data assimilation and Best-estimate model validation Activities in EURATOM-Fission Programs

D.G. Cacuci 
Nuclear Energy Division, Commissariat a l’Energie Atomique, France
and
Institute for Nuclear Technology and Reactor Safety, University of Karlsruhe, Germany
dan-gabriel cacuci@cea.fr and cacuci@ikr.uni-karlsruhe.de 

As is well known, a physical system and/or the result of an indirect experimental measurement are characterized by independent variables, dependent variables, and relationships between these quantities. Such relationships can be modelled mathematically in terms of: (a) linear and/or nonlinear equations that relate the system's independent variables and parameters to the system's state (i.e., dependent) variables; (b) inequality and/or equality constraints that delimit the ranges of the system's parameters; and (c) one or several quantities, customarily referred to as system responses (or objective functions, or indices of performance, or results of indirect measurements), which are to be analyzed as the parameters vary over their respective ranges. Mathematical models also include parameters whose actual values are not known precisely, but may vary within some ranges that reflect our incomplete knowledge or uncertainty regarding them. Furthermore, the methods needed to solve various equations numerically introduce themselves (numerical) errors. The effects of such errors and/or parameter variations must be quantified in order to assess the respective model’s range of validity. Moreover, the effects of uncertainties in the model’s parameters on the uncertainty in the computed results must also be quantified. Generally speaking, the objective of sensitivity analysis is to quantify the effects of parameter variations on computed results. Terms such as influence, ranking by importance and dominance, are all related to sensitivity analysis. On the other hand, the objective of uncertainty analysis is to assess the effects of parameter uncertainties on the uncertainties in computed results. Sensitivity and uncertainty analyses can be considered as formal methods for evaluating data and models because they are associated with the computation of specific quantitative measures that allow, in particular, assessment of variability in output variables and importance of input variables (see, e.g., Refs [1] and [2], and references therein).
Models of complex physical systems usually involve two distinct sources of uncertainties, namely: (i) stochastic uncertainty, which arises because the system under investigation can behave in many different ways, and (ii) subjective or epistemic uncertainty, which arises from the inability to specify an exact value for a parameter that is assumed to have a constant value in the respective investigation. Epistemic (or subjective) uncertainties characterize a degree of belief regarding the location of the appropriate value of each parameter. In turn, these subjective uncertainties lead to subjective uncertainties for the response, thus reflecting a corresponding degree of belief regarding the location of the appropriate response values as the outcome of analyzing the model under consideration. A typical example of a complex system that involves both stochastic and epistemic uncertainties is a nuclear power reactor plant: in a typical risk analysis of a nuclear power plant, stochastic uncertainty arises due to the many hypothetical accident scenarios which are considered in the respective risk analysis, while epistemic uncertainties arise because of the many uncertain parameters that underlie the estimation of the probabilities and consequences of the respective hypothetical accident scenarios. 
Sensitivity and uncertainty analysis procedures can be either local or global in scope. The objective of local analysis is to analyze the behaviour of the system response locally around a chosen point or trajectory in the combined phase space of parameters and state variables. On the other hand, the objective of global analysis is to determine all of the system's critical points (bifurcations, turning points, response maxima, minima, and/or saddle points) in the combined phase space formed by the parameters and dependent (state) variables, and subsequently analyze these critical points by local sensitivity and uncertainty analysis. The methods for sensitivity and uncertainty analysis are based on either deterministic or statistical procedures. In principle, both types of procedures can be used for either local or for global sensitivity and uncertainty analysis, although, in practice, deterministic methods are used mostly for local analysis while statistical methods are used for both local and global analysis. 
Most of methods currently used in the nuclear industry and technical safety organizations can be roughly decomposed in four sequential steps. The end-status of each step determines if the next step is performed or not. The first step is devoted to analyzing the code’s applicability and identifying uncertain parameters and important phenomena of the transient scenario under consideration. On this basis, the code’s predictive capability is verified against the model’s inventory, the model’s origin and background. Finally, one identifies the output variables (i.e., results) of interest for post-processing. The second step is devoted to reviewing the correlations that involve the uncertain parameters, and identifying the parameter that are potentially significant for the physical phenomena observed during the transient scenario under consideration. This (second) step thus corresponds to a “sensitivity analysis”. In the third step, experimental uncertainties of the initial and boundary conditions and expert judgment (supported, when available by the “code qualification test-experiment matrix”) are employed to obtain uncertainty bands and subjective probability density functions for the uncertain parameters. In the fourth step, the various uncertainties are combined probabilistically via “response surface methods” in order to derive “uncertainties” for the results of interest. Examining carefully all of these steps clearly reveals that a significant amount of conservatism remains embedded even in the “best-estimate” codes, and that the “code validation and qualification” processes are not yet entirely objective. In particular, it seems necessary to incorporate concepts of data adjustment and assimilation [3] into the construction, validation and qualification of best-estimate simulation tools.
The development of software modules for validation and verification of simulation programs, including global sensitivity and uncertainty analysis, is a generic issue of fundamental importance for the safe operation of all types of reactors, while allowing reductions in design margins. Numerical simulations comprise transversal activities common to all types of reactors, and also include specific issues concerning specific types of reactor systems. Besides performing specific experiments and developing specific simulation models, essential activities are “code verification” (“is the code solving the mathematical model correctly?”), “code validation” (“does the model represent reality?”), and “code qualification” (certifying that a proposed simulation/design methodology/system satisfies all performance and safety specifications). Code validation and qualification (V&Q) can be attained only by selected benchmarking, taking into account systematically (i.e., using sensitivities) all sources of uncertainties (computational, experimental, etc.). The validation and qualification of numerical simulations against experiments requires the inclusion of methods of sensitivity and uncertainty analysis together with data assimilation in the presence of both computational and experimental uncertainties. 
The development of standardized validated and qualified tools for numerical simulations is a key activity within the EURATOM-Fission Programs. This paper will highlight the activities and methods regarding sensitivity and uncertainty analysis, and best-estimate validated simulation code systems planned within the FP-6 EURATOM Coordination Action “Sustainable Nuclear Fission Technology Platform” (SNF-TP). The SNF-TP [4] proposes research structures and programs for developing a coherent European strategy for: (i) improving the performance of currently operating (Generation II) and future near-term (Generation III and III+) Light Water Reactors (LWR) while maintaining a high degree of safety, performing studies regarding the feasibility of novel designs, and establishing a unified approach of LWR life time extension; (ii) establishing a sustainable, closed fuel cycle for electricity production using innovative (Generation IV) fast neutron reactor systems in conjunction with partitioning and transmutation (P&T) technologies; (iii) establishing a commercially viable High Temperature Reactor (V/HTR) for process heat and hydrogen production; (iv) assuring adequate training to preserve and enhance the human competence in the nuclear field, and maintaining/renewing the infrastructure necessary for achieving sustainability of nuclear energy; and (v) implementing a Strategic Research Agenda for conceiving and coordinating the European research and development in nuclear science and technology. 

References

 [1] D.G. Cacuci, Sensitivity and Uncertainty Analysis: Theory, Volume 1, Chapman & Hall/CRC, Boca Raton, 2003.
[2] D.G. Cacuci, M. Ionescu-Bujor, and M.I. Navon, Sensitivity and Uncertainty Analysis: Applications to Large Scale Systems, Volume 2, Chapman & Hall/CRC, Boca Raton, 2005. 
[3] D.G. Cacuci, M.I. Navon, and M. Ionescu-Bujor, Computational Methods for Data Analysis and Assimilation, Chapman & Hall/CRC, Boca Raton, 2007. 
[4] “Sustainable Nuclear Fission Technology Platform (SNF-TP)”, EURATOM Contract Nr. FP6-036410, July 2006, European Commission, Brussels.