GPT-Free Sensitivity Analysis for Reactor Depletion and Analysis

Abstract

Introduced in this dissertation is a novel approach that forms a reduced-order model (ROM), based on subspace methods, that allows for the generation of response sensitivity profiles without the need to set up or solve the generalized inhomogeneous perturbation theory (GPT) equations. The new approach, denoted hereinafter as the generalized perturbation theory free (GPT-Free) approach, computes response sensitivity profiles in a manner that is independent of the number or type of responses, allowing for an efficient computation of sensitivities when many responses are required. Moreover, the reduction error associated with the ROM is quantified by means of a Wilks’ order statistics error metric denoted by the κ-metric.

Traditional GPT has been recognized as the most computationally efficient approach for performing sensitivity analyses of models with many input parameters, e.g. when forward sensitivity analyses are computationally overwhelming. However, most neutronics codes that can solve the fundamental (homogenous) adjoint eigenvalue problem do not have GPT (inhomogenous) capabilities unless envisioned during code development. Additionally, codes that use a stochastic algorithm, i.e. Monte Carlo methods, may have difficult or undefined GPT equations. When GPT calculations are available through software, the aforementioned efficiency gained from the GPT approach diminishes when the model has both many output responses and many input parameters. The GPT-Free approach addresses these limitations, first by only requiring the ability to compute the fundamental adjoint from perturbation theory, and second by constructing a ROM from fundamental adjoint calculations, constraining input parameters to a subspace. This approach bypasses the requirement to perform GPT calculations while simultaneously reducing the number of simulations required.

In addition to the reduction of simulations, a major benefit of the GPT-Free approach is explicit control of the reduced order model (ROM) error. When building a subspace using the GPT-Free approach, the reduction error can be selected based on an error tolerance for generic flux response-integrals. The GPT-Free approach then solves the fundamental adjoint equation with randomly generated sets of input parameters. Using properties from linear algebra, the fundamental k-eigenvalue sensitivities, spanned by the various randomly generated models, can be related to response sensitivity profiles by a change of basis. These sensitivity profiles are the first-order derivatives of responses to input parameters. The quality of the basis is evaluated using the κ-metric, developed from Wilks’ order statistics, on the user-defined response functionals that involve the flux state-space. Because the κ-metric is formed from Wilks’ order statistics, a probability-confidence interval can be established around the reduction error based on user-defined responses such as fuel-flux, max-flux error, or other generic inner products requiring the flux. In general, The GPT-Free approach will produce a ROM with a quantifiable, user-specified reduction error.

This dissertation demonstrates the GPT-Free approach for steady state and depletion reactor calculations modeled by SCALE6, an analysis tool developed by Oak Ridge National Laboratory. Future work includes the development of GPT-Free for new Monte Carlo methods where the fundamental adjoint is available. Additionally, the approach in this dissertation examines only the first derivatives of responses, the response sensitivity profile; extension and/or generalization of the GPT-Free approach to higher order response sensitivity profiles is natural area for future research.