Probabilities Of Basic Events Research Articles

Research on Risk Probability and Economic Safety Assessment Method and Application of Mechanical System

Form human-machine-environment, this paper analyzed the factor of risk assessment, and proposed how to calculate the failure probability of basic event including the human factor and the mechanical device. The accident serious degree was changed into the economic losses by the view of safety economics. Integrating the failure probability and the economic losses to determine the risk assessment items, system rank was evaluated by checking assessment items. After establishing the quantitative risk probability and economic risk assessment method, the example about crane travel mechanism was given to demonstrate the method correct and feasible.

Fuzzy set theoretic approach to fault tree analysis

Research in conventional fault tree analysis (FTA) is based mainly on failure probability of basic events, which uses classical probability distributions for the failure probability of basic events. In the present paper the probabilistic consideration of basic events is replaced by possibilities, thereby leading to fuzzy fault tree analysis. Triangular and trapezoidal fuzzy numbers are used to represent the failure possibility of basic events. Since a system may have to go through different operating conditions during the design or testing phase. Thus the failure possibility of a basic event will be assigned more than one fuzzy numbers by different experts under various operating conditions. It is also well established that the selection of a fuzzy number to represent a basic event is vital in fault tree analysis. Here we developed an algorithm to find a single fuzzy number for a basic event, wherein more than one fuzzy number is assigned to that particular event. Using this algorithm, we obtain a single fuzzy number, having least variance from all fuzzy numbers assigned to the concerned event. The adequate and appropriate means and procedure for the detection of basic events having key role in the occurrence of top event in system analysis become essential. Here we have put forward an approach to rank the basic events in accordance with their importance in the occurrence of top event. This approach can be widely used to improve the reliability and to reduce the operating cost of a system. The proposed techniques are discussed and illustrated by taking an example of a nuclear power plant. Keywords: Fault tree, Triangular and Trapezoidal fuzzy number, Fuzzy importance, Ranking of fuzzy numbers

SU‐GG‐T‐520: A Fault Tree Model and Analyses for Gamma Knife Stereotactic Radiosurgery

Purpose: To develop a fault tree model of Gamma Knife stereotactic radiosurgery (GKSRS) with Model 4C to identify critical errors leading to medical events. Methods and Materials: We constructed a fault tree with a medical event as the top event, i.e., the administered dose being different by more than 10% from the prescription dose. The fault tree consisted of 47 basic events and 30 logic gates (25 OR and 5 AND gates). The basic events were grouped into five types of errors: the hardware (15 events), software (8), human (22), and other (5) errors. The probabilities of basic events were subjectively estimated based on authors' experiences and some published data. By varying the probabilities of the five types of events, we calculated the probabilities of the medical event. Results: For a reference case we assumed that the probabilities of occurrences of hardware, software, human, and other types of errors were 10−7, 10−7, 10−6, and 10−7, respectively. The calculation yielded 20 events per a million treatments as the probability of the medical event. When we decreased the probability of the human errors (e.g., incorrect depth measurements, a failure in identifying treatment site, etc.) to 10−7, we found that the medical event occurred only 4 times per a million procedures. When the probabilities of other three types of errors were changed to 10−9, the medical event probability decreased only slightly to 18 per million. Conclusions: Our initial analysis showed that decreasing the human errors before and during GKSRS is the most effective among potential errors for decreasing the medical event. Most of the human errors can be prevented by performing secondary checks of the treatment parameters and setups. The model is being applied to evaluate the usefulness of patient‐specific secondary dose calculation software for prevention of hazardous errors in GKSRS procedures.

Uncertainty analysis in PSA with correlated input parameters

The probability of occurrence of top event of a fault tree in probabilistic safety assessment (PSA) is estimated from the probabilities of the basic events which constitute the fault tree. However the failure probabilities of basic events are subjected to statistical uncertainty. While analyzing the uncertainty of top event, the basic events are assumed uncorrelated or independent in most of the situation, but are not the case every time. Such statistical correlations are introduced into failure data by many causes. To handle the propagation of uncertainties in the presence of correlations, there are three methods: (i) Method of moments, (ii) P-box approach and (iii) Monte Carlo simulations. However the first two methods are difficult for implementation in large scale problems when partial correlations are present instead of fully correlated data. The paper presents a methodology based Monte Carlo simulation with Nataf Transformation of generating correlated random variables for uncertainty analysis in PSA. Computer code has been developed to implement the proposed methodology and a case study from NPP has been carried out.

An approach to the construction of a one top fire event PSA model

An internal fire event probabilistic safety assessment (PSA) model has been generally quantified by modifications of a pre-developed internal events PSA model. New accident sequence logics not covered in the internal events PSA model are separately developed to incorporate them into the fire PSA model. Previous studies on the changes of the one top internal event PSA model for the one top fire event PSA model have been limited to the equipment failures affected by the fire. In addition, they assumed that the probabilities of basic events associated with equipment or cables impacted by the fire are one. However, the probabilities of spurious operation events and human failure events affected by the fire might not be estimated as one. In this study, new modification rules were proposed for the construction of a one top PSA model for fire events by using a one top internal event PSA model. The proposed new modification rules can be applied to all the fire damage events for the fire-induced equipment failure events and spurious operation events, human error events impacted by a fire, regardless of whether they are estimated as one or not. Applications of the proposed modification rules to the compartment and scenario-level fires for the hypothetical plants were performed for demonstrating their appropriateness to the changes of the one top internal event PSA model to the one top fire event PSA model. In addition, quantification procedure with the one top fire event PSA model was presented and discussed.

Uncertainty reduction for improved mishap probability prediction: Application to level control of distillation unit

At all levels, the understanding of uncertainty associated with risk of major chemical industrial hazards should be enhanced. In this study, a quantitative risk assessment (QRA) was performed for a knockout drum in the distillation unit of a refinery process and then probabilistic uncertainty analysis was applied for this QRA. A fault tree was developed to analyze the probability distribution of flammable liquid released from the overfilling of a knockout drum. Bayesian theory was used to update failure rates of the equipment so that generic information from databases and plant equipment real life data are combined to gain all available knowledge on component reliability. Using Monte Carlo simulation, the distribution of top event probability was obtained to characterize the uncertainty of the result. It was found that the uncertainty of basic event probabilities has a significant impact on the top event probability distribution. The top event probability prediction uncertainty profile showed that the risk estimation is improved by reducing uncertainty through Bayesian updating on the basic event probability distributions. The whole distribution of top event probability replaces point value in a risk matrix to guide decisions employing all of the available information rather than only point mean values as in the conventional approach. The resulting uncertainty guides where more information or uncertainty reduction is needed to avoid overlap with intolerable risk levels.

Fault tree analysis for urban flooding

Traditional methods to evaluate flood risk generally focus on heavy storm events as the principal cause of flooding. Conversely, fault tree analysis is a technique that aims at modelling all potential causes of flooding. It quantifies both overall flood probability and relative contributions of individual causes of flooding. This paper presents a fault model for urban flooding and an application to the case of Haarlem, a city of 147,000 inhabitants. Data from a complaint register, rainfall gauges and hydrodynamic model calculations are used to quantify probabilities of basic events in the fault tree. This results in a flood probability of 0.78/week for Haarlem. It is shown that gully pot blockages contribute to 79% of flood incidents, whereas storm events contribute only 5%. This implies that for this case more efficient gully pot cleaning is a more effective strategy to reduce flood probability than enlarging drainage system capacity. Whether this is also the most cost-effective strategy can only be decided after risk assessment has been complemented with a quantification of consequences of both types of events. To do this will be the next step in this study.

Reliability Analysis of a conveyor system using hybrid data

AbstractThe quantification of a fault tree is difficult without an exact probability value for all of the basic events of the tree. To overcome this difficulty in this paper, we propose a methodology which employs ‘hybrid data’ as a tool to analyse the fault tree. The proposed methodology estimates the failure probability of basic events using the statistical analysis of field recorded failures. Under these circumstances, where there is an absence of past failure records, the method follows a fuzzy set based theoretical evaluation based on the subjective judgement of experts for the failure interval. The proposed methodology has been applied to a conveyor system. The results of the analysis reveal the effectiveness of the proposed methodology and the instrumental role played by the experience of experts in providing reliability oriented information. Copyright © 2006 John Wiley & Sons, Ltd.

06/01231 Seismic fragility analyses of nuclear power plant structures based on the recorded earthquake data in Korea: Sung, G. C. and Yang, H. J. Nuclear Engineering and Design, 2005, 235, (17–19), 1867–1874

Model uncertainties can have a significant impact on decisions regarding licensing basis changes. We present a methodology to identify basic events in the risk assessment that have the potential to change the decision and are known to have significant model uncertainties. Because we work with basic event probabilities, this methodology is not appropriate for analyzing uncertainties that cause a structural change to the model, such as success criteria. We use the risk achievement worth (RAW) importance measure with respect to both the core damage frequency (CDF) and the change in core damage frequency (ΔCDF) to identify potentially important basic events. We cross-check these with generically important model uncertainties. Then, sensitivity analysis is performed on the basic event probabilities, which are used as a proxy for the model parameters, to determine how much error in these probabilities would need to be present in order to impact the decision. A previously submitted licensing basis change is used as a case study. Analysis using the SAPHIRE program identifies 20 basic events as important, four of which have model uncertainties that have been identified in the literature as generally important. The decision is fairly insensitive to uncertainties in these basic events. In three of these cases, one would need to show that model uncertainties would lead to basic event probabilities that would be between two and four orders of magnitude larger than modeled in the risk assessment before they would become important to the decision. More detailed analysis would be required to determine whether these higher probabilities are reasonable. Methods to perform this analysis from the literature are reviewed and an example is demonstrated using the case study.

Is mapping a part of common cause failure quantification?

Abstract Important issues in any procedure used for estimating basic event probabilities of common cause failures (CCF) for probabilistic safety assessments (PSA) are: which plants and systems to use, how to combine them, and how to transform data from systems with different numbers of similar components to obtain CCF-rates for a specific group of components. These issues are addressed with focus on the last part called “mapping”. Certain parametric models are considered for transforming CCF event experience from data-source plants to the target plant, the plant of interest. Two sets of rules are reviewed and compared for transforming rates and assessment uncertainties from larger to smaller systems i.e. mapping down. Epistemic uncertainties are taken into account in the estimation. Mapping down equations are presented also for the alpha-factors and for the variances of CCF-rates. Consistent rules are developed for mapping up CCF-rates, uncertainties and alpha factors from smaller to larger systems. These rules are not limited to a binomial CCF model. Consistency requirements are severe and dictate certain limits to possible parametric values. Empirical alpha factors are used to quantify robust mapping ratios of complete CCF-rates. Mapping is critically analyzed and practical recommendations are made.

Including model uncertainty in risk-informed decision making

Model uncertainties can have a significant impact on decisions regarding licensing basis changes. We present a methodology to identify basic events in the risk assessment that have the potential to change the decision and are known to have significant model uncertainties. Because we work with basic event probabilities, this methodology is not appropriate for analyzing uncertainties that cause a structural change to the model, such as success criteria. We use the risk achievement worth (RAW) importance measure with respect to both the core damage frequency (CDF) and the change in core damage frequency (ΔCDF) to identify potentially important basic events. We cross-check these with generically important model uncertainties. Then, sensitivity analysis is performed on the basic event probabilities, which are used as a proxy for the model parameters, to determine how much error in these probabilities would need to be present in order to impact the decision. A previously submitted licensing basis change is used as a case study. Analysis using the SAPHIRE program identifies 20 basic events as important, four of which have model uncertainties that have been identified in the literature as generally important. The decision is fairly insensitive to uncertainties in these basic events. In three of these cases, one would need to show that model uncertainties would lead to basic event probabilities that would be between two and four orders of magnitude larger than modeled in the risk assessment before they would become important to the decision. More detailed analysis would be required to determine whether these higher probabilities are reasonable. Methods to perform this analysis from the literature are reviewed and an example is demonstrated using the case study.

Fault-tree modelling of computer system security

Quantitative assessment of the effect of security breaches on a computer system can be based on the following: specification of all foreseeable types of basic events and estimation of their probabilities of occurrence over a stated period of time; observation of the various types of security measures employed by the system; definition of the undesired top events resulting from security breaches, and estimation of the system’s vulnerability to each of these events as the cost incurred by the system if that event took place; mathematical modelling of the logical relations between the aforementioned entities. In this paper we adapt the fault-tree methodology of reliability engineering to the quantification of security exposure of computer systems. In this new context, a fault tree can be described as a logic diagram whose input represents breach events at various system levels, and whose vertices represent logic operations or gates. The root or output of the fault tree can be any of the undesired top events. We briefly survey algorithms for converting the switching (Boolean) expression of the indicator variable for the top event into a probability expression. Once the top event probability is determined, it can be multiplied by the system’s vulnerability to that event to yield a quantified value of the system’s exposure to it. We also handle the doubly stochastic problem of estimating the uncertainty in the top event probability by using an analytic exact formula relating the variance of the top event probability to the variances of the basic event probabilities. An example of a typical computer system is presented wherein numerical estimates are obtained for the top event probabilities and their variances and for the importance ranking of the various breach events.

Uncertainties and quantification of common cause failure rates and probabilities for system analyses

Simultaneous failures of multiple components due to common causes at random times are modelled by constant multiple-failure rates. A procedure is described for quantification of common cause failure (CCF) basic event probabilities for system models using plant-specific and multiple-plant failure-event data. Methodology is presented for estimating CCF-rates from event data contaminated with assessment uncertainties. Generalised impact vectors determine the moments for the rates of individual systems or plants. These moments determine the effective numbers of events and observation times to be input to a Bayesian formalism to obtain plant-specific posterior CCF-rates. The rates are used to determine plant-specific common cause event probabilities for the basic events of explicit fault tree models depending on test intervals, test schedules and repair policies. Three methods are presented to determine these probabilities such that the correct time-average system unavailability can be obtained with single fault tree quantification. Recommended numerical values are given and examples illustrate different aspects of the methodology.

An Intrinsic Order Criterion to Evaluate Large, Complex Fault Trees

A new efficient algorithm is introduced to evaluate (non) coherent fault trees, obtaining exact lower & upper bounds on system unavailability, with a prespecified maximum error. The algorithm is based on the canonical normal form of the Boolean function, but overcomes the large number of terms needed, by using an intrinsic order criterion (IOC) to select the elementary states to evaluate. This intrinsic order implies lexicographic (truth table) order. The criterion guarantees a priori that the probability of a given elementary system state is always greater than or equal to the probability of another state, for any set of basic probabilities. IOC is exclusively based on the positions of 0 & 1 in the binary n-tuples defining the elementary states. The algorithm does not require any qualitative analysis of the fault tree. The computational cost mainly depends on the basic event probabilities, and is related to system complexity, only because the Boolean function must be evaluated.

Improved accuracy in quantitative fault tree analysis

The fault tree diagram defines the causes of the system failure mode or 'top event' in terms of the component failures and human errors, represented by basic events. By providing information which enables the basic event probability to be calculated, the fault tree can then be quantified to yield reliability parameters for the system. Fault tree quantification enables the probability of the top event to be calculated and in addition its failure rate and expected number of occurrences. Importance measures which signify the contribution each basic event makes to system failure can also be determined. Owing to the large number of failure combinations (minimal cut sets) which generally result from a fault tree study, it is not possible using conventional techniques to calculate these parameters exactly and approximations are required. The approximations usually rely on the basic events having a small likelihood of occurrence. When this condition is not met, it can result in large inaccuracies. These problems can be overcome by employing the binary decision diagram (BDD) approach. This method converts the fault tree diagram into a format which encodes Shannon's decomposition and allows the exact failure probability to be determined in a very efficient calculation procedure. This paper describes how the BDD method can be employed in fault tree quantification.

Improved accuracy in quantitative fault tree analysis

The fault tree diagram defines the causes of the system failure mode or ‘top event’ in terms of the component failures and human errors, represented by basic events. By providing information which enables the basic event probability to be calculated, the fault tree can then be quantified to yield reliability parameters for the system. Fault tree quantification enables the probability of the top event to be calculated and in addition its failure rate and expected number of occurrences. Importance measures which signify the contribution each basic event makes to system failure can also be determined. Owing to the large number of failure combinations (minimal cut sets) which generally result from a fault tree study, it is not possible using conventional techniques to calculate these parameters exactly and approximations are required. The approximations usually rely on the basic events having a small likelihood of occurrence. When this condition is not met, it can result in large inaccuracies. These problems can be overcome by employing the binary decision diagram (BDD) approach. This method converts the fault tree diagram into a format which encodes Shannon's decomposition and allows the exact failure probability to be determined in a very efficient calculation procedure. This paper describes how the BDD method can be employed in fault tree quantification. © 1997 John Wiley & Sons, Ltd.

Using higher-level failure data in fault tree quantification

This paper presents a Bayesian method which can simultaneously combine basic event and statistically independent higher event-level failure data in fault tree quantification. Such higher-level data could correspond to train, subsystem or system failure events. In fact, because highest-level data are usually available for existing facilities, the method presented here allows such data to be propagated to lower levels. The method has two stages: (1) a top-down propagation scheme which allocates the higher event-level information to the basic events, at a cost of making them dependent; and (2) a scheme for sampling the probabilities of the dependent basic events. A simple example illustrates the performance of the method.

Uncertainty in fault tree analysis: A fuzzy approach

In fault tree analysis, the uncertainties in the failure probability and/or failure rate of system components or basic events can be propagated to find the uncertainty in the overall system failure probability. The conventional approach is Monte-Carlo simulation by assuming a probability distribution for the failure probability. In addition, a new methodology based on fuzzy set theory is also being used in the fault tree analysis for quantifying the basic event uncertainty and for propagating it. However, identification of the components which contribute maximum to the system failure probability is also important in fault tree analysis. Similarly, ranking the components based on their contribution of uncertainty to the uncertainty of the system failure probability is also very important. This paper presents a comparative study of probabilistic and fuzzy methodologies for top event uncertainty evaluation. Further, it explains a new approach to rank the system components or basic events depending on (1) their contribution to the top event failure probability and (2) their uncertainty contribution to the uncertainty of the top event based on fuzzy set theory.

Calculating exact top-event probabilities using Σ-Patrec

This paper presents a method for calculating top-event exact probability. This responds to a surge in the application of probabilistic risk assessment (PRA) techniques to ecological and weapon safety assessments. In these domains, basic event probabilities can be large; events characterizing human error and some natural phenomena are typical examples. The method described combines the /spl Sigma/II algorithm of Corynen and the pattern recognition scheme of Keen et al. The PC-based program that is developed using this method is called /spl Sigma/II-Patrec and computes the exact probability of top-event of a system fault-tree model as defined by its cut sets. The /spl Sigma/II module partitions and disjoints the cut sets and solves the resultant sub-models recursively. The pattern recognition module reduces the computational complexity by recognizing repeated sub-models in the calculation process and thus avoiding repeated evaluations. /spl Sigma/II-Patrec is designed to quantify the fault-tree models of both coherent and incoherent systems, and interfaces with the graphic package, SEATREE, for interactive generation of fault trees. In the anticipated case, the /spl Sigma/II-Patrec method of evaluation of exact top-event probability is polynomial in the number of cut sets; it can, however, be weakly exponential in the worst case. Either way, this method results in a substantial reduction in computation requirements compared to the inclusion-exclusion method. The algorithm is described through an example problem. The results of several experiments with large accident sequence are also presented.

Optimization of test and maintenance intervals based on risk and cost

A general procedure is presented for optimizing the test and maintenance intervals of safety related systems and components. The method is based on minimizing the total plant-level cost under the constraint that the total accident frequency (risk) remains below a set criterion. The measure of risk is the time-average value of an accident rate. The probabilities of component failures and other basic events are linear functions or inversely proportional to the test or maintenance intervals. Human errors and common cause failures are included in the formalism. Analytical results are obtained for single components and simple systems while a numerical procedure is given for obtaining optimal intervals for complex plants with multiple systems and initiating events.