Monte Carlo Simulation for Reliability and Availability analyses. November 2016

Transcription

1 Mone Carlo Simulaion for Reliabiliy and Availabiliy analyses November 206

2 The experimenal view

3 Conens 3 Sampling Random Numbers Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis of a componen Examples

4 The Hisory of Mone Carlo Simulaion Buffon Kelvin Gosse (Suden) Fermi, von Neumann, Ulam Neuron ranspor Sysem Transpor (RAMS) s 950 s 990 s

5 SAMPLING RANDOM NUMBERS

6 6 Example: Exponenial Disribuion Probabiliy densiy funcion: Expeced value and variance: ) ( e f T 2 ] [ ] [ T Var T E f T () () T f e

7 Sampling Random Numbers from F X (x) 7 Sample R from U R (r) and find X: X F X R Example: Exponenial disribuion F X X x x e R F e x x X F X R ln R

8 Sampling from discree disribuions sample an x0 x x k,,...,,... F P X x P X x k k i i0 R~ U[0,) k 8 Graphically:

9 Failure probabiliy esimaion: example 9 Arc number i Failure probabiliy P i Calculae he analyic soluion for he failure probabiliy of he nework, i.e., he probabiliy of no connecion beween nodes S and T 2- Repea he calculaion wih Mone Carlo simulaion

10 Analyic soluion 0 ) Minimal cu ses M 2 ={2,5} M 3 ={,3,5} M 4 ={2,3,4} 2) Nework failure probabiliy (rare even approximaion): 4 P[ X ] P[ M ] T 4 j P[ M ] p p P[ M ] p p j P[ M ] p p p P[ M ] p p p

11 Mone Carlo simulaion The sae of arc i can be sampled by applying he inverse ransform mehod o he se of discree probabiliies pi, pi of he muually exclusive and exhausive saes B 2 B p i B 3 B p i S R S RU[0,) 0 We calculae he arrival sae for all he arcs of he nework As a resul of hese ransiions, if he sysem falls in any configuraion corresponding o a minimal cu se, we add o N f The rial simulaion hen proceeds unil we collec N rials. We obain he esimaion of he failure probabiliy dividing N f by N

12 Mone Carlo simulaion 2 N number of rials N F number of rials corresponding o realizaion of a minimal cu se The failure probabiliy is equal o N F / N clear all; %arc failure probabiliies p=[0.05,0.025,0.05,0.02,0.075]; nf=0; n=e6; %number of MC simulaions for i=:n %sampling of arcs faul evens r=rand(,5); s=zeros(,5); rpm=r-p; for j=:5 if (rpm(,j)<=0) s(,j)=; end end %cu se check faul=s(,)*s(,4)+s(,2)*s(,5)+s(,)*s(,3)*s(,5)+s(,2)*s(,3)*s(,4); if faul >= nf=nf+; end end pf=nf/n; %sysem failure probabiliy For n=0 6, we obain PX [ T ]

13 SIMULATION OF SYSTEM TRANSPORT

14 Mone Carlo simulaion for sysem reliabiliy 4 PLANT = sysem of Nc suiably conneced componens. COMPONENT = a subsysem of he plan (pump, valve,...) which may say in differen exclusive (muli)saes (nominal, failed, sand-by,... ). Sochasic ransiions from sae-o-sae occur a sochasic imes. STATE of he PLANT a = he se of he saes in which he N c componens say a. The saes of he plan are labeled by a scalar which enumeraes all he possible combinaions of all he componen saes. PLANT TRANSITION = when any one of he plan componens performs a sae ransiion we say ha he plan has performed a ransiion. The ime a which he plan performs he n-h ransiion is called n and he plan sae hereby enered is called k n. PLANT LIFE = sochasic process.

15 Plan life: random walk 5 Random walk = realizaion of he sysem life generaed by he underlying sae-ransiion sochasic process.

16 Phase Space 6

17 Example: Sysem Reliabiliy Esimaion 7 R R C ( ) C ( ) 0, T M R R C ( ) C ( ), T M R R C ( ) C ( ), T M R R C ( ) C ( ) 0, T M Evens a componens level, which do no enail sysem failure ˆ C R () FT () M

18 Sochasic Transiions: Governing Probabiliies 8 T( ; k )d = condiional probabiliy of a ransiion a d, given ha he preceding ransiion occurred a and ha he sae hereby enered was k. C(k k ; ) = condiional probabiliy ha he plan eners sae k, given ha a ransiion occurred a ime when he sysem was in sae k. Boh hese probabiliies form he raspor kernel : K(; k ; k )d = T( ; k )d C(k k ; ) (; k) = ingoing ransiion densiy or probabiliy densiy funcion (pdf) of a sysem ransiion a, resuling in he enrance in sae k

19 SIMULATION FOR COMPONENT AVAILABILITY / RELIABILITY ESTIMATION 9

20 One componen wih exponenial disribuion of he failure ime 20 2 m Sae 2 Sae Sae X= ON Sae X=2 OFF

21 One componen wih exponenial disribuion of he failure ime 2 2 m Sae 2 Sae Sae X= ON Sae X=2 OFF

22 One componen wih exponenial disribuion of he failure ime 22 2 m Sae 2 Sae Sae X= ON Sae X=2 OFF

23 One componen wih exponenial disribuion of he failure ime 23 m 2 values h - m h - Sae X= ON Sae X=2 OFF Limi unavailabiliy: / m U / m/ 0.07

24 Mone carlo simulaion for esimaing he sysem availabiliy a ime 24 N ime inervals Δ If he componen fails in +Δ, he couner increases c A ( j ) = c A ( j ) +; oherwise, c A ( j ) = c A ( j ) Trial

25 Mone carlo simulaion for esimaing he sysem availabiliy a ime 25 anoher rial Trial Trial i-h

26 Mone carlo simulaion for esimaing he sysem availabiliy a ime 26 anoher rial Trial Trial i-h Trial M-h

27 Mone carlo simulaion for esimaing he sysem availabiliy a ime 27 The couner c A ( j ) adds unil M rials have been sampled G ( ) P{ X( ) } A j = unavailabiliy a ime j = mean value of c A ( j ) a ime j Trial Trial i-h Trial M-h G A c ( ) j A ( j ) M

28 Unavailabiliy Pseudo Code 28 %Iniialize parameers Tm=mission ime; N=number of rials; D=bin lengh; Time_axis=0:D:Tm; FOR i=:n %parameer iniializaion for each rial =0; failure_ime=0; repair_ime=0; sae=; while <Tm if sae= =+exprnd(/lambda); sae=2; failure_ime=; lower_b=minimum(find(ime_axis>=failure_ime)); couner(i,lower_b)=; Else =+exprnd(/mu); sae=; repair_ime=; upper_b=find(ime_axis<up_ime,,'las'); couner(i,lower_b+:upper_b)=; Endif End while End For unav=sum_channel(couner)/n_simulaions Fala de disponibilidad único Single ensayo rial Simulacion Mone Carlo Mone simulaion Carlo Fala Limi de unavailabiliy disponibilidad en esado esacionario Time Tiempo

29 SIMULATION FOR SYSTEM AVAILABILITY / RELIABILITY ESTIMATION 29

30 Phase Space 30

31 Iniial Indirec Mone Carlo: Example () 3 A C B Componens imes of ransiion beween saes are exponenially disribued i ( = rae of ransiion of componen i going from is sae j i o he sae m i ) j i m - Arrival AB ( ) - 2 A 3 ( B ) A 3 ( B ) A( B) 2 A( B) 3 A( B) 23

32 Iniial Indirec Mone Carlo: Example (2) 32 Arrival C 2 C 3 C C 2 C 3 2 C C 3 C 2 3 C C 4 C 42 C 43 The componens are iniially (=0) in heir nominal saes (,,) One minimal cu se of order (C in sae 4:(*,*,4)) and one minimal cu se of order 2 (A and B in 3: (3,3,*)).

33 33 SAMPLING THE TIME OF TRANSITION The rae of ransiion of componen A(B) ou of is nominal sae is: The rae of ransiion of componen C ou of is nominal sae is: The rae of ransiion of he sysem ou of is curren configuraion (,, ) is: We are now in he posiion of sampling he firs sysem ransiion ime, by applying he inverse ransform mehod: where R ~ U[0,) ) ( 3 ) ( 2 ) ( B A B A B A C C C C C B A,, ) ln(,, 0 R Analog Mone Carlo Trial

34 Sampling he kind of Transiion () 34 Assuming ha < T M (oherwise we would proceed o he successive rial), we now need o deermine which ransiion has occurred, i.e. which componen has undergone he ransiion and o which arrival sae. The probabiliies of componens A, B, C undergoing a ransiion ou of heir iniial nominal saes, given ha a ransiion occurs a ime, are: A,,,,,,, Thus, we can apply he inverse ransform mehod o he discree disribuion B, C R A C B,,,,,, C 0

35 Sampling he Kind of Transiion (2) 35 Given ha a componen B undergoes a ransiion, is arrival sae can be sampled by applying he inverse ransform mehod o he se of discree probabiliies B B 2 3, B B of he muually exclusive and exhausive arrival saes 0 B 2 B As a resul of his firs ransiion, a he sysem is operaing in configuraion (,3,). The simulaion now proceeds o sampling he nex ransiion ime 2 wih he updaed ransiion rae,3, B 3 B A B C 3 S R S R S U[0,)

36 Sampling he Nex Transiion 36 The nex ransiion, hen, occurs a where R ~ U[0,). ln( ),3, 2 R Assuming again ha 2 < T M, he componen undergoing he ransiion and is final sae are sampled as before by applicaion of he inverse rasform mehod o he appropriae discree probabiliies. The rial simulaion hen proceeds hrough he various ransiions from one sysem configuraion o anoher up o he mission ime T M.

37 Unreliabiliy and Unavailabiliy Esimaion 37 When he sysem eners a failed configuraion (*,*,4) or (3,3,*), where he * denoes any sae of he componen, allies are appropriaely colleced for he unreliabiliy and insananeous unavailabiliy esimaes (a discree imes j [0, T M ]); Afer performing a large number of rials M, we can obain esimaes of he sysem unreliabiliy and insananeous unavailabiliy by simply dividing by M, he accumulaed conens of C R ( j ) and C A ( j ), j [0,T M ]

38 Direc Mone Carlo: Example () 38 A C For any arbirary rial, saring a =0 wih he sysem in nominal configuraion (,,) we would sample all he ransiion imes: ln( i i m i 0 i R,m m i B i ) i m m i i A, B, C 2,3 2,3,4 for for i A, B i C where ~ R i, m U i [0,)

39 Direc Mone Carlo: Example (2) 39

40 MARA Monecarlo Availabiliy Reliabiliy Analysis code 40 A USER-FRIENDLY SOFTWARE FOR RELIABILITY ANALYSIS OF COMPLEX SYSTEMS

41 MARA Monecarlo Availabiliy Reliabiliy Analysis code 4. Design sysem srucure

42 MARA Monecarlo Availabiliy Reliabiliy Analysis code 42. Design sysem srucure 2. Cu-ses compuaion

43 MARA Monecarlo Availabiliy Reliabiliy Analysis code 43. Design sysem srucure 2. Cu-ses compuaion 3. Define sysem parameers

44 MARA Monecarlo Availabiliy Reliabiliy Analysis code 44. Design sysem srucure 2. Cu-ses compuaion 3. Define sysem parameers 4. Run Mone Carlo compuaion

45 bin-averaged Sisem Unavailabiliy MARA Monecarlo Availabiliy Reliabiliy Analysis code 45. Design sysem srucure 2. Cu-ses compuaion 3. Define sysem parameers 4. Run Mone Carlo compuaion 5. See Resuls 3.5 x ime

46 A real example of Indirec Simulaion PRODUCTION AVAILABILITY EVALUATION OF AN OFFSHORE INSTALLATION Zio, E., Baraldi, P., Paelli E. Assessmen of he availabiliy of an offshore insallaion by Mone Carlo simulaion. Inernaional Journal of Pressure Vessels and Piping 83 (2006)

47 The offshore producion plan 5 TC TC 0. MSm 3 0. MSm MSm3/d 50% 50% 2.2 MSm 3 /d Fuel Gas 25 b Gas Lif 60b 4.4 MSm 3 /d TEG 6 MW MSm 3 /d Flare Gas Expor 3.0 MSm 3 /d, 60b Wells Producion Gas : 5 MSm 3 /d Oil : Sm 3 / d Waer : 8000 Sm 3 /d 4.4 MSm 3 /d Separaion 7000 Sm 3 /d Sm 3 /d 0. MSm 3 TG 3 MW 50% Gas Lif 00b Oil Tr EC 6 MW 7 MW Oil expor Wa. Tr 0. MSm 3 TG 50% 3 MW Sea 7 MW Waer Inj.

48 Componen failures and repairs: TCs and TGs TC TG 0 0 m 0 m h h h h h h - m h h - m h h - Sae 0 = as good as new Sae = degraded (no funcion los, greaer failure rae value) Sae 2 = criical (funcion is los)

49 Componen failures and repairs: EC and TEG 53 EC TEG 0 2 m h h - m h h - Sae 0 = as good as new Sae 2 = criical (funcion is los)

50 Producion prioriy 54 When a failure occurs, he sysem is reconfigured o minimise (in order): he impac on he expor oil producion he impac on expor gas producion The impac on waer injecion does no maer

51 Producion prioriy: example 55 TC TC 0. MSm 3 0. MSm MSm3/d 50% 50% 2.2 MSm 3 /d Fuel Gas 25 b Gas Lif 60b 4.4 MSm 3 /d TEG 6 MW MSm 3 /d Flare Gas Expor 3.0 MSm 3 /d, 60b 4.4 MSm 3 /d Gas Lif 00b EC Oil expor Wells Producion Gas : 5 MSm 3 /d Oil : Sm 3 / d Waer : 8000 Sm 3 /d Separaion 7000 Sm 3 /d Sm 3 /d 0. MSm 3 TG 3 MW 50% Oil Tr 7 MW Wa. Tr 0. MSm 3 TG 50% 3 MW Sea Waer Inj.

52 Mainenance policy: reparaion 56 Only repair eam Prioriy levels of failures:. Failures leading o oal loss of expor oil (boh TG s or boh TC s or TEG) 2. Failures leading o parial loss of expor oil (single TG or EC) 3. Failures leading o no loss of expor oil (single TC failure)

53 Mainenance policy: prevenive mainenance 57 Only prevenive mainenance eam The prevenive mainenance akes place only if he sysem is in perfec sae of operaion Type of mainenance Frequency [hours] Duraion [hours] Turbo-Generaor and Turbo-Compressors Type 260 (90 days) 4 Type ( year) 20 (5 days) Type (5 years) 672 (4 weeks) Elecro Compressor Type

54 MARKOV APPROACH Number of componens = 6 Number of saes for componen = 2 or = 324 plan saes repair eam 29 new plan saes mainenance eam Non homogeneous Markov chain + + Markov approach oo complex MONTE CARLO APPROACH

55 Case A: perfec sysem and prevenive mainenances 62 P.Mainenance Type (TC,TG) P.Mainenance Type 2 (EC) P.Mainenance Type 3 (TC,TG) Mean Sd Oil [k m 3 /d] Gas [k Sm 3 /d] Waer [k m 3 /d]

56 The heoreical view

57 Conens 67 Sampling Evaluaion of definie inegrals Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis

58 Conens 68 Sampling Evaluaion of definie inegrals Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis

59 Buffon s needle 69 Buffon considered a se of parallel sraigh lines a disance D apar ono a plane and compued he probabiliy P ha a needle of lengh L < D randomly posiioned on he plane would inersec one of hese lines. P P Y Lsin fy ( y) D y [0, D] f ( ) [0, ] dy d L / D P D /2 A

60 Sampling (pseudo) Random Numbers Uniform Disribuion 70 cdf : U R r PR r r 0 pdf : u R r dur dr r

61 Sampling (pseudo) Random Numbers Uniform Disribuion 7 R~ U[0,) xi axi c mod m Where r i xi m a, c [0, m ] m? x x... x x Example: a = 5, c =, m = r 6 (5 2 ) mod 6 r 3 3 r 6 2 6

62 Sampling (pseudo) Random Numbers Generic Disribuion 72 Pr R r U r R F X x Sample R from U R (r) and find X: X F X R r R 0 X x Quesion: which disribuion does X obey? X x P F R P X Applicaion of he operaor F x o he argumen of P above yields P x X x PR F x F x Summary: From an R U R (r) we obain an X F X (x) X X

63 Example: Exponenial Disribuion 73 Markovian sysem wih wo saes (good, failed) hazard rae, = consan cdf pdf f F T T PT e d P T d e d Sampling a failure ime T R R r F e F T T F T R ln R

64 Example: Weibull Disribuion 74 hazard rae no consan CDF pdf T F P T e T f d P T d e d Sampling a failure ime T R R F r F e T T FT R ln R

65 Sampling by he Inverse Transform Mehod: Discree Disribuions sample an x0 x x k,,...,,... F P X x P X x k k i i0 R~ U[0,) k 75 P F R F F F F F R ~ U[0,) and FR ( r) r ( ) ( ) k k R k R k P F R F F F f P X x k k k k k k Graphically:

66 Conens 76 Sampling Evaluaion of definie inegrals Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis

67 MC Evaluaion of Definie Inegrals (D) Analog Case 77 G f b a g x f x dx x pdf f x 0 ; f xdx MC analog dar game: sample x from f(x) he probabiliy ha a sho his x dx is f(x)dx he award is g(x) Consider N rials wih resul {x, x 2,,x n }: he average award is G N g N N x i i

68 MC Evaluaion of Definie Inegrals (D) Example 78 2 G cos x dx By seing: f x, g xcos x 2 G E g( x) 2 2 E g ( x) cos x dx Var G Var g x E g x E g x N N 2 N ( ) ( ) 2 2 Var GN N 2 N 2 4 for N 0 hisories, xi ~ U[0,) g xi cos xi G , s 9.60 N 2 6 G N 2

69 MC Evaluaion of Definie Inegrals (D) Biased Case 79 The expression for G may be wrien MC biased dar game: sample x from f (x) he probabiliy ha a sho his x dx is f (x)dx he award is f x g x g x G f x N x x f G g D f D x f xdx g x f x g N dx x N i i

70 MC Evaluaion of Definie Inegrals (D) Biased Case: Example 80 2 G cos xdx The pdf f * (x) is: f ( x) a bx * 2 From he normalizaion condiion: * b f ( x) dx a bx dx a b 3( a ) 3 * 2 f ( x) a 3( a ) x 3 For he minimum value a.5 2 cos x G xdx x dx 2 2 cos (.5.5 ) x 0 0 f ( x) g ( x)

71 MC Evaluaion of Definie Inegrals (D) Biased Case: Example 8 Finally we obain: Var G N N N N i 4 * 2 i i 2.5.5xi for N 0 hisories, x ~ f g x G , s G N 4 cos x

72 Conens 82 Sampling Evaluaion of definie inegrals Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis

73 Mone Carlo simulaion for sysem reliabiliy 83 PLANT = sysem of Nc suiably conneced componens. COMPONENT = a subsysem of he plan (pump, valve,...) which may say in differen exclusive (muli)saes (nominal, failed, sand-by,... ). Sochasic ransiions from sae-o-sae occur a sochasic imes. STATE of he PLANT a = he se of he saes in which he N c componens say a. The saes of he plan are labeled by a scalar which enumeraes all he possible combinaions of all he componen saes. PLANT TRANSITION = when any one of he plan componens performs a sae ransiion we say ha he plan has performed a ransiion. The ime a which he plan performs he n-h ransiion is called n and he plan sae hereby enered is called k n. PLANT LIFE = sochasic process.

74 85

75 Sochasic Transiions: Governing Probabiliies 86 T( ; k )d = condiional probabiliy of a ransiion a d, given ha he preceding ransiion occurred a and ha he sae hereby enered was k. C(k k ; ) = condiional probabiliy ha he plan eners sae k, given ha a ransiion occurred a ime when he sysem was in sae k. Boh hese probabiliies form he raspor kernel : K(; k ; k )d = T( ; k )d C(k k ; ) (; k) = ingoing ransiion densiy or probabiliy densiy funcion (pdf) of a sysem ransiion a, resuling in he enrance in sae k

76 The von Neumann s Approach and he Transpor Equaion 87 The ransiion densiy (; k) is expanded in series of he parial ransiion densiies: n (; k) = pdf ha he sysem performs he nh ransiion a, enering he sae k. Then, (, k) n0 n (, k) 0 (, k) k ' 0 d' ( ', k') K(, k Transpor equaion for he plan saes ', k') 87

77 88 *) *,, ( ),, ( ), ( ), ( * 0 0 k k K k k K k d k k ),, ( *) *,, ( ),, ( ), ( ), ( 2 2 * 2 2 * k k K k k K d k k K k d k k k ),, ( ),, ( *) *,, ( ), ( 2 2 * * 2,...,, * 2 2 n n n k k k n n k k K k k K k k K d d d k n n n Iniial Condiions: (*, k*) Formally rewrie he parial ransiion densiies: Mone Carlo Soluion o he Transpor Equaion ()

78 MC Evaluaion of Definie Inegrals G f b a g x f x dx x pdf f x 0 ; f xdx 89 MC analog dar game: sample x = (, k ; 2, k 2 ;...) from f(x)= K(, k *, k*) K( 2, k2, k) K(, k n, kn ) he probabiliy ha a sho his x dx is f(x)dx he award is g(x)= Consider N rials wih resul {x, x 2,,x n }: he average award is G N g N N x i i

79 Conens 90 Sampling Evaluaion of definie inegrals Simulaion of sysem ranspor Simulaion for reliabiliy/availabiliy analysis

80 Mone Carlo Simulaion in RAMS 9 G( ) (, k) R (, ) d k 0 k Expeced value = subse of all sysem failure saes R k (,) = G() = unreliabiliy R k (,) = prob. sysem no exiing before from he sae k enered a < G() = unavailabiliy Mone Carlo soluion of a definie inegral: expeced value sample mean

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83 Quesions & Answers 94