Reliability estimation of passive systems using fuzzy fault tree approach

Abstract

In general, a power plant (nuclear, thermal, chemical etc.) consists of operating and emergency safety systems. These systems vary from very complex to simple systems. A system normally consists of active components and passive components. The failure of any operating system will lead to a change in the state of the plant. The availability of the plant depends on the successful operation of the operating systems and the operation of the components in the systems. In order to ensure the availability of the plant reliability of the systems/components should be ensured. In recent years most of the advanced nuclear reactors implement passive systems, aimed at improved safety and availability. In the traditional reliability analysis of passive systems the failure probability is estimated based on the actual components present in the system and their corresponding failure data information. However, the passive system may fail to fulfill its mission not only because of a consequence of classical mechanical failure of component (passive or active) of the passive system, but also due to the deviation from expected behavior due to physical phenomena mainly related to thermal hydraulic (called as virtual component, VC). Hence, one should consider the failure probability of the VC in the analysis. In this paper a methodology for performing passive system reliability, which combines the actual component failures and the failure of VC, has been proposed based on fuzzy fault tree approach. This methodology will eliminate the simulation based approach that is being adopted in the present day passive system reliability analysis. The methodology has been demonstrated with a case study on passive decay heat removal system of a typical nuclear power plant.

Appendix

1.1 Probability ↔ possibility transformations

Consider a random variable X which follows normal distribution with mean and standard deviation as μ and σ respectively. The density function of the variable is given as

$$ f_{X} \left( x \right)\,=\,\frac{1}{{\sigma\sqrt {2\pi}}}e^{{\left[{-\frac{1}{2}\left({\frac{x-\mu }{\sigma }} \right)^{2} } \right]}} \quad -\infty\,<\,x\,<\,+\infty $$

(A-1)

CDF of X can be given as:

$$ F_{X} (x)\,=\,\int\limits_{-\infty }^{x} {f_{X} \left( x \right)dx}\,=\,\Upphi \left( {\frac{x\,-\,\mu }{\sigma }} \right) $$

(A-2)

In the fuzzy membership function the maximum membership value is 1. For the normal distribution f _max will occur at mean value. Hence, the fuzzy membership function for the normal distribution can be given as follows:

$$ \begin{aligned} f_{\max } &\,=\,\frac{1}{{\sigma \sqrt {2\pi}}} \\ M_{X} \left( x \right) &\,=\,\frac{f}{{f_{\max }}}\,=\,e^{{\left[{-\frac{1}{2}\left( {\frac{x - \mu }{\sigma }} \right)^{2}} \right]}} \quad -\infty\,<\,x\,<\,+\infty \\ \end{aligned} $$

(A-3)

In general, for any distribution one can divide the range of the parameter into very small intervals and one can find the probability of occurrence of the parameter in each interval as follows:

$$ P(x_{i}\,\le\,X\,\le\,x_{i}\,+\,\Updelta x)\,=\,F(x_{i}\,+\,\Updelta x)\,-\,F(x_{i})\,=\,\Updelta F_{i} $$

(A-4)

Find the ΔF_max from the above information. Now membership function for each interval can be written as follows:

$$ \begin{aligned} M_{i} &\,=\,\frac{{\Updelta F_{i}}}{{\Updelta F_{\max}}} \\ M_{i} &\,=\,\frac{1}{{\Updelta F_{\max}}}\Updelta F_{i}\,=\,Cp_{i} \quad i\,=\,1, 2, \ldots n \\ \end{aligned} $$

(A-5)

In the above expression C is a constant depends on the distribution. The above expression can be used to transform possibility to probability distribution as follows:

$$ \begin{gathered} p_{i}\,=\,\frac{1}{C}M_{i} \quad i\,=\,1, 2 \ldots n \hfill \\ {\text{where}} \hfill \\ C=\,\sum\limits_{i\,=\,1}^{n} {M_{i}} \;{\text{and}} \hfill \\ \sum\limits_{i = 1}^{n} {p_{i}}\,=\,1 \hfill \\ \end{gathered} $$

(A-6)