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8/3/2019 Optimization of the Core Configuration Design Using a Hybrid
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Nuclear Engineering and Design 239 (2009) 27862799
Contents lists available at ScienceDirect
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/002954938/3/2019 Optimization of the Core Configuration Design Using a Hybrid
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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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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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A. Hedayat et al. / Nuclear Engineering and Design239 (2009) 27862799 2789
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
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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-
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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.
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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.
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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 (
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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.
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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
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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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