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Optimal operation of a hydroelectric reservoir Do, Tung Van 1987

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OPTIMAL OPERATION OF A HYDROELECTRIC RESERVOIR By TUNG VAN DO B.Sc, The University of Hue, Vietnam, 1973 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of C i v i l Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October 1987 © Tung Van Do, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of £j \/; / v^e-e^r-/' The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Qsjb. /S~ /?*7 DE-6(3/81) i i OPTIMAL OPERATION OF A HYDROELECTRIC RESERVOIR by TUNG VAN DO ABSTRACT This thesis examines the influence of d i f f e r e n t degrees of s e r i a l c o r r e l a t i o n in the streamflow records on optimal operation of a h y d r o e l e c t r i c r e s e r v o i r . This t h e s i s a l s o i n v e s t i g a t e s the p r a c t i c a l aspects of choosing d i f f e r e n t d e c i s i o n v a r i a b l e s , c o n s i d e r i n g e f f e c t s on ease of implementation, t o t a l benefit, and actual use for real-time operations. Stochastic dynamic programming was used to optimize the long-term operation of a hydroelectric project with a single r e s e r v o i r . R e s e r v o i r i n f l o w s were analyzed using monthly flow record f o r 58 years with the assumption that monthly in f l o w s are e i t h e r p e r f e c t l y c o r r e l a t e d , uncorre1ated, or p a r t i a l l y correlated. Reservoir l e v e l change and powerhouse discharge were considered as a l t e r n a t i v e decision variables f o r each of the three cases of i n f l o w s e r i a l c o r r e l a t i o n . The optimization r e s u l t s were then examined and compared to determine the s i g n i f i c a n c e of the c h o i c e of d e c i s i o n v a r i a b l e s and to explore the e f f e c t s of i n f l o w s e r i a l c o r r e l a t i o n on p r a c t i c a l operating decisions which might be i i i based on the results of the optimization. I t was found that (1) Case 2 i n which in f l o w s were assumed p e r f e c t l y c o r r e l a t e d and Case 3 with p a r t i a l l y correlated inflows produce, respectively, highest and lowest t o t a l expected return, (2) the difference in t o t a l expected re t u r n between cases depends l a r g e l y upon the p h y s i c a l c h a r a c t e r i s t i c s of the system, (3) the r e s e r v o i r l e v e l change decision case produces more conservative r e s u l t s than the discharge d e c i s i o n case, (4) the r e s u l t s from the reservoir l e v e l change decision are easier to use for r e a l -time operation than those from the discharge decision case, (5) d i f f e r e n t r e s u l t s w i l l be produced with d i f f e r e n t choice of decision variables. i v TABLE OF CONTENTS Abstract i i Table of contents iv L i s t of figures v i L i s t of tables v i i Acknowledgement v i i i 1. Introduction 1 1.1 Summary 1 1.2 Outline of stochastic dynamic programming and Markovian processes 4 1.3 Outline of the thesis 7 2. Literature Review 9 2.1 Basic theory of dynamic programming 9 2.2 Stochastic dynamic programming and applications 11 2.3 Chance - constrained dynamic programming and applications 19 2.4 Areas of future research 22 3. Problem Formulation 24 3.1 Introduction 24 V 3.2 Description of the physical system and model s p e c i f i c a t i o n s 26 3.3 Analysis of h i s t o r i c a l inflows 31 3.4 Stochastic dynamic programming formulation .... 45 4. Results of the Steady-state Optimization Analysis .. 51 4.1 Case 1 : Uncorrelated inflows 51 4.2 Case 2 : Perfectly correlated inflows 55 4.3 Case 3 : P a r t i a l l y correlated inflows 56 5. Real-time Operation Considerations 57 5.1 Reservoir inflow forecasting 57 5.2 Real-time optimization 60 6. Conclusions 63 7. References 64 Appendices 66 Appendix A. C a l c u l a t i o n of the objective f u n c t i o n c o e f f i c i e n t s 66 Appendix B. Optimization results for Case 1 .. 71 Appendix C. Optimization results for Case 2 .. 80 Appendix D. Optimization results for Case 3 .. 143 Appendix E. Computer program l i s t i n g 152 v i LIST OF FIGURES No. T i t l e Page 2.1 Optimal strategy structure 16 2.2 Optimal strategy structure for Russell's model .. 16 B.l Optimal monthly reservoir l e v e l s . Case 1, State change d e c i s i o n 78 B. 2 Optimal r e s e r v o i r l e v e l s . Case 1, Discharge decision 79 C. l Optimal reservoir l e v e l s . Case 2, State change decision 119 C. 2 Optimal r e s e r v o i r l e v e l s . Case 2, Discharge decision 131 D. l Optimal reservoir l e v e l s . Case 3, State change decision 150 D.2 Optimal r e s e r v o i r l e v e l s . Case 3, Discharge decision 151 v i i LIST OF TABLES No. T i t l e Page 3.1 Physical c h a r a c t e r i s t i c s of the system 27 3.II Histogram analysis for monthly inflows 34 3 . I l l Monthly inflow probability d i s t r i b u t i o n s 40 3.IV Monthly conditional p r o b a b i l i t y d i s t r i b u t i o n s 41 B.I Optimal monthly reservoir levels for Case 1 .... 72 B.II Optimal monthly powerhouse flows for Case 1 .... 73 B.III Summary of Case 1 results, state change decision 74 B. IV Summary of Case 1' results, discharge decision .. 76 C. I Optimal monthly reservoir l e v e l s . Case 2 81 C.II Optimal monthly powerhouse flows. Case 2 88 C. III Summary of Case 2 results, state change decision 95 C I V Summary of Case 2 resu l t s , discharge decision .. 107 D. I Optimal monthly reservoir l e v e l s . Case 3 144 D.II Optimal monthly powerhouse flows. Case 3 145 D.III Summary of Case 3 results, state change decision 146 D.IV Summary of Case 3 resu l t s , discharge decision .. 148 v i i i ACKNOWLEDGEMENTS I would l i k e to extend my si n c e r e a p p r e c i a t i o n to my thesis advisor, Professor Samuel 0. Russell, for his e f f o r t s and support, and to Professor W. F. Caselton for his h e l p f u l discussions. Very s p e c i a l thanks are due to Mr. C h a r l e s D. D. Howard, who u n f a i l i n g l y provided encouragement and support, including the provision of f i n a n c i a l assistance throughout my years at the University of Toronto and the University of B r i t i s h Columbia. F i n a l l y , I would l i k e to dedicate t h i s work to my friends and r e l a t i v e s who are struggling bravely to retain t h e i r dignity and humanity i n Vietnam. - 1 -1. INTRODUCTION 1.1 Summary This t h e s i s examines the problem of o p t i m i z i n g the o p e r a t i o n of a h y d r o e l e c t r i c p r o j e c t , w i t h a s i n g l e reservoir, using as an example the Powell River hydro plant in B r i t i s h Columbia. If a r e s e r v o i r empties each year and i f the i n f l o w s during s u c c e s s i v e time periods are independent of one another, then an optimal operating p o l i c y can be determined by s t o c h a s t i c dynamic programming, working backwards from the time when the r e s e r v o i r i s empty. However, i f the r e s e r v o i r does not empty each year, there i s no ready s t a r t i n g p o i n t for the o p t i m i z a t i o n ; and i f the flows i n s u c c e s s i v e time p e r i o d s are c o r r e l a t e d , the dynamic programming algorithm becomes much more complicated. The b a s i c approach i s to t r e a t the i n f l o w s as an example of a Markov process (Howard, 1960) and determine the optimal long term strategy by continuing stochastic dynamic programming over a s e r i e s of years u n t i l the r e s u l t s s t a b i l i z e . From the r e s u l t s , average long term values can be assigned to the various reservoir l e v e l s at the beginning of each month. These values can then be used, i n r e a l time, with s t o c h a s t i c dynamic programming to determine optimal -2-operating decisions. In t h i s t h e s i s , two d i f f e r e n t ways of f ormulating the stochastic DP, with d i f f e r e n t decision variables are used. A l s o the e f f e c t s from three d i f f e r e n t l e v e l s of s e r i a l c o r r e l a t i o n between s u c c e s s i v e i n f l o w s are examined; one with no s e r i a l c o r r e l a t i o n ; one with perfect c o r r e l a t i o n and one with p a r t i a l c o r r e l a t i o n . The time ho r i z o n of one year i s d i v i d e d i n t o twelve equal time steps of one month each. The reservoir inflows i n these time steps can be taken as a completely uncorrelated, p e r f e c t l y c o r r e l a t e d , or p a r t i a l l y c o r r e l a t e d sequence, depending on which of these t h r e e cases i s b e i n g investigated. For Case 1 of t h i s study the recorded monthly i n f l o w s fo r 58 years were analyzed to determine the p r o b a b i l i t y d i s t r i b u t i o n of the flows f o r each month of the year. The r e s u l t i n g p r o b a b i l i t y density functions of inflows were then dis c r e t i z e d into twelve increments and these are used as the input into the stochastic dynamic programming. The reservoir inflows can also be described as i f they are s e r i a l l y correlated and the degree of c o r r e l a t i o n could be one hundred p e r c e n t or l e s s . Case 2 i n t h i s study demonstrates an applicatio n of stochastic DP with p e r f e c t l y -3-c o r r e l a t e d i n f l o w s . As i n Case 1, the p r o b a b i l i t y d e n s i t y f u n c t i o n of the i n f l o w s was d i s c r e t i z e d and the expected value of the t o t a l benefit was maximized for the entire time horizon and over the twelve d i f f e r e n t l e v e l s of probability. Case 3 assumes that the s e r i a l c o r r e l a t i o n i s not p e r f e c t . For t h i s case, a t r a n s i t i o n matrix was used to describe the inflows between two successive months and the DP was r e v i s e d to i n c l u d e the current storage and the l a s t period inflow state variables. Powerhouse f l o w and r e s e r v o i r l e v e l change were considered as a l t e r n a t i v e decision variables for each case mentioned above. The optimization r e s u l t s were then analyzed and compared to determine the significance of t h i s choice of decision variables. R e a l - t i m e o p e r a t i o n w i t h f o r e c a s t i n f l o w s was considered, using the steady-state optimization r e s u l t s as a constraint for deriving the optimal operating strategy. - 4 -1.2 Outline of Markovian processes and stochastic DP Consider a system which at any p a r t i c u l a r time can be i n any one of N f i n i t e d i s c r e t e s t a t e s , S j _ , . . . , s N . In the d e t e r m i n i s t i c case, the system changes from one of these admissible states to another as a deterministic function of the i n i t i a l s t a t e s^. I f that change i s s t o c h a s t i c , the process i s ruled by a t r a n s i t i o n matrix P = (p^j)» where P i j = the p r o b a b i l i t y that the system i s i n s t a t e j at time t+1, given that i t was i n state i at time t. If the p r o b a b i l i t y that the system i s i n s t a t e i at time t t . is s^ and assuming the t r a n s i t i o n matrix P i s independent of time then, N t+1 V t , = Z_ P i i s i / J = 1, 2, N (1-1) i=l 0 s i = c i I f a l l of the t r a n s i t i o n p r o b a b i l i t i e s P^j are t p o s i t i v e , these functions s. converge as t -»° to quantities s^ which s a t i s f y the "steady-state" equation N s^ = P i j S ^ , j = 1, 2, N (1-2) -5-and the l i m i t i n g values are independent of the i n i t i a l state of the system (Bellman and Dreyfus, 1962). In more general situations in which decisions are made at each stage, the t r a n s i t i o n matrix can be w r i t t e n as follows : t , x t / v t+1 , t 7 P (k) = pj_j(k) = P s = S j | s = s i a n d decision kj (1-3) These are the t r a n s i t i o n p r o b a b i l i t i e s of a Markov chain which i s time-dependent and decision-dependent. I f B t ( s ^ , S j , k ) are the net b e n e f i t s during p e r i o d t (the system s t a r t s i n s t a t e s^ and ends i n s t a t e S j ) when d e c i s i o n k i s made, then the p o l i c y that maximizes the expected net benefits i s found by solution of f t ( s ^ ) = max k 21 p t i j ( k ) ^ B t ( s i , s j , k ) + f t + 1 ( S j ) (1-4) These are the b a s i c equations of s t o c h a s t i c DP i n Markov chains (Loucks, Stedinger, and Haith, 1981). To include the t r a n s i t i o n u n c e r t a i n t y , i t i s necessary to c a l c u l a t e , at each step, the expected net b e n e f i t s , or the net b e n e f i t s corresponding to some desired quantile of the d i s t r i b u t i o n , resulting from each decision. -6-Note that the time indexing used in equations from 1-1 to 1-4 i s forward but the computation i n s t o c h a s t i c DP progresses backwards. The forward time indexing was used throughout this thesis. -7-1.3 Outline of the thesis The thesis i s divided into three sections, introduction and l i t e r a t u r e review (Chapter 1 and 2), problem formulation and the o p t i m i z a t i o n a n a l y s i s (Chapter 3 and 4), r e a l - t i m e operation and conclusions (Chapter 5 and 6). Chapter 1 introduces the thesis and summarizes i t . This chapter a l s o o u t l i n e s s t o c h a s t i c dynamic programming and Markovian processes. Chapter 2 i s a r e v i e w of s t o c h a s t i c DP, chance-constrained DP, and applications proposed i n the l i t e r a t u r e . The trends of c u r r e n t research and the l i k e l y areas of interest for future research are also included here. Chapter 3 d e s c r i b e s the p h y s i c a l system which i s used here as a case study and the s p e c i f i c a t i o n s of the model. The a n a l y s i s of h i s t o r i c a l i n f l o w s i s described and the ap p l i c a t i o n of stochastic DP to the problem i s i l l u s t r a t e d . Chapter 4 describes i n d e t a i l the r e s u l t s of the steady-state optimization analysis for each case with both the r e s e r v o i r l e v e l change and powerhouse f l o w as a l t e r n a t i v e decision variables. Chapter 5 considers r e a l - t i m e operation using the optimization r e s u l t s from the steady-state model. Reservoir -8-i n f l o w f o r e c a s t i n g methods are described and a method by which steady-state r e s u l t s can be used to derive real-time operating decisions i s explained. Chapter 6 provides comments and o u t l i n e s research issues that arose from the computational experience gained during the course of th i s research. -9-2. LITERATURE REVIEW 2.1 Basic Theory of Dynamic Programming Dynamic programming, a method formulated and developed l a r g e l y by Richard Bellman (1957), i s a procedure f o r optimizing a multi-stage (sequential) decision problem. This type of problem can be d i v i d e d i n t o stages, with a p o l i c y d e c i s i o n r e q u i r e d at each stage. The stages, or s e q u e n t i a l c h a r a c t e r i s t i c s of the problem often are time periods, however the stages can a l s o be space regions or p h y s i c a l e n t i t i e s . Each stage has a f i n i t e number of possible states a s s o c i a t e d with i t . In general, the s t a t e s are the v a r i o u s p o s s i b l e c o n d i t i o n s of the system at that stage of the problem. The p o l i c y d e c i s i o n at each stage transforms the current s t a t e i n t o a s t a t e a s s o c i a t e d with the next stage. Given the current state, the DP determines an optimal p o l i c y for the remaining stages which i s independent of the p o l i c y adopted i n pre v i o u s stages. This i s based on the p r i n c i p l e of optimality stated by Richard Bellman (1957): "An optimal p o l i c y has the property that whatever the i n i t i a l state and i n i t i a l decision are, the remaining d e c i s i o n s must c o n s t i t u t e an optimal p o l i c y with r e g a r d to the s t a t e r e s u l t i n g from the f i r s t decision." Let f t ( s , x t ) be the maximizing/minimizing value of the objective function, given that the system starts i n state s -10-* at stage t and the decision x t i s selected. Let f t ( s ) be the maximum/minimum value of f t ( s , x t ) over a l l possible value of x^. The recursive r e l a t i o n s h i p has the form * f t ( s ) = max/min f t ( s , x t ) (2-1) x t x t * where f t ( s , x t ) would be w r i t t e n i n terms of s, x t, f t +-^(.) ( H i l l i e r and Lieberman, 1980). -11-2.2 Stochastic DP and Applications The i n f l o w s i n t o a r e s e r v o i r can be considered as a s t o c h a s t i c process i n which the c o n d i t i o n a l p r o b a b i l i t i e s for the i n f l o w s i n a p e r i o d t,namely I t , depends o n l y on the s t a t e of the process i n the previous period t-1 and on known p r o b a b i l i t i e s (Nemhauser, 1966) A stochastic process which has the property defined by the equation above i s s a i d to be a f i r s t order Markov process. The future behaviour of such a process depends on only i t s present s t a t e and not on i t s past h i s t o r y ; i t i s memoryless. To d e s c r i b e Markov p r o b a b i l i s t i c b e h a v i o u r , the f o l l o w i n g parameters of the process need to be defined (Benjamin and Cornell, 1970): - a d i s t r i b u t i o n on the i n i t i a l s t a t e s : the s t a t e i n which the system originates at 'time' 0: P ( l t | l t-1' ,I x) = p ( I t | I t _ x ) (2-2) 0 P [ s 0 for a l l i = l - the t r a n s i t i o n p r o b a b i l i t y p ^ j ( t ) : the p r o b a b i l i t y that the process w i l l be i n s t a t e j at time t g iven that i t was i n state i at the previous step: -12-P i j = P t s = j t - l S = 1 for a l l i , j pairs I f there are a f i n i t e number of s t a t e s at each stage and a f i n i t e number of stages, the p r o b a b i l i t y of t r a n s i t i o n from a s t a t e at one stage to a s t a t e at the next stage e x i s t s (Yeh, 1985). Howard (1960) introduced the concept of rewards into a Markov process. With t h i s concept a sequence of rewards i s generated as t r a n s i t i o n s are made from st a t e to s t a t e . The reward i s thus a random v a r i a b l e w i t h a p r o b a b i l i t y d i s t r i b u t i o n governed by the t r a n s i t i o n p r o b a b i l i t i e s of the Markov process. A t y p i c a l d i s c r e t i z e d s t o c h a s t i c DP when a p p l i e d to a r e s e r v o i r o p t i m i z a t i o n model has the f o l l o w i n g form (Yeh, 1985): t, max f t ( S t , I t _ i ) = max! z Rt (It=0 + f B(R t) ( S ^ n ,U) t + l ^ t + l ' x t (2-3) subject to s t + l = s t + * t " R t " e t / n,max f n ( S n ' J n - l * = m a x ) 21 P R n I n=° I n l I n - l | • B(R n), - 1 3 -in which f t ( S t , I t _ 1 ) = expected return from the optimal operation of the system which has t time periods to the end of the planning horizon; S t = storage at the beginning of time period t; I t = inflow during time period t; R t = release decision during time period t; B(R t) = r e t u r n (or reward) obtained consequent to r e l e a s i n g a q u a n t i t y of water "Rt" during time peri o d t. B can a l s o be a f u n c t i o n of the storage as i n the case of hydropower productions; p j ^ I t | l t _ 1 J = t r a n s i t i o n p r o b a b i l i t i e s connecting inflow I t i n the t time period with i n f l o w I ^ - i previous time period t-1; e t = evaporation loss during time period t; t = time p e r i o d index from 1 to n, eg., months numbered from the beginning of the planning horizon. The recursive equation uses the assumption that inflow during a g i v e n time p e r i o d i s r e l a t e d to the i n f l o w i n the p r e v i o u s time p e r i o d by a c o n d i t i o n a l p r o b a b i l i t y I t | l t _ - ^ . This can be interpreted as the p r o b a b i l i t y that I t has occurred during the time period t given the knowledge -14-that I ^ - i has occurred during the preceeding time period t - l . The right-hand side of the r e c u r s i v e equation states that f o r a g i v e n storage l e v e l S t at the beginning of the current time period t and a given i n f l o w I ^ - l during the preceeding time perio d t - l , one can f i n d a r e l e a s e R t such that the expected value of the sum of the immediate return and expected future r e t u r n i s maximized. Since the r e l e a s e R t i s a f u n c t i o n of S t and I t ( I t i s taken over a l l p o s s i b l e values i n the range I t=0 and I t = I t m a x with c o n d i t i o n a l p r o b a b i l i t y P t [ I t _ J )/ i t i s possible, with given values of S t and I t _ ^ , to search over a l l p o s s i b l e v a l u e s of R t and to choose the one that maximizes the expected return. An i m p o r t a n t c o n t r o v e r s y i n the l i t e r a t u r e of s t o c h a s t i c r e s e r v o i r operations concerns the appropriate s t a t i s t i c a l assumptions fo r the i n f l o w sequence. The a p p r o p r i a t e n e s s of a p a r t i c u l a r s t o c h a s t i c sequence assumption w i l l depend on the i n t e r v a l between d e c i s i o n times (Yakowitz, 1982). L i t t l e (1955) used the time lapse of two weeks and chose the Markov assumption that the conditional p r o b a b i l i t y d e n s i t i e s f o r the i n f l o w s should s a t i s f y (2-2). He defined the state to be the two-dimensional quantity x t = ( s t , I t_ x) -15-and the mass balance the same as i n the deterministic case, s t + l = m i n j s t + I t " R t ' C J where s t i s the state variable (reservoir water volume) at the beginning of the t period; I t represents the i n f l o w volume during the time period; the c o n t r o l v a r i a b l e R t represents the r e l e a s e of water during the t^^ 1 d e c i s i o n period, and C denotes the r e s e r v o i r capacity. With s t + ^ so determined, i t s p r o b a b i l i t y density can be found given Rt, x^ . = ( s t , It_-^) and the Markov t r a n s i t i o n p r o b a b i l i t y G e s s f o r d and K a r l i n (1958), t h r o u g h a d o p t i n g independence assumptions for the inflow process, were able to achieve some inventory-1 ike r e s u l t s . They assumed that the single-stage loss i s where x t i s the reservoir storage l e v e l , Rt i s the release and D t i s the known demand at time t and i s a constant of proportionality. The optimal strategy S t has the form shown i n Figure 2.1. The parameters R-^  and R 2 depend on stage t, reservoir capacity and the cost of d e f i c i t . R u s s e l l (1972) has extended Gessford and K a r l i n ' s (1958) model to i n c l u d e costs r e l a t e d to r e l e a s e s and p ( I t I I t _ i ) • (2-4) J -16-a v a i l a b l e reservoir storage f a l l i n g outside a certain range. Through a n a l y s i s of the a s s o c i a t e d dynamic programming f u n c t i o n a l equation, R u s s e l l has proved that at each d e c i s i o n time t the optimal s t r a t e g y i s nondecreasing and piecewise lin e a r , as shown i n Figure 2.2 Storage l e v e l x f c Figure 2.1. Optimal strategy structure PS CD m (d 0) .H u I—I to e •H -P o Slopes are either 45° or 0° l2 R3 R4 R5 R6 Storage l e v e l x. Figure 2.2, Optimal strategy structure for Russell's model -17-Butcher (1971) adopted the same model as L i t t l e (1955) and employed discrete stochastic dynamic programming to find optimal stationary strategy for operating the Watasheamu Dam near the C a l i f o r n i a - Nevada border. He noted that i f the termi n a l d e c i s i o n time i s f a r enough i n t o the future, then one obtains a " s t a t i o n a r y optimal strategy", i.e., S t(x) i s the same for a l l t. Su and D e i n i n g e r (1974) a p p l i e d the model and methodology of L i t t l e (1955) and used time dependence of the Markov t r a n s i t i o n p r o b a b i l i t i e s i n order to capture the effects of seasonality. Gablinger and Loucks (1970) noted that the optimal control problem can be solved by l i n e a r programming as well as dynamic programming i f s t a t e s and c o n t r o l s are discretized. But they concluded that the dynamic programming performed much b e t t e r f o r the reason t h a t the l i n e a r programming t r a n s c r i p t i o n r e q u i r e s many more s o l u t i o n variables than the dynamic programming formulation. Schweig and Cole (1968) adapted L i t t l e ' s (1955) model to a two-reservoir problem and employed discrete stochastic dynamic programming to compute an optimal strategy based on data from Lake Vyrnwy, Wales. They encountered severe c o m p u t a t i o n a l d i f f i c u l t i e s i n s p i t e of v e r y c o a r s e d i s c r e t i z a t i o n . -18-Arunkumar and Yeh (1973) applied a penalty function to the s t o c h a s t i c DP to maximize the f i r m energy output. They a l s o a p p l i e d the decomposition approach to a p a r a l l e l two reservoir system i n which a stationary p o l i c y for reservoir 1 i s fixed while the second reservoir i s optimized. Then the optimized p o l i c y of reservoir 2 replaces the i n i t i a l p o l i c y and r e s e r v o i r 1 i s optimized. The procedure was continued u n t i l the improvement between successive approximations of the reward f u n c t i o n was u n i f o r m l y bounded by a reasonably small number. Casel t o n and R u s s e l l (1976) developed a method to decompose a mixed hydrothermal power system i n t o separate subsytems and to o p t i m i z e these subsystems b e f o r e integrating operating p o l i c y for the system as a whole. The s u b o p t i m i z a t i o n r e s u l t s p r o v i d e i n s i g h t i n t o the complementary aspects of long-term operation of a l a r g e storage hydro project and thermal plant. -19-2.3 Chance-constrained DP and Applications Chance-constrained DP i s a l s o c a l l e d r e l i a b i l i t y -c o nstrained DP. Both terms imply 'chance of f a i l u r e ' and ' r e l i a b i l i t y ' . In a long-range reservoir operation analysis, the t r a d e - o f f between r e t u r n and the a s s o c i a t e d r i s k i s of prime concern. This problem can be solved with the penalty function approach in which one seeks to minimize N J(S) = H E r L t ( s t , s t + 1 , S t ( x t ) ) l (2-6) t=l L subject to the 'target r e l e a s e ' c o n s t r a i n t that R t D T whenever possible and the 'chance constraint' that s t+I t ^ D T , 1 ^ t < N] > 1 - p (2-7) where (3 i s a specified constant selected from a p r o b a b i l i t y d i s t r i b u t i o n . Askew (1974) used p r o b a b i l i s t i c DP with a p e n a l t y function, w: f t ( S t ) = max zL. P ( I F ) R t It Z p ( i t : B(R t) + r . f t + 1 ( S t + 1 ) - w (2-8) i n which r = ( l + i ) ~ ^ i s the discount f a c t o r with a discount rate i , w = 0 for Rt > DFC and w > 0 for Rt < D T. This f o r m u l a t i o n tends to reduce the expected net -20-benefit associated with releases that cause larger possible s h o r t f a l l s , and hence i t tends to r e s u l t i n a more conservative optimum release policy. While such a strategy s a t i s f i e s the chance c o n s t r a i n t , i t does not n e c e s s a r i l y achieve the minimizing value of J among a l l such constrained strategies (Yakowitz, 1982). Sniedovich and David (1975) reported that the true chance-constraints can be formulated i f the state variables x t i s augmented so that i n the l i f e of the system up to the time period t. Rossman (1977) applied Lagrangian d u a l i t y theory to the reservoir operations problem and constructed dual variables on the r e l i a b i l i t y c o n s t r a i n t s . The r e s u l t i n g f o r m u l a t i o n was solved with the following p r o b a b i l i s t i c DP procedure. I f there are k e v e n t u a l i t i e s for which there are desired r e l i a b i l i t y constraints, the p r o b a b i l i s t i c equation becomes: x t = ( s t , y t) where y t <• y i s defined as the expected number of f a i l u r e s f t (S t) = max k (2-9) - 2 1 -where A j and Y j are the Lagrangian m u l t i p l i e r and the maximum allowable occurrences for events j over the l i f e of the p r o j e c t N; B(S t,R t,I t) and G j ( S t , R t , I t ) are the net benefits and the r e l i a b i l i t y constraints such that Gj=l i f event j happens, Gj = 0 i f event j does not happen (Yeh, 1985). -22-2.4 Areas of Future Research Yeh (1985) recommended the f o l l o w i n g f i v e areas of future research f o r i n c o r p o r a t i o n of DP i n t o the o v e r a l l system approach: - continuation of research i n Incremental DP to include r i s k (or r e l i a b i l i t y ) and multiobjectives; - development of research i n D i f f e r e n t i a l DP procedures to f a c i l i t a t e nondifferentiable objective functions; - continuation of research i n real-time operations to incorporate adaptive f o r e c a s t s i n an i n t e r a c t i v e mode; - development of r e s e a r c h i n d e c o m p o s i t i o n (or p a r t i o n i n g ) to f a c i l i t a t e nonseparable o b j e c t i v e functions; and - i n i t i a t i o n of research i n semi-Markov programming for m u l t i y e a r r e s e r v o i r operation with an econometric forecast. The following issues which need further research arose during the course of the present research: - Neither the Markov dependence assumption nor the independence assumption r e f l e c t the p r o b a b i l i t y d i s t r i b u t i o n of r e a l reservoir inflow sequences. If the p r o b a b i l i t y d i s t r i b u t i o n s of cumulative inflows -23-are known, the difference between successive sequence dis t r i b u t i o n s can be determined and incorporated into stochastic DP; - The turbine dispatching optimization analysis should be incorporated into reservoir operation optimization to achieve the o v e r a l l optimal p o l i c y . -24-3. PROBLEM FORMULATION 3.1 Introduction R e s e r v o i r o p e r a t i o n s are u s u a l l y c o n t r o l l e d by operational decisions about the volume of water to release d u r i n g a time p e r i o d . The o p t i m i z a t i o n of l o n g - t e r m operations i n v o l v e s determining optimal r e l e a s e s f o r the successive time periods so that the expected value of t o t a l r e t u r n over a time horizon i s maximized w i t h i n p r a c t i c a l operating c o n s t r a i n t s . In the o p t i m i z a t i o n of r e a l - t i m e reservoir operation, the current reservoir storage and the i n f l o w during the previous time perio d are known, and the r e l e a s e d e c i s i o n has to be made at the beginning of the current period. As each time period progresses the decision can be revised i n the l i g h t of the actual experience during the previous time period. In t h i s chapter the problem of optimizing the long-term o p e r a t i o n of a s i n g l e r e s e r v o i r f o r hydropower w i t h s t o c h a s t i c i n f l o w i s formulated. The o b j e c t i v e of the optimization i s to maximize the t o t a l energy output with no constraint on minimum energy production. At each time period (which i s taken as one month i n t h i s study), the model searches f o r the optimal r e l e a s e using twelve p o s s i b l e discrete values of the inflow each associated with a l e v e l of p r o b a b i l i t y . Monthly i n f l o w s used i n the model are - 2 5 -assumed to have annual cycles, which means each year has an i d e n t i c a l set of monthly p r o b a b i l i t y d i s t r i b u t i o n s . The model i s run over a period of several years u n t i l i t reaches steady-state. The f i n a l r e s u l t s show the optimal long-term operations f o r var i o u s r e s e r v o i r storage l e v e l s at the beginning of each month. -26-3.2 D e s c r i p t i o n of the P h y s i c a l System and Model Specifications The system considered i n t h i s study i s the Powell Lake system which includes one reservoir i n a basin which extends i n e l e v a t i o n from near sea l e v e l to f i v e thousand feet. Water from the reservoir i s discharged through penstocks to a powerhouse which has f i v e t urbines with i n d i v i d u a l capacities varying from 3 MW to 30 MW, depending somewhat on the net head, and a t o t a l i n s t a l l e d capacity of 48 MW. The b a s i n has a v a r i e t y of h y d r o l o g i c a l regimes including permanent snow and ice f i e l d s . Runoff r e s u l t s from r a i n f a l l and snowmelt and on any given day both sources may be c o n t r i b u t i n g v a r y i n g amounts from d i f f e r e n t e l e v a t i o n ranges. Under certain conditions r a i n f a l l may be running o f f from the lower elevations, snowmelt may be occuring in the middle range, and at the highest e l e v a t i o n s snow may be accumulating on the ground (Charles Howard and Associates, 1987). Physical c h a r a c t e r i s t i c s of the system are i l l u s t r a t e d i n Table 3.1 The o b j e c t i v e of the model i s to maximize the average annual energy output over a long-term operation. The energy output during a time period i s a f u n c t i o n of r e l e a s e and 7 -27-Table 3.1. Physical cha r a c t e r i s t i c s of the system Watershed Area (sq. mi.) 580 Average annual p r e c i p i t a t i o n (in.) measured at the plant 43 Average runoff (cfs) 3300 Reservoir Area (sq. mi.) 45 Live storage capacity (ac-ft) 580000 Maximum elevation (ft) 285 Minimum elevation (ft) 265 Stage storage curve i s l i n e a r i n the range 265-285 f t with the slope 0.068395 ft/1000 cfs-days Tailwater elevation (ft) 100 Powerhouse Turbine c a p a c i t y and flow ra t e at max. e l e v a t i o n 285 f t : Turbine 1 Turbine 2 Turbine 3 Turbine 4 Turbine 5 Total 10 MW 3 MW 5 MW 5 MW 30 MW 48 MW 800 cfs 250 cfs 400 cfs 400 cfs 2400 cfs 3850 cfs - 2 8 -reservoir storage which, respectively, could be represented by power c o e f f i c i e n t s showing the value of powerhouse discharge i n terms of KW/cfs and storage c o e f f i c i e n t s i n d i c a t i n g the value of marginal r e s e r v o i r storage i n KW/cfs-days. A discount factor which varies with time i s applied to both energy output and storage ( p o t e n t i a l energy) based on the assumption that energy produced i n a near time period i s more v a l u a b l e than energy produced i n the future. Water which i s not r e l e a s e d i n a current time peri o d (stored i n the r e s e r v o i r ) i s assumed to be r e l e a s e d i n a subsequent time period, t h e r e f o r e , i t has l e s s value. The discount factor r t used i n the case study i s given by r t = 1 - i . t / 12 (3-1) where i i s the annual i n t e r e s t rate and t i s the time step which i s taken as one month i n th i s study. Therefore, one KW generated i n month t has higher value than one KW generated in month t+n by r t " rt+n = i - n / 1 2 An annual interest rate of 6% was used i n t h i s study. The r e s e r v o i r l i v e storage i s d i s c r e t i z e d i n t o 41 states with i d e n t i c a l increments which are equivalent to 0.5 - 2 9 -f t of drawdown. The minimum and maximum reservoir elevations (265 and 285 f t ) correspond to the l i v e storage of 0 and 292420 cfs-days, r e s p e c t i v e l y . The stage-storage curve of t h i s r e s e r v o i r happens to be approximately l i n e a r , and because of t h i s 0.5 f t of drawdown i s e q u i v a l e n t to a storage of 7310.5 cfs-days. The s t a t e s ( r e s e r v o i r storage) are defined as follows: S - L = 0 s41 = 2 9 2 4 2 0 cfs-days s± = ( i - 1) x 7310.5 cfs-days i = 2,3,...,40 If the s t a t e change i s the d e c i s i o n v a r i a b l e , r e l e a s e i n a time perio d w i l l be a f u n c t i o n of random i n f l o w s and reservoir storages at the beginning and the end of the time period. This r e l a t i o n s h i p i s governed by the c o n t i n u i t y equation: s t + l " s t = *t - r t < 3" 2 ) i n which q t and r t are, r e s p e c t i v e l y , i n f l o w and t o t a l r e l e a s e . Therefore, f o r the case i n which the r e s e r v o i r l e v e l change i s the decision variable, the highest reservoir storage l e v e l which can be reached at the end of time p e r i o d t given that the reservoir storage l e v e l at the beginning of time p e r i o d t i s s t and the i n f l o w q t i s g i v e n by - 3 0 -s t + i 4 q t + s t < rt = 0 ) (3-3) When powerhouse discharge i s chosen as the d e c i s i o n variable, i t i s dis c r e t i z e d into 17 values varying from zero to the maximum capacity of the hydro plant. The values have i d e n t i c a l increments which are n e a r l y e q u i v a l e n t to the volume of water i n 0.5 f t drawdown. T h i s p r o v i d e s a convenient b a s i s f o r comparison with the st a t e change decision case. The capacity of the plant i s 3850 cfs, which, to be consistent with storage volume units, corresponds to: Capacity = 3850 cfs number of days i n a time period As with the st a t e change d e c i s i o n case, the maximum powerhouse release i n a time period t can be computed using the continuity equation: i n which C t denotes the plant capacity i n time period t . t,max = min t' (3-4) -31-3.3 Analysis of H i s t o r i c a l Inflows The a n a l y s i s of a 58-year monthly i n f l o w r e c o r d provides the basis for the p r o b a b i l i t y d i s t r i b u t i o n s used i n the r e s e r v o i r o p e r a t i o n o p t i m i z a t i o n procedure. The frequency d i s t r i b u t i o n s were derived for each month and the r e s u l t s are shown in Table 3.II. Twelve p r o b a b i l i t y l e v e l s were chosen to represent the frequency d i s t r i b u t i o n for each month. They are, r e s p e c t i v e l y , 0.05, 0.05, 0.10, 0.10, 0.10, 0.10, 0.10, 0.10, 0.10, 0.10, 0.05, 0.05. The i n f l o w s associated with these p r o b a b i l i t i e s for each month are shown in Table 3.III. These twelve d i s t r i b u t i o n s f o r each of the twelve months are assumed to be p e r i o d i c with an annual cycle. That means that the d i s t r i b u t i o n s are i d e n t i c a l for the same month of various years and are d i f f e r e n t from month to month within a year. For Case 1 (no s e r i a l c o r r e l a t i o n between two successive months i s assumed), each time period has twelve d i s c r e t e i n f l o w s and the t r a n s i t i o n between s t a t e s i s independent of the inflows i n the previous time period. The f u l l dependence assumption i s used f o r Case 2 i n which i n f l o w s stay at the same p r o b a b i l i t y l e v e l f o r the e n t i r e time horizon. In other words, the s t o c h a s t i c model becomes twelve deterministic models with a common objective f u n c t i o n . Each of these d e t e r m i n i s t i c models has o n l y one -32-possible inflow associated with a p r o b a b i l i t y during a time period. It should be noted that t h i s i s a somewhat arbit r a r y scheme. F u l l dependency represents an extreme case when considering correlated flows and would not occur i n nature. For Case 3 i n which some l e v e l of dependence i s assumed for i n f l o w s from one time p e r i o d to the next i t i s necessary to determine the c o n d i t i o n a l p r o b a b i l i t y matrix. The p r o b a b i l i t y t h a t the event A has o c c u r r e d g i v e n the knowledge t h a t the event B has o c c u r r e d i s c a l l e d c o n d i t i o n a l p r o b a b i l i t y and i s denoted by p(A|B). The procedure i s best described by a numerical example as follows: For the f i r s t l e v e l of p r o b a b i l i t y (0.05) the lowest F e b r u a r y i n f l o w s i n the range of 13704 and 19750 c f s occurred i n the year 1922 and 1929 respectively. The inflows i n March of these two years were 27752 and 52159 c f s , respectively. The f i r s t number belongs to the range 22750 -32625 c f s which i s the second p r o b a b i l i t y l e v e l f o r March while the second number belongs to the s i x t h p r o b a b i l i t y l e v e l . Therefore, each has the c o n d i t i o n a l p r o b a b i l i t y of 0.5 and i n the other p r o b a b i l i t y l e v e l s the c o n d i t i o n a l p r o b a b i l i t y i s zero. Mathematically, - 3 3 -j , k=l IMar.l IFeb.J = °- 5 f o r 3 = 2 a n d 6 =0.0 f o r j=l,3,4,5,7,8,..,12 By doing t h i s i n a s i m i l a r way for a l l twelve l e v e l s of pro b a b i l i t y , a conditional p r o b a b i l i t y matrix can be found for each month. Table 3.IV shows the r e s u l t i n g twelve c o n d i t i o n a l p r o b a b i l i t y matrices for the twelve months of the y e a r . These a r e t h e c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s which were used i n the optimization model for Case 3. - 3 4 -TABLE 3.II. MONTHLY INFLOW HISTOGRAM ANALYSIS JANUARY ******* INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 11860 34429 23144 0. 269 12. 07 12. 07 0. 0000053 7 34429 56997 45713 0. 532 20. 69 32. 76 0. 0000092 12 56997 79566 68282 •0. 795 24. 14 56. 90 0. 0000107 14 79566 102135 90850 1. 057 13. 79 70. 69 0. 0000061 8 102135 124704 113419 1. 320 10. 34 81. 03 0. 0000046 6 124704 147272 135988 1. 582 8. 62 89. 66 0. 0000038 5 147272 169841 158557 1. 845 3. 45 93. 10 0. 0000015 2 169841 192410 181125 2. 108 3. 45 96. 55 0. 0000015 2 192410 214978 203694 2. 370 0. 00 96. 55 0. 0000000 0 214978 237547 226263 2. 633 3. 45 100. 00 0. 0000015 2 ************************************************************ MAX = 237547 MIN = 11860 MEAN = 85939 STD.DEV. = 50176 ************************************************************ FEBRUARY ******** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 10598 25550 18074 0. 260 5. 17 5. 17 0. 0000035 3 25550 40501 33025 0. 475 13. 79 18. 97 0. 0000092 8 40501 55453 47977 0. 690 22. 41 41. 38 0. 0000150 13 55453 70404 62929 0. 905 13. 79 55. 17 0. 0000092 8 70404 85356 77880 1. 120 17. 24 72. 41 0. 0000115 10 85356 100308 92832 1. 335 12. 07 84. 48 0. 0000081 7 100308 115259 107783 1. 550 6. 90 91. 38 0. 0000046 4 115259 130211 122735 1. 765 1. 72 93. 10 0. 0000012 1 130211 145162 137687 1. 980 1. 72 94. 83 0. 00000.12 1 145162 160114 152638 2. 195 5. 17 100. 00 0. 0000035 3 ************************************************************ MAX = 160114 MIN = 10598 MEAN = 69542 STD.DEV. = 33738 ************************************************************ - 3 5 -TABLE 3.II. (cont'd) MONTHLY INFLOW HISTOGRAM ANALYSIS MARCH ***** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 27752 47618 37685 0. 543 18. 97 18. 97 0. 0000095 11 47618 67484 57551 0. 830 36. 21 55. 17 0. 0000182 21 67484 87351 77418 1. 116 27. 59 82. 76 0. 0000139 16 87351 107217 97284 1. 403 10. 34 93. 10 0. 0000052 6 107217 127083 117150 1. 689 3. 45 96. 55 0. 0000017 2 127083 146949 137016 1. 976 1. 72 98. 28 0. 0000009 1 146949 166815 156882 2. 262 0. 00 98. 28 0. 0000000 0 166815 186682 176749 2. 549 0. 00 98. 28 0. 0000000 0 186682 206548 196615 2. 835 0. 00 98. 28 0. 0000000 0 206548 226414 216481 3. 122 1. 72 100. 00 0. 0000009 1 ************************************************ MAX = 226414 MIN = 27752 MEAN = 69349 STD.DEV. = 30302 ************************************************************ APRIL ***** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 40847 50993 45920 0. 492 1. 72 1. 72 0. 0000017 1 50993 61140 56067 0. 601 5. 17 6. 90 0. 0000051 3 61140 71286 66213 0. 710 12. 07 18. 97 0. 0000119 7 71286 81433 76359 0. 819 18. 97 37. 93 0. 0000187 11 81433 91579 86506 0. 928 13. 79 51. 72 0. 0000136 8 91579 101725 96652 1. 036 8. 62 60. 34 0. 0000085 5 101725 111872 106799 1. 145 12. 07 72. 41 0. 0000119 7 111872 122018 116945 1. 254 17. 24 89. 66 0. 0000170 10 122018 132165 127091 1. 363 5. 17 94. 83 0. 0000051 3 132165 142311 137238 1. 471 5. 17 100. 00 0. 0000051 3 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * MAX = 142311 MIN = 40847 MEAN = 93267 STD.DEV. = 23118 ************************************************************ - 3 6 -TABLE 3.II. (cont'd) MONTHLY INFLOW HISTOGRAM ANALYSIS MAY * * * INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 80534 94677 87605 0. 596 3. 45 3. 45 0. 0000024 2 94677 108820 101748 0. 692 8. 62 12. 07 0. 0000061 5 108820 122963 115891 0. 788 18. 97 31. 03 0. 0000134 11 122963 137106 130034 0. 884 12. 07 43. 10 0. 0000085 7 137106 151249 144177 0. 980 15. 52 58. 62 0. 0000110 9 151249 165391 158320 1. 077 8. 62 67. 24 0. 0000061 5 165391 179534 172463 1. 173 13. 79 81. 03 0. 0000098 8 179534 193677 186606 1. 269 10. 34 91. 38 0. 0000073 6 193677 207820 200749 1. 365 1. 72 93. 10 0. 0000012 1 207820 221963 214892 1. 461 6. 90 100. 00 0. 0000049 4 *************************************************** MAX = 221963 MIN = 80534 MEAN =147060 STD.DEV. = 34262 ************************************************************ JUNE * * * * INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 57788 75295 66542 0. 456 3. 45 3. 45 0. 0000020 2 75295 92803 84049 0. 575 5. 17 8. 62 0. 0000030 3 92803 110310 101557 0. 695 6. 90 15. 52 0. 0000039 4 110310 127818 119064 0. 815 22. 41 37. 93 0. 0000128 13 127818 145325 136571 0. 935 15. 52 53. 45 0. 0000089 9 145325 162832 154079 1. 055 10. 34 63. 79 0. 0000059 6 162832 180340 171586 1. 175 17. 24 81. 03 0. 0000098 10 180340 197847 189094 1. 295 8. 62 89. 66 0. 0000049 5 197847 215355 206601 1. 415 5. 17 94. 83 0. 0000030 3 215355 232862 224108 1. 534 5. 17 100. 00 0. 0000030 3 ***************************************************** MAX = 232862 MIN = 57788 MEAN =146058 STD.DEV. = 40692 ************************************************************ - 3 7 -TABLE 3.II. (cont'd) MONTHLY INFLOW HISTOGRAM ANALYSIS JULY **** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 32283- 50649 41466 0. 416 6. 90 6. 90 0. 0000038 4 50649 69016 59832 0. 600 22. 41 29. 31 0. 0000122 13 69016 87382 78199 0. 784 8. 62 37. 93 0. 0000047 5 87382 105748 96565 0. 968 20. 69 58. 62 0. 0000113 12 105748 124115 114931 1. 153 17. 24 75. 86 0. 0000094 10 124115 142481 133298 1. 337 8. 62 84. 48 0. 0000047 5 142481 160847 151664 1. 521 8. 62 93. 10 0. 0000047 5 160847 179213 170030 1. 705 1. 72 94. 83 0. 0000009 1 179213 197580 188397 1. 889 1. 72 96. 55 0. 0000009 1 197580 215946 206763 2. 074 3. 45 100. 00 0. 0000019 2 ****************************************** MAX = 215946 MIN = 32283 MEAN = 99714 STD.DEV. = 41375 ************************************************************ AUGUST ****** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 13673 23516 18595 0. 349 6. 90 6. 90 0. 0000070 4 23516 33360 28438 0. 534 17. 24 24.14 0. 0000175 10 33360 43203 38281 0. 719 15. 52 39. 66 0. 0000158 9 43203 53046 48125 0. 903 15. 52 55. 17 0. 0000158 9 53046 62890 57968 1. 088 15. 52 70. 69 0. 0000158 9 62890 72733 67811 1. 273 12. 07 82. 76 0. 0000123 7 72733 82576 77654 1. 458 1. 72 84. 48 0. 0000018 1 82576 92419 87498 1. 643 3. 45 87. 93 0. 0000035 2 92419 102263 97341 1. 827 6. 90 94. 83 0. 0000070 4 102263 112106 107184 2. 012 5. 17 100. 00 0. 0000053 3 ************************************************************ MAX = 112106 MIN = 13673 MEAN = 53267 STD.DEV. = 25263 ************************************************************ -38-TABLE 3.II. (cont'd) MONTHLY INFLOW HISTOGRAM ANALYSIS SEPTEMBER ********* INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 14529 28665 21597 0. 337 12. 07 12. 07 0. 0000085 7 28665 42801 35733 0. 558 22. 41 34. 48 0. 0000159 13 42801 56936 49869 0. 778 17. 24 51. 72 0. 0000122 10 56936 71072 64004 0. 999 13. 79 65. 52 0. 0000098 8 71072 85208 78140 1. 219 12. 07 77. 59 0. 0000085 7 85208 99344 92276 1. 440 6. 90 84. 48 0. 0000049 4 99344 113480 106412 1. 661 1. 72 86. 21 0. 0000012 1 113480 127615 120548 1. 881 3. 45 89. 66 0. 0000024 2 127615 141751 134683 2. 102 5. 17 94. 83 0. 0000037 3 141751 155887 148819 2. 322 5. 17 100. 00 0. 0000037 3 ************************************************************ MAX = 155887 MIN = 14529 MEAN = 64080 STD.DEV. = 37037 ************************************************************ OCTOBER ******* INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 17490 43935 30712 0. 250 6. 90 6. 90 0. 0000026 4 43935 70379 57157 0. 465 13. 79 20. 69 0. 0000052 8 70379 96824 83602 0. 680 10. 34 31. 03 0. 0000039 6 96824 123269 110046 0. 895 20. 69 51. 72 0. 0000078 12 123269 149714 136491 1. 110 22. 41 74. 14 0. 0000085 13 149714 176158 162936 1. 325 10. 34 84.48 0. 0000039 6 176158 202603 189381 1. 541 8. 62 93. 10 0. 0000033 5 202603 229048 215825 1. 756 0. 00 93. 10 0. 0000000 0 229048 255492 242270 1. 971 1. 72 94. 83 0. 0000007 1 255492 281937 268715 2. 186 5. 17 100. 00 0. 0000020 3 ************************************************************ MAX = 281937 MIN = 17490 MEAN =122926 STD.DEV. = 59749 ************************************************************ -39-TABLE 3.II. (cont'd) MONTHLY INFLOW HISTOGRAM ANALYSIS NOVEMBER ******** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 17360 52338 34849 0. 304 3. 45 3. 45 0. 0000010 2 52338 87315 69826 0. 610 27. 59 31. 03 0. 0000079 16 87315 122293 104804 0. 916 34. 48 65. 52 0. 0000099 20 122293 157270 139781 1. 221 22. 41 87. 93 0. 0000064 13 157270 192248 174759 1. 527 5. 17 93. 10 0. 0000015 3 192248 227225 209736 1. 832 3. 45 96. 55 0. 0000010 2 227225 262203 244714 2. 138 1. 72 98. 28 0. 0000005 1 262203 297180 279691 2. 444 0. 00 98. 28 0. 0000000 0 297180 332158 314669 2. 749 0. 00 98. 28 0. 0000000 0 332158 367135 349646 3. 055 1. 72 100. 00 0. 0000005 1 ************************************************************ MAX = 367135 MIN = 17360 MEAN =114460 STD.DEV. = 53369 ************************************************************ DECEMBER ******** INTERVAL PLOT PLOT PERCENTAGE PROB. NO. OF FROM TO POSN. POSN/AV FREQ. CUM. DENSITY VALUES 36333 54799 45566 0. 413 6. 90 6. 90 0. 0000037 4 54799 73264 64031 0. 581 15. 52 22. 41 0. 0000084 9 73264 91730 82497 0. 748 13. 79 36. 21 0. 0000075 8 91730 110195 100962 0. 916 17. 24 53. 45 0. 0000093 10 110195 128661 119428 1. 084 18. 97 72. 41 0. 0000103 11 128661 147126 137893 1. 251 6. 90 79. 31 0. 0000037 4 147126 165592 156359 1. 419 8. 62 87. 93 0. 0000047 5 165592 184057 174824 ' 1. 586 5. 17 93. 10 0. 0000028 3 184057 202523 193290 1. 754 0. 00 93. 10 0. 0000000 0 202523 220988 211755 1. 921 6. 90 100. 00 0. 0000037 4 ************************************************************ MAX = 220988 MIN = 36333 MEAN =110221 STD.DEV. = 44250 ************************************************************ - 4 0 -TABLE 3 . I l l MONTHLY INFLOW PROBABILITY DISTRIBUTIONS Cumu. Prob. Jan. Feb. Mar. Apr. May Jun. 0. 05 12500 15000 18500 52350 90150 71790 0. 10 20000 24500 27000 58670 98350 87550 0. 20 31800 36900 38250 66760 107660 105060 0. 30 42700 44970 43740 72120 115120 112870 0. 40 52500 54490 49230 77880 126400 121400 0. 50 61830 61960 54710 85240 136320 132680 0. 60 73360 71220 61030 96250 146440 147660 0. 70 89720 81160 68230 104770 161150 160390 0. 80 111170 89750 75430 111260 171410 170540 0. 90 138220 102850 91330 117610 184720 190250 0. 95 170990 117000 108220 127420 204640 207180 1. 00 226260 123720 216480 137240 214890 224110 Cumu. Prob. J u l . Aug. Sep. Oct. Nov. Dec. 0. 05 37000 16000 13500 25000 38500 41000 0. 10 47000 21000 20000 40000 47000 50500 0. 20 59500 26500 26600 63500. 59000 63000 0. 30 69500 32150 32910 81500 69000 74000 0. 40 80040 38500 40260 95070 78920 85000 0. 50 88910 44840 48460 107850 89060 96000 0. 60 98040 51190 58350 119810 99210 107000 0. 70 108690 57530 69250 131610 111800 119000 0. 80 123750 65560 83080 151480 127400 136500 0. 90 145060 90450 121480 185000 149000 163000 0. 95 171850 97660 135150 243140 190000 183000 1. 00 206760 107180 148820 268720 349650 211760 -41-TABLE 3.IV. MONTHLY CONDITIONAL PROBABILITY DISTRIBUTIONS JANUARY Prob. .05 .05 .10 . 10 . 10 . 10 .10 .10 .10 . 10 .05 .05 0.05 0 0 0 0 .33 .33 0 0 0 .33 0 0 0.05 0 0 0 0 .50 .50 0 0 0 0 0 0 0.10 0 .17 0 0 . 33 0 0 .17 .17 0 .17 0 0.10 0 .25 0 0 0 0 .75 0 0 0 0 0 0.10 0 0 0 0 .17 .17 0 0 .33 .17 0 .17 0.10 0 .25 0 0 .25 0 0 .25 0 . 25 0 0 0.10 .11 .11 .11 0 .11 0 .22 0 .11 0 .11 .11 0.10 0 0 0 0 .13 .38 0 .13 .13 .13 .13 0 0.10 0 0 0 .33 0 .17 .33 0 0 .17 0 0 0.10 0 0 0 0 .33 .33 0 0 .33 0 0 0 0.05 0 0 0 0 .33 0 .33 0 0 0 .33 0 0.05 0 0 .25 0 0 0 0 .25 .50 0 0 0 FEBRUARY Prob. .05 .05 . 10 .10 . 10 . 10 . 10 . 10 .10 .10 .05 .05 0.05 1 0 0 0 0 0 0 0 0 0 0 0 0.05 .25 .25 0 .25 0 0 0 0 0 0 0 .25 0.10 0 0 0 . 50 0 0 0 0 .50 0 0 0 0.10 0 0 0 0 .50 0 .50 0 0 0 0 0 0. 10 0 .20 . 10 . 10 .30 0 0 0 . 10 0 . 10 .10 0.10 0 0 .25 . 25 0 0 .25 .13 0 .13 0 0 0.10 0 0 .13 0 .13 0 .13 . 37 .13 .13 0 0 0.10 0 .25 0 0 .25 .50 0 0 0 0 0 0 0.10 0 .13 0 .13 0 .13 .25 .13 0 .13 .13 0 0.10 0 0 0 0 .20 0 .40 0 .20 .20 0 0 0.05 0 0 0 0 . 25 .25 0 0 0 0 0 . 50 0.05 0 0 0 0 0 0 0 0 .50 .50 0 0 MARCH Prob. .05 .05 . 10 . 10 . 10 .10 .10 . 10 . 10 .10 .05 .05 0.05 0 .50 0 0 0 .50 0 0 0 0 0 0 0.05 0 0 0 0 .20 .20 .20 0 .40 0 0 0 0.10 0 .25 0 .25 0 .25 0 .25 0 0 0 0 0.10 0 0 .17 0 0 .17 0 0 .33 .17 .17 0 0.10 0 .13 0 .13 0 .25 .13 . 13 .13 .13 0 0 0.10 0 0 .25 0 0 .50 0 0 0 .25 0 0 0.10 0 0 0 0 0 .25 .13 .13 .25 0 .13 .13 0.10 0 0 0 .40 0 0 .20 .20 .20 0 0 0 0. 10 0 0 0 0 0 0 0 .40 .40 .20 0 0 0.10 0 0 0 0 0 0 .20 0 .20 .40 .20 0 0.05 0 . 50 0 0 .50 0 0 0 0 0 0 0 0.05 0 0 0 0 0 0 .50 0 .25 0 .25 0 - 4 2 -TABLE 3.IV. MONTHLY CONDITIONAL PROBABILITY DISTRIBUTIONS APRIL Prob. .05 .05 .10 . 10 .10 .10 . 10 .10 . 10 .10 .05 .05 0.05 0 0 0 0 0 0 0 0 0 0 0 0 0.05 0 0 .50 .50 0 0 0 0 0 0 0 0 0.10 0 0 0 0 0 0 .50 0 0 .50 0 0 0.10 .25 0 0 0 .50 0 0 0 0 .25 0 0 0.10 0 .50 0 0 0 0 0 0 0 .50 0 0 0.10 0 0 .20 .10 .20 .20 .10 .10 .10 0 0 0 0.10 0 0 0 . 14 .14 0 .14 .14 0 .14 0 .29 0.10 0 0 0 .17 0 .67 .17 0 0 0 0 0 0.10 0 .17 0 0 .17 0 08 .17 08 .25 08 0 0.10 0 0 0 .17 0 0 0 0 .33 .17 .33 0 0.05 0 0 .25 0 0 . 50 0 0 0 0 0 .25 0.05 0 0 0 0 0 0 0 1 0 0 0 0 MAY Prob. .05 .05 . 10 .10 . 10 . 10 . 10 .10 .10 .10 .05 .05 0.05 1 0 0 0 0 0 0 0 0 0 0 0 0.05 0 0 0 .33 0 .33 0 0 0 .33 0 0 0.10 0 0 . 20 .20 .40 . 20 0 0 0 0 0 0 0.10 0 0 0 0 .17 0 .17 .33 .17 .17 0 0 0.10 0 . 14 0 0 . 14 0 .29 . 14 . 14 0 0 . 14 0.10 0 0 .13 .13 0 . 38 .25 0 .13 0 0 0 0.10 0 0 0 .20 .20 0 .20 .20 0 0 .20 0 0.10 0 0 0 0 .40 0 0 0 .20 .20 0 .20 0.10 0 0 .50 0 0 0 0 0 0 .50 0 0 0.10 .13 0 .25 0 0 0 .13 0 .38 .13 0 0 0.05 0 0 0 0 . 33 0 0 .33 0 0 0 .33 0.05 0 .33 0 0 0 0 0 0 .33 0 0 .33 JUNE Prob. .05 .05 .10 . 10 . 10 • 10 .10 .10 . 10 .10 .05 .05 0.05 0 0 0 0 0 .50 .50 0 0 0 0 0 0.05 0 0 0 .50 0 .50 0 0 0 0 0 0 0.10 .50 0 0 0 . 17 0 .33 0 0 0 0 0 0.10 0 0 .25 0 .50 .25 0 0 0 0 0 0 0.10 .13 .13 0 .13 .25 .13 .13 0 .13 0 0 0 0.10 0 0 0 0 0 0 .20 .20 .60 0 0 0 0.10 0 0 .14 0 .43 . 14 0 0 0 . 14 . 14 0 0.10 0 0 .40 0 .20 0 .20 0 0 0 .20 0 0.10 0 0 0 0 .13 0 .25 0 .38 .13 .13 0 0.10 0 0 0 . 17 0 .17 0 0 .33 .17 0 .17 0.05 0 0 0 0 0 0 0 0 1 0 0 0 0.05 0 0 0 0 0 0 0 0 0 .50 0 .50 -43-TABLE 3.IV. MONTHLY CONDITIONAL PROBABILITY DISTRIBUTIONS JULY Prob. .05 .05 .10 .10 . 10 . 10 . 10 . 10 .10 .10 .05 .05 0.05 .50 0 .25 0 0 0 0 .25 0 0 0 0 0.05 0 1 0 0 0 0 0 0 0 0 0 0 0.10 • 25. 0 .25 0 .25 .25 0 0 0 0 0 0 0.10 0 .33 .33 0 0 0 0 .33 0 0 0 0 0.10 .10 0 .20 .10 .20 .10 .10 . 10 .10 0 0 0 0.10 0 0 .50 0 0 .17 .17 0 0 .17 0 0 0.10 0 0 .13 0 .13 .25 0 .13 .13 .25 0 0 0.10 0 0 0 0 0 0 0 0 0 1 0 0 0.10 0 0 . 10 0 0 .10 . 10 .20 . 10 . 30 .10 0 0.10 0 0 0 0 .20 .40 0 0 .20 .20 0 0 0.05 0 0 0 0 0 0 0 .33 .33 0 0 .33 0.05 0 0 0 0 0 0 0 .33 0 0 .33 .33 AUGUST Prob. .05 .05 .10 . 10 . 10 . 10 .10 .10 . 10 .10 .05 .05 0.05 .25 0 .25 0 . 50 0 0 0 0 0 0 0 0.05 0 0 0 .50 .50 0 0 0 0 0 0 0 0.10 . 10 . 10 . 30 0 . 10 0 .20 .10 .10 0 0 0 0.10 0 0 0 1 0 0 0 0 0 0 0 0 0.10 .20 0 . 20 .40 0 0 0 0 .20 0 0 0 0.10 0 0 . 13 0 0 .25 .13 0 .25 0 0 .25 0.10 0 0 0 0 . 33 0 0 0 .33 0 .33 0 0.10 0 0 0 .13 .25 0 .25 .25 .13 0 0 0 0.10 0 0 0 0 0 0 0 .20 .40 .20 . 20 0 0.10 0 0 0 .13 0 .13 .25 .25 .13 0 .13 0 0.05 0 0 0 0 0 0 . 50 0 0 .50 0 0 0.05 0 0 0 0 0 0 0 0 0 .50 0 .50 SEPTEMBER Prob. .05 .05 . 10 . 10 . 10 . 10 . 10 . 10 .10 . 10 .05 .05 0.05 0 .33 0 .67 0 0 0 0 0 0 0 0 0.05 0 0 0 1 0 0 0 0 0 0 0 0 0.10 0 . 17 0 0 . 17 .17 . 17 0 .17 .17 0 0 0.10 .17 0 0 . 17 .33 0 0 .17 .17 0 0 0 0.10 .14 0 .14 0 .14 0 .14 0 .43 0 0 0 0.10 0 0 0 .33 .33 0 0 0 0 0 0 .33 0.10 0 0 0 0 0 .63 . 13 .13 0 0 .13 0 0.10 0 0 0 .33 0 .17 0 0 .33 0 .17 0 0. 10 .11 0 .11 .11 0 .11 .11 .11 .22 0 .11 0 0.10 0 0 0 0 0 0 .33 0 0 .33 0 .33 0.05 0 0 0 0 . 33 0 .33 0 .33 0 0 0 0.05 0 0 0 0 0 0 .33 0 .33 0 0 .33 -44-TABLE 3.IV. MONTHLY CONDITIONAL PROBABILITY DISTRIBUTIONS OCTOBER Prob. .05 .05 .10 . 10 . 10 . 10 .10 .10 . 10 . 10 .05 .05 0.05 .33 0 0 0 .33 0 0 .33 0 0 0 0 0.05 0 .50 0 0 .50 0 0 0 0 0 0 0 0.10 .50 0 0 0 0 0 0 0 .50 0 0 0 0.10 0 0 .13 .13 0 .25 .13 .13 0 .13 0 .13 0.10 0 .17 .17 . 17 .17 0 0 .33 0 0 0 0 0.10 0 0 .13 .13 0 0 .13 .25 .13 .25 0 0 0.10 0 .14 0 .14 0 . 14 0 0 .29 . 14 0 . 14 0.10 0 .33 .67 0 0 0 0 0 0 0 0 0 0.10 0 .09 0 0 .09 0 .09 .18 .18 . 18 .09 .09 0.10 0 0 0 0 0 0 .50 0 .50 0 0 0 0.05 0 0 .33 0 0 0 .33 0 .33 0 0 0 0.05 0 0 0 0 0 .33 0 0 .33 .33 0 0 NOVEMBER Prob. .05 .05 . 10 .10 . 10 . 10 .10 . 10 .10 . 10 .05 .05 0.05 0 0 0 0 0 . 50 0 0 0 .50 0 0 0.05 0 0 0 .20 .20 .20 0 .20 .20 0 0 0 0.10 0 0 .17 . 50 . 17 0 .17 0 0 0 0 0 0.10 0 0 .25 0 .25 0 0 .25 .25 0 0 0 0.10 .25 0 0 .25 .25 0 .25 0 0 0 0 0 0.10 0 0 0 0 .25 .25 0 0 0 .25 .25 0 0.10 0 0 0 0 0 0 . 20 .20 .40 . 20 0 0 0.10 0 .13 0 0 .13 .13 0 .25 .13 0 .13 .13 0.10 0 0 0 0 0 0 .11 .11 .22 .22 .33 0 0.10 0 0 0 .14 . 29 .14 .14 0 .14 .14 0 0 0.05 0 0 0 0 0 0 0 0 0 1 0 0 0.05 0 0 0 0 0 0 0 .67 .33 0 0 0 DECEMBER Prob. .05 .05 .10 . 10 . 10 . 10 . 10 . 10 .10 .10 .05 .05 0.05 0 0 0 0 0 0 1 0 0 0 0 0 0.05 0 0 0 0 0 0 0 0 1 0 0 0 0.10 0 0 0 .50 0 0 .50 0 0 0 0 0 0.10 0 0 .17 .17 .33 0 .33 0 0 0 0 0 0.10 0 0 .13 . 13 .13 0 0 .25 .25 .13 0 0 0.10 0 0 .20 0 0 0 .20 0 .20 .20 .20 0 0.10 0 0 .20 0 . 20 . 20 0 .20 .20 0 0 0 0.10 .25 0 0 .13 .13 .25 0 .25 0 0 0 0 0. 10 .11 .11 .11 0 0 0 .22 .11 .11 .11 .11 0 0.10 0 0 .14 0 .14 0 0 .14 0 0 .14 .43 0.05 0 .20 0 0 0 . 20 .40 . 20 0 0 0 0 0.05 0 0 0 0 0 0 0 0 0 0 0 1 - 4 5 -3.4 Stochastic Dynamic Programming Formulation It should be noted that i n stochastic DP, computations s t a r t at the end of the planning period and each set of computations looks forward i n time but f o r the next set of computations, a step backwards i n time i s taken. Since the forward time indexing was used i n t h i s study, the time r e l a t i o n of computations can be expressed as follows: Time progresses i n th i s d i r e c t i o n »-Time periods 1 2 ... t - l t t+1 Streamflows Ij_ l 2 Reservoir storage s^ s 2 s^ Release R l R2 I t - 1 I t It+1 s t - l s t s t + l st+2 R t - 1 R t Rt+1 Computations progress i n t h i s d i r e c t i o n Let "tj B = reservoir storage at the beginning time period t; = inflow at p r o b a b i l i t y l e v e l j during time period t; = reservoir release during time period t; = pr o b a b i l i t y at l e v e l j ; = return obtained as a consequence of operational decision; - 4 6 -P ^ t k ^ t - l , j j) = conditional p r o b a b i l i t y of inflow I t k i n t t h t i m e p e r i o d , a t pr o b a b i l i t y l e v e l k given inflow in previous time period t-1 i s I t-1, j at p r o b a b i l i t y l e v e l j ; = expected r e t u r n from the optimal operation of the system which has t time periods to the end of the time horizon. The s t o c h a s t i c DP fo r m u l a t i o n can be w r i t t e n f o r each case as follows: Case 1: Uncorrelated Inflows (a) Discharge d e c i s i o n : Powerhouse discharge was chosen as the d e c i s i o n v a r i a b l e and i t v a r i e d from 0 to the capacity of the plant. For a given p o s s i b l e v a l u e of inflow, the model searches for the maximum return among various discharges (each d i s c h a r g e produces a d i f f e r e n t r e t u r n ) . The expected return can then be determined for a l l possible inflows with t h e i r corresponding p r o b a b i l i t i e s . The procedure can be formulated using the no t a t i o n defined above: J - 4 7 -max j 2. Rt ( J f t ( S t ) = ax<L P j ^ B ( S t , I t j , R t ) + f t + 1 ( S t + 1 ) (3-5) Given the reservoir storage l e v e l S t at the beginning of of time period t, a p o s s i b l e value of i n f l o w I t j and a release Rt, the r e s u l t i n g reservoir storage at the end of time p e r i o d t may not be f a l l i n g on any pr e -d etermined s t a t e S t + 1 . T h e r e f o r e , the f u n c t i o n ^t+l^t+1^ which i s the maximum expected return from the n e a r e s t pr e-determined s t a t e St+-^ was chosen to associate with the calculated reservoir storage l e v e l . The expected return f t + l ^ s t + l ^ c a n a l s o b e interpolated between the two nearest pre-determined states. With the small i n t e r v a l adopted between reservoir storage states the error induced by the nearest state approximation was considered to be small enough to be acceptable. The constraints include: - continuity : S t +^-S t = I t-R t~P t - reservoir capacity : 0 ^  S t ^ s m a x - powerhouse capacity : 0 ^ Rt ^ in which P t i s the s p i l l . These constraints are the same for the other cases. (b) State change decision : When the state change i s used as used as the d e c i s i o n v a r i a b l e , the model, at a time period t and with a given possible inflow, computes the -48-release Rt and the return B for each possible reservoir storage volume i n time period t+1, and s e l e c t s the stat e which produces the maximum return. By doing t h i s i n a s i m i l a r way f o r a l l p o s s i b l e i n f l o w s , with the asso c i a t e d p r o b a b i l i t i e s , the expected r e t u r n w i l l be determined. The formulation i s : Case 2: Perfectly Correlated Inflows (a) Discharge decision : For each possible inflow, the model computes the maximum return over the entire time horizon (a deterministic analysis). The t o t a l expected return i s determined by taking the summation of the products of each i n d i v i d u a l d e t e r m i n i s t i c r e t u r n and the corresponding inflow probability. The model can be expressed mathematically f o r each pr o b a b i l i t y l e v e l as follows: ( b ) S t a t e change d e c i s i o n : S i m i l a r t o the d i s c h a r g e d e c i s i o n case, twelve d e t e r m i n i s t i c DP models with twelve p o s s i b l e i n f l o w s were s o l v e d with a common t+1 - 4 9 -C objective function. The maximum returns were found from among the d i f f e r e n t states: f t j ( S t j ) = max Jlp3  St+1,j B ( s t j ' I t j ' s t + l , j ) + f t + l , j ( s t + l , j } (3-8) Case 3: P a r t i a l l y Correlated Inflows There were two d i f f e r e n t types of p r o b a b i l i t y involved i n t h i s case. The f i r s t type was the monthly p r o b a b i l i t y d i s t r i b u t i o n f o r the i n f l o w s i n the previous time period. The second type was the conditional p r o b a b i l i t y d i s t r i b u t i o n for the i n f l o w s i n the current time period. The model computed the expected return for a l l of the possible inflows i n the current time peri o d c o n s i d e r i n g each of twelve possible inflows i n the previous time period. (a) Discharge decision: f t ^ t ^ t - l ) = maxjZ Pj Z P ( l t k | l t - i , j) Rt ( 3 k B ( s t ' I t k ' R t ) •*- f t + 1 ( s t + 1 , i t ) (3-9) (b) State change decision! f t ^ t ^ t - l ) = m a x < ? Pj Z P ( I t k | l t - i , j) >t+l( ^  B ( s t ' I t k ' s t + l ) + f t + l ( s t + l ' (3-10) -50-Th e r e t u r n i n the o b j e c t i v e f u n c t i o n was a f u n c t i o n of reservoir storage and powerhouse discharge. It can be written as B = r t.C Q.Q + r t + 1 . C s . S (3-11) where B = the return i n KW C Q = power c o e f f i c i e n t i n KW/cfs Q = powerhouse discharge i n cfs Cg = storage c o e f f i c i e n t i n KW/cfs-days S = storage volume i n cfs-days r t = discount factor i n time period t The c a l c u l a t i o n of the c o e f f i c i e n t C Q and Cg i s i l l u s t r a t e d i n Appendix A. -51-4. RESULTS OF THE OPTIMIZATION ANALYSIS 4.1 Case 1: Uncorrelated Inflows The r e s u l t s with state change as the decision variable are shown i n Table B.I. The f i r s t column i s the r e s e r v o i r l e v e l i n feet on the f i r s t day of a month. The other twelve columns indicate the optimal reservoir l e v e l s i n feet on the l a s t day of twelve months from January to December. For example, as seen i n the table, i f reservoir l e v e l on January 1 i s 275.0 f t , the sequence of optimal r e s e r v o i r l e v e l s on the l a s t day of each month f o r the whole year are 275.5, 276.5, 275.5, 276.0, 280.5, 284.5, 285.0, 283.0, 282.0, 283.5, 284.0, 285.0. The numbers i n the l a s t column are the t o t a l returns i n MW f o r the e n t i r e three year time h o r i z o n c o n s i d e r i n g the s t a r t i n g r e s e r v o i r l e v e l i n the f i r s t column. In a s i m i l a r format as i n Table B.I, the r e s u l t s with powerhouse d i s c h a r g e as the d e c i s i o n v a r i a b l e are i l l u s t r a t e d i n Table B.II except that the numbers i n the twelve time step columns are not r e s e r v o i r l e v e l s but the powerhouse flows i n 1000 c f s . U n l i k e the sta t e change decision case i n which the optimal reservoir operations for the e n t i r e year can be read d i r e c t l y from the s o l u t i o n table, the r e s u l t s for the discharge decision case show the optimal powerhouse release during a month corresponding to -52-the r e s e r v o i r l e v e l at the beginning of the month i n the f i r s t column. Therefore the expected reservoir l e v e l at the end of the month must be computed ( u s i n g the i n f l o w distribution) i n order to determine the optimal release for the next month. Tables B.III and B.IV summarize the r e s u l t s i n Tables B.I and B.II r e s p e c t i v e l y for f i v e d i f f e r e n t s t a r t i n g r e s e r v o i r l e v e l s and p r e s e n t them i n a more e a s i l y comparable format. Table B.III contains the f o l l o w i n g information: Col. 1: months of the year, from January to December Col. 2: monthly average reservoir inflow i n cfs-days. These va l u e s are shown f o r reference only. They were not used i n the optimization analysis. C o l . 3: optimal r e s e r v o i r l e v e l s i n feet at the end of each month given the s t a r t i n g r e s e r v o i r l e v e l at the beginning of January shown on the top. These l e v e l s are extracted from Table B.I. Col. 4: ending reservoir volume i n cfs-days associated with reservoir reservoir levels i n c o l . 3. Col. 5: expected powerhouse flow i n cfs-days given starting and ending r e s e r v o i r l e v e l s and twelve d i s c r e t e values of inflow for a given month. Col. 6: Expected energy i n MW-month computed using expected -53-powerhouse flow i n c o l . 5 and corresponding power c o e f f i c i e n t . Table B.IV contains the same information as that i n Table B.III except that the powerhouse flows in column 5 are the optimal solutions derived from Table B.II and the ending r e s e r v o i r l e v e l s i n column 3 are the expected values computed given a s t a r t i n g r e s e r v o i r l e v e l , a p o s s i b l e release and twelve discrete inflows. The f o l l o w i n g c h a r a c t e r i s t i c s of these r e s u l t s were noticed: - The t o t a l r e t u r n of the discharge d e c i s i o n case i s higher than that of the s t a t e change d e c i s i o n case (Tables B.I and B.II); - The reservoir l e v e l at the end of the year i s higher for the s t a t e change d e c i s i o n case than for the discharge decision case (Tables B.III and B.IV); - Although the discharge d e c i s i o n case has higher energy production for an assumed starting reservoir l e v e l , i t also has monthly minimum energy production which i s much l e s s than the s t a t e change d e c i s i o n (Tables B.III and B.IV); - The reservoir l e v e l at the end of December i s 285 f t ( f u l l supply l e v e l of the r e s e r v o i r ) i n the sta t e change decsion case for a l l starting r e s e r v o i r l e v e l s -54-on January 1 ( F i g u r e B . l ) , whereas the f i n a l r eservoir l e v e l i n the discharge decision case varies from 277.5 to 279 f t (Figure B.2). The discharge decision case always has a higher return because a f t e r r e l e a s i n g a given amount of water fo r power production, most of the s u r p l u s water can be stored i n the reservoir. In the state change decision case the release are generally less so that, after storing some amount of water to achieve a given reservoir l e v e l , a portion of the surplus water must be s p i l l e d because of the l i m i t e d powerhouse capacity. A t e s t run w i t h the model was made i n which the powerhouse capacity was set ten times bigger than the actual value. The o p t i m i z a t i o n r e s u l t s f o r t h i s f i c t i o n a l power plant indicate that the state change decision case produces higher returns than the discharge decision case. Therefore, i t can be concluded that for uncorrelated inflows d i f f e r e n t r e s u l t s w i l l be produced with d i f f e r e n t choices of decision v a r i a b l e s . The f a c t o r s which a f f e c t the t o t a l r e t u r n are s t o r a g e c a p a c i t y , powerhouse c a p a c i t y , and i n f l o w p r o b a b i l i t y d i s t r i b u t i o n s . -55-4.2 Case 2 : Perfectly Correlated Inflows Tables C.I and C.II show the o p t i m i z a t i o n r e s u l t s for Case 2 i n which the in f l o w s are assumed to be p e r f e c t l y correlated. These tables have the same format as the Tables B.I and B.II except that f o r each s t a r t i n g r e s e r v o i r l e v e l there are twelve d i f f e r e n t s o l u t i o n s f o r twelve p o s s i b l e inflow sequences. The t o t a l r e t u r n f o r the discharge d e c i s i o n case i s s l i g h t l y higher than that f o r the s t a t e change d e c i s i o n case. This i s probably because of the approximation which i s made when the expected r e t u r n at the nearest s t a t e i n the next time perio d i s added to the expected r e t u r n i n the current time period. Compared with the r e s u l t s of the Case 1 (uncor re 1 a ted i n f l o w s ) , Case 2 has higher t o t a l expected r e t u r n with the state change decision and lower t o t a l expected return with the discharge decision. When the actual powerhouse capacity i s r e p l a c e d by a very l a r g e number, however, both Case 1 and Case 2 produce the same t o t a l expected return. T a b l e s C.III and C I V summarize the o p t i m i z a t i o n r e s u l t s for three d i f f e r e n t s t a r t i n g reservoir l e v e l s and these results are shown graphically i n Figures C.3 and C.4. -56-4.3 Case 3 : P a r t i a l l y Correlated Inflows This case produced a lower t o t a l expected return compared wi t h the f i r s t two cases. T h i s i s at l e a s t p a r t i a l l y e x plained by d i f f e r e n c e s among the p r o b a b i l i t y d i s t r i b u t i o n s . In Case 3 the c o n d i t i o n a l p r o b a b i l i t y d i s t r i b u t i o n s f o r i n f l o w s are d i f f e r e n t from the monthly p r o b a b i l i t y d i s t r i b u t i o n s used i n Case 1 and Case 2 and these differences change the t o t a l expected return. As i n Case 1, the discharge d e c i s i o n case has higher t o t a l expected r e t u r n than the s t a t e change d e c i s i o n case. When the powerhouse c a p a c i t y i s repl a c e d by a very l a r g e number the opposite r e s u l t was obtained. Therefore, the same conclusion can be drawn which i s that d i f f e r e n t r e s u l t s w i l l be produced with d i f f e r e n t choices of decision variables. Tables D.I, D.II, D.III, and D.IV and Figures D.l and D.2 i l l u s t r a t e the results for Case 3. -57-5. REAL-TIME OPERATION CONSIDERATIONS 5.1 Reservoir Inflows Forecasting Given the considerable number of r a i n f a l l - r u n o f f models which have been developed to date, s e l e c t i n g a s u i t a b l e model for real-time use i s not an easy task. Generally, the r a i n f a 1 1 - r u n o f f models can be c l a s s i f i e d i n t o t h r e e d i f f e r e n t types (Wood and O'Connell, 1985): (a) D i s t r i b u t e d models. The models of catchment hydrology that are fi r m l y based i n our understanding of the physics of the hydrological process which control catchment response. With such models the o b j e c t i v e i s to use the equations of mass, energy, and momentum to describe the movement of water over the land surface and through the unsaturated and saturated zones. The descriptive equations fo r p h y s i c a l l y based models are i n general 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 that cannot be s o l v e d a n a l y t i c a l l y . Solutions must then be solved numerically at a l l points on a three-dimensional grid representation of a catchment system. (b) Conceptual models. These models are formulated on the b a s i s of a simple arrangement of a number of component processes, each of which i s a s i m p l i f i e d but p l a u s i b l e conceptual r e p r e s e n t a t i o n of one process element i n the -58-r a i n f a l 1 - r u n o f f process instead of using the r e l e v a n t equations of mass, energy, and momentum. (c) B l a c k box models. These models depend upon e s t a b l i s h i n g a r e l a t i o n s h i p between r a i n f a l l input and strearaflow output without attempting to describe any of the p h y s i c a l l y based t r a n s f e r f u n c t i o n s of the i n t e r n a l mechanisms whereby t h i s transformation takes place. Such models may include successful approaches as unit hydrogrpah, extreme frequency analysis, regression analyses, etc. The following factors should be considered i n selecting a suitable model for real-time forecasting: - The r e l i a b i l i t y of flow f o r e c a s t s . This i s l a r g e l y i n f l u e n c e d by the l e v e l of accuracy with which a mathematical model represents the response of a catchment to r a i n f a l l . It i s a l s o a f f e c t e d by the a v a i l a b i l i t y and r e l i a b i l i t y of r a i n f a l l forecasts i f the lead-time for which the flow forecast i s required exceeds the catchment 'lag'. - Update c a p a b i l i t y . I f r a i n f a l l and flow data are a v a i l a b l e i n real-time, flow forecasts can be updated at each time point as new data become a v a i l a b l e . This can be done e f f i c i e n t l y o n l y with models that have minimal computation requirements. -59-- Adaptive mode operation. For real-time forecasting i t i s necessary to have a model that can operate within the adaptive mode; that i s , the model output u t i l i z e s p revious observed outputs i n c a l c u l a t i n g current model output. -60-5.2 Real-time Optimization Using Steady-state Solution The operations that occur i n the current period and some subsequent periods are i n f l u e n c e d by the current conditions, and they are defined as the real-time operation. The i n f l o w s used f o r r e a l - t i m e o p e r a t i o n s are f o r e c a s t values based on the a c t u a l i n f l o w i n the period p r e v i o u s to the c u r r e n t p e r i o d . These i n f l o w s are represented by t h e i r p r o b a b i l i t y d i s t r i b u t i o n s . In the r e a l - t i m e operations, the d e c i s i o n makers need to know the optimal r e l e a s e d e c i s i o n f o r o n l y the current period. They have to make the immediate d e c i s i o n on the volume of water to release i n order to obtain a p r a c t i c a l l y good return without loosing future benefit. In other words, the r e a l - t i m e o p e r a t i o n s must s a t i s f y the c u r r e n t requirement and must be c o n s i s t e n t with the long-term optimization pol i c y . The steady-state optimal r e s e r v o i r l e v e l at a time p e r i o d i n the future can be used to c o n s t r a i n the ending reservoir l e v e l for real-time operation. Therefore, shorter time peri o d o p t i m i z a t i o n can be achieved by using the results from the longer time period optimization. Note that the optimal reservoir l e v e l s at a given time period are not r e a d i l y a v a i l a b l e from the discharge decision -61-r e s u l t s but they can be computed using the optimal releases and the i n f l o w p r o b a b i l i t y d i s t r i b u t i o n s . The r e s u l t s are the expected value of r e s e r v o i r l e v e l s . The sta t e change d e c i s i o n case, on the o t h e r hand, g i v e s the o p t i m a l reservoir l e v e l at any given time period and thi s makes i t much easier for real-time applications. Although the discharge d e c i s i o n case produced higher t o t a l expected return than the state change decision case by about 7 percent, i t a l s o has much l e s s minimum energy production. This i s normally an important c o n s t r a i n t i n hydropower o p e r a t i o n s , a l t h o u g h not i n the example considered. Since an immediate d e c i s i o n has to be made about the volume of water to release, the optimization r e s u l t in Case 2 which has the twelve d i f f e r e n t s o l u t i o n s f o r the f i r s t time period may be d i f f i c u l t to use. The recursive equation, therefore, should be modified to produce one si n g l e solution for the f i r s t time period. On the other hand, the optimal r e s e r v o i r l e v e l s from steady-state r e s u l t s f o r Case 1 and Case 3 are the expected values and they may not be the optimal s o l u t i o n f o r r e a l - t i m e operation i n a dry or wet year. In summary, the s t a t e change d e c i s i o n case g i v e s optimal reservoir l e v e l s d i r e c t l y for real-time applications -62-whereas some computation i s required for the discharge decision case. The re s u l t s from Case 2 in which inflows are assumed p e r f e c t l y correlated give more information than the other two cases but t h i s model does not r e a l i s t i c a l l y represent p r a c t i c a l situations. -63-6. CONCLUSIONS In t h i s study, the optimization of long-term operations of a hydropower reservoir i s undertaken using stochastic DP with d i f f e r e n t decision variables and d i f f e r e n t degrees of s e r i a l c o r r e l a t i o n of the inflows. The r e s u l t s from the s t a t e change d e c i s i o n case are more conservative than the r e s u l t s from discharge decision case i n terms of higher f i n a l reservoir l e v e l (which means less r i s k for future), and higher minimum energy production although at the cost of lower t o t a l expected return. The difference i n t o t a l expected return between Case 1 and Case 2 depends upon the physical c h a r a c t e r i s t i c s of the system such as maximum storage volume and powerhouse capacity. Case 3 has a d i f f e r e n t inflow d i s t r i b u t i o n and i t provided the least t o t a l expected return. The problem of s e r i a l c o r r e l a t i o n may be s o l v e d by i n c o r p o r a t i n g c u m u l a t i v e i n f l o w d i s t r i b u t i o n s i n t o s t o c h a s t i c DP rather than a n a l y z i n g i n f l o w s f o r each separate time period. Turbine dispatching optimization should be incorporated i n t o r e s e r v o i r operation o p t i m i z a t i o n problem to handle a change i n powerhouse c a p a c i t y with head and a change i n power c o e f f i c i e n t with generator u n i t s . -64-REFERENCES Arunkumar, S., and Yen, W. W-G., " P r o b a b i l i s t i c Models i n the Design and Operation of a Multi-purpose Reservoir System," Contribution No. 144, C a l i f o r n i a Water Resources Center, University of C a l i f o r n i a , Davis, December 1973. Askew, A., "Optimum Reservoir Operating P o l i c i e s and the Imposition of a R e l i a b i l i t y Constraint," Water Resources Reasearch, Vol. 10, No. 1, February 1974, pp. 51-56. Bellman, R., "Dynamic Programming", Princeton U n i v e r s i t y Press, Princeton, New Jersey, 1957. Bellman, R., and Dreyfus, S., "Applied Dynamic Programming", Princeton, New Jersey, 1962. Benjamin, J.R., and Cornell, C.A., "Probability, S t a t i s t i c s , and Decision for C i v i l Engineers", McGraw-Hill, New York, 1970. Butcher, W.S., "Sto c h a s t i c Dynamic Programming f o r Optimum Reservoir Operation," Water Resources B u l l e t i n , Vol. 7, No. 1, February 1971, pp. 115-123. Caselton, W.F., and R u s s e l l , S.O., "Long-term Operation of Storage Hydro Projects", Journal of the Water Resources Planning and Management D i v i s i o n , ASCE, V o l . 102, No. WR1, A p r i l , 1976, pp. 163-176. Charles Howard & Associates Ltd., "Lake Management Model", Technical Report, V i c t o r i a , B.C., June 1987. Gablinger, M., and Loucks, D.P., "Markov Models for Flow Regulation," Journal of the H y d r a u l i c s D i v i s i o n , ASCE, Vol. 96, No. HY1, January 1970, pp. 165-181. G e s s f o r d , J., and K a r l i n , S., "Optimal P o l i c y f o r Hydroelectric Operations," i n Studies i n the Mathematical Theory of Inventory and Production, edited by K.J. Arrow, S. K a r l i n , and H. Scarf, Standford U n i v e r s i t y Press, C a l i f o r n i a , 1958, pp. 179-200. H i l l i e r , F.S., and Lieberman, G.J., " I n t r o d u c t i o n to Operations Research", Holden-Day, Inc., San Francisco, 1980. -65-Howard, R.A., "Dynamic Programming and Markov Processes", MIT Press, Cambridge, Massachusettes, 1960. L i t t l e , J.D.C., "The Use of Storage Water i n a Hydroelectric System," Operations Research, V o l . 3, No. 2, May 1955, pp. 187-197. Loucks, D.P., S t e d i n g e r , J.R., and H a i t h , D.A., "Water Resource Systems Planning and Analysis", Prentice-Hall, Inc., New Jersey, 1981. Nemhauser, G.L., "Dynamic Programming", John Wiley, New York, 1966. Rossman, L., " R e l i a b i l i t y Constrained Dynamic Programming and Randomized Release Rules i n Reservoir Management," Water Resources Research, Vol. 13, No. 2, A p r i l 1977, pp. 247-255. 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Yakowits, S., "Dynamic Programming A p p l i c a t i o n s i n Water Resources," Water Resources Research, V o l . 18, No. 4, August 1982, pp. 673-696. Yeh, W. W-G., "Reservoir Management and Operations Models. A State-of-the-Art Review," Water Resources Research, Vol. 21, No. 12, December 1985, pp. 1797-1818. -66-APPENDIX A. CALCULATION OF THE OBJECTIVE FUNCTION COEFFICIENTS - 6 7 -APPENDIX A. Calculation of Objective Function Coefficients A.l Storage Coefficients The v a l u e of storage i n v o l v e s a t r a d e - o f f between maximizing the cu r r e n t head and r e s e r v i n g space to avoid f u t u r e s p i l l . The p e n a l t y f o r s p i l l i m p l i e d i n the maximization of the energy output i m p l i c i t l y accounts for the value of storage space. The v a l u e f o r head however can be a n o n - l i n e a r f u n c t i o n of storage although for the r e s e r v o i r e v a l u a t e d here t h i s i s not the case. A s p e c i a l c o e f f i c i e n t can be chosen to r e f l e c t the marginal value of storage based on the l o c a l slope of the.stage-storage curve and the conversion factor which converts discharge to power. Power generated i n any time period i s given by: P = K(H) . Q . H (A-1) i n which P i s power i n KW, K i s a conversion f a c t o r which v a r i e s with head, Q i s discharge i n c f s , and H i s head i n Value of K v a r i e s from 11.14 to 12.54 KW/cfs as head changes from 165 f t to 185 f t (measured data), or K v a r i e s from 11.14/165 = 0.067515 KW/cfs.ft to 12.54/185 = 0.067784 KW/cfs.ft. -68-Assuming a lin e a r change of K with head, then 3K / 3H = (0.067784 - 0.067515) / (185 - 165) = 1.345 x 10~ 5 Therefore, K(H) = 0.067515 + 1.345 x 10" 5 (H - 165) = 0.065296 + 1.345 * 10 _ 5H Equation (A-l) can be written as P = (0.065296 + 1.345 * 10~5H).HQ = (a + bH).HQ and, 3P 3P dp dP = dK + dH + dQ (A-2) 3K 3 H 3 Q = QH.dK + KQ.dH + KH.dQ or, dP = (bQH + KQ).dH + KH.dQ = (bH + K).Q.dH + KH.dQ The e f f e c t of storage on power generation i s 3 p 3H 2Q = (K + bH) . Q. + KH. (A-3) 3 S 3 S 3 S i n which 3 Q / 3 s can be assumed to be zero because a change i n storage does not a f f e c t Q. Thus, - 6 9 -3 P 3 H = (K + bH).Q. (A-4) 3 S 3 S where 3 H / 3 S i s the slope of the stage-storage curve and Q i s the powerhouse flow c a p a c i t y at the corresponding, head-dependent value of K. The stage-storage curve for th i s p a r t i c u l a r reservoir has a constant slope of 6.8395 * 1 0 - 5 f t / c f s - d a y s i n the range of head 165 - 185 f t . For p r a c t i c a l purposes, the values of H, Q and K were taken at t h e i r maximum: H = "max = 1 8 5 f t K = K m a x = 0.067784 KW/cfs.ft Q = Qmax = 3 8 5 0 c f s and, 3P 3S = 0.018504 KW/cfs-days 0.067784 + 1.345 * 10~ 5(185)| 3850(6.8395 * 10~5) Therefore, the storage c o e f f i c i e n t i s C s = 0.018504 KW/cfs-days A.2 Power Coefficients The power c o e f f i c i e n t i s the conversion f a c t o r K mentioned above. It accounts f o r changes i n e f f i c i e n c y of the u n i t s i n the powerhouse and v a r i e s from 11.14 to 12.54 -70-KW/cfs as the head changes from 165 f t to 185 f t . During a time period t, the mean r e s e r v o i r l e v e l i s determined as the average of r e s e r v o i r l e v e l s at the beginning and the end of the time period. I f H i s the mean reservoir l e v e l , the power c o e f f i c i e n t i s given by C Q = 0.07 H - 7.41 In t h i s powerhouse the tailwater i s independent of the powerhouse discharge and the r e f o r e does not i n f l u e n c e the analysis. -71-APPENDIX B. OPTIMIZATION RESULTS FOR CASE 1 -72-TABLE B . I . OPTIMAL MONTHLY RESERVOIR LEVELS (ft) CASE 1: UNCORRELATED INFLOWS - STATE CHANGE DECISION Elev . 1 2 3 4 Time step (month) 5 6 7 8 9 10 11 Return 12 MW 265. .0 265 .5 266. .0 266.0 268.5 271, .0 269. 5 267. . 5 266, .0 265. . 5 266. . 5 267. , 5 267. . 5 32558 265. , 5 266.0 266. .5 266, .5 269. .0 271. .5 270.0 268. .0 266. .5 266. ,0 267. ,0 268. ,0 268. .0 32647 266. .0 266, . 5 267. ,0 267. .0 269. 5 272, .0 270.5 268. . 5 267, ,0 266. ,5 267, , 5 268. .5 268. , 5 32736 266. , 5 267, .0 267, ,5 267. , 5 270. ,0 272.5 271.0 269. ,0 267. , 5 267. 0 268. .0 269. ,0 269. ,0 32826 267. .0 267 .5 268, .0 268.0 270, .5 273, .0 271.5 269.5 268. .0 267. .5 268, .5 269. .5 269. . 5 32915 267, . 5 268, .0 268. .5 268. .5 271, ,0 273, .5 272.0 270. .0 268. .5 268. ,0 269. .0 270. .0 270. .0 33004 268. .0 268 .5 269, .0 269, .0 271. .0 274, .0 272.5 270, .5 269. .0 268. .5 269, .5 270, . 5 270. , 5 33093 268. . 5 269, .0 269. .5 269, , 5 271. .0 274. .5 273.0 271. .0 269. .5 269. ,0 270. .0 271. ,0 271. .0 33182 269. ,0 269, . 5 270. .0 270. .0 271, ,0 275, .0 273.5 271, .5 270. .0 269. .5 270. .5 271, .5 271. . 5 33271 269. , 5 270, ,0 270. .5 270. .5 271. .0 275. .5 274.0 272. ,0 270. ,5 270.0 271. .0 272. ,0 272. ,0 33358 270. .0 270, .5 271. ,0 271. .0 271. . 5 276. .0 274.5 272. .5 271. .0 270. 5 271. . 5 272. .5 272. , 5 33445 270. , 5 271, .0 271. , 5 271. , 5 272. ,0 276, ,5 275.0 273. ,0 271. ,5 271. ,0 272. ,0 273. ,0 273. ,0 33532 271. .0 271. .5 272. ,0 272. ,0 272. ,5 277. .0 275.5 273. , 5 272. .0 271. ,5 272. ,5 273. .5 273. , 5 33619 271. ,5 272. ,0 27 2. ,5 272. .5 273. ,0 277. .5 276.0 274. .0 272. .5 272. ,0 273. .0 274, .0 274, .0 33705 272, .0 272, . 5 273. ,0 273. .0 273. . 5 278, .0 276.5 274, .5 273. .0 272. .5 273. .5 274. ,5 274, . 5 33792 272. . 5 273.0 273. .5 273. .5 273. .5 278. .5 277.0 275. .0 273. .5 273. ,0 274. .0 275. .0 275. .0 33878 273. .0 273. .5 274. ,0 274. ,0 274.0 279, .0 277.5 275. .5 273.5 273. .5 274. . 5 275, .5 275. .5 33964 273. .5 274.0 274. 5 274. .5 274. ,5 279. ,0 278.0 276.0 273. ,5 274. ,0 275. .0 276. ,0 276. .0 34050 274.0 274, , 5 275. .0 274. ,5 275. .0 279.0 278.5 276. .5 273. ,5 274. ,5 275, .5 276. ,5 276. . 5 34136 274. 5 275. .0 275. ,5 274. ,5 275. ,5 279. .0 279.0 277. .0 274.0 275. ,0 276. ,0 277. .0 277. ,0 34221 275. ,0 275. . 5 276. .0 274.5 275. .5 279. 5 279.5 277. , 5 274.5 275. ,5 276. 5 277. ,5 277, . 5 34305 275. 5 276. .0 276. ,5 275. ,0 276.0 280. ,0 280.0 278. ,0 275. ,0 276. 0 277. ,0 278.0 278. ,0 34390 276. .0 276. ,5 276. , 5 275. ,0 276. ,0 280. .5 280.5 278. . 5 275. ,0 276. ,5 277. , 5 278. , 5 278. . 5 34473 276. 5 277.0 277. ,0 275. .5 276. , 5 281. .0 281.0 279. .0 275, .5 277. ,0 278. ,0 279. .0 279. .0 34556 277. .0 277. . 5 277. .5 275, ,5 277. .0 281. .0 281.5 279. .5 276, .0 277. .5 278. .5 279. .5 279, . 5 34640 277. 5 278.0 278. ,0 276. ,0 277.5 281. .5 282.0 280.0 276, .5 278. .0 279. .0 280. ,0 280, .0 34722 278. ,0 278.5 278.5 276.5 278.0 281. .5 282.5 280. .5 277, .0 278.5 279, .5 280, . 5 280 .5 34804 278. 5 279. .0 278. .5 276. .5 278. ,5 282. .0 283.0 281. .0 277. .0 279. .0 280, .0 281. .0 281, .0 34887 279. 0 279. .5 279. ,0 277. .0 278. 5 282. .0 283.5 281. .5 277. .5 279. .5 280, .5 281. .5 281. . 5 34968 279. 5 280. ,0 279. 5 277. ,5 279. ,0 282. .5 284.0 282. ,0 278. .0 280. ,0 281. .0 281. ,5 282. .0 35049 280. 0 280. , 5 280. ,0 278. .0 279. .5 282. .5 284.5 282. . 5 278. ,5 280. . 5 281, , 5 282. .0 282. . 5 35129 280. 5 281. ,0 280. 0 278. .5 280. ,0 283. ,0 284.5 283. ,0 279. ,0 281. ,0 282. ,0 282. . 5 283. ,0 35206 281. 0 281. .5 280. .5 27 8, .5 280. .0 283. .0 285.0 283. .0 279. ,S 281. .5 282. .5 283, .0 283, . 5 35283 281. 5 282. ,0 281. ,0 278. .5 280. , 5 283. .5 285.0 283. .5 279. .5 281. .5 283. .0 283. .5 284. .0 35360 282. 0 282. , 5 281. , 5 279.0 281. ,0 283. .5 285.0 283. ,5 280. .0 282. ,0 283, .5 283, . 5 284, .5 35436 282. 5 283. ,0 282. 0 279. ,5 281. ,5 284.0 285.0 284. ,0 280.5 282. ,0 284. .0 283. .5 284. .5 35511 283. 0 283. .5 282. ,0 280. ,0 282. ,0 284.0 285.0 284, .5 281. ,0 282. ,0 284. . 5 284, .0 284, . 5 35586 283. 5 283. 5 282. 5 280. .0 282. ,0 284. .5 285.0 285. .0 281. .5 282. ,0 285. .0 284. ,0 285. .0 35659 284. 0 283. , 5 282. , 5 280. .5 282. , 5 284. ,5 285.0 285. ,0 282. .0 282. .0 285. .0 284. .5 285, .0 35733 284. 5 284. 0 283. 0 281. 0 283. 0 285. ,0 285.0 285.0 282. .5 282. ,5 285, ,0 285. .0 285. .0 35804 285. 0 284.5 283. ,5 281. ,5 283. ,0 285. ,0 285.0 285. ,0 283. ,0 283. ,0 285. .0 285. ,0 285. .0 35875 -73-TABLE B . I I . OPTIMAL MONTHLY POWERHOUSE FLOWS (1000 cfs) CASE 1: UtJCORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev . 1 2 3 4 5 6 7 8 9 10 11 12 MW 265. .0 0 .0 6 .7 0, .0 0, .0 0. .0 0, .0 0. ,0 0. ,0 0. .0 14, .9 36. . 1 0. .0 34787 265. . 5 0. .0 6. . 7 0. .0 0. ,0 0. .0 0. ,0 0. 0 0. 0 0. .0 22. .4 43. .3 0, .0 34874 266. .0 0, .0 6. .7 7, .5 0. ,0 0, .0 0. ,0 0. ,0 0. 0 0. .0 29, .8 43. .3 0, .0 34961 266. , 5 0. .0 6. .7 7. .5 0. .0 0. .0 0. 0 0. 0 0. 0 21. , 7 37. , 3 43. .3 0. ,0 35049 267. .0 0. .0 40. .4 7. .5 0. .0 7, ,5 0. ,0 0. ,0 0. ,0 21. .7 44, .8 43. .3 0, .0 35136 267. , 5 0. .0 40, .4 7. . 5 0, .0 14.9 0. ,0 0. 0 0. 0 21. .7 52. . 2 72, .2 0, .0 35223 268. .0 0, .0 40, .4 7. .5 0, .0 22, .4 0. ,0 0. ,0 0. 0 21. .7 59, .7 72. .2 0. .0 35310 268. , 5 0. .0 40. .4 37.3 0. ,0 29. .8 0. 0 0. 0 0. 0 21. ,7 67. .1 72. . 2 0. ,0 35397 269. ,0 0. .0 40, .4 37. .3 0. .0 37. .3 0. ,0 0. 0 0. 0 21. ,7 67. . 1 72. .2 0. .0 35484 269. .5 0. .0 40, .4 37. . 3 0. .0 44. .8 0. 0 0. 0 0. 0 21. .7 67, . 1 72. . 2 0, .0 35571 270. .0 7, .5 40, .4 37. .3 7. .2 52. .2 7. ,2 0. ,0 0. 0 21. .7 67. .1 79. .4 7. . 5 35658 270. . 5 14. .9 40. .4 67. . 1 14. .4 52. 2 14. 4 0. 0 0. 0 21. ,7 67. . 1 72. , 2 14. .9 35745 271. ,0 22. .4 40. .4 67. .1 43. .3 52. .2 36. ,1 0. 0 7. 5 28. ,9 67, .1 79. .4 22. ,4 35832 271. . 5 29. .8 40, .4 67. . 1 43. .3 52. .2 36. ,1 0. 0 14.9 86. .6 67. .1 86. .6 29. .8 35919 272. .0 37. .3 40, .4 67, .1 43. .3 82, ,1 36. ,1 0. ,0 22. ,4 86. .6 67, . 1 93. .8 37, . 3 36007 272. .5 44.8 40. ,4 67. . 1 43. .3 82. , 1 43. ,3 0. 0 29. 8 86. ,6 89, .5 101. .1 37. , 3 36094 273. ,0 52. . 2 40. .4 67, .1 50. ,5 82. ,1 50. ,5 29. 8 37. 3 86. .6 119. .4 108. .3 37. . 3 36183 273. 5 59. , 7 40. .4 67. . 1 57. ,8 82. 1 57. 8 37. 3 44.8 86.6 119. ,4 115. .5 37. .3 36271 274. .0 67, . 1 40, .4 67. .1 65. .0 82. .1 79. ,4 44. ,8 44. ,8 86. .6 119, .4 115. .5 37, . 3 36360 274. 5 74. .6 40. ,4 82. , l ' 101. , 1 119. ,4 101. 1 52. 2 44.8 86.6 119. .4 115. .5 37. . 3 36449 275. ,0 74. .6 40. .4 82. ,1 101. . 1 119. .4 101. ,1 59. ,7 44. 8 93. .8 119. .4 115. .5 37. . 3 36534 275. 5 74. ,6 40. .4 82. 1 101. .1 119. ,4 101. 1 67. 1 44. 8 101. ,1 119. .4 115. , 5 37. , 3 36620 276. .0 97. .0 40.4 82. .1 101. . 1 119. .4 101. .1 74. 6 44. ,8 108. 3 119. .4 115. .5 97, .0 36708 276. .5 104. .4 40. .4 104. .4 108. ,3 119. ,4 101. 1 82. 1 44. 8 115. ,5 119. ,4 115. , 5 104, .4 36798 277. 0 111. .9 40. .4 119. .4 115. , 5 119. .4 108. ,3 89. ,5 44.8 115. ,5 119, .4 115. .5 111. .9 36888 277. 5 119. ,4 40. .4 119. ,4 115. ,5 119. ,4 115. 5 97. 0 44.8 115.5 119. ,4 115. , 5 119. ,4 36978 278. .0 119. .4 40. .4 119. .4 115. .5 119. .4 115. ,5 104. ,4 104. .4 115. .5 119, . 4 115, .5 119, .4 37066 278. 5 119. .4 40. .4 119. ,4 115. , 5 119. .4 115. 5 111. 9 111. ,9 115. .5 119. .4 115.5 119. .4 37153 279. 0 119. .4 47. . 2 119. .4 115. ,5 119. .4 115. ,5 119. 4 119. 4 115. .5 119, . 4 115. .5 119, .4 37240 279. 5 119. ,4 67. .4 119. ,4 115. ,5 119. 4 115. 5 119. 4 119. 4 115. .5 119.4 115. . 5 119, ,4 37327 280.0 119. .4 67. .4 119. .4 115. ,5 119. ,4 115. 5 119. 4 119. 4 115. .5 119. .4 115. .5 119, .4 37414 280. 5 119. ,4 67. .4 119. ,4 115. ,5 119.4 115. 5 119. 4 119. 4 115. .5 119. .4 115. . 5 119, .4 37502 281. 0 119. .4 107. ,8 119. .4 115. .5 119. .4 115. 5 119. 4 119. 4 115. .5 119, . 4 115, .5 119, .4 37588 281. 5 119. .4 107. .8 119. ,4 115. ,5 119. 4 115. 5 119. 4 119. 4 115. , 5 119, .4 115. . 5 119, , 4 37675 282. 0 119. ,4 107. .8 119. ,4 115. ,5 119. ,4 115. 5 119. 4 119. 4 115. .5 119. .4 115, . 5 119, . 4 37758 282. 5 119. .4 107. .8 119. .4 115.5 119. .4 115. 5 119. 4 119. 4 115. ,5 119, .4 115, .5 119.4 37840 283. 0 119. .4 107. ,8 119. ,4 115. .5 119. .4 115. ,5 119. 4 119. 4 115. ,5 119, .4 115, . 5 119, .4 37922 283. 5 119. ,4 107. .8 119. ,4 115. 5 119. ,4 115. 5 119. 4 119. 4 115. , 5 119, . 4 115, , 5 119. .4 38003 284.0 119. , 4 107. ,8 119. ,4 115. ,5 119. ,4 115. 5 119. 4 119. 4 115.5 119. .4 115. . 5 119. .4 38077 284. 5 119. 4 107. ,8 119. 4 115. 5 119. 4 115. 5 119. 4 119. 4 115. ,5 119. .4 115. ,5 119. .4 38151 285. 0 119. .4 107. .8 119. .4 115. , 5 119. .4 115. ,5 119.4 119. 4 115. . 5 119.4 115. .5 119, .4 38224 - 7 4 -TABLE B.III. SUMMARY RESULTS - CASE 1, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t 1 85939 265.5 7311 65955 736 2 68999 266.5 21932 53655 601 3 69349 267. 5 36553 47959 541 4 93267 271.0 87726 40800 467 5 147060 277.0 175452 57207 673 6 146058 281.5 241247 73491 892 7 99714 283. 5 270489 67166 831 8 53267 281.5 241247 80741 998 9 64080 281.5 241247 60666 746 10 122926 283.0 263178 83637 1033 11 114460 284. 0 277799 79117 984 12 110221 285.0 292420 84545 1057 STARTING RESERVOIR LEVEL = 270 f t 1 85939 270.5 80416 65955 759 2 68999 271. 5 95037 53655 620 3 69349 272.5 109658 47959 558 4 93267 273.5 124279 76996 901 5 147060 279.0 204694 63908 762 6 146058 283. 5 270489 73491 902 7 99714 285.0 292420 73368 916 8 53267 283.0 263178 80741 1007 9 64080 282.0 248557 72363 895 10 122926 283. 5 270489 83637 1036 11 114460 284.0 277799 84507 1052 12 110221 285. 0 292420 84545 1057 STARTING RESERVOIR LEVEL = 275 f t 1 85939 275. 5 153521 65955 782 2 68999 276.5 168142 53655 639 3 69349 275.5 153521 75564 900 4 93267 276.0 160831 83711 996 5 147060 280. 5 226626 77067 930 6 146058 284.5 285110 79340 981 7 99714 285.0 292420 85065 1065 8 53267 283.0 263178 80741 1007 9 64080 282.0 248557 72363 895 10 122926 283.5 270489 83637 1036 11 114460 284. 0 277799 84507 1052 12 110221 285.0 292420 84545 1057 -75-TABLE B. III. SUMMARY RESULTS - CASE 1, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 280 f t 1 85939 280. 5 226626 65955 805 2 68999 280.0 219315 73428 896 3 69349 278.0 190073 88601 1074 4 93267 278.0 190073 90079 1085 5 147060 281.5 241247 88718 1080 6 146058 285.0 292420 84802 1053 7 99714 285. 0 292420 90473 1135 8 53267 283.0 263178 80741 1007 9 64080 282.0 248557 72363 895 10 122926 283. 5 270489 83637 1036 11 114460 284.0 277799 84507 1052 12 110221 285.0 292420 84545 1057 STARTING RESERVOIR LEVEL = 285 f t 1 85939 284.5 285110 77651 972 2 68999 283.0 263178 84737 1055 3 69349 280.0 219315 100297 1233 4 93267 279. 5 212005 95620 1164 5 147060 282.5 255868 93835 1150 6 146058 285. 0 292420 94203 1173 7 99714 285.0 292420 90473 1135 8 53267 283.0 263178 80741 1007 9 64080 282.0 248557 72363 895 10 122926 283. 5 270489 83637 1036 11 114460 284.0 277799 84507 1052 12 110221 285.0 292420 84545 1057 - 7 6 -TABLE B.IV. SUMMARY OF CASE 1 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t 1 85939 270.6 81617 0 0 2 68999 272.5 109558 40400 469 3 69349 272. 5 109164 67100 783 4 93267 275.8 157836 43300 510 5 147060 277.6 183760 119400 1428 6 146058 279.5 211876 115500 1396 7 99714 278.2 192615 119400 1446 8 53267 274.6 140979 104400 1246 9 64080 273. 1 118291 86600 1018 10 122926 273.3 121317 119400 1399 11 114460 272. 7 112261 115500 1351 12 110221 277.6 183624 37300 442 STARTING RESERVOIR LEVEL = 270 f t 1 85939 275.1 147223 7500 88 2 68999 277.0 175164 40400 481 3 69349 273.4 122469 119400 1415 4 93267 275. 7 156642 57800 683 5 147060 277.5 182566 119400 1427 6 146058 279.4 210682 115500 1395 7 99714 278.1 191481 119400 1445 8 53267 274.6 139845 104400 1246 9 64080 273.0 117157 86600 1018 10 122926 273.2 120182 119400 1398 11 114460 273.1 118023 108300 1268 12 110221 278. 0 189383 37300 443 STARTING RESERVOIR LEVEL = 275 f t 1 85939 275.5 152955 74600 884 2 68999 277.4 180897 40400 482 3 69349 273.8 128202 119400 1418 4 93267 275.6 155175 65000 768 5 147060 277.4 181098 119400 1426 6 146058 279.3 209215 115500 1395 7 99714 278.0 190087 119400 1444 8 53267 274.5 138451 104400 1245 9 64080 272.9 115764 86600 1017 10 122926 273.1 118789 119400 1397 11 114460 273.0 116699 108300 1268 12 110221 277.9 188062 37300 443 -77-TABLE B.IV. SUMMARY OF CASE 1 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 280 f t 1 85939 277.3 179845 119400 1444 2 68999 279.2 207786 40400 488 3 69349 275.6 154469 119400 1434 4 93267 274.9 145342 101100 1199 5 147060 276. 7 171265 119400 1421 6 146058 279.6 213689 101100 1219 7 99714 278.3 194337 119400 1447 8 53267 274.2 135201 111900 1335 9 64080 272. 7 112514 86600 1016 10 122926 274.9 145439 89500 1052 11 114460 274. 2 135177 115500 1364 12 110221 279.1 205679 37300 446 STARTING RESERVOIR LEVEL = 285 f t 1 85939 281.7 244833 119400 1484 2 68999 279.0 205374 107800 1317 3 69349 275.4 152177 119400 1432 4 93267 274.8 143050 101100 1198 5 147060 276.6 168973 119400 1419 6 146058 279.5 211490 101100 1218 7 99714 278.1 192248 119400 1446 8 53267 274.6 140612 104400 1246 9 64080 273.1 117924 86600 1018 10 122926 273. 3 120949 119400 1398 11 114460 272.7 111912 115500 1351 12 110221 277. 5 183274 37300 442 TIME PERIOD CMONTH} FIGURE B . l . CASE 1 RESULTS - STATE CHANGE DECISION TIME PERIOD CMONTH) FIGURE B.2. CASE j RESULTS - DISCHARGE DECISION -80-APPENDIX C. OPTIMIZATION RESULTS FOR CASE 2 -81-TABLE C . I . OPTIMAL MONTHLY RESERVOIR LEVELS (ft) CASE 2: PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time step (month) Return Elev . 1 2 3 4 5 6 7 8 9 10 11 12 MW 265.0 265.5 266.0 266.0 266.5 267.0 267.5 267.5 268.0 268.5 268.5 269.0 269.0 270.0 269.5 271.0 270.5 272.5 271.0 266.5 266.5 268.5 265.5 272.0 266.0 265.5 266.0 266.5 266.5 267.0 267.5 268.0 268.0 268.5 269.0 269.0 269.5 269.5 270.5 270.0 271.5 271.0 273.0 271.5 267.0 266.5 269.0 266.0 272.5 266.5 266.0 266.5 267.0 267.0 267.5 268.0 268.5 268.5 269.0 269.5 269.5 270.0 270.0 271.0 270.5 272.0 271.5 273.5 272.0 267.5 266.5 269.5 266.5 273.0 267.0 266.5 267.0 267.5 267.5 268.0 268.5 269.0 269.0 269.5 270.0 270.0 270.5 270.5 271.5 271.0 272.5 272.0 274.0 272.5 268.0 266.5 270.0 267.0 273.5 267.5 267.0 267.5 268.0 268.0 268.5 269.0 269.5 269.5 270.0 270.5 270.5 271.0 271.0 272.0 271.5 273.0 272.5 274.5 273.0 268.5 267.0 270.5 267.5 274.0 268.0 267.5 268.0 268.5 268.5 269.0 269.5 270.0 270.0 270.5 271.0 271.0 271.5 271.5 272.5 272.0 273.5 273.0 275.0 273.5 269.0 267.0 271.0 268.0 274.5 268.5 266.0 268.5 266.5 269.0 267.5 269.5 267.5 269.5 268.0 270.0 268.5 270.5 269.0 271.5 269.5 272.0 270.0 272.5 265.0 265.0 265.0 265.5 271.5 266.5 266.5 269.0 267.0 269.5 268.0 270.0 268.0 270.0 268.5 270.5 269.0 271.0 269.5 272.0 270.0 272.5 270.5 273.0 265.0 265.5 265.0 266.0 272.0 267.0 267.0 269.5 267.5 270.0 268.5 270.5 268.5 270.5 269.0 271.0 269.5 271.5 270.0 272.5 270.5 273.0 271.0 273.5 265.0 266.0 265.0 266.5 272.5 267.5 267.5 270.0 268.0 270.5 269.0 271.0 269.0 271.0 269.5 271.5 270.0 272.0 270.5 273.0 271.0 273.5 271.5 274.0 265.0 266.5 265.5 267.0 273.0 268.0 268.0 270.5 268.5 271.0 269.5 271.5 269.5 271.5 270.0 272.0 270.5 272.5 271.0 273.5 271.5 274.0 272.0 274.5 265.0 267.0 266.0 267.5 273.5 268.5 268.5 271.0 269.0 271.5 270.0 272.0 270.0 272.0 270.5 272.5 271.0 273.0 271.5 274.0 272.0 274.5 272.5 275.0 265.5 267.5 266.5 268.0 274.0 269.0 271.0 269.5 271.5 270.5 272.0 272.0 272.5 272.5 273.5 273.0 274.0 274.0 275.0 275.0 276.0 275.5 276.5 276.5 269.5 270.0 270.5 271.0 271.5 272.5 271.5 270.0 272.0 271.0 272.5 272.5 273.0 273.0 274.0 273.5 274.5 274.5 275.5 275.5 276.5 276.0 277.0 277.0 270.0 270.5 271.0 271.5 272.0 273.0 272.0 270.5 272.5 271.5 273.0 273.0 273.5 273.5 274.5 274.0 275.0 275.0 276.0 276.0 277.0 276.5 277.5 277.5 270.5 271.0 271.5 272.0 272.5 273.5 272.5 271.0 273.0 272.0 273.5 273.5 274.0 274.0 275.0 274.5 275.5 275.5 276.5 276.5 277.5 277.0 278.0 278.0 271.0 271.5 272.0 272.5 273.0 274.0 273.0 271.5 273.5 272.5 274.0 274.0 274.5 274.5 275.5 275.0 276.0 276.0 277.0 277.0 278.0 277.5 278.5 278.5 271.5 272.0 272.5 273.0 273.5 274.5 273.5 272.0 274.0 273.0 274.5 274.5 275.0 275.0 276.0 275.5 276.5 276.5 277.5 277.5 278.5 278.0 279.0 279.0 272.0 272.5 273.0 273.5 274.0 275.0 267.5 266.0 268.0 266.0 269.0 266.5 269.5 267.0 270.0 267.5 271.0 268.0 271.5 268.5 272.0 268.5 273.0 269.0 267.0 265.0 268.5 265.0 271.0 265.0 268.0 266.5 268.5 266.5 269.5 267.0 270.0 267.5 270.5 268.0 271.5 268.5 272.0 269.0 272.5 269.0 273.5 269.5 267.5 265.0 269.0 265.0 271.5 265.0 268.5 267.0 269.0 267.0 270.0 267.5 270.5 268.0 271.0 268.5 272.0 269.0 272.5 269.5 273.0 269.5 274.0 270.0 268.0 265.0 269.5 265.0 272.0 265.0 269.0 267.5 269.5 267.5 270.5 268.0 271.0 268.5 271.5 269.0 272.5 269.5 273.0 270.0 273.5 270.0 274.5 270.5 268.5 265.0 270.0 265.0 272.5 265.5 269.5 268.0 270.0 268.0 271.0 268.5 271.5 269.0 272.0 269.5 273.0 270.0 273.5 270.5 274.0 270.5 275.0 271.0 269.0 265.0 270.5 265.5 273.0 266.0 270.0 268.5 270.5 268.5 271.5 269.0 272.0 269.5 272.5 270.0 273.5 270.5 274.0 271.0 274.5 271.0 275.5 271.5 269.5 265.5 271.0 266.0 273.5 266.5 265.5 266.5 266.0 267.5 266.5 269.0 267.0 270.5 267.5 271.5 268.0 272.0 268.5 273.0 269.5 274.0 270.5 275.0 265.5 269.5 266.0 273.5 267.0 275.0 266.0 267.0 266.5 268.0 267.0 269.5 267.5 271.0 268.0 272.0 268.5 272.5 269.0 273.5 270.0 274.5 271.0 275.5 266.0 270.0 266.5 274.0 267.5 275.5 266.5 267.5 267.0 268.5 267.5 270.0 268.0 271.5 268.5 272.5 269.0 273.0 269.5 274.0 270.5 275.0 271.5 276.0 266.5 270.5 267.0 274.5 268.0 276.0 267.0 268.0 267.5 269.0 268.0 270.5 268.5 272.0 269.0 273.0 269.5 273.5 270.0 274.5 271.0 275.5 272.0 276.5 267.0 271.0 267.5 275.0 268.5 276.5 267.5 268.5 268.0 269.5 268.5 271.0 269.0 272.5 269.5 273.5 270.0 274.0 270.5 275.0 271.5 276.0 272.5 277.0 267.5 271.5 268.0 275.5 269.0 277.0 268.0 269.0 268.5 270.0 269.0 271.5 269.5 273.0 270.0 274.0 270.5 274.5 271.0 275.5 272.0 276.5 273.0 277.5 268.0 272.0 268.5 276.0 269.5 277.5 267.5 267.5 33384 268.0 268.0 269.0 269.0 269.5 270.0 270.0 270.5 271.0 271.5 271.5 272.0 272.5 273.0 273.5 274.0 267.5 268.0 270.0 269.5 281.0 271.0 268.0 268.0 33466 268.5 268.5 269.5 269.5 270.0 270.5 270.5 271.0 271.5 272.0 272.0 272.5 273.0 273.5 274.0 274.5 268.0 268.5 270.5 270.0 281.5 271.5 268.5 268.5 33546 269.0 269.0 270.0 270.0 270.5 271.0 271.0 271.5 272.0 272.5 272.5 273.0 273.5 274.0 274.5 275.0 268.5 269.0 271.0 270.5 282.0 272.0 269.0 269.0 33625 269.5 269.5 270.5 270.5 271.0 271.5 271.5 272.0 272.5 273.0 273.0 273.5 274.0 274.5 275.0 275.5 269.0 269.5 271.5 271.0 282.5 272.5 269.5 269.5 33704 270.0 270.0 271.0 271.0 271.5 272.0 272.0 272.5 273.0 273.5 273.5 274.0 274.5 275.0 275.5 276.0 269.5 270.0 272.0 271.5 283.0 273.0 270.0 270.0 33783 270.5 270.5 271.5 271.5 272.0 272.5 272.5 273.0 273.5 274.0 274.0 274.5 275.0 275.5 276.0 276.5 270.0 270.5 272.5 272.0 283.5 273.5 TABLE C.I. (cont'd) OPTIMAL MONTHLY RESERVOIR LEVELS (ft) CASE 2: Elev. PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time step (month) 1 2 3 4 5 6 7 8 9 10 11 12 268. 5 269. .0 269.0 271.5 274.0 272.5 270. , 5 269, ,0 268. . 5 269, . 5 270. , 5 270. . 5 269. ,0 269. .5 269.5 272.0 274. 5 273.5 271. ,0 269. ,0 269. ,0 270. . 5 271. ,0 271. ,0 270. .0 270. .5 270. 5 272. 5 275.0 275.0 272. .0 269. .5 269. . 5 272. .0 272. .0 272. ,0 270. .5 271, ,0 270. 5 272.5 275.5 275.5 272. . 5 270. ,0 270, .0 273. , 5 272. , 5 273. ,0 271, . 5 271, . 5 271.0 273.0 276. 5 276.0 273. ,0 270. , 5 270. . 5 274, . 5 273. .0 273. . 5 272. .0 272. .0 271.5 273.5 277.0 277.0 274. ,0 271. ,0 271. .0 275. .0 274. ,0 274. , 5 273, .0 272. .5 272.0 274. 5 278.0 278.0 274. ,5 271. .5 271. .5 276. .0 274. . 5 275. .0 274.0 273. 5 272.5 275.0 279.0 278.5 275. ,0 271. , 5 272. ,5 277. .0 275. .5 276. ,0 275. . 5 274. .0 273.0 275.5 279.0 279. 5 276. ,0 272. ,0 273. ,5 278. .0 276. . 5 277. .0 269. ,5 267, ,5 266.0 268.0 272. 5 273.0 270. ,0 266. 0 268. ,5 272. . 5 270. , 5 271. ,0 271. .5 268. ,5 267.0 268.5 273.5 274.0 271. .5 266. .5 269. .0 276. .5 273. .0 272. .5 275. .0 269, .0 274.5 269.5 274. 5 275.5 274. .0 267. ,0 270. ,0 278. .0 284. .0 274. ,0 269. .0 269, . 5 269. 5 272.0 274. 5 273.0 271. .0 269.5 269.0 270. .0 271, .0 271. ,0 269. ,5 270. .0 270.0 272. 5 275.0 274.0 271. , 5 269. , 5 269. .5 271. .0 271. . 5 271. , 5 270. 5 271, ,0 271.0 273.0 275.5 275.5 272. , 5 270. ,0 270. ,0 272, , 5 272. .5 272. .5 271. ,0 271. , 5 271.0 273.0 276.0 276.0 273. ,0 270. 5 270. ,5 274.0 273. ,0 273. , 5 272. .0 272. .0 271. 5 273.5 277.0 276.5 273. ,5 271. ,0 271. ,0 275, ,0 273. ,5 274. .0 272. ,5 272. ,5 272.0 274.0 277.5 277.5 274.5 271. 5 271. , 5 275. .5 274.5 275. ,0 27 3. ,5 273, .0 272.5 275.0 278.5 278.5 275. .0 272. .0 272. ,0 276, .5 275. .0 275, .5 274. 5 274.0 273.0 275.5 279.5 279.0 275. ,5 272. .0 273. .0 277. , 5 276. .0 276. , 5 276. .0 274.5 273. 5 275. 5 279.0 280.0 276. ,5 272. ,5 274. .0 278.5 277. .0 277. ;s 270.0 268.0 266.5 268.5 273.0 273.5 270. 5 266. ,5 269. ,0 273.0 271. .0 271. , 5 272. ,0 269. ,0 267.5 269.0 274.0 274.5 272. ,0 267. ,0 269. .5 277. .0 273. , 5 273. .0 275. ,5 269. ,5 275.0 270.0 275.0 276.0 274.5 267. .5 270. , 5 278. . 5 284. .5 274. , 5 269. , 5 270. .0 270.0 272.5 275.0 273.5 271. ,5 270. ,0 269. ,5 270, .5 271. . 5 271. . 5 270. 0 270. 5 270. 5 273.0 275.5 274. 5 272.0 270.0 270. 0 271. , 5 272.0 272. .0 271. ,0 271. .5 271. 5 273.5 276.0 276.0 273. ,0 270. ,5 270.5 273, .0 273. .0 273. .0 271. .5 272. ,0 271.5 273.5 276. 5 276.5 273. 5 271. 0 271. ,0 274. .5 273. , 5 274. ,0 272. , 5 272. , 5 272.0 274.0 277.5 277.0 274. ,0 271. ,5 271. .5 275, , 5 274. .0 274. .5 273. 0 273. ,0 272. 5 274.5 278.0 278.0 275. ,0 272. 0 272. ,0 276. .0 275. .0 275. , 5 274.0 273. ,5 273.0 275. 5 279.0 279.0 275. , 5 272. , 5 272. ,5 277. .0 275. ,5 276. .0 275. 0 . 274. 5 273.5 276.0 280.0 279.5 276. 0 272. 5 273. ,5 278. ,0 276. , 5 277. ,0 276. 5 275. ,0 274.0 275. 5 279.0 280.5 277. ,0 273. ,0 274. ,5 279. .0 277. .5 278. .0 270. 5 268.5 267.0 269.0 273.5 274.0 271. ,0 267. .0 269. ,5 273. .5 271. .5 272. .0 272. . 5 269. .5 268.0 269.5 274.5 275.0 272. , 5 267. .5 270. .0 277. .5 274. .0 273. .5 276. 0 270. ,0 275. 5 270. 5 275.5 276.5 275. ,0 268.0 271. .0 279. .0 285. ,0 275. .0 270. .0 270. , 5 270. 5 273.0 275.5 274.0 272. ,0 270. .5 270. .0 271. .0 272. ,0 272. .0 270.5 271. 0 271.0 273. 5 276.0 275.0 272. 5 270. 5 270. ,5 272. .0 272. , 5 272. . 5 271. 5 272. ,0 272.0 274.0 276. 5 276.5 273. , 5 271. ,0 271. ,0 273, . 5 273. .5 273. .5 272. 0 272. 5 272.0 274.0 277.0 277.0 274. ,0 271. 5 271. , 5 275. .0 274. ,0 274. . 5 273. 0 273. 0 272.5 274.5 278.0 277.5 274. , 5 272. ,0 272. .0 276. .0 274. , 5 275. .0 273. 5 273. 5 273.0 275.0 278.5 278.5 275. 5 272. .5 272. .5 276. .5 275. ,5 276, .0 274. 5 274. ,0 273. 5 276.0 279.5 279.5 276. ,0 273. ,0 273. .0 277, .5 276. .0 276. . 5 275. 5 275. 0 274.0 276.5 280. 5 280.0 276. 5 273.0 274. .0 278. ,5 277. .0 277. , 5 277. 0 275. 5 274. 5 275. 5 279.0 281.0 277. 5 273. 5 275. ,0 279. ,0 278. ,0 278. .5 271. 0 269. 0 267.5 269. 5 274.0 274.5 271. 5 267. 5 270. 0 274. ,0 272. ,0 272. ,5 273. 0 270. 0 268.5 270.0 275.0 275.5 273. 0 268.0 270. .5 278. ,0 274. .5 273. ,5 276. 5 270. 5 276.0 271.0 276.0 277.0 275. 5 268. 5 271. 5 279. ,5 285. ,0 275. ,5 270. 5 271. 0 271.0 273. 5 276.0 274.5 272. 5 271. 0 270. 5 271. ,5 272. .5 272. , 5 271. 0 271. 5 271.5 274.0 276. 5 275.5 273. 0 271. 0 271. 0 272. 5 273. .0 273. ,0 272. 0 272. ,5 272.5 274.5 277.0 277.0 274.0 271. 5 271. 5 274.0 274. .0 274. ,0 272. 5 273.0 272.5 274.5 277.5 277.5 274. 5 272.0 272.0 275. ,5 274.5 275.0 273. 5 273. 5 273.0 275.0 278.5 278.0 275. 0 272. 5 272. ,5 276, , 5 275. ,0 275. .5 274.0 274.0 273.5 275.5 279.0 279.0 276. 0 273. 0 273. 0 277. .0 276. .0 276. .5 275. 0 274. 5 274.0 276.5 280.0 280.0 276.5 273. 5 273. ,S 278. .0 276. .5 277. .0 276. 0 275. 5 274.5 277.0 281.0 280. 5 277. 0 273. 5 274. 5 279. ,0 277. .5 278. .0 277. 5 276. ,0 275.0 275.5 279.0 281.5 278. ,0 274. ,0 275. .5 279, .0 278, .5 279, .0 271. 5 269.5 268.0 270.0 274. 5 275.0 272. 0 268. 0 270. ,5 274. , 5 272. .5 273. .0 273. 5 270. ,5 269.0 270.5 275.5 276.0 273. 5 268. ,5 271. ,0 278.5 275, .0 274. .0 277. 0 271. 0 276.5 271.5 276.5 277.5 276.0 269. 0 272. ,0 280. .0 285. .0 276. .0 271. 0 271. .5 271.5 274.0 276.5 275.0 273. ,0 271. 5 271. ,0 272. .0 273. .0 273. .0 271. 5 272. 0 272.0 274.S 277.0 276.0 273. 5 271. 5 271. 5 273. ,0 273. ,5 273. ,5 272. 5 273. ,0 273.0 275.0 277.5 277.5 274.5 272. 0 272. ,0 274.5 274.5 274.5 273. 0 273. 5 273.0 275.0 278.0 278.0 275. 0 272. 5 272.5 276.0 275. .0 275. .5 274.0 274.0 273.5 275.5 279.0 278.5 275. ,5 273. ,0 273. .0 277, .0 275. .5 276. .0 274.5 274.5 274.0 276.0 279. 5 279.5 276. 5 273. 5 273. 5 277.5 276.5 277. .0 275. 5 275. ,0 274.5 277.0 280.5 280.5 277. ,0 274.0 274. .0 278.5 277. .0 277. .5 276. 5 276.0 275.0 277.5 281.5 281.0 277.5 274.0 275. ,0 279. ,5 278.0 278. .5 278. 0 276. 5 275.5 275.5 279.0 281.5 278.5 274.5 276. ,0 279. .0 279, .0 279. .5 272. 0 270. 0 268.5 270.5 275.0 275.5 272. 5 268. 5 271. ,0 275. ,0 272. , 5 273. ,5 274.0 271. 0 269.5 271.0 276.0 276.5 274. ,0 269. ,0 271. ,5 279. .0 275. ,5 274. .5 277. 5 271. 5 277.0 272.0 277.0 278.0 276. 5 269. 5 272. ,5 280. ,5 285. ,0 276. .5 Return MW 268.0 268.5 270.0 33862 33940 34019 34097 34176 34255 - 8 3 -TABLE C . I . (cont'd) OPTIMAL MONTHLY RESERVOIR LEVELS (ft) CASE 2: PERFECTLY .CORRELATED INFLOWS - STATE CHANGE DECISION Time step (month) Return Elev . 1 2 3 4 5 6 7 8 9 10 11 12 MW 34333 271 .0 271 . 5 272 .0 272 .0 274. 5 277.0 275, .5 273. .5 272, .0 271, .5 272, . 5 273. . 5 273.5 272, .0 272, . 5 272, . 5 275, .0 277. 5 276, .5 274. .0 272. , 0 272. .0 273, . 5 274. ,0 274.0 273 .0 273 . 5 273 . 5 275, . 5 278.0 278, .0 275. .0 272, .5 272, . 5 275. .0 275. ,0 275.0 273, . 5 274, .0 273, . 5 275, . 5 278. 5 278. .5 275. .5 273, .0 273. .0 276, , 5 275. , 5 276.0 274 . 5 274 . 5 274, .0 276, .0 279.5 279, .0 276. .0 273, . 5 273. .5 277. . 5 276. .0 276.5 275, .0 275, .0 274, . 5 276. .5 280.0 280. .0 277. ,0 274. .0 274. .0 278. ,0 277. ,0 277.5 276 .0 275 . 5 275, .0 277, . 5 281.0 281, .0 277. , 5 274, . 5 274. .5 279. ,0 277. , 5 278.0 277, .0 276, .5 275, .5 278. .0 281. 5 281. .5 278. ,0 274. . 5 275. . 5 280. ,0 278. . 5 279.0 278, .5 277 .0 275, .5 275, . 5 279.0 281. .5 279.0 275, .0 276. , 5 279. .0 279. .5 280.0 272. .5 270, . 5 269, .0 271, .0 275.5 276, .0 273.0 269. .0 271. .5 275. ,5 273. 0 274.0 274, . 5 271, . 5 270, .0 271, . 5 276.5 277, .0 274. .5 269. .5 272. ,0 279. 5 276. ,0 275.0 278. .0 272, .0 277. . 5 272, , 5 277. 5 278, ,5 277. ,0 270. .0 273. .0 281. 0 285. ,0 277.0 271 . 5 272, .0 272, .5 272, . 5 275. .0 277.5 276. .0 274. .0 272. .5 272. .0 273. ,0 274. ,0 274.0 272, , 5 273, .0 273.0 275. ,5 278.0 277. ,0 274. 5 272. , 5 272. ,5 274. ,0 274. ,5 274.5 273. . 5 274, .0 274, .0 276. .0 278.5 278.5 275. , 5 273. .0 273. .0 275. .5 275. .5 275.5 274, ,0 274.5 274, ,0 276.0 279.0 279. .0 276. ,0 273. .5 273. .5 277.0 276. ,0 276.5 275. .0 275, .0 274.5 276. .5 280.0 279. .5 276. .5 274, .0 274.0 278. ,0 276. .5 277.0 275. ,5 275. .5 275. ,0 277. ,0 280.5 280. .5 277. .5 274. .5 274.5 278. .5 277. ,5 278.0 276. , 5 276. .0 275. .5 278. .0 281.5 281, .5 278. .0 275. .0 275. .0 279. , 5 278. ,0 278.5 277. . 5 277. ,0 276. .0 278. .5 281.5 282. ,0 278. ,5 275. .0 276. .0 280. ,5 279. ,0 279.5 279. .0 277. . 5 275. .5 275. , 5 279.0 281. , 5 279. .5 275. .5 276. , 5 279. .0 280. ,0 280.5 273. .0 271. .0 269. .5 271. ,5 276.0 276. .5 273. ,5 269. .5 272. .0 276. ,0 273. ,5 274.5 275. ,0 272. .0 270. 5 272. ,0 277.0 277. .5 275. ,0 270. ,0 272. ,5 280. .0 276. ,5 275.5 278. , 5 272, .5 278. .0 273. ,0 278.0 279. .0 277. .5 270. , 5 273. . 5 281. , 5 285.0 277. 5 272. .0 272. . 5 273. .0 273. ,0 275. .5 278.0 276. . 5 274. , 5 273. ,0 272. .5 273. , 5 274.5 274. 5 273. ,0 273. .5 273. .5 276. .0 278.5 277. .5 275. ,0 273. ,0 273.0 274.5 275. ,0 275.0 274. .0 274. .5 274. .5 276. . 5 279.0 279. .0 276. ,0 273. , 5 273. ,5 276. ,0 276. ,0 276.0 274. ,5 275. .0 274. .5 276. .5 279.5 279. .5 276. ,5 274.0 274. .0 277. .5 276. ,5 277.0 275. . 5 275. , 5 275. .0 277. .0 280.5 280. ,0 277. ,0 274. , 5 274. . 5 278. .5 277. .0 277.5 276. ,0 276. .0 275. ,5 277. . 5 281.0 281. .0 278. ,0 275. .0 275. .0 279. .0 278. ,0 278.5 277. ,0 276. . 5 276. ,0 278. .5 282.0 282. ,0 278. ,5 275. , 5 275. . 5 280. .0 278. .5 279.0 278. ,0 277. .5 276. ,5 278. , 5 281.5 282. .5 279. ,0 275. .5 276. ,5 281. ,0 279. , 5 280.0 279. .5 278. .0 275. .5 275. .5 279.0 281. .5 280. ,0 276. .0 276. , 5 279. .0 280. .0 281.0 273. ,5 271. , 5 270. ,0 272. ,0 276.5 277. .0 274. 0 270.0 272. , 5 276. .5 274. ,0 275.0 275. , 5 272. . 5 271. ,0 272. .5 277.5 278. .0 275. ,5 270. .5 273. ,0 280. ,0 277. .0 276.0 279. ,0 273. ,0 278.5 273. ,5 278.5 279. .0 278. 0 271. .0 274. .0 282. .0 285. 0 278.0 272. , 5 273. ,0 273. .5 273. ,5 276. ,0 278.5 277. .0 275. ,0 273. , 5 273. ,0 274. .0 275. ,0 275.0 273. 5 274. ,0 274. ,0 276.5 279.0 278. .0 275. 5 273. . 5 273. , 5 275.0 275. .5 275. 5 274. , 5 275. .0 275. ,0 277. ,0 279.5 279. ,5 276. ,5 274. .0 274. .0 276. .5 276. ,5 276.5 275. 0 275. ,5 275. ,0 277. .0 280.0 280. ,0 277. 0 274. 5 274. , 5 278. ,0 277. ,0 277.5 276. ,0 276. .0 275. . 5 277. , 5 281.0 280. .5 277. ,5 275, .0 275. ,0 279. ,0 277. ,5 278.0 276. 5 276. ,5 276. .0 278. 0 281.5 281. .5 278. 5 275.5 275. .5 279. ,5 278. ,5 279.0 277. , 5 277. ,0 276. ,5 279. ,0 282.5 282. , 5 279. 0 276. .0 276. ,0 280. . 5 279. ,0 279.5 278. 5 278. .0 277. ,0 278. 5 281.5 283. 0 279. 5 276. ,0 277. ,0 281. 5 280. 0 280. 5 280. .0 278. , 5 275. .5 275. . 5 279.0 281. ,5 280. ,5 276. .5 276. ,5 279.0 280. .0 281.5 274. .0 272. ,0 270. 5 272. 5 277.0 277. 5 274. 5 270. 5 273.0 277. 0 274. .5 275.5 276. ,0 273. ,0 271. . 5 273. ,0 278.0 278.5 276. 0 271. .0 273. ,5 280. .5 277. ,5 276.5 279. 5 273. 5 279. 0 274.0 279.0 279. 5 278. 5 271. ,5 274. 5 282. 5 285.0 278.5 273. .0 273. 5 274. ,0 274. ,0 276. . 5 279.0 277. ,5 275. .5 274. .0 273. ,5 274. , 5 275. .5 275. 5 274.0 274.5 274. 5 277.0 279.5 278. ,5 276. 0 274. ,0 274. 0 275. 5 276. 0 276.0 275. ,0 275. , 5 275. ,5 277. , 5 280.0 280. .0 277. ,0 274. ,5 274. ,5 277. ,0 277. .0 277.0 275. 5 276. .0 275. ,5 277. 5 280. 5 280. 5 277. 5 275. ,0 275. 0 278. 5 277. 5 278.0 276. ,5 276. .5 276. ,0 278. ,0 281.5 281. ,0 278. .0 275. .5 275. ,5 279. .5 278. .0 278.5 277. 0 277. ,0 276. ,5 278. 5 282.0 282. 0 279. 0 276. ,0 276. ,0 280. 0 279. 0 279.5 278. ,0 277. .5 277. ,0 279. ,5 282.5 283.0 279. 5 276. .5 276. , 5 281. .0 279. .5 280.0 279. .0 278.5 277. ,5 278. 5 281.5 283.5 280. 0 276.5 277. 5 282. 0 280. ,5 281.0 280.5 278. ,5 275. .5 275. ,5 279.0 281. .5 281. 0 277. .0 276. , 5 279. ,0 280. ,0 281.5 274.5 272. ,5 271. 0 273. 0 277.5 278. ,0 274.5 271. 0 273. 5 277. .5 275. ,0 276.0 276. ,5 273. .5 272. ,0 273. ,5 278.5 279. ,0 276. 5 271. ,5 274.0 281. ,0 278. ,0 277.0 280. ,0 274.0 279. ,5 274. 5 279. 5 280. 0 279. 0 272. 0 275. 0 283. 0 285. 0 279.0 273. . 5 274.0 274.5 274. ,5 277. ,0 279.5 278.0 276. 0 274.5 274.0 275. ,0 276. ,0 276.0 274.5 275. .0 275. .0 277.5 280.0 279.0 276.5 274.5 274.5 276.0 276.5 276.5 275. ,5 276. .0 276.0 278. ,0 280.5 280.5 277. .5 275. .0 275. ,0 277. ,5 277. ,5 277.5 276. 0 276. ,5 276.0 278. 0 281.0 281. 0 278.0 275. ,5 275.5 279. ,0 278.0 278.5 277. .0 277. ,0 276.5 278.5 282.0 281. ,5 278. .5 276. ,0 276.0 280. .0 278.5 279.0 277. 5 277.5 277. 0 279.0 282.5 282.5 279. 5 276. .5 276.5 280. ,5 279. ,5 280.0 278. ,5 278. .0 277. ,5 280. ,0 282.5 283. ,5 280. 0 277. .0 277. .0 281. ,5 280.0 280.5 279. ,5 279.0 278.0 278. 5 281.5 284. 0 280. 5 277. ,0 278.0 282. ,5 281. .0 281.5 280. , 5 278. .5 275. ,5 275. ,5 279.0 281. ,5 281. 5 277. .5 276. .5 279. ,0 280. ,0 281.5 275. 0 273. ,0 271. 5 273. 5 278.0 278. ,5 275. 0 271. ,5 274.0 278.0 275. .5 276.5 277. ,0 274. ,0 272. ,S 274. ,0 279.0 279.5 277. 0 272. .0 274. . 5 281. , 5 278. .5 277.5 280. 5 274.5 280. 0 275. 0 280.0 280. 5 279. 5 272. ,5 275. , 5 283. 5 285.0 279.5 34411 34489 34568 34645 34723 -84-TABLE C . I . ( c o n t ' d ) OPTIMAL MONTHLY RESERVOIR LEVELS ( f t ) CASE 2: PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time s t e p (month) Return E l e v . 1 2 3 4 5 6 7 8 9 10 11 12 MW 274, .0 274 . 5 275 .0 275 .0 277 . 5 280 .0 278, .5 276, . 5 275, .0 274, .5 275 .5 276. .5 276. . 5 34800 275, .0 275, . 5 275, .5 278, .0 280, .5 279, . 5 277. .0 275. .0 275. .0 276, . 5 277. .0 277. ,0 276, .0 276, . 5 276 . 5 278, . 5 281, .0 281, .0 278. .0 275, .5 275. . 5 278, .0 278. .0 278. .0 276. .5 277, .0 276, . 5 278. .5 281. . 5 281. . 5 278. . 5 276. .0 276. ,0 279, . 5 278. , 5 279. ,0 277, .5 277, . 5 277, .0 279, .0 282. .5 282. .0 279. .0 276. .5 276. , 5 280, . 5 279. .0 279. , 5 278. .0 278.0 277. .5 279, . 5 283. .0 283. .0 280. .0 277. .0 277. .0 281. .0 280. .0 280. , 5 279, .0 278, .5 278.0 280, . 5 282. .5 284, .0 280. . 5 277, .5 277. ,5 282, .0 280. . 5 281. .0 280. .0 279. .5 278, . 5 278. .5 281. .5 284. . 5 281. .0 277. .5 278. ,5 283. .0 281. , 5 282. ,0 280, . 5 278, . 5 275, .5 275, . 5 279, .0 281. .5 282. .0 278, .0 276. . 5 279, .0 280. .0 281. ,5 275, . 5 273, . 5 272. .0 274, .0 278, .5 279.0 275. .5 272. ,0 274. ,5 278. .5 276. ,0 277. ,0 277. .5 274. . 5 273, .0 274. .5 279. .5 280. .0 277. . 5 272. .5 275. ,0 282. .0 279. ,0 278. ,0 281. .0 275. .0 280. . 5 275. .5 280. .5 281. .0 280. .0 273. .0 276. ,0 284, .0 285. ,0 280. ,0 274, . 5 275, .0 275, . 5 275, .5 278, .0 280, .5 279, .0 277. .0 275. .5 275. .0 276. .0 277. .0 277. .0 34878 275. . 5 276, .0 276. .0 278. . 5 281, .0 280.0 277. . 5 275. .5 275. . 5 277. .0 277. , 5 277. , 5 276, .5 277. .0 277, .0 279. .0 281, .5 281. .5 278. . 5 276.0 276. .0 278, .5 278. . 5 278. .5 277. .0 277, . 5 277. .0 279. .0 282. .0 282. .0 279. ,0 276.5 276. 5 280, .0 279. .0 279. , 5 278. .0 278. .0 277, .5 279. . 5 283. .0 282. . 5 279. .5 277. .0 277.0 281, .0 279.5 280. .0 278. 5 278. 5 278.0 280. ,0 283. ,5 283. 5 280. , 5 277. ,5 277. ,5 281. . 5 280. ,5 281. ,0 279. .5 279, .0 278, . 5 280, . 5 282. .5 284. .5 281. .0 278. .0 278. .0 282, .5 281. .0 281. . 5 280. . 5 280. .0 279. .0 278, . 5 281. ,5 285. .0 281. . 5 278. .0 279. ,0 283. . 5 282. .0 282. , 5 280. . 5 278, . 5 275, .5 275. .5 279. .0 281. .5 282. .0 278. . 5 276. .5 279, .0 280. .0 281. .5 276. .0 274.0 272. . 5 274.5 279. .0 279. .5 276. .0 272. .5 275.0 279. .0 276. .5 277. , 5 278. .0 275. .0 273. . 5 275, .0 280.0 280. . 5 278. .0 273. .0 275. .5 282 . 5 279. . 5 278. .5 281. ,5 275. , 5 281. ,0 276. ,0 281. ,0 281. .5 280. 5 273. , 5 276. . 5 284, . 5 285. ,0 280. . 5 275. ,0 275. . 5 276. .0 276. .0 278. .5 281. .0 279.5 277. .5 276. .0 275.5 276 .5 277, .5 277. . 5 34956 276. ,0 276. .5 276. . 5 279. .0 281. .5 280. . 5 278.0 276. .0 276. ,0 277, . 5 278. .0 278. ,0 277. .0 277. . 5 277. . 5 279, . 5 282. .0 282. .0 279. .0 276. .5 276. .5 279, .0 279. .0 279. .0 277. .5 278. ,0 277. ,5 279. , 5 282. ,5 282. ,5 279. ,5 277. ,0 277. ,0 280, .5 279. ,5 280. ,0 278. , 5 278. .5 278, .0 280. ,0 283. . 5 283. .0 280. .0 277. .5 277. ,5 281, . 5 280, ,0 280. .5 279. 0 279. ,0 278. , 5 280. . 5 283. .5 284. ,0 281. .0 278. ,0 278. ,0 282. .0 281. ,0 281. , 5 280. .0 279. .5 279, .0 280. . 5 282. .5 285. .0 281. . 5 278.5 278. .5 283, .0 281, .5 282. .0 281. ,0 280. . 5 279. .0 278. .5 281. ,5 285. ,0 282. ,0 278. ,5 279. . 5 284. .0 282. . 5 283. ,0 280. , 5 278. , 5 275. . 5 275. . 5 279.0 281. ,5 282. .0 278. .5 276. ,5 279, .0 280. .0 281. . 5 276. 5 274. , 5 273. ,0 275. ,0 279.5 280.0 276. ,5 273. ,0 275. ,5 279. .5 277. ,0 278. ,0 278. .5 275. . 5 274. .0 275. .5 280. .5 281. ,0 278. .5 273. .5 276. .0 283, .0 280. .0 279. ,0 282. 0 276. 0 281. , 5 276. , 5 281. ,5 282. ,0 281. ,0 274.0 277. ,0 285. .0 285. ,0 281. ,0 275. 5 276. .0 276. .5 276. .5 279. .0 281. .5 280. .0 278. .0 276. .5 276. .0 277, .0 278. .0 278, .0 35034 276. 5 277. .0 277. .0 279. ,5 282. .0 281. ,0 278. , 5 276. ,5 276. , 5 278, .0 278. .5 278. . 5 277. , 5 278. .0 278. .0 280. .0 282. .5 282. ,5 279. .5 277, .0 277. .0 279, .5 279. .5 279, .5 278. ,0 278.5 278. ,0 280. ,0 283. ,0 283. ,0 280. ,0 277. , 5 277. ,5 281. .0 280. ,0 280. 5 279. ,0 279. ,0 278. ,5 280. ,5 284. ,0 283. .5 280. .5 278. .0 278. .0 282, .0 280. .5 281. .0 279. 5 279. ,5 279. ,0 281. ,0 283. ,5 284. ,5 281. , 5 278. ,5 278. ,5 282. . 5 281. , 5 282. ,0 280. ,5 280. .0 279. .5 280. .5 282. .5 285.0 282. .0 279. .0 279. .0 283, .5 282. .0 282. .5 281. 5 281. ,0 279. ,0 278. , 5 281. ,5 285. ,0 282. .5 279. .0 280. ,0 284. .5 283.0 283. , 5 280. 5 278. , 5 275. .5 275. .5 279. .5 281. ,5 282. .0 278. .5 276. , 5 279, .0 280. .0 281. .5 276. 5 275. ,0 273. , 5 275. ,5 280. ,0 280. . 5 277.0 273. , 5 275. ,5 280, .0 277. , 5 278. ,5 279. 0 276. ,0 274. .5 276. ,0 281. .0 281. ,5 279. .0 274. ,0 276. ,5 283. . 5 280. .5 279. .5 282. 5 276. 5 282. ,0 277. ,0 282. ,0 282. ,5 281. . 5 274.5 277. ,5 285. .0 285. ,0 281. , 5 276. 0 276. 5 277. .0 277. ,0 279. , 5 282. ,0 280. , 5 278. ,5 277. ,0 276. ,5 277, .5 278. .5 278. . 5 35110 277. b 277. ,5 277. , 5 280. ,0 282. ,5 281. .5 279. .0 277. .0 277. 0 278. .5 279.0 279. .0 278. ,0 278. , 5 278. .5 280. ,5 283. ,0 283. .0 280. .0 277. .5 277. . 5 280, .0 280. .0 280, .0 278. 5 279. ,0 278. ,5 280. , 5 283. ,5 283. ,5 280. . 5 278. ,0 278. ,0 281. . 5 280. .5 281. ,0 279. .5 279. ,5 279. ,0 281. ,0 284. , 5 284.0 281. .0 278. .5 278. ,5 282, . 5 281. ,0 281. .5 280. 0 280. 0 279. ,5 281. 5 283. 5 285. ,0 282. ,0 279. ,0 279. 0 283. ,0 282. .0 282. . 5 281. .0 280. s 280. ,0 280. . 5 282. ,5 285. ,0 282. , 5 279. , 5 279. 5 284. .0 282. .5 283. ,0 282. 0 281. ,5 279. ,0 278. , 5 281. ,5 285. ,0 283. ,0 279. ,5 280. 5 285. .0 283. ,5 284.0 280. 5 278. 5 275. .5 276. .0 279. .5 281. ,5 282. .0 278. .5 276. .5 279.0 280.0 281. .5 277. .0 275. ,5 274.0 276. ,0 280. ,0 281. ,0 277. ,5 274.0 276. .0 280. .5 278.0 279. ,0 279.5 276. ,5 275. .0 276. , 5 281. ,5 282.0 279. ,5 274. ,5 277. ,0 284. .0 281. .0 280. .0 283. 0 277.0 282. ,5 277. 5 282. ,5 283. 0 282. ,0 275. ,0 278. ,0 285.0 285. ,0 282. ,0 276. 5 277. ,0 277. .5 277. . 5 280. ,0 282. ,5 281. ,0 279. ,0 277. .5 277. ,0 278. .0 279. .0 279. , 0 35187 277. ,5 278.0 278. .0 280. ,5 283. ,0 282. ,0 279. ,5 277. .5 277.5 279.0 279. .5 279. .5 278. ,5 279. ,0 279.0 281. .0 283. .5 283. .5 280. ,5 278.0 278.0 280.5 280.5 280. .5 279. ,0 279. ,5 279.0 281. ,0 284. ,0 284. ,0 281. ,0 278. .5 278.5 282. .0 281. .0 281. .5 280.0 280. ,0 279. .5 281. , 5 284. .5 284.5 281. .5 279. .0 279.0 283 .0 281, .5 282. .0 280. 5 280. 5 280.0 282. ,0 283. ,5 285.0 282. , 5 279.5 279. ,5 283, .5 282. .5 283. .0 281. ,5 281. ,0 280. , 5 280. ,5 282. ,5 285. .0 283. ,0 280. ,0 280. .0 284 .5 283. .0 283, .5 282. ,5 282. ,0 279.0 278. ,5 281. ,5 285. ,0 283. .5 280.0 281. .0 285 .0 284, .0 284.5 280. .5 278. .5 275. .5 276. .5 280. .0 281. .5 282. .0 278. .5 276. .5 279 .0 280 .0 281 . 5 277. 5 276.0 274. ,5 276. ,5 280. , 5 281. ,5 278. .0 274.5 276. .5 281 .0 278, .5 279, .5 280. ,0 277. ,0 275. .5 277. ,0 282. .0 282. ,5 280. .0 275. .0 277. , 5 284 .5 281. .5 280.5 283. 5 277. 5 283. ,0 278.0 283. ,0 283. ,5 282. .5 275. ,5 278. .5 285, .0 285, .0 282, . 5 -85-TABLE C . I . ( c o n t ' d ) OPTIMAL MONTHLY RESERVOIR LEVELS ( f t ) CASE 2: PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time s t e p (month) E l e v . 1 2 3 4 5 6 7 8 9 10 11 12 MW 277 . 0 277 . 5 278. .0 278. .0 280. , 5 283, .0 281. , 5 279. . 5 278. ,0 277. , 5 278, .5 279. . 5 279. . 5 35264 278 .0 278. . 5 278. , 5 281, .0 283. . 5 282. , 5 280. .0 278. 0 278. ,0 279. . 5 280. ,0 280. ,0 2 7 9 . 0 279. . 5 279. , 5 281. . 5 284, .0 284. .0 281. .0 278. ,5 278. , 5 281. .0 281. .0 281. .0 279 . 5 280. ,0 279, .5 281. ,5 284. , 5 284. , 5 281. , 5 279. 0 279. 0 282. , 5 281. 5 282. ,0 280. 5 280. . 5 280. .0 282. ,0 284. .5 285. .0 282. ,0 279. 5 279. , 5 283. , 5 282. ,0 282. .5 281 .0 281. ,0 280. , 5 282. ,0 283. , 5 285. ,0 283. ,0 280. 0 280. 0 284 .0 283. 0 283. , 5 2 8 2 . 0 281. .5 281, .0 280. . 5 282, . 5 285 .0 283. .5 280. ,5 280. ,5 285. .0 283. .5 284. .0 283 .0 282. .5 279. .0 278. .5 281. . 5 285. ,0 284. ,0 280. 5 281. , 5 285. .0 284. 5 285. ,0 280. 5 278. .5 275. .5 277. ,0 280, . 5 281. ,5 282, ,0 278. ,5 276. , 5 279. .5 280. ,0 281. .5 278 .0 276. .5 275. .0 277. .0 281. .0 282. ,0 278. , 5 275. 0 277. ,0 281. . 5 279. 0 280. ,0 280 . 5 277. . 5 276. .0 277. . 5 282. .5 283. .0 280. ,5 275. 5 278. ,0 285. ,0 282. ,0 281. .0 284 .0 278. ,0 283. .5 278. . 5 283. , 5 284. .0 283. ,0 276 .0 279. 0 285. .0 285. 0 283. .0 277 . 5 2 7 8 . 0 278. , 5 278. . 5 281. ,0 283. , 5 282. .0 280. .0 278. 5 278. ,0 279. ,0 280. ,0 280. .0 35340 278. 5 279. ,0 279. .0 281. . 5 284. .0 283. ,0 280. , 5 278. 5 278. 5 280. .0 280. 5 280. , 5 279 . 5 280. .0 280. ,0 282. ,0 284. , 5 284. , 5 281. , 5 279 . 0 279. ,0 281. , 5 281. , 5 281. ,5 280 .0 280. . 5 280. .0 282. .0 285. .0 285. .0 282. .0 279. 5 279. ,5 283. .0 282. .0 282. . 5 2 8 1 . 0 281. .0 280. .5 282. .5 284 .5 285. .0 282. ,5 2 8 0 . 0 280. .0 284 .0 282. .5 283, .0 281 .5 281 . 5 281. .0 282. ,0 283. . 5 285. ,0 283. 5 280. .5 280. .5 284. .5 283 . 5 284. ,0 282. 5 282. .0 281. , 5 280. . 5 282. .5 285. .0 284. .0 281. ,0 281. .0 285, .0 284. .0 284. .5 283 .5 283. ,0 279. ,0 278. , 5 281. . 5 285. ,0 284. 5 281 . 0 282. ,0 285. .0 285. ,0 285. .0 280. 5 278 . 5 275. . 5 277. ,0 281. .0 281. , 5 282. .0 278. ,5 276. ,5 280. .0 280. .0 281. .5 278 .5 277. ,0 275. , 5 277. ,5 281. . 5 282. ,5 279 .0 275. 5 277. ,5 282. .0 279. , 5 280. .5 2 8 1 . 0 278. ,0 276. .5 278. ,0 283. ,0 283. . 5 281. ,0 276. 0 278. ,5 285. .0 282. .5 281. .5 284 .5 278. 5 284. ,0 279. 0 284. ,0 284. 5 283. 5 276. 5 279. 5 285. ,0 285. ,0 283. . 5 278 . 0 2 7 8 . 5 279. .0 279, .0 281. .5 284. .0 282. .5 280. .5 279. ,0 278. .5 279, .5 280. .5 280, . 5 35416 279 .0 279. . 5 279. ,5 282. ,0 284. . 5 283. , 5 281. ,0 279. 0 279. ,0 280. .5 281. .0 281. .0 2 8 0 . 0 280. , 5 280, . 5 282. . 5 285. .0 285. .0 282. .0 2 7 9 . 5 279. ,5 282. .0 282. .0 282, .0 280 .5 281. ,0 280. . 5 282. . 5 285. .0 285. ,0 282. , 5 280 .0 280. ,0 283. .5 282. .5 283. .0 281 .5 281. , 5 281. .0 283. ,0 284. .5 285. ,0 283. .0 280. ,5 280. .5 284, .5 283. .0 283, .5 282 .0 282. 0 281. ,5 282. ,0 283. .5 285. ,0 2 8 4 . 0 281. 0 281. ,0 285. .0 284. .0 284. 5 2 8 3 . 0 282. ,5 281. .5 280. , 5 282. .5 285. .0 284. .5 281. ,5 281. .5 285, .0 284 .5 285, .0 2 8 4 . 0 283. 0 279. ,0 278. ,5 281. .5 285. ,0 285. ,0 281. 5 282. , 5 285. .0 285. .0 285. ,0 280. 5 278. .5 275. , 5 277. , 5 281. , 5 282. ,0 282. ,0 278. ,5 276. ,5 280, . 5- 280. .0 281, .5 279 .0 277. 5 276. ,0 278. ,0 282. ,0 283. ,0 279. ,5 276. 0 278. .0 282. .5 280. .0 281, .0 281 .5 278. . 5 277. .0 278. .5 283. .5 284. .0 281. .5 276. ,5 279. ,0 285, .0 283, .0 282 .0 2 8 5 . 0 279. 0 284. , 5 279. , 5 284. . 5 285. .0 284. ,0 277. ,0 280. ,0 285, .0 285. .0 284, .0 278 . 5 2 7 9 . 0 279. , 5 279. .5 282. .0 284. .5 283. .0 281. .0 279. , 5 279. .0 280, .0 281, .0 281, .0 35492 279 .5 280. 0 280. ,0 282. ,5 285. .0 284 .0 281. , 5 279. 5 279. , 5 281, .0 281, . 5 281, . 5 280 .5 281. ,0 281. ,0 283. ,0 285. ,0 285. .0 282. .5 280. ,0 280. .0 282, .5 282. . 5 282 .5 281 .0 281. 5 281. ,0 283. ,0 285. ,0 285. ,0 283. ,0 280. 5 280. , 5 284. .0 283. .0 283. . 5 2 8 2 . 0 282. ,0 281. ,5 283. , 5 284. . 5 285. ,0 283. ,5 281. .0 281. .0 285, .0 283. .5 284, .0 282 .5 282. 5 282. .0 282. .0 283. .5 285. ,0 284. , 5 281 . 5 281. , 5 285. .0 284. . 5 285, .0 283. 5 283. ,0 281. , 5 280. , 5 282. , 5 285. ,0 285. ,0 282. ,0 282. ,0 285, .0 285. .0 285. .0 284. 5 283. .0 279. .0 278. , 5 281. . 5 285 .0 285. .0 282. .0 283. ,0 285, .0 285, .0 285, .0 280 . 5 278. .5 275. .5 278. ,0 282. .0 282. .5 282. .0 278. .5 276. , 5 281 .0 280. .0 281 .5 279 .5 278 .0 276 .5 278 .5 282. . 5 283. ,5 280. ,0 276. ,5 278. ,5 283, .0 280. .5 281, .5 2 8 2 . 0 279. ,0 277. ,5 279 .0 284. .0 284 .5 282. ,0 277. .0 279 . 5 285, .0 283, .5 282 .5 285 .0 279. 5 285. ,0 2 8 0 . 0 285. .0 285. ,0 284. , 5 277. ,5 280 .5 285, .0 285, .0 284, . 5 2 7 9 . 0 279 .5 280. ,0 280. ,0 282. , 5 285. .0 283. ,5 281. ,5 280. ,0 279. ,5 280 . 5 281. . 5 281 . 5 35568 2 8 0 . 0 2 8 0 . 5 280. 5 283. 0 285. ,0 284 .5 282. .0 280. 0 280. ,0 281, . 5 282. .0 282. .0 2 8 1 . 0 281. ,5 281. ,5 233. ,5 285. ,0 285. ,0 283. .0 280. ,5 280. ,5 283 .0 283. .0 283 .0 281 .5 282. 0 281. 5 283 . 5 285. ,0 285. .0 283. ,5 281 . 0 281. ,0 284. 5 283, ,5 284. .0 2 8 2 . 5 282. .5 282. .0 284. .0 284, .5 285. .0 284. .0 281. ,5 281. ,5 285 .0 284, .0 284 .5 283 .0 283. .0 282. .5 282. ,0 283. .5 285. ,0 285. ,0 282. 0 282. .0 285, .0 285. .0 285, .0 2 8 4 . 0 283. .5 281. ,5 280. , 5 282. .5 285. .0 285. .0 282. ,5 282. .5 285, .0 285. .0 285 .0 2 8 4 . 5 283. .0 279. ,0 278. .5 282. .0 285. ,0 285. .0 282. 5 283. .5 285. .0 285. .0 285. .0 2 8 0 . 5 278. .5 276. ,0 278. , 5 282. . 5 283. .0 282. ,0 278. 5 277. .0 281. .5 280. .0 281, .5 2 8 0 . 0 278. ,5 277. ,0 279. ,0 283. .0 284. .0 280 .5 277. 0 279. 0 283. ,5 281. ,0 282. .0 2 8 2 . 5 279 .5 278. .0 279. , 5 284. , 5 285. .0 282. ,5 277. 5 2 8 0 . 0 285. .0 2 8 4 . 0 283. .0 2 8 5 . 0 280 .0 285. ,0 280. 5 285. ,0 285. ,0 285. 0 278 . 0 281 . 0 285. ,0 285. 0 285. ,0 2 7 9 . 5 2 8 0 . 0 2 8 0 . 5 280. .5 283. ,0 285. ,0 284. .0 282. ,0 280 . 5 2 8 0 . 0 281. ,0 2 8 2 . 0 282. ,0 35644 2 8 0 . 5 281. ,0 281. .0 283. .5 2 8 5 . 0 285. ,0 282. ,5 280. 5 2 8 0 . 5 2 8 2 . 0 282. ,5 282. .5 2 8 1 . 5 282. .0 282. ,0 2 8 4 . 0 285. .0 285. ,0 283. .5 281 . 0 281. ,0 283. .5 283. ,5 283. .5 282 .0 282 .5 282. ,0 284 .0 285 .0 285. .0 2 8 4 . 0 281 . 5 281. ,5 285. .0 284 .0 284 .5 2 8 3 . 0 283. .0 282. ,5 284. ,0 284. .5 285. ,0 284 .5 282. ,0 282. .0 285, .0 284. .5 285. .0 2 8 3 . 5 283. ,5 283. ,0 282. ,0 283. ,5 2 8 5 . 0 2 8 5 . 0 282. 5 282. ,5 285, .0 2 8 5 . 0 285, .0 2 8 4 . 5 2 8 4 . 0 281. ,5 280. .5 282. .5 285. .0 285. ,0 283. ,0 283. ,0 285 .0 285. .0 285 .0 284 .5 283. ,0 2 7 9 . 0 279. ,0 282. ,0 285. ,0 2 8 5 . 0 283. 0 2 8 4 . 0 285, .0 285, ,0 285, .0 280 .5 278. ,5 276. .5 279. .0 283. ,0 283. .5 282. ,0 278. ,5 277. ,5 281 . 5 280, . 5 281 .5 280 .5 279. 0 277. ,5 279. ,5 283. .5 284 .5 281. ,0 277. ,5 279. .5 2 8 4 . 0 281, , 5 282 .5 2 8 3 . 0 280. .0 278. .5 280. .0 285, .0 285. .0 283. .0 278. .0 2 8 0 . 5 285 .0 284 .5 283 .5 285 .0 280. ,5 285. .0 281. ,0 285. .0 285. ,0 285 .0 278 .5 281. .5 285 .0 285, .0 285 .0 -86-TABLE C . I . ( c o n t ' d ) OPTIMAL MONTHLY RESERVOIR LEVELS ( f t ) CASE 2: PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time s t e p (month) R e t u r n E l e v - 1 2 3 4 5 6 7 8 9 10 11 12 MW 280 .0 280 .5 281 .0 281 .0 281 .5 282 .0 282 .5 282 .5 283 .0 283 .5 2 8 3 . 5 284 .0 2 8 4 . 0 2 8 5 . 0 284 .5 284 .5 283 .0 280 .5 2 7 9 . 0 281 .0 279 .5 283 .5 280 .5 285 .0 281 .0 280 .5 2 8 1 . 0 281 .5 281 .5 282 .0 282 .5 2 8 3 . 0 283 .0 283 .5 2 8 4 . 0 2 8 4 . 0 284 .5 284 .5 2 8 5 . 0 2 8 5 . 0 284 .5 283 .0 280 .5 279 .5 281 .5 2 8 0 . 0 2 8 4 . 0 2 8 1 . 0 285 .0 2 8 1 . 5 2 8 1 . 0 281 .5 2 8 2 . 0 282 .0 282 .5 2 8 3 . 0 283 .5 283 .5 2 8 4 . 0 284 .5 284 .5 285 .0 285 .0 285 .0 2 8 5 . 0 284 .5 2 8 3 . 0 280 .5 2 8 0 . 0 282 .0 280 .5 284 .5 281 .5 285 .0 282 .0 281 .5 2 8 2 . 0 282 .5 282 .5 283 .0 2 8 3 . 5 2 8 4 . 0 284 .0 284 .5 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 284 .5 283 .0 2 8 1 . 0 280 .5 282 .5 2 8 1 . 0 2 8 5 . 0 2 8 2 . 0 285 .0 282 .5 282 .0 282 .5 2 8 3 . 0 283.6 2 8 3 . 5 2 8 4 . 0 284 .5 284 .5 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 284 .5 2 8 3 . 0 2 8 1 . 5 2 8 1 . 0 2 8 3 . 0 281 .5 2 8 5 . 0 2 8 2 . 5 285 .0 2 8 3 . 0 2 8 2 . 5 2 8 3 . 0 283 .5 2 8 3 . 5 2 8 4 . 0 2 8 4 . 5 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 284 .5 2 8 3 . 0 282 .0 281 .5 283 .5 2 8 2 . 0 2 8 5 . 0 2 8 3 . 0 285 .0 283 .5 281 .0 283 .5 281 .5 284 .0 282 .5 284 .5 282 .5 284 .5 283 .0 2 8 4 . 0 283 .5 282 .0 281 .5 280 .5 279 .0 279 .0 277 .0 279 .5 278 .0 280 .0 2 7 9 . 0 280 .5 285 .0 281 .5 281 .5 2 8 4 . 0 282 .0 284 .5 2 8 3 . 0 2 8 5 . 0 283 .0 285 .0 283 .5 284 .0 2 8 4 . 0 282 .0 281 .5 280 .5 279 .0 279 .5 277 .5 280 .0 278 .5 280 .5 279 .5 281 .0 2 8 5 . 0 2 8 2 . 0 282 .0 284 .5 282 .5 285 .0 283 .5 2 8 5 . 0 283 .5 285 .0 2 8 4 . 0 2 8 4 . 0 284 .0 2 8 2 . 0 281 .5 280 .5 2 7 9 . 0 2 8 0 . 0 2 7 8 . 0 280 .5 279 .0 281 .0 2 8 0 . 0 281 .5 285 .0 282 .5 282 .5 2 8 5 . 0 283 .0 285 .0 2 8 4 . 0 2 8 5 . 0 284 .0 285 .0 284 .5 2 8 4 . 0 284 .0 282 .0 281 .5 2 8 0 . 5 2 7 9 . 0 280 .5 278 .5 281 .0 279 .5 281 .5 280 .5 2 8 2 . 0 285 .0 283 .0 2 8 3 . 0 2 8 5 . 0 283 .5 2 8 5 . 0 2 8 4 . 5 2 8 5 . 0 284 .5 2 8 5 . 0 2 8 5 . 0 2 8 4 . 0 284 .0 2 8 2 . 0 281 .5 281 .0 279 .0 281 .0 2 7 9 . 0 281 .5 2 8 0 . 0 2 8 2 . 0 281 .0 282 .5 2 8 5 . 0 283 .5 2 8 3 . 5 2 8 5 . 0 2 8 4 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 4 . 0 2 8 4 . 0 2 8 2 . 0 2 8 1 . 5 281 .0 279 .5 281 .5 2 7 9 . 5 2 8 2 . 0 280 .5 282 .5 281 .5 2 8 3 . 0 285 .0 284 .0 2 8 5 . 0 284 .5 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 284 .5 2 8 5 . 0 283 .5 285 .0 282 .5 285 .0 282 .5 285 .0 283 .5 284 .0 284 .0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 284 .5 2 8 5 . 0 283 .5 285 .0 282 .5 285 .0 283 .0 285 .0 284 .0 284 .5 284 .5 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 284 .5 285 .0 283 .5 2 8 5 . 0 283 .0 285 .0 283 .5 285 .0 284 .5 284 .5 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 284 .5 2 8 5 . 0 283 .5 285 .0 283 .0 285 .0 2 8 4 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 284 .5 2 8 5 . 0 283 .5 285 .0 283 .5 2 8 5 . 0 284 .5 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 284 .5 2 8 5 . 0 2 8 4 . 0 285 .0 284 .0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 282 .5 281 .0 283 .0 281 .0 284 .0 281 .5 284 .5 282 .0 2 8 5 . 0 282 .5 285 .0 283 .0 2 8 5 . 0 283 .5 285 .0 283 .5 282 .0 278 .5 281 .5 278 .0 283 .5 278 .5 285 .0 279 .0 2 8 3 . 0 2 8 1 . 5 283 .5 281 .5 284 .5 282 .0 2 8 5 . 0 282 .5 2 8 5 . 0 283 .0 285 .0 283 .5 2 8 5 . 0 284 .0 285 .0 284 .0 282 .0 278 .5 282 .0 278 .5 284 .0 2 7 9 . 0 285 .0 279 .5 283 .5 2 8 2 . 0 284 .0 2 8 2 . 0 2 8 5 . 0 282 .5 285 .0 2 8 3 . 0 285 .0 283 .5 285 .0 284 .0 285 .0 284 .5 2 8 5 . 0 284 .5 2 8 2 . 0 278 .5 282 .5 2 7 9 . 0 284 .5 279 .5 285 .0 2 8 0 . 0 284 .0 282 .5 284 .5 282 .5 2 8 5 . 0 2 8 3 . 0 285 .0 283 .5 2 8 5 . 0 284 .0 285 .0 284 .5 285 .0 285 .0 2 8 5 . 0 285 .0 282 .0 278 .5 283 .0 279 .5 2 8 5 . 0 280 .0 285 .0 280 .5 284 .5 283 .0 2 8 5 . 0 2 8 3 . 0 2 8 5 . 0 283 .5 2 8 5 . 0 2 8 4 . 0 2 8 5 . 0 284 .5 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 282 .5 278 .5 2 8 3 . 5 280 .0 285 .0 2 8 0 . 5 2 8 5 . 0 281 .0 2 8 5 . 0 283 .5 285 .0 283 .5 2 8 5 . 0 2 8 4 . 0 285 .0 2 8 4 . 5 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 3 . 0 2 7 9 . 0 2 8 4 . 0 2 8 0 . 5 2 8 5 . 0 2 8 1 . 0 285 .0 281 .5 280 .5 281 .5 281 .0 282 .5 281 .5 2 8 4 . 0 282 .0 285 .0 282 .5 285 .0 283 .0 285 .0 283 .5 285 .0 284 .0 285 .0 278 .0 282 .0 280 .0 284 .5 281 .0 2 8 5 . 0 282 .0 285 .0 281 .0 282 .0 281 .5 283 .0 282 .0 284 .5 282 .5 285 .0 283 .0 2 8 5 . 0 283 .5 285 .0 284 .0 2 8 5 . 0 284 .0 285 .0 278 .5 282 .5 280 .5 285 .0 281 .5 2 8 5 . 0 282 .5 285 .0 281 .5 282 .5 2 8 2 . 0 283 .5 282 .5 2 8 5 . 0 283 .0 285 .0 283 .5 2 8 5 . 0 284 .0 285 .0 284 .5 2 8 5 . 0 284 .0 285 .0 2 7 9 . 0 283 .0 281 .0 2 8 5 . 0 282 .0 285 .0 2 8 3 . 0 285 .0 282 .0 283 .0 2 8 2 . 5 284 .0 2 8 3 . 0 2 8 5 . 0 283 .5 285 .0 2 8 4 . 0 2 8 5 . 0 284 .5 285 .0 284 .5 2 8 5 . 0 284 .0 285 .0 2 7 9 . 5 283 .5 281 .5 285 .0 282 .5 285 .0 283 .5 285 .0 282 .5 283 .5 2 8 3 . 0 284 .5 283 .5 285 .0 2 8 4 . 0 285 .0 284 .5 2 8 5 . 0 285 .0 285 .0 2 8 4 . 5 285 .0 284 .0 285 .0 2 8 0 . 0 284 .0 2 8 2 . 0 285 .0 2 8 3 . 0 2 8 5 . 0 2 8 4 . 0 2 8 5 . 0 2 8 3 . 0 284 .0 2 8 3 . 5 285 .0 2 8 4 . 0 2 8 5 . 0 2 8 4 . 5 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 2 8 4 . 5 2 8 5 . 0 2 8 4 . 0 2 8 5 . 0 2 8 0 . 0 2 8 4 . 5 282 .5 285 .0 283 .5 2 8 5 . 0 284 .5 285 .0 282 .5 282 .5 35720 283 .0 283 .0 284 .0 284.0 284 .5 285.0 285 .0 285 .0 285 .0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 281 .0 281 .5 282 .0 283 .0 285 .0 284 .0 285 .0 285 .0 2 8 3 . 0 2 8 3 . 0 35796 283 .5 283 .5 284 .5 284 .5 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 281 .5 282 .0 282 .5 283 .5 2 8 5 . 0 284 .5 285 .0 285 .0 283 .5 283 .5 35872 284 .0 284 .0 285 .0 285 .0 285 .0 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 282 .0 282 .5 283 .0 284 .0 285 .0 2 8 5 . 0 285 .0 285 .0 284 .0 284 .0 35947 284 .5 284 .5 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 285 .0 282 .5 283 .0 283 .5 284 .5 2 8 5 . 0 285 .0 285 .0 285 .0 284 .5 284 .5 36019 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 285 .0 285 .0 2 8 5 . 0 282 .5 283 .0 284 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 36092 285 .0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 285 .0 285 .0 2 8 5 . 0 2 8 5 . 0 285 .0 2 8 5 . 0 2 8 3 . 0 283 .5 284 .5 2 8 5 . 0 2 8 5 . 0 2 8 5 . 0 285 .0 285 .0 -87-TABLE C . I . ( c o n t ' d ) OPTIMAL MONTHLY RESERVOIR LEVELS ( f t ) CASE 2: PERFECTLY CORRELATED INFLOWS - STATE CHANGE DECISION Time s t e p (month) Return E l e v . 1 2 3 4 5 6 7 8 9 10 11 12 MW 283, .0 283 .5 284 .0 284 .0 285, .0 285.0 285 .0 285, .0 284 .0 283. . 5 284, . 5 285, .0 285, .0 36164 284, .0 284. . 5 284 .5 285, .0 285.0 285, .0 285. .0 284, .0 284. ,0 285. .0 285.0 285.0 285, .0 285 .0 285 .0 285, .0 285.0 285 .0 285, .0 284 . 5 284. . 5 285, .0 285, .0 285. .0 285, .0 285.0 285 .0 285. .0 285.0 285, .0 285. .0 285. .0 285. ,0 285. .0 285. .0 285. ,0 285, .0 285, .0 285 .0 284. .0 284. 5 285, .0 285. .0 285, .0 285. ,0 285. ,0 285. .0 285. .0 285. .0 285, .0 284.0 282. .0 284.0 285, .0 285. .0 285, .0 285. .0 285. .0 285, .0 285. .0 285, .0 285 .0 281 . 5 281. . 5 284.5 285, .0 285, .0 285, .0 284. . 5 285, .0 285, .0 285, .0 284.5 283, .0 279 .5 282. .0 285.0 285. .0 285. .0 285. .0 284. ,0 285. ,0 285. ,0 285. .0 282. .5 281, .5 280 .0 282. . 5 285.0 285. .0 283. . 5 279, . 5 280. , 5 285. .0 283. .5 284. .0 284. .0 282. .5 281 .0 283. ,0 285.0 285. .0 284. .5 281. .0 283. 0 285. ,0 285. ,0 285. .0 285, .0 283, .5 282 .0 283. .5 285.0 285, .0 285. .0 281, .5 284. .0 285, .0 285, .0 285. .0 285. .0 284. .0 285 .0 284. .5 285.0 285. .0 285. .0 282. .0 285. ,0 285. .0 285. .0 285. .0 283. . 5 284. .0 284, .5 284 .5 285. .0 285.0 285, .0 285. .0 284.5 284. .0 285. .0 285. .0 285. .0 36235 284, .5 285. .0 285, .0 285. .0 285.0 285. .0 285. .0 284. .5 284. .5 285.0 285.0 285. .0 285. .0 285, .0 285 .0 285. .0 285.0 285, .0 285. .0 285, .0 285. .0 285. .0 285. .0 285, .0 285. ,0 285. .0 285 .0 285. .0 285.0 285, .0 285. .0 285, .0 285. .0 285. .0 285, .0 285, .0 285, .0 285, ,0 285 .0 284. .0 284. 5 285, .0 285, .0 285, .0 285. .0 285, .0 285. .0 285 .0 285. .0 285. .0 284.0 282. ,0 284.5 285, .0 285.0 285. .0 285. .0 285.0 285.0 285. .0 285. .0 285, .0 281 . 5 282. ,0 285.0 285. .0 285, .0 285, .0 284.5 285. .0 285. .0 285, .0 284. . 5 283. .0 280, .0 282. , 5 285.0 285. .0 285. .0 285. .0 284. .0 285. .0 285. .0 285. .0 283. .0 282. .0 280. .5 283. .0 285.0 285, .0 284, .0 280, .0 281. .0 285, .0 284. .0 284, .5 284.5 283. .0 281 .5 283. .5 285.0 285. .0 285. .0 281, . 5 283. . 5 285. .0 285. .0 285, .0 285. .0 284. .0 282 .5 284.0 285.0 285. .0 285. .0 282, .0 284. . 5 285. .0 285. .0 285, .0 285. .0 284. 5 285. .0 285. ,0 285.0 285.0 285. .0 282. .5 285. .0 285. .0 285. .0 285. .0 284.0 284. 5 285. .0 285, .0 285. .0 285.0 285, .0 285. .0 285, .0 284. .5 285, .0 285. .0 285, .0 36303 285.0 285. ,0 285. .0 285. ,0 285.0 285. .0 285.0 285, .0 285.0 285. .0 285, .0 285. .0 285, .0 285. .0 285 .0 285. .0 285.0 285. .0 285. .0 285, .0 285. .0 285, .0 285. .0 285 .0 285. .0 285. .0 285, .0 285. ,0 285.0 285.0 285. .0 285. .0 285. .0 285. .0 285, .0 285. .0 285. ,0 285. .0 285. .0 284. .0 284.5 285. .0 285. .0 285, .0 285. .0 285, .0 285, .0 285 .0 285. .0 285. .0 284.0 282. ,0 285.0 285. .0 285. ,0 285, .0 285. .0 285. .0 285. .0 285, .0 285. ,0 285. .0 281. .5 282. ,5 285.0 285. .0 285. .0 285, .0 284. , 5 285. .0 285. .0 285 .0 284. .5 283. .0 280.5 283. ,0 285.0 285. .0 285. .0 285. .0 284. .0 285.0 285. .0 285, .0 283. .5 282, .5 281. .0 283. ,5 285.0 285. .0 284. 5 280, .5 281. .5 285, .0 284, .5 285 .0 285. ,0 283. ,5 282. .0 284. ,0 285.0 285. .0 285. ,0 282, .0 284. ,0 285. ,0 285, .0 285.0 285. ,0 284.5 283, .0 284. 5 285.0 285. .0 285. .0 282. . 5 285. .0 285, .0 285, .0 285, .0 285. 0 285. ,0 285.0 285. ,0 285.0 285. ,0 285. ,0 283. .0 285.0 285.0 285.0 285. .0 284.5 285. .0 285, .0 285, .0 285. .0 285.0 285. .0 285. .0 285, .0 285, .0 285, .0 285, .0 285. .0 36370 285. ,0 285. ,0 285. .0 285. ,0 285.0 285.0 285. .0 285. .0 285. .0 285. .0 285. .0 285. .0 285. ,0 285. .0 285, .0 285. ,0 285.0 285. .0 285. ,0 285, .0 285. .0 285, .0 285, .0 285. .0 285. ,0 285. ,0 285. .0 285. ,0 285.0 285. .0 285.0 285. .0 285.0 285.0 285. .0 285, .0 285. ,0 285. ,0 285. .0 284. ,0 285.0 285. ,0 285. .0 285. .0 285. .0 285. .0 285. .0 285, .0 285. .0 285.0 284.0 282. .5 285.0 285. .0 285. .0 285. .0 285. ,0 285, .0 285, .0 285. .0 285. ,0 285. .0 281, .5 283. .0 285.0 285. .0 285. .0 285. .0 284. . 5 285, .0 285, .0 285 .0 284. ,5 283. ,0 281. .0 283. ,5 285.0 285. .0 285. ,0 285. .0 284. ,0 285. .0 285; .0 285. .0 284. ,0 283. .0 281. .5 284. ,0 285.0 285. .0 285. .0 281. .0 282. .0 285. .0 285. .0 285. .0 285. 0 284.0 282, .5 284. 5 285.0 285. ,0 285. .0 282. .5 284. 5 285.0 285.0 285, .0 285. ,0 285. .0 283. .5 285. ,0 285.0 285. .0 285. .0 283. .0 285. .0 285, .0 285, .0 285 .0 285. ,0 285.0 285. .0 285. .0 285.0 285. .0 285. ,0 283. . 5 285. .0 285. .0 285, .0 285. .0 285. 0 285. ,0 285. .0 285, .0 285. ,0 285.0 285. .0 285. .0 285. .0 285. .0 285. .0 285. .0 285. .0 36437 285. 0 285. ,0 285. .0 285.0 285.0 285. ,0 285. ,0 285. ,0 285. ,0 285. .0 285. .0 285. .0 285. ,0 285. .0 285. .0 285. ,0 285.0 285. ,0 285. ,0 285. .0 285. ,0 285. .0 285. .0 285. .0 285. .0 285. ,0 2BS. .0 285. ;o 285.0 285. .0 285. .0 285.0 285.0 285.0 285. .0 285. .0 285. .0 285. .0 285. .0 284. ,0 285.0 285. .0 285. .0 285. .0 285. .0 285. .0 285. .0 285, .0 285. ,0 285. .0 284. .0 283. 0 285.0 285. ,0 285.0 285. .0 285. ,0 285. ,0 285. .0 285. .0 285. ,0 285. .0 281. .5 283. ,5 285.0 285. ,0 285. .0 285. .0 284. 5 285. .0 285, .0 285, .0 284.5 283. ,0 281. .5 284. 0 285.0 285. ,0 285. ,0 285. ,0 284.0 285. .0 285. .0 285, .0 284. .5 283. .5 282, .0 284.5 285.0 285. .0 285. .0 281. .5 282. .5 285. .0 285. .0 285, .0 285. ,0 284.5 283. .0 285. 0 285.0 285.0 285.0 283. .0 285. .0 285. .0 285. .0 285. .0 285. ,0 285. .0 284.0 285. ,0 285.0 285. .0 285. .0 283. .5 285. .0 285. .0 285, .0 285, .0 285.0 285.0 285. .0 285.0 285.0 285.0 285. .0 284. .0 285. .0 285. .0 285. .0 285.0 -88-TABLE C.II. OPTIMAL MONTHLY POWERHOUSE FLOW (1000 cfs) CASE 2: PERFECTLY CORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev. 1 2 3 4 5 6 7 8 9 10 11 12 MW 265.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33649 0.0 13.5 0.0 0.0 0.0' 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 22.4 43.3 0.0 119.4 74.1 89.5 115.5 119.4 115.5 119.4 89.5 115.5 119.4 115.5 119.4 119.4 101.1 104.4 115.5 119.4 115.5 119.4 97.0 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 104.4 115.5 119.4 115.5 119.4 265.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33732 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 . 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 29.8 43.3 0.0 119.4 80.9 97.0 115.5 119.4 115.5 119.4 97.0 115.5 119.4 115.5 119.4 119.4 107.8 111.9 115.5 119.4 115.5 119.4 104.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 111.9 115.5 119.4 115.5 119.4 266.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33814 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 37.3 43.3 0.0 119.4 87.6 104.4 115.5 119.4 115.5 119.4 104.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 111.9 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 266.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33894 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 44.8 43.3 0.0 119.4 94.3 111.9 115.5 119.4 115.5 119.4 111.9 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 267.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33972 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 7.5 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 52.2 43.3 0.0 119.4 101.1 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 267.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34050 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 0.0 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 . 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0;0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 • 14.9 20.2 0.0 0.0 0.0 0.0 0.0 0.0 79.4 59.7 43.3 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 -89-TABLE C.II. (cont'd) OPTIMAL MONTHLY POWERHOUSE FLOW (1000 cfs) CASE 2: PERFECTLY CORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev. 1 2 3 4 5 6 7 8 9 10 11 12 MW 268.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34129 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0' 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.5 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 22.4 20.2 0.0 0.0 7.5 0.0 0.0 0.0 79.4 67.1 43.3 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 268.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34208 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 29.8 20.2 0.0 7.2 14.9 7.2 0.0 0.0 79.4 74.6 43.3 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 269.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34286 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 7.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 37.3 20.2 0.0 14.4 22.4 14.4 0.0 7.5 79.4 74.6 43.3 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.S 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 269.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34365 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 14.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 44.8 20.2 0.0 21.7 29.8 101.1 0.0 14.9 79.4 22.4 50.5 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 270.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34443 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 22.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 • 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 52.2 20.2 0.0 28.9 37.3 101.1 0.0 22.4 79.4 29.8 57.8 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 270.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34521 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0,0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 7.5 b.O 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.5 0.0 0.0 0.0 6.7 29.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 59.7 20.2 0.0 36.1 44.8 101.1 0.0 29.8 79.4 37.3 65.0 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 -90-TABLE C.II. (cont'd) OPTIMAL MONTHLY POWERHOUSE FLOW (1000 cfs) CASE 2: PERFECTLY CORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev. 1 2 3 4 5 6 7 8 9 10 11 12 MW 271.0 0.0 40.4 0.0 0.0. 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34600 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 14.9 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14.9 0.0 0.0 0.0 6.7 37.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 7.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 21.7 0.0 0.0 0.0 67.1 20.2 0.0 43.3 52.2 101.1 0.0 37.3 79.4 44.8 72.2 0.0 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 271.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34679 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 22.4 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 22.4 0.0 0.0 0.0 6.7 44.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 14.9 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 7.5 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 7.5 50.5 59.7 101.1 7.5 44.8 79.4 52.2 79.4 7.5 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 272.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34757 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 29.8 0.0 0.0 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 29.8 0.0 0.0 0.0 6.7 52.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 22.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 0.0 14.9 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 14.9 57.8 67.1 101.1 14.9 52.2 79.4 59.7 86.6 14.9 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 272.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34836 0.0 13.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 37.3 0.0 7.5 36.1 0.0 0.0 0.0 0.0 36.1 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 37.3 0.0 0.0 0.0 6.7 59.7 7.2 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 29.8 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 7.2 22.4 0.0 0.0 0.0 21.7 0.0 0.0 0.0 0.0 20.2 22.4 65.0 74.6 101.1 22.4 59.7 79.4 67.1 93.8 22.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 • 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 273.0 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34915 0.0 13.5 0.0 0.0 7.5 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 0.0 14.9 36.1 0.0 0.0 0.0 0.0 36.1 O.'O 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 44.8 0.0 0.0 0.0 6.7 67.1 14.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 37.3 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 101.1 29.8 0.0 0.0 0.0 21.7 0.0 0.0 0.0 7.5 20.2 29.8 72.2 82.1 101.1 29.8 67.1 79.4 74.6 101.1 29.8 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 273.5 0.0 40.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 34994 0.0 13.5 0.0 0.0 14.9 0.0 0.0 0.0 0.0 0.0 43.3 0.0 0.0 40.4 0.0 0.0 0.0 79.4 0.0 0.0 0.0 0.0 0.0 0.0 0.0 33.7 0.0 7.2 22.4 36.1 0.0 0.0 0.0 0.0 36.1. 0.0 0.0 6.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 52.2 0.0 0.0 0.0 6.7 74.6 21.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 44.8 0.0 7.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 40.4 0.0 101.1 37.3 0.0 0.0 0.0 21.7 0.0 0.0 0.0 14.9 20.2 37.3 79.4 89.5 101.1 37.3 74.6 86.6 74.6 108.3 37.3 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 119.4 107.8 119.4 115.5 119.4 115.5 119.4 119.4 115.5 119.4 115.5 119.4 - S I -T A B L E C . I I . ( c o n t ' d ) O P T I M A L M O N T H L Y P O W E R H O U S E F L O W ( 1 0 0 0 c f s ) C A S E 2 : P E R F E C T L Y C O R R E L A T E D I N F L O W S - D I S C H A R G E D E C I S I O N T i m e s t e p ( m o n t h ) R e t u r n E l e v . 1 2 3 4 5 6 7 8 9 10 11 12 M W 2 7 4 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 0 7 2 0 . 0 1 3 . 5 0 . 0 0 . 0 2 2 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 3 . 7 0 . 0 1 4 . 4 2 9 . 8 3 6 . 1 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 5 9 . 7 0 . 0 0 . 0 0 . 0 6 . 7 8 2 . 1 2 8 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 5 2 . 2 0 . 0 1 4 . 9 0 . 0 0 . 0 0 . 0 0 . 0 7 . 5 7 . 2 0 . 0 0 . 0 4 0 . 4 0 . 0 1 0 1 . 1 4 4 . 8 0 . 0 0 . 0 0 . 0 2 1 . 7 0 . 0 0 . 0 7 . 5 2 2 . 4 2 0 . 2 4 4 . 8 8 6 . 6 9 7 . 0 1 0 1 . 1 4 4 . 8 8 2 . 1 9 3 . 8 7 4 . 6 1 1 5 . 5 4 4 . 8 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 2 7 4 . 5 0 . 0 4 0 . 4 0 . 0 0 . 0 7 . 5 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 1 5 1 0 . 0 1 3 . 5 0 . 0 0 . 0 2 9 . 8 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 3 . 7 0 . 0 2 1 . 7 3 7 . 3 3 6 . 1 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 6 7 . 1 0 . 0 0 . 0 0 . 0 . 6 . 7 8 9 . 5 3 6 . 1 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 5 9 . 7 0 . 0 2 2 . 4 0 . 0 0 . 0 7 . 5 0 . 0 1 4 . 9 1 4 . 4 0 . 0 7 . 5 4 0 . 4 7 . 5 1 0 1 . 1 5 2 . 2 7 . 2 0 . 0 0 . 0 2 1 . 7 0 . 0 0 . 0 1 4 . 9 2 9 . 8 2 7 . 0 5 2 . 2 9 3 . 8 1 0 4 . 4 1 0 1 . 1 5 2 . 2 8 9 . 5 1 0 1 . 1 7 4 . 6 1 1 5 . 5 5 2 . 2 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 2 7 5 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 1 4 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 2 2 8 0 . 0 1 3 . 5 0 . 0 0 . 0 3 7 . 3 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 7 . 5 0 . 0 0 . 0 0 . 0 3 3 . 7 0 . 0 2 8 . 9 4 4 . 8 4 3 . 3 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 7 4 . 6 0 . 0 0 . 0 0 . 0 6 . 7 9 7 . 0 4 3 . 3 7 . 5 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 6 7 . 1 7 . 2 2 9 . 8 0 . 0 0 . 0 1 4 . 9 0 . 0 2 2 . 4 2 1 . 7 0 . 0 1 4 . 9 4 0 . 4 1 4 . 9 1 0 1 . 1 5 9 . 7 1 4 . 4 0 . 0 0 . 0 2 1 . 7 0 . 0 0 . 0 2 2 . 4 3 7 . 3 3 3 . 7 5 9 . 7 1 0 1 . 1 1 1 1 . 9 1 0 8 . 3 5 9 . 7 9 7 . 0 1 0 8 . 3 7 4 . 6 1 1 5 . 5 5 9 . 7 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 2 7 5 . 5 0 . 0 4 0 . 4 0 . 0 0 . 0 2 2 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 3 0 5 0 . 0 1 3 . 5 0 . 0 0 . 0 4 4 . 8 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 1 4 . 9 0 . 0 0 . 0 0 . 0 3 3 . 7 7 . 5 0 . 0 5 2 . 2 5 0 . 5 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 7 . 5 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 8 2 . 1 0 . 0 0 . 0 0 . 0 6 . 7 1 0 4 . 4 5 0 . 5 1 4 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 7 . 5 0 . 0 7 4 . 6 1 4 . 4 3 7 . 3 7 . 2 0 . 0 2 2 . 4 0 . 0 2 9 . 8 2 8 . 9 0 . 0 2 2 . 4 4 0 . 4 2 2 . 4 1 0 1 . 1 6 7 . 1 2 1 . 7 0 . 0 0 . 0 2 1 . 7 0 . 0 0 . 0 2 9 . 8 4 4 . 8 4 0 . 4 6 7 . 1 1 0 8 . 3 1 1 9 . 4 1 1 5 . 5 6 7 . 1 1 0 4 . 4 1 1 5 . 5 1 0 4 . 4 7 2 . 2 5 9 . 7 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 2 7 6 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 2 9 . 8 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 3 8 1 0 . 0 1 3 . 5 0 . 0 0 . 0 5 2 . 2 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 0 . 0 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 2 2 . 4 0 . 0 0 . 0 0 . 0 3 3 . 7 1 4 . 9 0 . 0 5 9 . 7 5 7 . 8 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 1 4 . 9 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 8 9 . 5 0 . 0 0 . 0 0 . 0 6 . 7 1 1 1 . 9 5 7 . 8 2 2 . 4 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 4 . 9 0 . 0 8 2 . 1 2 1 . 7 4 4 . 8 1 4 . 4 0 . 0 2 9 . 8 0 . 0 3 7 . 3 3 6 . 1 0 . 0 2 9 . 8 4 0 . 4 2 9 . 8 1 0 1 . 1 7 4 . 6 2 8 . 9 0 . 0 0 . 0 2 1 . 7 0 . 0 0 . 0 3 7 . 3 5 2 . 2 4 7 . 2 7 4 . 6 1 1 5 . 5 1 1 9 . 4 1 0 1 . 1 7 4 . 6 1 1 1 . 9 1 1 5 . 5 1 1 1 . 9 7 9 . 4 5 9 . 7 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 2 7 6 . 5 0 . 0 4 0 . 4 0 . 0 0 . 0 3 7 . 3 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 5 4 5 9 0 . 0 1 3 . 5 0 . 0 0 . 0 5 9 . 7 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 4 3 . 3 0 . 0 0 . 0 4 0 . 4 7 . 5 0 . 0 0 . 0 7 9 . 4 0 . 0 0 . 0 0 . 0 2 9 . 8 0 . 0 0 . 0 0 . 0 3 3 . 7 2 2 . 4 0 . 0 6 7 . 1 6 5 . 0 0 . 0 0 . 0 0 . 0 0 . 0 3 6 . 1 0 . 0 0 . 0 6 . 7 2 2 . 4 0 . 0 7 . 5 0 . 0 0 . 0 0 . 0 0 . 0 9 7 . 0 0 . 0 0 . 0 0 . 0 6 . 7 1 1 9 . 4 6 5 . 0 2 9 . 8 7 . 2 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 8 9 . 5 2 8 . 9 5 2 . 2 2 1 . 7 0 . 0 3 7 . 3 0 . 0 4 4 . 8 4 3 . 3 0 . 0 3 7 . 3 4 0 . 4 3 7 . 3 1 0 1 . 1 8 2 . 1 3 6 . 1 0 . 0 0 . 0 2 1 . 7 7 . 5 0 . 0 4 4 . 8 • 5 9 . 7 1 0 7 . 8 8 2 . 1 1 1 5 . 5 1 1 9 . 4 1 0 1 . 1 8 2 . 1 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 8 6 . 6 5 9 . 7 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 0 7 . 8 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 1 1 5 . 5 1 1 9 . 4 TABLE C . I I . ( c o n t ' d ) OPTIMAL MONTHLY POWERHOUSE FLOW (1000 c f s ) CASE 2: PERFECTLY CORRELATED INFLOWS - DISCHARGE DECISION Time s t e p (month) Return E l e v . 1 2 3 4 5 6 7 8 9 10 11 12 MW 277.0 277.5 278.0 278.5 279.0 279.5 0. .0 40 .4 0, .0 0, ,0 44.8 0.0 0. .0 0. ,0 0. .0 0, .0 0. 0 0.0 35535 0. ,0 13, . 5 0, .0 0. ,0 67, . 1 0.0 0. 0 0. .0 0. .0 0. ,0 43. 3 0.0 0. .0 40, .4 14.9 0. .0 0, .0 79.4 0. ,0 0. ,0 0. ,0 37. . 3 0. ,0 0.0 0. ,0 33, . 7 29.8 0. ,0 74, .6 72.2 0. 0 0. 0 0. 0 0. 0 36. 1 0.0 0. ,0 6, .7 29. .8 0. ,0 14. .9 7.2 0. ,0 0. ,0 0. ,0 104. .4 7. 2 0.0 7. . 5 6, .7 119. .4 72. , 2 37. . 3 14.4 0. 0 0. 0 0. ,0 0. 0 0. 0 0.0 0. .0 0 .0 97. ,0 36. , 1 59, . 7 28.9 0. ,0 44. ,8 0. ,0 52. ,2 50. 5 0.0 44. ,8 40, .4 44. .8 101. , 1 89. .5 43.3 0. 0 7. 5 21. ,7 14. 9 0. 0 44.8 67. .1 107, .8 89. .5 115. , 5 119.4 101.1 89. 5 119. 4 86. 6 119. ,4 93. 8 82.1 119. 4 107. ,8 119. .4 115. 5 119. .4 115.5 119. 4 119. 4 115. 5 119. 4 115. 5 119.4 119. .4 107, .8 119. .4 115. ,5 119, .4 115.5 119. 4 119. .4 115. ,5 119. 4 115. 5 119.4 119. ,4 107. .8 119. ,4 115. ,5 119, .4 115.5 119. 4 119. 4 115. 5 119. 4 115. 5 119.4 0. ,0 40, .4 0. .0 0. ,0 52. .2 0.0 0. ,0 0. ,0 0. .0 0. ,0 0. 0 0.0 35612 0. 0 13. ,5 0. 0 0. 0 74.6 0.0 0. 0 0. 0 0. 0 0. 0 43. 3 0.0 0. ,0 40, .4 22. .4 0. .0 0, .0 79.4 0. .0 0. ,0 0. .0 44. ,8 0. 0 0.0 0. 0 33. . 7 37. .3 0. 0 82. . 1 79.4 0. 0 0. 0 0. ,0 7. 5 36. 1 0.0 0. ,0 6. .7 37. .3 0. ,0 22. .4 14.4 0. ,0 0. ,0 0. ,0 111. ,9 14.4 0.0 14. 9 6. ,7 119. 4 79. 4 44, .8 21.7 0. 0 7. 5 0. 0 0. 0 0. 0 0.0 0. ,0 0. ,0 104. ,4 43. ,3 67. . 1 36.1 0. 0 52. 2 0. ,0 59. .7 57. 8 0.0 52. 2 40. .4 52. ,2 101. ,1 97. .0 50.5 0. 0 14. 9 21. .7 22. 4 0. 0 52.2 74. 6 107, .8 97. .0 115. , 5 119, .4 101.1 97. ,0 119. .4 93. ,8 119. ,4 101. 1 89.5 119. 4 107. .8 119. 4 115. 5 119. .4 115.5 119. 4 119. 4 115. ,5 119. 4 115. 5 119.4 119. 4 107. .8 119. 4 115. 5 119, .4 115.5 119. 4 119. 4 115. ,5 119. .4 115. 5 119.4 119. 4 107. ,8 119. 4 115. 5 119. ,4 115.5 119. 4 119. 4 115. 5 119. 4 115. 5 119.4 0. 0 40. .4 0. ,0 0. ,0 59. .7 0.0 0. .0 0. ,0 0. .0 0. .0 0. .0 0.0 35689 0. 0 13. , 5 0. 0 0. 0 82. . 1 0.0 0. 0 0. 0 0. ,0 0. ,0 43. 3 0.0 0. 0 40. ,4 29. ,8 0. 0 7. ,5 79.4 0. 0 0. 0 0. ,0 52. .2 0. 0 0.0 0. 0 33. , 7 44.8 0. 0 89. , 5 86.6 0. 0 0. 0 0. 0 14. 9 36. 1 0.0 0. 0 6. ,7 44.8 0. .0 29. .8 21.7 0. ,0 0. ,0 0. ,0 119. .4 21. ,7 0.0 22. 4 6. ,7 119. 4 86. 6 52. . 2 28.9 0. 0 14. 9 0. .0 7. , 5 0. 0 0.0 0. 0 0. .0 111. 9 50. 5 74. .6 43.3 0. 0 59. 7 7. ,2 67. , 1 65. ,0 7.5 59. 7 40. ,4 59. 7 101. 1 104. .4 57.8 7. 5 22. 4 21. ,7 29. ,8 7. 2 59.7 82. 1 107. .8 104. 4 115. 5 119. .4 101.1 104.4 119. 4 101. 1 119. .4 108. 3 97.0 119. 4 107. .8 119. 4 115. 5 119. .4 115.5 119. 4 119. 4 115. ,5 119. 4 115. 5 119.4 119. 4 107. .8 119. 4 115. ,5 119. ,4 115.5 119. 4 119. 4 115. ,5 119. .4 115. ,5 119.4 119. 4 107. ,8 119. 4 115. 5 119. .4 115.5 119. 4 119. 4 115.5 119. .4 115. 5 119.4 0. 0 40. .4 0. 0 0. 0 67. .1 0.0 0. 0 0. 0 0. ,0 0. .0 0. ,0 0.0 35766 0. 0 13. ,5 0. 0 0. 0 89. , 5 0.0 0. 0 0. 0 0. 0 0. 0 43. 3 0.0 0. 0 40. .4 37. 3 0. ,0 14, .9 79.4 0. 0 0. ,0 0. ,0 59. .7 0. ,0 0.0 0. 0 33. , 7 52. 2 7. 2 22. .4 93.8 0. 0 0. 0 0. ,0 22. ,4 36. 1 0.0 7. 5 6. .7 52. 2 0. 0 37. , 3 28.9 0. 0 a. 0 0. ,0 119. .4 28. 9 0.0 29. 8 6. ,7 119.4 93. 8 52. , 2 36. 1 0. 0 22. 4 0. 0 14. 9 0. 0 7.5 0. 0 0. 0 119. 4 57. 8 82. , 1 50.5 0. 0 67. 1 14. .4 74. .6 72. 2 14.9 67. 1 40. ,4 67. 1 101. 1 111. .9 65.0 14.9 29. 8 21. ,7 37. 3 14. 4 67.1 89. 5 107. 8 111. 9 115. 5 119.4 108.3 111. 9 119. 4 108. 3 119.4 115. 5 104.4 119. 4 107. 8 119. 4 115. 5 119. ,4 115.5 119. 4 119. 4 115. 5 119. 4 115. 5 119.4 119. 4 107. .8 119. 4 115. 5 119.4 115.5 119. 4 119. 4 115.. 5 119. .4 115. 5 119.4 119. 4 107. 8 119. 4 115. 5 119. .4 115.5 119. 4 119. 4 115. 5 119. 4 115. 5 119.4 0. 0 40. .4 0. 0 0. 0 74. .6 0.0 0. 0 0. 0 0. ,0 0. .0 0. ,0 0.0 35843 0. 0 13.5 0. 0 0. 0 97. ,0 0.0 0. 0 0. 0 0. 0 0. 0 43. 3 0.0 0. 0 40. ,4 44. 8 7. 2 22. .4 79.4 0. 0 0. 0 0. 0 67. , 1 0. 0 0.0 0. 0 33. 7 59. 7 14. 4 29. ,8 101.1 0. 0 0. 0 0. 0 29. a 36. 1 0.0 14. 9 6. .7 59. 7 7. 2 44. .8 36.1 0. 0 0. 0 0. ,0 119. .4 36. 1 0.0 37. 3 6. ,7 119. 4 101. 1 59. ,7 43.3 0. 0 29. 8 0. .0 22. 4 0. 0 14.9 0. 0 0. .0 119. 4 65. 0 89. .5 57.8 7. 5 74. 6 21. ,7 82. ,1 79. 4 22.4 74.6 40.4 74.6 108. 3 119. 4 72.2 22. 4 37. 3 21. 7 44.8 21. 7 74.6 97. 0 107. ,8 119. 4 115. 5 119.4 115.5 119.4 119. 4 115. 5 119.4 115. 5 111.9 119. 4 107.8 119. 4 115. 5 119. .4 115.5 119. 4 119. 4 115. 5 119. 4 115.5 119.4 119. 4 107. .8 119.4 115. 5 119.4 115.5 119.4 119.4 115. ,5 119.4 115. 5 119.4 119. 4 107. 8 119. 4 115.5 119. .4 115.5 119. 4 119. 4 115.5 119. 4 115. 5 119.4 0. 0 40. ,4 0. 0 0. 0 82. ,1 0.0 0. 0 0. 0 0. .0 0. ,0 0. ,0 0.0 35920 0. 0 13. ,5 7. 5 0. 0 104. .4 7.2 0. 0 0. 0 0. 0 0. ,0 43.3 0.0 0. 0 40. .4 52. 2 14.4 29. .8 79.4 0. ,0 0. ,0 0. ,0 67. .1 7. 2 0.0 0. 0 33.7 67. 1 21. 7 37.3 108. 3 0. 0 0. 0 0. ,0 37.3 36.1 0.0 22. 4 6. ,7 67. 1 14.4 52. .2 43.3 0. 0 0. 0 0. .0 119. ,4 43. 3 7.5 44.8 6. ,7 119.4 108. 3 67.1 50.5 7. 5 37.3 0. ,0 29. ,8 7. 2 22.4 0.0 0. .0 119.4 72. 2 97, .0 65.0 14.9 82. ,1 28. ,9 89. .5 86. 6 29.8 82. 1 40. ,4 82. 1 115. 5 119. .4 79.4 29. 8 44. 8 21. ,7 52.2 28. 9 82.1 104. 4 107. ,8 119. 4 115. 5 119.4 115.5 119. 4 119. 4 115. .5 119. .4 115. 5 119.4 119. 4 107.8 119.4 115. 5 119. ,4 115. 5 119. 4 119. 4 115.5 119. ,4 115. 5 119.4 119. 4 107. ,8 119. 4 115. 5 119. ,4 115.5 119.4 119. 4 115. ,5 119. ,4 115. ,5 119.4 119.4 107. ,8 119. 4 115. 5 119. .4 115.5 119. 4 119. ,4 115. ,5 119. .4 115. 5 119.4 -53-TABLE C.II. (cont'd) OPTIMAL MONTHLY POWERHOUSE FLOW (1000 cfs) CASE 2: PERFECTLY CORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev. 1 2 3 4 5 6 7 8 9 10 11 12 MW 35997 280. .0 0, .0 40 . 4 0 .0 0, .0 89 . 5 0, .0 0.0 0. ,0 0. ,0 0. ,0 0. ,0 0. .0 0. .0 13. . 5 14, .9 0. .0 111, .9 14, .4 0.0 0. 0 0. 0 0. 0 43. 3 0. ,0 0, .0 40 .4 59 .7 21. .7 37, .3 79. .4 0.0 0. 0 0. ,0 67. , 1 14. .4 0. ,0 0. .0 33, . 7 74, .6 28, .9 44. .8 115. . 5 0.0 0. 0 0. 0 44. .8 36. 1 0. 0 29.8 6 .7 74 .6 21, . 7 59, .7 50. 5 7.5 7. 5 0. 0 119. .4 50. 5 14. ,9 52. .2 6, .7 119 .4 115, .5 74.6 57. .8 14.9 44. 8 0. .0 37. . 3 14. 4 29. .8 7, . 5 0 .0 119 .4 79, .4 104, .4 72. .2 22.4 89. .5 36. , 1 97. .0 93. .8 37. , 3 89. . 5 40, .4 89, .5 115. 5 119, .4 86.6 37.3 52. 2 21. .7 59. 7 36. 1 89. , 5 111. .9 107, .8 119. .4 115. . 5 119, .4 115. ,5 82.1 119. 4 115. 5 119. ,4 115. 5 119. ,4 119. .4 107, .8 119, .4 115. .5 119, .4 115. , 5 119.4 119. 4 115. 5 119. 4 115. 5 119. ,4 119. .4 107, .8 119, .4 115. . 5 119. .4 115. .5 119.4 119. 4 115. 5 119. 4 115. 5 119. ,4 119. .4 107, .8 119, .4 115. .5 119, .4 115.5 119.4 119. 4 115. 5 119. 4 115. 5 119. .4 280. .5 0. .0 40, .4 7 . 5 0. .0 97, .0 7. .2 0.0 0. ,0 0. ,0 0. ,0 0. ,0 0. .0 0. .0 13, . 5 22, . 4 0. .0 119. .4 21, .7 0.0 0. 0 0. .0 0. ,0 43. 3 0. .0 0. .0 40.4 67, .1 28.9 44.8 79.4 7.5 0. ,0 0. ,0 67. , 1 21. .7 0. .0 0. .0 33. .7 82. . 1 36. 1 52.2 50.5 0.0 0. 0 0. ,0 52. 2 36. 1 7. . 5 37. .3 6, . 7 82. . 1 28. .9 67. . 1 57. .8 14.9 14. 9 0. ,0 119. .4 57.8 22. .4 59. , 7 6, . 7 119. .4 50. . 5 82, . 1 65. ,0 22.4 52. 2 7. ,2 44.8 21. 7 37. , 3 14. .9 6, .7 119 .4 86. .6 111, .9 79. .4 29.8 97. ,0 0. .0 104. .4 101. .1 44. .8 97. .0 47. . 2 97, .0 115. .5 119. .4 93. .8 44.8 59. ,7 21. .7 67. . 1 43. ,3 97. .0 119. .4 107. .8 119, .4 115. .5 119. .4 115. ,5 89. 5 119. 4 115. , 5 119. .4 115. , 5 119. .4 119. ,4 107.8 119. .4 115. , 5 119. .4 115. ,5 119.4 119. 4 115. , 5 119. ,4 115. 5 119. .4 119. .4 107. .8 119, .4 115. .5 119. .4 115. .5 119.4 119. ,4 115. , 5 119. .4 115. ,5 119. .4 119. ,4 107, .8 119.4 115. , 5 119. .4 115. ,5 119.4 119. 4 115. ,5 119. 4 115. 5 119. .4 281. 0 0. .0 40. .4 7, .5 0. .0 104, .4 14. ,4 0.0 0. .0 0.0 7. .5 0. .0 0, .0 0. ,0 13. . 5 29. .8 0. ,0 119, .4 28. .9 0.0 0. 0 0. ,0 0. ,0 43. ,3 0, .0 0. .0 40. .4 74, .6 36. . 1 52. .2 79. .4 14.9 0. ,0 0. ,0 67. .1 28. ,9 7 , .5 0. ,0 33. . 7 89. 5 43. ,3 59. ,7 57. ,8 7.5 7. 5 0. ,0 59. 7 36. 1 14. .9 44. .8 6, .7 82. . 1 36. .1 74. ,6 65. .0 22.4 22. .4 0. .0 119. .4 65. ,0 29. .8 67. 1 13. , 5 119. .4 57. 8 89. , 5 72. ,2 29.8 59. 7 14. ,4 52. 2 28. 9 44. .8 22. .4 67, .4 119, .4 93. .8 119. .4 86. .6 37.3 104. ,4 0. ,0 111. .9 108. .3 44, .8 104. ,4 53. .9 104.4 115.5 119, .4 101. .1 52.2 67. 1 28. ,9 74. .6 50. ,5 104. .4 119. 4 107. .8 119, .4 115. .5 119, .4 115. .5 97.0 119. 4 115. ,5 119. .4 115. .5 119, .4 119. 4 107. .8 119, .4 115. ,5 119. ,4 115. ,5 119.4 119. 4 115. 5 119. 4 115. 5 119, .4 119. 4 107. .8 119. .4 115. , 5 119. .4 115. , 5 119.4 119. 4 115. ,5 119.4 115. ,5 119. .4 119. 4 107. .8 119. .4 115. ,5 119. .4 115. ,5 119.4 119. 4 115. 5 119. 4 115. 5 119. ,4 281. 5 7. 5 40. .4 7. . 5 0. ,0 111. ,9 21. ,7 0.0 0. 0 0. .0 14. ,9 0. ,0 0. ,0 7. 5 13. , 5 37, . 3 7. ,2 119. .4 36. , 1 0.0 0. 0 0. .0 0. .0 43. . 3 0. .0 0. ,0 40. .4 82. . 1 43. , 3 59.7 79. .4 22.4 7 . 5 0. .0 67. .1 36. ,1 14. .9 7. 5 33. .7 97, .0 50. , 5 67. , 1 65.0 14.9 14. 9 0. ,0 67. .1 36. 1 22. .4 52. 2 6. .7 82. . 1 43. , 3 82. . 1 72. ,2 29.8 29. .8 0. ,0 119. .4 72. ,2 37. , 3 74. 6 20. .2 119. .4 65. .0 97. ,0 79. ,4 37.3 67. 1 21. ,7 59. .7 36. 1 52. , 2 29. 8 67. .4 119. ,4 101. 1 119. .4 93.8 44.8 111. 9 7. ,2 119. .4 115. 5 52. .2 111. ,9 60. .6 111. .9 115. . 5 119. .4 108. 3 59.7 74.6 36. , 1 82. . 1 57. .8 I l l , .9 119. 4 107. .8 119. .4 115. , 5 119. .4 115.5 104.4 119. 4 115. , 5 119. .4 115. .5 119. .4 119. 4 107. .8 119. .4 115. 5 119. .4 115. ,5 119.4 119. 4 115. ,5 119. , 4 115. ,5 119.4 119. 4 107. .8 119. .4 115. , 5 119. .4 115. , 5 119.4 119. 4 115. , 5 119. .4 115. .5 119. .4 119. 4 107. 8 119. .4 115. 5 119.4 115. ,5 119.4 119. 4 115. .5 119. 4 115. ,5 119. ,4 282. 0 14. 9 40. .4 7. .5 7. .2 119. .4 28. ,9 0.0 0. 0 0. ,0 22. ,4 0. ,0 0. .0 14.9 13. .5 44.8 14.4 119. .4 43. .3 0.0 0. 0 0. .0 0. ,0 43. .3 7, .5 0. .0 40. ,4 89. .5 50. .5 67, . 1 79. ,4 29.8 14. 9 0. .0 67, . 1 43. ,3 22, .4 14. 9 33. .7 104. .4 57. 8 74. .6 72. .2 22.4 22. 4 0. ,0 74.6 36. , 1 29, .8 59. 7 13. ,5 82. . 1 50. .5 89. .5 79. .4 37.3 37. 3 0. ,0 119. .4 79. .4 44. .8 82. 1 27. ,0 119. .4 72. 2 104.4 86.6 44.8 74.6 28. 9 67. , 1 43. .3 59. . 7 37. 3 67. ,4 119. .4 108. 3 119. ,4 101. , 1 52.2 119. 4 14. 4 74. ,6 57. .8 59. . 7 119. 4 67. ,4 119. .4 115. 5 119. ,4 115. 5 67.1 82. 1 43. 3 89. 5 65. 0 119.4 119.4 107. .8 119. .4 115. . 5 119.4 115.5 111.9 119.4 115. 5 119.4 115. .5 119, .4 119. 4 107. .8 119. .4 115. .5 119. .4 115. ,5 119.4 119. 4 115. 5 119.4 115. 5 119. .4 119. 4 107, .8 119. .4 115. ,5 119. .4 115. , 5 119.4 119.4 115. 5 119.4 115. ,5 119. ,4 119.4 107. ,8 119. .4 115.5 119. ,4 115. 5 119.4 119. 4 115. 5 119. 4 115. 5 119. ,4 282. 5 22. 4 40. .4 7. .5 14. 4 119. .4 36. ,1 0.0 0. 0 0. 0 29. 8 7. 2 7. .5 22. 4 13. ,5 52.2 21. 7 119. .4 50. 5 7.5 7. 5 0. 0 7. 5 43. 3 14.9 0. .0 40.4 97, .0 57. .8 74. .6 79. .4 37.3 22. 4 0. 0 67. .1 50.5 29. .8 22. 4 33. ,7 111. .9 65. ,0 82. .1 79. 4 29.8 29. 8 0. 0 82. ,1 36. ,1 37, .3 67. ,1 20.2 82. .1 57. .8 97. .0 86. .6 44.8 44. 8 7. .2 119. .4 86. .6 44, .8 89. 5 94. ,3 119. ,4 79.4 111. ,9 93. 8 52.2 82. 1 36.1 74. 6 50. .5 67, . 1 44. 8 67. ,4 119. .4 115. ,5 119. .4 108.3 59.7 74.6 21. ,7 82. .1 65. .0 67. .1 119. 4 74.1 119. ,4 115. 5 119.4 115. 5 74.6 89. 5 50. ,5 97. ,0 72. .2 119, .4 119. 4 107. .8 119. .4 115. .5 119. .4 115. .5 119.4 119. .4 115. .5 119. .4 115, .5 119 .4 119. 4 107. .8 119. .4 115. ,5 119. ,4 115.5 119.4 119. .4 115. .5 119. .4 115, . 5 119 .4 119. 4 107. ,8 119. .4 115. ,5 119. .4 115. .5 119.4 119. .4 115. .5 119. .4 115, . 5 119 .4 119. 4 107. 8 119. ,4 115. 5 119. .4 115. .5 119.4 119. 4 115. ,5 119. ,4 115. . 5 119 .4 36074 36150 36224 36298 36367 - 9 4 -T A B L E C . I I . ( c o n t ' d ) O P T I M A L M O N T H L Y P O W E R H O U S E F L O W (1000 c f s ) C A S E 2: P E R F E C T L Y C O R R E L A T E D I N F L O W S - D I S C H A R G E D E C I S I O N T i m e s t e p ( m o n t h ) R e t u r n E l e v . 1 2 3 4 5 6 7. 8 9 10 11 12 M W 283 .0 29 .8 40, .4 7 .5 21. .7 119, .4 43. .3 7.5 0. .0 0, .0 37. .3 14, .4 14. .9 36436 29, .8 13. .5 59, .7 28. .9 119. .4 57. .8 14.9 14, .9 0. .6 14, .9 43. , 3 22. .4 0 .0 40, .4 104, .4 65. ,0 82. . 1 79. .4 44.8 29. .8 0. .0 67, . 1 57. ,8 37. . 3 -29. .8 33. .7 119, .4 72. .2 89. ,5 86. ,6 37.3 37. , 3 7. , 2 89. , 5 43. ,3 44. ,8 74, .6 27. .0 82. . 1 65. .0 104. ,4 93. .8 52.2 52. .2 14. .4 119. .4 93. ,8 52. , 2 97 .0 94, . 3 119, .4 86. .6 119. .4 101. .1 59.7 89. . 5 43. . 3 82. . 1 57. 8 74. ,6 52 .2 67, .4 119, .4 115. . 5 119. .4 115. .5 67.1 82. .1 28. .9 89, .5 72. .2 74. ,6 119, .4 80.9 119, .4 115. .5- 119. .4 115. , 5 82.1 97. .0 57. ,8 104. .4 79. ,4 119. , 4 119. .4 107, .8 119.4 115. . 5 119. .4 115. .5 119.4 119.4 115. .5 119. .4 115. . 5 119. ,4 119. .4 107. .8 119. .4 115. , 5 119. ,4 115. ,5 119.4 119, ,4 115. , 5 119. ,4 115. 5 119. ,4 119 .4 107, .8 119, .4 115. .5 119. .4 115. , 5 119.4 119, .4 115. . 5 119. .4 115. .5 119. .4 119. . 4 107. .8 119, .4 115. .5 119. .4 115. ,5 119.4 119, .4 115. , 5 119. .4 115. , 5 119. .4 283, . 5 37. .3 40. .4 7, .5 28. .9 119. .4 50. .5 14.9 7. .5 0. .0 44. .8 21. , 7 22. .4 36505 37. . 3 13. .5 67. . 1 36. , 1 119. ,4 65. ,0 22.4 22. ,4 0. ,0 22. .4 43. ,3 29. ,8 7, . 5 40.4 111. .9 72. .2 89. ,5 86. .6 52.2 37. , 3 7. .2 67. .1 65. ,0 37. .3 37, .3 33. .7 119.4 79. .4 97. .0 93. .8 44.8 44. .8 14. .4 97. .0 50.5 52. . 2 82, .1 94. .3 82, .1 72. .2 111. ,9 101. ,1 59.7 59. .7 21. . 7 119. .4 101. .1 59.7 104, .4 94, .3 119, .4 93. .8 119. .4 108. ,3 67.1 97. ,0 50. .5 89. ,5 65. ,0 82. . 1 59. .7 67, .4 119, .4 115. .5 119. ,4 115. .5 74.6 89. , 5 36. . 1 97. .0 79. ,4 82. . 1 119.4 87. .6 119. ,4 115. ,5 119. ,4 115. ,5 89.5 104.4 65.0 111. ,9 86. 6 119.4 119. .4 107 , .8 119, .4 115. .5 119. .4 115. ,5 119.4 119. .4 115. , 5 119. .4 115. .5 119. .4 119, .4 107. .8 119.4 115. ,5 119. ,4 115. ,5 119.4 119. ,4 115. . 5 119. .4 115. . 5 119, .4 119. .4 107. .8 119. .4 115. .5 119. ,4 115. .5 119.4 119. .4 115. , 5 119. ,4 115. .5 119. .4 119. ,4 107. ,8 119. .4 115. ,5 119. ,4 115. ,5 119.4 119. ,4 115. .5 119. .4 115.5 119. .4 284. .0 44. .8 40. ,4 7. .5 36. , 1 119.4 57. .8 22.4 14. .9 0. .0 52. ,2 28. .9 29. .8 36572 44. .8 13. ,5 67, .1 43. .3 119. ,4 72. .2 29.8 29. .8 7. .2 29. .8 43. .3 37. . 3 14. .9 40. ,4 119. .4 79. 4 97. .0 93. .8 59.7 44. .8 14. .4 67. . 1 72. .2 44. .8 44. .8 33. ,7 119. .4 86. 6 104.4 101. , 1 52.2 52. .2 21. .7 104. .4 57. 8 59. .7 89. . 5 94. .3 82, . 1 79. ,4 119. ,4 108. .3 67.1 67. , 1 28. .9 119. .4 108. ,3 67. .1 111. .9 94. ,3 119. .4 101. ,1 119.4 115. ,5 74.6 104. .4 57. 8 119.4 72. ,2 89. . 5 67. . 1 67. .4 119. ,4 115. S 119.4 115. .5 82.1 97. .0 43. ,3 104. .4 86. .6 89, .5 119. .4 94. ,3 119. .4 115. , 5 119. ,4 115.5 97.0 111. .9 72. .2 119. .4 93. .8 119. , 4 119. ,4 107. .8 119. .4 115. , 5 119.4 115. , 5 119.4 119. ,4 115. .5 119. .4 115. .5 119. .4 119.4 107. ,8 119. .4 115. ,5 119. .4 115. ,5 119.4 119.4 115. , 5 119. ,4 115. ,5 119. .4 119. .4 107. .8 119. .4 115. ,5 119. .4 115. .5 119.4 119. .4 115. .5 119. ,4 115. .5 119. .4 119. .4 107. .8 119. .4 115. 5 119.4 115. .5 119.4 119. .4 115, ,5 119. .4 115. . 5 119. .4 284. .5 52. . 2 40. ,4 59. .7 43. ,3 119. .4 65. .0 29.8 22. .4 7. .2 59. . 7 36. . 1 37, .3 36640 52. . 2 20. ,2 67, , 1 50. 5 119. 4 79. ,4 37.3 37. , 3 14.4 37. .3 43. .3 44. .8 22. .4 40.4 119. .4 86. .6 104.4 101. ,1 67.1 44.8 21. .7 67. .1 79. .4 52. .2 52. ,2 40. 4 119. ,4 93. 8 n r . ,9 108.3 59.7 59. ,7 28. ,9 111. ,9 65. ,0 67. . 1 97. .0 94. .3 82. .1 86. .6 119.4 115. .5 74.6 74. .6 36. , 1 119. .4 115. .5 74, .6 119. .4 94. .3 119. .4 108.3 119. 4 115. ,5 82.1 111. ,9 65. .0 119. .4 79.4 97. ,0 74. .6 67. .4 119. ,4 115. . 5 119. ,4 115. .5 89.5 104. .4 50. . 5 111. .9 93. .8 97. .0 119. 4 101. ,1 119. ,4 115. 5 119.4 115. 5 104.4 119. ,4 79. .4 119. ,4 101. ,1 119. .4 119. ,4 107. ,8 119. .4 115. ,5 119. 4 115. ,5 119.4 119. ,4 115. .5 119. .4 115. , 5 119. .4 119. ,4 107. 8 119. .4 115. ,5 119. ,4 115. 5 119.4 119. .4 115. .5 119. .4 115. .5 119. .4 119. .4 107. .8 119. ,4 115. , 5 119. .4 115. ,5 119.4 119, .4 115. . 5 119.4 115. .5 119. .4 119. .4 107.8 119. .4 115.5 119. 4 115. ,5 119.4 119. ,4 115. .5 119. ,4 115. , 5 119. ,4 285. .0 59. .7 40.4 67. .1 50. ,5 119. ,4 72. 2 37.3 29. .8 14. .4 67. .1 43. ,3 37. .3 36707 59. ,7 27. 0 67. ,1 57. 8 119. 4 86. 6 44.8 44. .8 21. ,7 44. ,8 50. . 5 52. . 2 29. 8 40. 4 119. .4 93. ,8 111. 9 108. .3 74.6 44.8 28. .9 67, . 1 86. .6 59, .7 59. .7 47. 2 119. .4 101. ,1 119. 4 115. ,5 67.1 67. ,1 36.1 119. .4 72. , 2 74.6 104. .4 94. ,3 82. .1 93. 8 119.4 115. .5 82.1 82. .1 43. .3 119. .4 79. .4 82. .1 119. ,4 94. ,3 119. .4 115. 5 119. 4 115.5 89.5 119. .4 72. .2 119. ,4 86. .6 104. .4 82. ,1 74.1 119. ,4 115. ,5 119.4 115. ,5 97.0 111. ,9 57. ,8 119. .4 101. , 1 104. .4 119.4 107. 8 119. .4 115. 5 119. 4 115. 5 111.9 119. ,4 86. 6 119. 4 108. .3 119. .4 119. .4 107. .8 119.4 115.5 119. .4 115. ,5 119.4 119. .4 115. .5 119. .4 115. .5 119. .4 119. .4 107.8 119. .4 115.5 119. 4 115. ,5 119.4 119. .4 115. , 5 119. ,4 115. .5 119. .4 119.4 107, .8 119. .4 115. ,5 119. 4 115. ,5 119.4 119. ,4 115. .5 119. .4 115. .5 119. .4 119. ,4 107. 8 119. ,4 115. 5 119.4 115. 5 119.4 119. .4 115. , 5 119. .4 115.5 119. .4 - 9 5 -TABLE C. III. SUMMARY RESULTS ; - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 1 1 85939 265.5 7311 5190 58 2 68999 266. 5 21932 379 4 3 69349 267. 5 36553 3879 44 4 93267 271.0 87726 1177 13 5 147060 277.0 175452 2424 29 6 146058 281.5 241247 5996 73 7 99714 284.0 277799 448 6 8 53267 285.0 292420 1379 17 9 64080 285. 0 292420 13500 169 10 122926 285.0 292420 25000 314 11 114460 285. 0 292420 38500 483 12 110221 285.0 292420 41000 514 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 2 1 85939 266.0 14621 5379 60 2 68999 267.5 36553 2569 29 3 69349 269.0 58484 5069 58 4 93267 273.0 116968 186 2 5 147060 279. 5 212005 3314 40 6 146058 285.0 292420 7135 88 7 99714 285.0 292420 47000 589 8 53267 285.0 292420 21000 263 9 64080 285.0 292420 20000 251 10 122926 285. 0 292420 40000 502 11 114460 285.0 292420 47000 589 12 110221 285. 0 292420 50500 633 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 3 1 85939 267.0 29242 2558 29 2 68999 269.5 65795 348 4 3 69349 272.0 102347 1698 20 4 93267 276.5 168142 966 11 5 147060 283. 5 270489 5313 65 6 146058 285.0 292420 83129 1038 7 99714 285. 0 292420 59500 746 8 53267 285.0 292420 26500 332 9 64080 285.0 292420 26600 334 10 122926 285.0 292420 63500 796 11 114460 285. 0 292420 59000 740 12 110221 285.0 292420 63000 790 -96-TABLE C. II I . SUMMARY RESULTS 1 - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 4 1 85939 267. 5 36553 6148 69 2 68999 270. 5 80416 1107 13 3 69349 273.0 116968 7188 83 4 93267 277.5 182763 6326 75 5 147060 285.0 292420 5463 67 6 146058 285.0 292420 112870 1415 7 99714 285. 0 292420 69500 872 8 53267 285.0 292420 32150 403 9 64080 285. 0 292420 32910 413 10 122926 285.0 292420 81500 1022 11 114460 285.0 292420 69000 865 12 110221 285.0 292420 74000 928 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 5 1 85939 268. 5 51174 1327 15 2 68999 272.0 102347 3317 38 3 69349 275.0 146210 5367 63 4 93267 280. 0 219315 4775 57 5 147060 284.5 285110 60606 748 6 146058 285. 0 292420 114090 1429 7 99714 285.0 292420 80040 1004 8 53267 285.0 292420 38500 483 9 64080 285.0 292420 40260 505 10 122926 285. 0 292420 95070 1192 11 114460 285.0 292420 78920 990 12 110221 285. 0 292420 85000 1066 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 6 1 85939 269.0 58484 3346 38 2 68999 273.0 116968 3476 40 3 69349 276. 5 168142 3537 42 4 93267 282.0 248557 4825 59 5 147060 283. 5 270489 114389 1416 6 146058 285.0 292420 110749 1383 7 99714 285.0 292420 88910 1115 8 53267 285.0 292420 44840 562 9 64080 285. 0 292420 48460 608 10 122926 285.0 292420 107850 1352 11 114460 285. 0 292420 89060 1117 12 110221 285.0 292420 96000 1204 - 9 7 -TABLE C.III. SUMMARY RESULTS - CASE 2, STATE CHABGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 7 1 85939 270. 0 73105 255 3 2 68999 274.5 138900 5426 63 3 69349 278. 5 197384 2546 30 4 93267 280.5 226626 67008 814 5 147060 282. 5 255868 117198 1441 6 146058 285.0 292420 111108 1384 7 99714 285.0 292420 98040 1229 8 53267 285.0 292420 51190 642 9 64080 284. 5 285110 65661 822 10 122926 285.0 292420 112500 1409 11 114460 285.0 292420 99210 1244 12 110221 285.0 292420 107000 1342 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 8 1 85939 271.0 87726 1994 23 2 68999 276. 5 168142 745 9 3 69349 279.0 204694 31678 381 4 • 93267 278. 5 197384 112081 1356 5 147060 281.5 241247 117287 1430 6 146058 285.0 292420 109217 1356 7 99714 285.0 292420 108690 1363 8 53267 285.0 292420 57530 721 9 64080 284.0 277799 83871 1049 10 122926 285.0 292420 116989 1463 11 114460 285.0 292420 111800 1402 12 110221 285. 0 292420 119000 1492 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 9 1 85939 272.5 109658 1513 17 2 68999 278. 5 197384 2024 24 3 69349 275.5 153521 119293 1429 4 93267 275.5 153521 111260 1321 5 147060 279. 5 212005 112926 1357 6 146058 283.5 270489 112056 1378 7 99714 284.0 277799 116440 1450 8 53267 280.5 226626 116734 1441 9 64080 278. 5 197384 112322 1365 10 122926 281.0 233936 114928 1399 11 114460 282.0 248557 112779 1387 12 110221 283.0 263178 119350 1476 - 9 8 -TABLE C.III. SUMMARY RESULTS - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 10 1 85939 266. 5 21932 116289 1302 2 68999 266.5 21932 102850 1157 3 69349 265. 0 0 113262 1268 4 93267 265.0 0 115500 1287 5 147060 269. 5 65795 118926 1344 6 146058 274.5 138900 115500 1343 7 - 99714 276.0 160831 119350 1415 8 53267 274.0 131589 119350 1413 9 64080 274. 5 138900 114170 1346 10 122926 279.0 204694 119206 1426 11 114460 281.0 233936 115500 1408 12 110221 284.0 277799 119137 1473 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 11 1 85939 268.5 51174 119350 1344 2 68999 269.0 58484 107800 1229 3 69349 268.0 43863 119350 1359 4 93267 268. 5 51174 115500 1313 5 147060 274.0 131589 119350 1382 6 146058 280. 0 219315 115500 1384 7 99714 283.5 270489 119350 1469 8 53267 282.0 248557 119350 1478 9 64080 283.0 263178 115500 1428 10 122926 285. 0 292420 119350 1488 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119350 1497 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 12 1 85939 272.0 102347 119350 1359 2 68999 273.0 116968 107800 1257 3 69349 279. 5 212005 119350 1424 4 93267 281.0 233936 115309 1408 5 147060 285.0 292420 119350 1480 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119350 1497 8 53267 284.0 277799 119350 1492 9 64080 285.0 292420 115500 1444 10 122926 285.0 292420 119350 1497 11 114460 285. 0 292420 115500 1448 12 110221 285.0 292420 119350 1497 - 9 9 -TABLE C. III. SUMMARY RESULTS - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 1 1 85939 275. 5 153521 5190 62 2 68999 276.5 168142 379 5 3 69349 277.5 182763 3879 46 4 93267 281.0 233936 1177 14 5 147060 285. 0 292420 31666 393 6 146058 285.0 292420 71790 900 7 99714 285.0 292420 37000 464 8 53267 285.0 292420 16000 201 9 64080 285. 0 292420 13500 169 10 122926 285.0 292420 25000 314 11 114460 285.0 292420 38500 483 12 110221 285.0 292420 41000 514 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 2 1 85939 276.0 160831 5379 64 2 68999 277. 5 182763 2569 31 3 69349 279.0 204694 5069 61 4 93267 283.0 263178 186 2 5 147060 285.0 292420 69108 862 6 146058 285.0 292420 87550 1098 7 99714 285.0 292420 47000 589 8 53267 285.0 292420 21000 263 9 64080 285.0 292420 20000 251 10 122926 285.0 292420 40000 502 11 114460 285.0 292420 47000 589 12 110221 285. 0 292420 50500 633 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 3 1 85939 277.0 175452 2558 30 2 68999 279.5 212005 348 4 3 69349 282.0 248557 1698 21 4 93267 285.0 292420 22897 285 5 147060 285.0 292420 107660 1350 6 146058 285.0 292420 105060 1317 7 99714 285.0 292420 59500 746 8 53267 285.0 292420 26500 332 9 64080 285. 0 292420 26600 334 10 122926 285.0 292420 63500 796 11 114460 285.0 292420 59000 740 12 110221 285.0 292420 63000 790 - 1 0 0 -TABLE C. III. SUMMARY RESULTS - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 4 1 85939 277.5 182763 6148 73 2 68999 280.5 226626 1107 13 3 69349 283.0 263178 7188 88 4 93267 285.0 292420 42878 535 5 147060 285.0 292420 115120 1444 6 146058 285.0 292420 112870 1415 7 99714 285. 0 292420 69500 872 8 53267 285.0 292420 32150 403 9 64080 285.0 292420 32910 413 10 122926 285.0 292420 81500 1022 11 114460 285. 0 292420 69000 865 12 110221 285.0 292420 74000 928 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 5 1 85939 278.5 197384 1327 16 2 68999 282.0 248557 3317 40 3 69349 285.0 292420 5367 67 4 93267 284.0 277799 92501 1157 5 147060 284.5 285110 119090 1487 6 146058 285.0 292420 114090 1429 7 99714 285.0 292420 80040 1004 8 53267 285. 0 292420 38500 483 9 64080 285.0 292420 40260 505 10 122926 285. 0 292420 95070 1192 11 114460 285.0 292420 78920 990 12 110221 285. 0 292420 85000 1066 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 6 1 85939 279.0 204694 3346 40 2 68999 283.0 263178 3476 43 3 69349 284. 0 277799 40089 499 4 93267 282.0 248557 114482 1420 5 147060 283. 5 270489 114389 1416 6 146058 285.0 292420 110749 1383 7 99714 285.0 292420 88910 1115 8 53267 285.0 292420 44840 562 9 64080 285.0 292420 48460 608 10 122926 285.0 292420 107850 1352 11 114460 285. 0 292420 89060 1117 12 110221 285.0 292420 96000 1204 -101-TABLE C. III. SUMMARY RESULTS ; - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 7 1 85939 280. 0 219315 255 3 2 68999 284.5 285110 5426 67 3 69349 281.5 241247 104893 1301 4 93267 280.5 226626 110871 1359 5 147060 282. 5 255868 117198 1441 6 146058 285.0 292420 111108 1384 7 99714 285.0 292420 98040 1229 8 53267 285.0 292420 51190 642 9 64080 284. 5 285110 65661 822 10 122926 285.0 292420 112500 1409 11 114460 285. 0 292420 99210 1244 12 110221 285.0 292420 107000 1342 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 8 1 85939 281.0 233936 1994 24 2 68999 283.0 263178 51918 640 3 69349 279.5 212005 119350 1465 4 93267 279.0 204694 112081 1360 5 147060 282.0 248557 117287 1434 6 146058. 285. 0 292420 115500 1436 7 99714 285.0 292420 108690 1363 8 53267 285.0 292420 57530 721 9 64080 284.0 277799 83871 1049 10 122926 285.0 292420 116989 1463 11 114460 285.0 292420 111800 1402 12 110221 285.0 292420 119000 1492 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 9 1 85939 280. 5 226626 30755 370 2 68999 279.5 212005 104371 1272 3 69349 276. 5 168142 119293 1437 4 93267 276.5 168142 111260 1329 5 147060 280.0 219315 119350 1440 6 146058 284.0 277799 112056 1382 7 99714 284. 5 285110 116440 1454 8 53267 281.0 233936 116734 1445 9 64080 279.0 204694 112322 1369 10 122926 281.5 241247 114928 1403 11 114460 282. 5 255868 112779 1391 12 110221 283.5 270489 119350 1480 - 1 0 2 -TABLE C.III. SUMMARY RESULTS - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Re s . E1 e. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 10 1 85939 276. 5 168142 116289 1383 2 68999 276.0 160831 107800 1286 3 69349 274. 0 131589 119350 1413 4 93267 274.0 131589 115500 1359 5 147060 278. 5 197384 118926 1418 6 146058 283. 5 270489 115500 1416 7 99714 285.0 292420 119350 1490 8 53267 283.0 263178 119350 1488 9 64080 283.0 263178 115500 1432 10 122926 285.0 292420 119350 1488 11 114460 285. 0 292420 115500 1448 12 110221 285.0 292420 119350 1497 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 11 1 85939 278.5 197384 119350 1428 2 68999 279.0 204694 107800 1305 3 69349 278.0 190073 119350 1442 4 93267 278. 5 197384 115500 1394 5 147060 284.0 277799 119350 1465 6 146058 285. 0 292420 115500 1444 7 99714 285.0 292420 119350 1497 8 53267 283. 5 270489 119350 1490 9 64080 284.5 285110 115500 1440 10 122926 285.0 292420 119350 1495 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119350 1497 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 12 1 85939 282.0 248557 119350 1442 2 68999 283.0 263178 107800 1333 3 69349 285.0 292420 119350 1488 4 93267 285.0 292420 115500 1448 5 147060 285. 0 292420 119350 1497 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119350 1497 8 53267 284.0 277799 119350 1492 9 64080 285. 0 292420 115500 1444 10 122926 285.0 292420 119350 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119350 1497 -103-TABLE C.III. SUMMARY RESULTS - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 1 1 85939 285.0 292420 12500 157 2 68999 285.0 292420 15000 188 3 69349 285.0 292420 18500 232 4 93267 285.0 292420 52350 656 5 147060 285.0 292420 90150 1130 6 146058 285.0 292420 71790 900 .7 99714 . 285.0 292420 37000 464 8 53267 285.0 292420 16000 201 9 64080 285.0 292420 13500 169 10 122926 285.0 292420 25000 314 11 114460 285. 0 292420 38500 483 12 110221 285.0 292420 41000 514 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 2 1 85939 285.0 292420 20000 251 2 68999 285.0 292420 24500 307 3 69349 285.0 292420 27000 339 4 93267 285.0 292420 58670 736 5 147060 285.0 292420 98350 1233 6 146058 285.0 292420 87550 1098 7 99714 285.0 292420 47000 589 8 53267 285. 0 292420 21000 263 9 64080 285.0 292420 20000 251 10 122926 285.0 292420 40000 502 11 114460 285.0 292420 47000 589 12 110221 285.0 292420 50500 633 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 3 1 85939 285. 0 292420 31800 399 2 68999 285.0 292420 36900 463 3 69349 285.0 292420 38250 480 4 93267 285.0 292420 66760 837 5 147060 285.0 292420 107660 1350 6 146058 285.0 292420 105060 1317 7 99714 285.0 292420 59500 746 8 53267 285.0 292420 26500 332 9 64080 285.0 292420 26600 334 10 122926 285.0 292420 63500 796 11 114460 285.0 292420 59000 740 12 110221 285.0 292420 63000 790 -104-TABLE C. II I . SUMMARY RESULTS 1 - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 4 1 85939 285.0 292420 42700 535 2 68999 285.0 292420 44970 564 3 69349 285.0 292420 43740 548 4 93267 285.0 292420 72120 904 5 147060 285. 0 292420 115120 1444 6 146058 285.0 292420 112870 1415 7 99714 285.0 292420 69500 872 8 53267 285.0 292420 32150 403 9 64080 285. 0 292420 32910 413 10 122926 285.0 292420 81500 1022 11 114460 285. 0 292420 69000 865 12 110221 285.0 292420 74000 928 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 5 1 85939 285.0 292420 52500 658 2 68999 285.0 292420 54490 683 3 69349 285.0 292420 49230 617 4 93267 284.0 277799 92501 1157 5 147060 284.5 285110 119090 1487 6 146058 285.0 292420 114090 1429 7 99714 285.0 292420 80040 1004 8 53267 285. 0 292420 38500 483 9 64080 285.0 292420 40260 505 10 122926 285.0 292420 95070 1192 11 114460 285.0 292420 78920 990 12 110221 285.0 292420 85000 1066 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 6 1 85939 285.0 292420 61830 775 2 68999 285.0 292420 61960 111 3 69349 284.0 277799 69331 867 4 93267 282.0 248557 114482 1420 5 147060 283. 5 270489 114389 1416 6 146058 285.0 292420 110749 1383 7 99714 285.0 292420 88910 1115 8 53267 285.0 292420 44840 562 9 64080 285.0 292420 48460 608 10 122926 285.0 292420 107850 1352 11 114460 285.0 292420 89060 1117 12 110221 285.0 292420 96000 1204 -105-TABLE C. III. SUMMARY RESULTS ; - CASE 2, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 7 1 85939 285.0 292420 73360 920 2 68999 285.0 292420 71220 893 3 69349 281.5 241247 112204 1393 4 93267 280.5 226626 110871 1359 5 147060 282.5 255868 117198 1441 6 146058 285.0 292420 ' 111108 1384 7 99714 285.0 292420 98040 1229 8 53267 285.0 292420 51190 642 9 64080 284. 5 285110 65661 822 10 122926 285.0 292420 112500 1409 11 114460 285.0 292420 99210 1244 12 110221 285.0 292420 107000 1342 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 8 1 85939 284.5 285110 97031 1215 2 68999 283.0 263178 103092 1284 3 69349 279.5 212005 119350 1465 4 93267 279.0 204694 112081 1360 5 147060 282.0 248557 117287 1434 6 146058 285.0 292420 115500 1436 7 99714 285.0 292420 108690 1363 8 53267 285.0 292420 57530 721 9 64080 284.0 277799 83871 1049 10 122926 285. 0 292420 116989 1463 11 114460 285.0 292420 111800 1402 12 110221 285.0 292420 119000 1492 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 9 1 85939 284. 5 285110 118481 1484 2 68999 283.0 263178 107800 1342 3 69349 280.0 219315 119293 1467 4 93267 279.5 212005 115500 1406 5 147060 283.0 263178 119350 1465 6 146058 285.0 292420 115500 1440 7 99714 285.0 292420 119350 1497 8 53267 281.5 241247 116734 1450 9 64080 279.5 212005 112322 1373 10 122926 281.5 241247 119350 1459 11 114460 282. 5 255868 112779 1391 12 110221 283.5 270489 119350 1480 -106-TABLE C. III. SUMMARY RESULTS i - CASE 2, STATE'CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 10 1 85939 285.0 292420 119350 1497 2 68999 284.5 285110 107800 1350 3 69349 282. 5 255868 119350 1484 4 93267 282.5 255868 115500 1428 5 147060 285.0 292420 119350 1486 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119350 1497 8 53267 283.0 263178 119350 1488 9 64080 283.0 263178 115500 1432 10 122926 285.0 292420 119350 1488 11 114460 285. 0 292420 115500 1448 12 110221 285.0 292420 119350 1497 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 11 1 85939 285.0 292420 119350 1497 2 68999 285. 0 292420 107800 1352 3 69349 284.0 277799 119350 1492 4 93267 284. 5 285110 115500 1442 5 147060 285.0 292420 119350 1495 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119350 1497 8 53267 283. 5 270489 119350 1490 9 64080 284.5 285110 115500 1440 10 122926 285.0 292420 119350 1495 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119350 1497 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 12 1 85939 285. 0 292420 119350 1497 2 68999 285.0 292420 107800 1352 3 69349 285.0 292420 119350 1497 4 93267 285.0 292420 115500 1448 5 147060 285.0 292420 119350 1497 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119350 1497 8 53267 284.0 277799 119350 1492 9 64080 285.0 292420 115500 1444 10 122926 285.0 292420 119350 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119350 1497 -107-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 1 1 85939 265.9 12500 0 0 2 68999 266.9 27500 0 0 3 69349 268. 1 45999 0 0 4 93267 271.7 98350 0 0 5 147060 277. 9 188500 0 0 6 146058 282.8 260290 0 0 7 99714 284.8 289790 7500 93 8 53267 283.9 275990 29800 372 9 64080 284.8 289490 0 0 10 122926 281.9 247390 67100 834 11 114460 284.6 285890 0 0 12 110221 284.8 289590 37300 467 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 2 1 85939 282.1 249890 59700 701 2 68999 282.8 260890 13500 167 3 69349 280.6 228190 59700 735 4 93267 284.6 286860 0 0 5 147060 283.2 265810 119400 1488 6 146058 285. 0 292420 57800 721 7 99714 285.0 292420 44800 562 8 53267 283.4 268620 44800 559 9 64080 284.7 288620 0 0 10 122926 284.9 291320 37300 467 11 114460 284. 7 287820 50500 633 12 110221 285.0 292420 44800 561 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 3 1 85939 285.0 292420 29800 353 2 68999 284.8 288920 40400 506 3 69349 279. 2 207770 119400 1472 4 93267 283.3 267330 7200 88 5 147060 284. 5 285490 89500 1115 6 146058 284.8 289450 101100 1265 7 99714 283.8 274350 74600 932 8 53267 282.5 256050 44800 556 9 64080 284. 3 282651 0 0 10 122926 284.1 279051 67100 838 11 114460 283. 2 265851 72200 898 12 110221 284.9 291551 37300 465 -108-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Re s. Vo1 Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 4 1 85939 283.8 274551 59700 704 2 68999 284.5 285821 33700 421 3 69349 279.4 210161 119400 1472 4 93267 282.8 260581 21700 266 5 147060 284.6 286201 89500 1114 6 146058 284.9 290771 108300 1356 7 99714 285.0 292420 67100 841 8 53267 282.6 257470 67100 836 9 64080 284.9 290380 0 0 10 122926 282.3 252480 119400 1485 11 114460 284. 5 285380 36100 449 12 110221 285.0 292280 67100 840 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 5 1 85939 281.4 240380 104400 1223 2 68999 284.7 288170 6700 83 3 69349 282.5 255300 82100 1021 4 93267 283.8 275380 57800 717 5 147060 284.3 282380 119400 1490 6 146058 284. 7 288280 115500 1444 7 99714 285.0 292420 74600 935 8 53267 282.0 248820 82100 1021 9 64080 284.8 289080 0 0 10 122926 283. 1 264750 119400 1488 11 114460 282.1 249870 93800 1160 12 110221 284.8 290070 44800 557 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 6 1 85939 280.9 232500 119400 1397 2 68999 284.2 280960 13500 167 3 69349 279.8 216270 119400 1472 4 93267 277.7 186010 115500 1398 5 147060 284. 0 277530 44800 549 6 146058 285.0 292420 115500 1444 7 99714 285.0 291830 89500 1122 8 53267 279.9 217270 119400 1476 9 64080 283.2 265730 0 0 10 122926 284.9 291480 82100 1024 11 114460 285.0 292420 86600 1086 12 110221 284.4 284020 104400 1307 -109-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 7 1 85939 284. 3 282780 74600 882 2 68999 284.6 286600 67400 843 3 69349 280.6 228230 119400 1477 4 93267 281.3 237880 86600 1061 5 147060 283. 1 264920 119400 1474 6 146058 285.0 292420 115500 1441 7 99714 285.0 292420 97000 1216 8 53267 280.8 231710 111900 1387 9 64080 284.8 290060 0 0 10 122926 284.9 290470 119400 1496 11 114460 284. 7 288580 101100 1266 12 110221 285.0 292420 97000 1215 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 8 1 85939 283.0 262740 119400 1405 2 68999 283.0 263000 80900 1003 3 69349 279.5 211830 119400 1466 4 93267 278.8 201100 115500 1401 5 147060 281.6 242850 119400 1457 6 146058 285.0 292420 108300 1345 7 99714 284.8 289210 111900 1402 8 53267 280.5 227340 119400 1478 9 64080 283.8 274890 21700 268 10 122926 284.6 287100 119400 1491 11 114460 285.0 292420 101100 1267 12 110221 285.0 292020 119400 1497 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 9 1 85939 284.4 283790 119400 1411 2 68999 283.2 265740 107800 1343 3 69349 280. 2 221770 119400 1469 4 93267 279.9 217530 115500 1408 5 147060 283.4 269540 119400 1469 6 146058 285.0 292420 115500 1442 7 99714 285. 0 292420 119400 1497 8 53267 281.3 238580 119400 1482 9 64080 279.1 206160 115500 1410 10 122926 281. 3 238240 119400 1457 11 114460 282.1 250140 ' 115500 1422 12 110221 283.3 267240 119400 1478 -110-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 10 1 85939 284. 6 286060 119400 1412 2 68999 284.2 281110 107800 1347 3 69349 282. 3 253040 119400 1483 4 93267 282.5 255150 115500 1427 5 147060 285.0 292420 119400 1487 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283.0 263470 119400 1489 9 64080 283.4 269450 115500 1434 10 122926 285.0 292420 119400 1491 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 11 1 85939 285.0 292420 119400 1414 2 68999 285.0 292420 107800 1352 3 69349 284. 2 281240 119400 1494 4 93267 285. 0 292420 115500 1445 5 147060 285.0 292420 119400 1497 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283. 5 270680 119400 1491 9 64080 284.9 290330 115500 1442 10 122926 285.0 292420 119400 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 265 f t - PROBABILITY LEVEL 12 1 85939 285.0 292420 119400 1414 2 68999 285.0 292420 107800 1352 3 69349 285. 0 292420 119400 1497 4 93267 285.0 292420 115500 1448 5 147060 285.0 292420 119400 1497 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 284.2 280200 119400 1494 9 64080 285.0 292420 115500 1445 10 122926 285.0 292420 119400 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 - I l l -TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 1 1 85939 275.9 158710 0 0 2 68999 274.1 133310 40400 478 3 69349 275.4 151810 0 0 4 93267 279.0 204160 0 0 5 147060 280.0 219710 74600 907 6 146058 284.9 291500 0 0 7 99714 284.9 291200 37300 468 8 53267 284.0 277400 29800 373 9 64080 284.9 290900 0 0 10 122926 282.0 248800 67100 834 11 114460 284.6 287300 0 0 12 110221 284.9 291000 37300 467 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 2 1 85939 282.2 251300 59700 722 2 68999 282.9 262300 13500 167 3 69349 280.7 229600 59700 735 4 93267 284. 7 288270 0 0 5 147060 283.3 267221 119400 1489 6 146058 284.8 289770 65000 811 7 99714 285.0 291971 44800 561 8 53267 283. 3 268171 44800 559 9 64080 284.7 288170 0 0 10 122926 284.9 290870 37300 467 11 114460 284.7 287370 50500 632 12 110221 285.0 292420 44800 561 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 3 1 85939 285.0 292420 29800 363 2 68999 284.8 288920 40400 506 3 69349 279. 2 207770 119400 1472 4 93267 283.3 267330 7200 88 5 147060 284.5 285490 89500 1115 6 146058 284.8 289450 101100 1265 7 99714 283.8 274350 74600 932 8 53267 282.5 256050 44800 556 9 64080 284. 3 282651 0 0 10 122926 284.1 279051 67100 838 11 114460 283. 2 265851 72200 898 12 110221 284.9 291551 37300 465 -112-TABLE CIV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 4 1 85939 283.8 274551 59700 725 2 68999 284.5 285821 33700 421 3 69349 279.4 210161 119400 1472 4 93267 282.8 260581 21700 266 5 147060 284.6 286201 89500 1114 6 146058 284.9 290771 108300 1356 7 99714 285.0 292420 67100 841 8 5-3267 282.6 257470 67100 836 9 64080 284.9 290380 0 0 10 122926 282.3 252480 119400 1485 11 114460 284.5 285380 36100 449 12 110221 285.0 292280 67100 840 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 5 1 85939 281.4 240380 104400 1260 2 68999 284. 7 288170 6700 83 3 69349 282.5 255300 82100 1021 4 93267 283.8 275380 57800 717 5 147060 284.3 282380 119400 1490 6 146058 284. 7 288280 115500 1444 7 99714 285.0 292420 74600 935 8 53267 282.0 248820 82100 1021 9 64080 284.8 289080 0 0 10 122926 283. 1 264750 119400 1488 11 114460 282.1 249870 93800 1160 12 110221 284.8 290070 44800 557 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 6 1 85939 280.9 232500 119400 1438 2 68999 284.2 280960 13500 167 3 69349 279.8 216270 119400 1472 4 93267 277.7 186010 115500 1398 5 147060 284.0 277530 44800 549 6 146058 285.0 292420 115500 1444 7 99714 285. 0 291830 89500 1122 8 53267 279.9 217270 119400 1476 9 64080 283. 2 265730 0 0 10 122926 284.9 291480 82100 1024 11 114460 285. 0 292420 86600 1086 12 110221 284.4 284020 104400 1307 -113-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 7 1 85939 284. 3 282780 74600 908 2 68999 284.6 286600 67400 843 3 69349 280. 6 228230 119400 1477 4 93267 281.3 237880 86600 1061 5 147060 283.1 264920 119400 1474 6 146058 285.0 292420 115500 1441 7 99714 285.0 292420 97000 1216 8 53267 280.8 231710 111900 1387 9 64080 284.8 290060 0 0 10 122926 284.9 290470 119400 1496 11 114460 284. 7 288580 101100 1266 12 110221 285.0 292420 97000 1215 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 8 1 85939 283.0 262740 119400 1447 2 68999 283.0 263000 80900 1003 3 69349 279.5 211830 119400 1466 4 93267 278.8 201100 115500 1401 5 147060 281.6 242850 119400 1457 6 146058 285. 0 292420 108300 1345 7 99714 284.8 289210 111900 1402 8 53267 280. 5 227340 119400 1478 9 64080 283.8 274890 21700 268 10 122926 284.6 287100 119400 1491 11 114460 285.0 292420 101100 1267 12 110221 285.0 292020 119400 1497 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 9 1 85939 284.4 283790 119400 1453 2 68999 283.2 265740 107800 1343 3 69349 280. 2 221770 119400 1469 4 93267 279.9 217530 115500 1408 5 147060 283.4 269540 119400 1469 6 146058 285.0 292420 115500 1442 7 99714 285. 0 292420 119400 1497 8 53267 281.3 238580 119400 1482 9 64080 279.1 206160 115500 1410 10 122926 281.3 238240 119400 1457 11 114460 282. 1 250140 115500 1422 12 110221 283.3 267240 119400 1478 -114-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 10 1 85939 284.6 286060 119400 1454 2 68999 284.2 281110 107800 1347 3 69349 282. 3 253040 119400 1483 4 93267 282.5 255150 115500 1427 5 147060 285. 0 292420 119400 1487 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283.0 263470 119400 1489 9 64080 283.4 269450 115500 1434 10 122926 285.0 292420 119400 1491 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 11 1 85939 285.0 292420 119400 1455 2 68999 285. 0 292420 107800 1352 3 69349 284.2 281240 119400 1494 4 93267 285.0 292420 115500 1445 5 147060 285.0 292420 119400 1497 6 146058 285. 0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283. 5 270680 119400 1491 9 64080 284.9 290330 115500 1442 10 122926 285.0 292420 119400 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 275 f t - PROBABILITY LEVEL 12 1 85939 285.0 292420 119400 1455 2 68999 285.0 292420 107800 1352 3 69349 285. 0 292420 119400 1497 4 93267 285.0 292420 115500 1448 5 147060 285. 0 292420 119400 1497 6 146058 285.0 292420 115500 1448 7 99714 285. 0 292420 119400 1497 8 53267 284.2 280200 119400 1494 9 64080 285.0 292420 115500 1445 10 122926 285.0 292420 119400 1497 11 114460 285. 0 292420 115500 1448 12 110221 285.0 292420 119400 1497 -115-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 1 1 85939 281.8 245220 59700 742 2 68999 280.0 219820 40400 495 3 69349 281. 3 238320 0 0 4 93267 284.9 290670 0 0 5 147060 282.9 261420 119400 1488 6 146058 284.8 289910 43300 540 7 99714 284.8 289610 37300 467 a 53267 283.9 275810 29800 372 9 64080 284.8 289310 0 0 10 122926 281.9 247210 67100 834 11 114460 284. 5 285711 0 0 12 110221 284.8 289410 37300 467 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 2 1 85939 282.1 249710 59700 743 2 68999 282.8 260711 13500 167 3 69349 280.6 228011 59700 735 4 93267 284.6 286681 0 0 5 147060 283.2 265631 119400 1488 6 146058 285. 0 292420 57800 721 7 99714 285.0 292420 44800 562 8 53267 283.4 268620 44800 559 9 64080 284.7 288620 0 0 10 122926 284.9 291320 37300 467 11 114460 284.7 287820 50500 633 12 110221 285.0 292420 44800 561 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 3 1 85939 285.0 292420 29800 374 2 68999 284.8 288920 40400 506 3 69349 279.2 207770 119400 1472 4 93267 283.3 267330 7200 88 5 147060 284. 5 285490 89500 1115 6 146058 284.8 289450 101100 1265 7 99714 283.8 274350 74600 932 8 53267 282.5 256050 44800 556 9 64080 284. 3 282651 0 0 10 122926 284.1 279051 67100 838 11 114460 283. 2 265851 72200 898 12 110221 284.9 291551 37300 465 -116-TABLE C. IV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Re s. Vo 1. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 4 1 85939 283.8 274551 59700 746 2 68999 284.5 285821 33700 421 3 69349 279.4 210161 119400 1472 4 93267 282.8 260581 21700 266 5 147060 284.6 286201 89500 1114 6 146058 284.9 290771 108300 1356 7 99714 285.0 292420 67100 841 8 53267 282.6 257470 67100 836 9 64080 284. 9 290380 0 0 10 122926 282.3 252480 119400 1485 11 114460 284. 5 285380 36100 449 12 110221 285.0 292280 67100 840 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 5 1 85939 281.4 240380 104400 1296 2 68999 284. 7 288170 6700 83 3 69349 282.5 255300 82100 1021 4 93267 283.8 275380 57800 717 5 147060 284. 3 282380 119400 1490 6 146058 284. 7 288280 115500 1444 7 99714 285.0 292420 74600 935 8 53267 282.0 248820 82100 1021 9 64080 284.8 289080 0 0 10 122926 283. 1 264750 119400 1488 11 114460 282.1 249870 93800 1160 12 110221 284.8 290070 44800 557 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 6 1 85939 280.9 232500 119400 1480 2 68999 284. 2 280960 13500 167 3 69349 279.8 216270 119400 1472 4 93267 277.7 186010 115500 1398 5 147060 284.0 277530 44800 549 6 146058 285.0 292420 115500 1444 7 99714 285.0 291830 89500 1122 8 53267 279.9 217270 119400 1476 9 64080 283. 2 265730 0 0 10 122926 284.9 291480 82100 1024 11 114460 285. 0 292420 86600 1086 12 110221 284.4 284020 104400 1307 -117-TABLE CIV. SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 7 1 85939 284. 3 282780 74600 934 2 68999 284.6 286600 67400 843 3 69349 280. 6 228230 119400 1477 4 93267 281.3 237880 86600 1061 5 147060 283. 1 264920 119400 1474 6 146058 285.0 292420 115500 1441 7 99714 285.0 292420 97000 1216 8 53267 280.8 231710 111900 1387 9 64080 284.8 290060 0 0 10 122926 284.9 290470 119400 1496 11 114460 284. 7 288580 101100 1266 12 110221 285.0 292420 97000 1215 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 8 1 85939 283.0 262740 119400 1489 2 68999 283.0 263000 80900 1003 3 69349 279.5 211830 119400 1466 4 93267 278.8 201100 115500 1401 5 147060 281.6 242850 119400 1457 6 146058 285. 0 292420 108300 1345 7 99714 284.8 289210 111900 1402 8 53267 280. 5 227340 119400 1478 9 64080 283.8 274890 21700 268 10 122926 284.6 287100 119400 1491 11 114460 285.0 292420 101100 1267 12 110221 285. 0 292020 119400 1497 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 9 1 85939 284.4 283790 119400 1495 2 68999 283.2 265740 107800 1343 3 69349 280. 2 221770 119400 1469 4 93267 279.9 217530 115500 1408 5 147060 283.4 269540 119400 1469 6 146058 285.0 292420 115500 1442 7 99714 285.0 292420 119400 1497 8 53267 281.3 238580 119400 1482 9 64080 279.1 206160 115500 1410 10 122926 281.3 238240 119400 1457 11 114460 282, 1 250140 115500 1422 12 110221 283.3 267240 119400 1478 -118-TABLE C I V . SUMMARY OF CASE 2 RESULTS, DISCHARGE DECSION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 10 1 85939 284.6 286060 119400 1495 2 68999 284.2 281110 107800 1347 3 69349 282. 3 253040 119400 1483 4 93267 282.5 255150 115500 1427 5 147060 285.0 292420 119400 1487 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283.0 263470 119400 1489 9 64080 283.4 269450 115500 1434 10 122926 285.0 292420 119400 1491 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 11 1 85939 285.0 292420 119400 1497 2 68999 285.0 292420 107800 1352 3 69349 284.2 281240 119400 1494 4 93267 285. 0 292420 115500 1445 5 147060 285.0 292420 119400 1497 6 146058 285.0 292420 115500 1448 7 99714 285.0 292420 119400 1497 8 53267 283. 5 270680 119400 1491 9 64080 284.9 290330 115500 1442 10 122926 285.0 292420 119400 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 STARTING RESERVOIR LEVEL = 285 f t - PROBABILITY LEVEL 12 1 85939 285.0 292420 119400 1497 2 68999 285.0 292420 107800 1352 3 69349 285. 0 292420 119400 1497 4 93267 285.0 292420 115500 1448 5 147060 285.0 292420 119400 1497 6 146058 285.0 292420 115500 1448 7 99714 285. 0 292420 119400 1497 8 53267 284.2 280200 119400 1494 9 64080 285.0 292420 115500 1445 10 122926 285.0 292420 119400 1497 11 114460 285.0 292420 115500 1448 12 110221 285.0 292420 119400 1497 -119-1 1 1 1 1 1 1 1 1 r - PROBABILITY LEVEL 2 260 [ ^ L _ 1 2 3 4 5 6 7 8 9 10 11 12 TIME PERIOD CMONTH) FIGURE C.lb. CASE 2 RESULTS - STATE CHANGE DECISION F I G U R E C . I c . C A S E 2 R E S U L T S - S T A T E C H A N G E D E C I S I O N -122-FIGURE C . l e . CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . i f . CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . l g . CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . l h . CASE 2 RESULTS - STATE CHANGE DECISION TIME PERIOD CMONTH) FIGURE C . l i . CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . t j . CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . Ik. CASE 2 RESULTS - STATE CHANGE DECISION FIGURE C . ! I . CASE 2 RESULTS - STATE CHANGE DECISION I H . I U-til > - J Q: H o > UJ CO UJ 290 ?85 280 L 275 270 L 265 260 1 2 3 4 5 6 7 8 9 TIME PERIOD CMONTH) 10 11 12 FIGURE C . 2 a . CASE 2 RESULTS - DISCHARGE DECISION I CM n h-U. v/ _l > U J —I a. H o > U J co U J a: 290 285 280 L 275 270 265 L 260 1 2 3 4 5 6 7 8 9 TIME PERIOD (MONTH) 1 0 1 1 1 2 FIGURE C . 2 b . CASE 2 RESULTS - DISCHARGE DECISION 290 h PROBABILITY LEVEL 3 260 I • , , , , 1 2 3 4 5 6 7 8 9 10 11 12 TIME PERIOD CMONTH) FIGURE C . 2 c . CASE 2 RESULTS - DISCHARGE DECISION -134-FIGURE C . 2 e . CASE 2 RESULTS - DISCHARGE DECISION FIGURE C . 2 f . CASE 2 RESULTS - DISCHARGE DECISION I n Y-hi > hi OC H O > co UJ 290 285 280 L 275 270 L 265 L 260 1 2 3 4 5 6 7 8 9 TIME PERIOD CMONTrO 10 11 12 FIGURE C . 2 g . CASE 2 RESULTS - DISCHARGE DECISION -138--139-FIGURE C . 2 J . CASE 2 RESULTS - DISCHARGE DECISION I H H I \~ U_ _ J L U > H o > a: UJ <o UJ QC 290 285 280 L J 275 £ 270 L 265 L 260 3 4 5 6 7 8 9 TIME PERIOD CMONTH) 10 11 12 FIGURE C . 2 k . CASE 2 RESULTS - DISCHARGE DECISION 268 1 2 3 4 5 6 1 7 8 9 10 11 12 TIME PERIOD CMONTH) FIGURE C . 2 1 . CASE 2 RESULTS - DISCHARGE DECISION - 1 4 3 -APPENDIX D. OPTIMIZATION RESULTS FOR CASE 3 -144-TABLE D . I . OPTIMAL MONTHLY RESERVOIR LEVELS (ft) CASE 3: PARTIALLY CORRELATED INFLOWS - STATE CHANGE DECISION Time step (month) Return Elev . 1 2 3 4 5 6 7 8 9 10 11 12 MW 265. .0 265. . 5 266, .0 266, . 5 268. .5 271, .0 269, .5 267, . 5 266, .0 265, . 5 266 .5 267, , 5 267, . 5 30149 265. .5 266, .0 266. . 5 267, .0 269. .0 271, . 5 270. .0 268. .0 266. .5 266. ,0 267, .0 268. .0 268. .0 30234 266, .0 266, . 5 267. .0 267, . 5 269. . 5 272. .0 270. . 5 268, . 5 267, .0 266, .5 267. 5 268. .5 268. . 5 30320 266. . 5 267, .0 267. .5 268.0 270. .0 272. .5 271. ,0 269, .0 267, .5 267. ,0 268. .0 269. .0 269. .0 30406 267. .0 267. 5 268. .0 268, . 5 270. . 5 273. .0 271. .5 269, . 5 268. .0 267. .5 268. . 5 269. . 5 269. . 5 30493 267. .5 268.0 268. .5 268. . 5 271. ,0 273. , 5 272.0 270. .0 268.5 268. ,0 269, .0 270. .0 270. .0 30579 268. .0 268. 5 269, .0 268, .5 271. . 5 274. .0 272. , 5 270, . 5 269. .0 268. ,5 269, .5 270. .5 270.5 30666 268. .5 269. .0 269. . 5 268. .5 272. .0 274. , 5 273. .0 271, ,0 269. ,5 269. ,0 270, .0 271. ,0 271. .0 30753 269. .0 269, .5 270, .0 268, .5 272. , 5 275. ,0 273. .5 271, .5 270. .0 269. .5 270, .5 271, .5 271. .5 30839 269. .5 270. .0 270. .5 269. .0 272. ,5 275. .5 274.0 272, .0 270. , 5 270. ,0 271, .0 272. .0 272. .0 30926 270. . 0 270, . 5 271. .0 269. .5 272. . 5 276. .0 274. .5 272, .5 271. .0 270. .5 271 . 5 272. .5 272. . 5 31013 270. , 5 271, .0 271. . 5 270. .0 272. , 5 276. , 5 275. .0 273. .0 271. , 5 271. ,0 272, .0 273. .0 273. .0 31099 271. .0 271, . 5 272. . 0 270. 5 272. . 5 277. .0 275. .5 273. .5 272. .0 271. .5 272 . 5 273. .5 273. .5 31185 271. . 5 272, .0 272. . 5 271. ,0 273. ,0 277. , 5 276.0 274. .0 272. , 5 272. ,0 273. .0 274. .0 274. .0 31270 272. ,0 272. , 5 273. .0 271. .0 273. , 5 278. .0 276. .5 274.5 273. .0 272. ,5 273. .5 274. .5 274. .5 31355 272. , 5 273. .0 273. , 5 271. ,5 273. , 5 278.5 277. .0 275. .0 273. .5 273. ,0 274. .0 275, .0 275, .0 31439 273. .0 273, .5 274. .0 272. .0 274. .0 279. .0 277. .5 275. .5 274. .0 273. ,5 274. .5 275. .5 275. .5 31524 273. ,5 274, .0 274. ,5 272. . 5 274. ,5 279. .0 278.0 276. .0 274. .5 274. ,0 275, .0 276.0 276. .0 31609 274. .0 274, . 5 275. .0 273. .0 275. ,0 279. .0 278. .5 276. . 5 274. .5 274. .5 275 .5 276. .5 276. .5 31693 274. .5 275. .0 275. ,0 273. , 5' 275. ,0 279. ,0 279. ,0 277. .0 274. ,5 275. ,0 276.0 277. .0 277. .0 31777 275. ,0 275. . 5 275. , 5 273. .5 275. , 5 279. , 5 279. .5 277. .5 274. .5 275. .5 276. .5 277. .5 277, .5 31861 275. 5 276, .0 276. .0 274. .0 276. ,0 280. 0 280. 0 278. .0 275. ,0 276. .0 277, .0 278. .0 278. .0 31945 276. .0 276. , 5 276. ,0 274. .5 276. ,0 280. . 5 280. , 5 278. .5 275. ,0 276. ,5 277. .5 278. .5 278. . 5 32029 276. 5 277. ,0 276. ,5 274. ,5 276. .5 280. 5 281. .0 279. .0 275. ,5 277. ,0 278, .0 279. ,0 279, .0 32111 277. 0 277. .5 276. . 5 275. ,0 277. ,0 281. .0. 281. ,5 279. . 5 276. ,0 277. ,5 278, .5 279. .5 279. .5 32191 277. 5 278. ,0 277. ,0 275. ,5 277. .5 281. ,5 282. 0 279. .5 276. .0 278. ,0 279, .0 280. .0 280. .0 32272 278. 0 278. .5 277. . 5 276. ,0 278. ,0 281. .5 282. ,5 280. .0 276. .5 278. ,5 279. 5 280. .5 280. . 5 32352 278. 5 279. ,0 278. .0 276. ,5 278. .5 282. 0 283. 0 280. , 5 277. .0 279. ,0 280. .0 281. .0 281. .0 32430 279. 0 279. . 5 278. , 5 277. ,0 279. ,0 282. ,0 283. , 5 281. .0 277. .5 279. , 5 280, . 5 281. .5 281, . 5 32508 279. 5 280. ,0 278. ,5 277. .5 279. 5 282. 5 283.5 281. , 5 278. ,0 280. ,0 281. .0 282. ,0 282.0 32586 280. 0 280. . 5 279. ,0 277. . 5 279. , 5 283. ,0 284. ,0 282. .0 278. .5 280. , 5 281. . 5 282. .5 282. .5 32664 280. 5 281. ,0 279. 5 277. .5 280. ,0 283. 0 284.5 282. . 5 279. ,0 280. .5 282. .0 283. ,0 283.0 32742 281. 0 281. , 5 280. ,0 278. ,0 280. 5 283. ,0 285. ,0 283. ,0 279. .5 280. ,5 282. .5 283. .0 283, .0 32820 281. 5 282. 0 280. 5 278. .5 281. 0 283.5 285. 0 283. , 5 280. ,0 280. . 5 283. .0 283. .5 283, .0 32898 282. 0 282. ,0 281. ,0 279. ,0 281. ,0 283. ,5 285. .0 284.0 280. .0 280.5 283 .5 284. .0 283, . 5 32973 282. 5 282.0 281. 0 279. 5 281. 5 284. 0 285.0 284. .5 280. ,5 280.5 284. .0 284. .0 284.0 33049 283. 0 282. , 5 281. ,5 280. .0 282. ,0 284.0 285. ,0 285. .0 281. .0 281. .0 284, .5 284. .0 284. .5 33123 283. 5 283. 0 282. 0 280. 5 282. 5 284. 5 285.0 285. .0 281. ,5 281. .5 284. .5 284. ,5 285. ,0 33195 284. 0 283. , 5 282. , 5 281. ,0 282. , 5 284. ,5 285. ,0 285. .0 282. ,0 282. ,0 285, .0 285. .0 285. .0 33265 284. 5 284. 0 283. 0 281. 5 283. 0 285. 0 285. 0 285. ,0 282. .5 282. ,5 285. .0 285. .0 285. .0 33336 285. 0 284. , 5 283. 0 281. , 5 283. .0 285. ,0 285. ,0 285. .0 283. .0 283. .0 285, .0 285. .0 285. .0 33405 -145-TABLE D.II. OPTIMAL MONTHLY POWERHOUSE FLOW (1000 cfs) CASE 3: PARTIALLY CORRELATED INFLOWS - DISCHARGE DECISION Time step (month) Return Elev. 1 2 3 4 5 6 7 8 9 10 11 12 MW 265. .0 0. .0 6. . 7 22. . 4 0. .0 0. ,0 0. ,0 0. .0 0. ,0 0. .0 0. .0 0. ,0 0. 0 31919 265. . 5 0. .0 6. . 7 29. .8 0. ,0 0. 0 0. 0 0. ,0 0. 0 0. 0 0. ,0 43. 3 0. 0 32005 266. .0 0. .0 6. , 7 37. . 3 0. ,0 0. ,0 0. 0 0. .0 0. 0 0. ,0 0. ,0 43. .3 0. 0 32091 266. . 5 0. ,0 6. , 7 44. .8 0. 0 0. 0 0. 0 0. ,0 0. 0 0. 0 7. , 5 43. 3 0. 0 32177 267. .0 0. .0 40. .4 52. . 2 0. .0 0. .0 0. ,0 0. .0 0. 0 0. ,0 14. .9 43. ,3 0. 0 32263 267. , 5 0. .0 40.4 59. .7 0. ,0 7. , 5 0. 0 0. ,0 0. 0 0. 0 22. ,4 43. .3 0. 0 32348 268. .0 0. .0 40. .4 67. , 1 0. ,0 14. ,9 0. 0 0. ,0 0. 0 0. ,0 29. .8 43. ,3 0. 0 32434 268. .5 0. .0 40. .4 67, . 1 0. ,0 22. ,4 O. 0 0. ,0 0. 0 0. 0 37. , 3 72. 2 0. 0 32520 269. .0 7. . 5 40. ,4 67. . 1 0. .0 29. .8 0. ,0 0, .0 0. ,0 0. ,0 44. .8 72. ,2 0. .0 32607 269. .5 14.9 40.4 67. ,1 0. ,0 37. ,3 0. 0 0. ,0 0. 0 21. ,7 52. . 2 72. 2 0. 0 32693 270. .0 22. .4 40. ,4 97. .0 7. , 2 44. ,8 0. 0 0, .0 0. 0 21. ,7 59. .7 72. .2 7. 5 32779 270. .5 29. .8 40. .4 104. .4 14. , 4 52. , 2 7. 2 0. .0 0. 0 21. ,7 67.1 72. , 2 14. 9 32866 271. .0 37. .3 40. .4 104. .4 21. . 7 52. ,2 14. .4 0. .0 0. ,0 21. ,7 67, . 1 72. .2 22. ,4 32953 271. , 5 44. .8 40. ,4 104. ,4 43. , 3 52. , 2 36. 1 0. ,0 0. 0 21. ,7 67. , 1 72. ,2 29. 8 33041 272. .0 52. .2 40.4 119. ,4 43. .3 74. ,6 36. 1 14. .9 0. ,0 86. ,6 67, , 1 72. .2 37. 3 33128 272. 5 59. .7 40. .4 119.4 43. , 3 82. , 1 36. 1 22. .4 7. 5 86. ,6 67. . 1 72. .2 37.3 33216 273. .0 67. . 1 40. .4 119. .4 50. .5 82. .1 43. 3 29. .8 14.9 86. 6 67, .1 79. ,4 37. ,3 33304 273. , 5 74. .6 40. ,4 119.4 57. ,8 82. 1 50. 5 37. , 3 22. 4 86. ,6 67. . 1 86. .6 37. 3 33393 274. .0 74. .6 40. ,4 119. .4 65. ,0 82. , 1 101. 1 44. .8 29. 8 86. 6 67. , 1 93. .8 37. 3 33481 274. . 5 74. .6 40. .4 119. .4 101. , 1 111. .9 101. 1 52. .2 37. 3 86. .6 67. . 1 101. .1 37. 3 33570 275. .0 82. . 1 40. .4 119.4 101. .1 119. .4 101. ,1 59. ,7 44. ,8 86. .6 119. .4 108. ,3 37. ,3 33658 275. ,5 82. . 1 40. ,4 119.4 101. , 1 119. ,4 101. 1 67. , 1 44. 8 86. 6 119. ,4 115. ,5 89. 5 33745 276. .0 82. . 1 40. ,4 119. .4 101. . 1 119. .4 101. ,1 74. .6 44. 8 86. .6 119, .4 115. .5 97.0 33832 276. .5 111. .9 40. .4 119. .4 101. , 1 119. ,4 101. 1 82. , 1 44. ,8 93. 8 119. .4 115. ,5 104. ,4 33921 277. .0 119. .4 40. ,4 119.4 108. ,3 119. .4 101. ,1 89. ,5 44. ,8 101. .1 119. .4 115. .5 111. ,9 34011 277. ,5 119. .4 40. ,4 119.4 115. , 5 119. ,4 108. 3 97. ,0 44. 8 108. ,3 119. .4 115. ,5 119. 4 34101 278. .0 119. .4 67, .4 119, .4 115. . 5 119. .4 115. .5 104. .4 44. .8 115. .5 119, .4 115. .5 119.4 34189 278. , 5 119. ,4 107. .8 119. .4 115. , 5 119. ,4 115. 5 111. .9 97. ,0 115.5 119. ,4 115.5 119. .4 34277 279. .0 119. .4 107. .8 119. .4 115. , 5 119. ,4 115. 5 119. .4 104. ,4 115. .5 119. .4 115. .5 119. .4 34365 279. .5 119. .4 107. ,8 119. .4 115. ,5 119. ,4 115. 5 119. .4 111. ,9 115. ,5 119. .4 115. .5 119.4 34454 280. .0 119. .4 107. .8 119. .4 115. . 5 119. .4 115. ,5 119. .4 119. .4 115. , 5 119, .4 115. .5 119. .4 34542 280. , 5 119.4 107. .8 119. .4 115. , 5 119. ,4 115. 5 119. ,4 119. 4 115. ,5 119. .4 115. ,5 119. .4 34630 281. .0 119. .4 107. .8 119. ,4 115. , 5 119. .4 115. ,5 119. .4 119. .4 115. ,5 119, .4 115. .5 119. .4 34718 281. ,5 119. ,4 107. ,8 119. ,4 115. , 5 119. ,4 115. 5 119. ,4 119. 4 115. , 5 119. .4 115.5 119. ,4 34807 282. .0 119. .4 107. .8 119, .4 115. . 5 119. .4 115. .5 119. .4 119. .4 115. .5 119, .4 115, .5 119. .4 34890 282. . 5 119. ,4 107. .8 119. .4 115. .5 119. , 4 115. ,5 119. .4 119. ,4 115. .5 119, .4 115. .5 119. .4 34974 283. ,0 119. .4 107. .8 119. , 4 115. . 5 119. ,4 115. ,5 119. .4 119. ,4 115. . 5 119, . 4 115, .5 119. .4 35057 283. . 5 119. .4 107. .8 119. ,4 115. , 5 119. ,4 115. 5 119. .4 119. 4 115. ,5 119. .4 115. .5 119. .4 35141 284. .0 119. .4 107. .8 119. .4 115. , 5 119. .4 115. , 5 119, .4 119. .4 115, .5 119, .4 115 .5 119. .4 35218 284. , 5 119. ,4 107. .8 119. .4 115. ,5 119. ,4 115. 5 119. .4 119. .4 115. .5 119. .4 115. .5 119. ,4 35294 285. ,0 119. .4 107. .8 119. .4 115. , 5 119. .4 115. , 5 119. .4 119. ,4 115. .5 119, .4 115, .5 119. ,4 35370 -146-TABLE D.III. SUMMARY RESULTS - CASE 3, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t 1 85939 265. 5 7311 65955 736 2 68999 266.5 21932 53655 601 3 69349 268.0 43863 41185 465 4 93267 271. 5 95037 40800 468 5 147060 277.5 182763 57207 675 6 146058 282.0 248557 73491 895 7 99714 284.0 277799 67166 833 8 53267 282.0 248557 80741 1001 9 64080 280. 5 226626 78211 960 10 122926 282.0 248557 83637 1027 11 114460 284.0 277799 67420 836 12 110221 285.0 292420 84545 1057 STARTING RESERVOIR LEVEL = 270 f t 1 85939 270.5 80416 65955 759 2 68999 271. 5 95037 53655 620 3 69349 271.0 87726 68793 796 4 93267 272. 5 109658 70042 813 5 147060 278.5 197384 57207 679 6 146058 283.0 263178 73491 900 7 99714 285.0 292420 67166 838 8 53267 283.0 263178 80741 1007 9 64080 281.0 233936 84060 1036 10 122926 282. 5 255868 83637 1030 11 114460 284.0 277799 73269 910 12 110221 285.0 292420 84545 1057 STARTING RESERVOIR LEVEL = 275 f t 1 85939 275.5 153521 65955 782 2 68999 276.0 160831 60506 720 3 69349 274. 5 138900 82143 974 4 93267 275.0 146210 83711 990 5 147060 279.5 212005 77067 925 6 146058 283.5 270489 79340 975 7 99714 285.0 292420 73368 916 8 53267 283.0 263178 80741 1007 9 64080 281.0 233936 84060 1036 10 122926 282.5 255868 83637 1030 11 114460 284.0 277799 73269 910 12 110221 285.0 292420 84545 1057 -147-TABLE D. II I . SUMMARY RESULTS - CASE 3, STATE CHANGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 280 f t 1 85939 280. 5 226626 65955 805 2 68999 279.5 212005 79277 966 3 69349 277.5 182763 88601 1071 4 93267 277.5 182763 90079 1082 5 147060 281.5 241247 82958 1008 6 146058 285.0 292420 84802 1053 7 99714 285.0 292420 90473 1135 8 53267 283.0 263178 80741 1007 9 64080 281.0 233936 84060 1036 10 122926 282.5 255868 83637 1030 11 114460 284.0 277799 73269 910 12 110221 285.0 292420 84545 1057 STARTING RESERVOIR LEVEL = 285 f t 1 85939 284.5 285110 77651 972 2 68999 283.0 263178 84737 1055 3 69349 280.0 219315 100297 1233 4 93267 279. 5 212005 95620 1164 5 147060 282.5 255868 93835 1150 6 146058 285.0 292420 94203 1173 7 99714 285.0 292420 90473 1135 8 53267 283.0 263178 80741 1007 9 64080 281.0 233936 84060 1036 10 122926 282.5 255868 83637 1030 11 114460 284.0 277799 73269 910 12 110221 285. 0 292420 84545 1057 -148-TABLE D. IV. SUMMARY OF CASE 3 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL = 265 f t 1 85939 270. 6 81617 0 0 2 68999 272.5 109558 40400 469 3 69349 268. 9 56863 119400 1378 4 93267 275.2 148836 0 0 5 147060 277.0 174760 119400 1423 6 146058 279.8 217009 101100 1221 7 99714 278.5 197491 119400 1449 8 53267 275.5 153255 97000 1162 9 64080 273.9 130567 86600 1024 10 122926 277.5 183195 67100 798 11 114460 276. 7 171045 115500 1385 12 110221 277.0 175308 104400 1250 STARTING RESERVOIR LEVEL = 270 f t 1 85939 274.1 132323 22400 261 2 68999 276.0 160263 40400 478 3 69349 , 272.4 107568 119400 1407 4 93267 275. 7 156241 43300 510 5 147060 277.5 182165 119400 1427 6 146058 279.9 217204 108300 1310 7 99714 278.5 197676 119400 1449 8 53267 275.5 153440 97000 1162 9 64080 273.9 130752 86600 1024 10 122926 277. 5 183361 67100 798 11 114460 276.7 171203 115500 1385 12 110221 277.0 175466 104400 1250 STARTING RESERVOIR LEVEL = 275 f t 1 85939 275.0 145727 82100 972 2 68999 276.9 173669 40400 481 3 69349 273.3 120974 119400 1414 4 93267 275.6 155147 57800 682 5 147060 277.4 181071 119400 1426 6 146058 279.8 216164 108300 1309 7 99714 278. 5 196689 119400 1448 8 53267 275.4 152453 97000 1162 9 64080 273.9 129765 86600 1023 10 122926 277.5 182473 67100 798 11 114460 276.7 170360 115500 1384 12 110221 276.9 174623 104400 1249 -149-TABLE D. IV. SUMMARY OF CASE 3 RESULTS, DISCHARGE DECISION Month Average Ending Ending Powerhouse Energy Inflow Res. Ele. Res. Vol. Flow MW STARTING RESERVOIR LEVEL' = 280 f t 1 85939 277.3 179845 119400 1444 2 68999 279.2 207786 40400 488 3 69349 275.6 154469 119400 1434 4 93267 .274.9 145342 101100 1199 5 147060 276.7 171265 119400 1421 6 146058 279.6 213689 101100 1219 7 99714 278. 3 194337 119400 1447 8 53267 275.3 150101 97000 1161 9 64080 273. 7 127413 86600 1022 10 122926 277.3 180356 67100 797 11 114460 276. 5 168349 115500 1383 12 110221 276.8 172612 104400 1248 STARTING RESERVOIR LEVEL = 285 f t 1 85939 281.7 244833 119400 1484 2 68999 279.0 205374 107800 1317 3 69349 275.4 152177 119400 1432 4 93267 274.8 143050 101100 1198 5 147060 276.6 168973 119400 1419 6 146058 279.5 211490 101100 1218 7 99714 278.1 192248 119400 1446 8 53267 278. 7 200212 44800 541 9 64080 275.2 148625 115500 1383 10 122926 275.4 151373 119400 1416 11 114460 274.6 140815 115500 1367 12 110221 279.4 211034 37300 447 298 260 L 1 _ _ — _ i _ — i i i i i i i i i 1 2 3 4 5 6 7 8 9 10 11 12 TIME PERIOD CMONTH) FIGURE D . I . CASE 3 RESULTS - STATE CHANGE DECISION 1 1 1 1 I 1 1 1 ,1 I I I 2 3 4 5 6 7 8 9 10 11 12 TIME PERIOD CMONTH) FIGURE D .2 . CASE 3 RESULTS - DISCHARGE DECISION APPENDIX E. COMPUTER PROGRAM LISTING -153-•********************************************************************•##******* '* OPTIMAL OPERATION * '* OF A HYDROELECTRIC RESERVOIR * '* (USING STOCHASTIC DYNAMIC PROGRAMMING} * • .a**************************************************************************** •TUNG VAN DO - SEPTEMBER 06, 1987 'stage = time, state = reservoir l e v e l 'decision variable = reservoir level change or powerhouse release 'uncorrelated, p e r f e c t l y correlated, or p a r t i a l l y correlated inflows ' MAIN PROGRAM OPTION BASE 1 : DEFINT I-N DIM PROB(12), NGO(36,41), QI(12,12), QPH(17) DIM BEFIT(36,41), NDM(12) DIM MGO(12,12,41), BFIT(41), CPROB(12,12,12) DEF FNPV(X) = 0.07 7.41 'power coef. (kw/cfs) 11 FF? = CHR$(12) 'form feed CLS:PRINT:BEEP PRINT "MENU 1 : DECISION VARIABLE" PRINT "1. State Change" PRINT "2. Powerhouse Flow" INPUT "Enter your choice (1, or 2) :";MENU1 IF MENU1 < 1 OR MENU1 > 2 THEN BEEP : GOTO 11 IF MENU1 = 1 THEN TIT1? = "OPTIMAL MONTHLY RESERVOIR LEVELS ( f t ) " DV$ = "STATE CHANGE DECISION" ELSE TIT1S = "OPTIMAL MONTHLY POWERHOUSE DISCHARGES (1000 c f s ) ' DV$ = "DISCHARGE DECISION" END IF VCAP = 292420 SSS = 20000 / VCAP CS = 0.018504 'reservoir capacity in cfs-days 'slope of stage-storage curve 'storage coef. (kw/cfs-days) PRINT:PRINT:BEEP PRINT "MENU 2 : SERIAL CORRELATION" PRINT "1. Uncorrelated Inflows" PRINT "2. Perfectly Correlated Inflows" PRINT "3. P a r t i a l l y Correlated Inflows" 13 INPUT "Enter your choice (1, 2, or 3) : IF MENU2 < 1 OR MENU2 > 3 THEN 13 ;MENU2 GOSUB IP 'read data OPEN F2$ FOR OUTPUT AS #2 IF MENU2 = 1 THEN TIT2$ = "UNCORRELATED INFLOWS - " + DVS IF MENU1 - 1 THEN GOSUB HDPUNC : GOSUB OP11 IF MENU1 = 2 THEN GOSUB QDPUNC : GOSUB OP12 END IF IF MENU2 = 2 THEN TIT2S = "PERFECTLY CORRELATED INFLOWS - " + DV$ IF MENU1 = 1 THEN GOSUB HDPCOR : GOSUB 0P21 IF MENU1 = 2 THEN GOSUB QDPCOR : GOSUB OP22 END IF IF MENU2 = 3 THEN TIT2S = "PARTIALLY CORRELATED INFLOWS - " + DVS IF MENU1 - 1 THEN GOSUB HDPPAR : GOSUB 0P31 IF MENU1 » 2 THEN GOSUB QDPPAR : GOSUB OP32 END IF PRINT END GOOD BYE ..." ! BEEP -154-'SUBROUTINE IP (read input data) IP: NY = 3 'number of years DATA 31,28,31,30,31,30,31,31,30,31,30,31 FOR I = 1 TO 12 : READ NDM(I) : NEXT I DATA .05, .05, .1, .1, .1, .1, .1, .1, .1, .1, .05, .05 'probabilities FOR I = 1 TO 12 : READ PROB(I) : NEXT I OPEN "MQIN.SDP" FOR INPUT AS #1 'read inflow data LINE INPUT #1,TS FOR I = 1 TO 12 FOR J = 1 TO 12 INPUT #1, QI(I,J) NEXT J NEXT I CLOSE #1 OPEN "QMATRIX.SDP" FOR INPUT AS #1 'read conditional inflow LINE INPUT #1,T$ FOR I = 1 TO 12 LINE INPUT #1,A$ FOR J = 1 TO 12 FOR K = 1 TO 12 : INPUT #1, CPROB(I,J,K) : NEXT K NEXT J NEXT I discount factor = 6% p.a. = 0.005 per month DF = 0.005 PRINT:BEEP:INPUT "Enter Filename for Output :";F2$ NTS = 12 * NY 'time steps (= 12 months * no. of years) QCAPD = 3850 'powerhouse capacity i n c f s RETURN -155-'SUBROUTINE HDPUNC (sdp, state change decision, uncorrelated inflows) HDPUNC: FOR T% = 1 TO NTS-1 : FOR I = 1 TO 41 : BEFIT(T%,I) = 0 : NEXT I : NEXT T% ' storage values at the la s t time step FOR I = 1 TO 41 S9 - ((I - 1) / 2) * 1000 / SSS BEFIT(NTS,I) = CS * S9 * (1 - (DF * NTS)) NEXT I ENGY9 = 0 51 53 FOR T% = NTS-1 TO 1 STEP -1 MON = T% MOD 12 IF MON = 0 THEN MON =12 DAY = NDM(MON) QCAP = DAY * QCAPD PRINT "TIME STEP #";T%;" " FOR I = 1 TO 41 HI = I / 2 + 264.5 SI = ((I - 1) / 2) * 1000 / SSS FOR J = 1 TO 41 H2 = J / 2 + 264.5 S2 = ((J - 1) / 2) * 1000 / SSS HAVG = 0.5 • (HI + H2) CQ = FNPV(HAVG) IF QI(MON,12) < (S2 - SI) THEN 53 ESUM = 0 FOR K = 1 TO 12 'probability levels IF QI(MON.K) (S2 - SI) THEN 51 'too h i for t h i s inflow RR = QI(MON.K) - S2 + SI 'res. release IF RR < QCAP THEN PHQ = RR ELSE PHQ = QCAP ENG = CQ * PHQ * (1 - (DF * T%)) +_ CS * (1 - (DF * (T% + 1))) * S2 + BEFIT(T%+1,J) ESUM = ESUM + ENG * PROB(K) 'expected value NEXT K IF ESUM > ENGY9 THEN ENGY9 = ESUM : NGO(T%,I) = J NEXT J BEFIT(T%,I) = ENGY9 : ENGY9 = 0 'total benefit at t NEXT I NEXT T% 'time steps, backwards 'month number ' # of days in t h i s month 'powerhouse capacity 'state subscript at t 'res. l e v e l at t 'storage at t 'state subscript at t+1 'res. l e v e l at t+1 'storage at t+1 'mean res. l e v e l 'power coef. 'can't reach t h i s state RETURN 'SUBROUTINE OP11 (print output, state change decision, uncorrelated) OP11: FOR Y% = 1 TO NY GOSUB TITLE 11 = (Y% - 1) * 12 + 1 12 = II + 11 FOR I « II TO 12 : PRINT #2,USING "#*####":I; : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT *2,"-"; : NEXT I : PRINT #2, FOR I = 1 TO 41 PRINT #2,USING "###.#";264.5 + 1 / 2 ; FOR Tt • II TO 12 PRINT #2,USING "*•#•.•";NGO(T*,I) / 2 + 264.5; NEXT T% PRINT #2,USING "******";BEFIT(II,I) / 1000 NEXT I NEXT Y% RETURN -156-1 SUBROUTINE QDPUNC (sdp, discharge decision, uncorrelated inflows) QDPUNC: FOR T% = 1 TO NTS-1 : FOR I = 1 TO 41 : BEFIT(T%,I) = 0 : NEXT I : NEXT T% FOR I = 1 TO 41 S9 = ((I - 1) / 2) * 1000 / SSS BEFIT(NTS,I) = CS * S9 * (1 - (DF * NTS)) NEXT I ENGY9 = 0 FOR T% = NTS-1 TO 1 STEP -1 'time steps, backwards MON = T% MOD 12 'month number IF MON = 0 THEN MON =12 DAY = NDM(MON) '# of days in this month QCAP = DAY * QCAPD 'powerhouse capacity PRINT "TIME STEP #";T%;" " FOR I = 1 TO 41 'state subscript at t HI = I / 2 + 264.5 'res. l e v e l at time t SI = ((I - 1) / 2) * 1000 / SSS 'storage at t FOR J = 1 TO 17 'ph flow subscript J l = J - 1 QPH(J) = J l * 240.625 * DAY 'powerhouse flow i n cfs IF QPH(J) > (QKMON.12) + SI) THEN 93 'can't release t h i s much ESUM = 0 FOR K = 1 TO 12 'probability index IF QPH(J) > (QI(MON,K) + SI) THEN 91 S2 = QI(MON.K) - QPH(J) + SI 'storage at t+1 IF S2 > VCAP THEN S2 = VCAP H2 = S2 * SSS / 1000 + 265 'res. l e v e l at t+1 IF H2 > 285 THEN H2 = 285 HAVG = (HI + H2) / 2 'average head CQ = FNPV(HAVG) 'power coef. TEM = (H2 + 0.25) * 2 'set res. l e v e l H2 = INT(TEM) / 2 12 = (H2 - 264.5) * 2 'res. l e v e l index at t+1 ENG = CQ * QPH(J) * (1 - (DF * T%)) +_ CS * (1 - (DF * (T% + 1))) * S2 + BEFIT(T%+1,12) ESUM = ESUM + ENG * PROB(K) 'expected value 91 NEXT K IF ESUM > ENGY9 THEN ENGY9 = ESUM : NGO(T%,I) = J NEXT J 93 BEFIT(T«,I) = ENGY9 : ENGY9 = 0 'tot a l benefit at t NEXT I NEXT T% 'SUBROUTINE OP12 (print output, discharge decisiqon, uncorrelated) OP12: FOR Y% = 1 TO NY GOSUB TITLE 11 = (Y% - 1) * 12 + 1 12 = II + 11 FOR I = II TO 12 -. PRINT #2,USING "M####";I; : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT #2,"-"; : NEXT I : PRINT *2, FOR I = 1 TO 41 PRINT #2,USING "###.#";264.5 + I / 2; FOR T% = II TO 12 IF T» > 12 THEN T l % - T% - ((Y% - 1) * 12) ELSE T l % = T% PRINT #2,USING "#•*#.•"r(NGO(T%,I)-1)*240.625*NDM(Tl%)/1000: NEXT T% PRINT #2,USING "######";BEFIT(11,1) / 1000 NEXT I NEXT Y% RETURN -157-'SUBROUTINE HDPCOR (sdp, state change decision, correlated inflows) HDPCOR: FOR J = 1 TO 41 : BFIT(J) = 0 : NEXT J FOR I « 1 TO 41 S9 = ((I - 1) / 2) * 1000 / SSS BEFIT(NTS.I) = CS * S9 * (1 - (DF * NTS)) NEXT I FOR K = 1 TO 12 FOR T% = 1 TO NTS-1: FOR I = 1 TO 41: BEFIT(T%,I) = 0: NEXT I: NEXT T% ENGY9 = 0 PRINT "PROB. LEVEL ";K FOR T% = NTS-1 TO 1 STEP -1 'time steps, backwards MON = T% MOD 12 'month number IF MON = 0 THEN MON =12 DAY = NDM(MON) '# of days in t h i s month QCAP = DAY * QCAPD 'powerhouse capacity FOR I = 1 TO 41 'state subscript ' HI = I / 2 + 264.5 'res. l e v e l at time t SI = ({I - 1) / 2 ) * 1000 / SSS 'storage at t FOR J = 1 TO 41 'state subscript at t+1 H2 = J / 2 + 264.5 'res. l e v e l at t+1 S2 = (( J - 1) / 2) * 1000 / SSS 'storage at t+1 IF QI(MON.K) < (S2 - SI) THEN 95 'can't reach thi s l e v e l RR = QI(MON.K) - S2 + SI 'res. release IF RR < QCAP THEN PHQ = RR ELSE PHQ = QCAP HAVG = (HI + H2) / 2 'average head CQ = FNPV(HAVG) 'power coef. ENG = PROB(K) * (CQ * PHQ * (1 - (DF * T%)) +_ CS * (1 - (DF * (T% + 1))) * S2) + BEFIT(T%+1,J) IF ENG > ENGY9 THEN ENGY9 = ENG IF T% <= 12 THEN MGO(K,T%,I) = J END IF NEXT J 95 BEFIT(T%,I) = ENGY9 : ENGY9 = 0 'total benefit at t NEXT I NEXT T% FOR IK = 1 TO 41 : BFIT(IK) = BFIT(IK) + BEFIT(1,IK) : NEXT IK NEXT K RETURN 'SUBROUTINE OP21 (print output, state change decision, correlated) OP21: GOSUB TITLE FOR I - 1 TO 12 : PRINT #2,USING "######"?Ir : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT #2,"-"; : NEXT I : PRINT #2, FOR I = 1 TO 41 PRINT #2,USING "###.#";264.5 + 1 / 2 ; FOR K » 1 TO 12 IF K > 1 THEN PRINT #2," "; FOR T% » 1 TO 12 PRINT #2,USING "####.#";MGO(K,T%,I) / 2 + 264.5; NEXT T% IF K =» 1 THEN PRINT #2, USING "###### ";BFIT(I) / 1000_ ELSE PRINT 12, NEXT K NEXT I RETURN - 1 5 8 -'SUBROUTINE QDPCOR (sdp, discharge decision, correlated inflows) QDPCOR: FOR I = 1 TO 41 : BFIT(I) = 0 : NEXT I FOR I = 1 TO 41 S9 - ((I - 1) / 2) * 1000 / SSS BEFIT(NTS,I) = CS * S9 * (1 - (DF * NTS)) NEXT I FOR K = 1 TO 12 FOR T% = 1 TO NTS-1: FOR I = 1 TO 41: BEFIT(T%,I) = 0: NEXT I: NEXT T% ENGY9 = 0 PRINT "PROB. LEVEL ";K FOR T* = NTS-1 TO 1 STEP -1 'time steps, backwards MON = T% MOD 12 'month number IF MON = 0 THEN MON = 12 DAY = NDM(MON) '# of days in t h i s month QCAP = DAY * QCAPD 'powerhouse capacity FOR I = 1 TO 41 'state subscript HI = I / 2 + 264.5 'res. lev e l at time t SI = ((I - 1) / 2) * 1000 / SSS 'res. storage at t FOR J = 1 TO 17 'ph flow subscript JI = J - 1 QPH(J) = JI * 240.625 * DAY 'powerhouse flow i n c f s IF QPH(J) > (QI(MON,K) + SI) THEN 97 ESUM = 0 S2 = QI(MON,K) - QPH(J) + SI 'storage at t+1 IF S2 < 0 THEN S2 = 0 IF S2 > VCAP THEN S2 = VCAP H2 = S2 * SSS / 1000 + 265 'res. l e v e l at t+1 IF H2 285 THEN H2 = 285 HAVG = (HI + H2) / 2 'average head CQ = FNPV(HAVG) 'power coef. TEM = (H2 + 0.25) * 2 '3et res. lev e l H2 = INT(TEM) / 2 12 = (H2 - 264.5) * 2 ENG = PROB(K) * (CQ * QPH(J) * (1 - (DF * T%)) +_ CS * (1 - (DF * (T% + 1))) * S2) + BEFIT(T%+1,12) IF ENG > ENGY9 THEN ENGY9 = ENG IF T% <= 12 THEN MGO(K,T%,I) = J END IF NEXT J 97 BEFIT(T%,I) = ENGY9 : ENGY9 = 0 'tot a l benefit at t NEXT I NEXT T% FOR IK = 1 TO 41 : BFIT(IK) = BFIT(IK) + BEFIT(l.IK) : NEXT IK NEXT K RETURN 'SUBROUTINE OP22 (print output, discharge decision, correlated) OP22: GOSUB TITLE FOR I = 1 TO 12 : PRINT #2,USING "*###**";I; : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT #2,"-"; : NEXT I : PRINT #2, FOR I = 1 TO 41 PRINT #2,USING "#•#.#";264.5 + 1 / 2 ; FOR K = 1 TO 12 IF K > 1 THEN PRINT #2," FOR T% = 1 TO 12 PRINT #2,USING "####.•";(MGO(K,T%,I)-1)*240.625*NDM(T%)/1000; NEXT T% IF K = 1 THEN PRINT #2,USING "••###•";BFIT(I) / 1000_ ELSE PRINT #2, NEXT K NEXT I RETURN -159-'SUBROUTINE HDPPAR (sdp, state change decision, p a r t i a l l y correlated inflows) HDPPAR: FOR T% = 1 TO NTS-1 : FOR I = 1 TO 41 : BEFIT(T%,I) = 0 : NEXT I : NEXT T% FOR I = 1 TO 41 S 9 = ( ( I - l ) / 2 ) * 1000 / SSS BEFIT(NTS,I) = CS * S9 * (1 - (DF * NTS)) NEXT I ENGY9 = 0 151 'time steps, backwards 'month number '# of days in th i s month 'powerhouse capacity 'res. l e v e l at t 'res. l e v e l at t+1 FOR T% = NTS-1 TO 1 STEP -1 MON = T% MOD 12 IF MON = 0 THEN MON =12 DAY = NDM(MON) QCAP = DAY * QCAPD PRINT "TIME STEP #";T%r" " FOR I = 1 TO 41 HI = I / 2 + 264.5 SI = ((I - 1) / 2) * 1000 / SSS FOR J = 1 TO 41 H2 = J / 2 + 264.5 S2 = ((J - 1) / 2) * 1000 / SSS HAVG = 0.5 * (HI + H2) CQ = FNPV(HAVG) ESUM2 = 0 FOR K = 1 TO 12 ESUM1 = 0 FOR L = 1 TO 12 IF QI(MON,L) < (S2 - SI) THEN 151 RR = QKMON, L) - S2 + SI 'res. release IF RR < QCAP THEN PHQ = RR ELSE PHQ = QCAP ENG = CPROB(MON,K,L)* PROB(K)* (CQ* PHQ* (1- (DF* T%)) + CS * (1 - (DF * (T% + 1))) * S2 + BEFIT(T%+1,J)] ESUM1 = ESUM1 + ENG NEXT L ESUM2 = ESUM2 + ESUM1 NEXT K IF ESUM2 > ENGY9 THEN ENGY9 = ESUM2 : NGO(T%,I) = J NEXT J BEFIT(T»,I) = ENGY9 t ENGY9 = 0 NEXT I NEXT T% 'mean res. l e v e l 'power coef. 'flow in previous month 'conditional prob. RETURN 'SUBROUTINE OP31 (print output, state change decision, p a r t i a l l y correlated) OP31: FOR Y% = 1 TO NY GOSUB TITLE 11 » (Y% - 1) * 12 + 1 12 = II + 11 FOR I = II TO 12 : PRINT #2,USING "#M#M";I; : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT #2,"-"; : NEXT I : PRINT #2, FOR I • 1 TO 41 PRINT #2,USING "##•.•";264.S + 1 / 2 ; FOR T% =• II TO 12 PRINT #2,USING "MM. t" ;NGO(T%, I) / 2 + 264.5; NEXT TS PRINT #2,USING "*•####";BEFIT(11,1) / 1000 NEXT I NEXT Y» RETURN -16C-'SUBROUTINE QDPPAR (sdp, d i s c h a r g e d e c i s i o n , p a r t i a l l y c o r r e l a t e d i n f l o w s ) QDPPAR: FOR T% = 1 TO NTS-1 : FOR I = 1 TO 41 : B E F I T ( T % , I ) = 0 : NEXT I : NEXT T% FOR I = 1 TO 41 S 9 = ( ( I - l ) / 2 ) * 1000 / SSS BEFITfNTS,I) = CS * S9 * (1 - (DF * NTS)) NEXT I ENGY9 = 0 'time s t e p s , backwards 'month number 191 '# o f days i n t h i s month 'powerhouse c a p a c i t y ' r e s . l e v e l i n d e x ' r e s . l e v e l a t time t ' r e s . s t o r a g e 'powerhouse f l o w index 'powerhouse f l o w i n c f s 'flo w i n p r e v i o u s month ' c o n d i t i o n a l prob. THEN 191 FOR T% = NTS-1 TO 1 STEP -1 MON = T% MOD 12 IF MON = 0 THEN MON = 1 2 DAY = NDM(MON) QCAP = DAY * QCAPD PRINT "TIME STEP #";T%;" " FOR I = 1 TO 41 HI = I / 2 + 264.5 SI = ( ( I - 1) / 2) * 1000 / SSS FOR J = 1 TO 17 J l = J - 1 QPH(J) = J l * 240.625 * DAY ESUM2 = 0 FOR K = 1 TO 12 ESUM1 = 0 FOR L = 1 TO 12 IF QPH(J) > (QI(MON,L) + SI) S2 = QI(MON,L) - QPH(J) + SI IF S2 > VCAP THEN S2 = VCAP H2 = S2 * SSS / 1000 + 265 IF H2 > 285 THEN H2 = 285 HAVG = (HI + H2) / 2 CQ = FNPV(HAVG) 'power c o e f . TEM = (H2 + 0.25) * 2 H2 = INT(TEM) / 2 12 = (H2 - 264.5) * 2 ENG= CPROB(MON,K,L)* PROB(K)* (CQ* QPH(J)* ( 1 - (DF*T%)) + CS * (1 - (DF * (T% + 1 ) ) ) * S2 + BEFIT(T%+1,I2)T ESUM1 = ESUM1 + ENG NEXT L ESUM2 = ESUM2 + ESUM1 NEXT K IF ESUM2 > ENGY9 THEN ENGY9 = ESUM2 : NGO(T%,I) = J NEXT J B E F I T ( T % , I ) = ENGY9 : ENGY9 = 0 NEXT I NEXT T% 'SUBROUTINE OP32 ( p r i n t o u t p u t , d i s c h a r g e d e c i s i o n , p a r t i a l l y c o r r e l a t e d ) OP32: FOR Y% = 1 TO NY GOSUB T I T L E 11 = (YS - 1) * 12 + 1 12 = I I + 11 FOR I - I I TO 12 : PRINT #2,USING "#*#•#•";I; : NEXT I PRINT #2," MW" FOR I = 1 TO 83 : PRINT #2,"-"; : NEXT I : PRINT #2, FOR I = 1 TO 41 PRINT #2,USING "###.#";264.5 +1/2; FOR T% » I I TO 12 I F T% > 12 THEN Tl« = T% - ( ( Y % - 1) * 12) ELSE T l % = T% PRINT #2,USING "###•.•";(NGO(T*, I)-1)*240.625*NDM(Tl«)/1000; NEXT T% PRINT #2,USING "#•####";BEFIT(11,1) / 1000 NEXT I NEXT Y% RETURN -161-'SUBROUTINE TIT L E ( p r i n t t i t l e ) T I T L E : PRINT #2,FF$ PRINT #2,TIT1$ PRINT #2,TIT2$ PRINT #2," Time s t e p (month)" PRINT #2,"Elev."; RETURN 

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