1. Prof. Enrico Zio Fault tree analysis Prof. Enrico Zio Politecnico di Milano Dipartimento di Energia

2. Prof. Enrico Zio

3. Fault Tree Analysis (FTA) Systematic and quantitative Systematic and quantitative Deductive Deductive AIM: 1.Decompose the system failure in elementary failure events of constituent components 2.Computation of system failure probability, from component failure probabilities from component failure probabilities

4. Prof. Enrico Zio

6. FT construction

7. Prof. Enrico Zio FT construction: Procedure steps 1. 1.Define top event (system failure) Electrical generating system E1E1E1E1 E2E2E2E2 G1G1G1G1 G2G2G2G2 G3G3G3G3 E1, E2 = engines E1, E2 = engines G1, G2, G3 = generators, each one is rated at 30 KVA is rated at 30 KVA

8. Prof. Enrico Zio FT construction: Procedure steps 1. 1.Define top event (system failure) 2.Decompose top event by identifying subevents which can cause it. At least two out of the three generators do not work

9. Prof. Enrico Zio FT construction: Procedure steps 1. 1.Define top event (system failure) 2. 2.Decompose top event by identifying subevents which can cause it 3. 3.Decompose each subevent in more elementary subevents which can cause it

10. Prof. Enrico Zio FT construction: Procedure steps 1. 1.Define top event (system failure) 2. 2.Decompose top event by identifying subevents which can cause it 3. 3.Decompose each subevent in more elementary subevents which can cause it 4. 4.Stop decomposition when subevent probability data are available (resolution limit): subevent = basic or primary event basic event

11. Prof. Enrico Zio FT construction: Procedure steps 1. 1.Define top event (system failure) 2. 2.Decompose top event by identifying subevents which can cause it 3. 3.Decompose each subevent in more elementary subevents which can cause it 4. 4.Stop decomposition when subevent probability data are available (resolution limit): subevent = basic or primary event

12. Prof. Enrico Zio FT Example 1

13. Prof. Enrico Zio FT event symbols

14. Prof. Enrico Zio FT gate symbols

15. Prof. Enrico Zio FT gate symbols

16. Prof. Enrico Zio FT Example 2: The System Hydraulic Control A Hydraulic Control B Actuators Linkage

17. Prof. Enrico Zio FT Example 2: The Fault Tree Hydraulic Control A Hydraulic Control B Actuators Linkage

18. Prof. Enrico Zio FT qualitative analysis

19. Prof. Enrico Zio FT qualitative analysis Introducing: X i : binomial indicator variable of i-th component state (basic event) Xi =Xi =Xi =Xi = 1 failure event true 0 failure event false  FT = set of boolean algebraic equations (one for each gate) => structure (switching) function  X T =  (X 1, X 2, …, X n )

20. Prof. Enrico Zio Fundamental Product  FT = set of boolean algebraic equations (one for each gate) => structure (switching) function  X T =  (X 1, X 2, …, X n )  An important theorem states that a structure function can be written uniquely as the union of the fundamental products which correspond to the combinations of the variables which render the function true. This is called the canonical expansion or disjunctive normal form of 

21. Prof. Enrico Zio Fundamental Product

22. Prof. Enrico Zio Structure function: Example 1 OR gate AND gate E1E1E1E1 E2E2E2E2 D

23. Prof. Enrico Zio Coherent structure functions A physical system would be quite unusual (or perhaps poorly designed) if improving the performance of a component (that is, replacing a failed component by a functioning one) caused the system to change from the success to the failed state. Thus, we restrict consideration to structure functions that are monotonically increasing in each input variable. These structure functions do not contain complemented variables; they are called coherent and can always be expressed as the union of fundamental products. The main properties of a coherent structure function are:

24. Prof. Enrico Zio FT qualitative analysis Reduce  in terms of minimal cut sets (mcs) cut sets = logic combinations of primary events which render true the top event minimal cut sets = cut sets such that if one of the events is not verified, the top event is not verified

25. Prof. Enrico Zio FT qualitative analysis FT = set of boolean algebraic equations (one for each gate) => structure (switching) function  X T =  (X 1, X 2, …, X n ) Boolean algebra to solve FT equations

26. Prof. Enrico Zio mcs: Example 1 3 minimal cut sets:

27. Prof. Enrico Zio mcs: Example 2 Hydraulic Control A Hydraulic Control B Actuators Linkage 5 minimal cut sets: M 1 =X L M 2 =X A X B M 3 =X A X HB M 4 =X HA X B M 5 =X HA X HB

28. Prof. Enrico Zio FT qualitative analysis: results 1. 1.mcs identify the component basic failure events which contribute to system failure 2. 2.qualitative component criticality: those components appearing in low order mcs or in many mcs are most critical

29. Prof. Enrico Zio FT quantitative analysis Compute system failure probability from primary events probabilities by: 1. 1.using the laws of probability theory at the fault tree gates Hydraulic Control A Hydraulic Control B Actuators Linkage p=0.1 p=0.1 p=0.1p=0.1 p=0.2 p=0.2 p=0.04p=0.05

30. Prof. Enrico Zio FT quantitative analysis Compute system failure probability from primary events probabilities by: 1. 1.using the laws of probability theory at the fault tree gates 2. 2.using the mcs found from the qualitative analysis It can be shown that:

31. Prof. Enrico Zio FT quantitative analysis: Example 2 Hydraulic Control A Hydraulic Control B Actuators Linkage 5 mcs: P(M 1 ) =P(X L =1)=0.01 P(M 2 )=P(X A X B =1)=0.1·0.1=0.01 P(M 3 )=P(X A X HB ) = 0.1·0.1=0.01 P(M 4 )= P(X HA X B =1)=0.1·0.1 = 0.01 P(M 5 ) =P(X HA X HB )=0.1·0.1=0.01

32. Prof. Enrico Zio FT: comments 1. 1.Straightforward modelization via few, simple logic operators; 2. 2.Focus on one top event of interest at a time; 3. 3.Providing a graphical communication tool whose analysis is transparent; 4. 4.Providing an insight into system behaviour; 5. 5.Minimal cut sets are a synthetic result which identifies the critical components.

33. Prof. Enrico ZioETA+FTA