a pwr reload optimisation code (xcore) using artificial neural networks and genetic algorithms

A PWR reload optimisation code (XCore) usingartificial neural networks and genetic algorithms

Adem Erdogana,* and Melih Geckinlib

aTAEK, Cekmece Nuclear Research and Training Center, P.K. 1 Havaalani, 34831-Istanbul, TurkeybIstanbul Technical University, Institute of Nuclear Energy, Maslak, 80626-Istanbul, Turkey

Received 29 January 2002; accepted 28 March 2002

Abstract

A computer program package has been developed, which supports the in-core fuel man-agement activities for pressurized water reactors. The package generates and recommends anoptimum-loading pattern to ensure safe and efficient reactor operation. The search for an

optimum fuel-loading pattern has been conducted by predicting several core parameters suchas the power distribution by means of an artificial neural network. This reduces the calcu-lation time and makes it possible to analyse more loading patterns in the same time interval

by increasing the probability of finding a desired optimum. A genetic algorithm method hasbeen implemented and used to automate the loading pattern generation. The code has beentested using the data from the PWR Almaraz Nuclear Power Station in Spain. # 2002Elsevier Science Ltd. All rights reserved.

1. Introduction

In spite of the apparent short-term decline in the commercial interest in nuclearpower, in-core fuel management continues to be among the main concerns of thecurrent nuclear power plant operators. The optimisation of fuel utilization increasesthe cost effectiveness of the overall plant operation. Therefore, nuclear engineersseek better ways to optimise in-core fuel management activities.As an optimisation problem, in-core fuel management includes safety-related

limitations as constraints while the cycle length is externally given in the planned

Annals of Nuclear Energy 30 (2003) 35–53

www.elsevier.com/locate/anucene

0306-4549/03/$ - see front matter # 2002 Elsevier Science Ltd. All rights reserved.

PI I : S0306-4549(02 )00041 -5

* Corresponding author.

E-mail address: [email protected] (A. Erdogan).

power demand. Some of the safety-related factors are the hot channel factor, mod-erator temperature coefficient and maximum allowable fuel assembly burnup. Inmost cases, it may not be possible to deal with economic parameters directly toreduce the cost. Thus, direct optimisation of economic parameters is usuallyreplaced by the optimisation of some other quantity related to reactor physics, forexample the uranium utilisation or cycle length.The optimum solution is based on the following parameters: (a) fresh fuel enrich-

ment, (b) number of fresh fuel assemblies to be inserted in the core, (c) loading pat-tern, and (d) burnable absorbers present in the assembly. Some of these parametersmay be fixed, for example the fuel enrichment. Normally, in most cases, factorsother than ‘‘loading pattern’’ are determined before the final optimisation is done. Inthis case, the problem is reduced to finding the optimum loading pattern (LP).The optimisation is performed stepwise. In the first step, an educated guess is

made on the number of fuel assembly types characterized by their enrichment andburnable poison content. Secondly, a LP is constructed using this proposed fuelassembly inventory. Then the power distribution is calculated for the given reactorcore configuration. These steps are repeated until the power distribution complieswith the constraints.The optimum LP should allow for a reasonably long cycle length without com-

promising the integrity of the reactor core components. During this iterative pro-cedure, the LP for each cycle influences the succeeding ones. These multi-cycle effectshave to be taken into account by performing the optimisation over several cycles.There is an important difference between a usual analytical optimisation problem

and an in-core fuel management optimisation. In the former case, the calculation ofthe objective function is performed using a simple formula. However, in our case theevaluation of the objective function in discrete-multivariable search space involveswhole reactor calculations based on the solutions of complicated equations such asthe neutron transport or diffusion, which is rather time consuming.In the present work, a computer code XCore has been developed. The code makes

use of computational intelligence techniques namely artificial neural networks andgenetic algorithms to help pressurized water reactor (PWR) in-core fuel manage-ment activities. The primary goal is to find optimum LPs that satisfy the reactorsafety requirements, and minimise the operating cost by reducing the initial amountof fuel charged to the core and by increasing burnup.The organisation of the paper is as follows: in Section 2 an artificial network to

predict the core power distribution together with its effective multiplication factor isgiven, in Section 3 an optimisation based on genetic algorithm is considered andvalidated, and in Section 4 the ideas developed in the previous sections are imple-mented in XCore code which offers an interactive tool to the core designer.

2. Prediction of power distribution and keff by a neural network

Artificial Neural Networks (ANN) have been successfully applied to predict var-ious parameters in nuclear fuel management. Kim et al. (1993) used an ANN to find

36 A. Erdogan, M. Geccckinli / Annals of Nuclear Energy 30 (2003) 35–53

optimal LPs. They predicted the power fraction and the effective multiplicationfactor (keff) with a maximum error of �6 and �0.3%, respectively. Their model wasbased on a simple core with three different assembly types, which used two networksto estimate the power fraction and the keff separately. Lysenko et al. (1999) used adynamic node architecture network to predict the keff with errors lower than 4.3%for their set test range. Their network estimated keff only. The method used query-based adaptation of training sets to reduce the network error. Sadighi et al. (2002)mapped the fuel management problem—with attributes similar to travelling sales-person problem—onto a Hopfield neural network accompanied by the simulatedannealing method which converged to a LP that complies best with their flat-fluxconstraint. They reported to have achieved a core LP with power peaking factorlower than that suggested by its commercial vendor.In our work, a feed-forward multi-layer neural network with a single hidden layer

has been used to estimate the core parameters (Erdogan, 2001). The use of a singlenetwork for estimating both the power fraction and keff results in relatively highererrors, but spares the time required to train two networks. Actually, higher errors donot compromise the output, since the main purpose is to develop a fast and roughestimator that provides the genetic algorithm parameters that are required to rankthe candidate solutions.The power distribution and keff, which are required during the search for a sui-

table LP are the parameters predicted by the ANN. The encoded fuel assembly dis-tributions in the core are mapped into vectors to be used as input to train thenetwork. The Almaraz NPP core configuration and data has been used to demon-strate the technique (Ahnert et al., 1990).Almaraz NPP is a pressurized water reactor (PWR) having 157 fuel assemblies in

its core. Seven different assembly types are defined with respect to their uraniumenrichment and burnable absorber contents. A core LP is constructed by selecting asuitable configuration from these assemblies available, which are presented inTable 1.Only 26 assembly locations need to be considered for the analysis, as the

nuclear reactor core possesses one-eighth symmetry. Fig. 1 illustrates an example

Table 1

Fuel assemblies defined for Almaraz NPP

Fuel

assembly

ID

Inventory

(quantity)

Enrichment

(%)

Burnable

absorber

rods

k1 Network input

value (raw)

xFAi ; i ¼ 1; 2; :::7:

120 2 2.6 20 0.959 0.1

116 5 2.6 16 0.988 0.2

112 1 2.6 12 1.016 0.3

316 1 3.1 16 1.046 0.4

312 1 3.1 12 1.054 0.5

121 11 2.1 – 1.073 0.6

131 5 3.1 – 1.174 0.7

A. Erdogan, M. Geccckinli / Annals of Nuclear Energy 30 (2003) 35–53 37

LP. Calculations are performed in two-dimensional (2D) geometry using the one-eighth core symmetry.The neural network consists of 26 input nodes plus a bias node, one for each

assembly location and 29 output nodes, one for each power fraction generated in thecorresponding fuel assembly plus three nodes for the (keff) of the reactor. The keff isrepresented by three output nodes to introduce redundancy in anticipation of stres-sing the importance of this parameter and hence to reduce the noise in the output.The final keff is the average of the values of those three particular output nodes. Thenumber of hidden layer nodes employed is 250, which has been determined bynumerical experimentation.The reactor core was modelled with the MCRAC-LEOPARD in-core fuel man-

agement package (Petrovic, 1990) to obtain the data required for the training of theneural network. Power distribution and keff’s have been calculated for 2000 ran-domly generated LPs. A set of 1000 patterns is evenly partitioned into disjoint net-work model training and validation subsets. The remaining 1000 LPs, named as thetest set, is put aside for the final evaluation of the best network architecture con-cerning its generalization capabilities (Haykin, 1999).For each of the 2000 training patterns, which were generated by the in-core fuel

management package, the following data are used to train the ANN: (a) locations ofthe fuel assemblies in the reactor core configuration, which constitutes the elements

Fig. 1. An example LP for Almaraz PWR NPP.


of the input vector; and (b) the corresponding power fraction distribution generatedby the LP, and the keff of the core configuration as the elements of the demand vector.The elements of the input vector is indexed by the assembly location in the core as

depicted in Fig. 1, whereas the value of each element of the input vector associateswith a particular fuel assembly composition as listed in Table 1. These input valuesmay be assigned randomly, but in this implementation they were ordered withrespect to their k values. For example, a LP given in Fig. 1 may be mapped into avector:

X ¼ 121 120 121 121 116 121 120 121 116 121 121 116 121 116 121½

112 121 116 312 131 121 316 131 131 131 131� ð1Þ

which with the help of Table 1 is encoded as:

X ¼ 0:6; 0:1; 0:6; 0:6; 0:2; 0:6; 0:1; 0:6; 0:2; 0:6; 0:6; 0:2; 0:6;½

0:2; 0:6; 0:3; 0:6; 0:2; 0:5; 0:7; 0:6; 0:4; 0:7; 0:7; 0:7; 0:7�:

ð2Þ

There are two requirements for the training patterns to be unique (Kim et al.,1993):

1. Different core LPs must be represented with different vectors.2. The difference due to the location in the core must be taken into account.

Fulfilling these requirements guarantees a unique input vector for every distincttraining pattern generated.During the training, the weight adjustment at a hidden node is proportional to the

corresponding components of the input vector. An assembly with a position ofhigher importance must make its presence felt by the weight that it is connectedthrough.In the light of the above argument, a kFA

eff method has been applied to incorporatethe effect of the location into the patterns during training. A separate effectivemultiplication factor kFAi

eff ;j i ¼ 1; 2; :::; 7; j ¼ 1; 2; :::; 26ð Þ was calculated for differentLPs constructed by locating a single fuel assembly of type i at core location j andleaving other locations empty (filled by water). When the calculations were per-formed for seven different fuel assemblies the total number of kFA

eff ’s was 182.Each fuel assembly member of the input vector x is multiplied by the corre-

sponding kFAeff value with respect to its position. This strategy includes the posi-

tion related effects to the input vector. The input vector X for our example LPbecomes:

X ¼ xFA6 kFA6

eff ;1xFA1 kFA1

eff ;2xFA6 kFA6

eff ;3 . . . xFA7 kFA7

eff ;26

h ið5Þ


where xFAn is the neural network value taken from Table 1 for assembly FAn locatedat core position n and defined as

xFAn ¼ xFA1 ; xFA2 ; xFA3 ; :::; xFA7� �

¼ 0:1; 0:2; 0:3; :::; 0:7½ �: ð6Þ

The elements of the output vector composed of power fractions for the fuelassemblies and the keff for the core has been normalized to the interval [0.1, 0.9],since, the range of activation function is confined into the interval [0.0, 1.0]. Theweights have been initialised with random numbers in the interval [�0.1, 0.1].The training iterations have been finalised when the cumulative sum of the

squared error E ¼ 12�

29k¼1 tk � okð Þ

2 has satisfied 0.005 error criteria, tk and ok beingthe target and output values at node k, respectively.The choice of momentum parameter, �, and training rate, �, and the number of

hidden layer nodes are critical during the learning process. Unfortunately, nostraightforward procedure exists to determine proper training parameters. Thetraining has been conducted for possible values and the near optimum parametershave been specified from the graphs drawn from the results. These trial training runshave been performed with a convergence criterion of 0.05 to obtain the results faster.While � was changed from 0.1 to 1.0, as a starting point � was fixed to 0.6 and the

number of hidden layer nodes were taken to be 150. The results are shown in Fig. 2.It can be easily seen that, for an � value of 0.9 the network learning is fastest andmost stable. A value greater than about 0.94 made the network unstable. With ahigher � value (>0.95) the network was unable to converge. An � value of 0.9 hasbeen used throughout the network training.Similarly, the training procedure was conducted for � values between 0.05 and 1.0

with a fixed � of 0.9, convergence criterion of 0.05 and 150 hidden layer nodes. The

Fig. 2. Number of iterations vs. momentum parameter (�) and training rate (�).


results are shown in Fig. 2. � Values from 0.4 to 0.6 converged fastest while 0.4was the most stable and values greater than 0.5 caused some oscillation. � valuesabove 1.0 prevented the network from convergence. An � value of 0.4 was chosen tobe the optimum parameter for the network training.To determine the optimum number of hidden layer nodes, several training sessions

were performed. � And � parameters were set to 0.9 and 0.4, respectively, and theconvergence criterion to 0.05. The number of hidden layer nodes for the networkwas changed from 50 to 500. The relationship between the number of hidden layernodes and the number of iterations is shown in Fig. 3. The network was unable tolearn the relationship for a number of hidden layer nodes that was less than 50.When the value was between about 250 and 400, the number of iterations was thefewest. A hidden layer node number of 250 was set for this work. Finally the singlelayer network with 250 hidden nodes was trained with the above selected trainingparameters and convergence criterion of 0.005. When the network converged, itwas tested using the remaining 1000 randomly generated LPs. The network pre-dicted the maximum power fraction with an average error of 4.39% and themaximum error was 14.3%. For the effective multiplication factor, keff, the resultswere 0.21 and 0.98% for the average and maximum errors, respectively. Theresults for the maximum power fraction prediction and effective multiplicationfactor are shown in Figs. 4 and 5. It is seen that both for the power and keff datathe linear fits approximately lie along the diagonal axis within �15 and �1.0%respectively.Finally, to show the significance of the position effect of the assemblies, the net-

work was trained for two different cases. In the first case, no effect was included, andthe encoded data is used as is. Then, the network was trained with the kFAeff -weightingmethod. Fractional power distribution for sample LPs have been generated for allcases. The results are shown in Table 2.

Fig. 3. Number of iterations to achieve E<0.05 vs. the number of hidden layer nodes.


The kFAeff -weighting method performed best with respect to the error in the maxi-mum power fraction. The cost of this improvement was a larger number of iterations.The calculation of the power distribution and keff takes about 1 min on a personel

computer (Intel Celeron 300 MHz CPU) using the conventional codes. On the otherhand, for a single LP, it takes about a fraction of a second with the neural network.

Fig. 4. Maximum power fraction: generated output vs. target values (error bounds are set with respect to

the exact estimator).

Fig. 5. Effective multiplication factor: generated output vs. target values.


In this work, we preferred to assign the tasks of estimating the keff and theassembly power sharing fraction to a single neural network rather than training aseparate neural network for each individual task. This approach spares the com-puter time at the expense of accuracy. Actually, this situation does not make anycompromises on the final output, since our main purpose is to construct a fast andrough estimator that provides parameters that are needed to carry out an opti-misation scheme based on the genetic algorithms. The final LPs generated will betested with licensed in-core fuel management codes.

3. Optimising fuel loading patterns by genetic algorithms

In this work, we used Genetic Algorithms (GAs) to search for the optimum LP.GAs are used to solve large combinatorial problems, which require judicious

searching through a large number of parameters using simultaneous processing(Mitchell, 1996). The main purpose is to evolve a population representing possiblesolutions to a given problem by using genetic operators: selection, crossover andmutation. This method, which is categorized as a subset of evolutionary algorithms,was developed by Holland. The work presented by Holland’s 1975 book Adaptationin Natural and Artificial Systems was the first attempt to put computational evolu-tion on a firm theoretical footing (Holland, 1992).In GAs possible solutions are represented as a string of symbols, which are called

chromosomes. In this biological metaphor, the symbols, which encode a particularelement of the chromosome, correspond to genes. Usually, a gene consists of 10s and00s or sequence of adjacent bits, but for some problems use of higher level alphabetsthan binary bits in encoding the genes proves to be more expedient.All possible solutions to a problem make up the search space. Each chromosome

represents a point in the search space among the candidate solutions.GAs apply selection, crossover and mutation operators to construct fitter solutions.

A GA processes populations of chromosomes by replacing unsuitable candidatesaccording to the fitness function. The fitness function determines how well the pro-cessed chromosome solves the problem.As a permutation problem the fuel management optimisation requires that the

crossover operator to be modified so that it produces chromosomes that do not vio-late the contents of the fuel inventory (a constraint which dictates that a fuel assembly

Table 2

The effect of position weighted coding of the fuel assemblies on the performance of the ANN (until

E<0.05)

Method Number of

iterations

Average error in maximum

power fraction (%)

None 870 10.03

kFAeff -Weighting method 1411 7.31


with a particular ID number is not be expended more than available in the fuelinventory). Chapot et al. (1999) used the re-ranking technique based on orderingfuel assembly ID’s in the chromosome and renumbering them with respect to theirorders.Initially, this was achieved by building a one-dimensional array of ID numbers of

the fuel assemblies (Parks, 1996). To have a different chromosome for each LP, fuelassembly ID numbers are exchanged with unique rank values. In this work, the fuelassembly types were given rank numbers with respect to the order of their k values.Fuel assemblies with the same ID number were given rank numbers with respect tothe processing order. The number of fuel assemblies for each type was taken fromTable 1. Fuel assemblies and their rank numbers are listed in Table 3.Fig. 6 illustrates the steps of this process for an example LP. The chromosome for

the LP is obtained by replacing all assembly ID numbers of the vector with theirassociated rank values given in Table 3.The next step is to generate a set of LPs to construct the initial population in order

to start the evolutionary computation process. Then members of the initial popu-lation representing this set of LPs have been used to breed with the crossover and

Fig. 6. The loading pattern and its corresponding chromosome.

Table 3

Fuel assembly types and their rank numbers

Assembly ID 131 121 312 316 112 116 120

Quantity available 5 11 1 1 1 5 2

Assigned rank no. 26–22 21–11 10 9 8 7–3 2–1


mutation operators to search for better candidates. In our work we generated 200candidates to initiate the search.A chromosome for each of the patterns was constructed and stored. Reactor core

calculations were performed for each LP with the neural network to predict thepower distribution and the keff.Each chromosome in the population was assigned a fitness value, which is the

measure of qualification for the respective LP. For a chromosome with higher fitnessvalue there is a higher probability of survival to the next generation. The fitness f iscalculated from:

f ¼ ck keff � koeff� �

� cp pmax � pomax

� �ð7Þ

where keff and pmax are the results from the neural network calculation and koeff andpomax are the reference values of keff and maximum power fraction, which are set tobe 1.0 and 1.5, respectively. The weighting coefficients ck and cp were taken as 0.3and 0.7, respectively. A higher power fraction weight ensures a higher contributionfrom the power fraction value, which is preferred since a lower maximum powerpeak is more desirable in an optimum LP.Fitness f, given with Eq. (7), results in a higher fitness when keff increases and pmax

decreases, in other words the optimum LP has the largest keff and the smallest pos-sible pmax for which the fitness function, f, attains the maximum value.LPs complying with the constraints that specify optimum, that is to say, patterns

that have pmax values less than pomax and keff values greater than koeff , were archived.The archive store was set to maximum size of 10, which contained LPs that wereacceptable before terminating the execution of the search process.The population is evolved by selecting pairs of LPs, which are referred as ‘‘par-

ents’’. The top one-third of the population was taken as parents, which were pairedrandomly, so that, the possibility of different chromosomes to get into the samegroup during each generation was increased.The genetic crossover and mutation operators are applied to the parent patterns to

start a new generation. A probability was defined to determine which operator to apply.We used the crossover operator with a 99% probability, which randomly exchanges agroup of fuel assemblies, or genes, between the selected pairs. The resulting offspringchromosomes are expected to have better fitness value compared to their parents,which means they have a higher probability to survive to the next generation. Other-wise, the offspring patterns cease to exist in the succeeding generations.A two-dimensional block-crossover operation has been defined and performed

throughout the search process (Parks, 1996); namely, a group of fuel assemblies,called a block, have been randomly chosen from a neighbouring group of assembliesin a pair of parents. This is illustrated in Fig. 7.A block for crossover consisted of any number of fuel assemblies between 4 and

10. In Fig. 7 the block size was taken as 5. A random walk starting from a selectedlocation determines the rest of the fuel assemblies to be included into the block.Having determined the size and location of a block to be used in the crossover

operation, the corresponding fuel assemblies are exchanged between the two LPs


representing ‘‘parents’’ (Fig. 7). The resulting LPs are the offspring produced by thecrossover operation. One further operation is required since the rank information ofthe patterns may include duplicates. To rectify this problem small random numbershave been added to rank values of patterns (DeChaine and Feltus, 1996), which arethen rearranged to obtain the final offspring.The mutation operator was applied individually to each of the parent chromo-

somes to ensure that the search does not get stack in local optima. The number ofpairs of genes to be mutated was determined randomly, which is typically between 1and 3.To affect the mutation operation, these pairs of genes are swapped. The fitness

value of the offspring is evaluated after application of the crossover and/or mutationoperators.One third of the LPs in the global population was replaced with the offspring

chromosomes. Population revision is necessary to eliminate failed offspring that fitpoorly to the environment. Then, the chromosomes in the population were rankedwith respect to their fitness values and only those surpassing the optimum LP cri-teria were archived.The evolution process is continued until the number of accumulated LPs has

reached the maximum allowed in the archive.Finally, the chromosomes are decoded back to their physical attributes as shown

in Fig. 6. This is performed by selecting a rank value from Table 3 and replacing itwith its ID counterpart.The flowchart for the genetic algorithm developed is given in Fig. 8.

3.1. A simplified case for testing and verification

To demonstrate and verify the technique, a simple tractable case was setup withthe number of fuel assembly types reduced to two. In total 24 fuel assemblies of type

Fig. 7. A selected block for the crossover operation between two parents.


121 and two fuel assemblies of type 116 were used to generate the patterns. For thisproblem, the number of possible LPs that can be generated is 325. Thus this sim-plified case provides a search space of only 325 possible LPs, which can be used totest and verify the GA.All LPs were calculated using MCRAC code to obtain the core parameters. For

this newly defined fuel inventory, the neural network was re-trained. The averagenetwork error for this case was about 2%.For this special case we limited the archive size to five LPs with pmax less than 1.6

and keff larger than 1.0. It may be noted that the value of the reference maximumpower fraction of 1.6, which is different from its previous value, was chosen becausein the totality of the population, there were exactly 10 LPs satisfying the criterion. Ittook seven generations to produce five LPs qualified to be a member of the solution set.The results obtained at the end of the search are listed in Table 4 giving the top 10 LPs.Fig. 9 illustrates the top 10 patterns of the test problem. The dark and light col-

ours represent ID=116 and ID=121 fuel assemblies, respectively. The Manhattandistance between the two assemblies of ID=116 are given in the same figure.It has been found that the distance between the two highly enriched fuel assem-

blies in so-called Manhattan distance measure plays an important role when per-forming the block-wise crossover operation. Smaller the Manhattan distanceindicates higher the probability of survival. The Manhattan distance is defined as thedistance between two points measured along the Cartesian axes. In a plane withpoint P1 at (x1,y1) and P2 at (x2,y2), it is ðx1 � x2Þ þ ðy1 � y2Þ (Mount, 1998).Results for the 500 search sessions are listed in Table 5. It can be seen that, LPs

with small Manhattan distances were selected more frequently as predicted by the

Fig. 8. The flow chart of the genetic algorithm.


schema theorem (Goldberg, 1989). The 46.4% of all selected patterns has a Man-hattan distance of 1 unit with 30.8 and 22.8% having Manhattan distances of 2 and3 units respectively.

3.2. Calculations using an inventory with seven assembly types

The search was repeated with the full seven different assembly types to demon-strate the technique for a realistic case. The GA was used to generate 15 LPs that

Fig. 9. Patterns of the genetic algorithm results for the simplified case.

Table 4

Genetic algorithm results for the simplified case

Top 10 loading patterns out of 325 Optimums selected by GA as archived

Pmax Keff Fitness Pmax Keff Fitness

1.493712 1.056560 0.02137 1.577696 1.062341 �0.03568

1.504018 1.053977 0.01338 1.493712 1.056560 0.02137

1.514983 1.058759 0.00714 1.542664 1.054219 �0.01421

1.542664 1.054219 �0.01421 1.596004 1.053885 �0.05104

1.550375 1.054102 �0.01900 1.514983 1.058759 0.00714

1.577696 1.062341 �0.03568

1.591533 1.059088 �0.04635

1.596004 1.053885 �0.05104

1.601356 1.052807 �0.05511

1.608914 1.060672 �0.05804


surpass the pmax<1.4 and keff>1.0 criteria. The results that were archived at the endof the calculation are presented in Table 6. In total 24 generations were enough toaccumulate the desired number of patterns in the archive.It took a few seconds to generate the optimum population, since calculations are

carried out with the fast neural network method.Fig. 10 shows the change of the average fitness value of the population as a func-

tion of generations, which increases monotonically. Gradually fitter LP s accumu-lates in the population and the time required registering one more pattern into thearchive decreases during this process.

3.3. Benchmark

It is interesting to compare the fitness of the LP presented in the benchmark byAhnert et al. (1990) for the Almaraz Nuclear Power Plant and some of the resultsobtained by the genetic algorithm (Table 7). The genetic algorithm generated someloading patterns that have higher fitness values.

Table 5

GA search results for 500 random trials for simplified case

pmax keff Fitness Number of times

being selected by GA

Manhattan

distance

1.577696 1.062341 �0.03568 97 1

1.542664 1.054219 �0.01421 93 2

1.596004 1.053885 �0.05104 68 1

1.493712 1.056560 0.02137 67 1

1.514983 1.058759 0.00714 61 2

1.504018 1.053977 0.01338 52 3

1.550375 1.054102 �0.01900 38 3

1.591533 1.059088 �0.04635 24 3

Table 6

Archived results of the sample calculations with genetic algorithm

pmax keff pmax keff

1.383089 1.052585 1.388094 1.034512

1.323069 1.054277 1.393167 1.048240

1.394021 1.031450 1.370616 1.032716

1.360441 1.043750 1.385096 1.036185

1.395731 1.056448 1.392338 1.044205

1.397488 1.047677 1.383126 1.029877

1.273009 1.052470 1.397182 1.035324

1.326133 1.044959


4. XCore code

The computer program package XCore has been developed to generate andrecommend optimum LPs complying with the restrictions imposed by the physicallimitations of the reactor core (Fig. 11).The program code was written in the programming language C++. It runs under

the LINUX operating system on personal computers. XCore also requires theX-Windows system.The XCore system consists of the following components: a graphical user inter-

face, LEOPARD cell calculation code, MCRAC whole core calculation code, aneural network algorithm, rule-based random-search module and a genetic algo-rithm. All modules run separately to enable the graphical user interface to pro-cess the events received from the X-server while the system is calculating in thebackground.The graphical user interface is responsible for displaying the core information in a

visual format. A quarter of the core is sketched on the main program window. Thefuel assemblies are displayed as colored boxes, which include the relevant numerical

Fig. 10. Average fitness vs. generation number for population for the realistic core.

Table 7

A PWR benchmark vs. GA results

pmax keff Fitness

PWR benchmark 1.281 1.013 0.157

GA

Sample 1 1.273 1.052 0.175

Sample 2 1.252 1.054 0.189


data. Different colors, which are graded to show some core parameters, representthree main distributions; namely (1) neutronics, (2) thermal-hydraulics, and (3) iso-tope information. In addition, the color maps extend into cycle life represented withtotal core burnup. Each time interval, which is called a step, can be displayed withcolor maps. The color map system also supports the multi-cycle design of the reactorcore.The reactor core may be designed from scratch using the tools of the graphical

user interface. Fuel assembly types are defined and assigned to any fuel assembly.Direct interactive accesses to the information about a fuel assembly is also availablefrom a screen menu.

Fig. 11. Main window of program XCore (one-eight symmetry is displayed on a quarter section of the

core layout).


A manual fuel shuffling mechanism has been implemented. The boxes representingfuel assemblies may be dragged and dropped to desired core locations. The loadingpattern obtained by shuffling fuel assemblies is re-calculated and a new color map isdisplayed automatically.XCore has the following structure in accordance with the operating sequence of its

flow diagram: When a BOC fuel assembly inventory is assigned, it trains a built-inartificial neural network which can predict the results of whole core calculations toaccelerate the subsequent LP search process by substantially lowering the compu-tational burden. Then the search process is carried out via a choice of methods i.e.either rule- or GA-based approaches are available. Finally the code suggests a list ofLPs for the core designer. Any selected LP is re-evaluated by direct use of the wholecore calculation codes.

5. Conclusions

A PWR fuel reload optimisation program package XCore was developed andtested successfully. The code couples the genetic algorithm’s stochastic optimisationcapability with the powerful learning and mimicking ability of artificial neural net-works for complex function approximation.The ANN predicts the power distribution and the keff to be used in the GA to

search for an optimum LP for a beginning of cycle (BOC) reactor core. The use ofan ANN as a core parameter estimator reduces the calculation time and makes itpossible to analyse more loading patterns in the same time interval by increasing theprobability of finding a desired optimum. The GA automates the search for LPs inregions of the space that is impractical by conventional methods.The code was verified with a simple and fictitious yet nontrivial test case of a small

search space. For validation its results were compared to those given for AlmarazNuclear Power Station. Our results conform very well with those LPs suggested forthe commercial nuclear reactor core.It should be emphasized that the system recommends candidate optimum LPs for

the BOC reactor core. The results provide a starting point for a more detailed ana-lysis including a cycle-length or burnup optimisation. It would be a quite straight-forward procedure to improve upon the capabilities of XCore further by includingburn-up and leakage-flux considerations in both the ANN and GA modules of thecode. For this purpose, a recurrent architecture, which will incorporate the dynamicbehaviour required by the depletion process, is under consideration. To completethe optimisation the objective function of the GA will be modified to include theburnup effects.

References

Ahnert, C., Aragones, J.M., Merino, F., 1990. PWR Benchmark Parameters, Almaraz Nuclear Power

Plant. IAEA, Vienna.


Chapot, J.L.C., Silva, F.C.D., Schirru, R., 1999. A new approach to the use of genetic algorithms to solve

the pressurized water reactor’s fuel management optimization problem. Annals of Nuclear Energy 26,

641–655.

DeChaine, M.D., Feltus, M.A., 1996. Fuel management optimization using genetic algorithms and expert

knowledge. Nuclear Science and Engineering 124, 188–196.

Erdogan, 2001. Application of Computational Intelligence Methods to In-Core Fuel Management. PhD

Thesis, Istanbul Technical University, Turkey.

Goldberg, D.E., 1989. Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-

Wesley, MA.

Haykin, S., 1992. Neural Networks, second ed.. Prentice Hall International.

Holland, J.H., 1992. Adaptation in natural and artificial systems. MIT Press, MA.

Kim, H.G., Chang, S.H., Lee, B.H., 1993. Optimal fuel loading pattern design using an artificial neural

network and a fuzzy rule-based system. Nuclear Science and Engineering 115, 152–163.

Lysenko, M.G., Wong, H., Maldonado, G.I., 1999. Predicting neutron diffusion eigenvalues with a query-

based adaptive neural architecture. IEEE Transactions on Neural Networks 10, 790–800.

Mitchell, M., 1996. An Introduction To Genetic Algorithms. MIT Press, MA.

Mount, D.M., 1998. ANN Programming Manual. Department of Computer Science, University of

Maryland, College Park, MD.

Parks, G.T., 1996. Multiobjective pressurized water reactor reload core design by nondominated genetic

algorithm search. Nuclear Science and Engineering 124, 178–187.

Petrovic B.G., 1990. In-core fuel management: PWR core calculations using MCRAC. In: Workshop on

Reactor Calculations for Applications in Nuclear Technology, Zagreb, Yugoslavia.

Sadighi, M., Setayeshi, S., Salehi, A.A., 2002. PWR Fuel Management Optimization Using Neural Net-

works. Annals of Nuclear Energy 29, 41–51.


a pwr reload optimisation code (xcore) using artificial neural networks and genetic algorithms

Documents