pwr loading pattern optimization using harmony search algorithm

Annals of Nuclear Energy 53 (2013) 288–298

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Annals of Nuclear Energy

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PWR loading pattern optimization using Harmony Search algorithm

N. Poursalehi, A. Zolfaghari ⇑, A. MinuchehrEngineering Department, Shahid Beheshti University, GC, P.O. Box 1983963113, Tehran, Iran

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 March 2012Received in revised form 22 June 2012Accepted 25 June 2012Available online 28 November 2012

Keywords:Harmony Search algorithmNodal expansion methodFitness functionLoading pattern optimization

0306-4549/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.anucene.2012.06.037

⇑ Corresponding author. Tel.: +98 21 22431596; faxE-mail address: [email protected] (A. Zolfagha

In this paper a core reloading technique using Harmony Search, HS, is presented in the context of findingan optimal configuration of fuel assemblies, FA, in pressurized water reactors. To implement and evaluatethe proposed technique a Harmony Search along Nodal Expansion Code for 2-D geometry, HSNEC2D, isdeveloped to obtain nearly optimal arrangement of fuel assemblies in PWR cores. This code consists oftwo sections including Harmony Search algorithm and Nodal Expansion modules using fourth degree fluxexpansion which solves two dimensional-multi group diffusion equations with one node per fuel assem-bly. Two optimization test problems are investigated to demonstrate the HS algorithm capability in con-verging to near optimal loading pattern in the fuel management field and other subjects. Results,convergence rate and reliability of the method are quite promising and show the HS algorithm performsvery well and is comparable to other competitive algorithms such as Genetic Algorithm and ParticleSwarm Intelligence. Furthermore, implementation of nodal expansion technique along HS causes consid-erable reduction of computational time to process and analysis optimization in the core fuel managementproblems.

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1. Introduction

Over the past four decades, a large number of algorithms havebeen developed to solve various engineering optimization prob-lems. Most of these algorithms are based on numerical linear andnonlinear programming methods that require substantial gradientinformation and usually seek to improve the solution in the neigh-borhood of a starting point. These numerical optimization algo-rithms provide a useful strategy to obtain the global optimum insimple and ideal models. Many real-world engineering optimiza-tion problems, however, are very complex in nature and quite dif-ficult to solve using these algorithms. In-core fuel management isone of the most challenging areas of nuclear engineering which in-volves the optimal arrangement of hundreds of fuel assemblies inreactor cores. The optimization of this arrangement is very impor-tant from economical point of view to make the nuclear powergenerating station competitive. An optimal nuclear reload designcan be defined as a configuration which has the maximum cyclelength for the given fuel inventory or uses the minimum amountof fissionable materials for the given cycle length while satisfyingsafety constraints such as limitation on power peaking factor.The main problem in the fuel assembly position determination isthe large number of possible combinations for the fuel loading pat-tern in the core. In addition, the fact that this is a nonlinear and dis-

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: +98 21 29902546.ri).

crete problem creates complications in the use of conventionaloptimization techniques (Babazadeh et al., 2009).

Several meta-heuristics or computational intelligence ap-proaches have been expanded in order to optimize this distributionfor the fuel assemblies in the core. Among these techniques, dy-namic programming (Wall and Fenech, 1965), direct search (Stout,1973), variational techniques (Terney and Williamson, 1982),backward diffusion calculation (Chao et al., 1986), reverse deple-tion (Downar and Kim, 1986; Kim et al., 1987), linear programming(Stillman et al., 1989), simulated annealing (Smuc et al., 1994;Mahlers, 1994), Ant Colony algorithm for maximizing boron con-centration (Machado and Schirru, 2002), Genetic Algorithms(Yamamoto, 1997; Mohseni et al., 2008), continuous GA for flattingof power distribution (Zolfaghari et al., 2009), discrete ParticleSwarm Optimization (Babazadeh et al., 2009), continuous ParticleSwarm Optimization (Khoshahval et al., 2010), Artificial Bee Col-ony algorithm along a Random Key in order to decode real to inte-ger number for the ICFMO problem of PWR reactor (Oliveira andSchirru, 2011; Safarzadeh et al., 2011), Cellular Automata formaximizing initial excess reactivity and minimizing power peakingfactor (Fadaei and Setayeshi, 2009) and Artificial Intelligence tech-niques like Artificial Neural Networks (ANNs) (Sadighi et al., 2002)are the ones most commonly used in the core fuel managementoptimization.

A further study based on hybrid algorithms was performed byseveral authors (Stevens, 1995; Erdog and Geçkinli, 2003; Fadaeiet al., 2009; Fadaei and Setayeshi, 2008). Artificial Intelligencetechniques like fuzzy logic Artificial Neural Networks (Kim et al.,

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N. Poursalehi et al. / Annals of Nuclear Energy 53 (2013) 288–298 289

1993) and knowledge-based systems (Galperin et al., 1989) havebeen successfully tested in order to accelerate search for the bestpositioning of the fuel assemblies in reactor cores. Another impor-tant study was performed by DeChaine and Feltus (1996). Theydeveloped the stochastic fuel management optimization using Ge-netic Algorithms and heuristic rules. Their Genetic Algorithm (GA)is used to develop many loading patterns using Haling Power Dis-tribution (HPD). By evaluating a very large number of core patternsusing an appropriate objective function, an optimum loading pat-tern may be found.

Recently (Geem et al., 2001) developed a new Harmony Search(HS) meta-heuristic algorithm that was conceptualized using themusical process of searching for a perfect state of harmony. Inthe HS algorithm, musicians try to find the musically pleasing har-mony by performing several improvisations. The quality of thegenerated harmonies is determined by aesthetic or artistic stan-dards. In the optimization process, global optimum solution maybe found by performing several iterations through different valuesof decision variables. The quality of the obtained solutions wasevaluated by Ayvaz, (2009). The harmony in music is analogousto the optimization solution vector, and musicians improvisationsare analogous to local and global search schemes in optimizationtechniques. The HS algorithm does not require initial values forthe decision variables. Furthermore, instead of a gradient search,the HS algorithm uses a stochastic random search that is basedon the harmony memory considering rate and the pitch adjustingrate. So that derivative information is unnecessary. Compared toearlier meta-heuristic optimization algorithms, the HS algorithmimposes fewer mathematical requirements and can be easilyadopted for various types of engineering optimization problems,(Lee and Geem, 2005). HS has been successfully applied to a widevariety of practical optimization problems such as pipe-networkdesign, structural optimization and combined heat and power eco-nomic dispatch problem (Chakraborty et al., 2009).

In this paper, we developed Harmony Search (HS) algorithm innuclear power plant field to optimize FAs arrangement of the reac-tor cores to satisfy an arbitrary objective function along constraints.In order to evaluate the performance of the method, the flatteningof relative power in a PWR core is considered as fitness functionalthough combination of other safety parameters can be taken asan objective function. For this purpose, we developed HarmonySearch Nodal Expansion Code (HSNEC2D) for the fuel managementoperation of the nuclear reactor core. In the code for treatment ofdiffusion equation a neutronic module which solves the two dimen-sional-multi group diffusion equation using second order of averagecurrent nodal expansion method (NEM), applying fourth degreeflux expansion (Putney, 1984), in order to decrease computationtime of the core calculation using one node per a FA is used.

The rest of the paper is organized in the following way; Section 2briefly outlines the HS algorithm and its components, Section 3gives description of a test case reactor and mapping reactor loadingpattern on the HS along mapping fitness function definition and inSection 4 the results are illustrated for two test cases and finallythe paper is concluded in Section 5.

2. Harmony Search optimization algorithm

In this section, a general view of Harmony Search algorithmis presented and its components and optimization steps arereviewed.

Fig. 1. Harmony of two notes with a frequency ratio of 2:3 and their waveform.

2.1. Music-based algorithms

Harmony Search is a music-inspired meta-heuristic optimiza-tion algorithm. It is inspired by the observation that the aim of mu-

sic is to search for a perfect state of harmony. This harmony inmusic is analogous to find the optimality in an optimization pro-cess. The search process in optimization can be compared to amusician’s improvisation process. This perfectly pleasing harmonyis determined by the audio aesthetic standard. The aesthetic qual-ity of a musical instrument is essentially determined by its pitch(or frequency), timbre (or sound quality), and amplitude (or loud-ness). Timbre is largely determined by the harmonic content whichis in turn determined by the waveforms or modulations of thesound signal. However, the generating harmonics will largely de-pend on the pitch or frequency range of the particular instrument.Different notes have different frequencies. For example, the note Aabove middle C (or standard concert A4) has a fundamental fre-quency of f0 = 440 Hz. As the speed of sound in dry air is aboutm = 331 + 0.6T m/s where T is the temperature in degrees Celsiusnear 0 �C. So at room temperature T = 20 �C, the A4 note has awavelength k ¼ mf0 � 0:7795 m. When we adjust the pitch, weare in fact trying to change the frequency. In music theory, pitchp is often represented as a numerical scale (a linear pitch space)using the following formula:

p ¼ 69þ 12log2f

440 Hz

� �;

or

f ¼ 440� 2ðp�69Þ=12

ð1Þ

which means that the A4 notes has a pitch number 69. In this scale,octaves correspond to size 12 and semitone corresponds to size 1.Furthermore, the ratio of frequencies of two notes which are an oc-tave apart is 2:1. Thus, the frequency of a note is doubled (orhalved) when it raised (or lowered) by an octave. For example, A2has a frequency of 110 Hz, while A5 has a frequency of 880 Hz.

The measurement of harmony when different pitches occurringsimultaneously, like any aesthetic quality, is somewhat subjective.However, it is possible to use some standard estimation for har-mony. The frequency ratio, pioneered by ancient Greek mathema-tician Pythagoras, is a good way for such estimations. For example,the octave with a ratio of 1:2 sounds pleasant when playing to-gether, so are the notes with a ratio of 2:3 which is illustrated inFig. 1. However, it is unlikely for any random notes such as thoseshown in Fig. 2 to produce a pleasant harmony (Yang, 2010).

2.2. The meta-heuristic Harmony Search algorithm

Like other meta-heuristic optimization algorithms, the mainphilosophy of the HS algorithm depends on a heuristic event. How-ever, HS does not get its philosophy from the natural processes; in-stead, it gets it from the musical improvisation which occurs whena group of musicians searches for a better state of harmony. Geemet al. (2001) first adapted this musical improvisation process intothe solution of engineering optimization problems. In this adapta-tion, each musician corresponds to a decision variable and possiblenotes in the musical instruments correspond to the possible valuesof the decision variables. When the musicians find the fantasticharmony from their memories, it means, a global optimum solu-tion is found through corresponding decision variables. Fig. 3 dem-onstrates the analogous relationship of the musical improvisationand optimization.

Fig. 2. Random music notes.

290 N. Poursalehi et al. / Annals of Nuclear Energy 53 (2013) 288–298

As can be seen from Fig. 3, each musician has several notes intheir memories. For example, if the first musician plays {Do} whilesecond and third musicians play {Sol} and {La} from their harmonymemories, {Do, Sol, La} makes a new harmony. If this harmony isbetter than the worst one in the harmony memory, it replacesthe worst one and this process is repeated until fantastic harmonyis found. On the other hand, in the optimization process, eachmusician is replaced with a decision variable and notes are re-placed with preferred values of decision variables. If the decisionvariables take {1.0}, {1.5}, and {1.7} from the harmony memory,a new solution vector {1.0,1.5,1.7} is obtained. If this solution vec-tor is better than the worst one stored in the harmony memory,this new solution vector replaces the worst one. This iterative pro-cess is repeated until the given termination criterion is satisfied(Geem, 2009).

2.2.1. Seeking harmony in music and engineering optimizationproblem

Similarly in engineering optimization, each decision variableinitially chooses any value within the possible range, togethermaking one solution vector. If all the values of decision variablesmake a good solution, that experience is stored in each variable’smemory and the possibility to make a good solution is also in-creased next time. When a musician improvises one pitch, he (orshe) has to follow any one of following three rules (Chakrabortyet al., 2009):

1. Playing any one pitch from his (or her) memory.2. Playing an adjacent pitch of one pitch from his (or her) memory.3. Playing totally random pitch from the possible range of pitches.

Similarly, when each decision variable chooses one value in theHS algorithm, it follows any one of three rules, i.e.

1. Choosing any one value from the HS memory (defined as mem-ory considerations).

2. Choosing an adjacent value of one value from the HS memory(defined as pitch adjustments).

Fig. 3. Analogy between musical im

3. Choosing totally random value from the possible range of val-ues (defined as randomization).

The three rules in HS algorithm are effectively directed usingtwo parameters, i.e. harmony memory considering rate (HMCR)and pitch adjusting rate (PAR), as stated later.

2.2.2. Basic elements of HS algorithmThe basic elements of HS scheme are:

� Harmony: Harmony is similar to the gene in GA. It is the set ofthe values of all the variables of the objective function.� Harmony memory (HM): the places where harmonies are

stored.� Harmony memory size (HMS): the number of places that HM

has the best harmony is stored in the 1st place and the rest har-monies are classified according to their performance. Definitionof HMS is an important part of the calibration of the model.� Maximum number of Iterations (MaxIter): it is the termination

criterion.

2.2.3. HS algorithm’s processThe HS algorithm consists of the following five steps (Lee and

Geem, 2005):

Step 1: Initializing the optimization problem and algorithmparameters.

Step 2: Initializing the harmony memory (HM).Step 3: Improvising a new harmony from the HM.Step 4: Updating the harmony memory.Step 5: Checking stopping criterion.

Step 1: Initializing the optimization problem and algorithmparameters

First, the optimization problem is specified as follows:

Minimize ðor MaximizeÞ f ð~xÞsubjected to xi 2 Xi; i ¼ 1; . . . ;N:

ð2Þ

where f ð~xÞ is a scalar objective function to be optimized;~x is the setof each design variable xi; Xi is the set of possible range of values foreach decision variable xi (continuous decision variable), that isxiL 6 Xi 6 xiU, where xiL and xiU are the lower and upper boundsfor each decision variable respectively, and N is the number of deci-sion variables. The HS algorithm parameters that are required tosolve the optimization problem are also specified in this step, i.e.

provisation and optimization.

Fig. 4. The Harmony Search algorithm optimization procedure.

Table 1Type and the number of FAs in the KWU PWR loading pattern.

Type 1.9 1.9 + BP 2.5 2.5 + BP 3.2 3.2 + BP

Number 11 1 2 9 4 4

Fig. 6. FAs position vector in the core.


the harmony memory size (HMS), harmony memory consideringrate (HMCR), pitch adjusting rate (PAR) and termination criterion(maximum number of searches). HMCR and PAR are parametersthat are used to improve the solution vector. Both are defined inStep 3.

Step 2: Initializing the harmony memory (HM)In Step 2, the ‘‘harmony memory’’ (HM) matrix, shown in Eq.

(3), is filled with randomly generated solution vectors and sortedby the values of the objective function, f ð~xÞ.

Fig. 5. Proposed KWU PWR loa

HM ¼

~x1

~x2

..

.

~xHMS

266664

377775: ð3Þ

Step 3: Improvising a new harmony from the HMIn this step, a new harmony vector~x0 ¼ ðx01; x02; x03; . . . ; x0NÞ is gen-

erated based on three rules: (1) memory consideration, (2) pitchadjustment, and (3) random selection. Generating a new harmonyis called ‘improvisation’ and the value of any decision variable x0i forthe new vector may be picked up from a corresponding value of achosen vector which is already existing in the current HM, i.e. from

ding pattern using HSNEC.

Fig. 7. Shekel’s Foxholes test values.

Table 2Results for Shekel’s Foxholes using HS optimization algorithm.

Experiment Fitness Searching number

Best results#1 499.002 80#2 499.002 119#3 499.002 123#4 499.002 141#5 499.002 149#6 499.002 152#7 499.002 166#8 499.002 228#9 499.002 273#10 499.002 287Average 499.002 171.8

Worst results#1 499.002 621#2 499.002 653#3 499.002 726#4 499.002 715#5 499.002 734#6 499.002 851#7 499.002 910#8 499.002 1121#9 499.002 1196#10 499.002 1555Average 499.002 908.2

All executions average 499.002 540


the set ~x11; . . . ;~xHMS

1

� �on the following way. At the beginning with

generating a random variable r and comparing it with a probabilityset value of HMCR, one decides to use HM matrix for decision vari-ables, x0i, or to put it away. The HMCR, which varies between 0 and1, is the rate of choosing one value from the previous values storedin the HM, while (1-HMCR) is the rate of randomly selecting a freshvalue from the possible range of values, i.e.

x0i xi 2 x1

i ; . . . ; xHMSi

� �with probability HMCR

xi 2 Xi with probability ð1-HMCRÞ

(ð4Þ

For example, an HMCR = 0.9 indicated that the HS algorithmwill choose the decision variable value from historically stored val-ues in the HM with a 90% probability or from the entire possiblerange with a 10% probability. An HMCR value of 1.0 is not recom-mended because it eliminates the possibility that the solution may

be improved by values not stored in the HM. This is similar to thereason why the genetic algorithm uses a mutation rate in the selec-tion process.

In the case of r 6 HMCR, one chooses randomly a vector fromHM and uses its corresponding value for x0i in our new harmonyvector~x0.

Furthermore, every component which is chosen by the memoryconsideration is further examined to determine whether it shouldbe pitch-adjusted. This operation uses the parameter PAR (which isthe rate of pitch adjustment) as follows for continues vari-ables:pitch adjusting decision for

x0i ¼x0i � randð0;1Þ � bw with probability PARx0i with probability ð1-PARÞ;

�ð5Þ

where bw is an arbitrary distance bandwidth (a scalar number) andrand() is a uniformly distributed random number between 0 and 1.Evidently Step 3 is responsible for generating new potential varia-tion in the algorithm. Thus, either the decision variable is perturbedwith a random number between 0 or bw with a probability PAR or itis left unchanged with probability (1-PAR).

For the case of r P HMCR, x0i is not elected from HM and is pickup randomly a fresh value from the possible range of values. TheHMCR and PAR parameters introduced in the Harmony Search helpthe algorithm find globally and locally improved solutions, respec-tively. Lee and Geem (2005) have recommended the HS parametervalues: HMCR range between 0.7 and 0.95; PAR values range be-tween 0.2 and 0.5; and HM values range between 10 and 50 to pro-duce good performance of the HS algorithm.

Aforementioned procedure is carried out to obtain all elementsof new harmony vector ~x0.

Step 4: Updating the harmony memoryIf the new harmony vector is better than the worst harmony in

the HM, judged in terms of the objective function value, the newharmony is included in the HM and the existing worst harmonyis excluded from the HM. This is actually the selection step ofthe algorithm where the objective function value is evaluated todetermine if the new variation should be included in the popula-tion, i.e. harmony memory.

Step 5: Checking stopping criterionIn Step 5, the computations are terminated when the termina-

tion criterion, the maximum number of samples or searching, issatisfied. If not, Steps 3 and 4 are repeated. The whole optimizationprocess is given in Fig. 4.

Fig. 8. Shekel’s Foxholes optimal value along the number of searching using HS algorithm.

Fig. 9. FAs relative power distribution of KWU designer loading pattern using HSNEC.

Table 3Fitness function for KWU PWR loading pattern optimization using HS algorithm.

Experiment HMS = 20 HMS = 40 HMS = 60

#1 0.5021 0.5065 0.5013#2 0.5082 0.5014 0.4995#3 0.5103 0.5070 0.5014#4 0.5034 0.5004 0.5034#5 0.5004 0.4995 0.5015#6 0.5028 0.5051 0.5028#7 0.5007 0.5014 0.5131#8 0.5120 0.5040 0.5000#9 0.5037 0.5130 0.5008


3. Description of a test case reactor

To demonstrate and evaluate the proposed method, BushehrPWR, old KWU design, is chosen as a test case. There are two fueltypes in this rector core, with burnable poison (BP) and withoutburnable poison. Each type has three different enrichments, 1.9%,2.5% and 3.2% (Norouzi et al., 2011) and their specifications are gi-ven in Table 1. It is found that the core has 6 different kinds of FA.KWU PWR core has 1/8 symmetric shape and 201 rectangular FA.The used core model and its representation in symmetry sectorof 45� are given in Fig. 5.

#10 0.4990 0.4995 0.5101

Best result 0.4990 0.4995 0.4995Worst result 0.5120 0.5130 0.5131Average 0.5043 0.5037 0.5034Std. dev. 0.0043 0.0043 0.0045The number of samples 12,000 15,000 18,000Time execution (min) 34 43 52

3.1. Mapping reactor loading pattern on the HS

There are 31 FA in one-eighth of the core; from Table 1 it isfound that they are classified in six kinds and are numbered from1 to 6. The name of each kind along its number in one-eighth of the

Fig. 10. KWU PWR fitness function along the number of searching using HSNEC (HMS = 20).



core is given in Table 1. For instance, from Table 1 it is revealed thatthere are 11 FA with 1.9% enrichment, first kind, in one-eighth ofthe core. Therefore, one needs to specify the position of 31 FA inthe sector. For every position in the sector, from the center of corein Fig. 5 to the left, a unique number from 1 to N is given in se-quence and they are put in a vector which is called the vector ofFA position or harmony vector, x0i. For every position in the har-mony vector, a kind of FA is specified. For instance, if the 21th po-sition in the harmony vector is occupied by 5 it means in thesector, Fig. 5, the position of 21 is allocated by 5th kind FA. There-fore, the numbers from 1 to 6 are distributed 31 times in a har-mony vector. The vector of FA positions, x0i, for our test case maybe considered as in Fig. 6 which is a 1 � N (N = N1 + N2 + . . . + N6)position vector and Ni is the number of each kind of FA (see Ta-ble 1). As illustrated, in the simple case for instance, the first N1positions in the vector is involved by first kind of FA, and the sec-ond N2 positions involved for the second type of FA and so forth.

For our test case reactor N = 31, N1 = 11, N2 = 1, N3 = 2, N4 = 9,N5 = 4, and N6 = 4. The scattering of FA kinds in the harmony vec-tor, loading pattern, may take a huge number of cases and in the HSalgorithm, our aim is to find a loading pattern, LP, which maximize(or minimize) our objective function. In the HS technique, wechoose randomly HMS loading pattern for construction of HM.

In a harmony vector, each element corresponds to the positionof a FA in the core. Therefore, harmony vector elements are notnon-integer and repeated numbers; these facts may be consideredas constrains. In the other word, in the HM generating step, eachelement of harmony vector takes integer and non-repeated num-ber from 1 to N. In the HS process of LP optimization, on the con-trary to process such as PSO (Meneses et al., 2009), the HStechnique is not deal with real generated numbers, then the decod-ing of real numbers to integers is not required.

Having established HM, loading patterns, one begins the pro-cess of obtaining new harmony vector, LP, based on the procedure


Table 4Comparison of fitness function and PPF of methods for KWU PWR loading pattern.

Method Ft PPF Average execution time (min)

Hopfield 0.737 1.26 –EICGA 0.528 1.23 120a

HS 0.499 1.216 29b

a PC CPU 2.4 GHz with 1 GB RAM.b PC CPU 2.4 GHz with 512 MB RAM.


described in the previous section and then evaluating the corre-sponding fitness function, f ð~xÞ.

3.2. Fitness function definition for fuel management optimization

One of the important keys in the reactor core design is selectionof optimized core loading pattern to satisfy safety considerations.

Fig. 13. FAs relative power distribution of Hop

Power peaking factor minimization is the fission interaction rateflatting by proper arrangement of FA in a core (Driscoll et al.,1990). In this project, flattening of FAs relative power distributionalong power peaking factor (PPF) minimization in a loading patternoptimization problem is chosen as a goal. In order to use HS algo-rithm for loading pattern optimization, the fitness function can bedefined as:

Ft ¼

Xn

i¼1

ðPi � 1Þ2; PPF < Prmax

1þ PPFPrmax

� ��Xn

i¼1

ðPi � 1Þ2; PPF P Prmax

8>>>><>>>>:

ð6Þ

where Pi and Prmax are the FA relative power and the maximum ofallowable FA relative power, respectively, which Prmax in this pro-ject is taken to be 1.3 and n is the number of FAs in the core. InEq. (6), the fitness function, Ft, is a kind of least square methodwhich provides a measure of deviation from power flattening.

field algorithm proposed loading pattern.

Fig. 14. FAs relative power distribution of EICGA algorithm proposed loading pattern.

Fig. 15. FAs relative power distribution of HS algorithm proposed loading pattern.


Indeed, whenever FA power is close to average power in the core,the relative power is approached to unity. Then, according to Eq.(6), Ft is a measure of power flatness. Loading pattern with less Ft

has more relative power flattening. In order to have effective searchprocess in the presence of a constraint, the fitness function isneeded to be carefully defined, which penalizes infeasible solutionswith appropriate severity. A simple form of fitness is expressed byconsidering an additional term to Eq. (6) whenever PPF is greaterthan Prmax. This means that when power peaking factor in a FA inthe core is higher than allowable limit and consequently, the con-sidered fitness function is increased. Then the accordant loadingpattern has lower chance to proceed further in computational pro-cedure and may be rejected in comparison to best results.

Loading pattern with fresh assemblies in the interior has thelowest leakage and largest burn up, but highest power peaking.Keeping a flat power distribution with low power peaking in a core,it is not unusual to allocate the positions of the periphery of the

core to higher enriched FA. This only causes decreasing searchspace. In the loading pattern optimization, we fixed higher en-riched FAs to periphery positions in the core. Although, this causesthe increasing of vessel flounce but one may consider the combina-tion of lowering flux in periphery and flattening as an objective(Norouzi et al., 2011).

4. Optimization results of HS algorithm

In order to present the ability of Harmony Search (HS) algo-rithm to solve optimization problems, two optimization tests areinvestigated. Prior to core loading pattern optimization, appliedHS algorithm is tested and validated against to known single globaloptimum problem, i.e. Shekel’s Foxholes problem. After validationof the algorithm, HS optimization algorithm is applied to KWUPWR loading pattern optimization and results are given.


4.1. Shekel’s Foxhole test function

Shekel’s Foxholes, introduced by (Shekel, 1971) and adapted formaximization by De Jong (1975), is a two-dimensional functionwith 25 peaks with different heights, ranging from 476.191 to499.002. The peaks are shown in Fig. 7 and the global optimumis located at (�32, �32). Shekel’s Foxholes function is defined by:

Zðx; yÞ ¼ 500� 1

0:002þP24

i¼01

1þiþðx�aðiÞÞ6þðy�bðiÞÞ6;

� 65:536 6 x; y 6 65:536; ð7Þ

where aðiÞ ¼ 16½ðimode 5Þ � 2� and bðiÞ ¼ 16½ði=5Þ � 2�.Due to high modality of Shekel’s Foxholes function, it is a diffi-

cult task to determine its points of local and global maximum andmaking it a great challenge for optimization algorithms. Table 2gives the results obtained by HS algorithm including 10 of bestand worst maximum values and used searching numbers. Fig. 8presents HS results for the successive independent 10 runs andshows characteristics between objective function and the numberof searching. In this test, HMS is set to 20. According to results, HSalgorithm obtained the optimal value in all tests. HS algorithm ob-tained successfully the optimal solution quickly and rejected localoptimal values (local maximums) and reached to global optimalpoint using different numbers of searching.

4.2. KWU PWR loading pattern optimization

Reactor core loading pattern optimization has been performedusing developed fuel management package, i.e. HSNEC2D for Bush-ehr NPP, old KWU design, which the published data is available. FArelative power distribution of KWU designer loading pattern usingNodal Expansion Code is presented in Fig. 9. Fig. 9 shows up-and-down of FAs relative powers of KWU designer loading pattern. Theobjective of loading pattern optimization for this problem is flat-tening of these relative powers according to minimization of de-fined fitness function, Eq. (6). HSNEC2D is used for variousharmony memory size (HMS) of 20, 40 and 60 to obtain the bestFAs. The developed nuetronic solver uses second order nodalexpansion method to solve the diffusion equation with two energygroups in octant symmetry of the core and one node per a FA withdimensions of 23 cm � 23 cm. Table 3 shows fitness function andcomputational time of 10 independent loading pattern optimiza-tion experiments using HSNEC2D with different HMSs. Accordingto Table 3, the disparities of various HMSs fitness values are trivialand all of these results are acceptable. Figs. 10–12 illustrate fitnessfunction along the number of searching. According to these figures,optimization experiments with different HMSs are converged welland reached to semi optimal fitness function but whatever HMS isgreater, the convergence is slower. In Table 4, the best obtained fit-ness function and PPF of HSNEC2D are compared with Hopfield(Sadighi et al., 2002) and EICGA (Norouzi et al., 2011) methods.The ability of HS algorithm to optimize the loading pattern is foundfrom the comparison of results. Reduction of computational timeusing HS algorithm and nodal expansion method (NEM) as a solveris considerable due to the merit of HS and nodal methods in whichuses coarse meshes in the calculation. However, results indicateHSNEC finds out fitness value associated with less PPF close toother published results with suitable computational time. The pro-posed FAs arrangement of HS algorithm and its FA relative powervalues in octant symmetry are given in Fig. 5. Moreover, the sche-matic view of relative power distributions of proposed loading pat-terns obtaining from Hopfield, EICGA and HS methods are alsoillustrated in Figs. 13–15, respectively.

At last, it should be mentioned that the HSNEC2D process can-not generate cross sections for fission fragments so the burn upcannot be carried out through the whole cycle.

5. Conclusions

In this paper the HS algorithm was mapped and implementedfor loading pattern optimization of PWR. The performance of HSshows that the algorithm is quiet promising and could be appliedto other optimization problems in nuclear reactor field. First testcase illustrates capability of HS to escape from local optimal. TheHS algorithm is flexible and very robust, at least for experimentsconsidered in this work. The great advantage of HS is significantgain in computational cost in comparison to competitive algo-rithms; as it is revealed from Table 4 the spent time for HS isapproximately a quarter of well known GA algorithm. Further-more, outcome of our experiments shows that in every experimentthe search approach to optimal value is quickly and on the averagethe final band width of search fitness values is narrow and veryclose to optimal value. Last but not least, HS is easy to implementand reliable.

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