Report

The 4 standard failure models -to be used in maintenance optimization, with focus on state modelling Professor Jørn Vatn 1 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 2 Failure progression 1 - Observable gradual failure progression Failure Critical failure progression Maintenance limit Time Tmaint Tcrit 3 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 4 Failure progression 2 - Observable “sudden” failure progression F Failure Critical failure progression P PFinterval 5 Time Examples: observable “sudden” failure progression Cracks in a train wheel Isolation resistance in a signalling cable 6 Failure progression 3 - Non-observable failure progression Failure Critical failure progression Time Tcrit 7 Failure progression 4 - Shock F Failure Critical failure progression P Time 8 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 9 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: 10 Degradation quantities of interest WP(t): Quantities that are direct performance measures ($!!!) E.g., the pumping capacity of a pump WI(t): Quantities that are only indicators of the degradation of the component E.g., the bearing temperature WD(t): Quantities that represent measurable degradation Examples are crack shape and size, corrosion level, geometrical defects (inclusive wear) WS(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 WP(t), WI(t) and WD(t) will be (probabilistic) modelled by the state variable Y(t) 11 Y(t) Challenges in failure modelling Failure limit 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) 12 Time Purpose of modelling – binary systems We want to establish a mathematical model describing the relation between the effective failure rate, E, and E(,l) the maintenance, i.e., the inspection interval, , and the intervention level, l l=6 l=3 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 13 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 2t 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 Xi 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 14 Markov model 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 Failure progression/crack size, Y(t) Critical failure progression (Breakage) F Detectable failure progression P t Tinit T det T crit PF interval 15 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 35 % 30 % 25 % 20 % 15 % 10 % 5% 0% Q = Q() 0 30 60 90 120 150 time [months] 16 Determining Q0 (simplified) TPF PF interval (random variable) PF Probability distribution function of TPF q Failure probability of one inspection Inspection interval Qt Failure probability for fixed value, TPF = t Q0 Failure probability of given strategy 17 The argument Assume PF-interval is fixed, i.e., TPF = t Let n = int(t/) Number of opportunities for inspection: P Best Worst F t n + 1 opportunities n opportunities Δ = −⋅ We get an extra inspection, if the first inspection after the «P» comes before time units, i.e., Probability of n+1 opportunities: Δ/ = t/ − n Qt(,q,t) = (n + 1 − t/) ⋅ qn + (t/ − n) ⋅ q(n+1) 0 , , PF = 0 , , PF 18 Cost elements - Optimization The most important cost elements are: The cost per inspection, CI The (unavailability) cost per system failure, CF The cost of repairing a system failure, CCM The cost of renewing the system upon a potential failure, CRC The total cost per unit time is then C() = CI/ + (CF+CCM)E() + CRC() The objective is now to minimize C() wrt maintenance interval and intervention level E() Q0 / (MTTF-E(TPF) ) E() (1-Q0 )/ (MTTF-E(TPF) ) = renewal rate 19 Failure progression, Y(t) The Wiener process Failure Failure limit, t = expected drift t Time 20 The shock model Accumulated damage, Y(t) Failure Xi damage caused by ith shock ith shock Time The shocks represent WS(t) The magnitude of the shock also represents WS(t) The impact Xi represents WD(t) 21 The Markov state model State Failure yr r yr-1 r-1 . …. : : 2 y2 1 y1 y0 0 T0 T1 ….. T2 22 Time Tr-1 State yr Model assumptions r : : y2 y1 y0 2 1 0 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 y0, y1,…, yr The values could either be numerical, or a qualitative description of a state or phenomenon The system starts in state y0, and jumps to a higher state (yi to yi+1) with a time independent intensity i There is generally a cost assossiated with being in state yi The system fault state is yr The system is inspected at intervals of length (offline) The system is renewed if Y(t) yl at an inspection 23 t Maintenance r-1 yr yl : : 0 y1 y0 l Maintenance limit y2 1 2 3 4 r 1 0 2 5 24 6 7 8 Time Par. Spec. Calculation Markov differential equations Introduce Pi(t) = Pr(the system is in state i at time t) Consider the change in a small time interval t: Standard Markov considerations gives: r Pi(t+t) = Pi(t)(1-it) + Pi-1(t) i-1t i-1 (*) i i i-1 Equation (*) could now be used to obtain the state probabilities, Pi(t), as a function of time by numerical integration 25 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-Pr(t)]dt in fact a sum … To verify our calculations we should verify the analytical result: MTTF = i=0:r-1MTTFi = i=0:r-11/i 26 Calculation procedure: with maintenance The system is inspected at intervals of length The system is renewed if Y(t) yl at an inspection (Fig.) The model is integrated as before, but when t equals , 2, 3,… special considerations are necessary Procedure 1. Define the initial conditions: P0(0) = 1, Pi(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 + Pr(t) 5. If t =, 2, 3,…, then let P0(t) = P0(t)+ il Pi(t), and Pi(t) = 0, il 6. Loop to Step 3 until t is sufficient large 7. System failure frequency now equals E(,l) = f/t 27 Essential source code in VBA Do While t < MaxT ‘ Main loop nFail = nFail + IntegrateDt(dt) P(0) = P(0) + P(r) Function IntegrateDt(dt As Single) For i = r To 1 Step -1 P(i) = P(i) * (1 - lam (i) * dt) _ P(r) = 0 + P(i - 1) * lam (i - 1) * dt t = t + dt Next If t > inspection Then P(0) = P(0) * (1# - lambda(0) * dt) inspection = inspection + tau IntegrateDt = P(r) nRenewal = nRenewal + Inspect(L, q) End Function End If Loop 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 28 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 Qij = 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 29 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. ) 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. ) 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 30 Calculation example MTTF without maintenance MTTF without maintenance yr r yr-1 r-1 . …. : : 2 y2 1 y1 y0 Failure 0 Time 31 Calculation example Input parameters: Input values MTTF r V=r-1/0 Intervention, l q Time horizon Result Output result v 120 8 0 MTTF-verify MTTF(,l) 8 12 4 0,05 4800 A(l) Ren. Rate MTBR 32 1,35 0,0294 119,98 2480,14 0,00040 0,01008 99,25 MarkovStateModel.xls 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 Effective failure rate, E(,l) 0,006 0,005 0,004 Intervention: l = 6 0,003 0,002 Intervention: l = 4 0,001 0 3 6 9 12 15 Inspection interval, 33 18 21 24 Cost elements - Optimization The most important cost elements are: The cost per inspection, CI The (unavailability) cost per system failure, CF The cost of repairing a system failure, CCM The cost of renewing the system at state l, CRC The total cost per unit time is then C(,l) = CI/ + (CF+CCM)E(,l) + CRC(,l) The objective is now to minimize C(,l) wrt maintenance interval and intervention level 34 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 35 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)=uvxv-1e-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 36 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 FT(t) = Pr(T<t) = Pr(Y(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/v2 - 1/(12v2) 37 Non-stationary gamma process The gamma process could be extended to a nonstationary 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 FT(t) = Pr(T<t) = Pr(Y(t) > ) = (v(t), u)/(v(t)) The expected time to failure, and variance in time to failure could be found by numerical methods 38 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 39 Exercise Verify E(T) u/v + 1/(2v) by numerical integration, i.e., E(T) = 0 R(t)dt 40 Non-stationary gamma process The gamma process could be extended to a nonstationary 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 FT(t) = Pr(T<t) = Pr(Y(t) > ) = (v(t), u)/(v(t)) The expected time to failure, and variance in time to failure could be found by numerical methods 41 Integration of the gamma process Let S|t,dt = Y(t+dt) - Y(t) be the degradation during a small time interval dt after time t S|t,dt ~ Ga(v(t+dt)-v(t),u) Further, let g(s | t, dt) denote the pdf of S|t,dt If the pdf of Y(t) is known, we may obtain the pdf of Y(t+dt) by a convolution argument: (| + ) = ( =0 − |) (|, ) (*) Assume the system is inspected every time unit, and renewed whenever Y > yM To find the effective failure rate, we integrate (*) from t = 0 to , and whenever t = k, probability mass is moved to 0 42