تجزیه و تحلیل حساسیت برای راستی آزمایی طراحی قابل اعتماد توربوست های هسته ای

In this paper, we present an application of sensitivity analysis for design verification of nuclear turbosets. Before the acquisition of a turbogenerator, energy power operators perform independent design assessment in order to assure safe operating conditions of the new machine in its environment. Variables of interest are related to the vibration behaviour of the machine: its eigenfrequencies and dynamic sensitivity to unbalance. In the framework of design verification, epistemic uncertainties are preponderant. This lack of knowledge is due to inexistent or imprecise information about the design as well as to interaction of the rotating machinery with supporting and sub-structures. Sensitivity analysis enables the analyst to rank sources of uncertainty with respect to their importance and, possibly, to screen out insignificant sources of uncertainty. Further studies, if necessary, can then focus on predominant parameters. In particular, the constructor can be asked for detailed information only about the most significant parameters.

Design verification is an essential step in the safe and reliable operation of large industrial structures, particularly when dealing with large and hazardous structures. This is generally undertaken through independent design assessment in order to make sure that the new machinery would operate safely and reliably. As it occurs before the acquisition of the assets, there is unavoidable lack of detailed information; moreover, the construction of a detailed mathematical model of the machine accounting for interaction with its operational environment is not feasible at this stage. As in other fields of structural design and safety, design verification engineering involves the consideration of uncertainty: among the large and heterogeneous sets of parameters lacking detailed information, it is typically useful to identify the most important contributors to exceedance of appropriate design criteria in order to focus additional information retrieval onto the critical model parameters, or, if necessary, change the design. Design verification should hence benefit from the large amount of on-going research devoted to developing uncertainty and sensitivity analysis within large industrial engineering and modelling [1], [2], [3] and [4]. The peculiar application motivating for this paper comes from the key case of design verification for nuclear turbosets. Before the acquisition of a turbogenerator, a key asset performs studies in order to verify that vibration levels remain in an acceptable range. But engineers lack the detailed information on the design of the new turboset. Uncertainties related to some model parameters could push the global behaviour of the turbogenerator into a critical zone, while other parameters have almost no influence on its dynamical behaviour. For vibration diagnostic purposes, when rotating machinery have been installed, then models are quite detailed and the behaviour is generally identified by experimental tests. This is the context generally treated in the literature [5], [6] and [7], where other sources of uncertainty, such as those related to unbalance, rotor misalignment and bearing clearances have to be accounted for. Some of these authors perform local sensitivity analysis (based on derivatives) for these models, for example, Petrov [7] and Chouchane et al. [8]. This approach is naturally limited as (i) it does not permit accounting for simultaneous variation of uncertain parameters and (ii) it is carried out with respect to a reference situation, although it is known that parameters interactions are generally important in dynamical problems as the one considered. Clearly, global sensitivity analysis is more appropriate for studying importance of model parameters in such complex dynamical systems. The objective of this paper is therefore to propose a modus operandi for design verification that allows one to take into account epistemic uncertainty in the preliminary models. This procedure, based on global sensitivity analysis, is rooted in the common conceptual framework for quantitative uncertainty management in industrial applications introduced by de Rocquigny et al. [1]. In the phase of design verification, after the bid offer, and before installing the machines in the machine hall, relatively simple mechanical models are considered. At this stage, uncertainty is principally due to lack of knowledge and thus of epistemic nature. This means that model output variability could be reduced by collecting appropriate additional information. Supplementary information could be introduced by questioning the manufacturer on a limited number of elements or by enhancing some aspects of the rather simple finite element models used in the verification phase. This is the rationale for undertaking global sensitivity analysis. Beyond that, there is generally a design criteria to verify in the context of design verification and thus there exits a threshold not to be exceeded by the model output or variable of interest. Therefore it seems interesting to complete the variance-based sensitivity analysis (implemented via the extended Fourier Amplitude Sensitivity Test [23]) by an importance measure accounting for the presence of the threshold. We have chosen Monte Carlo filtering along with the Smirnov statistics [2] in order gain complementary insights into the model behaviour. Other variance-based and non-variance-based sensitivity measures, providing useful complementary information to decision makers, are discussed in [9]. But, given the presence of the design criteria, Monte Carlo filtering seems the most appropriate technique in order to tackle this issue. Note that several alternative methods have been suggested in the literature for treating epistemic uncertainty, the latter include probabilistic and extra-probabilistic approaches, see for example [10]. In this paper, uncertainty will be modelled in a probabilistic setting. This is firstly because design verification requires communicable, easily-graspable and unequivoque procedures for decision-making, whereby a probabilistic setting encoding uncertainty with respect to observable quantities is an appropriate choice (cf. [1] and [10]). This approach is also convenient given the availability of both a “best-estimate” model, established by experienced engineers, and recognised information on the distribution of the input parameters. The information on the distributions of uncertainty in the inputs are related to their mathematical and physical properties in similar cases already tackled by established literature (e.g. [15]). Stiffness, for instance, is a positive variable and its inverse has to be finite. The stiffness of a spring might take values in the range [kmin, ∞[ where kmin is given by expert judgement while no upper value can be assigned. Indeed, with increasing stiffness, the spring tends to a fixed boundary condition (clamped), a possible but unlikely configuration. Besides, the parameters of the “best-estimate” model are generally interpreted as the mean or median values. Jaynes’ maximum entropy principle [11] allows us then to assign “objective” probability distributions based only on available information. The remainder of this paper is organised as follows. In Section 2, we give an overview over the study case and the probabilistic setting. We give a description of the deterministic “best-estimate” model used for turboset verification introducing variables of interest and design criteria. The topic of uncertainty quantification and modelling is also treated. In Section 3, the use of sensitivity analysis for design verification is addressed. We show how the uncertainty and sensitivity analysis are carried out in order to support decision-making in the design verification procedure.

In this paper we have presented an application of sensitivity analysis for design verification of nuclear turbosets. In the early verification phase, epistemic uncertainties are preponderant in turboset modelling. This is due, on the one hand, to interaction with supports and sub-structures in the machine hall and, on the other hand, to imperfect knowledge about the turbogenerator itself. The aim of the robust design verification procedure presented here is to account for uncertainties and to perform sensitivity analysis in order to support the decision-making process. In this context, uncertainty ranking is helpful for evaluating the criticality of the different design parameters. Furthermore, in design verification, there is generally a criterion related to the non-occurrence of a threshold event that has to be tested. This is why classical variance-based sensitivity analysis can be usefully completed by Monte Carlo filtering-based importance measures as introduced here. However, it does not allow for appreciating the interaction effects so that further research in that respect is still needed. For our particular study, the results confirm that the most important contributors to output variability are the parameters related to the supporting structure. Moreover, our analysis showed that (variance-based) interactions between parameters are very important. Uncertainty with respect to the variable of interest could be reduced by more accurately addressing the modelling of the supporting structure, for example by means of a dedicated finite element model. Beyond these results, further studies need to be done in order to accurately capture interaction between groups of variables. Dependence between factors (e.g., stiffness and damping of the bearings depend on the mass of the rotors), should also be included in the sensitivity analysis. In general, reliable design would valuably benefit from advanced sensitivity analysis in order to robustly improve the compromise between modelling, specification and data acquisition.