تجزیه و تحلیل حساسیت و کاهش ابعاد مدل مولد بخار برای تشخیص گرفتگی

Nuclear steam generators are subject to clogging of their internal parts which causes safety issues. Diagnosis methodologies are needed to optimize maintenance operations. Clogging alters the dynamic behaviour of steam generators and particularly the response of the wide range level (WRL – a pressure measurement) to power transients. A numerical model of this phenomenon has previously been developed. Its input variables describe the spatial distribution of clogging and its output is a discretization of the WRL dynamic response. The objective of the present study is to characterize the information about the clogging state of a steam generator that can be inferred from the observation of its WRL response. A methodology based on several statistical techniques is implemented to answer that question. Principal component analysis reveals that clogging alters the WRL response mainly in two distinct ways. Accordingly, the output can be summarized into a vector of dimension 2. A sensitivity analysis is carried out to rank the input variables by magnitude of influence. It has shown that they can be divided into two groups corresponding to the two sides of the steam generator. Finally, sliced inverse regression is used to reduce the input dimension from 16 to 2. A sampling issue that arises when the input dimension is high is addressed. The simplification of the original problem yields a diagnosis methodology based on response surface techniques.

Pressurized light water nuclear power plants mainly consist of two separated water loops that exchange heat. The water from the primary loop goes first through the reactor where it is heated by the nuclear reaction and then through heat exchangers called steam generators (SGs) where it transfers heat to the water of the secondary loop. Steam exits the SGs by their upper opening and then flows through the turbines. A SG consists of a cylindrical tank (approx. 20 m high and 3 m wide) that contains the secondary steam–liquid mixture. The primary water enters the SG at its bottom and goes through a bundle of U shaped tubes. Eight circular plates called tube support plates (TSPs) maintain the tube bundle. The tubes fit in circular holes drilled in the TSPs. These holes are surrounded by additional quatrefoil holes to let the secondary steam-liquid mixture flow through. A SG diagram can be found in Fig. 1. Full-size image (33 K) Fig. 1. Westinghouse type 51 steam generator. Figure options SGs internal elements foul with iron oxides carried by the secondary feed-water. This causes clogging of the quatrefoil holes that induces safety issues. Means to estimate TSP clogging are needed to optimize maintenance operations. The pressure difference measured between the steam dome and the bottom of the SG is called the wide range level (WRL). Previous studies [1] and [2] have shown that the shape of the WRL response curve to a power transient is altered by the clogging state of the TSPs and derived a diagnosis method that utilizing this link. The principle of the method is to compare measured response curves with simulations using with a mono-dimensional SG model. To assess the method's potential and make it reliable, it is necessary to characterise how much information about the clogging state can be inferred from the WRL response. This issue breaks down into three closely related questions: • how does TSP clogging affect the shape of the WRL response? • Are these effects different in nature and magnitude depending on the location inside the SG? • What is the simplest formulation of input and output variables that captures these effects? The methodology presented here to answer these questions relies on computer intensive statistical methods. As the CPU time for a transient simulation with the 1D SG model is around 5 min, large samples of response curves corresponding to different clogging configurations can be generated. Sensitivity analysis [3] and principal component analysis (PCA) [4] have been carried out to address the first two questions and the simplification of the output. The results suggested the use of a dimension reduction technique called sliced inverse regression (SIR) [5] to simplify the input. Along the process, bootstrap techniques were used to assess the robustness of the results and help with the interpretation. The SG numerical model and the statistical method that have been used are described in Section 2. The results are presented and discussed in Section 3.

A methodology combining several statistical techniques has been carried out with a 1D SG model. It allowed to characterize the information about the clogging state of a SG that can be inferred from its WRL response to a power transient. The study has shown that: • clogging affects the WRL response in two distinct ways. It alters its global slope and its curvature. • These effects depend on the leg of the SG and the elevation of the clogging sites. Clogging of the hot leg and cold leg have a different impact and the former is predominant. The higher is the clogging site in the SG, the greater is the magnitude of the alteration. • The WRL response curves can be summarised by vectors of size 2, each coordinate describing respectively the global slope and the curvature of the curves. The clogging state of individual half-TSPs cannot be identified by analysing the WRL response. The diagnosis actually consists in average clogging ratios of each leg. The low dimensions of the simplified input and output provide a convenient framework for future development of a diagnosis methodology. Two response surfaces, one for each direction of the e.d.r. subspace basis, can be built by swapping the input and output. Then, any measured WRL response can be projected on the PC basis yielding two coordinates. The average clogging of each leg is then indicated by the heights of the two response surfaces associated to the couple of coordinates. The methodology can be easily adapted to other diagnosis contexts. Here there are three remarks to serve that purpose. The derivation of the diagnosis method is mainly based on the dimension reduction achieved with PCA and SIR. However, the basis of the e.d.r. subspace that SIR outputs may not be the most pertinent for the diagnosis. The sensitivity analysis provides valuable insights on the role of each input variable and suggests meaningful combinations of the e.d.r. directions found with SIR. When the input dimension is high and it is suspected that important features may appear only for extreme values of the input variables, the SIR should be carried out with both a Gaussian sample and a uniform sample. Possible inconsistencies in the results can be caused by a too strong violation of the linearity hypothesis in the uniform sample case. In such a situation, the sensitivity analysis can be used to remove the least influential input variables prior to the SIR. Keeping only the 3–5 most prominent variables allows to build a Gaussian sample that covers a more reasonable portion of the hypercube domain. Finally, in situations where the output is more complex, that is, when there are more PCs with non-null eigenvalue, the reduction of the output dimension can be carried out in a more sophisticated way. One drawback of using the PC scores as the new output variables is that this choice is independent of the input. Using Hotelling's theory of most predictable variates, Li et al. [37] have proposed an extension of SIR that relies on the data to find the output projection basis.