A binary mixed integer coded genetic algorithm for ...

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

Do Quang Binh, Ngo Quang Huy and Nguyen Hoang Hai

A binary mixed integer coded genetic algorithm for multi-objective optimization of nuclear research reactor fuel reloading This paper presents a new approach based on a binary mixed integer coded genetic algorithm in conjunction with the weighted sum method for multi-objective optimization of fuel loading patterns for nuclear research reactors. The proposed genetic algorithm works with two types of chromosomes: binary and integer chromosomes, and consists of two types of genetic operators: one working on binary chromosomes and the other working on integer chromosomes. The algorithm automatically searches for the most suitable weighting factors of the weighting function and the optimal fuel loading patterns in the search process. Illustrative calculations are implemented for a research reactor type TRIGA MARK II loaded with the Russian VVR-M2 fuels. Results show that the proposed genetic algorithm can successfully search for both the best weighting factors and a set of approximate optimal loading patterns that maximize the effective multiplication factor and minimize the power peaking factor while satisfying operational and safety constraints for the research reactor. Binärkodierter gemischt-ganzzahliger genetischer Algorithmus zur multiobjektiven Optimierung der Umladung von Brennelementen eines Forschungsreaktors. In der vorliegenden Arbeit wird ein neuer Ansatz vorgestellt, der auf einem binärkodierten gemischt-ganzzahligen genetischen Algorithmus basiert in Verbindung mit der gewichteten Summenmethode für die multiobjektive Optimierung des Beladungsschemas der Brennelemente eines Forschungsreaktors. Der vorgeschlagene genetische Algorithmus arbeitet mit zwei Arten genetischer Operatoren: einem auf der Grundlage binärer Chromosomen und einem anderen auf der Grundlage ganzzahliger Chromosomen. Der Algorithmus sucht automatisch nach den am besten geeigneten Wichtungsfunktionen und dem optimalen Beladungsschema für die Brennelemente. Illustrative Berechnungen werden durchgeführt für einen Forschungsreaktor vom Typ TRIGA MARK II beladen mit russischen VVR-M2 Brennelementen. Die Ergebnisse zeigen, dass der vorgeschlagene genetische Algorithmus erfolgreich die besten Wichtungsfaktoren und eine Reihe optimaler Beladungsschemata sucht, wodurch der effektive Multiplikationsfaktor maximiert und der Leistungsspitzenfaktor minimiert wird, unter Einhaltung betrieblicher Beschränkungen und Sicherheitsauflagen des Forschungsreaktors.

1 Introduction The fuel reloading optimization problem contains a diversity of objectives. As pointed out by Turinsky [1], the objectives 79 (2014) 6

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include the maximization of end of cycle (EOC) reactivity, the maximization of discharged burnup, the minimization of power peaking, the minimization of feed enrichment, the minimization of the fresh fuel inventory, the minimization of the burnable poison inventory. Constraints include limits on discharged fuel burnup, maximum power peaking factor (PPF), minimum cycle length, moderator reactivity feedback, and so on. It is apparent that the fuel reloading problem is really a multi-objective optimization problem, where an improvement in one objective is often only gained at the cost of deterioration in other objectives. Traditional methods of fuel reloading that have been adopted to assure the flat radial power distribution over the reactor core are the ‘out-in’ pattern, and the checkerboard pattern for pressurized water reactors (PWRs) and boiling water reactors (BWRs) [2], and the ‘bi-directional axial’ method for the CANDU reactor [3]. Evolution of researches into in-core fuel management optimization problem results in the development of nuclear fuel management optimization capabilities for PWRs and BWRs ranging from the employment of experience-based rules to the usage of mathematical approaches. Kropaczek and Turinsky [4] combined simulated annealing with a reactor physics model based on second-order accurate generated perturbation theory to find near optimal loading patterns (LPs) for a variety of different objectives and constraints. Steven et al. [5] extended the use of simulated annealing combined with heuristics to search for optimal reload patterns for a three loop PWR. Lin et al. [6] applied the simple tabu method incorporated with heuristics to automatically search for feasible fuel LPs that minimize the nuclear enthalpy rise hot channel factor through the whole cycle for a PWR. De Chaine and Feltus [7, 8] utilized Genetic Algorithms (GAs) and expert knowledge to optimize fuel reloading patterns. Poon and Parks [9] and Parks [10] used GAs combined with generated perturbation theory to identify and offer a family of solutions lying on the trade-off surface between competing objectives, to maximize the EOC boron concentration, minimize discharge burn-up and minimize PPF. Recent research works on the problem of fuel loading pattern optimization have often utilized various evolutionary methods. Do et al. [11, 12] have developed an evolutionary algorithm based on a GA, the elitist strategy and heuristic rules, for a multi-cycle and multi-objective optimization of the refueling simulation for a CANDU reactor. Particle swarm optimization algorithm has been developed to optimize fuel core LP for a typical VVER-1000 and a PWR [13, 14]. An enhanced integer coded genetic algorithm was developed to design LPs for a PWR by Norouzi et al. [15]. The harmony search with different mutation based on pith adjustment was 1

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

applied to PWR reloading optimization [16]. A new quantum inspired evolutionary algorithm was developed to apply to the optimization of nuclear reactor core fuel reload [17]. The artificial bee colony algorithm was used to find an optimal configuration of fuel assemblies for a VVER-1000 reactor [18]. A new meta-heuristic optimization strategy combined with the firefly algorithm, which is based on the idealized behavior of the flashing characteristics of fireflies, was developed to optimize the nuclear reactor LP for PWRs [19]. It is realized from a brief overview of the past research works related to the nuclear reactor fuel reload optimization problem that a lot of methods for solving the problem have been discovered and evolutionary algorithms can work efficiently to search for solutions that optimize several objectives while satisfying operational and safety constraints. One of the most common methods for multi-objective optimization is the weighted sum method [20], in which the objective function is usually formulated in the form: F¼

X

wi gðPi Þ

ð1Þ

i

where wi is a weighting factor imposing importance on the objective Pi, g(Pi) is a function of Pi. A typical characteristic of the refueling optimization problem is computation time consuming because reactor calculation, which needs much time, is necessary part of the solution procedure. For time-consuming calculations, the weighting factors in the objective function are often determined by the Trial and Error method through preliminary examining the behavior of the search process and kept unchanged during the search [9, 10, 12]. It is obvious that keeping the weighting factors constant during the search process limits the search directions, so limits the search space, and hence the search may approach only to the local optimal solutions, not the global optima. Varying the weighting factors randomly in an orientated search may be a way to overcome the above limitation. In this work, we develop a new method based on GAs in conjunction with the weighted sum method to automatically determine the weighting factors in the objective function for the duration of the search process. The search scheme based on the new method also finds the weighting factors and the optimal fuel reloading patterns simultaneously. Because the weighting factors are not kept constant and determined in the search process, it is hoped that the search direction is flexible and the search may move towards a set of approximate global optimal solutions.

2 Optimization problem In this study, a typical fuel reload optimization problem for a research reactors is set up to search for fuel LPs that maximize the effective multiplication factor keff and minimize the power peaking factor PPF while satisfying operational and safety constraints. The two objectives are keff and PPF. According to the weighted sum method, the objective function of the optimization problem can be written as: F ¼ (ðkeff % 1Þ þ bðPPF0 % PPFÞ

ð2Þ

where ( and b are the weighting factors for keff and PPF, respectively; in the standard form, ( + b = 1; PPF0 is an input factor that is chosen so that PPF is always lower than it. The constraints include: kmin 4 keff 4 kmax ;

2

ð3Þ

PPF 4 PPFmax ;

ð4Þ

BU 4 BUmax ;

ð5Þ

where kmin and kmax are the limits of keff, PPFmax is the maximum limit of PPF, BU is the fuel burn-up, and BUmax is its maximum value. Decision variable of the optimization problem is a fuel LP that is just the core configuration consisting of fresh and burnt fuels at the beginning of cycle. Eqs. (3) and (4) are the reactivity and thermal limits. Constraint (5) defines the set of burnt bundles remained in the reactor core for the next cycle. The constraints define a region of feasible solutions of the problem. By writing the objective function in the form of Eq. (2), the problem of minimizing PPF is transformed into a maximization problem. In the standard form, sum of the weighting factors ( and b equals to 1, so when one of the factors is determined, the other is also defined. 3 Methodology In this work, we propose a new genetic algorithm that is based on the standard genetic operators with a search procedure that simultaneously finds the optimal fuel LPs and the weighting factors. The proposed genetic algorithm works with two types of chromosomes: integer chromosome and binary chromosome. The integer chromosome represents fuel LPs and the binary chromosome represents weighting factors. And the two different kinds of genetic operators are required to work on the two types of chromosomes. The coding procedures for the two types of chromosomes are described below. 3.1 Coding procedure for fuel LPs The genotype structure for fuel LPs is a one-dimensional integer chromosome. The coding procedure for fuel LPs is as described below: . Consider a reactor core consisting of N positions for fuel loading and the total number of fuel bundles (FBs) loaded in the core is also equal to N. . First, number all the core positions by integers from 1 to N. . Then assume a base LP by replacing the discharge FBs by fresh FBs. An FB in the base LP is then assigned with the same number as the core position into which the FB is loaded as seen in Fig. 1. . Encoding: an LP is encoded into a chromosome of length N that is a string of N integer numbers (i1 i2 ... iN), where ik = {1, ..., N} is the FB number. The position of gene ik in the chromosome defines the core position pos(k) into which the FB ik is loaded. . Decoding: the chromosome (i1 i2 ... iN) is decoded into an LP by loading FB number ik into position number pos(k) in the reactor core. 3.2 Coding procedure for weighting factors Weighting factors are coded into binary strings due to their value. The coding procedure for weighting factors is the same as conventional binary coding procedure for real numbers. Because the optimization problem in this study has two weighting factors and their sum is 1, only one of the weighting factors needs to be found in the search process. If the factor ( is found, the factor b is certainly defined by (1 – (). Below is the coding procedure for the factor (. 79 (2014) 6

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

Fig. 1. Base LP of the reactor at the end of cycle 1 XXX: fuel bundle number YYY: burnup of fuel bundle number XXX (%) (% of Dm5/m50, where Dm5 is the spent amount of 235U, and m50 is the original amount of 235U in a fresh fuel bundle)

The factor ( is encoded into a binary chromosome that is a string of bits 1 or 0. A chromosome is represented symbolically by the string (m1m2...ml), where each mi (i = 1, l) may take on a value 1 or 0. The length of the string is determined by the precision of ( and it’s limiting range. The length l is the minimum integer that satisfies the following formula: ð(max % (min Þ 3 10n 4 2l % 1

ð6Þ

where (min and (max are the minimum and maximum values of (, and n is the number of digits following the decimal point in the number that represents the value of (. As 0 £ ( £ 1, the above equation becomes: 10n 4 2l % 1

ð7Þ

The binary string is decoded into the real value of ( based on the following formula: ( ¼ (min þ

l l (max % (min X 1 X mi & 2l%i mi & 2l%i ¼ l 2 % 1 2l % 1 i¼1 i¼1

ð8Þ

3.3 Genetic operators for integer chromosomes In general, GAs work with three genetic operators – selection, crossover and mutation [21]. Selection carries better solutions into the next generation based on their fitness values. Crossover mixes parts of two parent solutions to create two different off-springs. Finally, mutation makes some small random changes in the solutions maintaining the diversity of population to prevent a premature convergence to local optima. In this study, the elitist strategy (ES) is used to preserve the best solutions during the search process. 79 (2014) 6

Selection is implemented by the Russian roulette wheel spin method modified by ES proposed by Park [8], which is performed by creating an archive that contains non-dominated solutions in terms of the two objectives. In our problem, a solution X with PPF1 and keff1 is dominated by a solution Y with PPF2 and keff2 if PPF2 £ PPF1 and keff2 ‡ keff1, and there exists at least PPF2 < PPF1 or keff2 > keff1. Any solution that is not dominated by others is regarded as a non-dominated solution. Every solution in the archive is directly transferred to a breeding pool for the next generation. Crossover is carried out by the one-point method. For onedimensional integer chromosomes, the one-point crossover is performed by two steps: two members of the breeding pool to be mated are randomly selected with a crossover probability, then the two chromosomes undergo crossing over as follows: an integer position k along the chromosome is selected uniformly at random between 1 and the chromosome length less one N – 1. Two new strings are created by swapping all characters between positions k + 1 and N inclusively. With the above coding procedure, a modification on the crossover operator is needed [22]. In case two parents have some genes with the same number, crossing these two parents over may create two off-springs which have some identical genes. This kind of chromosome is not suitable for the refueling problem. If this case occurs, small random numbers between 0 and 1 are added to the genes with the same value to make a difference between them, and then rank these genes again to make two new off-springs. Mutation is conducted by a binary shuffle of two genes in the chromosome. First, a chromosome to be mutated is randomly selected with a mutation probability. Then two uniformly selected genes of the chromosome are exchanged their positions. 3

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

3.4 Genetic operators for binary chromosomes Genetic operators for the binary chromosomes in this study are also the same as in the standard GAs. Selection is also performed by the roulette wheel spin method that chooses the fittest individuals in the current generation to create a breeding pool for the next generation. Crossover is performed by the one-point method that mixes parts of the two parent solutions in the breeding pool to create two off-springs. The one-point crossover is implemented as follows: first, a pair of strings in the breeding pool is randomly chosen with a certain probability; then a crossing site k between positions l and l – 1 is generated uniformly at random; finally, two new strings are created by swapping all the characters between positions k + 1 and l inclusively. Mutation is performed by randomly altering the value of a bit between 1 and 0 with a small probability. 3.5 Search scheme Search process progresses as follows: Step 1: Create initial populations for binary chromosomes (population type I) and for integer chromosomes (population type II) randomly. Step 2: Decode each chromosome in population type I in the current generation into a real number in the interval [0, 1] for (, then calculate b = 1 – (; and decode each chromosome in population type II in the current generation into a fuel LP. Step 3: Perform global reactor calculation to evaluate keff and PPF of each chromosome in population type II. Step 4: Check if all the chromosomes in population type II are archived or not in order to create the archive with nondominated solutions. Step 5: Select randomly a chromosome in population type I and a chromosome in population type II to form a population type III that consists of pairs of binary and integer chromosomes. Step 6: Determine the fitness function based on equation (2) with the values of (, b, keff and PPF corresponding to the selected pairs of binary and integer chromosomes in population type III. Step 7: Check the stopping condition. The search process stops if the population-average fitness value does not improve in several sequential generations or when the number of generations exceeds a specified value. Step 8: Perform the selection to update the breeding pool for the new generation of population type I (breeding pool I), then perform the selection to update the breeding pool for the new generation of population type II (breeding pool II). Step 9: Perform the crossover for individuals in breeding pool I, and then perform the crossover for individuals in breeding pool II. Step 10: Perform the mutation for every gene of each chromosome in breeding pool I, and perform the mutation for each chromosome in breeding pool II. Step 11: Update the new populations for the next generation and repeat the search process from Step 2. The archive of non-dominated solutions for population type II is created in the following way [22]. At the beginning, any non-dominated solution that satisfies all the constraints on keff and PPF is stored in the archive. When the archive is full, each trial solution is compared with existing members in the archive. If it dominates any member of the archive, that member is removed and the new solution is then added. If any member of the archive dominates a new solution, that so4

lution is not archived. If a solution neither dominates nor is dominated by any member of the archive, but is better than some solution in the archive in terms of the fitness value, it will replace for the member of the archive with the lowest fitness value. The size of the archive equals a quarter of the population size in this study. All the members of the archive are transferred directly to the breeding pool and the selection operator takes other members for the breeding pool from the current generation by the roulette wheel spin method. 4 Calculation results and discussions 4.1 Reactor calculation This study uses a research reactor located in Da Lat, Vietnam for illustrative calculations. This reactor is a TRIGA MARK II reactor type, loaded with the Russian fuel type VVR-M2. The reactor core consists of a 121-cell hexagonal lattice of FBs, controls rods, beryllium rods, irradiation channels, some beryllium blocks, and is surrounded by a graphite reflector. The reactor was loaded with fresh fuels with 36 % 235U enrichment in the first criticality. The first fuel reloading of the reactor was performed after it had been in operation of about 12 800 hr at nominal power of 500 kW. At that time, all the 89 existing FBs in the core were unchanged; only 11 beryllium rods at the core periphery were replaced by fresh FBs. This core configuration defines the base LP as seen in Fig. 1. Global reactor calculation in this study is performed by a 3D finite difference multi-group diffusion theory code CITATION [23]. The group constants for use in CITATION are generated by using WIMSD-5B [24]. Reactor calculations provide the values of keff and PFF for fuel LPs. 4.2 Optimization calculation The fitness function is defined based on equation (2) with the value of PPF0 = 1.40, which is the maximum value of PPF saved in preliminary investigations. Optimization calculation based on the proposed GA in this study is performed with the size of 30 for both population types. An integer chromosome has the length of 100, which is the total FBs loaded in the reactor core. A binary chromosome has the length of 17, representing a real number between 0 and 1 with 5 digits after the decimal point. The search process progresses through 200 generations. Genetic parameters are the same for both population types with crossover probability pc = 0.5, and mutation probability pm = 0.001. By the way, one can choose different values for pc and pm and the different population sizes for two population types, but the size of population type III always equals to the size of population type II. Results from a GA run are presented in the figures below. Figure 2 shows the change in the population average weighting factor ( with generation, indicating that the value of ( increases from a random value around 0.4 to an equilibrium value of 0.95. Figure 3 presents the distribution of ( value at the initial, the 100th and the final generations in a radar type chart, indicating the convergence of the population of ( in the search process. This result indicates that the search directs the change in the weighting factor ( to a stable value. Figures 4 and 5 present the changing direction of keff and PPF in the search process. These figures show that the search process maximizes the effective multiplication factor keff and minimizes the power peaking factor PPF simultaneously. The population average values of the objectives strongly change at the beginning stage, and after about 100 generations approach to the 79 (2014) 6

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

stable values with a relative large variation. However, the fitness value is more stable than the values of keff and PPF at the final stage as seen in Fig. 6. Figure 7 shows the convergence of the search for keff and PPF through the miniature of population in a two-objective diagram at the final genera-

tion. The results indicate that not only the two objectives but also the weighting factors converge to their best stable values in the final stage of the search process. The archive consists of 7 best LPs, which have the same fitness of 0.06397. The maximum value of keff of the solutions in the archive is keffmax =

Fig. 2. Change in the weighting factor a versus generation Fig. 5. Change in the power peaking factor PPF of population versus generation

Fig. 3. Distribution of the weighting factor a in the initial, 100th, and 200th generations in the radar type chart

Fig. 6. Change in the fitness versus generation

Fig. 4. Change in the effective multiplication factor keff of population versus generation

Fig. 7. Distribution of solutions at the initial, middle and final generations in a keff-PPF diagram

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

1.06541, and the minimum value of PPF is PPFmin = 1.34208. One of the best LPs in terms of the given objectives has the maximum keff of 1.06541 and PPF = 1.34296, and another has

keff = 1.06536 and the minimum PPF of 1.34208. The two LPs are shown in Figs. 8 and 9. It can be seen that the two LPs have a lot of positions loaded with the same FBs, especially

Fig. 8. The best LP with the maximum keff in the archive XXX: fuel bundle number YYY: fuel burnup of fuel bundle number XXX (%)

Fig. 9. The best LP with the minimum PPF in the archive XXX: fuel bundle number YYY: fuel burnup of fuel bundle number XXX (%)

6

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Do Quang Binh et al.: A binary mixed integer coded genetic algorithm

the positions at the center of the reactor core. The difference appears in positions next to the outermost ring of the core. All the archived LPs are much better than those of the fuel LP used in practice with keff = 1.0604 and PPF = 1.370. The gain in keff of the best LPs compared to the practical LP allows the reactor to operate about more than 2 000 hr longer at the nominal power. It is clear that these LPs are low-leakage LPs with the fresh FBs loaded in inner positions of the reactor core. 5 Conclusions A binary mixed integer coded genetic algorithm in conjunction with the weighted sum method was developed for multiobjective optimization of nuclear reactor fuel reloading. It has a capability of automatically finding the weighting factors in the objective function, and searching for the weighting factors and optimal fuel LPs simultaneously. The proposed algorithm can successfully search for the fuel LPs that maximize the effective multiplication factor keff and minimize the power peaking factor PPF while satisfying operational constraints for a research reactor. In this study, only two objectives are examined; however, the method does not impose any limit on the number of objectives. One can use this method to treat a lot of objectives conveniently. The best LPs found contained in the archive has advantages compared to the LP used in practice, and they are low-leakage LPs with fresh FBs loaded into inner positions of the reactor core. Acknowledgements This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 103.04-2012.61. (Received on 1 April 2014)

References 1 Turinsky, P. J.: Nuclear fuel management optimization: a work in progress. Nuclear Technology 151 (2005) 3 2 Bell, G. I.; Glasstone, S.: Nuclear reactor theory. Van Nostrand Reinhold Company, New York (1970) 3 IAEA Directory of nuclear reactors. Vol. IV (1962) 4 Kropaczek, D. J.; Turinsky, P. J.: In-core nuclear fuel management for pressurized water reactors utilizing simulated annealing. Nuclear Technology 95 (1991) 9 5 Stevens, J. G.; Smith, K. S.; Rempe, K. R.; Downar, T. J.: Optimization of pressurized water reactor shuffling by simulated annealing with heuristics. Nuclear Science and Engineering 121 (1995) 67 6 Lin, C.; Yang, J. L.; Lin, K. J.; Wang, Z. J.: Pressurized water reactor loading pattern design using the simple tabu search. Nuclear Science and Engineering 129 (1998) 61 7 De Chaine, M. D.; Feltus, M. A.: Nuclear fuel management optimization using genetic algorithms. Nuclear Technology 111 (1995) 109 8 De Chaine, M. D.; Feltus, M. A.: Fuel management optimization using genetic algorithms and expert knowledge. Nuclear Science and Engineering 124 (1996) 188 9 Poon, P. W.; Parks, G. T.: Application of genetic algorithms to incore fuel management optimization. Proceedings of the Joint Conference on Mathematical Methods and Supercomputing in Nuclear Applications, Karlsruhe, Germany, 1993, p. 777 10 Parks, G. T.: Multiobjective pressurized water reactor reload core design by non-dominated genetic algorithm search. Nuclear Science and Engineering 124 (1996) 178 11 Do, Q. B.; Choi, H.; Roh, G.: An evolutionary optimization of the refueling simulation for a CANDU reactor. IEEE Transactions on Nuclear Science 53 (2006) 2957, DOI:10.1109/TNS.2006.882369

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12 Do, Q. B.; Roh, G.; Choi, H.: Optimal refueling pattern search for a CANDU reactor using a genetic algorithm. Proceedings of the International Congress on Advance Power Plant 2006, Nevada, USA, 2006, p. 2422 13 Babazadeh, D.; Boroushaki, M.; Lucas, C.: Optimization of fuel core loading pattern design in a VVER nuclear power reactor using particle swarm optimization (PSO). Annals of Nuclear Energy 36 (2009) 923, DOI:10.1016/j.anucene.2009.03.007 14 Yadav, R. D. S.; Gupta, H. P.: Optimization studies of fuel loading pattern for a typical pressurized water reactor (PWR) using particle swarm method. Annals of Nuclear Energy 38 (2011) 2086, DOI:10.1016/j.anucene.2011.05.019 15 Norouzi, A.; Aghaie, M.; Mohamadi Fard, A. R.; Zolfaghari, A.; Minuchehr, A.: Nuclear reactor core optimization with Parallel Integer Coded Genetic Algorithm. Annals of Nuclear Energy 60 (2013) 308, DOI:10.1016/j.anucene.2013.05.013 16 Aghaie, M.; Nazari, T.; Zolfaghari, A.; Minucherhr, A.; Sharani, A.: Investigation of PWR core optimization using harmony search. Annals of Nuclear Energy 57 (2013) 1, DOI:10.1016/j.anucene.2013.01.046 17 Da Silva, M. H.; Schirru, R.: Optimization of nuclear reactor core fuel reload using the new quantum PBIL. Annals of Nuclear Energy 38 (2011) 610, DOI:10.1016/j.anucene.2010.09.010 18 De Oliveira, I. M. S.; Schirru, R.: Swarm intelligence of artificial bees applied to in-core fuel management optimization. Annals of Nuclear Energy 38 (2011) 1039, DOI:10.1016/j.anucene.2011.01.009 19 Poursalehi, N.; Zolfaghari, A.; Minuchehr, A.: Multi-objective loading pattern enhancement of PWR based on the discrete firefly algorithm. Annals of Nuclear Energy 57 (2013) 151, DOI:10.1016/j.anucene.2013.01.043 20 Marler, R. T.; Arora, J. S.: Survey of multi-objective optimization methods for engineering. Structural and Multidisciplinary Optimization 26 (2004) 369, DOI:10.1007/s00158-003-0368-6 21 Goldberg, D.F.: Genetic algorithms in search optimization and machine learning. Addison Wesley, Reading, Massachusetts (1989) 22 Do, Q. B.; Nguyen, P. L.: Application of a genetic algorithm to the fuel reload optimization for a research reactor. Applied Mathematics and Computation 187 (2007) 977, DOI:10.1016/j.amc.2006.09.024 23 Fowler, T. B.; Vondy, D. R.; Kemshell, F. B.: Nuclear reactor core analysis code: CITATION. ORNL-TM-2496, RSICC (1971) 24 AEA Technology: WIMSD – A neutronics code for standard lattice physics analysis. ANSWERS Software Service (1997)

The authors of this contribution Do Quang Binh (corresponding author) University of Technical Education Ho Chi Minh City 01 Vo Van Ngan Street, Thu Duc District Ho Chi Minh City, Vietnam E-mail: [email protected] Ngo Quang Huy University of Industry Ho Chi Minh City 12 Nguyen Van Bao street, Ward 4, Go Vap District Ho Chi Minh City, Vietnam E-mail: [email protected] Nguyen Hoang Hai Centre for Research and Development of Radiation Technology 202A Street 11, Linh Xuan Ward, Thu Duc District Ho Chi Minh City, Vietnam E-mail: [email protected]

Bibliography DOI 10.3139/124.110447 KERNTECHNIK 79 (2014) 6; page 1 – 7 ª Carl Hanser Verlag GmbH & Co. KG ISSN 0932-3902 7