Annals of Nuclear Energy 36 (2009) 1553–1559

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Annals of Nuclear Energy

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Advanced and flexible genetic algorithms for BWR fuel loading pattern optimization

Cecilia Martín-del-Campo a,*, Miguel-Ángel Palomera-Pérez b, Juan-Luis François a

a Departamento de Sistemas Energéticos, Facultad de Ingeniería, Universidad Nacional Autónoma de México, Paseo Cuauhnáhuac 8532, Jiutepec, Mor., 62550, Mexicob Instituto de Investigaciones en Matemáticas Aplicadas y Sistemas, Universidad Nacional Autónoma de México, Circuito Escolar, Ciudad Universitaria, 04510 DF, Mexico

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

Article history:Received 22 April 2009Received in revised form 19 July 2009Accepted 21 July 2009Available online 21 August 2009

0306-4549/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.anucene.2009.07.013

* Corresponding author. Tel.: +52 777 3194101; faxE-mail addresses: cmcm@fi-b.unam.mx (C. Martín

iimas.unam.mx (M.-Á. Palomera-Pérez), jlfl@fi-b.unam

This work proposes advances in the implementation of a flexible genetic algorithm (GA) for fuel loadingpattern optimization for Boiling Water Reactors (BWRs). In order to avoid specific implementations ofgenetic operators and to obtain a more flexible treatment, a binary representation of the solution wasimplemented; this representation had to take into account that a little change in the genotype must cor-respond to a little change in the phenotype. An identifier number is assigned to each assembly by meansof a Gray Code of 7 bits and the solution (the loading pattern) is represented by a binary chain of 777 bitsof length. Another important contribution is the use of a Fitness Function which includes a Heuristic Func-tion and an Objective Function. The Heuristic Function which is defined to give flexibility on the applicationof a set of positioning rules based on knowledge, and the Objective Function that contains all the param-eters which qualify the neutronic and thermal hydraulic performances of each loading pattern. Experi-mental results illustrating the effectiveness and flexibility of this optimization algorithm are presentedand discussed.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

In the core of the BWR studied in this work, there are 444 posi-tions in which fresh and burned assemblies are loaded to satisfy:(a) some heuristic restrictions, (b) the cycle energy production,(c) operational thermal limits and (d) reactivity constraints. Thestudy of different ways to solve these types of problems is highlyimportant in this day in age, because bigger cores are being usedin the advanced BWRs, thereby presenting a large combinatorialdimension. Several studies have described different genetic algo-rithm approaches, to solve the huge combinatorial problem relatedwith the loading pattern optimization for light water reactors,some of them being the following: the application of GAs to thepressurized water reactor In-Core fuel management optimizationby Poon and Parks (1993); the work of De chaine and Feltus(1995) for nuclear fuel management optimization using GAs alsofor the pressurized water reactor (PWR); the optimization of fuelreloading in BWRs of François and Lopez (1999), the flexible appli-cation of genetic algorithms to PWR reloading patterns optimiza-tion of Hongchun (2001); the development of Alim and Ivanov(2004), in which heuristic rules were embedded in GAs for PWRloading optimization; and the development of a BWR loading pat-tern design system based on modified genetic algorithms andknowledge, by Martín del Campo et al. (2004). The present work

ll rights reserved.

: +52 55 56161855.-del-Campo), mapp@uxmcc2.

.mx (J.-L. François).

is a continuation of this last paper in which GAs were used as anoptimization method to search for fuel loading patterns for the La-guna Verde Nuclear Power Plant Unit 1 (LVNPP-1) in Mexico; how-ever there are two main differences.

(1) The mathematical representation of each solution (loadingpattern) in the GA is different; the solution used in our previouswork had a direct representation which consisted of a list of alpha-numeric assembly identifications associated to the positions in thecore. This kind of representation was easy to apply, but it was nec-essary to build non-conventional crossover and mutation geneticoperators, based on heuristic rules, which were defined specificallyfor the case study using our own knowledge. It goes without sayingthat the genetic operators based on heuristics have little flexibilityand adaptability. Their efficiency depends strongly on the properapplication of the heuristic and, in the case of the application toloading patterns, it is necessary to redefine new crossover andmutation operators, particularly when a new region or a new fueltype is used. However, in order to avoid specific implementationsand to obtain a more flexible treatment, in the present work eachassembly is represented by a 7 bits Gray Code and the loading pat-tern is represented by a binary chain of 777 bits of length as will beexplained in Section 2.1.

(2) The use of a Fitness Function which includes: (a) the HeuristicFunction which quantifies the fulfillment of the heuristic rules toplace a variety of fresh and burned assemblies in different geomet-rical positions in the core and (b) the Objective Function which eval-uates the neutronic and thermalhydraulic parameters of theloading pattern. In our previous work the fitness function consisted

1554 C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559

only of the objective function which was evaluated when a solu-tion completely had satisfied a list of heuristic restrictions, withoutany flexibility.

Another minor difference is that in the previous study (Martíndel Campo et al., 2004) the optimization method was applied forthe fifth cycle of the LVNPP-1 using only one fresh fuel type, andthis time it was applied to the 10th cycle of the same unit includingtwo different types of fresh fuel.

The problem to be solved in the loading pattern task was to ob-tain the ‘‘best” distribution of fuel assemblies (FAs) in the core. ABWR, as found in Laguna Verde, has 444. Some of these FAs arefresh and others have been present in one or more cycles of oper-ation. To reduce the complexity of the loading pattern design andthe maneuvering of control rods during the cycle operation, a heu-ristic rule in BWR fuel management is to assign quarter core sym-metry, thus giving only 111 different positions to allocate the FAs.

2. Genetic algorithm approach

In the present work a traditional GA has been implemented aswas presented by Goldberg (1989).

The GA can be summarized as follows:

1. Encoding of the problem solution in a numerical representation.2. Creation of an initial population of individuals.3. Classification of the individuals in terms of their fitness.4. Selection of individuals that will mate according to their share

in the population global fitness.5. Genome crossovers and mutations that modify the composition

of the descendants.6. If the number of generations required is met, it stops, if not it

goes back to number three.

GAs have very good characteristics for solving complex combi-natorial problems, they do not require any functional derivativeinformation; they cover the search space in a relatively fast man-ner and work well with reduced search spaces. However, there isno proof that the optimum has been found.

This GA implementation uses a constant size population N; eachindividual consists of a binary chain of length l, and the classicalcrossover and mutation genetic operators are applied directly tothe genotype. This GA was implemented as follows: the first pop-ulation is randomly generated following a procedure described inSection 2.3. Then each individual is evaluated by means of the fit-ness function. After this, the algorithm creates new populations byrecombining the ‘‘best evaluated” individuals using the crossoverand mutation operators. Because all the application cases are basedon a quarter core symmetry, the optimization process was simpli-fied by using only 111 positions in the core. This GA was linked tothe 3-dimensional steady state Core Master PRESTO (CM-PRESTO,1993) code to simulate the loading patterns and to obtain theterms of the objective function.

2.1. Binary representation of the solution

In the previous work (Martín del Campo et al., 2004) the solu-tion was directly represented by a list of alphanumeric assemblyidentifications associated to the positions in the core. Using thiskind of representation, it was necessary to build non-conventionalcrossover and mutation genetic operators, based on heuristic ruleswhich were specifically defined for the case study using expertknowledge. Therefore, their effectiveness depends strongly on theproper application of the heuristic. It is a given that, the geneticoperators based on heuristics have little flexibility and adaptabilityto new configurations. If it is necessary to define a new region or a

new fuel type, as in the particular case of the application to loadingpatterns, it is necessary to build new crossover and mutation oper-ators. In order to avoid specific implementations and to obtain ageneral more flexible treatment, it was decided to use a binary rep-resentation of the solution. Such representation has to take into ac-count that, a little change in the genotype must correspond to alittle change in the phenotype, where the genotype is the binarysolution representation, and the phenotype is the direct represen-tation as is simulated using the CM-PRESTO.

For this simulator each fuel assembly can be represented bymeans of an alphanumeric chain for example #�###, where thefirst # is the cycle number indicating when the assembly wasintroduced into the core as fresh fuel; the � is a letter (A, B, C,. . .)that identifies the fuel type; and the last three ### identify eachassembly, that allows one to follow its life history in the reactor.Examples are 8P125, 9Q446.

In the direct representation of the solution, the loading patternis an array of 111 assembly alphanumeric identifications, wherethe assembly location in the array is related to the position inthe quarter core. In this work the classical crossover and mutationoperators were used and therefore a binary representation for thesolution was required. It should be remembered that a mutationimplies that a very little change in the genotype corresponds to avery little change in the phenotype. In reference to loading pat-terns, a mutation in one position implies that the current assemblymust be changed by another with ‘‘similar” characteristics. How-ever, what is the meaning of ‘‘similar”?

Definition. Two fuel assemblies are ‘‘similar” if they were intro-duced in the same cycle and are the same type. In addition, thedifference between their cumulated burnup should be close tozero.

This implies that similar fresh assemblies are identical. But, howsimilar are two burned assemblies? The answer to this question re-quires that we measure just how much their cumulated burnup isdifferent.

If B1 and B2 represent the burnup of two similar A1 and A2

assemblies, the absolute value of their difference |B1 � B2| deter-mines how much their burnup is different; the smaller this param-eter is, the more similar A1 is to A2. So the relation: 1

jB1�B2 jshows us

how similar the two assemblies are. For the purpose of this workthis relation is called the ‘‘Similarity Degree”.

The similarity degree is used to set up the binary representationas follows:

(1) The similar assemblies are clustered.(2) Each cluster is ranked in decreasing order based on the

assembly ‘‘similarity degree”, so the assemblies with highsimilarity are placed together.

(3) The clusters are grouped in a list based on their cycle num-ber, from the current cycle to the first cycle.

(4) A number from 1 to 111 is assigned to each assembly andthis number is coded by means of a 7 bits Gray Code and thiscode is the binary representation of the assembly. The solu-tion (the loading pattern) is represented by a binary chain of777 bits of length and there are 2777 candidate solutions inthe search space.

2.2. Heuristic restrictions for the fuel assembly’s positioning

Experience has shown that the use of certain heuristic restric-tions in the positioning of the different assemblies in the regionsof the core (see Fig. 1) avoids the evaluation of a huge quantityof solutions with a very low probability for being the optimal solu-tion. For fuel management purposes we can define three regions inthe core: (a) the Peripheral (P) positions, (b) the Control Cell Core

Fig. 1. Peripheral (P), In-Core (InC) and Control Cell Core (CCC) positions in thequarter core.

C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559 1555

(CCC) positions, and (c) the In-Core (InC) positions. The quartercore has 17 P, 24 CCC and 70 InC positions.

We can also classify the fuel assemblies into six types: (i) fresh(one or more types of fresh), (ii) low burnup (one or two cycles),(iii) medium burnup (two or three cycles), (iv) high burnup (threeor more cycles), (v) assemblies in P positions in previous cycles,and (vi) assemblies in the CCC positions in previous cycles.

In order to obtain feasible loading patterns we defined five heu-ristic restrictions Ri (see Table 1) that correlate the six types of FAsto the three regions in the core.

The heuristic restrictions in Table 1 permit one to discard non-feasible loadings patterns before their neutronic–thermalhydraulicsimulation.

2.3. First population

The following procedure was used to build each loading patternfor the first generation:

(1) Each of the 17 P positions is filled with assemblies selectedrandomly between the 25 assemblies with the highest bur-nup to satisfy R2. If the position has a symmetrical couple,a similar assembly is directly placed at its symmetrical posi-tion so as to satisfy R5. In order to comply with the R1, theseassemblies must not be selected in the future.

(2) The 24 CCC positions are filled by randomly choosing assem-blies with burnup that is higher than the average, and theselected assemblies must also satisfy R3 and R4. Again, inthe case of symmetrical positions, similar assemblies aredirectly selected. All the assemblies chosen are also dis-carded from the set of assemblies available for the loadingpattern.

Table 1Heuristic restrictions for the fuel assembly’s positioning.

Heuristicrestriction

Description

R1 The loading pattern is composed of different assemblies. Thismeans each available assembly is used in only one coreposition

R2 The assemblies with the highest burnup (with three or morecycles) must be located in the P positions

R3 Fresh assemblies cannot be placed in the predefined CCCpositions

R4 CCC positions do not accept fuel assemblies which have beenin CCC positions in any previous loading cycle

R5 Fuel assemblies must be distributed to form an 1/8 coresymmetry, meaning that similar or identical assemblies mustbe located in symmetrical positions. The couples: (1, 111),(2, 100), (8, 110), (80, 92) are examples of symmetricalpositions. See Fig. 1 to identify the couples in the 1/8symmetrical positions

(3) The rest of the assemblies in the InC positions are randomlyassigned taking into account that symmetrical positionsshould always contain similar assemblies.

N different loading patterns are created using the previous pro-cess and they become the first generation. This generation is eval-uated using the fitness function and the individuals with goodqualities have a higher probability of being selected for crossover.

2.4. Mathematical model of the fitness function

In the loading pattern design process, it is very important to de-fine a fitness function that takes into account the most importantparameters affecting the operation of the power plant. The fitnessfunction has a double objective: (a) to look for loading patternsthat fill the restrictions Ri listed in Table 1. The heuristic functionwas defined to comply with this and (b) to find the best loadingpattern in terms of its neutronic–thermohydraulic quality. Theobjective function was used to evaluate this quality.

2.4.1. The heuristic functionThe heuristic function is used to look for loading patterns which

better satisfy the restrictions listed in Table 1 by using the follow-ing equation:

HðxÞ ¼X5

i¼1

riNRiðxÞ ð1Þ

In this equation NRi(x) is the number of times that the individ-ual x misses the restriction Ri (explained in Table 1), and ri is aweight for this restriction. Note that when an individual fulfillsall the restrictions, H(x) is equal to zero. However, experience hasshown that to find an individual that absolutely satisfies all therestrictions (listed in Table 1) is very improbable. On the otherhand, some restrictions can be unmet without repercussion inthe neutronic behavior. Therefore a threshold value is introducedto give flexibility for finding individuals which partially satisfythe restrictions. For example, to tolerate 10 asymmetries (missesof the restriction R5), the following operations are made: to givea threshold u equal to �11, the weight r5 equal to �1 and all therest of the weights are equal to �11. A similar treatment can be ap-plied to give flexibility to any other restriction.

2.4.2. The objective functionIn order to determine the neutronic–thermohydraulic quality,

the loading pattern is simulated with the CM-PRESTO code, whichcalculates the values of the parameters in Table 2. The first sixparameters in Table 2 are obtained from a Haling End of Cycle(EOC) simulation using quarter core symmetry condition. How-ever, the Hot Excess of Reactivity (HER) is obtained from a hot fullpower calculation at the Beginning of Cycle (BOC). The ShutdownMargin (SDM) is obtained from full core calculations at cold zeropower conditions which are also found at BOC. Most of theseparameters have a quota (maximum, minimum, or both) whichmust be satisfied as constraint (see Table 2).

Using these parameters, the objective function O is calculatedfor each solution x with the Eq. (2), and the optimization searchto maximize O.

OðxÞ ¼ w1EnergyðxÞ þw2DMLHGRðxÞ þw3DOXMPGRðxÞþw4DMRNPðxÞ þw5DMCPR þw6DMFABðxÞþw7HERðxÞ þw8DSDMðxÞ ð2Þ

where

Table 2Parameters used in the objective function.

j Parameter Description Unit Constraint

1 Energy Cycle length MWd/t2 MLHGR Maximum linear heat generation rate W/cm <MLHGRmax

3 XMPGR Fraction of the limiting average lineal generation rate <XMPGRmax

4 MRNP Maximum relative nodal power <MRNPmax

5 MCPR Maximum critical power ratio >MCPRmin

6 MFAB Maximum fuel assembly burnup MWd/t <MFABmax

7 HER Hot Excess Reactivity at BOC %Dk/k >HERmin

<HERmax

8 SDM Shutdown Margin at BOC %Dk/k >SDMmin

1556 C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559

DMLHGRðxÞ ¼ MLHGRmax �MLHGRðxÞDXMPGRðxÞ ¼ XMPGRmax � XMPGRðxÞDMRNPðxÞ ¼MRNPmax �MRNPðxÞDMCPRðxÞ ¼MCPRðxÞ �MCPRmin

DMFABðxÞ ¼MFABmax �MFABðxÞHERðxÞ ¼ ðKEFFðxÞ � KEFFcritÞ=KEFFcrit

DSDM ¼ SDMðxÞ � SDMmin

In this study the KEFFcrit is equal to 1 and HER(x) must be be-tween HERmin and HERmax as a constraint. The weighing factorsw1 to w8 are obtained by means of experimentation with calcula-tions of the objective function for hundreds of loading patterns.When the value of the physical parameter j satisfies its correspond-ing constraint, the wj is equal to zero. This makes it possible not topenalize loading patterns that got good parameters.

2.4.3. The fitness functionFinally the fitness function F (x) is described by the following

equation:

FðxÞ ¼HðxÞ . . . if HðxÞ < u

OðxÞ . . . other case

�ð3Þ

where the term u is used as threshold, H(x) is the heuristic function,described by Eq. (1), which evaluates the fulfillment of the heuristicrestrictions by the solution x, and O(x) is the objective function, de-scribed in Eq. (2), which evaluates the neutronic–thermalhydraulicparameters of the solution x.

In order to evaluate the fitness function F(x), each individual x istransformed from its binary representation (genotype) to its pheno-type description as required by the CM-PRESTO 3-dimensional sim-

0

10

20

30

1 21 41 61 81

Generati

Indi

vidu

als

wit

h H

u

Fig. 2. Number of individuals that s

ulator. Therefore, the values of NRi(x) for each restriction Ri arecounted and H(x) is calculated using Eq. (1). If the value of H(x) sat-isfies the threshold u, with H(x) P u the individual x is simulatedwith the CM-PRESTO code to obtain all the parameters describedin Table 2, and finally the objective function O(x) is calculated usingEq. (2). This technique permits one to easily eliminate the loadingpatterns with non-feasible configurations thereby considerablyreducing the computing time. This is because a simulation withCM-PRESTO takes much more computing time than several ‘‘ifs”.In addition, the more the number of generations increases, the morethe algorithm ‘‘learns” to obtain solutions that satisfy the heuristicrestrictions Ri. Examples of non-feasible configurations are loadingpatterns with a large quantity of asymmetries, or with fresh assem-blies in P positions or in CCC positions, etc.

Fig. 2 shows an example of a run case (Study Case 1, describedlater); the graph displays the number of new individuals withH P u created in each generation of the optimization process. Inthis case there are 30 individuals in the population, and it wasfound that at the beginning of the process, or generations less than20, there are really very few new individuals created by the geneticoperators that satisfy H P u; however, when the generation num-ber increases there are more individuals that do satisfy H P u.

Fig. 2 additionally shows that on average there are nine newloading patterns in each generation that satisfy the H P u. This isa low number however, and this procedure guaranties that onlythe loading patterns which satisfy the restrictions fairly well willbe evaluated by the CM-PRESTO simulator, which requires a lotof computing time. Using this procedure it is possible to optimizethe fitness function by the exploration of a large number of individ-uals with the heuristic function H, where only the few which sat-isfy the threshold, are evaluated with the expensive objectivefunction O.

101 121 141 161 181

on Number

atisfy H P u in each generation.

Table 3Procedure to select the individuals that become parents to be crossed.

Step Action

1 A number between 1 and N is assigned to each one of theindividuals of the population

2 A random permutation of the N individuals is generated3 Pairs of individuals are formed by means of grouping two

consecutive individuals4 The individual with fitness better than its pair becomes a member

of the group of parents5 Steps 1–4 are repeated until the number of parents searched for, is

achieved

C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559 1557

For Case Study 1, Fig. 3 shows the evolution of the average valueof the objective function of all the individuals in the population foreach generation; the values for the best and the worst individualsare also shown.

2.5. Genetic operators

In this section we describe the genetic operators as they wereimplemented in the genetic algorithm.

2.5.1. SelectionThe selection of individuals to be crossed is done by means of a

tournament algorithm. In order to select the individuals that be-come pairs of parents which will be crossed, the algorithm followsthe steps shown in Table 3.

The pairs to be crossed are formed by two different individualsof the group of parents, that means, no individual crosses with it-self. This implies that in the population each individual is unique.The expected number of pairs of parents in which an individual xparticipates is determined by means of Eq. (4). It is important tonote that the best individual of the population will always reap-pear as many times as the procedure is repeated. As shown inthe examples of the study the best individual reappears in eachof the four pairs, whereas the worst appears in none.

EpðxÞ ¼ n 1� Nf ðxÞN � 1

� �ð4Þ

where Ep(x) is the expected number of pairs of parents in which theindividual x is involved, n is the number of times the procedure isrepeated, Nf (x) is the number of individuals with fitness functionbetter than F(x) and N is the number of individuals in thepopulation.

2.5.2. CrossoverThe crossover operator generates new individuals by recombin-

ing parts of two parents. The operator was applied to the genotype.The crossover consists of interchanging the sub-chains of the par-ents’ genotype. The crossover can be one point or multipoint. Forone point, point pi is randomly selected between bit 2 and bit776 and two crossover offspring are generated by combining thesub-chains. In this work the number of the crossover points is anexecution time dependent variable.

2.5.3. MutationAfter the crossover, the mutation operator is applied to the off-

spring. It consists of making a change in only 1 bit. The probability

26500

28500

30500

32500

34500

0 50Generat

Obj

ecti

ve F

unct

ion

Fig. 3. Objective function v

that one mutation occurs is function of the similarity between theindividuals in the population, and it is calculated by means of thefollowing equation:

Pm ¼ 1� Fmax � Fmin

Fmaxð5Þ

where Fmax and Fmin are the values of the fitness function of the bestand the worst individuals in the population.

Finally, the individuals obtained by using this method are eval-uated with the fitness function, and when H P u the objectivefunction is evaluated. These individuals are added to the currentpopulation and only the N best will be included in the nextgeneration.

The procedure is repeated until the number of generations re-quired is reached.

3. Experimental results

In order to demonstrate the advantages obtained with the fit-ness function and the application of the genetic algorithm de-scribed in the previous section, two case studies of loadingpatterns are presented here.

The reload was designed for a typical BWR core, loaded with444 fuel assemblies, as found in the core of the reactors of the La-guna Verde Nuclear Power Plant (LVNPP). The goal is to find theoptimal loading pattern which satisfies the reactivity and thermallimits maximizing the energy obtained during the cycle. The twocases contain 28 fresh fuel assemblies (FA) in the quarter core;the rest are 83 partially burned assemblies. Two types of freshFA, types AA and AB, were used in the loading pattern with thesame average 235U enrichment, but assembly AA contains higherconcentration of gadolinia than AB. Case 1 contains 10 AA and 18AB and Case 2 contains 12 AA and 16 AB. Case 2 contains higher con-centration of gadolinia than Case 1. The parameters of the genetic

100 150 200ion Number

BestAverageWorst

s. generation number.

1558 C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559

algorithm are: 30 individuals in the population and 200generations.

Table 4 shows the limiting value for each parameter and theweighting factors used in the objective function; these factors wereobtained by evaluating hundreds of loading patterns.

Several runs were executed to investigate the effectiveness ofthis algorithm; in Fig. 4 the evolution of the fitness function is pre-sented for the best solution in the population (30 individuals) ofeach generation (200 generations in total) for the two pairs ofcases. This means that four cases are reported, Case 1 and Case 10

used identical data but they have different series of random num-bers during the run. Also in Case 20 the same data is used as in Case2 but other series of random numbers were used. Table 5 showsthe neutronic and thermalhydraulic parameters for the ‘‘best” indi-vidual obtained in each case study. These optimization resultswere compared with the parameters of an actual loading patternof LVNPP-1, called ‘‘Actual Case” in Table 5. It can be seen thatthe four optimization cases produced more energy than the ‘‘Actual

Table 4Constraints and weighting factors.

j Parameter Limit value Units wj

1 Energy None MWd/t 2.752 MLHGRmax 400 W/cm 553 XMPGRmax 0.83 55004 MRNPmax 2.25 7005 MCPRmin 1.5 606 MFABmax 55,000 MWd/t 57 HERmin 1.5 %Dk/k 650

HERmax 2.5 %Dk/k8 SDMmin 1.5 %Dk/k 2100

31000

31500

32000

32500

33000

33500

34000

34500

35000

35500

0 50

Generat

Fit

ness

Fun

ctio

n

Case 1

Fig. 4. Evolution of the fitness function vs. generation numbe

Table 5Neutronic and thermalhydraulic parameters for the ‘‘best” individual in the 200th genera

Energy, MWd/t LHGR, W/cm XMPGR MR

Case 1 12,974 283.8 0.781 2.08Case 10 12,641 282.0 0.878 2.07Case 2 12,879 286.2 0.767 2.10Case 20 12,635 291.7 0.830 2.14Actual Case 11,932 253.5 0.806 1.84

Case” and this was the objective. In general, the thermal hydraulicparameters for the optimization cases are closer to the limiting val-ues than those of the ‘‘Actual Case”; nevertheless these limits werealways satisfied. The reactivity parameters in the ‘‘Actual Case”have more conservative values concerning the reactor safety, butin the four optimization cases the SDM and the HER were also sat-isfied. In summary all the constraints were satisfied and the energywas ‘‘maximized”.

In each of the four cases the obtained ‘‘best” solution satisfiedthe neutronic–thermalhydraulic constraints of Table 4. Cases 1and 10 obtained lower HER than Cases 2 and 20 and this is explainedby the higher concentration of gadolinia in the fresh FAs found inthe loading patterns of Cases 2 and 20.

Case 1 found the loading pattern with the highest fitness func-tion followed by Case 2, then Case 10 and then Case 20. We can seethat this algorithm, as it has been implemented, is capable of opti-mizing loading patterns; however, there is a dependence on theseries of random numbers during the run; making it necessary todo several runs to obtain a satisfactory search. However, the differ-ence in the Energy Cycle between Case 1 and Case 10 is only 2.6%and for Case 2 and Case 20 is only 1.9%. Another observation is thatall the individuals in the 200th generation are very similar and allsatisfy the constraints of Table 4.

The effectiveness of the algorithm can be explained by means ofFig. 5 in which, for the Case Study 1, the objective function of eachindividual that satisfied H P u was graphed (see Fig. 5). In sum-mary, in the 200 generations, a total of 5889 individuals were cre-ated by means of the genetic operators. They were evaluated withthe heuristic function (see Fig. 6) and we found a total of 4288 withH < u, giving only a total of 1601 that satisfy H P u that were con-sequently simulated with CM-PRESTO. This means that on average

100 150 200

ion Number

Case 2 Case 1' Case 2'

r for the best individual during the optimization process.

tion.

NP MCPR MFAB, MWd/t HER, %Dk/k SDM, %Dk/k

9 1.621 45,870 2.37 1.5171 1.654 46,760 2.30 1.7037 1.595 44,700 2.19 1.6907 1.617 46,380 2.10 1.5080 1.795 44,510 1.02 2.121

29500

30500

31500

32500

33500

34500

35500

0 20 40 60 80 100 120 140 160 180 200Generation Number

Obj

ecti

ve F

unct

ion

Fig. 5. Objective function vs. generation number for the offspring which satisfied H P u.

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

00 20 40 60 80 100 120 140 160 180 200

Generation Number

Heu

rist

ic F

unct

ion

Fig. 6. Heuristic function vs. generation number for the offspring which did not comply with H P u.

C. Martín-del-Campo et al. / Annals of Nuclear Energy 36 (2009) 1553–1559 1559

there were nine offspring individuals that compete with the 30individuals of the current population, and only the 30 best willform the next generation.

4. Conclusions

This work puts forth advances in the implementation of a ge-netic algorithm approach for fuel loading pattern optimization ofBoiling Water Reactors. In order to avoid any specific implementa-tion and to obtain a more flexible treatment, a binary representa-tion of the solution was used. An identifier number was assignedto each assembly by means of a Gray Code of 7 bits and the solu-tion was represented by a binary chain of 777 bits of length. Thiskind of representation allows for a little change in the genotypeto correspond to a little change in the phenotype. Another impor-tant contribution was the use of a fitness function which includes aheuristic function and an objective function. The heuristic functioncontains a set of positioning rules based on prior knowledge andthe objective function contains all the parameters which qualifythe neutronic and thermalhydraulic performance of each solution.A threshold value is used to give flexibility to the algorithm forfinding individuals which partially satisfy the heuristic restrictions,and it was possible to optimize the objective function by the explo-ration of a large number of individuals with the heuristic function

H, where only the few that satisfy the threshold are evaluated withthe expensive objective function O. This treatment gives flexibilityand avoids defining specific heuristic rules when there are changesin the number of types of fresh fuels, or in the number of differentcycles in which the burned fuels were introduced in the past, orwhen new regions in the reactor core are defined.

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