Uncertainty Propagation Methodology Research Articles

Methodology for neutronic uncertainty propagation and application to a UAM-LWR benchmark

This work covers an important point of the benchmark released by the expert group on Uncertainty Analysis in Modeling of Light Water Reactors (UAM-LWR). This ambitious benchmark aims to determine the uncertainty in LWR systems and processes in all stages of calculation, with emphasis on multi-physics (coupled) and multi-scale simulations. Specifically, in this work, a simplified BWR core is used to propagate the uncertainty of nuclear data. Due to a high computational cost in the analysis all fuel assemblies are modeled as fresh. The propagation is subdivided into two levels i) assembly level –with SCALE6.2.1 and SAMPLER module–, and ii) core level, –with PARCSv3.2 and DAKOTA 6.3–. The first level takes into account the uncertainties contained in the master library ENDF/B-VII.1 and as a result a problem-dependent neutronic library in NEMTAB format is obtained using TXT2NTAB code. This is a friendly Matlab code developed within this work. Finally, the uncertainty contained in the neutronic library is further propagated through PARCS. A different approach is presented in this work to propagate the uncertainty between codes. Following this approach only two neutronic libraries are generated, one with the average responses and the other with their standard deviations. Then, the standard deviation and a matrix of perturbation factors are used to perturb the main neutronic parameters. A parallel work is done to propagate the thermal-hydraulic parameters in a PWR core (Mesado et al., ).

Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

Stochastic partial differential equations (SPDEs) are ubiquitous in engineering and computational sciences. The stochasticity arises as a consequence of uncertainty in input parameters, constitutive relations, initial/boundary conditions, etc. Because of these functional uncertainties, the stochastic parameter space is often high-dimensional, requiring hundreds, or even thousands, of parameters to describe it. This poses an insurmountable challenge to response surface modeling since the number of forward model evaluations needed to construct an accurate surrogate grows exponentially with the dimension of the uncertain parameter space; a phenomenon referred to as the curse of dimensionality. State-of-the-art methods for high-dimensional uncertainty propagation seek to alleviate the curse of dimensionality by performing dimensionality reduction in the uncertain parameter space. However, one still needs to perform forward model evaluations that potentially carry a very high computational burden. We propose a novel methodology for high-dimensional uncertainty propagation of elliptic SPDEs which lifts the requirement for a deterministic forward solver. Our approach is as follows. We parameterize the solution of the elliptic SPDE using a deep residual network (ResNet). In a departure from traditional squared residual (SR) based loss function for training the ResNet, we introduce a physics-informed loss function derived from variational principles. Specifically, our loss function is the expectation of the energy functional of the PDE over the stochastic variables. We demonstrate our solver-free approach through various examples where the elliptic SPDE is subjected to different types of high-dimensional input uncertainties. Also, we solve high-dimensional uncertainty propagation and inverse problems.

Uncertainty Quantification and Propagation of Multiphysics Simulation of the Pressurized Water Reactor Core

In recent years, the demand to provide best-estimate predictions with confident bounds is increasing for the nuclear reactor performance and safety analysis. The Organisation for Economic Co-operation and Development Nuclear Energy Agency has been developing an international benchmark of the light water reactor (LWR) uncertainty analysis in modeling (UAM) for the examination of uncertainty quantification and propagation methodologies with various modeling and simulation code systems. The objective of the present work is to develop an uncertainty propagation mechanism based on the stochastic sampling method by taking into account the uncertainties of both basic nuclear data and fuel modeling parameters in the simulation of pressurized water reactors (PWRs) that can be incorporated in the conventional LWR simulation approach. More specifically, the Three Mile Island Unit 1–related exercises from the LWR-UAM benchmark have been modeled using the coupled TRACE/PARCS code system in the three-dimensional core representation. The input uncertainties of the neutronics simulation include few-group cross sections and kinetics parameters generated using the Sampler/Polaris sequence of SCALE 6.2.1. Several heat transfer–related variables for the fuel modeling were considered as sources of input uncertainty of the thermal-hydraulics simulations, including the thermal conductivity of fuel and cladding, fuel heat capacity, and the gap conductance. Dakota was used to sample input parameters of the coupled code system and to perform the uncertainty analysis. Two types of simulations were conducted: steady-state calculation at hot full-power condition and transient scenario initiated by the spatially asymmetric rod ejection accident. Quantities of interest for the steady-state calculation, including core multiplication factor and power peaking factors, were calculated with associated uncertainties. For transient calculations, best-estimate plus uncertainty results of the time evolution of core reactivity, core power, and peak fuel temperature were generated and analyzed. The Wilks’ formula was used to determine the necessary sample size to achieve a 95% confidence of the 95% limit of output quantities of interest. Although the uncertainty propagation and quantification method presented in this paper was developed for PWRs, it could be in general applicable to the multiphysics uncertainty quantification of other types of LWR cores.

Advanced Methodology for Uncertainty Propagation in Computer Experiments with Large Number of Inputs

In the framework of the estimation of safety margins in nuclear accident analysis, a quantitative assessment of the uncertainties tainting the results of computer simulations is essential. Accurate uncertainty propagation (estimation of high probabilities or quantiles) and quantitative sensitivity analysis may call for several thousand code simulations. Complex computer codes, as the ones used in thermal-hydraulic accident scenario simulations, are often too CPU-time expensive to be directly used to perform these studies. A solution consists in replacing the computer model by a CPU-inexpensive mathematical function, called a metamodel, built from a reduced number of code simulations. However, in case of high-dimensional experiments (with typically several tens of inputs), the metamodel building process remains difficult. To face this limitation, we propose a methodology which combines several advanced statistical tools: initial space-filling design, screening to identify the noninfluential inputs, and Gaussian process (Gp) metamodel building with the group of influential inputs as explanatory variables. The residual effect of the group of noninfluential inputs is captured by another Gp metamodel. Then, the resulting joint Gp metamodel is used to accurately estimate Sobol’ sensitivity indices and high quantiles (here 95% quantile). The efficiency of the methodology to deal with a large number of inputs and reduce the calculation budget is illustrated on a thermal-hydraulic calculation case simulating with the CATHARE2 code a loss-of-coolant accident scenario in a pressurized water reactor. A predictive Gp metamodel is built with only a few hundred code simulations which allows the calculation of the Sobol’ sensitivity indices. This Gp also provides a more accurate estimation of the 95% quantile and associated confidence interval than the empirical approach, at equal calculation budget. Moreover, on this test case, the joint Gp approach outperforms the simple Gp.

JEFF-3.3 covariance application to ICSBEP using SANDY and NDAST

Providing reliable estimates of the nuclear data contribution to the uncertainty of well-known integral benchmarks is fundamental to the validation and verification process for a nuclear data library. The Nuclear Energy Agency has produced and maintains the NDaST sensitivity tool, which integrates the DICE sensitivities and nuclear data covariances. This system has been used to rigorously and efficiently provide direct feedback to evaluators and streamline validation. For its future evolution and to identify high-priority development areas, NDaST is continuously compared against state-of-the-art codes that use different uncertainty propagation methodologies. In this work, NDaST was compared to the nuclear data sampling code SANDY for several ICSBEP criticality benchmarks using the JEFF-3.3 evaluated data. Despite excellent overall agreement for cross sections and fission neutron multiplcities, discrepancies due to processed covariance descriptions for angular distributions and prompt fission neutron spectra have identified areas where coordinated development of nuclear data covariance descriptions should be prioritised.

The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets

This paper outlines a methodology for Bayesian multimodel uncertainty quantification (UQ) and propagation and presents an investigation into the effect of prior probabilities on the resulting uncertainties. The UQ methodology is adapted from the information-theoretic method previously presented by the authors (Zhang and Shields, 2018) to a fully Bayesian construction that enables greater flexibility in quantifying uncertainty in probability model form. Being Bayesian in nature and rooted in UQ from small datasets, prior probabilities in both probability model form and model parameters are shown to have a significant impact on quantified uncertainties and, consequently, on the uncertainties propagated through a physics-based model. These effects are specifically investigated for a simplified plate buckling problem with uncertainties in material properties derived from a small number of experiments using noninformative priors and priors derived from past studies of varying appropriateness. It is illustrated that prior probabilities can have a significant impact on multimodel UQ for small datasets and inappropriate (but seemingly reasonable) priors may even have lingering effects that bias probabilities even for large datasets. When applied to uncertainty propagation, this may result in probability bounds on response quantities that do not include the true probabilities.

Uncertainty and sensitivity analysis in reactivity-initiated accident fuel modeling: synthesis of organisation for economic co-operation and development (OECD)/nuclear energy agency (NEA) benchmark on reactivity-initiated accident codes phase-II

Abstract In the framework of OECD/NEA Working Group on Fuel Safety, a RIA fuel-rod-code Benchmark Phase I was organized in 2010–2013. It consisted of four experiments on highly irradiated fuel rodlets tested under different experimental conditions. This benchmark revealed the need to better understand the basic models incorporated in each code for realistic simulation of the complicated integral RIA tests with high burnup fuel rods. A second phase of the benchmark (Phase II) was thus launched early in 2014, which has been organized in two complementary activities: (1) comparison of the results of different simulations on simplified cases in order to provide additional bases for understanding the differences in modelling of the concerned phenomena; (2) assessment of the uncertainty of the results. The present paper provides a summary and conclusions of the second activity of the Benchmark Phase II, which is based on the input uncertainty propagation methodology. The main conclusion is that uncertainties cannot fully explain the difference between the code predictions. Finally, based on the RIA benchmark Phase-I and Phase-II conclusions, some recommendations are made.

Uncertainty and sensitivity analysis for the simulation of a station blackout scenario in the Jules Horowitz Reactor

An uncertainty and sensitivity analysis for the simulation of a station blackout scenario in the Jules Horowitz Reactor (JHR) is presented. The JHR is a new material testing reactor under construction at CEA on the Cadarache site, France. The thermal-hydraulic system code CATHARE is applied to investigate the response of the reactor system to the scenario. The uncertainty and sensitivity study was based on a statistical methodology for code uncertainty propagation, and the ‘Uncertainty and Sensitivity’ platform URANIE was used. Accordingly, the input uncertainties relevant to the transient, were identified, quantified, and propagated to the code output. The results show that the safety criteria are not exceeded and sufficiently large safety margins exist. In addition, the most influential input uncertainties on the safety parameters were found by making use of a sensitivity analysis.

Nuclear data uncertainty propagation by the XSUSA method in the HELIOS2 lattice code

Uncertainty quantification has been extensively applied to nuclear criticality analyses for many years and has recently begun to be applied to depletion calculations. However, regulatory bodies worldwide are trending toward requiring such analyses for reactor fuel cycle calculations, which also requires uncertainty propagation for isotopics and nuclear reaction rates. XSUSA is a proven methodology for cross section uncertainty propagation based on random sampling of the nuclear data according to covariance data in multi-group representation; HELIOS2 is a lattice code widely used for commercial and research reactor fuel cycle calculations. This work describes a technique to automatically propagate the nuclear data uncertainties via the XSUSA approach through fuel lattice calculations in HELIOS2. Application of the XSUSA methodology in HELIOS2 presented some unusual challenges because of the highly-processed multi-group cross section data used in commercial lattice codes. Currently, uncertainties based on the SCALE 6.1 covariance data file are being used, but the implementation can be adapted to other covariance data in multi-group structure. Pin-cell and assembly depletion calculations, based on models described in the UAM-LWR Phase I and II benchmarks, are performed and uncertainties in multiplication factor, reaction rates, isotope concentrations, and delayed-neutron data are calculated. With this extension, it will be possible for HELIOS2 users to propagate nuclear data uncertainties directly from the microscopic cross sections to subsequent core simulations.

Manufacturing Data Uncertainties Propagation Method in Burn-Up Problems

A nuclear data-based uncertainty propagation methodology is extended to enable propagation of manufacturing/technological data (TD) uncertainties in a burn-up calculation problem, taking into account correlation terms between Boltzmann and Bateman terms. The methodology is applied to reactivity and power distributions in a Material Testing Reactor benchmark. Due to the inherent statistical behavior of manufacturing tolerances, Monte Carlo sampling method is used for determining output perturbations on integral quantities. A global sensitivity analysis (GSA) is performed for each manufacturing parameter and allows identifying and ranking the influential parameters whose tolerances need to be better controlled. We show that the overall impact of some TD uncertainties, such as uranium enrichment, or fuel plate thickness, on the reactivity is negligible because the different core areas induce compensating effects on the global quantity. However, local quantities, such as power distributions, are strongly impacted by TD uncertainty propagations. For isotopic concentrations, no clear trends appear on the results.

Derived Quantities Uncertainty Propagation in High Precision Battery Testing

The successful electrification of the transportation and stationary power markets requires batteries with 10–30 years of lifetime. Validations of such long time scales can take several years and require significant confidence in future life predictions. Battery life predictions across such timescales can be highly sensitive to uncertainty in the collection of primary parameters such as current, voltage and time. A methodology to quantify how these primary parameters propagate into the derived parameters of interest such as capacity and coulombic efficiency is presented. The recent development of high precision battery testing and equipment has met the need for accurate and precise assessment of measurement errors. As an example of the derived quantity uncertainty propagation methodology and approach developed in this paper, a prototype high current, high precision tester is also evaluated. The aim of this paper is to describe a standardized method for the precise quantification of battery derived parameter uncertainty, apply the method to the performance of a prototype high current, high precision tester (HPT) as a case study, and then to model the sensitivity of these precision results across a population of possible test conditions.

Bearing capacity of spatially random rock masses obeying Hoek–Brown failure criterion

ABSTRACTThis paper presents a probabilistic analysis to compute the probability density function of the bearing capacity of a strip footing resting on a spatially varying rock mass. The rock is assumed to follow the generalised Hoek–Brown failure criterion. The uniaxial compressive strength of the intact rock (σc) was considered as a random field and the geological strength index was modelled as a random variable. The uncertainty propagation methodology employed in the analysis is the sparse polynomial chaos expansion. A global sensitivity analysis based on Sobol indices was performed. Some numerical results were presented and discussed.

PIV uncertainty propagation

This paper discusses the propagation of the instantaneous uncertainty of PIV measurements to statistical and instantaneous quantities of interest derived from the velocity field. The expression of the uncertainty of vorticity, velocity divergence, mean value and Reynolds stresses is derived. It is shown that the uncertainty of vorticity and velocity divergence requires the knowledge of the spatial correlation between the error of the x and y particle image displacement, which depends upon the measurement spatial resolution. The uncertainty of statistical quantities is often dominated by the random uncertainty due to the finite sample size and decreases with the square root of the effective number of independent samples. Monte Carlo simulations are conducted to assess the accuracy of the uncertainty propagation formulae. Furthermore, three experimental assessments are carried out. In the first experiment, a turntable is used to simulate a rigid rotation flow field. The estimated uncertainty of the vorticity is compared with the actual vorticity error root-mean-square, with differences between the two quantities within 5–10% for different interrogation window sizes and overlap factors. A turbulent jet flow is investigated in the second experimental assessment. The reference velocity, which is used to compute the reference value of the instantaneous flow properties of interest, is obtained with an auxiliary PIV system, which features a higher dynamic range than the measurement system. Finally, the uncertainty quantification of statistical quantities is assessed via PIV measurements in a cavity flow. The comparison between estimated uncertainty and actual error demonstrates the accuracy of the proposed uncertainty propagation methodology.

Comparison of nuclear data uncertainty propagation methodologies for PWR burn-up simulations

Several methodologies using different levels of approximations have been developed for propagating nuclear data uncertainties in nuclear burn-up simulations. Most methods fall into the two broad classes of Monte Carlo approaches, which are exact apart from statistical uncertainties but require additional computation time, and first order perturbation theory approaches, which are efficient for not too large numbers of considered response functions but only applicable for sufficiently small nuclear data uncertainties. Some methods neglect isotopic composition uncertainties induced by the depletion steps of the simulations, others neglect neutron flux uncertainties, and the accuracy of a given approximation is often very hard to quantify. In order to get a better sense of the impact of different approximations, this work aims to compare results obtained based on different approximate methodologies with an exact method, namely the NUDUNA Monte Carlo based approach developed by AREVA GmbH. In addition, the impact of different covariance data is studied by comparing two of the presently most complete nuclear data covariance libraries (ENDF/B-VII.1 and SCALE 6.0), which reveals a high dependency of the uncertainty estimates on the source of covariance data. The burn-up benchmark Exercise I-1b proposed by the OECD expert group “Benchmarks for Uncertainty Analysis in Modeling (UAM) for the Design, Operation and Safety Analysis of LWRs” is studied as an example application. The burn-up simulations are performed with the SCALE 6.0 tool suite.

Effect of parameters correlation on uncertainty and sensitivity in dynamic thermal analysis of thermal protection blanket in service

In this paper, an uncertainty propagation methodology considering correlations among the input variables for heat transfer analysis through fibrous composite insulation material subjected to severe aerodynamic conditions is presented, based on Monte Carlo (MC) simulation with Cholesky decomposition. The effects of correlation coefficient on the thermal evolution uncertainty, failure probability, and Sobol sensitivity indexes are discussed. The results show that the thermal response outputs become more scattered after taking into account the dependence of variables. The failure probability increase with the rise of the correlation coefficient, and nearly 15% increase is observed when the temperature limit is 470 K under the current uncertainty level. Sobol sensitivity analysis demonstrates that the correlated group variables of aerodynamic thermal and pressure load have the most significant first order and total effect on the thermal response simulations. However, the mean fiber diameter and the correlated group variables of density, specific heat and thermal conductivity of virgin material have weak contributions to the first order effect, but have relatively strong contribution to the interaction effect with other variables. The correlation effect of input parameters causes different influences on the sensitivity indexes of different variables. In all cases, the weakest correlation gives the highest total sensitivity index. The provided analyses demonstrate that incorporation of correlation information of stochastic variables is essential in random variable characterization, thermal evolution uncertainty level quantification, parameter importance identification and risk assessment, and it is especially imperative for thermal protection components with high reliability requirement to prevent the occurrence of catastrophic event.

PROPAGATION OF NUCLEAR DATA UNCERTAINTIES FOR PWR CORE ANALYSIS

An uncertainty propagation methodology based on the Monte Carlo method is applied to PWR nuclear design analysis to assess the impact of nuclear data uncertainties. The importance of the nuclear data uncertainties for 235,238U, 239Pu, and the thermal scattering library for hydrogen in water is analyzed. This uncertainty analysis is compared with the design and acceptance criteria to assure the adequacy of bounding estimates in safety margins.

Study of Nuclear Decay Data Contribution to Uncertainties in Heat Load Estimations for Spent Fuel Pools

At the Paul Scherrer Institut (PSI), a methodology for nuclear data uncertainty propagation in CASMO-5M (C5M) assembly calculations is under development. This paper presents a preliminary application of this methodology to C5M decay heat calculations. Applying a stochastic sampling method, nuclear decay data uncertainties are first propagated for the cooling phase only. Thereafter, the uncertainty propagation is enlarged to gradually account for cross-section as well as fission yield uncertainties during the depletion phase. On that basis, assembly heat load uncertainties as well as total uncertainty for the entire pool are quantified for cooling times up to one year. The relative contributions from the various types of nuclear data uncertainties are in this context also estimated.

Probabilistic Analysis of Strip Footings Resting on Spatially Varying Soils and Subjected to Vertical or Inclined Loads

A probabilistic analysis of vertically and obliquely loaded strip footings resting on a spatially varying soil is presented. The system responses are the footing vertical and horizontal displacements. The deterministic computation of these system responses is based on numerical simulations using the software FLAC3D. Both cases of isotropic and anisotropic random fields are considered for the soil elastic properties. The uncertainty propagation methodology employed makes use of a nonintrusive approach to build up analytical equations for the two system responses. Thus, a Monte Carlo simulation approach is applied directly on these analytical equations (not on the original deterministic model), which significantly reduces the computation time. In the case of the footing vertical load, a global sensitivity analysis has shown that the soil Young's modulus E mostly contributes to the variability of the footing vertical displacement, the Poisson ratio being of negligible weight. The decrease in the autocorrelation distances of E has led to a smaller variability of the footing displacement. On the other hand, the increase in the coefficient of variation of E was found to increase both the probabilistic mean and the variability of the footing displacement. Finally, in the inclined loading case, the results of the probability of failure against exceedance of a vertical and/or a horizontal footing displacement are presented and discussed.

Uncertainty propagation in fatigue crack growth analysis using dimension reduction technique

This paper proposes a general probabilistic methodology for uncertainty propagation in fatigue crack growth analysis under both constant amplitude and variable amplitude loadings. A recently developed small time scale model is used to predict the deterministic fatigue crack growth curve (a-N curve). The dimension reduction technique is used for uncertainty propagation in fatigue crack growth analysis. The basic idea is to avoid direct simulations and focuses on the statistical moment behaviour of output random variables. A modified Chebyshev algorithm is presented to improve the approximation accuracy when calculating the integral points and associated weights for an arbitrary probabilistic distribution. Uncertainties of some material properties are considered as input random variables and propagated through the mechanical model. Prediction result of the proposed methodology is compared with the direct Monte Carlo Simulation (MCS) and is verified using experimental data of aluminium alloys under constant amplitude loading and block loading.

Bearing capacity of strip footings on spatially random soils using sparse polynomial chaos expansion

SUMMARYA probabilistic model is presented to compute the probability density function (PDF) of the ultimate bearing capacity of a strip footing resting on a spatially varying soil. The soil cohesion and friction angle were considered as two anisotropic cross‐correlated non‐Gaussian random fields. The deterministic model was based on numerical simulations. An efficient uncertainty propagation methodology that makes use of a non‐intrusive approach to build up a sparse polynomial chaos expansion for the system response was employed. The probabilistic numerical results were presented in the case of a weightless soil. Sobol indices have shown that the variability of the ultimate bearing capacity is mainly due to the soil cohesion. An increase in the coefficient of variation of a soil parameter (c or φ) increases its Sobol index, this increase being more significant for the friction angle. The negative correlation between the soil shear strength parameters decreases the response variability. The variability of the ultimate bearing capacity increases with the increase in the coefficients of variation of the random fields, the increase being more significant for the cohesion parameter. The decrease in the autocorrelation distances may lead to a smaller variability of the ultimate bearing capacity. Finally, the probabilistic mean value of the ultimate bearing capacity presents a minimum. This minimum is obtained in the isotropic case when the autocorrelation distance is nearly equal to the footing breadth. However, for the anisotropic case, this minimum is obtained at a given value of the ratio between the horizontal and vertical autocorrelation distances. Copyright © 2012 John Wiley & Sons, Ltd.