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

A generalized Brown-Proschan model for preventive and corrective maintenance laurent.doyen@iut2.upmf-grenoble.fr Jean Kuntzmann Laboratory Grenoble University France 1 of 28

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

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

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

II/ A maintenance data set PM and CM times of a subsystem of a fossil-fired thermal plant of EDF: 1179 2640 4101 5562 7023 8035 8329 8376 8393 8455 8484 8494 8605 8628 8641 8744 8903 9105 9660 9845 9846 9866 9919 9985 9987 10010 10363 10470 11021 11494 12851 12956 13662 14161 14217 14244 25 20 15 10 5 0 0 2000 4000 6000 8000 10000 12000 14000 Cumulative number of failures 5 of 28

III. Notations 0 PM PM c CM CM PM CM T m t 3 2 1 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 3 2 1 T 1 T 2 T 3 6 of 28

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

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

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

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

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

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

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

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

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

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

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

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

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

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

MLE combined with moment estimation: α = 1.84 10 6, β = 1.95, p = 0.602, log(lt ( α, β, p)) = 150.900 MLE computed with EM algorithm or direct maximization: ˆα = 1.87 10 6, ˆβ = 1.94, ˆp = 0.614, log(lt (ˆα, ˆβ, ˆp)) = 150.902 25 20 15 10 5 0 0 2000 4000 6000 8000 10000 12000 14000 π ˆθ m : 0.61 0.62 0.69 0.73 0.07 0.00 1.00 0.99 0.96 0.46 Wearing out state at time c failures surge ABAO ABAO AGAN AGAN AGAN 22 of 28

Quality of estimation: θ = (1.9e 6, 1.9, 0.61) } NEB: Normalized Empirical Bias estimated over NESD: Norm. Emp. Standard Deviation 60 000 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

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 1-7.8 e-2-5.4 e-2 5.8 e-1 5.8 e-1 2-9.3 e-2-5.8 e-2 5.9 e-1 5.8 e-1 3-9.4 e-2-5.8 e-2 5.9 e-1 5.7 e-1 4-6.8 e-2-5.8 e-2 6.0 e-1 5.6 e-1 5-4.3 e-2-5.6 e-2 5.3 e-1 5.2 e-1 6-3.7 e-1-6.1 e-2 6.4 e-1 5.8 e-1 7-1.3 e-2-5.8 e-2 2.8 e-1 4.6 e-1 8 7.2 e-3-5.8 e-2 3.9 e-1 5.1 e-1 9 9.4 e-4-5.4 e-2 4.0 e-1 5.2 e-1 10-8.5 e-3-5.5 e-2 4.7 e-1 5.5 e-1 Mean -7.5 e-2-5.7 e-2 5.1 e-1 5.4 e-1 Individual PM efficiency estimation is relevent after c 24 of 28

Failure intensity for θ = ˆθ 0.01 0.008 0.006 0.004 0.002 0 0.9 1 1.1 1.2 1.3 1.4 25 Cumulative failure intensity and number of failures for θ = ˆθ 20 15 10 5 0 0.9 1 1.1 1.2 1.3 1.4 25 of 28

Forecast reliability for θ = ˆθ 1 0.8 0.6 0.4 0.2 0 44 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 Forecast cumulative number of failures for θ = ˆθ x 10 4 40 36 32 28 24 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 x 10 4 26 of 28

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... 27 of 28

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