**5. Illustrative examples**

In order to validate the proposed approach for implementing the method of moments, two cases were tested.

#### **5.1 Case study 1**

The first case, taken from Chang et al. [8], introduces a fault tree (**Figure 4**) describing a generic top event "system failure," T, with seven basic events (X(1) to *Treatment of Uncertainties in Probabilistic Risk Assessment DOI: http://dx.doi.org/10.5772/intechopen.83541*

*Ci* <sup>¼</sup> *si Pi*

where *si* denotes the standard deviations of ith (i = 1, 2, 3, …*, n*) subsequent event of the sequence, *n* is the number of input events, and *sseq* is the standard

Many uncertainty distributions associated with the basic events of fault trees (reliability or failure probability data) often can be approximated in reliability and safety studies by log-normal functions. If a random variable ln(*x*) has a normal distribution, the variable *x* has then a log-normal distribution. The log-normal

<sup>2</sup><sup>π</sup> <sup>p</sup> exp �ð Þ ln ð Þ� *<sup>x</sup>* <sup>μ</sup>

<sup>¼</sup> <sup>χ</sup><sup>50</sup> χ5

where μ and σ are the mean and the standard deviation of ln(*x*), respectively

2σ<sup>2</sup> !

2

*EF* ¼ exp 1ð Þ *:*645 σ *:* (18)

*,* (16)

*,* (17)

*,* (19)

probability density function (pdf), *f(x)* is then given by Eq. (16) [30]:

*x*σ ffiffiffiffiffi

(i.e., these are the parameters of the "underlying" normal distribution). The error factor, *EF*, of a log-normal pdf is defined as Eq. (17):

> *EF* <sup>¼</sup> <sup>χ</sup><sup>95</sup> χ<sup>50</sup>

where χ95, χ50, and χ<sup>5</sup> are the 95th, 50th (median), and 5th percentiles,

*EF* is often used as an alternative to the standard deviation of "underlying" normal distribution, σ, for characterizing the spread of a log-normal distribution,

The mean, *P*, and standard deviation, *s*, of the log-normal variable, *x*, can be

σ2 2 � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2μ þ σ<sup>2</sup> ð Þ exp σ<sup>2</sup> ½ � ð Þ� 1

*P* ¼ exp μ þ

. (20)

Eqs. (4)–(20) are used for uncertainty propagation of log-normal pdf in fault

In order to validate the proposed approach for implementing the method of

The first case, taken from Chang et al. [8], introduces a fault tree (**Figure 4**) describing a generic top event "system failure," T, with seven basic events (X(1) to

*f x*ð Þ¼ <sup>1</sup>

and these two quantities are related by Eq. (18):

given by the following Eqs. (19) and (20), respectively:

*s* ¼

and event trees, as illustrated in the following examples.

**5. Illustrative examples**

**5.1 Case study 1**

**134**

moments, two cases were tested.

q

deviation of the accident sequence.

respectively.

**4.3 Propagation of log-normal distributions**

*Reliability and Maintenance - An Overview of Cases*

*,* (15)

X(7)), characterized by the log-normal distributions. This simple example was chosen in order to compare the results of the method of moments with the uncertainty propagation analyses using Monte Carlo simulation.

The log-normal distributions assigned to the basic events (represented by median and mean values of probabilities, error factors, and standard deviations) are shown in **Table 2**. An analysis of the fault tree shows that its minimal cut sets (MCSs) are X(1), X(6), X(7), X(2)X(4), X(2)X(5), X(3)X(4), and X(3)X(5), which are used to estimate the top event probability and propagate the uncertainties. The application of the method of moments is carried out in a bottom-up approach. Starting from basic events of the fault tree, the coefficients of variation of the intermediate events are estimated using Eqs. (4)–(7) for "OR" gates and Eqs. (8)–(11) for "AND" gates. This procedure is repeated interactively until the top event is reached, and its standard deviation is obtained. Considering that, in the same way as the basic events, the top event has also a log-normal distributions, Eqs. (16)–(20) are used to estimate the 5th percentile, median, and 95th percentile for the top event, as shown in **Table 3**. These estimates are slightly lower than the values obtained by Chang et al. [8] with the Monte Carlo simulation (percent

**Figure 4.** *Fault tree analysis for a generic top event "system failure" (adapted from Chang et al. [8]).*


**Table 2.**

*Basic event distribution for a generic top event "system failure" (χ<sup>50</sup> and* EF *values were taken from Ref. [8]).*


current work, using the method of moments, with the analyses of uncertainty propagation using Wilks method, Monte Carlo simulation, and Fenton-Wilkinson

*Treatment of Uncertainties in Probabilistic Risk Assessment*

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

(FW) method.

**Figure 6.**

**Basic events**

**Table 4.**

**137**

*Fault tree analysis for a nuclear power plant core melt.*

**Mean of log-normal pdf** *(P)*

**Error factor of log-normal pdf (***EF***)**

<sup>A</sup> 6.00 <sup>10</sup><sup>2</sup> <sup>5</sup> 7.60 <sup>10</sup><sup>2</sup> <sup>B</sup> 6.60 <sup>10</sup><sup>6</sup> <sup>5</sup> 8.36 <sup>10</sup><sup>6</sup> <sup>C</sup> 1.00 <sup>10</sup><sup>2</sup> <sup>5</sup> 1.27 <sup>10</sup><sup>2</sup> <sup>D</sup> 2.13 <sup>10</sup><sup>3</sup> <sup>5</sup> 2.70 <sup>10</sup><sup>3</sup> <sup>E</sup> 8.33 <sup>10</sup><sup>4</sup> <sup>5</sup> 1.06 <sup>10</sup><sup>3</sup> <sup>F</sup> 5.20 <sup>10</sup><sup>5</sup> <sup>5</sup> 6.59 <sup>10</sup><sup>5</sup> <sup>G</sup> 6.10 <sup>10</sup><sup>5</sup> <sup>5</sup> 7.73 <sup>10</sup><sup>5</sup> <sup>H</sup> 4.20 <sup>10</sup><sup>5</sup> <sup>5</sup> 5.32 <sup>10</sup><sup>5</sup> <sup>I</sup> 1.58 <sup>10</sup><sup>3</sup> <sup>5</sup> 2.00 <sup>10</sup><sup>3</sup> <sup>J</sup> 1.00 <sup>10</sup><sup>4</sup> <sup>5</sup> 1.27 <sup>10</sup><sup>4</sup> <sup>K</sup> 9.00 <sup>10</sup><sup>2</sup> <sup>5</sup> 1.14 <sup>10</sup><sup>1</sup> <sup>L</sup> 1.00 <sup>10</sup><sup>1</sup> <sup>5</sup> 1.27 <sup>10</sup><sup>1</sup> <sup>M</sup> 1.20 <sup>10</sup><sup>4</sup> <sup>5</sup> 1.52 <sup>10</sup><sup>4</sup>

*Basic event distribution for illustrative example (P and EF values were taken from Ref. [26]).*

**Standard deviation of lognormal pdf (***s***)**

**Table 3.**

*Comparison of top event probabilities obtained by Monte Carlo simulation and by method of moments.*

#### **Figure 5.**

*Comparison of pdf obtained by method of moments and by the Monte Carlo simulation for the top event of Figure 4.*

difference less than 4%). This good agreement can also be verified through the probability density function (obtained with Eq. (16)), as shown in **Figure 5**.

#### **5.2 Case study 2**

The second case study illustrates the application of the method of moments for assessing the uncertainty of a fault tree taken from a probabilistic safety analysis of a nuclear power plant (NPP). The fault tree shown in **Figure 6** was constructed using MCSs and basic event distributions provided by El-Shanawany et al. [26]. It represents a fault tree analysis for the top event "nuclear power plant core melt," taking into account loss of off-site and on-site power systems and failure of core residual heat removal. The basic events A, B, C, D, E, F, G, H, I, J, K, L, and M are related to off-site power system failure, operator errors, emergency diesel generators (EDGs) failures, pump failures, and common cause failures (CCFs). A detailed description of each one of these basic events is given in the caption of **Figure 6**. An accurate logical analysis of this drawn fault tree can demonstrate that its MCSs are ABC, ABD, ABE, ABF, ABH, ABI, ABJ, AFG, and AKLMH, which describes the illustrative example analyzed in the literature.

The log-normal distributions assigned to the basic events (represented by mean values of probabilities, error factors, and standard deviations) are shown in **Table 4**. Such distributions are also used in Ref. [26], to compare the results of this current work, using the method of moments, with the analyses of uncertainty propagation using Wilks method, Monte Carlo simulation, and Fenton-Wilkinson (FW) method.

**Figure 6.** *Fault tree analysis for a nuclear power plant core melt.*


**Table 4.**

*Basic event distribution for illustrative example (P and EF values were taken from Ref. [26]).*

difference less than 4%). This good agreement can also be verified through the probability density function (obtained with Eq. (16)), as shown in **Figure 5**.

*Comparison of pdf obtained by method of moments and by the Monte Carlo simulation for the top event of*

**Method 5th percentile Median 95th percentile** Monte Carlo simulation<sup>1</sup> 4.15 <sup>10</sup><sup>3</sup> 8.02 <sup>10</sup><sup>3</sup> 1.64 <sup>10</sup><sup>2</sup> Method of moments<sup>2</sup> 3.99 <sup>10</sup><sup>3</sup> 7.95 <sup>10</sup><sup>3</sup> 1.58 <sup>10</sup><sup>2</sup> % difference 3.8% 0.9% 3.5%

*Reliability and Maintenance - An Overview of Cases*

*Comparison of top event probabilities obtained by Monte Carlo simulation and by method of moments.*

The second case study illustrates the application of the method of moments for assessing the uncertainty of a fault tree taken from a probabilistic safety analysis of a nuclear power plant (NPP). The fault tree shown in **Figure 6** was constructed using MCSs and basic event distributions provided by El-Shanawany et al. [26]. It represents a fault tree analysis for the top event "nuclear power plant core melt," taking into account loss of off-site and on-site power systems and failure of core residual heat removal. The basic events A, B, C, D, E, F, G, H, I, J, K, L, and M are related to off-site power system failure, operator errors, emergency diesel generators (EDGs) failures, pump failures, and common cause failures (CCFs). A detailed description of each one of these basic events is given in the caption of **Figure 6**. An accurate logical analysis of this drawn fault tree can demonstrate that its MCSs are ABC, ABD, ABE, ABF, ABH, ABI, ABJ, AFG, and AKLMH, which describes the

The log-normal distributions assigned to the basic events (represented by mean

**Table 4**. Such distributions are also used in Ref. [26], to compare the results of this

values of probabilities, error factors, and standard deviations) are shown in

**5.2 Case study 2**

**Figure 5.**

*Figure 4.*

**136**

*1 Ref. [8]. 2*

*Current work.*

**Table 3.**

illustrative example analyzed in the literature.

The application of the method of moments is carried out in a similar way as in the first case study. Considering that the top event is also log-normally distributed, its 5th percentile, median, and 95th percentile are estimated. As can be seen in **Table 5**, the median values of the method of moments show a good agreement with Wilks method and are 25.8% and 20.4% greater than the results of Monte Carlo simulation and FW method, respectively. This is also illustrated in **Figure 7**, where the cumulative distribution function obtained by method of moments is compared with the data in the mentioned literature [26]. As can be seen, the results of the method of moments agree reasonably with the Wilks method, being slightly lower, moving toward the analyses of uncertainty propagation using Monte Carlo simulation, which is considered for many purposes to be close to the exact solution for simple models.

probabilities [31], as well as matrix computations for obtaining the standard devia-

This work addresses the uncertainty propagation in fault and event trees in the

performing a systematic PRA uncertainty treatment of risks and system reliabilities associated with complex industrial facilities, mainly in risk-based decision-making. Two illustrative examples using the method of moments for carrying out the uncertainty propagation in fault trees are presented, and their results are compared

approaches. The method of moments proved to be conceptually simple to be used. It confirmed findings postulated in literature, when dealing with simple and small systems. More complex systems will require the support of specialized reliability and risk assessment software, in order to implement the proposed approach.

The authors would like to thank the following institutions, which sponsored this work: Development Center of Nuclear Technology/Brazilian Nuclear Energy Com-

The authors are the only responsible for the printed material included in this

Development Center of Nuclear Technology/Brazilian Nuclear Energy Commission,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

scope of probabilistic risk assessment (PRA) of industrial facilities. Given the uncertainties of the primary input data (component reliability, system failure probabilities, or human error rates), the method of moments is proposed for the evaluation of the confidence bounds of top event probabilities of fault trees or event

sequence frequencies of event trees. These types of analyses are helpful in

with available analyses in literature using different uncertainty assessment

mission (CDTN/CNEN) and Brazilian Innovation Agency (FINEP).

Vanderley de Vasconcelos\*, Wellington Antonio Soares, Antônio Carlos Lopes da Costa and Amanda Laureano Raso

\*Address all correspondence to: vasconv@cdtn.br

CDTN/CNEN, Belo Horizonte, Brazil

provided the original work is properly cited.

tions along the trees, as proposed by Simões Filho [32].

*Treatment of Uncertainties in Probabilistic Risk Assessment*

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

**6. Final remarks**

**Acknowledgements**

**Conflict of interest**

**Author details**

paper.

**139**

Overall, uncertainty propagation using the method of moments in fault trees, as shown in the two case studies, or in event trees, is quite simple in small systems and does not require the specification of probability density functions of basic events but only their means and standard deviations. For more complex systems and large fault and event trees, computer implementation of the described bottom-up approach can be performed, for instance, using specialized computer software for obtaining the minimal cut sets and quantitatively assessing the top event


#### **Table 5.**

*Comparison of core melt frequency obtained by the method of moments with data from literature.*

#### **Figure 7.**

*Comparison of cumulative distribution function for core melt frequency obtained by the method of moments with data from literature [26].*

probabilities [31], as well as matrix computations for obtaining the standard deviations along the trees, as proposed by Simões Filho [32].
