A genetic algorithm for multigroup energy structure search
Introduction
The multigroup theory is an important approach to neutron cross section energy dependence representation in deterministic codes for neutron transport (or diffusion) equation solution (Duderstadt and Hamilton, 1976, Stacey, 2001). The energy space and the neutron cross section (XS) libraries are discretized on several intervals, called energy groups, and the generation of such libraries with small number of groups is a crucial step for core calculations. The choice of the energy groups’ boundaries is a key issue of this process, as all energy dependent quantities, including the neutron flux and current, are to be described according to the chosen discretization. Hence from a non-optimal energy structure (ES) may easily follow an inappropriate representation of the spectrum, possibly leading to significant deviation of the simulation results with respect to the real behavior.
Unfortunately, the methods currently employed to do such an important choice are neither user-friendly nor precise, as they are mostly based on expert judgement (Cacuci, 2010). This, of course, presumes a deep knowledge of the reactor physics of the considered system and a highly specific competence in the XS-related nuclear science, which might not be the case for many code users. This difficulty in the generation process promotes the creation (and the use) of “general-purpose” few-groups XS libraries, which energy structure, however, might be unsuitable to particular reactor design. In fact, as the neutron spectrum plays a major role in the energy discretization and homogenization procedure, XS libraries should be considered specific to the reactor they have been designed for, and should not be applied to systems too distant from the proper one. In fact, a XS library can adequately model a reactor transient, even with a limited number of groups, if it has been correctly designed for the considered system.
The solution to these issues would be an automatization of the energy boundaries selection which, combined with a XS-collapsing procedure, would make possible the creation of highly-specific XS libraries for each reactor starting from a unique and general fine-group library. Due to the high non-linearities involved, this automatic choice cannot be achieved through a deterministic procedure, but an evolutionary algorithm can overcome the problem and attain the goal.
Among the family of evolutionary algorithms, based on Darwinian selection and evolution principles, a genetic algorithm (GA) has been chosen for the present study. In GAs possible solutions are considered as individuals of a population, each characterized by a genotype and a phenotype; as time passes, the features, or genes, of successful solutions are perpetuated and, appropriately mixed, better individuals are found. The GA concept dates from the 1960s, when it has been proposed by Holland (1962) and is widely used in a number of fields.
Similar approaches to the multigroup energy mesh search, based on metaheuristic optimization, have been proposed by Mosca and Mounier, 2008, Mosca et al., 2011a, Mosca et al., 2011b and by Yi and Sjoden (2013). The studies are based on swarm algorithms rather than evolutionary ones and are applied to a predefined set of infinite homogeneous medium problems for fast reactors or to a single pin in a thermal system.
Having this in mind a GA has been developed and proposed in the paper to perform the energy groups’ boundaries search in neutron transport problems. The developed GA has then been joined together with the SIMMER-III code (Yamano et al., 2003, Kondo et al., 2000), as a complement to the XS collapsing extension already proposed in 2014 (Massone et al., 2014), and tested on 3 reactor systems: the Advanced Sodium Technological Reactor for Industrial Demonstration (ASTRID) developed in the framework of the European Sustainable Nuclear Industrial Initiative (ESNII+) (SNETP, 2014), the Working Horse (WH) core (Fiorini, 2009) of the European Sodium Fast Reactor (ESFR) (Andriolo, 2015), and the Molten Salt Fast Reactor (MSFR) considered for the Evaluation and Viability of Liquid Fuel Fast Reactor System (EVOL) project (EVOL, 2011). The obtained results are then analysed and compared to demonstrate the effectiveness of the approach and to study the physical implications underlying the GA choices.
Section snippets
First approach
The first attempt to produce an automated procedure for energy structure choice has been the transposition into an algorithm of the study on the flux spectrum, which is the basis of the current methods used for energy structure determination. Hence, adjacent energy groups characterized by similar values of the neutron population should be collapsed together, while avoiding averaging together peaks and valleys in the spectrum profile. Essentially, this represents a greedy approach (Cormen et
Algorithm configuration
For the test cases, the 72-groups libraries (Table 1) built by Rineiski et al. for fast reactor systems (Rineiski et al., 2011) are taken as SL; the objective will be finding ESs for the test systems with 11 energy groups. It is important specifying that this number is considered a compromise between computational time reduction and energy span of the groups, but it is in substance arbitrary; the GA and the XS collapsing tool can be applied with any number of groups chosen by the user. The
Test systems
The above described GA is applied to 3 nuclear reactor systems, in order to check its functioning and study the results.
The main test system is the ESNII+ ASTRID (Bortot et al., 2015). The others have been chosen in order to consider:
- •
One similar system: ESFR WH core (Fiorini, 2009);
- •
One completely different system: MSFR (EVOL, 2011).
Results and discussion
5 runs of the GA have been performed for each system to reduce the stochastic effect; both sorted and unsorted chromosomes have been used, in 3 and 2 calculations respectively (denoted as S1/2/3 and U1/2).
Possible improvements
The GA described in the paper is a first attempt to establish an automated tool for determining an optimal ES when performing a XS library collapsing. The results show that the procedure is effective in finding useful configuration, but this does not mean that it is efficient, i.e. that the convergence cannot be accelerated.
In the Section 3.1, it has been shown how important is the balance between exploration, i.e. the search of new good options among the whole solution space, and exploitation,
Conclusion
A genetic algorithm for the automatic choice of energy groups’ boundaries in neutron XS collapsing procedure has been developed and presented. The algorithm provides excellent results, correctly representing the physics of the test reactors, so proving that the approach is effective in finding appropriate energy structures for each test system.
The number of energy cuts to be set is fixed and chosen by the user, with the possible solutions represented as non-binary chromosomes. Appropriate
Acknowledgements
The authors deeply thank Dr. Kiefhaber and Dr. Vezzoni for their precious suggestions, and are profoundly grateful to Dr. Maschek and Prof. Ravetto for their invaluable support.
References (34)
- et al.
When the greedy algorithm fails
Discrete Optim.
(2004) - et al.
A comparative analysis of selection schemes used in genetic algorithms
Found. Genet. algorithms
(1991) - et al.
JANIS 4: An Improved Version of the NEA Java-based Nuclear Data Information System
Nucl. Data Sheets
(2014) - et al.
Molten Salt related extensions of the SIMMER-III code and its application for a burner reactor
Nucl. Eng. Des.
(2006) - et al.
Energy group structure determination using particle swarm optimization
Ann. Nucl. Eng.
(2013) Impact des combustibles sphere-pac innovants sur les performances de sûreté des réacteurs à neutrons rapides refroidis au sodium
(2015)- et al.
A comparison of selection schemes used in evolutionary algorithms
Evol. Comput.
(1996) Group Constants for Nuclear Reactor Calculations
(1964)- Bortot, S., Alvarez-Velarde, F., Fridman, E., Cruzado, I.G., Herranz, N.G., Mikityuk, K., Panadero, A.-L., Pelloni, S.,...
- (2010)