24
2.8 Modelling in Condition-Based Maintenance
The objective in developing a CBM model is to acquire knowledge about the equipment condition based upon its monitored parameters, so that any necessary maintenance
decisions can be made accordingly. Generally, models can be divided into two parts: the deterioration model and a decision model Frangopol et al., 2004. The deterioration
model is used to approximate and predict the actual deterioration process, taking into consideration age and condition-monitoring information. The decision model uses the
deterioration model to determine the optimal decision that we would like to decide in order to minimise a criterion of interest, such as cost, safety and others. Since the
current and future states of a system are unknown unless it is directly observed, a probabilistic approach is suitable for the model.
In this section, a general idea is given on how this uncertainty is modelled as a key element of CBM optimisation. Several approaches are being researched and applied in
the development of CBM models. One of the methods that have been widely used is Proportional Hazards Modelling PHM.
2.8.1 Proportional Hazards Model PHM
Introduced by Cox 1972, the Proportional Hazards PH model is used to identify significant covariates and to quantify their effects on survival using hazard function as
a function of covariate values and the working age time. The model has been widely
used in the biomedical field, for which medical research and drug trials are examples. Recently there has been an increasing interest in its application for reliability
engineering. The use of PH models in reliability assumes that the hazard function of a machine is the product of both a baseline hazard and a term containing explanatory
variables or covariates at time t as illustrated by Kumar 1996 in Figure 2-5.
25
Figure 2-5: Illustration of pattern of the hazard function and the baseline hazard function.
Kumar 1996 stated that effect of a covariate is to increase or to decrease the hazard function. Mathematically, the hazard function can be written as
, ,
t z
t h
t z
t h
2-1
where
t h
is the baseline hazard function, ,
t z
is the functional covariate term, t
z is a vector of covariates and
is a vector of covariate coefficients. These
covariates may be measurements of machine condition, such as the levels of metallic components in oil analysis or vibration amplitude.
may be estimated from the data to provide a quantitative measure of the importance of each covariate and their impact on
the hazard.
To use PHM, a stopping rule the interval for maintenance needs to be defined. In Jardine et al. 1997 and Jardine et al. 1998, the interval is defined as
d
T , where
d
. To find the optimum value of
d
, the cost model Jardine et al., 1998; Kobbacy et al., 1997 is used, and shown below
1 d
W d
Q C
d Q
C d
C
f p
2-2
26 where
p
C
is the mean cost of a preventive replacement.
f
C
is the mean cost of a failure replacement. d
Q is the probability that an item will fail before a preventive replacement.
d W
is the expected time between two consecutive replacements regardless of reason for replacement
– preventive or at failure.
The details of this formulation can be found in Jardine et al. 1998. The two most common techniques for estimating the PHM coefficients are i Cox‘s partial likelihood
method, which estimates the coefficients without making any assumptions about the form of the base hazard, and ii the maximum likelihood estimation MLE method, in
which an explicit form of base hazard e.g. Weibull hazard is assumed Gurvitz, 2005. Higher values of
will reduce the estimated survival time for a machine, so it can be viewed as a factor accelerating the failure rate of the machine Mann et al., 1995.
Jardine et al. 1998 consider the baseline hazard
t h
as a Weibull hazard function, which has the following form
1
t t
h
o
2-3
where and
are the scale and shape parameters. Once the optimal threshold level
d
is calculated, it is easy to utilize the optimal replacement rule where the replacement is taking place at the first time t for which
t d
t z
t z
ln 1
ln
2 2
1 1
2-4
This implies that PHM combines all the significant measurements into one single value with appropriate weights. The decision rule then suggests preventive replacement at the
time t when the combined covariate values reach the warning level, which depends on
t . Several authors have considered proportional hazards models for modelling the
27 condition monitoring process Jardine et al., 2001; Love and Guo, 1991; Vlok et al.,
2002. A list of applications of PHM in reliability studies can be found in Kumar 1996. The advantage of this method is that it includes both the age and the condition of the
equipment in the calculation of the hazard at time t Jardine et al., 1998. Needless to say, PHM assumes that the covariates will change the hazards that may be true in some
cases, but in reality it may be the state of the system that causes the change in the observed parameters and not vice versa, such as in the case of vibration monitoring.
This would be a significant problem if PHM were used in CBM, where the relationship between the monitored parameters and the underlying state of the system does not
follow the assumptions made in PHM.
2.8.2 Proportional Intensities Model PIM
Kategori