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Mapping Fault Tree into Fuzzy Algorithms for Nuclear Safety Analysis HANY SALLAM1 MOSTAFA AREF2 2 Operation Safety and Human Factors Department, Computer Science Department 1 Nuclear and Radiological Regulatory Authority, 2Faculty of computer and information sciences, Ain Shams University Egypt
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Abstract: - In this paper a new method for mapping fault tree into fuzzy algorithms is introduced. This method is based on the minimal cut sets of a fault tree. This method is characterized by its one-to-one mapping from fault tree (FT) to fuzzy algorithm (FA) and vice versa. So it is consider a canonical method to map FTs into FA. Fault tree analysis (FTA) is a risk analysis technique to assess likelihood (in a probabilistic context) of an accident. The objective data available to estimate the likelihood is often missing, and even if available, is subject to incompleteness and imprecision or vagueness. Without addressing incompleteness and imprecision in the available data, FTA and subsequent risk analysis give a false impression of precision and correctness that undermines the overall credibility of the process. To solve this problem, qualitative justification in the context of failure possibilities can be used as alternative for quantitative justification. To this end, fuzzy reliability approach can be used as solution. While the estimation of failure probabilities of rare events with high consequences is the hot topic of nuclear power plants probabilistic safety assessment, the proposed method is used to estimate the possibility of fission products release from nuclear power plants due to core damage.
Key-Words: - Fault Trees, Fuzzy Algorithms, Minimal Cut Sets, and Nuclear Safety difficult to obtain component failure data, which are specific to the plant under investigation [6]. Also, many modern systems are highly reliable and thus, it is often very difficult to obtain sufficient statistical data to estimate precise failure rates or failure probabilities [7]. Accurate failure statistics is crucial requirement for reliability estimation in NPPs failure. In a situation, wherein failure data may not be corrected accurately due to various reasons, it is more practical to employ linguistic terms to express data value for failure of a particular event [8]. Moreover, the inaccuracy associated with system models due to human errors is difficult to deal with solely by means of the conventional probabilistic reliability theory [9]. These fundamental problems with probabilistic reliability theory have led researchers to look for new models or new reliability theories which can complement the classical probabilistic definition of reliability. Fuzzy set theory can be used to deal with this issue. Therefore, fuzzy fault tree analysis FFTA algorithm is developed to deal with such issues [10][11] . It seems to be impractical to assign a single fuzzy number to the failure possibility of the basic events in a fault tree analysis [12]. In present paper we have categorized the failure probability of basic events into three fuzzy sets high, medium, and low.
1 Introduction A fault tree analysis (FTA) is an analytical technique to estimate failure probability of an undesired event. It is widely used as a deductive tool for probabilistic safety assessment (PSA) to assess how likely undesirable events to occur or what their probabilities are [1]. In this technique, an undesired state of the system is specified then the system is analysed based on its environment and operation to find out all possible ways in which the predefined undesired event may occur [2]. Boolean logic gates are used to logically represent a failure scenario in a tree structure. Boolean algebra mathematically calculates the probability of the undesired event to occur [3]. FTA can be implemented only if the reliability data of all basic events constructing the tree are well known in advance [4]. Reliability data, which are directly taken from the plant being analysed, are the most appropriate sources. It assumes that exact probabilities of events are given and sufficient failure data is available. The expert judgment method uses direct estimation of probabilities by specialists [5]. Since the estimation of failure probabilities of rare events with the high consequences is the focus of the nuclear power plants NPPs PSA, it is often very
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A cut set in a fault tree is a set of basic events whose (simultaneous) occurrence ensures that the top event occurs. A cut set is said to be minimal if the set cannot be reduced without losing its status as a cut set. The top event will therefore occur if all the basic events in a minimal cut set occur at the same time. A minimal cut set is said to fail when all the basic events occur (are present) at the same time. This paper is organized as follows, section two is a mathematical model for fault tree and minimal cut sets. Section three is the proposed method modelling. Section four is the study case. And finally the conclusion.
r
Q t q j 1
j ,i
t
(3)
2 Fault tree Mathematical model FTA is the most commonly used technique used for causal analysis in risk reliability studies. A cut set in a FT is a set of basic events whose simultaneous occurrence ensures that the top event occurs [13]. A fault tree can be modelled by a set of AND gates and OR gates connecting between basic events and intermediate events [1], as shown in Fig. 1 and Fig. 2 respectively where:
Fig.2 OR gate connecting m basic events It is assumed that all the r basic events in the minimal cut set j are independent.
AND gate with m basic event is given by: m
Q t q (t ) 0
(1)
j
j 1
Fig. 3 Minimal cut set
Fig. 4 Fault tree represented by minimal cut sets
3 Proposed Method
Fig.1 AND gate connecting m basic events
Step1: each basic event is fuzzified into by assigning three linguistic fuzzy numbers, Low, Medium, and High. Step2: the fault tree is represented by a set of minimal cut sets as shown in Fig. 4. Step3: minimal cut sets MCS j are mapped into
And OR gate with m basic events is given by:
Q t 1 1 q t m
0
j 1
j
(2)
fuzzy rule Rule j in the following form:
A minimal cut set fails if and only if the basic events in the set fail at the same time as shown in Fig.. 3. The probability that the cut set j fails at time t is:
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Rule j : if BE1 and BE2 and ...BEn then TE Step 4: experts’ opinion are added to these rules in
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the form of linguistic values describing each basic event as shown in Fig. 6:
x max
coa ( D )
Rule j : if BE1 is low and BE2 is high
x.
D
( x)
x min x max
and ...BEn is low then TE is high
(6) D
( x)
x min
Step 5: fuzzy inference by using min-max implication. Step 6: defuzzification, to convert fuzzy numbers into fuzzy possibility score FPS which represents the most possibility that an expert believe in the occurrence of a basic event [2].
A
A
B 1
1
The advantage of this method is that the probability of the top event can be calculated for any combination of linguistic values of basic events (high, medium …). With this method the range of expectation of the top event is increased by mk . Where k is the number of basic events and m is the number of linguistic values assigned to a given basic event, assuming that all basic events have the same set of linguistic values.
B
(a) Sets Intersection
(b) Sets Union
Fig.5 Fuzzy Operation (t-norms and co-norms) Fuzzy reliability approach combines fuzzy set theory with failure possibility theory to assess a component failure. Failure possibility is used to evaluate a component failure in qualitative natural language. Defuzzification is used to convert membership functions into fuzzy possibility score FPS [1], where a crisp score that represents the degree of experts belief of the most likely score to indicate that an event may occur .
3.1 Fuzzy Inference Fuzzy numbers aggregation techniques are used to aggregate the basic events justification into one justification to calculate component failure probabilities. The min-max reasoning method is used [14]. This method is based on t-norms and co-norms which are defined as follows as shown in Fig. 5:
3.3 Fuzzy Failure Possibility (FFP) a- t-norms, fuzzy intersection (fuzzy AND) Let A and B be two fuzzy sets in X the intersection of A and B is the fuzzy set D A B with membership function [8][16]:
D x min[ A x , B x ]
x X
To ensure compatibility between real number and fuzzy FPS, fuzzy possibility score or defuzzified fuzzy number is transformed into fuzzy failure probability FFP [1] [2]:
(4)
1 FFP 10 k 0 where
b- Co-norms, fuzzy union (fuzzy OR) Let A and B be two fuzzy sets in X the union of A and B is the fuzzy set D A B with membership function [8][16]: D x max[ A x , B x ] x X (5)
1 FPS K FPS
3.2 Defuzzification
FPS 0 (7) 1
3
2.301
A fuzzy failure possibility is defined as an error rate, which is obtained by dividing the frequency of error with the total chance that an event may have error [18].
Defuzzification is the process of representing a fuzzy set with crisp number [15]. The most commonly used defuzzification method is the centre of area method also this method is known as the centroid method [2]. The centre of the fuzzy set D is determined by:
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FPS 0
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Fig.6 Mapping Fault Tree into Fuzzy Algorithm Fig. 7 Fault Tree for Fission Products Release
4 Case Study
The consequent of each rule is calculated by applying min operator for the antecedent’s fuzzy sets according to equation (4). This represents the degree of fulfilment of the antecedent and by which the consequent will be scaled. The consequents fuzzy sets are aggregated by the fuzzy operator max according to equation (5) the output of the aggregation is a fuzzy set. To get a crisp value, a defuzzification process is required. The defuzzification method used is the center of area, equation (6). This method determines the center of area of the fuzzy set and returns the corresponding crisp value which in our case is 0.0643 for basic events value BE1:0.05, BE2:0.0718, BE3:0.0631, BE4:0.025 and BE5:0.0213 as shown in Fig. 8. The results are obtained by using fuzzy tool of MATLAB.
Fission products release is assumed to be hazardous and have harmful effect to human and environment. The occurrence some events may cause fission products and consequently radiation release. The fault tree for such case is shown in Fig. 7 this fault tree is a part of the fault tree presented in [6].The basic events and intermediate events are shown in Table 1. It is assumed that each basic event is fuzzified by assigning three linguistic fuzzy numbers to each basic event, Low, Medium, and High as shown in Table 2. The minimal cut set for this fault tree are: BE1 , ( BE 2, BE 3 ), and ( BE 4, BE 5 ). These three minimal cut sets are mapped into three fuzzy rules after adding expert’s opinions to be in the following form:
Table 2 Basic Events and Top Event Fuzzification
R1: if BE1is Medium then TP is Medium
Event R 2 : if BE 2 is High and BE 2 is Medium then TP is High
Type
BE1 triangle
R3: if BE 4 is Medium and BE5 is Low then TP is Low BE2 trapezoidal
Table 1: Top Event, Intermediate Events, Basic Events TE I1 I2 I3 BE1 BE2 BE3 BE4 BE5
BE3 triangle
Release of fission products due to core damage Physical damage to the core Thermal damage to the core Mechanical damage to the core Explosive damage to the core Safety signal fails power fails Power fails insufficient thermal removal Reactor on
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BE4 trapezoidal
BE5 trapezoidal
TE
285
trapezoidal
Fuzzy numbers Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High Low Medium High
Fuzzy sets 0.046 0.05 0.054 0.065 0.07 0.075 0.075 0.08 0.085 0.045 0.05 0.055 0.06 0.055 0.06 0.07 0.075 0.072 0.08 0.085 0.093 0.032 0.042 0.052 0.042 0.052 0.062 0.050 0.061 0.065 0.012 0.02 0.024 0.032 0.02 0.03 0.036 0.046 0.034 0.04 0.042 0.048 0.07 0.1 0.12 0.15 0.18 0.2 0.25 0.31 0.23 0.32 0.57 0.41 0.031 0.35 0.046 0.056 0.046 0.056 0.069 0.089 0.069 0.089 0.094 0.099
Recent Advances in Information Science
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The last step in our method is the calculation of the fuzzy failure possibility FFP based on the value of fuzzy possibility score FPS. According to equation (7) the value of FFP is 0.0036 which represents the error rate.
[3] J. B. Dugan, “ Fault-Tree Analysis of ComputerBased Systems,” Annual Reliability and Maintainability Symposium, January 2001. [4] Renjith, VR and Madhu, G and Nayagam, Lakshmana Gomathi V and Bhasi, AB (2010) Twodimensional fuzzy fault tree analysis for chlorine release from a chlor-alkali industry using expert elicitation. In: Journal of Hazardous Materials, 183 (1-3). pp. 103-110 [5] Dokas, I.M., D.A. Karras, and D.C. Panagiotakopoulos, Fault tree analysis and fuzzy expert systems: Early warning and emergency response of landfill operations. Environmental Modeling and Software, 2009. 24(1): p. 8-25.
Fig. 8 Fuzzy implication and defuzzification
5 Conclusion In this paper a new method for mapping fault tree into fuzzy algorithms based on minimal cut sets is presented to overcome insufficient and imprecise data Associated with rare events with high consequences. As fuzzy algorithms have the advantage of reasoning and deciding with incomplete and imprecise data. Minimal cut sets are mapped into a fuzzy algorithm by converting each cut set into a fuzzy rule. These fuzzy rules are modified by expert’s opinions to calculate the top event value. This method used to calculate the probability of release of radioactive products as result of core damage. The results show that this method can be widely used in nuclear power plants probabilistic safety assessment. This method is characterized by its simplicity beside it overcomes the problems of in complete and imprecise data which are required for probabilistic calculation.
[6] Tyagi S.K, Pandey D., Tyagi R., “Fuzzy set theoretic approach to fault tree analysis”, International Journal of Engineering, Science and Technology, Vol. 2, No. 5, 2010, pp. 276-283. [7] Julwan Hendry Purba, Jie Lu, Guangquan Zhang, “Fuzzy Failure Rate for Nuclear Power Plant Probabilistic Safety Assessment by Fault Tree Analysis,” Computational Intelligence Systems in Industrial Engineering Atlantis Computational Intelligence Systems Volume 6, 2012, pp. 131-154. [8] B.S. Mahapatra and G.S. Mahapatra, “Intuitionistic fuzzy fault tree analysis using intuitionistic fuzzy numbers”, International Mathematical Forum, 5, 21(2010), 1015 – 1024.
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[12] Li YF, Huang HZ, Liu Y, Xiao N, Li H. A new fault tree analysis method: fuzzy dynamic fault tree analysis. Eksploatacja i Niezawodnosc – Maintenance and Reliability 2012; 14 (3): 208-214.
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