Fault Tree Analysis Part 8 Probability Calculation RESULTS

Fault Tree Analysis Part 8 - Probability Calculation

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

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 and Usually, it is assumed that the basic events in a fault tree are independent. Thus,

PROBABILITY OF EVENTS CONNECTED BY AN “OR” GATE

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

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

Z AND X IF X and Y are Y Independent

AND-Gate Algorithm

Z OR X Y

OR-Gate Algorithm

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

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.

Example

3 HOT NITRIC ACID 1 TEMPERATURE SENSOR 2 HEAT EXCHANGER 8 TO REACTOR 5 4 AIR TO OPEN TRC 6 7 COOLING WATER SET POINT

+1 +1 FOU +1 LED ) +1 +1 -1 0 ) CK TU RS SO 0 +1 EN +1 +1 C (TR -1 MP. S -1 ERS V E R ST (TRC ) ED -10 +1 ) UCK A NM 0 (O (TE +1 +1 (CONTROL V ALVE REVERSED) 0 VALVE STUCK 0 (HX -1 NUA L) -1 +1 +1

CUT SET IMPORTANCE The importance of a cut set K is defined as Where, is the probability of the top event. as the conditional probability that the cut set the top event has occurred. may be interpreted occurs given that 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.

[ Example ] TOP OR OR 1 G 2 G 3 AND 2 3 OR AND G 5 3 G 6 OR G 4 G 7 AND 6 5 G 2 3 4 GATE CUT SETS 2 (1) (2) 5 (3 , 4) 4 (3 , 4) (5) 7 (1 , 3) (2 , 3) 3 (3 , 4 , 3) (3 , 5) 6 (6) (1 , 3) (2 , 3) 1 (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).

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 2 3 4 5 6 . 16. 2 1. 4 30 5. 5 1. 5 E-5 (. 125) 7 E-4 (6) 1. 1 E-4 (1) 5. 5 E-5 (. 5) 2. 4 E-6 3. 0 E-6 9. 8 E-4 3. 3 E-3 5. 5 E-4 2. 75 E-5 Now, compute the probability of occurrence for each cut set and top event probability. (1) 2. 4 E-6 (2) 3. 0 E-6 (6) 2. 75 E-5 (3, 4) 3. 23 E-6 (3, 5) 5. 39 E-7 3. 67 E-5

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

10 0 +1 +1 0 0 -1 (PUMP 1 SPEED = -10) FAILURE POWER 1. S 0 +10 0 -10 +1 0 SWITCH STUCK 0 +1 POWER 2 FAIL URE SPEED FAILURE PUMP 2 MECH. PUMP 2 0 -10 +1 -10 PUMP 1 SPEED PUMP 1 MECH. -10 FAILURE 0 1 POWER 1 0 FAILURE 1

G 1 AND Pump 1 Shut Down OR O R G 3 G 2 P 1 Mech Fail. Local Power 1 Failure 1 2 P 2 Mech Fail. 3 Pump 2 Not Started Local Power 2 Faiture Switch Stuck Local Power 1 Failure 4 5 2

OR 2 AND 1 AND 5 COMP 1 AND 3 1 4 q 1 1/3 4 Hr. 2 1/25 5 Hr. 3 1/5 1 Week 4 1/35 1 Week 5 1/10 3 Months

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

Unreliability Importances