A hierarchical Bayesian calibration framework for quantifying input uncertainties in thermal-hydraulics simulation models

In the framework of Best Estimate plus Uncertainty (BEPU) methodology, the uncertainties involved in simulations must be quantified to prove that the investigated design is reasonable and acceptable. The predictive uncertainties are usually calculated by propagating input uncertainties through the simulation model, which requires knowledge of the input uncertainties. However, such input uncertainty information in nuclear best-estimate Thermal-Hydraulics (TH) codes are not always available and are mostly based on expert judgment, especially for parameters in closure laws or empirical correlations that are used to describe complex two-phase flow and heat transfer phenomena. The physical model parameters used in empirical correlations cannot be directly measured nor have inherent physical meanings, so they may have large uncertainties that are unknown to code users. Obtaining statistical information of them, therefore, becomes crucial if the effects of these input parameters on model responses need to be studied for safety analysis. This work aims to develop a framework to quantify the input uncertainty of physical model parameters in the TH system code to address the ``lack of input uncertainty information'' issue. Bayesian calibration, or inverse Uncertainty Quantification (UQ), is the process of updating uncertainty distributions on the model inputs in a way that is consistent with observed data. The process of Bayesian calibration for nuclear system codes typically includes Sensitivity Analysis (SA), surrogate model construction, and posterior sampling by Markov Chain Monte Carlo (MCMC) algorithms. SA aims at screening out input parameters that have low impacts on Quantity of Interests (QoI), surrogate models are developed to replace the computationally expensive TH codes and improve the efficiency of the framework, and MCMC algorithms are used to approximate the posterior distributions of input parameters that are consistent with observed data, using Bayes' rule. The hierarchical Bayesian calibration framework developed in this work employs a multi-level structure for input parameters, which has a more realistic assumption, a more flexible structure, and is demonstrated to have the capability of avoiding overfitting. The framework uses an efficient No-U-Turn Sampler (NUTS) algorithm for posterior sampling, which requires no hand-tuning and thus has less user effect. The advantages of the hierarchical framework is demonstrated using toy examples as well as the measured steady-state void fraction data in the BFBT benchmark. More complex hierarchical structures including Gaussian Mixture Model (GMM) are also studied and compared. The framework has the capability of calibrating against transient experimental data. Artificial Neural Network (ANN) and Gaussian Processes (GP) with Principal Component Analysis (PCA) models are used as surrogate models for the high-dimension and high-correlation outputs in transients. The framework is demonstrated using the transient void distribution experiments in the PSBT benchmark. The effects of covariance information in time-dependent data are studied and it is shown that the covariance could substantially change the parameters' posterior distributions and can help capture the overall shape of the time-series in Bayesian calibration. The input uncertainties of physical model parameters involved in both steady-state and transient experiments are successfully quantified and validated using the novel hierarchical framework developed in this work. The resulting posterior distributions of inputs are necessary for future forward UQ and SA in reactor design and safety analysis