Transcript skfPhase1Session2-principles

EXAKT
SKF
Phase 1, Session 2
Principles
1
The CBM Decision supported by
EXAKT
Given the condition today, the asset mgr.
takes one of three decisions:
1. Intervene immediately and conduct
maintenance on an equipment at this
time, or to
2. Plan to conduct maintenance within a
specified time, or to
3. Defer the maintenance decision until the
next CBM observation
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Condition
The conventional CBM decision method
from Nowlan & Heap, (Moubray)
Detectable
indication of a
failing process
Potential
failure, P
Detection of the
potential failure
Net P-F
Interval
Functional
failure, F
CBM inspection interval:
< P-F Interval
P-F Interval
Working age
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Moubray (RCM II) addresses two extreme cases
Special case 1 – completely random (age independent,
dependent only on condition monitoring data)
Failures
occur on a
random
basis
PF
FF
PF detected at
least 2 months
before FF.
F2
0
1
Age (years)
F3
2
Many failure modes are both
age and condition indicator
dependent. (The age parameter
often summarizes the influence
of all those wear related factors
not explicitly included in the
hazard model.)
Inspections
at 2 month
intervals
F1
3
4
5
Special case 2 – completely age dependent
Tread depth
Tread depth when new
= 12 mm
Potential failure
= 3 mm
Functional failure
= 2 mm
Cross-section of
tire tread
P-F interval
At least 5000
km
PF
FF
0
10
20
Operating Age
(x 1000 km)
30
40
50
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The P-F Interval method
Assumes that:
1. The potential failure set point, P, of an
identifiable condition is known, and that
2. The P-F interval is known and is
reasonably consistent (or its range of
variation can be estimated), and that
3. It is practical to monitor the item at
intervals shorter than the P-F interval
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EXAKT has two ways of deciding
whether an item is in a “P” state
1. A decision based solely on failure
probability.
2. A decision based on the combination of
failure probability and the quantifiable
consequences of the failure, and
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The two methods
1. Age data
2. CM data
3. Cost data
Hazard Model
Transition Model
Cost and
Availability Model
RULE
Maintenance
Decision
Failure probability
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Assumptions and models
used in EXAKT
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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 T (>t) given the state of the
vector (process) at t.
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2nd assumption
 1
t

h(t , Z (t );  , ,  )    e
  
  0,  0,   ( 1 ,  2 , ,  m )
m
1
 i Zi (t )
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.
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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.
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Representative values
State 4
Representative value
State 3
Representative value
State 2
Representative value
State 1
Representative value
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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
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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.
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Making CBM decisions
Two ways
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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.
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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.
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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.
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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).
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( 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
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(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.
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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
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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
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Decision 1- Only Probability
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Decision 2-Probability and cost
minimization
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Decision 3-Probability and
availability maximization
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Decision 4-Probability and
profitability maximization
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