Optimal design of multi-reservoir systems for water supply
Hirad Mousavi
a,*, A.S. Ramamurthy
b,1a
Resource Planning Division, Seattle Public Utilities, Dexter Horton Building, 710 Second Avenue, 11th Floor, Seattle, WA 98104, USA b
Department of Building, Civil and Environmental Engineering, Concordia University, Montreal, Que., Canada H3G 1M8
Received 15 February 1999
Abstract
Several successful applications of optimal control theory based on the PontryaginÕs minimum principle have been recorded in literature for optimizing the operating policy of multi-reservoir systems. However, the application of optimal control theory in sizing multi-reservoir systems resulted in sub-optimal solution. In this study, an optimization model based on a new composite algorithm is introduced. This model applies optimal control theory and penalty successive linear programming as two promising techniques to optimize large and complex water supply systems. The epsilon constraint approach was implemented in the model in order to consider the two non-commensurate objectives of minimizing cost and water de®cit. The application of this model to a multi-reservoir system was compared to an existing dynamic programming model. The result of this study showed that the developed model is a very promising optimization method to design multi-reservoir systems regardless of their sizes. Ó 2000 Elsevier Science
Ltd. All rights reserved.
Keywords:Reservoirs; Water supply; Storage; Optimization; Design; Cost; Modeling; Multi-objective programming
1. Introduction
Water supply is probably the most important reser-voir use worldwide [18]. To satisfy the growing demand for water, reservoir storage is required to control the uneven temporal and spatial distribution of water and provide enough water to consistently meet the demand at speci®ed locations. However, due to ®nancial and environmental reasons, only a limited number of reser-voirs can be constructed in a river basin. Therefore, an optimal policy is needed to design a multi-reservoir system to accomplish the problem objectives (e.g., sup-plying water demand) at the minimum cost. A successful model should be able to take advantage of system fea-tures that lead to simpler mathematical formulations and of the proper choice of solution algorithms that overcome dimensionality problems [12].
Optimization models use mathematical programming techniques to ®nd the best possible solution based on a speci®ed performance function and some physical
con-straints. Mathematical programming includes several techniques such as dynamic programming (DP), non-linear programming (NLP), non-linear programming (LP), genetic algorithms (GAs), and optimal control theory (OCT). Among the family of mathematical program-ming methods, OCT still is generally not popular in water resources system analyses. Hiew [7] performed a comparative performance evaluation on several optimi-zation procedures. The criteria to compare were: accu-racy of results, rate of convergence, smoothness of resulting storage and release trajectories, computer time and memory requirements, robustness of the algorithm, and sensitivity to the initial solution. The result showed that optimal control theory and successive linear pro-gramming are the most promising techniques for opti-mizing nonlinear, non-convex objective functions of large hydrosystems. The same study showed that OCT is insensitive to initial solutions. This feature is very helpful in optimizing large optimization problems. OCT has been used successfully by some researchers in optimizing reservoir operations [1,15,16,21,22]. Mousavi [14] has shown that the OCT results in sub-optimal solution while designing the con®guration of multi-reservoir systems.
In the present study, an optimization model based on a new composite algorithm is introduced to size reser-voirs at minimum cost to meet a projected system ®rm
*
Corresponding author. Tel.: +1-206-615-0826; fax: +1-206-386-9147.
E-mail addresses: [email protected] (H. Mousavi), [email protected] (A.S. Ramamurthy).
1
Tel.: +514-848-7807; fax: +514-848-2809.
yield for water supply. A brief review of existing models is presented ®rst. This is followed by a presentation of the model formulation. The performance of the model will be evaluated by comparing its performance with an existing dynamic programming model.
2. Literature review
The subject of optimization in water resources sys-tems in the literature is mainly focused on reservoir operation. However, relatively few attempts have been made to design the optimal con®guration of reservoirs in a multi-reservoir system. Loucks et al. [11] presented the yield, chance-constrained, and the stochastic LP models to design a multi-reservoir system for water supply purpose. Discrete DP-based models were devel-oped by Fontane [6], Mays and Bedient [13], Bennet and Mays [3], Supangat [19], and Taur et al. [20] to deter-mine the best water storage strategies. Lall and Miller [10] and Khaliquzzaman and Chander [8] employed LP-based screening models that integrate simulation models to examine the system in detail.
A survey of these design models shows that each has some limitations in practical situations. These limitations are due to (1) the simplistic assumptions made in the models, (2) diculties in selecting initial solutions, or (3) the problem of dimensionality. It is felt that the devel-opment of a design model that is computationally ecient is still called for. Based on the studies performed by Hiew [7] and Zhang et al. [23], a design model based on a new composite method is introduced here. The developed model is called PSLP±OCT model, which integrates PSLP and OCT as the two promising optimization tech-niques. The proposed model presents a two-level model to determine optimum reservoir pattern and sizing that minimizes the water de®cit in the system at minimum cost. OCT is a method that applies the minimum principle of Pontryagin to optimize a system over time and space [4]. In this sense, it is like dynamic programming. However, it has many similarities with nonlinear pro-gramming techniques in terms of computational proce-dures. PSLP is the last generation of the successive linear programming which solves the NLP through successive use of LP. Baker and Lasdon [2] developed a simpli®ed version of PSLP that compares well with the conven-tional successive linear programming methods and copes better with nonlinear constraints. Zhang et al. [23] improved the PSLP algorithm and gave a convergence proof for it.
3. PSLP±OCT composite algorithm
A new methodology is introduced in this section which uses a composite optimization strategy. This
composite algorithm constitutes the PSLP±OCT tech-nique, which employs OCT and PSLP algorithms as promising optimization techniques. The PSLP±OCT is a general-purpose optimization technique that can be used inmixed typeoptimization problems consisting of static and dynamic (time dependent) control variables. In the conventional approach to optimize such problems, only static optimization techniques (e.g., NLP or PSLP) have to be used. The major drawback in these methods is their initial solution requirement. In the non-convex problems, this requirement could greatly aect the ®nal solution proposed by these methods. Therefore, the re-lated computer programs should be run several times to achieve the best possible local optimum solution.
The introduced PSLP±OCT method however, recog-nizes the dynamism of the problem and dierentiates the dynamic variables from the static ones. Therefore, based on the nature of the decision variables, the mixed type problem can be divided into two parts. The OCT opti-mizes the dynamic part and the PSLP optiopti-mizes the static part of the problem. This approach not only re-duces the computer execution time, but also alleviates and often eliminates the necessity for those programs to be run several times. That feature is due to the insensi-tivity of the OCT method to initial solutions and could be extremely helpful in optimizing large systems with non-convex, mixed type optimization problems. The objectives with nonlinear dynamic variables can be as-signed to the OCT part of the composite algorithm and the PSLP part optimizes the objectives with linear and nonlinear variables. The application of this new tech-nique in designing the multi-reservoir system is de-scribed in the following sections. That is, the problem is divided into two parts and each part is assigned to the corresponding component in the PSLP±OCT method. The proposed procedure can be used as a guidance to apply the PSLP±OCT method to other mixed type optimization problems.
4. PSLP±OCT design model formulation
the optimization problem to the OCT, which is a more promising technique in optimizing large-scale optimi-zation problems [7].
The structure of the PSLP±OCT model is generically illustrated in Fig. 1. Based on this ¯ow chart, the PSLP module at each iteration speci®es a set of reservoir ca-pacities. Then, the OCT module minimizes the water de®cit for such a system and transfers the optimized yield variables back to the PSLP module. Based on these results, the PSLP module selects the next move to pro-pose a new reservoir con®guration. The process in the PSLP module terminates when optimal reservoir ca-pacities are found or when a series of successive itera-tions fail to improve the solution. The essence of the composite model is that the reservoir yield and release over the operation period are dependent directly on the reservoir capacity and stream¯ow sequence. This de-pendence may be functionalized and evaluated inde-pendently by using OCT with respect to target reservoir capacities in the multi-reservoir system.
4.1. PSLP module formulation
The optimization problem considered in the PSLP module is given by the following equations:
minimize X
Nr
n1
fnc ;
X
Nd
j1
XT
t1
Dtj;max
(
ÿX
Nr
n1
ynt;j
!)
1
subject to reservoir capacity constraints and
Dtj;maxÿ
XNr
n1
ynt;jP0; 2
where Dj;maxt is the maximum predicted water demand
during monthtat areajandP
yn;jt is the summation of
the yields provided by candidate reservoirs to satisfy
Dj;maxt. T is the terminal period, fnc is the total cost of
reservoir n, which is assumed to be a continuous
func-tion of the reservoir capacity (xn)
fc
n AnxnBnx2n 3
PSLP module applies epsilon constraint algorithm as the most suitable generating method and a superior tech-nique for problems with dierent dynamic characteris-tics of objective functions [9]. That is, it keeps the cost minimization in the objective function and transfers the second objective to the equivalent constraint
minimize X
Nr
n1
fc
n 4
subject to
06xminn 6xn6xmaxn forn1;2;. . .;Nr; 5
06Dtj;maxÿ
XNr
n1
ynt;j6e t
j forj1;2;. . .;Nd;
fort1;2; :. . .;T; 6
wherexminn andx
max
n are the lower and upper bounds on
reservoir capacities andet
jis the monthly water de®cit at
theD#j. The upper bounds in Eq. (6) are determined by
epsilon constraint algorithm. Reservoir capacities
X x1;x2;. . .;xNr are decision variables in the PSLP
module. This module has a nonlinear objective function and a set of water supply constraints in terms of the decision variables of the PSLP module. The reservoir yield (yt
n;j) in Eq. (6) is a nonlinear implicit function of
the reservoir capacity xn and can be obtained through
4.2. OCT module formulation
OCT is used as the inner optimization module and considers other objectives in the design problem, the mass balance equation, and system constraints. The primary objective of the OCT module formulated is to supply water for dierent demand areas, using the pre-speci®ed set of optimal reservoir capacities by PSLP module. Minimizing the rapid variations (bang±bang) on reservoir yields and releases, and minimizing the storage dierences at the beginning and end of the op-timization period (terminal function) are among the secondary objectives that have been included in the problem formulation. The proposed formulation is dis-cretized over time
n are the yield and release (L
3) from
reservoirnto the demand areajduring the montht. Nd
is the total number of municipal, industrial, and
irriga-tion demand areas, Nr the number of candidate
reser-voirs in the system,Dt
jthe water requirement (L
3) at the
demand area jduring the montht,Wjy the yield weight
coecient, andWby
n andW
br
n are the bang±bang weight
coecients applied to yield and release variables, re-spectively. TheWfs
n is a weight coecient applied to the
terminal storage function, st
n is the storage (L
3) of the
reservoirnat the beginning of month t.
Reservoir yield and release are control variables and reservoir storage is the state variable in the formulation presented. Eq. (7) explains the desired criteria related to the reservoir yields and releases as the control variables. It consists of four terms. The ®rst term in Eq. (7) tries to minimize the water supply shortage for each demand area in the multi-reservoir system. The square term is used to minimize the dierence between water supply and demand in either way. The second and third terms are intended to avoid rapid variations on control vari-ables (yt
j and rtn [4]. Controlling the rapid variation
(bang±bang) of reservoir yields and releases over the time, makes the reservoir gate operation smoother and easier. The fourth term controls the terminal state of the system and is intended to provide storage volumes in the reservoirs, which are needed for the next operation pe-riod. Based on the order of magnitude and importance of each criterion in Eq. (7) to the designer, dierent
values can be assigned to the weight coecients (Wjy,
Wby
n , and W
br
n ). Quadratic forms were selected for the
last three terms to increase the convergence and speed of the search direction and line minimization algorithms.
4.2.1. Constraint equations
The related system constraints applied to the objec-tives described in the previous section are
(1) Continuity equation:
Eq. (8) is the discrete form of the continuity equation for reservoirnoverTmonths, whereSt
nis the storage (L
3) of
the reservoir n at the beginning of timetandQt
n is the
volume of the unregulated local in¯ow (L3) into
reser-voirnduring timet. Evaporation from the reservoir has
been assumed to be the only source of system loss.Kknis the element of the layout con®guration matrix of the
multi-reservoir system with Nr rows and columns
(KNrNr). Each row of this matrix shows the reservoir
number and each column shows the release number. The state of any element at thenth row and thekth column is set equal to 1, if the reservoirn receives thekth release. If reservoirndelivers thekth release, the related element
is set equal to )1. For the unconnected reservoirs, the
element is equal to 0.CJ
nis the element of the return ¯ow
matrix withNrrows andNdcolumns (CNrNd) that shows
demand areas with return ¯ow in¯uent to reservoir n.
Each row of this matrix shows the reservoir number and each column shows the upstream demand area that its
return ¯ow discharges into the reservoir n. The state of
any element at the nth row and the jth column is set
equal to 1, if the return ¯ow from demand area j is
¯owing into reservoir n and 0 for other cases. qm
J is a
coecient for return ¯ow from the demand area J
during timem.et
nis the evaporation per unit area (L
3/L2)
from reservoir n during montht. Cenand P
e
n are the
co-ecient and exponent in the surface area±storage
rela-tionships of the reservoirn. The third term (P
Kk nrtk)
speci®es both the release(s) from reservoirn and all the
in¯ows resulting from upstream reservoir releases to
reservoir n. The fourth term (P
yt
n;j) shows the total
yield supplied by the reservoir n to all demand areas
during montht.Ndis the number of demand areas in the
hydrosystem. The ®fth term in Eq. (8) determines the
summation of all return ¯ows that the reservoir n
re-ceives during the monthtfrom in¯uent areas. In the last
average reservoir capacities through the reservoir's area±storage relationship.
(2) Constraints on release based on the downstream minimum ¯ow needs and ¯ood control requirements
06rt
where njn is the element of the demand area matrix
(nNrNd) which is used to specify the demand areas
located in the downstream of a release. An element ofn
at columnjand rownis equal to 1, if the demand areaj
is located in the downstream of the releasen. The
ele-ments offare equal to 0 otherwise.fknis the element of
the upstream reservoir matrix withNrrows and columns
(fNrNr) and is used to specify the reservoirs located in
the upstream of a release. An element offat column k
and rown is equal to 1, if the related reservoir kis lo-cated in the upstream of the releasen. The elements off are equal to 0 otherwise.
The constraint in Eq. (9) controls the minimum and the maximum ¯ow downstream of each reservoir. The minimum ¯ow constraint (rminn ) can be other than zero to
consider instream recreation, navigation, and water
quality control. The maximum ¯ow constraintrmax
n will
prevent the system from being ¯ooded due to excess reservoir release and the upstream reservoir yields at-tributed to downstream demand areas.
(3) Reservoir yield constraints for water supply:
aDtj;min6
Eq. (10a) is to guarantee that the total water supplied to any demand areajis within a desirable range.Dtj;maxand
Dtj;min are the maximum and minimum water
require-ment of the demand areajat timet. Eq. (10b) speci®es
the lower and upper bounds on the yield supplied by
reservoir n for the demand area j. If the reservoir n is
located at downstream of the demand areaj or due to
any reason is not supposed to supply water for it, the
ynt;;minj andy t;max
n;j will be set equal to zero.
(4) Storage constraints based on physical limits
sminn 6stn16smaxn for n1;2;. . .;Nr;
n are lower and upper bounds on the
storage of reservoirn, respectively. Storage upper bound of each candidate reservoir can be selected by using the topographic map of the reservoir site. The lower bound
smin
n can be considered as the conservation (dead) storage
of reservoirn to provide a minimum storage for
recre-ation or the reservoir sedimentrecre-ation. A constraint to maintain prescribed ratios of minimum storage to res-ervoir storage at each site has also been considered
asminn Xnxn;
bpminn 6sminn 6pmaxn ; 12
where Xn is a pre-speci®ed ratio of the dead storage to
reservoir capacity xn. Therefore, the dead storage is a
function of the reservoir storage capacity.pmin
n andp
max
n
are minimum and maximum permissible dead storage bounds. Eq. (12b) ensures that the dead storage will not fall beyond the maximum and minimum permissible dead storage for the reservoirn.
5. PSLP module algorithm
The dimension of the problem assigned to PSLP module can be reduced to decrease the number of con-straints. Constraint (6) can be rearranged as
Dtj;maxP
The upper bound in Eq. (13) was already considered in Eq. (10a) and therefore is redundant. The lower bounds
on the reservoir yields in Eq. (13) is called the water
supply constraints,whichforce the PSLP module to meet water demands in dierent demand areas.
The Penalty Successive LP algorithm of Zhang et al. [23] is used to solve Eqs. (4), (5) and (13). The procedure to solve nonlinear optimization problems is given in [23]. However, the following explains the application of their algorithm to the problem described in PSLP module:
(1) De®ne the exact penalty function by adding the
nonlinear constraints to the objective function using pre-speci®ed penalty weights. The penalty weights in the exact penalty function are positive scalars that have to be in excess of the largest Lagrange multiplier (dual variable) value expected. The exact penalty function for the formulation proposed in the PSLP model would be as
j ), corresponding toNd demand
constraints constitutes the linearly constrained penalty (LCP) problem.
(2) De®ne the Approximating function Pl(X) by
re-placing all nonlinear parts in the exact penalty function by its ®rst order Taylor series approximation about a base pointX0 x1;0;x2;0;. . .;xNr;0
by a sequence of minimization of Pl(X) with an upper
bound on the step size. This leads to theApproximating
Problem
minimize Pl X 16
subject to the linear constraint (5) and the newdeviation constraintas
ÿx6 xnÿxn;06x forn1;2;. . .;Nr; 17
where x is the upper bound on the Taylor series step
size. The deviation constraint is to maintain a solution in the neighborhood of the current solution.
(3) Apply the ®rst order Taylor series expansion to linearize the nonlinear part of water supply constraints (13) about initial solutions (yt
n;j;0). According to the continuity Eq. (8), each yield is a function of its reservoir storage and all incoming return ¯ows
yt
Considering the fact that reservoir storage is a func-tion of the reservoir capacity, it can be stated that every yield is a function of all reservoir capacities as
yt
n;jH xi fori1;2;. . .;Nr: 19 Therefore, the Taylor series expansion of the constraint (13) is as
By applying the chain rule for dierentiation, a direct
relation between reservoir yield yt
n;j and capacities
xis1i;i1;2;. . .;Nr can be established. Ignoring
minor changes in evaporation losses in the continuity equation, theoyt
Eq. (21) implies that if a reservoir cannot supply yield to a demand area (maximum yield is zero), its
corre-sponding yield derivative with respect tox is zero. The
percent error in excluding the evaporation term in Eq. (21) is usually less than 0.1% [10] and therefore negli-gible.
(4) De®ne a linear program equivalent to the problem de®ned in step 2 as
subject to the linear constraints (5) and (17) and a new linearized water supply constraint as
for j1;2;. . .;Nd; fort1;2;. . .;T
In the above,pk andnk aredeviation variables, which
allow us to represent the water supply constraint (13) linearly in an LP algorithm.
Applying small penalty weights (Wjp) may result in an
infeasible solution. If, however, large penalty weights
are selected, the decision variables Xare forced to stay
too close to the feasible region for the majority of PSLP iterations. Consequently, it may cause slow convergence in some problems [2]. Therefore, based on the order of magnitude of Eqs. (4) and (13), one can select reason-able small penalty weights to start the problem. If the PSLP iterations have terminated with an infeasible so-lution in the original nonlinear problem i.e., the demand is not satis®ed fully, increase the weights and start again.
6. Oct module algorithm
In application of minimum principle, the system dy-namic Eq. (8) adjoins the objective functions by using a
set of Lagrange Multipliersk. The state±space inequality
constraint (11) are included by using a quadratic penalty
function g and a penalty weight ps to account for the
violation of constraints on state variables. The aug-mented objective function is called the Lagrangian
functionL.By the minimum principle of Pontryagin, the
necessary condition for L to be the minimum value is
that the dierential changes in L due to dierential
changes in control variables must be zero [4]
a oL
ost
n0 forn1;2;. . .;Nr; fort2;3;. . .;T;
b oL
osT1
n 0 forn1;2;
. . .;Nr;
8
> <
> :
24
oL oRt n
0 fort1;2;. . .;T for n1;2;. . .;Nr: 25
Eqs. (24a), (24b) and (25) are called the adjoint, trans-versality, andstationaryconditions respectively. To ®nd the optimal solution, Eqs. (7)±(12), (24) and (25) should be solved simultaneously. Practically speaking, solving these nonlinear equations is not an easy task and in some cases it may be possible to solve them only
nu-merically. Hence,direct solutionmethods of
mathemat-ical programming are considered instead. The ``double sweep'' algorithm [1] is used in this study to solve the objectives implemented in the OCT module.
7. Numerical studies of PSLP±OCT model
7.1. Developing computer programs
Two computer programs were developed based on the OCT and PSLP algorithms described earlier. The OCT code based on the double sweep algorithm was
developed. Polak±Ribiere conjugate gradient and
BrentÕs methods [17] were selected to improve estimates
of the control variables based on the gradient of the Lagrangian function. The PSLP code was also
devel-oped based on the simplex method to solvethe
equiva-lent linear problemof Eq. (22).
7.2. Case study application
Based on the fact that this study is oriented towards applications, the experimental approach is adopted to evaluate the accomplishment of the PSLP±OCT model. That is, the performance of the PSLP±OCT model in designing multi-reservoir systems was compared to an existing DP model. A problem from [19] was selected as a test problem to compare the performances of the devel-oped design models. This problem is based on the project of a graduate course CE-646 in Colorado State
Univer-sity (at Fort Collins) to develop water storage strategies for water supply. The CE-646 was selected because (1) the required data and its best-known DP solution are available in [19] and (2) the numerical problem fairly represents a large-scale multi-reservoir system. The CE-646 problem consists of six candidate reservoirs (Fig. 2). The DP solution to the CE-646 problem was selected as a benchmark to evaluate PSLP±OCT performance. The benchmark solution supplies the annual water demand of 51900 MCM and optimizes the multi-reservoirs
con-®guration at the $182.8´106cost. Both models have two
main objectives: to minimize the cost and water de®cit. Therefore, the success of PSLP±OCT model depends on the cost of the designed system and the corresponding level of water supply. The other objectives are minor objectives and are considered in the assessment only if the two main objectives are equally met.
In order to compare the performance of the PSLP± OCT model with the existing DP solution, the same hydrological data of two consecutive dry years is used to design the system. Following Supangat [19], all the dif-ferent demand areas are assumed to be located at the outlet of the most downstream reservoir. The same as-sumptions of zero minimum storage and return ¯ows are also applied to the system. The monthly in¯ows to the CE-646 multi-reservoir system is shown in Fig. 3 and
Fig. 2. The layout of the CE-646 test problem.
all other hydrologic and model input data are given in Appendix A. It should be emphasized that using dry year hydrologic data will over-design the system and in a real optimization problem a representative data should be used. That is, data records should be of length sat-isfactory to de®ne the model input parameters and the recorded data should not be drawn from unusually wet or dry periods.
8. PSLP±OCT model performance
To meet the annual water demand fully, zero de®cits (et
j) were considered in the water supply constraints and
proper weight coecients were assigned in the model. Very small/zero weight coecients were selected for those minor objectives that were not considered in the benchmark solution. The best storage penalty scheme in the OCT module was one with a maximum value of 100, initial value of 0.0001, and a ®ve to tenfold increase after each round of iteration. Several adjustments were made to ®nd the proper ®nal set of weight coecients in Eq. (7).
In spite of applying dierent combinations of weight coecients to the objective components in Eq. (7), the PSLP±OCT model, though close to the benchmark so-lution, was never able to fully supply the monthly water demands. Several model experiments showed that two factors are mainly important in selecting the proper yield weight coecients in the OCT module. These factors are: (1) the relative magnitudes of candidate reservoir capacities with respect to each other and (2) the ratio of the systemÕs hydrology (¯ow variability) to the demand levels. To survey the eect of reservoir sizes and their in¯ow on the OCT module performance, a closer look into the OCT algorithm is necessary. With an initial low penalty weight, the OCT algorithm sup-plies the demand fully at the expense of violating the storage constraints. As the storage penalty weights in-crease in subsequent iterations, the magnitudes of these feasibility violations decrease. At the ®nal solution with the largest penalty weight implemented, the multi-res-ervoir is supposed to supply the demand without any storage constraint infeasibilities. In the problem for-mulation (7), all the reservoirs are contributing to sup-ply a certain demand and hence share the same yield gradients. The yield gradients at each OCT iteration depend on the total water supplied at the previous iteration.
A multi-reservoir system like the CE-646 problem can be characterized as a system with high water demand levels and large dierences in the selected reservoir ca-pacities. In this system, small candidate reservoirs are incorporated with low in¯ows and large reservoirs with large ones. High demand levels impose large yield gra-dients in the OCT module. As mentioned in the previous
paragraph, large yield gradients in small reservoirs push the algorithm to supply yield at the expense of infeasible storage trajectories (i.e., negative storage) that stay in the infeasible region for most of the penalty iterations of the OCT algorithm. Therefore, even with moderate storage penalty weights, the small reservoir yields are more than what they can supply in reality and other (large) reservoirs in the system supply the remaining unsatis®ed demand. When the penalty weights are large enough, the yields of small reservoirs decrease and hence their storage goes back to the feasible region. This may change the ¯ow pattern (upstream reservoir releases) in the system and increase the unsatis®ed demand (yield gradient) which has to be supplied by other reservoirs in the system. The new situation for the gradient search techniques is like a new problem starting with a high penalty weight. This situation, as was mentioned in [7], leads to solution divergence and makes gradient search techniques slow and unstable.
To remove the eect of a small reservoir on large reservoir yield trajectories, a slightly dierent approach is required. In this approach, the OCT module can op-timize the water supply problem using one reservoir at a time. To accelerate the OCT convergence, dierent yield weight coecients, based on the ratio of ¯ow avail-ability to demand level, can be assigned to candidate reservoirs. Using smaller yield weight coecients for the small reservoirs with low in¯ow reduces their yield gradients and hence increases the OCT convergence. In this approach, the OCT module optimizes the objective (Eq. (7)) by using one reservoir at a time.
To do this, reservoirs are numbered sequentially from upstream to downstream. Once the PSLP module de-termines the reservoir capacities of the whole system at each outer (PSLP) iteration, the OCT module starts from the most upstream reservoir and considers one reservoir at a time to optimize its yield trajectories in order to minimize the water de®cit. Then, the remaining water de®cit is calculated and the OCT module uses the next downstream reservoir to supply the water de®cit. To accelerate the OCT convergence, the original yield
weight coecients (Wjy) can be changed into new
coef-®cients (Wny;j) that can be dierent for each candidate
reservoir n. Based on the new approach, for each
Eq. (7-R) is the revised version of the original water supply objectives. In this equation,dt
n;jis the water
de-mand assigned to thenth reservoir and is determined in
the sequential water supplying procedure in the OCT module and is shown in Fig. 4.
The new approach was adapted into the PSLP±OCT model and was applied to the CE-646 problem. The model was able to optimize the system successfully at
$160.83´106. The penalty weight Wp
1 of 80 was large
enough to meet the monthly demand levels in Eq. (13) and there was no diculty in assigning proper weights in the OCT module. Table 1 shows the assigned weight coecients for only the selected reservoirs in the OCT module. The weight coecients related to non-selected reservoirs are not reported, because they were just mass
balance nodes (in¯owout¯ow) to the OCT module
and could not aect the optimal result.
The PSLP±OCT model was able to supply water demand successfully at a proposed construction cost lower than the benchmark solution. The monthly res-ervoir storage of the selected resres-ervoirs and the corre-sponding total water supply are shown in Figs. 5 and 6. Zero values for monthly yields and releases, and reser-voir capacities are assumed as the initial solution for the PSLP and OCT modules. Using these initial solutions, the PSLP±OCT model required 68 s of execution time on a Pentium-Pro 180 Personal Computer to solve the CE-646 problem. It goes without saying that a better initial solution can further reduce the execution time. The low computer time requirement of the PSLP±OCT model shows that unlike DP models, it does not suer from the curse of dimensionality and consequently can be applied to large and complex multi-reservoir systems. It took only a few (3±4) iterations to construct a good approximation to the optimal solution. However, the gradient algorithm in PSLP converged slowly near the optimum. Consequently, a relatively large number of iterations (16 iterations in this example) were required to ®nd the true optimum, which resulted in only a small improvement in the objective function value. At the
optimal result, the terminal storage objective
ST1
n S
1
n though very close, was not fully met.
In-creasing the terminal storage weight coecient could be a remedy. However, this will aect the optimum yield trajectories as will be discussed in the weight coecient sensitivity analysis. A simpler remedy is to adjust cor-responding reservoir releases manually. These adjust-ments are unlikely to aect the optimality of the ®nal results since they are relatively small in magnitude.
9. Comparison of results
The layout designed and the total water supplied by the PSLP±OCT model together with the benchmark solution are shown in Table 2. The result shows that the PSLP±OCT model was able to design the system at a lower cost than the benchmark solution. The objective function of the PSLP module in this problem is convex and its nonlinear constraint is concave. Therefore, it is insensitive to initial solution. Compared to the bench-mark solution, the proposed PSLP±OCT model requires smaller reservoir capacities to supply the same level of water demand. This is due to the simpli®cations made Fig. 4. Flow chart of the sequential procedure in the OCT module.
Fig. 5. PSLP±OCT solution of optimum state and control trajectories of RES#4 for the CE-646 problem.
Fig. 6. PSLP±OCT solution of optimum state and control trajectories of RES#6 for the CE-646 problem.
Table 1
Selected weight coecients in the OCT module
Selected reservoirs
Weight coecients
Wn;y1 W by
n W
br
n W
fs
n RES#4 1.0e)7 0.0 1.0e)3 1.0e)5
by the decomposition technique and variable discreti-zations in the DP model. According to Supangat [19], the storage discretization was equal to 1% of the mean annual critical ¯ow to each reservoir (i.e., 200±800 MCM) and yield discretization is determined such that the maximum number of yield levels and their associated storage value is 30 [19, p. 121]. It is obvious that creasing the discretization intervals in the state and de-cision variables will enable the DP model to design the system more accurately. However, this approach may create the ``curse of dimensionality'' and consequently, be not applicable in large-scale multi-reservoir systems. The comparative performances of the two models show that the PSLP±OCT model has designs the system by a bit less than 14% and is a very promising optimization method to design multi-reservoir systems regardless of their sizes.
10. Sensitivity of the weight coecients
Appropriate selections of the weight coecients de-pend on the order of magnitude and the importance of all objectives. Among the four weight coecients (Wny;j;Wnbr;W
by
n andW
fs
n the yield coecient is the most
important one. One strategy to select the proper weight could be to choose yield weight coecients for each reservoir and then ®nd the appropriate weights for other objectives. Based on this strategy and considering the order of magnitude of the hydrologic data, yield weight coecients equal to one and zero weights for secondary objectives were selected in the beginning and the per-formance of the OCT module was examined. Then based on the obtained result, appropriate weights for all objectives were determined. For the CE-646 problem, 4± 5 adjustments were made for each weight coecient to ®nd the most appropriate coecients.
Larger terminal weight coecients Wnfs were not
used to meet the terminal storage conditions. This is due to the fact that larger weights (Wfs
n increase the eect of
Lagrange multipliers on the yield gradients and may
eventually reduce the reservoir yields. SmallWfs
n on the
other hand, increase the deviation of ®nal storage from its target storage. The same situation applies to the
bang-bang weight coecients. IncreasingWbr
n and W
by
n
will smooth the control trajectories over time. However, they generally lower the total water supply and hence lead to a higher system cost. Therefore, if in designing a
multi-reservoir system, having smooth control trajecto-ries is not as important as the system cost, the related weights should be kept as small as possible. While
bang-bang control on yieldWby
n will de®nitely reduce the total
system yield all the time, assigning reasonable bang± bang weights to the release may help the system to
supply water more eciently. However, largeWbr
n causes
the bang-bang term dominate the multiplier term in the release gradient. This may result in inappropriate (too large/small) releases and consequently, pushes the stor-age to stay in the infeasible region.
The penalty weights in the PSLP module have to be at least greater than the related decision variables in the dual problem. The PSLP±OCT model performance showed that the model is insensitive to the penalty weight Wjp as long as its value is greater than the
deci-sion variables in the dual problem. However, large Wjp
may keep the deviation variables in the basis for the majority of PSLP iterations which usually slow con-vergence. Therefore, the best strategy to ®nd the proper
Wjp, as Zhang et al. [23] mentioned, is to start the
problem with a small Wjp. If the current solution is
in-feasible in the PSLP module (monthly demands are not satis®ed), increase the weight and start again.
11. Extensions and comment
The present study has been limited to dealing with problems of deterministic approach: That is, the stream ¯ows are assumed to be known with certainty. A sto-chastic formulation is generally a more realistic repre-sentation of a hydrosystem since stream ¯ows have randomness and are stochastic in nature. However, in stochastic optimization models, the reliability of the designed system is not considered. Therefore, a proba-bilistic approach (e.g., chance-constrained/yield formu-lations) is recommended to be incorporated to the PSLP±OCT model. In the chance constraint approach, the system is optimized subject to a constraint that storages are greater than or equal to some base level with a speci®ed reliability [18]. The objective function with the yield probabilistic approach minimizes the re-quired reservoir costs to supply yields with a pre-speci-®ed reliability [11]. The probabilistic approach though adds to the complexity of the problem, will help the water resources engineer to design the multi-reservoir system with a desired reliability.
Table 2
Comparison of design model performances
Model Required storage size in reservoir no. (MCM)´10ÿ2 Design cost Annual yield
1 2 3 4 5 6 ´10ÿ6($) (MCM)´10ÿ2
PSLP±OCT ) ) ) 3.95 ) 247.50 160.80 519.00
The extension of PSLP±OCT to consider demand areas distributed over the whole watershed was not covered in this study, but it may be achieved without major changes in the program codes. This is due to the fact that the same decision variables are considered for single and multiple demand area cases.
The application of the epsilon constraint method provides the model with the capability to introduce trade-os between dierent possible water demand levels in the future and the optimized reservoir con®gurations. This feature lets decision-makers evaluate the sensitivity of each candidate reservoir to dierent water demand levels. The less sensitive reservoir has the priority in being built.
Appendix A. Input data to the CE-646 problem
· (Total months) T 24, (total reservoirs) Nr6,
(total demand areas)Nd1.
· Ratio of minimum storage Smin
n to reservoir storage
i.e.,Xn: 6´0.
· Upper and lower bounds on minimum storage Qmin
n
andQmax
n 6´0., 6´0.
· Upper bounds on reservoir capacity Xmax
n : 10780
15360 16030 1000 9500 48700.
· Lower bound on reservoir storageXmin
n : 6´0.
· Maximum water demand at area j during month t
Dt
j4; j1; t1;24.
7214.1 7162.2 6279.9 4100.1 3788.7 2595.0
1141.8 1453.2 2698.8 3944.4 5553.3 5968.5
7214.1 7162.2 6279.9 4100.1 3788.7 2595.0
1141.8 1453.2 2698.8 3944.4 5553.3 5968.5
· Upper and lower bounds on reservoir releasesRmax(n)
andRmin(n): 6´99,999, 6´0.
· Maximum penalty weightPmax: 100.
· Monthly evaporation rate m/month EVAP(n,t):
6
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