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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, Uni v e r s i t y 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 E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA May 1973 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p urposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f £L,(TCy/Q/o/QL &AJ<S-/A/67^/A> The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada Date ms)y /7 , /9~72> ABSTRACT The t h e s i s d eals w i t h a procedure f o r developing mathematical models of 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 generating u n i t s . H y d r a u l i c system components modelled are the penstock, r e s e r v o i r , surge tank, F r a n c i s t u r b i n e and the wicket gate a c t u a t o r . A m o d e l l i n g p h i l o -sophy i s suggested. The t h e s i s proposes that f o r a generator 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 breaker r e c l o s i n g , i t s c a p a c i t y to m a i n t a i n synchronism can be enhanced by t a k i n g advantage of waterhammer to sharp-l y reduce the t u r b i n e ' s h y d r a u l i c torque input to the generator d u r i n g the f i r s t few swings of the t i e - l i n e . R e s u l t s of computer s i m u l a t i o n s i n c l u d e d support t h i s c l a i m . i 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 The Turbine By-Pass Valve 7. COMPUTER SIMULATIONS 7.1 Introduction 7.2 The System Model. 7.3 Discussion of Simulation Results 7.4 Comments 8. SUPPLEMENTARY CONSIDERATIONS 8.1 The Draft Tube 8.2 Actuator Design 8.3 The Unit's Structural Strength 9. CONCLUSIONS 10. REFERENCES i i ACKNOWLEDGMENT The author wishes to express his gratitude to Dr. E. Sigurdson, his thesis supervisor. The suggestions, criticisms and help from the members of the committee, Dr. M. S. Davies, Dr. E. Ruus and Dr. Y. N. Yu are also appreciated. i i i 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 unit. The work was motivated by the poss 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 effect of gate motion and water-hammer. The objective of rapid torque control i s to reduce the hydraul-i c torque applied to the unit's rotating assembly during the f i r s t swing of the power system following line breaker reclosing. This w i l l reduce the p o s s i b i l i t y of pole slipping 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 electric power generating unit i s i n synchronism when under the influence of a severe transient i s the electric torque angle. Since actual f i e l d 1 2 p r o f i l e s are n o n - l i n e a r ' and r e q u i r e s e v e r a l F o u r i e r terms to 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 synchronism i s that the a i r gap torque a c t s i n a d i r e c t i o n opposite to torque angle growth. Once the torque changes s i g n , the r o t o r i s s a i d to have s l i p p e d a. p o l e . S l i p p i n g a pole once or s e v e r a l times may not be d e t r i m e n t a l to the u n i t ' s s t r u c t u r e and windings; however continued s l i p p i n g represents l o s s of synchronism and i n current Western p r a c t i c e the u n i t i s then 3 u s u a l l y removed from the 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 clai m s u n i t s i n the U.S.S.R. are at times kept running asynchronously and attempts are made to resynchronize without t r i p p i n g them.) The u n i t may of course l a t e r be resynchronized and reloaded; nonetheless the power system w i l l have operated momentarily at a net power d e f i c i t and customers i n p r o x i m i t y of the u n i t w i l l have experienced v o l t a g e and frequency v a r i a t i o n s u n t i l other u n i t s have been able to i n c r e a s e t h e i r generation. The type of severe t r a n s i e n t s being considered are those due to l i n e c i r c u i t breakers 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 or by other nearby u n i t s i n the 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 of i n t e r n a l f a u l t d e t e c t i o n or a c c i d e n t a l o p e r a t i o n by operators. The t r a n s i e n t s are c h a r a c t e r i z e d by t o t a l or near t o t a l l o s s of e l e c t r i c r e a c t i o n torque i n the generators. In the l i t e r a t u r e , torque angle has been given d e f i n i t i o n s ranging from the angle between the fundamentals of the a i r gap f l u x 4 and the r o t o r magnetomotive forc e to the angle between the bus v o l t a g e phasor and the d i r e c t or quadrature axes.^'^ 3 Adopting the l a t e r , the torque angle speed i s given by dG = n co - co (1-1) d T r s where co i s the speed of the r o t o r to which the 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 the number of p o l e p a i r s and co^ i s the speed of the bus v o l t a g e phasor at the u n i t ' s t e r m i n a l s or at some convenient r e f e r e n c e p o i n t . In equation (1-1) co^ _ i s an absolute v e l o c i t y w i t h respect to an i n e r t i a l r e f e r e n c e . U n i t s of speed f o r co^ and 8 are e l e c t r i c a l radians/second. A c c e l e r a t i o n of the rotor, assembly i s given by — = J . ( h y d r a u l i c torque input - magnetic r e a c t i o n torque - l o s s e s ) (1-2) From equations (1-1) and (1-2) we see th a t a c c e l e r a t i o n of the torque angle toward l o s s of synchronism i s d i r e c t l y i n f l u e n c e d by the hydrau-l i c torque a p p l i e d to the r o t a t i n g assembly by the t u r b i n e . The h y d r a u l i c and magnetic a i r gap torques are the two dom-inant f a c t o r s governing the behavior of the torque angle and t h e r e f o r e the u n i t ' s a b i l i t y to main t a i n synchronism o r , a f t e r s l i p p i n g a few p o l e s , to recapture synchronism. C o n t r o l of these torques i s thus of v i t a l importance i n guidi n g a u n i t through a t r a n s i e n t . In recent times much a t t e n t i o n has been d i r e c t e d toward the c o n t r o l of e l e c t r i c torque which i s achieved by the use of high 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 which f i e l d t e s t s have shown to 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, this thesis directs attention to the hydraulic torque applied to the rotating assembly by the turbine and proposes that rapid control of hydraulic torque may be feasible i n helping a unit maintain synchronism, particularly for plants with long penstocks, by taking advantage of waterhammer in the penstocks. To this end, this study emphasizes the hydraulic components of a hydro-electric unit. The f i r s t step in an investigation 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 limits i t s e l f to the presentation of an approach to be taken when fabricating a (mathematical) model and points out precautions to be taken i n their preparation. •Although some models are given, a complete model set i s not offered, for a model set is valid only for one particular genera-ting unit. To obtain values of coefficients and model simplifications i t i s further necessary to run f i e l d tests on a particular unit to verify the v a l i d i t y of numerous assumptions, model adequacy and simpli-fication. Field tests on existing units werenot part of this study. The work in this study however prepares the ground for f i e l d tests to confirm the models, which would be the next step. The development of this 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 in the s c r o l l case, the dynamics of the entire hydraulic system, particularly the penstock, affect 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 in 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 drive the gate or valve mechanism have an appreciable influence on the over-all 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 their use i n dynamic optimization methods for generating optimal control sequences. 6 2. COMMENTS ON MODELLING Before embarking upon modelling of h y d r a u l i c components, i t i s e s s e n t i a l to e s t a b l i s h f i r s t an approach philosophy to modelling. Model-l i n g can be c o n t r o v e r s i a l f o r there 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 devices having n o n l i n e a r and discontinuous c h a r a c t e r i s -t i c s . The f o l l o w i n g describes a suggested approach to be taken when de-v e l o p i n g models and i t i s used as a guide i n t h i s t h e s i s . The key word i n engineering i s p r e d i c t i o n . Our e f f o r t s are concerned w i t h the p r e d i c t i o n of e f f e c t s i n the 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 of a p h y s i c a l process, however, i t i s necessary f i r s t to o b t a i n a model of the p h y s i c a l process and then run the model at 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 of the model s t a t e i s the p r e d i c t i o n of 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 f u t u r e . P r e d i c t i o n i s the essence of engineering and the model of the p h y s i c a l process provides the means of 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 never be exact. There w i l l always be some f i n i t e e r r o r between the e v o l u t i o n of the model's s t a t e t r a j e c t o r y and that of 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 nonetheless can be reduced by the use of an accurate model of the process. Processes cont a i n an i n f i n i t e number of s t a t e v a r i a b l e s and parameters, and the l a r g e r the number of q u a n t i t i e s i n c o r p o r a t e d i n t o the model, the b e t t e r the p r e d i c t i o n . The s a l i e n t q u e stion i s how to l i m i t the number of e f f e c t s to be p r e d i c t e d by the model, and how to determine xtfhich parameters and s t a t e v a r i a b l e s should be i n c l u d e d . 7 This i s the dilemma of the model developer. At an e a r l y stage i n model development the time range of r e s -ponse must be e s t a b l i s h e d . The dominant i n f l u e n c e s f o r processes at low speeds may be q u i t e d i f f e r e n t from those d e s c r i b i n g the e v o l u t i o n of the s t a t e at h i g h speeds. For example, i f a h y d r a u l i c system c o n s i s t s of a ten m i l e t u n n e l , a surge tank and a 4000 foot penstock and the objec-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 generation, then the hy-d r a u l i c components may be s a t i s f a c t o r i l y modelled as lumped parameter systems. However, i f the o b j e c t i v e i s to 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 torque during a one second system swing, then the. penstock must be model-l e d 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 i n c e r t a i n p l a n t s be neglected. 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 i n c o r p o r a t e d i n t o the model and on the complexity, or s i m p l i c i t y , of 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 ng which p r e d i c t i o n i s sought i s between zero and four seconds. A f t e r t h i s , a u n i t w i l l have e i t h e r maintained or l o s t synchronism. The eventual need to compromise between model s i m p l i c i t y and accuracy has been i m p l i e d e a r l i e r . In so doing, cognizance must be taken of the f a c t that c e r t a i n parameters are more dominant than others. By s e n s i t i v i t y methods a sequence of dominant parameters can be estab-l i s h e d . R e l a t i v e dominance of parameters w i l l s h i f t as s e n s i t i v i t y methods are a p p l i e d along d i f f e r e n t s t a t e t r a j e c t o r i e s . With the rank of parameters i n hand, the c u t - o f f p o i n t must next be e s t a b l i s h e d . The proper procedure cannot be given at t h i s i n i -t i a l stage of model development but one can surmise the f o l l o w i n g argu-ment . 8 In the end, the overall cost function must have the units of dollars. Each additional 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 prediction and hence lead to an improved r e a l i s t i c minimum for the cost function and a truer optimal control. The reduced cost function is the benefit, which must be assigned a dollar value. The model complexity cut-off point i s then determined by the point in 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 in a lower, truer minimum cost value of the cost func-tion. In a sense this i s an embedded optimization problem. The above philosophy of approach to modelling leads to the following observations and considerations. In the over-all scope of a project, model development i s an iterative process. From f i e l d experience with devices one f i r s t begins with a best guess of the characteristics, and writes their interdepen-dences in shorthand form called an i n i t i a l mathematical model. These interdependencies are expressed as sets of differential and algebraic equations. After this, f i e l d tests must be run to verify that the model has captured the dominant characteristics of the device and i s able to predict i t s dynamic behaviour to the satisfaction of the user. As a result 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. The latter is true i f earlier neglected discontinuity effects of dead-time influence the behaviour sufficiently to require their inclusion in the model. Modelling of physical hardware is 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 limited to the i n i t i a l model level. Physical processes involving the transmission of energy through mass are distributed parameter in nature. When beginning model develop-ment for a device, i t is thus expedient to f i r s t view the device as a distributed parameter system. However, provided the effects of energy and mass transfer delays throughout the system's space have negligible effect on the end use of the model, the modelling equations may then be approximated by the simpler lumped parameter equations and represented by ordinary differential equations. Model accuracy of components or subsystems having lesser i n -fluence need not be as exact as those exhibiting the more dominant affects. Thus the dynamics of the hydraulic preamplifier are less important than those of the power servomotor appearing later in the cascade. Similarly, the surge tank need not be modelled too precisely for i t s influence i s less dominant. (Surge tank effects are important, however, for evaluation of pseudo-steady-state conditions for the optimal trajectory.) It is thus evident that in model development each device must be considered separately, f i r s t whether i t s effects warrant i t being included in the model, and secondly, the degree of exactness required of i t s model. The above dealt principally with dynamic models for optimal trajectory determination and, as mentioned, dynamic accuracy may not be essential for certain devices and hence omitted to simplify the dy-namic model. However, a precise static model describing the boundary conditions may s t i l l be required. For example, the f i n a l steady-state pressure head in the s c r o l l case may be required to establish the boun-dary value for that variable. (It is assumed that relieving 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 c o n d i t i o n s s i m p l i f i e s the a l g o r i t h m , as i t would i f a space f l o o d and s e l e c t procedure were used to f i n d the g l o b a l optimal t r a j e c t o r y . ) A modelling philosophy must be e s t a b l i s h e d to serve as a guide before proceding with the d e t a i l s of development. The f o l l o w i n g model development of h y d r a u l i c components of a h y d r o e l e c t r i c generating u n i t w i l l proceed i n the l i g h t of the above d i s c u s s i o n . 11 3. THE PENSTOCK Penstocks are f u l l flowing pressure conduits whose function i s to convey water from a r e s e r v o i r or an intermediate surge tank to a turbine at i t s downstream end. In h y d r o e l e c t r i c plant layout design, penstocks are kept as short as p r a c t i c a l i n order not only to reduce the cost of ma t e r i a l , i n s t a l l a t i o n and excavation for subsurface i n s t a l l a t i o n s but also to enhance the governing a b i l i t y of the turbine governors. A long penstock i s detrimental to unit response, p a r t i c u l a r l y i f the nominal pressure head i s low. Long penstocks e x h i b i t c h a r a c t e r i s t i c s detrimental to f a s t governing. The d i f f i c u l t y a r ises because of the time required f o r pres-sure waves to t r a v e l from ,the turbine gates to the r e s e r v o i r , informing i t that a change i n flow i s required at the turbine end. For a penstock 2000 feet long, the round t r i p delay i s approximately one second. Thus when a torque increase i s required the wicket gates increase t h e i r aper-tures to increase the flow of water. Although the flow of water does increase by a c e r t a i n percentage immediately, the pressure head reduces by a greater percentage so that the net torque i s indeed reduced instead of increased. The reduced torque e x i s t s u n t i l the wave of reduced pres-sure t r a v e l s to the r e s e r v o i r , where the pressure d i f f e r e n t i a l causes more water to flow i n t o the penstock and t h i s flow increase accompanied by the pressure increase of the returning wave i s f e l t at the turbine end. 9 In one extreme case t h i s requires 2.5 seconds . S i m i l a r l y , f o r a required torque decrease, i n reducing t h e i r aperture the gates cause the pressure head to r i s e and thus increase the 12 torque. As the pressure waves t r a v e l to and fro between the r e s e r v o i r and the turbine they eventually damp out and the torque i s reduced to i t s new lower value. In t h i s thesis i t i s proposed that the detrimental e f f e c t of water hammer can be used to advantage i n c o n t r o l l i n g the h y d r a u l i c t o r -que of a unit while i t i s experiencing a sharp t r a n s i e n t . Since the torque before the transient and a f t e r w i l l be i d e n t i c a l (the steady-state power generation set point i s assumed to remain constant during the tra n s i e n t period), the wicket gate c o n t r o l l e r would i n i t i a t e the following c o n t r o l sequences. During the f i r s t p o r t i o n of the system's o s c i l l a t i o n when a reduction i n torque i s required, the gates would be driven open to reduce the torque. During the second p o r t i o n of' the cycle when an i n -crease i n torque i s required, the gates would be returned to t h e i r i n i -t i a l p o s i t i o n and i n the course of t h e i r -motion increase the torque. This c o n t r o l sequence i s the reverse of that generated by conventional turbine speed-error governors. Conventional governors are designed to reduce the gate aperture on sensing a speed increase and to increase the aperture on sensing a speed decrease. A speed increase s i g n i f i e s an inc r e a s i n g torque angle, which must be decreased, not increased, to ensure synchronism. Conventional governor a c t i o n i s there-fore opposite to that required, and f o r t h i s reason present gate controls are designed to remain i n a c t i v e during transient periods. For an optimal c o n t r o l l e r to be capable of discern i n g when to i n i t i a t e reverse action, i t must of necessity possess an accurate model of the dynamics of both the actuator and the penstock. The time delay e f f e c t i s c r i t i c a l . That i s , the d i s t r i b u t e d parameter nature of the penstock becomes appreciable. 13 The modelling approach adopted considers the penstock metal and i t s contained f l u i d to be a d i s t r i b u t e d parameter system. A f t e r t h e i r d e r i v a t i o n , the set of p a r t i a l d i f f e r e n t i a l equations are reduced to a s e t of ordinary d i f f e r e n t i a l equations by the method of c h a r a c t e r -i s t i c s . 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 ordinary d i f f e r e n t i a l equations are converted i n t o d i f f e r e n c e equations, not 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 generated by the e a r l i e r conversion. Each equation i n the set of d i f f e r e n c e equa-t i o n s resembles a lumped parameter model of a reach of a penstock. The penstock can be considered to have been broken up i n t o reaches f o r each of which the lumped parameter model i s adequate. A p p l i c a t i o n of the method of c h a r a c t e r i s t i c s to penstock p a r t i a l d i f f e r e n t i a l equations i s w e l l documented i n reference 10. For convenience the p h y s i c a l model (Figure 1), the derived p a r t i a l d i f f e r -e n t i a l equations, the generated o r d i n a r y d i f f e r e n t i a l equations and t h e i r 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 given. A p p l i c a t i o n of the equation of motion and the equation of c o n t i n u i t y to an elemental f l u i d d i s c as defined i n Figure 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 d i f f e r e n t i a l equations: 9H at + v 9H 3x (3-1) + V 9V 3x H H I i + g — + FV | V I = 0 (3-2) where 'a', the v e l o c i t y of sound i n the water, i s given by a 2 K/p (3-3) 14 PA + d Fjgure 1. Free body diagram of elemental f l u i d d i s c i n a conduit. w = u n i t weight of water p = pressure head A = penstock area z = e l e v a t i o n above datum 15 and where V = v e l o c i t y of water i n f t / s e c , H = pressure head i n feet of water, K = bulk modulus for water (43.2 10^ l b / s q . f t ) , p = density of water, 2 g = a c c e l e r a t i o n of g r a v i t y , 32.2 f t / s e c E = e l a s t i c modulus for the penstock w a l l material (approximately 4.32 10 l b / s q . f t f o r s t e e l ) , F = f/2D = H fg/£V 2 where i s the l i n e f r i c t i o n head as measured by c o n t r o l l e r instrumentation, f = f r i c t i o n f a c t o r i n the Darcy-Weisbach formula, JI = length of the penstock D = pipe diameter, and t 1 = pipe w a l l thickness. According to the method of c h a r a c t e r i s t i c s , i n combining equations (3-1) and (3-2) with an unknown m u l t i p l i e r , any two r e a l and d i s t i n c t values f o r the unknown m u l t i p l i e r generate two ordinary d i f f e r e n t i a l equations i n the two unknowns, H and V, of 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 equations, and these new equations possess a l l the c h a r a c t e r i s t i c s of the former equations (3-1) and (3-2). Combining and as L = L 1 + X L 2 (3-4) and s o l v i n g f o r the unknown m u l t i p l i e r X y i e l d s X = + a/g (3-5) which are two r e a l , d i s t i n c t values giving the four equations below. 16 For A = + a/g, called the positive characteristic curve, we obtain + + * F V | V | = 0 ( 3 - 6 ) dt g dt g 1 ' ~ = V + a ( 3 - 7 ) For A = - a/g, the negative characteristic curve, we have £ - V - a (3-9) 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 is that they represent the direction of wave propagation; downstream propagation along the positive .characteristic and upstream for the negative characteristic. In practice, V i s usually less than 20 ft/sec and the velocity of sound greater than 3000 ft/sec and so equations ( 3 - 7 ) and ( 3 - 9 ) can be simplified to | | = + a . ( 3 - 1 0 ) ( 3 - 1 1 ) respectively. Conversion of the ordinary differential equations to difference form is f a c i l i t a t e d by equations ( 3 - 1 0 ) and ( 3 - 1 1 ) . These equations define the relation between the reach length and the time interval to be Ax = a • At ( 3 - 1 2 ) 17 I f our interconnected e l e c t r i c network contains o t h e r penstocks having d i f f e r e n t wave v e l o c i t i e s , then the number of reaches each pen-stock can have i s predetermined, and hence each penstock cannot be d i v i -ded i n t o equal length reaches. C o n s i d e r a t i o n must t h e r e f o r e be given to the e n t i r e i n t erconnected system and from i n s p e c t i o n of worst cases, 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. Thereafter each penstock's l e n g t h i s a r b i t r a r i l y adjusted to the nearest m u l t i p l e of i t s reach, as defined by = a^ • At, j = j * " ' 1 penstock' V a r i a b l e s H and V i n equations (3-6) and (3-8) are s o l v e d as f o l l o w s . Since the s o l u t i o n to equation (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 equation (3-10), the d i f f e r e n t i a l s dH and dV i n equation (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 equation (3-8) must l i e on i t s c h a r a c t e r i s t i c equation (3-11). The d i f f e r e n t i a l s dH and dV i n equation (3-8) at i n s t a n t s k and k+1 are: dH = H . . . - — H . . _ . i , k + l l + l , k and dV = V. . ,. - - V . . i , k + l l + l , k Figure 2 i l l u s t r a t e s the r e l a t i o n s h i p s as f u n c t i o n s of i and k. The above set 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 t h e i r r e s p e c t i v e equations. The r e s u l t s are: 18 xo x i - l x i xi+l xn DISTANCE FROM RESERVOIR Figure 2. The relationship between i and k for velocity V and pressure H on the x-t plane. 19 H- i a . 1 - H- i i + ~ ( v - i i n - V. . . ) + —FAt V. . . |V. . . I = 0 x,k+l i - l , k g i ,k+l l - l , k g l - l , k 1 i - l , k 1( 3 - 1 3 ) 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 ) Adding equations ( 3 -13 ) and (3 -14) e l i m i n a t e s V, , to give H. = ^ "{H. + H . + -(V. . . - V. . . . ) i , k + l 2 l - l , k i + I , k g l - l , k i + I , k + f F A t ( v i + i , k ! v i + i , k l " V i , k l v i - i , k i ) } ( 3 " 1 5 ) and s u b t r a c t i n g equations ( 3 -13 ) from ( 3 -14 ) e l i m i n a t e s H. , . to give V- , j.i = T W . ., . + V. . . + ^(H. . . - H. ,. . ) i , k + l 2 i + I , k l - l , k a i - l , k i + l , k " F A t ( v i + i , k l V i + i , ; J + v i - i . , k l v i - i , k l > } <3"16> where i = 0 , 1 , 2 , 3 , . . . , n - l , n represents the end of reach s t a t i o n s f o r the n reaches, and represent i n s t a n t s of time, 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 the o p t i m i z a t i o n problem i s being formulated. Equations (3 -15) and ( 3 -16 ) describe i n d i s c r e t i z e d form the dynamic behaviour of the water i n the penstock. Being d i s c r e t i z e d , they may be d i r e c t l y programmed on a d i g i t a l computer. Since each new V and H depends only on i t s adjacent neighbour, equations ( 3 -15 ) and (3 -16 ) cover the e n t i r e length of the penstock except the two ends, which r e q u i r e t e r m i n a l c o n s t r a i n t s f o r s o l u t i o n . Equations ( 3 -15 ) and (3 -16 ) t h e r e f o r e are the mathematical model of the penstock. They account f o r the 20 e l a s t i c i t y of water, the e l a s t i c i t y of the c o n f i n i n g penstock w a l l m a t e r i a l and f r i c t i o n of the f l o w i n g water. 21 4. PENSTOCK TERMINAL CONSTRAINTS Upstream t e r m i n a l c o n s t r a i n t s w i l l next be der i v e d f o r p l a n t c o n f i g u r a t i o n s w i t h a la r g e r e s e r v o i r or a simple surge tank. The downstream t e r m i n a l c o n d i t i o n represented by a h y d r a u l i c t u r b i n e w i l l be considered i n a l a t e r s e c t i o n . 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 of water of s u f f i c i e n t s u r f a c e area to ensure that i t s water surface e l e v a t i o n may be considered 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 namely, H , . = H 1 = H° (4-1) o,k+l o,k where H° i s the e l e v a t i o n of the r e s e r v o i r water s u r f a c e . For s i m p l i -c i t y the v e l o c i t y head i s not i n c l u d e d i n equation (4-1); i t can be considered to be an eq 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 the l i n e f r i c t i o n l o s s . Although t h i s w i l l r e s u l t i n a discrepancy i n the pres-sure p r o f i l e near the r e s e r v o i r end of the penstock, the e f f e c t w i l l not be appreciable at the turbine end. In the model, the penstock f r i c t i o n f a c t o r merely appears l a r g e r . D e r i v a t i o n of the upstream c o n d i t i o n s begins w i t h r e w r i t i n g equation (3-14) as H. ., = - V . ... + C (4-2) i , k + l g i , k + l o where C = H - . - - V . . + - FAt V... , IV,,. . I (4-3) o i + l , k g l + l , k g l + l , k 1 i + l j k 1 Expression (4-3) contains the delayed e f f e c t s which t r a v e l 22 .upstream through the f i r s t reach i n time At. C o n s t r a i n i n g H , - i n (4 -2 ) to remain constant 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 = f (H° - C ) (4 -3 ) o,k+1 a o and H o , k + l " H ° < 4" 4 ) Equations (4 -3 ) and (4 -4 ) provide the m i s s i n g values at s t a t i o n zero l o c a t e d at the r e s e r v o i r end of the penstock. 4 . 2 The Surge Tank In 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 miles from the powerhouse, i t i s necessary to break up the long l e n g t h of conduit i n t o two segments by p r o v i d i n g a free water surface 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. This v e s s e l i s c a l l e d a surge tank. The main types of surge tanks are the simple surge tank, the r e s t r i c t e d o r i f i c e surge tank, and the d i f f e r e n t i a l surge tank. The downstream segment which connects the surge tank to the p l a n t i s made as short 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 to provide a f r e e surface at 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 of a surge tank i s to decouple the two c o n d u i t s , 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 ort penstock and so enhancing the governor's a b i l i t y to r e g u l a t e speed. Surge tank dynamics need not be i n c l u d e d i n the dynamic model. However the high speed model must then be supported by a slower model 23 which 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 the s o l u t i o n t r a j e c t o r y . The slower model p r e d i c t s the new stea d y - s t a t e head at the surge tank, which can be considered by the penstock as a r e s e r v o i r of l i m i t e d s i z e whose surface e l e v a t i o n v a r i e s w i t h flow. I f the decoupling i s adequate the segment upstream of the tank can be considered as a lumped parameter system and modelled by o r d i n a r y d i f f e r e n t i a l equations. The equations are: A t H . = H + -r±- • (V. - V , ) At (4-5) o,k+l o,k A k o,k s V k + i • i < E„,k +i - co> < 4 - 6 ) V l - \ + ^ < H ° - H c , k - F \ l V k l ' - ( 4 - 7 ) Refer to Figure 3 f o r symbol d e f i n i t i o n s . represents the v e l o c i t y at i n s t a n t k of the mass of water i n the t u n n e l which i s considered to be a lumped parameter system. The accuracy of (4-5) and (4-7) can be improved by u s i n g average flows i n place of the values at the end of the previous time i n t e r v a l . In p r a c t i c e , however, the water s u r f a c e l e v e l does not a l t e r s i g n i f i c a n t l y and the rougher approximation may prove to 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 conduits may e x h i b i t reson-ant c h a r a c t e r i s t i c s . For p e r i o d i c motion of the w i c k e t gates at s u i t -able f r e q u e n c i e s , a resonance node may appear near the surge tank rendering i t i n e f f e c t i v e i n decoupling the two conduits. Studies should be made w i t h the tunnel modelled by d i s t r i b u t e d parameter equations to v e r i f y that the lumped parameter model o f equations (4-5), (4-6) and (4-7) i s adequate. The tunnel would be modelled i d e n t i c a l l y to the penstock. The r e s e r v o i r end of the tunnel i n v o l v e s equations (4-3) and (4-4). DATUM F i g u r e 3. Schematic layout of a h y d r a u l i c system showing r e s e r v o i r , t u n n e l , surge tank, penstock 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 J Hn,k+1 Ho,k+l A0 4f" = A* v i - A v i (4-10) s dt t n,k p o,k v H = - V . ,. + C (4-11) o,k+l g o,k+l o ^ ' H w. = " f v , a.-! + c (4-12) n,k+l g n,k+l n where C » H . . + - V . 1 - - F A t V . 1 V 1 1 n n-l,k g n-l,k g n-l,k n-l,k and i s obtained similarly to C . o Applying these four constraints and solving for the unknowns yields H i J . n = H i + T ^ ( A V . - A V . J! (4-13) o,k+l o,k A s t n,k p o,k' v ' H v , i = H , + (A V . - A V , ) (4-14) n,k+l n,k A: v t n,k p o,k' v ' s V o , k + l = \ < Ho,k +l " Co> ( 4 " 1 5 ) Vn,k +1 = " \ < H n,k +l - Cn> <*"16> Equations (4-13) to (4-16) are the difference equations for 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 for the unknowns in difference equation form. 26 The restricted o r i f i c e surge tank requires an additional constraint since i t s surface elevation w i l l not be identical to the pressure head at the junction. The restricting o r i f i c e w i l l exhibit a pressure drop as water i s forced to flow through i t " ^ . Further, the ori f i c e is usually not reciprocal i n that i t offers more resistance to inflow than to outflow. A differential surge tank can be viewed as a combination of a simple surge tank and a restricted 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 is used only on total load rejection, and the non-linearity does therefore not affect our modelling. In the foregoing, modelling equations for several penstock upstream terminal conditions were derived and items deserving special 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 is necessary because the turbine model i s a function of the gate position. 27 5. THE ACTUATOR Wicket gate servomotors c o n t r o l the wicket gates to r e g u l a t e the flow of water through the t u r b i n e . They respond to a weak e l e c t r i c s i g n a l fed i n t o an ele c t r o m e c h a n i c a l transducer. Between the transducer and servomotors one f i n d s a number of h y d r a u l i c devices which amplify and c o n d i t i o n the s i g n a l as i t passes from the transducer to the servo-12 motors . This c o l l e c t i o n of h y d r a u l i c devices w i l l f o r convenience be c a l l e d the actu a t o r . Of a l l the h y d r a u l i c stages i n an ac 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 the f i n a l h y d r a u l i c power stage. The dom-inant c h a r a c t e r i s t i c s of the h y d r a u l i c system are inherent i n i t . The f i n a l stage can be e i t h e r an i n t e g r a t o r or a power a m p l i f i e r . For sm a l l e r u n i t s i t i s of t e n a h y d r a u l i c i n t e g r a t o r . For l a r g e r u n i t s i n which l a r g e r a c t u a t i o n forces are r e q u i r e d to 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 torques, the f i n a l stage 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 to ensure the gates maintain 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 of f i n a l stage actuators i s the power a m p l i f i e r which 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 to i t s a c t u a t i n g s i g n a l . The i n t e -g r a t o r w i l l be considered f i r s t . 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 p i s t o n , 28 x = p o s i t i o n of p i l o t v a l v e and A = a constant. The above model i s v a l i d only i f the d r i v i n g f o r c e of the p i s t o n i s zero or n e g l i g b l e i n r e l a t o n to the pressure of the h y d r a u l i c ' f l u i d source. For a l l but the f i n a l stage, f i e l d measurements may show t h i s model to be adequate. For the f i n a l stage to be representable 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 to be g r o s s l y overdesigned, which economics p r o h i b i t . Indeed, i n d e s i g n i n g an a c t u a t o r , manufacturers estimate the magnitudes of the v a r i o u s types of r e s i s t i v e forces asso-c i a t e d w i t h a gate mechanism. They then design the a c t u a t o r w i t h j u s t enough motive c a p a c i t y to overcome these forces and a c c e l e r a t e the mech-anism mass at a speed that w i l l meet the customer's response s p e c i f i -c a t i o n s . The pressure d i f f e r e n c e across 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 necessary to d e r i v e a model which accounts 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 , consider the r e s i s t i v e forces to be overcome by the a c t u a t o r . The dominant forces are: - mechanism b e a r i n g s t i c t i o n , - mechanism b e a r i n g f r i c t i o n , - unbalanced h y d r a u l i c torque on the guide vanes, and - i n e r t i a of the mechanism mass, which l i e s mostly i n the r o t a t i o n a l i n e r t i a of the wicket gates about t h e i r b e a r i n g 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 determine 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 stiction and f r i c t i o n may be effected by unsatisfactory operation of the lubrication system. Figure A illustrates a typical actuator configuration showing the accumulator tank, the pi l o t valve and the integrator power piston. The accumulator tank i s partly f i l l e d with compressed a i r to provide pressure energy for the f l u i d . The lower portion contains hydraulic f l u i d . Since air dissolves in the o i l which flows into the actuator, the accumulator tank must be replenished with both o i l (from the sump) and a i r . The accumulator tank is a closed vessel 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 20% before the governor pumps start on low o i l level. It may sometimes be necessary for the controller model to take this variation into account in order to predict actuator response. Recall that our models are intended to be valid for a unit experiencing large signal actuation sequences. Pi l o t valve spools can be of varying types, such as overlapped, zero-lapped and underlapped. Wear can also alter the effective spool characteristics. For the sake of generality this 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 substituting the appropriate values defined above. F i g u r e 4. Schematic-diagram of an i n t e g r a t i n g w i c k e t gate a c t u a t i o n system complete 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 , servomotors and gate l i n k a g e mechanism. 31 The equations relating flow through spool ports and the port pressure drops are; q x = cjx+kx] /j.P s - P ; L| sgn (P g - u(x + kX) (5-2) q 2 = C 2|-x + kX| / |P g - p 2| sgn (P g - p 2 ) u(-x + kX) (5-3) q 3 = C3|-x + kX| / | P l - P e | sgn ( P l - P e) u(-x + kX) (5-4 q 4 = C j x + k | / |p2 - P e| sgn (p 2 - P e) u(x + kX) (5-5) where for (5-2) and (5-5) x $ D - kX and for (5-3) and (5-4) -x >, D - kX . The function u( • ) is 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. This is an approximation. If as a result of f i e l d tests the relation i s found to be markedly d i f -ferent, and i f this difference significantly affects our end result, namely, the torque control, then the relation should be modelled by a third 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 travel limits, however, must be re-tained. Pi l o t valve pistons often have travel limits, restricting their range to less than the spool port travel. Travel limits are usually not considered when modelling for small signal applications. However, in large signal applications the pilo t spools do travel their f u l l range and so travel limits must be included in the model. A set of constraints relating the pi l o t valve flows and pressures 32 on both, s i d e s of the p i s t o n s are considered next. They i n c l u d e the c o m p r e s s i b i l i t y of the f l u i d i n the p i s t o n chambers. Otherwise, when both ports are close d simultaneously as they w i l l be f o r the over-lapped spool and may be f o r the zero-lapped s p o o l , the pressures become ind e -terminate and the model f a i l s . This can be v i s u a l i z e d i f one considers the gate to be i n motion when the overlapped spool v a l v e reaches i t s center p o s i t i o n where the ports are a l l c l o s e d . I f c o m p r e s s i b i l i t y i s not accounted f o r , then an impulse f o r c e must act on the p i s t o n s to a r r e s t the momentum of the gate mechanism i n zero time. 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 pressure v a l u e s , although being very l a r g e , w i l l remain f i n i t e 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 bulk modulus d e f i n i t i o n dP - K ^ V w i t h respect to time gives dP K dV dt " dt <5"6> where K = bulk modulus of the o i l , which i s approximately 250,000 p s i , depending on the type of o i l and the amount of a i r d i s s o l v e d i n i t . V = combined volume of the commoned chambers of both c y l i n d e r s and a l l p i p i n g up to the spool v a l v e s , and P = the pressure i n the chambers. From Figure 4 the net i n f l o w i n t o c y l i n d e r chamber 1 i s dV 1 ^ I F = q l ' q 3 + A d t " K l ( P 1 _ P 2 } • ( 5 " 7 ) and i n t o chamber 2 33 dV- , - d T = q 2 - q 4 - A a f + K l ( p l - p 2 ) <5"8> where = p i s t o n leakage c o e f f i c i e n t which i t may be necessary to 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 pre-d i c t e d , s = displacement of the p i s t o n , ds — = v e l o c i t y of the p i s t o n , and A = combined face area of both p i s t o n s exposed to each chamber, S u b s t i t u t i n g equations (5-7) and (5-8) i n t o (5-6) gives d p 1 v ~dt = V-TTS { q i " q 3 + A f " K l ( P 1 " p 2 > } < 5- 9 ) dp? K He; { q 0 " q,. ~ A %+ K ( p P o ) } (5-10) dt V + A(S-s) "H2 H 4 dt 1^1 VT where V' = the sum of the volume i n a c y l i n d e r not s t r o k e d by the p i s t o n s and the volume of 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 spool v a l v e p o r t s , and S = p i s t o n t r a v e l range. The f i n a l elements of the a c t u a t o r to be modelled are the gates and the l i n k a g e s i n the gate mechanism. For convenience, t h e i r motion w i l l be t r a n s l a t e d to t h e i r l i n e a r e q u i v a l e n t referenced to the servomotor p i s t o n s h a f t . The r o t a t i o n a l i n e r t i a of 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 expressed 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 to the servomotor s h a f t . S i m i l a r l y , a l l forces i n c l u d i n g 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 their equivalent forces acting on the servomotor shaft, Applying Newton's Law of Motion to the equivalent lumped mass gives (5-11) "5T ' M _ 1 ( ( P i " *2> A " ( £ s + . £ v + V ( 5 - 1 2 > where eF f -s ";| B l| + e s g n ( 8 1 ) ( 5 ~ 1 3 ) fv= F v S l <5"14> M = equivalent mass on the shaft, s^ = velocity of the piston, and f, represents the effect of the hydraulic torque acting on h the guide vanes as the water flows through them. Equation (5-13) represents stiction effects present in mech-anism bearings when motion is near s t a n d s t i l l . .F represents the peak value of the starting f r i c t i o n . Parameter 'e' defines the gradient with which the stiction force decreases as velocity increases. The combina-tion of the absolute value and sign functions ensures that the stiction expression w i l l be an odd function. Equation (5-14) accounts for viscous f r i c t i o n present at greater velocities. For i n i t i a l model purposes, the viscous f r i c t i o n term is assumed to be linear in velocity. If f i e l d tests prove this assumption to be inadequate, higher power terms must then be appended to obtain satisfactory correlation. Friction 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 torque a c t i n g on the guide vanes to servomotor s t r o k e 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 order polynomial f i t should prove adequate. Values of torque must be obtained from f i e l d measurements. Figure 5 i n d i c a t e s that the torque at zero gate acts i n the opening d i r e c t i o n and at f u l l gate, acts to c l o s e the vanes. This design ensures that upon l o s s of h y d r a u l i c f l u i d pressure the gates w i l l auto-m a t i c a l l y d r i v e to about 20% gate and so l i m i t t u r b i n e power output. The a c t u a t o r model c o n s i s t s of equations (5-9) , (5-10), (5-11) and (5-12) . Equations (5-2)to (5-5) must be s u b s t i t u t e d i n t o equations (5-9) and (5-10) and equations (5-13) and (5-14) i n t o equation (5-12). To i l l u s t r a t e the r e d u c t i o n of the above model to the l i n e a r i n t e g r a t o r i n equation (5-1), assume a l l r e s i s t a n c e forces and the i n e r t i a of the gate mechanism to be zero. In a d d i t i o n , assume the p i l o t v a l v e s p o o l to have p e r f e c t zero l a p . N e g l e c t i n g f l u i d c o m p r e s s i b i l i t y con-s t r a i n t s , we have q l " q4 q 2 = q 3 and p 1 = p 2 . S o l v i n g f o r ds/dt, the servomotor p i s t o n v e l o c i t y , y i e l d s — r ~ i ds _v dt A *\ which i s e q u i v a l e n t to (5-1). P - P S „ 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 torque a c t i n g on each wicket 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 converted i n t o a power a m p l i f i e r by feeding back the servomotor s h a f t p o s i t i o n to the p i l o t v a l v e summing mechanism. In response to an a c t u a t i o n s i g n a l y, spool displacement x becomes v - V £ 1 + l2 where and a r e t n e mechanism arm lengths. The range of 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 necessary to impose s t a t e bounds on these v a r i a b l e s . In summary, a power a c t u a t o r i s a non-linear- device w i t h d i s -continuous f u n c t i o n s . For an economically designed 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 ap 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 forces cannot be neglected. For 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 forces can be neglected, and the l i n e a r i n t e g r a t o r model may be adequate. Models of power actuators w i t h overlapped spools i n t h e i r p i l o t v alves must i n c l u d e the e l a s t i c e f f e c t s of the f l u i d when the i n e r -t i a of the wicket gate mechanism i s a p p r e c i a b l e . Otherwise, i f the sp o o l closes w h i l e the gate mechanism i s i n motion, the impulse r e q u i r e d to i n s t a n t a n e o u s l y decelerate the mechanism to s t a n d s t i l l causes the model to f a i l because the pressures 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 is 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 collectively. There are two classes of turbines, impulse and reaction tur-bines. Examples of these are the Pelton Wheel and the Francis turbine respectively. These classes dif f e r in the manner they use to extract momentum from the pressurized water entering the confines of the turbines. Our attention w i l l be limited to the Francis turbine. Equations sought are those describing the penstock end constraints and those re-lating 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 is not totally converted into velocity head, as i s the case with impulse turbines. Only a portion is converted as water flows through the wicket gates. The runner and the gates therefore are not decoupled and so they behave like a compound o r i f i c e . A brief physical description of water flowing through a tur-bine follows. Water enters the turbine assembly at the entrance to the s c r o l l case, a circular duct of decreasing cross-sectional area which guides the incoming water in a circular path around the runner axis, con-verting the incoming water's linear 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 which f u r t h e r add angular momentum to the water. The guide vanes a l s o d i r e c t the flow i n a d i r e c t i o n that minimizes entry shock to the runner. Between the stay vanes and guide vanes and between the guide vanes and the runner, water flows 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 flows inward. F i n a l l y , having given up i t s angular momentum, i t leaves the confines of the .runner and discharges i n t o the t a i l r a c e through the d r a f t tube. Inf. seeking c o n s t r a i n t equations, 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 . However, the com-pound o r i f i c e was found d i f f i c u l t to model. The main " d i f f i c u l t y was due to f r e e v o r t e x motion t a k i n g p l a c e not only i n the plane of runner r o t a t i o n but al s o i n the v e r t i c a l plane which sweeps around the runner p e r i p h e r y . A p p l i c a t i o n of Flow P o t e n t i a l Theory r e q u i r e s conformal map-pin g i n three dimensions. Furthermore, to compute torque and o r i f i c e back pressure r e q u i r e s the t u r b i n e blades and gate blades to be confor-mallymapped i n t o s i m p l e r geometric shapes. The v o r t e x motion must als o be transformed. The expression would be complex which the end use may not j u s t i f y . Moreover, P o t e n t i a l Flow Theory does not account f o r shock and f r i c t i o n l o s s e s , f o r i n the theory f l u i d s are i n v i s c i d . A r e l a t i o n s h i p between pressure, v e l o c i t y , t u r b i n e speed, and s t r o k e can, nonetheless, be obtained by f i t t i n g a hypersurface to sets of values obtained from f i e l d measurements. A s a t i s f a c t o r i l y f i t t e d s u rface w i l l simulate the 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 that determine a t u r b i n e ' s c h a r a c t e r i s t i c s . Sets of data r e l a t i n g pressure, flow, s t r o k e and speed could not be found and so the c o n s t r a i n i n g equations considered 40 assume constant speed. In computer simulations presented l a t e r , the greatest speed deviation was l e s s than f i v e percent. 6.2 Turbine-Penstock End Constraints The v a r i a b l e s involved i n these end constraint equations are head, v e l o c i t y and stroke. An equation r e l a t i n g these v a r i a b l e s f o r the turbine i s 2 V . . = a + a. s. n + a_H , ., + a_ s H + a. s. (6-1) n,k+l o 1 k+1 2 n,k+l 3 k+1 n,k+l 4 k+1 Equation (6-1) i s a polynomial i n head and stroke and gives a reasonable f i t . The values of the c o e f f i c i e n t s depend on the un i t ' s design head, design flow and the c h a r a c t e r i s t i c s of the turbine's compound o r i f i c e . Values w i l l be given i n the following chapter. An equation r e l a t i n g head and flow f o r the end of the penstock i s obtained by r e w r i t i n g equation (3-13) with i = n, V . - £ (C - H , . . ) • . <6-2) n,k+l a n n,k+l where C = H . + — (V - . - FAt V , , | V , ,|) (6-3) n n-l,k g n-l,k n-l,k 1 n-l,k' Solving f o r head and v e l i c i t y i n equations (6-1) and (6-2) y i e l d s C - (a + a.s, + a.s2,..) H • .. = a n o 1 k+1 4 k+1 ,* n,k+l (6-4) — + a_ + a s . -g 2 3 k+1 Equations (6-2) and (6-4) give the values f o r pressure and v e l o c i t y f o r the penstock-turbine end condition. In the computer program, equation (6-4) must be computed f i r s t , and i t s value s u b s t i t u t e d i n t o (6-2). 41 6.3 Turbine Torque An equation expressing torque as a function of head and stroke can be obtained from a polynomial f i t to data r e l a t i n g these v a r i a b l e s . Data containing runner speed was not found. Runner speed w i l l therefore be assumed to be constant and equal to the unit's syn-chronous speed. An equation for torque i s 2 Tk+1 = b o + b l Sk+1 + b 2 Hn,k+1 + b 3 Sk+1 Hn,k+1 + b4 Sk+1 + b 5 \ + l Hn,k +1 ( 6 " 5 ) Again, the values of the c o e f f i c i e n t s b Q to b,. depend on the design head, design flow and the c h a r a c t e r i s t i c s of the compound o r i f i c e . Equation "(6-5) gives the h y d r a u l i c torque coupled from the water to the runner as a function of head and stroke at instants of time k. Together, t h i s h y d raulic torque and the e l e c t r i c torque acting on the rotor determine the unit's speed deviation during transient conditions. In making the f i t , the region near the rated power contour on a head against discharge p l o t must be f i t more p r e c i s e l y than that at lower power zones. Units are usually designed to accept 15% overload. This together with the fa c t that a i r gap torque i s greatest f o r the overload condition requires the f i t to be most precise along the 15% overload power contour, and decreasing toward lower powers. Below the 40% power contour, an e r r o r i n f i t of s e v e r a l percent may be acceptable. The region beyond 15% load i s also of l e s s e r concern. The p r e c i s i o n of f i t should follow the power contours, being most precise near the 42 overload r a t i n g . 6.4 The Turbine By-Pass Valve Our main concern i s the c o n t r o l of the h y d r a u l i c torque coupled i n t o the t u r b i n e runner. 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 by-pass valve may enable a u n i t to maintain synchronism f o r any l o a d . The e f -f e c t i v e area of a s m a l l valve would be approximately between one and two percent of the penstock's c r o s s - s e c t i o n a l area. 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-stock diameters upstream of the s c r o l l case entrance and i t by-passes water 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 water f l o w i n g down the penstock to be d i v e r t e d around the t u r b i n e without i t s energy being coupled i n t o the runner. In 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 place by penstock pressure a p p l i e d to a p i s t o n i n a double-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 pressure would be switched o f f and the c y l i n d e r vented to atmosphere, a l l o w i n g the penstock pressure a c t i n g on the needle valve's p o r t area 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 penstock pressure 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 , the needle v a l v e would close g r a d u a l l y i n order not to induce unwanted p o s i t i v e pressures. The equation f o r torque i s i d e n t i c a l to equation (6-5), because the head i s common to both the t u r b i n e and the v a l v e . The equation r e l a t i n g head and flow f o r the penstock-turbine i n t e r f a c e i s s i m i l a r to equation (6-4); the flow through the v a l v e must be added to equation (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 at the t u r b i n e ' s i n l e t . 43 The head and flow through the val v e are r e l a t e d by Q = ( C d A) y ^ i ? (6-6) The corresponding penstock v e l o c i t y due to t h i s flow i s n A V = — i = —£ /2eH . P A p A p ^ (6-7) A = v^gH • G P where A = C, A, the e f f e c t i v e v a l v e area, and A = e f f e c t i v e f u l l g d ' g o va l v e area G = val v e s t r o k e , 0 £ G <: 1. Thus A = A G g go Equation (6-7) i s not 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 to equation (6-1), a s u i t a b l e l i n e a r i z e d e q u i v a l e n t i s sought. Since the head during 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 of val v e o p e r a t i o n by approximately 20%, a reasonable l i n e a r i z a t i o n i n head can be obtained i n a s m a l l region below the u n i t ' s design head v a l u e . The l i n e a r i z a t i o n i s A G VP = 2 f J fo <H° + H> <6"8> where H ° = design head. Combining equations (6-1) and (6-8) y i e l d s 2 Vn,k+1 = a o + al Sk+1 + a2 Hn,k+1 + a 3 Sk+1 Hn,k+1 + a4 Sk+1 + a 5 (H° + H n,k +i> * G k + r ( 6" 9 ) where A a 5 2A P ZN| • Is The penstock end c o n s t r a i n t i s equation (6-2). S o l v i n g f o r head i n equations (6-9) and (6-2) y i e l d s 44 C - [a + a.s. + a.s.2,, + aqH°G, ] a n o 1 k+1 4 k+1 5 k+1 , n N H = (6-10) ! a 2 + VW1 + " 5 G k + l + ? where i s given by equation (6-3). Equations (6-10) and (6-2) give 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 penstock, the t u r b i n e and the by-pass v a l v e . Remarks Before applying surface 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 data curves must be confirmed; i t may.present the head d i f -ference 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 or that 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 at the entrance to the s c r o l l 19 case and at 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 of a polynomial f i t t e d to f i e l d data i s that 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 equation. In a n a l y t i c d e r i -v a t i o n s i t r e q u i r e s a conscious e f f o r t to ensure the dominant e f f e c t s have been i n c l u d e d . 45 7. COMPUTER SIMULATIONS 7..1 I n t r o d u c t i o n I t has been proposed that the a b i l i t y of a u n i t to maintain synchronism f o l l o w i n g a l i n e c l e a r i n g o p e r a t i o n could be enhanced by moving the wicket gates i n a reverse d i r e c t i o n to that i n c u r r e n t p r a c -t i c e f o r speed r e g u l a t i o n , and thus take advantage of waterhammer to reduce the h y d r a u l i c torque input to the machine u n t i l the f i r s t swing peak has passed. This chapter presents the r e s u l t s of computer s i m u l a t i o n s which, f o r the models used, suggest the t h e s i s p r o p o s i t i o n to be v a l i d . The c r i t e r i o n f o r improvement i s the i n c r e a s e i n p r e f a u l t s t e a d y - s t a t e a c t i v e power which a u n i t and t r a n s m i s s i o n l i n e can d e l i v e r to the i n f i n i t e bus and yet r e t a i n synchronism f o l l o w i n g l i n e c l e a r i n g . Loss of synchronism i s d i f f i c u l t to d e f i n e . In p r a c t i c e a r o t o r may s l i p poles s e v e r a l times before the u n i t i s t r i p p e d by p ro-t e c t i v e r e l a y s . For t h i s study l o s s of synchronism i s defined to occur wherever the torque angle (between the r o t o r and the i n f i n i t e bus) ex-ceeds 180 e l e c t r i c a l degrees w i t h i n f i v e seconds of breaker r e c l o s u r e . This allows f o r approximately f i v e system swings. 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 p l a y s an important r o l e i n determining the a b i l i t y of a u n i t to m a i n t a i n syn-chronism. Although the actuator reduces the h y d r a u l i c input torque through the f i r s t swing peak, i t i s the duty of 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 that i t can keep the u n i t i n synchronism 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 par-t i c u l a r l y a f f e c t the a b i l i t y of a u n i t to maintain synchronism. 46 It i s assumed that i f syncrhonism is maintained for five swings, optimal exciter control would be capable of returning the system to i t s original, prefault steady-state condition. The electric power transmission system shown in Figure 6 con-sists of a generator-transformer connecting the unit to a switchyard bus, and a pair of parallel transmission lines connecting the switchyard bus to an i n f i n i t e capacity power grid. The disturbance is assumed to be a balanced 3-phase line fault applied to one ofthe parallel lines. Line clearing is effected by 3-pole breakers, not single-pole breakers. This assumption obviates the need to model negative and zero sequence characteristics 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 identical 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 pas-sing 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 excitation. In practice, over-all power network generation scheduling would dictate the various generation requirements. For our study however, since our system is radially connected to the grid we shall i n s i s t that each radial 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 grid. The condition for each test therefore is 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 is determined by the gate position and by the i n f i n i t e bus voltage, defined to be 1 p.u. 47 0.13 p.u. x = 1.20 p.,u. x = 1,20 p..u. o- I n f i n i t e Bus Synchronous Generator Generator Transformer ^ S w i t chyard Transmission ^-Line Bus L i n e s Breakers F i g u r e 6. C o n f i g u r a t i o n of e l e c t r i c system simulated. 48 7.2 The System Model The h y d r o - e l e c t r i c power generating system selected for the simulation studies i s the si n g l e unit equivalent of the generators at the W.A.C. Bennett Dam, Hudson Hope, Canada. The t e c h n i c a l data was obtained from the design notes and the contract documents prepared f o r the p r o j e c t . Several quantities were a l t e r e d to emphasize t h e i r e f -f e c t s . Models with values 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 Hydraulic System The h y d r a u l i c system consists of a 2000 foot penstock between the r e s e r v o i r and the turbine. Although the a c t u a l length i s only about 1200 feet, the longer length was taken i n order to emphasize the e f f e c t of length on the system's behaviour. In the study, a 1000 foot length was also simulated. In s p i t e of the waterhammer wave t r a v e l time being halved, the o v e r a l l e f f e c t on the unit's power l i m i t was l e s s than one percent. F r i c t i o n l o s s i n penstocks generally l i e s below f i v e percent of design head. Inorder to emphasize the e f f e c t s of f r i c t i o n , a loss of ten percent i s assumed. The system parameters are: Design head, 500 feet , Design flow, 5850 cu. f t / s e c , Design power (turbine), 310,000 brake horsepower, Design e f f i c i e n c y (turbine), 94.0 % Design speed, 150 RPM, Rated head, 455 f e e t , Rated flow, 6750 cu. f t / s e c , 49 Penstock diameter, 18 feet, Friction loss at design flow, 50 feet W.C. System conditions are: Reservoir elevation 550 feet above tailrace elevation, Servomotor stroke at 70%. The equations for the penstock are: H i , k + 1 " 1 { H i - l , k + Hi+l,k + ( V i - l , k - V i + l , k ) / C ( 7 - X ) + F A t ( v i + i , k - v i - i , k ) / c } i,k+l 2 x+l,k l - l , k l - l , k i+I,k • - F A t < v i + i , k + vi-i,k» where i = l , 2 , . . . , 9 C = g/a At = time increment, a = 4000 f t . / s e c , the velocity of sound in the penstock, F = 0.0402/V2 n,o The absolute value function of velocity i s not used in each simulation since the flow i s always positive. At the penstock's reservoir end, the equations are: Ho,k+l - 5 5 0 ( 7 ' 3 )  Vo,k+l = C ( Ho,k " Hl,k> + V l , k " F A t V i , k ( 7 " 4 ) At the penstock's turbine end, the equations are: Hn,k+1 - <C H n_ 1 ) k + ( V ^ - F A t V 2 _ 1 > k ) - ( a ^ s ^ + a ^ ) { I + a2 + a3 \ + l } (7-5) 50 v t x i = C < H 1 1 " H i n ) + v 1 1 - FAt V 2 n . (7-6) n,k+l n - l , k n,k+l n - l , k n - l , k For a t u r b i n e w i t h a by-pass v a l v e , the equations f o r the t u r b i n e end of the penstock are: H ... = ( C H + V - FAt V 2 n,k+l n~l>k n - l , k n - l , k " ( a o + a l V l + a4 Sk+1 + a5 H ° G k + 1 ) } / -{ a 2 + a 3 S k - f l + a 5 Gk+1 + C } ( 7 ~ 7 ) Vn,k+1 = C ^ n - l . k " \ > k + l ) + V l . k " F A t V n - l , k ( 7 " 8 ) Sk+1 = S k + T~ U l , ( k + 1 ) ( 7 ' 9 ) a Values f o r the c o e f f i c i e n t s are: n = 10> the number of reaches i n the penstock, a = -12.952 o a = +44.605 a 2 = .01375 a 3 = .02365 a / = -22.935 4 a = .000704 A 5 8° A = by-pass valve e f f e c t i v e area, go T =2.5 seconds, servomotor f u l l s t r o k e t r a v e l p e r i o d a 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., the design head. 7.2.2 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 of a generator, a generator transformer and a p a r a l l e l set of t r a n s m i s s i o n l i n e s . The system para-meters expressed i n the per u n i t system are: D i r e c t - a x i s synchronous reactance, x^ = 0.90 p.u. Quadrature-axis synchronous reactance x^ = 0.55 p.u. 51 D i r e c t - a x i s transient reactance x' = 0.18 p.u. d D i r e c t - a x i s open-circuit time constant, = 7.76 second 9 6 9 Machine i n e r t i a , WR = 220 10 l b . f t Transmission l i n e reactance (each l i n e ) , x = 1.20 p.u. Generator transformer reactance x = .13 p.u. g 26 The equations f o r the t h i r d order synchronous machine are: d$ -X (X -X') "dT " * F ( t ) + -^7~ C ° S 6 ( t ) + U 2 ( t ) ( 7 ~ 1 0 ) do d d f - <o(t) - % ( 7 - 1 1 ) f t - " - c h " Telec> <7"12> where ^ (t) X' - X T e l e c = + -YTY1 -cs 6(t)} s i n e ( t ) " (7-13) do d d q and T ^ i s given by equation (6-5). In equation (7-10) and (7-13), the modified reactances are: 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 The e l e c t r i c a l v a r i a b l e s are: iji = d i r e c t axis f l u x linkages, r 6 = torque angle between d-axis and i n f i n i t e bus, The c o e f f i c i e n t s i n equation (6-5) are: 52 t>2 = -.000061 b 3 = +.007530 b. = +.153 4 b c = -.00368 5 ^ ( t ) , the generator 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 given by: u„ = 0.3 (OJ-U) ) + f o r -2 < u„ < .5 2 o Fo 2 = 10 f o r i n equation (7-14) .5, and = -2 f o r i n equation (7-14) $ -2. where C J q = 120TT radians/second, the e l e c t r i c a l speed, and Vp Q = 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 obtained from the i n i t i a l c o n d i t i o n s . The above e x c i t a t i o n s i g n a l generating f u n c t i o n was obtained by t r i a l and e r r o r . I t was found to be a simple yet e f f e c t i v e f u n c t i o n i n the s i m u l a t i o n s . For the synchronous generator, the machine parameters are as-sumed to be constant. E f f e c t s of magnetic s a t u r a t i o n are n eglected. S i n c e , w h i l e on the verge of a u n i t l o s i n g synchronism, f l u x e s and currents assume l a r g e v a l u e s , f i e l d t e s t s may prove neglect of s a t u r a t i o n 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 . Appropriate delays are i n s e r t e d i n t o the e x c i t e r and the ac-t u a t o r f o r improved r e a l - l i f e s i m u l a t i o n . For both, a 50 ms. delay i s 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 c l e a r i n g o p e r a t i o n i s r e q u i r e d . For the e x c i t e r , an a d d i t i o n a l 50 ms. i s allowed f o r the e x c i t e r to 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 (the e x c i t e r i s 2 4 assumed to be of the s t a t i c SCR c o n t r o l l e d t y p e ) . For the a c t u a t o r , an a d d i t i o n a l 150 ms. i s allowed f o r i t s s o l e n o i d s to react and the massive gates to begin motion. Thus delays of 100 and 200 ms. are 53 applied to the e x c i t e r and actuator r e s p e c t i v e l y . 7.3 Discussion of Simulation Results In the simulations, the hydraulic system was simulated alone f i r s t . The objective was to show that gate motion generates pressure waves which t r a v e l back and forth between the turbine and the r e s e r v o i r . A f t e r t h i s , the composite h y d r a u l i c - e l e c t r i c system was simulated. The objective was to determine the improvement i n the power l i m i t of the unit obtained by using waterhammer to advantage. For each simulation the disturbance to the system i s assumed to be a transmission l i n e f a u l t . One of the two l i n e s i s assumed to ex-perience a balanced three-phase f a u l t to ground. The l i n e breakers at both ends of the f a u l t e d l i n e t r i p simultaneously and, a f t e r 500 m i l l i s e c o n d s , reclose. The f a u l t i s assumed to have been cleared i n the i n t e r v a l . As a r e s u l t of the above disturbance, the system begins to swing. The power l i m i t i s determined by the maximum p r e f a u l t power generated by the u n i t f o r which a p a r t i c u l a r c o n t r o l sequence can keep the system from l o s i n g synchronism. The incremental distance and time f o r the penstock was 200 f e e t and 0.05 seconds r e s p e c t i v e l y . For the generator equations, the time increment was 0.01 seconds. The simulation run was 5.0 seconds. In the f i r s t set of composite system simulations a number of control sequences are applied to the wicket gates. In the second s e t , the h y d r a u l i c system i s modified to include a by-pass valve around the turbine. In t h i s set, only the by-pass valve i s c o n t r o l l e d ; the wicket gates are kept at r e s t . In the f i n a l set, 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 -stated. The c o n t r o l s i g n a l applied to the actuator i s obtained from a speed governor. The objective was to determine the power l i m i t obtained from a conventional governor and to compare the r e s u l t s to those obtained by 54 t a k i n g advantage of waterhammer. 7.3.1 The h y d r a u l i c system The r e s u l t s of the h y d r a u l i c system s i m u l a t i o n are given i n Figure 7. I t shows the pressure and torque response, to 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 both pressure and torque im-mediately 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 to 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 of a 0.2 second gate opening 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 to 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 the penstock i s one second, and • a c c o r d i n g l y the disturbance r e f l e c t i o n s are seen at the t u r b i n e end of the penstock at 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 pressure 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 d u p l i c a t e d 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 s m a l l d i s -turbances 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 f r i c -t i o n i n the penstock are not s i g n i f i c a n t . Trace 'b' i s the response to a long a c t u a t i o n sequence c o n s i s -t i n g of a 1.2 second gate opening d r i v e , a 0.4 second h o l d and a 1.2 second c l o s i n g d r i v e . In t h i s sequence, the a c t u a t o r s t r o k e reaches 0.94 of f u l l gate t r a v e l ( i n i t i a l gate i s 7/10). Again the presence of t r a v e l l i n g pressure waves can be seen. Although the gates are b e i n g d r i v e n open f o r a f u l l 1.2 seconds, the pressure and torque ab r u p t l y become constant at 1.0 second due to the 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 r e t u r n i n g from the r e s e r v o i r . In the f o l l o w i n g 1.2 to 1.6 seconds, during which the a c t u a t o r s t r o k e i s h e l d constant, the pressure and torque begin to r i s e as the r e f l e c t e d waves due to gate a c t u a t i o n one second ago continue r e t u r n i n g from the r e s e r v o i r . At 1.6 seconds, the 55 1.20 HYDRAULIC TORQUE 1.10 IN P.U. 1.00 0.90 0.80 PRESSURE HEAD IN FEET W.C. AT SCROLL CASE ENTRANCE ACTUATOR STROKE 3 4 5 Actuator at f u l l stroke ACTUATOR CONTROL SEQUENCE +1 stroke increasing 2 3 TIME IN SECONDS Figure 7. Ef f e c t s of water-hammer on pressure head and hy d r a u l i c torque due to gate stroking. 56 gates begin c l o s i n g and hence the slope of both pressure and torque r i s e more s t e e p l y due to the added l o c a l e f f e c t of the c l o s i n g gates. The r i s i n g pressure continues to 2.6 seconds, when f i n a l l y the gates have been returned to 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 subside 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 . For a c t u a t i o n sequences extending beyond the penstock's n a t u r a l wave t r a v e l time, the d i s t r i b u t e d parameter nature becomes l e s s obvious. Trace 'a' o f f e r s a s m a l l torque decrease i n i t i a l l y and has p o s i t i v e torque pulses t h e r e a f t e r at one second i n t e r v a l s . Trace 'b' o f f e r s a very l a r g e i n i t i a l r e d u c t i o n which i s f o l l o w e d by a l o n g , de-caying p o s i t i v e torque p e r i o d . In our a p p l i c a t i o n to generator 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 pressure waves w i l l develop a l a r g e i n i t i a l torque r e d u c t i o n . Succeeding peaks should not be excessive and, i f pos-s i b l e , be i n synchronism w i t h the a c c e l e r a t i n g torque requirements of the u n i t ' s r o t a t i n g assembly as the e l e c t r i c system o s c i l l a t e s a f t e r l i n e r e c l o s u r e . I t i s thus evident that as a r e s u l t of gate motion, pressure waves are generated and that they t r a v e l to and f r o i n the penstock, g r a d u a l l y 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 Figure 8 shows the r e s u l t s of a s e r i e s of a c t u a t o r c o n t r o l sequences a p p l i e d to 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 coding i s given to f a c i l i t a t e i n t e r p r e t a t i o n of the con-t r o l f u n c t i o n p l o t . Each c o n t r o l sequence has an i n i t i a l 0.2 second a c t u a t i o n delay, as e x p l a i n e d e a r l i e r . Time TI s i g n i f i e s the end of the gate opening d r i v e and time T2 represents the l e n g t h of the h o l d p e r i o d . 57 - F u l l gate power at 1.22 p.u. PREFAULT STEADY-STATE POWER IN P.U. 1.22 1.20 1.18 1.16 1.14 1.12 1.10 1.08 1.06 1.04 1.02 1.00 J L T2= 0 1 .2 .3 .4 .5 .6 .7 .8 . 9 1 . 0 VARIABLE T l IN SECONDS ACTUATOR CONTROL SEQUENCE DEFINITION +1 0 -1 r T l = end of p o s i t i v e drive T2 = pause period J L 0 .2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0 TIME IN SECONDS Figure 8. P l o t of steady-state power (cost function ) values as a function of several actuator c o n t r o l sequences. 58 The gate c l o s i n g p e r i o d equals the opening p e r i o d . This r e t u r n s the gate to i t s o r i g i n a l p o s i t i o n as r e q u i r e d i n order to 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 defined by T l = .5 and T2 = .2 i s the most fav o u r a b l e . I t enables the system to maintain synchronism w i t h an i n i t i a l p r e f a u l t power throughput of 1.166 p.u. Compared to the dead ac t u a t o r value ( T l = .2, T2 = 0) of 1.112 p.u., which 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 alone, t h i s favourable c o n t r o l sequence has i n c r e a s e d the power l i m i t by .054 p.u. or approximately 5%. Actuators having 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 i n c r e a s e t h i s l i m i t maximum f u r t h e r . The above r e s u l t s are v a l i d f o r an a c t u a t o r having the c a p a c i t y to d r i v e the gates at the r a t e of 40% f u l l s t r o k e per second. Engineering and manufacturing cost and t e c h n i c a l f a c t o r s 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 . One foreseeable t e c h n i c a l f a c -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 spools and ports begin to wear and backlash 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 prove 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 v f a c t o r c o n t r o l and f a s t 20 ms. response to any one of 256 values.) The importance 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 requirements was mentioned e a r l i e r . The decrease i n power l i m i t f o r values of T l between .4 and .7 seconds i s due to the pressure peaks being out of phase w i t h that r e q u i r e d . Indeed, the out of phase pressures can reduce the power l i m i t to values below that f o r a dead a c t u a t o r (e.g. the f u n c t i o n having T l = .6, T2 = .2). A gradual increase i n the power l i m i t s appears between T l = .7 to .8 as the pressure wave f r o n t again becomes i n phase w i t h the second swing of the e l e c t r i c system. This i n c r e a s e however i s l i m i t e d by the 59 gates being d r i v e n to f u l l gate during these long a c t u a t i o n sequences. Furthermore, as the p r e f a u l t l i m i t i n c r e a s e s , the corresponding i n i t i a l 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 gate, l e a v i n g l e s s room f o r the act u a t o r to move and take advantage of waterhammer. The above computer r e s u l t s show th a t reverse a c t u a t i o n of the wic k e t gates can indeed i n c r e a s e a u n i t ' s a b i l i t y to keep synchronism during a f a u l t .. At the greater power l e v e l s , as the i n i t i a l gate p o s i t i o n approaches f u l l gate, the act u a t o r ' s manoeuvering space decreases and the e f f e c t i v e n e s s of the reverse a c t u a t i o n scheme di m i n i s h e s . I t w i l l t h e r e -f o r e not be p o s s i b l e f o r the power l i m i t to reach the f u l l gate power value. (For our study, which i s v a l i d f o r a s c r o l l case pressure head of 500 f e e t W.C., t h i s f u l l gate l i m i t i s 1.22 p.u. This 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 va l v e Figure 9 shows the power l i m i t s o b t a i n a b l e using by-pass valves of s e v e r a l s i z e s . The val v e i s assumed to f a l l open i n 0.2 seconds and c l o s e i n 4.0 seconds. A val v e having an e f f e c t i v e area of about three square fee t enables a u n i t to r a i s e i t s power l i m i t to n e a r l y 20% overload. This i m p l i e s that f o r u n i t s whose ov e r l o a d i s l e s s than 20%, the by-pass va l v e w i l l always be able to keep the oanit i n synchronism. When l o s s of synchronism occurred, i n thie s i m u l a t i o n s , i t took p l a c e on the peak of the t h i r d o r f o u r t h swing, and not on the f i r s t or second swing as was the case w i t h reverse, gate a c t u a t i o n . This could be avoided by g i v i n g the e x c i t e r a l a r g e r f i e l d f o r c i n g c a p a b i l i t y or g i v i n g the val v e a longer c l o s i n g p e r i o d . F u l l gate power at 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 of by-pass valves of several s i z e s on a unit's steady-state power throughput. y—Full gate power at 1.22 p.u. PREFAULT 1.2-STEADY-STATE POWER IN P.U. . 1.1 -1 ! 1 1 1 ' ° 1 1 1 I 1 -.75 -.60 -.45 -.30 -.15 0 .15 .30 .45 . .60 .75 GOVERNOR GAIN IN SECONDS PER RADIAN Figure 10. A f f e c t of conventional governor gain on power throughput. 61 7.3.4 The speed governor Figure 10 shows the power l i m i t obtainable by the system using a conventional, negative speed-feedback governor. I t i s known from l i n e a r , small s i g n a l analysis and from f i e l d experience that the feedback s i g n a l must be negative to r e a l i z e stable speed, re g u l a t i o n and to enable load sharing between units. Figure 10 gives a p l o t of the power l i m i t s of a governor with s e v e r a l values of both p o s i t i v e and negative feedback. These are applied f o r only f i v e seconds during the t r a n s i e n t . The r e s u l t s show that for small gain values the power l i m i t indeed increases with i n c r e a s i n g negative speed feedback. The l e v e l i n g of the slope for gains greater than .30 i n magnitude are due to the actuator's c o n t r o l bounds. As the gain exceeds .30, the control becomes almost bang-bang i n nature and so the e f f e c t of l a r g e r values i s l e s s pronounced. The maximum power l i m i t of 1.127 p.u. f o r gains i e s s than -.30 i s much l e s s than 1.166 obtained by reverse actuation. These r e -s u l t s i n d i c a t e that, f o r a unit t r y i n g to maintain synchronism a f t e r f a u l t c l e a r i n g and l i n e reclosure, actuation sequences described i n Figure 8 can allow a greater power l i m i t than that given by the conven-t i o n a l speed-feedback governor. 7.4 Comments The foregoing r e s u l t s apply to an actuator having the capa-c i t y s p e c i f i e d e a r l i e r . Larger c a p a c i t i e s w i l l enable the power l i m i t to be r a i s e d beyond the s i g n i f i c a n t 5% increase. The e f f e c t i v e n e s s of reverse gate actuation depends on the length of time the transmission l i n e breakers remain byen before r e c l o s i n g . For shorter breaker open periods, the units inherent power l i m i t w i l l 62 5 6 be greater than for longer periods-' . However, during these shorter periods the effectiveness of the high speed actuator w i l l be reduced. It has l e s s time to move the gates (at t h e i r maximum speed) to reduce the h y d r a u l i c torque, using waterhammer. The e f f e c t i v e n e s s of the scheme proposed also depends on the geometric layout of the hyd r a u l i c system. The effectiveness i s more pronounced on hydraulic systems having long penstocks and low pressure heads. On systems having short penstocks and high heads the e f f e c t of reverse actuation may be n e g l i g i b l e or even be undesirable. The above i s due to two key f a c t o r s . The more obvious f a c t o r i s time. In longer penstocks, the longer time required f o r the pressure wave to t r a v e l to the r e s e r v o i r and back provides more time f o r gate motion to take e f f e c t . The second f a c t o r i s the r a t i o of the change i n pressure to the change i n v e l o c i t y , the change being i n i t i a t e d by gate movement. This r a t i o i s a/g, where 'a' i s the speed of the pressure waves i n the penstock and 'g', the constant 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 . To i l l u s t r a t e the above, consider the graph i n Figure 11. The ordinates have been normalized and read i n percentage 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 represent the a/g r a t i o f o r hy d r a u l i c systems having d i f f e r e n t r a t i o s of head to v e l o c i t y . Our reference point f o r th i s discussion i s given by the coordinates of 100% design head and 80% discharge. The torque developed at t h i s point i s 85% of f u l l gate. The curve represents the 85% torque contour. Segment 1 represents the a/g r a t i o of a hy d r a u l i c system having a low head and a high v e l o c i t y . As can be seen, moving the gates from 0.7 to 0.9 of f u l l stroke w i t h i n the time i t takes the pressure 63 130 40 60 80 PERCENT DISCHARGE 100 . 120 140 F i g u r e 11. The e f f e c t of r a p i d gate motion 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 from 85% to 64%, a 21% reduction. Shortly a f t e r t h i s the gates must be returned to t h e i r i n i t i a l s e t t i n g . I f the gates are kept at 0.9 stroke, then, as the pressure waves return and the transient condition subsides, the torque w i l l s t a -b i l i z e at the i n t e r s e c t i o n of 0.9 stroke and 100% design head. This value, being about 95%, i s greater than the i n i t i a l 64%. Segment 2 represents the a/g r a t i o of a system having a high head and low v e l o c i t y . Moving the gates to 0.9 stroke, as before, causes the torque to increase to 92%. This i s of course not desired. For such a system, the preferred c o n t r o l sequence i s to drive the gates closed. Closing the gates to 0.5 stroke reduces the torque to 78%. Hydraulic systems whose a/g segment i s p a r a l l e l to the 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 become steeper at lower gate stroke values. This implies that f o r a system o f f e r i n g some advantage at rated conditions the b e n e f i t i s reduced at lower stroke values. At f u l l gate, the advantage w i l l be greatest. This i s the region where i t i s needed most, f o r the units are loaded to t h e i r maximum power values. The r a t i o a/g i s the conversion c o e f f i c i e n t r e l a t i n g changes i n v e l o c i t y to changes i n pressure head. V e l o c i t y Is r e l a t e d to k i n e t i c energy and pressure, to p o t e n t i a l energy. The ef f e c t i v e n e s s of the scheme thus depends on the r a t i o of p o t e n t i a l energy to k i n e t i c energy of the water i n the system. In the above discussion, the gate movements must be made within the time required for the pressure waves to t r a v e l to the r e s e r v o i r (or surge tank) and back to the turbine. 65 8. SUPPLEMENTARY CONSIDERATIONS The models of the h y d r a u l i c components derived above are the i n i t i a l models only. The next step i s to 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. They' are not r e p r e s e n t a t i v e of a l l u n i t s ; each one should be se p a r a t e l y i n v e s t i g a t e d . 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 c o n s i d e r a t i o n s should be given to the d r a f t tube, the a c t u a t o r and the u n i t ' s s t r u c t u r e . 8.1 The D r a f t Tube The dynamics of the d r a f t tube were not considered f o r these are u s u a l l y much s h o r t e r than those of penstocks and hence l e s s i n f l u -e n t i a l on the t u r b i n e ' s behaviour. Nonetheless, i f f i e l d t e s t s show that the a p p l i c a t i o n of r a p i d c o n t r o l s to the wicket gates causes the d r a f t tube water column to separate, i t may be necessary to i n c l u d e the e f f e c t s of the d r a f t tube. Since d r a f t tubes are short i n comparison to penstocks they may be modelled as a r i g i d water column. The procedure i s s i m i l a r to tha t used i n modelling the tunnel between the surge tank and the r e s e r -v o i r . 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 penstock w i t h one or two reaches. I t w i l l be necessary to i n c l u d e the e f f e c t s of the 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 lo n g , the v e l o c i t i e s are hig h and the 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 respect to the e l e v a t i o n of the t u r b i n e ' s shroud r i n g . 8.2 Actuator 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 necessary to design actuators 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 linear and to control wicket gates in response to an analog signal at the transducer input. This signal i s derived from the unit speed error of the speed governor. The governor and actuator have time lags and travel limits to ensure that a faulty signal w i l l not drive the massive gates erratically. During a transient condition, the required control w i l l , i t is 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 directly 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 signal to the solenoids controlling 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 special designs may be required'before the concepts proposed can by" fully realized. 8.3 The Unit's Structural Strength Before applying rapid actuation sequences to the wicket gates, the strength of the unit's supporting structure should be investigated to confirm that the unit can cope with additional forces. For example, rapid gate motion, by adjusting the jet direction, w i l l increase the entrance shock losses at the runner inlet and this may induce unwanted vibrations into the runner blades. Rapid control may also introduce undesirable side effects such as vibrations in the supporting structure not anticipated during turbine design. Cavitation on runner blades w i l l also be increased. The reduction of hydraulic torque by allowing water to pass by implies that the water leaving the runner has a large angular velocity. 67 The design of the concrete i n the d r a f t tube should be checked to v e r i f y that i t can withstand, frequent exposure to these pressures. When the c o n t r o l requires, an in c r e a s e i n torque, excessive torque e x t r a c t i o n w i l l cause the s t a t i c pressure of the water l e a v i n g the runner to be below atmospheric pressure. As a r e s u l t water column s e p a r a t i o n may occur. This should be avoided because when the surge r e t u r n s , the momentum of the water behind the c o l l a p s i n g vacuum may rupture the runner head cover and damage the 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 before implimenting 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 of h y d r o - e l e c t r i c generating 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 long penstocks can be increased by a p p l y i n g s p e c i a l c o n t r o l sequences to 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 obtained by a l l o w i n g the c o n t r o l sequence to d r i v e the gates i n a d i r e c t i o n that w i l l , due to 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 only a short p e r i o d of time f o l l o w i n g a severe t r a n s i e n t disturbance on the system. Waterhammer can t h e r e f o r e be used to advantage i n r a i s i n g the power l i m i t of gener-a t i n g u n i t s . 69 REFERENCES 1. S.V. Ahamed and E.A. Erelyi, "Nonlinear Theory of Salient Pole Machines", IEEE Trans, on Power Apparatus and Systems, Vol. PAS 85, No. 1, January, 1966. 2. E.A. Ere l y i , S.V. Ahamed and R.E. Hopkins, "Nonlinear Theory of Syn-chronous Machines On-Load", IEEE Trans, on Power Apparatus and Sys- tems, Vol. PAS 85, No. 7, July 1966. 3. Venekov, Transient Phenomena in Electric Power Systems, 1964. 4. Fitzgerald and Kingsley, Ele c t r i c Machinery, Chptr. 3, McGraw-Hill, 1952. 5. W.D. Stevenson, Elements of Power Systems Analysis, McGraw-Hill, 1962. 6. G.W. Stagg and A.H. El-Abiad, Computer Methods in Power System Ana- l y s i s , McGraw-Hill, 1968. 7. R.M. Shier and A.L. Blythe, "Field Tests of Dynamic Stability Using a Stabilizing Signal and Computer Program Verification", IEEE Trans. on Power Apparatus and Systems, Vol. PAS 87, No. 2, February^ 1968. 8. Dandeno, Karas, McClymont, Watson, "Effect of High-Speed Rectifier Excitation Systems on Generator Stability Limits", IEEE Trans, on  Power Apparatus and Systems, Vol. PAS 87, No. 1, January 1968. 9. L.O. Long, ."Governor Aspects of the Coteau Creek Project", for pre-sentation to the Power Technical Group, Vancouver Section of the IEEE, Ap r i l 1967. 10. V.L. Streeter, Fluid Mechanics, McGraw-Hill, 1966. 11. G.R. Rich, Hydraulic Transients, Dover, 1963. 12. A.W. Fuller, "Transistorized Electric-Hydraulic Governor for Hydraulic Turbnes", Publication PMCC 65-11, Woodward Governor Company, Rockford, I l l i n o i s . 13. E. Mosonyi, Water Power Development, Vol. 1, Hungary 1963. 14. Selecting Hydraulic Reaction Turbines, Engineering Monograph No. 20, United States Department of the Interior, Bureau of Reclamation, Denver, 1966. 15. J. Parmakian, Waterhammer Analysis, Dover 1963. 16. A.E. Aeberli, "Governing of Water Turbines", Water Power, October, 1967. 70 17. L. Bergeron, Waterhammer in Hydraulics and Wave Surges in E l e c t r i c i t y , Wiley, 1961. 18. L.M. Hovey, "Optimum Adjustment of Governors in Hydro Generating Stations", Engineering Journal, (Canada), pg. 64, November 1960. 19. Paynter, "Surges and Waterhammer Problems", Trans, of ASCE, Vol. 118, 1953. 20. Daughherty, Ingersol, Fluid Mechanics, 1954. 21. Streeter, Handbook of Fluid Dynamics, 1961. 22. R. Walters, Hydraulic and Electro-hydraulic Servo Systems, I l i f f e , London, 1967. 23. H.M. Morris, Applied Hydraulics in Engineering, Ronald Press, 1963. 24. P.A. Wooldrige, A.L. Blythe, "Considerations Affecting The Design Philosophy of Solid State Exciters", IEEE Trans, of Power Apparatus  and Systems, Vol. PAS 87, No. 5, May 1968. 25. Design Notes, Portage Mountain Hydro-electric Power Development, International Power and Engineering Consultants Ltd., 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 British Columbia, 1968. 

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