Modelling imperfect maintenance and repair of components under - - PowerPoint PPT Presentation

Modelling imperfect maintenance and repair of components under competing risk

Helge Langseth and Bo H. Lindqvist {helgel, bo}@math.ntnu.no. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

MMR’02 – p.1

Motivation: OREDA data

→

Four different failure mechanisms:

• Deformation
• Leakage
• Breakage
• Other mechanical

→

Failures classified as

• Critical failures
• Degraded failures

→

Periodic PM (8–12 months period) History of (fictitious) component GV −384 Time Event Fail.mech. Severity Into service — — 314 Failure Vibration Critical 8.760 (Periodic) PM — — 17.520 (Periodic) PM — — 18.314 Failure Corrosion Degraded 20.123 Out of service — —

Ambition: To estimate the “effect” of maintenance.

→

Models for Preventive Maintenance (PM) (scheduled/unscheduled) (w/ competing risks)

→

Several failure mechanisms (w/ competing risks)

→

Critical and degraded failures

→

Imperfect repair

MMR’02 – p.2

Database definitions

Events:

→ Failures (possibly several failure mechanisms - or root

causes)

→ Maintenance (Preventive (PM) or Corrective)

Distribution of time between events:

→ Random (failures, unscheduled PM) → Deterministic/periodic (scheduled PM)

Note: Duration of maintenance and repair are assumed negligible.

MMR’02 – p.3

Notation and definitions

Notation:

→ Event times: T0 = 0, T1, T2, . . . → Inter-event times: Yi = Ti − Ti−1, i = 1, 2, . . . → Failure counts: N(t) = No. failures in (0, t].

Y1 T1 Y2 T2 Y3 T3 t

Basic hazard rate: ω(t) = hazard for unit of “age” t, ω(t) = f(t)/R(t). Ω(t) = accumulated hazard up to time t, Ω(t) = t

0 ω(u) du.

MMR’02 – p.4

Modelling PM

Motivation:

Performance Degraded Perfect Critical Critical Failure t

Preventive maintenance(PM):

→ Basically at fi xed intervals of length τ. → Non-scheduled PM due to casual observation of evolving

critical failure – classifi ed as “degraded failure”.

MMR’02 – p.5

Modelling unscheduled PM

Z = PM (unscheduled) X = tCritical t

Competing times:

→ X = (potential) time to critical failure, hazard ω(x). → Z = (potential) time to (unscheduled) PM.

Observed time to next event:

→ Y = min(X, Z).

Cooke’s Random signs model:

→ q = probability of catching a critical failure by PM. → 1 − q = probability that degraded failure develops to

critical failure.

→ {Z < X} ⊥

⊥ X ⇒ P(Z < X|X = x) = P(Z < X) = q.

MMR’02 – p.6

Repair alert: Defi ning P(Z ≤ z|Z<X, X=x)

Assumption:

→ The component emits a “repair alert signal” r(·) with

R(z) = z

0 r(u) du s.t.

P(Z ≤ z|Z < X, X = x) = R(z) R(x), 0 ≤ z ≤ x, for some positive and continuous function r(·). Remarks:

→ R(·) – and therefore r(·) – is identifi able under the

assumptions above.

→ We use r(t) = ω(t) in our model (saves a number of

parameters).

MMR’02 – p.7

Wrap-up: Unscheduled PM

Assumptions:

→ Random signs: qx = P(Z < X|X = x) = P(Z < X) = q. → The repair alert model with R(z) = Ω(z), i.e.,

P(Z ≤ z|Z < X, X = x) = Ω(z) Ω(x), 0 ≤ z ≤ x . Motivation: The maintenance model should reflect the real behavior of the maintenance crew. We have f(z|Z < X, X = x) ∝ ω(z) when R(z) = Ω(z). This is a coarse, but maybe also a somewhat reasonable approximation.

MMR’02 – p.8

Modelling time to next event

Joint density of Z, X, conditional on Z < X: f(z, x|Z < X) = ω(z) Ω(x) · ω(x)e−Ω(x) , z < x .

Random Signs: f(x|Z < X) = f(x) Repair alert: f(z|Z < X,X = x)

Marginal density of Z, conditional on Z < X: f(z|Z < X) = ∞

z

ω(z) Ω(x) ω(x)e−Ω(x) dx = ω(z) Ie(Ω(z)) . Ie(·) is the exponential integral, Ie(x) = ∞

x exp(−u)/u du.

Distribution of time to next event: f(y) = (1 − q) ω(y) exp{−Ω(y)} + q ω(y) Ie(Ω(y))

MMR’02 – p.9

Several failure mechanisms

Assume component is subject to k competing risks (failure mechanisms (FMs), or root causes). We treat these as independent risks. We have k = 4 in the OREDA-data, where the FMs are:

→ Deformation → Leakage → Breakage → Other

. . . Zk X1 Xk Z1 Dependent Independent Y1 Y Yk

MMR’02 – p.10

Modelling of repairable systems

λ(t|Ft−) t T2 T1

Conditional intensity at time t, given previous history of the process, λ(t|Ft−) = lim∆t↓0

Pr(failure in [t,t+∆t)|Ft−) ∆t

. Perfect repair: λ(t|Ft−) = ω

• t − TN(t)
• where t − TN(t) is

time since last event. Minimal repair: λ(t|Ft−) = ω(t). Imperfect repair: λ(t|Ft−) = ω

• ΞN(t) + t − TN(t)
• where Ξj

is the “effective age” of unit immediately after repair no. j.

MMR’02 – p.11

Brown & Proschan (1983)

Repair is

→ Perfect with prob. p. → Minimal with prob. 1 − p.

λ (t|Ft−) = ω

• ΞN(t) + t − TN(t)
• Ξ1

Y1 Ξ3 Ξ2 T3 Y3 T2 Y2 T1 t

Ξi =                       

• prob. p

Yi

• prob. p· (1−p)

Yi−1 +Yi

• prob. p· (1−p)2

Yi−2 +Yi−1 +Yi

• prob. p· (1−p)3

... Y1 +... +Yi

• prob. (1 −p)i

MMR’02 – p.12

Full model

Building blocks:

→

k independent FMs.

• Time to critical failure is distributed s.t. ωi(·) may vary over FMs.

→

Each FM modelled as a competition between Critical Failure and unscheduled PM (motivated by Degraded Failure).

• q = probability of “

PM wins”.

• Repair alert with R(t) = Ω(t).

→

Imperfect repair (Brown & Proschan, 1983):

• pi = probability of perfect repair for FMi at PM.
• πi = probability of perfect repair for FMi at critical failure (corrective

maintenance). Observations:

→

“ Winning”FM i and type of event (PM or Critical Failure).

→

Times between events. Model parameters: q,(pi,πi,ωi(·), i = 1,...,k)

MMR’02 – p.13

OREDA: Gas turbine data

→

23 mechanical units.

→

Totally 603.690 operating hrs.

→

Four failure mechanisms (FMs).

→

22 failures:

• 8 critical.
• 14 degraded.

→

78 periodic PM events (8–12 months period). Deformation Leakage Breakage Other # Crit. 1 4 1 2 # Degr. 2 8 4 ^ pi .6 .3 1 .8 ^ πi, πi def = pκ

i with ^

κ = 1· 10−2. 1 1 1 1 Our model: MTTFF(Naked) 4.0· 105 7.7· 104 1.2· 106 1.8· 105 OFR model: MTTFF(Naked) 6.0· 105 1.5· 105 6.0· 105 3.0· 105 We used ωi(t) = µi in these calculations.

MMR’02 – p.14

On the identifiability

The competing failure mechanisms: Individual failure mechanisms are in general non-identifiable, but identifiability holds under assumption of independent failure mechanisms. The Brown & Proschan’s model: Repair is

→

Perfect with probability p

→

Minimal with probability 1−p Let the models be indexed by (p,F), and suppose that data are (only) Y1,Y2,... Then it holds that:

→

p is not identifiable if “ F”is exponential (Whitaker and Samaniego, 1989).

→

If 0 < q < 1 is required in our model, then “ F”cannot be exponential ⇒ Identifiability is OK. Random signs & Repair alert:

→

Must assume random signs class of models (can be falsified, but cannot be proved right).

→

Repair alert function R(·) identifiable

MMR’02 – p.15

Summary

→ Estimated parameters (pi, πi, q) give information on:

• Quality of maintenance, e.g. for comparison between

systems or plants.

• Appropriateness of sub-models.

→ Modest demands of data (available in most reliability data

banks):

• Time To Failure.
• Failure Mechanism and Severity.
• PM program.

→ Estimation of parameters by Maximum Likelihood.

MMR’02 – p.16

The end...

MMR’02 – p.17

Alternative modelling of time to PM

→

The suggested model connects the time for unscheduled PM to the given failure mechanism (much in a “ black box”fashion).

→

More careful modelling the degradation process:

• Discrete Markov models
• Discrete semi-Markov models
• Continuous processes (e.g. Gaussian

processes, gamma processes) References: Cooke (1996), Hokstad and Frøvig (1996), Aalen and Gjessing (1999), Lindqvist and Amundrustad (1998), Limnios (MMR 2000)

MMR’02 – p.18