Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: I. Effects of Imprecisely Known Microscopic Total and Capture Cross Sections

The subcritical polyethylene-reflected plutonium (PERP) metal fundamental physics benchmark, which is included in the Nuclear Energy Agency (NEA) International Criticality Safety Benchmark Evaluation Project (ICSBEP) Handbook, has been selected to serve as a paradigm illustrative reactor physics system for the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) that was developed by Cacuci. The 2nd-ASAM enables the exhaustive deterministic computation of the exact values of the 1st-order and 2nd-order sensitivities of a system response to the parameters underlying the respective system. The PERP benchmark is numerically modeled in this work by using the deterministic multigroup neutron transport equation discretized in the spatial and angular independent variables. Thus, the numerical model of the PERP benchmark developed includes the following imprecisely known uncertain parameters: 180 group-averaged total microscopic cross sections, 21,600 group-averaged scattering microscopic cross sections, 120 fission process parameters, 60 fission spectrum parameters, 10 parameters describing the experiment’s nuclear sources, and six isotopic number densities. Thus, the numerical simulation model for the PERP benchmark comprises 21,976 uncertain parameters, which implies that, for any response of interest, there are a total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to the model parameters. Computing these sensitivities exactly represents the largest sensitivity analysis endeavor ever carried out in the field of reactor physics. Only 241,483,276 are distinct from each other, and some of these turned out to be zero due to the symmetry of the 2nd-order sensitivities. The numerical results for all of these sensitivities, together with discussions of their major impacts, will be presented in a sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”. This work is the first in this sequence, presenting formulas of general use for neutron transport problems, along with the numerical results that were produced by these formulas for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic total cross sections. For comparison, this work also presents formulas of general use and numerical results for the 180 first-order and 32,400 second-order sensitivities of the PERP leakage response with respect to the neutron transport model’s group-averaged isotopic capture cross sections. It has been widely believed hitherto that, for reactor physics systems modeled by the neutron transport or diffusion equations, the second-order sensitivities are all much smaller than the first-order ones. However, contrary to this widely held belief, the numerical results that were obtained in this work prove, for the first time ever, that many of the 2nd-order sensitivities are much larger than the corresponding 1st-order ones, so their effects can become much larger than the corresponding effects stemming from the 1st-order sensitivities. For example, the 2nd-order sensitivities of the PERP leakage response cause the expected value of this response to be significantly larger than the corresponding computed value. The importance of the 2nd-order sensitivities increases as the relative standard deviations for the cross sections increase. For the extreme case of fully correlated cross sections, for example, neglecting the 2nd-order sensitivities would cause an error as large as 2000% in the expected value of the leakage response and up to 6000% in the variance of the leakage response. The significant effects of the mixed 2nd-order sensitivities underscore the need for reliable values for the correlations that might exist among the total cross sections, which are unavailable at this time. The 2nd-order sensitivities with respect to the total cross sections also cause the response distribution to be skewed towards positive values relative to the expected value. Hence, neglecting the 2nd-order sensitivities could potentially cause very large non-conservative errors by under-reporting of the response variance and expected value.

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ON THE IMPORTANCE OF SECOND-ORDER RESPONSE SENSITIVITIES TO NUCLEAR DATA IN REACTOR PHYSICS UNCERTAINTY ANALYSIS

EPJ Web of Conferences ◽

10.1051/epjconf/202124715021 ◽

2021 ◽

Vol 247 ◽

pp. 15021

Author(s):

Dan G. Cacuci

Keyword(s):

Cross Sections ◽

Neutron Transport ◽

Nuclear Data ◽

Expected Value ◽

Second Order ◽

Model Parameters ◽

Response Distribution ◽

Reactor Physics ◽

Adjoint Sensitivity ◽

Total Cross Sections

This invited keynote presentation compares the relative importance of 1st-order versus 2nd-order sensitivities of the leakage response of an OECD/NEA benchmark (polyethylene-reflected plutonium sphere) to the nuclear data characterizing this benchmark. The imprecisely known parameters underlying the neutron transport computational model for this benchmark include 180 group-averaged total microscopic cross sections, 21600 group-averaged scattering microscopic cross sections, 60 parameters describing the fission process, 30 parameters describing the fission spectrum, 10 parameters describing the system’s sources, and 6 isotopic number densities. Thus, this benchmark comprises 21886 1st-order sensitivities of the leakage response with respect to the model parameters, and 478,996,996 2nd-order sensitivities, of which 239,509,441 are distinct. The exact deterministic computation of all of these 1st- and 2nd-order sensitivities was made possible by the application of the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) developed by Cacuci. Thousands (out of the 32 400 elements) of the 2nd-order sensitivities of the leakage response with respect to the total cross sections turned out to be significantly larger than the largest corresponding 1st-order sensitivities, contrary to some previously held beliefs in the reactor physics community. Hence, it will be shown that neglecting the 2nd-order sensitivities to total cross sections would cause very large non-conservative errors by under-reporting the response’s variance and expected value. The 2nd-order sensitivities also cause the response distribution to be skewed towards positive values relative to the expected value, which, in turn, is significantly larger than the computed value of the leakage response. The result presented in this paper also underscore the need for obtaining reliable cross section covariance data, which are not available at this time.

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Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark. VI: Overall Impact of 1st- and 2nd-Order Sensitivities on Response Uncertainties

Energies ◽

10.3390/en13071674 ◽

2020 ◽

Vol 13 (7) ◽

pp. 1674 ◽

Cited By ~ 3

Author(s):

Dan G. Cacuci ◽

Ruixian Fang ◽

Jeffrey A. Favorite

Keyword(s):

Sensitivity Analysis ◽

Cross Sections ◽

Relative Sensitivity ◽

Second Order ◽

Model Parameters ◽

Adjoint Sensitivity Analysis ◽

Reactor Physics ◽

Adjoint Sensitivity ◽

First Order ◽

Analysis Methodology

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the 1st-order and unmixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity to the isotopic number densities for the two fissionable isotopes have large values, which are comparable to, or larger than, the corresponding sensitivities for the total cross sections. Furthermore, several 2nd-order unmixed sensitivities for the isotopic number densities are significantly larger than the corresponding 1st-order ones. This work also presents results for the first-order sensitivities of the PERP benchmark’s leakage response with respect to the fission spectrum parameters of the two fissionable isotopes, which have very small values. Finally, this work presents the overall summary and conclusions stemming from the research findings for the total of 21,976 first-order sensitivities and 482,944,576 second-order sensitivities with respect to all model parameters of the PERP benchmark, as presented in the sequence of publications in the Special Issue of Energies dedicated to “Sensitivity Analysis, Uncertainty Quantification and Predictive Modeling of Nuclear Energy Systems”.

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Illustrating Important Effects of Second-Order Sensitivities on Response Uncertainties in Reactor Physics

Journal of Nuclear Engineering ◽

10.3390/jne2020012 ◽

2021 ◽

Vol 2 (2) ◽

pp. 114-123

Author(s):

Dan G. Cacuci

Keyword(s):

Cross Sections ◽

Large Scale ◽

Practical Method ◽

Expected Value ◽

Second Order ◽

Response Distribution ◽

Reactor Physics ◽

Adjoint Sensitivity ◽

Total Cross Sections ◽

The Mean

This paper illustrates the relative importance of the largest first- and second-order sensitivities of the leakage response of an OECD/NEA reactor physics benchmark (a polyethylene-reflected plutonium sphere) to the benchmark’s underlying total cross sections. It will be shown that numerous 2nd-order sensitivities of the leakage response with respect to the total cross sections are significantly larger than the largest corresponding 1st-order sensitivities. In particular, the contributions of the 2nd-order sensitivities cause the mean (expected) value of the response to differ appreciably from its computed value and also cause the response distribution to be skewed towards positive values relative to the mean. Neglecting these large 2nd-order sensitivities would cause very large non-conservative errors by under-reporting the response’s variance and expected value. The results presented in this paper also underscore the need for obtaining reliable cross section covariance data, which are currently unavailable. Finally, comparing the CPU-times needed for computations, this paper demonstrates that the Second-Order Adjoint Sensitivity Analysis Methodology is the only practical method for computing 2nd-order sensitivities exactly, without introducing methodological errors, for large-scale systems characterized by many uncertain parameters.

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Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: V. Computation of Mixed 2nd-Order Sensitivities Involving Isotopic Number Densities

Energies ◽

10.3390/en13102580 ◽

2020 ◽

Vol 13 (10) ◽

pp. 2580 ◽

Cited By ~ 1

Author(s):

Ruixian Fang ◽

Dan G. Cacuci

Keyword(s):

Sensitivity Analysis ◽

Cross Section ◽

Cross Sections ◽

Second Order ◽

Adjoint Sensitivity Analysis ◽

Number Density ◽

Reactor Physics ◽

Adjoint Sensitivity ◽

Energy Group ◽

Analysis Methodology

This work applies the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to compute the mixed 2nd-order sensitivities of a polyethylene-reflected plutonium (PERP) benchmark’s leakage response with respect to the benchmark’s imprecisely known isotopic number densities and the other benchmark imprecisely known parameters, including: (i) the 6 × 180 mixed 2nd-order sensitivities involving the total microscopic cross sections; (ii) the 6 × 21,600 mixed 2nd-order sensitivities involving the scattering microscopic cross sections; (iii) the 6 × 60 mixed 2nd-order sensitivities involving the fission microscopic cross sections; and (iv) the 6 × 60 mixed 2nd-order sensitivities involving the average number of neutrons produced per fission. It is shown that many of these mixed 2nd-order sensitivities involving the isotopic number densities have very large values. Most of the large sensitivities involve the isotopic number density of 239Pu, and the microscopic total, scattering or fission cross sections for the 12th or 30th energy groups of 239Pu or 1H, respectively. The 2nd-order mixed sensitivity of the PERP leakage response with respect to the isotopic number density of 239Pu and the microscopic total cross section for the 30th energy group of 1H is the largest of the above mentioned sensitivities, attaining the value −94.91.

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Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of an OECD/NEA Reactor Physics Benchmark: I. Computed Sensitivities

Journal of Nuclear Engineering ◽

10.3390/jne2030024 ◽

2021 ◽

Vol 2 (3) ◽

pp. 281-308 ◽

Cited By ~ 1

Author(s):

Ruixian Fang ◽

Dan Gabriel Cacuci

Keyword(s):

Uncertainty Analysis ◽

Cross Sections ◽

Relative Sensitivity ◽

Numerical Results ◽

Reactor Physics ◽

Adjoint Sensitivity ◽

Total Cross Sections ◽

Analysis Methodology ◽

Total Leakage ◽

Sensitivity And Uncertainty Analysis

This work extends the investigation of higher-order sensitivity and uncertainty analysis from 3rd-order to 4th-order for a polyethylene-reflected plutonium (PERP) OECD/NEA reactor physics benchmark. Specifically, by applying the 4th-order comprehensive adjoint sensitivity analysis methodology (4th-CASAM) to the PERP benchmark, this work presents the numerical results of the most important 4th-order sensitivities of the benchmark’s total leakage response with respect to the benchmark’s 180 microscopic total cross sections, which includes 180 4th-order unmixed sensitivities and 360 4th-order mixed sensitivities corresponding to the largest 3rd-order ones. The numerical results obtained in this work reveal that the number of 4th-order relative sensitivities that have large values (e.g., greater than 1.0) is far greater than the number of important 1st-, 2nd- and 3rd-order sensitivities. The majority of those large sensitivities involve isotopes 1H and 239Pu contained in the PERP benchmark. Furthermore, it is found that for most groups of isotopes 1H and 239Pu of the PERP benchmark, the values of the 4th-order relative sensitivities are significantly larger than the corresponding 1st-, 2nd- and 3rd-order sensitivities. The overall largest 4th-order relative sensitivity S(4)σt,6g=30,σt,6g=30,σt,6g=30,σt,6g=30=2.720×106 is around 291,000 times, 6350 times and 90 times larger than the corresponding largest 1st-order, 2nd-order and 3rd-order sensitivities, respectively, and the overall largest mixed 4th-order relative sensitivity S(4)σt,630,σt,630,σt,630,σt,530=2.279×105 is also much larger than the largest 2nd-order and 3rd-order mixed sensitivities. The results of the 4th-order sensitivities presented in this work have been independently verified with the results obtained using the well-known finite difference method, as well as with the values of the corresponding symmetric 4th-order sensitivities. The 4th-order sensitivity results obtained in this work will be subsequently used on the 4th-order uncertainty analysis to evaluate their impact on the uncertainties they induce in the PERP leakage response.

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Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: II. Effects of Imprecisely Known Microscopic Scattering Cross Sections

Energies ◽

10.3390/en12214114 ◽

2019 ◽

Vol 12 (21) ◽

pp. 4114 ◽

Cited By ~ 7

Author(s):

Fang ◽

Cacuci

Keyword(s):

Sensitivity Analysis ◽

Cross Sections ◽

Second Order ◽

The Other ◽

Adjoint Sensitivity Analysis ◽

Adjoint Sensitivity ◽

First Order ◽

Scattering Cross ◽

Analysis Methodology ◽

Scattering Cross Sections

This work continues the presentation commenced in Part I of the second-order sensitivity analysis of nuclear data of a polyethylene-reflected plutonium (PERP) benchmark using the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM). This work reports the results of the computations of the first- and second-order sensitivities of this benchmark’s computed leakage response with respect to the benchmark’s 21600 parameters underlying the computed group-averaged isotopic scattering cross sections. The numerical results obtained for the 21600 first-order relative sensitivities indicate that the majority of these were small, the largest having relative values of O (10−2). Furthermore, the vast majority of the (21600)2 second-order sensitivities with respect to the scattering cross sections were much smaller than the corresponding first-order ones. Consequently, this work shows that the effects of variances in the scattering cross sections on the expected value, variance, and skewness of the response distribution were negligible in comparison to the corresponding effects stemming from uncertainties in the total cross sections, which were presented in Part I. On the other hand, it was found that 52 of the mixed second-order sensitivities of the leakage response with respect to the scattering and total microscopic cross sections had values that were significantly larger than the unmixed second-order sensitivities of the leakage response with respect to the group-averaged scattering microscopic cross sections. The first- and second-order mixed sensitivities of the PERP benchmark’s leakage response with respect to the scattering cross sections and the other benchmark parameters (fission cross sections, average number of neutrons per fission, fission spectrum, isotopic atomic number densities, and source parameters) have also been computed and will be reported in subsequent works.

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Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: III. Effects of Imprecisely Known Microscopic Fission Cross Sections and Average Number of Neutrons per Fission

Energies ◽

10.3390/en12214100 ◽

2019 ◽

Vol 12 (21) ◽

pp. 4100 ◽

Cited By ~ 6

Author(s):

Cacuci ◽

Fang ◽

Favorite ◽

Badea ◽

Rocco

Keyword(s):

Cross Sections ◽

Nuclear Data ◽

Second Order ◽

Adjoint Sensitivity Analysis ◽

Response Distribution ◽

Adjoint Sensitivity ◽

Total Cross Sections ◽

Scattering Cross ◽

Analysis Methodology ◽

Scattering Cross Sections

The Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) is applied to compute the first-order and second-order sensitivities of the leakage response of a polyethylene-reflected plutonium (PERP) experimental system with respect to the following nuclear data: Group-averaged isotopic microscopic fission cross sections, mixed fission/total, fission/scattering cross sections, average number of neutrons per fission (), mixed /total cross sections, /scattering cross sections, and /fission cross sections. The numerical results obtained indicate that the 1st-order relative sensitivities for these nuclear data are smaller than the 1st-order sensitivities of the PERP leakage response with respect to the total cross sections but are larger than those with respect to the scattering cross sections. The vast majority of the 2nd-order unmixed sensitivities are smaller than the corresponding 1st-order ones, but several 2nd-order mixed relative sensitivities are larger than the 1st-order ones. In particular, several 2nd-order sensitivities for 239Pu are significantly larger than the corresponding 1st-order ones. It is also shown that the effects of the 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s parameters underlying the average number of neutrons per fission, , on the moments (expected value, variance, and skewness) of the PERP benchmark’s leakage response distribution are negligible by comparison to the corresponding effects (on the response distribution) stemming from uncertainties in the total cross sections, but are larger than the corresponding effects (on the response distribution) stemming from uncertainties in the fission and scattering cross sections.

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Comprehensive Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) Applied to a Subcritical Experimental Reactor Physics Benchmark: IV. Effects of Imprecisely Known Source Parameters

Energies ◽

10.3390/en13061431 ◽

2020 ◽

Vol 13 (6) ◽

pp. 1431 ◽

Cited By ~ 2

Author(s):

Ruixian Fang ◽

Dan Gabriel Cacuci

Keyword(s):

Sensitivity Analysis ◽

Cross Sections ◽

Source Parameters ◽

Second Order ◽

Adjoint Sensitivity Analysis ◽

Response Distribution ◽

Adjoint Sensitivity ◽

Scattering Cross ◽

Analysis Methodology ◽

Scattering Cross Sections

By applying the Second-Order Adjoint Sensitivity Analysis Methodology (2nd-ASAM) to the polyethylene-reflected plutonium (PERP) benchmark, this work presents results for the first- and second-order sensitivities of this benchmark’s leakage response with respect to the spontaneous fission source parameters. The numerical results obtained for these sensitivities indicate that the 1st-order relative sensitivity of the leakage response to the source parameters for the two fissionable isotopes in the benchmark are all positive, signifying that an increase in the source parameters will cause an increase in the total neutron leakage from the PERP sphere. The 1st- and 2nd-order relative sensitivities with respect to the source parameters for 239Pu are very small (10−4 or less). In contradistinction, the 1st-order and several 2nd-order relative sensitivities of the leakage response with respect to the source parameters of 240Pu are large. Large values (e.g., greater than 1.0) are also displayed by several mixed 2nd-order relative sensitivities of the leakage response with respect to parameters involving the source and: (i) the total cross sections; (ii) the average neutrons per fission; and (iii) the isotopic number densities. On the other hand, the values of the mixed 2nd-order relative sensitivities with respect to parameters involving the source and: (iv) the scattering cross sections; and (v) and the fission cross sections are smaller than 1.0. It is also shown that the effects of the 1st- and 2nd-order sensitivities of the PERP benchmark’s leakage response with respect to the benchmark’s source parameters on the moments (expected value, variance and skewness) of the PERP benchmark’s leakage response distribution are negligibly smaller than the corresponding effects (on the response distribution) stemming from uncertainties in the total, fission and scattering cross sections.

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Optimal Capacitors Placement of Distribution Systems Using 2nd Order Power Loss Sensitivity Analysis and Hierarchical Clustering

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.986-987.377 ◽

2014 ◽

Vol 986-987 ◽

pp. 377-382 ◽

Cited By ~ 1

Author(s):

Hui Min Gao ◽

Jian Min Zhang ◽

Chen Xi Wu

Keyword(s):

Sensitivity Analysis ◽

Hierarchical Clustering ◽

Distribution System ◽

Power Flow ◽

Distribution Systems ◽

Reactive Power ◽

Digital Simulation ◽

Second Order ◽

First Order ◽

The Impact

Heuristic methods by first order sensitivity analysis are often used to determine location of capacitors of distribution power system. The selected nodes by first order sensitivity analysis often have virtual high by first order sensitivities, which could not obtain the optimal results. This paper presents an effective method to optimally determine the location and capacities of capacitors of distribution systems, based on an innovative approach by the second order sensitivity analysis and hierarchical clustering. The approach determines the location by the second order sensitivity analysis. Comparing with the traditional method, the new method considers the nonlinear factor of power flow equation and the impact of the latter selected compensation nodes on the previously selected compensation location. This method is tested on a 28-bus distribution system. Digital simulation results show that the reactive power optimization plan with the proposed method is more economic while maintaining the same level of effectiveness.

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