Calculation of Probability and Expected Frequency of the Undesired Events from Minimal Cut Sets

After finding the minimal cut sets the procedure may be continued with fault tree evaluation, e.g. calculation of the expected frequency of the undesired event (Top event). This is calculated by forming the expectation of the structure function given in Eq. F-1. With eliminating of the powers of binary variables in the Eq.F-1, which are equal to the binary variables themselves (law of idempotencies):

The general relationship of the structure function can be described as:

y(x) = Sk -ZZkxj +S S SkixJxk + • +(-1)L-1-k1-'* k (F-5)

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. 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. F-3 is evaluated for each of them separately using unavailabilities for the basic events [86]. The corresponding expected frequency of the undesired event is obtained by multiplying the unavailability with the frequency of the initiating event.

Uncertainties of the reliability data are propagated through the fault tree. This is performed by a Monte Carlo calculation, with the used of a lognormal distribution, described in the following section. Based on the Monte Carlo simulation, the program calculates a failure rate (and analogous to unavailability) using the following equation [86]:

Where zp and vp are random numbers uniformly distributed 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, e.g. for the unreliability according to:

With the standard deviation:

Where, P is the total number of trials and qs,p is the unreliability of the system calculated in trial p.

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