DECLARATION 2- PUBLICATIONS
5.4 Results and discussion
5.4.1 Parameter estimates and reliability modelling
Table 5-1 presents the results of the MOM extracted from Section 4.7.1. In Section 4.7.1, it was shown that Kolmogorov Smirnov (K-S) test was implemented to test the hypothesis that the data really came from the purported Weibull distribution. The test accepted the results for the specified parameters (that is, β and η) of the Weibull distribution. The parameter estimates also fitted within the confidence intervals; hence the null hypothesis could not be rejected.
Table 5-1: Estimates of parameters β and η by the MOM
Weibull Parameters β η [x105 hrs.]
Estimated parameter values 3.4988 3.2786
Standard error (se) 0.6697 0.3025
Confidence interval (2.5%), (97.5%) (2.9048),( 4.5105) (2.7049), (3.828)
In this section, equations (5.1) to (5.3) have been applied, based on the values of β and η from Table 5-1, to generate plots of the CDF, the hazard rate and the PDF. The plots are presented and annotated in Figure 5-5. Figure 5-5 is referred to as a comparative model because it can be used to compare or benchmark results from other, but similar asset populations.
Figure 5-5: Comparative model of (a) CDF and hazard rate; and (b) PDF
0 10 20 30 40 50 60 70
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 (a) CDF and hazard rate h(t) at =3.4988,=327860 hrs.
Technical life, x [yr.]
h(t) [Magnitude] and CDF [Probability]
CDF hazard rate
0 10 20 30 40 50 60 70
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035
0.04 (b) PDF for =3.4988,=327860 hrs.
Probability density
Technical life, x[yr.]
A
B C
D
The CDF is a failure distribution function; hence it can be used to represent the failure probability [35]. It stands for the chance that the equipment will have a lifespan of up to time x [14]. Thus, Figure 5-5 (a) can be used as a model from which the failure probabilities to be used for risk modelling can be extracted. It can also be used to compare with or benchmark against failure probability trends of similar assets currently in operation. Point A on Figure 5-5 (a) signifies the end of life, whereby the CDF = 1 and the asset age is 67 years. Point B corresponds to point A, where h (x) = 0.384. On the other hand, the PDF shows the chance of equipment failing at the age of x. Point D on Figure 5-5 (b) (the PDF curve) corresponds to the steepest slope on the CDF plot (point C) and it occurs at 34 years of age. In terms of risk mitigation, point C indicates that refurbishment of these transformers should be done before the age of 34 years. The scale parameter (η) represents the age at which 63% of equipment will have failed [35], [81]. For these transformers, η is 327860 hours or 37 years. The probabilities in Figure 5-5 (a) can be used as inputs into the FMECA within the RCM.
The FMECA is normally conducted in two stages, namely: the first stage involving current actions (measures) and the second one after proactive measures are taken to reduce the risk [26].
Table 5-2 (a) shows the first stage, whereas Table 5-2 (b) presents the second stage. The computed failure probabilities are incorporated in the second stage of the FMECA. For example, from Figure 5-5, at 30 years the probability is 0.387. This is inserted in column F of Table 5-2 (b), and the risk priority number (RPN) is given as the product of columns F, G and H (see column I). By comparing the RPN under the current actions (business as usual) with that after proactive actions are taken, the asset manager can determine whether the strategies implemented were effective in mitigating the risk or not.
Table 5-2 (a): FMECA Stage 1 [from system to failure mode]
A B C D
System Function Functional failure Failure mode
Transmission transformer
To transmit power
Failure to transmit power
Decrease of mechanical, thermal and electrical strength
Table 5-2 (b): FMECA Stage 2 [from current measures to risk priority number (RPN)]
E F G H I
Current measures to
rectify failure Probability Severity (1-10)
Detection (1-10)
RPN (F x G x H) Planned
maintenance 0.387 3.35 3 3.889
The heading of column E in Table 5-2 (b) will change to reflect the type of strategy applied.
For example, if CBM is conducted as the proactive measure that is needed to reduce the RPN, the heading will change to proactive measure and the CBM will be listed under it as shown in Table 5- 2 (c). Table 5-2 (c) shows that the RPN reduces to 1.34 (from 3.889 to 1.34), which means the measures that were taken were effective.
Table 5-2 (c): FMECA after applying proactive measures
E F G H I
Proactive measures to
rectify failure Probability Severity (1-10)
Detection (1-10)
RPN (F x G x H)
CBM 0.2 3.35 2 1.34
The severity (consequence) depends on the impact of the equipment on system reliability, criticality of customer and type of public service served [76]. Detection shows the relative difficulty (in terms of the skill needed) in which a failure mode is detected on a scale of 1 to 10 [9]. For an objective risk analysis, the RPN must be computed for each failure mode that can be identified. It is worth noting that probabilities and detection are likely to change, whereas the severity will always remain the same.
It is also worth noting that stage one of the FMECA can be recorded in the RCM information worksheet according to [105]. A typical RCM worksheet would also give details of the failure effects and whether the failures are evident or hidden. Since the LV network has numerous components and most of which are small, the best practice is to group the small components and then to do a failure finding exercise for the group, rather than to deal with the individual components. Thereafter the degree of failure is quantified. If the components go through partial failure before total failure, the probability of failure could be calculated from the time of partial failure to the total failure.
Different assets will have different failure probabilities due to the varying operating environments they are subjected to. For example, the following values of failure probabilities at the equipment age of 30 years were derived using the Inverse Power Law and the Arrhenius model [59]: transmission transformers, 25%; distribution transformers, 21%; switch gears, 14%; contact breakers, 37%; and load interrupters, 20%. This kind of information can be compared with the results that are plotted in Figure 5-5.
Section 5.4.2 demonstrates how the Markov process analyses transitional probabilities and the MTTFF. In the process, failure and repair rate data is applied to the state-space model that was presented in Figure 5-4 (b). The fundamental purpose of this section is two-fold. First, to develop a model that simplifies the risk analysis, using only a few data sets from a short time span. Second, to apply the MTTFF in conducting the RCM. This will help to model the impact of maintenance strategies on costs.
5.4.2 Simulation of MTTFF and transient probabilities