Journal of Hydrology 390 (2010) 66–74

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Journal of Hydrology

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Constrained genetic algorithms for optimizing multi-use reservoir operation

Li-Chiu Chang a, Fi-John Chang b,*, Kuo-Wei Wang b, Shin-Yi Dai b

a Department of Water Resources and Environmental Engineering, Tamkang University, Taipei, Taiwan, ROCb Department of Bioenvironmental Systems Engineering, National Taiwan University, Taipei, Taiwan, ROC

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 March 2010Received in revised form 14 June 2010Accepted 18 June 2010

This manuscript was handled by G. Syme,Editor-in-Chief, with the assistance ofSoroosh Sorooshian, Associate Editor

Keywords:Reservoir operationConstrained genetic algorithms (CGA)Penalty strategyEcological base flow

0022-1694/$ - see front matter � 2010 Elsevier B.V. Adoi:10.1016/j.jhydrol.2010.06.031

* Corresponding author. Tel.: +886 2 23639461; faxE-mail address: changfj@ntu.edu.tw (F.-J. Chang).

To derive an optimal strategy for reservoir operations to assist the decision-making process, we propose amethodology that incorporates the constrained genetic algorithm (CGA) where the ecological base flowrequirements are considered as constraints to water release of reservoir operation when optimizing the10-day reservoir storage. Furthermore, a number of penalty functions designed for different types of con-straints are integrated into reservoir operational objectives to form the fitness function. To validate theapplicability of this proposed methodology for reservoir operations, the Shih-Men Reservoir and itsdownstream water demands are used as a case study. By implementing the proposed CGA in optimizingthe operational performance of the Shih-Men Reservoir for the last 20 years, we find this method pro-vides much better performance in terms of a small generalized shortage index (GSI) for human waterdemands and greater ecological base flows for most of the years than historical operations do. We dem-onstrate the CGA approach can significantly improve the efficiency and effectiveness of water supplycapability to both human and ecological base flow requirements and thus optimize reservoir operationsfor multiple water users. The CGA can be a powerful tool in searching for the optimal strategy for multi-use reservoir operations in water resources management.

� 2010 Elsevier B.V. All rights reserved.

1. Introduction

Reservoirs can increase the reliability of water supplies in pro-moting livelihoods, raising agricultural productivity, and reducingthe farmers’ vulnerability to droughts. Although the annual precip-itation is abundant in Taiwan, the frequency and intensity of ex-treme hydrological events (such as floods and droughts) varyhighly, which obviously restricts the effective use of water andcauses insufficient water supply. To overcome the problem ofinsufficient water supply during periods of low flow, attentionhas been focused on improving water resources management,especially in optimization of reservoir operations (Chang andChang, 2001; Chang et al., 2005; Chen et al., 2007). The majorityof researches in water resources are devoted to developing waysto meet human water demands, while the explicit inclusion ofenvironmental flow targets has received less attention. Lately,we, as water resources engineers, often face criticisms regardingwater resources management due to the lack of careful assess-ments of environmental impacts caused by reservoir operations.In an attempt to minimize these negative impacts, the related envi-ronmental aspects must be taken into account in the managementof water storage facilities (Bauer and Olsson, 2008; Galizia Tundisi

ll rights reserved.

: +886 2 23635854.

and Matsumura-Tundisi, 2003; Gibbins et al., 2001; McCartneyet al., 2005; Suen and Eheart, 2006).

In the last decades, the minimum environmental flow targetgained the attention of scientists, engineers, and managers (Pusey,1998; Stalnaker and Arnette, 1976; Tennant, 1976). The target wasdesigned to maintain the minimum stream flows for sustainingaquatic ecosystem. It provided a single value that could be easilyimplemented as another objective to be optimized in water man-agement programs. The implementation of a single flow, however,was found to eliminate the flow variability sustaining healthy anddiverse aquatic communities. Lately, there has been a great deal ofeffort devoted to estimating environmental flow requirements andto the identification of the need for maintaining ecological flows toprotect freshwater biodiversity (Richter et al., 1996; Poff et al.,1997; Lytle and Poff, 2004). We now understand the importanceof mimicking components of the natural flow regime as targetsfor managing flow magnitude, frequency, and timing variables(Arthington et al., 2006; Chang et al., 2008, 2009). The flow regime,however, contains too many variables, which makes it very diffi-cult to implement in water resources management. Moreover,translating general hydrologic-ecological principles and knowl-edge into specific management rules for particular river basinsand reaches has remained a daunting challenge (Poff et al.,2003). Harman and Stewardson (2005) also mentioned that onlya few published studies explicitly described the link between envi-ronmental flow targets and dam operating rules. To resolve the

L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74 67

river management-driven issues and bridge the gap between asimple minimum stream flow target and the great complexity ofthe natural flow regime, we propose a realistic and feasible ap-proach that incorporates the human demands with the most essen-tial aspects of natural flow variability (i.e., the seasonal base flowrequirements) as the key targets in searching for the optimal strat-egy of reservoir operation management.

In recent decades, evolutionary computation techniques, suchas genetic algorithms, have become popular in global optimizationapplication of reservoir planning and management in science andengineering (Cai et al., 2001; Huang et al., 2002; Suen and Eheart,2006; Chen and Chang, 2007; Kerachian and Karamouz, 2007;Chang, 2008; Chaves and Chang, 2008; Chen and Chang, 2009).Most developed algorithms are capable of handling complex plan-ning and management problems. Evolutionary computation tech-niques have received a great deal of attention regarding theirpotential in optimization techniques for complex systems. Never-theless, they have not really produced a significant breakthroughin complex nonlinear problems because they do not address the is-sues of constraints in a systematic way. Only recently have severalmethods been proposed for handling nonlinear constraints by evo-lutionary algorithms for numerical optimization in water resourcesproblems (Chang, 2008; Katsifarakis and Petala, 2006).

This study proposes a constrained genetic algorithm (CGA) formulti-use water resources management that incorporates humanneeds and ecological sustainability requirements. We take thereservoir’s downstream ecological seasonal demands into consider-ation and intend to minimize the influence of hydraulic engineeringstructures on the river aquatic environment. First, the ecologicalbase flow requirements are considered as constraints to water re-lease of reservoir operation. Next, for each constraint, a reasonablepenalty function is established in a constrained genetic algorithmadopted to search for the feasible solutions of reservoir operationsin different yearly hydrological events in order to achieve theecological base flow requirements of the Tahan River in wet anddry seasons. In summary, the CGA method is implemented toidentify reference flows and assess the impacts on generalizedshortage index (GSI) of human needs and ecological base flowrequirements.

Fig. 1. The architecture of the CGA search process.

2. Constrained genetic algorithms

The genetic algorithm proposed by John Holland (Holland, 1975)has been broadly utilized to solve different optimization problems.The GA mimics some processes observed in natural evolution. Itstarts with a randomly generated initial population and progressesto improve the fitness of solutions through iterations by imple-menting operators including scheme selection (reproduction),crossover and mutation operators. Though the GAs are very effi-cient in searching for the optimal solutions of unconstrained prob-lems, they still encounter some difficulties in solving constrainednonlinear optimization problems. Most optimization problems,however, are constrained problems. In other words, the searchspace is restricted to some limited subspace for optimization. Forhighly constrained problems, it is quite common to get mostly infe-rior solutions of small fitness values or infeasible solutions duringthe early stage of the search processes. It would usually cause earlyconvergences or inefficient searches with ignorance of the infeasi-ble and inferior solutions.

When the GAs are applied to nonlinear constrained problems,constraint handling becomes critical. During the past few years,several methods, such as rejection, repair, and penalization strate-gies, have been presented for handling constraints when using GAsto solve constrained optimization problems (Deep and Dipti, 2008).The advantage of the rejection and repair approaches is that only

feasible solutions are generated. For highly constrained problems,these approaches, unfortunately, could make the search processextremely difficult because of the large number of infeasible solu-tions in population, which certainly restricts the evolution pro-cesses of the GAs and makes the search process inefficient infinding a solution without any constraint violation. The most pop-ular approach to handling constraints in the GA community is touse penalty functions (Chang, 2008). A penalty function convertsa constrained optimization model to an unconstrained one to suitthe application of evolutionary techniques. The penalty cost is afunction designed to penalize infeasible solutions by reducing theirfitness values in proportion to the number and the magnitude ofconstraint violations and is also a penalty multiplier to effectivelyguide the search direction towards promising solutions in the ge-netic search (see Fig. 1).

The penalty functions for problems with inequality constraintsare divided into two categories: the exterior and the interior func-tions. For the interior penalty functions, the constraints act as bar-riers to forcing the generated search points to be always within thefeasible domain (Abedian et al., 2006). This is the reason why theinterior penalty functions could not be incorporated into GA opti-mization methods. Nevertheless, for the exterior penalty functions,a penalty value is added to the violated solutions by taking thenumber of violated constraints and their distance to the feasibledomain into account. Lately, researchers have proposed someeffective penalty functions to improve the efficiency of the GAs(Deep and Dipti, 2008; Katsifarakis and Petala, 2006; Michalewiczand Schoenauer, 1996; Richardson et al., 1989). However, there is

68 L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74

no general guideline for designing effective penalty functions andthese methods are quite problem dependent.

In general, an optimization problem can be written in nonlinearprogramming form:

min f ð~xÞ ð1Þ

subject to gið~xÞ 6 0 i ¼ 1;2; . . . ; l ð2Þhjð~xÞ ¼ 0 j ¼ 1;2; . . . ;m ð3Þxl

k 6 xk 6 xukð~xÞ ¼ 0 k ¼ 1;2; . . . ;n ð4Þ

where f ð~xÞ is the objective function with n decision variables (~x),gið~xÞ is the ith inequality constraint, and hjð~xÞ is the jth equality con-straint. xl

k and xuk are lower and upper bounds on the kth variable xk .

To easily implement the GA in solving an optimization problem,we usually adopt the external penalty functions to convert a con-strained optimization problem into an unconstrained one. Thereare two possible ways to construct the fitness function with pen-alty terms. The easiest way to construct a fitness function is toadd penalty values associated with those infeasible solutions tothe objective function.

Fð~xÞ ¼f ð~xÞ x 2 feasible regionf ð~xÞ þ pð~xÞ x R feasible region

�ð5Þ

where pð~xÞ is the penalty function. However, there is no straightfor-ward way to find a suitable and effective penalty function, and atroublesome trial and error procedure must be conducted in generalto obtain a suitable set of penalty functions. In this study, we de-voted a large amount of effort to combining different types of pen-alty functions. Our experiences showed that the multiplicationform, overall, performs more effectively in obtaining reasonablesolutions. Consequently, the corresponding penalty function of Eq.(6) is used to construct the fitness function.

pð~xÞ ¼Yl

i¼1

1þ ci maxðgið~xÞ; 0Þð Þ �Ymj¼1

1þ djjhjð~xÞj� �

ð6Þ

where ci and di are the penalty weights of the inequality and equal-ity constraints, respectively.

3. Case study

3.1. Study area

The Tahan River is 135 km long with an average channel slopeof 1/37, and its watershed covers an area of 1163 km2. The wa-tershed has received an annual average of 2410 mm of rainfall inthe last 40 years; however, the rainfalls were uneven—over 70% oc-curred between May and October (typhoon seasons). The Shih-Men Reservoir, located upstream of the Tahan River, satisfies themultiple purposes of water supply including irrigation, hydro-power generation, industrial and domestic water uses, flood con-trol, and recreation. The watershed of the Shih-Men Reservoircovers an area of 763.4 km2 with an effective storage capacity of2.35 � 108 m3. The water supply for human demands, such ashydropower generation, agriculture and industry uses togetherwith drinking water, is under pressure nowadays. This studymainly concentrates on how to arrange water release from the res-ervoir to best satisfy human demands and ecological base flowrequirements in the downstream area of the Shih-Men Reservoir.Fig. 2 shows the study area of watershed, with the thick line indi-cating the downstream area and the operation system of the Shih-Men Reservoir and the Tahan River. Fig. 3 presents the time seriesof annual flows and wet and drought seasonal flows during theperiod of 1984–2003. We can find great variability in those series,

and the flow in the wet season is much greater than the flow in thedrought season.

3.2. Ecological base flow requirements

Generally, the minimum flow designates a minimum range ofwater flow that prevents the regional ecosystem from significantharm, and above which water is available for reasonable beneficialuse. The minimum base flow requirements should consider theability of aquatic communities to remain healthy. Recently, a min-imum base flow requirement was proposed for the water resourcesmanagement by the Water Resources Agency in Taiwan. We, how-ever, like to note that an obtained optimal minimum base flowmay not be appropriate for establishing the flow regime of the en-tire river. Furthermore, the optimal minimum base flow could onlyprovide a viable research starting point for discussion, yet not nec-essarily be adopted as the minimum flow required for the river.

The determination of minimum base flow may be based on awide variety of existing information, including biological and topo-graphical information, scientific literature, water flow data, andoutput from computer simulation models. There are methods forgenerating the minimum base flow, such as: (1) using historicalflow records to identify the natural flow over time; (2) investigat-ing the hydraulic geometry of the river; (3) using local and tradi-tional knowledge to establish flow; and (4) matching flow levelsin order to maintain aquatic species. Since most of these methodswere developed according to site-specific requirements, there areno universal methods that can be used to transfer (Bauer and Ols-son, 2008). Table 1 provides minimum base flow requirementsbased on the habitat methods (0.6–4 cms) and the historical flowmethod (Q95) (Chang, 2004) in the Tahan River where data wereacquired from previous studies. In this study, we also computethe minimum 10-day average flows in dry and wet seasons onthe basis of historical data of the last 40 years (1964–2003). It ap-pears that the minimum 10-day average flows for dry and wet sea-sons are 6.75 cms and 12.28 cms, respectively. We regard thesevalues as the minimum ecological base flow requirements that willbe deemed as constraints to objectives in searching for the optimalreservoir operation strategy of water resources management.

4. The reservoir operation formulation

4.1. Objective function

We consider the human demand and ecological base flowrequirements of the downstream area in order to avoid long-termwater deficiency and overly focus on short-term water deficiency.These two goals, however, should not be considered as two inde-pendent objectives since they are not always in conflict. In thisstudy, we consider the ecological base flow requirements as con-straints when optimizing the reservoir operations. So, the objectivefunction is the human demand shown in the Eq. (7). Our goal is tooptimize the objectives while complying with relevant regulationsand physical constraints. The objective function should primarilymeet different user requirements and prevent from serious watershortages that would affect the downstream environment and eco-system and eventually severely harm flora and fauna. Conse-quently, the amount of deficiency in each time step is squaredfor rapidly increasing the function value, which can apparently dis-play the volume of water shortage within a specific 10-day period.Meanwhile, a continuous water shortage might be inconvenient forall users and also result in a cumulative effect on the ecosystem.Therefore, the objective function is defined cumulatively as thesquare of the deficit in each period multiplied by the number ofcontinuous periods without water release, shown in Eq. (7).

Ecological base flow

requirements

Human consumption

Taiwan China

Tanshui River Keelung River

Hsintien River

Tahan River

Tahan River

Shihmen

W E

S

N

Reservoir

Weir

Study area

30°N

120°E

Fig. 2. The locations of the Shih-Men Reservoir and the Tahan River.

L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74 69

min f ð~SÞ ¼minX36

i¼1

max 0;Di � Rdi

Di

� �� �2

� ni

" #( )ð7Þ

where Di and Rdi are the human water demand and downstreamwater release in the ith 10-day period. The purpose of the variable,ni; is to enlarge the effect of continuous deficiency and to avoid anoccurrence of a continuous deficiency.

4.2. Constraints

The reservoir operations should follow the operating regula-tions and comply with physical constraints, such as reservoir waterbalance and the permissible bounds of water release and storage,as well as the final target of water storage. The operation of theShih-Men Reservoir is guided by M-5 rule curves. A detail of theM-5 rule curves and its operating rules can be found in Changet al. (2005). The constraints shown in Eqs. (8)–(14) are relatedto the water balance equations, the upper and lower bounds on re-lease and storage, the ecological base flow requirements, and gen-eral standards of operating procedures, respectively.

Si ¼ Si�1 þ Ii � ðRdi þ RciÞ ð8Þ

0 6 Si 6 Smax ð9Þ

Si � SUi ; i ¼ 1 � 15 ðwet seasonÞ ð10Þ

Si � SLi ; i ¼ 16 � 36 ðdry seasonÞ ð11Þ

S36 � 0:9S0 ð12Þ

Rdi �max½Ei;Di� ð13Þ

Si; Ii;Di; Ei;Rdi;Rci P 0 ð14Þ

where Si, Ii, Di, Ei, Rdi and Rci are the storage, inflow, downstream hu-man water demand, ecological base flow requirements, down-stream water release (including the release to ecological baseflow), and water release of other canals in the ith 10-day period,respectively; SU

i and SLi are the upper and lower storage limits of

M-5 rule curve; Smax is the maximum storage; � and � similar to> and < can be violated slightly. Eq. (8) is the mass balance equation.Eq. (9) is the storage capacity. Eqs. (10) and (11) are the storage lim-its with respect to dry and wet seasons where the storage should bemaintained close to the upper limit in a wet season and the reser-voir should hold its storage higher than the lower limit curve to pre-vent from a serious drought in a dry season. Eq. (12) confines thecondition at the final stage of reservoir storage. Eq. (13) meansthe downstream water release must exceed human water demand

500

Fig. 3. The time series of annual flows and wet and drought seasonal flows during the period of 1984–2003.

Table 1The estimated minimum ecological base flow requirements for the Tahan River.

No. Ecological base flowrequirements

Methods References

1 0.6–4 cms Habitat methods, considering the needs of two kinds of fishes (1986–2000) Water Resources Agency (Chang, 2004)2 8.96 cms Historical flow method: using input and flow of side calculation Q95 (1965–

2002)Northern Region Water Resources Office (Chang,2004)

3 Dry: 6.75 cms Mean annual minimum 10-day flow during the dry and wet periods within1964–2004

This studyWet: 12.28 cms

70 L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74

and ecological base flow requirements. Eq. (14) shows the negativecondition.

4.3. Fitness function and penalty function

To implement the constrained GA, the constraints must be inte-grated into the objective function f ð~SÞ to form the fitness functionFð~SÞ through some appropriate penalty function pð~SÞ. The fitnessfunction is defined in Eq. (16).

Fð~SÞ ¼ f ð~SÞpð~SÞ ð15Þ

pð~SÞ ¼ 1þXnc

j¼1

Xnx

i¼1

gjið~SÞ; i ¼ 1;2; . . . ;nx; j ¼ 1;2; . . . ; nc ð16Þ

gjið~xÞ ¼0 x 2 feasible regiongjið~xÞ x R feasible region

(ð17Þ

where nx is the number of genes, i.e., 36 genes (variables); nc is thenumber of constraints; gji is a penalty function of the jth constraintof the ith variable.

The constraints are divided into three categories: boundaryconstraints, soft constraints and hard constraints. Boundary con-straints, Eqs. (9) and (14), confine the search space to the designed

ranges. Soft constraints, Eqs. (10)–(13), are allowed to be violatedand punished by the penalty terms, which will slightly decreasethe fitness values when reaching a low (tiny) degree of violation,but significantly decrease the fitness value when reaching a highdegree of violation. That means these constraints are not strictlycomplied with to certain degrees. The violation degrees of soft con-straints will guide the GA towards a feasible solution space in thesearch process. The corresponding constants of penalty terms can-not be set to large values; otherwise, it will be difficult to find theoptimal fitness values.

In the computation process, if various genes have reached thelimits of conditions, then we have to apply the respective penaltiesaccording to the degrees of limit violations. This fitness would rap-idly decrease. This penalty mechanism can be more effectivethrough reducing the probability of choosing a worse solutionand suitably respond to the decision-making process of choosingan infeasible solution.

We consider the differences between constraints and their de-grees of violations. The penalty function of each limit is set througha number of trial and error processes and presented as follows:

(1) When the value of water release is smaller than zero, whichis apparently an infeasible solution, the penalty mechanismfor violating the physical condition is to multiply the

L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74 71

obtained negative release value by an appropriate penaltyvalue (see Eq. (18)). This set value is crucial to the searchfor different hydrological events. In wet or normal hydrolog-ical years, the penalty value should be set to a large value toobtain a convergent solution earlier. However, in droughtyears, the search algorithm cannot obtain a convergent solu-tion because all genes do not satisfy the boundary con-straints; therefore, a smaller value should be consideredfor the penalty value.

g1ið~SÞ ¼ �1000Ri if Ri < 0; i ¼ 1 � 36 ð18Þ

(2) During wet seasons, we want to hold the water at a certainlevel to avoid a water shortage in the next drought period. Asa consequence, when the water storage is below the corre-sponding upper storage limit of M-5 rule curve, the penaltyfunction is set to two-times the difference of water storageand the corresponding upper storage limit of M-5 rule curve(Si < SU

i ) as shown in Eq. (19).

g2ið~xÞ ¼ ðSUi � SiÞ � 2 i ¼ 1 � 15 ð19Þ

(3) During dry seasons, if water storage is lower than the corre-sponding lower storage limit of the M-5 rule curve (Si < SL

i ),the penalty function is set as follows:

g3ið~SÞ ¼ ðSLi � SiÞ � 2; i ¼ 16 � 36 ð20Þ

(4) To prevent a shortage of water supply, the water storageshould be maintained at a certain level all the time. Whenthe water storage is below the critical lower limit of M-5 rulecurve, the penalty function is set at four-times the differenceof water storage and the critical lower limit of the M-5 rulecurve as follows:

g4ið~SÞ ¼ ðSCLi � SiÞ � 4; i ¼ 1 � 36 ð21Þ

where SCLi is the critical lower limit storage of M-5 rule curve

in the ith 10-day period.

(5) To maintain the initial water storage of each year at a similar

condition, we should set the initial condition similar to thecondition at the end of the previous year. The penalty func-tion will be applied if the storage at the end of the year is10% less than that of at the beginning of the year. If the waterstorage at the end of the year is below 50% of the lower limitof the M-5 rule curve, a penalty function is implemented asfollows:

g5ið~SÞ ¼ ðS0 � SiÞ � 5; i ¼ 36 ð22Þ

g5ið~SÞ ¼ 0:5SL36 � Si

� �� 5; i ¼ 6 ð23Þ

(6) For maintaining the ecological base flow requirements, wealso set a penalty function if the volume of water release isless than that of the ecological base flow demand. The pen-alty function is set as follows:

g6ið~SÞ ¼ ðEi � RdiÞ � 10; i ¼ 1 � 36 ð24Þ

The coefficients defined in the above functions are basedmainly on a subjective manner through a great number of trialand error processes for effectively obtaining suitable solutions.

4.4. Setting parameters of the genetic algorithm

The constrained GA is applied to searching for the optimal levelof water storage within 1 year according to the designed objectiveand constraints. The inflow records of the last 20 years (1984–2003) are used. A 10-day period, which is a traditional referencetime frame in Chinese agricultural society, is used as a time step.According to this scale, each month has three time steps, and henceevery year has 36 time steps. Consequently, in each year, there are36 water storages (variables) that need to be identified and the cor-responding water release histogram can then be obtained. The ini-tial value of water storage in each year is set the same as the valueof water storage at the end of the previous year, and a year-over-year search is continuously operated and recorded for 20 years(1984–2003). The related GA’s parameters used in this study arelisted in Table 2.

5. Results

5.1. Evaluation indicator

The CGA search results are compared with the historical opera-tions, i.e. the simulation results based mainly on the M-5 operatingrule curves which only considered human demand. The general-ized shortage index (GSI) proposed by Hsu (1995) is used to pres-ent the status of frequency and intensity of water shortage. Itsformula is given as follows.

DPD ¼XðDDRð%Þ � NDCÞ ð25Þ

GSI ¼ 100N

XN

i¼1

DPDi

100� DYi

�� k

ð26Þ

where DDR is 10-day deficit rate, i.e., total deficit in a period/waterdemand in the period � 100%; NDC is the number of 10-days in acontinuous deficit; DYi is the number of 10-day periods in a year(=36); N is the number of sample years; k is a constant, usuallyset as 2.

5.2. Evaluating different hydrological events

To investigate the suitability and reliability of the proposedCGA, we consider three cases of ecological base flow requirements:Case I – 4 cms, Case II – 8.94 cms, and Case III – 6.85 cms for dryseasons and 12.28 cms for wet seasons, respectively. The annualhistorical operations and CGA search results of 1984–2003 are pre-sented in Tables 3 and 4 and Fig. 4, respectively, and the 20-yearmean values of those results are summarized in Table 5.

Table 3 represents the water shortages of human demand asso-ciated with historical operations and CGA results based on differ-ent ecological base flow requirements. We found that most of thewater shortages of human demand in the drought years weremuch larger than those in wet years. Table 4 represents the watershortages of ecological requirements associated with historicaloperations and CGA results based on different ecological base flowrequirements. We also found (1) the water shortages of ecologicalrequirements would only occur in four drought years as the eco-logical base flow requirement was set to 4 cms; (2) the watershortages of ecological requirements would increase dramaticallyas the ecological base flow requirement increased to 8.94 cms;(3) for the ecological base flow requirements of dry and wet sea-sons (6.75 cms for dry seasons/12.28 cms for wet seasons), thewater shortages of ecological requirements in most of the simula-tion years would be slightly smaller than those of the ecologicalbase flow requirements set to 8.94 cms. This phenomenon is emi-nent in the years of 1986, 1990, 1992, 1996, 2001 and 2002. Those

Table 2Parameters of the CGA.

GAparameters

Setting value

Codingmethod

Real coded

Populationsize

1000

Generation >500Selection Roulette wheel selection with elite strategyCrossover Linear interpolation methodCrossover

rate0.8

Mutation rate 0.1Constraint

methodPenalty function

Stop criteria 1. The improving rate of fitness is smaller than 0.02.2. The difference of fitness between consecutive genera-

tions is smaller than 100.

Table 3The water shortages of human demand associated with historical operations and CGAresults based on different ecological base flow requirements.

Type Year CGA results Historical operations

I II III (dry & wet)

Drought 1988 1.5 0.8 0.1 501993 60 63 65 741995 63 59 58 891996 3 1 0.2 541998 4 11 0.7 441999 0 0 0 472002 52 66 62 402003 64 67 69 82

Normal 1986 7 11 8 681987 0 0 0 681991 0 0 0 301992 4 0.6 0.5 28

Wet 1984 0 0 0 181985 0 0 0 451989 0 0 0 351990 34 36 29 301994 0 0.1 0 131997 0 0 0 562000 0 0 0 312001 36 49 36 41

Remark: three different ecological base flow requirements: I – 4 cms; II – 8.94 cms;III – 6.75 cms in dry seasons and 12.28 cms in wet seasons.

72 L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74

years were marked in Table 4, and their dry season inflows are lessthan average, while their wet season inflows are larger than theaverage; (4) all water shortages of ecological requirements ob-tained from the CGA method are dramatically decreased as com-pared with those obtained from historical operations. The annualresults of GSI obtained by the CGA and historical operations areshown in Fig. 4. It demonstrates that (1) the GSI values obtainedfrom the constrained GA for all cases are much smaller than thoseof historical operations; (2) most of the GSI values obtained fromCGA are very small (close to zero) and only four larger GSI values(greater than 5.0) happen in very dry years (i.e. 1993, 1995,2002, and 2003); and (3) the corresponding GSI values slightly in-creased in only a few years even though the ecological base flowrequirements greatly increased from Cases I (4 cms), to II(8.94 cms), up to III (6.75 cms for dry seasons/12.28 cms for wetseasons) by using CGA method. In summary, these results basedon historical operations indicate that (1) the operations could not

meet the water demands of human demand and would result inlarge GSI and water shortages in an almost 20-year simulationperiod; and (2) the water release to the downstream of the TahanRiver would approximately provide 4–8.94 cms for maintainingecological base flow requirements in different seasons. As weclosely examine the CGA search results, quite similar results areobtained for these three different ecological base flow require-ments. These results indicate that by means of the CGA search,the water release from a reservoir could provide approximately9 cms for downstream ecological base flow and would not causeany significant water shortage in most of the years.

Table 5 gives the mean values of different indices within theirperiods for the CGA and historical operations. It is obvious thatthe mean values of GSI of historical operations for the last 20 yearsare larger than 20 and those of GSI generated by the CGA methodfor Cases I–III are all smaller than 2. Moreover, the mean values ofwater shortages of three different ecological base flow require-ments in historical operations are 2, 33, and 30 MCM, respectively,while those of shortages generated by the CGA method are 1, 29,and 23 MCM, respectively. It appears the CGA search results interms of both GSI and water shortages of human demand and eco-logical base flow requirements are all much smaller than the re-sults obtained from historical operations in the past 20 years.

6. Conclusion

Evolutionary computation techniques have great potential inoptimizing complex systems. This study proposes a constrainedgenetic algorithm (CGA) for multi-use reservoir management thatincorporates human demand requirements and ecological baseflow requirements. The ecological base flow requirements are con-verted as constraints to the CGA. To implement the CGA, the con-straints are integrated into the objective function configuredwith appropriate penalty functions to form the fitness function.Constraints are allowed to be violated and can be punished bythe penalty terms. The violation degrees of soft constraints wouldguide the GA towards a feasible solution space in the search pro-cess. To incorporate these penalty functions into the objectivefunction to form the fitness function, our experience indicatesthe multiplication form of the penalty function, in general, is moreeffective in obtaining reasonable solutions. The CGA search couldbe very efficient by using the penalty mechanism which is capableof reducing the probabilities of choosing worse solutions and cansuitably guide the search direction towards promising solutionsduring genetic search.

The CGA techniques and advanced system analysis methodshave been implemented to search for the feasible solutions of theShih-Men Reservoir operation in twenty different yearly hydrolog-ical events, and to assess the impacts of water shortages on ecolog-ical base flow requirements and human demand. According toprevious studies and historical flows of the Tahan River, differentecological base flow requirements have been set for the down-stream area: Case I – 4 cms; Case II – 8.94 cms; and Case III –6.75 cms for dry seasons and 12.28 cms for wet seasons. Basedon historical records of reservoir water release, the water releasecould usually provide more than 4 cms flow for human demandand downstream ecological base flow requirements. But as the goalof water release is set to 8.94 cms (Case II), the reservoir would faceserious water shortages, i.e., large values of water shortage, indrought years if the traditional M-5 rule curves are used. On theother hand, the values of GSI would dramatically decrease as theproposed constrained GA is used in search of optimal reservoiroperations. Moreover, the reservoirs could provide more waterfor ecological base flows if CGA method is used. We demonstratethat the constrained GA can effectively incorporate the constraints

Table 4The water shortages of ecological flow requirements associated with historical operations and CGA results based on different ecological base flow requirements.

Type Year Hydrological characteristics CGA results Historical operations

Dry Wet Annual I II III I II III

Drought 1988 � � � 0 23 27 0 34 331993 � � � 4.6 93 97 5.6 78 601995 � � � 13 85 83 8.7 96 801996� � + o 0 20 1 0 30 291998 � + o 0 22 24 0 34 371999 + � � 0 0 0 0 16 152002� � � � 4 88 65 5.6 55 622003 � � � 4 98 102 14.3 115 127

Normal 1986� � + + 0 25 0 0 23 181987 o + + 0 0 0 0 15 101991 + � + 0 0 0 0 7 111992� � + + 0 13 5 0 19 16

Wet 1984 + + + 0 0 0 0 15 161985 + + + 0 0 0 0 30 201989 + + + 0 0 0 0 10 51990� � + + 2 52 18 0 22 81994 o + + 0 0.5 0 0 14 111997 + + + 0 0 0 0 24 182000 + o + 0 0 0 0 10 162001� � + + 0.6 64 33 0.7 21 11

Remark: (1) three different ecological base flow requirements: I – 4 cms; II – 8.94 cms; III – 6.75 cms in dry seasons and 12.28 cms in wet seasons. (2) +: inflow greater thanaverage; �: inflow less than average; o: inflow equal to average.

Fig. 4. The GSI values obtained from CGA search results and historical operations during 1984–2003.

L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74 73

into the objective function to efficiently search for the optimalsolutions and the Shih-Men Reservoir has great potential toincrease its water supply capability for both human demand andecological base flow requirements if advanced mathematic pro-gramming techniques, such as CGA, could be used and properoperational strategy is implemented.

Acknowledgment

This study is sponsored by the Water Resources Planning Insti-tute, Water Resource Agency, Taiwan, Republic of China. Theauthors are indebted to the editors and reviewers for their valuablecomments and suggestions.

Table 5Summarized table of the CGA results and historical operations.

Type (average) (1) GSI (2) Human shortage (3) Ecological shortage

I II III I II III I II III

CGA resultsDrought 3.60 3.69 3.90 31 34 32 3.0 54 50Normal 0.05 0.07 0.04 3 3 2 0.0 10 1Wet 0.76 0.99 0.63 9 11 8 0.3 15 620 years 1.80 1.90 1.80 16 18 16 1.0 29 23

Historical operationsDrought 29.2 60 4.0 57 55Normal 26.4 48 0.0 16 14Wet 8.5 34 0.1 18 1320 years 20.3 47 2.0 33 30

(1) GSI.(2) Human shortage: water shortage (MCM) of human demand.(3) Ecological shortage: water shortage (MCM) of three different ecological base flow requirements; I – 4 cms; II – 8.94 cms; III – 6.75 cms in dry seasons and 12.28 cms in wetseasons.

74 L.-C. Chang et al. / Journal of Hydrology 390 (2010) 66–74

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