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Application of Cuckoo Search algorithm to Loading Pattern Optimization problems
Annals of Nuclear Energy 139 (2020) 107214
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Annals of Nuclear Energy journal homepage: www.elsevier.com/locate/anucene
Application of Cuckoo Search algorithm to Loading Pattern Optimization problems Anderson Alvarenga de Moura Meneses a,⇑, Patrick Vasconcelos da Silva a, Fernando Nogueira Nast a, Lenilson Moreira Araujo a, Roberto Schirru b a b
Federal University of Western Pará, Laboratory of Computational Intelligence, Av. Vera Paz, s/n, Salé, CEP 68.035-110 Santarém, PA, Brazil Federal University of Rio de Janeiro, COPPE, Nuclear Engineering Program, Brazil
a r t i c l e
i n f o
Article history: Received 6 September 2019 Received in revised form 11 November 2019 Accepted 15 November 2019
Keywords: Loading Pattern Optimization Nuclear Power Plant Optimization metaheuristics Pressurized water reactor Cuckoo Search Nuclear fuel
a b s t r a c t The Loading Pattern Optimization (LPO) is related to important goals in a Nuclear Power Plant (NPP) operation such as the extension of the cycle according to safety margins. The LPO is a combinatorial problem of relevance and interest for Nuclear Engineering. Optimization metaheuristics have been efficient in solving the LPO. The recent metaheuristic Cuckoo Search (CS) is based on the brood parasitism of some cuckoo species, combined with the behavior of the Lévy flight of some birds. In the present work the results of the application of CS to the LPO using IAEA-3D and BIBLIS-2D benchmarks are presented, as well as the application of CS in the optimization of 7th cycle of Angra 1 NPP, in Brazil. The results are compared to the metaheuristics Artificial Bee Colony and Population-Based Incremental Learning. Statistical analyses show that CS is the most robust algorithm for the set of instances selected for tests. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The Loading Pattern Optimization (LPO) is related to important goals in a Nuclear Power Plant (NPP) operation such as the extension of the operation cycle by determining the position of the Fuel Assemblies (FAs) in the reactor core, within adequate safety margins (Levine, 1986). The LPO is a prominent problem of economic interest, as well as a complex combinatorial problem, classified as NP-hard (Papadimitriou and Steiglitz, 1982), with a considerable number of feasible solutions, sub-optimal solutions, disconnected feasible regions, high dimensionality, complex and timeconsuming evaluation of candidate solutions using Reactor Physics codes (Stevens et al., 1995). Computational intelligence methods such as Optimization Metaheuristics (OMH) are efficient in solving combinatorial problems such as the LPO. Several OMHs have been successful when applied to LPO, including: Simulated Annealing (SA; Parks, 1990; Stevens et al., 1995), Genetic Algorithm (GA; Chapot et al., 1999; Poon and Parks, 1992), Tabu Search (TS; Hill and Parks, 2015; Lin et al., 1998), Population-Based Incremental Learning (PBIL; Caldas and Schirru, 2008; Machado, 2005), Ant Colony Optimization (ACO; de Lima et al., 2008; Machado and Schirru, 2002), ⇑ Corresponding author. E-mail address: [email protected] (A.A.M. Meneses). https://doi.org/10.1016/j.anucene.2019.107214 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
Particle Swarm Optimization (PSO; Meneses et al., 2009), Artificial Bee Colony (ABC; de Oliveira and Schirru, 2011), and Cross-Entropy (CE; Meneses and Schirru, 2015). In the present work, CS was applied to the benchmarks IAEA-3D (Argonne National Laboratory, 1977) and BIBLIS-2D (Poursalehi et al., 2013), using the PARCS v3.0 code (U.S. NRC Core Neutronics Simulator; Downar et al., 2009a,b; Joo et al., 1998) as well as to the 7th cycle of Angra 1 NPP, located in Brazil, using the RECNOD code (Chapot, 2000). The remaining of the present article is organized as follows: Section 2 contains a review of previous works; in Section 3 the theoretical background is presented; Section 4 describes the methodology used to solve the LPO problems; Section 5 presents the results; the discussion is presented in Section 6; and finally, Section 7 is devoted to the conclusions. 2. Related works The CS algorithm was first presented by Yang and Deb (2009). The CS is based on the strategy of reproduction of some cuckoo species combined with the behavior of the Lévy Flight of some birds, and is a swarm intelligence algorithm (Engelbrecht, 2007). Ouaarab et al. (2014) applied a CS to the Traveling Salesman Problem (TSP). In such work the Improved Discrete CS (DCS) uses a fraction of intelligent cuckoos to explore other areas of the search
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space, using the local search heuristic 2-opt (Johnson and McGeoch, 1997). DCS and its improved version were applied to 41 instances of the problem and compared to the Genetic Simulated Annealing-Ant Colony System with PSO Techniques (GSAACS-PSOT; Chen and Chien, 2011), as well as to the Discrete PSO (DPSO). Improved DCS obtained superior results in comparison to GSA-ACS-PSOT and DPSO for the TSP. CS has also been applied to a nuclear reload problem by Yarizadeh-Beneh et al. (2016), in order to optimize the reload of the first cycle of the reactor located in the Bushehr NPP in Iran, using Random Keys (RK; Bean, 1994). For thermal-neutronic calculations, CS was coupled to PARCS reactor physics code, WIMSD-5B, and COBRA-5B, obtaining results that outperform those obtained in the original reload by a designer, extending the cycle in up to 7 days. Yarizadeh-Beneh et al. (2016) applied CS to the LPO of a real-world WWER NPP, and calculations for beginning of cycle (BOC) and end of cycle (EOC) states. In the present work the Critical Boron Concentration is calculated at the equilibrium of Xenon in the RECNOD code (7th cycle of Angra 1 NPP), and at the BOC in the IAEA-3D and BIBLIS-2D LPO problems. For the application of CS to the Angra 1 – 7th cycle LPO, Silva et al. (2017) performed a preliminary investigation. In the present work, the CS was also applied to two well-known benchmark problems (IAEA-3D and BIBLIS-2D), being compared favorably with the results of two competitive metaheuristics for the LPO, namely the ABC and PBIL algorithms. The results of the present work are compared to the results described by Meneses et al. (2018), who reported the application of the PSO, CE, PBIL, and ABC algorithms to LPO problems based on the data of the benchmarks BIBLIS-2D (Poursalehi et al., 2013) and IAEA-3D (Argonne National Laboratory, 1977), both of them modelled with the Reactor Physics code PARCS as well as the Angra 1 7th Cycle LPO problem, which was modelled with the RECNOD code (Chapot et al., 1999). According to Meneses et al. (2018) the best algorithms were ABC and PBIL respectively for IAEA-3D and for BIBLIS-2D. For the problem of Angra 1 NPP modeled with the reactor physics code RECNOD the algorithms PBIL, ABC, and CE achieved the best results. 3. Theoretical background 3.1. Loading Pattern Optimization (LPO) The LPO aims to determine LPs that maximize energy production satisfying safety requirements (Levine, 1986). The LPO is an NP-hard problem, which means that its complexity grows nonpolynomially, with a discontinuous and multimodal search space (de Lima, 2005). The substitution of burned FAs is necessary when it is not possible to keep the reactor running at nominal power, or more precisely after a period called the operating cycle. After each cycle the FAs present different neutronic characteristics. For example, the FAs are burned in different ways depending on the exposure time and position in the reactor core, with the accumulation of fission products such as Xenon and Samarium. Consequently, only some FAs are reused in the next cycle, and approximately 1/3 or 1/4 of the FAs are replaced by new ones (Hill and Parks, 2015; Meneses et al., 2010). Thus, forming an LP by fresh and burned FAs characterizes the LPO as a combinatorial problem. The benchmark LPO problems used for tests in the present work are described below. 3.1.1. IAEA-3D and BIBLIS-2D benchmarks IAEA-3D and BIBLIS-2D are theoretical problems taken as reference for the validation of numerical methods applied to the
neutron diffusion theory. In the present work both reactors were modeled considering the symmetry of ¼ in PARCS code, and during the OMH search new candidate solutions (LPs) are formed with octant symmetry. FAs on symmetry lines are not permuted with FAs off symmetry lines as described by Meneses et al. (2018). Despite being a two-dimensional reactor, BIBLIS-2D presents a much higher level of complexity than the IAEA-3D regarding the LPO, because of the number of candidate LP solutions. The number of candidate solutions for the BIBLIS-2D reactor is approximately 4:0 1016 whereas for the IAEA-3D reactor the total number is 288,288. More information about the geometry, boundary conditions, and nuclear parameters of those two benchmarks is given by Meneses et al. (2018). 3.1.2. 7th cycle of Angra 1 Nuclear Power Plant (Brazil) The Angra 1 NPP is located in the Rio de Janeiro state, Brazil, with a 626 MW PWR designed by Westinghouse and operated by Eletronuclear. Table 1 shows the data for the 7th cycle of Angra 1 NPP. The symmetry of 1/8 Angra 1 PWR yields 21 FAs for permutation. In our implementations, the ten quartet elements (on symmetry lines) are not permuted with the ten octet elements (offsymmetry lines). Considering that the central FA is not permuted, the total number of possible solutions is 10! 10! ffi 1:3 1013 . 3.2. Cuckoo Search algorithm The Cuckoo Search (CS) algorithm is a nature-inspired OMHs within the swarm intelligence paradigm, and was initially proposed by Yang and Deb (2009) for the solution of multimodal problems. CS is based on the breeding strategy of some cuckoo species, the so-called brood parasitism. In this strategy the cuckoo places its eggs in nests of birds of other species, so that the hosts raise the offspring after the egg hatching. The evolution of some cuckoo species has occurred in such a way that some females are experts at mimicking the color and pattern of host bird eggs, thus reducing the chances of their eggs being rejected (Payne, 2005). Occasionally, host birds discover these parasitic eggs and discard them or build a new nest elsewhere. In addition, the algorithm is improved by using the Lévy flight instead of simple random isotropic search (Brown et al., 2007). The Lévy flight represents a random search model characterized
Table 1 Burnup and kinf data for the 7th cycle of Angra 1 NPP simulated by RECNOD code. FA
Burnup (MWD/TU)
kinf
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
9603 13,045 7882 13,006 0 13,012 14,650 8622 13,181 0 14,068 13,115 13,135 0 0 11,404 7873 0 0 13,285
1.069 0.906 1.087 0.906 1.187 0.906 1.037 1.079 0.903 1.193 1.026 0.906 0.904 1.188 1.194 1.050 1.099 1.191 1.188 0.907
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by steps that obey a power law distribution and has its length defined by the Lévy distribution (Eq.1), which has infinite mean and infinite variance (Yang and Deb, 2009). Researches show that the behavior of some predators in the search for preys also follows the characteristics of Lévy flights (Yang and Deb, 2013). This model is commonly represented by small steps followed by big ones (Brown et al., 2007; Shlesinger et al., 1995). A symmetrical Lévy stable process {x} (Mantegna, 1994) is a stochastic process which has a probability density function L(x) in integral form that can be expressed as a series expansion. It can be shown that the leading term of the L(x) series is proportional to x -1-b and
LðxÞ jxj1b ;
0 < b 2;
ð1Þ
where b is the Lévy index. It is worth to notice that Cauchy distribution and Gaussian distribution are special cases of the Lévy distribution respectively with b = 1 and b = 2. In addition, Gutowski (2002) discusses the restrictions in b values. According to Yang (2014), a simple description of the CS metaphor is that each candidate solution is represented by an egg laid by one cuckoo in a randomly selected nest. Modelling more complicated versions is also possible. In that simple case, in a search space with d dimensions, at iteration t the quality of n eggs (that is, n candidate solution vectors xti where i 2 {1, . . ., n} where each vector has d components) is then evaluated with a fitness function. The main idea is that the best candidate solutions are kept and the worst ones are replaced. The CS uses a balanced combination of local search and global search. The local search (or local random walk according to Yang, 2014) is given by
xtþ1 ¼ xti þ a1 s Hðpa 2Þ xtj xtk i
ð2Þ
where xti is the ith candidate solution at iteration t; a1 is the scaling factor and in the present work empirically defined as 1, according to the results of preliminary tests; s is the step size, which is a random number over the range ½0; 1; Hð Þ is the Heaviside step function; the commutation parameter pa 2 (0, 1) is the probability of an egg being discovered by a host bird; is a random real number obtained from a uniform distribution over the range ½0; 1; and xtj and xtk are two different solutions randomly selected from current solutions, represented with the indices j and k 2 {1, . . ., n}. The Hadamard multiplication operator represents the entrywise product between matrices or vectors with the same dimensions. The global search is given by
xtþ1 ¼ xti þ a2 LðbÞ i
ð3Þ
where a2 in the present work is defined as 0.01 so that the algorithm does not make very large jumps, as recommended by Yang and Deb (2010). LðbÞ is a number drawn from the Lévy distribution (Eq. (1)), with b = 1.5 in the present work as also implemented by Yang and Deb (2010). Computationally LðbÞ can be determined in several ways, but the most direct way is the use of the so-called Mantegna Algorithm for a stable and symmetric Lévy distribution (Mantegna, 1994; Yang, 2010; Yang and Deb, 2013), which can be summarized as
LðbÞ
u jv j b
1
ðxtbest xti Þ
ð4Þ
where xtbest is the candidate solution with the best fitness evaluation, determined by the CS algorithm until the iteration t. In addition, u and m are obtained from the normal distribution, that is,
u Nð0; r2u Þ and
ð5Þ
v
Nð0; r2v Þ
ð6Þ
with
ru
8 91=b
rv ¼ 1
ð7Þ
where C is the standard Gamma distribution. The CS algorithm is shown in Fig. 1. 3.2.1. Random Keys (RK) In the CS for the LPO problem feasible combinatorial candidate solutions are obtained with RK. The RK model was proposed by Bean (1994) and consists of transforming vectors of real numbers into possible solutions of combinatorial problems. In Fig. 2 the RK steps are described for a hypothetical five-dimensional problem (for an LP with five FAs): (a) a real vector is obtained by the CS algorithm; (b) an integer vector is associated with the real vector; (c) the real vector is sorted; (d) an associated permutation of the integer vector is obtained; (e) finally the permutation will represent an LP (candidate solution) to be evaluated. Fig. 3 depicts the RK method for generating a combinatorial solution for the LPO: (a) one candidate solution with 20 dimensions is generated by the CS algorithm; (b) since in our implementations symmetry lines FAs are not permuted with FAs off symmetry lines, the 20 keys are split in two groups (the order of the numbers is also represented, as in Fig. 2 – step b); (c) each real group is sorted and then the integer numbers change their positions; (d) the reference LP is taken as a base and each FA is represented by an integer number; (e) finally, symmetry lines FAs are shuffled following the first group order, as well as FAs off symmetry lines according to the second group order. 4. Methodology 4.1. LPO with the Cuckoo Search algorithm The IAEA-3D and BIBLIS-2D benchmarks were implemented with the PARCS reactor physics code using the same configurations presented in Meneses et al. (2018), that is, a combination of the Nodal Expansion Method (Finnemann et al., 1977) and Analytical Nodal Methods (Smith, 1979) with 2 2 spatial discretization in each FA. Critical Boron search is performed without considering thermal-hydraulic feedback. The optimization parameter used as a safety constraint for the IAEA-3D and BIBLIS-2D benchmarks is the peak power factor F q , defined as the quotient between the maximum power of each FA and the average power of the whole core. All the benchmarks were simulated with a null initial critical Boron concentration. Under such conditions, the original LPs of the benchmarks IAEA-3D and BIBLIS-2D yield respectively the eigenvalues keff = 1.029096 and keff = 1.025368, which are in accordance to the reference values. Under the same conditions, when the critical Boron search is performed, the values obtained are respectively CB = 344.70 ppm and CB = 325.71 ppm. For a comparison between CS and the ABC and PBIL algorithms described by Meneses et al. (2018), the CB values yielded by the PARCS code were multiplied by 0.01 so that CB ranges from 0 to less than 2000 ppm, which are typical values of a real PWR, for which the absolute Boron Worth is approximately 7–10 pcm/ppm (see Oka et al., 2014, p. 196 and p. 197). The nuclear parameters produced by the RECNOD code are the Boron Concentration (C B ) in the Xenon equilibrium, so that, according to Chapot (2000), 4 ppm correspond to 1 Effective Full Power Day (EFPD), used to estimate the duration of the cycle (optimization criterion) and the Maximum Normalized Power of FA (P rm )
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Fig. 1. Pseudocode of the CS algorithm.
Fig. 2. Random Keys example.
used as a safety parameter (constraint), corresponding to the F q of the PARCS code. In our implementations the objective function for the LPO according to the parameters generated by the reactor physical codes is to minimize
1 CB
ð8Þ
subject to
Sconstr S0
ð9Þ
where Sconstr is a security constraint variable (F q for IAEA-3D and BIBLIS-2D, as well as P rm for Angra 1 7th Cycle) and S0 is a real number. Prm = 1.395 for Angra 1 7th Cycle assures the compliance with technical safety requirements according to Chapot (2000), and was also used in other works (Chapot et al., 1999; Meneses et al., 2009; Meneses and Schirru, 2015). Fq = 2.200 for IAEA-3D and Fq = 1.400 for BIBLIS-2D were arbitrated by Meneses et al. (2018) according to preliminary tests and also used in the present work. The aggregated fitness function (considering that the values of Sconstr are always greater than reciprocal of the Boron concentration) is
( Fitness ¼
where k = 1 ppm is a constant used to maintain the fitness dimensionless. The discontinuity in S0 does not represent a drawback for the performance of OMHs, which do not depend on derivatives. Meneses et al. (2009), Meneses and Schirru (2015), and Meneses et al. (2018) also used Eq. (10) as fitness function for the OMHs PSO, CE, PBIL, and ABC. Caldas and Schirru (2008) used another discontinuous fitness function for the OMH Parameter-Free PBIL (FPBIL). Hill and Parks (2015) give examples of objective functions used in the LPO problem. In this sense, the fitness function could include other parameters or be otherwise formulated (e.g. see Caldas and Schirru, 2008; Yarizadeh-Beneh et al., 2016). However, given that in the present work we compare CS to ABC and PBIL, the fitness function is the same as the one used by Meneses et al. (2018) for a fair comparison. The CS code and all interfaces used to calculate the fitness functions calling the executable files of reactor physics codes were implemented in C language with the Microsoft Visual Studio Community Edition 2017 IDE on a Windows 10 operating system. The Windows 7 operating system was used to run the tests. The computers used were two HPÒ desktop computers with an IntelÒ Core i7-3440 (3.40 GHz) processor and 8.0 GB of RAM. The parameters used in CS are shown in Table 2. Fifty independent tests (i.e., with different seeds) were performed.
k C1B ; ifScontr S0 Sconstr ; otherwise
ð10Þ
4.2. Statistical analyses Since OMHs are stochastic algorithms and some variability is expected in the results, the same algorithm can perform differently depending on the problem. Wolpert and Macready (1997) presented No Free Lunch (NFL) theorems that address the average performance of general-purpose algorithms concerning all possible problems, which was proven identical for static and timedependent problems, among other considerations. In other words, in such conditions there is no ‘‘always-the-best” algorithm. However, as pointed out by Joyce and Herrmann (2018), there are still successful OMHs, used in specific sets of problems, which possess certain structures and characteristics, for which some algorithms are better suited and therefore such algorithms perform better than others on average in those problems. So, the better an OMH’s
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Fig. 3. RK example for configurating a combinatorial candidate solution (LP) for the 7th cycle of Angra 1 NPP.
Table 2 Parameters of the CS algorithm, in each problem.
Nests pa Iterations Evaluations
RECNOD
BIBLIS
IAEA
100 25% 500 100,000
100 25% 500 100,000
25 25% 20 1000
overall performance in a set of problems, the more robust such OMH is in that set of problems. In order to compare if there are significant differences between the algorithms, statistical tests were performed. If certain assumptions are fulfilled, such statistical differences can be assessed by the parametric test Analysis of Variance (ANOVA; see Rutherford, 2011). Such assumptions are: (a) samples randomly obtained; (b) normal distribution of the results; (c) results obtained independently; and (d) homogeneity of variances. The use of a pseudorandom number generator guarantees the fulfilment of assumption (a). Regarding the assumption (c), according to García et al. (2009) the algorithms’ runs are independent events because the occurrence of one run does not modify the probability of the another run since those events have different initial seeds for the pseudorandom number generation. In the present work the assumption (b) was assessed with the Shapiro-Wilk test and the assumption (d), with the Levene’s test. In the one-way ANOVA the F-statistic is the ratio of betweengroups to within-groups variations. The null hypothesis is that there is no significant difference between the groups. A p-value less
than a cut-off level indicates that there is significant difference between the groups. The ANOVA only verifies if there is an overall statistical difference between the groups, thus a pairwise comparison between the OMHs is needed (that is, a post-hoc test) and in our work the Tukey’s studentized range (Honestly Significant Difference, HSD) test was used. In the Shapiro-Wilk test, the null hypothesis is that the sample comes from a normal distribution. A p-value less than a cut-off value indicates that the sample does not come from a normal distribution and thus the parametric ANOVA cannot be used, that is, the assumption (b) is not fulfilled. The null hypothesis of Levene’s test is that the variances are homogeneous. If a p-value is less than a cut-off level then the assumption (d) is not fulfilled and ANOVA cannot be used either (see also García et al., 2009). Since such population parameters assumptions do not hold in many cases for comparing algorithms, some authors discuss the application of nonparametric procedures for machine learning algorithms (e.g. Demšar, 2006; García et al., 2010) and OMHs (e.g. García et al., 2009). Thus, the Kruskal-Wallis test (see Sheskin, 2003), which is the nonparametric equivalent of ANOVA, was performed in all cases, whether the assumptions concerning population parameters were fulfilled (IAEA-3D) or not (BIBLIS-2D and Angra 1 7th Cycle), as described in section 5. The Kruskal-Wallis test statistic H can be approximated by a v2 distribution and if the p-value is less than a cut-off level then there is statistical difference among the mean ranks of the groups. Such as ANOVA, the Kruskal-Wallis test verifies if there is an overall statistically significant difference and therefore a pairwise post-hoc test is also needed. Thus the Dunn’s
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test was used (Dmitrienko et al., 2007), complementing the application of the Kruskal-Wallis test. The statistical analyses were performed with SAS/STATÒ University Edition, using the UNIVARIATE, GLM, and NPAR1WAY procedures, as well as a Dunn’s test macro (Dmitrienko et al., 2007). A cut-off value 0.05 for p was adopted in the statistical tests. 5. Computational results The results of the CS are compared to the results of the algorithms ABC (Karaboga, 2005) and PBIL (Baluja, 1994) obtained in previous work (Meneses et al., 2018). ABC and PBIL were selected because they were the best algorithms for the instances IAEA-3D, BIBLIS-2D, and Angra 1 – 7th cycle for 100,000 evaluations. The best LPs found by CS for each problem are shown in Fig. 4. The CS average running times for each instance were: (a) 29.94 min (0.50 h) for IAEA-3D; (b) 272.27 min (4.54 h) for BIBLIS-2D; and (c) 212.82 min (3.55 h) for Angra 1 7th cycle. 5.1. IAEA-3D LPO problem The descriptive statistics of the CB results for the IAEA-3D LPO problem is presented in Table 3. In Fig. 5 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations.
The final CB results of the ABC, PBIL, and CS algorithms were normal (Shapiro-Wilk test, respectively with p-values 0.4194, 0.1459, and 0.3712). According to Levene’s test, CB variances were homogeneous (p = 0.2761). An ANOVA was then performed and a statistically significant difference between the means was found (F ffi 7.99; p = 0.0005). The Tukey’s studentized range (HSD) posthoc test was also performed (with a = 0.05), indicating that ABC and CS were the best algorithms for the IAEA-3D LPO problem, with no statistically significant difference between them (see Table 4). Both ABC and CS outperformed PBIL. The Kruskal-Wallis test (v2 ffi 16.07; p = 0.0003) and the Dunn’s test (see Table 5) were also performed. Both tests corroborate the ANOVA’s results. 5.2. BIBLIS-2D LPO problem The descriptive statistics of the CB results for the BIBLIS-2D LPO problem is presented in Table 6. In Fig. 6 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations. The final CB results of the ABC, PBIL, and CS algorithms were non-normal (Shapiro-Wilk test, respectively with p < 0.0001, p = 0.0008, and p < 0.0001) and a Kruskal-Wallis test was then performed indicating a statistically significant difference between the mean ranks (v2 = 17.3; p = 0.0002). The Dunn’s post-hoc test for
Fig. 4. Best LPs found by CS for: (a) IAEA-3D LPO problem (CB = 546.88; Fq = 2.193; FAs with control rods, with number 4, are not swapped; number 1 represents reflector); (b) BIBLIS-2D LPO problem (CB = 563.49; Fq = 1.382; number 3 represents reflector); (c) Angra 1 PWR 7th cycle (CB = 1406; Prm = 1.395; the central FA is not swapped).
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A.A.M. Meneses et al. / Annals of Nuclear Energy 139 (2020) 107214 Table 3 Descriptive statistics of CB results (in ppm) for the IAEA-3D LPO problem.
Average Standard Dev. Median Maximum Minimum a b
Table 6 Descriptive statistics of CB results (in ppm) for the BIBLIS-2D LPO problem.
ABCa
PBILa
CSb
511.89 11.56 511.23 546.88 485.80
502.79 14.79 500.81 543.14 474.09
511.92 12.95 510.00 546.88 488.59
Average Standard Dev. Median Maximum Minimum a
Meneses et al. (2018). Present work.
b
ABCa
PBILa
CSb
472.26 121.85 515.22 559.70 81.85
524.67 24.16 525.69 558.35 439.56
528.44 37.83 538.70 563.49 393.60
Meneses et al. (2018). Present work.
Fig. 5. Comparison between PBIL, ABC and CS for the IAEA-3D LPO problem. The bars represent a 95% confidence interval for the average (considering the Student’s t-distribution for 50 tests).
Fig. 6. Comparison between PBIL, ABC, and CS for the BIBLIS-2D LPO problem. The bars represent a 95% confidence interval for the averages (considering the Student’s t-distribution for 50 tests).
Table 4 Tukey’s Studentized range (HSD) post hoc test results for the IAEA-3D LPO problem (a = 0.05).
The final CB results of the ABC and PBIL algorithms were normal (Shapiro-Wilk test, respectively with p-values 0.0893 and 0.2687), whereas the final results of the CS algorithm were non-normal (p = 0.0004). Thus, a Kruskal-Wallis test was performed indicating a statistically significant difference between the mean ranks (v2 = 7.08; p = 0.0290). The Dunn’s post-hoc test for multiple comparisons was also performed, indicating that CS and ABC were the best algorithms for the Angra 1 7th Cycle LPO problem, with no statistically significant difference between them (see Table 9). CS outperformed PBIL, although ABC did not outperformed PBIL.
Comparison number
Group comparison
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
** **
**Indicates a statistically significant difference between algorithms.
Table 5 Dunn’s post hoc test results for the IAEA-3D LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
1.84 31.04 29.20
20.8015 20.8015 20.8015
** **
**Indicates a statistically significant difference between algorithms.
multiple comparisons was also performed, indicating that CS and PBIL were the best algorithms for the BIBLIS-2D LPO problem, with no statistically significant difference between them (see Table 7). Both CS and PBIL outperformed ABC. 5.3. Angra 1 7th cycle LPO problem The descriptive statistics of the CB results for the Angra 1 7th Cycle LPO problem is presented in Table 8. In Fig. 7 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations.
6. Discussion As mentioned by Joyce and Herrmann (2018), notwithstanding the NFL theorems, and as computational experiments point out, for specific sets of problems there are algorithms that on average perform better than others because of also specific characteristics. For the LPO problem some remarkable results have been obtained by ABC algorithm (de Oliveira and Schirru, 2011; Safarzadeh et al., 2011; Meneses et al., 2018), as well as by PBIL algorithm (Caldas and Schirru, 2008; Meneses et al., 2018). Yarizadeh-Beneh et al. (2016) showed the efficiency of the CS algorithm for a WWER NPP, and in the present work we applied CS to the LPO based on the benchmarks IAEA-3D and BIBLIS-2D, as well to the Angra 1 7th Cycle problem, and in all of the three cases statistical analyses showed that CS is among the best algorithms. Yang (2014) discusses an interesting characteristic of the CS showing its similarities to Differential Evolution (Storn and Price, 1997), PSO, and SA in the sense that CS can be seen as an efficient combination of those algorithms. Yang (2014) also reports that the
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Table 7 Dunn’s post hoc test results for the BIBLIS-2D LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
35.98 21.08 14.90
20.8015 20.8015 20.8015
** **
**Indicates a statistically significant difference between algorithms.
Table 8 Descriptive statistics of CB results (in ppm) for the Angra 1 7th Cycle LPO problem.
Average Standard Dev. Median Maximum Minimum a b
ABCa
PBILa
CSb
1307 48 1311 1435 1224
1303 45 1304 1400 1209
1324 61 1328 1406 1073
Meneses et al. (2018). Present work.
And such characteristics of the CS algorithm for global optimization are essential for the search in the LPO as real vectors are mapped into combinatorial solutions with the RK. The RK approach for efficient global optimizers in continuous spaces have enabled good performances for several algorithms. And as new global optimizers are developed with efficient search in continuous spaces, notably the application of RKs makes them also efficient in the LPO problem. 7. Conclusion In the present work the CS algorithm for the optimization of the IAEA-3D, BIBLIS-2D, and Angra 1 7th Cycle LPO problems (Meneses et al., 2018) was implemented. The algorithms selected for comparison were ABC and PBIL. Statistical analyses showed that: (a) for the IAEA-3D problem, CS and ABC were the best algorithms; (b) for the BIBLIS-2D LPO problem, CS and PBIL were the best algorithms; and (c) for the Angra 1 7th Cycle LPO problem, CS and ABC were the best algorithms. Thus, in all of the three LPO problems tested, CS was among the best algorithms, reaching the best overall performance, therefore being the most robust of the three algorithms compared. In future works we will include other instances for algorithms comparison, implement the Improved DCS for the LPO, and optimize multi-cycle problems. CRediT authorship contribution statement Anderson Alvarenga de Moura Meneses: Conceptualization, Methodology, Formal analysis, Writing - original draft. Patrick Vasconcelos da Silva: Software, Validation, Investigation, Writing - original draft. Fernando Nogueira Nast: Software, Investigation, Visualization. Lenilson Moreira Araujo: Formal analysis, Writing original draft. Roberto Schirru: Conceptualization, Methodology, Writing - review & editing, Supervision. Declaration of Competing Interest
Fig. 7. Comparison between PBIL, ABC and CS for the Angra 1 7th cycle LPO problem. The bars represent a 95% confidence interval for the averages (considering the Student’s t-distribution for 50 tests).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements
Table 9 Dunn’s post hoc test results for the Angra 1 7th Cycle LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
18.66 2.49 21.15
20.8015 20.8015 20.8015
**
**Indicates a statistically significant difference between algorithms.
CS algorithm’s global convergence is guaranteed, which is corroborated by He et al. (2018), with a validation with numerical analysis. According to Yang (2014), the efficiency of the CS algorithm is due to the balance of global and local search processes, which are controlled by the parameter pa . The typical value pa = 0.25 also used in our tests means that such control enables the global search to be performed in approximately 3/4 of the search. This fact as well as the usage of Lévy flights makes the CS a remarkable global optimizer.
R.S. acknowledges CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) and FAPERJ (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Brazil). The authors acknowledge CNEN (Comissão Nacional de Energia Nuclear – participant of the CAMP agreement with the USA NRC), in the terms of the agreement of mutual cooperation with the Federal University of Western Pará for using the PARCS v3.0 code. The authors would like to thank the reviewers for their valuable comments and suggestions. References Argonne National Laboratory, 1977. Benchmark Problem Book (Supplement 2). Technical Report ANL-7416. Baluja, S., 1994. Population-Based Incremental Learning: A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning. Technical Report. School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania. Bean, J.C., 1994. Genetic algorithms and random keys for sequencing and optimization. ORSA J. Comput. 6, 154–160. https://doi.org/10.1287/ijoc.6.2.154. Brown, C.T., Liebovitch, L.S., Glendon, R., 2007. Lévy flights in Dobe Ju/’hoansi foraging patterns. Hum. Ecol. 35, 129–138. https://doi.org/10.1007/s10745006-9083-4.
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Meneses, A.A.M., Araujo, L.M., Nast, F.N., da Silva, P.V., Schirru, R., 2018. Application of metaheuristics to Loading Pattern Optimization problems based on the IAEA3D and BIBLIS-2D data. Ann. Nucl. Energy 111, 329–339. https://doi.org/ 10.1016/j.anucene.2017.09.008. Meneses, A.A.M., Gambardella, L.M., Schirru, R., 2010. A new approach for heuristics-guided search in the In-Core Fuel Management Optimization. Prog. Nucl. Energy 52, 339–351. https://doi.org/10.1016/j.pnucene.2009.07.007. Meneses, A.A.M., Machado, M.D., Schirru, R., 2009. Particle swarm optimization applied to the nuclear reload problem of a pressurized water reactor. Prog. Nucl. Energy 51, 319–326. https://doi.org/10.1016/j.pnucene.2008.07.002. Meneses, A.A.M., Schirru, R., 2015. A cross-entropy method applied to the in-core fuel management optimization of a pressurized water reactor. Prog. Nucl. Energy 83, 326–335. https://doi.org/10.1016/j.pnucene.2015.04.007. de Oliveira, I.M.S., Schirru, R., 2011. Swarm intelligence of artificial bees applied to in-core fuel management optimization. Ann. Nucl. Energy 38, 1039–1045. https://doi.org/10.1016/j.anucene.2011.01.009. Oka, Y., Uchikawa, S., Suzuki, K., 2014. Light water reactor design. In: Oka, Y. (Ed.), Nuclear Reactor Design. Springer. https://doi.org/10.1007/978-4-431-548980_3. Ouaarab, A., Ahiod, B., Yang, X.-S., 2014. Improved and discrete cuckoo search for solving the travelling salesman problem. In: Yang, X.-.S. (Ed.), Cuckoo Search and Firefly Algorithm. Springer, pp. 63–84. Papadimitriou, C., Steiglitz, K., 1982. Combinatorial Optimization: Algorithms and Optimization. Prentice-Hall. Parks, G.T., 1990. An intelligent stochastic optimization routine for nuclear fuel cycle design. Nucl. Technol. 89, 233–246. https://doi.org/10.13182/NT90A34350. Payne, R.B., 2005. The Cuckoos. Oxford University Press. Poon, P.W., Parks, G.T., 1992. Optimizing PWR reload core designs. In: Manner, R., Manderick, B. (Eds.), Parallel Problems Solving from Nature II, pp. 371–380. Poursalehi, N., Zolfaghari, A., Minuchehr, A., 2013. Development of a high order and multi-dimensional nodal code, ACNEC3D, for reactor core analysis. Ann. Nucl. Energy 55, 211–224. https://doi.org/10.1016/j.anucene.2012.10.022. Rutherford, A., 2011. ANOVA and ANCOVA: A GLM Approach. 2nd Ed. John Wiley and Sons, Inc. Safarzadeh, O., Zolfaghari, A., Norouzi, A., Minuchehr, H., 2011. Loading pattern optimization of PWR reactors using Artificial Bee Colony. Ann. Nucl. Energy 38, 2218–2226. https://doi.org/10.1016/j.anucene.2011.06.008. Shlesinger, M.F., Zaslavsky, G.M., Frisch, U., 1995. Lévy Flights and Related Topics in Physics. Springer-Verlag. Sheskin, D.J., 2003. Handbook of Parametric and Nonparametric Statistical Procedures. Chapman & Hall/CRC. Silva, P.V., Nast, F.N., Schirru, R., Meneses, A.A.M. Busca Cuco com Voo de Lévy Aplicado ao Problema de Recarga de Combustíveis em Reatores Nucleares. In: Proceedings of the International Nuclear Atlantic Conference 2017, Belo Horizonte, Brazil, October 22–27, 2017 (in Portuguese). Smith, K.S., 1979. An Analytic Nodal Method for Solving the Two-group, Multidimensional, Static and Transient Neutron Diffusion Equations. Massachusetts Institute of Technology (Ph.D. thesis). Stevens, J.G., Smith, K.S., Rempe, K.R., Downar, T.J., 1995. Optimization of pressurized water reactor shuffling by simulated annealing with heuristics. Nucl. Sci. Eng. 121, 67–88. https://doi.org/10.13182/NSE121-67. Storn, R., Price, K., 1997. Differential evolution – a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11, 341–359. https://doi.org/10.1023/A:1008202821328. Wolpert, D.H., Macready, W.G., 1997. No free lunch theorems for optimization. IEEE T. Evolut. Comput. 1, 67–82. https://doi.org/10.1109/4235.585893. Yang, X.-S., 2010. Nature-inspired Metaheuristic Algorithms. Luniver Press. Yang, X.-S., Deb, S., 2009. Cuckoo search via Lévy flights. In: World Congress on Nature & Biologically Inspired Computing. IEEE, pp. 210–214. https://doi.org/ 10.1109/NABIC.2009.5393690. Yang, X.-S., Deb, S., 2010. Engineering optimization by cuckoo search. Int. J. Math. Model. Numer. Optim. 1, 330–343. https://doi.org/10.1504/ IJMMNO.2010.03543. Yang, X.-S., Deb, S., 2013. Multiobjective cuckoo search for design optimization. Comput. Oper. Res. 40, 1616–1624. https://doi.org/10.1016/j.cor.2011.09.026. Yang, X.-S., 2014. Cuckoo search and firefly algorithm: overview and analysis. In: Yang, X.-S. (Ed.), Cuckoo Search and Firefly Algorithm: Theory and Applications. Springer, pp. 1–26. https://doi.org/10.1007/978-3-319-02141-6_1. Yarizadeh-Beneh, M., Mazaheri-Beni, H., Poursalehi, N., 2016. Improving the refueling cycle of a WWER-1000 using cuckoo search method and thermalneutronic coupling of PARCS v2. 7, COBRA-EN and WIMSD-5B codes. Nucl. Eng. Des. 310, 247–257. https://doi.org/10.1016/j.nucengdes.2016.10.006.
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Application of Cuckoo Search algorithm to Loading Pattern Optimization problems Anderson Alvarenga de Moura Meneses a,⇑, Patrick Vasconcelos da Silva a, Fernando Nogueira Nast a, Lenilson Moreira Araujo a, Roberto Schirru b a b
Federal University of Western Pará, Laboratory of Computational Intelligence, Av. Vera Paz, s/n, Salé, CEP 68.035-110 Santarém, PA, Brazil Federal University of Rio de Janeiro, COPPE, Nuclear Engineering Program, Brazil
a r t i c l e
i n f o
Article history: Received 6 September 2019 Received in revised form 11 November 2019 Accepted 15 November 2019
Keywords: Loading Pattern Optimization Nuclear Power Plant Optimization metaheuristics Pressurized water reactor Cuckoo Search Nuclear fuel
a b s t r a c t The Loading Pattern Optimization (LPO) is related to important goals in a Nuclear Power Plant (NPP) operation such as the extension of the cycle according to safety margins. The LPO is a combinatorial problem of relevance and interest for Nuclear Engineering. Optimization metaheuristics have been efficient in solving the LPO. The recent metaheuristic Cuckoo Search (CS) is based on the brood parasitism of some cuckoo species, combined with the behavior of the Lévy flight of some birds. In the present work the results of the application of CS to the LPO using IAEA-3D and BIBLIS-2D benchmarks are presented, as well as the application of CS in the optimization of 7th cycle of Angra 1 NPP, in Brazil. The results are compared to the metaheuristics Artificial Bee Colony and Population-Based Incremental Learning. Statistical analyses show that CS is the most robust algorithm for the set of instances selected for tests. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction The Loading Pattern Optimization (LPO) is related to important goals in a Nuclear Power Plant (NPP) operation such as the extension of the operation cycle by determining the position of the Fuel Assemblies (FAs) in the reactor core, within adequate safety margins (Levine, 1986). The LPO is a prominent problem of economic interest, as well as a complex combinatorial problem, classified as NP-hard (Papadimitriou and Steiglitz, 1982), with a considerable number of feasible solutions, sub-optimal solutions, disconnected feasible regions, high dimensionality, complex and timeconsuming evaluation of candidate solutions using Reactor Physics codes (Stevens et al., 1995). Computational intelligence methods such as Optimization Metaheuristics (OMH) are efficient in solving combinatorial problems such as the LPO. Several OMHs have been successful when applied to LPO, including: Simulated Annealing (SA; Parks, 1990; Stevens et al., 1995), Genetic Algorithm (GA; Chapot et al., 1999; Poon and Parks, 1992), Tabu Search (TS; Hill and Parks, 2015; Lin et al., 1998), Population-Based Incremental Learning (PBIL; Caldas and Schirru, 2008; Machado, 2005), Ant Colony Optimization (ACO; de Lima et al., 2008; Machado and Schirru, 2002), ⇑ Corresponding author. E-mail address: [email protected] (A.A.M. Meneses). https://doi.org/10.1016/j.anucene.2019.107214 0306-4549/Ó 2019 Elsevier Ltd. All rights reserved.
Particle Swarm Optimization (PSO; Meneses et al., 2009), Artificial Bee Colony (ABC; de Oliveira and Schirru, 2011), and Cross-Entropy (CE; Meneses and Schirru, 2015). In the present work, CS was applied to the benchmarks IAEA-3D (Argonne National Laboratory, 1977) and BIBLIS-2D (Poursalehi et al., 2013), using the PARCS v3.0 code (U.S. NRC Core Neutronics Simulator; Downar et al., 2009a,b; Joo et al., 1998) as well as to the 7th cycle of Angra 1 NPP, located in Brazil, using the RECNOD code (Chapot, 2000). The remaining of the present article is organized as follows: Section 2 contains a review of previous works; in Section 3 the theoretical background is presented; Section 4 describes the methodology used to solve the LPO problems; Section 5 presents the results; the discussion is presented in Section 6; and finally, Section 7 is devoted to the conclusions. 2. Related works The CS algorithm was first presented by Yang and Deb (2009). The CS is based on the strategy of reproduction of some cuckoo species combined with the behavior of the Lévy Flight of some birds, and is a swarm intelligence algorithm (Engelbrecht, 2007). Ouaarab et al. (2014) applied a CS to the Traveling Salesman Problem (TSP). In such work the Improved Discrete CS (DCS) uses a fraction of intelligent cuckoos to explore other areas of the search
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space, using the local search heuristic 2-opt (Johnson and McGeoch, 1997). DCS and its improved version were applied to 41 instances of the problem and compared to the Genetic Simulated Annealing-Ant Colony System with PSO Techniques (GSAACS-PSOT; Chen and Chien, 2011), as well as to the Discrete PSO (DPSO). Improved DCS obtained superior results in comparison to GSA-ACS-PSOT and DPSO for the TSP. CS has also been applied to a nuclear reload problem by Yarizadeh-Beneh et al. (2016), in order to optimize the reload of the first cycle of the reactor located in the Bushehr NPP in Iran, using Random Keys (RK; Bean, 1994). For thermal-neutronic calculations, CS was coupled to PARCS reactor physics code, WIMSD-5B, and COBRA-5B, obtaining results that outperform those obtained in the original reload by a designer, extending the cycle in up to 7 days. Yarizadeh-Beneh et al. (2016) applied CS to the LPO of a real-world WWER NPP, and calculations for beginning of cycle (BOC) and end of cycle (EOC) states. In the present work the Critical Boron Concentration is calculated at the equilibrium of Xenon in the RECNOD code (7th cycle of Angra 1 NPP), and at the BOC in the IAEA-3D and BIBLIS-2D LPO problems. For the application of CS to the Angra 1 – 7th cycle LPO, Silva et al. (2017) performed a preliminary investigation. In the present work, the CS was also applied to two well-known benchmark problems (IAEA-3D and BIBLIS-2D), being compared favorably with the results of two competitive metaheuristics for the LPO, namely the ABC and PBIL algorithms. The results of the present work are compared to the results described by Meneses et al. (2018), who reported the application of the PSO, CE, PBIL, and ABC algorithms to LPO problems based on the data of the benchmarks BIBLIS-2D (Poursalehi et al., 2013) and IAEA-3D (Argonne National Laboratory, 1977), both of them modelled with the Reactor Physics code PARCS as well as the Angra 1 7th Cycle LPO problem, which was modelled with the RECNOD code (Chapot et al., 1999). According to Meneses et al. (2018) the best algorithms were ABC and PBIL respectively for IAEA-3D and for BIBLIS-2D. For the problem of Angra 1 NPP modeled with the reactor physics code RECNOD the algorithms PBIL, ABC, and CE achieved the best results. 3. Theoretical background 3.1. Loading Pattern Optimization (LPO) The LPO aims to determine LPs that maximize energy production satisfying safety requirements (Levine, 1986). The LPO is an NP-hard problem, which means that its complexity grows nonpolynomially, with a discontinuous and multimodal search space (de Lima, 2005). The substitution of burned FAs is necessary when it is not possible to keep the reactor running at nominal power, or more precisely after a period called the operating cycle. After each cycle the FAs present different neutronic characteristics. For example, the FAs are burned in different ways depending on the exposure time and position in the reactor core, with the accumulation of fission products such as Xenon and Samarium. Consequently, only some FAs are reused in the next cycle, and approximately 1/3 or 1/4 of the FAs are replaced by new ones (Hill and Parks, 2015; Meneses et al., 2010). Thus, forming an LP by fresh and burned FAs characterizes the LPO as a combinatorial problem. The benchmark LPO problems used for tests in the present work are described below. 3.1.1. IAEA-3D and BIBLIS-2D benchmarks IAEA-3D and BIBLIS-2D are theoretical problems taken as reference for the validation of numerical methods applied to the
neutron diffusion theory. In the present work both reactors were modeled considering the symmetry of ¼ in PARCS code, and during the OMH search new candidate solutions (LPs) are formed with octant symmetry. FAs on symmetry lines are not permuted with FAs off symmetry lines as described by Meneses et al. (2018). Despite being a two-dimensional reactor, BIBLIS-2D presents a much higher level of complexity than the IAEA-3D regarding the LPO, because of the number of candidate LP solutions. The number of candidate solutions for the BIBLIS-2D reactor is approximately 4:0 1016 whereas for the IAEA-3D reactor the total number is 288,288. More information about the geometry, boundary conditions, and nuclear parameters of those two benchmarks is given by Meneses et al. (2018). 3.1.2. 7th cycle of Angra 1 Nuclear Power Plant (Brazil) The Angra 1 NPP is located in the Rio de Janeiro state, Brazil, with a 626 MW PWR designed by Westinghouse and operated by Eletronuclear. Table 1 shows the data for the 7th cycle of Angra 1 NPP. The symmetry of 1/8 Angra 1 PWR yields 21 FAs for permutation. In our implementations, the ten quartet elements (on symmetry lines) are not permuted with the ten octet elements (offsymmetry lines). Considering that the central FA is not permuted, the total number of possible solutions is 10! 10! ffi 1:3 1013 . 3.2. Cuckoo Search algorithm The Cuckoo Search (CS) algorithm is a nature-inspired OMHs within the swarm intelligence paradigm, and was initially proposed by Yang and Deb (2009) for the solution of multimodal problems. CS is based on the breeding strategy of some cuckoo species, the so-called brood parasitism. In this strategy the cuckoo places its eggs in nests of birds of other species, so that the hosts raise the offspring after the egg hatching. The evolution of some cuckoo species has occurred in such a way that some females are experts at mimicking the color and pattern of host bird eggs, thus reducing the chances of their eggs being rejected (Payne, 2005). Occasionally, host birds discover these parasitic eggs and discard them or build a new nest elsewhere. In addition, the algorithm is improved by using the Lévy flight instead of simple random isotropic search (Brown et al., 2007). The Lévy flight represents a random search model characterized
Table 1 Burnup and kinf data for the 7th cycle of Angra 1 NPP simulated by RECNOD code. FA
Burnup (MWD/TU)
kinf
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
9603 13,045 7882 13,006 0 13,012 14,650 8622 13,181 0 14,068 13,115 13,135 0 0 11,404 7873 0 0 13,285
1.069 0.906 1.087 0.906 1.187 0.906 1.037 1.079 0.903 1.193 1.026 0.906 0.904 1.188 1.194 1.050 1.099 1.191 1.188 0.907
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by steps that obey a power law distribution and has its length defined by the Lévy distribution (Eq.1), which has infinite mean and infinite variance (Yang and Deb, 2009). Researches show that the behavior of some predators in the search for preys also follows the characteristics of Lévy flights (Yang and Deb, 2013). This model is commonly represented by small steps followed by big ones (Brown et al., 2007; Shlesinger et al., 1995). A symmetrical Lévy stable process {x} (Mantegna, 1994) is a stochastic process which has a probability density function L(x) in integral form that can be expressed as a series expansion. It can be shown that the leading term of the L(x) series is proportional to x -1-b and
LðxÞ jxj1b ;
0 < b 2;
ð1Þ
where b is the Lévy index. It is worth to notice that Cauchy distribution and Gaussian distribution are special cases of the Lévy distribution respectively with b = 1 and b = 2. In addition, Gutowski (2002) discusses the restrictions in b values. According to Yang (2014), a simple description of the CS metaphor is that each candidate solution is represented by an egg laid by one cuckoo in a randomly selected nest. Modelling more complicated versions is also possible. In that simple case, in a search space with d dimensions, at iteration t the quality of n eggs (that is, n candidate solution vectors xti where i 2 {1, . . ., n} where each vector has d components) is then evaluated with a fitness function. The main idea is that the best candidate solutions are kept and the worst ones are replaced. The CS uses a balanced combination of local search and global search. The local search (or local random walk according to Yang, 2014) is given by
xtþ1 ¼ xti þ a1 s Hðpa 2Þ xtj xtk i
ð2Þ
where xti is the ith candidate solution at iteration t; a1 is the scaling factor and in the present work empirically defined as 1, according to the results of preliminary tests; s is the step size, which is a random number over the range ½0; 1; Hð Þ is the Heaviside step function; the commutation parameter pa 2 (0, 1) is the probability of an egg being discovered by a host bird; is a random real number obtained from a uniform distribution over the range ½0; 1; and xtj and xtk are two different solutions randomly selected from current solutions, represented with the indices j and k 2 {1, . . ., n}. The Hadamard multiplication operator represents the entrywise product between matrices or vectors with the same dimensions. The global search is given by
xtþ1 ¼ xti þ a2 LðbÞ i
ð3Þ
where a2 in the present work is defined as 0.01 so that the algorithm does not make very large jumps, as recommended by Yang and Deb (2010). LðbÞ is a number drawn from the Lévy distribution (Eq. (1)), with b = 1.5 in the present work as also implemented by Yang and Deb (2010). Computationally LðbÞ can be determined in several ways, but the most direct way is the use of the so-called Mantegna Algorithm for a stable and symmetric Lévy distribution (Mantegna, 1994; Yang, 2010; Yang and Deb, 2013), which can be summarized as
LðbÞ
u jv j b
1
ðxtbest xti Þ
ð4Þ
where xtbest is the candidate solution with the best fitness evaluation, determined by the CS algorithm until the iteration t. In addition, u and m are obtained from the normal distribution, that is,
u Nð0; r2u Þ and
ð5Þ
v
Nð0; r2v Þ
ð6Þ
with
ru
8 91=b
rv ¼ 1
ð7Þ
where C is the standard Gamma distribution. The CS algorithm is shown in Fig. 1. 3.2.1. Random Keys (RK) In the CS for the LPO problem feasible combinatorial candidate solutions are obtained with RK. The RK model was proposed by Bean (1994) and consists of transforming vectors of real numbers into possible solutions of combinatorial problems. In Fig. 2 the RK steps are described for a hypothetical five-dimensional problem (for an LP with five FAs): (a) a real vector is obtained by the CS algorithm; (b) an integer vector is associated with the real vector; (c) the real vector is sorted; (d) an associated permutation of the integer vector is obtained; (e) finally the permutation will represent an LP (candidate solution) to be evaluated. Fig. 3 depicts the RK method for generating a combinatorial solution for the LPO: (a) one candidate solution with 20 dimensions is generated by the CS algorithm; (b) since in our implementations symmetry lines FAs are not permuted with FAs off symmetry lines, the 20 keys are split in two groups (the order of the numbers is also represented, as in Fig. 2 – step b); (c) each real group is sorted and then the integer numbers change their positions; (d) the reference LP is taken as a base and each FA is represented by an integer number; (e) finally, symmetry lines FAs are shuffled following the first group order, as well as FAs off symmetry lines according to the second group order. 4. Methodology 4.1. LPO with the Cuckoo Search algorithm The IAEA-3D and BIBLIS-2D benchmarks were implemented with the PARCS reactor physics code using the same configurations presented in Meneses et al. (2018), that is, a combination of the Nodal Expansion Method (Finnemann et al., 1977) and Analytical Nodal Methods (Smith, 1979) with 2 2 spatial discretization in each FA. Critical Boron search is performed without considering thermal-hydraulic feedback. The optimization parameter used as a safety constraint for the IAEA-3D and BIBLIS-2D benchmarks is the peak power factor F q , defined as the quotient between the maximum power of each FA and the average power of the whole core. All the benchmarks were simulated with a null initial critical Boron concentration. Under such conditions, the original LPs of the benchmarks IAEA-3D and BIBLIS-2D yield respectively the eigenvalues keff = 1.029096 and keff = 1.025368, which are in accordance to the reference values. Under the same conditions, when the critical Boron search is performed, the values obtained are respectively CB = 344.70 ppm and CB = 325.71 ppm. For a comparison between CS and the ABC and PBIL algorithms described by Meneses et al. (2018), the CB values yielded by the PARCS code were multiplied by 0.01 so that CB ranges from 0 to less than 2000 ppm, which are typical values of a real PWR, for which the absolute Boron Worth is approximately 7–10 pcm/ppm (see Oka et al., 2014, p. 196 and p. 197). The nuclear parameters produced by the RECNOD code are the Boron Concentration (C B ) in the Xenon equilibrium, so that, according to Chapot (2000), 4 ppm correspond to 1 Effective Full Power Day (EFPD), used to estimate the duration of the cycle (optimization criterion) and the Maximum Normalized Power of FA (P rm )
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Fig. 1. Pseudocode of the CS algorithm.
Fig. 2. Random Keys example.
used as a safety parameter (constraint), corresponding to the F q of the PARCS code. In our implementations the objective function for the LPO according to the parameters generated by the reactor physical codes is to minimize
1 CB
ð8Þ
subject to
Sconstr S0
ð9Þ
where Sconstr is a security constraint variable (F q for IAEA-3D and BIBLIS-2D, as well as P rm for Angra 1 7th Cycle) and S0 is a real number. Prm = 1.395 for Angra 1 7th Cycle assures the compliance with technical safety requirements according to Chapot (2000), and was also used in other works (Chapot et al., 1999; Meneses et al., 2009; Meneses and Schirru, 2015). Fq = 2.200 for IAEA-3D and Fq = 1.400 for BIBLIS-2D were arbitrated by Meneses et al. (2018) according to preliminary tests and also used in the present work. The aggregated fitness function (considering that the values of Sconstr are always greater than reciprocal of the Boron concentration) is
( Fitness ¼
where k = 1 ppm is a constant used to maintain the fitness dimensionless. The discontinuity in S0 does not represent a drawback for the performance of OMHs, which do not depend on derivatives. Meneses et al. (2009), Meneses and Schirru (2015), and Meneses et al. (2018) also used Eq. (10) as fitness function for the OMHs PSO, CE, PBIL, and ABC. Caldas and Schirru (2008) used another discontinuous fitness function for the OMH Parameter-Free PBIL (FPBIL). Hill and Parks (2015) give examples of objective functions used in the LPO problem. In this sense, the fitness function could include other parameters or be otherwise formulated (e.g. see Caldas and Schirru, 2008; Yarizadeh-Beneh et al., 2016). However, given that in the present work we compare CS to ABC and PBIL, the fitness function is the same as the one used by Meneses et al. (2018) for a fair comparison. The CS code and all interfaces used to calculate the fitness functions calling the executable files of reactor physics codes were implemented in C language with the Microsoft Visual Studio Community Edition 2017 IDE on a Windows 10 operating system. The Windows 7 operating system was used to run the tests. The computers used were two HPÒ desktop computers with an IntelÒ Core i7-3440 (3.40 GHz) processor and 8.0 GB of RAM. The parameters used in CS are shown in Table 2. Fifty independent tests (i.e., with different seeds) were performed.
k C1B ; ifScontr S0 Sconstr ; otherwise
ð10Þ
4.2. Statistical analyses Since OMHs are stochastic algorithms and some variability is expected in the results, the same algorithm can perform differently depending on the problem. Wolpert and Macready (1997) presented No Free Lunch (NFL) theorems that address the average performance of general-purpose algorithms concerning all possible problems, which was proven identical for static and timedependent problems, among other considerations. In other words, in such conditions there is no ‘‘always-the-best” algorithm. However, as pointed out by Joyce and Herrmann (2018), there are still successful OMHs, used in specific sets of problems, which possess certain structures and characteristics, for which some algorithms are better suited and therefore such algorithms perform better than others on average in those problems. So, the better an OMH’s
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5
Fig. 3. RK example for configurating a combinatorial candidate solution (LP) for the 7th cycle of Angra 1 NPP.
Table 2 Parameters of the CS algorithm, in each problem.
Nests pa Iterations Evaluations
RECNOD
BIBLIS
IAEA
100 25% 500 100,000
100 25% 500 100,000
25 25% 20 1000
overall performance in a set of problems, the more robust such OMH is in that set of problems. In order to compare if there are significant differences between the algorithms, statistical tests were performed. If certain assumptions are fulfilled, such statistical differences can be assessed by the parametric test Analysis of Variance (ANOVA; see Rutherford, 2011). Such assumptions are: (a) samples randomly obtained; (b) normal distribution of the results; (c) results obtained independently; and (d) homogeneity of variances. The use of a pseudorandom number generator guarantees the fulfilment of assumption (a). Regarding the assumption (c), according to García et al. (2009) the algorithms’ runs are independent events because the occurrence of one run does not modify the probability of the another run since those events have different initial seeds for the pseudorandom number generation. In the present work the assumption (b) was assessed with the Shapiro-Wilk test and the assumption (d), with the Levene’s test. In the one-way ANOVA the F-statistic is the ratio of betweengroups to within-groups variations. The null hypothesis is that there is no significant difference between the groups. A p-value less
than a cut-off level indicates that there is significant difference between the groups. The ANOVA only verifies if there is an overall statistical difference between the groups, thus a pairwise comparison between the OMHs is needed (that is, a post-hoc test) and in our work the Tukey’s studentized range (Honestly Significant Difference, HSD) test was used. In the Shapiro-Wilk test, the null hypothesis is that the sample comes from a normal distribution. A p-value less than a cut-off value indicates that the sample does not come from a normal distribution and thus the parametric ANOVA cannot be used, that is, the assumption (b) is not fulfilled. The null hypothesis of Levene’s test is that the variances are homogeneous. If a p-value is less than a cut-off level then the assumption (d) is not fulfilled and ANOVA cannot be used either (see also García et al., 2009). Since such population parameters assumptions do not hold in many cases for comparing algorithms, some authors discuss the application of nonparametric procedures for machine learning algorithms (e.g. Demšar, 2006; García et al., 2010) and OMHs (e.g. García et al., 2009). Thus, the Kruskal-Wallis test (see Sheskin, 2003), which is the nonparametric equivalent of ANOVA, was performed in all cases, whether the assumptions concerning population parameters were fulfilled (IAEA-3D) or not (BIBLIS-2D and Angra 1 7th Cycle), as described in section 5. The Kruskal-Wallis test statistic H can be approximated by a v2 distribution and if the p-value is less than a cut-off level then there is statistical difference among the mean ranks of the groups. Such as ANOVA, the Kruskal-Wallis test verifies if there is an overall statistically significant difference and therefore a pairwise post-hoc test is also needed. Thus the Dunn’s
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test was used (Dmitrienko et al., 2007), complementing the application of the Kruskal-Wallis test. The statistical analyses were performed with SAS/STATÒ University Edition, using the UNIVARIATE, GLM, and NPAR1WAY procedures, as well as a Dunn’s test macro (Dmitrienko et al., 2007). A cut-off value 0.05 for p was adopted in the statistical tests. 5. Computational results The results of the CS are compared to the results of the algorithms ABC (Karaboga, 2005) and PBIL (Baluja, 1994) obtained in previous work (Meneses et al., 2018). ABC and PBIL were selected because they were the best algorithms for the instances IAEA-3D, BIBLIS-2D, and Angra 1 – 7th cycle for 100,000 evaluations. The best LPs found by CS for each problem are shown in Fig. 4. The CS average running times for each instance were: (a) 29.94 min (0.50 h) for IAEA-3D; (b) 272.27 min (4.54 h) for BIBLIS-2D; and (c) 212.82 min (3.55 h) for Angra 1 7th cycle. 5.1. IAEA-3D LPO problem The descriptive statistics of the CB results for the IAEA-3D LPO problem is presented in Table 3. In Fig. 5 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations.
The final CB results of the ABC, PBIL, and CS algorithms were normal (Shapiro-Wilk test, respectively with p-values 0.4194, 0.1459, and 0.3712). According to Levene’s test, CB variances were homogeneous (p = 0.2761). An ANOVA was then performed and a statistically significant difference between the means was found (F ffi 7.99; p = 0.0005). The Tukey’s studentized range (HSD) posthoc test was also performed (with a = 0.05), indicating that ABC and CS were the best algorithms for the IAEA-3D LPO problem, with no statistically significant difference between them (see Table 4). Both ABC and CS outperformed PBIL. The Kruskal-Wallis test (v2 ffi 16.07; p = 0.0003) and the Dunn’s test (see Table 5) were also performed. Both tests corroborate the ANOVA’s results. 5.2. BIBLIS-2D LPO problem The descriptive statistics of the CB results for the BIBLIS-2D LPO problem is presented in Table 6. In Fig. 6 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations. The final CB results of the ABC, PBIL, and CS algorithms were non-normal (Shapiro-Wilk test, respectively with p < 0.0001, p = 0.0008, and p < 0.0001) and a Kruskal-Wallis test was then performed indicating a statistically significant difference between the mean ranks (v2 = 17.3; p = 0.0002). The Dunn’s post-hoc test for
Fig. 4. Best LPs found by CS for: (a) IAEA-3D LPO problem (CB = 546.88; Fq = 2.193; FAs with control rods, with number 4, are not swapped; number 1 represents reflector); (b) BIBLIS-2D LPO problem (CB = 563.49; Fq = 1.382; number 3 represents reflector); (c) Angra 1 PWR 7th cycle (CB = 1406; Prm = 1.395; the central FA is not swapped).
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A.A.M. Meneses et al. / Annals of Nuclear Energy 139 (2020) 107214 Table 3 Descriptive statistics of CB results (in ppm) for the IAEA-3D LPO problem.
Average Standard Dev. Median Maximum Minimum a b
Table 6 Descriptive statistics of CB results (in ppm) for the BIBLIS-2D LPO problem.
ABCa
PBILa
CSb
511.89 11.56 511.23 546.88 485.80
502.79 14.79 500.81 543.14 474.09
511.92 12.95 510.00 546.88 488.59
Average Standard Dev. Median Maximum Minimum a
Meneses et al. (2018). Present work.
b
ABCa
PBILa
CSb
472.26 121.85 515.22 559.70 81.85
524.67 24.16 525.69 558.35 439.56
528.44 37.83 538.70 563.49 393.60
Meneses et al. (2018). Present work.
Fig. 5. Comparison between PBIL, ABC and CS for the IAEA-3D LPO problem. The bars represent a 95% confidence interval for the average (considering the Student’s t-distribution for 50 tests).
Fig. 6. Comparison between PBIL, ABC, and CS for the BIBLIS-2D LPO problem. The bars represent a 95% confidence interval for the averages (considering the Student’s t-distribution for 50 tests).
Table 4 Tukey’s Studentized range (HSD) post hoc test results for the IAEA-3D LPO problem (a = 0.05).
The final CB results of the ABC and PBIL algorithms were normal (Shapiro-Wilk test, respectively with p-values 0.0893 and 0.2687), whereas the final results of the CS algorithm were non-normal (p = 0.0004). Thus, a Kruskal-Wallis test was performed indicating a statistically significant difference between the mean ranks (v2 = 7.08; p = 0.0290). The Dunn’s post-hoc test for multiple comparisons was also performed, indicating that CS and ABC were the best algorithms for the Angra 1 7th Cycle LPO problem, with no statistically significant difference between them (see Table 9). CS outperformed PBIL, although ABC did not outperformed PBIL.
Comparison number
Group comparison
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
** **
**Indicates a statistically significant difference between algorithms.
Table 5 Dunn’s post hoc test results for the IAEA-3D LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
1.84 31.04 29.20
20.8015 20.8015 20.8015
** **
**Indicates a statistically significant difference between algorithms.
multiple comparisons was also performed, indicating that CS and PBIL were the best algorithms for the BIBLIS-2D LPO problem, with no statistically significant difference between them (see Table 7). Both CS and PBIL outperformed ABC. 5.3. Angra 1 7th cycle LPO problem The descriptive statistics of the CB results for the Angra 1 7th Cycle LPO problem is presented in Table 8. In Fig. 7 the performance of the CS, ABC, and PBIL algorithms are compared over the evaluations.
6. Discussion As mentioned by Joyce and Herrmann (2018), notwithstanding the NFL theorems, and as computational experiments point out, for specific sets of problems there are algorithms that on average perform better than others because of also specific characteristics. For the LPO problem some remarkable results have been obtained by ABC algorithm (de Oliveira and Schirru, 2011; Safarzadeh et al., 2011; Meneses et al., 2018), as well as by PBIL algorithm (Caldas and Schirru, 2008; Meneses et al., 2018). Yarizadeh-Beneh et al. (2016) showed the efficiency of the CS algorithm for a WWER NPP, and in the present work we applied CS to the LPO based on the benchmarks IAEA-3D and BIBLIS-2D, as well to the Angra 1 7th Cycle problem, and in all of the three cases statistical analyses showed that CS is among the best algorithms. Yang (2014) discusses an interesting characteristic of the CS showing its similarities to Differential Evolution (Storn and Price, 1997), PSO, and SA in the sense that CS can be seen as an efficient combination of those algorithms. Yang (2014) also reports that the
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Table 7 Dunn’s post hoc test results for the BIBLIS-2D LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
35.98 21.08 14.90
20.8015 20.8015 20.8015
** **
**Indicates a statistically significant difference between algorithms.
Table 8 Descriptive statistics of CB results (in ppm) for the Angra 1 7th Cycle LPO problem.
Average Standard Dev. Median Maximum Minimum a b
ABCa
PBILa
CSb
1307 48 1311 1435 1224
1303 45 1304 1400 1209
1324 61 1328 1406 1073
Meneses et al. (2018). Present work.
And such characteristics of the CS algorithm for global optimization are essential for the search in the LPO as real vectors are mapped into combinatorial solutions with the RK. The RK approach for efficient global optimizers in continuous spaces have enabled good performances for several algorithms. And as new global optimizers are developed with efficient search in continuous spaces, notably the application of RKs makes them also efficient in the LPO problem. 7. Conclusion In the present work the CS algorithm for the optimization of the IAEA-3D, BIBLIS-2D, and Angra 1 7th Cycle LPO problems (Meneses et al., 2018) was implemented. The algorithms selected for comparison were ABC and PBIL. Statistical analyses showed that: (a) for the IAEA-3D problem, CS and ABC were the best algorithms; (b) for the BIBLIS-2D LPO problem, CS and PBIL were the best algorithms; and (c) for the Angra 1 7th Cycle LPO problem, CS and ABC were the best algorithms. Thus, in all of the three LPO problems tested, CS was among the best algorithms, reaching the best overall performance, therefore being the most robust of the three algorithms compared. In future works we will include other instances for algorithms comparison, implement the Improved DCS for the LPO, and optimize multi-cycle problems. CRediT authorship contribution statement Anderson Alvarenga de Moura Meneses: Conceptualization, Methodology, Formal analysis, Writing - original draft. Patrick Vasconcelos da Silva: Software, Validation, Investigation, Writing - original draft. Fernando Nogueira Nast: Software, Investigation, Visualization. Lenilson Moreira Araujo: Formal analysis, Writing original draft. Roberto Schirru: Conceptualization, Methodology, Writing - review & editing, Supervision. Declaration of Competing Interest
Fig. 7. Comparison between PBIL, ABC and CS for the Angra 1 7th cycle LPO problem. The bars represent a 95% confidence interval for the averages (considering the Student’s t-distribution for 50 tests).
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements
Table 9 Dunn’s post hoc test results for the Angra 1 7th Cycle LPO problem (a = 0.05). Comparison number
Group comparison
Difference in average ranks
Cutoff at alpha = 0.05
Significant difference = **
1 2 3
ABC-CS ABC-PBIL CS-PBIL
18.66 2.49 21.15
20.8015 20.8015 20.8015
**
**Indicates a statistically significant difference between algorithms.
CS algorithm’s global convergence is guaranteed, which is corroborated by He et al. (2018), with a validation with numerical analysis. According to Yang (2014), the efficiency of the CS algorithm is due to the balance of global and local search processes, which are controlled by the parameter pa . The typical value pa = 0.25 also used in our tests means that such control enables the global search to be performed in approximately 3/4 of the search. This fact as well as the usage of Lévy flights makes the CS a remarkable global optimizer.
R.S. acknowledges CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico, Brazil) and FAPERJ (Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro, Brazil). The authors acknowledge CNEN (Comissão Nacional de Energia Nuclear – participant of the CAMP agreement with the USA NRC), in the terms of the agreement of mutual cooperation with the Federal University of Western Pará for using the PARCS v3.0 code. The authors would like to thank the reviewers for their valuable comments and suggestions. References Argonne National Laboratory, 1977. Benchmark Problem Book (Supplement 2). Technical Report ANL-7416. Baluja, S., 1994. Population-Based Incremental Learning: A Method for Integrating Genetic Search Based Function Optimization and Competitive Learning. Technical Report. School of Computer Science, Carnegie Mellon University, Pittsburgh, Pennsylvania. Bean, J.C., 1994. Genetic algorithms and random keys for sequencing and optimization. ORSA J. Comput. 6, 154–160. https://doi.org/10.1287/ijoc.6.2.154. Brown, C.T., Liebovitch, L.S., Glendon, R., 2007. Lévy flights in Dobe Ju/’hoansi foraging patterns. Hum. Ecol. 35, 129–138. https://doi.org/10.1007/s10745006-9083-4.
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