Application of genetic algorithms in optimization of SFR nuclear reactor design

It is a fundamental tool that assesses each individual from the population. Its formula depends strictly on the given situation, and in general, its value is higher for better individuals and lower for worse ones.

The selection operation is carried out as a draw from a collection of individuals where the probability of drawing is proportional to the FF of each individual.

It consists of a random selection of two individuals from the population and a random selection of the place of their cut. Next, the individuals are swapped for their end parts and reunited. Supposing that A₁ and A₂ are a pair of randomly chosen parents, the following equations can be arrived at: A1=01101; A2=11000. {{\rm{A}}_1} = 01101;\,\,{{\rm{A}}_2} = 11000.

Next, we need to determine the crossover point; it should be a random number between 1 and l − 1, where l is the length of the chromosome. Suppose the drawn point is 2, then the child strings A’₁ and A’₂ will take the following form: A'1=01|000; A'2=11|101. {\rm{A}}{{\rm{'}}_1} = 01|000;\,\,{\rm{A}}{{\rm{'}}_2} = 11|101.

Once this is done for all the parents, the individuals in the population are replaced with their children.

This operation is realized by changing a random bit in the individual, with the probability set as a mutation rate.

The implementation of our algorithm consisted of storing numerical values of the optimized parameters as one large concatenated binary string in such a way that each individual parameter corresponded to a given subfragment of the entire string. With this solution, the algorithm at each iteration of the crossover operation exchanged substrings, which were associated with the given parameters of the models. Evaluation of a given solution consisted of reading values of coded parameters and then running simulation models with these values. The algorithm read results from output files and then used them to calculate the FF.

The end result was that those chromosomes whose binary-coded parameters generated models with better results more likely entered the next generation.

The fuel assembly and the reactor core models were created and then simulated with the SERPENT2 computer code, which was developed by Finnish VTT. It was applied to solve neutronics problems with the ENDF-VII library. It is a 3D continuous energy neutron transport Monte Carlo code with isotopic fuel depletion [19, 20, 21].

In this work, we used a 3D homogeneous cylindrical core model, which was a simplified version of the SFR oxide core with 3600 MWth power and oxide fuel, defined in the Organization for Economic Cooperation and Development/Nuclear Energy Agency (OECD/NEA) benchmark [1].

For SFR reactors, it is possible that a positive void reactivity coefficient, the so-called sodium void reactivity effect (SVRE), may appear due to the removal of sodium as an effect of gas bubble entrainment or leak. This is caused by neutron spectrum hardening, which leads to an increased fission reaction rate. The SVRE can reduce core safety characteristics to the level which can potentially be dangerous during some transients. In consequence, the SVRE should be minimized during the design process [3]. The concept of utilizing boron shields in sodium reactor designs as a measure to reduce SVRE is presented in other works [22, 23, 24]. This design feature was studied in this paper. We placed boron shields around the reactor core in the form of rings partially filled with boron carbide. It is worth mentioning that boron carbide (B₄C) is the most commonly used absorber in fast reactors because of its (a) relatively high neutron absorption cross sections for fast neutrons, especially with low B-10 isotope enrichment, and (b) relative low price with comparative ease of fabrication [24].

The model is schematically shown in Fig. 2. The shade of green is related to the percentage enrichment of B-10 isotope in boron carbide, and the lighter the green colour, the greater the enrichment.

Fig. 2

The goal was to find such a design of shields which at the same time minimizes the SVRE and does not interfere with the ability of the core to be critical. In order to achieve that, our algorithm had to manipulate selected parameters of the boron shields: –

bottom elevation

Critical simulations were performed for two core states: normal and voided without sodium in the active part of the core. The resulting k_eff values for each case will then be used to calculate sodium void reactivity (SVR), which was determined with the formula: (1) SVR=kvoided−knormalkvoided⋅knormal*105 {\rm{SVR}} = {{{k_{{\rm{voided}}}} - {k_{{\rm{normal}}}}} \over {{k_{{\rm{voided}}}} \cdot {k_{{\rm{normal}}}}}}*{10^5} where SVR is sodium void reactivity and k is the effective multiplication factor. The goal is to minimize SVRE, and this means that the selected FF must increase as the SVR value decreases. The simplest example is FF = 1/SVR. However, selecting such a FF would mean that the most preferred cores would be those that have k_voided the same as k_normal or very similar. As a result, the algorithm would place the thick shields as close as possible to the active part of the core, thus minimizing k_voided but also k_normal as well. The eigenvalue k_normal <1.0 would mean that the core is not able to maintain chain reaction. In consequence, the FF must also maximize k_normal at the same time. The applied FF is therefore presented as follows: (2) FF=knormalSVR−C {\rm{FF}} = {{{k_{{\rm{normal}}}}} \over {{\rm{SVR}}}} - C

In addition, we added constant C, which is given by the formula below: (3) C={0for n=00.9*min(fc)for n≠0 C = \left\{{\matrix{0 \hfill & {{\rm{for}}\,n = 0} \hfill \cr {0.9*\min ({f_c})} \hfill & {{\rm{for}}\,n \ne 0} \hfill \cr}} \right. where f_C is the value of the FF of the weakest core in the initial generation. The constant C was determined by an experimental method. It makes the individuals distinguishable from the roulette method so that the best ones will be selected for crossing over.

The aim of this simulation was to obtain a fuel mixture with the highest possible initial k_eff and at the same time the least amount of long-lived radioactive waste. In this task, we used a 3D model of a single fuel rod with periodic border conditions, whose geometry was derived from the OECD/NEA benchmark [1], and it is shown in Fig. 3.

Fig. 3

A simplified mixed oxide (MOX) fuel model was used as a basis for the mixture, in which the atomic ratio of isotopes of plutonium in the mixture was changed. During simulation, the initial content of oxygen, depleted uranium, and plutonium did not change and was found to be 2/3, 4/15, and 1/15, respectively.

In the simulations, the five most stable isotopes of plutonium were taken into account, that is, those with mass numbers from 238 to 242. The chromosome was 80 bits long, exactly 16 bits per one plutonium isotope, the value from it is read by dividing it into five equal parts, each corresponding to one of the isotopes. Then, each part is converted into an integer; finally, the isotope sum is normalized to a given value, which in this case is 1/15 of the total fuel quantity. In the next step, the control program creates input files for the SERPENT code based on the values obtained in the previous step (Fig. 1).

Three actinides, americium, curium, and neptunium, were selected as the elements associated with long-lived radioactive waste, and their concentration in spent fuel was studied. The simulation was carried out for a burnup of 100 MWd/kgU, which corresponds to six years of operation in the reactor.

The developed FF is given by Eqs. (4) and (5). (4) fC=(keff,Mactinides)=keffMactinides {f_C} = ({k_{{\rm{eff}}}},{{\rm{M}}_{{\rm{actinides}}}}) = {{{k_{{\rm{eff}}}}} \over {{{\rm{M}}_{{\rm{actinides}}}}}} (5) Mactinides=MAm+MCm+MNp {{\rm{M}}_{{\rm{actinides}}}} = {{\rm{M}}_{{\rm{Am}}}} + {{\rm{M}}_{{\rm{Cm}}}} + {{\rm{M}}_{{\rm{Np}}}}

Maximizing such a function will lead to maximizing the effective neutron multiplication factor and minimizing the concentration of selected actinides.

The population consisted of 20 individuals with a mutation level equal to 1%, and the number of generations equal to 11. Simulation results for the following generations are presented in Table 1. In Figs. 4 and 5, the evolution of the SVR and eigenvalues is presented. Figure 6 presents the evolution of the arrangement of the boron shields in the core.

Fig. 4

Fig. 5

Fig. 6

As one can see from the results of the simulations, the algorithm has managed to obtain an optimized core in which the SVR has dropped noticeably (Fig. 4); the SVRE was reduced from 2420 pcm to 2130 pcm (~12% difference) – see Table 1. Unfortunately, k_normal has fallen slightly below 1, which means that the reactor could not work in this configuration without introducing additional modifications. Apparently, the FF used in this simulation gave too little weight, maximizing the k_normal while minimizing the SVR. The algorithm favoured individuals with the lowest possible SVR and k_normal <1 rather than individuals with lower SVR and k_normal >1 (Table 1). However, we can see that FF steadily increases; hence, the algorithms operate as designed, and the applied FF is the main limitation. In future research, the core design has to be updated with more fissile mass and updated FFs. Unfortunately, due to the limited scope of the project and necessary computing power, we could not redo simulations with more appropriate FF.

The simulations were conducted with the following parameters of the GA: –

35 Generations

One can see in Fig. 7 that the value of the FF stabilizes around the 15th generation and remains stable until the end of the simulation, with a small leap in the 33rd generation. The best individual comes from the 35th generation.

Fig. 7

For comparison, before using the GA, simulations were performed for the mixture described in the OECD/NEA publication [1]. In consequence, the following results were obtained: the mass of selected actinides for the burnup simulation described earlier was ascertained at M_actinide = 4.14 g and effective neutron multiplication factor at k_eff = 1.331(0.002). The parameters for the best individual obtained using the GA are as follows: the mass of selected actinides was ascertained at M_actinide = 1.16 g and effective neutron multiplication factor at k_eff = 1.315(0.002).

Figures 8 and 9 show the evolution of tested values for an individual with the highest FF in a given generation.

Fig. 8

Fig. 9

As a result of the program, a mixture of plutonium isotopes was obtained, the details of which are presented in Table 2.

The obtained mixture is dominated by the isotope of plutonium, Pu-238, followed by Pu-239. The algorithm resulted in the appearance of some quantities of Pu-240 and Pu-241 isotopes as well, and in minimization of the quantity of Pu-242 isotope.

The obtained mixture is the result of optimization using a simple FF containing a small number of parameters. The presented solution should be treated with a proper margin of confidence. In reality, for nuclear fuel, several other factors play a role, and the obtained mixture is characterized by isotopic inventory, which is an academic case.

By collecting data from each generation, it was possible to calculate Pearson’s linear correlation coefficient for the concentration of a given isotope and the content of selected actinides in the burnout fuel. The calculated values are presented in Table 3.

A negative value of the coefficient indicates a decrease in the mass of selected actinides, with an increase in the share of a given isotope. A positive value, on the other hand, indicates an increase in their mass when increasing the isotope content.