Application of generalised linear model for time-dependent trend assessment—A case study for the ageing PSA network

doi:10.1016/j.ress.2008.11.005

Reliability Engineering & System Safety

Volume 94, Issue 6, June 2009, Pages 1021-1029

https://doi.org/10.1016/j.ress.2008.11.005 Get rights and content

Abstract

The paper presents the results of a case study on “Investigation of component age-dependent reliability models” conducted under European Commission Joint Research Centre Ageing Probabilistic Safety Assessment Network Task 4 activities. Several Generalised Linear Model cases were applied and investigated in relation to both continuous and discrete data. The Fisher's χ² minimisation approach was used for goodness of fit testing and parameters elaboration. Finally, uncertainty analysis was done for parameters estimation and model extrapolations. The results were analysed and compared with other approaches. Engineering considerations are beyond the scope of the present study.

Introduction

One of the tasks of the European Commission Joint Research Centre (EC JRC) Network on the Use of Probabilistic Safety Assessment (PSA) for Evaluation of Ageing Effects for the Safety of Energy Facilities (Ageing PSA Network) [1] is to investigate which methods and approaches are suitable for identifying ageing, taking into account available PSA reliability data for safety-important nuclear power plant (NPP) components.

The initial reliability data which are usually available for analysis could be presented in two ways:

•
individual times to failure data: times to failure and censoring times,
•
binned data, also called readout data: failure intensities or failure probabilities on demand per bin.

In the first case, various models and techniques are available for trend analysis and parameter estimation. The difficulties relate to the censoring of data, which is usually an issue for PSA components. The results of maximum likelihood estimation are not reliable for strongly censored data [2] and quite complicated for data censored by interval.

For binned data, trend analysis is much simpler and could be done even without specific statistical software packages.

This article presents examples of using Fisher's χ² minimisation method for verifying model validity, formulating parameters and evaluating uncertainties. The approach was applied to several specific cases of Generalised Linear Model. The study was performed by State Technical University for Nuclear Power Engineering and EC JRC Institute for Energy, within the context of Ageing PSA Network activities.

Section snippets

Task specification

The study set out to demonstrate methods of building up and assessing component age-dependent reliability models for binned data.

The following tasks were performed:

•
verification of model validity,
•
parameters estimation,
•
characterisation of uncertainties for estimated parameters and whole model,
•
assessment of possible extrapolation and uncertainties of extrapolation.

Initial data sets

Sometimes exact times of failure are not known—only an interval of time within which the number of failures are recorded. The ordered by component age sequence of intervals with the corresponding number of failures and the accumulated exposure time during the bin (interval) constitutes the binned data (also known as readout data).

To demonstrate the method applicability and to compare the results with other case studies, it was proposed to use several sets of binned data on failure intensities

Models applied

For the continuous failure intensity function, it was proposed to apply the following statistical models:

1.
Constant failure intensity: $ϕ (\overset{\Rightarrow}{θ}; t) = Const$ ;
2.
Linear failure intensity: $ϕ (\overset{\Rightarrow}{θ}; t) = θ_{1} + θ_{2} t$ ;
3.
Log-linear or exponential failure intensity: $\ln ϕ (\overset{\Rightarrow}{θ}; t) = θ_{1} + θ_{2} t$ or $ϕ (\overset{\Rightarrow}{θ}; t) = θ_{1}^{*} \exp (θ_{2} t)$ , where $θ_{1}^{*} = \ln θ_{1}$ ;
4.
Power-low (Weibull) failure intensity model: $ϕ (\overset{\Rightarrow}{θ}; t) = θ_{1} t^{θ_{2}}$ .

For models 2–4, the parameter θ₂>0 means a positive trend in time, i.e. the component failure intensity increases with the age of the component.

Proposed approach

The

Identification of the components susceptible to ageing

In total 37 data sets were analysed. The results of the study are presented in table and graph forms.

Examples of graphical interpretation of fitted models and data uncertainties are given at Fig. 1.

The results were analysed in three stages:

•
At the first stage, the component groups for which one or more proposed models fit well with the data were selected. It was decided to consider all models where the p-value is more then 0.1.
•
Secondly, component groups for which the best-fitted model is

Conclusions and recommendations

(1)
Proposed approach made it possible to identify the component groups with increasing failure intensity and to choose best-fitted reliability model.
Overall, the results of the screening show that a positive ageing trend could be assumed for 10 out of 37 component groups examined. In most of the cases (7 out of 10) the log-linear model was identified as best for the interpolation of data.
For 2 out of 37 component groups, the log-linear model with positive ageing parameter was identified as well

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Demonstration of statistical approaches to identify components ageing by operational data analysis—a case study for the ageing PSA network
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Blombach J, Brahmstaedt K-U, Camarinopoulos L. Does ageing of NPPs require the incorporation of time dependent failure...

There are more references available in the full text version of this article.

Cited by (3)

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