*Treatment of Uncertainties in Probabilistic Risk Assessment DOI: http://dx.doi.org/10.5772/intechopen.83541*

are not exactly known because of the lack of data, variability of plants,

• Modeling uncertainty—inadequacy of conceptual, mathematical, numerical,

• Uncertainty about completeness—systematic expert reviewing can minimize

The main focus of this work is the treatment of uncertainties regarding numerical values of the parameters used in fault and event trees in the scope of PRA and their propagation in these models. If a probability density function (pdf) is provided for the basic events (e.g., normal, log-normal, or triangular), a pdf or confidence bounds can be obtained for an FT top event or an ET scenario sequence.

There are several available methods for propagating uncertainties such as analytical methods (method of moments and Fenton-Wilkinson (FW) method), Monte Carlo simulation, Wilks method (order statistic), and fuzzy set theory. They are different from each other, in terms of characterizing the input parameter uncertainty and how they propagate from parameter level to output level [16].

The analytical methods consist in obtaining the distribution of the output of a model (e.g., fault or event trees) starting from probability distribution of input parameters. An exact analytical distribution of the output however can be derived

The method of moments is another kind of analytical method where the calculations of the mean, variance, and higher order moments are based on approximate models (generally using Taylor series). As the method is only an approximation, when the variance in the input data are large, higher order terms in the Taylor expansion have to be included. This introduces much more complexity in the analytical model, especially for complex original models, as in the case of PRAs [19]. The Monte Carlo simulation estimates the output parameter (e.g., probability of the top event of an FT) by simulating the real process and its random behavior in a computer model. It estimates the output occurrence by counting the number of times an event occurs in simulated time, starting to sample the pdf from the input

The fuzzy set theory is used when empirical information for input data are limited and probability theory is insufficient for representing all type of uncertainties. In this case, the so-called possibility distributions are subjectively assigned to input data, and fuzzy arithmetic is carried out. For uncertainty analysis in FTAs, instead of assuming the input parameter as a random variable, it is considered as a

The Wilks method is an efficient sampling approach, based on order statistics,

which can be used to find upper bounds to specified percentiles of the output

fuzzy number, and the uncertainty is propagated to the top event [21].

only for specific models such as normal or log-normal distributions [17]. The Fenton-Wilkinson (FW) method is a kind of analytical technique of approximating a distribution using log-normal distribution with the same moments. It is a moment-matching method for obtaining an exact analytical distribution for the output (closed form). This kind of closed form is helpful, when more detailed uncertainty analyses are required, for instance, in parametric studies involving uncertainty importance assessments, which require re-estimating the overall

the difficulties in assessing or quantifying this type of uncertainty.

processes or components, and inadequate assumptions.

**3. Methods of uncertainty propagation used in PRA**

and computational models.

*Reliability and Maintenance - An Overview of Cases*

uncertainty distribution many times [18].

data [20].

**130**

distribution. Order statistics are statistics based on the order of magnitudes and do not need assumptions about the shape of input or output distributions. According to the authors' knowledge, this method has been of little use in the field of reliability modeling and PRA, although it is used in other aspects of NPP safety, such as uncertainty in input parameters associated with the loss-of-coolant accident (LOCA) phenomena [22].

The mentioned methods for uncertainty propagation have many differences and similarities, advantages and disadvantages, as well as benefits and limitations. **Table 1** summarizes a comparison of these methods.

A brief discussion about the comparison of the mentioned methods is given as follows.

The method of moments is an efficient technique that does not require the specification of the probabilistic distributions of the basic event probabilities. It is difficult to be applied to complex fault trees with many replicated events [23]. This can be solved with the use of computer codes that automatically get the minimal cut sets (MCSs) of the fault trees. It is a simple method, easily explainable and suited for screening studies, due to inherent conservatism and simplicity [24].

The Monte Carlo simulation is computationally intensive for large and complex systems and requires pdf of input data. It has the disadvantage of not readily revealing the dominant contributors to the uncertainties. With current computer technology and availability of user-friendly software for Monte Carlo simulation, computational cost is no longer a limitation.

The fuzzy set theory does not need detailed empirical information like the shape of distribution, dependencies, and correlations. Fuzzy numbers are a good representation of uncertainty when empirical information is very scarce. It is inherently conservative because the inputs are treated as fully correlated [25].

The Fenton-Wilkinson (FW) method improves the understanding of the contributions to the uncertainty distribution and reduces the computational costs involved, for instance, in conventional Monte Carlo simulation for uncertainty


#### **Table 1.**

*Comparison of methods for uncertainty propagation.*

estimation. It is applicable only when the uncertainties in the basic events of the model are log-normally distributed. FW estimates are most accurate in the central range, and the tails of the distributions are poorly represented. The Wilks method requires relatively few samples and is computationally inexpensive. It is useful for providing an upper bound (conservative) for the percentiles of the uncertainty distribution. However, its calculated values are less accurate than the FW estimates over practically the entire range of the distribution. For both Wilks and FW methods, the greatest errors are found in the low tails of the distributions, but in almost all reliability applications the high tails are of more interest than the low tails [26].

*C*0 *<sup>i</sup>* <sup>¼</sup> *si* 1 � *Pi*

given by Eq. (8):

"AND" gate.

event tree.

**133**

(15), respectively:

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …*, n*) input, *n* is the number of input events, and *sor* is the standard deviation of the output of "OR" gate. For an "AND" gate of a fault tree, the probability of output event, *Pand*, is

*Pand* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup>*

where *Pi* denotes the probability of ith (i = 1, 2, 3, …, *n*) independent events

The uncertainty propagation through the "AND" gate is given by Eq. (9). It calculates the coefficient of variation of output, *Cand*, as function of the coefficients

*and* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup>*

*i*¼1

*Cand* <sup>¼</sup> *sand Pand*

> *Ci* <sup>¼</sup> *si Pi*

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …, *n*) input, *n* is the number of input events, and *sand* is the standard deviation of output of the

Uncertainty propagation in an event tree is similar (or analogous) to uncertainty propagation of an "AND" gate of a fault tree. The frequency of occurrence of each

*i*¼1

*Fseq* <sup>¼</sup> *<sup>λ</sup>* � <sup>Y</sup>*<sup>n</sup>*

where λ is the frequency of occurrence of the initiating event and *Pi* denotes the probabilities of ith (i = 1, 2, 3, …, *n*) subsequent independent events leading to the accident scenario and *n* is the number of input events. These values can be obtained from fault trees constructed for each ith event or system failure of the

The uncertainty propagation through the accident sequence is given by Eq. (13) that provides the coefficient of variation of accident sequence, *Cseq*, as function of the coefficients of variation of subsequent events, *Ci*, according to Eqs. (14) and

> <sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> *i*

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

*seq* <sup>¼</sup> <sup>Y</sup>*<sup>n</sup> i*¼1

*Cseq* <sup>¼</sup> *sseq Fseq*

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup> *i*

(or basic events) and *n* is the number of input events.

*Treatment of Uncertainties in Probabilistic Risk Assessment*

*DOI: http://dx.doi.org/10.5772/intechopen.83541*

**4.2 Method of moments applied to ETA**

accident scenario, *Fseq*, is given by Eq. (12),

of variation of inputs, *Ci*, according to Eqs. (10) and (11) [29]:

<sup>1</sup> <sup>þ</sup> *<sup>C</sup>*<sup>2</sup>

*i*¼1

*,* (7)

*Pi,* (8)

� �*,* (9)

*,* (10)

*,* (11)

*Pi,* (12)

� �*,* (13)

*,* (14)
