1. A fault tree – Based Bayesian network construction for the failure rate assessment of a complex system 46th ESReDA Seminar May 29-30, 2014, Politecnico di Torino, Torino, Italy Reliability Assessment and Life Cycle Analysis of Structures and Infrastructures’ El-Hassene Ait Mokhtar a, Radouane Laggoune a, Alaa Chateauneuf b a Research Unit “Modeling and Optimisation of Systems (LaMOS)”, University of Bejaia, Algeria b Pascal Institute, University of Blaise Pascal, France

2. Contents Introduction Problem position Complex systems modeling (FT  BN) Industrial application Conclusion & perspectives

3. Introduction Technological progress have made that modern systems have become more complex, especially regarding their structure. This made it difficult to estimate the reliability characteristics, in particular the failure rate of such systems. Simple systems - For series system, the failure of any component (element) leads to the whole system failure. - The failure of a parallel system is only caused by the failure of all its components. When it concerns a complex system, it is more difficult to evaluate the impact of the failure of one component because of the different interactions between its components. Definition (complex system) A complex system is a structured set of independent heterogeneous devices that are connected and communicate with each other in order to perform a function.

4. Introduction Several methods can be used for the reliability assessment of complex systems (systems with complex structure, configured as networks) such as: - Bayesian Network, - Petri Network, - Fault (Failure) Tree. Bayesian networks (BN) are native from artificial intelligence, they are very useful since they allow a quantitative and qualitative representation of the different links and dependencies between all the system components. The system structure, modeled as BN, shows clearly the conditional dependencies between the variables, whereas the conditional probabilities allow their quantification.

5. Introduction Bayesian networks and traditional methods used in reliability Unlike the fault tree method, the BNs permit to take into consideration the events with several consequences; Unlike the Markov Chain method, the BNs allow the modeling of systems configured in networks with several causalities and with a very large number of variables; Unlike the Petri networks method, the BNs Permit to treat the low frequency events (example: an accident is a low frequency event but it may have critical consequences on the system, the operators and/or environment.)

6. Problem position It was established that modeling of complex systems using Bayesian network is very expensive (CPU time). Therefore, we introduce the fault tree method, which is more explicit and less complicate to establish, in order to facilitate the Bayesian network construction.

7. Complex systems modeling To establish a Bayesian network, two major phases are required. The construction of the graph structure The evaluation of the probability tables These two phases need the use of important means. Some papers have shown that it is very difficult to define all the links and dependencies between the variables (components). For this reason we have introduced the fault tree method to establish the BN,

8. Complex systems modeling Mapping fault tree to Bayesian network The transition, ‘’Fault tree’’ to ‘’Bayesian network’’, is made thanks to a transition algorithm, which can be used if the following assumptions are established.  The events must be binary (example: working/not working);  The events are statistically independent;  The relations between the events are represented by logical gates;  The root of the FT is the undesired event to be analyzed.

9. Complex systems modeling Mapping fault tree to Bayesian network The transition steps from the FT to BN are: 1. For each leaf node (primary event or component) in the FT, create a root node in the BN. If more than one leaf on the FT represents the same event, create just one root node in the BN; 2. Assign to the root node in the BN the prior probability of the related leaf node in the FT; 3. For each logical gate in the FT, create a node in the BN; 4. Connect the nodes in the BN as the corresponding nodes are connected in the FT; 5. For each logical gate in the FT, assign the conditional probability tables to the related nodes in the BN.

10. Complex systems modeling AB FBA F ABF 000 011 101 111 Logical gate ‘’OR’’

11. Complex systems modeling AB F BA F ABF 000 010 100 111 Logical gate ‘’AND’’

12. Complex systems modeling Creation of the probability tables For the logical gates, it is very simple to create their corresponding CPT. But for the primary events or components it is not as simple as the gates. In fact, to create prior probabilities we need to transform the failure rate used for FT into a probability in BN.

13. Industrial application For the validation purpose, this methodology is applied to an industrial system. It consists of a turbo-pump located in a pumping station of the Haoud-El-Hamra-Bejaia pipeline (Algerian company of petroleum SONATRACH). Group turbo-pump

14. Industrial application Fault tree of the studied system

15. Industrial application Bayesian network of the studied system According to the maintenance department of the company, it is to note that the “breakdown state” of the turbo-pump is caused by three (03) main causes: the failure of one of its components (according to the fault tree of the system), the operator competency (OP) and the external environment like the climate conditions (AC). Initial Bayesian network

16. Industrial application Bayesian network of the studied system

17. Industrial application Probabilities of the Bayesian network The prior probability tables of the root nodes are created from the failure rate of the corresponding component, knowing that: For the transformation of the failure rate into a failure probability we have taken (with the collaboration of the experts of the maintenance department) equal to one (01) failure per two (02) days (0.5 failures per day). With: MUT is the Mean Up Time

18. Industrial application Failure probability of the nodes Node Failure probability Non failure probability A10,186%99,814% A20,086%99,914% AT0,186%99,814% GT0,186%99,814% PP0,374%99,626% VV0,28%99,72% CC0,186%99,814% CY0,374%99,626% MS0,374%99,626% BR0,086%99,914% GP0,108%99,892% SR0,054%99,946% DCS0,186%99,814% OP0,5%99,5% CD0,8%99,2% Conditional probability tables (CPTs) The CPTs of the nodes “NP”, “TB” and “DP” are obtained from the transition of the logical gates “AND” and “OR”. For the node “S” which represents the system, its CPT is estimated with concordance of the maintenance department. ACTPOP Node "S" 01 00010 00110 0100,070,93 0110,050,95 10010 10110 1100,020,98 1110,0010,999

19. Industrial application We have to note that the execution time of this algorithm is substantially minimal. So, it is recommended to use this method even for the complex systems with a great number of components and complex linking or interactions. The failure rate of the system is: With:

20. Conclusion The main objective of this paper is the failure rate assessment of a complex system. The fault tree method allows a better and efficient method to define the interactions of the complex systems component. An algorithm allowing the transition of the fault tree to Bayesian network is also shown. The Bayesian networks are used thanks their interesting properties; They permit an exact calculation of probabilities; The computation time is minimal even for the complex systems with a great number of components; We can associate the expert judgment with the statistical data.

21. Thank you for your attention !