Reformulation and Efficient Implementation of the Generalized Perturbation Theory Suited for Monte Carlo Eigenvalue Calculations

Abstract

Sensitivity coefficients of reactor performance parameters, such as the effective multiplication factor, k, and power distribution to input parameters, are key factors in the fields of the sensitivity and uncertainty (S/U) analysis, safety assessments, nuclear data adjustments, etc. In order to estimate the k sensitivity from the Monte Carlo (MC) particle transport calculations, the MC adjoint-weighted perturbation (AWP) methods based on the iterated fission probability (IFP) concept have been successfully applied for k uncertainty quantifications due to the nuclear data uncertainty. Recently, there have been several approaches to extend the MC AWP methods to estimate sensitivities of a ratio of responses in the form of ratios of linear or bilinear functions based on generalized perturbation theory (GPT). In addition, we derive a GPT formulation adequate to estimate sensitivities of general MC tallies such as volume flux, reaction rates, etc., not the form of a ratio of linear or bilinear functions by making the best of the fact that MC standard tally estimates are normalized to be per a fission source neutron in the MC eigenvalue calculations. From the generalized adjoint function equation with an adjoint source for the MC general tally, the physical meaning of the generalized adjoint function for a MC tally is derived. Next, it is shown that the derived MC GPT formulation can be used to estimate the k sensitivity by comparing with the MC AWP k-sensitivity estimation formulation. As the equivalence of the AWP method and the first-order DOS/FSP for the k sensitivity estimation, it is proven that the derived GPT formulation is equivalent to the first-order DOS/FSP for the general tally. However, the current MC adjoint-weighted tally techniques and the new proposed GPT-based MC AWP methods require a memory amount which is proportional to the numbers of the adjoint-weighted tallies and histories per cycle to store history-wise tally estimates during the convergence of the k-adjoint or the generalized adjoint function. Especially the MC adjoint-weighted perturbation (AWP) calculations for the nuclear data sensitivity and uncertainty (S/U) analysis suffer from the huge memory consumption to realize the IFP concept. In order to reduce the memory requirement drastically, we present a new adjoint estimation method of which the memory usage is irrelevant to the numbers of histories per cycle by applying the IFP concept for the MC Wielandt calculations. The new algorithms for the adjoint-weighted kinetics parameter estimation and the AWP calculations in the MC Wielandt method are implemented in a Seoul National University MC code, McCARD and its validity is demonstrated in critical facility problems. From the comparison of the nuclear data S/U analyses, it is demonstrated that the memory amounts to store the sensitivity estimates in the proposed method become negligibly small.