Sparse Polynomial Chaos expansion for advanced nuclear fuel cycle sensitivity analysis

Highlights

• Sparse Polynomial Chaos expansion methodology applied to an advanced fuel cycle.

• Estimation of the Sobol indices and high degree interactions.

• Large reduction of the computational requirements for sensitivity analyses.

• Methodology applicable by any institution using any fuel cycle code.

Abstract

Uncertainty and sensitivity analyses are required in fuel cycle analyses for studying the viability of electronuclear scenarios by means of their response to variations in the input parameters. This implies that nuclear fuel cycle simulators have to be extended for dealing with this kind of problems in order to become reliable tools. However, given their complexity, the excessive computational effort these analyses demand constraint the ability of the codes to solve the problem without making any approximation. In this work, we propose to use a methodology based on a sparse Polynomial Chaos expansion that, given the low computational demand, makes it possible to be used for sensitivity analyses in any fuel cycle code. This has been applied to a European collaborative advanced scenario where the results have been compared with previous studies with the aim of showing the potential that this technique has when applied to fuel cycle studies.

Introduction

Nuclear fuel cycle simulators are being extensively used to evaluate different strategies mainly in mid and long term with the aim of assessing, among others, the possibility of implementation of advanced fast reactor technologies or the sustainability potential of the fuel cycle in the sense of safety, non-proliferation, economic competitiveness and waste management (Salvatores et al., 2008; Bianchi, 2010; OECD/NEA, 2013; IAEA, 2018). The fuel cycle scenarios, which cover all the technologies involved from the front-end to the back-end, are usually described as a collection of parameters that, given the correct operations (irradiation, reprocessing, decay, etc.), provide a result in terms of a set of indicators, such as the amount of waste, the gallery length required in the final repository or the amount of natural resources needed for fuel fabrication.

In recent studies, a special effort has been done so that the scenario analyses do not only provide the result of the assessment but also an estimation of the uncertainties in the output indicators, due to the assumptions made in the models, the inherent uncertainties in the input parameters or nuclear data. These uncertainties may be crucial for the evolution of the fuel cycle scenario since they can lead to the unavailability of certain materials when they are needed, introducing unexpected changes in the fuel cycle with strong consequences in terms of technology implementation or economic competitiveness.

Under this topic, in order to propagate the uncertainties from the input specifications to the desired observables as well as to quantify the impact of each variable, the research done in the recent years cover different degrees of approximation (OECD/NEA, 2017; Thiollière et al., 2018; Skarbeli and Álvarez-Velarde, 2019). The simplest techniques, involving parametric studies and local methods, have a narrow range of validity since they make strong assumptions about the nature of the underlying mathematical dependence that the input parameter space has with a certain output indicator (Cacuci, 2003). In the case of parametric studies, linear dependency is assumed between the input and the output parameters, while results of local methods, based on a first order approximation, are constrained to the vicinity of the reference values (true linear dependence does not exist when there is irradiation of nuclear fuel). The obvious benefit of these techniques is that their calculation only require a few evaluations. On the other hand, global theories (like the Sobol’s variance decomposition) do not impose any restriction, allowing the capture of non-linear relationships along the whole input parameter space as well as interactions (Saltelli et al., 2007).

However, the excessive computational cost these global theories require usually makes them non affordable from a practical point of view, so only a small subset of variables can be selected for this kind of studies. This variable selection can be done relying again on local approximations, like performing a sensitivity search over the whole input parameter space for finding the most relevant variables in a first order approximation. Nevertheless, the computational demand for estimating the Sobol indices with this hybrid methodology is still very high. For example, in the EULER cluster available at CIEMAT (consisting on Xeon E5450 and X5570 quadcores with 2 GB of RAM per core that can execute approximately 100 cases in 20 min), the estimation of the Sobol indices (with more than 100000 simulations) required around two weeks of calculations for an advanced nuclear fuel cycle scenario.

One possibility for dealing with this situation is to approximate the computational exhaustive simulation with a simpler model. This approach, known as surrogate model or metamodel, include splines, artificial neural networks and support vector machine approximations, among others (see for example de Boor, 2001, Basheer and Hajmeer, 2000, Clarke et al., 2004). Another way of generating the surrogate model is by using the Polynomial Chaos expansion (Ghanem and Spanos, 1991, Xiu and Karniadakis, 2003, Sudret, 2008). The advantage of this choice, as it will be explained in Section §4, is that it allows for the easy deduction of the statistical information as it has already been proven by previous works within the nuclear community. In Williams (2006), they are used for solving a stochastic transport equation. Transient scenarios have also been studied with this methodology by means of the point kinetics approximation. For example, Gilli et al. (2012) studies a reactivity insertion which was lately explored by Wu and Kozlowski (2017) too but for performing inverse uncertainty quantification. In Perkó et al. (2014), the case of an unprotected loss of flow is analyzed by using an adaptive basis which is constructed depending to the contributions to the variance.

In this work, the Polynomial Chaos expansion has been applied to an advanced European nuclear fuel cycle with pressurized water reactors (PWR) and accelerator driven subcritical systems (ADS) focused on the advanced management of the transuranic inventories. The expansion coefficients have been determined by regression, in which a regularization term was included in order to build an adaptive sparse basis. The Sobol indices have been calculated with this technique, and the results have been compared with previous studies in order to show the improvements of this new methodology.