Optimization of the loading pattern of the PWR core using genetic algorithms and multi-purpose fitness function
, oraz | 25 lis 2021
Article Category: Original Paper
Data publikacji: 25 lis 2021
Zakres stron: 147 - 151
Otrzymano: 18 sty 2021
Przyjęty: 07 lut 2021
DOI: https://doi.org/10.2478/nuka-2021-0022
Słowa kluczowe
© 2021 Wojciech Kubiński et al., published by Sciendo
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
Most of the nuclear power reactors operate on the basis of cycles that involve periodically replacing part of the fuel to introduce additional reactivity to compensate for the loss of fissile material and the creation of fission products. In a new cycle, some of the assemblies with high burnup are replaced with the fresh ones, and some of the old assemblies are set in a new configuration (the so-called shuffling). This process requires reactor shutdown and opening the vessel, which results in enhanced costs due to operations and interruption in power production. The selection of the first core loading pattern and further shuffling schemes are therefore important in terms of the economy of the core’s operation [1, 2].
The core usually contains several hundred fuel assemblies that differ in their isotopic composition, enrichment, amount of burnable absorbers (BAs), and/or other parameters, which may additionally vary with time. For a typical pressurized water reactor (PWR), the number of fuel assemblies used is in the range of 120–250 [1]. Assuming only 10 different types of assemblies and 200 of them in the core (20 for each type), one can get 200!/(10 · 20!) ≈ 10355 possible configurations. Finding the optimal core fuel loading scheme is, therefore, an extremely complex problem and requires a special approach.
One of the possible techniques for solving the above problem may be the use of genetic algorithms (GAs). GAs are optimization tools based on genetics and Darwin’s theory of evolution [2, 3]. GAs are based on a population of encoded chromosomes (set of characters) containing information about the optimized system. These chromosomes evolve over time and approach the optimal solution. Due to the use of random elements (mutation, crossing), GAs are resistant to get stuck within a local optimum, which is a significant advantage in solving complex, multidimensional problems.
GAs have been successfully used in many fields, such as pattern recognition, data mining, and image processing [4]. Additionally, they were also used in the field of nuclear engineering. In the available literature, loading pattern [1], core design [5, 6], BA arrangement [7, 8], online refueling [9], and thorium loading [10] were considered. However, due to the required computing power, these optimizations were typically simplified and focused on one parameter or used significantly simplified models.
GA operates on generations of chromosomes (usually 50–100 per population) for which three main genetic operators are used: selection, crossover, and mutation [2, 3].
Selection is related to the assessment of the chromosome. For this purpose, a fitness function (FF) is defined, which is to be maximized during the operation of the algorithm. Based on the FF value, chromosomes are selected for further GA steps. In general, the higher the FF value, the greater the probability of that chromosome to survive. Crossover is the process of gene exchange between chromosomes. Usually, two chromosomes (parents) are selected randomly; they exchange subarrays between each other and create new specimens (offspring). Mutation involves random replacement of a given gene with another.
In this study, the chromosome is defined as 1/4 of the PWR core containing a set of numbers that reflects the current configuration of the core (Fig. 1). The algorithm operates on 1/4 of the configuration and then mirrors it symmetrically to build the whole core. The assumption of symmetry allows a significant reduction in the number of possible solutions and decreases the time needed for optimization.
Fig. 1
Chromosome representing 1/4 of the optimized core.
In this work, the 1000 MWe PWR defined in MIT BEAVRS benchmark was applied [11]. The first fuel cycle core design is presented in Fig. 2, and we limited the choice of fuel assemblies to nine types which were present during this cycle. It covers fuel assemblies with enrichment of 1.6%, 2.4%, and 3.1% and the number of BA rods per assembly equals 0, 6, 12, 15, 16 or 20. BAs, made of borosilicate glass, are placed in control rods guide tubes in assemblies without control rods. Neutronic calculations were performed with the PARCS core simulator [13, 14]. The model, validation, and test details are described in [15], and core definition and detailed design are available in benchmark definition document [11].
Fig. 2
The BEAVRS core loading pattern for Cycle 1 (based on [12]).
In this work, we have introduced the PPF into our algorithm and FFs, as it was not studied in our previous research [12]. It is the parameter that describes quantitatively the uniformity of the heat sources (also neutron flux) in the core [16]. The total nuclear PPF is defined by Eqs. (1) and (3) [15].
In the PARCS code [13, 14], the neutronic solution is based on large nodes with XY dimensions similar to assemblies ~20 cm × 20 cm × 20 cm. The so-called pin-power reconstruction is necessary to find detailed location of hot channels. In this work, this approach was not applied, as it is beyond the scope of this study. The code estimates hot channel on the basis of the available nodalization and an assembly is treated as a single cooling channel in the context of Eqs. (1)–(3). The PARCS calculates all peaking parameters, but in this report we focused on the optimization of the total PPF only.
Typically the PPF should be minimized to avoid large discrepancies in neutron flux/power between different parts of the core, both radially and axially. Lower PPF will lead to more uniform fuel depletion and uniform coolant temperature distribution both in axial and radial directions, which can lead to more economical use of the fissile material in the core.
Two simulations were performed and a reference BEAVRS calculation. The first one applies the algorithm using a simplified form of FF. In this part, the only goal of the algorithm was to minimize the PPF that determines the non-uniformity of the power distribution. Thus, the FF took a simple form (Eq. (4)):
In the second part, it was decided to use multi-purpose FF and optimize two parameters of the core’s operation: PPF and cycle length. Therefore, the goal was also to minimize the PPF, but at the same time, extend the length of the cycle. In this part, FF took the form (Eq. (5)):
For each of the FFs, 500 generations containing 100 chromosomes were performed (50 000 simulations in total). The mutation level in both cases was 2%. Figure 1 shows the relative change in the FF over generations.
From Fig. 3, one can see that FF increases very fast at the beginning of the simulation; then, the changes are smaller. After about 300 steps, regardless of the case, the FF reaches a maximum value. Then, due to the mutation, the algorithm does not converge to a specific value but oscillates around it.
Fig. 3
Relative change of FF over generations for cases 1 and 2.
As a result of the simulations, two optimized core configurations were obtained containing fuel assemblies from the BEAVRS benchmark. Figure 4 shows a comparison of the obtained configurations (Figs. 4b,c) with the BEAVRS core configuration (Fig. 4a). One can see that assemblies with greater enrichment are arranged at the edges of the core for each configuration to minimize the fall of the flux at the outer boundary. Then in the center are alternately arranged assemblies with different enrichments but usually containing a greater amount of burnable absorber (BA) rods to flatten the distribution of the flux in the central part of the core.
Fig. 4
BEAVRS core configuration (a), and optimal configurations obtained for Case 1 (b) and Case 2 (c).
The characteristics of the obtained configurations and the BEAVRS core are presented in Table 1. It contains an indicator of the amount of fissile material used (the average enrichment calculated as the sum of the enrichments in the assemblies divided by their number), the number of BA rods, the initial
Characteristics of the obtained configurations and the BEAVRS core
| Average enrichment (%) | No. of BA rods | Initial |
PPFmax | Cycle length (days) | |
|---|---|---|---|---|---|
| BEAVRS | 2.36 | 1268 | 1.08 | 1.88 | 333.6 |
| Case 1 | 2.76 | 1392 | 1.13 | 1.82 | 451.7 |
| Case 2 | 2.78 | 1356 | 1.13 | 1.89 | 512.5 |
The maximal
| BEAVRS | Case 1 | Case 2 | |
|---|---|---|---|
| 1.33 | 1.26 | 1.34 | |
| 1.21 | 1.26 | 1.17 |
As the parameter optimized for both cases of the algorithm’s operation was PPF, it was decided to plot the normalized radial power distribution at the beginning and end of the cycle (EOC) for the BEAVRS and the cores obtained by the algorithm (Fig. 5a–f).
Fig. 5
Radial power distribution (normalized to average) at the BOC and the EOC for the BEAVRS core (a, d), Case 1 core (b, e) and Case 2 core (c, f).
As can be seen from Fig. 5, the power distribution for BEAVRS is characterized by an initial peak in the center of the core that spreads at the EOC towards the outer boundary. The maximum value of
As part of the study, two optimizations were performed, the first of which aimed at minimizing the PPF and the second at minimizing the PPF, while extending the fuel cycle. As a result of the simulations, two configurations were obtained, both characterized by a longer fuel cycle compared to the original configuration. In Case 1, the maximum PPF value was reduced from 1.88 to 1.82 and the cycle was extended by 118 days. In Case 2, a slightly higher PPF (1.89) was obtained, while the cycle was extended by 179 days. Of course, other core parameters have also changed. The cores in Cases 1 and 2 use more fissile material (average enrichment 2.76% and 2.78% vs. 2.36% in the original configuration) hence also have a higher initial
The above-mentioned examples show the successful operation of the algorithm. As part of the simulations, it was possible to optimize the objectives included in the multi-purpose FF, i.e., flattening the power distribution and extending the cycle. To propose a comprehensively optimized configuration that could be used in a nuclear power plant, future simulations may include more extensive FFs that would take into account other constraints such as maximum average enrichment, acceptable