A generalized Brown-Proschan model for preventive and corrective maintenance. Laurent DOYEN.

Transcription

1 A generalized Brown-Proschan model for preventive and corrective maintenance Jean Kuntzmann Laboratory Grenoble University France 1 of 28

2 I. Introduction The dependability of complex repairable systems depends strongly on the efficiency of maintenance actions. Corrective Maintenance (CM): After a failure, put the system into a state in which it can perform its function again. Preventive Maintenance (PM): When the system is operating, intends to slow down the wear process. Basic maintenance effect assumptions: As Bad As Old (ABAO): restores the system in the same state it was just before maintenance. As Good As New (AGAN): restores the system as if it were new. Reality is between these two extreme cases: Imperfect maintenance. 2 of 28

3 Brown-Proschan model [83], CM is : AGAN with probability p, ABAO with probability 1 p. repair effects (AGAN or ABAO) are mutually independent and independent of already observed failure times. CM effects can be characterized with random variables: { 1 if the i th CM is AGAN B i = 0 if the i th CM is ABAO Statistical studies when the {B i } i 1 are known: Whitaker-Samaniego[89], Hollander-Presnell-Sethuraman [92], Kvam-Singh- Whitaker [02], Bathe-Franz [96], Agustin-Pena[99],... 3 of 28

4 Practical purposes : {B i } i 0 are hidden variables. p = 0 : ABAO p characterizes maintenance efficiency: p = 1 : AGAN 0 < p < 1 : imperfect Joint assessment of CM efficiency and intrinsic wear-out: Lim [98]; Lim-Lie[00]; Lim-Lu-Park[98]; Langseth-Lindqvist [04]. Presentation aim: Generalize the BP model to PM effects Assess PM efficiency and intrinsic wear-out when PM effects are unknown Compute reliability indicators. 4 of 28

5 II/ A maintenance data set PM and CM times of a subsystem of a fossil-fired thermal plant of EDF: Cumulative number of failures 5 of 28

6 III. Notations 0 PM PM c CM CM PM CM T m t C 1 C 2 C 3 C 4 u 1 = 0 u 2 = 0 u 3 = 1 u 4 = 0 τ 1 τ 2 τ 3 N t T 1 T 2 T 3 6 of 28

7 IV. Model assumptions λ(t) the failure rate of the new unmaintained system. Λ(t) = t 0 λ(s) ds, Maintenance durations are not taken into account, PM are done at deterministic times, all the PM times are known CM are done at random times, CM are observed over [c, T ] (c < T ) CM effects are ABAO, PM effects follow a Brown-Proschan model, i.e. PM effects are : mutually independent and independent of previous failure times, AGAN with probability p, ABAO with probability 1 p. B i = { 1 if the i th PM is AGAN 0 if the i th PM is ABAO P (B i = 1 T i, B i 1 ) = P (B i = 1) = p 7 of 28

8 V. Joint assessment of intrinsic wear-out and PM efficiency a/ Maximum likelihood method Likelihood associated to a single observation of the failure process over [c, t]: L t (θ) = f T nt N t =n t (t nt ) P (N t = n t ) Maximum Likelihood Estimator (MLE): ˆθ = arg max L T (θ) = arg max log(l T (θ)) θ θ Notations: L c (θ) = 1 L τ,t (θ) = f T nt nτ,n t τ=n t n τ (t nτ +1 τ,..., t nt τ) the likelihood associated to the system new in τ and observed over[c τ, t]. 8 of 28

9 Let us denote : D m t = (1 p) m t m c τ m t j t λ(t j τ m ) Two equivalent equations for likelihood recursive computation: L t (θ) = m t m=0 p 1l {m 0} D m t forward computation algorithm. L t (θ) = e (Λ(t τ m) Λ((c τ m ) τ m )) L c τm (θ) p 1l {τ t} Dτ 0 m e (Λ(τ m c) Λ(c)) L τm,t(θ) τ {τ 1,...,τ mt,t} backward computation algorithm. 9 of 28

10 Proof of the equation used for the forward computation: {B m = 1, {B j = 0} m<j mt }{{} τ m is the last AGAN PM since t } 0 m mt is a partition of the probability space. Given B m = 1, the CM process can be divided into 2 independent processes. L t (θ) = [ m t p 1l {m 0} (1 p) m t m }{{} m=0 P (B m =1,{B j =0} m<j mt ) c τ m t j t λ(t j τ m ) e (Λ(t τ m) Λ((c τ m ) τ m )) }{{} Given B m =1,{B j =0} m<j mt, CM times after τ m follow a NHPP initialized in τ m ] L c τm (θ) }{{} Given B m =1, maintenances before τ m follow a BP PM-ABAO CM model 10 of 28

11 b/ MLE combined with moment estimation : λ(t) = αβt β 1 Power Law Process: E[N T ] = α(t β c β ) α β = N T /(T β c β ). BP model, E[N t ] = α S t, where S t = m τ τ {τ m c+1,...,τ mt,t} k=0 p 1l {k 0} (1 p) m τ k ((τ τ k ) β ((τ mτ c) τ k ) β ) MLE combined with moment estimation of α: α β,p = N T S T E[ α β,p ] = α ( β, p) = arg max (β,p) log(l T ( α β,p, β, p)) and α = α β, p Dimension reduction of the likelihood maximization space. 11 of 28

12 c/ Individual PM efficiency assessment p characterized the average global PM efficiency. The m th PM effect can be characterized by: π θ m = P (B m = 1 N T = n T, T nt = t nt ) which verifies: π θ m = p L c τ m (θ) L τm,t (θ) L T (θ) It can naturally be estimated by: π ˆθ m L τm (ˆθ): intermediate computing values of the forward algorithm L τm,t (ˆθ): intermediate computing values of the backward algorithm 12 of 28

13 d/ Expectation-Maximization algorithm Complete likelihood: L c t (θ) = f T nt N t =n t,b nt =b nt (t nt ) P (N t = n t B nt = b nt )P (B nt = b nt ) EM algorithm: Expectation (E) step: Compute Q(θ θ k ) = E θ k [log(l c t (θ)) N] Maximization (M) step: θ k+1 = arg max θ θ k k local likelihood maxima Q(θ θ k ) 13 of 28

14 Complete likelihood: L c t (θ) = N t n=1 λ(a tn ) e t c λ(a s ) ds m t m=1 p B m (1 p) 1 B m where A s is the virtual age, ie the time elapsed since the last perfect PM m s ]τ m, τ m+1 ] [c, T ] λ(a s ) = [λ(s τ m i )] 1l {Bτm i } i=0 where Bτ i m = at time τ m, the last AGAN PM time is τ m i 14 of 28

15 Complete likelihood: E θ [log(l c t (θ)) N]=Q(p θ) {}}{ L c t (θ) = N t n=1 λ(a tn ) e t c λ(a s ) ds }{{} E θ [log(. ) N]=Q 2 (λ θ) m t m=1 p B m (1 p) 1 B m }{{} E θ [log(. ) N]=Q 1 (p θ) where A s is the virtual age, ie the time elapsed since the last perfect PM m s ]τ m, τ m+1 ] [c, T ] λ(a s ) = [λ(s τ m i )] 1l {Bτm i } i=0 where Bτ i m = at time τ m, the last AGAN PM time is τ m i 15 of 28

16 Complete likelihood: E θ [log(l c t (θ)) N]=Q(p θ) {}}{ L c t (θ) = N t n=1 λ(a tn ) e t c λ(a s ) ds }{{} E θ [log(. ) N]=Q 2 (λ θ) m t m=1 p B m (1 p) 1 B m }{{} E θ [log(. ) N]=Q 1 (p θ) where A s is the virtual age, ie the time elapsed since the last perfect PM m s ]τ m, τ m+1 ] [c, T ] λ(a s ) = [λ(s τ m i )] 1l {Bτm i } i=0 where Bτ i m = at time τ m, the last AGAN PM time is τ m i Q 2 function of πm,i [1l θ = Eθ {B i τm } N T = n T, T nt = t nt [ ] Q 1 function of πm,0 θ = πθ m = E θ B m N T = n T, T nt = t nt ] 16 of 28

17 E step: Compute for 0 m m T and 0 i m, π 0,0 (θ) = 1 π θ m,i = Eθ [1l {B i τm } N T = n T, T nt = t nt ] for m {1,..., m T }, π θ m,0 = πθ m for 0 m < m T and 0 i m, π θ m+1,i+1 = πθ m,i p1+1l {m>i} (1 p) i (c τ m i )<t j τ m+1 λ(t j τ m i ) e (Λ((c τ m+1) τ m i ) Λ((c τ m i ) τ m i )) L c τ m i (θ)l τm+1,t (θ) L T (θ) 17 of 28

18 M step: λ(t) = αβt β 1 p k+1 = β k+1 = arg max β [ mt ] m=1 πθ k m,0 /m T τ {τ m c+1,...,τ mt,t } with S k (β) = [ n T (log(n T β) log(s k (β))) + (β 1) m τ i=0 τ {τ m c+1,...,τ mt,t } π θ k m τ,m τ i m τ i=0 (c τ mτ )<t j τ π θ k m τ,m τ i ] log(t j τ i ) ((τ τ i ) β ((c τ mτ ) τ i) β ) α k+1 = n T /S k (β k+1 ) 18 of 28

19 VI. Reliability indicators Failure intensity: λ t = lim t 0 1 t P (N t+ t N t T Nt, N t ) λ t = m t m=0 Dm C Kt e (Λ(t τ m) Λ((c τ m ) τ m ) L c τm (θ) m t m=0 Dm C Kt λ(t τ m ) e (Λ(t τ m) Λ((c τ m ) τ m ) L c τm (θ) Cumulative failure intensity: Λ t = t 0 λ s ds K t Λ t = log L Ck (θ) log L t (θ) L C + (θ) L k 1 C + (θ) Kt k=1 19 of 28

20 Reliability: P (T NT +1 > s T NT, N T ) = exp( (Λ s Λ T )) Expected cumulative number of failures: E[N s T NT, N T ] = N T + + m s i=m T +1 [ mt (Λ(s τ i ) + E[N τ i T NT, N T ] N T )p(1 p) m s i ] (Λ(s τ i ) Λ(T τ i ))(1 p) m s m T πm θ T,m T i i=0 20 of 28

21 VII. Application The BP PM-ABAO CM model is implemented in MARS (Maintenance Assessment of Repairable Systems): a free software developped by LJK (Grenoble university) and EDF (French electricity utility). 21 of 28

22 MLE combined with moment estimation: α = , β = 1.95, p = 0.602, log(lt ( α, β, p)) = MLE computed with EM algorithm or direct maximization: ˆα = , ˆβ = 1.94, ˆp = 0.614, log(lt (ˆα, ˆβ, ˆp)) = π ˆθ m : Wearing out state at time c failures surge ABAO ABAO AGAN AGAN AGAN 22 of 28

23 Quality of estimation: θ = (1.9e 6, 1.9, 0.61) } NEB: Normalized Empirical Bias estimated over NESD: Norm. Emp. Standard Deviation simulations θ : MLE estimator when the B i are known Initial intensity: λ(t) = αβt β 1 or λ(t) = β/η(t/η) β 1 EM is more robust than direct likelihood maximization NEB NESD α η β p α η β p θ 590% 36% 8.2% -9.4% 3300% 130% 35% 50% ˆθ 580% 37% 9.2% -12% 3300% 130% 36% 51% θ 18000% -24% -12% 0% 5000% 35% 22% 25% 23 of 28

24 Quality of individual PM efficiency estimation: Empirical mean Empirical standard deviation m th PM π θ m B m p B m π θ m B m p B m e e e e e e e e e e e e e e e e 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 Mean -7.5 e e e e-1 Individual PM efficiency estimation is relevent after c 24 of 28

25 Failure intensity for θ = ˆθ Cumulative failure intensity and number of failures for θ = ˆθ of 28

26 Forecast reliability for θ = ˆθ Forecast cumulative number of failures for θ = ˆθ x x of 28

27 VIII. Prospects Maintenance times optimization Consider a more general distribution over ], 1] for the B i. Develop confidence intervals and tests for the BP model of 28

28 References Agustin MZN and Peˆna EA. Order statistics properties, random generation, and goodness-of-fit testing for a minimal repair model. Journal of American Statistical Association 1999; 94: Bathe F and Franz J. Modeling of repairable systems with various degrees of repair. Metrika 1996; 43: Brown M and Proschan F. Imperfect repair. Journal of Applied Probability 1983; 20: Kumar U and Klefsjö B. Reliability analysis of hydraulic systems of LHD machines using the power law process model. Reliability Engineering and System Safety 1992; 35: Kvam PH, Singh H and Whitaker LR. Estimating distributions with increasing failure rate in an imperfect repair model. Lifetime Data Analysis 2002; 8: Hollander M, Presnell P and Sethuraman J. Nonparametric methods for imperfect repair models. The Annals of Statistics 1992; 20: Langseth H and Lindqvist BH. A maintenance model for components exposed to several failure mechanisms and imperfect repair. In Mathematical and Statistical Methods in Reliability, B.H. Lindqvist, K.L. Doksum (eds). World Scientific: 2004; Lim JH, Lu KL, Park DH. Bayesian imperfect repair model. Communications in Statistics - Theory and Methods 1998; 27: Lim TJ. Estimating system reliability with fully masked data under Brown-Proschan imperfect repair model. Reliability Engineering and System Safety 1998; 59: Lim TJ and Lie CH. Analysis of system reliability with dependent repair modes. IEEE Transactions on reliability 2000; 49: Whitaker LR and Samaniego FJ. Estimating the reliability of systems subject to imperfect repair. American Statistical Association 1989; 84: Journal of 28 of 28