Implementation of the unscented transformation with low rank approximation in uncertainty analysis during large-break loss of coolant accident

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Highlights

  • Singular value decomposition and Unscented Transform algorithms are combined.

  • Revealing the active subspace thanks to Low Rank Approximation algorithm.

  • Uncertainty quantification is performed during LB-LOCA without scram in PWR.

  • SCALE 6.2 is used in calculating the reactivity coefficients and covariance matrix.

  • Response variables during accident are computed by ATHLET thermal-hydraulic code.

  • Both SVD/UT and LRA/UT sampling significantly reduce the computational time.

Abstract

The Low Rank Approximation (LRA) and Unscented Transform (UT) are integrated to produce a new algorithm having the capability to decrease the time required for the uncertainty quantification during Loss of coolant accident (LOCA) in Pressurized Water Reactors (PWR). The LRA is an efficient technique used in reducing computational cost due to its ability to perform dimensionality reduction by revealing the active or important degrees of freedom and calculate the basis of the so-called active subspace basing on the Singular Value Decomposition (SVD). For further reduction in the computational time; the UT algorithm is also implemented to generate a set of sigma points, these sigma points are the representatives of the whole probability distribution (the UT is restricted to Gaussian distribution). The main safety parameter is the maximum cladding temperatures during the accident which are computed by ATHLET thermal-hydraulic code. The reactivity coefficients and the covariance matrix are calculated using the SCALE 6.2 code. The present calculation model has 14-dimensions, therefore the number of sigma points needed for the SVD/UT technique is 29, and can be minimized to 5 sigma points only if the LRA/UT is used where two singular values are sufficient to reproduce/span the space thanks to the strong correlations between the reactivity coefficients.

Introduction

According to the recommendations of the IAEA to perform Uncertainty and Sensitivity analysis (U&S) when using Best Estimated codes (called Best Estimated Pulse Uncertainty BEPU) (IAEA, 2002), it was necessary to develop the methodologies to be used in the uncertainty quantification during normal and accident conditions (Boyack et al., 1990, Boyack et al., 1989).

In general, there are two approaches used for uncertainty analysis. The first one is the deterministic method which uses the adjoint-based perturbation theory to compute sensitivity coefficients for a given response, these sensitivities are then multiplied by the covariance matrices (sandwich formula) to obtain the response uncertainties. The perturbation theory is divided into two types: the Classical Perturbation Theory (CPT) and Generalized Perturbation Theory (GPT) (Gandini, 1967). Although the adjoint-based perturbation theory computes sensitivities efficiently it cannot be applicable to all codes.

The second technique is the statistical random sampling method which perturbs the entire input data simultaneously, so that the total uncertainty in all responses, due to all data uncertainties is obtained. Two main types of technique can be distinguished; simple random sampling (SRS) and stratified sampling (such as Latin Hypercube Sampling (LHS)) (McKay et al., 1979). Statistical sampling methods can be used to infer sensitivity coefficients using Sobol methods, and this may require a large number of simulations to obtain a full set of sensitivity coefficients (Sobol, 1993).

Although the sampling-based uncertainty analysis is relatively simple it is computationally expensive since calculations have to be performed N-times (where N is the sampling size), where the sample size depends on many parameters such as the minimum acceptable level of precision, confidence level, and etc (Janssen, 2013).

The focus is directed to reduce the computational time required by the statistical sampling methods to compute the uncertainty of the nuclear reactor parameter during accident conditions with the required statistical precision. In our present study, a new method has been developed to reduce the computational time based on the Unscented Transform (UT) algorithm and singular value decomposition (SVD). The SVD and UT algorithms are combined to generate a minimal set of carefully chosen sample points, these sample points completely capture the true mean and covariance of the input variable though the detailed probability distribution is not evaluated, we named this method SVD/UT. For further reduction in the computational time; the Low Rank Approximation (LRA) is also implemented by revealing the active subspace and reducing the order of covariance matrix (Eckart and Young, 1936), the second method is named LRA/UT.

Although the LRA is used before in nuclear engineering field however it was used individually, in this work the LRA technique is added to UT algorithm to get a further reduction to the computational time required for the UQ during accident conditions in NPP.

In order to verify our method, the uncertainty calculated by the Unscented Transform algorithm based on the singular value decomposition (SVD/UT) and the Low Rank Approximation (LRA/UT) are compared with the normal random Sampling (RS).

This paper focuses on the uncertainty quantification in Pressurized Water Reactors during a Large Break Loss of Coolant Accident (LB-LOCA) scenario without control rod insertion, which is an Anticipated Transient Without Scram (ATWS) and classified by the regulatory body as a beyond design basis accident. During the LB-LOCA there are remarkable negative reactivity coefficients as a result of the large void fraction produced during the accident (due to the rapid depression in the primary coolant).

The ATWS scenario is very important in our studies because the large negative reactivity guarantees that the reactor will be safely shutdown even if the control rods failed to insert. Accordingly, the variable with uncertainty is the coolant density reactivity coefficients and the response variables are the peak cladding temperatures during the accident.

The reactivity coefficients of coolant density have uncertainty due to various reasons, e.g., nuclear data and approximations adopted in numerical methods. Among sources of uncertainty for reactivity coefficients of coolant density, the uncertainty of nuclear data is only considered. The degree of uncertainty of the reactivity coefficient depends on coolant density since the sensitivities of nuclear data are quite different for lower and higher-coolant density. Furthermore, the uncertainties for different coolant density are not independent; they are correlated since the source of uncertainty is the same (nuclear data). Therefore, not only the variance of reactivity coefficients of coolant density at various coolant density conditions but also their covariance should be evaluated for realistic analysis.

Thus, the means (average) and relative covariance matrix of coolant density reactivity are calculated by SCALE 6.2 code, using TSAR and TSURFER modules considering the uncertainty of nuclear data (SCALE, 2017). The thermal-hydraulic calculations are performed using ATHLET code (ATHLET, 2017) for calculating the peak cladding temperatures, where the reactivity coefficients are entered as input tables. A PYTHON script has been developed to compute the perturbed values of reactivities using normal random, SVD/UT, and LRA/UT sampling, linking different calculation steps, generate and run N-size ATHLET inputs, reading ATHLET output files, extracting the results, and perform interpolation and U&S analysis.

The paper is organized as follows: The Unscented Transform (UT) algorithm based on the singular value decomposition (SVD) and the Low Rank Approximation (LRA) are explained in detail in Section 2. In Section 3, the nominal ATHLET calculations as well as the uncertainty estimation of input reactivities on the peak cladding temperature are presented. Finally, the conclusion is summarized in Section 4.

Section snippets

Theory

Using the normal random sampling method (RS), the minimum size, N, of a random sample such that the probability that the maximum of the sample is larger than a given α-percentile can be calculated using Wilks estimator as follow (Wilks, 1942, Sanchez-Saez et al., 2018):1-αN-N(1-α)αN-1β,where β is the confidence level. For example: the sample size N should be greater than 115 runs for 98% confidence interval with 95% probability. In this studies Wilks formula is only used for motivation, it

Nominal ATHLETE calculations

The detailed description of the German-type PWR of 1300 MW(e) is presented in Table 1 (Kessler et al., 2014).

The thermal-hydraulic calculations are performed using the ATHLET code. The simulation assumes that the break occurs in the cold leg of a single loop (loop-2) while the other three circuits are lumped in one intact loop (loop-1) as shown in Fig. 6.

First, let us explain the LB-LOCA scenario and its consequences on the Pressurized Water Reactor (PWR). The accident scenario assumes a large

Conclusions

The present paper discusses how to reduce the calculation time required by the statistical sampling technique in calculating uncertainty. First, the singular value decomposition (SVD) and the Unscented Transform (UT) algorithms were combined to generate a set of carefully chosen sample points. Due to the strong correlation between the physical data, the computational time was drastically reduced by revealing the active subspace thanks to the Low Rank Approximation (LRA) algorithm. Both SVD/UT

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement

Basma Foad: Conceptualization, Methodology, Writing - original draft, Writing - review & editing. Akio Yamamoto: Conceptualization, Methodology, Writing - review & editing, Supervision. Tomohiro Endo: Conceptualization, Methodology, Writing - review & editing, Supervision.

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