IAEA Training in level 1 PSA and PSA applications BasicBasic LevelLevel 1.1. PSAPSA coursecourse forfor analystsanalysts BooleanBoolean AlgebraAlgebra andand PSAPSA quantificationquantification Boolean Algebra and PSA quantification ContentContent zz WhatWhat isis aa set?set? zz DealingDealing withwith abstractabstract setssets zz SystemSystem failurefailure statesstates zz BooleanBoolean Algebra.Algebra. OperationsOperations andand LawsLaws zz BooleanBoolean equationequation && CutCut-sets-sets zz ExampleExample Slide 2. Boolean Algebra and PSA quantification WHATWHAT ISIS AA SET?SET? zz AA setsetis is aa collectioncollection ofof itemsitems thatthat havehave somethingsomething inin commoncommon zz TheThe universaluniversal setset(E (E oror 1)1) isis thethe setset thatthat containscontains allall possiblepossible itemsitems zz TheThe membershipmembershipof of aa setset isis expressedexpressed asas xx MM (hence(hence nonnon-- membershipmembership byby xx M)M) zz AA setset XX isis aa subsub-set-set of of setset YY ifif allall thethe elementselements ofof XX areare alsoalso elementselements ofof Y:Y: XX YY zz TheThe nullnull setset( (or or 0)0) isis thethe setset thatthat containscontains nono elementselements atat allall zz TheThe complementarycomplementary setsetof of AA (( )) isis thethe setset thatthat containscontains allall thethe elementselements thatthat dodo notnot belongbelong toto setset AA Slide 3. Boolean Algebra and PSA quantification DEALINGDEALING WITHWITH ABSTRACTABSTRACT SETSSETS zz THETHE IDEAIDEA OFOF AA SETSET ISIS VERYVERY GENERAL:GENERAL: OneOne cancan dealdeal withwith setssets ofof physicalphysical thingsthings oror withwith setssets ofof abstractabstract thingsthings ThisThis lecturelecture willwill dealdeal withwith SETSSETS OFOF SYSTEMSYSTEM STATESSTATES Slide 4. Boolean Algebra and PSA quantification SETSSETS OFOF SYSTEMSYSTEM STATESSTATES T MVA XVB SVC PA NVA PB NVB XVC XVD XVE SYS S A ST U N / K F A S S S S S S S S S S S 1 S S S S S S S S S F F F 2 F S S S S S S S S S F F 3 S . S S S F S S S S S S S ni F S S S F F S S S S S S ni+1 F S S S F F F S S S S S ni+2 F . S S S S S F F S S S S nj F . F S S S S S S S S S S nk F F F S S S S S S S S S nk+1 F . Slide 5. Boolean Algebra and PSA quantification BOOLEANBOOLEAN ALGEBRAALGEBRA ININ PSAPSA zz BooleanBoolean AlgebraAlgebra isis aa simplesimple methodmethod ofof findingfinding thethe minimalminimal combinationscombinations ofof failuresfailures thatthat causecause systemsystem failure:failure: MINIMALMINIMAL CUTCUT SETSSETS zz ThisThis isis thethe firstfirst stepstep inin thethe assessmentassessment ofof thethe systemsystem failurefailure probabilityprobability Slide 6. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THETHE UNION:UNION: VX VY VX VY SYSTEM FINAL STATE S S 1 = SUCCESS SF2 = FAILURE FS3 = FAILURE FF4 = FAILURE Slide 7. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THETHE UNION:UNION: E = { system states } = { 1,2,3,4 } E A B A = { system states that contain 3 4 2 failure of VX } = { 3,4 } B = { system states that contain 1 failure of VY } = { 2,4 } A U B A U B = { system states that contain all the failures of VX OR the failures of VY } = = { 2,3,4 } Slide 8. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THETHE UNION:UNION: A U B is normally written as A SYSTEM + B and in fault tree notation FAILURE it is represented by an OR gate P (TOP) = P(A) + P(B) - P(AB) Using the rare event FAILURE OF FAILURE OF VX VY approximation: P(AB) <<< 1 A B P(TOP) = P(A) + P(B) Slide 9. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THETHE INTERSECTION:INTERSECTION: VX VZ SYSTEM FINAL STATE VX S S 1 = SUCCESS SF 2 = SUCCESS F S 3 = SUCCESS F F 4 = FAILURE VZ Slide 10. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THE INTERSECTION: THETHE INTERSECTION:INTERSECTION: E = { system states } = { 1,2,3,4 } E A C A = { system states that contain failure of VX } = 3 4 2 { 3,4 } C = { system states that contain 1 failure of VZ } = { 2,4 } A C A C = { system states that contain failure of VX AND failure of VZ } = { 4 } Slide 11. Boolean Algebra and PSA quantification OPERATIONSOPERATIONS ANDAND LAWSLAWS THETHE INTERSECTION:INTERSECTION: SYSTEM A C is normally FAILURE written as A × C and in fault tree notation it is represented by an AND gate FAILURE OF FAILURE OF P (TOP) = P(A) x P(C) VX VZ A C (Assuming that A and C are independent) Slide 12. Boolean Algebra and PSA quantification OPERATIONS AND LAWS COMBINATIONS OF UNIONS AND INTERSECTIONS COMBINATIONSCOMBINATIONS OFOF UNIONSUNIONS ANDAND INTERSECTIONSINTERSECTIONS VX VY VZ SYSTEM FINAL STATE S S S 1 = SUCCESS S S F 2 = SUCCESS VX VY S F S 3 = SUCCESS F S S 4 = SUCCESS S F F 5 = FAILURE F S F 6 = FAILURE F F S 7 = SUCCESS VZ F F F 8 = FAILURE Slide 13. Boolean Algebra and PSA quantification OPERATIONS AND LAWS COMBINATIONS OF UNIONS AND INTERSECTIONS E = { system states } = E { 1,2,3,4,5,6,7,8 } A B A = { system states that contain 4 7 3 failure of VX } = { 4,6,7,8 } 8 B = { system states that contain 6 5 failure of VY } = { 3,5,7,8 } 1 C = { system states that contain 2 failure of VZ } = { 2,5,6,8 } C (A U B) C = { system failure states } = { system states that contain failure of VX OR VY, AND failure of VZ } = ({ 4,6,7,8 } (A U B) C U { 3,5,7,8 }) { 2,5,6,8 } = { 3,4,5,6,7,8 } { 2,5,6,8 } = { 5,6,8 } Slide 14. Boolean Algebra and PSA quantification OPERATIONS AND LAWS DISTRIBUTIVE LAW (A U B) C = (A C) U (B C) SYSTEM SYSTEM FAILURE FAILURE FAILURE OF FAILURE OF FAILURE OF FAILURE OF VX AND VZ VY AND VZ VX OR VY VZ C FAILURE OF FAILURE OF FAILURE OF FAILURE OF FAILURE OF FAILURE OF VX VY VX VZ VY VZ A B A C B C Slide 15. Boolean Algebra and PSA quantification OPERATIONS AND LAWS ABSORPTION LAW A U (A B) = A A B A B = C C A B C C U A = A Slide 16. Boolean Algebra and PSA quantification OPERATIONS AND LAWS LAWS OF THE BOOLEAN ALGEBRA COMMUTATIVECOMMUTATIVEA A ++ BB == BB ++ AA AA ××B B == BB ××A A ASSOCIATIVEASSOCIATIVEA A ++ BB ++ CC == (( AA ++ BB )) ++ CC == AA ++ (( BB ++ CC )) AA ××B B ××C C == (( AA ××B B )) ××C C == AA ××( ( BB ××C C )) DISTRIBUTIVEDISTRIBUTIVEA A ××( ( BB ++ CC )) == (( AA ××B B )) ++ (( AA ××C C )) IDEMPOTENTIDEMPOTENTA A ++ AA == AA AA ××A A == AA NULLNULL SETSET AA ++ 00 == AA AA ××0 0 == 00 UNIVERSALUNIVERSAL SETSETA A ++ 11 == 11 AA ××1 1 == AA ABSORPTIONABSORPTIONA A ++ (( AA ××B B )) == AA Slide 17. Boolean Algebra and PSA quantification BOOLEANBOOLEAN EQUATIONEQUATION && CUTCUT SETSSETS TheThe combinationscombinations ofof systemsystem failuresfailures cancan bebe obtainedobtained fromfrom aa FAULTFAULT TREETREE MODELMODEL zz TheThe solutionsolution ofof thethe faultfault treetree isis thethe BOOLEANBOOLEAN EQUATIONEQUATION zz TheThe BooleanBoolean equationequation isis aa combinationscombinations ofof UNIONSUNIONS (+)(+) andand INTERSECTIONSINTERSECTIONS ((×):×): A1A1 ××A2 A2 ××....×....× Ai Ai+ + B1B1 ××B2 B2 ××....×....× Bi Bi+ + ......…… zz AA MINIMALMINIMAL CUTCUT SETSET isis thethe minimalminimal combinationcombination ofof failuresfailures thatthat leadlead toto toptop gategate failurefailure Slide 18. Boolean Algebra and PSA quantification EXAMPLEEXAMPLE D B C E Slide 19. Boolean Algebra and PSA quantification EXAMPLE FAULT TREES zz TheThe followingfollowing faultfault treestrees areare intendedintended toto illustrateillustrate thethe conceptsconcepts presentedpresented earlierearlier andand havehave beenbeen simplifiedsimplified asas follows:follows: TheyThey dodo notnot showshow thethe specificspecific failurefailure modesmodes ofof thethe componentscomponents TheyThey dodo notnot showshow humanhuman failurefailure eventsevents TheyThey dodo notnot showshow commoncommon causecause failuresfailures Slide 20. Boolean Algebra and PSA quantification EXAMPLE FAULT TREES Slide 21. Boolean Algebra and PSA quantification EXAMPLE FAULT TREES Slide 22. Boolean Algebra and PSA quantification EXAMPLE Minimal Cutsets D = (M VA + TANK) . (XVB + TANK) = (MVA . XVB) + (MVA . TANK) + (TANK . XVB) + (TANK . TANK) = (MVA . XVB) + TANK C = SVC + D = SVC + (MVA . XVB) + TANK B = (PA + NVA + C) . (PB + NVB + C) = (PA . PB) + (PA . NVB) + (PA . C) + (NVA . PB) + (NVA . NVB) + (NVA . C) + (C . PB) + (C . NVB) + (C . C) = (PA . PB) + (PA . NVB) + (NVA . PB) + (NVA . NVB) + C = (PA . PB) + (PA . NVB) + (NVA . PB) + (NVA . NVB) + SVC + (MVA . XVB) + TANK TOP = A1 . A2 . A3 TOP = (XVC + B) . (XVD + B) . (XVE + B) = [(XVC . XVD) + (XVC . B) + (B . XVD) + (B . B)] . (XVE + B) = [(XVC . XVD) + B] . (XVE + B) = (XVC . XVD . XVE) + (XVC . XVD . B) + (B . XVE) + (B . B) = (XVC . XVD . XVE) + B = XVC . XVD . XVE + PA . PB + PA . NVB + MINIMAL CUTSETS NVA . PB + NVA . NVB + SVC + Do they make sense???? MVA . XVB + TANK Slide 23..