Probability Evaluation of Fault Trees

In the event that the probability of the occurrence of basic fault events is given, it is possible to calculate the probability of occurrence of the top fault event. This can only be calculated by first calculating the probability of occurrence of the output fault events of the intermediate and lower logic gates such as the OR and the AND gates. 

The probability of occurrence of the OR gate output fault event is expressed by [9] 

P Xj) = the probability of occurrence of AND gate input fault event Xy for  

When the probabilities of occurrence of primary/basic fault events are known, the occurrence probability of the top event can be calculated. This can only be achieved by first calculating probabilities of occurrence of the output fault events of all the intermediate and lower logic gates (e.g., AND and OR gates). 

A fault tree for the top fault event dark room. 

Similarly, the occurrence probability of the AND gate output fault event, B, is given by [3] 

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Answer Review the plant s design to determine how radioactive water could get from the plant to the river. Some ways are i) through the heat exchanger and through the condenser, ii) from the closed circuit water into the service water, iii) from the spent fuel storage pool, and iv) from the sump. Prepare fault trees or adapt existing fault trees to determine the probability of each of these release paths. Obtain reliability data for the components that are involved and evaluate the fault trees to determine the probability of each type of failure. For those pathways with a probabilit >7/y,... 

Tliis cliapter is concerned willi special probability distributions and tecliniques used in calculations of reliability and risk. Tlieorems and basic concepts of probability presented in Cliapter 19 are applied to llie determination of llie reliability of complex systems in terms of tlie reliabilities of their components. Tlie relationship between reliability and failure rate is explored in detail. Special probability distributions for failure time are discussed. Tlie chapter concludes with a consideration of fault tree analysis and event tree analysis, two special teclmiques lliat figure prominently in hazard analysis and llie evaluation of risk. 

HAZAN, on the other hand, is a process to assess the probability of occurrence of such accidents and to evaluate quantitatively the consequences of such happenings, together with value judgments, in order to decide the level of acceptable risk. HAZAN is also sometimes referred to as Probabilistic Risk Assessment (PRA) and its study uses the well-established techniques of Fault Tree Analysis and/or Event Tree Analysis ... 

Table 9.22 contains the data for evaluating the fault trees of Fig. 9.44 and Table 9.23 the corresponding minimal cut sets and their evaluation in terms of probabilities. 

As stated earlier, the fault tree is a model of the system fault state. There are qualitative and quantitative tools to evaluate the tree. Qualitative analysis of fault trees is conducted through the use of cut sets and simple Boolean algebraic manipnlation. Trees are quantified by applying probabilities or frequencies of occurrence of each event fault. The event faults are then combined through Boolean manipulation, and the top-event probabUity is determined. You may wish to review a math book and become familiar with Boolean algebra and probability theory. The U.S. Nuclear Regulatory Commission s Fault Tree Handbook (Roberts et al., 1981) and NASA s Fault Tree Handbook with Aerospace Applications (Stamatelatos et al, 2002) are excellent references as well. 

Han, S.H. Lim, H.-G. (2012). Top event probability evaluation of a fault tree having circular logics by using Monte Carlo method. Nuclear Engineering and Design, 243(0) 336-340. 

Risk analysis is required to evaluate the accident frequency and consequences. In railway industry. Safety Risk Model (SRM) is used to estimate system risk, SRM consists of Fault Tree Analysis (FTA) and Event Tree Analysis (ETA). Fault tree estimates accident frequency considering system failure logic (Muttram 2002). It calculates top event frequency or probability using minimal cut sets. Basic events in fault tree describe the component failures they can model revealed repairable failure, revealed unrepairable failure and unrevealed repairable failure with periodic inspection (Andrews Moss 2002). The above failure models for basic event are not enough to consider the effects of maintenances on risk as these models cannot describe multi-level repairsor inspections in details. 

The same probabilistic model checking techniques are used for computing the probability of the top-level event in fault trees. They can be extended to computing probabilities for dynamic fault trees [6]. Akin to checking the correctness of FDIR measures, we use the same probabilistic techniques to evaluate FDIR performance. For example, in addition to checking whether a fault is detected or not, we compute the probability of detection in case of fault recovery, we compute the probability that the system will recover from a fault. 

A fault tree is a graphical form of a Boolean equation, but the probability of the top event (and lesser events) can be found by substituting failure rates and probabilities for these iwo-staie events. The graphical fault tree is prepared for computer or manual evaluation by pruning" it of less significant events to focus on more significant events. Even pruned, the tree may be so large that it IS intractable and needs division into subtrees for separate evaluations. If this is done, care must be taken to insure that no information is lost such as interconnections between subtrees. 

The accident sequence frequencies are quantified by linking the system fault tree models together as indicated by the event trees for the accident sequence and quantified with plant-specific data to estimate initiator frequencies and component/human failure rates. The SETS code solves the fault trees for their minimal cutsets the TEMAC code quantitatively evaluates ihe cm sols and provides best estimates of component/event probabilities and frequencies. 

In this study detailed fault trees with probability and failure rate calculations were generated for the events (1) Fatality due to Explosion, Fire, Toxic Release or Asphyxiation at the Process Development Unit (PDU) Coal Gasification Process and (2) Loss of Availability of the PDU. The fault trees for the PDU were synthesized by Design Sciences, Inc., and then subjected to multiple reviews by Combustion Engineering. The steps involved in hazard identification and evaluation, fault tree generation, probability assessment, and design alteration are presented in the main body of this report. The fault trees, cut sets, failure rate data and unavailability calculations are included as attachments to this report. Although both safety and reliability trees have been constructed for the PDU, the verification and analysis of these trees were not completed as a result of the curtailment of the demonstration plant project. Certain items not completed for the PDU risk and reliability assessment are listed. 

The use of event trees is sometimes limiting for liazard analysis because it may lack die capability of quantifying die potendal of die event occurring. Tlie analysis may also be incomplete if all inidal occurrences are not identified. Its use is beneficial in examining, rather dian evaluating, die possibilities and consequences of a failure. For this reason, a fault tree analysis (FTA) should supplement diis, to establish die probabilities of die event tree branches. Tliis topic was introduced in a subsection of Cliapter 16. 

Probability analysis Way to evaluate the likelihood of an event occurring. By using failure rate data for equipment, piping, instruments, and fault tree techniques, the frequency (number of events per unit time) can be quantitatively estimated. 

The starting point in event tree analysis is the initiating event. The quantitative evaluation of the event tree requires condition probabilities. These may be based on reliability data, historical records, experience, or from fault trees. 

For the initiating event, frequencies per operation year are determined, as a rule, by evaluation of statistics, or, if no experience is available, by justified assessments. The probability of non-availablities of safety installations or measures is determined by fault trees or immediately by values derived from operational experiences. The total frequencies of the event sequences are then calculated by multiplication of the frequency of the initiating event with those of the non-availabilities. 

This approach is illustrated by the development of event trees and fault tree analysis. In fault tree analysis, the probability of an accident is estimated by considering the probabihty of human errors, component failures, and other events. This approach has been extensively applied in the field of risk analysis (Gertman and Blackman 1994). THERP (Swain and Guttman 1983) extends the conditioning approach to the evaluation of human reliability in complex systems. 

While MORT is based on the fault tree method of system safety analysis, its logic diagram does not require statistical entries and computations for event probabilities. MORT is presented as an incident investigation methodology and as a basis for safety program evaluation. 

After analyzing the preceding two classes of failures those dependent failures which are due to a common (shared) cause (CCF) remain to be explained. The common cause may be a design or construction flaw or a maintenance error, e.g. unsuitable lubricants used for pump bearings. CCFs are introduced into the fault tree in addition to the independent failures of the components involved. Probabilities are assigned to them using model-based evaluations of operating experience. 

The quantification is done on the basis of a qualitative analysis, which is reflected by the fault tree of Fig. 11.3, and its evaluation in terms of probabilities. The fault tree of Fig. 11.3 has the following minimal cut sets... 

The quantitative evaluation of the fault tree is done by analyzing the structure of levels using the Boolean algebra to determine the minimum combinations of terminal events or minimum sets of faults that lead to the undesired event. Its probability is calculated based on the probabilities of the basic events and undeveloped events to be evaluated from available historical data. 

There exist different methods like Fault Tree Analysis (FTA), Event Tree Analysis (ETA) and Monte Carlo Simulation (MCS) that can be applied and combined for the purpose of evaluating the frequency and probability of initiating events. However, the MCS can be handled much easier in order to accoimt for bormdary conditions like stochastic dependence, time dependence and physical impact (Hauschild Meyna 2007). The MCS has been apphed successfully for PSA in order to assess the safety of nuclear power plants (Devooght Smidts 1996, Woltereck2001) and especially for taking into account uncertain input data (BfS 2005). 

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