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: History of (fictitious) component GV − 384 • Deformation Time Event Fail.mech. Severity • Leakage 0 Into service — — • Breakage 314 Failure Vibration Critical • Other mechanical 8.760 (Periodic) PM — — Failures classified as → 17.520 • (Periodic) PM — — Critical failures • 18.314 Failure Corrosion Degraded Degraded failures 20.123 Out of service — — Periodic PM ( 8 – 12 months → period) Ambition: To estimate the “effect” of maintenance . Critical and degraded failures → Models for Preventive Maintenance (PM) → (scheduled/unscheduled) (w/ competing risks) Imperfect repair → Several failure mechanisms (w/ competing risks) → 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: T 0 = 0, T 1 , T 2 , . . . → Inter-event times: Y i = T i − T i − 1 , i = 1, 2, . . . → Failure counts: N ( t ) = No. failures in ( 0, t ] . t T 1 0 T 2 T 3 Y 1 Y 2 Y 3 Basic hazard rate: ω ( t ) = hazard for unit of “age” t , ω ( t ) = f ( t ) /R ( t ) . � t Ω ( t ) = accumulated hazard up to time t , Ω ( t ) = 0 ω ( u ) du . MMR’02 – p.4

Modelling PM Motivation: Performance Perfect Degraded Critical t Critical Failure 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 t 0 X = t Critical Competing times: Z = PM (unscheduled) → 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 � z R ( 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: q x = 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 : Random Signs: f ( x | Z < X ) = f ( x ) f ( z, x | Z < X ) = ω ( z ) Ω ( x ) · ω ( x ) e − Ω ( x ) , z < x . Repair alert: f ( z | Z < X,X = x ) Marginal density of Z , conditional on Z < X : � ∞ ω ( z ) Ω ( x ) ω ( x ) e − Ω ( x ) dx = ω ( z ) Ie ( Ω ( z )) . f ( z | Z < X ) = 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 Dependent Independent X 1 → Leakage Y 1 → Breakage Z 1 . → Other . Y . X k Y k Z k MMR’02 – p.10

Modelling of repairable systems t T 1 0 T 2 λ ( t | F t − ) Conditional intensity at time t , given previous history of the Pr ( failure in [ t,t + ∆t ) | F t − ) process, λ ( t | F t − ) = lim ∆t ↓ 0 . ∆t � � Perfect repair: λ ( t | F t − ) = ω t − T N ( t ) where t − T N ( t ) is time since last event. Minimal repair: λ ( t | F t − ) = ω ( t ) . � � Imperfect repair: λ ( t | F t − ) = ω Ξ N ( t ) + t − T N ( t ) where Ξ j is the “effective age” of unit immediately after repair no. j . MMR’02 – p.11

Brown & Proschan (1983) Repair is � � λ ( t | F t − ) = ω Ξ N ( t ) + t − T N ( t ) → Perfect with prob. p . → Minimal with prob. 1 − p . t T 1 0 T 2 T 3 Y 1 Y 2 Y 3 Ξ 1 Ξ 2 Ξ 3   0 prob. p      prob. p · ( 1 − p ) Y i      prob. p · ( 1 − p ) 2 Y i − 1 + Y i Ξ i =  prob. p · ( 1 − p ) 3 Y i − 2 + Y i − 1 + Y i       ...     prob. ( 1 − p ) i Y 1 + ... + Y i MMR’02 – p.12

Full model Building blocks: k independent FMs. → • Time to critical failure is distributed s.t. ω i ( · ) 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): → • p i = probability of perfect repair for FM i at PM. • π i = probability of perfect repair for FM i at critical failure (corrective maintenance). Observations: “ Winning”FM i and type of event (PM or Critical Failure). → Times between events. → Model parameters: q, ( p i ,π i ,ω i ( · ) , i = 1,...,k ) MMR’02 – p.13

OREDA: Gas turbine data 23 mechanical units. 22 failures: → → • 8 critical. Totally 603.690 operating hrs. → • 14 degraded. Four failure mechanisms (FMs). → 78 periodic PM events ( 8 – 12 months period). → Deformation Leakage Breakage Other # Crit. 1 4 1 2 2 8 0 4 # Degr. p i ^ .6 .3 1 .8 π i , π i def = p κ κ = 1 · 10 − 2 . ^ i with ^ 1 1 1 1 Our model: � 4.0 · 10 5 7.7 · 10 4 1.2 · 10 6 1.8 · 10 5 MTTFF(Naked) OFR model: � 6.0 · 10 5 1.5 · 10 5 6.0 · 10 5 3.0 · 10 5 MTTFF(Naked) We used ω i ( t ) = µ i 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) Y 1 ,Y 2 ,... 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 ( p i , π i , 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