During the normal operation of a nuclear reactor, the nuclear fuel in the reactor’s core is continuously being “burned”, i.e., the fissionable material, usually uranium or plutonium, is continuously being consumed and depleted. Hence, the majority of nuclear reactors is operated in cycles of 18-24 months since they must be periodically refueled.
This refueling outage is a complicated and expensive procedure that usually necessitates halting the reactor and opening the reactor pressure vessel. The fuel depletion is not homogeneously distributed throughout the core, and, usually, a third of the most depleted fuel assemblies (FAs) are replaced in each refueling. The loaded fresh FAs, together with the remaining partially burned ones, are rearranged to form a new core configuration. The arrangement of the nuclear FAs in the core is called a “Loading Pattern,” or simply, LP. The new core configuration should fulfill several objectives, which may be conflicting. For example, maximizing the energy production until the subsequent refueling outage while still satisfying all safety limitations and operational constraints.
This challenge of finding the best LP, such that the resulting core satisfies the safety and operational requirements in the best possible way, is a classical discrete optimization problem. This problem is characterized by a huge search space and is a multi-objective, nonlinear, nonconvex, NP-hard combinatorial problem. In a standard pressurized water reactor, there are typically about 200 different FAs, and the number of possible different LPs is approximately 200! Each LP constitutes a different nuclear core, which has to be evaluated, analyzed, and characterized using complicated calculations and intensive computer simulation. Assuming it takes 1 second to evaluate a single core, it would take 10360 years to traverse the entire search space. Considering that the age of the universe is only ~1010 years, one must find another method to solve this problem.
A well-known method used for addressing this kind of optimization problem is the evolutionary algorithm, more specifically, the genetic algorithm (GA). The GA is a search tool adapted for huge search spaces, where more direct ways, like gradient-based methods, are not applicable. The GA is based on the concept of the Darwinian evolution — survival of the fittest. The concept that underlies the GA consists of taking a population of solutions (individuals), which covers a portion of the search space, and performing on it a process of “evolution,” thus, moving the population through the search space in search of optimal solutions.
In our case, the evolutionary process is mimicked by considering each LP as an individual in the population, choosing the better ones as parents (selection) according to how much they “fit” the optimization objectives, mating (crossover) them, and finally mutating them to breed offspring solutions. As this process is reiterated throughout the generations, and since the better (fittest) individuals are favored as parents, superior offspring solutions emerge.
In a study published recently in Annals of Nuclear Energy, Dr. Erez Gilad and Ph.D. student Ella Israeli, from the Unit of Nuclear Engineering at Ben-Gurion University of the Negev (BGU), address some of the critical problems encountered when using GA for optimization of core design (LP). For example, many studies in this field employ basic and even obsolete GA implementations, disregarding important and relevant problem-related information, such as the geometrical structure of the core, or impose unnecessary symmetry restrictions to cut down on algorithm runtime. In their research, Dr. Gilad and Ms. Israeli develop, implement, and evaluate novel GA methods using different case studies of LP design for nuclear reactors.
In their study, the researchers from BGU developed improved crossover and mutation operators that consider the chromosomes’ representation as permutations. An LP of a nuclear reactor core is essentially a two-dimensional array containing materials of different types, e.g., fuel, absorber, reflector. It is represented by a “chromosome” whose “genes” represent the different locations and types of the FAs in the core. The chromosome representation is chosen to be a permutation of the core structure in order to preserve the predetermined quantities of the different materials and elements of the core. This representation gives simple and intuitive physical meaning to the genetic variance of the population, i.e., low genetic variance indicates that many chromosomes are similar in the sense that they position similar FAs in similar locations in the core.
Another novel feature of the algorithm is the consideration of the geometric nature of the problem by accounting for the spatial structure of the core. The researchers develop a new geometric crossover operator that utilizes this information and its implementation exhibits excellent results. Crossover is the genetic operator responsible for the creation of new solutions out of selected parents. It swaps gene segments between two parent chromosomes, mixing their genetic data to create offspring. The adaptive crossover operator developed for this study is based on swapping rectangular segments (of varying sizes) of neighboring FAs between two selected LP parents.
The BGU researchers also developed highly adaptive mutation strategies based on the instantaneous genetic variance of the population. The algorithm continuously monitors the genetic diversity of the population and automatically changes the mutation rate according to the level of homogeneity of the population. The more homogeneous the population is, the higher the mutation rate is. This feature is shown to dramatically enhance the algorithm performances.
Finally, the researchers from BGU challenge the traditional assumption of symmetrical core design, which dominantly underlies the entire field of LP optimization. This assumption is made because the different primary coolant loops of the nuclear reactor must maintain similar thermal-hydraulic condition during nominal operation, imposing symmetry on the power distribution in the reactor core. Moreover, symmetrical LP is much more intuitive, and nuclear engineers rely to some extent on their intuition and experience in designing core LPs. These symmetry constraints are removed once other types of reactors or critical facilities are considered (research reactors, SMRs). This study demonstrates an important and unconventional conclusion, that in some cases the best LPs are not symmetric and can be very surprising and counter-intuitive.
This study is of true inter-disciplinary nature in the sense that a combination of expertise in both evolutionary algorithms and nuclear reactor physics is required. This field of research is active and relevant, but the successful application of modern evolutionary algorithms for solving such problems is only just beginning.
These findings are described in the article entitled Novel genetic algorithm for loading pattern optimization based on core physics heuristics, recently published in the journal Annals of Nuclear Energy. This work was conducted by E. Israeli and E. Gilad from Ben-Gurion University of the Negev and is partially funded by the Israeli Ministry of Energy, contract number 216-11-008.