Failures analysis of systems modeled by Mixed Fault Trees

Sahbi Ghachem, Ayachi Errachdi, Kamel Benothman and Mohamed Benrejeb Unit of research LARA Automatic National School Engineers of Tunis

BP37, le Belvédère, 1002 Tunis, Tunisia sahbi.ghachem@yahoo.fr, errachdi_ayachi@yahoo.fr, kamel.benothman@enim.rnu.tn, Mohamed.benrejeb@enit.rnu.tn

Abstract— This paper presents a comparative study of availability evaluation of systems modeled by a Mixed Fault Tree (MFT). This MFT contain a static sub-trees and dynamic sub-trees. These sub-trees can be independent or dependent. To evaluate the availability of a system from its fault tree, three methods are applied here: the first uses the Classic Bayesian Networks (CBN) for static sub-trees and Dynamic Bayesian Networks (DBN) for dynamic sub-trees, the second being proposed, bases on the Binary Decision Diagrams (BDD) for static sub-trees and Dynamic Bayesian Networks (DBN) for dynamic sub-trees and the third being proposed uses the Binary Decision Diagram (BDD) for static sub-trees and Markov Chains (MC) for dynamic sub-trees . The three methods are compared in both cases; independence and dependence of sub-trees.

Keywords— Fault Tree, Binary Decision Diagram, Bayesian Network, Markov Chain, functional dependence.

I. INTRODUCTION The improvement of industrial products quality requires

an increased reliability of production systems. The desire to have a high reliability of a system is one of the principal objectives of advanced technologies in particular into aerospace. There are several availability evaluation techniques of industrial systems, like Fault Tree (FT), Consequences Tree (CT), Success Diagram (SD) and Failures Modes Effects Analysis (FMEA), etc [8, 9]. In this paper, we are interested in Fault Tree (FT), tool very much used for modeling and systems evaluation which having the static and/or dynamic aspect. Three types are distinguished: Static Fault Tree (SFT) [8], Dynamic Fault Tree (DFT) [4, 6] and Mixed Fault Tree (MFT) [13]. When the Fault Tree size is significant and complex, the availability evaluation of system is very difficult to determine. For that, reduction techniques of Trees, which hold in account the various relations between the events, are

used. Among these techniques we quote, Binary Decision Diagrams (BDD) [7], Bayesian Networks (BN) [3] and Markov Chains (MC) [5]. The BDD are generally used for Static Fault Trees (SFT) [7, 10, 11], Markov Chains (MC) for Dynamic Fault Trees (DFT) [1, 5] and Bayesian Networks (BN) for two types of trees (Classic Bayesian Networks (CBN) and Dynamic Bayesian Networks (DBN)) [2, 3]. In this paper, we are interested in evaluation of Mixed Fault Trees. Three methods of availability evaluation are applied to a same example. The First is containing Bayesian Networks (CBN and DBN) [12], the second which we propose use BDD and DBN and the third use BDD and MC. The evaluations are carried out in cases of independence and dependence of basic events and sub-trees.

II. EXAMPLE OF MIXED FAULT TREE

The example of fig.1 consists of two sub trees; static sub tree and a dynamic sub tree. The static sub tree has a functional dependency; A depends on B and C depends on B. It has also a repeated event (B). The dynamic sub tree consists of three dynamics gates (CSP, AND, PAND), it has a functional dependency; E depends on D, G depends on F and I depend on H.

Fig. 1. Mixed Fault Tree

III. THE MFT EVALUATION BY BAYESIAN NETWORKS Mixed Fault Tree evaluation generally uses the

Bayesian Networks methods. For static sub-tree, the CBN method is employed [3, 12]. For dynamic sub-tree, the DBN method is used [2, 12]. A. Static sub-tree evaluation by CBN

The CBN associated to static sub-tree is given by fig.2.

Fig. 2. CBN associated to static sub-tree

With P(A)=0.4, P(B)=0.2, P(C)=0.1, P(C/B)=0.45. The occurrence probability of event E5 is:

P(X)=P(E5)=0.0468 (1)

B. Dynamic sub-tree evaluation by DBN The DBN associated to dynamic sub-tree is presented in

fig.3. The failure rate of each basic event (D, E, F, G, H and I) is shown in table I.

Fig. 3. DBN associated to dynamic sub-tree

TABLE I. FAILURE RATE OF BASIC EVENT

Basic event Failure rate (10-3)/hour

D 4

E 4

F 5

G 5

H 5

I 1

Table II present the various occurrence probabilities of event E6 for different time values.

TABLE II. OCCURRENCE PROBABILITY OF EVENT E6

C. MFT evaluation The equivalent MFT is presented by fig.4.

Fig. 4. Equivalent Mixed Fault Tree

The occurrence probabilities of events X and Y are:

P(X)=P(E5)=P1=0.0468, P(Y)=P(E6)=P2 (see table II)

The principal gate, which binds two events X and Y, is a dynamic gate OR; to calculate the occurrence probability of top event (TE), the DBN is used. The evaluation is carried out in the case of dependence and independence of sub-trees.

1) Sub-trees independence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table III.

TABLE III. OCCURRENCE PROBABILITY OF (TE) IN INDEPENDENCE CASE

2) Sub-trees dependence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table IV.

TABLE IV. OCCURRENCE PROBABILITY OF (TE) IN DEPENDENCE CASE

IV. MFT EVALUATION BY BDD AND DBN The recent methods of evaluation treat the Dynamic

Fault Trees (DFT) and Static Fault trees (SFT) separately. For MFT analysis, the proposed method consists in

evaluating static sub-trees by BDD [11] and dynamic sub-trees by DBN, in case of independence and dependence.

A. Static sub-tree evaluation by BDD The optimal BDD with quantitative aspect associated to

the static sub-tree is given in fig.5.

Fig. 5. BDD associated to static sub-tree

be =P(B), ce =P(C), c/be P(C/B),= bi P(B),=1- ci P(C),=1-

The exit probability of BDD is:

P(X)=P(E5)=P(C/B) P(B) =0.09⋅ (2)

B. Dynamic sub-tree evaluation by DBN The occurrence probabilities of event Y are calculated

previously in section III.B.

C. MFT evaluation The principal gate, which binds two events X and Y, is a

dynamic gate OR; thus we must use the DBN to calculate the occurrence probability of Top Event (TE). Fig.6 presents the equivalent Mixed Fault Tree.

Fig. 6. Equivalent Mixed Fault Tree

1) Sub-trees independence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table V.

TABLE V. OCCURRENCE PROBABILITY OF (TE) IN INDEPENDENCE CASE

2) Sub-trees dependence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table VI.

TABLE VI. OCCURRENCE PROBABILITY OF (TE) IN DEPENDENCE CASE

V. THE MFT EVALUATION BY BDD AND MC The proposed method consists in evaluating static sub-

trees by the BDD and dynamic sub-trees by the MC, in the case of independence and dependence.

A. Static sub-tree evaluation by BDD The occurrence probability of event E5 is calculated

previously in section IV.A.

B. Dynamic sub-tree evaluation by MC The MCs associated to CSP gate, PAND gate and AND

gate are presented respectively in fig.7, fig.8 and fig.9. The MC associated to dynamic sub-tree is presented in fig.10.

The name of each basic event in MC designed that event its in normal function (D, E, F, G, H, I) but if each event have this symbol (‘) that mean failure function (D’, E’, F’, G’, H’, I’).

The failure rate of each basic event (D, E, F, G, H and I) is shown in table I.

Fig. 7. MC associated to CSP gate

Fig. 8. MC associated to PAND gate

Fig. 9. MC associated to AND gate

Fig. 10. MC associated to dynamic sub-tree

Table VII present the various occurrence probabilities of event E6 for different time values. These probabilities are calculated by this formula:

D E F G H I-( ) tP(Y)=P(E6)=1-e λ λ λ λ λ λ+ + + + + (3)

TABLE VII. OCCURRENCE PROBABILITY OF E6

C. MFT evaluation

The principal gate, which binds two events X and Y, is a dynamic gate OR; thus we must use the MC to calculate the occurrence probability of Top Event (TE). Fig.11 presents the equivalent Mixed Fault Tree.

Fig. 11. Equivalent Mixed Fault Tree

The occurrence probabilities of events X and Y are:

P(X)=P(E5)=P1=0.0468, P(Y)=P(E6)=P2 (table VII)

1) Sub-trees independence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table VIII.

TABLE VIII. OCCURRENCE PROBABILITY OF (TE ) IN INDEPENDENCE CASE

2) Sub-trees dependence case: the various occurrence probabilities of Top Event (TE) for different time values, is presented in table IX.

TABLE IX. OCCURRENCE PROBABILITY OF (TE) INDEPENDENCE CASE

VI. CONCLUSION The occurrence probability of static sub-tree Top Event

(TE) calculated by BDD method is different from that calculated by CBN method, this is due to simplifications made by BDD method during the stage of modeling. The reduction of the size Fault Tree (FT) makes it possible to give a better availability evaluation.

In the independence case of sub-trees, the MFT evaluation by BN method (CBN+DBN) gave different results to those determined by proposed methods (BDD+DBN) and (BDD+MC). But, in the dependence case of sub-trees, the BN method (CBN+DBN) gave same results with those determined by proposed method (BDD+DBN), and different results to those determined by (BDD+MC). This shows the effectiveness of the Markov Chain (MC) and DBN to evaluate the availability of system in the dependence case.

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