Optimization of Multireservoir Systems Operation Using Modified Direct Search Genetic Algorithm

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Optimization of Multireservoir Systems Operation UsingModified Direct Search Genetic Algorithm

Alireza B. Dariane1 and Shervin Momtahen, Ph.D.2

Abstract: A direct search method using genetic algorithms �DSGA�, which seeks to directly find optimal parameters for prescribedoperating policies, is utilized for optimization of multireservoir operational problems and several modifications are presented. Theproblems presented consist of 3, 7, and 16 reservoirs, respectively, from the Greater Karoon system in Iran. For the three-reservoirproblem, the DSGA method is used to obtain optimal linear operating policies and has proven to be very effective in both objectivefunction values and computational time in comparison to the more traditional optimization models based on dynamic programming.However, the model must be modified to optimize larger problems successfully. The first set of proposed modifications is primarily forenhancing the efficiency of the genetic algorithm �GA� used in the model and reducing its sensitivity to GA parameters such as theprobability of mutations and size of generations. The more robust modified DSGA is then applied to the seven-reservoir problem to obtainoptimal linear policies as well as two forms of piecewise linear ones, which achieves better objective values. The other modificationsapplied to the model are a Fourier series approximation which defines the seasonal variation of policy parameters and a stepwise GA,which employs varying lengths of simulations for fitness evaluations in different generations. These modifications reduce the time ofcomputations significantly. As a final point, the fine-tuned modified DSGA model optimizes the 16-reservoir problem in less than 20 h,which is a significant time period. Computational time is estimated to increase geometrically �second order� with the number of reservoirs.

DOI: 10.1061/�ASCE�0733-9496�2009�135:3�141�

CE Database subject headings: Water resources; Water management; Optimization models; Computation; Simulation models.

Introduction

Over the past decades, optimization of reservoir systems opera-tion has been a major field of water resource studies. Progress hasbeen achieved due to the improvements in mathematical modelsand optimization methods and advancements in computer tech-nology and calculating tools. Despite these achievements, optimi-zation of an integrated system of reservoirs remains a dauntingtask.

In a review on the optimization strategies of reservoir systems,Labadie �2004� classifies the strategies into four distinct catego-ries: implicit stochastic optimization �ISO�; explicit stochastic op-timization �ESO�; real-time optimal control with forecasting; andheuristic programming methods. Genetic algorithms �GA� areconsidered as a suitable means of optimization within heuristicprogramming methods and, in turn, have the ability to be linkeddirectly with trusted simulation models.

GA performs optimization through a process analogous to “themechanics of natural selection and natural genetics” in the field ofbiological sciences �Goldberg 1989�. GA should have the capa-bility to optimize the most complicated problems if the most com-plicated creatures are optimized using simple genetic operators.

1Associate Professor of Civil Engineering, K.N. Toosi Univ. of Tech-nology, Tehran, Iran. E-mail: [email protected]

2Independent Researcher, 1696 Deer’s Leap Place, Coquitlam, BC,Canada V3E 3C8. E-mail: [email protected]

Note. Discussion open until October 1, 2009. Separate discussionsmust be submitted for individual papers. The manuscript for this paperwas submitted for review and possible publication on August 3, 2006;approved on November 5, 2008. This paper is part of the Journal ofWater Resources Planning and Management, Vol. 135, No. 3, May 1,
2009. ©ASCE, ISSN 0733-9496/2009/3-141–148/$25.00.
JOURNAL OF WATER RESOURCE

J. Water Resour. Plann. Mana

Despite the novelty of using GA method in optimization ofreservoir systems over the past years, it has been reported as avery successful method compared to other more traditionaldynamic-programming-based methods. Esat and Hall �1994� ap-plied GA for optimization of a known four-reservoir system,which is frequently used as a benchmark in evaluating water re-source optimization models. They compared their GA model witha dynamic programming �DP� and assessed GA to have a betterperformance. Wardlaw and Sharif �1999� used GA in the deter-ministic optimization of a reservoir system and stated it to be asuccessful and robust method. Sharif and Wardlaw �2000� alsoextended their earlier work to multireservoir systems.

Oliveira and Loucks �1997� used GA in optimization of multi-reservoir systems operating rules and evaluated this method as apractical and robust method for estimating the operating policiesof complex reservoir systems. Cai et al. �2001�, Chen �2003�, andTung et al. �2003� also applied GA successfully for solving largenonlinear water resource management problems or obtaining rulecurves and their final evaluation proved this method to be a pow-erful tool.

Momtahen and Dariane �2007� employed GA for finding opti-mal operating rules having a variety of predefined forms. Thestudy compared them with the rules obtained from conventionalmethods such as stochastic dynamic programming �SDP� and dy-namic programming and regression �DPR�. As a result, some evensimple GA operation rules proved far more effective not only overa historical calibration time series, but also over the verificationtime series generated by statistical models.

Application of GA in reservoir systems can generally be cat-egorized into two types: “period-of-record release” and “policyparameters” optimization �Labadie 2004�. The works of Sharif
and Wardlaw �2000� and Oliveira and Loucks �1997� are consid-
S PLANNING AND MANAGEMENT © ASCE / MAY/JUNE 2009 / 141

ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

ered to be representative of these types of methods. The first typeseems to be mainly inspired from DP models and has the sameframework: dividing a large problem into several smaller ones,discretization of state and decision variables, etc. Accordingly,this type inherits some weaknesses of DP such as the “curse ofdimensionality,” especially when it is used for optimization ofmultireservoir systems. The second type is a direct search ap-proach using GA as its optimization method. In this type, thedecision variables are the parameters of a prescribed operatingrule or policy. The number of decision variables, in this type, islimited and generally much smaller than the first type. The shorterchromosomes make the computations of the second type moreeffective and applicable in multireservoir systems. Therefore, GAis utilized for optimization of parametric policies.

In this paper, the work of Momtahen and Dariane �2007� onthe direct search approach using genetic algorithms �DSGA� isextended to multireservoir systems. Moreover, some modifica-tions are proposed to enhance the computational efficiency of themethod, which is crucial in optimization of multireservoir sys-tems. The Greater Karoon system is used as an example to illus-trate the method and its modifications. This 16-reservoir system isoptimized in three steps. In the first step, DSGA is applied to athree-reservoir problem of the system and compared with SDPand DPR models in terms of computational performance and ob-jective values. Some modifications on GA operators are also pro-posed and evaluated. In the second step, the modified model isapplied to a seven-reservoir problem. Three operating policieswith different forms are obtained and their performances are com-pared. Two more innovative modifications are also presented andproven to be advantageous. In the third step, the final modifiedmodel is applied to the whole 16-reservoir system as a real worldproblem to demonstrate the capabilities and applicability of themodel. Moreover, the increase rate of calculations is estimated inthis final step.

All of the computations are performed on a simple personalcomputer with a Pentium IV, 2,800 MHz processor and 256Mbytes random-access memory.

Greater Karoon System

The Greater Karoon system, located in Southeast Iran, is one ofthe largest and most vital hydrosystems in the region �schemati-cally illustrated in Fig. 1�. Seven out of the 16 reservoirs are inthe process of construction with the remainder of the reservoirsalready designed and undergoing construction in the years tocome. The main purposes of the system are hydropower genera-tion, water supply �both domestic and agricultural�, and floodprevention. Water demands are assumed to be concentrated asthree lumped demands and are all located in the downstream ofthe basin.

The objective function of the problem is assumed to be theminimization of the overall operating losses or costs. Optimalreleases of the reservoirs should be defined as the results of opti-mization based on the states of the system, which are specified byreservoir storages and system inflows. The operating losses aredefined as the deviations from target storages and water demands.The corresponding loss function �Eq. �1�� consists of two parts:The first part �Eq. �2�� is related to the water supply purpose andthe second part �Eq. �3�� is related to hydropower generation andflood preventing purposes. For the purposes of water supply, thereleased water is first allocated to domestic demand which is pri-
ority and thereafter, for irrigation purposes. The hydropower gen-
142 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT


eration and flood preventing purposes are considered using targetstorages which have already been obtained from comprehensivedesign studies of each reservoir and are based on both structuralfeatures and hydrological aspects. The equations are as follows:

Min Zt = �t=1

N

�W1Z1,t + W2Z2,t� �1�

Z1,t = �j=1

ND

Wlj�Dtj − RDt

j�2 + We�SPt�2 �2�

Z2,t = �i=1

NR

Wsi�TSti − St

i�2 �3�

where N=length of operating time series; t=period number �from1 to N�; ND=number of water demands; and NR=number ofsystem reservoirs. Also, Dt

j = jth demand volume in period t;RDt

j =water released to meet Dtj; SPt=excess water spilling out

of the system; Sti=storage of reservoir i in period t; and TSt

i

=corresponding target storage, all in volumetric units. W1, W2,Wl, We, and Ws=weights of the objective function correspondingto water supply, hydropower generation, water deficits, excesswater, and deviation from target storages, respectively. In thisproblem, W2 is assumed to be 3 /7 �number of system demandsdivided by the number of existing reservoirs� and all otherweights are considered equal to 1 to show the same importancefor different purposes. Practically, these weights should be ad-justed based on the viewpoints of experts and decision makersusing multicriteria decision making procedures.

A 40-year monthly time series �from October 1955 to Septem-ber 1995� is available to define the system inflows entering eachof the reservoirs and also at the intersection of the Karoon andDez Rivers. Having the highest average monthly inflow, May isconsidered as the beginning of simulations when all the reservoirs

Fig. 1. Schematic of the Greater Karoon 16 reservoir system

are assumed to be initially full.

© ASCE / MAY/JUNE 2009

ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

Three-Reservoir Problem

The Karoon 1, Gotvand, and Dez Reservoirs, indicated, respec-tively, by the numbers 9, 11, and 16 in Fig. 1, are selected todefine the three-reservoir problem. This small multireservoirproblem is mainly studied to compare the performance of theDSGA approach with the more traditional ones. DPR, SDP, andGA with the linear form of operating policies �GAL� models areused to optimize the problem, representing ISO, ESO, and DSGAapproaches, respectively.

DPR Model

Using the deterministic dynamic programming �DDP� over thewhole simulation time series, the optimal values of the decisionvariables, which are the reservoir storages at the end of eachperiod, are obtained in the first stage of this model. The set ofoptimal decision values and their corresponding states of the sys-tem �the reservoir storages at the beginning of the period and theinflows� make up the state-decision vectors. Linear regression isthen used to fit operating rules to these vectors. This method wasinitially proposed by Young �1967� and then extended by Bhaskarand Whitlatch �1980�.

The general form of recursive relation for the DDP model usedin this research is

f tn�St� = Min

St+1

�Zt + f t+1n−1�St+1�� 4�

where t refers to the within-year period �month� and varies from 1to 480 for 40 years of monthly time series and n=stage variableof the backward DP, which is 1 when t=480. Zt=the system lossof Eq. �1� corresponding to period t.

The active storages �or state variables� of Reservoirs 9, 11, and16 are divided into 15, 6, and 12 discretizations respectively. Foreach month of the year and for each one of 1,080 �15�6�12�state combinations, 39 or 40 optimal decision values are gener-ated, corresponding to the 39- or 40-year data �39 values formonths from October to April because simulations start fromMay as noted before�. Therefore, 43,200 �1,080 state combi-nations�40 years� or 42,120 �1,080�39 years� optimal decision-state vectors are available for each month to achieve monthlypolicies. The volume discretization resolution is considered to beapproximately the same for the three reservoirs. Reservoir opera-tions obtained by the DDP model, as explained earlier, can beconsidered as the global solution in a discrete space to the long-term reservoir operation problem given a complete inflow fore-cast. Therefore, the objective value of the DDP model isconsidered as a reference which provides a lower bound on theachievable objective function value.

A linear function as shown in Eq. �5� is considered as the formof operating rules for each reservoir in the DPR model

Rti = at

iSti + bt

i�Iti + Qt

i� + ctiSt

sys + dtiQt

sys + eti �5�

where Rti=release of reservoir i in month t and It

i=water releasedfrom the upstream reservoirs to reservoir i, respectively. S andQ=initial storage and inflow �less the release from upstream res-ervoirs�, and sys is a superscript used to refer to the whole sys-tem. a, b, c, d, and e=policy parameters obtained by fitting thepolicy to the state-decision vectors.

As shown in Table 1, the loss function value of the DPR modelis 732.4�106 units, much greater than the losses from DDP�175.9�106�. It does not essentially mean that DPR performs
poorly because the loss value of DPR is obtained from its oper-


ating policies which can be used for future operations rather thanjust an optimization with perfect predictions of inflows like DDP.In other words, the loss value of DPR is only comparable to thatof the operating policies obtained from the other models. Ignoringthe insignificant CPU time required for establishing the regres-sions of DPR, the optimization process and consequently the timeused in DPR and DDP would be the same, as shown in Table 1.

SDP Model

This model is used as the representative of the ESO strategies.The model is applied frequently for optimization of reservoir sys-tems �e.g., Butcher 1971; Torabi and Mobasheri 1973; Stedingeret al. 1984�. The recursive relation of this backward moving SDPmodel is

f tn�St,Qt� = Min

St+1�Zt + �

Qt+1

P�Qt+1�Qt� . f t+1n−1�St+1,Qt+1�� 6�

where Zt is the same as Eq. �4�, t refers to the within-year oper-ating period �month�, and n to the total number of periods remain-ing before reservoir operation terminates. Qt=inflow pattern inperiod t and P=conditional probability, which is estimated fromthe observed inflow patterns of the historical time series.

A backward optimization process is iterated in annual cycles toachieve a steady state solution. The reservoir storage at the be-ginning of each period, St, and the system inflow, Qt, are the statevariables, whereas the reservoir storage at the end of the period,St+1, is considered as the decision variable. As previously indi-cated, Reservoirs 9, 11, and 16 are discretized into ten, six, andnine intervals, respectively, and three discretizations are consid-ered for both inflows.

The calculation results of the SDP model are shown in thesecond row of Table 1. These results reveal that the SDP with theobjective function value of 656.4�106 performs slightly betterthan DPR, but yet it is 3.7 times higher than the global solutionobtained by DDP �175.9�106�. Moreover, CPU time has de-creased substantially to about 5 h, which is less than half of DPR.

GAL model

A linear rule similar to the one used in DPR is considered as thepolicy structure of the GAL model. Its parameters, which are thedirect decision variables of the model, are estimated using aDSGA method rather than DDP. As the monthly linear rule ofEq. �5� contains five parameters �a, b, c, d, and e�, the GALmodel has 60 decision variables for each reservoir and 180 deci-sion variables altogether. Tournament selection, arithmetic cross-over, and arithmetic mutation are considered as the conventionaloperators of this real-coded GA �see Gen and Cheng 1997�. Foreach one of the policy parameters, a gene is designated and each

Table 1. Optimization Results of the Three-Reservoir Problem

Optimizationmethod Model

Objectivefunction��106�

CPU time�hh:mm:ss�

Differencefrom DDP

�%�

ISO DPR 732.4 12:10:04 316.3

ESO SDP 656.4 05:14:38 273.1

DSGA GAL 437.8 01:17:28 148.9

Deterministic DP 175.9 12:10:04 0.0

chromosome contains 60 genes.


ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

In Fig. 2�a�, the evolution trends of the model are shown forthree population sizes of generations. The population size of 200proves to perform the best in terms of both final objective valuesand computational times. The best values for the probabilities ofcrossover and mutation operators are 0.8 and 0.05, respectively,which are found by a trial and error procedure.

Because of the inherent randomness of GA operators, resultsof the GAL model are not exactly the same in different runs of themodel, and a single run should not be used as a good base formodel evaluation. Therefore, average results of five to ten runsare considered here for the evaluation of GAL performance andfor comparison purposes.

Moreover, there are several methods for terminating the evo-lution of GAL. In this paper, stabilization of fitness values isassumed as the termination criterion to define the processing timeof the runs. The stabilization is defined to take place when theimprovement within the last 500 evaluated solutions �chromo-somes� stays below a level of 50.0 units.

The results show that GAL performs much better than the twoother models. It reaches an objective value of 437.8�106 inabout 1 h of CPU time, much less than the other methods, andprovides an improved objective value.

Comparison of Models

The objective function values obtained using the above-mentioned three models and their corresponding computationaltimes are shown in Table 1. Given a complete inflow forecast, theobjective value of the DDP �175.9�106� is considered as the bestpossible operating loss of the system and a comparison base forthe objective values of the operating policies. The loss function ofGAL operating policies �437.8�106� is much less than that ofDPR �732.4�106� and SDP �656.4�106�. Moreover, the CPUtime of GAL is only 1 h and 17 min where that of SDP and DPRare about 12 and 5 h, respectively. These results prove the GAL to

Conventional GA

800

750

700

650

600

550

500

450

400

00:00

00:10

00:20

00:30

00:40

00:50

01:00

01:10

01:20

01:30

01:40

01:50

Millions

ObjectiveFunction

Population=100

Population=200

Population=300

(a)

Fig. 2. Objective function evolution trend of G

be a promising model for optimization of larger systems.



Modifications in GA

Although the GAL model and the DSGA method prove to havethe best performance in optimization of the three-reservoir prob-lem, some modifications can still be presented to enhance theirperformance and to reach better objective function in less com-putational time. On the other hand, the performance of the DSGAmethod is sensitive to its parameters and probabilities, especiallyits population size and mutation probability. In fact, the maindrawbacks of the DSGA approach are the sensitivity problemalong with diversity of results in different runs. Some suggestionsare proposed here to reduce these drawbacks.

Besides the conventional operators and methods of real-coded GAs, several other ones are proposed and proved to bemore efficient. Readers can refer to GA textbooks such as Deb�2001� for detailed explanations. After trying several options andoperators, elitist tournament selection, blend crossover or BLX�Eshelman and Shaffer 1993�, and normal distribution mutation�Schwefel 1981� are selected as the components of the modifiedGAL model. These operators are reported to be at least as effec-tive as other operators in the literature �see Deb 2001�. The com-ponents of the modified GA in comparison with the conventionalone are listed in Table 2.

First and foremost, normal distribution mutation is defined as

y = x + N�0,�� 7�

where x and y=initial and muted values of a gene; N�0,��=random value generated using a normal distribution with aver-

Table 2. List of GA Modifications

Characteristic Conventional GA Modified GA

Generation methodof initial population

Uniformdistribution

Normal distribution

Size of offspring 1�size of parents 5�size of parents

Selection operator Tournament Elitist tournament

Crossover operator Arithmetic BLX-0.5

Mutation operator Arithmetic Normal distributionusing updating rule

Modified GA

00:00

00:10

00:20

00:30

00:40

00:50

01:00

01:10

01:20

01:30

01:40

01:50

02:00

( )

Parent pop.=20

Parent pop.=50

Parent pop.=100

(b)

odels: �a� conventional GA; �b� modified GA

02:00

AL m


ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

age of 0; and �=standard deviation of the normal distribution,which can be considered as the power of the mutation. This typeof mutation is extensively used in evolutionary strategies, anothertype of evolutionary algorithm very similar to the real-coded GA.

Moreover, the following updating rule suggested by Schwefel�1981� is applied to the modified GA to adjust the value of � ingenerations of chromosomes:

�k+1 = cd . �k if Ps �15

�k/cd if Ps �15

�k otherwise Ps = 15

�8�

where k=generation number; Ps=percentage of successful muta-tions; and Cd=0.817 as suggested by Schwefel �1981�. Success-ful mutations are those which improve the fitness correspondingto the mutated chromosome. The initial value of � is estimatedexperimentally.

Mutation has a crucial role in finding true optimal solutionsusing real-coded GA. The updating rule of Eq. �8� can signifi-cantly reduce the sensitivity of the modified GA to its parametersand probabilities which are generally defined based on previousresearch experiments or by a trial and error procedure.

Table 3 and Fig. 2 illustrate the results of the modified and theconventional GAL models. The final objective functions and thecalculation times of three population sizes of parent generationsare presented in Table 3 and the evolution trend for each one ofthe population sizes during the computations is displayed in Fig.2. It is clear that the loss functions and the CPU times of themodified GAL model are substantially less than those of the con-ventional one. Moreover, the trends of different population sizesare much closer in the modified model �Fig. 2�. Therefore, themodifications prove to be considerably effective to reduce thesensitivity of the model to its parameters such as population sizes,and consequently to increase the robustness of the model.

Seven-Reservoir Problem

This problem of the Greater Karoon system consists of sevenconstructed or under construction reservoirs and is considered asa sufficient problem to demonstrate the values of the DSGA ap-proach in optimization of large reservoir systems and to describesome other effective modifications. The Karoon 4, Karoon 3,Karoon 1, Godarlandar, Gotvand, Bakhtiari, and Dez Reservoirs,which are numbered as 3, 7, 9, 10, 11, 15, and 16, respectively, inFig. 1 are selected. Dynamic-programming-based models, such as

Table 3. Comparison of Conventional and Modified GAL ModelsPerformance

GA typeParent

population


CPU.time�h:mm:ss�

Conventional 100 570.68 0:13:56

200 437.85 1:17:28

300 448.05 1:45:30

Modified 20a 447.43 0:16:31

50a 433.23 0:26:33

100a 433.73 0:40:41

Note: h=hours; mm=minutes; and ss=seconds.aSize of offspring=5�size of parents.

SDP and DPR, are not able to readily solve such problems prop-



erly because of the large dimensions of state and decision vari-ables. Therefore, DSGA is the only approach utilized here tooptimize this problem.

In addition to the linear policy of Eq. �4�, two forms of piece-wise linear rules are also considered as the other policy forms ofsystem operation. The operating rules of the first form, which ismentioned as the PL policy, are defined by a piecewise linearfunction �three segments� for each reservoir, as shown in Fig. 3.In Fig. 3, y, the dependent variable, is the release of the reservoirand x, the independent variable, is a linear polynomial of statevariables described as

xti = St

i + ati�It

i + Qti� + bt

iStsys + ct

iQtsys �9�

where the variables are the same as in Eq. �5�.A close examination of the rule reveals each linear piece of the

PL policy to be tantamount to a linear rule of Eq. �5� which isconnected to at least another linear piece.

The second form of piecewise linear policies, which here iscalled “2S,” is a two-stage operating policy similar to the oneused by Oliveira and Loucks �1997�. In the 2S policy, the releasesof the reservoirs are determined using two groups of piecewiselinear rules. The dependent and independent variables of the firstand the second rule groups are respectively defined as

Step 1: y = f�x� where y = Rtsys, x = St

sys + Qtsys �10�

Step 2: y = f�x� where y = St+1iR , x = St+1

sys �11�

where f represents a piecewise linear rule; t=time period;iR=index of reservoirs; and sys refers to the whole system.

In the first step of the 2S policy, total release by the wholesystem �Rt

sys� is obtained from Eq. �10� based on total availablewater in the system, which is the whole system storage �St

sys� plusthe whole system inflow �Qt

sys�. Then, the system storage in thenext period �St+1

sys� is calculated using the continuity equation. Inthe second step, St+1

sys is used in Eq. �11� as the independent vari-able to define all the reservoir storages of the next period as theoperating decisions. In each operating period, the parameters ofthe Step 1 piecewise linear rule are determined for the wholesystem and a group of rules for Eq. �11� is determined for eachreservoir.

Eight and six broken lines are used for defining the rules ofEqs. �10� and �11�, respectively, and policy parameters are deter-mined using the modified GA method. Optimization results ofDSGA models are shown in Table 4 for linear �LR�, simple piece-wise linear �PL�, and two-stage piecewise linear �2S� policies.The second column of Table 4 represents the number of variablesor parameters of each policy. The value of the loss function and

Fig. 3. Typical piecewise linear function

the termination times are given in the last two columns. The evo-


ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

lution trends of the objective function for these optimizations areshown in Fig. 4. Based on these results, the PL policy is signifi-cantly better �27% lower objective function value� than the LRpolicy. On the other hand, 2S policy gains only 5.0% improve-ments in the objective function when using twice the time of thePL method.

Sample results of the 2S policy are illustrated in Figs. 5 and 6.Fig. 5�a� shows the operating rules of the whole system related toEq. �10�, and Fig. 5�b� is the resulting rule for the Dez Reservoirrelated to Eq. �11�. Both sample rules are for October. Fig. 6compares the amount of releases for the Dez Reservoir and itstarget releases for the last 10 years of simulation, as a sample ofthe optimized 2S rule operation.

Table 4. Optimization Results of Different Policy Forms for the Seven-Reservoir System

Model

Numberof

variables


CPU.time�h:mm:ss�

LR 420 750.511 1:16:08

PL 756 545.955 1:23:42

2S 1,200 519.923 2:55:23

Note: h=hours; mm=minutes; and ss=seconds.

2,000

1,800

1,600

1,400

1,200

1,000

800

600

400

00:00

00:15

00:30

00:45

01:00

01:15

01:30

01:45

02:00

02:15

02:30

02:45

03:00

Millions

Time (hh:mm)

ObjectiveFunction

Linear (LR)

Piecewise Linear (PL)

2-Step PL (2S)

Fig. 4. Evolution trends of the objective function values for differentpolicies in the seven-reservoir problem

Step 1: Rule of the whole system

0

500

1000

1500

2000

2500

0 2000 4000 6000 8000 10000 12000 14000

Stsys+Qt

sys

Rtsys

(a)

Fig. 5. Sample rule of the 2S policy: �a� Step 1



Further Modifications

Two modification methods of using the Fourier series and simu-lation with varying length �here called stepwise GA� are proposedto further improve the computational efficiency of the DSGA andto also reduce its processing times. In the first method, a Fourierseries is used to define seasonal variations of policy parameters�Momtahen and Dariane 2007�. The Fourier series is defined as acontinuous function of operating time for each policy parametersuch that values of the parameter can be calculated for each pe-riod of time. For example at

i in Eq. �9� is defined using five termsof Fourier series as

ati = ai�t� = A0

i + A1i cos�2�t/T� + B1

i sin�2�t/T�

+ A2i cos�4�t/T� + B2

i sin�4�t/T� �12�

where T=total number of within-year operating periods and A0,A1, A2, B1, and B2=Fourier multipliers. By determining thesemultipliers, at

i can be calculated for each within-year time periodt. Therefore, instead of 12 monthly separate policy parameters ofat

i, only five parameters �Fourier multipliers� must be defined,which causes a considerable reduction in the length of chromo-somes. This makes the problem simpler and the GA much moreeffective, especially in large systems with long chromosomes.

In the stepwise GA, using varying length in simulations as thesecond improvement method, the fitness evaluation of GA chro-mosomes is modified. In fact, in earlier generations, a preciseevaluation is not necessary and a rough evaluation could be ac-complished using only part of the simulation data. This substan-tially cuts down the computational time, which is very importantin large systems. In the early generations of the GA, it would be

Step 2: Rule of Dez reservoir

0

500

1000

1500

2000

2500

3000

3500

4000

0 2000 4000 6000 8000 10000 12000St+1

sys

t+1

(b)

hole system�; �b� Step 2 �one of the reservoirs�

0

200

400

600

800

1000

1200

1400

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

Operating period

Actual ReleasesTarget Releases

Release(mcm)

Fig. 6. Target releases of Dez Reservoir and its actual releasesresulting from the 2S policy

SiR

�the w


ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

sufficient to evaluate the fitnesses imprecisely using just a smallpart of whole simulation period, as GA mainly explores the searchspace rather than exploiting the solution. The simulation period isthen extended gradually in the later generations until the wholelength is used to determine the optimal solution. Steps of thismethod are as follows:1. Consider an initial simulation length �N0

k� for evaluating fit-nesses of the first generation �k=1�—10–20% of wholesimulation length �N� is proposed for N0

k.2. Evaluate the chromosomes of this generation using simula-

tion length of N0k and make the offspring generation by this

evaluation.3. Determine the objective function of the best parent using all

N periods of simulation.4. Go to the next generation �k=k+1�, evaluate the chromo-

somes using simulation length of N0k, make the new genera-

tion of offspring, and determine objective function of thebest parent using the whole simulation periods.

5. If the objective function of the whole simulation periods isweaker than that of previous generations, increase N0

k by �N�with maximum amount of N�. A proposed value for �N is10% of N.

6. If N0k is less than N then go to Step 4. Otherwise exit these

steps and continue normal GA procedures using the wholesimulation periods.

Fig. 7 and Table 5 show optimization results of the GA-2Smodel obtained using the normal DSGA method and its modifiedversion. These results confirm the effectiveness of the above-mentioned modifications so that the processing time is reduced to41% of the normal method using both modifications together.Also, the objective function is improved when the Fourier series

Table 5. Results of More Modifications in the GAL Model with 2S Poli

MethodNumber ofvariables


Simple 1,200 519.92

Fourier 500 495.61

Varied simulation 1,200 528.37

Fourier and varied simulation 500 497.69

1,400

1,300

1,200

1,100

1,000

900

800

700

600

500

400

00:00

00:15

00:30

00:45

01:00

01:15

01:30

01:45

02:00

02:15

02:30

02:45

03:00

Millions

Time (hh:mm)

ObjectiveFunction

Simple

With varied simul.

Fourier

Fourier and varied Sim.

Fig. 7. Comparison of evolution trend between simple and modifiedmethods



method is used. This is because improvement of GA optimizationperformance is achieved by decreasing the length of chromo-somes especially in large systems. On the other hand, the stepwiseGA method produces acceleration especially in the beginning ofits computations and reduces the processing time. The effective-ness of this method is more considerable when a simpler termi-nation criterion is used for any reason.

Due to the above-mentioned results, both Fourier series andstepwise GA methods are used to apply the selected optimizationmethod on a very large 16-reservoir problem. Both PL and 2Spolicies are used as two optimization models of this problem.None of these models can be definitely selected as the superior.Although the objective function of the 2S model �386.8 millionunits� is slightly less than that of the PL model �398.7 millionunits�, the PL is better in terms of computational time and parsi-mony of parameters. In fact, the computational time for the 2Smodel is more than 10 h whereas that of the PL model is onlyabout 5 h.

The models are also used in optimization of some other con-figurations of the system with 3, 5, 7, and 15 reservoirs. Fig. 8shows the computational time of different system configurations.The computation time of these models is estimated to increasewith the second power of the number of the reservoirs. This rateis favorable in comparison with the exponential rate of increase incomputational time for DP-based models. The proposed modelsare therefore evaluated to be applicable even for larger and morecomplex real multireservoir systems.

The proposed DSGA models is a step forward in obtaining anideal operating optimization model for multireservoir systemswhich has been one of the major challenges in water resourcesmanagement. However, it seems that further improvements areeasily achievable with the DSGA method. For example, compu-

Calculatedtime

�h:mm�

Objective function at

30 min 60 min 120 min

2:55 1,104.13 666.37 542.08

1:16 529.34 499.75 489.91

2:02 617.02 559.09 527.45

1:13 514.38 500.18 492.80

y = 2.4778x1.9796

y = 1.8783x1.8247

0

100

200

300

400

500

600

700

0 5 10 15 20

System reservoirs

Com

putationalTime(min)

Simple PL

Double Stage PL

Power (Double Stage PL)

Power (Simple PL)

Fig. 8. Rate of computational time in the models

cy


ge. 2009.135:141-148.

Dow

nloa

ded

from

asc

elib

rary

.org

by

KM

UT

T K

ING

MO

NG

KU

T'S

UN

IV T

EC

H o

n 04

/27/

14. C

opyr

ight

ASC

E. F

or p

erso

nal u

se o

nly;

all

righ

ts r

eser

ved.

tational time can be reduced using parallel computations in GA.Moreover, multiobjective optimizations are better handled usingthis method. Any type of objective functions, even inseparableones—such as a system’s reliability—can be optimized by thismethod. Optimized policies are therefore capable to minimize op-erating loss functions and maximize system reliability at the sametime. Further research would reveal more advantages of DSGA.

Summary and Conclusion

The application of DSGA is evaluated in this paper for optimiza-tion of simple to complex multireservoir systems. Some modifi-cations are proposed to improve the effectiveness of the methodand the modifications are evaluated using the problems of theGreater Karoon multireservoir system. As the first problem, athree-reservoir system is considered and results of a DSGA modelnamed GAL is compared with DPR and SDP, traditional optimi-zation models. GAL model is then improved using some modifi-cations in the operators of GA. One of the importantmodifications is the utilization of normal distribution mutationwith an updating rule for power of the mutation. In addition to theimprovement of the final solution of the model, the modificationsdecreased the sensitivity of results to GA parameters such aspopulation size. The modified model is then used for optimizationof a seven-reservoir system as the second problem. In addition tothe linear policy, two types of piecewise linear policies are opti-mized and compared in this system. Two other modification meth-ods are proposed to decrease the computational time of theoptimization process, an issue which is very important in model-ing of large systems. These use the Fourier series for definingperiodical variation of policy parameters and stepwise GA, byvarying length of simulation periods in GA generations. Thesemodifications have reduced the computational time of the modelsby about 60%.

Two resulting models using all modifications are finally ap-plied to the whole Greater Karoon system with 16 reservoirs as areal large multireservoir system. For computational time, an in-creased rate of these models is estimated to be about second orderof number of system reservoirs.

The proposed models which are the combinations of proposedmodified DSGA method and piecewise linear policies are evalu-ated as very robust and powerful. They seem to be a step towardthe ideal optimization model for operation of multireservoir sys-tems. The models are capable to optimize large and complex mul-tireservoir systems with all details of simulation models. Nosystem simplification is required for optimization and the modelscan optimize any type of objective functions, even nonseparableones such as reliability functions. The optimization processes arecompleted in a reasonable time, and the time reduces more byutilizing GA parallel computations. Further research on the



method is highly recommended especially on optimization ofmultiobjective problems.

References

Bhaskar, N. R., and Whitlatch, E. E. �1987�. “Comparison of reservoirlinear operating rules using linear and dynamic programming.” WaterResour. Bull., 23�6�, 1027–1036.

Butcher, W. S. �1971�. “Stochastic dynamic programming for optimumreservoir operation.” Water Resour. Bull., 7�1�, 115–123.

Cai, X., McKinney, D. C., and Lasdon, L. S. �2001�. “Solving nonlinearwater management models using a combined genetic algorithm andlinear programming approach.” Adv. Water Resour., 24�1�, 667–676.

Chen, L. �2003�. “Real coded genetic algorithm optimization of long-term reservoir operation.” Environ. Urbanization, 39�5�, 1157–1165.

Deb, K. �2001�. Multi-objective optimization using evolutionary algo-rithms, Wiley, Chichester, U.K., 81–165.

Esat, V., and Hall, M. J. �1994�. “Water resources system optimizationusing genetic algorithms.” Hydroinformatics ‘94: Proc., 1st Int. Conf.on Hydroinformatics, Balkema, Rotterdam, The Netherlands, 225–231.

Eshelman, L. J., and Shaffer, J. D. �1993�. Proc., 2nd Workshop on Foun-dations of Genetic Algorithms, Vail, Colo., Morgan Kaufmann, 187-202.

Gen, M., and Cheng, R. �1997�. Genetic algorithms and engineeringdesign, Wiley, New York.

Goldberg, D. E. �1989�. Genetic algorithms in search, optimization andmachine learning, Addison-Wesley, Reading, Mass.

Labadie, J. W. �2004�. “Optimal operation of multireservoir systems:State-of-the-art review.” J. Water Resour. Plann. Manage., 130�2�,93–111.

Momtahen, Sh., and Dariane, A. B. �2007�. “Direct search approach usinggenetic algorithm for optimization of water reservoir operating poli-cies.” J. Water Resour. Plann. Manage., 133�3�, 202–209.

Oliveira, R., and Loucks, D. P. �1997�. “Operating rules for multireservoirsystems.” Water Resour. Res., 33�4�, 839–852.

Schwefel, H. P. �1981�. Numerical optimization of computer models,Wiley, Chichester, U.K.

Sharif, M., and Wardlaw, R. �2000�. “Multireservoir systems optimizationusing genetic algorithms: Case study.” J. Comput. Civ. Eng., 14�4�,255–263.

Stedinger, J., Sule, B., and Loucks, D. �1984�. “Stochastic dynamic pro-gramming models for reservoir operation optimization.” Water Re-sour. Res., 20�11�, 1499–1505.

Torabi, M., and Mobasheri, F. �1974�. “A stochastic dynamic program-ming model for the optimum operation of a multi-purpose reservoir.”Water Resour. Bull., 9�6�, 1089–1099.

Tung, C., Hsu, S., Liu, C. M., and Li, Sh., Jr. �2003�. “Application of thegenetic algorithm for optimizing operation rules of the LiYutan reser-voir in Taiwan.” J. Am. Water Resour. Assoc., 39�3�, 649–657.

Wardlaw, R., and Sharif, M. �1999�. “Evaluation of genetic algorithms foroptimal reservoir system operation.” J. Water Resour. Plann. Man-age., 125�1�, 25–33.

Young, G. K. �1967�. “Finding reservoir operating rules.” J. Hydr. Div.,93�6�, 297–321.


ge. 2009.135:141-148.

Optimization of Multireservoir Systems Operation Using Modified Direct Search Genetic Algorithm

Documents

Transcript of Optimization of Multireservoir Systems Operation Using Modified Direct Search Genetic Algorithm