Journal of Hydrology (2008) 354, 65–74

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Guiding rational reservoir flood operation usingpenalty-type genetic algorithm

Li-Chiu Chang *

Department of Water Resources and Environmental Engineering, Tamkang University, Taiwan, 151 Ying-Chuan Road,Tamsui, Taipei County 25137, Taiwan, ROC

Received 24 July 2007; received in revised form 10 January 2008; accepted 25 February 2008

00do

*

KEYWORDSPenalty strategy;Genetic algorithms;Flood control;Reservoir operation

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Tel.: +886 2 26258523; faxE-mail address: changlc@m

r ª 200.021

: +886 2ail.tku.e

Summary Real-time flood control of a multi-purpose reservoir should consider decreasingthe flood peak stage downstream and storing floodwaters for future usage during typhoonseasons. This study proposes a reservoir flood control optimization model with linguisticdescription of requirements and existing regulations for rational operating decisions. Theapproach involves formulating reservoir flood operation as an optimization problem andusing the genetic algorithm (GA) as a search engine. The optimizing formulation isexpressed not only by mathematical forms of objective function and constraints, but alsoby no analytic expression in terms of parameters. GA is used to search a global optimumof a mixture of mathematical and nonmathematical formulations. Due to the great numberof constraints and flood control requirements, it is difficult to reach a solution withoutviolating constraints. To tackle this bottleneck, the proper penalty strategy for each param-eter is proposed to guide the GA searching process. The proposed approach is applied to theShihmen reservoir in North Taiwan for finding the rational release and desired storage as acase study. The hourly historical data sets of 29 typhoon events that have hit the area in lastthirty years are investigated bye the proposedmethod. To demonstrate the effectiveness ofthe proposed approach, the simplex method was performed. The results demonstrated thata penalty-type genetic algorithm could effectively provide rational hydrographs to reduceflood damage during the flood operation and to increase final storage for future usages.ª 2008 Elsevier B.V. All rights reserved.

Introduction

Taiwan is located in the North Pacific Ocean subtropical jetstream monsoon district with an annual average rainfall

8 Elsevier B.V. All rights reserved

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around 2500 mm with high seasonal variability. Typhoon-re-lated rainfall has been recorded at over 1000 mm/day. Thelongest river in Taiwan is only 186 km long, but it drops fromnear 4000 m to sea level over that distance. Because thesteep mountainous landform makes most of the rainfall flowstraight into the ocean within a few hours, the rivers cannotkeep enough water for use as a stable source of water

.

Encoding variables

Reproduction

Selection

Fitness Evaluation

InitializeChromosome

Predetermineparameters of GA

Crossover

Mutate

Converge?

No

StopYes

Gene Pool

Penalty Strategy

Figure 1 The flowchart of solving the optimization problemusing the penalty-type GA.

66 L.-C. Chang

supply. Reservoirs have become the most important andeffective floodwater storage facilities. The over-concen-trated rainfall often exceeds the capacity of the reservoirs;consequently, reservoir operation plays a critical role insmoothing out a tremendous temporal variable inflow formitigating possible damage and to store floodwaters for sta-ble utilization of water resources – it certainly ensures damsafety during typhoons. Especially, with the gradual de-crease of reservoir capacity, the real-time flood control ofreservoir systems obviously is a more and more urgent anddifficult task.

A variety of optimization models have been developed forthe operation of reservoir systems (Yeh, 1985; Wurbs, 1993;Labadie, 2004). Many researchers havemade efforts for floodcontrol of reservoir systems by using optimization tech-niques, including linear programming (LP), dynamic program-ming (DP) and genetic algorithms (GA). Linear Programming,developed during the second world war, is an importantmethod of reservoir management and operation models(Windsor, 1973; Windsor, 1975; Labadie et al., 1986; Randallet al., 1997; Needham et al., 2000). Dynamic Programming,originally developed by Bellman (1957), effectively exploitsthe sequential decision structure to provide an optimizationprocedure for reservoir operation (Trezos and Yeh, 1987;Shim et al., 2002; Chang et al., 2002). However, the curseof dimension is a significant obstacle to solve optimizationproblems by using DP when the dimension of decision vari-ables is large (Needham et al., 2000; Reis et al., 2005). West-phal et al. (2003) demonstrated that decision supportsystems (DSS) can produce reasonable flood control operatingschedules. Karbowski et al. (2005) calculated water releasesbased on two-variant inflow forecasts and some expertknowledge and applied this to real-time operation of a waterretention reservoir during flooding. Genetic algorithms arewell known global search techniques for solving nonlinearreservoir optimization problems such as reservoir manage-ment (Chang and Chen, 1998; Cai et al., 2001; Chang et al.,2005b), reservoir operating rules (Oliveira and Loucks,1997; Chang et al., 2005a), real-time reservoir operation(Chang and Chang, 2001; Akter and Simonovic, 2004), Multi-reservoir systems optimization (Wardlaw and Sharif, 1999;Sharif and Wardlaw, 2000; Reis et al., 2005), etc.

With the aim of reducing the gap between academic re-search and real-world implementation, the present paperproposes a penalty-type genetic algorithm to find a rationalrelease hydrograph in reservoir flood operation for fulfillingthe objectives as well as complying with legal, linguisticdescription of requirements and other physical restrictions.Analysis will be based on the Shihmen reservoir, Taiwan,which is a multiple purpose reservoir with over forty yearsof records. The research will address the methodology ofincorporating the constraints into the traditional GA search-ing process and the benefit of efficiency and effectivenessof the proposed method for flood operation.

Methodology

The architecture of genetic algorithms

Genetic Algorithms, developed by Holland (Holland, 1975;Goldberg, 1987), are analogous to Darwinian natural selec-

tion, which combines an artificial survival of the fittestand the natural genetic operators to attempt to find thebest solution in a given solution space. GA has received agreat deal of attention regarding its potential as an optimi-zation technique for complex problems.

GA tends to mimic some of the processes observed innatural evolution. It codes decision variables of an optimi-zation problem to one artificial chromosome in the GA pop-ulation, which is defined as one of the potential solutions.The fitness of each chromosome is based on a measure ofthe objective function to evaluate the probability of sur-vival. The GA search starts with an initial randomly gener-ated population and progresses to improve the evaluationof solutions through iterations by implementing GA opera-tors as shown in Fig. 1, including selection scheme, repro-duction, crossover operators and mutation operators.

The selection operation is preferentially to choose chro-mosomes with a higher fitness for survival or producingoffspring in the gene pool of the next generation. Here selec-tion was performed by roulette wheel selection, with eachslice of a roulette wheel proportioned to fitness values. Forhighly constrained problems, inferior solutions of small fit-ness values or infeasible solutions are in the majority during

Figure 2 Shihmen reservoir and its watershed.

Guiding rational reservoir flood operation using penalty-type genetic algorithm 67

the early stage of the searching process. It would cause earlyconvergenceor inefficient search if the infeasible and inferiorsolutions were neglected or ignored. For avoiding this situa-tion, roulette selection is used to prevent disregarding infor-mation from infeasible solutions with small fitness values.

Information of candidate solutions is exchanged betweentwo random positions through crossover. Crossover is usu-ally applied to selected pairs from the gene pool with aprobability equal to a given probability of crossover (Pc).Single-point crossover or multi-point crossover is selectedat random and two parent chromosomes are interchangedat the point(s). During mutation, a random real, selectedwith a probability equal to a very low probability of muta-tion (Pm), is to jump to a random point within the searchspace.

Penalty functions

For most real-world optimization problems, the generalproblems are constrained optimization problems. SinceGAs are usually designed for finding the optimum solutionto complex unconstrained optimization problems, they haveto be adapted to handle constraints. It is difficult to explic-itly account for constraints by using GAs and to maintainfeasible solutions in the population.

Several proposed methods for handling constraints can beroughly classified as rejecting strategy, repairing strategy,modifying genetic operator strategy and penalizing strategy(Gen and Cheng, 1996). The advantage of the first threeapproaches is that no infeasible solutions are generated.However, they can make the searching process extremelydifficult. Infeasible solutions may have a large portion inpopulation for highly constrained problems and GAs will be-come inefficient for finding a solution without any constraintviolation. Therefore, a penalizing strategy is properly usedto consider infeasible solutions and to effectively guide thesearching direction towards promising solutions in the genet-ic search. A number of effective penalty functions have beenproposed to improve the efficiency of GAs (Richardson et al.,1989; Homaifar et al., 1994; Michalewicz, 1999). Unfortu-nately, there is no general guideline for designing efficientpenalty functions and these methods are quite problem-dependent.

The most common methods for using GAs in handlingconstraints involve the use of penalty functions. Thus,real-world optimization problems can be written as generalnonlinear programming:

min fð~xÞ ð1Þsubject to gið~xÞ 6 0 i ¼ 1; 2; . . . ; l ð2Þ

hjð~xÞ ¼ 0 j ¼ 1; 2; . . . ;m ð3Þxlk 6 xk 6 xukð~xÞ ¼ 0 k ¼ 1; 2; . . . ; n ð4Þ

where fð~xÞ is the objective function with n decision vari-ables (~x), gið~x) is the ith inequality constraint, and hjð~x) isthe jth equality constraints. xlk and xuk are lower and upperbounds on the kth variable xk. In general, the search spacecontains a feasible region and infeasible region. If a solution~x complies with all constraints and variable bounds it iscalled a feasible solution; otherwise, an infeasible solution.Penalty strategy transforms the constrained problem into anequivalent unconstrained problem by penalizing infeasible

solutions. There are two possible ways to construct theevaluation function with penalty terms. The easiest way isby adding a penalty for those infeasible solutions into theobjective function shown as follows:

Fð~xÞ ¼fð~xÞ x 2 feasible region

fð~xÞ þ pð~xÞ x 62 feasible region

�ð5Þ

where pð~xÞ is called the penalty function. If no violation oc-curs, pð~xÞ will be zero; otherwise pð~xÞ will be some positivevalues.

The other way is the multiplication form as follows:

Fð~xÞ ¼fð~xÞ x 2 feasible region

fð~xÞpð~xÞ x 62 feasible region

�ð6Þ

For minimization problems, if no violation occurs, pð~x) isone; otherwise pð~x) will be bigger than one. The additivepenalty-type has received much more attention than themultiplicative type in the GA community (Yeniay, 2005).However, finding a suitable and effective penalty functionfor a large number of constraints could be troublesome.

In this study, Eq. (6) is used to construct the fitness func-tion. The following penalty function is used.

pð~xÞ ¼Yli¼1ð1þ ci maxðgið~xÞ; 0ÞÞ �

Ymj¼1ð1þ djjhjð~xÞjÞ ð7Þ

where ci and di are the penalty weights for the inequalityand equality constraints, respectively.

Application

The Shihmen reservoir, located upstream of the Tahan river(Fig. 2), satisfies multiple purposes of irrigation, hydro-power generation, industrial and domestic water uses, floodcontrol, and recreation. The Tahan river has a length of135 km with an average channel slope of 1/37, and its wa-tershed covers an area of 1163 km2. The watershed receivesan average of 2410 mm of rainfall in last fourty-years; how-ever, the rainfall is uneven – over 70% occurs between May

68 L.-C. Chang

and October (typhoon season). The watershed of the Shih-men reservoir covers an area of 763.4 km2 with an effectivestorage capacity of 2.35 · 108 m3. The crest elevation of thedam is at 252.1 m, the top of the flood control pool is at anelevation of 250 m, the normal maximum water level of theconservation pool is at 245 m, and the dead water level is at195 m. There are six tainter floodgates of ogee-type con-crete spillway, each 14 m · 10.61 m with minimum dis-charge of 600 m3/s and maximum discharge up to11,400 m3/s. Furthermore, there are two tunnel spillwayswith a minimum discharge of 200 m3/s and maximum dis-charge 2400 m3/s.

The reservoir flood operation problem formulation

Real-time reservoir flood operation is a continuous and in-stant decision-making process based on relevant operatingrules, policy and water laws. The decision variables arethe release values for the flood operation period of the res-ervoir. Hence, a determination of water levels and releasesof a reservoir that satisfies various operating needs andrestrictions is the main purpose of flood operation. The reg-ular operation of the Shihmen reservoir is guided by M-5operating rule curves, whose upper limit curves can provideoperators with the target water level for flood operation ofthe different months. The constraints are related to thewater balance equations, the upper and lower bounds on re-lease and storage, and general standards of operatingprocedure.

Objective functionIn this study, four reservoir flood operation goals are consid-ered: (1) to ensure the dam safety; (2) to postpone the ar-rival time of flood peak; (3) to minimize the release peak foralleviating the downstream damage and (4) to save floodwa-ters and keep the final water level close to the desired stor-age for future water demand. The optimal operation shouldalways ensure dam safety while the final water level shouldaim to reach the target level, and the release peak shouldbe minimized and postponed for alleviating the downstreamdamage. The goal is to optimize the objectives while com-plying with relevant regulations and physical constraints.

Flood operation often faces a conflict between minimiz-ing damages incurred by floodwaters downstream (where itis required to keep space for flood control) and maximizingthe final storage for future use (where it is desirable to storefloodwaters as much as possible). These two goals, how-ever, should not be considered as two independent objec-tives because they are not always in conflict. Whether ornot there is conflict is highly dependent on the initial stateof the water level. If the initial water level is low, the oper-ator can conserve the floodwaters to reach the final targetstorage and to decrease the release peak as much as possi-ble for minimizing damage downstream. In such a case, it isnot appropriate to consider the optimal reservoir floodoperation problem as a multiple-objective problem. More-over, the main index in evaluating the effect of flood oper-ation is the extent of reduction of the flood peak (TheReport of the World Commission on Dams, 2000). Therefore,the objective function is set to minimize the release peak,while the final target storage is taken as a constraint.

The objective function is defined as

min fðOÞ ¼ maxt2½1;T�ðOðtÞÞ ð8Þ

where t is the time step and T the time horizon; O, decisionvariables, are the release values of the reservoir.

ConstraintsBesides taking dam safety as a constraint, the Shihmen res-ervoir flood control operations have to comply with operat-ing regulations, regarding the reservoir water balance, thepermissible bounds of release and storage, as well as the fi-nal target storage. These constraints are explained asfollows:

SðtÞ ¼ Sðt� 1Þ � OðtÞ þ IðtÞ ð9ÞSmin 6 SðtÞ 6 Smax ð10Þ0 6 OðtÞ 6 Omax ð11ÞSðTÞ ¼ Sn ð12Þ

where S(t) and I(t) are the effective storage and inflow attime step t, respectively; Smin, Smax, Sn are the minimum,maximum and target storage; the corresponding water lev-els of Smin, Smax are 195 and 251 m; Omax is the reservoirmaximum release.

According to relevant operating regulations and waterlaws, an inflow hydrograph over a flood operating periodcan be divided into three stages (shown in Fig. 3).

(1) In the coming of a typhoon: the reservoir level shouldbe kept below the upper limit curve of rule curves.

(2) In the early stage of flood peak arrival: the waterrelease should be kept less than or equal to the inflowand getting greater during the period between subse-quent time that the inflow is from 600 m3/s to 90% ofthe flood peak of recession curve. In addition, the rateof increase of the release cannot exceed the highestrate of increase of the inflow.

Oðt� 1Þ 6 OðtÞ ð13ÞOðtÞ 6 IðtÞ ð14ÞOðtÞ � Oðt� 1Þ 6 DImax ð15Þ

where DImax is the maximum increment inflow rate.

(3) After the flood peak: Unless there is still a risk of over-flow, the release should be kept smaller than theflood peak and getting smaller during the period.The reservoir must be appropriately operated to con-serve floodwaters in order to reach the final targetwater level.

OðtÞ 6 Oðt� 1Þ ð16ÞOðtÞ 6 Imax ð17Þ

where Imax is the peak flow discharge.These equations obey the normal operating procedure of

the spillway floodgates: the opening of floodgates should begradually larger and larger in the early stage of the floodpeak arrival while that of floodgates should be graduallysmaller and smaller after the flood peak.

The proposed reservoir optimization model might havetoo many constraints to form a large-scale programming.For example, if an event has 40 h duration and the flood

0

1000

2000

3000

4000

5000

6000

7000

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55

In th

e co

min

g of

a ty

phoo

n

After flood peak In the early stage of the flood

peak arrival

Q=600cms

90% Q max

Figure 3 Three stages of reservoir inflow hydrograph over operating period.

Guiding rational reservoir flood operation using penalty-type genetic algorithm 69

peak arrives at the 16th hour, then there are 216 constraints(121 = 40 · 3 + 1 Eqs. (9)–(12), 45 = 15 · 3 Eqs. (13)–(15),50 = 25 · 2 Eqs. (16), (17)).

The penalty-type GA and simplex method appliedfor optimal operation hydrograph

The proposed reservoir flood control optimization model isnot a typical linear programming though the objective func-tion and constraints look like linear. The objective functionhas a simple but discrete function with unspecified time.This is mainly because there still exist linguistic flood con-trol requirements and operating regulations that cannotbe easily expressed in mathematical form or satisfied byan analytic expression. In this case study, postponing theflood peak is one of the flood control goals, so the arrivaltime of the release peak, which directly affects the objec-tive function (Eq. (8)) and some constraints (Eqs. (13)–(16)), cannot be set beforehand. That means that Eqs.(13)–(16) cannot be explicitly expressed. Moreover, Eqs.(13) and (16) represent that the release would be gettinggreater or smaller and not increase or decrease abruptly,but it is not very clear to describe the release difference be-tween two successive times. Thus, the proposed model can-not be taken as a linear programming, and it is also difficultto use any conventional approach to find the optimalsolution.

Due to the flexible searching procedure of the penalty-type GA, we can use it to deal with the aforementionedproblems. First, we do not need to prescribe the arrivaltime of the release peak because it can be determined fromeach potential solution, and then Eqs. (8) and (13)–(16) be-come explicit forms. Second, the arrival time of the releasepeak can be postponed through penalizing the fitness ofthose chromosomes for an earlier arrival time. Third, thenear optimal releases obtained from the penalty-type GAwould be getting greater or smaller. When the optimum ex-actly rides on the borders of feasible and infeasible regions(Eqs. (13) and (16)), the GA search seldom gets the exactsolutions but a near optimal solution due to those geneticoperators, such as a crossover exchange of the informationof two candidate solutions between two random positions ofthese two solutions.

Before the penalty-type GA is implemented, all of theconstraints must be combined with the objective functionto form the fitness function through some appropriate pen-alty functions. Here the fitness function is defined asfollows:

FðOÞ ¼1� fðOÞ

ImaxQTi¼1pi

ð18Þ

The constraints are divided into bound constraints, soft con-straints and hard constraints. Bound constraints, Eqs. (11),(14) and (17), confine the search space to the designedranges. Soft constraints, Eqs. (12), (13), (15) and (16), areallowed to violate and be punished by the penalty terms(Eq. (7)) to slightly decrease the fitness values within low(tiny) degree of violation, and to significantly decreasethe fitness value with high degree of violation. That meansthese constraints are not strictly complied within certaindegree. The soft constraints can be used to present the lin-guistic control requirements. Their penalty terms are de-fined as Eqs. (19)–(22), respectively.

pi ¼ 1þ jHðtendÞ � Hnj � 2 ð19Þ

pi ¼ 1þmaxð0;Oðt� 1Þ � OðtÞÞ10

ð20Þ

pi ¼ 2 ð21Þ

pi ¼ 1þmaxð0;OðtÞ � Oðt� 1ÞÞ10

ð22Þ

The violation degree of soft constraints would guide the GAtowards a feasible solution space in the searching process.The corresponding constants of penalty terms cannot beset at large values; otherwise, the fitness value would bevery small after cumulative multiplication. A hard con-straint, Eq. (10), is not violated by being given severe pun-ishment to let the probability of survival be zero or verylow. Both sides of Eq. (10) are set to the same penalty termdefined as in Eq. (23).

pi ¼ 10000 ð23Þ

In addition, Eq. (9) is not a constraint, just used to calculatestorages from the release outflows, so it should never beviolated. In order to postpone the arrival time of the floodpeak, we would not penalize the fitness value of Eq. (20)

70 L.-C. Chang

as the rate of the release still continued increasing after theflood peak; Eq. (22) would be used only the rate starteddecreasing.

For comparison with the effectiveness of the penalty-type GA for searching the rational flood releases, the sim-plex method is implemented. However, the simplex methodcannot be solved due to the discrete function type unlessthe arrival time of release flow peak is set. Here we set itafter the flood peak without any constraint violation.

The available data

To prove the reliability and effectiveness of the proposedmodel by using the penalty-type GA search engine, the his-torical hourly inflow and operation hydrograph of the Shih-men reservoir were collected and used to evaluate theperformance. Twenty-nine typhoon events were collectedfrom the past thirty-one years (1970–2000), published bythe Water Resources Agency (WRA), Taiwan. These typhoonshave different peak, duration, initial water level, and are di-vided into three different categories according to theamount of floodwater shown in Table 1; ‘‘small’’ meansthe total floodwater is less than 70% of the capacity of Shih-men reservoir; ‘‘medium’’ is between 70% and 100%; the

Table 1 Collected hourly data of typhoons (1970–2000)

Event Date Duration (h) Total floodw

Fran 1970/9/6–9/8 51 66,574Agnes 1971/9/18–9/19 40 50,146Bess 1971/9/22–9/24 53 86,846Betty1 1972/8/16–8/18 58 109,000Betty2 1975/9/22–9/23 25 17,315Billie 1976/8/9–8/11 33 55,019Irving 1979/8/14–8/16 42 45,247Percy 1980/9/18–9/18 23 18,061June 1981/6/20–6/22 31 26,383Maury 1981/7/19–7/21 38 36,975Andy 1982/7/28–7/30 30 22,361Cecil 1982/8/9–8/11 28 22,423Holly 1984/8/18–8/20 51 60,643Nelson 1985/8/22–8/24 44 77,506Wayne 1986/8/24–8/25 36 38,040Abby 1986/9/18–9/20 36 39,779Gerald 1987/9/9–9/10 25 20,519Sarah 1989/9/11–9/13 68 92,229Yancy 1990/8/18–8/20 43 62,314Abe 1990/8/30–8/31 42 64,371Dot 1990/9/7–9/9 52 60,472Tim 1994/7/10–7/11 19 12,924Doug 1994/8/7–8/9 37 54,865Fred 1994/8/20–8/22 37 43,410Seth 1994/10/10–10/11 38 57,640Herb 1996/7/31–8/2 55 110,820Amber 1997/8/29–8/30 27 21,569Zeb 1998/10/15–10/17 54 82,512Xangsane 2000/10/31–11/2 30 29,321

remaining events belong to the ‘‘large’’. A large flood can fillup Shihmen reservoir twice; therefore, how to generate a ra-tional operation release during the flood is particularlyimportant.

Results

The performances of the penalty-type genetic algorithm,the simplex method and historical operations for the releasehydrograph in the Shihmen reservoir flood operation werepresented based on the criteria of the decrease rate of floodpeak (Rp), the final release flow in the end of operation(Oend), the postponed arrival time of flood peak (ETp), thehighest water level during the period of operation (Hmax)and the final water level (Hend).

The decrease rate and the postponed arrival time offlood peak are defined as follows:

Rp ¼ 1� Opeak

Ipeakð24Þ

ETp ¼ tOpeak� tIpeak ð25Þ

where Ipeak is the inflow peak and Opeak is the release peak;tIpeak and tOpeak

are the arrival time of the inflow peak and re-lease peak, respectively.

aters (cms) Peak (cms) Water level (m) Category

Initial Target

3681 223.54 240 Large2431 209.23 240 Medium5172 238.71 240 Large5665 237.22 240 Large1153 232.80 240 Small5488 219.87 240 Medium2006 225.04 240 Small1236 228.66 240 Small1441 221.42 235 Small2217 232.38 236 Small1262 220.80 236 Small1248 237.89 240 Small2166 241.02 240 Medium4906 233.91 240 Large1773 239.81 240 Small2129 239.09 240 Small1444 241.62 240 Small2959 237.82 240 Large4344 231.88 240 Medium3604 243.33 240 Medium2593 242.08 240 Medium1171 228.88 236 Small3065 235.82 240 Medium2440 238.82 240 Small3138 239.24 245 Medium6363 231.38 240 Large1351 241.47 240 Small4643 239.92 245 Large1852 237.43 245 Small

Guiding rational reservoir flood operation using penalty-type genetic algorithm 71

Oend should be confined to a low value, so that the finalwater level can be easily kept after the flood operation.ETp > 0 means the flood peak is postponed. The high valueof the Hmax will cause the high risk of dam failure.

The results of 29 events obtained by these two methodsand the historical reservoir operation are organized in Table2 and discussed as follows. It appears the historical opera-tions, which were obtained from experienced reservoiroperators, could nicely control and postpone the flood peakduring the typhoon events, but the final storage of 10 eventsdid not reach the target. In the simplex method, optimalsolutions for the decrease rate of flood peak (the first crite-rion shown in Table 2) were obtained, but the release peakwould arrive much earlier than the flood peak, from 1 to19 h ahead. Besides, it has the problems of high final releaseand unreasonably high water level (i.e. Hmax is very high) inmany cases. In fact, there are over 15 events in which thefinal releases are more than 1000 cms, and 9 events thattheir highest water level exceeded the normal maximum

Table 2 Comparison of the flood operating results among opera

Events Operator GA

(1) (2) (3) (4) (5) (1) (2)

Fran 0.61 790 9 245 244.26 0.55 148Agnes 0.95 63 2 241 241.45 0.90 89Bess 0.25 796 3 245 243.71 0.30 186Betty1 0.16 856 3 247 238.77 0.27 86Betty2 0.90 100 1 241 240.78 0.87 143Billie 0.84 817 11 245 243.57 0.68 30Irving 0.54 835 10 243 241.57 0.58 85Percy 0.93 15 1 238 238.4 1.00 1June 0.89 98 3 236 235.81 0.81 70Maury 0.61 726 10 242 240.97 0.35 211Andy 0.90 85 5 234 234.33 1.00 2Cecil 0.37 735 7 244 243.56 0.33 650Holly 0.01 1009 5 247 240.51 0.17 938Nelson 0.09 720 2 245 238.72 0.38 300Wayne 0.19 1024 9 246 243.43 0.21 513Abby 0.34 970 10 246 242.95 0.23 722Gerald 0.35 861 4 245 244.03 0.32 883Sarah 0.23 879 1 246 241.24 0.17 117Yancy 0.53 984 5 246 242.53 0.48 71Abe 0.15 1039 3 248 241.18 0.20 421Dot 0.16 796 3 245 239.34 0.19 370Tim 0.89 110 1 236 235.53 0.97 34Doug 0.13 918 3 245 241.78 0.32 329Fred 0.34 936 6 242 239.31 0.30 379Seth 0.09 869 2 247 243.97 0.32 331Herb 0.15 732 4 245 242.55 0.35 37Amber 0.36 451 10 244 243.56 0.14 884Zeb 0.01 530 1 245 243.47 0.36 252Xangsane 0.11 494 2 244 243.14 0.61 210

(1) The decrease rate of flood peak (Rp).(2) The final release flow in the end of operation (cms) (Oend).(3) The postponed arrival time of flood peak (h) (ETp).(4) The highest water level during the period of operation (m) (Hmax).(5) The final water level (m) (Hend).

water level (245 m), four of them even higher than thetop of the flood control pool (250 m). On the other hand,the penalty-type GA would produce satisfied and reasonableresults for all the five criteria (i.e. Rp, Oend, ETp, Hmax andHend). It appears the simplex method can only provide theoptimal solutions for single criterion (i.e. Rp) but othermay not be reasonable, while the penalty-type GA indeedprovides rational results.

For smaller flood events, the two methods can obtain theoptimal solution and store floodwaters as much as possibleto reach or be near the target shown in Fig. 4. As theamount of floodwater is not enough to fill the reservoir,the penalty-type GA could converge to allow the soft con-straint (the target) to be slightly unsatisfied, while the sim-plex method has problem to reach the final water leveltarget, such as typhoon Percy and Andy in Table 2.

For a rational release hydrograph, the release hydrographof the penalty-type GA, as historical operation, would getgreater during the rising limb and smaller during the fall limb

tors, the penalty-type GA and the simplex method

Simplex

(3) (4) (5) (1) (2) (3) (4) (5)

6 240 240 0.78 826 �7 241 2405 240 240 0.95 129 �19 240 2403 245 240 0.66 1783 �6 251 2404 247 240 0.61 0 �7 251 2401 240 240 0.88 140 �8 240 2403 240 240 0.88 682 �5 240 2405 240 240 0.77 464 �17 240 2402 239 238.7 1.00 0 �8 238.7 238.76 235 235 0.89 160 �12 235 2354 237 236 0.61 868 �6 238 2367 235 235 1.00 0 �15 235 235.13 241 240 0.46 680 �7 240 2406 244 240 0.36 1396 �5 245 2404 244 240 0.64 1759 �7 246 2405 242 240 0.36 1141 �2 242 2407 243 240 0.45 1162 �5 243 2407 243 241.4 0.24 1099 �1 242 2405 244 240 0.49 1522 �6 246 2406 243 240 0.71 1247 �6 243 2404 245 240 0.46 1942 �5 249 2403 245 240 0.44 1443 �6 246 2402 236 236 0.97 33 �8 236 2365 244 240 0.55 1383 �10 244 2403 242 240 0.50 1227 �6 244 2404 245 245 0.59 1296 �8 248 2457 245 240 0.69 1994 �7 251 2403 242 240 0.23 1040 �2 242 2403 248 245 0.67 1523 �5 251 2452 245 245 0.76 445 �12 245 245

0

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ow (c

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(a) Reservoir inflow and outflow hydrographs

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Figure 4 A comparison of outflow and water level hydro-graphs using operators, GA and the simplex method (typhoonTim).

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Figure 5 A comparison of outflow and water level hydro-graphs using operators, GA and the simplex method (typhoonBetty1).

72 L.-C. Chang

in most cases, while the simplex method makes a strong con-trast. The release hydrographs of the simplex method inmany cases are not applicable because its large amount of re-lease decrease may hinder closing of several gates within ashort time, such as the release of typhoon Betty1 decreasesfrom 2184 to 294 cms in the end of operation shown inFig. 5a, or the high final release (Oend) would cause the reser-voir not to keep the target water level after the flood opera-tion (Fig. 6a). In the penalty-type GA, the release wouldgradually change during the searching process and, more-over, the final water level also tended to reach the target.Hence, the penalty strategy of the final water level and rateof the release can restrict the highest water level; otherwise,the final water level would be far away from the target.

For the highest water level during a flood, the penalty-type GA produces the highest water level (Hmax) of less than250 m in all cases, while the simplex method could not sat-isfy this goal in some cases. There have been six typhoons(Bess, Betty1, Abe, Seth, Herb and Zeb) whose the highest

water level was higher than or near 250 m (shown in Table2 and Figs. 5b and 6b for typhoon Betty1 and Herb, respec-tively). It raises the risk of the dam failure and would cer-tainly not be accepted. It certainly would increase the riskof dam failure and would not be acceptable for flood controlin this circumstance.

Table 3 summarizes the average results of three catego-ries of typhoons (small, medium, and large) by both meth-ods. Apparently, the penalty-type GA can postpone thearrival time of a peak and obtain much lower final release(in terms of more rational release hydrograph) than that ofthe simplex method. It is, however, difficult or misleadingto compare the decrease rates of a peak of GA with that ofoperators due to the different final water levels. A high finalwater level means that more floodwater is stored in a reser-voir. Most large typhoon events have smaller outflow peak bypenalty-type GA than by operators. The lower releases dur-ing the final stage of a flood would keep the water level low-er within the range of safety during the flood operation.

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Figure 6 A comparison of outflow and water level hydro-graphs using operators, GA and the simplex method (typhoonHerb).

Table 3 Comparison of three models on the average ofthree categories of typhoons

Events Operator GA Simplex

(1) (2) (3) (1) (2) (3) (1) (2) (3)

Small 0.63 510 6 0.60 266 4 0.71 513 �9Medium 0.36 812 4 0.41 322 5 0.62 1190 �8Large 0.21 758 3 0.34 161 5 0.65 1344 �6(1) The decrease rate of flood peak.(2) The release flow in the end of operation (cms).(3) The postponed arrival time of flood peak (h).

Guiding rational reservoir flood operation using penalty-type genetic algorithm 73

Conclusion

For exploring the real-world reservoir flood operation, theflood control requirements and existing operating regulationsare taken into account. Although the simplex method can beused to obtain the optimal solution through simplified floodoperating decision model, the release hydrograph does not

satisfy the requirements of real reservoir flood operation,especially some linguistic description of operating require-ments. The results (e.g. Fig. 5 in typhoon Betty1) obtainedby the simplex method do show those unsatisfied conditions.

Soft constraints play an important role in the reservoirflood operating decision model for searching rational out-flow hydrographs. In this study, the penalty-type GA, whichimplements the proper penalty functions for those violatingconstraints (both soft and hard constraints) into the objec-tive function, is proposed that can effectively combine themathematical model with linguistic description of require-ments for searching the rational outflow hydrographs of res-ervoir flood control. With the proper penalty functions, theynot only guide the GA towards a feasible solution space butalso possess some properties to reinforce its searching capa-bility. Soft constraints are allowed to be unsatisfied withincertain degree that still conforms to the linguistic descrip-tion of requirements. The results show that the penalty-type GA provides an adequate, effective and robust wayfor searching rational reservoir flood operating hydrographs.

Acknowledgements

The author thankfully acknowledges the financial supportfor this research by National Science Council, Taiwan, ROC(Grant No. NSC 94-2313-B-032-004).

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