1. 1 The 4 standard failure models -to be used in maintenance optimization, with focus on state modelling Professor Jørn Vatn

2. 2 Situations and maintenance tasks 1.Observable gradual failure progression Inspect at regular intervals (or with shorter and shorter intervals) Replace when degradation is high 2.Observable “sudden” failure progression Inspect at regular intervals Replace if failure progression is detected 3.Non-observable failure progression Replace based on age 4.Shock Perform functional test to identify hidden failures

3. 3 1 - Observable gradual failure progression

4. 4 Examples, observable gradual failure progression The break disks on a train The wear on a railway rail The corrosion on a pipe Cracks in an airplane structure The level of degradation determines the next inspection, and whether a repair action is required

5. 5 2 - Observable “sudden” failure progression

6. 6 Examples: observable “sudden” failure progression Cracks in a train wheel Isolation resistance in a signalling cable

7. 7 3 - Non-observable failure progression

8. 8 4 - Shock

9. 9 Multistate systems Multistate systems are described by performance measures We use a state variable, Y(t), to describe the state of the system at time t, e.g., Performance (pump capacity, compressor efficiency etc) For binary systems Y(t) reduces to take only the values 0 and 1; Y(t) = 1 represents a functioning state, and Y(t) = 0 represents a fault state Y(t) is a random quantity, i.e. expressed in probabilistic terms, involving model parameters

10. 10 Content of the state variable Y(t) Y(t) was introduced as a performance variable However, we will let Y(t) be more general, and Y(t) will be used to express the state of the system at time t, i.e.; the direct performance of the system, capacities etc., or a direct measure of wear, or an indication of wear or increased failure probability We use W(t) as a general quantity that simply is related to degradation of the system:

11. 11 Degradation quantities of interest W P (t): Quantities that are direct performance measures ($!!!) E.g., the pumping capacity of a pump W I (t): Quantities that are only indicators of the degradation of the component E.g., the bearing temperature W D (t): Quantities that represent measurable degradation Examples are crack shape and size, corrosion level, geometrical defects (inclusive wear) W S (t): Stressors that influence the degradation process Examples could be the cyclic loads and corrosive medium The stressors them selves do not measure the likelihood of failure, but is important for the forecasting of the failure progression W P (t), W I (t) and W D (t) will be (probabilistic) modelled by the state variable Y(t)

12. 12 Challenges in failure modelling How to measure Y(t)? For quantities that could be measured: Use the quantity directly, i.e., crack length Transformations, for example FFT (Fast Fourier Transform) Non measurable quantities Define patterns for similarity comparison What is the relation between the readings from the measurements and the real physical state? Reliability of the measurement techniques To model failure (fixed failure limits rarely exist) To model failure, we generally specify the failure probability as a function of the value of the state variable, i.e., p = p(y) A simplification would be to assume that a failure occurs the first time the state variable reaches a fixed limit (failure limit)

13. 13 Purpose of modelling – binary systems We want to establish a mathematical model describing the relation between the effective failure rate, E, and the maintenance, i.e., the inspection interval, , and the intervention level, l E = E ( ,l) Establish a cost model: PM cost  (inspection interval) -1 Renewal cost increases with a restrictive intervention level CM cost/unavailability cost increases with increasing inspection interval CM cost/unavailability cost decreases with a restrictive intervention level Example  

14. 14 Classes of probabilistic models used PF model   Failure progression is defined between a potential failure (P) and a failure (F) The Wiener process   During an arbritary time interval  t, the “failure progression” is increased by a normally distributed quantity with mean  t and variance  2  t A failure occurs the first time the failure progression passes the critical value  The Gamma process Similar to the Wiener process, but the increments are gamma distributed The shock model   The system is exposed to shocks, and each shock causes a damage X i When the accumulated damage increases, so does also the failure probability The Markov state model   The failure progression is approximated by a discrete set of states The transitions between the sates are assumed to follow a Markov process The model is very flexible, and allows for modeling a large range of situations Markov model 

15. 15 The PF model The objective of the inspection is to detect e.g., a crack (potential failure) before it develops to a breakage (critical failure) The time from a crack is detectable (P) until the e.g., the rail breakage is a fact (F), is denoted the PF interval 

16. 16 Variation in the PF interval The length of the PF interval is assumed to vary from time to time cracks can be initialised in different places of the component crack propagation depends on several different factors such as load, structure quality, temperature etc The cracks that propagate very fast represent the largest risk of not being detected by the ultrasonic inspection The objective of the modelling is to obtain the probability, Q, of not detecting the crack in due time as a function of the inspection interval  Q = Q(  ) 

17. 17 Determining Q 0 (simplified) T PF PF interval (random variable)  PF Probability distribution function of T PF qFailure probability of one inspection  Inspection interval Q t Failure probability for fixed value, T PF = t Q 0 Failure probability of given strategy

18. 18 The argument

19. 19 Cost elements - Optimization The most important cost elements are: The cost per inspection, C I The (unavailability) cost per system failure, C F The cost of repairing a system failure, C CM The cost of renewing the system upon a potential failure, C RC The total cost per unit time is then C(  ) = C I /  + (C F +C CM )  E (  ) + C RC  (  ) The objective is now to minimize C(  ) wrt maintenance interval and intervention level E (  )  Q 0 / (MTTF-E(T PF ) )  E (  )  (1-Q 0 )/ (MTTF-E(T PF ) ) = renewal rate

20. 20 The Wiener process 

21. 21 The shock model  The shocks represent W S (t) The magnitude of the shock also represents W S (t) The impact X i represents W D (t)

22. 22 The Markov state model 

23. 23 Model assumptions The state variable, Y(t), describes the state of the system at time t, Y(t) is a random quantity The state variable could take one of the values y 0, y 1,…, y r The values could either be numerical, or a qualitative description of a state or phenomenon The system starts in state y 0, and jumps to a higher state (y i to y i+1 ) with a time independent intensity i There is generally a cost assossiated with being in state y i The system fault state is y r The system is inspected at intervals of length  (offline) The system is renewed if Y(t)  y l at an inspection

24. 24 Maintenance CalculationPar. Spec.

25. 25 Markov differential equations Introduce P i (t) = Pr(the system is in state i at time t) Consider the change in a small time interval  t: Standard Markov considerations gives: P i (t+  t) = P i (t)(1- i  t) + P i-1 (t) i-1  t (*) Equation (*) could now be used to obtain the state probabilities, P i (t), as a function of time by numerical integration

26. 26 The easy situation: no maintenance If no maintenance is carried out then integrate equation (*) starting from the initial state Mean time to failure is given by: MTTF =  t=0:  R(t) dt =  t=0:  [1-P r (t)]dt in fact a sum … To verify our calculations we should verify the analytical result: MTTF =  i=0:r-1 MTTF i =  i=0:r-1 1/ i

27. 27 Calculation procedure: with maintenance The system is inspected at intervals of length  The system is renewed if Y(t)  y l at an inspection (Fig.)Fig. The model is integrated as before, but when t equals , 2 , 3 ,… special considerations are necessary Procedure 1. Define the initial conditions: P 0 (0) = 1, P i (0) = 1, i > 0 2. Set f = 0, t = 0,  t = sufficient small 3. Integrate Equation (*) one step, and let t = t +  t 4. Let f = f + P r (t) 5. If t = , 2 , 3 ,…, then let P 0 (t) = P 0 (t)+  i  l P i (t), and P i (t) = 0, i  l 6. Loop to Step 3 until t is sufficient large 7. System failure frequency now equals E ( ,l) = f/t

28. 28 Do While t < MaxT ‘ Main loop nFail = nFail + IntegrateDt(dt) P(0) = P(0) + P(r) P(r) = 0 t = t + dt If t > inspection Then inspection = inspection + tau nRenewal = nRenewal + Inspect(L, q) End If Loop Function IntegrateDt(dt As Single) For i = r To 1 Step -1 P(i) = P(i) * (1 - lam (i) * dt) _ + P(i - 1) * lam (i - 1) * dt Next P(0) = P(0) * (1# - lambda(0) * dt) IntegrateDt = P(r) End Function Function Inspect(L As Integer, q As Single) rr = 0 For i = L To r - 1 rr = rr + P(i) * (1 - q) P(0) = P(0) + P(i) * (1 - q) P(i) = P(i) * q Next i DoInsp = rr End Function Essential source code in VBA

29. 29 Specification of model parameters In principle we need to specify all transition rates, i.e. 0, 1,…, r-1 We also need the probability of erroneous classification Qi  j = Pr(Classify into state i when the real state is j) In order to get numerical values (estimates) of the model parameters, we utilise: Experience data Expert and engineering judgements Degradation modelling, i.e. fracture mechanics, FEM etc For r > 4-5 this will be a huge number of parameters We want to simplify the parameter specification procedure  

30. 30 Simplified parameter specification We specify the parameters in the situation without maintenance, i.e. What will the mean time to failure (MTTF) be if no maintenance is carried out? (Fig.  )Fig.  Is the transition rate between states constant, or increasing? If it is increasing then we specify the ratio: V = r-1 / 0 = how much faster failure progression is just before failure compared to initially (Fig.  )Fig.  We also need to specify The number of states in the model (r ) The probability q that an inspection does not reveal that the system is in a critical state Calculation example 

31. 31 MTTF without maintenance 

32. 32 Calculation example Input parameters: Result MarkovStateModel.xls

33. 33 The effect of maintenance We have established (by means of the Excel model) the relation between maintenance (  and l) and i) the effective failure rate, E ( ,l), and ii) the renewal rate  ( ,l) Example results

34. 34 Cost elements - Optimization The most important cost elements are: The cost per inspection, C I The (unavailability) cost per system failure, C F The cost of repairing a system failure, C CM The cost of renewing the system at state l, C RC The total cost per unit time is then C( ,l) = C I /  + (C F +C CM )  E ( ,l) + C RC  ( ,l) The objective is now to minimize C( ,l) wrt maintenance interval and intervention level

35. 35 Extension of the Markov model More advanced maintenance strategies could be applied Reducing inspection interval as we approach the maintenance limit, l Conduct non perfect repair before the maintenance limit Models have been developed for hydro power plant

36. 36 The gamma process Stationary gamma process Background: X is said to be gamma distributed with shape parameter v, and scale parameter u if the PDF is given by: Ga(x|v,u)=u v x v-1 e -ux /  (v) Let Y(t) be the degradation level at time t Y(t) follows a stationary gamma process if Y(0) = 0 Y(s) - Y(t) ~ Ga([s-t ]v,u), s>t Y(t) has independent increments

37. 37 Mean time to failure in the gamma process Assume that the component fails as soon as the failure progression exceeds the value  Let T denote the time to failure It follows that F T (t) = Pr(T  ) =  (vt,  u)/  (vt) Where  (a, x) is the incomplete gamma function Welte (2008) reports the following: E(T)   u/v + 1/(2v) Var(T)   u/v 2 - 1/(12v 2 )

38. 38 Non-stationary gamma process The gamma process could be extended to a non- stationary process by letting the shape parameter be a function of time, i.e., v(t) is the shape function, and we have: Y(0) = 0 Y(s) - Y(t) ~ Ga(v(s)-v(t),u), s>t Y(t) has independent increments The CDF now reads F T (t) = Pr(T  ) =  (v(t),  u)/  (v(t)) The expected time to failure, and variance in time to failure could be found by numerical methods

39. 39 Comparison – Discrete model, vs gamma process For the discrete model we need to fix the number of states If the degradation is continuous, this seems not very natural, hence a gamma process is more appealing Degradation rate In the discrete model, the degradation rate (in terms of transition rates) depends on the state of the system, and not on the age (time) In a gamma process the degradation rate could also be modelled by a non-constant value, but degradation rate depends on the age, and not on the state

40. 40 Exercise Verify E(T)   u/v + 1/(2v) by numerical integration, i.e., E(T) = 0   R(t)dt

41. 41 Non-stationary gamma process The gamma process could be extended to a non- stationary process by letting the shape parameter be a function of time, i.e., v(t) is the shape function, and we have: Y(0) = 0 Y(s) - Y(t) ~ Ga(v(s)-v(t),u), s>t Y(t) has independent increments The CDF now reads F T (t) = Pr(T  ) =  (v(t),  u)/  (v(t)) The expected time to failure, and variance in time to failure could be found by numerical methods

42. 42 Integration of the gamma process