2007 International Nuclear Atlantic Conference - INAC 2007 Santos, SP, Brazil, September 30 to October 5, 2007 ASSOCIAÇÃO BRASILEIRA DE ENERGIA NUCLEAR - ABEN ISBN: 978-85-99141-02-1 THE PBIL ALGORITHM APPLIED TO A NUCLEAR REACTOR DESIGN OPTIMIZATION Marcelo D. Machado 1, Jose A. C. C. Medeiros 1, Alan M. M. de Lima 1 and Roberto Schirru 1 1 Instituto Alberto Luiz Coimbra de Pós-Graduação e Pesquisa de Engenharia (COPPE/UFRJ) Programa de Engenharia Nuclear - Laboratório de Monitoração de Processos (PEN/LMP) Av. Horácio Macedo, 2030 Bloco G Sala 206 Centro de tecnologia Cidade Universitária CEP 21941-914 Rio de Janeiro, RJ marcelo@lmp.ufrj.br canedo@lmp.ufrj.br alan@lmp.ufrj.br schirru@lmp.ufrj.br ABSTRACT The Population-Based Incremental Learning (PBIL) algorithm is a method that combines the mechanism of genetic algorithm with the simple competitive learning, creating an important tool to be used in the optimization of numeric functions and combinatory problems. PBIL works with a set of solutions to the problems, called population, whose objective is create a probability vector, containing real values in each position, that when used in a decoding procedure gives subjects that present the best solutions for the function to be optimized. In this work a new form of learning for algorithm PBIL is developed, having aimed at to reduce the necessary time for the optimization process. This new algorithm will be used in the nuclear reactor design optimization. The optimization problem consists in adjusting several reactor cell parameters, such as dimensions, enrichment and materials, in order to minimize the average peak-factor in a 3-enrichement zone reactor, considering some restrictions. In this optimization is used the computational code HAMMER, and the results compared with other methods of optimization by artificial intelligence. 1. INTRODUCTION Evolutionary algorithm (EA) is a generic term used to indicate any algorithm of optimization based on a population and that uses operators inspired by biological mechanisms of evolution, such as the selection, reproduction and mutation. Candidates the solution of the optimization problem are individuals of this population and a fitness determines the capacity of one determined individual to survive or not. The evolution of the population is carries through repeated applications of the operators used by the algorithm. The evolutionary algorithms do not need derivatives, as some traditional methods of optimization, no knowledge concerning the search space, what it allows that they are used in almost all the types of optimization problems. EAs are often viewed as a global optimization method although convergence to a global optimum is only guaranteed in a weak probabilistic sense. However, one of the strengths of EAs is that they perform well on "noisy" functions where there may be multiple local optima. EAs tend not to get "stuck" on a local minima and can often find a globally optimal solutions. EAs are well suited for a wide range of combinatorial and continuous problems, though the different variations are tailored towards specific domains.
The main objective of this work is to investigate the use of a New Population Based Incremental Learning algorithm (NPBIL) in a nuclear core optimization problem. The remainder of the paper is organized as follows. In the next section the New Population Based Incremental Learning algorithm is presented. In section 3 the reactor design optimization problem is described. In section 4, the implementation of the algorithms is briefly described and the results are shown. Finally, in section 5, the concluding remarks are made. 2. THE NEW PBIL ALGORITHM The algorithm Population-Based Incremental Learning (PBIL)[1] is a method that combines the mechanisms of the genetic algorithm and simple competitive learning, generating an important tool to be used in the optimization of numerical functions and combinatorial problems. The PBIL works with a set of solutions of the problem, called population, of form codified in chains of bits. Being that the form most common of used codification is the binary one, or either, using 0 and 1. The objective of the PBIL is to create a vector probability, contends real values in each position of the chain of bits, that to the being used in a decoding process, generates individuals that represent the best solutions it function to be optimized. Initially, the values of the probability vector are initialized to 0.5, this determines that the probability of generation of value "0" or "1" in each position is equal. Sampling from this vector reveals random solution vectors in the initial population. The manner in which the updates to the probability vector occur is similar to the weight update rule in supervised competitive learning networks, or the update rules used in Learning Vector Quantization (LVQ). The values in vector probability go being gradually modified of the initial value of 0.5 for next values 0.0 or 1.0, in order to represent the best individuals found in the population, for each generation. During the search process, to each generation, the values of the vector probability are modified with the following rule: P(i) = P(i) x ( 1.0 Lr ) + ( X(i) x Lr ) (1) where Lr = Learning rate P(i) = value of probability in position i X(i) = value of position i in the best vector of the population In this work, an alteration in the value of the learning rate is considered to be used in the update of the vector probability. In new algorithm, called NPBIL, the value of the learning rate is modified of proportional form to the value of fitness of individual in relation to best individual present throughout all the generations[2]. Thus, the learning rate of one determined individual k is defined of the following form:
Lr (k) = Lr x Best_Fitness / Fitness(k) (2) where Lr = Learning rate Lr (k) = Learning rate of individual k Best_Fitness = best fitness in all generations Fitness = fitness of individual k and the update of the vector probability is: P(i) = P(i) x ( 1.0 Lr(k) ) + ( X( k, i) x Lr(k) ) (3) where P(i) = value of probability in position i Lr (k) = Learning rate of individual k X(k, i) = value of position i in the individual k 3. PROBLEM DESCRIPTION Consider a cylindrical 3-enrichment-zone PWR, with typical cell composed by moderator (light water), cladding and fuel. The problem consists in adjusting several reactor cell parameters, such as dimensions, enrichment and materials, in order to minimize the average peak-factor in a 3-enrichment-zone reactor[3]. The design parameters that may be changed in the optimization process, as well as their variation ranges are shown in Table 1. Table 1. Parameters Range Parameter Symbol Range Fuel Radius (in.) Rf 0.2 to 0.5 Cladding Thickness (in.) c 0.01 to 0.1 Moderator Thickness (in.) Re 0.01 to 0.3 Enrichment of Zone 1 (%) E1 2.0 to 5.0 Enrichment of Zone 2 (%) E2 2.0 to 5.0 Enrichment of Zone 3 (%) E3 2.0 to 5.0 Fuel Material Mf U-Metal or UO2 Cladding Material Mc Zircaloy-2, Aluminum or Stainless-304
4. IMPLEMENTATION AND RESULTS 4.1. Implementation The new algorithm NPBIL is characterized by four parameters: The population size, number of samples to generate based upon each probability vector before an update. This was kept constant at 70 individual The learning rate, which is changed by the equation (2) in each generation. Ten values of these parameter, for the accomplishment of the tests, had been chosen. The third is the number of vectors to update the learning rate, only the best three vectors were used to this update. The last is the number of generations. This parameter was kept at a constant 1000. Each individual of the population of algorithm NPBIL was shaped in a binary chain contends eight substrings of 7 bits, representing each one of the parameter used in the solution of the problem. These substrings had been decoded in its entire values and later adjusted for the band of values allowed for each parameter The algorithm NPBIL is responsible for the generation of solutions for the optimization problem that are sent to the Reactor Physics code HAMMER[4], that evaluates each one of this solution and calculates the power-peaking, average thermal flux and the effective multiplication factor. These results are combined in a value of fitness that, if all constraints are satisfied, has the value of the average peak factor, that it must be minimized. Otherwise, it is penalized proportionally to the discrepancy on the constraint. 4.2. Results Table 2 shows the results obtained by NPBIL in comparison to the Standard Genetic Algorithm (SGA)[5], the Great Deluge Algorithm (GDA) [6] and standard Population_Based Incremental Learning (PBIL) [2]. The results of SGA and GDA are the presented in reference [6] for 100,000 iterations.
Table 2. Results for 10 experiments Experiment SGA GDA PBIL NPBIL #1 1.3185 1.2806 1.3757 1.3197 #2 1.3116 1.2913 1.3222 1.2801 #3 1.3300 1.2856 1.3198 1.2793 #4 1.3294 1.2891 1.3155 1.2818 #5 1.3595 1.2863 1.2972 1.2827 #6 1.3562 1.2845 1.3284 1.2783 #7 1.3372 1.2897 1.3294 1.2796 #8 1.3523 1.2842 1.3243 1.2837 #9 1.3614 1.2895 1.3298 1.2807 #10 1.3467 1.2827 1.3239 1.2812 5. CONCLUSIONS The new developed algorithm NPBIL, showed a simple optimizer and capable to carry a effective search in complex spaces without the necessity of knowledge about search space and in a reduced computational time in relation to the others artificial intelligence techniques. The good results gotten with algorithm NPBIL in comparison with the Standard Genetic Algorithm and the Great Deluge Algorithm, demonstrate the capacity of the use of this new algorithm in the solution of complex numerical problems. To future work, we are developing a model that makes possible the use of algorithm NPBIL in the solution of combinatorial problems. ACKNOWLEDGMENTS The authors are DTI holders supported by RENIA project sponsored by CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico). REFERENCES 1. BALUJA, S., CARUANA, R., Removing the genetics from de standard genetic algorithm, Technical Report CMU-CS-95-141, May (1995). 2. MACHADO, M. D., Um novo algoritmo evolucionário com aprendizado LVQ para a otimização de problemas combinatórios como a recarga de reatores nucleares, Tese de M. Sc., COPPE/UFRJ, Rio de Janeiro, Brasil, Abril (1999).
3. PEREIRA, C.M.N.A., SCHIRRU R. And MARTINEZ A.S., Basic Investigations Related to Genetic Algorithms in Core Designs, Annals of Nuclear Energy, vol. 26, pp.173-193 (1999). 4. SUICH J.E., HONEC H.C., The HAMMER System Heterogeneous Analysis by Multigroup Methods of Exponentials and Reactors, Savannah River Laboratory, Aiken, SC, USA, USA (1967). 5. GOLDBERG, D.E., Genetic Algorithms in Search, Optimization & Machine Learning, Addison-Wesley, Reading, MA, USA (1989). 6. SACCO W. F., OLIVEIRA C., PEREIRA C.M.N.A., The Great Deluge Algorithn Applied to a Nuclear Reactor Core Design Optimization Problem, Anals of International Nuclear Atlantic Conference 2005 (2005).