Optimization of the Core Configuration Design Using a Hybrid

8/3/2019 Optimization of the Core Configuration Design Using a Hybrid

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Nuclear Engineering and Design 239 (2009) 27862799

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Nuclear Engineering and Design

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Optimization of the core configuration design using a hybrid

artificial intelligence algorithm for research reactors

Afshin Hedayat a,c,, Hadi Davilu a, Ahmad Abdollahzadeh Barfrosh b, Kamran Sepanloo c

a Department of Nuclear Engineering and Physics, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, P.O. Box 15875-4413, Tehran, Iranb Department of Computer Engineering, Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, P.O. Box 15875-4413, Tehran, Iranc Reactor Research and Development School, Nuclear Science and Technology Research Institute (NSTRI), End of North Karegar Street, P.O. Box 14395-836, Tehran, Iran

a r t i c l e i n f o

Article history:Received 18 April 2009

Received in revised form 5 August 2009

Accepted 23 August 2009

a b s t r a c t

To successfully carry out material irradiation experiments and radioisotope productions, a high thermalneutron fluxat irradiationbox over a desired life time ofa core configurationis needed. Onthe otherhand,

reactor safety and operational constraints must be preserved during core configuration selection. Two

main objectives and two safety and operational constraints are suggested to optimize reactor core con-

figuration design. Suggested parameters and conditions are considered as two separate fitness functions

composed of two mainobjectives and two penalty functions.This is a constrained and combinatorial type

of a multi-objective optimization problem. In this paper, a fast and effective hybrid artificial intelligence

algorithm is introduced and developed to reach a Pareto optimal set. The hybrid algorithm is composed

of a fast and elitist multi-objective genetic algorithm (GA) and a fast fitness function evaluating system

based on the cascade feed forward artificial neural networks (ANNs). A specific GA representation of

core configuration and also special GA operators areintroduced andused to overcome thecombinatorial

constraints of this optimization problem. A software package (Core Pattern Calculator 1) is developed

to prepare and reform required data for ANNs training and also to revise the optimization results. Some

practicaltest parametersand conditionsare suggestedto adjust mainparameters of the hybrid algorithm.

Results show that introduced ANNs can be trained and estimate selected core parameters of a research

reactor very quickly. It improves effectively optimization process. Final optimization results show thata uniform and dense diversity of Pareto fronts are gained over a wide range of fitness function values.

To take a more careful selection of Pareto optimal solutions, a revision system is introduced and used.

The revision of gained Pareto optimal set is performed by using developed software package. Also some

secondary operational and safety terms are suggested to help for final trade-off. Results show that the

selected benchmark case study is dominated by gained Pareto fronts according to the main objectives

while safety and operational constraints are preserved.

2009 Elsevier B.V. All rights reserved.

1. Introduction

Research reactors are used for material researches and

radioisotope productions, performing neutron radiography, semi-

conductor doping and neutron activation analysis, education

and training, and also extracting a wide range of neutronbeam spectrum (IAEA Technical Report, 2007). The optimiza-

tion of the core design is necessary to use effectively of

research reactor utilizations. Some specific devices are devel-

oped and being operated presently (IAEA Technical Report,

2007). But also on the other hand, an appropriate cal-

Corresponding author at: Department of the Nuclear Engineering and Physics,

Amirkabir University of Technology (Tehran Polytechnic), 424 Hafez Avenue, P.O.

Box 15875-4413, Tehran, Iran.

E-mail addresses: [email protected], [email protected](A. Hedayat).

culation method is needed to optimize core configuration

design.

Although each of the specific radioisotope productions or mate-

rial irradiation tests in a research reactor needs to some specific

utilization conditions, generally to successfully carry out material

irradiation experiments and radioisotope productions, a high valueof thermal neutron flux at irradiation boxes over a desired life time

of a core configuration is needed. These criteria are independent

and conflict with each other. On the other hand, reactor safety and

operational constraints must be preserved during core configura-

tion selection. Safety and operational constraints can be obtained

from safety analysis studies (Hamidouche et al., 2003; Hedayat et

al., 2007; IAEA Technical Document, 1992; Woodruff, 1984).

In this paper, two main objectives and two safety and

operational constraints are suggested to optimize reactor core con-

figuration design. The suggested optimization criteria and also

safetyand operational constraints are separately dependent on the

0029-5493/$ see front matter 2009 Elsevier B.V. All rights reserved.

doi:10.1016/j.nucengdes.2009.08.027
http://www.sciencedirect.com/science/journal/00295493http://www.elsevier.com/locate/nucengdesmailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.nucengdes.2009.08.027http://dx.doi.org/10.1016/j.nucengdes.2009.08.027mailto:[email protected]:[email protected]://www.elsevier.com/locate/nucengdeshttp://www.sciencedirect.com/science/journal/00295493


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A. Hedayat et al. / Nuclear Engineering and Design239 (2009) 27862799 2787

Nomenclature

cd crowding distance

D normalized distance

fc crossover fraction

fk values of the kth objective function (fitness func-

tion)

fp Pareto front fraction

Fi non-dominated frontsF(x) objective function, fitness function

Flux thermal neutron flux density (at the irradiation box)

Flux0 constant value

g(x) inequality constraint

i, j counting number

k objective index

K number of objectives

Kout0 constant value

n counting number

nc number of generation defined for convergence cri-

teria

N number of solutions, population size

Pt population at tth generation

PPF power peaking factor (radial)Qt offspring population at tth generation

Rt combined population at tth generation

S feasible area in decision space, spread function

t generation index, time index

vc defined convergence limit for the average change of

spread values

x, y a solution

z objective value, fitness value

Z feasible area in the criterion space

Greek letters

reactor reactivity neutron flux densities

Subscripts

eff effective

E Euclidean distance

i, j counting number

in control rod absorbers are completely in the core

k kth objective function, kth fitness value

max maximum

out control rod absorbers are completelyoutof the core

rank rank of a individual in the population

t total tth generation

th thermal

Superscripts

min minimum

max maximum

core configuration design. This is a constrained and combinatorial

type of a multi-objective optimization problem.

Genetic algorithms (GA) are a popular meta-heuristic method

that is particularly well-suited for this class of problems (Konak

et al., 2006). Traditional genetic algorithms (Holland, 1975) are

customized to accommodate multi-objective problems by using

specialized fitness functions and introducing methods to promote

solution diversity. There are two general approaches to solve a

multi-objective optimization.

One is combining the individual objective functions into

a single composite function by using weighted-sum approach

(Zadeh, 1963) or moving all but one objective to the constraint

set.

The second general approach is determination of an entire

Pareto optimal solution set (Censor, 1977; Cunha and Polak, 1967)

or a representative subset. A Pareto optimal set is a set of solutions

that are non-dominated with respect to each other. Pareto optimal

solution sets are often preferred to single solutions because they

can be practical when considering real-life problems since the final

solution of the decision-maker is always a trade-off (Konak et al.,

2006).

The first approach needs additional procedure to adjust com-

bination weights for each desired conditions. On the other hand,

the second approach results a Pareto optimal set providing differ-

ent conditions. Then research reactor operators can expertly select

compatible core configuration. In this paper the second approach,

determination of an entire Pareto optimal solution set is used to

solve this multi-objective optimization problem.

A fast and effective hybrid artificial intelligence algorithm is

introduced and developed to reach a Pareto optimal set. This algo-

rithm is composed of a fast and elitist multi-objective genetic

algorithm (Deb et al., 2002) and a fast fitness function evaluating

system based on the cascade feed forward ANNs (Hedayat et al.,

2009). The core of the IAEA 10 MW LEU benchmark problem (IAEA

Technical Document, 1980) is used to introduce and test suggestedoptimizationmethod for the core configuration design of a research

reactor.

Although according to performed tests ( Jones et al., 2002),

GA are the most popular heuristic approach to multi-objective

design and optimization problems, but getting desired results from

population-based methods like GA usually needs many large com-

putations. On the other hand, to obtain objective function values

in each epoch, safety and neutron core parameters must be cal-

culated repeatedly. The best method to calculate accurately these

parameters is solving diffusion equation by core calculation codes

such as CITATION (Fowler et al., 1971). These codes solve the

diffusion equation by numerical methods. A combination of iter-

ative numerical methods and evolutionary process steps increases

the total time of the optimization process. The very large time ofcomputation process can restrict the effective using of these meth-

ods.

The first step to use a multi-objective genetic algorithm for the

core configuration design effectively, is replacing a new method

to approximate the fitness function values during optimization

process. This method should be effectively faster than iterative

numerical methods. Feed forward neural networks are a well

known fast predictor in nuclear industry (Kim et al., 1993; Mazrou

and Hamadouche, 2004; Hedayat et al., 2009). In this research, a

fast estimation systemof fitness function parameters is introduced

and developed by using cascade feed forward ANNs (Hedayat et al.,

2009).

A wide variety of completely different core arrangements are

needed to train and test developed ANNs. Needed parametersshould be extracted from diffusion theory calculations. They must

be converted to a compatible format to feed used ANNs. Doing this

manually takes a long time while some human errors are possible.

On theother hand,to take a more careful selectionof Paretooptimal

solutions, a revision systembasedon diffusiontheory calculation is

needed. In this research, a software package (Core Pattern Calcula-

tor1) is developedand used to prepare andreform requireddatafor

ANNs training and validation, and also to revise the optimization

results.

Some practical test parameters and conditions are suggested to

adjust main parameters of the hybrid algorithm. A specific GA rep-

resentation according to core configuration design and also special

GA operators are developed to overcome the combinatorial con-

straints. Final optimization results show that a uniform and dense


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2788 A. Hedayat et al. / Nuclear Engineering and Design239 (2009) 27862799

spread out of Pareto fronts are gained over a wide range of fitness

functions values.

In order to take, a more careful selectionof Paretofront, they are

revised by the developed software; and also some operational and

safety terms are suggested to take a more effective irradiation and

higher safety margins of selected Pareto optimal solutions during

final trade-off as a fine tuning method.

2. Multi-objective optimization

Someoptimization problemsmay formulatewith morethan one

objective, since a single-objective with several constraints maynot

adequately representthe problem being faced.If so,there isa vector

of objectives (Eq. (1)) that must be traded off in some way.

F(x) =

F1 (x) , F2 (x) ,...,Fm (x)

(1)

The relative importance of these objectives is not generally

known until the systems best capabilities are determined and

trade-offs between the objectives fully understood. As the num-

ber of objectives increases, trade-offs are likely to become complex

and less easily quantified. Thus, requirements for a multi-objective

design strategy must enable a natural problem formulation to be

expressed, and be able to solve the problem and enter preferences

into a realistic design problem. A multi-objective optimization

problem can be represented formally as follows:

Min {z1 = f1(x), z2 = f2(x), . . . , zq = fq(x)} (2)

s.t.gix0. i =1,2,. . ., mwherexRn isavectorofn decision variables,

f(x) an objective function, and gi(x) inequality constraint functions

which form an area of feasible solutions. The feasible area in deci-

sion space is noted by the set S (Eq. (3)), as follows:

S =

xRn|gi (x) 0, i = 1, 2, . . . , m

(3)

The multi-objective optimization problem can be shown in both

of decision space and criterion space (Gen and Cheng, 1999). S (Eq.

(3)) is used todenote the feasible region in the decision space andZ

(Eq. (4)) is used to denote the feasible region in the criterion space.Z=

zRq|z1 = f1 (x) , z2 = f2 (x) , . . . , zq = fq (x) ,xS

(4)

wherezRq is a vectorof values ofq objective functions. In the otherword, Z is the set of images of all points in S; and S is confined to

the Rn.

Note that becausef(x) isa vector, ifany ofthe componentsoff(x)

are competing, there is no unique solution to this problem. Instead,

the concept of non-inferiority (Zadeh, 1963) that also called Pareto

optimality (Censor, 1977; Cunha and Polak, 1967) must be used to

characterize the objectives. A non-inferior solution is one in which

an improvement in one objective requires a degradation of another.

In the case of multiple objectives, there does not necessarily

exist a solution that is best with respect to all objectives. Because

of incommensurability and confliction among objectives, a solu-

tion may be best in one objective but worst in another. Therefore,

there usually exist a set of solutions for the multi-objective case

which cannot simply be compared with each others. For such solu-

tions, called non-dominated solutions or Pareto optimal solutions,

no improvement is possible in any objective function without sacri-

ficing at least one of another objective functions. For a given point

zoZ, it is a non-dominated solution if and only if there does notexist another point zZsuch that for the minimization case:

zk < zok

for somek{1, 2, 3, . . . , q} (5)

zl = zol

forall l /= k

A solution is said to be Pareto optimal if it is not dominated by

any other solution in the solution space. The set of all feasible non-

dominatedsolutions in decision space(S) is referred toas thePareto

optimal set, and for a given Pareto optimal set, the corresponding

objective function values in the objective space or criterion space

(Z) are called the Pareto fronts.

The ultimate goal of a multi-objective optimization algorithmis

to identify solutionsin thePareto optimal set. However, identifying

the entire Pareto optimal set, for many multi-objective problems,

is practically impossible due to its size.

In addition, for many problems, especially for combina-

torial optimization problems, proof of solution optimality is

computationally infeasible. Therefore, a practical approach to

multi-objective optimization is to investigate a setof solutions (the

best-knownParetoset) thatrepresentthe Pareto optimalset as well

as possible. With these concerns in mind, a multi-objective opti-

mization approach should achieve the following three conflicting

goals (Zitzler et al., 2000):

The best-known Pareto front should be as close as possible to the

true Pareto front. Ideally, the best-known Pareto set should be a

subset of the Pareto optimal set. Solutions in the best-known Pareto optimal set should be uni-

formly distributed and diverse over of the Pareto fronts in order

to provide the decision-maker a true picture of trade-offs. The best-known Pareto front should capture the whole spectrum

of the Pareto front. This requires investigating solutions at the

extreme ends of the objective function space (criterion space).

Fora given computational time limit,the first goal is best served

by focusing the search on a particular region of the Pareto fronts.

On the contrary, the second goal demands the search effort to be

uniformly distributed over the Pareto fronts. The third goal aims at

extending of the Paretofronts at both ends, exploring new extreme

solutions (Konak et al., 2006).

3. Multi-criteria consideration for research reactor core

configuration designs

The most effective wayto increase utilization of a research reac-

tor is placement optimization of fuel assemblies in the core to

maximize neutron flux densities in the reactor channels used for

neutron physics researches, radioisotope productions and neutron

transmutationdoping of silicon (Mahlers, 1997). On theotherhand,

a designed core configuration must have the longest possible life

time while thesafety issues arekept.This is a constrained andcom-

binatorial type of a multi-objective problem. The most important

choice as the first optimization objective is the maximization of

the thermal neutron flux densities in the desired flux trap; and the

second is the maximization of the core configuration life time. It is

possible ifKeff-out, effective multiplication factor when the control

rod absorbers are completely out of the core, is maximized. Two

safety and operational conditions are selected to operate reactor

safely. Theradial power peaking factor(PPF)of each optimizedcon-

figuration must preserve safety limits; also control fuel assemblies

must be capable to shutdown reactor safely. These constraints can

be differentfor each type of research reactors.Two other secondary

conditions are suggested to help for a fine tuning study during final

trade-off.

4. Based case study

In this study, in order to validate the reactor physic calculations

used to generate needed data, the IAEA benchmark problem (IAEA

Technical Document, 1980) is chosen. Considered limits (Section 3)

were not defined for the main benchmark problem directly (IAEA

Technical Document, 1980). So two typical values are chosen to


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Table 1

Main benchmark problem operating conditions (IAEA Technical Document, 1980).

Core material

Nuclear fuel MTR

Fuel element Plate-type clad in Al

Coolant Light water (downward forced

flow)

Moderator Light water

Reflector Graphite-light water

Fuel specificationsFuel material UAlXAl

Fuel enrichment 20 w/o U-235

390 g U-235 per fuel element

(23/17 plates)

72 w/o of uranium in the

UAlXAl

only U-235 and U-238 in the

fresh fuel

Xenon-state Homogeneous Xenon content

corresponding to

average-power-density

Fuel element dimensions

Length (cm) 8.00

Width (cm) 7.60

Height (cm) 60.0

Number of plates SFE/CFE 23/17

Fuel plate dimensions

Plate meat (mm) 0.51

Width (cm) active/total 6.30/6.65

Height (cm) 60.0

Water channel between plates (mm) 2.23

Plate clad thickness (mm) 0.38

Core thermal hydraulics

Water temperature (C) 20

Fuel temperature (C) 20

Pressure at core height (bar) 1.7

solve and test optimization algorithm. The PPF must lower be than

1.9 (Mazrou and Hamadouche, 2006); and Keff-in, the effective mul-

tiplication factor when the control rod absorbers are completely in

the core, must be lower than 1 for capability of the reactor shut-down.

The core of the IAEA 10 MW LEU benchmark research reac-

tor (Table 1) has an arrangement of 56 elements containing 21

standard MTR-type fuel elements (SFE) of 23 plates each and four

control fuel elements (CFE) with 17 plates. Eight boxes of graphite

reflector (G) are located on both sides of the core. The core is sur-

rounded by water (W) and one flux trap is located in the center of

the core. (Fig. 1)

Burn up effects are considered using cell calculations accord-

ing to primary definitions of selected benchmark case study. The

WIMSD5(MTR PC V3.0 user manual, 2006) code is used to generate

the cross sections as a function of fuel burn ups. After condensation

and homogenization of macroscopic cross sections and scatter-

ing matrixes by POS WIMS program (POS WIMS V2.5 user manual,2006), the macroscopic cross section handler (HXS) program (HXS

V4.1 user manual, 2006) is used to handle macroscopic cross sec-

tions in library form (Hedayat et al., 2009).

The three-dimensional and three group diffusion calculation is

performed with the CITVAP V3.2 code (CITVAP V3.2 user manual,

2006).

According to core calculation studies performedby CITVAPV3.2

code (CITVAP V3.2 user manual, 2006) in this research, the maxi-

mum thermal neutron flux densities are extracted from the water

channels (core positions labeled by Win Fig. 1). In order to obtain

a sufficiently large searchspace forthe best neutron fluxtrap (main

irradiation box), a slight modification is introduced in the initial

core arrangement (Fig. 1). The four water channels (W) surrounded

the fuel assemblies at edges, can be located between fuel assem-

Fig. 1. IAEA 10 MW benchmark (LEU) BOC core.

blies. Each of them can have the maximum thermal neutron fluxdensities as the primary central flux trap. Then the fivewater chan-

nels are replaced by five flexible irradiation boxes (I.B.) to get the

best location of final flux trap (main irradiation box).

The referenced absorber material, pure aluminum, in the pri-

mary benchmark problem (IAEA Technical Document, 1980) has

too small absorption cross section to simulate realistic control rod

materials. So to have a more effective and realistic control rod

study while the primary definitions for the benchmark problem

(IAEA Technical Document, 1980) is kept, the (1/V) absorber type

is selected as the material absorber for the control rods in cell cal-

culations. But it still has too small neutron absorption cross section

to consider a operational shutdown margin as a constraint. To con-

sidera large shutdown marginfor a researchreactora more realistic

composite of absorber materials with large neutron absorptioncross sections such as Ag, In, Cd is needed. Also during reactor

operational tasks, the total reactivity of the fuel assemblies will be

reduced; and some reactor poisons such as, Xenon and Sumarium

will be produced. So reactor shutdown margin will be increased

inherently safe during reactor operation. In this paper, the two

reflector rows are considered fixed and 30 remaining positions are

selected to introduce and test the optimization algorithm.

5. Develop a software package to prepare required data for

ANNs training and revise the optimization results

A wide variety of completely different core arrangements are

needed to train effectively used ANNs (Hedayat et al., 2009). There

is a main difference between nuclear research reactors and nuclear


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Fig. 2. The GUI of the Core Pattern Calculator 1.

power reactors. Research reactor assemblies can have a wide

variety of fuel burn ups during their life time. Then to get real sim-

ulations using ANNs, and also to decrease required training data

sets, they should be provided separately for each of BOC states. It

can be performed according to respective core components includ-

ing deferent batch types of standard fuel assemblies, control fuel

assemblies, and irradiation channels. Needed parameters should

be extracted from diffusion theory calculations. They must be con-

verted to a compatible format to feed used ANNs (Hedayat et al.,

2009). Doing this manually takes a long time while some human

errors are possible. Respective calculations of Pareto frontrevisions

must be automatic and accurate too.

In this research, a software package (Core Pattern Calculator 1)

is developed and used. Core Pattern Calculator 1 is programmed

by using Borland Delphi 2006. The random state of the software is

used to create data sets necessary to train and test used ANNs; and

also the recalculation state of it is used to revise the optimization

results. Fig. 2 shows GUI of the Core Pattern Calculator 1.

An integer type of codingalgorithm is selected to representeach

core pattern. This representation is compatible with combinato-

rial optimizations too. Many strings composed of specific integer

numbers are chosen randomly to form different core configura-

tions. For each different state, Core Pattern Calculator 1 software

uses CITVAP V3.2 code (CITVAP V3.2 user manual, 2006) to extract

Fig. 3. The main diagram of c reating desired data.


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Table 2

Final learning and prediction properties of used ANNs.

Final ANNs parameters Keff-out Keff-in PPF th-max

Number of training data sets 500 500 500 500

Number of testing data sets 200 200 200 200

Learning rate 0.025 0.025 0.025 0.025

Momentum coefficient 0.86 0.86 0.86 0.86

Number of epochs 250 252 227 228

Estimated time for training 25 s 26 s 25 s 28 s

Average relative prediction error 0.006 0.027 0.0510 0.1063Maximum relative prediction error 0.023 0.098 0.233 0.506

Simulation time for 200 core configurations 0.033 s 0.029 s 0.027 s 0.034 s

needed core parameters. During calculation process, CITVAP V3.2

code uses macroscopic cross sections library provided by HXS V4.1

program (HXS V4.1 user manual, 2006). Then extracted parame-

ters are stored on a local data base table. Borland Data Base Engine

(BDE) wasused tostoreand read data from thelocaldatabase. Fig.3

shows the main diagram of creating desired data.

6. Evaluate the fitness functions using cascade feed

forward ANNs

If fitness functions can be estimated in very short time, GA(Holland, 1975) can be effectivelyused for presentedtype optimiza-

tion. In this research, a very fast estimation system for suggested

core parameters is developed by using cascade feed forward ANNs

(Hedayat et al., 2009).

The gradient descent method with momentum weight/bias

learning rule algorithm (Rumelhart et al., 1986a,b) is used to train

ANNs (Hedayat et al., 2009). To adjust the used ANNs architectures

and training parameters, a vast study was performed. It includes

the effects of variation of hidden neurons, hidden layers, activation

functions, learning and momentum coefficients, and also the num-

berof trainingdatasetson thetraining andsimulationresults. Some

experimental convergence criteria were defined and used to study

them. Then a comparison selection rule was used to adjust desir-

able conditions (Hedayat et al., 2009). Table 2 shows final adjustedpropertiesof introducedANNs.Final training andsimulation results

(Hedayat et al., 2009) show that introduced ANNs can be trained

and estimate selected core parametersof the research reactors very

quickly. It improves effectively optimization process of the core

configuration design.

7. Multi-objective optimization using genetic algorithms

Evolutionary computations have been developed for difficult

optimization problems since 1960s. The best-known algorithms

in this class include genetic algorithms developed by Holland

(Holland, 1975). GA are as powerful and broadly applicable for

stochastic searches and optimization techniques (Gen and Cheng,

1999). They are inspired by the evolutionist theory explaining the

origin of species (Holland, 1975).

In the GA terminology, a solution vector xX is called an indi-vidual or a chromosome. Chromosomes are made of discrete units

called genes. Each gene controls one or more features of the chro-

mosome. GA operate with a collection of chromosomes, called a

population. The genetic algorithm uses three main types of oper-

ators at each step to create the next generation from the current

population:

Selection operators which select the individuals, called parents,

to select chromosomes for crossover. Crossover operators which combine two parents to form off-

spring or children for the next generation.

Mutation operators which apply random changes to individual

parents to form offspring.

Different selection rules and reproduction operators are devel-

oped in GA to solve and optimize engineering and design problems

(Gen and Cheng, 1999).

The procedure of a generic GA (Holland, 1975) is given as fol-

lows:

Step 1: set t= 1 and randomly generate N solutions (chromo-

somes) to form the first population (P1). Step 2: evaluate the fitness of solutions in Pt. Step 3: operate crossover to generate new chromosomes called

offspring or children for creation new population Pt+1 as follows:

Step 3.1: select two solutionsx andy from Pt based on the fitness

values. Step 3.2: using a crossoveroperatorto generate offspringand add

them to Pt+1. Step 3.3: if the population size is satisfied, terminatethe loop and

go to the next step, else go to step 3.1.

Step 4: operate mutation: mutate each solution xPt+1 with apredefined mutation rate.

Step 5:if thestopping criterionis satisfied,terminatethe searchand return the current population, else set t= t+ 1; go to step 2.

Being a population-based approach, GA are well-suited to solve

multi-objective optimization problems. A generic single-objective

GA can be modified to find a set of multiple non-dominated solu-

tions in a single run. According to performed tests, GA are the most

popular heuristic approachto multi-objective design and optimiza-

tion problems (Jones et al., 2002).

The ability of GA to simultaneously search different regions of

a solution space makes it possible to find a diverse set of solutions

for difficult problems with non-convex, discontinuous, and multi-

modal solutions spaces. The operators of GA mayexploit structures

of good solutions with respect to different objectives to create new

non-dominated solutions in unexplored parts of the Pareto fronts.In addition, most multi-objective GA do not require the user to

prioritize, scale, or weight objectives (Konak et al., 2006).

8. Representation of the considered optimization strategy

In this paper, a non-dominated sorting-based multi-objective

genetic algorithm (NSGA-II) method (Deb et al., 2002) is used to

solve presented optimization problem. Simulation results (Deb et

al., 2002) on difficult test problems show that the NSGA-II (Deb et

al., 2002) in most problems, is able to find much better spread of

solutions and better convergence near the true Pareto optimal front

compared to the other elitist multi-objective evolutionary algo-

rithms (Zitzler, 1999; Knowles and Corne, 1999) that pay special

attention to creating a diverse Pareto optimal front. Also the NSGA-


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II (Deb etal.,2002) does not need additional adjustments for fitness

sharing parameters (Horn et al., 1994).

A specific GA representation and also GA operators are devel-

oped and used to optimize the core configuration design. Safety

and operational constraints are considered using penalty functions

(Gen and Cheng, 1996). Also each chromosome must introduce an

available reactor core configuration. This means that reactor opera-

torscan arrange eachgained pattern.It is a combinatorialconstraint

that should be considered too. Then some components of the used

optimization algorithm (Deb et al., 2002) must be developed or

changed to overcome the combinatorial constraint.

8.1. The non-dominated sorting genetic algorithm II (NSGA-II)

The non-dominated sorting-based genetic algorithm II (NSGA-

II) is a fast and elitist based multi-objective genetic algorithm (Deb

et al., 2002). NSGA-II uses crowding distance approach by using

crowded comparison operator () to reach a uniform spread of

solutions along the best-known Pareto front. The main advantage

of theusing crowding distance is that a measure of population den-

sityaround a solution is computedwithoutrequiringa user-defined

parameter such as share (niche size) or the kth closest neighbor

(Konak et al., 2006).Elitism strategy (Whitley, 1989) is used to preserve some or a

part of the best solutions during GA optimizations. NSGA-II (Deb

et al., 2002) is a fast and elitist multi-objective genetic algorithm

which preserves all previous and new generated population to

choose new population. In the other words, a pure elitism (Deb

et al., 2002) was introduced by comparing current population

with previously found best non-dominated solutions to use for the

NSGA-II. Fitness assignment is based on a non-domination ranking

method (Deb et al., 2002).

A usual binary tournament selection rule (Goldberg et al., 1989)

was developed to use for the NSGA-II based on crowding distance

method (Deb et al., 2002). Developed tournament selection tech-

nique called the crowded comparison operator (Deb et al., 2002).

It is used to select parents for offspring creation and also to choosethe next generation population from previous population and gen-

erated population by reproduction operators.Crowdedcomparison

operator () guidesthe selectionprocess atthe various stagesof the

algorithmtoward a uniformlyspread outPareto optimal front (Deb

et al., 2002). It is defined as follows:

Step 1: rank the every individual in the population according to

the non-dominated fronts (irank). Step 2: calculate the crowding distance for each of individual in

the population (cdi). Step 3: between two selected solutions in the same non-

dominated front (rank), the solution with a higher crowding

distance is selected. Otherwise, the solution with the lowest rank

is selected.

The crowding distance (Deb et al., 2002) is used for crowded

comparison operator and is defined as follows:

Step 1: rank the population and identify non-dominated fronts

F1, F2,. . ., FR. For each front j = 1,. . ., R repeat steps 2 and 3. Step 2: for each objective function k, sort the solutions in Fj in the

ascending order. Let l= |Fj| and x [i,k] represent the ith solution in the sorted list

with respect to the objective function k. Assign: cdk(x[i,k]) = for i =1, l

cdk(x[i,k]) = zk(x[i+1,k]) zk(x[i1,k])/zmaxk

zmink

, for i =

2, . . . , l 1

Step 3: to find the total crowding distance cd(x) of a solution

x, sum the solutions crowding distances with respect to each

objective, i.e., cd(x) =

kcdk(x).

NSGA-II uses a fixed population size of N. In generation t, The

next population Pt+1 of size Nis created from non-dominated fronts

F1, F2,. . ., FR are identified in the combined population Rt= PtUQt of

size2N. Where Ptis all of the previous population (generation t)and

Qt is thenew offspring population. The next population Pt+1

is filled

starting from solutions in F1, then F2, and so on as follows. Let k be

the index of a non-dominated front Fk that |F1UF2U. . .UFk| Nand|F1UF2U. . .UFkUFk+1| > N. First,all solutionsin fronts F1, F2,. . ., Fk arecopied to Pt+1; and then to choose exactly Npopulation members,

the solutions of the last front (Fk+1) are sortedby using the crowded

comparison operator in descending order andchoose the best solu-

tions needed to fill all population slots. This approach makes sure

that all non-dominated solutions (F1) are included in the next pop-

ulation if |F1|< N, and the secondary selection based on crowding

distance promotes diversity (Deb et al., 2002).

The complete procedure of NSGA-II is given as follows:

Step 1: set t= 0 and create a random parent population P0 of size

N. Step 2: the population is sorted based on the non-domination;

and each solution is assigned a rank. Step 3: theusual binarytournament selection (basedon crowded

comparison operator), crossover, and mutation operators are

used to create an offspring population Q0 of size N. Step 4: P1 = Q0. Step5:thepopulation(Pt) is sorted basedon the non-domination;

and each solution is assigned a rank. Step 6: theusual binarytournament selection (basedon crowded

comparison operator), crossover, and mutation operators are

used to create an offspring population Qt of size N. Step 7: set Rt= PtUQt. Step 8: using the fast non-dominated sorting algorithm, identify

the non-dominated fronts F1, F2,. . ., Fk in Rt.

Step 9: calculate crowding distance of the solutions in Fi. Step 10: fill up Pt+1 from Rt as follows: Case 1: if|Pt+1|+ |Fi| N, then set Pt+1 = Pt+1UFi. Case 2:if|Pt+1|+ |Fi|> Nthen addthe least crowded N |Pt+1| solu-

tions from Fi to Pt+1. Step 11: if the stopping criterion is satisfied then stop algorithm

and return Pt+1; else t= t+ 1 and go to step 5.

8.2. Develop a genetic representation of solutions to the problem

To introduce a developmental genetic representation for differ-

ent type core configuration of research reactors, each chromosome

defined by n genes. Each gene of a chromosome can have a partic-

ular and non-repeated integer number (i) between the 1 and n. n is

the maximum available number of core positions. Each gene rep-resents one of the core positions. In the other words, the reactor

assembly type at the ith core position can be identified with the ith

gene value.Then allof theused core components canhave different

specifications.

A simple decoding procedure is introduced and used. Similar

reactor components such as fuel assemblies with same burn up

are classified as a same category labeled by same integer numbers.

This simple decoding process is compatible to feed data for ANNs

estimations (Hedayat et al., 2009), and can be different for each

reactor cycle according to reactor utilities.

For this case study (Section 4), 30 positions on the core gird-

plate are chosen for core configuration optimization. Therefore

each chromosomecan onlyhave30 genes; and each gene ofa chro-

mosome can only have a non-repeated integer number between 1


8/14


Fig. 4. An example of the used chromosome and decoded form of it.

and 30. Defined case study has 21 standard fuel elements (SFE)

classified as the three different fuel batch types, four control fuel

elements, and five flexible irradiation box (I.B.) described in Sec-

tion 4. Fig. 4 shows a sample chromosome (genotype) and decoded

form of it (phenotype) for this case study (Section 4). Fig. 5 shows

the corresponding core configuration of the sample chromosome.

8.3. Creation of initial population

To create initial population, N initial chromosomes randomlycreated; where Nis the population size number. Each chromosome

composed of 30 genes. Genetic values (gene values) are randomly

created using particular integer numbers (Section 8.2).

8.4. Fitness functions definition

According to the reactor core configuration design considera-

tions (Section 3), twomain objectives andtwo problem constraints

are suggested for optimization strategy. They are as follows:Two

main objectives:

Select the best flux trap with the maximum thermal neutron flux

densities.

Select the core configuration design with the maximum life time.

Two safety and operational constraints:

Preserve the limit for PPF. Preserve the capability of control rods for the reactor shutdown.

Main objectives are considered by two separate fitness func-

tions; and safety constraints are satisfied by two penalty functions

Fig. 5. Corresponding core configuration of the sample chromosome.

(Gen and Cheng, 1996). The other main design constraint is the

capability to arrange each gained core configuration according to

the available core components. It is classified as a combinatorial

constraint satisfied using specific genetic representation (Section

8.2) and GA specific operators (Section 8.5). Maximization goal

(Section 3) is changed to a standard minimization problem (Sec-

tion 2) duringoptimization process bymultiplyingin a minus.Used

fitness functions are introduced as follows:

F(1)=

flux flux0

flux0+

penalty 1+

penalty 2 (6)

F(2) = kout kout0

kout0+ penalty 1 + penalty 2 (7)

where F is the fitness function, flux and kout are respectively the

maximum thermal neutron flux densities of the water channels

and effective multiplication factor when all control rod absorbers

are completely out of the core. Flux0 and Kout0 can be used to

change primary values to the relative changes according to refer-

ence or desired values. In this research, they are selected 11014

for Flux0 and 1 for Kout0. A penalty value (penalty 1) is added to

each of fitness functions if the PPF of an individual is larger than

the 1.9 (Mazrou and Hamadouche, 2004). And also a penalty value

(penalty 2) is added to each of fitness functions when Keff-in, the

effective multiplication factor when the control rod absorbers arecompletely in the core, is larger than 1. The second penalty pre-

serves the shutdown capability. In order to speed up optimization

process effectively, four suggested core parameters are estimated

by using four trained ANNs (Section 6).

8.5. Genetic operators

The four main genetic operators are used for optimization. They

are selection, elitism, crossover, and mutation. Developed selec-

tion and elitism strategy used in NSGA-II (Deb et al., 2002) are

introduced in Section 8.1. They are completely compatible with

presented optimization problem.

Crossover operator selects genes randomly from a pair of

chromosomes (individuals) in the current generation and thencombines them to form a child (offspring). A wide variety of

usual crossover operators (Gen and Cheng, 1999) is developed for

ordinary optimizations. But this is a combinatorial optimization

problem; and ordinary operators may produce some illegal chro-

mosomes. Primary studies are performed to select a compatible

crossover operator. Three popular crossover operators includ-

ing single point crossover, two points crossover, and scattered

crossover (Gen and Cheng, 1999) are selected. During crossover

operations, some gene values may be repeated and some of them

maybe non-cited in produced offspring.A special repairing method

is introduced to change illegal chromosomes to a legal one.

In order to introduce a developmental model each gene of

a chromosome can have a non-repeated unique integer number

within a specified interval (Section 8.2). Different types of reac-


9/14


tor core components such as standard fuel assemblies, control fuel

assemblies, andirradiation channels canbe located atthe core gird-

plate. They can be different at each specific life time of a research

reactor. They can be classified into same categories according to

their specification. Forexample fuel assemblies with thesame burn

up arelabeled with thesame integer number. Duringdecodingpro-

cess for each chromosome, gene values map to the initial labeled

category. So to keep the primary nature of the crossover opera-

tor during repairing process, repeated values are replaced with the

non-cited values which have the smallest absolute distance from

it. This method decreases changes during the repairing process

according to the final decoding process. Primary studies show that

thetwo pointscrossover is more compatible with this optimization

problem. It leads to more spread out diversity of populations.

Mutation operator applies random changes to individual par-

ents to create an offspring. Mutation adds to the diversity of a

population and thereby increases the likelihood that the algo-

rithmwill generate individuals withbetter fitness values. Similar to

crossover operator, ordinary defined mutation operators (Gen and

Cheng,1999) cannot be usedfor a combinatorialoptimizationprob-

lem. Introduced mutation procedure select randomly two genes of

a selected chromosome for mutation; and then replace them with

each other. Primary studies show that this operator is very suitable

and compatible with the core configuration optimization. Also itdoes not need any repairing process.

8.6. Main parameters of the used multi-objective genetic

algorithm

Two creation operators,crossoverand mutation,are usedto cre-

ate an offspring population Qtof size N. Each operatorshareto make

offspring population is specified by crossover fraction (fc).fc can be

a real number in a interval between the 0 and 1. In other words, in

each generation a fraction (fc) of offspringpopulation (Qt) is created

by the crossover operator while remaining offspring population is

created by the mutation operator.

The NSGA-II (Deb et al., 2002) does not need any adjustments

for fitness sharing parameters such as niche size parameter (Hornet al., 1994). It is a pure elitism GA that does not use any external

archive (Deb et al., 2002). So the population size (N) is the most

important GA parameter of the used algorithm.

There are two main parameters that can be adjusted according

to some practical tests. They are the number of population size (N)

and crossover fraction (fc).

8.7. Define a convergence criteria

A number of the best solutions belonging to the first rank are

selected as the Pareto front solutions. This number is specified by

a fraction of total population size (fp); and it limits the maximum

number of Pareto front solutions that are selected from the first

rank solutions using the crowded comparison operator.Appropriate convergence criteria must be defined to terminate

the algorithm after a sufficient number of generations. It is defined

by a limiting convergence value (vc) specification for the average

change of a spread function values over a specified number of gen-

eration numbers (nc). It means that if the average change of the

spread function values over a specified number of generations (nc)

is less than the introduced limiting value (vc), algorithm will be

terminated.

In this paper, two different spread functions are introduced

according to desired application. The first spread function used

for convergence criteria, named relative crowding spread func-

tion, is the change in crowding distance measures (Section

8.1) of individuals with respect to the previous generation.

This convergence process based on crowding distance mea-

sures preserves population diversity without requiring additional

parameters.

9. Adjusting the primary optimization parameters

The two primary parameters, the number of population size (N)

and crossover fraction (fc), must be adjusted to increase algorithm

performances. The main optimization goal is obtaining the best

possible Pareto front solutions at a reasonable time. The real Paretofront solutions cannot be specified for this problem. So a sufficient

number of Pareto fronts must be gained. They must have a uniform

spread out diversity over a wide range of fitness function values.

Maintenance of diversity is preserved by the crowded compari-

son operator during optimization process (Deb et al., 2002). After

convergence approached, a different spread function is introduced

for a fine tuning study. It is used to adjust the two optimization

parameters (N, fc). The second spread function named normalized

Euclidean distance spread function. The normalized Euclidean dis-

tance formulates as follows:

DE =

Kk=1

(fk(xi+1) fk(xi)

fmaxk

fmink

)2 (8)

where DE is the normalized Euclidean distance between the two

solutions; fmaxk

and fmink

are the maximum and minimum values of

the kth objective function (fk) respectively.

The chosen Pareto front solutions at the final generation are

sorted in an ascending order. Normalized Euclideandistancespread

function is equal to the variance, the square of the standard devia-

tion,of the normalized Euclideandistancesbetweenthe finalPareto

front solutions.

SNED =1

J

Jj=1

(Dj D)2 (9)

where Dj is the jth normalized Euclidean distance between the jth

and j + 1th Pareto front solutions.After primarily studies, listed test parameters (9: ae) are sug-

gested to adjust the number of population size and crossover

fraction.

a. Average values of the final generation number.

b. Average values of the final Pareto front number.

c. Average needed time for optimization process.

d. Average values of the minimum of each fitness function.

Fig. 6. The averagenumbersof generations and Pareto front solutionsas a function

of population size.


10/14


Fig. 7. The average used times as a function of population size.

Fig.8. Theaverage variances of Euclideanmeasuresas a functionof population size.

e. Average values of the variance of normalized Euclidean distances(Eq. (9)).

WhileGA have random operations, optimization process of each

different state is repeated 20 times; andthe average values of them

are chosen to study GA parameter adjusting.

It shouldbe noted that suggestedparameters (9:ae) were used

to test developed crossover and mutation operators capabilities

previously; results of that described in the Section 8.5.

Fig. 9. The averagenumbersof generations and Pareto front solutionsas a function

of crossover fraction.

Fig. 10. The average used times as a function of crossover fraction.

Fig. 11. The average variances of Euclidean measures as a function of crossover

fraction.

These studies are performed by using MATLAB 2008 on a PC(Pentium IV PC).

9.1. Adjusting the number of population size (N)

After primary studies 0.25, 0.8, 104 and 50 values are chosen

sequentially for defined convergence parameters (fp, fc, vc, and nc)

in the previous section (Sections 8.6 and 8.7).

A wide spread interval (50:1000) with an increasing step num-

ber 50 is used to adjust population size number (N) according to

Fig. 12. Average of the crowding distance measures as a function of the generation

number.


11/14


Fig. 13. Crowding spread values as a function of the generation number.

the suggested GA test parameters (9: ae). Each case of them is

repeated 20 times; and the average values of them are studied.

Studies result that if the population size is grater than 50, it

does not have an important effect on the range of fitness functions.

Figs. 68 show the results of the remaining tests (ac, e). Fig. 6

shows that the number of generations decreases effectively until a

population size of 500. Although the maximum number of Pareto

front solutionsstarts todecrease after a population size of 650, with

respect to the chosen ratio fraction for the Pareto front solutions

(fp), Pareto front capacity is unfilled after a population size of 500.

Fig. 8 shows that the variance of the Euclidean distance decreases

effectively until a population size of 400. After that, it is approxi-

mately constant and oscillates over a small range. According to the

comparison discussion, a population size of 500 can be suitable for

this GA optimization. Fig. 7 shows that the used time for optimiza-

tion process based on a population size of 500 is acceptable. Then a

population size of 500 is chosen for the final optimization problem.

9.2. Adjusting the crossover fraction (fc)

After primary studies 0.25, 500, 103 and 10 values are chosen

sequentially for described convergence parameters (fp, N, vc and nc)

in the previous section (Sections 8.6 and 8.7).

Theinterval (0:1) with an increasing step number0.05is used to

adjust crossover fraction (fc) accordingto thesuggested test param-

eters (9: ae). Each case of them is repeated 20 times; and the

average values of them are studied.

Fig. 14. Final Pareto fronts.

Same as a previous case (Section 9.1) studies result that the

crossover fraction does not have an important effect on the range

of fitness functions. Figs. 911 show the results of the remaining

tests (ac, e).They show that themost effective parameterto adjust

fc

is the variance of Euclidean distance. They punctuate that the

population size is selected so sufficiently for GA optimization pro-

cess previously. Also they show that the both of developedcreation

operators, crossover and mutation, are compatible with this prob-

lem type (optimization of reactor core configurations); but also

crossover fraction has effect on the Euclidean spread of the final

Pareto fronts. Fig. 11 shows that a value of 0.95 is suitable for the

crossover fraction.

10. Final optimization

Studies (described in Section 9) show that the suitable values

of the number of population size (N), and the crossover fraction

(fc) are respectively 500and 0.95 for this optimization process. The

0.5, 104, and10 values arechosenrespectively fordefined conver-

gence criteria (fp, vc and nc) at final optimization. They are enough

for algorithm convergence criteria.

Figs.12 and 13 show the average of the crowding distance mea-

sures, and relative crowding spread values as a function of the

generation number. Fig. 14 shows the final Pareto fronts.

Figs. 1214 show that optimization converged after 1108 gen-

erations; and also a uniform and dense spread out of Pareto fronts

over a wide range of fitness function values are approached. The

Table 3

Corresponding core parameters of the gained configurations providing two central water channels.

Configuration Optimization objectives Optimization constraints

Keff-out th-max (n/scm2)1014 Keff-in (


12/14


Fig. 15. Examples of the gained core configurations providing two central water channels.

average of crowding distance measures between the final Pareto

fronts is 0.0012. The final optimization process takes about 2 h on

a PC (Pentium IV PC).

11. Pareto front revisions

ANNs (Hedayat et al., 2009), that are used to estimate fitness

functions (Section 6), decrease effectively required time of opti-

mization process; while it leads to some errors during parameter

estimations. Then to select the most desired Pareto front solutions,

a completerevisionof Paretofronts valuesis needed. Itmustbe per-

formed by an accurate corecalculation model. All Pareto optimalset

solutions are recalculated by the developed software package (Core

Pattern Calculator 1). The most important core parameters includ-

ing Keff-out, Keff-in, out, in, PPF, th-max are calculated accurately

and stored on a local data base table. Also a symbolic scheme of

each desired core configuration is available to help for more accu-

rate selection. The maximum allowed number of Pareto optimal

solutions is chosen enough large to enhance final trade-off. Recal-

culated solutions can be filtered to reduce selection choices or to

preserve additional constraints such as maximum excessreactivity.

Two main objectives are introduced for core configuration

design optimization.They are the maximization of the thermal neu-

tron flux densities in the flux trap and the maximization of the core

Fig. 16. Examples of the gained core configurations providing higher neutron flux densities and pattern life time.


13/14


configuration life time by increasing the Keff-out. A relative com-

parison must be performed to select the best solution. It must be

performed according to each desired research reactor utility and

request (IAEA Technical Report, 2007). For example some gained

Pareto optimal configurations provide two central water channels

near the center of the core (Fig. 15; Table 3). They can be suitable to

locate some special material test facilities (IAEA Technical Report,

2007) or to take larger irradiation volumes.

In almostall of theresearchreactor applications(IAEATechnical

Report, 2007) maximization of thermal neutron flux densities is

the most important parameter while configuration life time must

be sufficient to each application. Table 4 shows some of Pareto

fronts that have relative enhancements to the reference core con-

figuration (described in Section 4) with respect to the both main

objectives (th-max , Keff-out). Fig. 16 shows corresponding core con-

figurations. Table 4 shows that the referencedcore configuration is

dominatedby gainedsolutions according to the problem objectives

(th-max , Keff-out). Also chosen safety and operational constraints arepreserved according to the (PPF, Keff-in) values.

Two secondary safety and operational conditions of control fuel

assemblies are suggested to help for final trade-off or take a more

fine tuning study. They are as follows:

Relative equal worth reactivity of each control rod according to

its position. A sufficient distance between the flux trap (irradiation box) and

control fuel assemblies.

The first condition is suggested to increase reactor safety; while

the secondcondition is used to decrease thermal neutron loses and

transients during radioisotope productions.

It must be noted that suggested constraint (Section 3) are guar-

anteed that the reactor can be shutdown by control rods, and also

the length of irradiation capsules can be considered in which con-

trol rod variations have not significant effects on the neutron flux

shape over them during radioisotope production periods (IAEA

Technical Report, 2007).Thetwo secondaryconditionscan be preservedby fixing control

fuel assemblies and irradiation box in a symmetrical form same as

the reference core configuration (Section 4). It can be possible if

corresponding genes are omitted from each chromosome. But it

limits search space (decision space); and may lead to loss of some

useful specific core configurations.

The suggested conditions can be used directly or relatively in

final selection too. To satisfy secondary conditions each quarter

of core can approximately have one control fuel assembly where

a minimum distance (composed of a standard fuel assembly) is

located between each control fuel assembly and the selected irra-

diation box. Each appropriate configuration that does not preserve

these criteria can be enhanced by using a simple repairing method

according to the suggested conditions. It can be possible if some

of control fuel assemblies (usually one or two) are replaced by the

same standard fuel assemblies to satisfy secondary conditions too;

in which same means assembly batches with the nearest fuel burn

up.

12. Conclusion

Four main core parameters are suggested to optimize core

configuration design of research reactors. This is a constrained

and combinatorial type of multi-objective optimization problems.

Maximization of thermal neutron flux densities at flux trap and

maximization of core configuration life time are chosen as two

separate objectives. The safety limit for PPFr (radial power peak-

ing factor) and capabilities of control fuel assemblies for reactor

shutdown are introduced by two separate penalty functions as the

problem constraints.

A fast and effective hybrid artificial intelligence algorithm are

introduced and used to optimize core configuration of research

reactors. It is composed of a fast multi-objective genetic algorithm

and trained ANNs (Hedayat et al., 2009) to estimate fitness func-

tions very quickly.

A non-dominated sorting-based multi-objective genetic algo-

rithm (NSGA-II) method (Deb et al., 2002) is developed to solve

presented optimization problem. It is a fast and elitist based multi-

objective genetic algorithm (Deb et al., 2002). Also it does not need

additional adjustments for fitness sharing parameters (Horn et al.,

1994).

Training and validation data sets must be prepared to use for

ANNs (Hedayat et al., 2009). Doing this manually takes a long time

while some human errors are possible. In this research, A software

package (Core Pattern Calculator 1) is developed and used for this

purpose. The random state of developed software is used to create

data sets which are necessary to train and test ANNs.

Main parameters of structural and learning properties of intro-

duced ANNs were adjusted according to some practical tests

separately. Total required times, the number of epochs, and the

number of necessary data sets to train ANNs are decreased effec-

tively (Hedayat et al., 2009). ANNs training and simulation results(Hedayat et al., 2009) show that introduced ANNs can be trained

and estimate suggested core parameters of the research reactor

very quickly. It improves optimization process of the core configu-

ration design effectively.

A developmental genetic representation and genetic operators

including crossover, and mutation are introduced to overcome

combinatorial constraints. Values of the main algorithm parame-

ters including population size, and crossover fraction are adjusted

according to the suggested test parameters.

Final Pareto fronts have a uniform and dense spread over a

wide range of fitness functions. The estimation by ANNs leads to

some errors. Then to select the most desired Pareto front solu-

tion, a complete revision of Pareto fronts values is needed. It must

be performed by an accurate core calculation model. Pareto opti-mal set solutions are recalculated by the recalculation state of the

developed software package (Core Pattern Calculator 1). The final

results show that the main objectives of the reference core config-

uration are dominated by Pareto front solutions while suggested

safety and operational constraints are kept. Some core configura-

tions with two central water channels are gained in the final Pareto

optimal set. They can be appropriate forsome specific material test

facilities (IAEA Technical Report, 2007).

Although control fuel assemblies can shutdown reactor accord-

ingto thesuggested constraint andalsothey do nothave significant

effects during radioisotope productionperiods,two secondary con-

ditions are suggested to help for final trade-off or improve each

selected configuration as a fine tuning study. They can be used

directly or relatively by a simple repairing method.To have a more effective and realistic control rod study while

the primary definitions for the benchmark problem (IAEATechnical

Document, 1980) is kept, the (1/V) absorber type is selected as the

material absorber for the control rods. But to calculate Keff-in, mul-

tiplication factor when control absorber rods are completely in the

core, realistically some more real conditions should be considered.

Absorber material such as Ag, In, Cd should be used in cell calcu-

lations. It is clear that to have a realistic simulation of absorber

materials, properties such as nuclear resonances must be modeled

carefully because, they can have large effects on the final ANNs

estimations results. On the other hand, they have so large neu-

tron absorption cross sections to consider a desired and operational

shutdown margin as a primarysafety constraint. Thenthe limitcon-

sideration for the capability of reactor shutdown can be change to


14/14


a desired shutdown margin using operational absorbers with large

neutron absorption cross sections. Also to increase safety margins,

some other safety and operational constraints such as the maxi-

mum excess reactivity can be considered thorough optimization

process or during final trade-off.

To get the highest thermal neutron flux densities, the one fixed

andcentral fluxtrap is replaced by fiveflexible irradiation boxes.In

each of the different core configurations, one of them has the max-

imum neutron flux densities; and it is chosen as a desired flux trap.

This method leads to a wide variety of search space during opti-

mization process, while it also results in larger prediction errors.

Because it saves a lot of time, gained ANNs errors can be acceptable

for a population-based multi-objective optimization algorithms

(like used NSGA-II). To decrease final prediction error of th-max,a fixed flux trap same as the reference BOC flux trap can be chosen

while it decreases search space.

In order to reduce time and tasks of fuel reloading process sym-

metrical and fixed positions for the control fuel assemblies can be

used.Also accordingto specific anddesired utilitiesof eachresearch

reactor one or more water channel elements can be chosen as a

fixed andspecified flux trap (irradiation box) too. These criteria can

be satisfied byomittingeach corresponding gene. Itmeansthatcor-

responding fixed positions are not explored during optimization.

This leads to smaller errors of ANN predictions, lesser core reload-ing tasks, and more simple trade-offs; but this reduces and limits

decision space.

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Optimization of the Core Configuration Design Using a Hybrid