An adaptive relaxed cellular automata method for reliability...

Scientia Iranica A (2017) 24(5), 2294{2306

Sharif University of TechnologyScientia Iranica

Transactions A: Civil Engineeringwww.scientiairanica.com

An adaptive relaxed cellular automata method forreliability-based hydropower operation of reservoirs

M. Azizipour� and M.H. Afshar

School of Civil Engineering, Iran University of Science and Technology, Tehran, Iran.

Received 5 December 2015; received in revised form 6 April 2016; accepted 18 June 2016

KEYWORDSReliability;Cellular automata;Genetic algorithm;Reservoir operation;Hydropower.

Abstract. This paper presents a novel cellular automata Approach as the solution toreliability-based reservoir hydropower operation problems. The method is based on theobservation that a low value of the penalty parameter would lead to partial enforcementof the constraints. Therefore, in this method, the constraints of the chance-constrainedoperation problem, namely operational and reliability constraints, are dealt with di�erently.A high enough value of the penalty parameter is used for the �rst set while a lower thanenough value is used for the second set, leading to strong enforcement of the �rst set of theconstraint and partial ful�llment of the second set. Since the proper value of the penaltyparameter to be used for the reliability constraints is not known a priori, an adaptivemethod is proposed to �nd the proper value. The proposed model is used for operation ofDez reservoir in Iran and the results are presented and compared with those of a GeneticAlgorithm. Hydropower operation is considered over short, medium, and long-term periodsto demonstrate the e�ciency and e�ectiveness of the proposed method for problems ofdi�erent scales. The proposed model has proven to produce results superior to the resultsof GA with much reduced computational e�ort.© 2017 Sharif University of Technology. All rights reserved.

1. Introduction

The growth of population in recent decades has ledto the increased development and construction of hy-dropower projects. Optimal use of water to producepower from an available hydropower plant is an im-portant issue. Many attempts have been made byresearchers to achieve the best possible performance ofthe reservoir system to produce �rm energy. Variousoptimization techniques have been used for optimalreservoir operation represented by a set of releasesor storages. Nonlinear programming (NLP) has beenwidely used to solve diverse reservoir operation prob-

*. Corresponding author. Tel.: +1 530 219-7507E-mail addresses: m [email protected] (M. Azizipour);[email protected] (M.H. Afshar)

doi: 10.24200/sci.2017.4151

lems with various objectives [1,2]. An improvedDynamic Programming (DP) has been proposed forhydropower operation [3]. A Dynamic Programming-Linear Programming (DP-LP) algorithm was proposedin [4] for optimization of multi-reservoir hydropowersystems operation. An extensive review of DP applica-bility and limitations was presented in [5]. StochasticDynamic Programming (SDP) has also been usedfor solving reservoir operation problems consideringuncertainty of the in ow to the reservoirs [6-8].

In the last decade, Evolutionary Algorithms (EA)have been introduced and widely used to optimizereservoir operation. Many di�erent forms of reservoiroperation problems have been solved by genetic algo-rithm [9-12]. Ant Colony Optimization (ACO) [13,14],Particle Swarm Optimization (PSO) [15], SimulatedAnnealing (SA) [16,17], and Honey Bees Mating Op-timization (HBMO) [18] have also been suggested andused to achieve optimal operating policies for reservoir

M. Azizipour and M.H. Afshar/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2294{2306 2295

systems. A comprehensive survey of the use of EAscan be found in [19]. Recently, Afshar and Shahidi [20]used a Cellular Automata (CA) method for large-scale optimal operation of single reservoirs, concludingthat CA outperformed heuristic search methods to �ndnear optimal solution in much less computational time.The method was later extended for multi-reservoirsystems with the same �ndings that CA could yieldbetter solution than heuristic search methods couldwith much reduced computational time [20]. None ofthese studies, however, have considered the reliabilityof reservoir operation.

While all the above-mentioned approaches wereused for optimal operation of reservoirs disregardingthe importance of reliability, some researchers pro-posed methods to solve chance-constrained reservoirmanagement problems, which included one or moreprobabilistic constraints. Askew [21] used SDP toderive optimum reservoir operating policy consideringreliability constraints. A DP procedure was developedin [22] to solve reliability-constrained reservoir opera-tion problems. Simonovic and Marino [23] proposeda three-level solution algorithm for a multipurposereservoir and solved reservoir management problem byreliability programming. A reliability programmingtechnique was introduced for multiple-multipurposereservoir system [24]. Marino and Mohammadi [25]applied a model based on Chance-Constrained LinearProgramming (CCLP) and DP to determine the opti-mum monthly releases from a multipurpose reservoir.Sreenivasan and Vedula [26] used CCLP formulationfor a multipurpose reservoir to determine the maximumannual hydropower produced while meeting irrigationdemand at a speci�ed reliability.

In this study, a novel Relaxed Cellular Automata(RCA) method is developed to determine optimalhydropower operating policy of a single reservoir for aspeci�ed reliability. In the proposed method, the con-straints of the operation problem, namely operationaland reliability constraints, are enforced di�erently. Ahigh enough value of the penalty parameter is used forthe operational constraints while a lower than enoughvalue is used for the reliability constraints, leading tofull enforcement of the �rst set and partial ful�llmentof the second set. In this way, the satisfaction ofthe reliability constraints is relaxed compared to theoperational constraints and, hence, the use of the termrelaxed cellular automata. Since the proper value ofthe penalty parameter to be used for the reliabilityconstraints is not known a priori, an adaptive methodis proposed to �nd the proper value. For this, theproblem is �rst solved for the optimal operation usinga zero value of the penalty parameter. The valueof the penalty parameter is then adjusted using thereliability of the optimal operation obtained. Ateach iteration, the penalty parameter increases if the

current reliability is less than the target reliabilityand decreases otherwise. The proposed model is usedfor optimal hydropower operation of Dez reservoir inIran and the results are presented and compared withthose of the GA. The short, medium, and long-termhydropower operations are considered to demonstratee�ciency and e�ectiveness of the proposed method forhydropower operation problems of di�erent scales. Themodel is shown to be superior to the GA in any regardssuch as computational e�ort and optimality of the �nalsolution.

2. Chance constrained hydropower operationof a single reservoir

The single-objective operation of reservoirs can beeither bene�t-based or reliability-based. In a bene�t-based operation, the reservoir is operated such thatthe net bene�t of the operation is maximized whilethe reliability-based operation is concerned with max-imizing the reliability of meeting a set of pre-de�neddemands over the operation period. While simulta-neous consideration of bene�t and reliability requiresthe use of bi-objective optimization methods, a simplerand more practical problem is often encountered inwhich prede�ned target reliability is to be met in abene�t-based reservoir operation. In these problems,commonly referred to as chance-constrained reservoiroperation problems, the objective is to �nd a set ofreleases, or storages, such that a prede�ned pattern ofdemands, water or energy, is observed with a prede�nedreliability while optimizing the objective function overthe operation period. Di�erent operational policiesrepresented by di�erent objectives are de�ned and usedfor operation of single or multi-reservoir systems.

Here, the chance-constrained hydropower opera-tion of a single reservoir is de�ned by an operationpolicy that minimizes the total positive deviation of theproduced power from the total capacity of hydroelectricplant (TCH) de�ned as:

Min F =NXt=1

�1� Pt

TCH

�t = 1; :::; N; (1)

subject to:

St+1 = St +Qt �Rt � Lt t = 1; :::; N; (2)

Smin � St � Smax t = 1; :::; N + 1; (3)

Rmin � Rt � Rmax t = 1; :::; N; (4)

Pr [P � TCH] � RT ; (5)

where Rt, St, Lt, and Qt are water release fromreservoir, water storage, evaporation losses, and in- ow to reservoir at period t, all in Million Cubic

2296 M. Azizipour and M.H. Afshar/Scientia Iranica, Transactions A: Civil Engineering 24 (2017) 2294{2306

Meters (MCM), respectively; N is the total numberof operation periods; Smin and Smax are minimumand maximum allowable reservoir storage, respectively;Rmin and Rmax are minimum and maximum waterrelease from reservoir, respectively; and RT is thetarget reliability. Here, Pt is the power generatedby the hydroelectric plant at period t (MW); otherparameters were de�ned earlier. The power producedby the plant is calculated by:

Pt=min��

ge1000�PF� Coef

��(RtHt);TCH

�: (6)

with:

Ht =�Et + Et+1

2

�� ET ; (7)

where g is gravity acceleration, e is e�ciency of thehydroelectric plant, PF is the Plant Factor, Ht is thee�ective head of hydroelectric plant, Et is the elevationof water in reservoir at period t; Coef is a coe�cientthat converts the release volume to discharge, and ETis the tail water elevation of hydroelectric plant. Theelevation of water in reservoir at any period could becalculated from volume-elevation curve of the reservoir.Here, a volume-elevation curve in the form of:

Et = a� S3t + b� S2

t + c� St + d; (8)

is used where a, b, c; and d are constant coe�cientsobtained by �tting Eq. (8) to the available data.

3. Cellular automata

The theory of CA as a self-reproducing model was�rst developed during the 1950s by Stan Ulam [27].Thatcher [28], Codd [29], and Burks [30] as well asother researchers made their own contributions tothe method, leading to the presentation of CA asa global simulation method. CA has been used tomodel di�erent problems such as pollutants dispersionin the atmosphere [31], heat ow [32], absorption-desorption phenomenon [33], inter-di�usion of atoms oftwo materials [34], the solidi�cation process [35], anddrug therapies for HIV infection [36]. Chen et al. [37]focused on the development and application of CAapproach for ecological and ecohydraulics modeling.Li et al. [38] developed an integrated model, whichcombined a two-dimensional hydrodynamic modulewith a vegetation evolution module and a �sh habitatmodule. They used an unstructured CA in the vegeta-tion module. Ghimire et al. [39] used two-dimensionalCA for ood modeling. Austin et al. [40] presented ane�cient and accurate conceptual CA-based simulatorfor sewer networks simulation. Sanchez et al. [41]

proposed an integrated cellular automata evolutionary-based approach to derive a drainage network layoutbased on future land use scenarios.

While CA was originally proposed as a simulationmethod to reproduce complex processes, it is now beingwidely used to solve optimization problems of di�erentengineering disciplines due to its interesting featuressuch as simplicity and computational speed.

CA, as an optimization technique, has been �rstused in structural optimization. Kita and Toyoda [42]used CA for shape and topology optimization of contin-uum structures. Tatting and Gurdal [43] employed CAfor design of 2D structures. Afshar and Faramarzi [44]used CA for size optimization of some truss structuresand compared the results with those of Fully StressedDesign (FSD) method and showed that CA was moree�cient than FSD. Faramarzi and Afshar [45] proposeda hybrid CA-LP method for size optimization of trussstructures. The optimization problem was recast interms of the nodal displacement and internal forcesand iteratively solved in two phases. In the �rstphase, the nodal displacements of the structure wereobtained using a CA approach while the internalforces were determined in the second phase by anLP method. Farmarzi and Afshar [46] extended theCA-LP method for simultaneous size and topologyoptimization of truss structures. The optimizationproblem was solved in two phases. The optimal topol-ogy of structures was obtained in the �rst phase andthe optimal sizing of structure was carried out in thesecond phase using the CA-LP method of Faramarziand Afshar [45].

The �rst application of the CA to water resourcesproblem was proposed by Keedwell and Khu [47], inwhich a CA was used to produce initial solutions fora GA model for optimal design of water distributionnetwork. Guo et al. [48] hybridized a Non-dominatedSorting GA (NSGAII) with CA and used it for multi-objective design of both water distribution and stormsewer networks. The �rst use of CA as a stand-aloneoptimizer in water resources problems was by Guo etal. [49] who proposed a CA model for the optimaldesign of storm-sewer networks with the objective ofminimizing the capital cost of the sewer network andtotal ooding volume in the system. While most ofthe research used an ad-hoc transition rule based onengineering judgments and physical characteristics ofthe considered problem, Afshar and Shahidi [20] werethe �rst to propose a CA model with a mathematicallyderived transition rule and used it as optimal solutionto reservoir operation problems. More recently, Af-shar [50] extended the model of Afshar and Shahidi [20]for optimal hydropower operation of multi-reservoirsystems. The objective function and constraints ofproblem were projected on each cell to derive thetransition rule represented by an optimization sub-


problem on each period, which was solved using anNLP solver. The method was used for solving the well-known four-reservoir and ten-reservoir problems andthe results were compared with those obtained by GAand PSO methods, indicating superiority of the methodto heuristic search methods.

Application of CA with mathematically derivedtransition rule for optimal design of sewer networkswas �rst introduced in [51]. The nodes of the networkwere used as CA cells with the corresponding elevationas cell states. Afshar and Rohani [52] extended themethod of Afshar et al. [51] to a hybrid two-stageCA method for optimal design of sewer networks.The problem of optimal design of sewer network wasdecomposed into two sub-optimization problems fornodal cover depths and pipe diameters of the network,which were both solved using CA method. Comparisonof the results with those obtained by GA, PSO, andACO demonstrated the e�ciency and e�ectiveness ofthe proposed hybrid method.

4. Proposed Adaptive Relaxed CellularAutomata (ARCA) method

The optimization problem in hand, chance-constrainedhydropower operation of a single reservoir, contains twosets of constraints of di�erent nature. The operationalconstraints, represented by Eqs. (2)-(4) and (6)-(8), areof explicit nature and, therefore, can be easily enforcedwhile searching for the optimal solution. However,the reliability constraint, represented by Eq. (5), is ofimplicit type and its imposition requires an iterativemethod. The �rst step for the application of CA toreliable reservoir operation problem is to recast thereliability constraints into more explicit form for easierhandling. De�ning A0 as a subset of A = f1; 2; 3; :::; Ngwith N 0 members, the reliability constraint is writtenas:

Pt � TCH; t 2 A0: (9)

subject to the condition that N 0 is equal to or greaterthan N � RT , where N 0 represents the total numberof periods for which the corresponding reliability con-straints are realized ensuring an operation of desiredtarget reliability, RT . The reliability constraint de-�ned by Eq. (9) is now in more useful form for theapplication of CA as it is explicitly de�ned in termsof the reservoir release as the decision variable of theproblem.

The second step for the application of CA tothe problem in hand is to recast the problem intoan unconstrained optimization problem. For this, apenalty method is used in which the constraints ofproblem are embedded into the objective function.This leads to the following unconstrained optimizationproblem de�ned as:

Min PF=F+��NXt=1

(CVo)2t +��

NXt=1

(CVR)2t ; (10)

where F stands for the original objective function,PF is the penalized objective function of the prob-lem, (CVo)t represents the violation of operationalconstraints de�ned as:

(CVo)j =max f(CVo1)j ; (CVo2)j ; (CVo3)j ; (CVo4)jg :(11)

With:

(CVo1)j = maxf(Sj � Smax); 0:0g;(CVo2)j = maxf(Smin � Sj); 0:0g;(CVo3)j = maxf(Rj �Rmax); 0:0g;(CVo4)j = maxf(Rmin �Rj); 0:0g:

(CVR)j is the violation of reliability constraint de�nedas:

(CVR)j = maxf(TCH� Pj); 0:0g; (12)

and � and � are penalty coe�cients for operational andreliability constraints, respectively.

Application of CA to any optimization problemrequires four primary components of the CA model,namely, cells, cell state, cell neighborhood, and thetransition or updating rule, to be de�ned. For theoperation problem in hand, the cells are taken asdiscrete points in time representing the beginning andthe end of each period of the operation. Therefore,the cell state, representing the decision variables of theoptimization problem, is taken as the reservoir storageat these discrete points. The surrounding cells areconsidered as the neighborhood cells. A schematicrepresentation of cell neighborhood is illustrated inFigure 1 as used in [20].

The transition rule for an arbitrary cell, j, isderived by requiring that the problem objective func-tion (Eq. (10)) should be minimized with respect tothe cell state, Sj , while all other cell states are keptconstant. Assuming pre-de�ned arbitrary initial values

Figure 1. Schematic representation of internal andboundary cells with their neighborhoods: (a) Leftboundary cell, (b) internal cell, and (c) right boundarycell.


for all cell states denoted by Skj , j = 1; 2; :::; N , theupdated value of the jth cell state Sk+1

j is obtained bysolving the following sub-optimization problem de�nedon the neighborhood of cell j for hydropower operationproblem:

Min(PF)j =�TCH� P k+1

j�1�2

+ (TCH� P k+1j )2

+ ��(CVO)2

j�1 + (CVO)2j

�k+1

+ �

��(CVR)2

j�1 + (CVR)2j

�k+1

: (13)

Here, P k+1j�1 and P k+1

j are the updated values of powerat periods j � 1 and j, respectively.

Rewriting Eq. (11) in terms of �Sj = Sk+1j � Skj

and its analytical solution leads to the updating rule,as shown in Box I, with:

@Pj�1

@Sj=�

ge1000��PF� Coef

��@Rj�1

@SjHj�1

+@Hj�1

@SjRj�1

�; (15)

@Pj@Sj

=�

ge1000��PF� Coef

��@Rj@Sj

Hj +@Hj

@SjRj�; (16)

@Hj

@Sj=

12

�@Ej@Sj

�=

12� �3aS2

j + 2bSj + C�

(17)

�j =�

1000� PF� Coef� TCHg � e

�� dHj=dsj

H2j

!; (18)

where BO and BR are binary variables with zerovalues if the solution at iteration k is feasible regardingoperational and reliability constraints, respectively,and with unit values otherwise.

It is a well-known fact that the value of thepenalty coe�cient in the penalty method determinesthe level of constraint satisfaction. With a zero valuefor the penalty coe�cient, the constraints would notbe taken into account as if an unconstrained problemis solved. With a large enough value of the penaltyparameter, for which any infeasible solution has a totalcost greater than any feasible solution, the constraintsare totally enforced. For any value between zeroand large enough value of the penalty parameter, theconstraints are partially enforced depending on howclose the penalty parameter is to the proper value.This means that assuming a proper value for theoperational penalty parameter, �, and a zero valuefor the reliability penalty parameter, �, would lead toan operationally feasible solution with the reliabilityconstraint totally disregarded. Increasing the value ofthe reliability penalty parameter, �, would logicallylead to operationally feasible solutions of higher reli-ability. To assess the e�ect of the reliability penaltyparameter on the reliability of the �nal solution, a seriesof experiments are carried out for the case of operationover 60 months of the test case used by [20] with a�xed value of � = 100 and the results are produced inTable 1.

It is seen that increasing the reliability penaltycoe�cient, �, leads to solutions of higher reliabilityas expected while a zero value of the parameter leadsto solutions in which no reliability constraints areconsidered. This experiment suggests that the penaltyparameter, �, could gradually increase until the desiredtarget reliability is reached. This process, however,could be time consuming since no prior knowledge ofthe proper magnitude of the penalty coe�cient is inhand.

An adaptive method is, therefore, proposed to�nd the proper value of the reliability penalty parame-ter, �. Having �xed the operational penalty parameter

�Sj =1�

@Pj�1@Sj

�2+�@Pj@Sj

�2+ �� [(Bo)j�1 + (Bo)j ] + � � [(BR)j�1 � �j�1 + (BR)j � �j ]

��@Pj�1

@Sj(TCH� P kj�1) +

@Pj@Sj

(TCH� P kj ) + �� [(CVo)j�1 + (CVo)j ]k + �

� [(CVR)j�1 + (CVR)j ]k�

(14)

Box I


Table 1. The e�ect of reliability penalty parameter on the reliability of the operation.

Without evaporation losses With evaporation losses� Reliability � Reliability

0.0 0.25 0.0 0.230.2 0.25 0.2 0.320.4 0.30 0.4 0.320.6 0.35 0.6 0.430.8 0.61 0.8 0.48

value, the problem is �rst solved using a zero value ofthe penalty coe�cient, �, leading to a solution with areliability, R, lower than the target reliability, RT . Ateach iteration, the penalty coe�cient is adjusted usingthe following relation:

�new =��old + (RT �Rold)

�; (19)

where Rold is the reliability of the solution obtainedusing the previous value of the reliability penaltyparameter �old. For a penalty coe�cient less thanproper value, the reliability of the solution Rold isless than the target reliability, leading to an increasein the updated value of the penalty parameter �new

and vice-versa. The iterative process of updating iscontinued until the current reliability is equal to thetarget reliability.

5. Model application and results

In this section, e�ciency and e�ectiveness of theproposed model are illustrated for chance-constrainedhydropower operation of \Dez" reservoir in the south-ern district of Iran. To assess the e�ect of the problemscale on the performance of the model, three di�erentoperation periods of 5, 20, and 40 years are consideredhere for two cases of with and without evaporation

losses. The average annual in ow to the reservoir isestimated at 5900 MCM. The active storage volume ofreservoir is 2510 MCM. The minimum and maximumadmissible storage volumes are 830 and 3340 MCM,respectively, while maximum and minimum water re-lease volumes are considered to be 1000 MCM and 0,respectively. The coe�cients of volume-elevation curvede�ned by Eq. (8) are taken a = 249:833, b = 0:05872,c = �1:37�10�5, and d = 1:526�10�9. The plant factorof hydroelectric plant is 0.417, its e�ciency is 90%, andthe tail water elevation of hydropower plant is 172 m.

The proposed ARCA model is used to solvethe chance-constrained hydropower operation of Dezreservoir over three di�erent operation periods withdi�erent target reliabilities for both cases of withand without evaporation losses. Table 2 presents thesolution cost, CPU time, reliability of the solutionobtained by the proposed model, number of adaptive it-erations required, and the �nal value of the operationalpenalty parameter, �, for di�erent target reliabilities ofhydropower operation without evaporation losses. Itis seen that the proposed method generally convergesin a few iterations and, therefore, it is able to �ndthe optimal solution within few seconds, illustratingthe e�ciency of the proposed method for the problem.Figures 2-4 show the evolution of the reliability penaltyparameter during the adaptive process starting with a

Table 2. Results of hydropower operation for periods of 60, 240, and 480 months without evaporation losses.

Months Targetreliability

Solutioncost

Computationaltime (sec)

ReliabilityNo. of

iterationsFinal penalty

coe�cient

600.6 7.41 0.16 0.6 6 0.750.65 7.39 0.18 0.65 14 1.030.7 7.38 0.22 0.7 27 1.47

2400.6 21.21 1.39 0.6 7 0.270.65 20.89 1.32 0.6 13 0.430.7 20.76 1.24 0.7 16 0.63

4800.6 45.99 2.44 0.6 10 0.440.65 45.71 2.15 0.65 14 0.620.7 45.55 2.1 0.7 22 0.86


Figure 2. Evolution of reliability penalty coe�cient forhydropower operation, without evaporation losses, over 60months with target reliability of 0.6.

Figure 3. Evolution of reliability penalty coe�cient forhydropower operation, without evaporation losses, over240 months with target reliability of 0.6.

Figure 4. Evolution of reliability penalty coe�cient forhydropower operation, without evaporation losses, over480 months with target reliability of 0.6.

zero value leading to solution reliabilities of 0.25, 0.42,and 0.4 at the start of iterative process for operationperiods of 60, 240, and 480 months, respectively. Inall cases, the penalty parameter is seen to rise from theinitial value of zero to the �nal value in a few iterationsbefore the �nal �ne tuning of the penalty parameter.The monthly produced power over 60 months withtarget reliability of 0.7 for the case of disregardingevaporation losses is presented in Figure 5.

The results of the proposed ARCA model forhydropower operation of Dez reservoir, consideringevaporation losses, over di�erent operation periodsand target reliabilities are shown in Table 3. Evolu-tion of the reliability penalty parameter during theiterative process for hydropower operation is shownin Figures 6-8, starting with a zero value leading tosolution reliabilities of 0.23, 0.39, and 0.39 at the startof iterative process for operation periods of 60, 240,and 480 months, respectively. Figure 9 shows themonthly produced power for the case of operation over60 months and target reliability of 0.7.

The problem under consideration is also solvedhere using a real-coded GA model, in which the storagevolumes are considered as the decision variables. Asensitivity analysis was carried out to determine theproper values of the GA parameters. As a result, atournament selection with random tournament size, asingle-point crossover with probability of 1 and randomweighted averaging after the crossover site, and a 1-bit mutation procedure with probability of 1=l, with lbeing the length of individual chromosome, are used toproduce the o�-springs. Population size is set at 50,100, and 200 for operation periods of 60, 240, and 480months, respectively, and exhaustive maximum num-bers of generations equal to 30000, 60000, and 90000

Figure 5. Monthly produced optimum power resultingfrom ARCA for hydropower operation, withoutevaporation losses, over 60 months with target reliabilityof 0.7.


Table 3. Results of hydropower operation for periods of 60, 240, and 480 months with evaporation losses.

Months Targetreliability

Solutioncost

Computationaltime (sec)

Reliability No. ofiterations

Final penaltycoe�cient

600.6 9.19 0.68 0.6 9 0.930.65 9.18 0.72 0.65 15 1.270.7 9.18 0.41 0.7 31 1.92

2400.6 25.97 3.15 0.6 11 0.430.65 25.84 2.67 0.6 11 0.600.7 25.80 2.57 0.7 20 0.85

4800.6 56.72 7.59 0.6 14 0.580.65 56.58 7.31 0.65 11 0.770.7 56.51 5.25 0.7 23 1.07

Figure 6. Evolution of reliability penalty coe�cient forhydropower operation, with evaporation losses, over 60months with target reliability of 0.6.



Figure 9. Monthly produced optimum power resultingfrom ARCA for hydropower operation, with evaporationlosses, over 60 months with target reliability of 0.7.


Table 4. Results of GA for hydropower operation, without evaporation losses, for periods of 60, 240, and 480 months.

Operationperiod

Targetreliability

Generation Total cost SD Average computationtime (sec)

Minimum Maximum Average

60

0.620000 7.73 9.34 8.54 0.47 17125000 7.72 8.92 8.40 0.60 21430000 7.72 8.70 8.35 0.50 257

0.6520000 8.13 9.25 8.72 0.36 15125000 7.86 9.25 8.54 0.69 18930000 7.80 8.68 8.29 0.44 227

0.720000 8.32 9.59 9.03 0.42 13325000 8.15 9.39 8.86 0.62 16630000 8.07 9.23 8.68 0.58 199

240

0.650000 23.10 25.80 24.21 1.22 161955000 23.09 25.77 24.11 1.35 178160000 23.09 25.77 24.08 1.35 1943

0.6550000 22.54 35.33 25.16 6.05 153655000 22.52 34.62 25.02 6.39 168960000 22.51 34.12 24.94 6.13 1843

0.750000 22.68 26.32 24.30 1.63 201555000 22.61 26.17 24.24 1.79 221760000 22.56 26.09 24.17 1.77 2418

480

0.680000 50.02 52.75 50.97 0.79 1386985000 50.01 52.73 50.89 0.82 1473690000 50.00 52.72 50.84 0.83 15603

0.6580000 48.68 52.74 51.06 1.17 1491785000 48.63 52.71 50.96 1.16 1585090000 48.61 52.32 50.85 1.09 16782

0.780000 50.20 52.63 51.69 0.82 1727085000 50.19 52.55 51.59 0.79 1834990000 50.11 52.33 51.51 0.78 19428

for operation periods of 60, 240, and 480 months areused, respectively, to make sure of the GA convergence.

Results of the GA runs including the maximum,minimum, and average solution costs of ten runs usingrandomly generated initial guesses, standard devia-tion of the solution, and the CPU times required toget the solution at di�erent generations for di�erenttarget reliabilities are shown in Tables 4 and 5 forthe hydropower operations, with and without evap-oration losses, respectively. The results obtained atdi�erent generations are also shown in these tables to

demonstrate the convergence of GA. Small standarddeviations of the solutions for both cases show that theGA model has converged on its best capacity.

Performance of the proposed ARCA method andthat of GA are shown in Table 6 for the case ofhydropower operation, without evaporation losses. Inthe table, the solution costs and the ratio of thecomputational times required by the GA and ARCA(tGA=tARCA) are shown for di�erent target reliabilitiesand operation periods. It is seen that the proposedARCA model produces solutions 6%, 9%, and 8%


Table 5. Results of GA for hydropower operation, with evaporation losses, for periods of 60, 240, and 480 months.

Operationperiod

Targetreliability

Generation Total cost SD Average computationtime (sec)

Minimum Maximum Average

60

0.620000 9.69 11.55 10.55 0.55 22925000 9.62 11.49 10.45 0.93 28630000 9.51 11.42 10.28 0.96 343

0.6520000 10.02 11.51 10.69 0.56 24625000 9.93 11.44 10.63 0.76 30830000 9.64 11.25 10.45 0.80 369

0.720000 9.92 11.86 11.04 0.53 22525000 9.91 11.78 10.91 0.93 28230000 9.90 11.73 10.77 0.92 338

240

0.650000 29.37 31.68 30.57 1.03 330555000 28.90 32.52 30.49 1.62 363560000 29.32 31.54 30.35 1.11 3966

0.6550000 28.69 38.10 31.10 4.37 270255000 28.62 37.84 30.86 4.81 297260000 28.51 37.60 30.65 4.75 3243

0.750000 28.90 32.52 30.49 1.62 309755000 28.86 32.49 30.41 1.82 340760000 28.71 32.34 30.30 1.82 3716

480

0.680000 61.67 65.30 63.45 1.32 2031485000 61.66 65.07 63.29 1.20 2158390000 61.66 64.79 63.17 1.15 22853

0.6580000 61.69 65.09 63.73 1.29 1938485000 61.68 65.00 63.61 1.25 2059690000 61.58 64.98 63.51 1.26 21807

0.780000 62.34 65.26 64.01 0.88 2137985000 62.30 65.08 63.85 0.82 2271590000 62.26 64.59 63.75 0.74 24052

cheaper than the solutions obtained by GA with acomputational e�ort 1257, 1581, and 7817 times lessthan those required by GA for the operation periods of60, 240, and 480 months, respectively.

Table 7 compares the results obtained by theproposed ARCA method with those of GA consideringdi�erent target reliabilities for hydropower operationconsidering evaporation losses. It is again seen thatthe proposed ARCA model has been able to producesuperior solutions to those of GA for all operation

periods and all target reliabilities. In fact, the solutionproduced by the ARCA is on average 5%, 12%, and9% cheaper than the �nal solutions obtained by GAwith much reduced computational e�ort for operationperiods of 60, 240, and 480 months, respectively. Itis interesting to note that the computational e�ortrequired by the proposed ARCA model to get a bettersolution is at least 2000 times less than that requiredby GA, emphasizing the computational e�ciency of theproposed method.


Table 6. Comparison of the results obtained by GA andARCA for hydropower operation, without evaporationlosses, for periods of 60, 240, and 480 months.

Operationperiod

Targetreliability

Total cost(tGA=tARCA)

Best solutionof GA

ARCA

600.6 7.72 7.41 16060.65 7.80 7.39 12610.7 8.07 7.38 905

2400.6 23.09 21.21 13980.65 22.51 20.89 13960.7 22.56 20.76 1950

4800.6 50.00 45.99 63950.65 48.61 45.71 78060.7 50.11 45.55 9251

Table 7. Comparison of the results obtained by GA andARCA for hydropower operation, with evaporation losses,for periods of 60, 240, and 480 months.

Operationperiod

Targetreliability

Total cost(tGA=tARCA)

Best solutionof GA

RCA

600.6 9.51 9.19 21440.65 9.64 9.18 20500.7 9.90 9.18 1536

2400.6 29.32 25.97 28530.65 28.51 25.84 24570.7 28.71 25.80 2997

4800.6 61.66 56.72 93660.65 61.58 56.58 101430.7 62.26 56.51 11453

6. Conclusions

A novel cellular automata approach was developedfor an e�cient and e�ective solution to reliability-based hydropower operation of reservoirs. The methodis based on di�erent treatments of the operationaland reliability constraints of the problem. A highenough value of the penalty parameter is used forthe operational constraints while a lower than enoughvalue is used for the reliability constraint, leadingto complete ful�llment of the �rst set of constraintsand partial enforcement of the second set. Since theproper value of the penalty parameter for the reliability

constraint is not known a priori, an adaptive methodis introduced to �nd its proper value. For this, theoptimal operation problem is �rst solved using a zerovalue of penalty parameter and is adjusted consideringthe reliability of the optimal operation obtained. Ateach iteration, the penalty parameter decreases if thecurrent reliability is greater than the target reliabilityand increases otherwise. The proposed method isapplied to optimal hydropower operation, for bothcases of with and without evaporation losses, of Dezreservoir in Iran considering various target reliabilitiesfor periods of 5, 20, and 40 years and the results arepresented and compared with those obtained by a GA.The results indicate the superiority of the proposedmethod to the GA in both e�ciency and e�ectivenessof the method.

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Biographies

Mohammad Azizipour received his BS degree inCivil Engineering from Shahid Chamran University,Ahwaz, Iran, in 2008, and his MS degree in WaterEngineering from the same university in 2010. Heis currently senior PhD student in Water ResourcesManagement.

Mohammad Hadi Afshar received his BS degreefrom the Engineering Faculty at Tehran University,Iran, in 1984, and his MS and PhD degrees fromthe University College of Swansea, UK, in 1989 and1993, respectively, all in Civil Engineering. He iscurrently Associate Professor of Civil Engineering atIran University of Science and Technology.

Dr. Afshar worked as part-time researcher withMahab Ghods Consulting Engineers from 1996 to 1999,where he managed two important research projectson the Pashakola Dam Break and Wave Energy Es-timation in the Chahbahar region. Moreover, hehas managed eleven research projects on the optimaldesign of water distribution networks, sewer networks,pipeline systems, ood controlling systems, reservoiroperation, spillway ow, and spillway design, leading tothe development of OPTNET software for the optimaldesign of water distribution networks.

Dr. Afshar is author or coauthor of over 90research articles published in national and internationaljournals and 70 conference papers, and has translatedseveral books into the Persian language. He has alsosupervised about 40 postgraduate students, three ofwhom are PhD candidates. His research interestsinclude mathematical modeling and optimization inhydraulic structures and water engineering.

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