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A Metropolis algorithm combined with Nelder–Mead Simplex applied to nuclear reactor core design
Available online at www.sciencedirect.com annals of
NUCLEAR ENERGY Annals of Nuclear Energy 35 (2008) 861–867 www.elsevier.com/locate/anucene
A Metropolis algorithm combined with Nelder–Mead Simplex applied to nuclear reactor core design Wagner F. Sacco
a,*
, Hermes Alves Filho a, Ne´lio Henderson a, Cassiano R.E. de Oliveira
b
a
b
Depto. de Modelagem Computacional, Instituto Polite´cnico, Universidade do Estado do Rio de Janeiro, R. Alberto Rangel, s/n, P.O. Box 972285, Nova Friburgo, RJ 28601-970, Brazil Nuclear and Radiological Engineering Program, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 2 February 2007; accepted 12 September 2007 Available online 24 October 2007
Abstract A hybridization of the recently introduced Particle Collision Algorithm (PCA) and the Nelder–Mead Simplex algorithm is introduced and applied to a core design optimization problem which was previously attacked by other metaheuristics. 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 three-enrichment-zone reactor, considering restrictions on the average thermal flux, criticality and sub-moderation. The new metaheuristic performs better than the genetic algorithm, particle swarm optimization, and the Metropolis algorithms PCA and the Great Deluge Algorithm, thus demonstrating its potential for other applications. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The Particle Collision Algorithm (PCA) (Sacco and de Oliveira, 2005; Sacco et al., 2006a) is a Metropolis-based algorithm (Metropolis et al., 1953) that was introduced as an alternative to Simulated Annealing (Kirkpatrick et al., 1983). The main motivation behind the PCA was that in spite of being very powerful, simulated annealing is too sensitive to the choice of free parameters, such as, for example, the annealing schedule and initial temperature (Carter, 1997). The PCA does not rely on user-supplied parameters to perform the optimality search, being thus more robust. This algorithm is loosely inspired by the physics of nuclear particle collision reactions (Duderstadt and Hamilton, 1976), particularly scattering and absorption. Thus, a particle that hits a high-fitness ‘‘nucleus’’ is ‘‘absorbed’’ and explores the boundaries. On the other
*
Corresponding author. Tel.: +55 22 2528 8545; fax: +55 22 2528 8536. E-mail address: [email protected] (W.F. Sacco).
0306-4549/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2007.09.006
hand, a particle that hits a low-fitness region is scattered to another region of the search space. In this article, we introduce a hybridization of the Particle Collision Algorithm and the Nelder–Mead Simplex algorithm (Nelder and Mead, 1965). The aim is to perform a wide search in the solution space using a stochastic optimization algorithm (the PCA) and then scan the promising areas with a deterministic local search technique (Nelder– Mead Simplex). This searching is performed iteratively until a certain number of fitness function evaluations is reached. This hybrid algorithm, called the Nelder–Mead Particle Collision Algorithm (NMPCA), is applied to a nuclear core design optimization problem that was introduced by Pereira et al. (1999), and has subsequently been attacked by other authors (Pereira and Lapa, 2003; Sacco et al., 2004; Sacco et al., 2006a; Domingos et al., 2006). The NMPCA is compared with the genetic algorithm (GA) (Holland, 1975; Goldberg, 1989), particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), the Metropolis-based Great Deluge Algorithm (GDA) (Dueck, 1993; Sacco et al., 2006a), and the PCA.
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The remainder of the paper is organized as follows. In the next section the original particle collision algorithm is outlined, the Nelder–Mead Simplex algorithm is described in detail and the new algorithm is presented. Section 3 presents the reactor design optimization problem. In Section 4, the implementation of the algorithm is briefly described. Next, the results are shown. Finally, in Section 6, the conclusions are made. 2. The PCA combined with Nelder–Mead Simplex 2.1. The PCA The PCA resembles in its structure that of simulated annealing: first an initial configuration is chosen; then there is a modification of the old configuration into a new one. The qualities of the two configurations are compared. A decision then is made on whether the new configuration is ‘‘acceptable’’. If it is, it serves as the old configuration for the next step. If it is not acceptable, the algorithm proceeds with a new change of the old configuration. The pseudo code description of the PCA for maximization problems is shown in Fig. 1. The ‘‘stochastic perturbation’’ in the beginning of the loop consists in random variations in each variable’s values within their ranges. If the quality or fitness of the new configuration is better than the fitness of the old configuration, then the ‘‘particle’’ is ‘‘absorbed’’, there is an exploration of the boundaries searching for an even better solution. Function
Generate an initial solution Old_Config For n = 0 to # of iterations Generate a stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config Exploration ( ) Else Scattering ( ) End If End For Exploration ( ) For n = 0 to # of iterations Generate a small stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config End If End For return
‘‘Exploration’’ performs this local search, generating a small stochastic perturbation of the solution inside a loop. The ‘‘small stochastic perturbation’’ is similar to the previous stochastic perturbation, but each variable’s new value is kept within the boundaries of the original value. Otherwise, if the quality of the new configuration is worse than the old configuration’s, the ‘‘particle’’ is ‘‘scattered’’. The scattering probability (pscattering ) is inversely proportional to its quality. A low-fitness particle will have a greater scattering probability. For further details on the canonical PCA, please refer to Sacco et al. (2006a). 2.2. The Nelder–Mead Simplex algorithm The Nelder–Mead Simplex method (Nelder and Mead, 1965; Dennis and Woods, 1987; Kelley, 1999) is a local search algorithm to obtain the solution of the problem: Min f ðxÞ ; ð1Þ x2X where X Rn is a neighborhood of an initial point. This algorithm belongs to a class of methods called direct search methods (Brent, 1973) that tries to solve problem (1) using only the objective function values. The choice of a direct search method avoids the calculation of derivatives of the objective function, being effective in problems where function f ðxÞ is discontinuous. In a given iteration of the Nelder–Mead algorithm, n þ 1 points, denoted by x1 ; . . . ; xn ; xnþ1 , are used to compute trial steps. In all iterations, we will always consider x1 ; . . . ; xn ; xnþ1 so that f ðx1 Þ 6 6 f ðxn Þ 6 f ðxnþ1 Þ holds. A trial step is accepted or rejected based on the function value of the trial point and on the three values f ðx1 Þ, f ðxn Þ and f ðxnþ1 Þ. Geometrically, at each iteration, this n þ 1 points may be thought of as the vertices of an simplex in Rn , S ¼ ½x1 ; . . . ; xn ; xnþ1 Rn . If n ¼ 2, then S ¼ ½x1 ; x2 ; x3 R2 is a triangle in the Euclidian plan, for example. Thus, xnþ1 is the vertex of the simplex that has the largest value of f (worst vertex). Trial steps are generated by the operations of reflection, expansion, contraction, and shrinkage. A reflected vertex is computed by reflecting the worst vertex through the centroid of the remaining vertices as xr ¼ ð1 þ aÞx axnþ1 ;
where a > 0 is the reflection coefficient, and x is the centroid given by
Scattering ( ) Fitness( New _ Config ) Best Fitness If pscattering > random (0, 1) Old_Config := random solution Else Exploration ( ); End if pscattering = 1 −
return
Fig. 1. PCA’s pseudo code.
ð2Þ
x ¼
n 1X xi : n i¼1
ð3Þ
The reflected vertex is accepted if f ðx1 Þ 6 f ðxr Þ < f ðxn Þ, and the next iteration begins with the simplex defined by S ¼ ½x1 ; . . . ; xn ; xr , where xr was not ordered with respect to the other vertices. If f ðxr Þ < f ðx1 Þ, then the trial step generated an acceptable point and the step is expanded. In this case, the expansion vertex is computed as
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
863
Fig. 2. Nelder–Mead Simplex.
xe ¼ cxr þ ð1 cÞx;
ð4Þ
where c > 1 is the expansion coefficient. If f ðxe Þ < f ðx1 Þ, then xe is accepted. Otherwise, xr is accepted. Thus, if f ðxr Þ < f ðxn Þ, then either the reflected or expanded vertex is accepted and the next iteration begins. But, if f ðxn Þ 6 f ðxr Þ, then the internal contraction vertex is computed as xc ¼ bxnþ1 þ ð1 bÞx;
ð5Þ
where b ¼ 1=2 is the contraction coefficient. Otherwise, the external contraction vertex is computed as ^xc ¼ bxr þ ð1 bÞx:
ð6Þ
The contraction vertex is accepted if it has a lower function value than xn . Finally, if both the reflection vertex and the contraction vertex are rejected, then the simplex is shrunk. In this case, each vertex xi , except x1 , is replaced by ðx1 þ xi Þ 8i ¼ 2; . . . ; n þ 1: ð7Þ 2 Values f ðxi Þ are computed and sorted along with f ðxi Þ. This final procedure determines the new simplex S k ¼ ½x1 ; . . . ; xn ; xnþ1 with which the next iteration starts. In this article, we used a ¼ 1 and c ¼ 2. The algorithm of the Nelder–Mead Simplex (NMS) method is described in Fig. 2 (Kelley, 1999).
Generate an initial solution Old_Config For n = 0 to # of iterations Generate a stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config Simplex ( ) Else Scattering ( ) End If End For Simplex ( ) Apply Nelder-Mead Simplex as described in Figure 2. return Scattering ( ) Fitness( New _ Config ) Best Fitness If pscattering > random (0, 1) Old_Config := random solution Else Exploration ( ); End if pscattering = 1 −
xi ¼
2.3. The PCA combined with Nelder–Mead Simplex The principle behind our hybrid metaheuristic is quite simple: the PCA explores the search space and when a
return
Fig. 3. NMPCA’s pseudo code.
better-than-previous solution is found, it is used as an initial point for Nelder–Mead Simplex. Function ‘‘Exploration’’ in the original algorithm was replaced by the simplex method in the NMPCA, as shown in Fig. 3.
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W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
3. Problem description As the main objective of this article is to compare the hybridized PCA with metaheuristics employed in previous works, the same problem is addressed (Pereira et al., 1999; Sacco et al., 2004, 2006a; Domingos et al., 2006). It will be briefly described here: consider a cylindrical 3-enrichmentzone PWR, with typical cell composed by moderator (light water), cladding and fuel. Fig. 4 illustrates such reactor. The design parameters that may be changed in the optimization process, as well as their variation ranges are shown in Table 1. The objective of the optimization problem is to minimize the average peak-factor, fp , of the proposed reactor, considering that the reactor must be critical ðk eff ¼ 1:0 1%Þ and sub-moderated, providing a given average flux /0 . Then, the optimization problem can be written as follows: Minimize fp ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ Subject to : /ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ ¼ /0 ; 0:99 6 k eff ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ 6 1:01; dk eff > 0; dV m Rf min 6 Rf 6 Rf max ;
ð8Þ ð9Þ ð10Þ ð11Þ
Dc min 6 Dc 6 Dc max ;
ð12Þ
Remin 6 Re 6 Remax ; E1 min 6 E1 6 E1max ;
ð13Þ ð14Þ
E2 min 6 E2 6 E2
ð15Þ
max ;
Fuel
R1
R2
Cladding
R3
h
Re
Δc Rf
Moderator
Fig. 4. (a) The reactor and (b) its typical cell.
Table 1 Parameters range Parameter
Symbol
Range
Fuel radius (cm) Cladding thickness (cm) Moderator thickness (cm) Enrichment of Zone 1 (%) Enrichment of Zone 2 (%) Enrichment of Zone 3 (%) Fuel material Cladding material
Rf Dc Re E1 E2 E3 Mf Mc
0.508–1.270 0.025–0.254 0.025–0.762 2.0–5.0 2.0–5.0 2.0–5.0 {U-Metal or UO2 } {Zircaloy-2, Aluminum or Stainless-304}
E3 min 6 E3 6 E3 max ; M f ¼ fUO2 or U-metalg;
ð16Þ ð17Þ
M c ¼ fZircaloy 2; Aluminum or stainless 304g;
ð18Þ
where V m is the moderator volume and the min and max subscripts refer to the lower and upper limits of the parameters ranges, given in Table 1. 4. Method application 4.1. Simulation tools 4.1.1. Optimization algorithm The NMPCA implementation was adapted from the PCA in Sacco et al. (2006a). The NMPCA was set up for 100,000 iterations, so that the results were obtained with the same computational effort of the canonical PCA. Each execution of 100,000 iterations took 10h30 min in a Pentium IV 3.8 GHz PC with 1 Gb RAM, as the reactor physics code is the system’s bottleneck. 4.1.2. Reactor physics code The HAMMER system (Suich and Honeck, 1967) was used for cell and diffusion equations calculations. It performs a multigroup calculation of the thermal and epithermal flux distribution from the integral transport theory in a unit cell of the lattice /ð~ rÞ ¼
Z
0
eRt j~r~r j 4p j~ r ~ r0 j
2
Sð~ r0 Þ d3 r0 :
ð19Þ
The integral transport equation for scalar flux /ð~ rÞ is solved for all sub-regions of the unit cell, being the neutron source Sð~ rÞ isotropic into the energy group under consideration. The transfer kernel in Eq. (19) is related to the collision probabilities for a flat isotropic source in the initial region. The solution is initially performed for a unit cell in an infinite lattice. The integral transport calculation is followed by a multigroup Fourier transfer leakage spectrum theory in order to include the leakage effects in the previous calculation and to proceed with the multigroup flux-volume weighting. Using the four group constants obtained from the mentioned procedure, a one-dimensional multi-region reactor calculation is performed. The diffusion equation is, then, solved to perform standard criticality calculation ~ g ðrÞr/ ~ g ðrÞ þ Rt;g ðrÞ/g ðrÞ rD 4 X 1 ¼ vg Rfg0 ðrÞ þ Rsg0 g ðrÞ /g0 ðrÞ: k eff g0 ¼1
ð20Þ
The flux /g ðrÞ is calculated assuming normalized source density. Eq. (20) is solved using finite difference method with a constant mesh width.
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
4.2. Fitness function The fitness function was developed in such a way that, if all constraints are satisfied, it has the value of the average peak factor, fp . Otherwise, it is penalized proportionally to the discrepancy on the constraint. Such penalization factors should be set up by the expert, according to the requirements and the priorities of the problem.
f ¼
8 > f ; > > p > > > > fp þ r1 Dk eff ; > > > > > > fp þ r2 D/; > > > > 0 > < fp þ r3 DDVkeffm ;
Dk eff 6 0:01; Dk eff > 0:01; Dk eff 6 0:01;
> > fp þ r1 Dk eff þ r2 D/; > > > 0 > > > fp þ r1 Dk eff þ r3 DDVkeffm ; > > > > > D0 k eff > > > fp þ r2 D/ þ r3 DVm ; > > 0 : fp þ r1 Dk eff þ r2 D/ þ r3 DDVkeffm ;
new hybrid algorithm outperformed the other metaheuristics in terms of fitness value, average and standard deviation. Table 4 shows the best configurations obtained by the SGA (in Pereira et al., 1999) with 300 individuals until convergence, by PSO (in Domingos et al., 2006) with 100 individuals, by GDA and PCA in 100,000 fitness function evaluations (Sacco et al., 2006a), and by the new hybrid
D/ 6 0:01/0 ;
D0 k eff DV m 0
>0
D/ 6 0:01/0 ;
D k eff DV m
>0
D/ > 0:01/0 ;
D0 k eff DV m
>0
0
D/ 6 0:01/0 ;
D k eff DV m
<0
Dk eff > 0:01;
D/ > 0:01/0 ;
D0 k eff DV m
>0
Dk eff > 0:01;
D/ 6 0:01/0 ;
D0 k eff DV m
<0
Dk eff 6 0:01;
Dk eff 6 0:01; Dk eff > 0:01;
0
D/ > 0:01/0 ;
D k eff DV m
<0
D/ > 0:01/0 ;
D0 k eff DV m
<0
5. Results Table 2 shows the results obtained in ten executions of the NMPCA. The last column displays the number of fitness function evaluations necessary to reach the optimum. Note that in 5 out of 10 executions the NMPCA reached the optimum in less than 50,000 fitness function evaluations. Table 3 shows the results obtained by NMPCA in comparison with those obtained by the SGA, by the GDA, by the standard PCA, and by PSO. For all algorithms, except the latter, each execution took 100,000 fitness function evaluations. In the case of PSO, Domingos et al. (2006) mention that the population consisted of 100 individuals, but the number of generations is not informed, which means that we don’t know the number of fitness function evaluations for each execution. The
865
ð21Þ
Table 3 Comparison with other metaheuristics Experiment
SGAa
PSOb
GDAa
PCAa
NMPCA
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std. dev.
1.3185 1.3116 1.3300 1.3294 1.3595 1.3562 1.3372 1.3523 1.3614 1.3467 1.3402 0.0175
1.2774 1.2773 1.2767 1.2767 1.2767 1.2767 1.2769 1.2770 1.2770 1.2770 1.2769 0.0003
1.2806 1.2913 1.2856 1.2891 1.2863 1.2845 1.2897 1.2842 1.2895 1.2827 1.2864 0.0035
1.2827 1.2876 1.2964 1.2874 1.2829 1.2791 1.2975 1.2865 1.2908 1.2845 1.2875 0.0059
1.2764 1.2768 1.2765 1.2765 1.2764 1.2768 1.2770 1.2769 1.2765 1.2769 1.2767 0.0002
a b
Sacco et al. (2006a). Domingos et al. (2006).
Table 2 NMPCA results for ten experiments Experiment
Rf (cm)
Dr (cm)
Dm (cm)
E1 (%)
E2 (%)
E3 (%)
Ef
Mc
Fitness
Fitness evaluations
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10
0.7272 0.6706 0.7175 0.7338 0.7359 0.7402 0.6529 0.6603 0.7388 0.7037
0.1909 0.1771 0.1893 0.1862 0.1950 0.1567 0.1269 0.1726 0.1845 0.1618
0.7577 0.7134 0.7501 0.7585 0.7618 0.7547 0.6806 0.7044 0.7608 0.7304
2.7187 2.7477 2.7275 2.6796 2.7241 2.5037 2.5800 2.7396 2.6578 2.5923
2.8519 2.8820 2.8617 2.8109 2.8578 2.6218 2.7058 2.8731 2.7886 2.7190
4.9449 4.9657 4.9577 4.8761 4.9610 4.5359 4.6437 4.9465 4.8357 4.6896
U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal
SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304
1.2764 1.2768 1.2765 1.2765 1.2764 1.2768 1.2770 1.2769 1.2765 1.2769
12820 30189 78407 39067 63355 93927 93569 32183 28393 53451
Average
0.7081
0.1741
0.7373
2.6671
2.7973
4.8357
–
–
1.2767
52536.1
866
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
Table 4 Comparison with previously published best results
Objectives and constraints
Parameters
a b c
SGAa
PSOb
GDAc
PCAc
NMPCA
Fitness Minimum average peak factor Average flux k eff
1.3100 1.3100
1.2767 1.2767
1.2806 1.2806
1.2791 1.2791
1.2764 1.2764
8:02 105 1.000
8:07 105 0.990
7:95 105 0.990
8:06 105 0.991
8:08 105 1.000
Rf (cm) Dr (cm) Dm (cm) E1 (%) E2 (%) E3 (%) Mf Mc
0.5621 0.1770 0.6581 2.756 4.032 4.457 U-metal Stainless-304
0.7459 0.1647 0.7620 2.5364 2.6608 4.6067 U-metal Stainless-304
0.5913 0.0638 0.5992 2.1485 2.2585 3.8590 U-metal Stainless-304
0.5497 0.1450 0.6111 2.7953 2.9469 5.0000 U-metal Stainless-304
0.7272 0.1909 0.7577 2.7187 2.8519 4.9949 U-metal Stainless-304
Pereira et al. (1999). Domingos et al. (2006). Sacco et al. (2006a).
metaheuristics also in 100,000 fitness function evaluations. Note that the configurations obtained by PSO and by the NMPCA are quite similar, which may suggest that both reached the same region of the search space. 6. Conclusions With this work, we show that a hybridization of stochastic optimization and deterministic optimization methods can be quite effective, as the former promote a thorough exploration of the search space and the latter exploit its promising areas. We do believe that the future in optimization lies in hybrid algorithms. In fact, there have been many recent efforts in this research field (see, for example, Resende and Werneck, 2006; Menon et al., 2006; Liao and Tsao, 2006). Moreover, we ratify the conclusion of Sacco et al. (2006a), who recommended that the PCA should be applied to other optimization problems in the nuclear engineering field. We are planning to apply both the PCA and the NMPCA to the nuclear core reload optimization problem (Poon and Parks, 1992), and also to a nuclear power plant surveillance tests optimization (Sacco et al., 2006b). In the future, we intend to hybridize other metaheuristics with deterministic algorithms, as for example particle swarm optimization and the Nelder–Mead Simplex. Acknowledgement Wagner F. Sacco is supported by FAPERJ (Fundac¸a˜o Carlos Chagas Filho de Amparo a` Pesquisa do Estado do Rio de Janeiro) under postdoctoral Grant E-26/ ´ vel 3). 152.661/2005 (Fixac¸a˜o de Pesquisador, Nı References Brent, R.P., 1973. Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ.
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NUCLEAR ENERGY Annals of Nuclear Energy 35 (2008) 861–867 www.elsevier.com/locate/anucene
A Metropolis algorithm combined with Nelder–Mead Simplex applied to nuclear reactor core design Wagner F. Sacco
a,*
, Hermes Alves Filho a, Ne´lio Henderson a, Cassiano R.E. de Oliveira
b
a
b
Depto. de Modelagem Computacional, Instituto Polite´cnico, Universidade do Estado do Rio de Janeiro, R. Alberto Rangel, s/n, P.O. Box 972285, Nova Friburgo, RJ 28601-970, Brazil Nuclear and Radiological Engineering Program, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0405, USA Received 2 February 2007; accepted 12 September 2007 Available online 24 October 2007
Abstract A hybridization of the recently introduced Particle Collision Algorithm (PCA) and the Nelder–Mead Simplex algorithm is introduced and applied to a core design optimization problem which was previously attacked by other metaheuristics. 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 three-enrichment-zone reactor, considering restrictions on the average thermal flux, criticality and sub-moderation. The new metaheuristic performs better than the genetic algorithm, particle swarm optimization, and the Metropolis algorithms PCA and the Great Deluge Algorithm, thus demonstrating its potential for other applications. Ó 2007 Elsevier Ltd. All rights reserved.
1. Introduction The Particle Collision Algorithm (PCA) (Sacco and de Oliveira, 2005; Sacco et al., 2006a) is a Metropolis-based algorithm (Metropolis et al., 1953) that was introduced as an alternative to Simulated Annealing (Kirkpatrick et al., 1983). The main motivation behind the PCA was that in spite of being very powerful, simulated annealing is too sensitive to the choice of free parameters, such as, for example, the annealing schedule and initial temperature (Carter, 1997). The PCA does not rely on user-supplied parameters to perform the optimality search, being thus more robust. This algorithm is loosely inspired by the physics of nuclear particle collision reactions (Duderstadt and Hamilton, 1976), particularly scattering and absorption. Thus, a particle that hits a high-fitness ‘‘nucleus’’ is ‘‘absorbed’’ and explores the boundaries. On the other
*
Corresponding author. Tel.: +55 22 2528 8545; fax: +55 22 2528 8536. E-mail address: [email protected] (W.F. Sacco).
0306-4549/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.anucene.2007.09.006
hand, a particle that hits a low-fitness region is scattered to another region of the search space. In this article, we introduce a hybridization of the Particle Collision Algorithm and the Nelder–Mead Simplex algorithm (Nelder and Mead, 1965). The aim is to perform a wide search in the solution space using a stochastic optimization algorithm (the PCA) and then scan the promising areas with a deterministic local search technique (Nelder– Mead Simplex). This searching is performed iteratively until a certain number of fitness function evaluations is reached. This hybrid algorithm, called the Nelder–Mead Particle Collision Algorithm (NMPCA), is applied to a nuclear core design optimization problem that was introduced by Pereira et al. (1999), and has subsequently been attacked by other authors (Pereira and Lapa, 2003; Sacco et al., 2004; Sacco et al., 2006a; Domingos et al., 2006). The NMPCA is compared with the genetic algorithm (GA) (Holland, 1975; Goldberg, 1989), particle swarm optimization (PSO) (Kennedy and Eberhart, 1995), the Metropolis-based Great Deluge Algorithm (GDA) (Dueck, 1993; Sacco et al., 2006a), and the PCA.
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The remainder of the paper is organized as follows. In the next section the original particle collision algorithm is outlined, the Nelder–Mead Simplex algorithm is described in detail and the new algorithm is presented. Section 3 presents the reactor design optimization problem. In Section 4, the implementation of the algorithm is briefly described. Next, the results are shown. Finally, in Section 6, the conclusions are made. 2. The PCA combined with Nelder–Mead Simplex 2.1. The PCA The PCA resembles in its structure that of simulated annealing: first an initial configuration is chosen; then there is a modification of the old configuration into a new one. The qualities of the two configurations are compared. A decision then is made on whether the new configuration is ‘‘acceptable’’. If it is, it serves as the old configuration for the next step. If it is not acceptable, the algorithm proceeds with a new change of the old configuration. The pseudo code description of the PCA for maximization problems is shown in Fig. 1. The ‘‘stochastic perturbation’’ in the beginning of the loop consists in random variations in each variable’s values within their ranges. If the quality or fitness of the new configuration is better than the fitness of the old configuration, then the ‘‘particle’’ is ‘‘absorbed’’, there is an exploration of the boundaries searching for an even better solution. Function
Generate an initial solution Old_Config For n = 0 to # of iterations Generate a stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config Exploration ( ) Else Scattering ( ) End If End For Exploration ( ) For n = 0 to # of iterations Generate a small stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config End If End For return
‘‘Exploration’’ performs this local search, generating a small stochastic perturbation of the solution inside a loop. The ‘‘small stochastic perturbation’’ is similar to the previous stochastic perturbation, but each variable’s new value is kept within the boundaries of the original value. Otherwise, if the quality of the new configuration is worse than the old configuration’s, the ‘‘particle’’ is ‘‘scattered’’. The scattering probability (pscattering ) is inversely proportional to its quality. A low-fitness particle will have a greater scattering probability. For further details on the canonical PCA, please refer to Sacco et al. (2006a). 2.2. The Nelder–Mead Simplex algorithm The Nelder–Mead Simplex method (Nelder and Mead, 1965; Dennis and Woods, 1987; Kelley, 1999) is a local search algorithm to obtain the solution of the problem: Min f ðxÞ ; ð1Þ x2X where X Rn is a neighborhood of an initial point. This algorithm belongs to a class of methods called direct search methods (Brent, 1973) that tries to solve problem (1) using only the objective function values. The choice of a direct search method avoids the calculation of derivatives of the objective function, being effective in problems where function f ðxÞ is discontinuous. In a given iteration of the Nelder–Mead algorithm, n þ 1 points, denoted by x1 ; . . . ; xn ; xnþ1 , are used to compute trial steps. In all iterations, we will always consider x1 ; . . . ; xn ; xnþ1 so that f ðx1 Þ 6 6 f ðxn Þ 6 f ðxnþ1 Þ holds. A trial step is accepted or rejected based on the function value of the trial point and on the three values f ðx1 Þ, f ðxn Þ and f ðxnþ1 Þ. Geometrically, at each iteration, this n þ 1 points may be thought of as the vertices of an simplex in Rn , S ¼ ½x1 ; . . . ; xn ; xnþ1 Rn . If n ¼ 2, then S ¼ ½x1 ; x2 ; x3 R2 is a triangle in the Euclidian plan, for example. Thus, xnþ1 is the vertex of the simplex that has the largest value of f (worst vertex). Trial steps are generated by the operations of reflection, expansion, contraction, and shrinkage. A reflected vertex is computed by reflecting the worst vertex through the centroid of the remaining vertices as xr ¼ ð1 þ aÞx axnþ1 ;
where a > 0 is the reflection coefficient, and x is the centroid given by
Scattering ( ) Fitness( New _ Config ) Best Fitness If pscattering > random (0, 1) Old_Config := random solution Else Exploration ( ); End if pscattering = 1 −
return
Fig. 1. PCA’s pseudo code.
ð2Þ
x ¼
n 1X xi : n i¼1
ð3Þ
The reflected vertex is accepted if f ðx1 Þ 6 f ðxr Þ < f ðxn Þ, and the next iteration begins with the simplex defined by S ¼ ½x1 ; . . . ; xn ; xr , where xr was not ordered with respect to the other vertices. If f ðxr Þ < f ðx1 Þ, then the trial step generated an acceptable point and the step is expanded. In this case, the expansion vertex is computed as
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
863
Fig. 2. Nelder–Mead Simplex.
xe ¼ cxr þ ð1 cÞx;
ð4Þ
where c > 1 is the expansion coefficient. If f ðxe Þ < f ðx1 Þ, then xe is accepted. Otherwise, xr is accepted. Thus, if f ðxr Þ < f ðxn Þ, then either the reflected or expanded vertex is accepted and the next iteration begins. But, if f ðxn Þ 6 f ðxr Þ, then the internal contraction vertex is computed as xc ¼ bxnþ1 þ ð1 bÞx;
ð5Þ
where b ¼ 1=2 is the contraction coefficient. Otherwise, the external contraction vertex is computed as ^xc ¼ bxr þ ð1 bÞx:
ð6Þ
The contraction vertex is accepted if it has a lower function value than xn . Finally, if both the reflection vertex and the contraction vertex are rejected, then the simplex is shrunk. In this case, each vertex xi , except x1 , is replaced by ðx1 þ xi Þ 8i ¼ 2; . . . ; n þ 1: ð7Þ 2 Values f ðxi Þ are computed and sorted along with f ðxi Þ. This final procedure determines the new simplex S k ¼ ½x1 ; . . . ; xn ; xnþ1 with which the next iteration starts. In this article, we used a ¼ 1 and c ¼ 2. The algorithm of the Nelder–Mead Simplex (NMS) method is described in Fig. 2 (Kelley, 1999).
Generate an initial solution Old_Config For n = 0 to # of iterations Generate a stochastic perturbation of the solution If Fitness(New_Config) > Fitness(Old_Config) Old_Config := New_Config Simplex ( ) Else Scattering ( ) End If End For Simplex ( ) Apply Nelder-Mead Simplex as described in Figure 2. return Scattering ( ) Fitness( New _ Config ) Best Fitness If pscattering > random (0, 1) Old_Config := random solution Else Exploration ( ); End if pscattering = 1 −
xi ¼
2.3. The PCA combined with Nelder–Mead Simplex The principle behind our hybrid metaheuristic is quite simple: the PCA explores the search space and when a
return
Fig. 3. NMPCA’s pseudo code.
better-than-previous solution is found, it is used as an initial point for Nelder–Mead Simplex. Function ‘‘Exploration’’ in the original algorithm was replaced by the simplex method in the NMPCA, as shown in Fig. 3.
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3. Problem description As the main objective of this article is to compare the hybridized PCA with metaheuristics employed in previous works, the same problem is addressed (Pereira et al., 1999; Sacco et al., 2004, 2006a; Domingos et al., 2006). It will be briefly described here: consider a cylindrical 3-enrichmentzone PWR, with typical cell composed by moderator (light water), cladding and fuel. Fig. 4 illustrates such reactor. The design parameters that may be changed in the optimization process, as well as their variation ranges are shown in Table 1. The objective of the optimization problem is to minimize the average peak-factor, fp , of the proposed reactor, considering that the reactor must be critical ðk eff ¼ 1:0 1%Þ and sub-moderated, providing a given average flux /0 . Then, the optimization problem can be written as follows: Minimize fp ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ Subject to : /ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ ¼ /0 ; 0:99 6 k eff ðRf ; Dc ; Re ; E1 ; E2 ; E3 ; M f ; M c Þ 6 1:01; dk eff > 0; dV m Rf min 6 Rf 6 Rf max ;
ð8Þ ð9Þ ð10Þ ð11Þ
Dc min 6 Dc 6 Dc max ;
ð12Þ
Remin 6 Re 6 Remax ; E1 min 6 E1 6 E1max ;
ð13Þ ð14Þ
E2 min 6 E2 6 E2
ð15Þ
max ;
Fuel
R1
R2
Cladding
R3
h
Re
Δc Rf
Moderator
Fig. 4. (a) The reactor and (b) its typical cell.
Table 1 Parameters range Parameter
Symbol
Range
Fuel radius (cm) Cladding thickness (cm) Moderator thickness (cm) Enrichment of Zone 1 (%) Enrichment of Zone 2 (%) Enrichment of Zone 3 (%) Fuel material Cladding material
Rf Dc Re E1 E2 E3 Mf Mc
0.508–1.270 0.025–0.254 0.025–0.762 2.0–5.0 2.0–5.0 2.0–5.0 {U-Metal or UO2 } {Zircaloy-2, Aluminum or Stainless-304}
E3 min 6 E3 6 E3 max ; M f ¼ fUO2 or U-metalg;
ð16Þ ð17Þ
M c ¼ fZircaloy 2; Aluminum or stainless 304g;
ð18Þ
where V m is the moderator volume and the min and max subscripts refer to the lower and upper limits of the parameters ranges, given in Table 1. 4. Method application 4.1. Simulation tools 4.1.1. Optimization algorithm The NMPCA implementation was adapted from the PCA in Sacco et al. (2006a). The NMPCA was set up for 100,000 iterations, so that the results were obtained with the same computational effort of the canonical PCA. Each execution of 100,000 iterations took 10h30 min in a Pentium IV 3.8 GHz PC with 1 Gb RAM, as the reactor physics code is the system’s bottleneck. 4.1.2. Reactor physics code The HAMMER system (Suich and Honeck, 1967) was used for cell and diffusion equations calculations. It performs a multigroup calculation of the thermal and epithermal flux distribution from the integral transport theory in a unit cell of the lattice /ð~ rÞ ¼
Z
0
eRt j~r~r j 4p j~ r ~ r0 j
2
Sð~ r0 Þ d3 r0 :
ð19Þ
The integral transport equation for scalar flux /ð~ rÞ is solved for all sub-regions of the unit cell, being the neutron source Sð~ rÞ isotropic into the energy group under consideration. The transfer kernel in Eq. (19) is related to the collision probabilities for a flat isotropic source in the initial region. The solution is initially performed for a unit cell in an infinite lattice. The integral transport calculation is followed by a multigroup Fourier transfer leakage spectrum theory in order to include the leakage effects in the previous calculation and to proceed with the multigroup flux-volume weighting. Using the four group constants obtained from the mentioned procedure, a one-dimensional multi-region reactor calculation is performed. The diffusion equation is, then, solved to perform standard criticality calculation ~ g ðrÞr/ ~ g ðrÞ þ Rt;g ðrÞ/g ðrÞ rD 4 X 1 ¼ vg Rfg0 ðrÞ þ Rsg0 g ðrÞ /g0 ðrÞ: k eff g0 ¼1
ð20Þ
The flux /g ðrÞ is calculated assuming normalized source density. Eq. (20) is solved using finite difference method with a constant mesh width.
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
4.2. Fitness function The fitness function was developed in such a way that, if all constraints are satisfied, it has the value of the average peak factor, fp . Otherwise, it is penalized proportionally to the discrepancy on the constraint. Such penalization factors should be set up by the expert, according to the requirements and the priorities of the problem.
f ¼
8 > f ; > > p > > > > fp þ r1 Dk eff ; > > > > > > fp þ r2 D/; > > > > 0 > < fp þ r3 DDVkeffm ;
Dk eff 6 0:01; Dk eff > 0:01; Dk eff 6 0:01;
> > fp þ r1 Dk eff þ r2 D/; > > > 0 > > > fp þ r1 Dk eff þ r3 DDVkeffm ; > > > > > D0 k eff > > > fp þ r2 D/ þ r3 DVm ; > > 0 : fp þ r1 Dk eff þ r2 D/ þ r3 DDVkeffm ;
new hybrid algorithm outperformed the other metaheuristics in terms of fitness value, average and standard deviation. Table 4 shows the best configurations obtained by the SGA (in Pereira et al., 1999) with 300 individuals until convergence, by PSO (in Domingos et al., 2006) with 100 individuals, by GDA and PCA in 100,000 fitness function evaluations (Sacco et al., 2006a), and by the new hybrid
D/ 6 0:01/0 ;
D0 k eff DV m 0
>0
D/ 6 0:01/0 ;
D k eff DV m
>0
D/ > 0:01/0 ;
D0 k eff DV m
>0
0
D/ 6 0:01/0 ;
D k eff DV m
<0
Dk eff > 0:01;
D/ > 0:01/0 ;
D0 k eff DV m
>0
Dk eff > 0:01;
D/ 6 0:01/0 ;
D0 k eff DV m
<0
Dk eff 6 0:01;
Dk eff 6 0:01; Dk eff > 0:01;
0
D/ > 0:01/0 ;
D k eff DV m
<0
D/ > 0:01/0 ;
D0 k eff DV m
<0
5. Results Table 2 shows the results obtained in ten executions of the NMPCA. The last column displays the number of fitness function evaluations necessary to reach the optimum. Note that in 5 out of 10 executions the NMPCA reached the optimum in less than 50,000 fitness function evaluations. Table 3 shows the results obtained by NMPCA in comparison with those obtained by the SGA, by the GDA, by the standard PCA, and by PSO. For all algorithms, except the latter, each execution took 100,000 fitness function evaluations. In the case of PSO, Domingos et al. (2006) mention that the population consisted of 100 individuals, but the number of generations is not informed, which means that we don’t know the number of fitness function evaluations for each execution. The
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ð21Þ
Table 3 Comparison with other metaheuristics Experiment
SGAa
PSOb
GDAa
PCAa
NMPCA
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10 Average Std. dev.
1.3185 1.3116 1.3300 1.3294 1.3595 1.3562 1.3372 1.3523 1.3614 1.3467 1.3402 0.0175
1.2774 1.2773 1.2767 1.2767 1.2767 1.2767 1.2769 1.2770 1.2770 1.2770 1.2769 0.0003
1.2806 1.2913 1.2856 1.2891 1.2863 1.2845 1.2897 1.2842 1.2895 1.2827 1.2864 0.0035
1.2827 1.2876 1.2964 1.2874 1.2829 1.2791 1.2975 1.2865 1.2908 1.2845 1.2875 0.0059
1.2764 1.2768 1.2765 1.2765 1.2764 1.2768 1.2770 1.2769 1.2765 1.2769 1.2767 0.0002
a b
Sacco et al. (2006a). Domingos et al. (2006).
Table 2 NMPCA results for ten experiments Experiment
Rf (cm)
Dr (cm)
Dm (cm)
E1 (%)
E2 (%)
E3 (%)
Ef
Mc
Fitness
Fitness evaluations
#1 #2 #3 #4 #5 #6 #7 #8 #9 #10
0.7272 0.6706 0.7175 0.7338 0.7359 0.7402 0.6529 0.6603 0.7388 0.7037
0.1909 0.1771 0.1893 0.1862 0.1950 0.1567 0.1269 0.1726 0.1845 0.1618
0.7577 0.7134 0.7501 0.7585 0.7618 0.7547 0.6806 0.7044 0.7608 0.7304
2.7187 2.7477 2.7275 2.6796 2.7241 2.5037 2.5800 2.7396 2.6578 2.5923
2.8519 2.8820 2.8617 2.8109 2.8578 2.6218 2.7058 2.8731 2.7886 2.7190
4.9449 4.9657 4.9577 4.8761 4.9610 4.5359 4.6437 4.9465 4.8357 4.6896
U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal U-Metal
SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304 SS-304
1.2764 1.2768 1.2765 1.2765 1.2764 1.2768 1.2770 1.2769 1.2765 1.2769
12820 30189 78407 39067 63355 93927 93569 32183 28393 53451
Average
0.7081
0.1741
0.7373
2.6671
2.7973
4.8357
–
–
1.2767
52536.1
866
W.F. Sacco et al. / Annals of Nuclear Energy 35 (2008) 861–867
Table 4 Comparison with previously published best results
Objectives and constraints
Parameters
a b c
SGAa
PSOb
GDAc
PCAc
NMPCA
Fitness Minimum average peak factor Average flux k eff
1.3100 1.3100
1.2767 1.2767
1.2806 1.2806
1.2791 1.2791
1.2764 1.2764
8:02 105 1.000
8:07 105 0.990
7:95 105 0.990
8:06 105 0.991
8:08 105 1.000
Rf (cm) Dr (cm) Dm (cm) E1 (%) E2 (%) E3 (%) Mf Mc
0.5621 0.1770 0.6581 2.756 4.032 4.457 U-metal Stainless-304
0.7459 0.1647 0.7620 2.5364 2.6608 4.6067 U-metal Stainless-304
0.5913 0.0638 0.5992 2.1485 2.2585 3.8590 U-metal Stainless-304
0.5497 0.1450 0.6111 2.7953 2.9469 5.0000 U-metal Stainless-304
0.7272 0.1909 0.7577 2.7187 2.8519 4.9949 U-metal Stainless-304
Pereira et al. (1999). Domingos et al. (2006). Sacco et al. (2006a).
metaheuristics also in 100,000 fitness function evaluations. Note that the configurations obtained by PSO and by the NMPCA are quite similar, which may suggest that both reached the same region of the search space. 6. Conclusions With this work, we show that a hybridization of stochastic optimization and deterministic optimization methods can be quite effective, as the former promote a thorough exploration of the search space and the latter exploit its promising areas. We do believe that the future in optimization lies in hybrid algorithms. In fact, there have been many recent efforts in this research field (see, for example, Resende and Werneck, 2006; Menon et al., 2006; Liao and Tsao, 2006). Moreover, we ratify the conclusion of Sacco et al. (2006a), who recommended that the PCA should be applied to other optimization problems in the nuclear engineering field. We are planning to apply both the PCA and the NMPCA to the nuclear core reload optimization problem (Poon and Parks, 1992), and also to a nuclear power plant surveillance tests optimization (Sacco et al., 2006b). In the future, we intend to hybridize other metaheuristics with deterministic algorithms, as for example particle swarm optimization and the Nelder–Mead Simplex. Acknowledgement Wagner F. Sacco is supported by FAPERJ (Fundac¸a˜o Carlos Chagas Filho de Amparo a` Pesquisa do Estado do Rio de Janeiro) under postdoctoral Grant E-26/ ´ vel 3). 152.661/2005 (Fixac¸a˜o de Pesquisador, Nı References Brent, R.P., 1973. Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ.
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