Annals of Nuclear Energy 54 (2013) 134–140

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Annals of Nuclear Energy

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Quantum behaved Particle Swarm Optimization with Differential Mutationoperator applied to WWER-1000 in-core fuel management optimization

Mostafa Jamalipour a,⇑, Reza Sayareh a, Morteza Gharib b, Farrokh Khoshahval c, Mahmood Reza Karimi a

a Faculty of Electrical and Computer Engineering, Kerman Graduate University of Technology, Kerman, Iranb Faculty of Engineering and Physics, Amirkabir University of Technology, Tehran, Iranc Engineering Department, Shahid Beheshti University, Tehran, Iran

a r t i c l e i n f o

Article history:Received 23 July 2012Received in revised form 1 November 2012Accepted 2 November 2012Available online 13 December 2012

Keywords:Differential MutationIn-core fuel managementParticle Swarm OptimizationQuantum Particle Swarm OptimizationWWER-1000

0306-4549/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.anucene.2012.11.008

⇑ Corresponding author. Tel.: +98 342 6226611; faxE-mail addresses: m.jamalipour@student.kgut.ac

(M. Jamalipour).

a b s t r a c t

This paper presents a new method using Quantum Particle Swarm Optimization with Differential Muta-tion operator (QPSO-DM) for optimizing WWER-1000 core fuel management. Genetic Algorithm (GA) andParticle Swarm Optimization (PSO) have shown good performance on in-core fuel management optimi-zation (ICFMO). The objective of this paper is to show that QPSO-DM performs very well and is compa-rable to PSO and Quantum Particle Swarm Optimization (QPSO). Most of the strategies for ICFMO arebased on maximizing multiplication factor (keff) to increase cycle length and minimizing power peakingfactor (Pq) in order to improve fuel integrity. PSO, QPSO and QPSO-DM have been implemented to fulfillthese requirements for the first operating cycle of WWER-1000 Bushehr Nuclear Power Plant (BNPP). Theresults show that QPSO-DM performs better than the others. A program has been written in MATLAB tomap PSO, QPSO and QPSO-DM for loading pattern optimization. WIMS and CITATION have been used tosimulate reactor core for neutronic calculations.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

In-core fuel management optimization (ICFMO) problem, load-ing pattern design optimization problem or nuclear reactor reloadproblem are denominations for the problem associated to the nu-clear reactor fuel reloading operation. It is a problem studied formore than three decades and several techniques had been usedin this optimization problem. Its principal characteristics arenon-linearity, multimodality, discrete solutions with nonconvexfunctions, disconnected feasible regions and high dimensionality(Stevens et al., 1995).

In most nuclear reactors, fuels have been arranged in a way thatleads to the maximum value of effective multiplication factor (keff)and minimum value of power peaking factor (Pq). For this aim thevalue of Pq must be kept lower than a predetermined value tomaintain fuel integrity and keff must be maximized under theseconstraints to extract maximum energy and cycle length (Lamarsh,1965).

During decades the core fuel reload optimization problem wassolved manually by the experts that used their knowledge andexperience to build configurations of the reactor nucleus, and test-ing them to verify if economic aspects and safety restrictions of the

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: +98 342 6228018..ir, mostafa6134@gmail.com

plant are satisfied. A large number of computational methods havebeen used for core fuel reload optimization. Swarm intelligence is anew range of computational algorithms that inspired by the collec-tive behavior of social insect colonies and other animal societies.Nowadays metaheuristic algorithms have opened a new era inoptimization fields. Genetic Algorithm (GA) and Particle SwarmOptimization (PSO) have shown good performance on ICFMO.

Several metaheuristics or computational intelligence methodshave been expanded to optimize reactor core loading pattern.Some of them are Linear Programing (Okafor and Aldemir, 1988),Dynamic Programing (Wall and Fenech, 1965), Direct Search(Motoda et al., 1975), Simulated Annealing (Kirkpatrick et al.,1983; Smuc et al., 1994), Hopfield Neural Network along with Sim-ulated Annealing (Sadighi et al., 2002), Genetic Algorithm (Poonand Parks, 1992; Yamamuto, 1997), Cellular Automata (Fadaeiand Setayeshi, 2009), Discrete Particle Swarm Optimization(Meneses et al., 2009; Babazadeh et al., 2009) Continuous ParticleSwarm Optimization (Khoshahval et al., 2010).

In this paper a new method using Quantum Particle SwarmOptimization with Differential Mutation operator (QPSO-DM) hasbeen applied for ICFMO. This method represents a new loading pat-tern for WWER-1000 reactor core for the first operation cycle inwhich the multiplication factor has been maximized in order toincrease the cycle length and power peaking factor has been min-imized for a decent operation of fuel rods. In this research, the corecalculations have been performed by CITATION-LD2 in order to

Fig. 1. Description of velocity and position updates in PSO for a two dimensionalparameter space.

Fig. 2. The pseudo code of the general PSO algorithm.

M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140 135

solve multi-group diffusion equation in 3-D. Furthermore WIMS-D5 has been used to generate average group constants for differentfuel assemblies. A program has been written in MATLAB to mapPSO, QPSO and QPSO-DM for loading pattern optimization.

2. Description of Particle Swarm Optimization and quantumbehaved Particle Swarm Optimization

2.1. Particle Swarm Optimization (PSO)

PSO is an evolutionary computation technique motivated by thesimulation of social behavior (Kennedy and Eberhart, 1995).Namely, each individual (agent) utilizes two important kinds ofinformation in decision process. The first one is their own experi-ence. That is, they have tried the choices and know which statehas been better so far, and they know how good it was. The secondone is other agent experiences. That is, they have knowledge ofhow the other agents around them have performed. Namely, theyknow which choices their neighbors have found are most positiveso far and how positive the best pattern of choices was. In the PSOsystem, each agent makes his decision according to his own expe-riences and other agent’s experiences. The system initially has apopulation of random solutions. Each potential solution, called aparticle (agent), is given a random velocity and is flown throughthe problem space. The agents have memory and each agent keepstrack of its previous (local) best position (called the Pbest) and itscorresponding fitness. There exists a number of Pbest for the respec-tive agents in the swarm and the agent with greatest fitness iscalled the global best (Gbest) for the swarm. Each particle is treatedas a point in an n-dimensional swarm. The i-th particle is repre-sented as Xi = (xi1, xi2, . . ., xin). The best previous position of the i-th particle (Pbesti) that gives the best fitness value is representedas Pi = (Pi1, Pi2, . . ., Pin). The best particle among all particles in thepopulation is represented by Pg = (Pg1, Pg2, . . ., Pn). The velocity forparticle i (i.e., the rate of the position change) is represented asVi = (vi1, vi2, . . ., vin).

The particles are manipulated according to the followingequations:

vkþ1i ¼ wkvk

i þ c1r1ðPki � xk

i Þ þ c2r2ðPkg � xk

i Þ ð1Þ

xkþ1i ¼ xk

i þ vkþ1i ð2Þ

where i = 1, 2, . . . , N, and N is the size of the population, w is theinertia weight which is defined by

wk ¼ wmax �wmax �wmin

kmax

� �k ð3Þ

where wmax is initial weight, wmin is final weight, k is the iterationnumber and kmax is the maximum iteration number. c1 and c2 aretwo positive constants, called the cognitive and social parametersrespectively. Eberhart and Shi (1998) have illustrated that wmax,wmin and ci are equal to 0.9, 0.4, and 2.0 and do not depend onthe problems. r1 and r2 are random numbers uniformly distributedwithin the range (0, 1). Eq. (1) is used to determine the i-th parti-cle’s new velocity vkþ1

i , in each iteration, while Eq. (2) providesthe new position of the i-th particle xkþ1

i , adding its new velocityvkþ1

i , to its current position xki . Fig. 1 shows the description of veloc-

ity and position updates of particle for a two dimensional space.Fig. 2 shows the pseudo code of the general PSO algorithm.

2.2. Quantum behaved Particle Swarm Optimization (QPSO)

The main disadvantage of PSO is that global convergence cannotbe guaranteed (Bergh, 2001). The concept of Quantum behavedPSO (QPSO) was developed to address this disadvantage and it

was first reported at conferences such as (Sun et al., 2004, 2005).The convergence of the PSO algorithm may be achieved if each par-ticle converges to its local attractor, pi = (pi1, pi2, . . . , pin) defined atthe coordinate (Clerc and Kennedy, 2002)

pki ¼ u � Pk

i þ ð1�uÞ � Pkg ð4Þ

where u e (0, 1). It can be seen that pi is a stochastic attractor ofparticle i that lies in a hyper-rectangle with Pi and Pg. In fact, whenthe particles are converging to their own local attractors, their per-sonal best positions, local attractors and the global best positionswill all converge to one point, leading the PSO algorithm toconverge.

In the quantum model of PSO, the state of a particle is depictedby wave function w, instead of position and velocity. The dynamicbehavior of the particle is widely divergent from that of the particlein traditional PSO systems in which the exact values of x and v can-not be determined simultaneously. We can only learn the probabil-ity of the particle’s appearing in position x from probability densityfunction |w|2. The form of which depends on the potential field theparticle lies in. Assuming that at iteration k, particle i moves in n-dimensional space with a d potential well at pk

i . Correspondingly,the wave function at iteration k + 1 is

wðxkþ1i Þ ¼

1ffiffiffiffiffiLk

i

q exp � jxkþ1i � pk

i jLk

i

!ð5Þ

Hence, the probability density function Q is a double exponen-tial distribution as follows:

Qðxkþ1i Þ ¼ jwðxkþ1

i Þj2 ¼ 1

Lki

exp �2jxkþ1

i � pki j

Lki

!ð6Þ

And thus the probability distribution function F is

Fðxkþ1i Þ ¼ exp �2

jxki � pk

i jLk

i

!ð7Þ

Fig. 4. The pseudo code of the general QPSO-DM algorithm.

Table 1Main characteristics of Fuel assemblies with Burnable Absorber Rods (BAR)concentration.

FAtype

AverageenrichmentU-235

Quantity offuel rods

BAR characteristics

Fuelrodtype

Fuelrodtype

Availabilityof BAR in FA

Quantityof BAR inFA

Boroncontent,g/cm3

136 M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140

where Lki is the standard deviation of the double exponential distri-

bution. Using Monte Carlo method, the position xki can be obtained

at iteration k + 1 as

xkþ1i ¼ pk

i �Lk

i

2ln

1u

� �ð8Þ

where u is a random number uniformly distributed in (0, 1) and Lki is

evaluated as

Lki ¼ 2 � b � jMbestk � xk

i j ð9Þ

where Mbest, called mean best position, is defined as the mean va-lue of the Pbest positions of all particles (Sun et al., 2005). That is,

Mbestk ¼ 1N

XN

i¼1

Pki ¼

1N

XN

i¼1

Pki1;

1N

XN

i¼1

Pki2; . . . ;

1N

XN

i¼1

Pkin

!ð10Þ

where N is the population size and Pi is the personal best position ofparticle i. Hence, the position of the particle updates according tothe following equation:

xkþ1i ¼ pk

i þ b � ðMbestk � xki Þ � ln

1u

� �if h > 0:5 ð11Þ

xkþ1i ¼ pk

i � b � ðMbestk � xki Þ � ln

1u

� �if h < 0:5

h is a random number distributed uniformly within (0, 1). Consider-ing that the number of iterations and population size are commonrequirements in every evolutionary algorithm, b, called Contrac-tion–Expansion coefficient, is the only parameter in QPSO algo-rithm. It can be turned to control the convergence speed of thealgorithms which is less than 1.7 (Sun et al., 2005).

QPSO is very easy to be understood and implemented and it hasalready been tried and tested in various standard optimizationproblems with excellent results (Sun et al., 2004). Fig. 3 showsthe pseudo code of the general QPSO algorithm.

2.3. Quantum behaved Particle Swarm Optimization with DifferentialMutation operator (QPSO-DM)

The Differential Evolution (DE) developed by Storn and Price(1997), is a stochastic search algorithm based on population coop-eration and competition of individuals and has been successfullyapplied to solve optimization problems particularly involvingnon-smooth objective functions. The optimization process in DEis carried out by combing the simple arithmetic operators withthe classical evolution operators of mutation, crossover and selec-tion to evolve from a randomly generated population to a finalsolution. The mutation operation is defined by (Lu et al., 2009)

xkþ1i ¼ xk

j þ ð1� YÞ � ðxkl � xk

mÞ þ Y � ðPkg � xk

j Þ ð12Þ

where j, l, m are random integers uniformly selected from the set{1, 2, . . . , N} and i – j – l – m, in other word the indices are mutu-ally different. Y is defined by

Fig. 3. The pseudo code of the general QPSO algorithm.

Y ¼ kcur

kmaxð13Þ

Where kcur is the current iteration and kmax is the number of max-imum iterations. Fig. 4 shows the pseudo code of the generalQPSO-DM algorithm.

3. WWER-1000 Reactor core

The WWER-1000 reactor is a Russian type Pressurized WaterReactor (PWR) that chemically purified water with boric acid isused for coolant and moderator. WWER-1000 core is composedof 163 fuel assemblies (FA) having hexagonal form, placed on ahexagonal grid with a constant step of 23.6 cm. Each fuel assemblyconsists of 311 fuel rods. UO2 is used for fuel rods with the averageenrichment of 1.6%, 2.4% and 3.62% for the first operating cycle ofreactor. Main characteristics of Fuel assemblies with Burnable Ab-sorber Rods (BAR) concentration are given in Table 1. Fig. 5 showsWWER-1000 core model prepared for core calculation.

4. Mapping PSO, QPSO and QPSO-DM for loading patternoptimization

We mapped PSO, QPSO and QPSO-DM on WWER-1000 BushehrNuclear Power Plant (BNPP) using MATLAB R2011b software.WIMS-D5 and CITATION-LD2 were used to simulate reactor core.WIMS-D5 was utilized to obtain multi group constants like absorp-

1 2

16 1.60 311(1.6)

– – – –

24 2.40 311(2.4)

– – – –

36 3.62 245(3.7)

66(3.3)

– – –

24B20 2.40 311(2.4)

– + 18 0.02

24B36 2.40 311(2.4)

– + 18 0.036

36B36 3.62 245(3.7)

66(3.3)

+ 18 0.036

Fig. 5. WWER-1000 core model prepared for core calculation.

M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140 137

tion and scatter cross sections and so on. These multi group con-stants were used in CITATION-LD2 to calculate multiplication fac-tor (keff) and power density distribution.

There are 163 fuel assemblies in the reactor core, thus it is reallysignificant to find a permissible arrangement for these fuel assem-blies considering economic and safety margins. As far as there ex-ists a large number of fuel assemblies, finding an optimized loadingpattern is a time consuming difficult task. Therefore we need toconsider one-sixth of the core for simplicity and place this symme-try in other one-sixths of the core to simulate total core. 28 fuelassemblies are available in symmetry of one-sixth of the core,but only 20 ones are allowed to be considered.

In this procedure, a position vector which is divided into 20cells is used for fuel assemblies placement. Randomly selectednumbers from 1 to 20 are placed in these cells. These numberswhich are nominated for fuel assemblies numbers, are updatedaccording to the algorithm instructions and are placed in thesecells. This method is being used until the best fitness function isreached or the number of iteration finishes. Fig. 6 shows a positionvector proposed for loading pattern optimization and Fig. 7 illus-trates Loading pattern proposed by the designer of BNPP.

When the mentioned method is used, two problems might oc-cur during optimization process. The first one is, the updated par-ticle might have non-integer numbers and the second one is, thereprobably are repeating numbers. The first problem can be easilysolved using rounded numbers after updating. A strategy has beenimplemented to amend the second problem. In this process, the

Fig. 6. A position vector proposed fo

program identifies repeating numbers in the updated particleand then substitutes them with the numbers that do not exist inupdated one from the primary particle. This strategy helps us toomit repeating numbers for arranging an appropriate pattern anddoes not make any problems during optimization process. It hasbeen illustrated in Fig. 8.

5. Objective function and optimization result

In the fuel management, one of the main objectives is powerpeaking factor minimization which can highly affect the safety ofplants. Power peaking factor minimization is the fission interactionrate flattening by proper arranging fuel assemblies in the reactor(Driscoll et al., 1990). Another main objective is multiplication fac-tor maximization that improves the economic aspects of nuclearpower reactors in order to increase the cycle length. Both thepower peaking factor and multiplication factor should be opti-mized in order to obtain a permissible arrangement for safetyand economic aspects. For implementing these criteria, we needto specify a proper fitness function in which the power peaking fac-tor and multiplication factor can be optimized. This fitness func-tion can be defined as

ff ¼ 1jkeff � 1j þ Pq ð14Þ

Where keff is the multiplication factor and Pq represents the powerpeaking factor. It is obvious that increasing in keff and decreasing

r loading pattern optimization.

Fig. 7. Loading pattern proposed by the designer of BNPP.

Fig. 8. Adapting position vector process example.

Table 2Parameters used for PSO, QPSO and QPSO-DM.

PSO QPSO & QPSO-DM

Number of particles: 20 Number of particles: 20kmax = 40 kmax = 40c1 = 2, c2 = 2 b = 1.2wmax = 0.9, wmin = 0.4 Table 3

Fitness value results using PSO, QPSO and QPSO-DM for loading pattern optimizationover 20 particles and 40 iterations.

Experiment PSO QPSO QPSO-DM

1 6.583822 6.569660 6.5829562 6.594785 6.574903 6.5653583 6.574665 6.592901 6.5743704 6.578516 6.582146 6.5909025 6.568300 6.576150 6.5557646 6.604714 6.577648 6.5808057 6.598644 6.577074 6.6027478 6.567492 6.582838 6.5583019 6.574360 6.593305 6.57662310 6.582974 6.563712 6.56593111 6.582656 6.556748 6.57425712 6.600386 6.593843 6.56908913 6.591288 6.579187 6.55876214 6.597211 6.587068 6.58930915 6.589248 6.571279 6.57837816 6.583102 6.555816 6.56052417 6.563319 6.586078 6.56699818 6.579019 6.569228 6.56643619 6.564735 6.587520 6.55941720 6.573506 6.605214 6.573184Best 6.563319 6.555816 6.555764Worst 6.604714 6.605214 6.602747Average 6.582637 6.579116 6.572506Std. dev. 0.009982 0.009894 0.009848

Fitness values which were resulted in a value more than referred value for keff andless than for Pq have been shown in bold format.

138 M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140

in Pq cause the fitness function to be minimized. Although no limi-tations have been specified for keff and Pq in this fitness function,theses parameters have been optimized choosing adequate particlesand iterations. As far as its very important to choose the number ofparticles and iterations, a sensitivity analysis has been carried outusing many experiments. The best results were obtained using 20particles and 40 iterations. PSO, QPSO and QPSO-DM have beentested for this work. The parameters used for these algorithms havebeen illustrated in Table 2. Twenty experiments have been done toshow their performances. The results show that QPSO-DM performsbetter than the others. Optimization results for PSO, QPSO andQPSO-DM using 20 particles and 40 iterations have been shownin Table 3. Fitness values which were resulted in a value more thanwhat has been mentioned in reference for keff and less than for Pq

(FSAR of BNPP-1, 2007) have been shown in bold format. For in-stance, in experiment 13, we have obtained the ff = 6.5913 in whichkeff = 1.1883 and Pq = 1.2807 for PSO, considering economic aspectmore, we can choose experiment 16 which leads to keff = 1.1893and Pq = 1.3006 (ff = 6.5831), however both of the experimentscan be implemented for loading pattern optimization. Table 4shows the best results obtained using PSO, QPSO and QPSO-DM.

Fig. 9 illustrates fitness value versus iteration number for thebest performance for PSO, QPSO and QPSO-DM implemented for

loading pattern optimization. Fig. 10–12 shows the best loadingpattern proposed using PSO, QPSO and QPSO-DM respectively.

Power density distribution for total core, proposed by the de-signer for the first operating cycle of BNPP has been shown inFig. 13. In Fig 14–16 the best power density distribution proposed

Fig. 9. Fitness Function versus iterations for the best result using PSO, QPSO andQPSO-DM.

Fig. 10. The best loading pattern proposed using PSO.

Fig. 11. The best loading pattern proposed using QPSO.

Fig. 12. The best loading pattern proposed using QPSO-DM.

Table 4The best results using PSO, QPSO and QPSO-DM for loading pattern optimization.

Method Fitness function Pq keff

Designer [FSAR] 6.679103 1.35 1.187648PSO 6.583102 1.300649 1.189306QPSO 6.582146 1.328623 1.190348QPSO-DM 6.569089 1.320111 1.190513

M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140 139

using PSO, QPSO and QPSO-DM have been shown respectively.They have been presented to show the effect of the fuel loadingpattern on the core power distribution.

Fig. 13. Power density distribution proposed by the designer of BNPP.

6. Conclusion

In this paper, a new algorithm, Quantum behaved ParticleSwarm Optimization using Differential Mutation operator (QPSO-DM) has been applied for in-core fuel management optimization(ICFMO) problem for the first operation cycle of Bushehr NuclearPower Plant (BNPP). A fitness function has been defined to test thisalgorithm for optimizing multiplication factor and power peakingfactor. Maximized multiplication factor increases the cycle lengthand minimized power peaking factor provides a better safety mar-

gin. GA and PSO have shown good performance on ICMFO. The re-sult of this method shows that QPSO-DM reaches a betteroptimized loading pattern and is comparable to PSO and Quantumbehaved Particle Swarm Optimization (QPSO).

Fig. 15. Power density distribution proposed using QPSO.

Fig. 16. Power density distribution proposed using QPSO-DM.

Fig. 14. Power density distribution proposed using PSO.

140 M. Jamalipour et al. / Annals of Nuclear Energy 54 (2013) 134–140

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