Nuclear reliability: system reliabilty

Transcription

1 Nuclear reliability: system reliabilty Dr. Richard E. Turner December 3, 203

2 Goal of these two lectures failures are inevitable: need methods for characterising and quantifying them LAST TIME: characterising component failures (fine level of granularity) TODAY: characterising system failures (coarse level of granularity) COURSE TEXT BOOK: Reliability and Risk Assessment, Andrews and Moss. Chapters 5 8.

3 Last time Three definitions of availability, key measures suitable for evaluating different systems (continuous/on demand) for different objectives (safety/productivity) Focused on continuously operational repairable systems: availability: A(t) = probability component works at time t unavailability: Q(t) = A(t) = probability does not work at time t reliability: R(t, t ) = probability work continuously between t & t unreliability: F (t, t ) = R(t, t ) probability fail somewhen between t & t Related properties of the Markov chain (easy to measure) to availability/reliability (important to know/predict/estimate) Today: relating system unavailability/unreliability/etc. to component unavailability/unreliability/etc.

4 Fault trees W water tank pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pipe P P2 pipe 2 P N T pump nozzle

5 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pipe P P2 pipe 2 P N T pump nozzle

6 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 T: failure of system on demand P N T pump nozzle

7 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 P N T T: failure of system on demand pump nozzle to nozzle from nozzle N

8 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 P N T T: failure of system on demand pump nozzle to nozzle from nozzle N to pump pump fails to start P

9 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 P N T T: failure of system on demand pump nozzle to nozzle from nozzle N to pump pump fails to start P from pipe from pipe 2

10 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 P N T T: failure of system on demand pump nozzle to nozzle from nozzle N from pipe to pump pump fails to start P from pipe from pipe 2 2

11 Fault trees W water tank reliablity = pressure vessel pressure vessel cooling system: normal ) temperature of pressure vessel > threshold 2) water drawn down two pipes to pump 3) pump sends water to nozzle 4) nozzle sprays pressure vessel to cool it pressure vessel cooling system: fault tree pipe P P2 pipe 2 P N T T: failure of system on demand pump nozzle to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

12 Fault trees Tips for fault tree construction: - external boundary - limit of resolution (internal boundary) - clear definition of events pressure vessel cooling system: fault tree T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

13 Fault trees Desire: top event probability p(t) in terms of event probabilities: p(w) p(p) p(p2) p(p) p(n) top event probability gives system: unavailability/unreliability/failure rate/etc. Key step: determine the necessary and sufficient causes for top failuer event to occur Tips for fault tree construction: - external boundary - limit of resolution (internal boundary) - clear definition of events pressure vessel cooling system: fault tree T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

14 Boolean representation of fault trees

15 Boolean representation of fault trees addition-like

16 Boolean representation of fault trees addition-like

17 Boolean representation of fault trees addition-like commutative

18 Boolean representation of fault trees addition-like commutative

19 Boolean representation of fault trees addition-like commutative

20 Boolean representation of fault trees addition-like commutative associative

21 Boolean representation of fault trees addition-like commutative associative

22 Boolean representation of fault trees addition-like commutative associative multiplication-like

23 Boolean representation of fault trees addition-like commutative associative multiplication-like

24 Boolean representation of fault trees addition-like commutative associative multiplication-like commutative

25 Boolean representation of fault trees addition-like commutative associative multiplication-like commutative

26 Boolean representation of fault trees addition-like commutative associative multiplication-like commutative

27 Boolean representation of fault trees addition-like commutative associative multiplication-like commutative associative

28 Question

29 Question

30 Question

31 Question

32 Question

33 Question

34 Question / gates distributive too

35 Other logic gates and symbols exclusive K-out-of-N x K K = NOT transfer symbol basic event top event or event description

36 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

37 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

38 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

39 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

40 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

41 Simplifying fault trees using Boolean logic T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

42 Simplifying fault trees using Boolean logic absorption law T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

43 Simplifying fault trees using Boolean logic absorption law T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

44 Simplifying fault trees using Boolean logic absorption law T: failure of system on demand to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

45 Simplifying fault trees using Boolean logic absorption law T: failure of system on demand if any one of these sets of events occurs the top event will also occur minimal cutsets to nozzle from nozzle N from pipe pipe blocked in tank to pump pump fails to start from pipe from pipe 2 P P W 2

46 Definition of a minimal cutset cutset: collection of basic events that result in top event occurring e.g. if T = N + P + W + P.P 2 = K = N.P and K = W.P are cutsets minimal cutset: necessary and sufficient cutset T = K + K K N K i = X.X X Li e.g. if T = N + P + W + P.P 2 = K = N, K 2 = P, K 3 = W, K 4 = P.P 2 for independent events, most of probability mass will be on low order cutsets pathset: set of basic events that result in top event not occurring computer programs (e.g. MOCUS) can be used to find cutsets efficiently

47 Calculating top event probability in terms of cutset probability

48 Calculating top event probability in terms of cutset probability

49 Calculating top event probability in terms of cutset probability 0

50 Calculating top event probability in terms of cutset probability

51 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion

52 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion Example: cooling system

53 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion Example: cooling system

54 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion Example: cooling system

55 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion Example: cooling system

56 Calculating top event probability in terms of cutset probability Inclusion-exclusion expansion Example: cooling system

57 The need for approximations Number of terms in an inclusion-exclusion expansion is if impossible (even for a computer) to enumerate need methods for approximating the top event probability first consider systems with independent events Inclusion-exclusion expansion Example: cooling system

58 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion Inclusion-exclusion expansion Example: cooling system

59 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion Inclusion-exclusion expansion Example: cooling system

60 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion Inclusion-exclusion expansion Example: cooling system

61 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion Inclusion-exclusion expansion Example: cooling system

62 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion Inclusion-exclusion expansion Example: cooling system

63 Upper and lower bounds plot successive terms in the inclusion-exclusion expansion progressively tighter lower/upper bounds on (if independent) Inclusion-exclusion expansion Example: cooling system

64 Upper and lower bounds = rare event upper bound (holds for dependent events too) plot successive terms in the inclusion-exclusion expansion progressively tighter lower/upper bounds on (if independent) Inclusion-exclusion expansion Example: cooling system

65 A tighter upper bound: minimal cutset upper bound P (T ) = P (at least one minimal cutset occurs) = P (no minimal cutsets occur) Now bound P (no minimal cutsets occur): P (no minimal cutsets occur) N P (minimal cutset n does not occur) n= Equality when a) events independent and b) no event occurs in more than one cutset. Using this to bound P (T ): N N P (T ) P (minimal cutset n does not occur) = ( P (K n )) n= n= E.g. for two cutsets P (T ) ( P (K ))( P (K 2 )) = P (K ) + P (K 2 ) P (K )P (K 2 ) P (K ) + P (K 2 ) Generally: N N P (T ) ( P (K n )) P (K n ) n= n= P (T ) minimal cutset upper bound rare event upper bound

66 A tighter upper bound: minimal cutset upper bound P (T ) = P (at least one minimal cutset occurs) = P (no minimal cutsets occur) Now bound P (no minimal cutsets occur): P (no minimal cutsets occur) N P (minimal cutset n does not occur) n= Equality when a) events independent and b) no event occurs in more than one cutset. Using this to bound P (T ): N N P (T ) P (minimal cutset n does not occur) = ( P (K n )) n= n= E.g. for two cutsets P (T ) ( P (K ))( P (K 2 )) = P (K ) + P (K 2 ) P (K )P (K 2 ) P (K ) + P (K 2 ) Generally: N N P (T ) ( P (K n )) P (K n ) n= n= P (T ) minimal cutset upper bound rare event upper bound

67 A tighter upper bound: minimal cutset upper bound P (T ) = P (at least one minimal cutset occurs) = P (no minimal cutsets occur) Now bound P (no minimal cutsets occur): P (no minimal cutsets occur) N P (minimal cutset n does not occur) n= Equality when a) events independent and b) no event occurs in more than one cutset. Using this to bound P (T ): N N P (T ) P (minimal cutset n does not occur) = ( P (K n )) n= n= E.g. for two cutsets P (T ) ( P (K ))( P (K 2 )) = P (K ) + P (K 2 ) P (K )P (K 2 ) P (K ) + P (K 2 ) Generally: N N P (T ) ( P (K n )) P (K n ) n= n= P (T ) minimal cutset upper bound rare event upper bound

68 A tighter upper bound: minimal cutset upper bound P (T ) = P (at least one minimal cutset occurs) = P (no minimal cutsets occur) Now bound P (no minimal cutsets occur): P (no minimal cutsets occur) N P (minimal cutset n does not occur) n= Equality when a) events independent and b) no event occurs in more than one cutset. Using this to bound P (T ): N N P (T ) P (minimal cutset n does not occur) = ( P (K n )) n= n= E.g. for two cutsets P (T ) ( P (K ))( P (K 2 )) = P (K ) + P (K 2 ) P (K )P (K 2 ) P (K ) + P (K 2 ) Generally: N N P (T ) ( P (K n )) P (K n ) n= n= P (T ) minimal cutset upper bound rare event upper bound

69 A tighter upper bound: minimal cutset upper bound P (T ) = P (at least one minimal cutset occurs) = P (no minimal cutsets occur) Now bound P (no minimal cutsets occur): P (no minimal cutsets occur) N P (minimal cutset n does not occur) n= Equality when a) events independent and b) no event occurs in more than one cutset. Using this to bound P (T ): N N P (T ) P (minimal cutset n does not occur) = ( P (K n )) n= n= E.g. for two cutsets P (T ) ( P (K ))( P (K 2 )) = P (K ) + P (K 2 ) P (K )P (K 2 ) P (K ) + P (K 2 ) Generally: N N P (T ) ( P (K n )) P (K n ) n= n= P (T ) minimal cutset upper bound rare event upper bound

70 Independent events in fault trees: example bounds

71 Independent events in fault trees: example bounds rare event

72 Independent events in fault trees: example bounds rare event minimal cutset = ground truth (indep./non-repeated)

73 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode

74 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode idea: introduce a repreat common cause event ( ) that captures the dependencies

75 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode idea: introduce a repreat common cause event ( ) that captures the dependencies

76 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode idea: introduce a repreat common cause event ( ) that captures the dependencies

77 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode idea: introduce a repreat common cause event ( ) that captures the dependencies rare event ground truth:

78 Dependent events in fault trees: common cause failures rare event minimal cutset = ground truth (indep./non-repeated) the two pipes ( and ) have a common failure mode idea: introduce a repreat common cause event ( ) that captures the dependencies rare event ground truth: minimal cutset = / ground truth (repeated events)

79 Summary fault trees describe system failure in terms of components and logic gates fault trees can be simplified by computer algorithms that traverse the tree and simplify the logic e.g. using the adsorption law cutsets can be identified which are collections of failures that result in a system failure availability (and other measures) can be computed from minimal cutsets in real systems bounds are the best we can hope for dependent component failures can be captured using shared repeated events