optimization of multireservoir systems by genetic algorithm

Water Resour Manage (2011) 25:1465–1487DOI 10.1007/s11269-010-9755-0

Optimization of Multireservoir Systemsby Genetic Algorithm

Onur Hınçal · A. Burcu Altan-Sakarya ·A. Metin Ger

Received: 12 May 2010 / Accepted: 5 December 2010 /Published online: 23 December 2010© Springer Science+Business Media B.V. 2010

Abstract Application of optimization techniques for determining the optimal oper-ating policy of reservoirs is a major issue in water resources planning and manage-ment. As an optimization Genetic Algorithm, ruled by evolution techniques, havebecome popular in diversified fields of science. The main aim of this study is toexplore the efficiency and effectiveness of genetic algorithm in optimization of multi-reservoirs. A computer code has been constructed for this purpose and verifiedby means of a reference problem with a known global optimum. Three reservoirsin the Colorado River Storage Project were optimized for maximization of energyproduction. Besides, a real-time approach utilizing a blend of online and a posterioridata was proposed. The results obtained were compared to the real operational dataand genetic algorithm was found to be effective and can be utilized as an alternativetechnique to other traditional optimization techniques.

Keywords Genetic algorithm · Optimization · Reservoirs · Real-time

1 Introduction

For 5,000 years dams have served the mankind, among other purposes, ensuring anadequate supply of water by storing water in times of surplus and releasing it in

O. HınçalHeadquarters, Soyak Construction and Trading Co., Büyükdere Cad. No:38,Mecidiyeköy, Istanbul, Turkey

A. B. Altan-Sakarya (B)Department of Civil Engineering, Middle East Technical University, Ankara, Turkeye-mail: [email protected]

A. Metin GerIstanbul Aydın University, Istanbul, Turkey

1466 O. Hınçal et al.

times of scarcity is the main usage. Today there are more than 45,000 large dams inthe world contributing to the management of scarce water resources and mitigatingdevastating floods and catastrophic droughts. Dams regulate the natural runoff dueto seasonal variations and climatic irregularities, to meet the demand for irrigatedagriculture, power generation, domestic and industrial supply and navigation. Theyalso serve recreational purposes, boost tourism, aquaculture and fisheries, and canenhance environmental conditions. Dams contribute greatly to the world’s foodproduction in providing water for irrigation. Many of them generate clean renewablepower without Carbon dioxide (CO2) emissions. In spite of the large investmentsmade in dams and reservoirs worldwide, many are still operated on the basis ofexperience, rules of thumb or static rules established at the time of construction.Thus, there are many areas where even a small improvement in the operating policiescan lead to large benefits for many consumers.

Optimization of reservoir operation is an area that has attracted extensive re-search over the years. Optimization in design, planning and implementation of waterresources systems have always been an intensive research area. Optimization ofwater resources systems is related not only to the physical structures and theirfunctional characteristics but also to the criteria by which the system is operated.A reservoir operation problem can be considered as a decision making problemhaving many constraints. Optimizing reservoir operations incorporate allocation ofresources, development of stream flow regulation strategies, operating rules andreal-time release decisions in its bodily constitution. A reservoir regulation plan,which is sometimes referred as operating procedure or release policy, is a set ofrules quantifying the amount of water to be stored, released or withdrawn from areservoir or system of reservoirs, under various conditions. This study intended tobuild an operational model to ease the decisions about the optimal volumes to bestored or released from the reservoirs in question, i.e. the operational decisions.

Multi-reservoir operation/management planning is a complex task involving manyvariables, objectives, and decisions. The complexities of the multiple reservoir systemcompel that the release decisions are determined by means of optimization and/orsimulation models. Most of the optimization methods are constructed upon the basisof mathematical modeling. So far, optimization methods have been implemented forboth planning purposes and for real time operation. Real time reservoir operationdeals with the optimal operation of an existing reservoir system and decisions aboutthe releases have to be made in reasonably short time periods. The condition thatthe storages at the end of the time period considered for optimization is greaterthan or equal to the target ending minimum storages is considered in optimal policydefinition. This condition is applied to future operations. In other words, it is desiredto establish the optimum release policy over the release periods specified, which shallresult in a set of target ending minimum storages in the final policy period that makessure of being adequate for future system operations.

Several approaches have been developed for optimization of reservoir operations,defining reservoir operating rules and many different techniques have been studiedwith regards to this optimization problem. Numerous optimization models havebeen proposed and reviewed by many scientists. Historical background of reservoiroperation optimization techniques has been given below.

For a long period, dynamic programming (Bellman 1957), was a powerful ap-proach in the optimization of reservoir operation. Young (1967) developed optimal

Optimization of Multireservoir Systems by Genetic Algorithm 1467

operating rules for a single reservoir using dynamic programming. Larson (1968)proposed a study embracing a four-reservoir problem by making use of incrementaldynamic programming. Hall et al. (1969), using a different form of incrementaldynamic programming, studied a two-reservoir system. Heidari et al. (1971) devel-oped a model, setting off from the proposal of incremental dynamic programming,which is called discrete differential dynamic programming. Howson and Sancho(1975) generated a progressive optimality algorithm for optimization of reservoiroperation policies. Loucks and Dorfman (1975) showed that chance constrainedmodels on reservoir planning and operation are overly conservative and generateoperational rules that exceed the prescribed reliability levels. Murray and Yakowitz(1979) have developed an effective technique, differential dynamic programming, foroptimization of multi-reservoir systems, without any requirement for discretizationstate and decision variables. Ahmed and Lansey (2001) proposed a method based onthe parameter iteration method of Gal (1979) involving quadratic approximation offuture benefits and parameterization of operating policies for hydropower systems.Cheng and Chau (2001) presented a fuzzy iteration methodology which gives theobjective weight and the relative membership degree of alternatives at the same timefor reservoir flood control. A three-person multi-objective conflict decision modelwas presented by Cheng and Chau (2002) for reservoir flood control and applied toFengman Reservoir in China. Labadie (2004) performed an extensive compilation onthe optimal operation of multi-reservoir models. Cheng and Chau (2004) presentedthe outcome of natural programming about the flood control management system forreservoirs in China. Liu et al. (2006) proposed and used the dynamic programmingneural-network simplex (DPNS) model in order to derive refill operating rules inreservoir planning and management. Folded Dynamic Programming (FDP) waspresented to develop optimal reservoir operation policies for flood control by Kumaret al. (2010) and applied to Hirakud Reservoir in Mahanadi, India, considering theobjective of obtaining optimal policy for flood control. Recently, Liu et al. (2010)derived optimal refill rules for multi-purpose reservoir aiming the maximization ofutilization benefits considering the flood control safety.

Genetic algorithm was firstly developed by Holland (1975) over the course of the1960s and 1970s and finally popularized by one of his students, David Goldberg,who was able to solve a difficult problem involving the control of gas pipelinetransmission for his dissertation (Goldberg 1989). Holland was the first to try todevelop a theoretical basis for genetic algorithms through his schema theorem. Thework of De Jong (1975) showed the usefulness of the genetic algorithm for functionoptimization and made the first concerted effort to find optimized genetic algorithmparameters.

Pioneers of genetic algorithm, Goldberg (1989) and Michalewicz (1992) presentedsatisfying introductions and several papers give general overviews of genetic algo-rithm. Cieniawski et al. (1995) studied the multi-objective optimal location of a net-work of ground-water monitoring wells under conditions of uncertainty by benefitingfrom genetic algorithm. Davidson and Goulter (1995) used genetic algorithms tooptimize the layout of rectilinear branched distribution (natural gas/water) systems.A study similar to that of Wang (1991), for the automatic calibration of conceptualrainfall-runoff models, has been reported by Francini (1996), who made use ofa genetic algorithm combined with a local search method; sequential quadraticprogramming.


Esat and Hall (1994) applied a genetic algorithm to the four-reservoir problem.The objective of this problem was to maximize the benefits from power generationand irrigation water supply, having constraints on both storages and releases fromthe reservoirs. Fahmy et al. (1994) applied genetic algorithm to a reservoir system,and compared performance of the genetic algorithm approach with that of dynamicprogramming. Raman and Chandramouli (1996) used an artificial neural network forinferring optimal release rules conditioned on initial storage, inflows, and demands.Oliveira and Loucks (1997) used a genetic algorithm to evaluate operating rulesfor multireservoir systems and indicated that optimum reservoir operating policiescan be determined by means of genetic algorithms. Cai et al. (2001) describe anapplication of genetic algorithms to solving large-scale nonlinear reservoir operationproblems over multiple periods. Chandramouli and Raman (2001) extended thestudy of Raman and Chandramouli (1996), developing operating rules for multireser-voir systems. Sharif and Wardlaw (2000) presented a real case study in Brantas Basinin Indonesia for the optimization of the system using genetic algorithm. Ahmed andSarma (2005) presented a genetic algorithm model for finding the optimal operatingpolicy of a multi-purpose reservoir, located on the river Pagladia, a major tributaryof the river Brahmaputra. A chaos genetic algorithm is presented by Cheng et al.(2008) to overcome premature optimum and increase the convergence speed. Thedeveloped model was applied for the monthly operation of a hydropower reservoir.

In the present study, the multi-reservoir system optimization problem is for-mulated which incorporates the decision variables, the objective function and theconstraints. Following the extensive study on genetic algorithm and its applicationsin optimization problems, genetic algorithm aiming to optimize the mathematicalproblem under consideration has been constructed. It is configured to include all thenecessary operators, the conditional statements in order to meet the constraints andmost importantly to find out the optimum solution remaining in strict compliancewith the objective function specified. Then, the computer code in which the abovementioned stages are all embedded and employed has been developed in FortranLanguage. Pursuing configuration of the code, a verification process has beenadministered by making use of the four reservoir problem having a known globaloptimum solution which has already been adopted as a reference problem in pastresearches focusing on optimal reservoir system operation. A sensitivity analysis hasbeen applied to the optimization problem in order to evaluate the effects of thevariables employed in the genetic algorithm.

A real case study followed this verification stage. A multi-reservoir system in theUnited States has been picked out as a real case. The data pertaining to the multi-reservoir system have been acquired, a real-time optimization has been applied andthe real case study has been performed onto this system.

2 Problem Definition

The purpose of optimal operating policy is to specify how water is managed through-out the system. Optimal operating policy serves to reach maximum benefit from thesystem satisfying the flow requirements and system demands. In this study, benefit isconsidered to be the energy gained throughout the system. Decision variables are thereleases from each reservoir at each time interval. The aim is to find out the optimum


combination of releases which will lead to generate maximum energy throughout thesystem. There are upper and lower boundaries for releases and storages. Besides, thestorages at the end of periods considered are to be equal to or above the target endingminimum storages. These limitations form the constraints of the problem. Anotherconstraint of the problem is that continuity equation is to be satisfied throughout thewhole system. Generally expressing the objective function as the maximization oftotal energy produced by all reservoirs at all times is defined as,

MaximizeI∑

i=1

T∑

t=1

(Energyi,t

)(1)

where I is the number of reservoirs and T is the total time period. Equation 1is subject to the following constraints. Continuity should be satisfied which isdefined as,

Si, t+1 = Si, t + Ii, t − Ri, t for i = 1, ..., I and for t = 1, ..., T (2)

where Si,t, Ii,t and Ri,t are the storage, inflow and releases for the ith reservoir at thetth time step, respectively.

Storages will be equal to or below maximum and equal to or above minimumstorages,

Si,min ≤ Si,t+1 ≤ Si,max for i = 1, ..., I and for t = 1, ..., T (3)

Similarly, releases will be between maximum and minimum release values,

Ri,min ≤ Ri,t ≤ Ri,max for i = 1, ..., I and for t = 1, ..., T (4)

Ending storage will be equal to or above the target ending minimum storages,

Si,T ≥ di,T for i = 1, ..., I (5)

where di,T is the target ending minimum storage for the ith reservoir at the Tth timestep and T is the ending time for the problem under consideration.

Continuity equation is readily satisfied, since the storages are computed by makinguse of continuity equation given in Eq. 2. Other constraints are embedded into theobjective function as a penalty function. Thus, constrained optimization problemtakes the form of an unconstrained optimization problem. The purpose lying beneaththe fact that constrained problem is transformed into an unconstrained problem isto be able to handle the problem by means of Genetic Algorithm. The constrainton storages defined by Eq. 2 is embedded into the objective function as penalty

terms which areI∑

i=1

T∑t=1

c1[min(0, (Si,max − Si,t))]2 andI∑

i=1

T∑t=1

c2[min(0, (Si,max − Si,t))]2.

Similarly,I∑

i=1c3[min(0, (di,T − Si,T))]2 is embedded into Eq. 1 as penalty term of the

constraint on the ending storages. Here, the deviations from maximum, minimumstorages and target ending minimum storages are penalized by square of deviationfrom constraints. c1, c2 and c3 are constants. Those constants act as a tuner of theweight of the penalty term in order for them to be in the order of magnitude of theobjective function.


3 Genetic Algorithms

Genetic algorithm is a search algorithm based on mechanics of natural selection andnatural genetics (Goldberg 1989). As the name implies, genetic algorithm is basedon principles of natural evolution and survival of the fittest. In genetic algorithms,a population of candidate solutions to the problem is employed. Genetic algorithmssimultaneously consider multiple candidate solutions to the problem and proceed bymoving this population of solutions toward a global optimum.

Genetic algorithm has a main generational process cycle. This cycle is drivenmainly by generation number. Within this cycle, an initial population is created;each individual is coded so as to be represented numerically; then each individual ofpopulation is assigned a fitness value which is a parameter with respect to which eachindividual is evaluated whether or not to live in subsequent generations. Evaluationand selection of individual which will be awarded to live in subsequent generationsare handled by means of genetic operators, selection, crossover and mutation.

Genetic algorithms start by generation of an initial population which is constitutedby individuals called chromosomes (or strings). Population size depends on thenature of the problem, but typically contains several hundreds or thousands ofpossible solutions. Traditionally, the population is generated randomly, covering theentire range of possible solutions (the search space). Given upper and lower boundsfor each chromosome, they are created randomly so as to remain within its upperand lower constraints. The principle is to maintain a population of chromosomes,which represent candidate solutions to the problem that evolves over time through aprocess of competition and controlled variation. Each chromosome in the populationhas an assigned fitness to determine which chromosomes are used to form newones in the competition process which is called selection. The new ones are createdusing genetic operators such as crossover and mutation. Fitness values are expectedto improve indicating creation of better individuals in new generations. A generalflowchart of a genetic algorithm indicating the processes within the algorithm is givenin Fig. 1.

3.1 Coding

Coding, mapping from phenotypes to genotypes, is performed in a number of wayssuch as binary coding, gray coding, e-coding and real coding. However, most commoncoding mechanisms are binary and real coding. In binary coding the chromosomesare expressed as binary strings. The most commonly used representation of chromo-somes in the genetic algorithm is that of the single-level binary string by making useof 0’s and 1’s.

The use of real-valued genes in genetic algorithms is claimed by Wright (1991), tooffer a number of advantages in numerical function optimization over binary coding.In real coded genetic algorithms, each individual is coded as a vector of floating pointnumbers (real numbers) having the same length as that of the solution vector. Realcoding allows the domain knowledge to be easily integrated into the Real CodedGenetic Algorithms. Goldberg (1991) and Eshelman and Schaffer (1993) leave to theuser the decision for choosing one of these coding mechanisms, suggesting that eachone of them has suitable properties for different types of fitness functions. On theother hand, other authors such as Michalewicz (1992) defend the use of real coding,


Fig. 1 Flowchart of a geneticalgorithm

START

Generate Initial Population (Randomly)

Inputs, Objective Function, Constraints

Calculate State Variables, Calculate Fitness Values

Termination Criteria

Last Generation?

STOP

Yes

No Selection Operator

Crossover Operator

Mutation Operator

Elitist Operator Retain k% of population

Formation of Next Generation

showing their advantages with respect to the efficiency and precision reached ascompared to the binary one.

3.2 Fitness Function

The fitness value of the chromosome is considered to be a grade for the evaluation ofthis member of the population whether or not to pass to the next generation. Fitnessvalues are calculated by making use of the objective function; hence fitness value ofa chromosome can be taken into consideration as the objective function value of thismember.

3.3 Selection

Selection is the survival of the fittest within the genetic algorithm. In the stage ofapplication of selection operator, the chromosomes that will be awarded to live in thesubsequent generation are determined. Those chromosomes selected to live in thesubsequent generation form the mating pool from which the parents of the new gen-eration undergoes the process of crossover. One of the techniques used as a selection


operator is the “Roulette Wheel Selection” operator. In this technique, for eachand every chromosome, the ratio of the fitness value of the chromosome to thetotal of the fitness values of the chromosome of the whole population is calculatedand this parameter computed for each chromosome is considered for this memberof the population as the probability of survival into the next generation. Roulettewheel selection enables chromosomes with higher fitness values to have a greaterprobability of survival. In addition, the number of chromosomes in a population iskept constant for each generation and hence the selection operator will generate anew population of the same size. This implies that chromosomes with higher fitnessvalues will eventually dominate the population (Ansari and Hou 1999).

To select a chromosome for selection, the roulette wheel is “spun,” and thechromosome corresponding to the slice at the point where the wheel stops on isgrabbed as the one to survive in the offspring generation. The algorithm of rouletteselection may be generalized in steps as follows:

1. Fitness of each individual, fk, in a population size of N and sum of them arecalculated.

2. A real random number, ran( ), within the range [0,1] is generated and s is set tobe equal to the multiplication of this random number by the sum of the fitnessvalues, s=ran ( ) x fsum

3. Minimal K is determined such thatK∑

k=1fk ≥ s, and the Kth individual is selected

in the next generation.4. Steps 2 and 3 are repeated until the number of selected individuals becomes

equal to the population size, N.

3.4 Crossover

Next genetic operator to be applied to the generation is the crossover operator.Crossover operator is a method for sharing information between chromosomes.Selected parents reproduce the offspring by performing a crossover operation onthe chromosomes. It has always been regarded as the main search operator ingenetic algorithms because it exploits the available information in previous samplesto influence future searches. The performance of real coded genetic algorithms ona particular problem will be strongly determined by the degrees of exploration andexploitation associated to the crossover operator being applied.

3.5 Mutation

One further operator in genetic algorithm is the mutation operator which does play arole of local random search within the framework of the generational process cycle.Mutation is a random process where once the genes are replaced by another toproduce a new genetic structure. In genetic algorithms, mutation is randomly appliedwith low probability and modifies elements in the chromosomes. Usually consideredas a background operator, the role of mutation is often seen as providing a guaranteethat the probability of searching any given chromosome will never be zero and actingas a safety net to recover good genetic material that may be lost through the actionof selection and crossover.


4 Construction of the Code

4.1 Generation of Initial Population

Operating rules prescribe how water is to be released or stored during the subsequentmonth based on current state of the system. A chromosome (individual) representingall reservoirs at all time steps has been constructed having the following form:

[Nvar] = [R1,1, R2,1, . . . , RI,1; . . . ; R1,t, R2,t, . . . , RI,t; . . . . . . .; R1,T , R2,T , . . . , RI,T

]

(6)

where I is the number of reservoirs in the system considered, t is an index specifyinga time period, T is the total number of time periods into which the time horizon isdivided. [Nvar] is the set of genes forming a chromosome of the population. Eachchromosome contains IxT genes. Each gene within chromosome represents releasemade from a reservoir at a specific time period and can take up any value betweenthe upper and lower bounds of releases. Nvar is the total number of genes in achromosome. Number of genes in a chromosome is defined by the product of numberof reservoirs and the total number of time periods considered in the system. Therange of time scale is considered “month” to be in compliance with the referenceproblem verified and since the set of data acquired pertaining to the Case Study wasreceived in terms of months.

How genes are arranged in a chromosome is of high importance. There are twobasic approaches.

1. Grouping releases by time step; such that the chromosomes contained in Tgroups of I genes representing the release from each reservoir in a particulartime step;

2. Grouping releases by reservoir; I groups of T genes with each group containingthe time series of releases from an individual reservoir.

Objective is to find a gene sequence that yields the best chromosome generatingthe maximum energy. In order for the genetic algorithm to be initialized, Nip

chromosomes are identified. Nip is the population size of the problem. Therefore,a matrix of Nip rows and Nvar columns considered. Each row of the initial populationin Eq. 7 represents a chromosome (individual) of the population. Initially, with anidentified number of individuals, i.e. population size of Nip, random numbers aregenerated to form a matrix of Nip × Nvar.

[Initial

Population

]

Nip×Nvar

= (Ri,max − Ri,min) ×[

Randomnumbers

]

Nip×Nvar

+ Ri,min (7)

where, Ri,max and Ri,min are the maximum and minimum values that the variable mayassume for reservoir i, respectively. Nip is the total number of chromosomes in apopulation which is an input.

4.2 Calculation of State Variables

After generation of initial population which is composed of individuals containingreleases (decision variables), calculation of storages (state variables) comes next.


Storage for each and every gene of the individuals is computed making use ofcontinuity equation which is the equality constraint of the problem. Usage of Eq. 2in calculation of storages ensures that continuity equation is satisfied for every genecreated. However, this does not enable the state variables (storages) determined byusing the continuity equation be within their boundaries. The inequality constraintsproviding storages remain within their limits are satisfied by incorporating the relatedpenalty terms into the objective function.

4.3 Calculation of Fitness Values

Next step is the computation of fitness values. Fitness assigned to each gene hasdirect influence on eligibility for each chromosome to live in the next generation.Fitness value is the bodily constitution of objective function and the penalty termsoriginating from violation of the constraints, if exists. Constraints are embeddedinto the objective function as penalty terms in order to penalize the violation of theconstraints related to storages.

4.4 Genetic Algorithm Operators

In this phase, genetic algorithm operators; selection, crossover, and mutation opera-tors are implemented onto the population. Using the selection operator, the mates,whose child to live in the subsequent generation are selected. Among the selectionoperators mentioned before, roulette wheel selection operator, recommended for itssuperiorities over the remaining ones has been used as the selection operator. Afterthe fitness values and the sum of the fitness values in the generation are computed,roulette wheel selection, mentioned in previously has been placed within the code.The higher the fitness value of an individual in the current population, the higher itsprobability of being selected as one of the mates whose children will live in the nextgeneration is. Selection probability is the ratio of the fitness of the individuals in thepopulation to the sum of fitness of each individual in the population.

The computer code has been developed so as to enhance the comparison of thecrossover techniques, arithmetic crossover, average crossover, random crossover andBLX-α, with different values of α. Finally, in the genetic algorithm code constructed,mutation is randomly applied with low probability, typically in the range 0.001 and0.02 to modify the genes of some individuals.

5 Verification of the Code

In order to verify the code, the four-reservoir problem which was formulated andfirst solved by Larson (1968), and elaborated further by Heidari et al. (1971), hasbeen used. The fact that this problem has a known global optimum made it eligiblefor verification.

The four-reservoir problem permits to test the performance of genetic algorithmsagainst a known global optimum and to perform sensitivity analysis. There are fourreservoirs in the system, the layout of which is shown in Fig. 2. Details given byHeidari et al. (1971) with regards to the four reservoir system may be summarized asfollows.


Fig. 2 Layout of the reservoirsfor the system considered

Supplies from the system are used for hydropower generation and for irrigation.The objective is to maximize the benefit from the system over 12 two-hour operatingperiods. The objective function is explicated as

Maximize4∑

i=1

12∑

t=1

(bi,t · Ri,t

) +12∑

t=1

(b 5,t · R4,t

)(8)

where bi,t is the unit return due to activity i, i = 1, 2, . . .5 during a period t.There are a total of five activities in the above criterion; four generation activities(b 1,t, b 2,t, b 3,t, b 4,t) and one irrigation activity (b 5,t). There are inflows to the first andsecond reservoirs only, and these are 2 and 3 units, respectively, in all time periods.The initial storage in all reservoirs is 5 units.

Constraints on reservoir storages for all times are:

0 ≤ S1, S2, S3 ≤ 10 (9)

0 ≤ S4 ≤ 15 (10)

Constraints on releases for all times are as follows:

0 ≤ R1 ≤ 3 (11)

0 ≤ R2, R3 ≤ 4 (12)

0 ≤ R4 ≤ 7 (13)

Continuity equations for each reservoir, i over each time period, t is as follows:

Si, t+1 = Si, t + Ii, t − Ri, t (14)


In accordance with the layout of the four reservoir problem, continuity equationthroughout the system may be expressed as follows:

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

S1,t+1

S2,t+1

S3,t+1

S4,t+1

⎫⎪⎪⎪⎬

⎪⎪⎪⎭=

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

S1,t

S2,t

S3,t

S4,t

⎫⎪⎪⎪⎬

⎪⎪⎪⎭+

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

I1,t

I2,t

I3,t

I4,t

⎫⎪⎪⎪⎬

⎪⎪⎪⎭+

⎡

⎢⎢⎢⎣

−1 0 0 00 −1 0 00 1 −1 01 0 1 −1

⎤

⎥⎥⎥⎦

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

R1,t

R2,t

R3,t

R4,t

⎫⎪⎪⎪⎬

⎪⎪⎪⎭(15)

Additionally, the target ending minimum storages, di’s are as follows:

d1 = d2 = d3 = 5 (16)

d4 = 7 (17)

and

gi(Si,12, di

) = −40[Si,12 − di

]2 for Si,12 ≤ di (18)

gi(Si,12, di

) = 0 for Si,12 > di (19)

where gi(Si,12, di) is a function that reflects a penalty to the system when the finalstate of the ith component of the system at time T is Si,12 instead of the desiredminimum state di. Such a penalty function is necessary to meet the requirementsrelated to the target ending minimum storages. The desired state vectors of theinitial and final stages for i = 1, 2, 3, 4 are assumed. To consider the limita-

tions on storages, the following penalty terms,I∑

i=1

T∑t=1

c1[min(0, (Si,max − Si,t))]2 and

I∑i=1

T∑t=1

c2[min(0, (Si,max − Si,t))]2 are embedded into the objective function

The inputs of the computer code created for the optimization of reservoir manage-ment by Genetic Algorithm may be listed as population size, number of generations,crossover technique, probability of crossover, and probability of mutation.

5.1 Comparison of Results

Adopting the four reservoir problem as an appropriate reference model for verifica-tion, objective function and constraints indicated before has been studied andexamined for testing the performance of the computer code constructed for theoptimization of multi-reservoir systems by genetic algorithm.

The computer code has been run to observe the effect of considered differentcrossover techniques, namely arithmetic crossover, average crossover, randomcrossover and BLX-α (with different values of α) techniques. The known globaloptimum for the four-reservoir problem was given by Wardlaw and Sharif (1999)as 401.3 units. Energy was given as product of benefit constants and release. Basedon the above mentioned input parameters, the computer code has been run andknown global optimum has been achieved. The output storages and releases obtainedafter execution of the code fit perfectly to those stated for the four reservoirproblem by Wardlaw and Sharif (1999). The fact that the target ending minimum


storages are satisfied, another constraint of the four reservoir model examined, hasalso been checked and confirmed for each reservoir location. Besides, the resultsobtained after optimization by the utilization of the computer code revealed that theinequality constraints defined in the four reservoir system have been met without anyviolation. Furthermore, as expected, it was confirmed that Central Processing Unit(CPU) time increases both with increasing generation number and with increasingpopulation size.

5.2 Sensitivity Analysis

Sensitivity analysis has been performed to achieve the influence of the change inthe input parameters on fitness. In the light of the recommended values for inputparameters; i.e. population size, generation number, probability of crossover, prob-ability of mutation and the results of sensitivity analysis, the following set for inputparameters were employed:

Population size: 5000,Generation number: 5000,Probability of crossover = 0.70,Probability of mutation = 0.02

The variation of the fitness values obtained after test runs for different crossovertechniques, namely, arithmetic crossover, random crossover, average crossover andBLX-α Crossover technique for different values of α = 0.10, 0.25 and 0.50 is shown inFig. 3. After exploration of the influence of different crossover techniques examinedand given in Fig. 3, it is seen that BLX-α Crossover technique exhibits a fasterconvergence with respect to that of the other crossover techniques. Fitness valuesdetermined by execution of code for different crossover techniques are given for

350

355

360

365

370

375

380

385

390

395

400

405

0 500 1000 1500 2000 2500

Generation Number

Fit

ness

Arithmetic CrossoverAverage CroosverBLX-α Crossover technique (with α=0.50)BLX-α Crossover technique (with α=0.25)BLX-α Crossover technique (with α=0.10)Random Crossover401.30 (known global optimum)

Fig. 3 Influence of crossover technique on fitness


Table 1 Fitness values for different crossover techniques

Crossover technique Generation number

2000 3000 4000 5000

Arithmetic crossover 400,589 400,887 400,942 401,006Average crossover 400,912 401,064 401,190 401,199BLX-α crossover (α = 0.50) 400,812 401,102 401,278 401,289BLX-α crossover (α = 0.25) 401,014 401,188 401,282 401,299BLX-α crossover (α = 0.10) 401,177 401,278 401,294 401,301Random crossover 399,470 400,610 400,988 401,234

generation numbers, 2000, 3000, 4000, 5000, in Table 1. This fact lead us to preferBLX-α Crossover technique with α = 0.1 as it gives the maximum objective functionvalue for all generation numbers considered.

5.2.1 Sensitivity to Crossover Probability

Firstly, sensitivity analysis has been performed with respect to the probability ofcrossover. Used input parameters were mutation probability of 0.02 and BLX-αCrossover technique (with α = 0.10). In order to see the effect of change in crossoverprobability on proportion of maximum fitness for different sets of population sizeand generation numbers. As shown in Fig. 4, crossover probability seems to haveno significant effect on fitness for the range covered. Besides, as also demonstratedby Fig. 4, with increasing generation number, the amplitude of fluctuations becomessmaller. Moreover, as generation number increases, the bandwidth which is formedby the change in population size, becomes narrower. Those variations indicate thatthe change in proportion of maximum fitness becomes insignificant with increasinggeneration number.

0,98

0,985

0,99

0,995

1

0,5 0,55 0,6 0,65 0,7 0,75 0,8 0,85 0,9

Crossover Probability

Pro

port

ion

of M

axim

um F

itnes

s

Pop=1000;Generation #:1000 Pop=1000;Generation #:3000 Pop=1000;Generation #:5000



Fig. 4 Effect of crossover probability on fitness


Variation of Proportion of Maximum Fitness against Generation Number

0,98

0,985

0,99

0,995

1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Generation Number

Pro

port

ion

of m

axim

um f

itne

ss

P:1000 P:1500

P:2000 P:2500

P:3000 P:3500

P:4000 P:4500

P:5000 P:5500

P:6000 P:6500

P:7000 P:7500

P:8000

Fig. 5 Effect of population size and generation number on fitness

5.2.2 Sensitivity to Population Size and Generation Number

Sensitivity to population size and generation number has also been investigated.Input parameters used were crossover probability of 0.70, mutation probability of0.02 and BLX-α Crossover technique (with α = 0.10). Fitness is again expressed asa proportion of the known optimum for the four-reservoir problem. Variation ofproportion of maximum fitness was examined against generation number for a series

0,93

0,94

0,95

0,96

0,97

0,98

0,99

1

0,00 0,05 0,10 0,15 0,20 0,25 0,30

Mutation Probability

Pro

port

ion

of M

axim

um F

itnes

s


Pop=3000;Generation #1000 Pop=3000;Generation #:3000 Pop=3000;Generation #:5000


Fig. 6 Effect of mutation probability on fitness


of different population sizes. As demonstrated in Fig. 5, the proportion of maximumfitness increases while the effect of population size on the proportion of maximumfitness becomes less appreciable.

5.2.3 Sensitivity to Mutation Probability

The effect of change in mutation probability has been investigated for different setsof population size and generation numbers. Input parameters used were crossoverprobability of 0.70 and BLX-α Crossover technique (with α = 0.10). Variation ofproportion of maximum fitness was examined against mutation probability for aseries of different population sizes and generation numbers. Irrespective of the pop-ulation size and generation number, the proportion of maximum fitness decreasessignificantly for mutation probability larger than 0.06 as depicted in Fig. 6. Formutation probability between 0.02 and 0.06, effect of mutation probability on theproportion of maximum fitness is insignificant.

6 Development of Real Time Approach, Application and Discussion of Results

6.1 Definition of Problem in the Colorado River Storage Project (CRSP)

Colorado River Storage Project (CRSP) was examined, and realized operational datafor the Blue Mesa, the Morrow Point and the Crystal Reservoirs were compared tothose obtained from the optimization code developed. General description of CRSPand information related to the multi-reservoir system comprising of aforementionedreservoirs are given in Hınçal (2008). Realized operational data pertaining to thetime period between 2002 and 2006, together with the characteristics of the reservoirswere obtained from the US Bureau of Reclamation, Water Resources Group, SaltLake City Office. The data includes all of the constraints, operational data; inflows,releases, power generated, water levels in the reservoirs, current status of the damsand reservoirs.

The objective function utilized in the four reservoir problem incorporates the con-straints specifically determined for that problem solely and therefore this objectivefunction cannot flexibly be applied to any other reservoir. Setting off from this idea,a more general objective function which can be applied to other real case problemshas been attempted to be formed. The objective function formulation of Barros et al.(2003) has been adopted in this study.

Maximize∑

t

∑

i

(ξi,t Ri,t

)(20)

where, ξ i,t is the energy production function in MWh.s/m3; such that:

ξi,t = εi�Hi,t = εi(HFi,t − HTi,t

)(21)

with εi is the specific productibility in MWh.s/m4. HFi,t is the reservoir upstreamwater level and HTi,t is the tailwater level in m.

Energy versus �Hi,t Ri,t values pertaining to the past data acquired from CRSPhas been plotted for each reservoir examined. Slope of the line fitted to those plotted


data reveals the specific productibility for three reservoirs. Upstream water levelis a function of the storage value and by means of the stage-area-capacity curvesobtained the relationship between the upstream water level, HF and the storagevalues, S are determined. Then, the energy formulation as a function of release andstorage for each reservoir location in the CRSP within a specified time are obtained.Additionally, constraints on reservoir storages, Si for all times, constraints on re-leases, Ri (m3/s) for all times, continuity equation for each reservoir over each timeperiod, t and the target ending minimum storages at the end of first year examined,are considered. Setting the boundary and initial conditions, the objective function,penalty terms and henceforth the fitness function; identifying the remaining inputsof the problem in the light of the outcomes of the sensitivity analyses; code has beenexecuted for different comparison approaches. The details of the study can be foundin Hınçal (2008).

The program has been run with the following set of values of genetic algorithmparameters: Initial Population: 5,000, Generation Number: 5,000, Probability ofCrossover: 0.70, Probability of Mutation: 0.02, Roulette Wheel Selection Operatorand BLX-α (α = 0.10) Crossover Technique.

6.2 Comparison of Approaches

Energy production of the CRSP has been compared to those determined by applica-tion of genetic algorithm with conventional approach and real-time approach.

6.2.1 Conventional Approach

As an initial consideration, developed optimization code has been executed takinginto account a period of 1-year (12 months of 2005). Secondly, year 2006 wasoptimized separately by means of the developed code. Available operational dataincluded 12 months in 2005 and 11 months in 2006.

One further consideration was optimization of both of the separately explored1-year periods one at a time; in other words, considering a 2-years period. Theresults obtained after optimization have been compared to those achieved in realizedoperational results.

6.2.2 Real-Time Approach

A real-time approach was attempted in the final stage for the multi-reservoir systemconsidered in the CRSP. The main goal of this real-time approach was intendedto ensure real-time optimization with respect to energy maximization of the multi-reservoir system by making use of the developed code.

Firstly, a period of 1 year (2005) is optimized by utilizing the code considering thepast occurred operational data. Optimized solution with respect to energy maximiza-tion criterion formed a template baseline, housing the historical background of theconditions concerning the system being examined.

This template baseline is used for future real-time optimizations. Second year(2006) is optimized by using this approach. In this approach, optimization is refreshedevery month. At the end of each month, inflow value becomes known and theoccurred inflow value is set equal to the inflow in first month of the second year(month 13). Release in this month is assumed to be the same as in the first month


of the baseline. Then, continuity equation is applied to determine the storage at theend of month 13. Storage at the end of month 13 is checked so as not to violate itsconstraints. In case of constraint violation, release in month 13 is adjusted so thatthe storage at the end of month 13 remains within its upper and lower boundaries.Storage at the end of month 13 is set as the target ending minimum storage of theup-to-date template baseline. Besides, the initial storage of the up-to-date templatebaseline is replaced by the storage which is the successor of the initial storage, in thetemplate baseline. The template baseline is then shifted and the code is run with theinputs of the shifted template baseline. This template baseline is shifted every monthfollowing the same flow mechanism until the end of the period considered.

As demonstrated in Fig. 7, a brief flow scheme including the steps of the approachmay be summarized as follows:

– Set inflow in time period 13 equal to the value known at the end of this month.– Assume that the release in time period 13 is the same as in the 1st month of the

baseline, (See 1 in Fig. 7)– Compute the storage at timepoint 13 (end of January of the second year) using

continuity equation. Check whether the storage computed is within the upperand lower boundaries of storages. If it is not, adjust the release in time period 13which was assumed to be the release in the 1st month of the baseline is adjustedso that the storage constraints will not be violated (See 2 in Fig. 7 and Eqs. 22–26)Assume R13 = R1

S13 = S12 + I13 − R13 (22)

If S13 > Smax then,

�S = S13 − Smax (23)

R13 = R1 + �S (24)

R1,

I1

R2,

I2

R3,

I3

R4,

I4

R5,

I5

R6,

I6

R7,

I7

R8,

I8

R9,

I9

R10

, I10

R11

, I11

R12

, I12

Time periods 1 2 3 4 5 6 7 8 9 10 11 12

BASELINE Jan.

Feb.

Mar

.

Apr

.

May

.

Jun.

Jul.

Aug

.

Sep.

Oct

.

Nov

.

Dec

.

Time points 0 1 2 3 4 5 6 7 8 9 10 11 12

S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12

R1,

I1

R2,

I2

R3,

I3

R4,

I4

R5,

I5

R6,

I6

R7,

I7

R8,

I8

R9,

I9

R10

, I10

R11

, I11

R12

, I12

R13

, I13

R14

, I14

R15

, I15

R16

, I16

R17

, I17

R18

, I18

R19

, I19

R20

, I20

R21

, I21

R22

, I22

R23

, I23

R24

, I24

R1,

I1

R2,

I2

R3,

I3

R4,

I4

R5,

I5

R6,

I6

R7,

I7

R8,

I8

R9,

I9

R10

, I10

R11

, I11

R12

, I12

R1,

I13

R14

, I14

R15

, I15

R16

, I16

R17

, I17

R18

, I18

R19

, I19

R20

, I20

R21

, I21

R22

, I22

R23

, I23

R24

, I24

Time periods 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Months Jan.

Feb.

Mar

.

Apr

.

May

.

Jun.

Jul.

Aug

.

Sep.

Oct

.

Nov

.

Dec

.

Jan.

Feb.

Mar

.

Apr

.

May

.

Jun.

Jul.

Aug

.

Sep.

Oct

.

Nov

.

Dec

.

Time points 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 S16 S17 S18 S19 S20 S21 S22 S23 S24

R1,

I1

R2,

I2

R3,

I3

R4,

I4

R5,

I5

R6,

I6

R7,

I7

R8,

I8

R9,

I9

R10

, I10

R11

, I11

R12

, I12

1 2 3 4 5 6 7 8 9 10 11 12

UPDATED BASELINE Jan.

Feb.

Mar

.

Apr

.

May

.

Jun.

Jul.

Aug

.

Sep.

Oct

.

Nov

.

Dec

.

0 1 2 3 4 5 6 7 8 9 10 11 12

S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12

1

2

4

3

Fig. 7 Illustration of real-time approach


If S13 > Smin then,

�S = Smin − S13 (25)

R13 = R1 − �S (26)

– Set storage at timepoint 13 as the target ending minimum storage of the up-to-date template baseline (See 3 in Fig. 7)

– Replace storage in the 0th timepoint of the up-to-date template baseline by thestorage in the 1st timepoint of the template baseline (See 4 in Fig. 7)

– Shift up-to-date template and run the code considering the inputs of the shiftedup-to-date template baseline.

– Follow the same procedure and shift the template for the remaining months.

6.3 Comparison and Results

Maximized energy amounts determined by using conventional approach and realtime approach, and real operational energy amounts, as well, are shown in Table 2and Fig. 8.

Occurred energy amounts were considered as a reference line in order tofigure out the improvement and/or approximation to the occurred/generated energyamount in the multi-reservoir system in the CRSP.

From the investigations of the multi-reservoir system in the CRSP for differentconsiderations, it is evident that:

– Optimizing a 1-year period, year 2005; energy of 627.88 GWh was achieved;indicating an improvement of 1.1% when compared to that gained throughrealized/produced energy of 620.97 GWh,

– In the next 1-year period, year 2006 was optimized and an energy amount of694.3 GWh was determined which meant an improvement of 1.27% compared tothe reference realized energy production,

– Two separately examined periods, years 2005 and 2006 have been optimizedconsidering a single period of 2 years. Energy amount achieved after optimizationof this multi-reservoir system considering 2-years period was 1,356.03 GWhwhich is by 3.8% improved from the energy actually realized/generated.

Table 2 Comparison of maximized energy amounts

Total energy generated (kWh)

Year Year Years 2005 Percentage2005 2006 and 2006 (%)

Conventional approachesYear 2005 627,880 627,880Year 2005 and Year 2006, separately 627,880 694,302 1,322,182 101.2Year 2005 and Year 2006, combined 644,685 711,341 1,356,026 103.8

Real-time approach 627,880 639,223 1,267,103 97.0Real operational data 620,971 685,627 1,306,598 100.0


Cumulative Energy VariationCRSP - Multi Reservoir System

0,00E+00

2,00E+08

4,00E+08

6,00E+08

8,00E+08

1,00E+09

1,20E+09

1,40E+09

1,60E+09

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

Months

Cum

ulat

ive

Ene

rgy

(kW

h)

Real Operation Conventional Approach (1-12 months) Conventional Approach (12-23 months)

Conventional Approach (1-23 months) Real Time Approach

Fig. 8 Comparison of cumulative energy for 2005–2006 in CRSP with respect to different consider-ations in optimization

– When above mentioned real-time approach is performed; energy obtained in2-years period is 1,267.1 GWh exhibiting an approximation of 3% to the realized/generated energy amount. It is to be noted that conventional approach is a poste-riori, while real time approach proposed is online and is heavily dependent on thetemplate baseline. In the event that the template baseline is formed embracinga long period, it is very likely that it will give better results. The realized valuesinclude tacit operational knowledge which have not been reflected on 1 year longdata which have been used to establish the template.

– As the period considered for optimization increases, improvement in the amountof optimized energy rises. Energy amount received from optimization of 2-yearsperiod is by 2.6% higher than the sum of the optimized energy amounts obtainedthrough optimization of two separate 1-year period.

7 Conclusion

Following the problem definition and the mathematical model defining the optimiza-tion problem with its objective, initial and boundary conditions was formed. Then,this framework was adapted to genetic algorithm which would be employed in theoptimization process. After configuration of this adaptation, a computer code inFortran programming language was constructed to solve this optimization problemby means of processors. This code would include the steps and principles which werenecessary for genetic algorithm.

The verification of the code was done by a previously studied and proven refer-ence multi-reservoir model with a known global optimum. The mathematical modelof this reference system was embedded with its objective function, initial and bound-ary conditions, into the code constructed. Results achieved through employment ofthe code well-fit to the known global optimum of the system. Hence, the code has


been verified. Beside verification, a sensitivity analysis was performed to see how thevariation in the output of the model was with respect to the controlling parametersin the system.

Following verification process, the code was attempted to be employed in a realcase multi-reservoir system under operation. The Blue Mesa, the Morrow Point andthe Crystal Reservoirs within the Colorado River Storage Project in the U.S. Datawhich would be required in the optimization process have been obtained. Moreover,operational data belonging to the same period has also been determined. Sincethe objective function used in the verification was solely valid for the consideredreference model, objective function has been modified to be applicable in theCRSP. Computer code was executed and the energy produced in the system wasoptimized by using two different approaches, conventional and real-time approach.In conventional approach, past data were utilized in optimization for 1 year and2 years periods. Two years period has been considered in two different cases; 2-yearstime period as a sole time horizon, and in the second case 2-years time period wasoptimized in two separate 1-year time period. In the real-time approach past datacontributed in formation of a template baseline which is continuously updated inaccordance with real-time data. As expected, the comparison of the results revealedthat the energy amounts optimized by using conventional approach were higherthan the energy produced in real operation. On the other hand, by using real-timeapproach, a close approximation to the real operational data has been achieved.

While conventional approaches make use of a priori data which belongs tooccurred time periods, in real-time approach a combination of a priori and posteriordata are used. A priori data constitutes a template baseline which will be updatedby means of so called posterior real-time data. Template baseline is constructedbenefiting from the past data. This baseline reflects the behavior of the flow regimein the considered system. In future researches, it can be further improved by beingconstructed upon past data belonging to a longer period of time. It is recommendedfor future researches that a learning capability is brought in this approach so as tocover a long period. With this study, it has been shown that genetic algorithms cansuccessfully be applied in optimization of reservoir operations.

The optimal values of population size, generation number, probability of cross-over, and probability of mutation suggested by the sensitivity analysis of GeneticAlgorithm model applied to this multi-reservoir project were employed and theobjective function considered was to optimize the energy generation. Further studiesmay be oriented to accommodate other objective functions such as, to optimize theamount of water to be released in order to meet irrigation demands for irrigation pur-pose reservoirs. In the mathematical model constructed, water losses that may occurdue to evaporation and seepage in the ground were not considered. Adding thosefacts into the mathematical model may contribute for improvements in this work.

References

Ahmed I, Lansey K (2001) Optimal operation of multi-reservoir systems under uncertainty. In:Phelps D (ed) Proc, world water and environmental resources congress. Environmental andWater Resources Institute of ASCE, Reston

Ahmed JA, Sarma AK (2005) Genetic algorithm for optimal operating policy of a multipurposereservoir. Water Resour Manage 19(2):145–161


Ansari N, Hou E (1999) Computational intelligence for optimization. Kluwer Academic, BostonBarros M, Tsai F, Yang S-L, Lopes J, Yeh W (2003) Optimization of large-scale hydropower system

operations. J Water Resour Plan Manage 129(3):178–188Bellman R (1957) Dynamic programming. Princeton University Press, PrincetonCai X, McKinney D, Lasdon L (2001) Solving nonlinear water management models using a combined

genetic algorithm and linear programming approach. Adv Water Resour 24(6):667–676Chandramouli V, Raman H (2001) Multireservoir modeling with dynamic programming and neural

networks. J Water Resour Plan Manage 127(2):89–98Cheng CT, Chau KW (2001) Fuzzy iteration methodology for reservoir flood control operation.

J Am Water Resour Assoc 37(5):1381–1388Cheng CT, Chau KW (2002) Three-person multi-objective conflict decision in reservoir flood con-

trol. Eur J Oper Res 142(3):625–631Cheng CT, Chau KW (2004) Flood control management system for reservoirs in China. Environ

Model Softw 19(12):1141–1150Cheng CT, Wang WC, Xu DM, Chau KW (2008) Optimizing hydropower reservoir operation using

hybrid genetic algorithm and chaos. Water Resour Manage 22(7):895–909Cieniawski SE, Eheart JW, Ranjithan S (1995) Using genetic algorithms to solve a multiobjective

groundwater monitoring problem. Water Resour Res 31(2):399–409Davidson JW, Goulter IC (1995) Evolution program for the design of rectilinear branched distribu-

tion systems. J Comput Civ Eng, ASCE 9(2):112–121De Jong KA (1975) An analysis of the behavior of a class of genetic adaptive systems. Doctoral

dissertation, University of Michigan, Ann Arbor, MIEsat V, Hall MJ (1994) Water resources system optimization using genetic algorithms. In: Proc. 1st

int. conf. on hydroinformatics. Balkema Rotterdam, The Netherlands, pp 225–231Eshelman LJ, Schaffer JD (1993) Real-coded genetic algorithms and interval-schemata. In: Whitley

LD (ed) Foundations of genetic algorithms 2. Morgan Kaufmann Publishers, San Mateo, pp 187–202

Fahmy HS, King JP, Wentzel MW, Seton JA (1994) Economic optimization of river managementusing genetic algorithms. Paper No. 943034, ASAE 1994 Int. Summer Meeting, Am. Soc. ofAgricultural Engrs., St. Joseph, Mich

Francini M (1996) Use of a genetic algorithm combined with a local search method for the automaticcalibration of conceptual rainfall-runoff models. J Hydrol Sci 41(1):21–40

Gal S (1979) Optimal management of a multi-reservoir water supply system. Water Resour Res15(4):737–749

Goldberg D (1989) Genetic algorithms in search optimization and machine learning. Addison-Wesley, Reading

Goldberg DE (1991) Real coded genetic algorithms, virtual alphabets, and blocking. Complex Syst5:139–167

Hall WA, Harboe W, Yeh WW-G, Askew AJ (1969) Optimum firm power output from a tworeservoir system by incremental dynamic programming. Water Resour Ctr Contrib, Universityof Calif 130:273–282

Heidari M, Chow V, Kotovic P, Meredith D (1971) Discrete differential dynamic programmingapproach to water resources system optimization. Water Resour Res 7(2):273–282

Hınçal O (2008) Optimization of multireservoir systems by genetic algorithm. PhD Dissertation,Middle East Technical University, Ankara

Holland JH (1975) Adaptation in natural and artificial systems. MIT, CambridgeHowson HR, Sancho NGF (1975) A new algorithm for the solution of multistate dynamic program-

ming problems. Math Program 8(1):104–116Kumar DN, Baliarsingh F, Raju KS (2010) Optimal reservoir operation for flood control using folded

dynamic programming. Water Resour Manage 24(6):1045–1064Labadie JW (2004) Optimal operation of multireservoir systems: state-of-the-art-review. J Water

Resour Plan Manage 130:93–111Larson RE (1968) State increment dynamic programming. Elsevier Science, New YorkLiu P, Guo S, Xiong L, Li W, Zhang H (2006) Deriving reservoir refill operating rules by using the

proposed DPNS Model. J Water Resour Manage 20(3):337–357Liu X, Guo S, Liu P, Chen L, Li X (2010) Deriving optimal refill rules for multi-purpose reservoir

operation. Water Resour Manage. doi:10.1007/s11269-010-9707-8Loucks D, Dorfman P (1975) An evaluation of some linear decision rules in chance-constrained

models for reservoir planning and operation. Water Resour Res 11(6):777–782

http://dx.doi.org/10.1007/s11269-010-9707-8


Michalewicz Z (1992) Genetic algorithms + data structures = evolution programs. Springer,New York

Murray DM, Yakowitz S (1979) Constrained differential dynamic programming and its applicationto multireservoir control. Water Resour Res 15(5):1017–1027

Oliveira R, Loucks D (1997) Operating rules for multireservoir systems. Water Resour Res33(4):839–852

Raman H, Chandramouli V (1996) Deriving a general operating policy for reservoirs using neuralnetwork. J Water Resour Plan Manage 122(5):342–347

Sharif M, Wardlaw R (2000) Multireservoir systems optimization using genetic algorithms: casestudy. J Comput Civ Eng 14(4):255–263

Wardlaw R, Sharif M (1999) Evaluation of genetic algorithms for optimal reservoir system opera-tions. J Water Resour Plan Manage 125:25–33

Wang QJ (1991) The genetic algorithm and its application to calibrating conceptual rainfall–runoffmodels. Water Resour Res 27(9):2467–2471

Wright AH (1991) Genetic algorithms for real parameter optimization. In: Rawlins E (ed) Founda-tions of genetic algorithms. Morgan Kaufmann, Massachusetts, pp 205–218

Young GK (1967) Finding reservoir operating rules. J Hydraul Div, ASCE 93(6):297–321

optimization of multireservoir systems by genetic algorithm

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