Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue6.2000:

Advances in Water Resources 23 (2000) 613±624

Optimal design of multi-reservoir systems for water supply
Hirad Mousavi a,*, A.S. Ramamurthy b,1
a

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 multireservoir 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 reservoir 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 reservoirs 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., supplying water demand) at the minimum cost. A successful
model should be able to take advantage of system features 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*

Corresponding author. Tel.: +1-206-615-0826; fax: +1-206-3869147.
E-mail addresses: hirad.mousavi@ci.seattle.wa.us (H. Mousavi),
ram@civil.concordia.ca (A.S. Ramamurthy).

1
Tel.: +514-848-7807; fax: +514-848-2809.

straints. Mathematical programming includes several
techniques such as dynamic programming (DP), nonlinear programming (NLP), linear programming (LP),
genetic algorithms (GAs), and optimal control theory
(OCT). Among the family of mathematical programming methods, OCT still is generally not popular in
water resources system analyses. Hiew [7] performed a
comparative performance evaluation on several optimization procedures. The criteria to compare were: accuracy 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 programming are the most promising techniques for optimizing 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 reservoirs at minimum cost to meet a projected system ®rm

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 5 3 - 6

614

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

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 systems 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 developed by Fontane [6], Mays and Bedient [13], Bennet and
Mays [3], Supangat [19], and Taur et al. [20] to determine the best water storage strategies. Lall and Miller
[10] and Khaliquzzaman and Chander [8] employed LPbased 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) diculties in selecting initial solutions, or (3)
the problem of dimensionality. It is felt that the development of a design model that is computationally ecient
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 techniques. 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 programming techniques in terms of computational procedures. 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 conventional 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 technique, which employs OCT and PSLP algorithms as
promising optimization techniques. The PSLP±OCT is a
general-purpose optimization technique that can be used
in mixed type optimization 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 a€ect the ®nal
solution proposed by these methods. Therefore, the related computer programs should be run several times to
achieve the best possible local optimum solution.
The introduced PSLP±OCT method however, recognizes the dynamism of the problem and di€erentiates 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 optimizes the dynamic part and the PSLP optimizes the
static part of the problem. This approach not only reduces 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 insensitivity 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 assigned 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 technique in designing the multi-reservoir system is described 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 term ``design model'' is often used vs. operation
model [11]. The main objectives of the design model
formulated are to determine a set of optimal reservoir
capacities that supply water for di€erent demand areas
at the minimum cost. In practice, some secondary objectives may be considered in the design problem, which
may be mandated by law for political, ecological, environmental, or operational reasons. To adapt the problem objectives and constraints to the model, they are
split into two parts. A small part of the optimization
problem is assigned to the PSLP and the OCT handles
the major computational part of the optimization
problem. This strategy reduces the computational burden of the PSLP module and assigns the major part of

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

615

Fig. 1. Flow chart of the PSLP±OCT model.

the optimization problem to the OCT, which is a more
promising technique in optimizing large-scale optimization 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 capacities. 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 propose a new reservoir con®guration. The process in the
PSLP module terminates when optimal reservoir capacities are found or when a series of successive iterations 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 dependence may be functionalized and evaluated independently by using OCT with respect to target reservoir
capacities in the multi-reservoir system.

nˆ1

subject to reservoir capacity constraints and
Dtj;max ÿ

Nr
X

t
yn;j
P 0;

…3†

PSLP module applies epsilon constraint algorithm as the
most suitable generating method and a superior technique for problems with di€erent dynamic characteristics 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

Nr
X
fnc

…4†

nˆ1

subject to
max
0 6 xmin
n 6 xn 6 xn

for n ˆ 1; 2; . . . ; Nr ;

Nr
X
t
yn;j
6 etj

…5†

for j ˆ 1; 2; . . . ; Nd ;

nˆ1

The optimization problem considered in the PSLP
module is given by the following equations:
!)
(
Nd X
Nr
Nr
T
X
X
X
c
t
t
fn ;
yn;j
Dj;max ÿ
…1†
minimize
jˆ1 tˆ1

fnc ˆ An  xn ‡ Bn  x2n

0 6 Dtj;max ÿ

4.1. PSLP module formulation

nˆ1

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 function of the reservoir capacity (xn )

…2†

nˆ1

where Dj;maxt is the maximumPpredicted water demand
during month t at area j and
yn;jt is the summation of

for t ˆ 1; 2; : . . . ; T ;

…6†

and xmax
are the lower and upper bounds on
where xmin
n
n
reservoir capacities and etj is the monthly water de®cit at
the D#j. The upper bounds in Eq. (6) are determined by
epsilon constraint algorithm. Reservoir capacities
X ˆ ‰x1 ; x2 ; . . . ; xNr Š 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 (ytn;j ) in Eq. (6) is a nonlinear implicit function of
the reservoir capacity xn and can be obtained through
the OCT module.

616

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

last three terms to increase the convergence and speed of
the search direction and line minimization algorithms.

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 di€erent demand areas, using the prespeci®ed set of optimal reservoir capacities by PSLP
module. Minimizing the rapid variations (bang±bang)
on reservoir yields and releases, and minimizing the
storage di€erences at the beginning and end of the optimization period (terminal function) are among the
secondary objectives that have been included in the
problem formulation. The proposed formulation is discretized over time

4.2.1. Constraint equations
The related system constraints applied to the objectives described in the previous section are
(1) Continuity equation:
s1n ˆ xn

for n ˆ 1; . . . ; Nr ;

Snt‡1 ˆ stn ‡ Qtn ‡

kˆ1

‡

Nd
X
J ˆ1
J6ˆj

minimize f1 ˆ

Nd
X
jˆ1

‡

8
>
>
>
<

Nr
P

Wnby
nˆ1

"

2

Wjy 4

Nd 
TP
ÿ1P

T
X
tˆ1

t‡1
yn;j

tˆ1 jˆ1

Nr
X

t
yn;j

nˆ1

ÿ

t
yn;j

2

!2 3
ÿ Dt 5

#

‡

9
>
>
>
=

TP
ÿ1ÿ
>
ÿ

2 >

2
>
>
>
;
: Wnbr
rnt‡1 ÿ rnt ‡ Wnfs snT ‡1 ÿ s1n >
tˆ1

2

6 J
4Cn

ÿ etm Cne

j

;
…7†

t
where yn;j
and rnt are the yield and release (L3 ) from
reservoir n to the demand area j during the month t. Nd
is the total number of municipal, industrial, and irrigation demand areas, Nr the number of candidate reservoirs in the system, Dtj the water requirement (L3 ) at the
demand area j during the month t, Wjy the yield weight
coecient, and Wnby and Wnbr are the bang±bang weight
coecients applied to yield and release variables, respectively. The Wnfs is a weight coecient applied to the
terminal storage function, stn is the storage (L3 ) of the
reservoir n at 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 di€erence between water supply
and demand in either way. The second and third terms
are intended to avoid rapid variations on control variables (yjt and rnt † [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 period. Based on the order of magnitude and importance
of each criterion in Eq. (7) to the designer, di€erent
values can be assigned to the weight coecients (Wjy ,
Wnby , and Wnbr ). Quadratic forms were selected for the

Nr
X
ÿ



t
X
mˆ1

stn

Nd

X
t
Kkn rkt ÿ
yn;j
jˆ1

0

B tÿm‡1
@qJ
e
t‡1 pn

‡ Sn
2

Nr
X
iˆ1
i6ˆn

13

m C7
yi;J
A5

for t ˆ 1; . . . ; T :

…8†

Eq. (8) is the discrete form of the continuity equation for
reservoir n over T months, where Snt is the storage (L3 ) of
the reservoir n at the beginning of time t and Qtn is the
volume of the unregulated local in¯ow (L3 ) into reservoir n during time t. Evaporation from the reservoir has
been assumed to be the only source of system loss. Kkn is
the element of the layout con®guration matrix of the
multi-reservoir system with Nr rows and columns
(KNr Nr ). Each row of this matrix shows the reservoir
number and each column shows the release number. The
state of any element at the nth row and the kth column is
set equal to 1, if the reservoir n receives the kth release.
If reservoir n delivers the kth release, the related element
is set equal to )1. For the unconnected reservoirs, the
element is equal to 0. CJn is the element of the return ¯ow
matrix with Nr rows and Nd columns (CNr Nd ) 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. qmJ is a
coecient for return ¯ow from the demand area J
during time m. etn is the evaporation per unit area (L3 /L2 )
from reservoir n during month t. Cne and Pne are the coecient and exponent in the surface area±storage
relaP
tionships of the reservoir n. The third term ( Kkn  rkt )
speci®es both the release(s) from reservoir n and all the
in¯ows resulting from upstream
releases to
P reservoir
t
) shows the total
reservoir n. The fourth term ( yn;j
yield supplied by the reservoir n to all demand areas
during month t. Nd is 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 receives during the month t from in¯uent areas. In the last
term, the evaporation during month t is related to the

617

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

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
0 6 rnt

for k ˆ 1; 2; . . . ; Nr ; for t ˆ 1; 2; . . . ; T ;
(
)
Nd
Nr 

X
X
k t
j
min
t
fn …yk;j
nn
rn 6 rn ‡
6 rnmax ;

…9†

where njn is the element of the demand area matrix
(nNr  Nd ) which is used to specify the demand areas
located in the downstream of a release. An element of n
at column j and row n is equal to 1, if the demand area j
is located in the downstream of the release n. The elements of f are equal to 0 otherwise. fkn is the element of
the upstream reservoir matrix with Nr rows and columns
(fNr  Nr ) and is used to specify the reservoirs located in
the upstream of a release. An element of f at column k
and row n is equal to 1, if the related reservoir k is located in the upstream of the release n. The elements of f
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 (rnmin ) can be other than zero to
consider instream recreation, navigation, and water
quality control. The maximum ¯ow constraint rnmax will
prevent the system from being ¯ooded due to excess
reservoir release and the upstream reservoir yields attributed to downstream demand areas.
(3) Reservoir yield constraints for water supply:
…a† Djt;min 6

Nr
X

t
yn;j
6 Djt;max

for j ˆ 1; 2; . . . ; Nd ;

nˆ1

…b†

t;min
yn;j

t
6 yn;j
6

for t ˆ 1; 2; . . . ; T ;
t;max
yn;j
:

…10†

Eq. (10a) is to guarantee that the total water supplied to
any demand area j is within a desirable range. Djt;max and
Djt;min are the maximum and minimum water requirement of the demand area j at time t. 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 area j or due to
any reason is not supposed to supply water for it, the
t;min
t;max
and yn;j
will be set equal to zero.
yn;j
(4) Storage constraints based on physical limits
smin
n

6 snt‡1

6 smax
n

for n ˆ 1; 2; . . . ; Nr ;
for t ˆ 1; 2; . . . ; T ;

…a† smin
ˆ Xn  xn ;
n

…12†

min
max
…b† pmin
n 6 sn 6 pn ;

kˆ1

jˆ1

ation or the reservoir sedimentation. A constraint to
maintain prescribed ratios of minimum storage to reservoir storage at each site has also been considered

…11†

and smax
are lower and upper bounds on the
where, smin
n
n
storage of reservoir n, respectively. Storage upper bound
of each candidate reservoir can be selected by using the
topographic map of the reservoir site. The lower bound
can be considered as the conservation (dead) storage
smin
n
of reservoir n to provide a minimum storage for recre-

where Xn is a pre-speci®ed ratio of the dead storage to
reservoir capacity xn . Therefore, the dead storage is a
and pmax
function of the reservoir storage capacity. pmin
n
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 reservoir n.

5. PSLP module algorithm
The dimension of the problem assigned to PSLP
module can be reduced to decrease the number of constraints. Constraint (6) can be rearranged as
Dtj;max P

Nr
X
nˆ1



t
yn;j
P Dtj;max ÿ etj ˆ Dtj

for j ˆ 1; 2; . . . ; Nd ;

…13†

for t ˆ 1; 2; . . . ; T :

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, which force the PSLP module to meet
water demands in di€erent 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 prespeci®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
P …X † ˆ

Nr
X
ÿ

An xn ‡ Bn x2n

nˆ1

‡

Nd
X
jˆ1

Wjp

T
X
tˆ1




"

max 0;

Dtj

Nr
X
t
yn;j
ÿ
nˆ1

!#

:

…14†

Nd penalty weights (Wjp ), corresponding to Nd demand
areas, have been used for the nonlinear constraints to
consider the scaling of each water supply constraint. The
resulting optimization problem has a nonlinear objective
function subject to the linear constraints (Eq. (5)).
The exact penalty function together with the linear

618

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

constraints constitutes the linearly constrained penalty
(LCP) problem.
(2) De®ne the Approximating function Pl(X) by replacing all nonlinear parts in the exact penalty function
by its ®rst order Taylor series approximation about a
base point X0 ˆ ‰x1;0 ; x2;0 ; . . . ; xNr ;0 Š
Nr h

i
X
An xn;0 ‡ Bn x2n;0 ‡ …An ‡ 2Bn xn;0 †…xn ÿ xn;0 †
Pl…X † ˆ
nˆ1

‡

(
Nd
X

Wjp

max

tˆ1

jˆ1

"

T
X

 0; Dtj ÿ

Nr 
X
nˆ1

t
oyn;j
t
yn;j;0
‡ …xn ÿ xn;0 †
oxn

#)

:
…15†

Pl(X) is a good approximation to P(X) if the step size
(xn ) xn;0 ) is not large. Thus the P(X) can be minimized
by a sequence of minimization of Pl(X) with an upper
bound on the step size. This leads to the Approximating
Problem
minimize Pl…X †

…16†

subject to the linear constraint (5) and the new deviation
constraint as
ÿx 6 …xn ÿ xn;0 † 6 x

for n ˆ 1; 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
t
). According to the
(13) about initial solutions (yn;j;0
continuity Eq. (8), each yield is a function of its reservoir
storage and all incoming return ¯ows
1
0
Nd X
Nr X
t
X
B
t
mC
yi;J
ˆ f @…stn †;
…18†
yn;j
A:
iˆ1 J ˆ1 mˆ1
i6ˆn

J 6ˆj

Considering the fact that reservoir storage is a function of the reservoir capacity, it can be stated that every
yield is a function of all reservoir capacities as
t
ˆ H …xi †
yn;j

for i ˆ 1; 2; . . . ; Nr :

…19†

Therefore, the Taylor series expansion of the constraint
(13) is as
!
t
Nr
Nr
Nr
X
X
X
oyn;j
t
t
…xi ÿ xi;0 †
yn;j 
yn;j;0 ‡
P Dtj ;
ox
i
…20†
iˆ1
nˆ1
nˆ1
for j ˆ 1; 2; . . . ; Nd ;

for t ˆ 1; 2; . . . ; T :

By applying the chain rule for di€erentiation, a direct
t
† and capacities
relation between reservoir yield …yn;j
1
…xi ˆ si ; i ˆ 1; 2; . . . ; Nr † can be established. Ignoring
minor changes in evaporation losses in the continuity
t
=oxi can be directly evaluated as
equation, the oyn;j


t
t;max
oyn;j
Zi;n
if yn;j
P 0;
ˆ
0
otherwise;
oxi
8
if i ˆ n;
>
< 1N
t
Pd
P
Zi;n ˆ
otherwise:
Cn;J qJtÿm‡1
>
: J ˆ1
mˆ1

…21†

J 6ˆj

Eq. (21) implies that if a reservoir cannot supply yield to
a demand area (maximum yield is zero), its corresponding yield derivative with respect to x is zero. The
percent error in excluding the evaporation term in Eq.
(21) is usually less than 0.1% [10] and therefore negligible.
(4) De®ne a linear program equivalent to the problem
de®ned in step 2 as
!
jT
Nd
Nr
X
X
X
p
Wj
‰…An ‡ 2Bn xn;0 †xn Š ‡
pk
minimize
jˆ1

nˆ1

kˆ…jÿ1†T ‡1

…22†

subject to the linear constraints (5) and (17) and a new
linearized water supply constraint as
for

j ˆ 1; 2; . . . ; Nd ;

for t ˆ 1; 2; . . . ; T

Nr
Nr X
X

…Zi;n xi † ‡ pk ÿ nk ˆ

nˆ1 iˆ1

Nr
Nr
X
X
nˆ1

iˆ1

Zi;n xi;0 ÿ

t
yn;j;0

!

‡ Dtj ;
…23†

where k ˆ j  t.
In the above, pk and nk are deviation 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 X are 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 reasonable small penalty weights to start the problem. If the
PSLP iterations have terminated with an infeasible solution in the original nonlinear problem i.e., the demand
is not satis®ed fully, increase the weights and start again.

6. Oct module algorithm
The OCT that is being used in this study is based on
the discrete minimum principle of Pontryagin. To consider di€erent objectives in optimizing multi-reservoir
systems, the multi-objective programming approach is
selected to avoid the monetary quanti®cation of the
social and political impacts resulting from choosing a
certain policy. The rationale for this preference is based
on the implied roles for the analyst and decision-makers
in this approach [5].

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

In application of minimum principle, the system dynamic Eq. (8) adjoins the objective functions by using a
set of Lagrange Multipliers k. 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 augmented objective function is called the Lagrangian
function L. By the minimum principle of Pontryagin, the
necessary condition for L to be the minimum value is
that the di€erential changes in L due to di€erential
changes in control variables must be zero [4]
8
for n ˆ 1; 2; . . . ; Nr ;
…a† ooLstn ˆ 0
>
<
for t ˆ 2; 3; . . . ; T ;
…24†
>
: …b†
oL
ˆ 0 for n ˆ 1; 2; . . . ; Nr ;
o sT ‡1
n

oL
ˆ 0 for t ˆ 1; 2; . . . ; T
oRtn

for n ˆ 1; 2; . . . ; Nr :

…25†

Eqs. (24a), (24b) and (25) are called the adjoint, transversality, and stationary conditions 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 numerically. Hence, direct solution methods of mathematical 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 developed based on the simplex method to solve the equivalent linear problem of Eq. (22).

Fig. 2. The layout of the CE-646 test problem.

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 CE646 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 ´ 106 cost. 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 different demand areas are assumed to be located at the
outlet of the most downstream reservoir. The same assumptions 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

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 developed design models. This problem is based on the project
of a graduate course CE-646 in Colorado State Univer-

619

Fig. 3. In¯ows to CE-646 Multi-reservoir system.

620

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

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 satisfactory 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
(etj ) were considered in the water supply constraints and
proper weight coecients were assigned in the model.
Very small/zero weight coecients 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 coecients in
Eq. (7).
In spite of applying di€erent combinations of weight
coecients to the objective components in Eq. (7), the
PSLP±OCT model, though close to the benchmark solution, 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 coecients 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 e€ect 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 supplies the demand fully at the expense of violating the
storage constraints. As the storage penalty weights increase in subsequent iterations, the magnitudes of these
feasibility violations decrease. At the ®nal solution with
the largest penalty weight implemented, the multi-reservoir is supposed to supply the demand without any
storage constraint infeasibilities. In the problem formulation (7), all the reservoirs are contributing to supply 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 di€erences in the selected reservoir capacities. In this system, small candidate reservoirs are
incorporated with low in¯ows and large reservoirs with
large ones. High demand levels impose large yield gradients 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 e€ect of a small reservoir on large
reservoir yield trajectories, a slightly di€erent approach
is required. In this approach, the OCT module can optimize the water supply problem using one reservoir at a
time. To accelerate the OCT convergence, di€erent yield
weight coecients, based on the ratio of ¯ow availability to demand level, can be assigned to candidate
reservoirs. Using smaller yield weight coecients 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 determines 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 coecients (Wjy ) can be changed into new coefy
®cients (Wn;j
) that can be di€erent for each candidate
reservoir n. Based on the new approach, for each reservoir n, the Eq. (7) is rewritten as
"
#
Nd
T 
2
X
X
y
t
t
Wn;j
ÿ dn;j
yn;j
minimize f1 ˆ
tˆ1

jˆ1

‡ Wnby
‡ Wnbr

"

Nd 
T ÿ1 X
2
X
t‡1
t
yn;j
ÿ yn;j
tˆ1 jˆ1

T ÿ1
X
tˆ1

‡

Wnfs

ÿ

rnt‡1 ÿ rnt

ÿ T ‡1

2
sn ÿ s1n :


2

#

…7-R†

621

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

Eq. (7-R) is the revised version of the original water
t
supply objectives. In this equation, dn;j
is the water demand assigned to the nth 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 W1p of 80 was large
enough to meet the monthly demand levels in Eq. (13)
and there was no diculty in assigning proper weights in
the OCT module. Table 1 shows the assigned weight
coecients for only the selected reservoirs in the OCT
module. The weight coecients related to non-selected
reservoirs are not reported, because they were just mass
balance nodes (in¯ow ˆ out¯ow) to the OCT module
and could not a€ect 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 reservoir storage of the selected reservoirs and the corresponding total water supply are shown in Figs. 5 and 6.
Zero values for monthly yields and releases, and reservoir 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 su€er
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
…SnT ‡1 ˆ Sn1 † though very close, was not fully met. Increasing the terminal storage weight coecient could be
a remedy. However, this will a€ect the optimum yield
trajectories as will be discussed in the weight coecient
sensitivity analysis. A simpler remedy is to adjust corresponding reservoir releases manually. These adjustments are unlikely to a€ect the optimality of the ®nal
results since they are relatively small in magnitude.

Fig. 4. Flow chart of the sequential procedure in the OCT module.

Table 1
Selected weight coecients in the OCT module
Selected
reservoirs

Weight coecients
y
Wn;1

Wnby

Wnbr

Wnfs

RES#4
RES#6

1.0e ) 7
1.0

0.0
0.0

1.0e ) 3
1.0e ) 4

1.0e ) 5
1.0e ) 2

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.

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 benchmark 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

622

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

Table 2
Comparison of design model performances
Model
PSLP±OCT
Bench mark

Required storage size in reservoir no. (MCM) ´ 10ÿ2

Design cost
ÿ6

1

2

3

4

5

6

´10

)
)

)
)

)
)

3.95
4.00

)
)

247.50
288.00

160.80
182.80

by the decomposition technique and variable discretizations 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 decreasing the discretization intervals in the state and decision 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 coecients
Appropriate selections of the weight coecients depend on the order of magnitude and the importance of
all objectives. Among the four weight coecients
y
; Wnbr ; Wnby and Wnfs † the yield coecient is the most
(Wn;j
important one. One strategy to select the proper weight
could be to choose yield weight coecients 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
coecients equal to one and zero weights for secondary
objectives were selected in the beginning and the performance 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 coecient to
®nd the most appropriate coecients.
Larger terminal weight coecients …Wnfs † were not
used to meet the terminal storage conditions. This is due
to the fact that larger weights (Wnfs † increase the e€ect of
Lagrange multipliers on the yield gradients and may
eventually reduce the reservoir yields. Small Wnfs on the
other hand, increase the deviation of ®nal storage from
its target storage. The same situation applies to the
bang-bang weight coecients. Increasing Wnbr and Wnby
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

($)

Annual yield
(MCM) ´ 10ÿ2
519.00
519.00

multi-reservoir system, having smooth control trajectories is not as important as the system cost, the related
weights should be kept as small as possible. While bangbang control on yield Wnby 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 eciently. However, large Wnbr 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 storage 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 decision 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 convergence. 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 infeasible 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 stochastic formulation is generally a more realistic representation 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 probabilistic approach (e.g., chance-constrained/yield formulations) 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 required 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.

623

H. Mousavi, A.S. Ramamurthy / Advances in Water Resources 23 (2000) 613±624

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.
Reservoir
number
1
2
3
4
5
6

0

0:2103 0:3737 0:4120 0:2798 0:1410 0:0717

C
B
B 0:0312 0:0250 0:0247 0:0260 0:0388 0:1030 C
C
B
C:
6B
C
B
B 0:2103 0:3737 0:4120 0:2798 0:1410 0:0717 C
A
@
0:0312 0:0250 0:0247 0:0260 0:0388 0:1030

Upper and lower bounds on
reservoir yields

Coecients of reservoir cost
functions An  zXn ‡ Bn  Xn2

Coecients of
evaporation function

t;max
yn;j

t;min
yn;j

A

B

cen

pne

24 ´ 50,000
24 ´ 50,000
24 ´ 50,000
24 ´ 50,000
24 ´ 50,000
24 ´ 50,000

24 ´ 0.
24 ´ 0.
24 ´ 0.
24 ´ 0.
24 ´ 0.
24 ´ 0.

0.0188
0.01565
0.01963
0.0043
0.0218
0.0071

)1.162e ) 07
2.302e ) 7
2.785
1.818e ) 6
)3.969e ) 9
)2.844e ) 8

0.410
1.000
0.33
1.730
0.475
0.740

0.70
0.64
0.67
0.51
0.72
0.74

The application of the epsilon constraint method
provides the model with the capability to introduce
trade-o€s between di€erent 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 di€erent 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 ˆ 24, (total reservoirs) Nr ˆ 6,
(total demand areas) Nd ˆ 1.
· Ratio of minimum storage Snmin to reservoir storage
i.e., Xn : 6 ´ 0.
Q
· Upper
and lower bounds on minimum storage min
n
Qmax
and n 6 ´ 0., 6 ´ 0.
· Upper bounds on reservoir capacity Xnmax : 10780
15360 16030 1000 9500 48700.
· Lower bound on reservoir storage Xnmin : 6 ´ 0.
· Maximum water demand at area j during month t
‰Dtj Š4; j ˆ 1; t ˆ 1; 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 releases Rmax (n)
and Rmin (n): 6 ´ 99,999, 6 ´ 0.
· W y …n; j†: 1.e ) 7 1. 1. 1. 1. 1.
· W by …n†: 6 ´ 0.
· W br …n†: 1.e ) 3 0. 0. 0. 0. 1.e ) 4.
· W fs …n†: 1.e ) 5 0. 0. 0. 0. 1.e ) 2.
· Return ¯ow fraction RHO…j; t†: 24 ´ 0.
· Maximum penalty weight Pmax : 100.
· Monthly evaporation rate m/month EVAP(n,t):

1

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