Evolutionary simulated annealing for fuel loading optimization of VVER-1000 reactor
Introduction
In-core fuel management (ICFM), i.e. fuel loading optimization, is an important task in designing a nuclear reactor core. This is known as a complicated multi-objective problem, in which several objectives have different variation trends. The problem of fuel loading pattern (LP) optimization has received attention from the beginning of nuclear reactor technology with the application of various optimization methods. A typical LWR consists of about 150–200 fuel assemblies, and therefore, the possible number of fuel LPs would be very large even when geometrical symmetry and constraints are considered. In the ICFM problem, two main objectives are typically considered: (1) maximization of fuel cycle length and (2) minimization of power peaking factor. The cycle length maximization means increasing the efficiency of fuel utilization, and consequently minimization of fuel cycle cost. The first objective is evaluated via maximizing the effective multiplication factor, . Whereas, minimization of power peaking factor results in the decrease of fuel failure possibility and the enhancement of safety margin. In general, the objectives are combined in a fitness function for searching optimal solutions.
Many efforts have been contributed to solve the problem of ICFM with the development and application of a number of optimization methods. Most of the methods are based on the simulation of natural systems such as Simulated Annealing (SA) (Mahlers, 1995, Stevens et al., 1995, Yamamoto, 1997, Lee et al., 2001, Rogers et al., 2009, Park et al., 2009, Zameer et al., 2014), Genetic Algorithms (GA) (DeChaine and Feltus, 1995, Parks, 1996), Evolution Method (Axmann, 1997), Particle Swarm Optimization (PSO) (Babazadeh et al., 2009, Yadav and Gupta, 2011, Jamalipour et al., 2013, Zameer et al., 2020), Tabu Search Algorithm (Lin et al., 1998, Hill and Parks, 2015), Ant Colony Optimization (ACO) (de Lima et al., 2008, Machado and Schirru, 2002), Gravitational Search Algorithm (GSA) (Mahmoudi et al., 2016), Inverse Analysis (Chao et al., 1986, Tran et al., 2009), Differential Evolution (DE) (Sacco et al., 2013, Charles and Parks, 2019, Phan et al., 2020), and so on. Although many attempts have been done, it is still a complicated task (Moghanloo and Mahmoudi, 2016). The SA method has been soon applied for the problem of ICFM. The advantage of the SA method is an ability to escape local optima due to an acceptance probability of a worse solution. However, the disadvantage of the SA method is a slow convergence. Due to a slow convergence, the number of calculated LPs in the SA search process is usually large. The main problem of the original SA is that the convergence speed decreases rapidly when the search process approaches around a global optimum. This results in a large number of candidate solutions to be evaluated around this point. The performance of the SA method depends on the neighborhood structure and/or the generation of a new trial solution (Smuc et al., 1994). Adaptive Simulated Annealing (ASA) method was developed to enhance the convergence speed by improving the annealing schedule and the generation of new trial LPs (Smuc et al., 1994, Kropaczek et al., 1994, Mirza et al., 2000, Lee et al., 2001, Rogers et al., 2009). The generation of a new trial LP in the ASA method is binary or ternary exchange combined with one of the two following strategies. The first strategy is known as ”return to the best” (Smuc et al., 1994, Kropaczek et al., 1994). It means that if the current best LP does not change after a number of trial LPs, the current best LP is reused as a base LP. The second one is the application of a transition probability matrix or restriction LP lists. A transition probability matrix approach was developed to make the annealing system adaptive by recording the impact of rejected solutions and bias the system away such solutions in the search space (Mirza et al., 2000). The lists include trial and base LPs, which have been previously examined. When a new trial LP is generated, it is compared with the lists. If the new trial LP is already included in the lists, it will not be used or will be reused with a descending probability.
In the present work, a novel evolutionary simulated annealing (ESA) method has been developed for the problem of fuel loading optimization. The ESA method is improved by using a crossover of two base LPs for generating a new trial LP, instead of binary or ternary exchanges in the original SA and ASA. This crossover is similar to that used in GA. Numerical calculations have been performed based on a VVER-1000 MOX fuel core in comparison with the performance of the SA and ASA methods. Two objective functions (OFs) were used to evaluate the performance of the ESA method in comparison with the SA and ASA methods. The first one aimed at comparing the possibility in reproducing a reference core LP of the three methods. The second one was applied to find the optimal LP by maximizing the value and flattening the radial power distribution. A statistical significance test, so-called Mann–Whitney U Test, was also conducted to compare the performance among the three methods.
Section snippets
SA and ASA methods
Simulated Annealing is originally based on the phenomenon of crystal vibration in annealing metal, which has been soon applied to fuel loading optimization of nuclear reactors. The SA method has ability to escape local optima by an acceptance probability of a worse solution. The procedure of the original SA method applied for the problem of ICFM can be summarized as follows:
- 1.
Starting with an initial trial LP.
- 2.
Core physics calculation of the trial LP is performed, and the objective function (OF)
VVER-1000 MOX benchmark core
Numerical calculations for LP optimization have been performed based on a VVER-1000 benchmark core loaded with 30% MOX fuel. The VVER-1000 benchmark core was proposed by OECD/NEA for studying the neutronics performance of a mixed UO2-MOX core, and verifying computational codes and methods (Gomin et al., 2006). The reference 1/6th core configuration consists of 28 fuel assemblies, including 19 UO2 fuel assemblies and 9 MOX fuel assemblies as shown in Fig. 3 (Gomin et al., 2006). The 19 UO2 fuel
Objective functions
Objective function used in the problem of LP optimization is usually a combination of several objectives such as maximization of cycle length, flattening of power distribution and constraint of power peaking factor (PPF), etc. A common objective function is used to maximize the at the beginning of cycle (BOC) and minimize the PPF (Babazadeh et al., 2009, Jamalipour et al., 2013, Moghanloo and Mahmoudi, 2016). Flatness of power distribution and the constraints of PPF were also included in
Conclusions
The ESA method has been developed and applied for the problem of fuel LP optimization of VVER-1000 reactor. The ESA method is improved by using a crossover of two base LPs for generating a new trial LP similar to that used in GA. The LPO-V code was developed for core physics calculations and LP optimization of VVER-1000 reactor. Verification calculations have been conducted based on VVER-1000 MOX fuel benchmark core and compared with MCNP4c calculations. Calculations for optimizing fuel LP of
CRediT authorship contribution statement
Viet-Phu Tran: Methodology, Formal analysis, Writing - original draft. Giang T.T. Phan: Formal analysis. Van-Khanh Hoang: Methodology, Formal analysis. Pham Nhu Viet Ha: Formal analysis. Akio Yamamoto: Methodology, Supervision. Hoai-Nam Tran: Conceptualization, Methodology, Supervision, Writing - original draft, Writing - review & editing.
Declaration of Competing Interest
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.
Acknowledgments
This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant 103.04-2020.06.
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