Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: II. Computed Response Uncertainties

Ruixian Fang,Dan Gabriel Cacuci

doi:10.3390/jne3010001

Abstract

This work quantifies the impact of the most important 4th-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) reactor physics benchmark with respect to the benchmark’s 180 group-averaged microscopic total cross sections, on the expected value, variance and skewness of the benchmark’s leakage response. This work shows that, as the standard deviations of the cross sections increase, the contributions of the 4th-order sensitivities to the response’s expected value and variance become significantly larger than the corresponding contributions stemming from the 1st-, 2nd- and 3rd-order sensitivities. Considering a uniform 5% relative standard deviation for all microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value and variance of the PERP leakage response amount to 56% and 52%, respectively. Considering 10% uniform relative standard deviations for the microscopic total cross sections, the contributions from the 4th-order sensitivities to the expected value increase to nearly 90%. Consequently, if the computed value L(a) were considered to represent the actual expected value of the leakage response and the 4th-order sensitivities were neglected, the computed value would represent the actual expected value with an error of 3400%. Furthermore, uniform relative standard deviations of 5% and larger (10%) for the microscopic total cross sections cause the higher-order sensitivities to contribute increasingly higher amounts to the response standard deviation: the contributions stemming from the 4th-order sensitivities are larger than the contributions stemming from the 3rd-order sensitivities, which in turn are larger than those stemming from the 2nd-order sensitivities, which are themselves larger than the contributions stemming from the 1st-order sensitivities. This finding evidently underscores the need for computing sensitivities of order higher than first-order. The results obtained in this work also indicate that the 4th-order sensitivities produce a positive response skewness, causing the leakage response distribution to be skewed towards the positive direction from its expected value. Increasing the parameter standard deviations tends to decrease the value of the response skewness, causing the leakage response distribution to become more symmetrical about the mean value. The results presented in this work highlight the finding that the microscopic total cross section for hydrogen (H) in the lowest (“thermal”) energy group is the single most important parameter among the 180 microscopic total cross sections of the PERP benchmark, as it contributes most to the various response moments.

Highlights

The previous works [1,2] on 2nd-order and 3rd-order uncertainty analysis for a polyethylene-reflected plutonium (PERP) OECD/NEA reactor physics benchmark [3] revealed that the uncertainties stemming from the 3rd-order sensitivities of the benchmark’s leakage response with respect to the total cross sections are significantly larger than the uncertainties stemming from the 2nd-order sensitivities, which, in turn, were larger than the uncertainties stemming from the 1st-order sensitivities
These results indicate that a Taylor series expansion of the leakage response up to 3rd-order sensitivities to the total cross sections may be an inadequately accurate representation of the leakage response distribution, which implies that the contributions stemming from 4th-order sensitivities would need to be quantified in order to assess their relative importance in contributing to the overall uncertainties induced in the leakage response
By considering 1%, 5% and 10% values of the relative standard deviations (RSD) for each of the normally-distributed and uncorrelated microscopic total cross sections, the contributions stemming from the 4th-order sensitivities to the various response moments were compared with those stemming from the 1st, 2ndand 3rd-order ones

Summary

Introduction

The previous works [1,2] on 2nd-order and 3rd-order uncertainty analysis for a polyethylene-reflected plutonium (PERP) OECD/NEA reactor physics benchmark [3] revealed that the uncertainties stemming from the 3rd-order sensitivities of the benchmark’s leakage response with respect to the total cross sections are significantly larger than the uncertainties stemming from the 2nd-order sensitivities, which, in turn, were larger than the uncertainties stemming from the 1st-order sensitivities. As has been discussed in [1,2] for the 2nd-order and 3rd-order uncertainty analyses of the leakage response of the PERP benchmark, the second-order moments (i.e., the standard deviations and correlations) for the group-averaged microscopic total cross sections are unavailable for this benchmark. When the group-averaged microscopic total cross sections are uncorrelated and normally-distributed, the variance of the leakage response for the PERP benchmark takes on the following particular form:. When the group-averaged microscopic total cross sections are uncorrelated and normally-distributed, the 3rd-order moment of the leakage response for the PERP benchmark takes on the following particular form:. Using Equations (16)−(31), the effects of the fourth-order sensitivities on the response’s expectation, variance and skewness can be quantified by considering uniform standard deviations of 1% (small), 5% (moderate), and 10% (large) of the group-averaged microscopic total cross sections, respectively

The Effects of the Fourth-Order Sensitivities on the Response Expectation

The Effects of the Fourth-Order Sensitivities on the Response’s Variance

Conclusions