Differential, criticality and Birnbaum importance measures: An application to basic event, groups and SSCs in event trees and binary decision diagrams

Recent works [Epstein S, Rauzy A. Can we trust PRA? Reliab Eng Syst Safety 2005; 88:195–205] have questioned the validity of traditional fault tree/event tree (FTET) representation of probabilistic risk assessment problems. In spite of whether the risk model is solved through FTET or binary decision diagrams (BDDs), importance measures need to be calculated to provide risk managers with information on the risk/safety significance of system structures and components (SSCs). In this work, we discuss the computation of the Fussel–Vesely (FV), criticality, Birnbaum, risk achievement worth (RAW) and differential importance measure (DIM) for individual basic events, basic event groups and components. For individual basic events, we show that these importance measures are linked by simple relations and that this enables to compute basic event DIMs both for FTET and BDD codes without additional model runs. We then investigate whether/how importance measures can be extended to basic event groups and components. Findings show that the estimation of a group Birnbaum or criticality importance is not possible. On the other hand, we show that the DIM of a group or of a component is exactly equal to the sum of the DIMs of the corresponding basic events and can therefore be found with no additional model runs. The above findings hold for both the FTET and the BDD methods.     

Use of importance measures in risk-informed regulatory applications

The use of importance measures to analyze PRA results is discussed. Commonly used importance measures are defined. Some issues that have been identified as potentially limiting their usefulness are addressed, namely: there is no simple relationship between importance measures evaluated at the single component level and those evaluated at the level of a group of components, and, as a result, some of the commonly used importance measures are not realistic measures of the sensitivity of the overall risk to parameter value changes; and, importance measures do not typically take into account parameter uncertainties which raises the question of the robustness of conclusions drawn from importance analyses. The issues are explored in the context of both ranking and categorization of structures, systems, and components (SSCs) with respect to risk-significance and safety-significance for use in risk-informed regulatory analyses.     

Development of systems reliability analysis code SECOM-2 for seismic PSA

An integrated code system SECOM-2, developed at the Japan Atomic Energy Research Institute (JAERI), has the following functions for systems reliability analysis in seismic probabilistic safety assessments (PSAs): (1) calculation of component failure probability, (2) extraction of minimal cut sets (MCSs) from a given fault tree (FT), (3) calculation of frequencies of accident sequences and core damage, (4) importance analysis with several measures with consideration of unique parameters of seismic PSAs, (5) sensitivity analysis, and (6) uncertainty analysis. This paper summarizes the special features of SECOM-2 to perform the analyses mentioned above. At JAERI, using an integrated FT which represents seismically induced core damage due to all initiating events as a system model to calculate core damage frequency of a nuclear power plant, SECOM-2 can calculate conditional point estimate probabilities of system failures, losses of safety functions, and core damage as a function of earthquake motions. The point estimate is computed by a method which gives an exact numerical solution using the Boolean arithmetic model method. As for consideration of correlation of component failure, which has been an important issue in seismic PSAs, a new technique based on direct FT quantification by a Monte Carlo simulation is being added to SECOM-2. Adding this technique, the core damage frequency can be calculated not only with the upper bound approximation based on MCSs but also with a near exact solution taking into account the correlation among all components. This paper also presents the preliminary results of a seismic PSA of a generic BWR plant in Japan performed at JAERI to demonstrate the functions of the SECOM-2 code.     

Issues related to structural aging in probabilistic risk assessment of nuclear power plants

Structural components and systems have an important safety function in nuclear power plants. Although they are essentially passive under normal operating conditions, they play a key role in mitigating the impact of extreme environmental events such as earthquakes, winds, fire and floods on plant safety. Moreover, the importance of structural components and systems in accident mitigation is amplified by common-cause effects. Reinforced concrete structural components and systems in NPPs are subject to a phenomenon known as aging, leading to time-dependent changes in strength and stiffness that may impact their ability to withstand various challenges during their service lives from operation, the environment and accidents. Time-dependent changes in structural properties as well as challenges to the system are random in nature. Accordingly, condition assessment of existing structures should be performed within a probabilistic framework. The mathematical formalism of a probabilistic risk assessment (PRA) provides a means for identifying aging structural components that may play a significant role in mitigating plant risk. Structural condition assessments supporting a decision regarding continued service can be rendered more efficient if guided by the logic of a PRA.     

Designing a risk-informed balanced system by genetic algorithms: Comparison of different balancing criteria

This paper deals with the use of importance measures (IMs) for the risk-informed optimization of system design and operation. It builds on previous work by the authors in which IMs are incorporated in the formulation of a genetic algorithm (GA) multi-objective optimization problem to drive the design towards a solution which is ‘balanced’ in the importance values of the components. This allows designing systems that are optimal from the point of view of economics and safety, without excessively low- or unnecessarily high-performing components.Different definitions of IMs quantify the risk- or safety-significance of components according to specific views of their role in the system: depending on the optimization problem at hand (e.g. system design optimization and/or maintenance strategy optimization) the use of one IM definition as a balancing criterion may be more appropriate than another.In this regard, a comparison of the Fussell-Vesely (FV), Birnbaum (B) and risk achievement worth (RAW) IMs is performed, with respect to their appropriateness for the optimization of test/maintenance intervals.The RAW is found inappropriate for the purpose, since this measure relates to the defense of the system against the failure of components, which is independent on how often the component is tested.Instead, the use of the FV or B measures allows allocating test/maintenance activities according to the importance of the components they relate to, in agreement with the principle of the risk-informed philosophy of avoiding unnecessary regulatory burdens and defining more efficient inspection and maintenance activities.     

PRA importance measures for maintenance prioritization applications

Various importance measures are standardly calculated in a standard probabilistic risk assessment (PRA). Approaches are developed in this paper for using two of these measures—the minimal cutset contribution and the risk reduction importance—for prioritizing the risk importances of maintenances. One approach which is developed prioritizes maintenances based on the risk importance of the associated contributor. The second approach prioritizes maintenances based on the risk impact if the maintenance is ineffective. The core damage frequency is used as the risk measure for prioritization. The demonstration studies, which are carried out using the Surry Plant PRA, indicate that the two approaches give similar results if appropriate cutoff criteria are used. As an additional evaluation, risk unimportant maintenances are identified using the risk increase importance, or risk achievement worth, calculated in the PRA.     

A fast method of finding minimal cutsets in a risk monitor

This paper introduces a new fast method to find minimal cutsets to calculate CDF (core damage frequency) in a risk monitor. The new method, called McFarm (a Missing Cutsets Finding Algorithm for Risk Monitor), is a fast and accurate algorithm to find minimal cutsets to calculate CDF by changing the fault tree to a simpler one. McFarm shows good results especially for non-safety systems out-of-service (OOS) cases and for OOS cases of supporting systems, such as the diesel generator and component cooling water, which are located at many places and lower levels in the fault tree structure. The fast calculation achieved in the OOS cases of supporting systems is due to adopting the ‘Super OOS AND gate’ concept. McFarm can be used for multiple-equipment OOS as well as single-equipment OOS. McFarm was tested by using two risk-monitor models for Korean nuclear power plants (i.e. for YGN 3,4 and for UCN 3,4) with KIRAP. However, McFarm can be used for the risk monitors which use any quantification methods such as CAFTA, IRRAS, etc. Because McFarm just changes the fault tree to a simpler one, thereafter any quantification methods can be applied to the simplified fault tree.     

Development of a new quantification method for a fire PSA

For an internal fire analysis, fire scenarios are developed carefully and quantified in a sequential and iterative way in a traditional fire Probabilistic Safety Assessment (PSA). However, there has been no proven explicit method to avoid these iterative quantifications till now. This study presents the Jung's Single Top And Run (JSTAR) method that facilitates a simultaneous single quantification of all fire scenarios. The JSTAR method could be employed at the fire PSA phases of a quantitative screening or detailed analysis. Using the JSTAR method, accurate fire risks of a fault tree that has many negates could be calculated by avoiding the frequent house event propagations of the fire scenario conditions. Furthermore, the proposed JSTAR method is a simple and explicit method to build a single-top external event PSA model for a risk-monitoring system.The JSTAR method could be implemented easily by developing a small automatic conversion tool. Depending on the maintenance policy of a fire PSA model, a single-top fire PSA model that is created by the conversion tool could be maintained permanently or it could be temporarily generated and discarded. The use of the JSTAR method is recommended for all external event PSAs such as an internal flooding risk analysis.     

A new importance measure for risk-informed decision making

In this paper, we introduce a new importance measure, the differential importance measure (DIM), for probabilistic safety assessment (PSA). DIM responds to the need of the analyst/decision maker to get information about the importance of proposed changes that affect component properties and multiple basic events. DIM is directly applicable to both the basic events and the parameters of the PSA model. Unlike the Fussell–Vesely (FV), risk achievement worth (RAW), Birnbaum, and criticality importance measures, DIM is additive, i.e. the DIM of groups of basic events or parameters is the sum of the individual DIMs. We discuss the difference between DIM and other local sensitivity measures that are based on normalized partial derivatives. An example is used to demonstrate the evaluation of DIM at both the basic event and the parameter level. To compare the results obtained with DIM at the parameter level, an extension of the definitions of FV and RAW is necessary. We discuss possible extensions and compare the results of the three measures for a more realistic example.     

Accounting for components interactions in the differential importance measure

A limitation of the importance measures (IMs) currently used in reliability and risk analyses is that they rank only individual components or basic events whereas they are not directly applicable to combinations or groups of components or basic events. To partially overcome this limitation, recently, the differential importance measure (DIM), has been introduced for use in risk-informed decision making. The DIM is a first-order sensitivity measure that ranks the parameters of the risk model according to the fraction of total change in the risk that is due to a small change in the parameters’ values, taken one at a time. However, it does not account for the effects of interactions among components. In this paper, a second-order extension of the DIM, named DIM^II, is proposed for accounting of the interactions of pairs of components when evaluating the change in system performance due to changes of the reliability parameters of the components. A numerical application is presented in which the informative contents of DIM and DIM^II are compared. The results confirm that in certain cases when second-order interactions among components are accounted for, the importance ranking of the components may differ from that produced by a first-order sensitivity measure.