Y x ZZk Z Z Z Kj Lk kL

An upper bound of the occurrence probability of the system is equal to the expectation of the first term on the right side of the Eq.3, a lower to that of the first two terms, a close upper if the three terms are evaluated, etc. Since the used probabilities are usually small numbers, in most cases the evaluation of the upper bound is a satisfactory approximation to the true result. If initiating events are taken into account, the Eq. 3 is evaluated for each of them separately using unavailability for the basic events. The corresponding expected frequency of the undesired event is obtained by multiplying the unavailability with the frequency of the initiating event.

For a stand-by component subjected to maintenance, the unavailability of the component (basic event) is given Eq. F-11 (Appendix F). By integrating the Eq.F-11 over the time interval between two inspections, the average unavailability is given by:

Where, dt is the time between inspections, T is the mean time to failure for components i. It is the inverse of the failure rate, l, i.e. X=1/Tt. If the component is not an object of maintenance its reliability and unreliability coincide, i.e., q;(t)=u;(t).

Uncertainties of the reliability data are propagated through the fault tree by a Monte Carlo calculation, where lognormal or rectangular distribution can be used. The corresponding probability density functions (pdf) for failure rate (l) (and analogous for unavailability, u) is given in Eq. F-14 (Appendix F). Based on the Monte Carlo methods, the program calculates a failure rate (and analogous for an unavailability) using the following equation:

Where Zp and Vp are random numbers uniformly distribute in the interval 0, 1. Several trials are carried out from which the mean value and standard deviation of the probability of occurrence of the undesired event are calculated, in Eq. F-7 and Eq. F-8 (Appendix F), respectively. Equipment Reliability Data

Evaluation of a fault tree requires the input of reliability data such as failure frequencies and probabilities for technical components and human actions as well as frequencies of occurrence for initiating events including external events. Mathematical description behaviour of component, reliability data on process plant, human error, and its uncertainties are described in Appendix F. The reliability data used in FTA are mostly a mixture of plant-specific data, generic data, and estimates.

An ideal situation is to have valid historical data from identical equipment in the same application. But in most cases, plant-specific (e.g. hydrogen plants) data are unavailable, because of the limited historical database of equipment failures. Only a small number of hydrogen technologies, systems and components are currently in operation. To overcome these problems generic failure rate data as surrogates for or supplements to plant-specific data have been used in the study. Because of the uncertainties inherent in risk analysis methodology, generic failure rate data are frequently adequate to identify the major risk contributors in a process or plant [8].

However, selecting appropriate generic data requires understanding and judgment. In many cases, the analyst can find a number of generic data points that might be used for a QRA. Data points chosen for use must provide the level of confidence necessary without creating an unacceptable tolerance uncertainty [2]. The uncertainties of data selection can be reduced by learning as much as possible about data sets, including the taxonomy and equipment boundaries used; the type, design, and construction of the equipment; the process medium; plant operation and maintenance programs; and failure modes. After data have been selected and combined with other generic data or plant-specific data to a single data point, judgment must still be exercised in their use. The analyst may choose to use the generic data directly if the equipment description, process conditions, and failure mode of the data sources are similar to the equipment being studied [8].

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