1. Fault Tree Analysis Part 8 - Probability Calculation

2. RESULTS OF PROBABILITY CALCULATIONS 1) The probability of the top event. 2) The “importance” of the cut sets and primal events.

3. PROBABILITY OF EVENTS CONNECTED BY AN “AND” GATE In general, if events X and Y are probabilistically dependent, then Where, is the probability that X occurs given that Y occurs. If events X and Y are probabilistically independent, then Usually, it is assumed that the basic events in a fault tree are independent. Thus, and

4. PROBABILITY OF EVENTS CONNECTED BY AN “OR” GATE

5. PROBABILITY OF EVENTS CONNECTED BY A m-OUT-OF-n VOTING GATE Assume then where

6. SHORT-CUT CALCULATION METHODS Information Required Approximation of Event Unavailability When time is long compared with MTTR and, the following approximation can be made, Where, is the MTTR of component j.

7. Z AND X Y IF X and Y are Independent

8. AND-Gate Algorithm

9. Z OR X Y

10. OR-Gate Algorithm

11. COMPUTATION OF ACROSS LOGIC GATES 2 INPUTS 3 INPUTS n INPUTS AND GATES OR GATES

12. COMPUTING TOP EVENT PROBABILITY 1.Compute q (= ) for each primal Event. 2.Compute the Probability or Failure Rate for each Cut Set (QK). Use the “AND” Equation. 3.Compute the Top Event Probability or failure rate. Use the “OR” Equation.

13. Example

14. HEAT EXCHANGER HOT NITRIC ACID TEMPERATURE SENSOR TO REACTOR 1 3 2 8 AIR TO OPENTRCSET POINT 5 6 COOLING WATER 4 7

15. +1 0 (HX FOULED) +1 0 VALVE STUCK +1 -1 (CONTROL VALVE REVERSED) +1 +1 +1 0 (TEMP. SENSOR STUCK) -10 +1 -1 (TRC REVERSED) 0 (TRC STUCK) 0 (ON MANUAL) +1

22. CUT SET IMPORTANCE The importance of a cut set K is defined as Where, is the probability of the top event. may be interpreted as the conditional probability that the cut set occurs given that the top event has occurred. PRIMAL EVENT IMPORTANCE The importance of a primal event is defined as or Where, the sum is taken over all cut sets which contain primal event.

23. [ Example ] TOP OR 1 2 G2 AND 3OR AND5 34 G3 G4 G5 OR 6AND G7 G2 3 G6 GATECUT SETS 2 5 4 7 3 6 1 (1) (2) (3, 4) (3, 4) (5) (1, 3) (2, 3) (3, 4, 3) (3, 5) (6) (1, 3) (2, 3) (1) (2) (3, 4) (3, 5) (6) (1, 3) (2, 3) Hence, the minimal cut sets for this tree are : (1), (2), (6), (3, 4) and (3, 5).

24. As an example, consider the tree used in the section on cut sets. The cut sets for this tree are (1), (2), (6), (3,4),(3,5). The following data are given from which we compute the unavailabilities for each event. 1.16 1.5E-5 (.125) 2.4E-6 2.2 1.5E-5 (.125) 3.0E-6 3 1.4 7E-4 (6) 9.8E-4 4 30 1.1E-4 (1) 3.3E-3 5 5 1.1E-4 (1) 5.5E-4 6.5 5.5E-5 (.5) 2.75E-5 Now, compute the probability of occurrence for each cut set and top event probability. (1) 2.4E-6 (2) 3.0E-6 (6) 2.75E-5 (3,4) 3.23E-6 (3,5) 5.39E-7 3.67E-5

25. THE COMMON–MODE FAILURES WITHIN FAULT TREES S 1 2 3 SWITCH PUMP 2 (STAND – BY) POWER 2 POWER 1 PUMP 1 (RUNNING) Shared Power Source

26. PUMP 2 SPEED S PUMP 1 SPEED PUMP 1 MECH. FAILURE POWER 1 FAILURE POWER 2 FAILURE PUMP 2 MECH. FAILURE +100 10 0 (PUMP 1 SPEED = -10) SWITCH STUCK -10 0101 0101 0 -10 +1 0 -10 +1 0 +10 POWER 1. FAILURE 0 0

27. AND G1 OR P1 Mech Fail. Local Power 1 Failure G2 Pump 1 Shut Down OR P2 Mech Fail. Local Power 2 Faiture Switch Stuck G3 Pump 2 Not Started 12345 Local Power 1 Failure 2

28. GATEMIN CUT SETS G2 G3 G1 (1), (2) (2), (3), (4), (5) (1, 2), (1, 3), (1,4), (1,5) (2, 2), (2, 3), (2,4), (2, 5) (1, 2), (1, 3), (1, 4), (1, 5) (2), (2, 3), (2, 4), (2, 5)

29. OR 2 AND 15 13 14 COMPq 1234512345 1/3 1/25 1/5 1/35 1/10 4 Hr. 5 Hr. 1 Week 3 Months

30. Cut Set (2) (1, 3) (1, 4) (1, 5) 1 / 25 Yr. 1 / 762 Yr. 1 / 5333 Yr. 1 / 120 Yr. TOP Event Unavailability Importances

31. Unreliability Importances