Appendix A. Example from practice–calculation of risk importance measures by a PRA computer code

This section describes, as an example “from the practice”, how the risk importance measures for basic events are calculated by RiskSpectrumÒ. Similar examples can be found with other PRA tools. RiskSpectrumÒ calculates the following importance measures for the basic events [39]: “Fussel-Vesely importance”, “risk decrease factor (RDF)”, “fractional contribution”, “risk increase factor (RIF)”. (The quotation marks are used to point out RS terminology.) Additionally, “sensitivity calculations” are done which can be related to the reliability importance as discussed in this monograph.

Using the symbols from the Theory Manual [39], the “Fussel-Vesely importance” for the basic event “i” is calculated as:

where QTOP refers to top event probability and the term QTOP (MCS including i) refers to the top event probability based only on the minimal cutsets which include basic event “i”.

The RDF (which corresponds to the RRW) is calculated as:

where Qi refers to the probability of basic event “i”. The term QTOP (QTOP = 0) represents the top event probability recalculated with probability Qi set to zero.

The “fractional contribution” is calculated from RDF as:

It is noted that this equation corresponds to (5-18). Basically, the measures “Fussel-Vesely importance” and “fractional contribution”, as applied to the basic events, have the same meaning. They are distinguished because RiskSpectrumÒ calculates them in different ways. The first one is calculated from actual contribution, the second one from its complement. In practical applications, calculated values are the same or very close to each other. (This would remain to be the case as long as all the basic events included in minimal cutsets have small probabilities.)

Finally, the RIF (which corresponds to the RAW) is calculated as:

where the term QTOP(Qi = 1) refers to the top event probability recalculated with probability Qi set to 1.

“Sensitivity calculations” for basic event “i” are done by dividing and multiplying the probability Qi by a sensitivity factor and recalculating the corresponding top event probabilities. (Sensitivity factor can have any value larger than 1, with default set to 10.) Calculated, also, is the “sensitivity value” defined as:

where QTOP,U,i and QTOP,L,i refer to top event probability recalculated with Qi multiplied and divided by the sensitivity factor, respectively. Reported are all three values, Si, QTOP,U,i and QTOP,L,i. It is noted that these “sensitivity calculations” can be related to the reliability importance discussed in this monograph as:

where σ refers to the “sensitivity factor”.

The importance measures, IiR, IiF and IiI can be calculated for different basic event groups in the same way as for individual basic events, where the probabilities are set to zero or to 1 for all basic events which belong to the group.

The importance measures, IiR, IiF and IiI are also calculated for the parameters which are used to calculate basic events probabilities: probabilities of failure on demand, failure rates, frequencies, repair rates (mean times to repair), test intervals and mission times. The formulas correspond to the above:

The term QTOP,BEST refers to the top event probability recalculated with parameter set to the “best theoretically possible” value (zero for all parameters). The term QTOP,WORST refers to the top event probability recalculated with parameter set to the “worst theoretically possible” value (1 for probability of failure on demand, the infinity for all others). The IiF is calculated from IiR in the same manner as above.