Assumptions and models
used in EXAKT
1
First assumption
An item's state of health is encoded within
measurable condition indicators (which, of
course, is the underlying premise of CBM).
Z(t) = (Z1(t), Z2(t), ... , Zm(t))
(eq. 1)
Each variable Zi(t) in the vector contains the value of
a certain measurement at that discreet moment, t
We would like to predict failure time T (>t) given the
state of the vector (process) at t.
2
2nd assumption

h(t , Z (t );  , ,  ) 

t
 
 
 1
 Z (t )

1 i i
e
m
where
  0,  0,   ( 1 ,  2 , ,  m )
where β is the shape parameter, η is the scale parameter, and γ
is the coefficient vector for the condition monitoring variable
(covariate) vector. The parameters β, η, and γ, will need to be
estimated in the numerical solution.
3
3rd assumption
In EXAKT, it is assumed that Z(d)(t) follows a nonhomogeneous Markov failure time model described by
the transition probabilities
Lij(x,t)=P(T>t, Z(d)(t)= Rj(z)|T>x, Z(d)(x)= Ri(z))
eq. 3)
(
where:
x is the current working age,
t (t > x) is a future working age, and
i and j are the states of the covariates at x and t respectively
Z(d)(t) is the vector of “representative” values of each condition indicator.
It is the probability that the item survives until t and the state of Z(d)(t) is j given
that the item survives until x and the previous state, Z(d)(x) , was i. The
transition behavior can then be displayed in a Markov chain transition
probability matrix, for example, that of the next slide.
4
Transition probability matrix
Table 1: Transition probability matrix
T,P
Future
T,P
Curren
t
1,1
1,2
1,3
2,1
2,2
2,3
3,1
3,2
3,3
1,1
.467
.176
2e-4
.162
.188
3.2e-3
2.5e-4
2.7e-3
0
1,2
.42
.184
2.4e-4
.16
.23
5e-3
3e-4
3.8e-3
0
1,3
.36
.178
7.7e-4
.16
.268
.029
3e-4
4.4e-3
0
2,1
.409
.167
2.2e-4
.183
.232
4.8e-3
3.6e-4
3.9e-3
0
2,2
.35
.175
2.9e-4
.18
.282
7.8e-3
4e-4
5.3e-3
0
2,3
.26
.164
1.6e-3
.16
.334
.066
4e-4
5.8e-3
0
3,1
.338
.163
2.5e-4
.19
.291
6.4e-3
1.5e-3
1.3e-2
0
3,2
.31
.171
2.9e-4
.188
.32
8.2e-3
1.2e-3
1.2e-2
0
3,3
0
0
0
0
0
0
0
0
1
5
Representative values
State 4
Representative value
State 3
Representative value
State 2
Representative value
State 1
Representative value
6
4th assumption
The combined PHM and transition
models
For a short interval of time, values of transition
probabilities can be approximated as:
Lij(x,x+Δx)=(1-h(x,Ri(z))Δx) • pij(x,x+ Δx)
(eq. 6)
Equation 6 means that we can, in small steps, calculate the future
probabilities for the state of the covariate process Z(d)(t). Using the hazard
calculated (from Equation 2) at each successive state we determine the
transition probabilities for the next small increment in time, from which we
again calculate the hazard, and so on.
7
Making CBM decisions
Two ways
8
CBM decisions based on
probability
The conditional reliability function can be expressed as:
R(t  x i )  P(T  t  T  x Z ( x)  i )   Lij ( x t )
(eq. 7)
j
The “conditional reliability” is the probability of survival to t given that
1.failure has not occurred prior to the current time x, and
2.CM variables at current time x are Ri(z)
Equation 7 points out that the conditional reliability is equal to
the sum of the conditional transition probabilities from state i to
all possible states.
9
Remaining useful life (RUL)
Once the conditional reliability function is calculated we can obtain
the conditional density from its derivative. We can also find the
conditional expectation of T - t, termed the remaining useful life
(RUL), as

E(T  t  T  t Z (t ))   R( x  t Z (t )) dx
(eq. 8)
t
In addition, the conditional probability of failure in a
short period of time Δt can be found as
d 

P t  T  T  t | t , Z
t   1  Rt  t | t , Z d  t 
(eq. 9)
For a maintenance engineer, predictive information based on current CM
data, such as RUL and probability of failure in a future time period, can be
valuable for risk assessment and planning maintenance.
10
CBM decisions based on
economics and probability
Control-limit policy:
perform preventive maintenance at Td if Td < T; or
perform reactive maintenance at T if Td ≥ T,
Where:
(Eq. 10)
Td=inf{t≥0:Kh(t,Z(d)(t))≥d}
Where:
K is the cost penalty associated with functional failure, h(t,Z(d)(t) is
the hazard, and d (> 0) is the risk control limit for performing
preventive maintenance. Here risk is defined as the functional
failure cost penalty K times the hazard rate.
11
The long-run expected cost of maintenance (preventive
and reactive) per unit of working age will be:
( d ) 
C p (1  Q(d ))  C f Q(d )
W (d )

C p  KQ(d )
(eq. 11)
W (d )
where Cp is the cost of preventive maintenance, Cf = Cp+K is the cost of
reactive maintenance, Q(d)=P(Td≥T) is the probability of failure prior to a
preventive action, W(d)=E(min{Td,T}) is the expected time of
maintenance (preventive or reactive).
12
( d ) 
C p (1  Q(d ))  C f Q(d )
C p  KQ(d )
W (d )
W (d )
Best CBMpolicy
Let d* be the value of d that minimizes the right-hand side of Equation
11. It corresponds to T* = Td*. Makis and Jardine in ref. 3 have shown
that for a non-decreasing hazard function h(t,Z(d)(t), rule T* is the best
possible replacement policy (ref. 4).
Equation 10 can be re-written for the optimal control limit policy as:
T*=Td*=inf{t≥0:Kh(t,Z(d)(t))≥d*}
(eq. 12)
For the PHM model with Weibull baseline distribution, it can be interpreted as (ref. 2))
m
T   min{t  0    i Z i (t )     (   1) ln t}
(eq. 13)
i 1
Where
  ln

 
  d
K
The numerical solution to Equation 13, which is described in detail in (Ref. 7) and
13
(Ref. 8).
The “warning level” function
 Z t 
i
i
g t       1 ln t 
*
Working age
• Plot the weighted sum of the value of the significant CM variables
(covariates). On the same coordinate system plot the function g(t).
• The combined graph can be viewed as an economical decision
chart.
• Shows whether the data suggests that the component has to be
renewed.
14
5th assumption
In the decision chart, we approximate the value of
m
  Z t 
i 1
d 
i
i
m
by
  Z t 
i 1
i
i
Decision chart
15
Numercal example
A CBM program on a fleet of Nitrogen compressors
monitors the failure mode “second stage piston ring
failure”. Real time data from sensors and process
computers are collected in a PI historian. Work
orders record the as-found state of the rings at
maintenance.
The next four slides illustrate 4 EXAKT optimal CBM decisions
16
Decision 1- Only Probability
17
Cost and probability
18
Decision 2-Probability and cost
minimization
19
Decision 3-Probability and
availability maximization
20
Decision 4-Probability and
profitability maximization
21