Estimation of the failure probability of a thermal-hydraulic passive system by means of Artificial Neural Networks and quadratic Response Surfaces

In this paper, Artificial Neural Network (ANN) and quadratic Response Surface (RS) empirical regression models are used as fast-running surrogates of a thermal-hydraulic (T-H) system code to reduce the computational burden associated with the estimation of the functional failure probability of a T-H passive system. The ANN and quadratic RS models are constructed on a limited-size set of input/output data examples of the nonlinear relationships underlying the original T-H code; once built, these models are used for performing, in an acceptable computational time, the numerous system response calculations needed for an accurate uncertainty propagation and failure probability estimation. An application to the functional failure analysis of an emergency passive decay heat removal system in a simple steady-state model of a Gas-cooled Fast Reactor (GFR) is presented. rature concerning the application of surrogate metamodels in reliability problems. In (Liel et al. 2009), polynomial Response Surfaces (RSs) are employed to evaluate the failure probability of structural systems; in (Arul et al. 2009, Fong et al. 2009, Mathews et al. 2009), linear and quadratic polynomial RSs are employed for performing the reliability analysis of T-H passive systems in advanced nuclear reactors; in (Cardoso et al. 2008), Artificial Neural Networks (ANNs) are trained to provide local approximations of the failure domain in structural reliability problems; in (Marrel et al. 2009, Storlie et al. 2009), various regression models (including Gaussian metamodels) are built to calculate global sensitivity indices for a complex hydrogeological model simulating radionuclide transport in groundwater. In this work, the possibility of using Artificial Neural Networks (ANNs) and quadratic Response Surfaces (RSs) to reduce the computational burden associated to the functional failure analysis of a natural convection-based decay heat removal system of a Gas-cooled Fast Reactor (GFR) (Pagani et al. 2005) is investigated. To keep the practical applicability in sight, a small set of input/output data examples is considered available for constructing the ANN and quadratic RS models: different sizes of the (small) data sets are considered to show the effects of this relevant practical aspect. The comparison of the potentials of the two regression techniques in the case at hand is made with respect to the estimation of i) the Probability Density Function (PDF) of the temperature of the naturally circulating coolant in the passive system, ii) the 95th percentile of the naturally circulating coolant temperature and iii) the functional failure probability of the passive system. The paper organization is as follows. In Section 2, the concepts of functional failure analysis for T-H passive systems are synthetically summarized. Section 3 briefly presents the problem of empirical regression modeling. In Section 4, the case study of literature concerning the passive cooling of a GFR is presented. In Section 5, the results of the application of ANNs and quadratic RSs to the functional failure analysis of the T-H passive system of Section 4 are reported. Conclusions are provided in the last section. 2 FUNCTIONAL FAILURE ANALYSIS OF T-H PASSIVE SYSTEMS The basic quantitative steps of the functional failure analysis of a T-H passive system are (Bassi & Marquès 2008): 1 Detailed modeling of the passive system response by means of a deterministic, best-estimate (typically long-running) T-H code. 2 Identification of the parameters/variables, models and correlations (i.e., the inputs to the T-H code) which contribute to the uncertainty in the results (i.e., the outputs) of the best estimate T-H calculations. 3 Propagation of the uncertainties through the deterministic, long-running T-H code in order to estimate the functional failure probability of the passive system. Step 3. above relies on multiple (e.g., many thousands) evaluations of the T-H code for different combinations of system inputs; this can render the associated computing cost prohibitive, when the running time for each T-H code simulation takes several hours (which is often the case for T-H passive systems). The computational issue may be tackled by replacing the long-running, original T-H model code by a fast-running, surrogate regression model (properly built to approximate the output from the true system model). In this paper, classical three-layered feed-forward Artificial Neural Networks (ANNs) (Bishop 1995) and quadratic Response Surfaces (RSs) (Liel et al. 2009) are considered for this task. 3 RESPONSE SURFACES AND ARTIFICIAL NEURAL NETWORKS Let us consider a generic meta-model to be built for performing the task of nonlinear regression, i.e., estimating the nonlinear relationship between a vector of input variables x = {x1, x2, ..., xj, ..., i n x } and a vector of output targets y = {y1, y2, ..., yl, ..., o n y }, on the basis of a finite (and possibly small) set of input/output data examples (i.e., patterns), Dtrain = {(xp, yp), p = 1, 2, ..., Ntrain} (Zio 2006). It can be assumed that the target vector y is related to the input vector x by an unknown nonlinear deterministic function μy(x) corrupted by a noise vector ( ) x ε , i.e., ( ) ( ) ( ) x ε x μ x y y + = (1) In the present case of T-H passive system functional failure probability assessment the vector x contains the relevant uncertain system parameters/variables, the nonlinear deterministic function μy(x) represents the complex, long-running T-H mechanistic model code (e.g., RELAP5-3D), the vector y(x) contains the output variables of interest for the analysis and the noise ( ) x ε represents the errors introduced by the numerical methods employed to calculate μy(x); for simplicity, in the following we assume ( ) x ε = 0 (Storlie et al. 2009). The objective of the regression task is to estimate μy(x) in (1) by means of a regression function f(x, w) depending on a set of parameters w to be properly determined on the basis of the available data set Dtrain; the algorithm used to calibrate the set of parameters w is obviously dependent on the nature of the regression model adopted, but in general it aims at minimizing the mean (absolute or quadratic) error between the output targets of the original T-H code, yp = μy(x), p = 1, 2, ..., Ntrain, and the output vectors of the regression model, yp = f(xp, w ), p = 1, 2, ..., Ntrain. Once built, the regression model f(x, w) can be used in place of the T-H code to calculate any quantity of interest Q, such as the 95 percentile of a physical variable critical for the system under analysis (e.g., the fuel cladding temperature) or the functional failure probability of the passive system. In this work, the capabilities of quadratic Response Surface (RS) and three-layered feed-forward Artificial Neural Network (ANN) regression models are compared in the computational tasks involved in the functional failure analysis of a T-H passive system. In extreme synthesis, quadratic RSs are polynomials containing linear terms, squared terms and possibly two-factors interactions of the input variables (Liel et al. 2009); the RS adaptable parameters w are usually calibrated by straightforward least squares methods. ANNs are computing devices inspired by the function of the nerve cells in the brain (Bishop 1995). They are composed of many parallel computing units (called neurons or nodes) interconnected by weighed connections (called synapses). Each of these computing units performs a few simple operations and communicates the results to its neighbouring units. From a mathematical viewpoint, ANNs consist of a set of nonlinear (e.g., sigmoidal) basis functions with adaptable parameters w that are adjusted by a process of training (on many different input/output data examples), i.e., an iterative process of regression error minimization (Rumelhart et al. 1986). The particular type of ANN employed in this paper is the classical three-layered feed-forward ANN trained by the error back-propagation algorithm.