modelling of predictive maintenance for a periodically inspected system

The decision rule for the predictive checking is formulated and the probabilities of correct and incorrect decisions are derived.. The effectiveness of the predictive maintenance is eval

Procedia CIRP 59 ( 2017 ) 95 – 101

2212-8271 © 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/)

Peer-review under responsibility of the scientific committee of the The 5th International Conference on Through-life Engineering Services (TESConf 2016) doi: 10.1016/j.procir.2016.09.032

ScienceDirect

The 5th International Conference on Through-life Engineering Services (TESConf 2016)

Modelling of predictive maintenance for a periodically inspected system

Ahmed Razaa, Vladimir Ulanskyb,*

a Department of the President’s Affairs, Overseas Projects and Maintenance, P.O Box: 372, Abu Dhabi, UAE

b National Aviation University, 1 Kosmonavta Komarova Avenue, Kiev 03058, Ukraine

* Corresponding author Tel.: +38-0632754982 E-mail address: vladimir_ulansky@nau.edu.ua

Abstract

Predictive maintenance includes condition monitoring and prognosis of future system condition where maintenance decision-making is based on the results of prediction In this paper, the modelling of predictive maintenance is conducted It is assumed that the system is periodically checked

by using imperfect measuring equipment The decision rule for the predictive checking is formulated and the probabilities of correct and incorrect decisions are derived The effectiveness of the predictive maintenance is evaluated by the average availability and downtime cost per unit time The mathematical models are proposed to calculate the maintenance indicators for an arbitrary distribution of time to failure The proposed approach is illustrated by determining the optimal number of predictive checks for a specific stochastic deterioration process Numerical example illustrates the advantage of the predictive maintenance compared to the corrective maintenance

© 2016 The Authors Published by Elsevier B.V

Peer-review under responsibility of the Programme Committee of the 5th International Conference on Through-life Engineering Services (TESConf 2016)

Keywords: Availability; Corrective maintenance; Decision rule; False failure; Predictive checking; Prognostics; Remaining useful life; Undetective failure

1 Introduction

Currently, the most promising strategy of maintenance for

various technical systems and production lines is the predictive

maintenance (PM), which can be applied to any system if there

is a deteriorating physical parameter like vibration, pressure,

voltage, or current that can be measured This allows to

recognize approaching troubles, to predict wear or accelerating

aging and to prevent failure through the repair or replacement

of the affected component Predictive maintenance is based on

the prognostic and health management technology, which

supposes that the remaining useful life of equipment can be

predicted However, due to uncertainty of prognostics there

could be wrong decisions regarding the remaining time to

failure The growing interest to PM is evident from the large

number of publications related to various mathematical models

and implementation techniques Let us consider some

references related to the modelling of the PM

In [1], a PM policy for a continuously deteriorating system

subject to stress is developed Condition-based maintenance

policy is used to inspect and replace the system according to the

observed deterioration level A mathematical model for the maintained system cost is derived In [2], the predictable reactive maintenance policies are studied based on a fatigue crack propagation model of the wind turbine blade considering random shocks and dynamic covariates In [3], a PM method is developed to determine the most effective time to apply maintenance to equipment and study its application to a real semiconductor etching chamber The PM decision is based on the likelihood of the predicted health condition, which exceeds

a certain maintenance threshold In [4], the costs model is analysed where the costs include small repair cost, PM cost and productive loss An optimal model of PM strategy is further proposed to overcome the shortcoming of PM model with identical period In [5], a PM structure for a gradually deteriorating single-unit system is considered The decision model enables optimal inspection and replacement decision in order to balance the cost engaged by failure and unavailability

on an infinite horizon In [6], a data-driven machine prognostics approach is considered to predict machine’s health condition and describe machine degradation A PM model is constructed

to decide machine’s optimal maintenance threshold and

© 2016 The Authors Published by Elsevier B.V This is an open access article under the CC BY-NC-ND license

( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

Peer-review under responsibility of the scientifi c committee of the The 5th International Conference on Through-life Engineering Services (TESConf 2016)

maintenance cycles In [7], a PM model for the deteriorating

system with semi-Markov process is proposed A method to

determine the best inspection and maintenance policy is

developed In [8], a discriminant function is developed on the

basis of the representation of the observed system degradation

process as a discrete parameter Markov chain In [9], a multiple

classifier machine learning methodology for PM is considered

The proposed PM methodology applies dynamical decision

rules to maintenance management

It should be noted that all considered models of PM do not

take into account the probabilities of the correct and incorrect

decisions made by the results of the predictive checks (PCs)

In this paper, a new PM model is developed for determining

optimal periodicity of PCs A decision rule is proposed for

inspecting the system condition, which is based on the

evaluation of the remaining time to failure Based on this

decision rule, general expressions are derived for calculating

the probabilities of correct and incorrect decisions made by the

results of the PCs The effectiveness of the PM is evaluated by

such indicators as average availability and average downtime

cost per unit time

Nomenclature

PC predictive check

PDF probability distribution function

PM predictive maintenance

2 Decision rule

Assume that the state of a system is completely determined

by the value of the parameter X (t), which is a non-stationary

random process with continuous time The system should

operate over a finite horizon T and is checked with prediction

of condition at discrete time kτ (݇ ൌ ͳǡ ܰതതതതത) When the system

state parameter exceeds threshold FF, the system passes into

the failed state The measured value of X(t) at time kτ is

expressed as follows:

Z k W X k W  Y k W , (1)

where Y(kτ) is the measurement error of the system state

parameter at time kτ

Assume that random variable Ξ (Ξ ≥ 0) denotes the failure

time of a system with probability distribution function (PDF)

ω(ξ) Let Ξk be a random assessment of Ξbased on the results

of the PC at time kτ

Random variables Ξ and Ξk are the smallest roots of the

following stochastic equations:

X t  FF (2)

Z k W  FF (3)

Let ξj,k be the realisation of Ξk for the j-th system Then,

when carrying out the PC at the instant kτ the following

decision rule is used: if ξj,k ≥ (k+1)τ, the system is judged to be

suitable for operation in the time interval [kτ, (k+1)τ]; if ξ j,k <

(k+1)τ, the system is judged as unsuitable for operation in the time interval [kτ, (k+1)τ]

By the results of the PC at time kτ the following decisions are made: to allow the j-th system to be used until the next PC

at the instant (k + 1)τ if ξ j,k ≥ (k+1)τ; to restore the j-th system

if ξj,k < (k+1)τ

The mismatch between the solutions of (2) and (3) results in the appearance of one of the following mutually exclusive

events by the results of the PC at the instant kτ:

1

1

k

i i

h k W ; ! ­ ® k  W § ¨ ; !  i W · ¸ ½ ¾

2

1

k

i

h k W ; !  ­ ® k W ; d  k W ª «; !  i W º » ½ ¾

3

1

k

i i

h k W ­ ® k W  ; d k  W § ¨ ; !  i W · ¸ ½ ¾

4

1

k

i

h k W ­ ° ® k W  ; d  k W ; d  k W ª «; !  i W º » ½ ° ¾

5

1

1

k

i i

h k W ; d ­ ® k W ª « ; !  i W º » ½ ¾

¯ ¿ (8)

6

1

k

i

h k W ; d ­ ® k W ; d k  W ª « ; !  i W º » ½ ¾

Events h1(kτ), h4(kτ) and h6(kτ) correspond to the correct decisions by the results of the PC at time kτ Event h2(kτ) is the

joint occurrence of two events: the system is suitable for use

over the interval [kτ, (k+1)τ] and by the results of the PC it is judged as unsuitable We define event h2(kτ) as a ‘false failure’ Events h3(kτ) and h5(kτ) we define as ‘undetected failure 1’ and

‘undetected failure 2’, respectively

3 System states

Let us consider the stochastic process S(t), which

characterizes the state of the system at an arbitrary instant of

time t: S1, if at time t, the system is used as intended and is in the operable state; S2, if at time t, the system is used as intended and is in an inoperable state (unrevealed failure); S3, if at time

t, the system is not used for its intended purpose because the

PC is carried out; S4, if at time t, the system is not used for its intended purpose because event h2 has occurred and a ‘false

corrective repair’ is performed; S5, if at time t, the system is not used for its intended purpose because either h3 or h6 event has occurred and a ‘true corrective repair’ is performed

Further we assume that process S(t) is the regenerative

stochastic process When determining maintenance efficiency indicators we use a well-known property of the regenerative stochastic processes [10], which is based on the fact that the

fraction of time for which the system is in the state S i (݅ ൌ ͳǡ ͷതതതതത)

is equal to the ratio of the average time spent in the state S i per

regeneration cycle to the average cycle duration

We designate T i as the time spent by the system in the state

Si ( ݅ ൌ ͳǡͷതതതത ) Obviously, T i is a random variable with the

expected mean time M[T i] The average duration of the

regeneration cycle is given by

0

1

i

i

M T ¦ M T (10)

4 Probabilities of correct and incorrect decisions

Let us define the conditional probabilities of correct and

incorrect decisions made at carrying out the PC under the

assumption that failure of the system occurs at time ξ and

The conditional probability of a ‘false failure’ by the results

of carrying out the PC at time ντ (ߥ ൌ ͳǡ ݇ െ ͳതതതതതതതതതത) is formulated

as follows:

1

U S

i



The conditional probability of the event ‘operable system is

correctly judged unsuitable’ by the results of carrying out the

PC at time kτ (݇ ൌ ͳǡ ܰതതതതത) is formulated as follows:

ν 1

U O



The conditional probability of the event ‘undetected failure

1’ by the results of carrying out the PC at time kτ is formulated

as follows:

ν 1

k

S O

The conditional probability that ‘suitable system is correctly

judged suitable’ by the results of carrying out the PC at time ντ

is formulated as follows:

1

S S

i

The conditional probability that inoperable system is

correctly judged as unsuitable by the results of carrying out the

PC at time jτ (݆ ൌ ݇ ൅ ͳǡ ܰതതതതതതതതതതത) is formulated as follows:

j

U I



The conditional probability of the event ‘undetected failure

2’ by the results of carrying out the PC at time Nτ (N = T/τ ‒ 1)

is formulated as follows:

1

N

i

S I

i

To determine the probabilities (12)-(17), we introduce the conditional joint PDF of random variablesȩതതതതതതത, under the ଵǡ ȩ௞ condition that Ξ = ξ, which we denote as ȳ଴ሺɌതതതതതത פ Ɍሻ ଵǡ Ɍ௞

As can be seen from (3), Ξk is a function of random variables

Ξ and Y(kτ) The presence of Y(kτ) in (3) results in appearing a

random measurement error Λi with respect to the time to failure

Ξ at time point iτ, which is defined as follows:

, 1

i

/ (18) Between random variables Ξ (0 <Ξ < Ğ) and Λi (−Ğ < Λi

< Ğ) exists additive relationship Therefore, random variable

Ξi is defined over a continuous range from −Ğ to +Ğ

Now probabilities (12)−(17) can be determined by using the well-known formula for calculating the probability of hitting a random point ȩതതതതതതത to the known area: ଵǡ ȩ௞

2

U S

QW W

 f f

f

2

k

U O

k

W W

 f f

f

( 1)τ 2

S O

k k

W W

f f f



( ν 1)τ ντ 2τ

S S

f f f



τ 2τ

j

U I

j

 f f

f

( 1)τ τ 2τ

S I

N N



Let us denote the conditional PDF of random variables

Ȧଵǡ Ȧ௞ തതതതതതതത under the condition that Ξ = ξ as݂ஃ൫ɉതതതതതതത פ Ɍ൯ ଵǡ ɉ௞ When Ξ = ξ, random variables ȩതതതതതതത can be represented as ଵǡ ȩ௞

Applying to (25) the theorem of continuous multivariate transformation, we obtain

0 ξ ,ξ ξ1 k f/ ξ1 ξ,ξk ξ ξ

The substitution of (26) to (19) gives

ντ 2τ

U S

 f f /

f

Assuming that a i = u i − ξ (݅ ൌ ͳǡ ݇തതതതത) in (27), we get

ντ-ξ 2τ-ξ

U S

 f f

/

f

A similar change of variables in (20)-(24) results in

τ-ξ 2τ-ξ

k

U O

k

 f f

/

f

( 1)τ-ξ τ-ξ 2τ-ξ

S O

f f f

/



(ν 1)τ -ξ ντ -ξ 2τ-ξ

S S

f f f

/



τ-ξ 2τ-ξ

j

U I

j

/

f

( 1)τ-ξ τ-ξ 2τ-ξ

S I



5 Expected up and down times

The mean time of staying the system in the state S1 is

determined as follows:

1

k

W

Q

W

- 

ª

«

«¬

1

k T

S S

¼ (34)

The mean time spent by the system in the state S2 is

2

k

U I

k k j k

W

-





T

d

Z - - (35)

The mean time spent by the system in the state S3 is

3

k

W Q W

1

N

j k



º

¼

¦

1

1

N

k N

W

³ d

Z - -, (36) where τpc is the mean time of a PC

The mean time of staying the system in the state S4 is

τ,(ν 1)τ ; ντ

k

k k

W Q W

1

N

U S k T

¼

¦

where τfr is the mean time of a ‘false corrective repair’

The mean time of staying the system in the state S4 is

1

k N

k k

W W

«¬ ¦ ³

0

d

W

¼

³ , (38) where τtr is the mean time of a ‘true corrective repair’

6 Indicators of maintenance efficiency

When the mean times ܯሾܶതതതതതതതതതതതതതതതത are known we can ଵሿǡ ܯሾܶହሿ identify any of the maintenance efficiency indicators Let us consider some of indicators

Average availability is one of the key performance indicators used in nuclear power plants [11], aviation and military systems [12] For this model the average availability is

A M T M T (39) Average total down time (TDT) can be defined as

M TDT T M T  M T M T (40) Average downtime cost per unit time is expressed as

> @ 5 > @

0 2

[ ]

i

M C ¦ C M T M T (41)

7 Optimal number of predictive checks

The problem of determining the optimal number of the PC

Nopt depends on the selected optimization criterion

By the criterion of maximum average availability N opt is

determined by solving the following problem:

max

opt

N

N Ÿ A N (42)

If the criterion of the minimum average downtime cost per

unit time is used, then

min

opt

N

N Ÿ M C N ª ¬ º ¼ (43)

8 Example

8.1 Deterioration process modelling

Assume that the deterioration process of a one-parameter

system is described by the monotonic stochastic function

X t A  A t, (44)

where A0 is the random initial value of X(t) and A1 is the random

rate of parameter deterioration defined in the interval from 0 to

Ğ It should be noted that a linear model of a stochastic

deterioration process was used in many previous studies for

describing real physical deterioration processes For example,

the linear regressive model studied in [13] describes a change

in radar output voltage with time, and a linear model was used

to represent a corrosion state function in [14]

Let us determine the conditional PDF ݂ஃ൫ɉതതതതതതത פ Ɍ൯ for the ଵǡ ɉ௞

stochastic process given by (44)

If ܻሺ߬ሻǡ ܻሺ݇߬ሻതതതതതതതതതതതതതതത are independent random variables, then

0

f

1

k

i i

where f(a0) is the PDF of the random variable A0, ψ(y i) is the

PDF of the random variable Y(iτ), and ω ξ a 0 is the

conditional PDF of the random variable Ξ under the condition

that A0 = a0

Let us prove relation (45) Since ܻሺ߬ሻǡ ܻሺ݇߬ሻതതതതതതതതതതതതതതത are

independent, then for the stochastic process (44) we can write

1

i

f a – f a (46)

Designating Y i = Y(iτ) (i = 1, 2, ), we solve the stochastic

equations

0 1Ξ

A A FF (47)

0 1Ξ +i i

A A Y FF (48)

in respect to Ξ and Ξi

FF A0 A1

;  (49)

i FF A Yi A

;   (50)

By substitution of (49) and (50) into (18) we obtain

1

i Y A i

/  (51)

Solving (47) in respect to A1 gives

A FF  A ; (52) Substituting (52) to (51), we get

/  ;  (53)

For any values Y i = y i , A0 = a0 and Ξ = ξ, the random variable

Λi with a probability of 1 has only one value Therefore, conditional PDF of random variables Ȧതതതതതതതത with respect to ଵǡ Ȧ௞

ଵǡ ௞

തതതതതതത, A0 and Ξ is the Dirac delta function:

 1 1 0 0

1

i

Using the multiplication theorem of PDFs, we find the joint PDF of the random variables Ȧതതതതതതതത, ଵǡ Ȧ௞ തതതതതതത, Ξ and Aଵǡ ௞ 0

λ ,λ , , ,ξ,1 k 1 k 0 1, ,ξ,k 0 λ ,λ1 k 1, ,ξ,k 0

Taking into account (54), expression (55) is converted to

1

k

i

Since random variables തതതതതതത are assumed to be ଵǡ ௞

independent and not dependent on Ξ and A0 then

1

i

f y y a a – y (57)

Substituting (57) into (56), we obtain

λ ,λ , , ,ξ,1 1 0 ξ, 0 ψ δ λ ξ 0

k

Integrating (58) by the independent variables തതതതതതത, we get ଵǡ ௞

1

i

f

f

We represent the joint PDF of the random variables Ξ and

A0 as follows:

ω ξ, a f a ω ξ a (60)

Taking into account (60), expression (59) takes the form

1

0

k

i

f

f

u

– ³

(61)

Consider the integral standing under the sign of the product

in (61) Given the properties of the delta function, we can write

0

f

f

Substituting (62) into (61) gives

1

k k

i

Integrating the PDF (63) over the variable a0, we determine

1 0

k k

i

Using the multiplication theorem of PDFs, we find

λ ,λ ξ1 k λ ,λ ,ξ ω ξ1 k 

f/ f (65)

Finally, by substitution of (64) to (65) we obtain (45)

8.2 Numerical calculations

As shown in [13], if the output voltage of a certain type of

radar transmitter exceeds the threshold FF = 25 kV, it needs

maintenance to avoid breakdown Let A1 be a normal random

variable and A0 = a0 In this case, the PDF of the random

variable Ξ is given by [15]

2 2

3 3

1 1

2σ

2 σ

t

t t

S



where m = E[A] and ߪଵൌ ξܸܽݎሾܣଵሿ

Assume T = 3000 h, τ pc = τfr = 3 h, τtr = 10 h, m1 = 0.002 kV/h, σ1 = 0.00085 kV/h, a0 = 16 kV, and σy = 0.25 kV

Let us now find the optimal number of PCs N opt that maximizes the system’s average availability

Figure 1 shows the average availability versus number of checks for two different types of maintenance Curve 1

corresponds to the PM with N opt = 3, τopt = 750 h and A(N opt) = 0.989 Curve 2 corresponds to the maintenance based on the periodic operability checking This type of maintenance can be

classified as corrective maintenance Here, N opt = 16, τopt =

176.5 h and A(N opt) = 0.923

Thus, the use of the PM instead of the corrective maintenance results in increasing average availability and decreasing the number of optimal checks

Fig 1 Average availability versus number of checks: (1) predictive checks;

(2) operability checks

9 Conclusions

In this paper, we have described a mathematical model of predictive maintenance based on prognostics and health management New approach has been proposed to determining the optimal periodicity of predictive checking, which is based

on the use of the PDF of random errors in measurement of remaining useful life New equations have been derived for calculating the probabilities of correct and incorrect decisions made by the results of a predictive checking Mathematical expressions have been derived to determine the mean times spent by the system in various states during operation and maintenance for an arbitrary distribution of time to failure On the example of a linear random process of degradation, the procedure of determining the joint PDF of random errors in the measurement of the remaining useful life has been shown By numerical calculations it has been shown that predictive maintenance is unconditionally more efficient than corrective maintenance because it provides a higher average availability

at a smaller number of checks

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Fig Average availability versus number of. .. calculations it has been shown that predictive maintenance is unconditionally more efficient than corrective maintenance because it provides a higher average availability

at a smaller number of checks...

of maintenance programs through the use of performance indicators Europian Comission Report: EUR 22602 EN 2006

[12] Enhanced aircraft platform availability