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Optimal hydraulic operation of a complex two reservoir hydro-electric system Okun, Michael Howard 1970

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OPTIMAL HYDRAULIC OPERATION OF A COMPLEX TWO RESERVOIR HYDRO-ELECTRIC SYSTEM by MICHAEL HOWARD OKUN .A.Sc., The University of B r i t i s h Columbia, 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the 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, 1970 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Depar tment o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l no t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Depar tment o f CIV/L £A/6-//U//l/tr The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8, Canada Date d)cf<?6<?S /3'. /$7P A B S T R A C T This thesis presents an example of the use of dynamic programming for optimizing the hydraulic operation of a complex two re s e r v o i r hydro-e l e c t r i c system i n order to provide maximum firm power output. Records of natural inflows are used over a winter drawdown period for study pur-poses and flow forecast procedures are discussed. The system, owned by the Aluminum Company of Canada Ltd. (ALCAN), Arvida, P.Q., i s described, including i t s operations and r e s t r i c t i o n s . Also described are the procedures used and the assumptions made for a d a i l y simulation computer program which i s used by ALCAN and was made avai l a b l e for purposes of th i s study. A de s c r i p t i o n of dynamic program-ming including d e f i n i t i o n s of the normally used terms "objective function, "stages" and "states" i s given and i s followed by a discussion of the l i m i tations of the ALCAN program for d i r e c t use i n the dynamic programming algorithm. A new s i m p l i f i e d simulation routine that cuts computer costs d r a s t i c a l l y yet gives reasonable r e s u l t s i s described. Assumptions of the new routine are stated and i t s operation i s described. The dynamic programming computer routine i s explained, followed by a short descrip-t i o n of the implications of system r e s t r i c t i o n s on the routine. Comparisons between ALCAN simulation output and the dynamic programming routine output are made and the method of obtaining maximum firm power i s indicated. Tests are made to determine the s e n s i t i v i t y of the "sta t e " increment to computer costs and program accuracy. i i i For operational purposes, the dynamic programming routine can be used to determine optimal reservoir drawdown rule curves which can be then fed into the ALCAN simulation f o r d a i l y operational purposes. TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES . . • v i Chapter I. INTRODUCTION 1 II . THE ALCAN SYSTEM 3 I I I . DYNAMIC PROGRAMMING 10 IV. DYNAMIC PROGRAMMING AND THE ALCAN DAILY SIMULATOR . . 16 V. RESULTS 24 VI. SUMMARY 28 BIBLIOGRAPHY 30 i v LIST OF TABLES Table Page I RESERVOIR AND POWERHOUSE DATA 4 v LIST OF FIGURES Figure Page 1 ALCAN Saguenay Generation System 31 2 Hydraulic Flow Chart - Saguenay Generation System . . . 32 3 Logic Diagram - ALCAN Daily Simulator 33 4 Ty p i c a l Snowmelt Runoff Hydrograph for Saguenay D i s t r i c t 34 5 a Simple Rule Curve 35 5 b Defining the Rule Curve 35 6 I l l u s t r a t i o n of Dynamic Programming Computational Scheme 36 7 T y p i c a l Discharge-Efficiency Curve 37 8 Logic Diagram f o r Subroutine Model . . . 38 9 Logic Diagram - Main Dynamic Programming Routine . . . . 39 10 Comparison of DYPRO with Alcan Daily Simulation . . . . 40 11 Method of Obtaining Maximum Firm Power 41 v i A C K N O W L E D G E M E N T S The author expresses s i n c e r e g r a t i t u d e to h i s s u p e r v i s o r , Mr. S. 0. R u s s e l l , f o r h i s val u a b l e guidance and encouragement during the research, development and pre p a r a t i o n of t h i s t h e s i s . The author i s a l s o g r a t e f u l to Mr. R. J . S i l v e r , Hydro Engineer, and h i s s t a f f of the Aluminum Company of Canada, L i m i t e d , A r v i d a , P.Q., f o r t h e i r a s s i s t a n c e and co-operation i n p r o v i d i n g i n -formation on the operation of t h e i r h y d r o - e l e c t r i c complex and supply-i n g a copy of t h e i r d a i l y s i m u l a t i o n computer program. v i i CHAPTER I INTRODUCTION The f i e l d of study known as "operations research" or "systems a n a l y s i s " has produced many promising optimization techniques, some of which have, i n f a c t , been i n existence for some time. U n t i l recently most of these techniques have not been applied to any great extent on r e a l c i v i l engineering problems. A considerable research e f f o r t i s required to bridge the gap between theory and p r a c t i c e . This involves deciding which of the a v a i l a b l e techniques i s most appropriate for the problem i n question; adapting the formal technique to meet the require-ments of the problem; and e x p l i c i t l y d efining the "objective function" to be optimized. This thesis examines the a p p l i c a t i o n of dynamic programming to maximize the firm energy output (over the reservoir drawdown period) of a complex h y d r o - e l e c t r i c system. This involves defining the "objective fun c t i o n " and breaking down the problem into a number of "stages" and "states" s u i t a b l e f o r dynamic programming. The author was fortunate i n being able to obtain data from the Aluminum Company of Canada Limited (ALCAN), Arvida, Quebec, i n the form of a large FORTRAN computer program (together with relevant hydrological i n f o r -mation) which simulates the d a i l y operation of t h e i r e n t i r e Saguenay D i s t r i c t h y d r o - e l e c t r i c system. This program was developed over a period of several years to a s s i s t i n the e f f i c i e n t operation of the system. It incorporates 1 2 most of the company's a v a i l a b l e knowledge of and experience with the system. ;: Since there are two large storage re s e r v o i r s within the system, simulation by i t s e l f cannot f i n d the most e f f i c i e n t operating procedure although i t can be used to test proposed procedures quickly. It thus o f f e r s an opportunity to determine at l e a s t near optimal procedures by a process of t r i a l and erro r . On the other hand, by using dynamic program-ming i t becomes possible to eliminate the t r i a l and error process inherent i n simulation and d i r e c t l y obtain the most e f f i c i e n t operating procedure. Furthermore, power demand can be incremented u n t i l a l l a v a i l a b l e storage i s used, r e s u l t i n g i n maximum firm power production by the system. In both the ALCAN simulation and i n the present study i t was assumed that, for operating purposes,. forecast data on flows would be a v a i l a b l e . This i s probably a reasonable assumption i n that the optimiza-t i o n i s c a r r i e d out only over the drawdown period i n winter months when natural flows are low and reasonably pr e d i c t a b l e , and water released from storage makes up a large portion of the t o t a l flow. For study purposes, f i x e d percentages of the long-term average recorded flow data were used. This thesis contains a d e s c r i p t i o n of the ALCAN hyd r o - e l e c t r i c system i n the Saguenay D i s t r i c t and the assumptions made i n the ALCAN d a i l y simulator program; a b r i e f d e s c r i p t i o n of the dynamic programming algorithm which, was used; a d e s c r i p t i o n of how the ALCAN system and the dynamic pro-gramming algorithm were made compatible; and a discussion of the r e s u l t s and some suggestions?for future work. CHAPTER II THE ALCAN SYSTEM Layout The Aluminum Company of Canada's hy d r o - e l e c t r i c generation system i n the Saguenay D i s t r i c t of Quebec includes three storage reservoirs and six powerhouses with an i n s t a l l e d capacity of 2687 megawatts. (Fig. 1). The system uses water from the 30,000 sq. mi. Lake St. John watershed and has an average annual fi r m energy output of 1940 megawatts. Water impounded i n Lake Manouane passes through the Bonnard Canal and a natural channel to the Passes-Dangereuses r e s e r v o i r . Flow then con-tinues through three powerhouses i n serie s on the Peribonka River to Lake St. John. Most of the outflow from Lake St. John passes through the I s l e Maligne powerhouse or over the spillway, where i t then rej o i n s the flow from a small separate channel which circumnavigates the powerhouse. This t o t a l flow i s now the Saguenay River, which, downstream of I s l e Maligne divides into two channels, each supplying water to one of the two f i n a l powerhouses (Shipshaw and Chute-a-Caron) i n the system. Flow a l l o c a t i o n between these two powerhouses has been optimized by ALCAN to give most e f f i c i e n t output for any p a r t i c u l a r flow. (It i s understood that dynamic programming has been used for t h i s ) . The-complete ALCAN system i s i l l u -strated schematically i n F i g . 2. In the ALCAN computer simulation program, hereafter referred to as 3 4 the ALCAN d a i l y simulator, assumptions are made for each p a r t i c u l a r func-t i o n i n the system. Table I presents data for various locations while the following paragraphs indi c a t e the key assumptions f o r each power-house or flow control l o c a t i o n . Each rese r v o i r or powerhouse function i s represented by a separate FORTRAN SUBROUTINE i n the ALCAN simulator. TABLE I RESERVOIR AND POWERHOUSE DATA (a) STORAGE RESERVOIR LIVE STORAGE CAPACITY BCF MAF Passes-Dangereuses 180 4.13 Lake Manouane 90 2.06 Lake St. John 140 3.21 410 9.40 (b) AVERAGE NET INSTALLED TURBINE NUMBER OF L O C A T I O N HEAD ( f t . ) CAPACITY (MW) TURBINES Shipshaw 210 896 12 Chute-a-Caron 160 224 4 I s l e Maligne 90 - 110 402 12 Chute-a-la Savane 110 210 5 Chute-du-Diable 110 205 5 Chute-des-Passes 470 - 640 750 5 5 The Simulator and Its Functions When using the simulator over the drawdown period, approximately December 1 to A p r i l 15, i t i s assumed that a l l the natural inflows are known or can be forecast by December 1. This assumption may cause some error i n simulator output during the higher inflows to Lake St. John, but the problem i s outside the scope of t h i s study. For study purposes past records of flow are used. A l l powerhouses have the capacity to pass water through the;tur-bines or s p i l l water over the spillway at the dam. Powerhouse flows are passed through the turbines at best e f f i c i e n c y . Also, for a l l l o c a t i o n s , the r e l a t i o n s h i p between any two required parameters, such as volume and elevation f o r a r e s e r v o i r , i s expressed as a tenth order polynomial. H i s t o r i c a l l y , the Lake Manouane re s e r v o i r at the upstream end of the system has been operated during the drawdown period by predetermining i t s e n t i r e discharge schedule. Its operation i s controlled by two gates whose settings are usually changed infrequently, thus giving near constant or at l e a s t predictable discharges over several weeks. Further, there i s no powerhouse at the o u t l e t of Lake Manouane. For these reasons, Manouane outflows are assumed to provide known inflows to the Peribonka system down-stream. The elevation of the Passes-Dangereuses rese r v o i r i s usually re-s t r i c t e d to a one hundred foot v a r i a t i o n . Powerhouse flows at Chute-des-Passes are passed through the turbines at best e f f i c i e n c y f o r the a v a i l a b l e head while any excess flow i s s p i l l e d over the dam. Tailwater l e v e l does not a f f e c t the Diable system downstream. 6 Although Chute-du-Dlable Is treated as a constant-head plant f o r most of the season, i t has a small forebay storage which can be drained within 10 days. In a low inflow period Diable then becomes a v a r i a b l e -head plant. Tailwater elevation i s independent of Savane generation down-stream. Chute-a-la Savane i s a constant-head plant. Its power output, however, i s affected by Lake St. John which at higher elevations backs up to the Savane tailwater, thus reducing the gross head at the power plant. Lake St. John i s a 400 square mile res e r v o i r having a p r a c t i c a l elevation v a r i a t i o n of 13 f e e t . Considerable natural inflow and storage release are routed through a unique natural outlet system. The major out-l e t channel i s a narrow gorge which l i m i t s flow according to the elevation of Lake St. John and determines the forebay elevation of the I s l e Maligne powerhouse. The minor hydraulic outlet i s c a l l e d the " l i t t l e discharge" and although i t s flow can be varied, i t i s generally held constant at 400 c f s . Restricted flow i n the gorge poses system d i f f i c u l t i e s only i f high flows are required at low lake l e v e l s or i f flow becomes c r i t i c a l . I s l e Maligne i s a variable-head power plant (maximum v a r i a t i o n about 20 feet) whose flow i s r e s t r i c t e d to the capacity of the gorge and whose head i s dependent on the head loss through the gorge. Maximum head loss here i s about 10 fe e t . The I s l e Maligne tailwater elevation does not a f f e c t downstream power generation. The f i n a l two powerhouses i n the system, Chute-a-Caron and Ship-shaw, are h y d r a u l i c a l l y i n p a r a l l e l . Both are run-of-the-river plants, but Shipshaw accepts a l l the flow up to a c e r t a i n point due to i t s higher head. 7 \ . When both plants are operating, the flow i s s p l i t for maximum power at best e f f i c i e n c y . Simulator Methodology F i g . 3 i s a l o g i c diagram of the hydraulic c h a r a c t e r i s t i c s of the ALCAN d a i l y simulator. For a normal day, computation begins by accepting the power demand and the natural inflows. Lake Manouane outflow i s then computed, followed immediately by a Lake St. John storage change, which i s predetermined by the i n i t i a l Lake St. John rule curve fed into the simulator. Then the Peribonka r i v e r discharge to Lake St. John i s compu-ted, thus giving both the values of discharges (and t h e i r corresponding power outputs) through the Peribonka plants and out of the Lake St. John r e s e r v o i r . Saguenay discharges and power plant outputs are calculated next, I s l e Maligne being considered f i r s t . I f t o t a l scheduled generation i s not met, a separate routine determines the necessary flow adjustment at Sa-vane.by computing s t a t i o n (powerhouse) e f f i c i e n c i e s i n terms of megawatts per cubic foot per second (MW/cfs) of water through the turbines; and c a l -culations i t e r a t e again at Savane. This computational scheme continues un-t i l the desired convergence of scheduled and generated power i s obtained. I f , however, convergence to a s p e c i f i e d tolerance i s not obtained within 10 i t e r a t i o n s , the power output from the tenth i t e r a t i o n i s accepted. For a t y p i c a l study over the drawdown period, the d a i l y simulator requires about 63 seconds of machine time for execution on the Univ e r s i t y of B r i t i s h Columbia (U.B.C.) IBM 360 Model 67 computer. This corresponds 8 to a "normal p r i o r i t y " cost of about $6.40. The degree of optimization achieved i s measured as the "carryover" l e f t i n the upper r e s e r v o i r s . Hydrology and Forecast Procedure Annually, the Lake St. John watershed i s characterized by a t y p i -c a l snowmelt runoff hydrograph. (Fig. 4). In general, the date of the spring "break-up" (the date when the flow begins to r i s e towards the peak) i s not known u n t i l at most two weeks beforehand. During the drawdown period, low temperatures obviate any problems associated with forecasting r a i n f a l l runoff. Natural inflows do, however, range from n e g l i g i b l e to s i g n i f i c a n t . Natural flows represent between about 35 per cent of the t o t a l flow i n the Peribonka i n December to less than 10 per cent i n March. For the Saguenay system the corresponding f i g -ures are 60 per cent i n December to 30 per cent i n March. For the ALCAN d a i l y simulator, the natural inflows over the draw-down period are assumed to be known. The v a l i d i t y of t h i s assumption i s best i l l u s t r a t e d by summarizing the forecast procedure and i t s r a t i o n a l e : 1. Winter flows are estimated as a f i x e d percentage of long-term normal flows; 2. At the onset of f r e e z i n g , flows during the following months are l a r g e l y a function of the e x i s t i n g l e v e l s of var-ious small lakes i n the watershed and the l e v e l of the ground-water table; these c r i t e r i a are applied by about December 1 to give a f a i r l y r e l i a b l e estimate of flows as a percentage of normal flows; 9 3. The r a t i o of the natural runoff on December 1 to the long-term average runoff on that date was found to be nearly equal to the r a t i o of the t o t a l runoff during the winter months to the long-term average winter runoff; 4. Occasional r a i n f a l l can a f f e c t the forecast but t h i s i s generally ignored. CHAPTER I I I DYNAMIC PROGRAMMING General Observations If from time A to time B a system must undergo a transformation, but the "best" transformation (the optimal path to follow) i s unknown, then dynamic programming i s a convenient technique which may aid i n de-f i n i n g the required transformation. Formally, the unknown path i s an optimal p o l i c y determined from a set of decisions made i n sequence. This chapter attempts to b r i e f l y describe the a p p l i c a t i o n of dynamic programming to determine optimal rule curves f o r maximum fir m energy output for the two active storage reservoirs i n the ALCAN system. In order to provide the most fir m energy from any system, the energy resource must be most e f f i c i e n t l y used, from the o v e r a l l system viewpoint. In the present case, the energy resource i s stored water, since natural inflows pass f r e e l y through the system anyway. Then i t follows that f o r any given time period, a minimum of stored water should be used while s t i l l meeting the power demand schedule. In t h i s manner, water i s a v a i l a b l e at a l a t e r date when natural inflows become minimal. Also the head on the rese r v o i r plants remains high as long as possible. Dynamic programming can be used for f i n d i n g , f o r each r e s e r v o i r , a r u l e curve which would meet the load demand but use the minimum amount of water. 10 11 Computational Scheme Since the a n a l y t i c function f o r a rule curve i s generally not known, the curve must be described by defining several f i n i t e points. (Fig. 5a). I f , as shown i n F i g . 5b, a series of h o r i z o n t a l and v e r t i -c a l l i n e s c a l l e d respectively states and stages are drawn through the defined points, then from the r e s u l t i n g g r i d a point i s defined by s p e c i -fying a stage and a corresponding state. Assuming the f i n a l elevation (state) i s known at the l a s t period i n time (stage), there are t h e o r e t i -c a l l y an i n f i n i t e number of paths to reach this f i n a l known state from the immediately preceding stage. In r e a l i t y , of course, only a pre-selected number of such paths can be examined, one of which i s part of or s a t i s f a c t o r i l y close to the optimal rule curve. Lines 0-1, 0-2, 0-3, and 0-4 of F i g . 6 are examples of such paths. The dynamic programming technique usually r e f e r s to the f i n a l stage (timewise) as the zeroth stage, i . e . , no stages l e f t to examine, and the second l a s t stage (timewise) as the f i r s t stage, i . e . , one stage l e f t to examine, etc. This terminology i s adopted for the remainder of this t h e s i s . For each state at stage 1, then, there may be several d i s t i n c t paths to reach t h i s state from stage 2. For example, r e f e r r i n g to stage 2, l i n e s 1-1, 1-2, 1-3 and 1-4 of F i g . 6 represent possible paths to reach state 1 of stage 1. A s i m i l a r set of paths can be drawn for each state of stage 1. The problem becomes one of deciding which state to end up at i n stage 1, and which state to s t a r t from i n stage 2; this problem i s inherent 12 for any two stages from the zeroth to the Nth or f i n a l stage. The value of the so-called objective function f or any f e a s i b l e path i s the basis for making each such dec i s i o n . Hence, a current account of the objective function i s maintained during computation. For the ALCAN problem, the objective function i s simply the t o t a l amount of stored water used to date, s t a r t i n g on the date at the zeroth stage. The o v e r a l l problem i s thus to minimize the objective function f or the whole drawdown period, while meeting the power demand schedule and observing the system r e s t r i c t i o n s . Recursion Formula How can t h i s be accomplished without examining every possible path from stage N to stage zero? The answer l i e s i n Bellman's p r i n c i p l e of optimality [1], which e s s e n t i a l l y says that for a given state regard-less of the p o l i c y selected i n previous stages, there must s t i l l be an optimal p o l i c y f o r the remaining stages. With t h i s i n mind, the problem of s e l e c t i n g the optimal path from a given stage and state i s reduced to applying the following recursion formula: f *(s) = min{ C + f *(x ) } n = 1, 2, 3, . . ., N (1) n s ,x n-1 n n where; n i s the number of stages (time periods) to go; these must proceed i n order, N i s the t o t a l number of stages, 13 s i s the current state (reservoir l e v e l ) of the system, x^ i s the decision v a r i a b l e , i . e . , where the system w i l l go next, or s p e c i f i c a l l y , which l e v e l the re s e r v o i r w i l l be at by the next stage, Q s,x^ i s the cost charged to the system, i . e . , the volume of water used i n making decision x^, i f the system i s i n state s with n stages to go, f , (x ) i s the minimum value of the objective func-n-1 n J t i o n with n-1 stages to go, given the system i s i n state s and the decision i s x , n * f (s) = minif (s,x ) }, which i s the minimum value of n n n the objective function with n stages to go, given the system i s i n state s and the decision i s x . This i s obtained by n comparing a l l possible values of the sum of C (the return s ,x n from the current stage) and f .. (x ) (the return from a l l n — 1 n subsequent stages), and s e l e c t i n g the minimum value of th i s sum. In order to formally adapt t h i s concept to the ALCAN system, and noting that since there are two reservoirs i n s e r i e s , i t i s possible to define the state of the system as the l e v e l of one r e s e r v o i r . Then for a given stage, when one change i n re s e r v o i r l e v e l i s s p e c i f i e d and i n -flows and power demand are known, the elevation change of the other reser-v o i r w i l l be a unique value. This n a t u r a l l y assumes the two re s e r v o i r 14 l e v e l changes are f e a s i b l e ( i . e . , the power demand can be met within the system r e s t r i c t i o n s ) . Also, i t should be noted that i f a decision i s s p e c i f i e d f o r one r e s e r v o i r to lower i t s elevation by x f e e t , then the objective function need only be the l e v e l of the other r e s e r v o i r . P a r a l l e l Studies For purposes of comparison, i t i s worth describing the more out-standing features of two somewhat s i m i l a r studies that have been made using dynamic programming. H a l l andRoefs [3] concluded that " . . . dynamic programming can be s u c c e s s f u l l y applied to complex rese r v o i r operation studies." This conclusion arose from studying a multipurpose 3-reservoir system i n which uses of the system other than hydropower production were treated as bound-ary conditions (they were not optimized). The objective function as such was not e x p l i c i t l y defined but the problem was to maximize monetary re-turns by developing an optimal operating p o l i c y for a p a r t i c u l a r hydrology. Monthly time periods were assumed throughout the analysis. From the hydro-power point of view the system was s i g n i f i c a n t l y simpler than the ALCAN system described herein and was e s s e n t i a l l y reduced to a s i n g l e r e s e r v o i r system. The r e s u l t s did i n f e r , however, that for long range planning pur-poses of the defined system, the dynamic programming algorithm produced an optimal operating p o l i c y . H a l l et at. [4] studied a system more c l o s e l y resembling the ALCAN system except that the reservoirs were connected h y d r a u l i c a l l y i n p a r a l l e l , water supply and flood control storage releases were s p e c i f i e d , and there 15 were only two powerhouses Cone at the outlet of each r e s e r v o i r ) . The a l -gorithm employed was a sort of pseudo-dynamic programming technique, c a l l e d incremental dynamic programming. D e t a i l s of the method are not given herein, but the theme of the algorithm i s captured i n the follow-ing quotation [4]: ". . . the algorithm s t a r t s with a f e a s i b l e i n i t i a l p o l i c y ; i . e . , a sequence of states through which the sys-tem must pass i n each month, and then analyzes only those new p o l i c i e s which are close to the i n i t i a l p o l i c y . " At any point i n the monthly computation procedure there are two state and two decision v a r i a bles (a two-dimensional problem) which are respective-ly the amount of storage i n each r e s e r v o i r at the beginning of the month and the t o t a l release from each rese r v o i r during the month. Although th i s approach appears to be useful f o r long-term planning of large sys-tems, i t does not guarantee an optimal s o l u t i o n . In summary, the f i r s t of the two references examined dealt with an apparently complex system but for computation purposes reduced i t to a system much simpler than was possible with the ALCAN system. The other study used a procedure which could be extremely useful under c e r t a i n c i r -cumstances but does not guarantee an optimal s o l u t i o n . CHAPTER IV DYNAMIC PROGRAMMING AND THE ALCAN DAILY SIMULATOR P r a c t i c a l Limitations Employment of the dynamic programming computational scheme out-lined i n the previous chapter implies the necessity for a v e r s a t i l e simu-l a t i o n routine which w i l l accept for any given time period natural i n -flows, power demand and a s p e c i f i c elevation change of one r e s e r v o i r , and w i l l output the t o t a l storage release required ( i f there i s a f e a s i b l e solution) and the elevation of the second r e s e r v o i r . I m p l i c i t i n such a v e r s a t i l e routine i s the constraint that i t should use a minimum of computer time. Referring once again to F i g . 6, i t can be observed that i f an operating p o l i c y i s to be derived over an average drawdown period of about 140 days, then the simulation routine w i l l be subject to repeated use. The immense number of computations i n -volved i s best i l l u s t r a t e d by an example using the e x i s t i n g ALCAN d a i l y simulator. If there were seven stages and 10 allowable states per stage, assuming 20 days per stage, then about 10,000 runs through the simulator would be required. The r e s u l t i n g cost on the U.B.C. computer f o r such a single study, which would y i e l d only one optimal r u l e curve for each reser-v o i r , was estimated to be i n the order of $500. Examination of the e x i s t i n g ALCAN d a i l y simulator with a view to incorporating i t i n the dynamic programming technique resulted i n three major observations: 16 17 1. D a i l y simulation was too detailed — this was e s p e c i a l l y true when considering actual operation i n view of the uncertainty about natural inflows; 2. The i t e r a t i v e scheme of readjusting flows to meet demands by changing the flows through the e n t i r e system was very time consuming, assuming one r e s e r v o i r l e v e l (state) would be preselected for each use of the simulator; 3. The empirical r e l a t i o n s , mostly tenth degree poly-nomials r e l a t i n g two variables i n FORTRAN double p r e c i s i o n , consumed too much machine time. General Modifications Items (1) and (2) above are basic to the ALCAN simulator; and since i t became apparent that problems and delays would a r i s e i n the course of making changes to the simulator, i t was decided that a new, s i m p l i f i e d simulation routine should be written. The new simulator routine, referred to from t h i s point on as "MODEL," incorporated two basic features that are quite d i f f e r e n t from those used i n the ALCAN d a i l y simulator. These two features are: (a) the use of discharge-efficiency curves for each power plant; (b) s e l e c t i o n of the Passes-Dangereuses res e r v o i r as the state v a r i a b l e ; t h i s implies readjustment of flows only i n the Saguenay system ( i . e . , downstream of the Lake St. John reservoir) to meet the required power 18 demand when a l e v e l of Passes-Dangereuses (the state) i s s p e c i f i e d . Generation of the disc h a r g e - e f f i c i e n c y curves (Fig. 7) was accomplished by writing a simple routine which employed various e x i s t -ing routines from the ALCAN simulator. S p e c i f i c a l l y , f o r constant-head plants, s t a t i o n discharge was varied by incremental amounts and corres-ponding power generation was calculated using the supplied FORTRAN subrou^ ti n e s ; r e s u l t i n g e f f i c i e n c i e s were calculated using the standard formula, s G : where n = s t a t i o n generating e f f i c i e n c y i n per cent, Q g = s t a t i o n discharge i n c f s , H„ = gross head i n feet (this i s discussed further below), K = conversion factor (constant), P = power output i n megawatts. For variable-head plants, a s i m i l a r approach was used except that gross head was also a v a r i a b l e . This resulted i n discharge-efficiency curves for various gross head values. The only noteable l i m i t a t i o n of t h i s generated data i s that i t i s only v a l i d f o r a fixed number of turbines a v a i l a b l e . Except during trash rack cleaning or unforeseeable turbine repairs i t i s considered that t h i s i s not a serious l i m i t a t i o n . ? 1 9 S p e c i f i c Modifications The o v e r a l l e f f e c t of developing the MODEL routine to replace' the ALCAN d a i l y simulator can be conveniently summarized by sta t i n g the new assumptions f o r each function of the system. At a l l relevant l o c a -tio n s , for a preselected time i n t e r v a l , the d a i l y natural inflows are summed and the average i s calculated and used as the natural inflow for that i n t e r v a l . Time i n t e r v a l s are assumed constant for a given study. Lake Manouane assumptions are s i m i l a r to those of the ALCAN d a i l y simulator. The only difference i s that rather than d i r e c t l y using the routine which calculates Manouane d a i l y outflows, i t i s employed i n d i r e c t -l y by averaging d a i l y flows over each preselected time i n t e r v a l for the whole drawdown study. Consequently these average flows are entered as data into the MODEL routine. Volume-elevation data f o r the Passes-Dangereuses res e r v o i r i s stored i n arrays; precise volume f or a given elevation i s calculated using a l i n e a r i n t e r p o l a t i o n routine. This avoids the use of the afore-mentioned polynomial-type r e l a t i o n s h i p which r e l a t e s these two var i a b l e s i n the ALCAN simulator. Gross head at the Chute-des-Passes powerhouse i s defined as reser-v o i r e l e v a t i o n minus a constant tailwater elevation. This i s , of course, averaged over the preselected time i n t e r v a l . A two-way l i n e a r i n t e r p o l a -t i o n routine i s required to calculate power output. F i r s t , using the average gross head, the appropriate discharge-efficiency curve i s selected, i n t e r p o l a t i n g between a head increment of f i v e per cent of the t o t a l allow-able head v a r i a t i o n . Then using the s t a t i o n discharge, e f f i c i e n c y i s deter-20 mined from a second i n t e r p o l a t i o n . Power i s then calculated using equa-t i o n (2). The i n t e r p o l a t i o n routine coupled with the stored discharge-e f f i c i e n c y data omitted the need for a major portion of the time consum-ing polynomial equations f o r a l l the powerhouse c a l c u l a t i o n s . Chute-du-Diable i s treated as a constant-head plant, ignoring i t s small storage c a p a b i l i t i e s . Hence gross head i s a constant forebay e l e -v a t i o n l e s s a constant tailwater elevation. Power output i s again c a l -culated from s t a t i o n e f f i c i e n c y , which i s obtained by l i n e a r i n t e r p o l a t i o n of d i s c h a r g e - e f f i c i e n c y data. Although the new assumptions at Chute-a-la-Savane are somewhat s i m i l a r to those at Chute-des-Passes, the computational procedure for c a l -c ulating power output i s s l i g h t l y d i f f e r e n t . In contrast to Chute-des-Passes, gross head i s defined as a constant forebay elevation minus a var-i a b l e tailwater elevation, u n t i l the elevation of Lake St. John that ceases to a f f e c t Savane tailwater i s reached. Otherwise power c a l c u l a t i o n s pro-ceed s i m i l a r to those at Chute-des-Passes. The problem of incorporating the unique outlet r e s t r i c t i o n s of Lake St. John into the s i m p l i f i e d simulator was overcome by coupling the e x i s t i n g routines governing these r e s t r i c t i o n s to the I s l e Maligne power c a l c u l a t i o n routine during computation of the I s l e Maligne discharge-efficiency-head data. These data were thus derived s i m i l a r to the Chute-des-Passes data, except that the "discharge" refers to the t o t a l discharge from Lake St. John, rather than the I s l e Maligne s t a t i o n discharge. Gross head at I s l e Maligne i s defined as Lake St. John elevation less a constant tailwater elevation. The usual i n t e r p o l a t i o n s are again made and the head increment between discharge-efficiency curves i s about nine per cent of the t o t a l allowable head v a r i a t i o n . 21 Since during the drawdown period the Shipshaw plant accepts almost a l l of the flow i n the Saguenay River, discharge-efficiency data were derived with Shipshaw and Chute-a-Caron power routines coupled together. The only inconvenience with t h i s approach i s that i f Saguenay discharge i s high enough to warrant power generation at Chute-a-Caron ( r e c a l l that Shipshaw accepts a l l the flow up to a ce r t a i n point due to i t s higher head), then a c a l c u l a t i o n must be made afterwards to determine the o p t i -mal, a l l o c a t i o n of the combined generation. Constant forebay elevation at Chute-a-Caron minus a constant tailwater elevation at Shipshaw repre-sents the gross head for power c a l c u l a t i o n s . In the ALCAN simulator, Shipshaw forebay elevation i s computed d i r e c t l y from Chute-a-Caron f o r e -bay elevation. Dynamic Programming Routine F i g . 8 i s a l o g i c diagram of the MODEL routine used i n the dyna-mic programming algorithm. Input to MODEL consists of natural inflows, power demand, i n i t i a l and f i n a l elevations of the Passes-Dangereuses reser-v o i r , and the i n i t i a l elevation of Lake St. John. (Recall that " i n i t i a l " here refers to the end of the time period). The routine then outputs rese r v o i r storage changes, power generations and flow at each s t a t i o n , and f i n a l and average elevations of both r e s e r v o i r s . Not i l l u s t r a t e d but note-worthy i s the fa c t that a f e a s i b l e s o l u t i o n on the Saguenay system i s only considered up to a,point j u s t beyond the beginning of . spillway discharge (maximum power) since excess s p i l l a g e below Lake St. John usually means nothing more than waste from the hydro-power point of view. 22 A l o g i c diagram of the o v e r a l l dynamic programming routine i s shown i n F i g . 9. Several s a l i e n t points not yet considered are out-l i n e d i n the following paragraphs. F i r s t l y , among the various elevations, dates, e f f i c i e n c i e s , etc., that are used i n the o v e r a l l routine, i s a r e s t r i c t i n g envelope of elevations and corresponding dates at the Passes-Dangereuses reser-v o i r . The area outside t h i s envelope represents a non-feasible s o l u t i o n range and i s precluded from consideration. This eliminates any necessity f o r the dynamic programming routine having to examine u n r e a l i s t i c reser-v o i r drawdowns. Points defining the envelope were determined by running the ALCAN simulator under extreme hydrological (low flow) and e l e c t r i c a l (high power demand) conditions, and then adjusting these values by a safety f a c t o r . The MODEL routine indicates that for every dynamic programming i t e r a t i o n , an i n i t i a l guess must be made at the t o t a l discharge out of Lake St. John. Such a guess was determined by reading into the dynamic programming routine a curve of Lake St. John discharge against t o t a l power generation from the three Saguenay River powerhouses. Points for the curve were determined by taking output data from the ALCAN simulator at various elevations of Lake St. John on d i f f e r e n t dates. ( O r i g i n a l l y , several curves were obtained, one f o r each new Lake St. John elevation. The curves were, however, close enough that t h e i r average value could be used). Another key point, e s p e c i a l l y f o r computer storage purposes, i s that r e l a t i v e l y few of the f e a s i b l e objective function values need actu-a l l y be stored i n the computer. In other words, when the system i s at 23 a t y p i c a l stage and state, only the minimum value of the objective func-t i o n and consequently other relevant variables. >need be stored. This follows d i r e c t l y from Bellman's p r i n c i p l e of optimality. R e s t r i c t i o n s Physical r e s t r i c t i o n s within the system are included i n the dyna-mic programming routine. These r e s t r i c t i o n s a r i s e from ALCAN operating p o l i c i e s , agency regulations, and p h y s i c a l capacities or l i m i t s . D e t a i l s of such r e s t r i c t i o n s are not given, but t h e i r e f f e c t s are worth consider-a t i o n . Although i t may not be immediately obvious, system r e s t r i c t i o n s are a blessing i n solving dynamic programming problems. (This has already been observed with the use of the Passes-Dangereuses envelope). In simple terms t h i s stems from the f a c t that with increasing r e s t r i c t i o n s , there must be a decreasing f e a s i b l e s o l u t i o n range. For example, the Passes-Dangereuses rese r v o i r can vary over a 170 foot elevation range, but i s operationally l i m i t e d to a 100 foot range. If i t were allowed to vary over the extra 70 f e e t , then many more states would have to be examined at every stage, thus increasing the computer time s u b s t a n t i a l l y . CHAPTER V RESULTS Program Checkout The most important requirement with any computer program i s to check the v a l i d i t y of the r e s u l t s . In th i s case the re s u l t s obtained by the dynamic programming routine (hereafter.referred to as DYPRO) were checked against r e s u l t s using the ALCAN d a i l y simulator. This was accomplished by feeding the optimal Lake St. John drawdown schedule out-put from the DYPRO routine into the ALCAN d a i l y simulator (using, of course, the same natural inflows and power demand). For t h i s t h e s i s , the time period or stage i n t e r v a l used i n the DYPRO routine was 20 days, and the study period was assumed to be from November 27 to A p r i l 15 i n c l u s i v e (140 days). In most cases, a near constant power demand was assumed adequate for study purposes. This assumption i s probably not u n r e a l i s t i c s ince the maj or portion of the demand on the system comes from aluminum smelter plants that operate s i x days per week. F i g . 10 i l l u s t r a t e s optimal r u l e curves from the DYPRO routine for both re s e r v o i r s f or 90 per cent of the long-term average natural i n -flows, an average power demand of 2200 MW, and a state increment of one foot.. Also p l o t t e d are the ru l e curves from the ALCAN simulator f o r iden-t i c a l conditions, except that the Lake St. John drawdown schedule for t h i s case was supplied by ALCAN as part of the o r i g i n a l data. The DYPRO rou-24 25 tine did not work when the s t a r t i n g elevation (on A p r i l 15) of the Passes-Dangereuses res e r v o i r was given exactly the same value as that i n the ALCAN simulator. This was for two reasons: (1) the Lake St. John r u l e curve supplied by ALCAN allowed elevations below the four foot minimum s p e c i f i e d by DYPRO ;:"and (2) the near f l a t portion of the required Passes-Dangereuses drawdown curve caused problems as a r e s u l t of the sen-s i t i v i t y of DYPRO to the state increment (explained l a t e r ) . Consequently the s t a r t i n g elevation of the Passes-Dangereuses r e s e r v o i r i n the DYPRO routine was lowered s l i g h t l y to allow more stored water to come from Passes-Dangereuses. Unfortunately at the time of w r i t i n g , no s p e c i f i c "best" rule curves for drawdown were a v a i l a b l e from ALCAN for comparison with the DYPRO routine. Hence, these optimal-rule curves (Fig. 10) from the DYPRO routine could only be checked out as previously mentioned; that i s , by inputting the Lake St. John drawdown schedule to the ALCAN simula-tor . No s i g n i f i c a n t discrepancies were found. Maximum Firm Power The method of obtaining the maximum firm power output from the system f o r a set of given system conditions i s i l l u s t r a t e d i n F i g . 11. With f i n a l (end of study) Lake St. John and Passes-Dangereuses elevations given and power demand and natural inflows as shown, optimal rule curves were obtained by successively increasing the power demand u n t i l the DYPRO routine f a i l e d , whereupon maximum firm power output was reached or ex-ceeded. Although i t i s not i l l u s t r a t e d , i t i s i n t e r e s t i n g to note that as 26 power output increases, the f e a s i b l e s o l u t i o n range decreases. In other words, the higher the demand on the system, the fewer r e s e r v o i r drawdown paths there are to meet the demand. One fortunate consequence of t h i s i s that computer costs decline with increasing power demand. S e n s i t i v i t y to State Increment; Computer Costs With a l l conditions f i x e d except the state increment (allowable un i t change i n elevation of the Passes-Dangereuses r e s e r v o i r ) , the DYPRO routine was run with a state.increment (DY8) of from two feet to seven feet i n c l u s i v e at one foot i n t e r v a l s . It was found that by increasing the increment from two feet to four feet, the machine execution time on the U.B.C. computer was cut from 72.6 sec. ($6.80) to 28.9 sec. ($2.97) r e s u l t i n g i n a 56 per cent reduction i n cost. No appreciable diffe r e n c e i n output was observed between these cases. The only obvious trend i n the output while increasing DY8 was that the t o t a l volume of water used from storage increased s l i g h t l y , i n d i c a t i n g s l i g h t l y lower system e f f i -ciency. When DY8 was seven f e e t , the machine execution time was 17.0 sec. ($1.85). It was observed that the DYPRO routine was quite s e n s i t i v e to the state increment when the Passes-Dangereuses elevation (state variable) remained r e l a t i v e l y constant over several 20. day periods. This can be explained assuming a large state increment of, say, f i v e f e e t . I f , at a given stage and state, the maximum return (least amount of water used from storage) i s obtained by changing the Passes-Dangereuses res e r v o i r elevation by only one-half foot, then i t i s apparent that the f i v e foot 27 elevation changes being examined would not only miss the maximum return value, but also would f a i l to allow the system to meet the required power demand. Referring again to F i g . 6, i f the system i s at stage 1 and state 1, then path 1-1 may not allow the system to meet the power demand while path 1-2 may cause excessive s p i l l a g e throughout the system. Hence, no s o l u t i o n would be obtained f o r that point. F i g . 10 further i l l u s t r a t e s t h i s point since the DYPRO routine f a i l e d for a l l attempted state increments greater than one foot. It appears that the costs l i s t e d i n the previous paragraphs j u s t i f y the use of the MODEL routine over the ALCAN d a i l y simulator i n the DYPRO routine. In f a c t , one en t i r e run over the drawdown period using MODEL i n the DYPRO routine was found to be cheaper than running the ALCAN program alone, with no optimization. On the other hand, the cost of developing the o v e r a l l dynamic programming and MODEL routines (estimated at about $1200 i n machine time plus about four man months of the author's time) should be noted. CHAPTER VI SUMMARY Simulating the operation of a complex h y d r o - e l e c t r i c system with s i x powerhouses and two reservoirs was s i m p l i f i e d to the point where one-dimensional dynamic programming could be used to obtain the maximum firm power output of the system. This involved using one r e s e r v o i r l e v e l as the " s t a t e " v a r i a b l e and ( e s s e n t i a l l y ) the other r e s e r v o i r l e v e l as the objective function, while extending the simulation time period from one day to 20 days. Results were checked and i n most cases did not appear to be p a r t i c u l a r l y s e n s i t i v e to the state increment. I t was found that i n order to apply dynamic programming e f f e c t i v e -l y , some s k i l l was required i n defining the objective function and i n s i m p l i f y i n g the computations (to keep the method computationally f e a s i b l e ) while keeping them s u f f i c i e n t l y d e t a i l e d that r e s u l t s are meaningful. The l e v e l of refinement used i n t h i s study i s believed to represent an appropriate compromise. The dynamic programming (DYPRO) routine developed can be used to determine Lake St. John optimal rule curves for the e n t i r e drawdown season and those can then be fed into the ALCAN simulator for d a i l y operational purposes. With minor changes, the DYPRO routine could be extended for use during any time period. Knowledge of inflows over the drawdown period was assumed i n the DYPRO routine; t h i s i s considered reasonable under the circumstances. The 28 2 9 method, however, could r e a d i l y be extended f or use with uncertain i n -flows. It could be used to develop a re l a t i o n s h i p between forecast flows, forecast accuracy and f i r m power, which would enable decisions to be made and reser v o i r r u l e curves to be developed f o r maximum amount of firm power with any desired degree of r i s k . B I B L I O G R A P H Y H i l l i e r , H. S., Lieberman, G. J . Introduction to Operations Re search. San Francisco, C a l i f . : Holden-Day, Inc., 1967. Bellman, R. E., Dreyfus, S. E. Applied Dynamic Programming. Princeton, N.J.: Princeton University Press, 1962. H a l l , W. A., Roefs, T. G. "Hydropower Project Output Optimi^.:. za t i o n , " Journal of the Power Division, ASCE, Vol. 92, No. P01 (January, 1966), pp. 67-79. H a l l , W. A., Harboe, R. C , Yeh, W. W.-G. "Optimum Firm Power Output From a Two Reservoir System by Incremental Dynamic Programming," Contribution No. 130} U n i v e r s i t y of C a l i f o r -n i a Water Resources Centre, October 1969. Private correspondence with Mr. R. J . S i l v e r , Hydro Engineer, Aluminum Company of Canada Limited, Shipshaw v i a Jonquiere P.Q. 30 FIG.i. ALCAN SAGUENAY GENERATION SYSTEM 3 2 ^ ^ 0 ^ 4 R | L A K E M A N O U A N E I QSTOR 9 | C H U T E DU D I A B L E T 0 ^V-> |CHUTE A LA SA.VANE| NOTES• — ore natural inflows. — Symbols are those used in ALCAN simulator. — QTURB refers to turbine flow. — SPILL refers to spillway flow, except at station 5. — QSTN refers to station flow. — QSTOR refers to storage release or increase. FIG. 2 . HYDRAULIC F L O W C H A R T - S A G U E N A Y GENERATION S Y S T E M S P E C I F Y D I M E N S I O N O F 9 A R R A Y S E X P L I C I T R E A L S P E C I F I C A T I O N E X P L I C I T I N T E G E R S P E C I F I C A T I O N M E G A W A T T S S C H E D U L E D IS E Q U A L T O T H A T F O R F I R S T P E R I O D IN S T U D Y S U B R O U T I N E I N T R O ' DEFINE UNITS AVAILABLE INITIAL RESERVOIR VOLUMES, LAKE MANOUAN AND Dl ABLE DUMP SCHEDULES ANO SPECIFY INFLOWS IN PERCENT OF NORMAL ^DEFINE OTHER CONTROL PARAMETERS, S U B R O U T I N E D U M P 5 > DEFINE LAKE ST. JOHN DUMP SCHEDULE FOR ENTIRE STUDY C O M P U T E D A I L Y I N F L O W S I N T O E A C H R E S E R V O I R S P E C I F Y N E W G E N E R A T I O N S C H E D U L E F O R D A Y IN Q U E S T I O N S U B R O U T I N E D U M P 9 \ COMPUTE LAKE MANOUAN SPILLAGE ANO BALANCE R E S E R VOIR J S U B R O U T I N E D U M P 7 > COMPUTE CHANCE IN CHUTE -DU - D1ADLE STORAGE ANO BALANCE RESERVOIR J S P E C I F Y P E R I O D I D E N T I F I C A T I O N C O D E COMPUTE CHANCE IN STORAGE END OF DAY AND AVERAGE ELEVATION FOR LAKE ST. JOHN RESERVOIR COMPUTE P E f l t B O N K A RIVER DISCHARGES COMPUTE L A K E ST. JOHN DISCHARGE S U B R O U T I N E \ P O W E R 6 > C O M P U T E S A V A N E / G E N E R A T I O N / f S U B R O U T I N E \ P O W E R 7 C O M P U T E D l A B L E G E N E R A T I O N i S U B R O U T I N E P O W E R 8 C O M P U T E C H U T E - D E S - P A S S E S / G E N E R A T I O N / S U B R O U T I N E 0 3 M A X . f C O M P U L I S L E MALIGNE DISCHARGE) \ FOR MAXIMUM / \ISLE MALIGNE STATION/ \ r, E N E N A I I O N / / S U B R O U T I N E \ L S P L I T 5 >J C OM PUT E SPLIT OF DISCHARGE V B ET WEEN LITTLE ANO / \ GRAND DISCHARGE/ C O M P U T E I S L E M A L I G N E F O R E B A Y E L E V A T I O N . F O R E B A Y S T O R A G E A N D 1. M . S T A T I O N D I S C H A R G E S U B R O U T I N E P O W E R 3 C O M P U T E I S L E M A L I G N E G E N E R A T I O N C O M P U T E S A G U E N A Y D I S C H A R G E S U B R O U T I N E P O W E R 12 ( C O M P U T E S H I P S H A W \ A N 0 C H U T E - A - C A R O N i \ G E N E R A T I O N / C O M P U T E F I N A L R E S E R V O I R V O L U M E S / S U B R O U T I N E \ f E F F I C \ COMPUTE STATION GENERATING . V CFFICIE NCIE 5 / ( W R I T E O U T P U T F O R D A Y ) I N I T I A T E S T U D Y F O R N E X T D A Y G O O N 01 ro N E X T Y I CdvKd I ran OWING. No. C - Sn - JiOO- SH, A l M M l n U B . CB. dl Conade, Lid. I S U B R O U T I N E G O R G E C O M P U T E G O R G E D R O P FIG. 3. L O G I C DIAGRAM - A L C A N DAILY S I M U L A T O R L A K E ST. JOHN WATERSHED AVERAGE RUNOFF FOR SEVEN DAY PERIODS. Bosed upon long term average Lake St. John watershed runoff. ( 1944? 1968incI.) (Copied from DWG. No. D-SK-2107-SH Aluminium Co. of Canada ). J_ 0 50 100 150 200 250 300 350 1 JAN. I FEB. I MAR. I APRIL I MAY I JUNE I JULY I AUG. I SEPT. I OCT. I NOV. I DEC. I TIME TIG. 4. T Y P I C A L SNOWMELT RUNOFF HYDROGRAPH FOR SAGUENAY DISTRICT. 35 STAGES T I M E F I G . 5 b . D E F I N I N G T H E R U L E C U R V E . DEC. STAGE N B. TIME •DYNAMIC PROGRAMMING -CONVENTION etc. STAGE 2 I STAGE I APRIL ZEROTH "STAGE F I G . 6 . I L L U S T R A T I O N OF D Y N A M I C P R O G R A M M I N G C O M P U T A T I O N A L S C H E M E LO ON 0 I 38 C O M M O N A R R A Y S , V A R I A B L E S R E A L , I N T E G E R , D I M E N S I O N S P E C I F I C A T I O N S A. C O M P U T E V O L U M E C H A N G E P A S S E S - D A H G E R E U S E S C O M P U T E S T A T I O N D I S C H A R G E , 0 8 A V E R A G E H E A 0 . H 8 C A L C U L A T E 0 M 8 , M A X . D I S C H A R G E F O R M A X . P O W E R , F C R H E A D H 8 < T Q 8 < 0 M 8 *. N O ^ C O M P U T E M A X . P O W E R F O R Q M S . S P I L L E X C E S S | Y E S C O M P U T P O W E R F O R Q 8 , H 8 i C O M P U T E D I S C H A R G E F O R D I A 8 L E P O W E R IS M A X . P O W E R . S P I L L E X C E S S C O M P U T E P O W E R F O R 0 7 C O M P U T E D I S C H A R G E 0 6 F O R S A V A N E P O W E R I S M A X . P O W E R . S P I L L E X C E S S U S I N G S T A R T O F P E R I O D E L E V A T I O N , E L E V S F O R L A K E S T . J O H N , • C O M P U T E P O W E R F O R Q 6 C O M P U T E P O W E R F R O M P E R I B O N K A S Y S T E M ; C O M P U T E R E C U I R E D P O W E R F R O M S A G U E N A Y S Y S T E M . G U E S S I N I T I A L L A K E S T . J O H N D I S C H A R G E , S P I L L 5 C O M P U T E P O W E R A T I S L E -M A L I G N E F O R S P I L L 5 , E L E V 5 A C O M P U T E S A G U E N A Y R I V E R D I S C H A R G E , Q S A G R E T U R N TO MAIN' C O M P U T E P O W E R A T S H I P S H A W / C H U T E A C A R O N F O R Q S A G R E C O M P U T E S A V A N E G E N E R A T I O N F O R E L E V 5 A , IF E L E V 5 A > II F T . C O M P U T E S Y S T E M G E N E R A T I O N M W S Y S T R E T U R N TO MAIN S U C C E S S F U L G E N E R A T I O N R E T U R N TO M A I N U N S U C C E S S F U L G E N E R A T I O N A D J U S T S P I L L 5 A C C O R D I N G T O S T A T I O N G E N E R A T I O N E F F I C I E N C I E S O N S A G U E N A Y , ( M W / C F S ) H<A) F I G . 8. L O G I C D I A G R A M F O R S U B R O U T I N E M O D E L 39 COMMON, REAL, INTEGER, DIMENSION SPECIFICATIONS _2_ READ DYNAMIC PROGRAMMING DATA START/ END DA TES, DISCHARGE EFFICIENCY ARRAYS, STARTING RESERVOIR ELEVATIONS MAX. STATION FLOWS, VOLUME ELEVATION ARRAYS, STATE INCREMENT 0Y8, ETC. I REAO NATURAL INFLOWS AND MANOUANE OUTFLOWS FOR ENTIRE STUDY J g . FS (I + I ) = FDUMMY, I.E. NEW LOW VALUE OF OBJECTIVE FUNCTION FOUND FOR STAGE I+l, STATE Y82 INITIALIZE OBJECTIVE FUNCTION • O I STORE NEW ELEVATIONS,STATION GENERATIONS COMPUTE NATURAL INFLOWS FOR THIS PERIOD BY AVERAGING OVER THE NO. OF DAYS IN PERIOD. STAGE « I I — ® SET ELEVATION OF PASSES - OANGER-EUSES.YBZ. FOR END OF PERIOD I STAG E ! I • I ) TO. MIN.ALLOWABLE I SET ELEVATION OF PASSES-DANGER EUSES, Y8I, FOR START OF PERIOD (STAGE * I) TO MIN.ALLOWABLE I INITIALIZE OBJECTIVE FUNCTION, FS ( I • I) FOR STAGE I + l, STATE Y82 TO HIGH VALUE, I. E. F S (I + I)«1.0 E 20 I SET STARTING ELEVATION LAKE ST. JOHN, ELEV5,EQUAL TO THAT FROM STAGE I AND STATE Y8I I SUBROUTINE MODEL /"COMPUTE TCTAL STORAGE VOLUME QSTORE REQUIRED FOR Y82. Y8I ELEV5.CHECK SYSTEM RESTRICTIONS .COMPUTE FINAL ELEVATION LAKE ST. JOHN, ELE V 5 IF GENERATION IS SUCCESSFUL IGEN • I IF GENERATION IS NOT SUCCESSFUL, IGEN = O OUTPUT OATES AND CORRESPONDING RESERVOIR ELEVATIONS AND STATION GENERATIONS NEW VALUE OF OBJECTIVE FUNCTION, FDUMMY = QSTORE + FS(I) FOR STAGE I+l, STATE Y82 F I G . 9 . L O G I C D I A G R A M M A I N D Y N A M I C P R O G R A M M I N G R O U T I N E . t— u. 2 i < • > . UJ < 140 ALCAN RULE. CURVE ( Input) DYPRO RULE CURVE (Output) NOTES= — Natural inflows 90%of long term average . _ All turbines available. — State increment =lft. — DYPRO did not accept Lake St.John elevations less than 4.0ft. 20 40 60 80 TIME ( DAYS) 100 120 •140 ALCAN RULE CURVE (Output) DYPRO RULE CURVE (Output) P O W E R D E M A N D S C H E D U L E D A T E S C H E D U L E D G E N E R A T I O N 271 1 2200 MW 17 12 2200 60 1 22 00 2 601 2200 1 5 02 2200 7 03 2 2 0 0 2 703 19 3 6 I 0 NOV. 27 20 4 0 60 80 TIME (DAYS ) 100 120 140 ARIL 15 FIG. 10. COMPARISON OF DYPRO WITH A L C A N D A I L Y S I M U L A T I O N NOTES - DY8 - 3ft-- Runoff is 3 0 % of longterm overage. • "Breok-up 'assumed April 15-- Passes-Dangereuses full Nov. 27. - AH turbines available at all stations. Estimated Max.Firm Power = 2 350MW for Period. Covering Nov. 27 to March 6 ; 2 I50MW from March 7 to March 26 ; 2 050MW from March 27 to April 15. 2 300MW 0 NOV. 27 60 TIME (DAYS) 100 MARCH 6 140 APRIL 15 FiG.M. METHOD OF OBTAINING MAXIMUM FIRM POWER. 

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