Stochastic modelling and optimization of water resources systems

02704255/82/020117-2oso3.00/0 CoPYrieht @ 1982 Pergamon Press Ltd.

Marhematicnl Modelling. Vol. 3. pp. 117-136. 1982 Printed in the USA. All rights reserved.

STOCHASTIC MODELLING AND OPTIMIZATION WATER RESOURCES SYSTEMS

OF

GOVINDASAMI NAADIMUTHU Department of Industrial Engineering FairleighDickinson University Teaneck, New Jersey 07666, USA

E. STANLEY LEE Department of Industrial Engineering Kansas State University Manhattan, Kansas 66506, USA Communicated

by Richard Bellman

Abstract-This work involves the simultaneous optimization of the initial design and operating policy over the life of multipurpose multireservoir water resources systems receiving stochastic inflows. The approach is based on the division of the reservoir into two imaginary water storage pools, namely, the conservation and flood pools. Based on this treatment, the optimization problem is stated using the concepts of Lagrange multipliers and parameter optimization. Two nonlinear programming techniques, namely, the generalized reduced gradient technique and the gradient projection technique, combined independently with Markovian decision are proposed to solve such a problem. To illustrate the use of the proposed techniques, the Walnut River Basin in southeastern Kansas, is employed in this work.

INTRODUCTION

Much attention and effort have been placed upon the planning, design, and operation of water resources systems, because of their importance to national welfare. The multiple uses of a water resources system can be approximately classified into three major types: withdrawal consumptive use, nonwithdrawal nonconsumptive use, and withholding (or retardation) use. Irrigation and urban water supply belong to the first type, hydroelectric power generation and recreation to the second type, and flood control to the third type. The multipurpose water resources system may be investigated under the assumptions of steady-state operating conditions, dynamic or unsteady state operating conditions, dynamic or unsteady state operating conditions, and stochastic conditions. All three types of problems are complicated for large water resources systems. However, the degree of complications and the nature of difficulties are different. Of the above three types of problems, the stochastic type has so far received only some, but not much, attention. However, in real life, water resources systems are stochastic in nature. The purpose of this work is twofold: one is to build a realistic but computationally amenable stochastic model which includes the benefits or losses of the complementary and/or competing water uses; the other is to develop efficient computational procedures to optimize such a model. Several papers have appeared on the modelling and optimization of water reservoirs. Cocks [71, Loucks [17, 191, Revelle et al. [23], Yeh et al. [32,33], Houck and Cohon [34,35], and Nelson et al. [36] applied linear programming to solve reservoir optimization problems. Kirshen [371, Nieh [38], and Windsor [39] used mixed integer programming 117

118

GOVINDASAMINAADIMUTHUANDE.STANLEYLEE

optimization models to reservoir design. Jonch-Clausen [40], and Koris and Nagy [41], employed quadratic programming for the optimal regulation of reservoirs. Turgeon [42], and Chu and Yeh [43], determined optimal reservoir operations through nonlinear programming. Bather [2], Buras [3,4], Butcher [5], Fukao and Nureki [ll, 121, Hall et al. [13], Loucks [18]; Meier and Beightler [20], Mobasheri et al. [21], Strupczewski [25], Tzvetanov [27], Young [28], Tauxe et al. [44,45], Beard and Chang [46], Jensen et al. [47], Sniedovich [48], Arunkumar [49], Bogardi et al. [50], Klemes [51], and Glanville [52] resorted to dynamic programming to obtain reservoir operating policies. Prekopa et al. [53] applied stochastic programming to design serially linked reservoirs. Eichert and Davis [54], Maidment and Chow [55], Keifer et al. [56], Riley and Scherer [57], Deb [58], Stedinger and Bell-Graf [59], Kaczmarek et al. [60], Hall [61], Bogardi et al. [62], Thompstone et al. [63], Croley II and Rao [64], and Franke and Leipold [65] dealt with systems analysis in the management of water reservoirs. Chow [6], Hufschmidt and Fiering [14], Law [16], Young et al. [29], Major and Lenton [66], White and Christodoulou [67], Eichert [68], Singh [69], Sniedovich [70], and Wright [71] discussed water resources simulation procedures. Sung [72], Olenik [73], Mades and Tauxe [74], Musselman and Talavage [75], Goicoechea et al. 1761, Ambrosino et al. [77], Loucks [78,81], Sakawa[79], Neuman and Krzysztofowicz [80], and Passy [82] considered multiobjective analysis in water resources planning. Fiering [lo], Langbein [151, and Thomas and Watermeyer [26] used queueing theory in the design of water resources systems. Morel-Seytoux [83], and Laufer and Morel-Seytoux [84] employed decision theory for reservoir optimization. Ikada and Yoshikawa [85] presented a game theory approach to water resources planning. Gal [9] formulated a parameters iteration method for optimal reservoir management. Arunkumar and Chou [86], and Akileswaran et al. [87], devised heuristic approaches for the optimal control of reservoirs. All the above publications treated a multipurpose reservoir to be a single lumped pool of water for modelling and optimization purposes. Also, they did not consider the simultaneous treatment of the various water benefits, along with the losses when the demands are not met. This paper finds the best initial design and the optimal operating policy over the life of multipurpose multireservoir systems, without the unrealistic assumption of a single lumped pool of water. The objective is to optimize the flood and various water usage benefits/losses simultaneously along with the capital and operating costs. A multipurpose reservoir generally serves two basically different usages, namely the prevention of flood damages during excessively high inflow periods and the conservation of water for various purposes, or uses during drought periods. Accordingly, it is proposed to consider a multipurpose reservoir to be consisting of two water storage pools, namely, the flood pool and the conservation pool. Thus, the flood pool is for the storage of accumulated water during flooding situations; and the conservation pool is for the storage of water for various purposes such as water supply, water quality control, recreation, etc. Apart from the above two pools, there is another pool known as the sedimentation pool for sedimentation purposes. It may be noted that the sedimentation pool is the bottom pool, the conservation pool is the middle pool and the flood pool is the top pool. The sedimentation pool volume is determined by the soil condition and the probable sedimentation rate. Thus, the sedimentation pool capacity is a given constant and need not be considered for modelling and optimization purposes. Then the best initial design involves finding the optimal conservation and flood pool capacities. In general, the operating policy is designed to meet the following requirements: (i) to maintain a nearly constant conservation pool level to meet the various water demands; (ii) to maintain a nearly zero flood pool level, except during flooding situations, to store future flood waters; and (iii) to maintain a minimum release so that sufficient water is maintained in the channel below the reservoir for the conservation of fish and wildlife.

Stochastic modelling and optimization of water resources systems

119

Then the optimal operating policy is to determine the actual optimal values of the decision and state variables. The major difhculty in water resources modelling and optimization lies in the stochastic nature of the inflows to the reservoirs. Because of this, stochastic modelling and optimization should be used. The stochastic inflows may be “independent” or “serially In this paper “first order” or “lag one” serially correlated infIows are correlated.” considered. This means that the inflow of each month is dependent only on the intJow of the previous month, forming a Markov chain. The stochastic optimization procedures advocated in this work are the two versions of the nonlinear programming approach, namely, the generalized reduced gradient (GRG) technique [ 11 and the gradient projection (GP) technique [24], each separately combined with Markovian decision. The basic idea of combining the GRG/GP technique with Markovian decision is to convert the probabilistic nature of the Markovian decision problem into an equivalent deterministic model and then solving it by the GRG/GP technique. The tool used here for smoothing out the probabilistic nature of the problem is the expected value criterion, which is based primarily on the law of large numbers. In finding the optimum of the problem, the whole life of the reservoir must be the useful life of a reservoir is assumed to be considered. In general, 100 years. Furthermore, in order to consider the flood benefits or losses, the duration of a stage must be in the order of the duration of the high intensity rainfalls. This duration is generally in the order of hours or days. Thus, the number of stages to be optimized is extremely large. This large problem is impractical to solve in terms of computer requirements. Moreover, data collection on a daily or monthly basis would be extremely time-consuming and impractical. Two approximations are used to overcome this difficulty. The first approximation is to use a typical one year duration to represent the 100 years duration. In other words, it is assumed that this one year represents the average of the 100 years. This obviously is only an approximation. The use of stochastic inflows compensates for some of the approximation. The second approximation concerns the inflow or runoff rate. It may be noted that although only one year duration is considered, the number of stages can still be very large if hourly or daily inflows are used. In this work, monthly inflows or runoffs are used. Thus, only twelve stages or months are needed to solve the problem. However, the monthly inflows cannot consider flood benefits or losses. In order to consider the flood effects, certain assumptions are made. The computation for all the benefits and costs, except for the flood benefits, is carried out on a monthly basis. But, the flooding which is generally caused by a relatively short duration high intensity rainfall is treated differently. To devise a feasible approach to handle flood, it should be noted that flood damage depends primarily on the maximum overflow. This flood damage seldom occurs twice or more times in a month. Even if flooding occurs twice within a relatively short time span, the added flood damage for the second occurence is small if the amount of rainfall or overflow in the second flood is not larger than that in the first one. Based on these observations, an approximate scheme is devised to handle the flooding situation by using the monthly runoff data. It is assumed that flooding can occur only once a month. Also, based on rough estimates for the Walnut river basin, it is assumed that this flooding is caused by a high intensity rainfall of two days duration. In other words, the excess water must be released within this short duration to avoid damage. Furthermore, it is assumed that if flooding occurs in a month, all the runoff for that month comes from the high intensity rainfall of the two days duration. No rainfall occurs in the other days of the month. It is obvious that the above assumptions are not completely realistic. However, the

120

GOVINDASAMINAADIMUTHUANDE.STANLEYLEE

degree of accuracy for a given problem can be improved, if more accurate data like daily or hourly rainfall records are available. To illustrate the application of the above concepts and methods, the Walnut River Basin in southeastern Kansas is employed in this work. The data used for the above system are furnished by the Corps of Engineers, Tulsa, Oklahoma District [8] and the Kansas Water Resources Board. An IBM 360/50 computer with a Fortran compiler is used in the computations.

WALNUT Description

RIVER BASIN

of the system

Walnut River is a tributary of the Arkansas River (see Fig. 1). There are four major tributaries of Walnut River, namely, West Branch, Whitewater River, Little Walnut Creek, and Timber Creek. The basin contains an area of 1955 square miles. Three reservoirs were suggested for construction for the purposes of flood control, water supply, water quality control, recreation, and fish and wildlife conservation. The three reservoirs are El Dorado on Walnut River, Towanda on Whitewater River, and Douglas on Little Walnut Creek. It appears that groundwater supply in this area is limited. Droughts and extended periods of low flow have caused serious water shortages in the basin. On the other hand, fifty-six storms with precipitation averaging 3.5 inches or more have occurred during the period January, 1922, through December, 1961. An average of one flood occurs on the main stream and major tributaries each year. Thus, flood control, water supply, and quality are suitable purposes for the reservoirs. The problem is to find the conservation pool and the flood pool capacities of each of the three reservoirs so as to maximize the net benefit over the expected life. The various costs and benefits data are furnished by the Corps of Engineers, Tulsa, Oklahoma District. These data are correlated into benefit and cost equations by the use of polynomial regression [22].

MODELLING Optimization

problem

The problem is to find the expected values of the conservation pool capacity, the flood pool capacity, and the operating policy during the life of each of the reservoirs so as to maximize the various benefits minus costs. Material balance equation Let a water resources system consisting following performance equation:

of a single reservoir

be represented

x,(n) + xf(n) = x,(n - 1) + x,(n - l)+ F(n) - R(n) with the initial condition x,(O) =

xf

Xf(O)= 0,

by the

(1)

Stochastic modelling and optimiition

of water resources systems

121

I

‘. : t

Y 2;

t

122

GOVINDASAMINAADIMUTHUANDE.STANLEYLEE

where n = specific month, 1. . . 12; x,(n) = state variable representing the conservation pool level in 10’ acre-feet at the end of month n ; x,(n) = state variable representing the flood pool level in 10’ acre-feet at the end of month n; R(n) = control or decision variable representing the amount of water release in ld acre-feet during during month n ; F(n) = inflow in lo3 acre-feet into the reservoir due to rainfall and other discharges during month n ; xt = desired conservation pool volume in lo3 acre-feet. Now introducing

the Markov process in Eq. (l), we obtain

x,0’, n) + x,(i, n) = EMn - l)l+ E[xf(n - l)l+ FO', n) - RO’,n), j = 1. . .4; n = 1 . . . 12

(2)

with the initial condition

J%(O)1 = x: E[x@)l = 0, where j = specific inflow’ class, 1 . . .4; E = expectation; xc(j, n) = new state variable representing the conservation pool level in 10’ acre-feet, for intlow class j at the end of month n ; x,(j, n) = the flood pool level in lo3 acre-feet, for the inflow class j at the end of month n; R(j, n) = new control or decision variable representing the amount of water release in ld acre-feet, for inflow class j during the month n ; F(j, n) = lag-one (or) first-order (or) one-step serially correlated random variable representing the mean in 10’ acre-feet of inflow class j into the reservoir due to rainfall and other discharges during month n. In Eq. (2), for any n, E[x,(n - l)] and E[xr(n - l)] are known. The inflow rate into the reservoir is stochastically derived from the past rainfall record. The other three variables in Eq. (2), namely, xc(j, n), xr(j, n), and R(j, n) are unknown. Of these three unknowns, one can be obtained by optimization and another one can be obtained by solving Eq. (2). In order to determine the third one, the conservation and flood pools are considered separately using weighting factors (Lagrange multipliers). Constraints For any given reservoir, because of hydrological reasons, the total reservoir capacity z (constant sedimentation pool volume plus conservation pool volume plus flood pool volume) should have an upper limit. Thus

a,

n) =

&nax,

j = 1 . . .4; n = 1. . . 12.

(3)

Stochastic

modelling and optimization

The lower limits on the conservation

123

of water resources systems

and flood pool volumes are given as

X,(j,n)zX,,i., x&j, n) 2 0.

j = 1...4;n

= l...

12 (4)

Because of spillway and outlet works design and channel capacity, there exists a maximum release rate. In order to conserve water quality and fish and wildlife in the channel, a minimum release rate must also be imposed. Thus, RminC R(j, n) 5 R$,,, where R$,, represents

j = 1 . . .4; n = 1 . . . 12,

(5)

the maximum amount that can be released in two days.

Objective fmcfion It is required to maximize the following expected

value of the objective

function:

$i = E(G(1, E[x,(O)l, E[x,(O)l, F(i, O)}), i = 1 . . .4 12

4

=~I=H[x,(j,n),xr(j,n),F(j,n),Rdi,n)lp(i,j,n), n=lj=! I2

i=1...4

4

=~,~[US~j,n)+u~(i,n)+UR(R~)lp(i,i,n),

i=1...4

where Gil, E[x,(O)], E[xf(0)], F(i, 0)) = value of the objective function for a reservoir starting with month 1 {Jan), initial expected conservation pool level E[x,(O)], initial expected flood pool level E[x,(O)] and initial inflow F(i, 0), i = 1 . . .4; p(i, j, n) = the first-order transition probability of moving from the ith class inflow of the (n - 1)th month, denoted by F(i, n - l), to the jth class inflow of the nth month, denoted by F(j, n); H = value of the objective function for inflow class j and month n; Us(j, n) = WSB for inflow class j and month n; U,(j, n) = WQB for inflow class j and month n; UR(j, n) = RB for inflow class j and month n; p&t) = WSB coefficient for month n; p,(n) = WQB coefficient for month n; RR(n) = RB coefficient for month n ; Js(j) = annual WSB for inflow clase j; J,(j) = annual WQB for inflow class j; JR(j) = annual RI3 for inflow class j; WSB = water supply benefits; WQB = water quality benefits; RB = recreation benefits.

124

GOVINDASAMINAADIMUTHUANDE.STANLEYLEE

Net objective function After introducing the flood benefits and the costs in the objective expected value of the net objective function is given by ci = $i + E(JF) - RPk,NE(Cc) - E(Oc), i = 1 . . .4,

function,

the

(7)

where E(&) = E(flood benefits) = f(max{E[x,(n)]}); n E(Cc) = E(capita1 cost) = f(max{E[x,(n)] n E(Oc) = E(OMR cost) = f(m;x{E[x,(n)]

+ E[xf(n)l}); + E[x,(n)]});

f( ) = function; N

R&N

= capital recovery

factor = (:y$‘_

*;

where k = interest rate; N = expected economic reservoirs).

life of the reservoir

(assumed

to be 100 years

for all the

Modified objective function The expected

value of the modified objective

function is given as

~i=t-A”~,E[x,(n)-x:1’-p”~,E[xf(n)-O]’, 12 4 = &-A "X,z kO',n)xYp(i,j,n) =

i=I...4

12 4 -Cc mZ,sbf(i,

= I

n)-012p(i,j,n),

i = 1.. .4,

(8)

where A = a weighting factor corresponding to the Lagrange multiplier in nonlinear programming; p = a weighting factor (similar to A) corresponding to the Lagrange multiplier in nonlinear programming. The parameters A, p, and xi are used to maintain the optimal conservation pool level xE as close to x: as possible and to maintain the optimal flood pool level at nearly zero except during flooding periods. Thus, these parameters also separate the conservation and flood pools functionally. By using very high values of A and p, the optimal conservation pool volume would always be nearly the same as the desired volume x:’ and the optimal flood pool volume would be nearly zero except during flooding periods. Operating policy Let S(j, n) = E[x,(n - l)] + E[x,(n - l)] + F(j, n) - R(j, n).

(9)

Stochastic modelling and optimization of water resources systems

125

Then the operating policy may be written as (a) If so’, n) Ix: xco’, n) = so’, n) x,0’, n) = 0. (b) If So’, n) > x:’ XJj, n) = x: xro’, n) = so’, n) - x:. OPTIMIZATION This section deals with the data input, preliminary calculations, and discussion for the independent optimization of each reservoir methods. Runof

and the detailed plan by the two proposed

data

The intlow rate into the reservoir depends on the amount of rainfall, drainage area, and hydrology. The inflow rate F(j, n) in the material balance equation is stochastically derived from the past rainfall record. The sample record of monthly and annual flows at the El Dorado reservoir site for the forty-year period October 1921 through September 1961 is shown in Table 1. Inflow classes For any reservoir, the number of inflow classes or intervals is to be so chosen as to cover a wide range of runoff rates and at the same time to make a compromise between (i) too large a computer memory as well as time due to too many classes and (ii) too obscure stochastic properties due to too few inflow classes. Here four inflow classes are used for each reservoir of the Walnut River Basin. Each of the class intervals has the same percentage of the ranked observations of the inflow data for each reservoir. The inflow intervals and their expected values for each reservoir of the Walnut River Basin are depicted in Table 2. Transition

probability

The fist-order transition probability of moving from the ith class inflow of the (n - 1)th month, denoted by F(i, n - l), to the jth class inflow of the nth month, denoted by F(j, n), is p(i, j, n). This probability can be calculated from the actual operating inflow data (Tables 1 and 2) as follows: p (i, j, n ) = Numerator/Denominator, where Numerator

= number of inflows in class j of the nth month corresponding in class i of the (n - 1)th month Denominator = total number of inflows in class i of the (n - 1)th month.

to the inflows

In this case, there are i = j = 4 inflow classes. Also, F(i, n - 1) = F(j, n), i = j = 1 . . .4,

2610

Mean

1980

437 1520 745 326 2000 1890 2480 IO80 582 2420 175 105 567 362 3980 362 276 159 2870 5060 4730 437 1790 2170 327 557 24,600 824 2510 1600 202 192 229 121 50 1990 928 5520 3020

-

-

140

-

113 593 1110 1970 552 2460 1470 18,400 429 562 2390 221 116 768 951 4590 278 293 163 5210 1800 6520 599 1830 5430 359 189 31,000 1310 853 2390 299 192 92 83 34 1370 890 5200 1220

1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 l%l

5050

3640 1150 7010 495 395 5680 5890 4340 695 1040 2000 284 168 510 436 3200 1880 418 274 1550 3080 2160 27,500 15,200 4600 11,400 3600 11,200 746 2430 12,800 954 165 76 85 69 29,700 1520 23,700 9920

Mar

Feb

Jan

Year

17,400 10,800 8310 1920 1000 5840 3310 16,300 11,200 2440 1200 1500 9330 42,700 1140 10,500 34,200 459 8600 3150 6340 35,000 23,300 6390 2110 15,600 336 12,000 1140 44,700 8410 14,000 522 1620 35 35,600 2660 10,200 15,200 49,500 11,900

10,320

-

May

21,100 1490 8630 1980 1720 58,100 5540 14,600 669 2940 1360 230 2520 883 304 5360 2760 915 8950 5400 17,300 1630 80,100 51,500 3030 41,000 848 15,500 891 4680 10,800 3510 251 67 126 5640 7760 3070 8450 11,200

Apr

8200

14,ooo 6240 1450 22,200 15,900 15,400 5360 3300 1090 6650 21,400 5680 6610 42,200 1430 1170 267 2190 593 28,900 3570 1690 5450 6290 10,270

-

4180

670 907 4250 103 864 33,100 2130 3090 204 452 1030 6620 22 700 0 1930 3320 3750 316 6450 3620 719 2470 1520 471 468 3000 1000 46,300 6440 611 245 I 16 66 519 1390 7660 13,000 7890

-

Au9

4340

601 931 521 623 13,300 5620 590 547 3890 1910 346 1330 1510 230 442 2380 974 76 1000 16,500 20,900 383 1220 27,700 586 114 713 3850 5450 20,300 139 56 0 314 2 2580 7770 1210 2660 24,300

-

Se9

4580

I18 234 4070 907 356 32,000 8010 781 433 555 236 215 268 1560 2730 93 10 285 220 54 124 15,600 12,800 549 3970 24,600 671 104 368 2770 1530 4220 II7 I25 II2 3920 0 831 1030 41,400 6170 -

Ott

in acre-feet at El Dorado dam site.

Jul

flows

13,900 7350 1830 201 I80 1930 7710 19,200 312 1000 9100 445 79 1530 31 16,400 1220 492 368 5170 7750 4740 8070 6950 366 1550 30,800 9440 8810 74,100 1000 428 69 75 377 6020 22,600 26,900 3690 25,900

1820 52,400 8030 1220 792 15,100 44,300 11,600 2130 16,400 11,900 148 1720 16,100 I41

Jun

Table I. Estimated monthly and annual

3730

sl:8 3020 438 1050 5440 2310 62,700 712 3380 12,600 I81 102 5010 6220 38 341 374 59 1580 11,900 2660 360 1290 1760 1360 124 II00 770 790 6380 277 244 55 66 6 3200 1450 2740 2320 -

Nov

2400

129 764 1470 459 217 2850 1650 9800 420 1210 3200 373 139 1240 1510 684 431 310 105 1220 4220 16,000 684 28,200 1610 375 437 595 914 820 2970 432 259 53 87 32 I510 1020 5000 2710 -

Dee

69,560

22,260 78,130 63.5 IO 118,700 75,220 211.800 40;010 21,490 1879 8752 1526 84,950 82.3 IO 103.200 94,070 139.200

23,380 74,450 13,840 57,390 59,900 13,150 24,200 100,200 113,200 72,880 182,500 144,200

317 65,490 84,620 43,020 10.880 59,420 141,800 146,100 92,120 25,750 43,360 32,520 11,460

Annual

5

Y 2

z m

2 6 E S S

z :! r 3

E

Stochastic

modelling and optimization

Table 2. Information

Inflow Class 1 2 3 4

0.000-0.432 0.432-1.530 1.530-5.840 5.840-80.100

systems

Douglas

Towanda

Expected Value 10’ (acre-ft)

Class InterTal

Expected Value 10’ (acre-ft)

(acLft) 0.000-0.781 0.781-2.710 2.710-10.500 10.500-137.500

0.216 0.981 3.685 42.970

127

on the inflow classes of the Walnut River Basin.

El Dorado Class Interval 10’ (acre-ft)

of water resources

0.3905 1.7455 6.6050 74.0000

Class Interval 10’ (acre-ft) 0.00CL0.436 0.436-1.630 1.630-5.890 5.890-98.700

Expected Valye (aA!-ft) 0.218 1.033 3.760 52.295

where F(i, n - 1) and F(j, n) represent the means of the respective inflow classes and stages (months). There are twelve transition probability matrices representing the transition from December to January, January to February.. . November to December. A typical transition probability matrix to represent the transition from the (n - 1)th stage to the nth stage can be represented by Table 3. A sample transition probability matrix representing the transition from January to February for a typical reservoir is shown in Table 4. Parameter

optimization

It is to be noted that x: is essentially an unknown parameter before the optimum is obtained. It is true that the desired level x: can be approximately obtained by engineering and hydrological studies. However, the true optimum value for xf cannot be obtained except by optimization calculations. This parameter optimization problem can be solved by two different approaches. The first approach can be called the trial-and-error approach. A series of values for xt are assumed. An optimization problem is solved for each assumed value of xt. That value of xf corresponding to the maximum expected value of the modified objective function is the optimal desired value of x?. The second approach is to consider both the releases and the xi as control variables and solve this optimization problem directly. In this work, the former approach is employed. Three different values are employed for xf of each reservoir. These values are listed in Table 5. The values in set 2 are the proposed design capacities for the conservation pools of the three reservoirs, by the Corps of Engineers, Tulsa, Oklahoma. Global optimum The proposed approaches do not guarantee global optimum if each reservoir has several local optima. It should be pointed out that there exists no optimization technique Table 3.

i

‘,:,,‘)th

F(1, F(2, F(3, F(4,

F(l, n) n n n n

-

1) 1) 1) 1)

F(2, n)

F(3, n)

p(i, i. n,)

F(4, n)

128

GOVINDASAMI

NAADIMUTHU AND E. STANLEY

LEE

Table 4. Sample transition probability matrix for the El Dorado Reservoir. January-February

Table 5. Desired conservation

pool volume, x:‘.

x:‘, lo3 acre-ft

1 2 3

which can guarantee the optima. To overcome this be used. If the results, by optimum is most probably

El Dorado

Towanda

Douglas

65.0 74.9 85.0

34.0 46.5 55.0

67.0 77.3 87.0

optimum of a general nonlinear problem with several local difficulty partially, several different sets of starting points can using these different sets, converge to the same optimum, this the global optimum.

Problem details The optimization problem has 144 constraints for each reservoir; 48 of these constraints are material balance equations which are equality constraints [Eq. (2)] and the remaining 96 are bounds [Eq. (5)]. There are 48 control variables, namely, R(j, n,), j = 1 . . .4; n = 1 . . . 12. Markouian

decision combined with the GP/GRG

technique

Either of these methods solves only one problem at a time. Thus, in both the methods, there is provision for the calculation of only one value of the objective function in a single computer run, corresponding to the specified initial inflow class. Hence the value of the objective function, for each combination of the specified initial inflow class, the xf value and the initial approximation for the control variables, has to be calculated in each separate computer run. Therefore the total number of trials for each reservoir by MDGP (Markovian decision combined with the GP technique), or MDGRG (Markovian decision combined with the GRG technique) is (the number of classes) x (the number of x? values) x (the number of initial approximations) = 4 x 3 x 2 = 24. COMPUTATIONAL

EXPERIENCE

AND DISCUSSION

OF RESULTS

Table 6 shows the sample results for the flooding situation of the El Dorado Reservoir using MDGRG. The same results using MDGP are given in Table 7. In both the above cases, the starting conservation pool level is taken to be the desired conservation pool volume of the reservoir and the starting flood pool level is taken to be zero. It may be noted that for the initial inflow class 1 of the El Dorado Reservoir using MDGP, 53 iterations and 259 functional evaluations are needed with the initial ap proximation for releases being 10,000 acre-feet, while only 8 iterations and 9 functional

129

Stochastic modclling and optimization of water resources systems

Table 6. Results for the flooding situation of El Dorado Reservoir by the Markovian decision approach combined with the GRG technique. Starting conservation pool level = desired conservation pool volume. Starting flood pool level = 0.

xf, set 1 1 2 2 3 3

Initial Approx. for Rti, n) (10’ acre-ft) 10 5 10 5 10 5

Expected Optimal Flood Control Pool Capacity (ld acre-ft)

Expected Optimal Net Objective Function (IO’ Dollars)

No. of Iterations

No. of Functional Evaluations

Computer Time (hours)

22.48 22.72 22.48 22.72 22.48 22.72

292.3 1 293.52 287.20 288.37 280.05 281.17

11 7 11 7 12 7

115 53 122 53 124 53

0.213 0.173 0.222 0.174 0.226 0.175

evaluations are needed with the initial releases being 5000 acre-feet. Similar results are obtained for the same reservoir with the initial inflow classes 2, 3, and 4. Also, the same experience is acquired for all four intlow classes of the Douglas reservoir using MDGP. This appears to be caused by the irregularity of the objective function with the initial releases being 10,000 acre-feet. This irregularity may appear as sharp ridges or very flat regions [30,31]. Actual computational experience tells us that due to this irregularity, the maximum step-size is continuously reduced by the problem until the step-size is so small that no significant improvement in the objective function can be made. Thus, the true optimum is never obtained with high accuracy by this approach, when the initial approximation for releases is 10,000 acre-feet. It is interesting to note that this difliculty is not encountered when this approach is used for the Towanda Reservoir. Also, MDGRG never experienced such a problem for any of the reservoirs. It may be noted from the results that the optimal desired conservation pool volume is the set 1, x! value for the El Dorado Reservoir. Similarly, the optimal desired conservation pool volumes for the Towanda and Douglas reservoirs have been found to be set 3 and set 1 values, respectively. Typical convergence rates for the optimal net objective function, the norm of the projected gradient and the norm of the reduced gradient are shown in Figs. 2 through 5. Although the initial approximations are far removed from the optimal values for illustrative purposes, only a resonable number of iterations are required by both the MDGP and MDGRG methods. The optimal operating variables for a typical case are listed in Table 8. Table 7. Results for the flooding situation of El Dorado Reservoir by the Markovian decision approach combined with the GP technique. Starting conservation pool level = desired conservation pool volume. Starting flood pool level = 0.

d x,, set

Initial Approx. for Rti, a) (10’ acre-ft)

1 1 2 2 3 3

10 5 IO 5 10 5

Expected Optimal Flood Control Pool Capacity (10’ acre-ft)

Expected Optimal Net Objective Function (10’ Dollars)

No. of Iterations

No. of Functional Evaluations

Computer Time (hours)

22.53 22.72 22.52 22.72 22.52 22.72

292.51 293.52 287.38 288.37 280.21 281.17

53 8 53 8 53 8

259 9 259 9 259 9

0.319 0.036 0.322 0.032 0.317 0.031

130

GOVINDASAMI NAADIMUTHU AND E. STANLEY LEE

_.-_I

4 initial - lo

-R(j,n)

1

I

4

2

I

1

I

4 6 ITERATION NUMBER

8

10

Fig. 2. Convergence rate of net optimal benefits for Towanda Reservoir with inflow class 1 and x:’ set 1 by the Markovian decision approach combined with the GRG technique.

---I

9

I

4 R(jpn)initial- 10

-

-- R(j.n)initial= 15 x 103

---

-, L___, I

_-_

I___.,

‘- -

Cl

3

5

7

9

--I ---

1

ITERATION NUMBER Fig. 3. Convergence set 1.

rate of the norm of the reduced gradient for Towanda Reservoir with inflow class 1 and xf

Stochastic

modelling and optimization

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131

- lo4

-R(jen)i,itial -R(j,n)

systems

initial = l5 ' lo3

ITERATION NTJDIBER Fig. 4. Convergence rate of net optimal benefits for Towanda Reservoir with inflow class 1 and xf set 1 by the Markovian decision approach combined with the GP technique.

_-__---

-RR(j,n)

initial

----R(jtn)ir,ftial

/

2

4

-

= lo4 - 15 x lo3

6

8

10

1

I’BERATIONNOMEER Fig. 5. Convergence x: set 1.

rate of the norm of the projected

gradient for Towanda Reservoir with inflow class I and

Dec.

Jan. Feb. March April May June July Aug. Sept. Oct. Nov.

Month, n 2.23 0. 7.83 18.85 22.48 13.35 12.83 5.43 8.02 14.16 2.84 4.25

61.38 59.61 60.98 64.24 64.67 63.36 63.09 61.42 61.26 62.75 61.28 60.43

6.33 5.18 10.20 16.92 17.25 12.59 12.32

:: 13:79 6.79 7.60

Eb,(~)l

EMn)l

E[R(n)l

El Dorado

6.80 6.50 20.00 18.66 18.08 13.66 13.60 9.22 11.60 17.87 7.47 9.27

E[R(n)l 31.19 30.97 34.00 34.00 34.00 33.59 33.45 32.81 33.37 33.44 33.43 33.04

EbSn)l

Towanda

4.91 5.12 50.97 48.60 46.29 29.25 29.48 13.36 22.63 45.74 5.94 13.36

Ebt(n)l 6.94 6.20 12.50 20.00 16.98 12.85 14.88 9.44 11.54 18.13 9.09 8.22

HWn)l

62.40 60.90 63.33 67.00 65.96 65.52 64.72 62.71 62.95 65.58 63.49 62.43

Ebh)l

pool

3.34 3.55 13.10 28.62 24.22 15.63 18.49 8.00 12.00 24.21 6.86 5.76

Ebh)l

conservation

Douglas

Table 8. Optimal operating variables for the flooding situation with inflow class I and x :’ set I. Starting level = desired conservation pool volume. Starting flood pool level = 0.

Stochastic

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133

The MDGP method requires 132, lo3 bytes of memory while MDGRG requires 298, 10’ bytes of memory in an IBM 360/50 computer for each combination of the xg value, the inflow interval and the initial approximation for releases.

CONCLUSION Since a multipurpose reservoir generally serves two basically different usagesnamely, the prevention of flood damages during excessively high inflow periods and the conservation of water for various purposes like water supply, water quality, recreation, etc .-this work was based on the imaginary division of the reservoir into flood and conservation pools. This paper illustrated the use of the two nonlinear programming techniques-namely, the generalized reduced gradient technique and the gradient projection technique, each independently combined with Markovian decision (MDGRG and MDGP)-in the planning, design, and operation of the three reservoirs of the Walnut river basin receiving stochastic inflows. Both MDGRG and MDGP presented no convergence difficulties. The results of both approaches are in good agreement. Although for each method, two different initial approximations were used, both of them converged to the same optimal solution. REFERENCES 1. J. Abadie and J. Carpentier, Generalization of the Wolfe reduced gradient method to the case of nonlinear constraints, R. Fletcher, ed., Optimization, pp. 37-17, Academic Press, NY (1969). 2. J. A. Bather, Optimal regulation policies for finite dams. J. Sot. Ind. Appl. Math. 10, 395 (1%2). 3. N. Buras, Conjunctive use of dams and aquifers. J. Hydraulics Div. ASCE 89, 111 (1%3). 4. N. Buras, A three-dimensional optimization problem in water resources engineering. Operat. Res. Q. 16, 419 (1965). 5. W. S. Butcher, Stochastic dynamic programming for optimum reservoir operation. Water Resources Bull. 7, 115-123 (1971). 6. V. T. Chow, Systems approaches in hydrology and water resources, Progr. Hydrol. 1.490-509 (1%9). 7. K. D. Cocks, Discrete stochastic programming. Manage. Sci. Theory 15, 72-89 (1968). 8. Corps of Engineers, Suruey Report on Walnut Rioer, Kansas, U.S. Army Engineer District, Tulsa, Oklahoma. 9. S. Gal, Optimal management of a multireservoir water supply system. Water Resources Res. 15, 737-749 (1979). 10. M. B. Fiering, Queueing theory and simulation in reservoir design. Trans. ASCE 127, 1114 (1%2). 11. T. Fukao, and R. Nureki, Applications of dynamic programming. Inform. Process. Japan 2.4 (1962). 12. T. Fukao and R. Nureki, Successive approximations methods of dynamic programming in reservoir control problems. Bull. Electrotech. Lab. Tokyo 26, 18 (1%2). 13. W. A. Hall, W. S. Butcher, and A. Esogbue, Optimization of the operation of a multiple-purpose reservoir by dynamic programming. Water Resources Res. 4, 471 (1%8). 14. M. M. Hufschmidt and M. B. Fiering, Simulation Techniques for Design of Water Resources Systems, MacMillan, London (1%7). 15. W. B. Langbein, Queueing theory and water storage. J. Hydraulics Dio. ASCE 84 (1958). 16. F. Law, Integrated use of diverse resources, J. Inst. Water Eng. 19, 413 (1965). 17. D. P. Loucks, Computer models for reservoir regulation. J. Sanitary Eng. Dia. ASCE 94, 657 (1968). 18. D. P. Loucks, Stochastic methods for analysis of river basin systems, Research Project Technical Completion Report, Water Resources and Marine Sciences Center (1969). 19. D. P. Loucks, Some comments on linear decision rules and chance constraints. Water Resources Res. 6, 668-671 (1970). 20. W. L. Meier and C. S. Beightler, An optimization method for branching multistage water resources systems. Water Resources Res. 3, 645 (1%7). 21. F. Mobasheri, A. Klinger. and M. Torabi, A stochastic multi-level optimization technique for design of a multi-purpose water resource project, Calif. Water Resources Center, Tech. Completion Report. 22. G. Naadimuthu, Stochastic water resources modeling and optimization, Ph.D. Thesis, Kansas State University, Manhattan, Kansas, 1974. 23. C. Revelle, E. Joeres, and W. Kirley. The linear decision rule in reservoir management and design. I. Development of the stochastic model. Water Resources Res. 5.767-777 (1%9).

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