Multipurpose reservoir operation

Journal of Hydrology, 69 (1984) 1--14

1

Elsevier Science Publishers B.V., Amsterdam -- Printed in The Netherlands [1] MULTIPURPOSE RESERVOIR OPERATION 1. M o n t h l y Model f or a Single Reservoir

MIGUEL A. MARINO and BEHZAD MOHAMMADI Department of Land, Air and Water Resources and Department of Civil Engineering, University of California, Davis, CA 95616 (U.S.A.)

(Received September 20, 1982; revised and accepted April 11, 1983)

ABSTRACT MariKo, M.A. and Mohammadi,: B., 1984. Multipurpose reservoir operation, 1. Monthly model for a single reservoir. J. Hydrol., 69: 1--14. The Becker and Yeh monthly operation model, which maximizes the energy output of a reservoir system, has been extended to allow for maximization of both water and energy from the system. The model is a combination of parametric linear programming (used for month-by-month optimization) and dynamic programming (used for optimization over the 1-yr. operation period). The resulting operating policy is updated as new monthly inflow forecasts become available, thus incorporating the stochasticity of inflows. An iterative procedure is used to reduce the computation time and computer storage requirements. This efficiency allows the use of minicomputers which are less expensive than mainframe computers. The use of the model is illustrated for the Shasta reservoir (California Central Valley Project). INTRODUC~ON

During the past t w o decades, the systems analysis approach has been used extensively in developing techniques to aid decision making in planning and design of water-related projects. T he const ruct i on o f new c o m p l e x waterresource facilities during this period has required t hat t h e y be operat ed efficiently in th e short-term so t hat the long-term goals of the project (decided at the planning and design stages o f the project) are achieved. As a result, water-resource systems analysts have devoted considerable a t t e n t i o n to th e problems of reservoir m a na ge m ent and operation. The devel opm ent o f high-speed digital c o m p u t e r s and the advancement of mathematical programming techniques have provided the necessary tools for reservoir m a n a g e m e n t and Operation modellers: Although m a n y reservoir operation models have been pr opos e d (e.g., Gilbert and Shane, 1982; Shane and Gilbert, 1982), we will briefly discuss some o f those employing deterministic o p t imizatio n techniques. Fults and H a nc ock (1972) developed a daily o p e r atio n mo d e l for releases f r om t he Shasta and Trinity reservoirs in California, U.S.A. T he model yields dally releases t h a t maximize t he m o n t h l y

energy generated by both reservoirs while satisfying other constraints of the system. The model is formulated as a two-dimensional forward dynamic programming (DP) and, due to dimensional considerations, is limited to four reservoirs. Becker and Yeh (1974) developed a linear programming--dynamic programming (LP--DP)methodology for the operation of a multiple-reservoir system to determine the m o n t h l y releases. The model maximizes the excess energy produced by the system during the year. The solution algorithm becomes computationally expensive when more than one primary purpose is considered. Chu and Yeh (1978) developed an hourly operation model for a single reservoir. The solution algorithm consists of nonlinear duality theorems and Lagrangian procedures. The algorithm requires an initial (starting) solution and its convergence slows after the first few iterations. Turgeon (1981) applied the principle of progressive optimality developed by Howson and Sancho (1975) to determine the optimal hourly releases for a system of reservoirs over a 1-week period. That method also requires an initial (starting) solution that affects the computation efficiency of the operational model. An extension and modification of the LP--DP approach of Becker and Yeh (1974) to reservoir operation is demonstrated in Part 1 of the presented two papers. The extended model treats both power generation and water supply for municipal and industrial (M & I) use as primary purposes of the reservoir. This extension necessitates the modification of the DP model to a two-dimensional DP formulation. Furthermore, a new solution algorithm is developed to improve the efficiency of the model. Those modifications reduce the storage and computation time requirements, thus allowing the use of minicomputers. The algorithm uses parametric LP and an iterative solution procedure. It is based on a procedure presented in Marifio and Mohammadi (1981). The algorithm is constructed so that it can be applied to multireservoir systems as well. The model presented in Part i is illustrated for the Shasta reservoir of the California Central Valley Project. In Part 2 a daily operation model for maximization of m o n t h l y water and energy o u t p u t from a system of two parallel reservoirs (Shasta and Folsom) will be given. It should be noted that the operation of a reservoir consists of two major steps: forecast and control. The model presented herein is developed solely for the control of a reservoir. The forecast of inflows is outside the scope of these two papers.

THEORY Fig. 1 shows the schematic of a single multipurpose reservoir. It is desirable to obtain the optimum m o n t h l y releases from this existing multipurpose reservoir system such that the annual hydroelectric energy and annual municipal and industrial (M & I) water releases are maximized. Further, the m o n t h l y releases must meet the m o n t h l y demands and shall not

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consideration, 51 ; and (2) the a m o u n t of extra M & I water released over the c o n tr act level during the m o n t h under consideration, 62. The solution algorithm requires that 51 and 62 be discretized; therefore, the decision variables are the n u m b e r of increments of excess energy and excess M & I water over their respective c ont r act levels during the current month. Those decision variables are the same as the increments added to the demand levels of energy and M & I water constraints (parametric constraints) in the LP model. The state variables are: (1) the cumulative sum of the increments of excess energy up to the current m o n t h , Ai; and (2) the cumulative sum of the increments of excess M & I water up to the current m ont h, A 2. In o t h e r words, the state variables are the cumulative sum of the decision variables. The objective function of the DP model is represented by a weighted sum of the releases made for power generation and M & I water use. The forward recursive relationship is given by (see also the Notation for symbols used in this paper): ft+l(A],A2)

=

m a x {(R t ' 0 Q q - M t ' J 3 t ) + f t ( A l , & 2 ) } , 5~, 5 2

t =

where

0,1,2,...,11

(1)

Rt and Mt are m o n t h l y releases made for power p r o d u c t i o n and M & I

NOTATION The following symbols are used in this paper AFE AFW C D DR

annual firm energy annual firm M & I water capacity of powerplant monthly releases made for downstream requirements monthly downstream requirements monthly evaporation rate e reservoir inflow I monthly releases made for M & I water use M monthly release made for power generation R monthly releases made for other purposes and/or spill R' Rmax penstock capacity reservoir storage at the beginning of month S Smin minimum required reservoir storage Smax maximum allowable reservoir storage t index representing months a weight reflecting the relative importance of M & I w a t e r releases Wm a weight reflecting the relative importance of power generation mr firm energy distribution c o e f f i c i e n t firm M & I water distribution c o e f f i c i e n t cumulative increments of energy over the contract level A1 A2 cumulative increments of M & I water over the contract level increments of energy over the contract level increments of M & I water over the contract level 82 rate of energy generation per unit of water released w a t e r use, r e s p e c t i v e l y ; c~t a n d fit are r e s p e c t i v e l y d e m a n d d i s t r i b u t i o n c o e f f i c i e n t s f o r e n e r g y a n d w a t e r ; ft (A1, A2) is t h e m a x i m u m w e i g h t e d s u m o f e n e r g y a n d w a t e r releases, a n d f0 ( A I , A 2 ) = 0. F o r t h e last m o n t h , t h e s o l u t i o n t h a t yields m a x i m u m excess e n e r g y a n d excess M & I w a t e r is selected. T h e DP o b j e c t i v e f u n c t i o n implies a b e n e f i t f u n c t i o n w i t h a c o n s t a n t increase (i.e. linear) similar t o t h a t o f Bec.ker a n d Y e h ( 1 9 7 4 ) . H o w e v e r , o t h e r t y p e s o f b e n e f i t f u n c t i o n s c a n be i n c o r p o r a t e d in t h e DP j u s t as easily. A t t h e e n d o f t h e last m o n t h , t h e c o m p l e t e set o f release policies is t r u n c a t e d t o c o n f o r m t o a n a c c e p t a b l e e n d i n g storage. An o p t i m a l p o l i c y c a n t h e n be t r a c e d b a c k t o t h e first m o n t h . T h e o p t i m a l o p e r a t i n g p o l i c y m i n i m i z e s u n a v o i d a b l e spillings f r o m t h e reservoir. W a t e r n o t spilled c a n be u s e d f o r b o t h excess p o w e r g e n e r a t i o n a n d M & I w a t e r releases. F u r t h e r savings in c o m p u t a t i o n t i m e a n d storage r e q u i r e m e n t c a n be o b t a i n e d b y using an iterative s o l u t i o n p r o c e d u r e . As m e n t i o n e d earlier, t h e size o f t h e i n c r e m e n t s p l a y s an i m p o r t a n t role in t h e c o m p u t a t i o n t i m e r e q u i r e m e n t s . T h e iterative s o l u t i o n p r o c e d u r e solves t h e p r o b l e m b y using large i n c r e m e n t s a t first a n d r e d u c i n g t h e size o f i n c r e m e n t s f o r s u b s e q u e n t i t e r a t i o n s until t h e r e q u i r e d p r e c i s i o n ( i n c r e m e n t sizes) is r e a c h e d . T h e p r o c e d u r e c a n be s u m m a r i z e d as follows: Step 1: C h o o s e a large i n c r e m e n t size f o r e a c h reservoir p u r p o s e (as

OPEN A LOOP S E - \ .{QUENCING THE NUMBER} OF ITERATIONS /

COMPUTE INCREMENT SIZES FOR BOTH M&I WATER AND ENERGY FOR THE CURRENT ITERATION (THE INCREMENT SIZES DECREASE WITH NUMBER OF ITERATIONS)

ACCORDING TO THE LP-DP PROCEDURE

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I CALL TRACER THIS SUBROUTINE TRACES BACK THE OPTIMA~T SOLUTION FROM THE LAST MONTH TO THE FIRST MONTH SUCH THAT THE ~NNUAL EXCESS WATER AND ENERGY ARE MAXIMIZED

FOR THE NEXT ITERATION, SET THE MONTHLY 1 ENERGY AND M&I WATER EQUAL TO THE OLD| CONTRACT LEVELS PLUS THE EXTRA INCRE- | MENTS OBTAINED IN SUBROUTINE TRACER |

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Fig. 3. Descriptive flow chart of the iterative procedure.

compared to the required increment sizes) and solve the problem using the LP--DP approach. Step 2: The optimal solution obtained in step 1 is used as the contract level for the next iteration. The new contract level equals the old contract level plus the extra increments obtained from the optimal solution found in step 1. Step 3: Solve the problem using the new contract level (found in step 2) and smaller increment sizes. Repeat step 2 until the required accuracy (i.e. increment sizes) is reached. Fig. 3 is a descriptive flow-chart of the iterative procedure. Fig. 4 is a graphical illustration of a hypothetical problem with one primary purpose. Fig. 4a shows the possible increments over the contract level for the primary purpose. Fig. 4b gives the cumulative increments over the contract level for the two different increment sizes. Paths A and C represent the maximum possible increments over the contract level when the increment sizes are 6 (1) and ~i(2), respectively. Path B is the optimal path when the increment size is 5 (1). It also represents the contract level for the next iteration when the increment size is ~i(2).

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EXAMPLE The m o n t h l y operation model discussed in the previous section is applied to the Shasta reservoir o f the California Central Valley Project. The reservoir has a m a x i m u m capacity of 4600 kiloacre-feet (kAF) or 5670" 106 m 3 (Mm 3 ) and a m i n i m u m required storage of 600 kAF or 740 Mm 3 , set for power production and recreation. Data from Hall et al. (1969) were used to develop mathematical models for relations between area and storage, energy rate and storage, and power plant capacity and storage using the leastsquares-fit method. The surface area (required for the computation of evaporation losses from the reservoir) was f o u n d to be: AREAt -- -- 3.30" 10-TSt2 + 4 . 2 5 " 1 0 - 3 S t + 2.79

(2)

where AREA is the surface area of the reservoir in ha; and S is the reservoir storage in Mm a. The coefficient of correlation was 0.9987. The power plant capacity in megawatts (MW) is approximated by three functions as follows:

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(3)

8 TABLE t Seasonal data for the Shasta reservoir (Hall et al., 1969) Month

M & I water distribution coefficient

Evaporation .1 rate (ft. month- l)

Flood .2 control (kAF)

Mandatory .2 releases (kAF)

Firm energy distribution coefficient

Oct. Nov. Dec. Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep.

0.06 0.06 0.05 0.05 0.06 0.08 0.07 0.09 0.12 0.14 0.13 0.09

0.170 0.070 0.022 0.040 0.047 0.090 0.215 0.340 0.484 0.715 0.635 0.450

3,993 3,993 3,993 3,993 3,993 4,193 4,493 4,493 4,493 4,493 4,493 4,493

240 232 160 160 144 141 137 144 144 144 144 144

0.072 0.056 0.040 0.048 0.056 0.056 0.056 0.056 0.080 0.160 0.180 0140

Total

1.00

3.278

1,934

10 0 0

• l 1 ft. month -1 ----0.305 m month -1; ,2 l k A F --- 1.233 • 106 m 3. w h e r e C is t h e c a p a c i t y o f t h e p o w e r p l a n t i n MW; a n d S is i n M m 3. V a l u e s o f s t o r a g e b e l o w 6 1 6 M m a are n o t c o n s i d e r e d s i n c e t h e m i n i m u m r e q u i r e d s t o r a g e is 7 4 0 M m a . T h e c o r r e l a t i o n c o e f f i c i e n t s f o r t h e last t w o s e g m e n t s were 0.995 and 0.9985, respectively. The rate of energy production per unit o f w a t e r r e l e a s e , ~t, i n m e g a w a t t - h o u r s ( M W - h r . ) p e r M m 3, is a p p r o x i m a t e d by the equation:

}t = 20.68(St -- 200) 0.3433

(4)

The coefficient of correlation was 0.9675. The evaporation losses from the reservoir, et, are computed by multiplying the surface area of the reservoir (eq. 2) and the evaporation rate (from Table I). The penstock capacity in Mm3 per month is given by: Rmax t = 24(DAYSt)(Ct)/}t

(5)

in which DAYS is the number of days of the month considered. Table I gives seasonal data for the Shasta reservoir. The reservoir is assumed to have a firm energy contract of 8" 10 s MW-hr.-yr_I and a firm water contract of 3700 Mm3-yr.-I . Development of these contract levels is based on the lowest recorded inflow, thus reducing the possibility of shortages. The beginning- and end-of-year reservoir storages are assumed to be 3700 Mm3. The model is run for the water-year 1970. When forecasting models are used to predict the flows, the monthly flows for the next 12 months should be known in advance. Since inflows cannot be forecast with certainty for the next 12 months, the results of the operating model should

be updated as new information on inflow forecast becomes available. This would require t h a t the operating policy be adjusted several times during the year to account for errors involved in the forecasts. Therefore, the stochastic nature of inflow is considered via update of the operating policy.

MODEL FORMULATION The LP model for m o n t h - b y - m o n t h optimization of releases can now be formulated by minimizing the sum of releases from the reservoir: MinR t +R' t

(t=1,...,12)

(6)

subject to the following constraints: Firm power:

~tRt >/ AFE'c~ t + 51

(7)

Firm M & I water: Mt >~ AFW'~t-}-52

(8)

Downstream requirements: Dt >/ DRt

(9)

Definition of Mr and Dt (continuity at point X in Fig. 1): D t + M t = R t +R't

(10)

Continuity equation: S t - I + I t - - ( R t + R ' t ) - - e t = St

(11)

Flood control and recreation~power-production considerations: St >/ Smint

and

St ~ Smaxt

(12), (13)

Penstock capacity: Rt ~ Rmaxt

(14)

where R t = m o n t h l y releases made through the power plant (Mm 3 month-1 ); R 't = m o n t h l y releases made for other purposes and/or spill (Mm 3 m o n t h - 1); ~t = rate of power production per unit of water released through the power plant in m o n t h t (kW-hr.-Mm-3); AFE = annual firm power (contract level) (kW-hr.-yr- 1 ); s t = firm power distribution coefficient; 51 = increments of power (over the contract level) (kW-hr.); M t = m o n t h l y releases made for M & I water demand ( M m 3 m o n t h - l ) ; AFW = annual firm water (contract level) (Mm3-yr_ 1 ); fit = firm M & I water distribution coefficient; 5 2 = increments of M & I water (over the contract level)(Mm3); D t = m o n t h l y releases made for downstream requirements ~Mm3 m o n t h - 1); Dirt = m o n t h l y downstream requirements (Mm 3 m o n t h - 1); St = storage at the end of m o n t h

T A B L E II

3,000 240 240 180 180 420 142 58 420 331 0 85 0

Dec.

2,907 3 , 0 2 4 232 160 232 160 180 150 180 150 412 310 138 105 45 32 412 310 531 745 0 0 93 73 0 0

Nov. 3,459 546 160 250 150 796 284 38 796 1,131 0 246 100

Jan.

Units are in k A F and 1,000 MW-hr. ~ 1 k A F = 1,233" 106 m3 i.

Beginning storage D o w n s t r e a m releases Downstream requirements M & I w a t e r releases M & I water demand E n e r g y releases Energy produced Energy demanded Totalreleases Inflow Spill Excess energy Excess M & I w a t e r

Oct.

Month

M o n t h l y o p e r a t i n g policy b a s e d o n t h e given c o n t r a c t levels

3,993 838 144 300 180 748 282 45 1,138 1,139 389 237 120

Feb. 3,993 141 141 363 240 504 190 45 504 707 0 145 123

Mar. 4,193 137 137 210 210 347 133 45 347 649 0 88 0

Apr. 4,489 144 144 431 270 575 225 45 575 589 0 181 161

May

Jul.

4,493 4,368 144 144 144 144 360 420 360 420 504 564 198 219 64 128 504 564 394 264 0 0 134 91 0 0

Jun.

Sep. 4,051 3 , 7 3 0 144 144 144 144 390 270 390 270 534 414 202 152 144 112 534 414 231 226 0 0 58 40 0 0

Aug.

3,530 3,014 1,934 3,504 3,000 6,128 2,271 800 6,518 7,141 389 1,471 504

Year

3,000 431 240 300 180 731 248 58 731 331 0 190 120

2,596 431 232 300 180 731 235 45 731 531 0 190 120

Nov. 2,394 162 160 250 150 412 128 32 412 745 0 96 100

Dec.

Feb.

2,727 3,358 449 203 160 144 250 300 150 180 699 503 229 178 38 45 699 503 1,331 1,139 0 0 190 133 100 120

Jan.

Units are in kAF and 1,000 MW-hr. (1 kAF = 1.233" 106m3).

Beginning storage Downstream releases Downstream requirements M & I water releases M & I water demand Energy releases Energy produced Energy demanded Total releases Inflow Spill Excess energy Excess M & I water

Oct.

Month

Optimum monthly operating policy

TABLE HI

3,993 141 141 390 240 531 200 45 531 707 0 155 150

Mar. 4,167 137 137 210 210 347 133 45 347 649 0 88 0

Apr. 4,463 144 144 437 270 581 227 45 581 589 0 183 167

May 4,461 144 144 527 360 671 263 64 671 394 0 199 167

Jun. 4,169 144 144 587 420 731 279 128 731 268 0 151 167

Jul.

3,686 144 144 557 390 701 256 144 701 231 0 112 167

Aug.

3,199 144 144 270 270 414 144 112 414 226 0 32 0

Sep. 3,000 2,674 1,934 4,376 3,000 7,050 2,519 800 7,050 7,141 0 1,719 1,376

Year

12 t (Mm 3 ); I t = m o n t h l y deterministic inflows (Mm 3 month-1 ); e t :: evaporation from the reservoir during m o n t h t (Mm 3 m o n t h -~ ); Smint = minimum storage of reservoir for m o n t h t (Mm 3); Smaxt = m a x i m u m storage of reservoir for m o n t h t (Mm 3); and R m a x t = penstock capacity for m o n t h t (Mm 3 m o n t h -1 ). Table II shows the operating policy based on the given contract levels for power and M & I water (this will be referred to as policy A). This solution is obtained by a simple m o n t h - b y - m o n t h optimization without, any addition to the c o n t r a c t levels (i.e. power and water constraints are treated as nonparametric constraints). Table III shows the resulting optimal operating policy for the Shasta reservoir. The policy gives an annual energy surplus of 17.2" l 0 s MW-hr. and an annual M & I water surplus of 1376 kAF (1696 Mm 3 ) over contract levels. The excess water and energy during the summer months, when the demands for both water and energy are higher, are significant amounts. Several observations can be made when the results in Tables II and III are compared: (1) The annual energy p r o d u c e d from the optimal operating policy is 2 . 4 7 " 1 0 s MW-hr. more than policy A; (2) The o p t i m u m release policy yields 872 kAF ( 1 0 7 5 M m 3) of water more than policy A; (3) The distribution of water and energy p r o d u c e d by the o p t i m u m policy provides more water and energy during high-demand months; and (4) The optimal policy results in no water spilled from the reservoir while policy A spills 3 8 9 k A F ( 4 8 0 M m 3) of water during February because of high reservoir storage c o n t e n t during the previous m o n t h . Policy A results in 530 kAF (653 Mm 3) more water stored in the reservoir at the end of September than the optimal policy. This 530 kAF (653 Mm 3) o f water and the 389 kAF (480 Mm 3) of water which was spilled during the year under policy A are used by the optimal policy to yield more M & I water and power, If the simple LP model were to be used for the next year, then at the end of next year, again, excess water would remain in the reservoir which could have yielded extra benefits during the year. Also, most of the 530 kAF (653 Mm 3) of excess water stored in the reservoir under policy A will likely be spilled during the next water year. This is because during the next 5 or 6 m ont hs of operation the inflows will increase while the demands will decrease. Therefore, m os t of the stored 530 kAF (653 Mm 3 ) of water under policy A will eventually be wasted. The main advantage of the optimal policy is that it uses the extra water stored in the reservoir during the year to prevent any waste of t hat water.

SUMMARY AND CONCLUSIONS The m o n t h l y operation model of Becker and Yeh (1974} which maximized energy o u t p u t o f the system has been e x t e n d e d to allow for maximization of b o t h water and energy f r om the system. The model is a combination of LP

13 (used for period-by-period optimization) and DP (used for optimization over the entire operation period). At every stage of the DP, a series of LP's are solved. Since the contract levels of water and energy are usually based on conservative estimates of natural inflows, the system is likely to be capable of providing more than these contract levels. To allow for the extra water and energy production, the values of the right-hand side of the contract constraints (in the LP model) are parametrically increased from the contract levels to the maximum possible levels in each period. To select the " b e s t " beginning-of-period reservoir storage (more than one beginning storage will be available because of the parametric LP solution in previous period), a forward DP is used such that the water and energy produced during the operation period (month or year) are maximized. The efficiency of the algorithm is improved through the use of parametric LP (reduces c o m p u t a t i o n time) and an iterative solution procedure (reduces c o m p u t a t i o n time and core-storage requirements). These efficiency measures allow the use of minicomputers, which are more suitable for frequent updating purposes (because of their lower cost). The computations for the model illustrated in this paper were made using a PDP ®-11/23. The computation time was 11 min. and 22 s. The program was written in FORTRAN IV and required 18 K words of core-storage memory. Also, a disk drive unit was used to store the intermediate results on scratch disk files. The reservoir operation model can easily be adapted for use in multireservoir systems. The iterative solution procedure allows for consideration of more than t w o primary purposes with relatively low computation time. Computation time increases linearly as reservoirs are added and increases geometrically as more primary purposes (objectives) are considered.

ACKNOWLEDGMENT This research was supported in part by CSRS Project CA-D*-LAW-4116-H. Additional funding for the research leading to this report was supported by the Office of Water Research and Technology, U.S.D.I., under the Annual Cooperative Program of Public Law 95-467, and by the University of California Water Resources Center, as part of Office of Water Resources and Technology Project No. A-088-CAL and Water Resources Center Project UCAL-WRC-W-617. Contents of this publication do n o t necessarily reflect the views and policies of the Office of Water Research and Technology, U.S. Department of the Interior, nor does mention of trade names or commercial products constitute their endorsement or recommendation for use by the U.S. Government.

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