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The 4 standard failure models

-to be used in maintenance optimization, with focus on state modellingProfessor Jørn Vatn

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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

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1 - Observable gradual failure progression Fa

ilure

pro

gres

sion

TimeTcrit

Failure

Tmaint

Critical failure progression

Maintenance limit

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

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2 - Observable “sudden” failure progressionF

ailu

re p

rogr

essi

on

Time

FailureCritical failure progression

F

PPF-

interval

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Examples: observable “sudden” failure progression

Cracks in a train wheel Isolation resistance in a signalling cable

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3 - Non-observable failure progressionFa

ilure

pro

gres

sion

TimeTcrit

FailureCritical failure progression

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4 - ShockFa

ilure

pro

gres

sion

Time

FailureCritical failure progression

F

P

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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

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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:

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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)

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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)

Time

Y(t)

Failure limit

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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

E(,l)

l=6l=3

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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

Markov model

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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)

PF interval

t

Critical failureprogression(Breakage)

Detectablefailureprogression

Tinit T critT det

P

F

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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 notdetecting the crack in due timeas a function of the inspectioninterval Q = Q()

0 %

5 %

10 %

15 %

20 %

25 %

30 %

35 %

0 30 60 90 120 150time [months]

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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

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The argument Assume PF-interval is fixed, i.e., TPF = t Let n = int(t/) Number of opportunities for inspection:

We get an extra inspection, if the first inspection after the «P» comes before time units, i.e.,

Probability of n+1 opportunities:

P Ft

Best n + 1 opportunities

Worst n opportunities

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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

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The Wiener process

Failure

Time

Failu

re p

rogr

essi

on, Y

(t)

Failure limit,

t

t = expected drift

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The shock modelFailure

Time

Acc

umul

ated

dam

age,

Y(t)

ith shock

Xi damage caused by ith shock

The shocks represent WS(t)

The magnitude of the shock also represents WS(t)

The impact Xi represents WD(t)

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The Markov state model

y1

y2

yr-1

yr

1

2

r

r-1

T1 T2 Tr-1Time

State

y0 0

T0 …..

::

Failure

…..

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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 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

y1

y2

yr

12

r

State

y0 0

::

t

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Maintenance

y1

y2

yl

yr

1

2

r

l

Time

y0 0

::

Maintenance limit

32 4 5 6 87

0

r-1

1

CalculationPar. Spec.

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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:

Pi(t+t) = Pi(t)(1-it) + Pi-1(t) i-1t (*)

Equation (*) could now be used to obtain the state probabilities, Pi(t), as a function of time by numerical integration

i

r

i-1

i-1

i

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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

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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 small3. Integrate Equation (*) one step, and let t = t + t4. Let f = f + Pr(t)5. If t =, 2, 3,…, then let P0(t) = P0(t)+ il Pi(t), and Pi(t) = 0, il6. Loop to Step 3 until t is sufficient large7. System failure frequency now equals E(,l) = f/t

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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

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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

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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

Calculation example

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MTTF without maintenance

y1

y2

yr-1

yr

1

2

r

r-1

Timey0 0

::

Failure

…..

MTTF without maintenance

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Calculation example

Input parameters: Result

MarkovStateModel.xls

Input values MTTF 120 r 8

V=r-1/0 8 12 Intervention, l 4 q 0,05 Time horizon 4800

Output result v 1,35 0 0,0294 MTTF-verify 119,98 MTTF(,l) 2480,14 A(l) 0,00040 Ren. Rate 0,01008 MTBR 99,25

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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 resultsEffective failure rate, E(,l)

0

0,001

0,002

0,003

0,004

0,005

0,006

3 6 9 12 15 18 21 24

Inspection interval,

Intervention: l = 6

Intervention: l = 4

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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

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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

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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

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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)

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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 readsFT(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

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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

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Exercise

Verify E(T) u/v + 1/(2v) by numerical integration, i.e., E(T) = 0

R(t)dt

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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 readsFT(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

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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: (*) 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