optimal operation of multi-reservoir system by multi-elite guide particle

Electrical Power and Energy Systems 48 (2013) 58–68

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Electrical Power and Energy Systems

journal homepage: www.elsevier .com/locate / i jepes

Optimal operation of multi-reservoir system by multi-elite guide particleswarm optimization

Rui Zhang, Jianzhong Zhou ⇑, Shuo Ouyang, Xuemin Wang, Huifeng ZhangSchool of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, PR China

a r t i c l e i n f o

Article history:Received 5 September 2012Received in revised form 20 November 2012Accepted 25 November 2012Available online 7 January 2013

Keywords:Multi-reservoir systemHydropower generationOptimal operationParticle swarm optimizationMulti-elite guideConstraint handling

0142-0615/$ - see front matter Crown Copyright � 2http://dx.doi.org/10.1016/j.ijepes.2012.11.031

⇑ Corresponding author. Tel.: +86 02 78 7543 127.E-mail address: [email protected] (J. Zhou)

a b s t r a c t

With increasingly scarce of fossil fuel resources required for energy demand, hydropower has becomeone of the most important energy resources. As a renewable and sustainable energy, hydropower systemhas developed rapidly during recent decades. The unprecedented rate of expansion and developmentscale of hydropower has posed a challenge to the operation of multi-reservoir system (OMRS). The opti-mal operation of multi-reservoir system is a large-scaled, non-linear and multi-stage problem which sub-ject to a series of hydraulic constraints. The main objective of OMRS is to find out the optimal hourlywater discharge rate of each hydro station in the multi-reservoir system to minimize the power deficitand distribute the uniformly deficit if any. In order to solve OMRS problem effectively, in this paper, amulti-elite guide particle swarm optimization (MGPSO) is proposed by introducing archive set into stan-dard particle swarm optimization. External archive set which can preserve elite solutions along the evo-lution process is employed to provide multi-elite flying directions for particles. Meanwhile, an effectiveconstrain handling method is proposed to handle the operational constraints of OMRS problem. This pro-posed method is applied to a multi-reservoir system consisting of 10 cascaded hydro plants for casestudy. Compared with several previous methods, the simulation results of MGPSO can get better solu-tions with smaller energy deficit, which proves it is an alternative method to deal with OMRS problem.

Crown Copyright � 2012 Published by Elsevier Ltd. All rights reserved.

1. Introduction

Due to increasingly scarce of fossil fuel resources required forenergy demand, hydropower has become one of the most impor-tant energy sources. In fact it is the only clean energy source thatcan be commercially developed on a large scale in present time[1]. Almost 20% of electricity generation is hydropower in theworld [2]. Compared with other energy sources, hydropower en-joys exceptional advantages: Hydropower uses water to generateelectricity without producing any pollution; the fuel of reservoiris essentially infinite and will not be depleted during the produc-tion; moreover, hydropower has the unique ability to change out-put quite fast, which makes it good at meeting changing demandsfor electricity and maintain the balance between supply and de-mand [3]. However, as the rate of expansion and scale of construc-tion developed quite rare in the world, it has become a tremendouschallenge to the operation of multi-reservoir system (OMRS) [4–6].

Generally, the main objective of OMRS is to determine the waterdischarge rate so as to maximize the hydro generation that satisfiesvarious complicated constraints [7]. Research on optimal operationfor hydropower system started in 1940s [8] and has been strongly

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.

influenced by operational theory and commuter science technolo-gies [9–11]. During recent decades, many research and applicationwere proposed to solve this problem, such as linear programming[12], non-linear programming [13] and dynamic programming[14]. Since 1990s, various artificial intelligence and population-based algorithms, mainly comprising artificial neural networks[15], fuzzy rule-based method [2], artificial bee colony algorithm[16], quantum-inspired evolutionary algorithm[17], genetic algo-rithm [18], differential evolution algorithm [19] and particleswarm optimization [20–22], have been used to the optimal oper-ation of multi-reservoir system and received various degrees ofsuccess in practical hydro operation.

However, the operation of multi-reservoir system is a high-dimensional, non-linearity, multi-stage and stringent constraintoptimal problem [23]. The operation decision of current stage willinfluence the future outcomes all over the system. Moreover, thelimited water resource makes the optimal operation much morecomplicated. The computation complexity during the solution pro-cess will increase with the number of plants and constraints.Therefore, the engineering practice requires an efficient and qual-itative technique to meet the demand of optimal operation so thatthe optimization results can be applied to practical hydro system[24]. However, the conventional methods still have some draw-backs, such as the curse of dimensionality, mass memory demand,

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R. Zhang et al. / Electrical Power and Energy Systems 48 (2013) 58–68 59

increasing computation time and infeasible solutions. Therefore,there are strong demands on a general and practical optimizationmethod to deal with these problems during the optimizationprocess.

Particle swarm optimization (PSO) [25] is one of the most rep-resentative algorithms among modern artificial intelligence tech-niques. As a population-based evolutionary algorithm, PSO hasbeen successfully applied to a lot of optimization problems inhydropower system [26–28]. However, the problem of prematureconvergence caused by falling into local optima still exists in stan-dard PSO algorithm. Thus, in this paper, a multi-elite guide particleswarm optimization (MGPSO) is proposed to solve the problem ofoptimal operation for multi-reservoir system (OMRS). The externalarchive set is used to preserve elite solutions along the evolutionprocess and provide multi-elite guide for particles. In this way,MGPSO algorithm can maintain the diversity of population and re-duce the possibility of local optima. To deal with the complicatedconstraints of the OMRS problem, a two-phase constraints han-dling method is proposed in this paper. Finally, this novel strategyis applied to a multi-reservoir system. Compared with currentmethods, the proposed MGPSO has a competitive performance innot only simulation results but also the convergence efficiency,which demonstrates MGPSO is an effective approach to solvehigh-dimensional and complicated optimization of reservoirdispatching.

The paper is organized as follows: Section 2 introduces the for-mulation of optimal operation for multi-reservoir system (OMRS).In Section 3, we describe the proposed MGPSO algorithm in details.The practical solution methodology of the MGPSO for OMRS prob-lem is presented in Section 4. In Section 5, the proposed MGPSO isapplied to the multi-reservoir system and compared the results ofcase studies with other methods. Section 6 concludes the paper fol-lowed by acknowledgements.

2. Problem formulation

The multi-reservoir system consists of cascaded hydropowerstations in different basins. The optimal operation of this probleminvolves the best use of limited water resource over the scheduleperiod.

2.1. Objective function

The purpose of multi-reservoir system operation is to figure outthe optimal results of hourly water discharge rate of hydro stationsso as to minimize the energy shortages while subjecting to a seriesof hydraulic and operational constraints. Mathematically, the oper-ation of multi-reservoir system can be expressed as a constrainedoptimization problem:

min J ¼ 12

XT

t¼1

Dt �XN

i¼1

Pi;t

" #2

; Pi;t ¼ ciqi;t ð1Þ

where Dt is the lord demand of electrical power system in scheduleperiod t. Pi,t represents the power generation of ith hydro station attth period. qi,t is the water release and ci is the generation coefficientfor each hydro station, respectively.

2.2. Constraints

Most reservoirs are proposed for multi-purpose and a lot ofrequirements during the practical operation have to be satisfied,such as the limits of power generation, water discharge rate, andstorage volume.

(1) Power balance constraint

XN

i¼1

Pi;t þ Et ¼ Dt þ PL;t ð2Þ

where Et presents the power shortages of the whole hydrosystem. PL,t is the total transmission line losses at the tthschedule period.

(2) The continuity equation for the hydro plant network

Vi;tþ1 ¼ Vi;t þ Ii;t � Q i;t � Si;t þXNui

k�1

ðQ k;t þ SkÞ" #

Dt ð3Þ

where Ii,t, Si,t are the inflow rate and spillage of the ith reser-voir at the tth period, Nui is the number of upstream plants di-rectly above the ith hydro unit and Qi,t is the water dischargerate.

(3) Hydro plant discharge rate limits

Q mini;t 6 Qi;t 6 Q max

i;t ð4Þ

where Qmini;t and Qmax

i;t are the lower and upper water dischargerate limits of the ith reservoir at tth period.

(4) Reservoir storage volume limits

Vmini;t 6 Vi;t 6 Vmax

i;t ð5Þ

where Vmini;t and Vmax

i;t are the minimum and maximum storagevolume of the ith hydro unit at tth period, respectively.

(5) Initial and terminal reservoir storage volumes

Vi;0 ¼ VBegini ; Vi;T�1 ¼ VEnd

i ; i ¼ 1;2; . . . ;N ð6Þ

where VBegini and VEnd

i are the initial and terminal reservoir storagelimits of ith hydro unit.

3. Multi-elite particle swarm optimization

3.1. Overview of particle swarm optimization

PSO (particle swarm optimization) [25] is a population-basedoptimization method in which every solution is called particle. InPSO model, particles fly in the feasible space to look for the optimalsolution. During the flight, each particle adapts its position accord-ing to its own experience as well as the experience of the wholeswarm. The search direction of one particle is decided by its historyexperience and the experience of its neighbors. In the searchingspace a particle’s status can be characterized by velocity and posi-tion, which are evolved by following method.

Let x and m denote a particle’s position and corresponding flightvelocity in the n-dimensional search space. Xi = (xi1,xi2, . . . ,xin)stands for the position of the ith particle of the population, andVi = (vi1,vi2, . . . ,vin) means the velocity of the ith particle. The bestposition of the ith particle and the global best position among allthe particles are represented as pi = (pi1,pi2, . . . ,pin) and pg = (pg1,pg2, . . . ,pgn). The evolutionary strategy of the ith particle is showedas follows:

v tþ1ij ¼ w � v t

ij þ c1 � r1 � ðptij � xt

ijÞ þ c2 � r2� � ðptgj � xt

ijÞxtþ1

ij ¼ xtij þ v t

ij

(ð7Þ

where x is the inertia weight factor; c1, c2 is acceleration constants,r1 and r2 are random values in the range of [1]; mt

ij and xtij are velocity

and position of ith particle at tth iteration, respectively.Particularly, the inertia weight factor x is a significant parame-

ter which influences the performance of PSO’s evolution. A largevalue of x is benefit for global search while a small one is suitablefor accurate local search in the current region. In order to preventfrom trapping into local optimal region, an adaptive inertia weight

Begin

Update : Evaluate each particle’s fitness and use the updating method renew the archive set

MG

PS

OA

lgorithm

Iterati

Constraints handling : If the solutions are outside the corridor, change them into the feasible ones.

Initialize evolution population, archive set and parameters. Set G = 0

Initialization

Generation : Use the multi-elite guide strategy to update the positions and velocities

60 R. Zhang et al. / Electrical Power and Energy Systems 48 (2013) 58–68

method is applied to adjust the inertia weight as objective valuechanges. The adaptive inertia weight can be expressed as follows:

w ¼wmin þ ðwmax�wminÞ�ðf�fminÞ

favg�fminf < favg

wmax f P favg

8><>: ð8Þ

where the wmax and wmin are the maximum and minimum values ofthe inertia weight factor. f is the objective value of current particle.favg and fmin are presented as the average and minimum objectivevalues of population, respectively.

3.2. Multi-elite guide based on archive set

Particle swarm optimization carries through its evolution byparticles’ own experience as well as the experience of the wholeswarm. It has been characterized by rapid convergence and goodglobal search capability. However, the problem of prematureconvergence always occurs when dealing with non-convex andmulti-constraint optimal problems. In particularly, the diversityof particles will reduce dramatically in the later stage of evolution,which may lead the evolution into local optimal space. Therefore,in this paper, a multi-elite guide method based on archive set isproposed by introducing archive set into PSO’s evolution so as toovercome this drawback.

Archive set (denoted as Q) was first introduced in [29] to remainthe non-dominated individuals found along the evolutionary pro-cess in multi-objective optimization problems. This method pro-vides the best solutions for individual evolution instead oforiginal population to accelerate algorithm convergence. Thus,we propose a multi-elite guide strategy based on archive set.According to preserve the elite particles along the evolution pro-cess, the individuals can fly under the guide of multi-elite solutionsso as to improve the searching speed as well as maintain the diver-sity of evolutionary population in later period.

Similarly, the multi-elite guide strategy can be decomposed intotwo stages: firstly, archive set provides multi-elite guide for origi-nal population evolution. Secondly, update the archive set by theelite solutions found along the evolution process. Fig. 1 showsthe relationship between evolution population and archive set.

Hence, archive set provides multi-elite guide for original popu-lation evolution in this way: For the ith particle, randomly select asolution in Q as the ith particle’s global best position. In this waythe diversity of population can be maintained and the possibilityof local optima will be reduced. Meanwhile, in order to strengthenexchanges and cooperation among particles, the best position ofthe ith particle is randomly selected in archive set expect the pre-vious global best. As a result, the ith particle can not only studyfrom individual’s best position but also from others’, which willimprove the searching ability in the local region. The formulationof selection is as follows:

pg ¼ Q rand1rand1 ¼ Random½0;NQ � 1�

pi ¼ Q rand2rand2 ¼ Random½0;NQ � 1� \ rand2–rand1

(ð9Þ

Update archive set

Provide multi-elite

guide

X1 X2 Xi XN-1 XN

Q1 Q2 Qi

Original Population

Archive Set

1QNQ − QNQ

Fig. 1. Multi-elite guide strategy based on archive set.

where Qrand1and Qrand2

are the rand1th and rand2th solution in ar-chive set Q. Thus, the generated strategy of PSO can be modified as:

v tþ1i;j ¼ w � v t

i;j þ c1 � r1 � ðQtrand2 ;j

� xti;jÞ þ c2 � r2� � ðQt

rand1 ;j� xt

i;jÞxtþ1

i;j ¼ xti;j þ v t

i;j

(

ð10Þ

Meanwhile, the archive set should update in the iterative pro-cess of the algorithm. For a newly generated particle Xi, the proce-dure of update for archive set Q is as follows:

Step 1: If the archive set Q is empty or the number of solutionsin Q has not reach NQ, add the newly generated particle Xi into Qdirectly and go to Step 4. Otherwise, go to Step 2.Step 2: Evaluate the objective value of current solutions in Q andfind out the minimum Qmin. Then compare the objective valueof Xi with Qmin.Step 3: If the objective value of Xi is bigger than that of Qmin,replace Qmin by Xi. Otherwise, the particle Xi cannot be addedinto archive. Go to Step 4.Step 4: Update of Q for particle Xi is terminated.

4. Optimal operation of multi-reservoir system by using MGPSO

The details of the proposed MGPSO for solving optimal opera-tion of multi-reservoir system are introduced in this section.

4.1. Initialization

The evolutionary population consists of NP particles and eachparticle is made of a series of decision variables. For the problem

G = G + 1

G >= GenerationNum

Termination : Output the best solutionin archive set

on

No

Yes

Fig. 2. The flow chart of MGPSO method for optimal operation for OMRS problem.


of optimal operation for multi-reservoir system (OMRS), water dis-charge rates are set as the decision variables. Therefore, one parti-cle can be expressed as follows:

X ¼

q1;1 q2;1 � � � qN;1

q1;2 q2;2 � � � qN;2

..

. ...

qi;t...

q1;T q2;T � � � qN;T

2666664

3777775 ð11Þ

The initialization of NP particles is in this way: qi,t is randomlygenerated between Qi,max and Qi,min while Vi,t is calculated by thecontinuity equation of hydro plant network. Check the generatedparticles and adjust if a particle violates the constraints.

Fig. 4. Convergence trajectory of different method for Test 1.

4.2. Constraint handling method

Since OMRS problem contains a series of inequality and equalityconstraints, it is quite significant to handle these constraints effi-ciently. The multi-reservoir system consists of a lot of stations withdifferent locations and operation characteristics. An upstreamplant will highly influence the operation of the next downstreamplant. The latter, however, also influence the upstream station byits effect on the tail water elevation and effective head. Moreover,the constraints are more complicated if these hydro plants are lo-cated on different streams.

4.2.1. Handling the inequality constraintsCompared with equality constraints, the inequality constraints

are easier to be handled. In OMRS problem, water discharge rateand storage volume limits are inequality constraints. When theparticles are generated, inequality constraints may be violated. Atthis time, we just make the violated particles equal to the bound-aries of the limits and change them into the feasible solutions:

Fig. 5. Independent repeated tests by MGPSO for Test 1.

qi;t ¼Q max

i;t ; if qi;t > Q maxi;t

Q mini;t ; if qi;t < Q min

i;t

(ð12Þ

To be mentioned, when the storage volume limits is violated,the violated volume can be adjusted by changing the water releaseof current schedule period. By water continuity equation, theseparticles can also be changed into feasible ones.

R2

R3

R4R5 R6

R1R7

R8

R9

R10

Fig. 3. The 10-reservoir hydropower system.

Table 1Simulation results of Test 1 by different methods.

DE PSO MGPSO

Initial value of objective 4000.33 3832.52 3723.32Optimization result 190.56 84.25 64.35Energy deficit 2.53 2.53 2.53

4.2.2. Handling the equality constraintsIn OMRS problem, the terminal reservoir storage constraints

and system power balance constraints are equality constraints

Table 2Hourly water discharge rate of Test 1 obtained by MGPSO.

Hour Res1 Res2 Res3 Res4 Res5 Res6 Res7 Res8 Res9 Res10

1 1.74 2.20 1.10 3.00 2.56 2.36 11.78 1.30 1.65 10.612 2.09 1.52 0.76 3.48 2.02 2.47 13.76 1.34 1.59 12.693 2.09 2.32 1.03 3.97 3.27 2.40 12.54 1.68 1.27 16.424 2.09 2.28 1.09 3.81 2.90 1.79 10.96 1.43 1.50 14.955 1.43 2.12 0.56 3.22 2.86 1.98 10.00 1.68 1.38 12.526 1.61 1.82 0.85 3.17 2.65 2.31 8.43 1.68 1.56 10.227 1.50 1.81 1.03 2.24 2.72 2.15 7.54 1.67 1.61 7.778 2.09 1.96 0.91 1.85 1.98 2.10 4.56 1.38 1.27 7.489 1.44 1.51 0.63 1.52 1.43 1.58 4.32 1.70 1.19 5.02

10 2.04 2.48 0.88 2.19 2.00 1.78 6.33 1.14 1.29 5.6611 2.09 1.81 0.61 1.83 2.72 2.17 6.10 1.19 1.66 9.4912 2.09 1.51 0.54 2.08 3.13 2.43 9.25 1.36 1.36 9.9313 1.66 1.83 0.52 3.10 2.59 2.47 9.44 1.67 1.25 13.1514 1.62 1.69 0.64 3.53 3.40 1.69 11.48 1.54 1.68 14.8515 1.96 1.97 0.91 2.97 3.27 1.60 13.92 1.61 1.65 16.0816 1.66 2.28 1.09 3.88 3.45 1.54 11.81 1.38 1.66 13.9417 1.45 2.47 1.09 3.39 3.40 2.41 10.81 1.68 1.44 11.0218 1.99 2.25 0.89 2.49 2.17 2.42 8.88 1.58 1.32 9.9619 2.07 2.37 0.39 3.00 1.53 2.14 7.61 1.49 1.66 7.6720 1.53 2.19 0.48 1.67 1.47 2.00 6.45 1.14 1.66 6.16

Fig. 6. Hourly reservoir storages of each reservoir obtained by MGPSO for Test 1.


which related to schedule periods and relative positions. Since theinitial and terminal reservoir storage constraints are temporal cou-pled, in this paper, we proposed a novel two-phase constraintshandling method by adjusting the reservoir volumes of hydroplants. The procedure is as follows:

Step 1: Set the hydro station i = 1, and the iteration number ofphase one count_1 = 0;Step 2: For the ith hydro plant, calculate the violation of the ter-minal reservoir storage constraint:

deltaVðiÞ ¼ Vi;T � VEndi ð13Þ

If |deltaV(i)| < e1 or count_1 > Num_1, the phase one adjustmentis done and go to Step 5 directly. Otherwise, go to step 3. e1 is theprecision of first phase adjusting and Num_1 is the maximum iter-ation number. Here we set e1 = 0.5 and Num_1 = 5.

Step 3: Adjust the water discharge rate of each schedule periodby the average violation of terminal reservoir storage con-straints. The details are as follows:


qi;t ¼ qi;t þ avgV ; avgV ¼ deltaVðiÞN

ð14Þ

Check qi,t if modified qi,t violate the constraint of water dis-charge rate, use the inequality constraints handling method tomake it equal to the boundaries. Then go to step 4.

Step 4: Calculate newly ViT and count_1 = count_1 + 1, if

count_1 > Num_1, go to Step 5 to implement the phase twoadjustment, otherwise, go back to Step 2.Step 5: Calculate the violation of the terminal reservoir storageconstraint. Set t = 1.Step 6: Add the violation into the water discharge rate of tthschedule period. Thus, adjust the water discharge rate in thefollowing way:


DE PSO MGPSO


qi;t ¼ qi;t þ deltaVðiÞ ð15Þ

Check qi,t and if modified qi,t violates the constraint of water dis-charge rate, make it equal to the boundaries.

Step 7: Evaluate the solution by calculating the objective values.If |deltaV(i)| < e2 or t > T, the phase two adjustment is done, then




1 1.57 0.95 0.12 4.02 5.36 3.92 8.77 1.72 0.59 12.642 3.91 2.96 1.85 2.83 4.60 2.69 7.54 2.61 2.52 15.203 2.81 2.08 1.22 3.05 2.77 1.67 13.80 2.85 2.30 15.094 2.75 3.75 1.60 3.58 1.92 2.04 9.83 1.43 2.41 14.555 2.13 3.12 0.68 5.02 3.16 2.09 11.73 1.15 0.80 9.696 1.25 3.41 0.04 0.98 3.71 3.98 8.79 2.53 1.48 8.927 1.80 3.32 0.20 2.27 3.27 2.60 10.81 1.92 1.25 3.638 0.16 0.01 0.06 0.07 2.28 1.34 7.77 0.20 0.38 8.099 0.09 0.38 0.71 0.27 0.37 0.60 5.34 1.03 1.54 6.8310 0.59 1.63 0.01 3.85 1.89 0.66 7.40 0.14 2.30 5.9411 1.87 2.53 1.31 1.87 1.20 1.37 7.21 1.04 1.90 8.7012 1.08 2.06 0.24 3.88 2.37 2.10 9.99 1.15 1.09 9.5113 2.53 0.36 1.12 5.89 1.67 2.52 10.09 1.70 2.26 11.5614 0.85 0.84 1.30 4.44 4.24 0.89 10.67 1.34 1.23 16.2015 2.10 1.97 1.42 2.38 2.08 2.05 15.21 1.68 1.84 14.9516 2.46 0.15 1.58 1.69 2.70 1.64 11.68 1.68 3.21 15.1717 2.98 4.04 0.73 3.39 1.37 2.95 9.82 1.47 1.01 11.2518 3.31 3.59 0.48 2.73 2.56 1.65 9.21 0.56 0.03 9.6719 1.38 1.80 1.29 2.27 3.01 2.94 5.54 2.86 1.44 8.7620 0.65 1.47 0.06 1.97 0.97 2.11 4.87 0.61 0.10 9.40


go the Step 8; otherwise, Set t = t + 1and go back to Step 6. Heree2 is set as 0.001.Step 8: i = i + 1, if i = N, then the adjustment of reservoir storageis done; otherwise, go back to Step 2.

4.3. Procedures of MGPSO method for optimal operation of multi-reservoir system

The procedure of optimal operation for multi-reservoir systemby using MGPSO is presented as follows and flow chart is shownin Fig. 2.

Fig. 9. Hourly reservoir storages of each re

Table 5Parameter sensitivity analysis.

Parameter Standard Parameter test

NP NQ

NP 20 10 30 20 20NQ 5 5 5 3 10c1, c2 2 2 2 2 2Vmax 0.5 0.5 0.5 0.5 0.5GenerationNum 200 200 200 200 200

Optimization Result 61.98 62.40 62.24 62.38 61.5Energy deficit 2.48 2.48 2.48 2.48 2.48Computing time 1.84 1.03 2.71 1.93 2.10

Step 1: Initialization. Randomly generate the evolution popula-tion. Initialize the archive set and the parameters of the algo-rithm. The generation number G = 0.Step 2: Update archive set. Evaluate each particle’s fitness bycalculating objective values and use the updating method men-tioned in Section 3.2 to renew the archive set.Step 3: Generation. Use the multi-elite guide strategy men-tioned in Section 3.2 to update the positions and velocities ofall the particles in original population.Step 4: Constraints handling. Use the constraint handlingmethod mentioned in Section 4.2 to deal with infeasible parti-cles. If the solutions are outside the corridor, change them intothe feasible solutions. Then evaluate the newly generatedparticles.Step 5: If the generation number G = GenerationNum (the maxi-mum of the iteration number), the search is terminated andoutput the best solution in archive set as the final result. Other-wise, set G = G + 1 and go back to Step 3.

5. Case studies

The proposed MGPSO method has been applied to a multi-res-ervoir system which consists of 10 cascaded hydropower stationsto determine the hourly optimal operation.

5.1. Description of multi-reservoir system

In order to prove the effectiveness of the proposed method, inthis paper, a multi-reservoir system which consists of 10 cascadedhydropower stations is considered. The diagrammatic location isshown in Fig. 3 and schedule horizon is 20 h. The detailed data ofthis multi-reservoir system is given in Appendix A. Appendix

servoir obtained by MGPSO for Test 2.

c1, c2 Vmax GenerationNum

20 20 20 20 20 205 5 5 5 5 51.5 3 2 2 2 20.5 0.5 0.1 5 0.5 0.5200 200 200 200 100 500

9 61.72 63.14 759.24 82.09 62.45 61.932.48 2.48 2.48 2.48 2.48 2.481.83 2.05 1.87 2.16 0.97 4.22


Table A1 lists the energy demand and the water inflows over thewhole operation horizon. Appendix Table A2 shows the initialand terminal reservoir storage volumes, constraints of water dis-charge rate and reservoir storage volume. Appendix Table A3 isthe power generation coefficient for each hydro station.

5.2. Parameters setting

The parameters setting influence the efficiency of an algorithma lot. Especially for a practical problem, the choice of the parame-ters needs tests and trials. In our study, the parameters of MGPSOare selected as follows: the size of the population NP = 20, the sizeof archive set NQ = 5, the acceleration constants of the PSO c1 = 2,c2 = 2, the maximum and minimum values of the inertia weightfactor wmax = 0.9, wmin = 0.4, respectively. The maximal and mini-mum velocity are Vmax = 0.5 and Vmin = �Vmax. Maximal generationof the algorithm GenerationNum is set as 200.


DE PSO MGPSO




In order to verify the effectiveness of the proposed method, thestandard PSO and differential evolution algorithm (DE) is alsoimplemented to deal with OMRS problem. The parameter settingof PSO is totally the same as MGPSO. While DE is as follows: thenumber of iterations is set as 200, mutation parameter F = 0.5,the crossover constant CR = 0.9, the size of the population NP = 20.

5.3. Simulation results and analysis

In this section, the proposed MGPSO is applied to the OMRSproblem. The scheduling horizon is 20 h. With the data given inAppendix A, all the algorithm have been coded in Eclipse Win32on a PC (Core 2, 250 GB, 2 GHz) to solve the optimal operation ofmulti-reservoir system.

5.3.1. Case study 1In this case study, the terminal storage of the multi-reservoir

Vend is [6,6,3,8,8,7,15,6,5,15] and set the parameters as in Sec-tion 5.2. Table 1 shows the simulation results obtained by PSO,DE and MGPSO. The energy deficit of each time interval is uni-formly distributed. So we define it as an identical average valuefor all time intervals. The formula for calculating energy deficit re-

ferred to in the results is 1T

PTt¼1 Dt �

PNi¼1ciqi;t

� �. Fig. 4 shows the

convergence trajectories of three methods. 100 times of indepen-dent tests are done by MGPSO and solutions are drawn in Fig. 5.The operation scheme of MGPSO which figures out the hourlywater discharge rate and storage volume is presented in Table 2and Fig. 6, respectively.

The data listed in Table 1 shows that the proposed MGPSO canget a better operational scheme with fewer energy shortages. Com-pare with the solutions obtained by DE and PSO, the result ofMGPSO can reduce the energy shortages over complete planninghorizon by 126.21 and 19.9, which means the proposed methodis able to deal with high-dimensional and multi-constraint optimi-zation problem. As shown in Fig. 4, it is observed from the conver-gence trajectory that the progress of MGPSO in reducing theobjective values is very rapid in the initial stage and then stabilizesto a constant small value during steady stage. Meanwhile, after 100times of independent tests by MGPSO, the simulation results are allbetween 64.1 and 65.0 in Fig. 5. Therefore, the robustness ofMGPSO has been proved. Moreover, from Table 2 and Fig. 6 wecan see the water discharge rate and storage volume of MGPSOscheme are all in the boundaries of constraints, which



1 0.51 0.93 0.95 4.97 3.92 3.97 8.84 2.09 3.09 11.882 1.71 3.66 1.87 3.84 3.58 2.70 8.37 2.98 4.09 14.173 3.52 2.72 0.96 5.48 5.22 1.82 13.33 1.77 2.10 13.524 3.11 1.95 0.82 4.70 1.78 1.80 11.80 0.97 2.01 13.655 0.72 3.68 0.85 4.55 2.22 2.42 10.99 2.41 1.43 10.166 3.34 2.91 0.76 2.11 2.04 4.21 9.72 0.07 1.19 8.287 2.60 2.73 0.35 1.47 4.61 1.21 9.69 0.17 0.68 6.218 0.94 0.30 0.17 0.21 2.00 0.95 8.41 0.19 0.24 7.489 0.04 0.31 0.14 0.24 0.65 0.87 3.72 0.62 0.59 8.72

10 0.09 0.14 0.23 2.18 1.09 1.60 7.59 1.19 1.02 7.1811 3.03 1.09 0.15 1.42 1.12 3.39 6.77 0.80 0.55 9.7112 0.68 3.26 1.21 1.17 2.35 1.58 9.84 1.59 0.66 9.9113 3.64 2.79 1.39 4.19 3.70 1.23 9.80 1.97 2.51 10.5014 0.84 0.37 1.40 4.35 2.62 1.76 9.23 2.85 1.28 17.2115 2.66 1.86 1.12 2.32 2.04 1.94 13.68 2.10 2.36 16.2716 2.07 2.25 1.51 2.46 3.45 2.27 11.39 1.73 1.73 14.0717 3.52 2.79 0.72 4.71 3.31 1.85 10.88 1.68 0.69 10.2018 2.06 2.75 0.97 1.86 1.95 2.55 9.05 1.81 0.42 10.0019 0.51 2.02 0.24 4.06 0.91 3.49 5.51 2.35 1.91 9.1620 0.65 1.90 0.19 0.14 2.94 0.22 7.42 0.30 1.13 7.41


demonstrates the proposed constraint handling method could sat-isfy the practical requirement of multi-reservoir system.


Vend = [6,6,3,8,8,7,15,6,5,15] and the parameters are set as theSection 5.2. Table 3 shows the simulation results obtained by dif-ferent methods. Fig. 7 shows the convergence trajectories of PSO,DE and MGPSO. 100 times of independent tests are done by MGPSOand solutions are drawn in Fig. 8. The operation scheme of MGPSOwith hourly water discharge rate and storage volume is presentedin Table 4 and Fig. 9. In order to assess the effect of the parameterssetting, a parameter sensitivity analysis shown in Table 5 is imple-mented in this section.

The data listed in Table 3 shows that the proposed MGPSO canget a better operational scheme with fewer energy shortages. Com-pare with the solutions obtained by DE and PSO, the result ofMGPSO can reduce the energy shortages by 71.49 and 35.44, whichmeans the proposed method is able to deal with high-dimensionaland multi-constraint optimization problem. As shown in Fig. 7, it isobserved that the progress of MGPSO in reducing the objective val-

Fig. 12. Hourly reservoir storages of each re

Table 8Hourly water discharge rate of Test 3 without constraint handling.

Hour Res1 Res2 Res3 Res4 Res5

1 1.85 3.19 �0.26 4.33 5.412 2.87 3.52 2.06 3.74 4.073 3.19 1.96 1.87 3.49 4.014 1.78 0.79 1.10 4.89 0.675 1.29 4.75 3.06 5.53 2.896 3.47 2.51 0.89 1.81 3.187 2.22 2.95 0.24 2.73 3.858 0.58 �0.24 �0.68 1.37 0.249 1.47 �0.52 �0.56 �0.70 0.76

10 �0.06 3.02 0.53 2.26 1.1111 1.32 1.15 1.11 �0.25 1.8212 2.79 0.22 0.64 2.99 2.2513 1.04 4.58 1.22 0.88 4.3914 1.15 2.61 2.50 1.98 4.4815 2.49 0.98 2.69 2.99 4.0416 2.16 3.13 0.85 5.11 3.2017 2.40 3.46 1.73 3.67 0.1618 0.56 3.33 �0.24 1.13 1.9119 1.46 1.46 �0.05 3.20 2.6820 �0.21 1.01 0.97 1.44 1.24

ues is very rapid in the initial stage and then stabilizes to a smallvalue during steady stage. After 100 times of independent tests,the simulation results of MGPSO are all between 61.70 and 62.05in Fig. 8, which proves that MGPSO is a robust algorithm. More-over, from Table 4 and Fig. 9 we can see the water discharge rateand storage volume of MGPSO scheme are all in the boundariesof constraints. This simulation demonstrates the proposed con-straints handling method could satisfy the practical requirementfor multi-reservoir system.

Besides, a parameter sensitivity analysis is implemented basedon this section. The main parameters of the proposed method arepopulation size NP, archive set size NQ, acceleration constants ofPSO c1, c2, maximal velocity Vmax and maximal generation Genera-tionNum. We tried different values of these parameters to evaluatethe influence of each parameter. From the test result shown in Ta-ble 5 we can see, computing time will change with the populationsize. However, bigger or small NP both leads to an increase ofobjective value. When NP is bigger, it will be hard to converge inlimited iterations. On the contrary, a smaller NP is not enough tomaintain the diversity of population to reach global optima. Mean-while, the values of optimization result and computing time show

servoir obtained by MGPSO for Test 3.

Res6 Res7 Res8 Res9 Res10

3.01 7.42 1.09 3.12 13.273.10 10.10 1.41 2.30 13.802.51 14.74 �1.07 3.22 14.583.93 7.57 1.81 2.04 18.552.88 8.47 2.42 2.52 10.754.47 8.48 1.34 2.52 8.890.77 11.94 0.36 0.40 5.110.22 9.22 1.19 �1.16 8.56�0.34 4.89 1.35 �0.13 8.69�0.15 8.76 0.20 0.01 6.89

1.34 7.36 0.93 1.46 11.352.78 10.13 1.79 0.75 9.944.16 11.91 0.76 1.70 9.844.16 9.62 0.84 1.99 15.511.31 13.39 2.78 2.20 16.703.84 10.33 3.37 2.28 12.962.47 11.51 0.25 1.98 11.50�0.30 11.87 0.27 0.58 10.64

2.62 6.59 2.02 1.33 9.902.26 8.26 2.05 1.48 5.97


that the parameter NQ and c1, c2, have a tiny effect on the algo-rithm. The change of Vmax presents a similar tendency as NP. That’sbecause the algorithm cannot globally search under a small Vmax

and a big one cannot perform local search. Furthermore, from thetest of maximal generation GenerationNum we can figure out theobjective value reduces rapidly and bigger GenerationNum will in-crease computing time greatly.


Vend is [5.98,5.98,2.98,7.98,7.98,6.98,14.99,5.99,4.99,14.99] andset the parameters as the Section 5.2. Table 6 shows the simulationresults obtained by PSO, DE and MGPSO. Fig. 10 is the convergencetrajectories of the three methods. 100 times of independent testsare done by MGPSO and solutions are drawn in Fig. 11. The opera-tion scheme of MGPSO which figures out the hourly water dis-


DE PSO MGPSO



Fig. 14. Convergence trajectory for function evaluations for Test 4.

charge rate and storage volume is presented in Table 7 andFig. 12, respectively. In order to assess the effect of constraint han-dling, the hourly water discharge rate obtained without constrainthanding is listed in Table 8.

The data listed in Table 6 shows that the proposed MGPSO canget a better operational scheme with fewer energy shortages. Com-pare with the solutions obtained by DE and PSO, the result ofMGPSO can reduce the energy shortages over complete planninghorizon by 36.15 and 9.25, which means the proposed method isable to deal with high-dimensional and multi-constraint optimiza-tion problem. As shown in Fig. 10, it is observed from the conver-gence trajectory that the progress in reducing the objective valuesis very rapid in the initial stage and then stabilizes to a constantsmall value during steady stage. Meanwhile, after 100 times ofindependent tests by MGPSO, the simulation results are all be-tween 62.70 and 63.10 in Fig. 11. Therefore, the robustness ofMGPSO has been proved. Moreover, from Table 7 and Fig. 12 wecan see the water discharge rate and storage volume of MGPSOscheme are all in the boundaries of constraints. However, the oper-ation scheme obtained without constraint handling is infeasibleand unacceptable. As shown in Table 8, the water discharge ratesin bold type violate the constraints. Tables 7 and 8 demonstratethe constraints handling method proposed in this paper could sat-isfy the practical requirement of multi-reservoir system.




1 1.15 1.89 1.01 3.37 5.25 2.06 9.19 1.05 1.41 12.932 1.77 2.47 1.19 4.20 3.39 3.92 9.73 2.96 0.32 14.463 2.27 2.95 0.89 3.65 4.31 2.37 12.07 2.24 2.41 15.774 3.37 2.09 1.49 4.33 2.43 2.23 9.07 2.27 2.92 15.035 2.40 3.93 0.07 5.95 2.14 1.37 10.48 1.52 0.99 10.926 1.95 2.14 0.53 0.50 3.92 4.19 11.18 0.27 1.76 7.757 1.38 3.93 1.98 1.57 3.69 0.85 9.18 0.51 1.12 6.258 0.94 0.47 0.04 1.31 1.62 2.14 7.70 0.04 0.09 7.439 0.87 0.10 0.13 0.05 1.59 0.21 4.78 0.91 0.52 7.83

10 1.64 2.00 0.17 1.68 0.86 0.92 6.64 1.85 1.67 6.7711 1.56 0.99 0.44 1.80 2.81 3.03 7.48 1.87 2.22 8.0512 0.28 1.89 0.33 3.38 2.95 2.15 10.40 2.31 1.85 8.8413 2.37 1.03 0.92 5.14 1.78 3.85 11.58 1.14 1.51 10.0414 1.73 1.57 1.06 4.12 2.30 0.87 10.88 2.76 2.28 15.2415 3.13 1.84 1.37 2.93 2.14 1.49 12.89 1.80 1.94 16.8616 2.55 1.78 2.08 3.57 2.71 1.92 11.28 0.98 2.33 14.2917 3.19 3.75 1.22 1.82 1.87 1.95 10.10 1.58 1.77 11.8118 1.74 2.88 0.48 2.64 2.51 1.38 9.07 1.19 0.16 10.7119 1.52 1.34 0.28 3.67 2.34 3.34 6.55 1.84 0.56 8.7120 0.58 1.52 0.49 1.22 1.02 1.71 6.75 0.66 1.98 7.20

Fig. 16. Hourly reservoir storages of each reservoir obtained by MGPSO for Test 4.

Table A1The inflows and power lord demand of the 10-reservoir system.

Schedule index I1 I2 I3 I5 I6 I8 D

1 0.50 0.40 0.80 1.50 0.32 0.71 80.02 1.00 0.70 0.80 2.00 0.81 0.83 90.0



Vend = [5.99,5.99,2.99,7.99,7.99,6.99,14.99,5.99,4.99,14.99] andset the parameters as the Section 5.2. Table 9 shows the simulationresults obtained by different methods. Fig. 13 shows the conver-gence trajectories of DE, PSO and MGPSO. The comparison for thesame computational effort is presented in Fig. 14. 100 times ofindependent tests are done by MGPSO and solutions are drawnin Fig. 15. The operation scheme of MGPSO is presented in Table 10and Fig. 16, respectively.

The data listed in Table 9 shows that the proposed MGPSO canget a better operational scheme with fewer energy shortages. Com-pare with the solutions obtained by DE and PSO, the result ofMGPSO can reduce the energy shortages by 23.50 and 15.97, whichmeans the proposed method is able to deal with high-dimensionaland multi-constraint optimization problem. In order to comparethe results with the same computational effort, all the methodsare run for the same number of iterations and objective functionevaluations respectively. As shown in Figs. 13 and 14, it can befound that the progress of MGPSO in reducing the objective valuesis very rapid in the initial stage and then stabilizes to a constantsmall value during steady stage. Meanwhile, after 100 times ofindependent tests by MGPSO, the simulation results are all be-tween 45.75 and 46.15 in Fig. 15. Therefore, the robustness ofMGPSO has been proved. Moreover, from Table 10 and Fig. 16 wecan see the water discharge rate and storage volume of MGPSOscheme are all in the boundaries of constraints, which demon-strates the proposed constraints handling method could satisfythe practical requirement for multi-reservoir system.

3 2.00 2.00 0.80 2.50 1.53 1.00 100.04 3.00 2.00 0.80 2.50 2.16 1.25 90.05 3.50 4.00 0.80 3.00 2.31 1.67 80.06 2.50 3.50 0.80 3.50 4.32 2.50 70.07 2.00 3.00 0.80 3.50 4.81 2.80 60.08 1.25 2.50 0.80 3.00 2.24 1.87 50.09 1.25 1.30 0.80 2.50 1.63 1.45 40.0

10 0.75 1.20 0.80 2.50 1.91 1.20 50.011 1.75 1.00 0.80 2.00 0.80 0.93 60.012 1.00 0.70 0.80 1.50 0.46 0.81 70.013 0.50 0.40 0.80 1.50 0.32 0.71 80.014 1.00 0.70 0.80 2.00 0.81 0.83 90.015 2.00 2.00 0.80 2.50 1.53 1.00 100.016 3.00 2.00 0.80 2.50 2.16 1.25 90.017 3.50 4.00 0.80 3.00 2.31 1.67 80.018 2.50 3.50 0.80 3.50 4.32 2.50 70.019 2.00 3.00 0.80 3.50 4.81 2.80 60.020 1.25 2.50 0.80 3.00 2.24 1.87 50.0

6. Conclusions

The operation of multi-reservoir system (OMRS) is a high-dimensional, multi-stage and multi-constraint optimal problem.In order to provide an effective and scientific tool to support theoperation of current and prospective multi-reservoir system, thispaper presents a multi-elite guide particle swarm optimization(MGPSO) algorithm. Besides, a novel constraint handling methodis proposed to handle the equality and inequality constraints inthe OMRS problem. The proposed method is implemented to amulti-reservoir system consisting of 10 cascaded hydro plants.The result of case studies shows that the proposed approach pro-vides a competitive performance in simulation solutions as wellas algorithm convergence. Compared with previous algorithms,

the result obtained by MGPSO is much closer to global optimum,which proves the proposed method can ensure the precision of glo-bal optimization. Meanwhile, after multiple tests of independentrepetition, simulation results are stable and in the neighborhoodof the minimum. Moreover, the scheme obtained by MGPSO is inthe boundaries of constraints. It demonstrates the proposed con-straints handling method could satisfy the practical requirementfor multi-reservoir system. To sum up, the proposed MGPSO isnot only with global optimum capability but also strong robust-ness. Therefore, it can be concluded that MGPSO has the potentialto deal with different optimization problem of hydropower system.

Acknowledgements

This work is supported by the research funds of University andcollege PhD discipline of China (No. 20100142110012) (Ph.D. Pro-grams Foundation of Ministry of Education of China), the Project ofSpecial Research Foundation for the Public Welfare Industry of theMinistry of Science and Technology and the Ministry of Water Re-sources of China (No. 201001080) and the National Natural ScienceFoundation for Young Scholars of China (No. 51109086).

Table A2The limits of the 10-reservoir system.

Unit 1 2 3 4 5 6 7 8 9 10

Vmax 12.0 17.0 6.0 19.0 19.1 14.0 30.1 13.16 7.9 30.0Vmin 1.0 1.0 0.3 1.0 1.0 1.0 1.0 1.0 0.5 1.0Qmax 4.0 4.5 2.12 7.0 6.43 4.21 17.1 3.1 4.2 18.9Qmin 0.005 0.005 0.005 0.005 0.006 0.006 0.01 0.008 0.008 0.01Vbegin 6.0 6.0 3.0 8.0 8.0 7.0 15.0 6.0 5.0 15.0Vend 6.0 6.0 3.0 8.0 8.0 7.0 15.0 6.0 5.0 15.0

Table A3The power generation coefficient for each hydro station.

Unit 1 2 3 4 5 6 7 8 9 10

c 1.1 1.4 1.0 1.1 1.0 1.4 2.6 1.0 1.0 2.7


Appendix A

See Table A1–A3.

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