Bayesian Calibration of Thermal-Hydraulics Model with Time-Dependent Output
The currently adopted practice for uncertainty quantification of thermal-hydraulics code predictions is done through statistical sampling where the code is evaluated multiple times using different values of input parameters that are randomly generated according to their distributions. However, the distributions describing the uncertainties of many important input parameters are often not known a priori and their estimation often rely solely on expert judgment. Furthermore, many of these parameters are not physically observable and often are used within the code as tuning parameters to fit the prediction to a specific dataset during code development and/or assessment phases. In this work, a Bayesian calibration framework is adopted to quantify the uncertainty of model parameters on the basis of experimental data. Within the framework, prior beliefs about the parameters are assumed and then updated by the available experimental data resulting in posterior distributions of the parameters. The new distributions represent the uncertainty of the calibrated parameters as informed by the available data. Four complications often arise in applying the framework to complex computational models. The first is to derive pertinent error information from multivariate time-dependent output. The large number of observations in time-dependent ouput can result in a computationally intractable problem if each of the highly correlated observations is to be considered individually. The second is to acknowledge that the models are only approximations of reality and thus subject to biases, often referred to as model discrepancy or epistemic uncertainty, while experiment data are also affected by measurement errors. The third is to ensure convergence of the calibration process despite non-linearity and interactions in the sensitivity of the output to the uncertain parameters. Fourth is the increased computation requirement to evaluate the code several thousands of times with different sampled parameters values. The first complication has already been addressed in a previous work through principal component analysis, while the Bayesian calibration approach presented in this work addresses the three other issues. The proposed Bayesian calibration employs a metamodel based on Gaussian Process regression to efficiently emulate the actual code runs, and a Markov Chain Monte Carlo algorithm to draw the samples from the resulting complex posterior distribution. To take into account model discrepancy/epistemic uncertainty and measurement errors, the formulation of the likelihood function that determines the posterior parameters distribution includes terms for biases and experiment data uncertainty. The method is demonstrated and verified for the calibration of a multi-parameter reflood model in the thermal-hydraulics system code TRACE. The calibration is based on experimental data from the reflooding experiment conducted in the FEBA separate effect test (SET) facility. The resulting calibration method is demonstrated to be very fast, accurate and converging for several (5) parameters. The results obtained from Bayesian calibration also helped identifying correlations between parameters. Finally, the updated (posterior) model parameter uncertainties is verified to result in narrower temperature prediction uncertainties while still consistent with the experimental data and the modelled bias term.