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Modular Optimization Framework (MOF) is a flexible tool for nuclear engineering optimization problems. More information about MOF and its applications can be found in [1] and [2]

General Definitions

The optimization of nuclear engineering problems typically involves maximizing/minimizing some quantity while meeting safety related constraints. The constraint maximization problem can be described in general through the following equation, where $f_0(x)$ is the maximization quantity, $f_i(x)$ is the quantity on which the $i^{th}$ constraint is applied with threshold $\alpha_i$.

$$ \begin{aligned} \mathrm{max.} \quad & f_0(x) \\ \mathrm{s.t.} \quad & f_i(x)\leq a_i, \quad i=1, \dots m \end{aligned} $$

The minimization problem can be easily transposed to a maximization problem and thus this definition covers both cases. This is a general definition that can be adapted for some specific nuclear engineering optimization problem. We will provide an example for the first cycle core loading pattern optimization. In this example, the decision variable $x \in \mathcal{X}^d$ is a complete loading pattern that includes $d$ fuel assemblies $x=[x_1, \dots x_d]$ that represent the whole reactor core. For a typical four-loop PWR, $d=193$ if no symmetry is assumed. If an octant symmetry is assumed then it can be reduced to $d=31$. Each loading pattern can take values from the discrete set $\mathcal{X}^d$ that depends on the available inventory of fresh fuel assembly designs. For an inventory of $p$ fuel assembly designs, the total number of combinations that provide unique loading patterns is $p^d$. The function to be maximized is $f_0: \mathcal{X}^d \rightarrow \mathcal{R}$ and a typical example is the energy production or the cycle length. The constraints that need to be met are represented by the functions $f_i: \mathcal{X}^d \rightarrow \mathcal{R}$ and some examples are the feedback coefficients, power peaking factors, and maximum boron concentration. Such optimization problem that involves multiple objectives (maximization/minimization and constraints) is usually treated through a fitness functions that maps the desired objectives into one effective quantity. The fitness function can have any functional form but the most widely used is a linear combination of the objectives with weights $w_i$ for each objective as defined in the following equation. The $max$ operator adds a penalization for every constraint that is not satisfied of which the value depends on the degree of violation.

$$ \mathrm{max.} \quad f(x) = w_0 f_0(x) - \sum^{m}_i w_i , \mathrm{max}(0, f_i(x)-a_i) $$

These definitions will be used throughout this wiki to present the various aspects of MOF and its optimization algorithms.

MOF Architecture

MOF aims at allowing a flexible management of optimization algorithms, optimization problems and codes.

References

[1] B. Andersen, G. Delipei, D. Kropaczek, and J. Hou, MOF: A Modular Framework for Rapid Application of Optimization Methodologies to General Engineering Design Problems, arXiv:2204.00141, 2022

[2] G. Delipei, J. Mikouchi-Lopez, P. Rouxelin and J. Hou, Reactor Core Loading Pattern Optimization with Reinforcement Learning, The International Conference on Mathematics and Computational Methods Applied to Nuclear Science and Engineering (Accepted), 2023.