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1 2009 5
2
3 WinBUGS D/G () λ...50 i
4 7.2 pd σ µ µ Thinning µσthinning ii
5 PSA A PSA A A A A.4 PSA B B.1 NUCIA B B.3 NUCIA C D E PSA E.1 PSA E.2 PSA F PSA iii
6 () iv
7 1 PSA 13 PSA PSA 18 PSA PSA ( 2) 2 NUCIA [1] PSA [2] [3] EF 2 NUCIA PSA NUCIA PSA 1
8 NUCIA PSA NUCIA PSA PSA PSA A /
9 2 2.1 NUCIA PSA ABWR / PSA NUCIA NUCIA PSA XNUCIA Y NUCIA p Y Xp/ p X / (MCMC)
10 1, 1 2, 2 m, m 4 LogNorm( m, m ) 1 2 m Beta(,) p p 1 p 2 p m 1 X 1 Poisson( 1 T 1 Y 1 Bin(p 1, X X 2 Poisson( 2 T 2 Y 2 Bin(p 2, X 2 m m X m Poisson( m T m Y m Bin(p m, X m i i p i i T i i X i i Y i i ( µ, σ ) LogNorm Poisson (,T ) λ ( α, β ) Beta Bin ( p, X ) 2-1
11 /4 [h] 5 [] [1/h] 5% [1/h] [1/h] 95% [1/h] EF * E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E
12 /4 [h] 5 [] [1/h] 5% [1/h] [1/h] 95% [1/h] EF *6 6 BWR 0 3.6E E E E E E E E E E E E E E E E E E E E E E E E E (PWR) 0 2.2E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E / 1 3.4E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E * E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E / 0 1.9E E E E E () 0 1.9E E E E E E E E E E / 0 2.4E E E E E ( 0 2.4E E E E E E E E E E (BWR) 0 4.4E E E E E (PWR) 0 1.2E E E E E PLR MG E E E E E RPS,CRDM MG 0 1.3E E E E E (PLR) 2 6.7E E E E E E E E E E
13 /4 [h] 5 [] [1/h] 5% [1/h] [1/h] 95% [1/h] EF * E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E * E E E E E * E E E E E E E E E E E E E E E * E E E E E E E E E E * E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E / 3 4.4E E E E E E E E E E ( 4 2.4E E E E E E E E E E E E E E E E E E E E E E E E E / 4 5.9E E E E E E E E E E / 8 7.5E E E E E E E E E E / 2 3.0E E E E E E E E E E / 5 2.0E E E E E
14 /4 [h] 5 [] [1/h] 5% [1/h] [1/h] 95% [1/h] EF * E E E E E / 1 5.6E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E NUCIA() 6.EF 2 (= 7. µ WinBUGSµ
15 % 95% EF *2 *1 [1/d] [1/d] [1/d] [1/d] 9 DG E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E NUCIA() 2.EF 2 (=
16 ipsanuciap i PSAX i Y i Bin(p i,x i ) f i ( p, X ) Y ~ Bin yi xi yi ( y ; x, p ) = C p ( 1 p ) i i i i x i i y i i i (3.1) X i i Yi i pi i NUCIA PSA p i a) (10 ) [3] b) (16 ) [2]
17 3.1.2 p i p i p i Beta( α, β ) p i α β p i Z W V p i Beta( α, β ) 3-1 [2][3] 3-1 p i Z/W Beta( α, β ) Wp i W W=Vp i 0.4 Beta( α, β ) Beta ( 4, 6) Beta( 4, 6) (W) 10 (Z) Beta ( 4,6) Beta(4,6) W(=V)=513Z=201 Jeffreys 0.4 W 11
18 Beta(4,6) X i Y i directed graph p i Beta ( 4,6) 12
19 p i X i Y i i p i i X i i Y i i α,β i ( p X ) Y ~ Bin, i i
20 a) it i X i λ i X i ~ Poisson( λ T ) ( λiti ) f ( xi; λi, Ti ) = exp( λiti ) x! i i (3.2) i xi b) λ i µσ lnλ i f ( λi ) = exp (ln λ ) 2 i µ σλ 2π 2σ (3.3) i < i 0 λ < < µ < σ > 0 c) µσ (a µ,b µ )(a σ,b σ ), 1 f ( µ ) = aµ µ bµ (3.4) bµ aµ 1 f ( σ ) = aσ σ bσ (3.5) b a σ σ d)
21 b a a b i T i i X i i λ i i X i i T i i µ,σ- a,b X i ~ Poisson ( λ T ) i i
22 3.2.2 a) id i ix i pd i X f i ~ Bin ( pd, D ) i i xi Di xi ( x ; pd, D ) = C pd ( 1 pd ) i i i Di Xi i i (3.6) b) pd i NUREG/CR-6823 [4] µσlogit(p) logit( pdi ) µ ( ) = exp σ π ( ) f pd (3.7) i 2 pd 2 σ i 1 pdi logit( pd i ) = ln[ pd /( 1 pd )] < µ < σ > 0 i i c) µσ (3-4),(3-5) d)
23 a b a b c i pd i D i X i i pd i i X i i D i i µ,σ- a,b pd = logit 1 ( c) = e /(1 + e i i ( pd D ) X ~ Bin, i i c i c i )
24 a b a b p i T i i i X i Y i i p i i X i i Y i i λ i i T i i µ,σ- a,b α,β
25 µ σ E ( x) = ( b + a) / 2 2 Var( x) = ( b a) /12 a µ µ = E( µ ) b = E( µ ) + 3Var( µ ) 3Var( µ ) (3.8) a σ σ = E( σ ) b = E( σ ) + 3Var( σ ) 3Var( σ ) (3.9) µσ λ max λ min 90 E(σ)E(µ) [6] E( σ ) = ln( λ E( µ ) = ln λ max / λmin ) / max 1.645E( σ ) = ln λ max λ λ min λ max min (3.10) Var(µ)Var(µ) 10 5 Var(σ)(3.9)σa σ b σ a σ σ>0 σ(0.1,3)a σ =0.1, b σ =3 (EF)1.2 < EF <
26 a µ µ = E( µ ) b = E( µ ) + 3Var( µ ) 3Var( µ ) (3.11) E ( µ ) = ln λ λ Var ( µ ) = 10 max min a σ b σ = 0.1 = 3 (3.12) a µ b µ (3.8) pd max /(1 pdmax) E( σ ) = ( m n) /3.29= ln / 3.29 pdmin /(1 pdmin ) pd max pdmin E( µ ) = m 1.645E( σ ) = ln 1 pd max 1 pdmin (3.13) m = logit( pd ) = ln( pd /(1 pd )) n = logit( pd max min ) = ln( pd max min /(1 pd pd min pd max max min )) Var(µ) 10 a σ =0.1, b σ =3 a µ µ = E( µ ) b = E( µ ) + 3Var( µ ) 3Var( µ ) (3.14) E ( µ ) pd x pd 1 pdmax 1 pd = ma ln min min 20
27 Var ( µ ) = 10 aσ = 0.1 bσ = 3 (3.15) a) Y i Max( λi,mle) 0.5 p i p EXP,i Max(λi,MLE) i=my m X m = Ym / pexp, m X m λ max Beta( α, β ) α /( α + β ) Beta 4,6 ( ) p = E( Beta(4,6)) = 4 /(4 + 6) = 0. 4 ( 3.16) EXP, i m λ max λ max = Ym /( 0.4 T m ) = λm, MLE 2.5 (3.17) Y m m T m m λ m m,mle 21
28 b) Y i Min(λ i,mle ) 0.5 Min(λ i,mle )λ min λ min = Min( λ i, ) (3.18) MLE c) 1) µ f µ ) = b µ 1 a ( µ µ aµ µ b (3.19) i) a µ b µ = ln λ λ max = ln λ λ min max min λmax = Max( λ i, MLE) 2.5 ( i = 1,2, L, n) λ = Min( λ ( i = 1,2, L, n) min i,mle) λ : i i,mle ii) a µ b µ pd = ln pd 1 pd max = ln pd 1 pd max max max = Max( pd i, MLE pd 1 pd min min 30 pd min 30 1 pd min ) 2.5 ( i = 1,2, L, n) max + pd min = Min( pd i, MLE) ( i = 1,2, L, n) 22
29 pd, i MLE : i 2) σ f 1 a ( σ ) = aσ σ bσ bσ σ a 0.1, b = 3 (3.20) σ = σ 23
30 WinBUGS Ver [5] D/G 3.0E-7 [/h] 3.4E-11 [/h] 4.5E-4 [/d] 1.9E-6 [/d] 4.2 (3.19)(3.20) 4-2 D/G 4-2 a µ b µ a σ b σ 1.9E+1 8.5E E+1 1.4E E+1 9.9E E+1 3.0E
31 4.3 WinBUGS WinBUGS [5] 4-1~ 4-4 WinBUGS / 4-1~ [6] =1hr =1 =0 25
32 model { for(i in 1:N){ lambda[i] ~ dlnorm(mu,tau) nu[i] <- lambda[i]*t[i] x[i] ~ dpois(nu[i]) p[i] ~ dbeta(alpha,beta) y[i] ~ dbin(p[i],x[i]) } mu ~ dunif(amu,bmu) sigma ~ dunif(asigma,bsigma) tau <- 1/(sigma*sigma) total <- sum(x[1:n-1]) } list( # ******************************************* # set evidences # ******************************************** # hours list : last data is dummy t=c( , , , , , , , , , , , ,968142,868784, , , , , , , ,816102, , , , , , , , , , , , , , , , , , , , ,651150, , , , , , ,1), # events list : last data is dummy y=c( 1,3,0,3,0,1,3,1,1,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,3,3,0,0,0,0,2,1,1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0), # ******************************************* # set paramters # ******************************************** # data collection probability parameters alpha=4, beta=6, # hyper parameters for failure rates amu=-1.9e+1, bmu=-8.5, asigma=0.1, bsigma=3, # the number fo plants N=50 ) #************************************************* # initial value for keep away compile error #************************************************* list( x=c( 2,4,1,4,1,2,4,2,2,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,4,4,1,1,1,1,3,2,2,1,1,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1) ) 4-1 BUGS model { for(i in 1:N){ lambda[i] ~ dlnorm(mu,tau) nu[i] <- lambda[i]*t[i] x[i] ~ dpois(nu[i]) p[i] ~ dbeta(alpha,beta) y[i] ~ dbin(p[i],x[i]) } mu ~ dunif(amu,bmu) sigma ~ dunif(asigma,bsigma) tau <- 1/(sigma*sigma) total <- sum(x[1:n-1]) } list( # ******************************************* # set evidences # ******************************************** # hours list : last data is dummy t=c( , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , ,1), # events list : last data is dummy y=c( 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0), # ******************************************* # set paramters # ******************************************** # data collection probability parameters alpha=4, beta=6, # hyper parameters for failure rates amu=-2.5e+1, bmu=-1.4e+1, asigma=0.1, bsigma=3, # the number fo plants N=50 ) #************************************************* # initial value for keep away compile error #************************************************* list( x=c( 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1) ) 4-2 BUGS 26
33 model { for(i in 1:N){ m[i] ~ dnorm(mu,tau) p[i] <- exp(m[i])/(1+exp(m[i])) x[i] ~ dbin(p[i],n[i]) pd[i] ~ dbeta(alpha,beta) y[i] ~ dbin(pd[i],x[i]) } mu ~ dunif(amu,bmu) sigma ~ dunif(asigma,bsigma) tau <- 1/(sigma*sigma) total <- sum(x[1:n-1]) } list( # ******************************************* # set evidences # ******************************************** # demands list : last data is dummy n=c( 595,628,601,672,663,723,755,695,573,584, 638,474,359,324,508,419,477,554,346,414, 456,280,542,354,697,513,298,258,1015,1178, 1429,1358,1300,1149,1125,1612,1730,1120,984,1786, 1997,535,343,2038,1928,1944,1998,556,806, 1), # events list : last data is dummy y=c( 0,0,0,0,0,0,2,1,0,0, 2, 0,0,0,1,0,1,0,0,1, 0,0,0,0,0,2,0,0,0,0, 0, 0,0,1,0,2,0,0,3,0, 1,0,0,0,0,1,0,0,1,0), # ******************************************* # set paramters # ******************************************** # data collection probability parameters alpha=4, beta=6, # hyper parameters for failure rates amu=-1.2e+1, bmu=-9.9e-1, asigma=0.1, bsigma=3, # the number fo plants N=50) #************************************************* # initial value for keep away compile error #************************************************* list( x=c( 1,1,1,1,1,1,3,2,1,1, 3,1,1,1,2,1,2,1,1,2, 1,1,1,1,1,3,1,1,1,1, 1,1,1,2,1,3,1,1,4,1, 2,1,1,1,1,2,1,1,2,1) ) 4-3 D/G BUGS model { for(i in 1:N){ m[i] ~ dnorm(mu,tau) p[i] <- exp(m[i])/(1+exp(m[i])) x[i] ~ dbin(p[i],n[i]) pd[i] ~ dbeta(alpha,beta) y[i] ~ dbin(pd[i],x[i]) } mu ~ dunif(amu,bmu) sigma ~ dunif(asigma,bsigma) tau <- 1/(sigma*sigma) total <- sum(x[1:n-1]) } list( # ******************************************* # set evidences # ******************************************** # demands list : last data is dummy n=c( 5153,7310,7002,7819,7714,9043,9429,8686,7156,7300, 7976,5911,4489,4048,6354,10658,12219,6042,3785,3065, 7355,2483,9709,7953,6548,3174,1376,1202,2964,3122, 5432,5162,4941,3207,3098,5108,5462,3507,3080, 3621, 4037,2133,1358,3452,3254,4355,4498,2022,4560,1), # events list : last data is dummy y=c( 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0, 0,0,0,0,0,0,0,0,0,0), # ******************************************* # set paramters # ******************************************** # data collection probability parameters alpha=4, beta=6, # hyper parameters for failure rates amu=-1.4e+1, bmu=-3.0, asigma=0.1, bsigma=3, # the number fo plants N=50) #************************************************* # initial value for keep away compile error #************************************************* list( x=c( 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1, 1,1,1,1,1,1,1,1,1,1) ) 4-4 BUGS 27
34 % ) [7] D/G WinBUGSMC error 5% NUREG/CR-6823 [4] ~ 4-7 / )MC error5% 28
35 2 3 / 29
36 MCMC 3 MCMC
37 MCMC 3 MCMC
38 4-6 D/G 32 2 MCMC 3 MCMC
39 MCMC 3 MCMC
40 total sample: total sample: total sample: D/G total sample:
41 [2] EF 30 (5.1) [2] / / median mean T T = median = median T B 2 exp( σ / 2) = median T 1 ln EFT exp (5.1) medianb median T mean T EF T 30 35
42 5.2 D/G [2] D/G NUCIA21 D/G D/G D/G D/G
43 5-3 D/G 37
44 Plant 36Plant41 2 / 30 / D/G (NUCIA21 ) ( ) ( 2) () 0.5h h D/G No. [h] E E E E E E E E E E E f(t;θ) θ ( ) ( ) T F T; θ = T f t; θ dt (5.2) T R( T; θ ) 1 F( T;θ ) 0 = (5.3) L(θ) 38
45 l ( ) = f ( t ) L ;θ m θ i F( Tj; ) R( Tk ; θ ) i= 1 j = 1 n θ (5.4) k = 1 t i l T j m T k n 5-1 T LC T RC f F R ν ν () t = νλt 1 exp( λt ) ν () t = 1 exp( λt ) ν () t = exp( λt ) (ν, λ>0) (5.5) (5.6) (5.7) L(ν,λ) 39
46 11 ν ( ν, λ) νλ ( 1 ν L t exp λt ) = i i i ν [ ( )] 4 ν [ ( )] 4 ν [ ( )] ν 1 exp 0.5 λ 1 exp 2 λ exp 0.5 λ [ exp( 2 λ) ] 9224 (5.8) t i, i=1~ (5.8)(ν,λ) WinBUGS WinBUGS / R(t),F(t)WinBUGS zeros trick zeros trick I X00 ( I ) X 0 ~ Poisson (5.9) L zt (I) 0 zt ( I ) L = I 0 exp 0! ( I ) = exp ( I ) (5.10) I=-ln() (5.10) (5.6) T LC1 (=0.5h)N LC1 (=4)T LC2 (=2.0h)N LC2 (=4) I = ln = N NLC 1 NLC 2 ([ F( T )] [ F( T )] ) LC1 LC1 ln 1 LC 2 ν ν ( exp( λt )) N ln( 1 exp( λt )) LC1 LC 2 LC 2 (5.11) (5.7) T RC1 (=0.5h)N RC1 (=33090)T RC2 (=2.0h)N RC2 (=9224) 40
47 I N RC1 N RC ([ R( T )] [ R( T )] ) 2 = ln = N = N R C RC RC1 ν ν ( exp( T ) N ln( ( λt ) 1 ln exp ν RC1 1λT + N RC 2 λ RC1 RC 2 RC 2 ν RC 2λTRC 2 (5.12) λ( t) = νλ t ( ν 1) (5.13) τ λ 1 τ ( ν 1) ( τ ) λ() dt = λτ = τ 0 t (5.14) WinBUGS
48 model; { # Prior distributions for Weibull parameters v ~ dgamma(0.001,0.001) lambda ~ dgamma(0.001,0.001) # Likelihood for the complete data for( i in 1 : NCOM ) { TCOM[i] ~ dweib(v,lambda) } # Likelihood for the left censored data # zeros trick C < XLC <- 0 XLC ~ dpois(ilc) ILC <- C - NLC[1] * log( 1- exp( - lambda * pow(tcen[1],v) )) - NLC[2] * log( 1- exp( - lambda * pow(tcen[2],v) )) # Likelihood for the right censored data #zeros trick XRC <- 0 XRC ~ dpois(irc) IRC <- C + NRC[1] * lambda * pow(tcen[1],v) + NRC[2] * lambda * pow(tcen[2],v) # 24-hour mean failure rate FR24 <- lambda*pow(24,v-1) } DATA list( # Complete data TCOM = c(4.17e-3, 5.56E-3, 1.67E-2, 1.67E-2, 3.33E-2, 5.00E-2, 1.67E-1, 1.83E-1, 4.33E-1, 1.07,1.83), NCOM = 11, ) # Censoring data TCEN[] NLC[] NRC[] END INITS list(v=0.5,lambda=1.e-3) list(v=1.0,lambda=1.e-4) 5-2 WinBUGS D/G
49 % 5% 95% 97.5% EF E E E E E E (FR24) λ(lambda) 5.03E E E E E E-4 ν(v) E E E FR24 chains 1:2 sample: E E E E E lambda chains 1:2 sample: E v chains 1:2 sample: ν λ 5.49E E-4 24 [/h] 1.20E E WinBUGS 43
50 6 3 NUCIA PSA χ 2 90 B 44
51 /4 45 [h] 11 [1/h] [1/h] 90% [1/h] EF *5 6 [] 5% 95% [1/h] EF *7 [1/h] [1/h] 8 EF [1/h] E E E E E E E E % 434% E E E E E % 7% 2 6.2E E E E E E E E % 411% E E E E E E E E % 829% 2 3.7E E E E E E E E % 457% 1 1.8E E E E E E E E % 171% 2 9.7E E E E E E E E % 649% 1 3.1E E E E E E E E % 285% 6 6.8E E E E E E E E % 2221% 8 7.5E E E E E E E E % 227% 2 1.3E E E E E E E E % 102% E E % 100% 9 9.1E E E E E E E E % 3281% 0 9.1E E E E E E E % 72% 2 9.1E E E E E E E E % 375% 0 9.1E E E E E E E % 72% 1 9.1E E E E E E E E % 139% 0 3.4E E E E E E E % 58% 0 3.4E E E E E E E % 58% 0 3.4E E E E E E E % 58% 0 3.4E E E E E E E % 58% 0 3.4E E E E E E E % 58% E E E E E E E E % 418% E E E E E E E E % 1205% 1 4.9E E E E E E E E % 226% 1 4.9E E E E E E E E % 226% E E E E E E E E % 929% E E E E E E E E % 1031% 3 1.0E E E E E E E E % 573% 0 1.0E E E E E E E % 77% 0 1.0E E E E E E E % 77% 0 1.0E E E E E E E % 77% 1 6.5E E E E E E E E % 174% 4 6.5E E E E E E E E % 1328% 0 6.5E E E E E E E % 82% 1 6.5E E E E E E E E % 174% 3 1.5E E E E E E E E % 532% 4 1.5E E E E E E E E % 1043% 0 1.5E E E E E E E % 94% 1 1.5E E E E E E E E % 180% 0 1.7E E E E E E E % 64% 0 1.7E E E E E E E % 64% 0 1.7E E E E E E E % 64% 0 1.7E E E E E E E % 64% E E E E E E E E % 216%
52 /4 46 [h] 11 [1/h] [1/h] 90% [1/h] EF *5 6 [] BWR 0 3.6E E E E E E E % 122% 0 3.6E E E E E E E % 122% 0 3.6E E E E E E E % 122% 0 3.6E E E E E E E % 122% 0 3.6E E E E E E E % 122% (PWR) 0 2.2E E E E E E E % 114% 6 1.3E E E E E E E E % 757% E E E E E E E E % 222% 0 1.3E E E E E E E % 66% 1 1.3E E E E E E E E % 125% 1 1.3E E E E E E E E % 125% / 1 3.4E E E E E E E E % 172% 7 6.0E E E E E E E E % 1559% E E % 104% 1 3.9E E E E E E E E % 138% 0 3.9E E E E E E E % 67% 0 3.9E E E E E E E % 67% 0 3.9E E E E E E E % 67% 0 3.9E E E E E E E % 67% * E E E E E E E E % 128% E E E E E E E % 143% 2 1.6E E E E E E E E % 695% 0 6.5E E E E E E E % 96% 0 6.5E E E E E E E % 96% 0 5.4E E E E E E E % 94% 0 5.4E E E E E E E % 94% 0 5.4E E E E E E E % 94% / 0 1.9E E E E E E E % 93% () 0 1.9E E E E E E E % 93% 0 1.9E E E E E E E % 93% / 0 2.4E E E E E E E % 101% ( 0 2.4E E E E E E E % 101% 2 2.4E E E E E E E E % 464% (BWR) 0 4.4E E E E E E E % 107% (PWR) 0 1.2E E E E E E E % 125% PLR MG E E E E E E E E % 381% RPS,CRDM MG 0 1.3E E E E E E E % 81% (PLR) 2 6.7E E E E E E E E % 561% 1 1.9E E E E E E E E % 162% [1/h] 5% [1/h] [1/h] 95% [1/h] EF *7 8 EF 9
53 /4 [h] 11 [1/h] [1/h] 90% [1/h] EF *5 6 [] 9 7.1E E E E E E E E % 1379% E E E E E E E E % 329% 1 7.1E E E E E E E E % 186% 5 6.2E E E E E E E E % 514% 0 3.4E E E E E E E % 78% 1 3.4E E E E E E E E % 167% * E E E E E E E E % 581% * E E E E E E E % 78% 1 1.5E E E E E E E E % 168% 3 1.5E E E E E E E E % 520% * E E E E E E E % 90% 0 3.7E E E E E E E % 90% * E E E E E E E E % 440% 0 8.3E E E E E E E % 66% E E E E E E E E % 1476% E E E E E E E E % 1330% 0 6.9E E E E E E E % 60% 0 6.9E E E E E E E % 60% 0 4.4E E E E E E E % 67% / 3 4.4E E E E E E E E % 471% E E E E E E E % 198% ( 4 2.4E E E E E E E E % 980% 0 1.3E E E E E E E % 98% 3 1.3E E E E E E E E % 636% 3 2.4E E E E E E E E % 592% 1 5.9E E E E E E E E % 175% / 4 5.9E E E E E E E E % 473% 0 7.5E E E E E E E % 84% / 8 7.5E E E E E E E E % 448% 0 3.0E E E E E E E % 76% / 2 3.0E E E E E E E E % 398% 0 2.0E E E E E E E % 82% [1/h] / 5 2.0E E E E E E E E % 1027% 5% [1/h] [1/h] 95% [1/h] EF*7 8 EF 9
54 /4 [h] 11 [1/h] E E E E E E E / E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E % 100% E E % 100% E E % 100% EF 2 ( = 0EF= % EF *5 5% [1/h] 6 [] [1/h] [1/ h] [1/ h] [1/h] 7.EF 2 ( = 8. 9.EFEF 10. µ WinBUGSµ % [1/h] EF * % 410% 507% 507% 497% 338% 580% 320% 784% 432% 401% 681% 332% 734% 342% 622% 926% 543% EF 9 94% 227% 73% 73% 145% 2021% 154% 1067% 76% 900% 531% 157% 320% 68% 102% 209% 100% 100%
55 *1 [1/d] 90% [1/d] [1/d] EF *2 *3 [1/d] 5% [1/d] 95% [1/d] [ 1/d] EF *4 *5 EF E E E E E E % 131% DG E E E E E E E % 580% E E E E E E E % 550% E E E E E E E % 2328% E E E E E E E % 103% E E E E E E E % 144% E E E E E E E % 2583% E E E E E E E % 206% E E E E E E E % 80% E E E E E E E % 1946% E E E E E E E % 504% E E E E E E % 75% E E E E E E % 83% E E E E E E E % 103% E E E E E E E % 169% E E E E E E E % 94% E E E E E E % 136% E E E E E E % 102% E E E E E E % 101% E E E E E E E % 334% E E E E E E E % 184% E E E E E E E % 587% E E E E E E E % 191% E E E E E E % 91% ( 2.EF = 0EF= EF 2 (= 5. 6.EFEF
56 7 () () 7.1 λ λ ( µ σ ) λ ~ Lognormal, (λ,µ,σ>0) (7.1) µlogλσlogλ µ ex p( µ)=λµλ σ ex p (1.645σ)=λEF=(λ 95%)/(λ)σ λ 95% 7.2 pd pd ( µ,σ ) logit( pd ) ~ Normal 0<pd<1, µ,σ> 0 (7.2) pd logit( pd) = log 1 pd µlogit( pd)σ logit(pd) pd logit( pd) r = 1 pd ( ) µ,σ r ~ Lognormal (7.3) µlog r σlog r pd << 1 r pd pd ( µ,σ ) pd ~ Lognormal (7.4) 50
57 µlog pd σlog pd a) µ exp(µ)=pdµpd σ exp(1.645σ)=pdef=(pd 95%)/(pd) σ pd 95% µσ/ EF µ/ µ (/hr) E E E E E E E E E E E E E E E E-05 µ DG E E E E E E E E E E E E E E E E-01 51
58 7-2 σ EF () σ EF σ σ σµ Unif a b = 1 b + a (b a) 2 ( σ µ ) ~ (, ) = (7.5) b a 2 12 σ Unif ( 0.1,3) σ EF Unif 0.1, ~139 ( ) 1 ( 0.01,1.5 ) 2 ( 1,3.3) Unif ~12 Unif ~228 Unif ~228 3 ( 0.01,3.3) WinBUGS C 52
59 7-4(1) 1 53
60 7-4(2) 1 54
61 7-4(3) D/G 1 55
62 7-4(4) 1 56
63 7-5(1) 2 57
64 7-5(2) 2 58
65 7-5(3) D/G 2 59
66 7-5(4) 2 60
67 61 7-6(1) 3
68 7-6(2) 3 62
69 7-6(3) D/G 3 63
70 7-6(4) 3 64
71 7.3.4 a) σ σ 1 σ b) 1 1) λ i ) 0-65
72 c) 2-1 d) σ σ - σ σ 66
73 sigma sample: sigma sample: sigma sample: sigma sample: D/G 7-2(1) σ 67
74 sigma sample: sigma sample: sigma sample: sigma sample: D/G 7-2(2) 1 σ 68
75 sigma sample: sigma sample: sigma sample: sigma sample: D/G 7-2(3) 2 σ 69
76 sigma sample: sigma sample: sigma sample: sigma sample: D/G 7-2(4) 3 σ 70
77 7.4 µ µ 4 D/G µσ Unif a b = 1 b + a ( b a) ( µ σ ) ~ (, ) = b a (7.6) µ µ 3/4 20%
78 7-7 µ DG Unif(-19,-8.5) Unif(-25,-14) Unif(-12,-0.99) Unif(-14,-3.0) 1 Unif(-25,-14.5) Unif(-27.8,-16.8)* Unif(-18.0,-6.99) Unif(-16,-5.0) 2 Unif(-15.5,-5.0) Unif(-23,-12) Unif(-8.0,3.01) Unif(-12,-1.0) 3 Unif(-21.6,-5.9) Unif(-27.6,-11) Unif(-14.8,1.81) Unif(-16.8,-0.2) ) WinBUGS WinBUGS C 72
79 7-8(1) 1 73
80 7-8(2) 1 74
81 75 7-8(3) D/G 1
82 7-8(4) 1 76
83 77 7-9(1) 2
84 7-9(2) 2 78
85 79 7-9(3) D/G 2
86 7-9(4) 2 80
87 (1) 3
88 7-10(2) 3 82
89 7-10(3) D/G 3 83
90 7-10(4) 3 84
91 7.4.4 a) 1) µµµ mu sample: mu s ample: DG µ 2) µµµ µ 0 mu sample: mu s ample: µ b) 1 1) % 5% EF EF 7-5 µ σ 7-6 σσ σ EF σ
92 0 mu sample: mu sample: DG µ µσ 2) 0 EF 7-7 µ 7-4 µ µµ µ EF 86
93 mu sample: mu sample: µ 1 b) 2 1) 1 EF 7-8 µ (1)µ µµ EF mu sample: mu sample: DG µ 2 2) 0 EF 7-8 µ 7-4 µ µµ µef 87
94 mu sample: mu sample: µ 2 c) 3 1) 1 µ 7-10 µ EF 1 µ1 µ µ mu sample: mu sample: DG µ 3 2) 0 µ 7-11 µµ EF 0 µ 88
95 mu sample: mu sample: µ µ 0 µ 0.5 µ 2 µ a) NRCNUREG/CR-6928 [8] 1998~2002 NRCPRAstandardized plant analysis risk(spar)nureg/cr-6928 µ 1) NUREG/CR-6928 NUREG/CR-6928 EPIX Jeffreys 6.53E-10 [/h][/h/ft] EF 18.8 LLLower allowable limit 1.30E-5[/d] EF ) 7-12 µ µ
96 µ µ µ 7-11 NUREG/CR-6928 EF 6.53E E [/h] [/h] 1.30E E [/d] [/h] -2 µ µ 19.0µ E-9λ[/h]2.0E µ E-11λ[/h]8.3E-7 D/G 12.0µ E-6pd[/d]3.7E µ E-7pd[/d]5.0E-2 90
97 7-12 µ NUREG/CR-6928 b) µ µ 7-13 µ- lnλa µ b µ λ max λ min E(µ) Var(µ) µλ max λ min 0.5 λ max λ min λ min λ min λ min Feasibility Study λ max 91
98 µλ max λ min - µ µ PSA 92
99 - λi µ- λj µ ln λ Var(µ) =10 a µ b µ aµ = E(µ) 3Var( µ ) b µ = E( µ ) + 3Var( µ ) a µ E(µ) bµ µ λ 5-5%95% E(µ) E( σ ) = ln(95% / 5% ) / 3.29 E( µ ) = ln95% 1.645E( σ ) 5% E(µ) ln(λ min ) 5% 95% 95% ln λ ln(λ max /p exp ) 95% p exp 0.4 T Y [hour] =Y/T [/hour] E+6 1.0E-7 λ min E+5 4.0E-5 λ max : E+5 5.0E-6 Y= µ 93
100 µ 1 0 µ µ µ 1/2 µ 1 WinBUGS µ 94
101 8 8.1 p i PSAFFFFp i NUCIAPSAFF ipsaffx i y i NUCIA y ~ Binomial i ( p, x ) yi xi yi f ( y ) = C p ( 1 p ) i x i y i i i i 0 y x x i,y i 0 p 1 (8.1) i i < i p i ( 4,6) p i ~ Beta (8.2) 8-1 f ( p ) i 1 4 = p 1 1 i i B ( 4,6) ( p ) 6 1 B(4,6) (8.3) dbeta(x, 4, 6) p pd( ) 8-1 PSA NUCIA PSA
102 , y i =0 x i 0 x i ( x, y, p λ,θ t) f ( θ ) ( p ) f ( θ ) f ( x λ,t ) f ( y p, x ) f, f i λ i i i i i i i (8.4) i i i i () () x i xx i y i ( NUCIA) yy i p i pp i λi λλ i θ i t i ( ) tt i i ix i pifull conditional distribution( fcd ) (8.4)x i p i (8.5) f ( x, p y,, θ, t ) f ( x λ, t ) f ( y p, x ) f ( p ) i i i λ (8.5) i i i i i i i i i (8.5) 3 (8.6) (8.7) (8.8) f ( λ, ) = Poisson( x ; λ t ) xi ( λ i t ) exp( λ t ) i i i x i i t i i i i = (8.6) xi 96
103 xi y f ( y i p i, x i ) ial ( y i ;, x i ) = p x i i yi = Binom pi i ( 1 pi ) (8.7) yi f p i = Beta i = i i (8.8) B 4,6 ( ) ( p ;4,6) ( p ) 4 1 ( 1 p ) xi (8.6)(8.7) x i (8.6)t i = 2E6[h] λ i =5E-7[ /h] x i 8-2 (8..11)y i =0 x i 8-3 ( 8.8) p i ~Beta (4,6) 8-2 (8.6)x i λ i t i =5E-7[ 1/h] 2E 6 [h] =1 8-3 (8.7)x i y i =0, pi~beta(4,6) 97
104 p 8.4 (8.5)x i p i λit i=5e-7 [1/h] 2E6 [h] = 1y i =0pi~Beta( 4,6) ( 8.5)x i pi 0 0~ piyixi ip i full conditional dist ribution, (fcd)(8.4) p i (8.9) f ( p x,,, θ ) f ( ) f ( y p, x ) i i y i i,t i p i i λ (8.9) i i (8.9) 2 f f 4-1 ( ) = Beta( ;4,6) ( 1 p ) 6 i i 1 p (8.10) p i p i yi xi yi ( y p, ) = Binomial ( ; p, x ) p ( 1 p ) i (8.11) i x i y i i i i i (8.12) p i ( 8.9)(8.13) 98
105 f yi xi yi ( p x, y,, θ, t ) ( 1 p ) f i i i λ (8.12) i i p i ( p x,,, θ, t ) Beta( p ; y + 4, y + 6) i i y i i i = i i x i i i λ (8.13) WinBUGSp i (8.13)fcdx i p i (8.13)p i (yi+4)/(x i + 10) y 0 0 x i i p i 0.4 y i 0 y i x i y i /xi 0.4 p i p i y i p i / D/ G 4 p i
106 8-1 5% 95% y i y i + xi p[1] p[2] p[3] p[4] p[5] p[6] p[7] p[8] p[9] p[10] p[11] p[12] p[13] p[14] p[15] p[16] p[17] p[18] p[19] p[20] p[21] p[22] p[23] p[24] p[25] p[26] p[27] p[28] p[29] p[30] p[31] p[32] p[33] p[34] p[35] p[36] p[37] p[38] p[39] p[40] p[41] p[42] p[43] p[44] p[45] p[46] p[47] p[48] p[49] x i ( 4 ) ( +10) 100
107 8-2 5% 95% y i y i + xi p[1] p[2] p[3] p[4] p[5] p[6] p[7] p[8] p[9] p[10] p[11] p[12] p[13] p[14] p[15] p[16] p[17] p[18] p[19] p[20] p[21] p[22] p[23] p[24] p[25] p[26] p[27] p[28] p[29] p[30] p[31] p[32] p[33] p[34] p[35] p[36] p[37] p[38] p[39] p[40] p[41] p[42] p[43] p[44] p[45] p[46] p[47] p[48] p[49] x i ( 4 ) ( +10) 101
108 8-3 DG 5% 95% y i y i + xi p[1] p[2] p[3] p[4] p[5] p[6] p[7] p[8] p[9] p[10] p[11] p[12] p[13] p[14] p[15] p[16] p[17] p[18] p[19] p[20] p[21] p[22] p[23] p[24] p[25] p[26] p[27] p[28] p[29] p[30] p[31] p[32] p[33] p[34] p[35] p[36] p[37] p[38] p[39] p[40] p[41] p[42] p[43] p[44] p[45] p[46] p[47] p[48] p[49] x i ( 4 ) ( +10) 102
109 8-4 5% 95% y i y i + xi p[1] p[2] p[3] p[4] p[5] p[6] p[7] p[8] p[9] p[10] p[11] p[12] p[13] p[14] p[15] p[16] p[17] x i ( 4 ) ( +10) p[18] p[19] p[20] p[21] p[22] p[23] p[24] p[25] p[26] p[27] p[28] p[29] p[30] p[31] p[32] p[33] p[34] p[35] p[36] p[37] p[38] p[39] p[40] p[41] p[42] p[43] p[44] p[45] p[46] p[47] p[48] p[49]
110 8.3 pi Beta(4,6)pi Beta( 4,6) Beta 4, ( ) 1 Beta ( 120,180) Beta ( 7,3) Beta ( 37,13)
111 1 Beta( 120,180) Beta( 7,3) Beta( 37,13)
112 a µ b µ (1) 2 D/G a µ b µ 2.0E+1 8.7E+0 2.5E+1 1.4E+1 1.2E+1 1.2E+0 1.4E+1 3.3E+0 8-6(2) 3 D/G a µ b µ 2.0E+1 8.8E+0 2.5E+1 1.4E+1 1.2E+1 1.3E+0 1.4E+1 3.3E
113 8-7(1) 1 107
114 8-7(2) 1 108
115 8-7(3) D/G 1 109
116 8-7(4) 1 110
117 8-8(1) 2 111
118 8-8(2) 2 112
119 8-8(3) D/G 2 113
120 8-8(4) 2 114
121 8-9(1) 3 115
122 8-9(2) 3 116
123 8-9(3) D/G 3 117
124 8-9(4) 3 118
125 /EF 2 / 6 7 EF 3 2 // // a) 1 / / EF / EF / EF Y X Y=0 Y=3 X D / / 1) / EF / EF 2) / EF / EF Y=3 b) 2 D/G EF 6 / 119
126 / 7 EF 8-8 X EF c) / d) / 120
127 8-6 X Y=3 8-7 X Y=0 121
128 9 9.1 MCMC 3 a) 1) Brooks Gelman Rubin WinBUGS ) Geweke (e.g. 5%) b) 0 2 thinning over-relaxing 9-1 thinning BUGS µ µ0,µ1, µ2, µ3, µ4, µ5, µ6, µ7, µ8, µ9, µ10, µ11, µ12, µ13, µ14, µ15, µ16, µ17, µ18, thinning=5 thinning µ0, µ5, µ10, µ15, µ1, µ6, µ11, µ16, µ2, µ7, µ12, µ17, µ3, µ8, µ13, µ18, 5%95% etc. c) 5%95% etc. MC 5% 122
129 alpha0 chains 1: iteration (a) alpha0 chains 1: iteration (b) 9-1 [WINBUGS Manual] (a) 9-2 (b) 9.2 WinBUGS MCMC a) 7 D/G 5 Gelman and Rubin (1992) Brooks and Gelman (1998) 123
130 WinBUGS bgr-diag chain 5 1) 9-3(1)µσλ BGR pooledchain withinchain R R chain R mu chains 1: sigma chains 1:5 iteration iteration mu chains 1: sigma chains 1:5 iteration iteration lambda[50] chains 1: lambda[50] chains 1: iteration iteration 9-3(1) µσλ BGR (2) 124
131 mu lambda[50] lag sigma lag lag 9-3(2) 2) µσλ BGR 9-3(3) mu chains 1: iteration sigma chains 1: iteration lambda[50] chains 1: iteration mu chains 1: sigma chains 1:5 iteration iteration lambda[50] chains 1: iteration 9-3(3) µσλ BGR 125
132 (4) mu lag sigma lag lambda[50] lag 9-3(4) 3) D/G D/G µσλ(p) BGR 9-3(5) 126
133 mu chains 1: iteration sigma chains 1: iteration mu chains 1: sigma chains 1:5 iteration iteration p[50] chains 1: p[50] chains 1: iteration iteration 9-3(5) D/G µσλ BGR (6) D/G mu lag sigma lag p[50] lag 9-3(6) D/G 127
134 4) µσλ(p) BGR 9-3(7) mu chains 1: sigma chains 1:5 iteration p[50] chains 1:5 iteration iteration mu chains 1: sigma chains 1:5 iteration p[50] chains 1:5 iteration iteration 9-3(7) µσλ BGR (8) 128
135 mu lag sigma lag p[50] lag 9-3(8) b) R D/G µσµσ Thinning thinning thinning Thinning D/G 4 thinning1 thinning=10 µ σ a) µ(mu)σ(sigma) λ(lambda) thinning=1 thinning= (1) 129
136 mu sigma lag lag mu lag sigma lag lambda[50] lag lambda[50] lag thinning=1 thinning=10 9-4(1) thinning thinning=1 thinning=10 9-4(1) thinning 130
137 9-4(1) 131
138 b) µ(mu)σ(sigma)λ(lambda) thinning=1 thinning=10 9-4(2) mu sigma lag lambda[50] lag lag mu lag sigma lag lambda[50] lag thinning=1 thinning=10 9-4(2) thinning thinning=1 thinning=10 9-4(2) thinning 132
139 9-4(2) 133
140 c) D/G D/G µ(mu)σ(sigma) thinning=1 thinning=10 9-4(3) sigma lag mu lag sigma lag mu lag p[50] lag p[50] lag thinning=1 thinning=10 9-4(3) thinning D/G D/G thinning=1 thinning=10 9-4(3) thinning 134
141 9-4(3) D/G 135
142 d) µ(mu)σ(sigma) thinning=1 thinning=10 9-4(4) mu lag mu lag sigma lag sigma lag p[50] lag p[50] lag thinning=1 thinning=10 9-4(4) thinning thinning=1 thinning=10 9-4(4) thinning 136
143 9-4(4) 137
144 9.3.2 µσ thinning Thinning µσ thinning thinning=1 thinning= MC error thinning 9-5 µσ thinning thinning 10 µσ (λ,p)thinning thinning thinning=
145 10 21 FF EF ) EF ) EF 1 EF 3 1 D/G a) =4.8E-08/hEF=60.0 =2.9E-06/hEF=4.3 b) 4.7E-05/dEF=51.7 D/G 1.5E-03EF= NUCIA
146 PSA RISKMAN
147 a) b) c) D/G d)
148 a) b) c) D/G d) a) b) c) D/G d)
149 10.3 EF EF EF EF=60. EF= EF EF 5 EF EF / 3 PSA 10 EF EF
150 11 21 (CDF)BWRPSA PWR CDFBWRPWR 1982 ~ [2 ] [3] EF 11.1 BWR PWR CDF CDF CDF a) BWR 21 CDF TQUV CDF 21 D/G CDF ,000 CDF 1 CDF 10-6 CDF b) PWR 21 CDF TRIP CDF TRIP ATWS
151 TQUX TQUV LOCA LOCA TB TW TC 11-1 BWR 145
152 RECIRC INJ ISO SUP TRIP SPRAY HEAT ECCS ECCS PWR 146
153 TQUX TQUV LOCA LOCA TB TW TC 11-3 BWR 147
154 RECIRC INJ ISO SUP TRIP SPRAY HEAT ECCS ECCS PWR 148
155 10, BWR 149
156 11-1 BWR 10,
157 10, BWR 151
158 12 PSA PSA A.3 A PSA NUCIA PSA 20 function failure, FFFF PSA FF NUCIA PSA PSA E FF PSAFF JEAC FF FF 2 CDF 1 PSA FF PSA 12.2 FF PSA 152
159 a), PSA b) D/G c) = 153
160 NUCIA PSA 154
161 13 NUCIA PSA PSA PSA 155
162 [1] [2] ( ) 13 2, P00001,() [3] PSA 9 3, [4] C.L.Atwood, et al., Handbook of Parameter Est imation for Probabil istic Risk Assessment, NUREG/CR-6823, USNRC, September [5] su.cam.ac.uk/bugs/ [6] Meng Yue, Tsong-Lun Chu, Estimation offailure Rates of Digital Components Using a Hierarchical Bayesian Method, PSAM-8th, May [7] [8] S.A.Eide, et al, Industry-Average PerformancEFor Components and Initiating Events at U.S. Commercial Nuclear Power Plants, NUREG/CR-6928, US NRC, February
163 A PSA A.1 A.2 A.3 3 A.4 PSA 157
164 A PSA PSA PSA 18 PSA PSA PSA PSA 19 PSA PSA PSA PSA 3. a. NUREG/CR-6823, Handbook of Parameter Estimation for PRA,NRC, 158
165 2002 ASME, St andard for PRA for Nuclear Power Plant Applications, ASME RA-Sb-2005 b. PSA 1PSA PSA 4. ( ) 1-2 PSA PSA PSA H20.10 H20.11 H20.12 CDF 159
166 A () ( ) FBR GE ) 2 2
167 A.3 3 PSA :2518:15 ( ) ()JAEA)(), NELJAEA,TEPSYS, GE (JNES) 17 MHI()JAEA CTINELJAEA 1-1 PSA 1-2 PSA 1-3 PSA 1-4 C() 1-5 E() (1) (2)
168 1-3 (3) C() PSA P13SC Xi λi Beta(4,6) Pi SPAR EF SPAR (4) E() PSA P13SC (5) 1-4,
169 2 PSA :3016:30 () ()JAEA)(), NELJAEA,TEPSYS, GE MHI 17 ()JAEA CTIJAEATEPSYS 2-1 PSA 2-2 PSA µ σ PSA C() (1) (2) 163
170 (3) 2-3 pd pd (4) µ 2-4-1µ µ µ µ µ 0.5 µ 0.5 µ (5) σ 2-4-2σ σ σ 2-3 Y=0 X 5 Y=0 X
171 1/ (6) 2-5 (7) PSA 2-6 PSA PSA PSA FF (8) 2-8 / (9)
172 EF EF CDF BWR CDF (10) 1 166
173 PSA :3016:30 ( ) ()JAEA)(), NEL)TEPSYS, () GE ()MHI 16 ()JAEAJAEA CTIJAEA 3-1 PSA () pd () µ (1) (2) (3)
174 14 19 FF (4) pd () 3-3 pd () pd (5) 3-4 (6) µ 3-5 µ µ µ EPIX µ
175 (7) 3-6 PWR (8) 3-7 Winbugs DG (9) 3-8 FF 169
176 FF WEB (10) 170
177 A.4 PSA - ), - 0 µ,σ µ, σ 0 SPAR NUREG/CR EF SPAR L07005, 20 6 NUREG/CR
178 8.1 Y i X i p i λ i 2-3 X i Y i X i Y i =0 X i 5 Y i =0 X i X i λ i T i λ i =1/1000T i =1000Y i =0 X i p i ~Beta(4,6) = NUCIA 1 9 PSA 172
179 X Y NUCIA X Y, PSA PSA
180 µ µ 0.5 b) Beta(4,6)[ 0.4] µ µ µ µ µ µ a) µ 0.5 µ PSA FF FF CDF 1 EF EF EF PSA EF 5.2 D/G CDF 174
181 CDF BWR CDF EDG EDG 3-7 D/G D/G 175
182 FF 3-8 PSA PSA 176
183 B B.1 NUCIA NUCIA PSA NUCIA P SA NUCIA NUCIA NUCIA PSA MPFF FF NUCIA NUCIA NUCIA B.2 B.2.1 PSA PSA a) 6.2 NUREG/CR
184 B.2.2 ASME PRA ASMEPRAPSA ASME HLR-DA-B Grouping components into a homogeneous population for parameter estimation shall consider both the design, environmental, and service condition of the components in the as-built and as-operated plant. DA-B2 DO NOT INCLUDE outliers in the definition of a group (e.g. do not group valves that are never tested and unlikely to be operated with those that are tested or otherwise manipulated frequently) DA-B2 Category When warranted by sufficient data, USE appropriate hypothesis tests to ensure that data from grouped components are from compatible population. B.2.3 NRC PSA SPAR NUREG/CR-6928 NUREG/CR-6928Industry-Average Performance for Components and Initiating Events at U.S. Commercial Nuclear Power Plants U.S. NRC February 2007 NUREG/CR-6928 PSA EPIX 95% EF 178
185 NUREG/CR-6928 a) NUCIA NUCIA Positive Displacement Pump Emergency Diesel Generator Combustion Turbine Generator Hydro Turbine Generator Cooling Tower Fan Circuit Breaker b) NUREG/CR-6928 Service Water System Non-Service Water System d) NUREG/CR-6928 (standby)/(running/alternating) BWR RCIC HPCI HPCS PSA FT 179
186 B.3 NUCIA NUCIA PSA B-1 B.2 PSA NUCIA B.3.1 a) PSA NUREG/CR-6928 BWR Positive Displacement Pump NUCIA NUCIA NUCIA MG NUREG/CR-6928 b) NUCIA B-2 NUCIA B-2 B-1 PSA c) NUCIA 180
187 4 PSA B.3.2 B.3.1 a) b) 1) NUCIA 2) PSA 3) c) 1) 4 2) 3) 181
188 B-1 NUCIA PSA 182
189 B-2(1) NUCIA BWR 183
190 B-2(2) NUCIA PWR 184
191 C C-1 4 (µ,σ) (WinBUGS dflat())µ-σ µ-σ (µ,σ) C-1 0 C-1 0 / C-1 0 / =0 (µ,σ) MCMC 185
192 ( 0) (=0) DG ( 0) (=0) C-1 186
193 D Y X p Y ~ Bin f ( Y; p, X ) ( p, X ) Y X Y = X CYp (1 p) (D.1) p Beta ( 4,6) p=0.5 Y 3 X g( X p, Y ) f ( Y p, X ) g( X ) (D.2) p=0.5y=3 g( X p = 0.5, Y = 3) X 3 X 3 C 30.5 (1 0.5) g( X ) (D.3) g(x) X D-1 Y 3 p X D-2 D-2 D-1 p X MCMC p p MCMCD-2 Beta p i ( 4,6) p0.1~0.9 PY=3 (i=1,2,,9)p 1 =0.1p 2 =0.2p 3 =0.3 p 9 =0.9 P i [X] i D-3 D-3 Beta( 4,6) D-2 p=0.3~0.6 (p 3 ~p 6 ) D-3 D-4 187
194 P i [ X ] i (D.4) D-4 Beta( 4,6) Y=3 X 1 Beta( 120,180) p D-4 X D-2 p=0.4 1 X D-5 D-5 Y=0 X D-6 188
195 D-1 p=0.5y=3 X D-2 p=0.10.9y=3 X 189
196 D-3 p i =0.10.9Y=3 P [ X] i i D-4 Beta(4,6) X Y=3 190
197 D-5 X Y=3 D-6 X Y=0 191
198 E PSA E.1 2 PSA PSA E.2 1 PSA E.1 PSA E.2 PSA 192
199 E.1 PSA 193
200 194
201 195
202 196
203 197
204 198
205 199
206 200
207 E.2 PSA 201
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