ac-  delay i s clearing  the e x c i t e r , an a d d i t i o n a l 50 ms.  i s allowed  the e x c i t e r t o e s t a b l i s h f u l l range f i e l d f o r c i n g ( t h e e x c i t e r i s 24  assumed to be of the s t a t i c SCR an a d d i t i o n a l 150  ms.  controlled type).  For the  i s a l l o w e d f o r i t s s o l e n o i d s to r e a c t  m a s s i v e gates to b e g i n m o t i o n .  Thus d e l a y s o f 100  and  actuator, and  200 ms.  the are  53 a p p l i e d t o the e x c i t e r and a c t u a t o r 7.3  D i s c u s s i o n of S i m u l a t i o n In  first.  Results  the s i m u l a t i o n s , the h y d r a u l i c system was  The o b j e c t i v e was  waves which  respectively.  to show t h a t gate motion generates p r e s s u r e  t r a v e l back and f o r t h between the t u r b i n e and the  After this, The o b j e c t i v e was  For  reservoir.  the composite h y d r a u l i c - e l e c t r i c system was s i m u l a t e d .  to determine the improvement i n the power l i m i t o f the  u n i t o b t a i n e d by u s i n g waterhammer t o  advantage.  each s i m u l a t i o n the d i s t u r b a n c e to the system i s assumed  to be a t r a n s m i s s i o n l i n e  One  o f the two  p e r i e n c e a b a l a n c e d three-phase f a u l t  to ground.  fault.  b o t h ends o f the f a u l t e d l i n e reclose.  simulated alone  The  lines  i s assumed t o ex-  The l i n e b r e a k e r s a t  t r i p s i m u l t a n e o u s l y and, a f t e r 500  f a u l t i s assumed to have been c l e a r e d i n the  milliseconds,  interval.  As a r e s u l t o f the above d i s t u r b a n c e , the system b e g i n s t o swing. The power l i m i t i s determined by by the u n i t f o r which from l o s i n g  the maximum p r e f a u l t power g e n e r a t e d  a p a r t i c u l a r c o n t r o l sequence  can keep the system  synchronism.  The i n c r e m e n t a l d i s t a n c e and time f o r the p e n s t o c k was f e e t and 0.05  seconds r e s p e c t i v e l y .  time increment was In  0.01  the f i r s t  c o n t r o l sequences  seconds.  200  F o r the g e n e r a t o r e q u a t i o n s , the  The s i m u l a t i o n run was  5.0  seconds.  s e t o f composite system s i m u l a t i o n s a number o f  are a p p l i e d t o the w i c k e t g a t e s .  I n the second s e t , the  h y d r a u l i c system i s m o d i f i e d to i n c l u d e a by-pass v a l v e around the In  turbine.  t h i s s e t , o n l y the by-pass v a l v e i s c o n t r o l l e d ; the w i c k e t gates are  kept at r e s t . stated.  The  I n the f i n a l s e t , the o r i g i n a l h y d r a u l i c system i s r e i n c o n t r o l s i g n a l a p p l i e d t o the a c t u a t o r i s o b t a i n e d from a  speed g o v e r n o r .  The o b j e c t i v e was  to determine the power l i m i t o b t a i n e d  from a c o n v e n t i o n a l governor and to compare the r e s u l t s t o those o b t a i n e d by  54  t a k i n g advantage o f waterhammer. 7.3.1  The h y d r a u l i c system The r e s u l t s o f the h y d r a u l i c system s i m u l a t i o n are g i v e n i n  F i g u r e 7.  I t shows the p r e s s u r e and torque response, t o two extreme w i c k e t  gate a c t u a t i o n sequences.  A r e d u c t i o n i n b o t h p r e s s u r e and t o r q u e im-  m e d i a t e l y a f t e r the i n i t i a l a c t u a t i o n i s e v i d e n t . Trace 'a' i s the response t o a s h o r t a c t u a t i o n sequence c o n s i s t i n g o f a 0.2  second gate o p e n i n g d r i v e , a 0.2  second h o l d , and a  0.2  second c l o s i n g d r i v e , r e t u r n i n g the gates t o t h e i r o r i g i n a l p o s i t i o n .  The  h a l f c y c l e p e r i o d f o r t h e p e n s t o c k i s one second, and • a c c o r d i n g l y the d i s t u r b a n c e r e f l e c t i o n s are seen a t the t u r b i n e end of the p e n s t o c k a t one second i n t e r v a l s .  The r e f l e c t i o n s decay g r a d u a l l y as the p r e s s u r e and  torque r e t u r n to t h e i r o r i g i n a l v a l u e s .  The t r a p e z o i d a l wave-shape i s  w e l l duplicated i n successive r e f l e c t i o n s i n d i c a t i n g that f o r small d i s t u r b a n c e s the n o n - l i n e a r h y d r a u l i c impedance of the t u r b i n e and the  fric-  t i o n i n the p e n s t o c k a r e n o t s i g n i f i c a n t . Trace 'b' i s the response t o a l o n g a c t u a t i o n sequence c o n s i s t i n g o f a 1.2  second gate o p e n i n g d r i v e , a 0.4 second h o l d and a  second c l o s i n g d r i v e . 0.94 of  I n t h i s sequence,  o f f u l l gate t r a v e l ( i n i t i a l  the a c t u a t o r s t r o k e r e a c h e s  gate i s 7/10).  t r a v e l l i n g p r e s s u r e waves can be seen.  1.2  A g a i n the p r e s e n c e  A l t h o u g h the gates a r e b e i n g  d r i v e n open f o r a f u l l 1.2 seconds, the p r e s s u r e and t o r q u e a b r u p t l y become c o n s t a n t a t 1.0 second due t o the r e t u r n i n g from the r e s e r v o i r .  p o s i t i v e l y r e f l e c t e d wave f r o n t  I n the f o l l o w i n g 1.2  t o 1.6  seconds,  d u r i n g w h i c h the a c t u a t o r s t r o k e i s h e l d c o n s t a n t , the p r e s s u r e and t o r q u e b e g i n t o r i s e as the r e f l e c t e d waves due t o gate a c t u a t i o n second ago c o n t i n u e r e t u r n i n g from the r e s e r v o i r .  A t 1.6  one  seconds, the  55  1.20 HYDRAULIC TORQUE IN P.U.  1.10  1.00  0.90 0.80  PRESSURE HEAD IN FEET W.C. AT SCROLL CASE ENTRANCE  3  4  Actuator  at f u l l  5 stroke  ACTUATOR STROKE  ACTUATOR CONTROL SEQUENCE  stroke increasing  +1  2  3  TIME IN SECONDS F i g u r e 7.  E f f e c t s o f water-hammer on p r e s s u r e head and h y d r a u l i c due t o g a t e s t r o k i n g .  torque  56  gates b e g i n c l o s i n g and hence the s l o p e o f b o t h p r e s s u r e and  torque  r i s e more s t e e p l y due t o the added l o c a l e f f e c t o f the c l o s i n g g a t e s . The r i s i n g p r e s s u r e c o n t i n u e s t o 2.6  seconds, when f i n a l l y the gates  have been r e t u r n e d t o t h e i r o r i g i n a l p o s i t i o n .  A f t e r 2.6  seconds,  the  waves g r a d u a l l y s u b s i d e and c o n d i t i o n s r e t u r n to t h e i r o r i g i n a l v a l u e s . F o r a c t u a t i o n sequences  e x t e n d i n g beyond the p e n s t o c k ' s n a t u r a l wave  t r a v e l t i m e , the d i s t r i b u t e d parameter n a t u r e becomes l e s s o b v i o u s . Trace 'a' o f f e r s a s m a l l t o r q u e decrease i n i t i a l l y and p o s i t i v e torque p u l s e s t h e r e a f t e r at one second i n t e r v a l s .  Trace  has 'b'  o f f e r s a v e r y l a r g e i n i t i a l r e d u c t i o n w h i c h i s f o l l o w e d by a l o n g , dec a y i n g p o s i t i v e torque p e r i o d .  I n our a p p l i c a t i o n to g e n e r a t o r s t a b i l i t y  f o l l o w i n g l i n e r e c l o s u r e , our endeavour i s to c o n t r o l the a c t u a t o r so t h a t the i n i t i a l and r e f l e c t e d p r e s s u r e waves w i l l develop torque reduction.  a large  initial  S u c c e e d i n g peaks s h o u l d not be e x c e s s i v e and, i f p o s -  s i b l e , be i n s y n c h r o n i s m w i t h the a c c e l e r a t i n g t o r q u e r e q u i r e m e n t s o f the u n i t ' s r o t a t i n g assembly line  as the e l e c t r i c system o s c i l l a t e s  after  reclosure. I t i s thus e v i d e n t t h a t as a r e s u l t o f gate m o t i o n , p r e s s u r e  waves are g e n e r a t e d and t h a t they t r a v e l t o and f r o i n the p e n s t o c k , gradually decaying a f t e r s e v e r a l r e f l e c t i o n s . 7.3.2  The composite  >  system  F i g u r e 8 shows the r e s u l t s o f a s e r i e s o f a c t u a t o r c o n t r o l sequences  a p p l i e d t o the composite h y d r a u l i c - e l e c t r i c system.  Sequence  d e f i n i t i o n and c o d i n g i s g i v e n t o f a c i l i t a t e i n t e r p r e t a t i o n o f the cont r o l function plot.  Each c o n t r o l sequence has an i n i t i a l 0.2  a c t u a t i o n d e l a y , as e x p l a i n e d e a r l i e r .  second  Time TI s i g n i f i e s the end o f  the gate o p e n i n g d r i v e and time T2 r e p r e s e n t s the l e n g t h o f the h o l d p e r i o d .  57  - F u l l g a t e power a t 1.22 p.u. 1.22  1.20  1.18 PREFAULT STEADYSTATE  1.16  1.14  POWER IN P.U.  1.12  1.10  1.08 T2= 0  1.06  1.04  1.02  1.00 1  J  L  .2  .3  .4  VARIABLE  ACTUATOR CONTROL SEQUENCE DEFINITION  +1  r  0 -1 .2  .4  .6 TIME  F i g u r e 8.  Tl  .6  .7  .8  . 9 1 . 0  IN SECONDS  T l = end o f p o s i t i v e d r i v e T2 = pause p e r i o d  J 0  .5  .8 IN  L 1.0  1.2  1.4  1.6  1.8  2.0  SECONDS  P l o t o f s t e a d y - s t a t e power ( c o s t f u n c t i o n ) v a l u e s o f s e v e r a l a c t u a t o r c o n t r o l sequences.  as a f u n c t i o n  58  The  gate c l o s i n g p e r i o d e q u a l s  gate to i t s o r i g i n a l p o s i t i o n  the opening p e r i o d .  This returns  the  as r e q u i r e d i n o r d e r t o r e t u r n the e n t i r e  system to i t s p r e f a u l t c o n d i t i o n s .  As shown on the p l o t , the c o n t r o l  sequence d e f i n e d by T l = .5 and T2 = .2 i s the most f a v o u r a b l e .  It  e n a b l e s the system to m a i n t a i n s y n c h r o n i s m w i t h an i n i t i a l p r e f a u l t power throughput o f 1.166 T2 = 0) o f 1.112  p.u.  p.u.,  Compared to the dead a c t u a t o r v a l u e  .054  p.u.  or approximately  i n c r e a s e d the power l i m i t  5%.  A c t u a t o r s h a v i n g l a r g e r power a m p l i f i e r c a p a c i t i e s can t h i s l i m i t maximum f u r t h e r . having  .2,  w h i c h i s the u n i t ' s l i m i t w i t h e x c i t e r c o n t r o l  a l o n e , t h i s f a v o u r a b l e c o n t r o l sequence has by  (Tl =  The  above r e s u l t s are v a l i d f o r an  increase actuator  the c a p a c i t y to d r i v e the gates a t the r a t e o f 40% f u l l s t r o k e  p e r second.  Engineering  and m a n u f a c t u r i n g  w i l l l i m i t the l a r g e s t u s e f u l c a p a c i t y .  c o s t and One  technical factors  foreseeable t e c h n i c a l fac-  t o r would be a c t u a t o r j i t t e r as the p r e a m p l i f i e r s p o o l s and p o r t s to wear and b a c k l a s h appears.  ( D i g i t a l v a l v e s i n t h i s a p p l i c a t i o n may  p r o v e to be s u i t a b l e because of t h e i r p r e c i s e v a l v e C f a s t 20 ms.  response t o any one  The  begin  o f 256  v  f a c t o r c o n t r o l and  values.)  i m p o r t a n c e of s y n c h r o n i z i n g h y d r a u l i c torque peaks w i t h  a c c e l e r a t i n g torque r e q u i r e m e n t s was  mentioned e a r l i e r .  i n power l i m i t f o r v a l u e s o f T l between .4 and  decrease  .7 seconds i s due  p r e s s u r e peaks b e i n g out o f phase w i t h t h a t r e q u i r e d . phase p r e s s u r e s  The  t o the  Indeed, the out  of  can reduce the power l i m i t t o v a l u e s below t h a t f o r a  dead a c t u a t o r (e.g. the f u n c t i o n h a v i n g T l = .6, T2 = . 2 ) . A gradual to  i n c r e a s e i n the power l i m i t s appears between T l =  .7  .8 as the p r e s s u r e wave f r o n t a g a i n becomes i n phase w i t h the second  swing o f the e l e c t r i c system.  T h i s i n c r e a s e however i s l i m i t e d by  the  59  gates b e i n g d r i v e n t o f u l l gate d u r i n g these l o n g a c t u a t i o n sequences. F u r t h e r m o r e , as the p r e f a u l t l i m i t i n c r e a s e s , t h e c o r r e s p o n d i n g  initial  gate p o s i t i o n a l s o i n c r e a s e s towards f u l l g a t e , l e a v i n g l e s s room f o r the a c t u a t o r t o move and take advantage o f waterhammer. The  above computer r e s u l t s show t h a t r e v e r s e a c t u a t i o n o f t h e  w i c k e t gates can i n d e e d i n c r e a s e a u n i t ' s a b i l i t y during  t o keep s y n c h r o n i s m  a f a u l t .. At t h e g r e a t e r power l e v e l s , as the i n i t i a l  approaches f u l l  gate p o s i t i o n  g a t e , t h e a c t u a t o r ' s manoeuvering space d e c r e a s e s and the  e f f e c t i v e n e s s o f t h e r e v e r s e a c t u a t i o n scheme d i m i n i s h e s .  I tw i l l  f o r e n o t be p o s s i b l e f o r t h e power l i m i t t o r e a c h t h e f u l l  gate power  value.  ( F o r o u r s t u d y , w h i c h i s v a l i d f o r a s c r o l l case p r e s s u r e  o f 500 f e e t W.C.,  there-  head  t h i s f u l l gate l i m i t i s 1.22 p.u. T h i s l i m i t v a r i e s  w i t h the r e s e r v o i r l e v e l . ) 7.3.3  The t u r b i n e by-pass v a l v e F i g u r e 9 shows t h e power l i m i t s o b t a i n a b l e u s i n g by-pass v a l v e s  of several sizes. and  The v a l v e i s assumed t o f a l l open i n 0.2 seconds  c l o s e i n 4.0 seconds.  A valve having  an e f f e c t i v e a r e a o f about  t h r e e square f e e t e n a b l e s a u n i t t o r a i s e i t s power l i m i t t o n e a r l y 20% overload.  T h i s i m p l i e s t h a t f o r u n i t s whose o v e r l o a d i s l e s s than 20%,  the by-pass v a l v e w i l l always be a b l e t o keep t h e oanit i n s y n c h r o n i s m . When l o s s o f s y n c h r o n i s m o c c u r r e d , i n thie s i m u l a t i o n s , i t took p l a c e on t h e peak o f t h e t h i r d o r f o u r t h s w i n g , and n o t on t h e f i r s t o r second swing as was the case w i t h reverse, gate a c t u a t i o n . c o u l d be a v o i d e d by g i v i n g t h e e x c i t e r a l a r g e r f i e l d or  g i v i n g the v a l v e a l o n g e r c l o s i n g p e r i o d .  This  forcing capability  F u l l g a t e power a t 1.22 p.u.  PREFAULT STEADY-STAT POWER IN P.  0  1.0  2.0  3.0  BY-PASS VALVE EFFECTIVE AREA IN SQ. FT. FLgure 9.  A f f e c t o f by-pass v a l v e s o f s e v e r a l s i z e s on a u n i t ' s s t e a d y - s t a t e power throughput.  y — F u l l g a t e power a t 1.22 p.u. PREFAULT 1.2STEADY-STATE POWER IN P.U.  .  1  -.75  !  -.60 -.45  1  -.30  1.1 -  1 ' ° 1  -.15  0  1  1  1  .15  .30  .45  I  . .60  1  .75  GOVERNOR GAIN IN SECONDS PER RADIAN F i g u r e 10. A f f e c t o f c o n v e n t i o n a l  governor g a i n on power  throughput.  61  7.3.4  The  speed governor  F i g u r e 10 shows the power l i m i t a conventional, negative  speed-feedback governor.  s m a l l s i g n a l a n a l y s i s and must be n e g a t i v e  o b t a i n a b l e by  from  f i e l d experience  I t i s known from t h a t the  to r e a l i z e s t a b l e speed, r e g u l a t i o n and  s h a r i n g between u n i t s .  the system u s i n g  feedback s i g n a l  t o enable l o a d  F i g u r e 10 g i v e s a p l o t of the power l i m i t s  governor w i t h s e v e r a l v a l u e s  o f b o t h p o s i t i v e and n e g a t i v e  i n c r e a s i n g negative g r e a t e r than .30 As  speed feedback.  the  -.30  maximum power l i m i t  i s much l e s s than 1.166  with  c o n t r o l bounds. nature  i s l e s s pronounced. of 1.127  o b t a i n e d by  p.u.  f o r gains  iess  reverse actuation.  s u l t s i n d i c a t e that, f o r a u n i t t r y i n g to maintain f a u l t c l e a r i n g and  results  f o r gains  c o n t r o l becomes almost bang-bang i n  so the e f f e c t o f l a r g e r v a l u e s The  to the a c t u a t o r ' s  The  increases  The l e v e l i n g o f the s l o p e  i n magnitude are due  the g a i n exceeds .30,  and  the power l i m i t i n d e e d  of a  feedback.  These a r e a p p l i e d f o r o n l y f i v e seconds d u r i n g the t r a n s i e n t . show t h a t f o r s m a l l g a i n v a l u e s  linear,  than  These r e -  synchronism  after  l i n e r e c l o s u r e , a c t u a t i o n sequences d e s c r i b e d  F i g u r e 8 can a l l o w a g r e a t e r power l i m i t than t h a t g i v e n by  in  the  conven-  the  capa-  t i o n a l speed-feedback governor. 7.4  Comments The  foregoing r e s u l t s apply  city specified earlier. t o be  Larger  c a p a c i t i e s w i l l e n a b l e the power l i m i t  r a i s e d beyond the s i g n i f i c a n t 5% The  to an a c t u a t o r h a v i n g  increase.  e f f e c t i v e n e s s o f r e v e r s e gate a c t u a t i o n depends on  l e n g t h o f time the t r a n s m i s s i o n l i n e b r e a k e r s For shorter breaker  the  remain byen b e f o r e r e c l o s i n g .  open p e r i o d s , the u n i t s i n h e r e n t power l i m i t  will  62  be  g r e a t e r than  periods It  5 6 for longer periods-' .  However, d u r i n g these  shorter  the e f f e c t i v e n e s s o f the h i g h speed a c t u a t o r w i l l be reduced.  has l e s s  time to move the gates  the h y d r a u l i c torque, The geometric  u s i n g waterhammer.  e f f e c t i v e n e s s o f the scheme proposed a l s o depends on the  l a y o u t o f the h y d r a u l i c system.  pronounced on heads.  ( a t t h e i r maximum speed) t o reduce  h y d r a u l i c systems h a v i n g  On systems h a v i n g  The e f f e c t i v e n e s s i s more  l o n g p e n s t o c k s and low p r e s s u r e  s h o r t p e n s t o c k s and h i g h heads the e f f e c t o f  r e v e r s e a c t u a t i o n may be n e g l i g i b l e o r even be u n d e s i r a b l e . The is  time.  above i s due to two key f a c t o r s .  In longer penstocks,  The more obvious  factor  the l o n g e r time r e q u i r e d f o r the p r e s s u r e  wave to t r a v e l t o the r e s e r v o i r and back p r o v i d e s more time f o r gate motion to take e f f e c t . pressure  The second f a c t o r i s the r a t i o o f the change i n  t o the change i n v e l o c i t y ,  the change b e i n g i n i t i a t e d by gate  movement. T h i s r a t i o i s a/g, where 'a' i s the speed o f the p r e s s u r e waves i n the p e n s t o c k and 'g', the c o n s t a n t To i l l u s t r a t e The  of g r a v i t a t i o n a l a c c e l e r a t i o n .  the above, c o n s i d e r the graph i n F i g u r e 11.  o r d i n a t e s have been n o r m a l i z e d  and r e a d i n p e r c e n t a g e u n i t s .  The  s t r a i g h t l i n e segments l a b e l l e d 1 and 2 r e p r e s e n t the a/g r a t i o f o r h y d r a u l i c systems h a v i n g  d i f f e r e n t r a t i o s o f head t o v e l o c i t y .  Our  r e f e r e n c e p o i n t f o r t h i s d i s c u s s i o n i s g i v e n by the c o o r d i n a t e s o f 100% d e s i g n head and 80% d i s c h a r g e . 85% o f f u l l  gate.  The curve r e p r e s e n t s  Segment 1 r e p r e s e n t s having  The torque  at this point i s  the 85% torque  contour.  the a/g r a t i o o f a h y d r a u l i c system  a low head and a h i g h v e l o c i t y .  from 0.7 t o 0.9 o f f u l l  developed  As can be seen, moving the gates  s t r o k e w i t h i n the time i t takes  the p r e s s u r e  63  130  40 PERCENT  F i g u r e 11.  60  80  100  . 120  140  DISCHARGE  The e f f e c t o f r a p i d g a t e m o t i o n on t u r b i n e  torque.  64  waves to t r a v e l to the r e s e r v o i r and back, reduces the torque to  64%, a 21% r e d u c t i o n .  to  their i n i t i a l If  Shortly after this  the gates  must be r e t u r n e d  setting.  the gates  are kept a t 0.9 s t r o k e , then,  waves r e t u r n and the t r a n s i e n t c o n d i t i o n s u b s i d e s , bilize  from 85%  as the p r e s s u r e  the torque w i l l  a t the i n t e r s e c t i o n o f 0.9 s t r o k e and 100% d e s i g n head.  value, being  sta-  This  about 95%, i s g r e a t e r than the i n i t i a l 64%.  Segment 2 r e p r e s e n t s head and low v e l o c i t y . causes the torque  the a/g r a t i o o f a system h a v i n g  Moving the gates  t o i n c r e a s e to 92%.  a high  t o 0.9 s t r o k e , as b e f o r e ,  T h i s i s o f course  not desired.  F o r such a system, the p r e f e r r e d c o n t r o l sequence i s t o d r i v e the gates closed.  C l o s i n g the gates  to 0.5 s t r o k e reduces the torque  t o 78%.  H y d r a u l i c systems whose a/g segment i s p a r a l l e l to t h e 85% torque  contour  o f f e r no advantage.  For a p a r t i c u l a r system, i t s power contours at  lower gate s t r o k e v a l u e s .  become  steeper  T h i s i m p l i e s t h a t f o r a system o f f e r i n g  some advantage a t r a t e d c o n d i t i o n s the b e n e f i t i s reduced a t lower stroke values.  A t f u l l gate,  the advantage w i l l be g r e a t e s t .  the r e g i o n where i t i s needed most, f o r the u n i t s a r e l o a d e d  This i s to their  maximum power v a l u e s . The in velocity  r a t i o a/g i s the c o n v e r s i o n  to changes i n p r e s s u r e  energy and p r e s s u r e ,  head.  t o p o t e n t i a l energy.  coefficient relating  changes  V e l o c i t y Is r e l a t e d to k i n e t i c The e f f e c t i v e n e s s o f the  scheme thus depends on the r a t i o o f p o t e n t i a l energy to k i n e t i c  energy  o f the water i n the system. In the above d i s c u s s i o n , the gate movements must be made w i t h i n the time r e q u i r e d f o r the p r e s s u r e waves to t r a v e l t o the r e s e r v o i r (or  surge tank) and back t o the t u r b i n e .  65  8.  SUPPLEMENTARY CONSIDERATIONS  The models o f the h y d r a u l i c components d e r i v e d above a r e the i n i t i a l models o n l y .  The n e x t s t e p i s t o v e r i f y by f i e l d t e s t s the r e -  l a t i v e importance of the parameters. u n i t s ; each one s h o u l d be s e p a r a t e l y  They' a r e n o t r e p r e s e n t a t i v e o f a l l investigated.  Before i n s t a l l i n g a c o n t r o l l e r , the f o l l o w i n g  considerations  s h o u l d be g i v e n t o t h e d r a f t tube, the a c t u a t o r and t h e u n i t ' s 8.1  structure.  The D r a f t Tube The dynamics o f the d r a f t tube were n o t c o n s i d e r e d f o r these  are  u s u a l l y much s h o r t e r than those o f p e n s t o c k s and hence l e s s  e n t i a l on t h e t u r b i n e ' s b e h a v i o u r .  influ-  N o n e t h e l e s s , i f f i e l d t e s t s show  t h a t the a p p l i c a t i o n o f r a p i d c o n t r o l s t o t h e w i c k e t gates causes t h e d r a f t tube w a t e r column t o s e p a r a t e , i t may be n e c e s s a r y t o i n c l u d e t h e e f f e c t s of the d r a f t tube. S i n c e d r a f t tubes a r e s h o r t i n comparison t o p e n s t o c k s they may be m o d e l l e d as a r i g i d w a t e r column.  The p r o c e d u r e i s s i m i l a r t o  t h a t used i n m o d e l l i n g the t u n n e l between the s u r g e tank and t h e r e s e r voir.  A l t e r n a t i v e l y they may be t r e a t e d as a p e n s t o c k w i t h one o r two  reaches.  I t w i l l be n e c e s s a r y t o i n c l u d e t h e e f f e c t s o f t h e d r a f t tube  i n i n s t a l l a t i o n s where the d r a f t tube i s l o n g , the v e l o c i t i e s a r e h i g h and t h e t a i l - w a t e r e l e v a t i o n i s low w i t h r e s p e c t t o t h e e l e v a t i o n o f t h e t u r b i n e ' s shroud 8.2  ring.  A c t u a t o r Design To r e a l i z e r a p i d gate a c t u a t i o n , i t may be n e c e s s a r y to d e s i g n  a c t u a t o r s s p e c i f i c a l l y f o r t h i s purpose.  Conventional actuators are  66  designed to be l i n e a r and to control wicket gates i n response to an analog s i g n a l at the transducer  input.  This s i g n a l i s derived from the  unit speed error of the speed governor.  The  governor and actuator have  time lags and t r a v e l l i m i t s to ensure that a f a u l t y s i g n a l w i l l not drive the massive gates e r r a t i c a l l y .  During a transient condition, the  required control w i l l , i t i s surmised, be extremal and therefore to provide a high speed of response a redesign may  be needed to enable  hydraulic f l u i d to be ported d i r e c t l y into the wicket gate servomotors. This arrangement, however, requires safety devices  to disable the con-  t r o l l e r whenever i t applies a faulty s i g n a l to the solenoids c o n t r o l l i n g flow to the ports.  Otherwise an uncontrolled high speed gate closure  w i l l rupture the penstock by causing waterhammer beyond design  limits.  Conventional actuators may" not have the capacity to move the gates at the required speed and therefore s p e c i a l designs may  be  required'before  the concepts proposed can by" f u l l y r e a l i z e d . 8.3  The Unit's S t r u c t u r a l Strength Before applying rapid actuation sequences to the wicket  the strength of the unit's supporting  structure should be investigated  to confirm that the unit can cope with a d d i t i o n a l forces.  For example,  rapid gate motion, by adjusting the j e t d i r e c t i o n , w i l l increase entrance shock losses at the runner i n l e t and this may v i b r a t i o n s into the runner blades.  Rapid control may  also be  the  induce unwanted also  introduce  undesirable side e f f e c t s such as vibrations i n the supporting not anticipated during turbine design.  gates,  structure  Cavitation on runner blades w i l l  increased. The reduction of hydraulic torque by allowing water to pass  by implies that the water leaving the runner has a large angular v e l o c i t y .  67  The d e s i g n o f the c o n c r e t e i n the d r a f t tube s h o u l d be checked t o v e r i f y t h a t i t can w i t h s t a n d , f r e q u e n t exposure t o these p r e s s u r e s . When t h e c o n t r o l r e q u i r e s , an i n c r e a s e i n t o r q u e , e x c e s s i v e  torque  e x t r a c t i o n w i l l cause t h e s t a t i c p r e s s u r e o f t h e w a t e r l e a v i n g t h e r u n n e r to be below a t m o s p h e r i c may o c c u r .  pressure.  As a r e s u l t w a t e r column s e p a r a t i o n  T h i s s h o u l d be a v o i d e d because when t h e surge r e t u r n s , t h e  momentum o f the w a t e r b e h i n d  the c o l l a p s i n g vacuum may r u p t u r e the r u n n e r  head c o v e r and damage t h e s e a l s .  S t r u c t u r a l s t r e n g t h must be ensured  b e f o r e i m p l i m e n t i n g r a p i d gate a c t u a t i n g c o n t r o l systems.  68  9.  CONCLUSIONS The power l i m i t o f h y d r o - e l e c t r i c g e n e r a t i n g u n i t s w i t h F r a n c i s  t u r b i n e s and l o n g p e n s t o c k s c a n be i n c r e a s e d by a p p l y i n g s p e c i a l c o n t r o l sequences t o t h e i r w i c k e t gate a c t u a t o r s .  The i n c r e a s e i s o b t a i n e d by  a l l o w i n g t h e c o n t r o l sequence t o d r i v e t h e gates i n a d i r e c t i o n t h a t w i l l , due t o waterhammer, reduce the h y d r a u l i c torque  coupled  i n t o the t u r b i n e .  The r e d u c t i o n e x i s t s and i s r e q u i r e d f o r o n l y a s h o r t p e r i o d o f time f o l l o w i n g a severe  t r a n s i e n t d i s t u r b a n c e on t h e system.  Waterhammer  can t h e r e f o r e be used t o advantage i n r a i s i n g t h e power l i m i t o f generating units.  69  REFERENCES 1.  S.V. Ahamed and E.A. E r e l y i , "Nonlinear Theory of Salient Pole Machines", IEEE Trans, on Power Apparatus and Systems, V o l . PAS No. 1, January, 1966.  85,  2.  E.A. E r e l y i , S.V. Ahamed and R.E. Hopkins, "Nonlinear Theory of Synchronous Machines On-Load", IEEE Trans, on Power Apparatus and Systems, V o l . PAS 85, No. 7, July 1966.  3.  Venekov, Transient Phenomena i n E l e c t r i c Power Systems,  4.  F i t z g e r a l d and Kingsley, E l e c t r i c Machinery, Chptr. 3, McGraw-Hill, 1952.  5.  W.D.  6.  G.W. Stagg and A.H. El-Abiad, Computer Methods i n Power System Anal y s i s , McGraw-Hill, 1968.  7.  R.M. Shier and A.L. Blythe, " F i e l d Tests of Dynamic S t a b i l i t y Using a S t a b i l i z i n g Signal and Computer Program V e r i f i c a t i o n " , IEEE Trans. on Power Apparatus and Systems, V o l . PAS 87, No. 2, February^ 1968.  8.  Dandeno, Karas, McClymont, Watson, " E f f e c t of High-Speed R e c t i f i e r E x c i t a t i o n Systems on Generator S t a b i l i t y Limits", IEEE Trans, on Power Apparatus and Systems, V o l . PAS 87, No. 1, January 1968.  9.  L.O. Long, ."Governor Aspects of the Coteau Creek Project", for presentation to the Power Technical Group, Vancouver Section of the IEEE, A p r i l 1967.  1964.  Stevenson, Elements of Power Systems Analysis, McGraw-Hill, 1962.  10.  V.L. Streeter, F l u i d Mechanics, McGraw-Hill, 1966.  11.  G.R.  12.  A.W. F u l l e r , "Transistorized E l e c t r i c - H y d r a u l i c Governor f o r Hydraulic Turbnes", Publication PMCC 65-11, Woodward Governor Company, Rockford, Illinois.  13.  E. Mosonyi, Water Power Development, V o l . 1, Hungary 1963.  14.  Selecting Hydraulic Reaction Turbines, Engineering Monograph No. 20, United States Department of the I n t e r i o r , Bureau of Reclamation, Denver, 1966.  15.  J . Parmakian, Waterhammer Analysis, Dover  16.  A.E. A e b e r l i , "Governing of Water Turbines", Water Power, October, 1967.  Rich, Hydraulic Transients, Dover,  1963.  1963.  70  17.  L. Bergeron, Waterhammer i n Hydraulics and Wave Surges i n E l e c t r i c i t y , Wiley, 1961.  18.  L.M. Hovey, "Optimum Adjustment of Governors i n Hydro Generating Stations", Engineering Journal, (Canada), pg. 64, November 1960.  19.  Paynter, "Surges and Waterhammer Problems", Trans, of ASCE, V o l . 118, 1953.  20.  Daughherty, Ingersol, F l u i d Mechanics, 1954.  21.  Streeter, Handbook of F l u i d Dynamics, 1961.  22.  R. Walters, Hydraulic and Electro-hydraulic Servo Systems, London, 1967.  23.  H.M. Morris, Applied Hydraulics i n Engineering, Ronald Press, 1963.  24.  P.A. Wooldrige, A.L. Blythe, "Considerations A f f e c t i n g The Design Philosophy of S o l i d State E x c i t e r s " , IEEE Trans, of Power Apparatus and Systems, V o l . PAS 87, No. 5, May 1968.  25.  Design Notes, Portage Mountain Hydro-electric Power Development, International Power and Engineering Consultants L t d . , Vancouver, Canada, 1965.  26.  Dawson, "Notes on Models of Synchronous Machines", Dept. of E l e c t r i c a l Engineering, University of B r i t i s h Columbia, 1968.  Iliffe,  

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