DECLARATION 2- PUBLICATIONS
5.4 Results and discussion
5.4.2 Simulation of MTTFF and transient probabilities
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
Table 5-4: MTTFF matrix for transition from States P1 to P4
MTTFF for case #1 (in the cell P1-P4) MTTFF for case #2 (in the cell P1-P4)
P1 P2 P3 P4 P1 P2 P3 P4
P1 1.42 62.2 99.1 302.34 P1 1.5813 50.476 57.42 164.51 P2 15.55 5.69 106.77 271.66 P2 14.42 5.535 64.17 140.9 P3 13.58 67.89 10.37 260.76 P3 13.2 56 6.8786 140.48 P4 21.25 37.22 65.19 41.5 P4 19.95 32.39 40.14 24.08
Figure 5-6 presents plots of the transient probabilities, corresponding to the data in Table 5-4, for cases # 1 and 2. For both cases, it was assumed that the system started in State 1. The annotation in Figure 5-6 shows the complements of the uptime state, P1 (that is, 1-P1), which represent the risk of system failure. The complement of the uptime state for case # 1 and 2 are, respectively, 0.296 and 0.37.
Figure 5-6: Transient probabilities for cases 1 and 2
Tables 5-5 to 5-7 and Figures 5-7 to 5-9 present results that have been simulated for different cases. The two-component model (Figure 5-4) has been applied, assuming that situations can arise when one component is repaired much faster than the other; or one component fails at a much
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(a)Transient behavior for: F1=0.0171; F2=0.0114; R1=0.0685;R2=0.0833
Time steps
Probability and unavailability of states
State 1 State 2 State 3 State 4 1-State 1 1- State 2 1-State 3 1-State 4
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
(b)Transient behavior for: F1=0.022; F2=0.02; R1=0.077;R2=0.087
Time steps
Probability and unavailability of states
St-ate 1 State 2 State 3 State 4 1-State 1 1- State 2 1-State 3 1-State 4 Complement of the up-time
state (P1), i.e., 1-P1
Complement of the up-time state (P1), i.e., 1-P1
higher rate than the other. Then, the impacts of these scenarios on the MTTFF and costs are evaluated. In practice, the repair and failure rates for all similar equipment are assumed to be constant. This is the major flaw of the application of the Markov processes in industry [81].
Figure 5-7 shows a special case where the combined effects of the failure and repair rates for the system result in State 2 being predominantly the uptime state. Although the system started in State 1, the initial state is short-lived and the system dwells in State 2, where component # 2 has the highest limiting transient state probability at 0.8294 or 82.9%. The MFPT matrix and MTTFF associated with Figure 5-7 are presented in Table 5-5.
Figure 5-7: Transient probabilities for case #3
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Transient behavior for: F1=0.057; F2=0.0017; R1=0.0087;R2=0.037
Time steps
Probability and unavailability of states
St-ate 1 State 2 State 3 State 4 1-State 1 1- State 2 1-State 3 1-State 4
Table 5-5: MFPT matrix and MTTFF associated with Figure 5-7 MTTFF [x 103] (in the cell P1-P4)
P1 P2 P3 P4
P1 0.009 0.0181 2.0169 0.6280
P2 0.1183 0.0012 2.0595 0.6215
P3 0.0927 0.0351 0.1719 0.2578
P4 0.1353 0.0286 1.6893 0.0262
Figure 5-8 presents plots of case # 4 where component #2 (from the two component model, Figure 5-4) has a lower failure rate and a higher repair rate than component # 1. In this case, State 2 has a higher transient state probability than State 1. Its corresponding MFPT matrix and the MTTFF are presented in Table 5-6. This scenario may occur if component # 2 is newer than component # 1.
Hence, it follows that the complement of the transient probability of component # 2 is 0.136 or 13.6%, which presents the lowest failure probability in the system. In terms of best practice asset management, component # 2 is critical to the continued reliability of the system, hence all efforts must be made (by the asset manager) to ensure that it is in the up-state.
Figure 5-8: Transient probabilities for case #4
Table 5-6: MFPT matrix and MTTFF associated with Figure 5-8 MTTFF [x 103] (in the cell P1-P4)
P1 P2 P3 P4
P1 0.0076 0.0176 3.8633 0.6794
P2 0.1150 0.0012 3.9610 0.6645
P3 0.0178 0.0180 1.6512 0.5911
P4 0.1154 0.0031 3.8726 0.2520
Figure 5-9 shows plots of case # 5, which is a scenario where λF1 = λF2 = 0.017123 and μR1 = μR2 = 0.068493. This scenario leads to the following limiting transient state probabilities: P1=0.64, P2=P3=0.16, and P4 = 0.04; and the corresponding MTTFF is 204.4. The MFPT matrix and the MTTFF for this case is presented in Table 5-7, whereby the MTTFF is in the cell P1-P4. It can be
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Transient behavior for: F1=0.057; F2=0.0017; R1=0.0087;R2=0.037
Time steps
Probability and unavailability of states
St-ate 1 State 2 State 3 State 4 1-State 1 1- State 2 1-State 3 1-State 4
shown that a similar case where λF1 = λF2 = 0.011416 and μR1 = μR2 = 0.08333 yields better results where P1= 0.77, P2 = P3 = 0.106, and P4 = 0.02; and the MTTFF is 451.1.
Figure 5-9: Transient probabilities for case #5
Table 5-7: MFPT matrix and MTTFF associated with Figure 5-9.
MTTFF (in the cell P1-P4)
P1 P2 P3 P4
P1 1.5625 65.7010 65.7010 204.4051
P2 16.4250 6.2501 73.0010 175.2046
P3 16.4250 73.0010 6.2501 175.2046
P4 23.7250 43.8005 43.8005 25.0006
5.4.2.2 General discussion of CDF, h (t), PDF and MTTFF plots
In general, Figure 5-5 is a comparative or benchmarking model that applies historical failure data to model risk of component failure, if the asset population has adequate data for statistical analysis. On the other hand, Figures 5-6 to 5-9 model risk of component failure even with a very
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Transient behavior for: F1=0.017123; F2=0.17123; R1=0.068493;R2=0.068493
Time steps
Probability and unavailability of states
St-ate 1 State 2 State 3 State 4 1-State 1 1- State 2 1-State 3 1-State 4
limited amount of failure data and from as short a time horizon as one year. Figures 5-6 to 5-9 have shown that, depending on the failure and repair rates, it is possible for a two component redundant system to behave like a series system. When the system behaves in this way, it will portray a Poisson (distribution) effect; that is, its failure will be catastrophic or of the binary form (existing as either 1 or 0). In other ways, the uptime state of such a system will basically depend on the uptime state of a single component. This single component, therefore, becomes the most critical one for a sustained system reliability. For example, at a point where λ1 = μ1 = 0.01112 and λ2 = μ2 = 0.01211, P1 = P2 = P3 = P4 = 0.9768 at the start of simulation (i.e., at t = 0), and the limiting states are P1 = P2 = P3 = P4 = 0.2718; whereas the MTTFF is 172.5. This is a kind of Poisson distribution, displayed by Figure 5-10, of the following form:
! 0 ) ) (
(
e
x
x e t t
P
x
x t t(5.20) where x = times the failures are occurring in the interval (0, t), similar to what is alluded to in [92].
This Poisson distribution analogy or concept is also similar to what was demonstrated in modelling of transformer spares to account for catastrophic failures, as advanced in [65].
Figure 5-10: Transient behavior for a case acting as a Poisson process
0 10 20 30 40 50 60 70 80
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Transient behavior for: F1=0.01112; F2=0.01211; R1=0.01112; R2=0.01211
Time steps
Probability: P1 to P4 and unavailability: 1-P1 to 1-P4
P1 = P2 = P3 = P4
Section 5.4.3 demonstrates how the transient probabilities can be applied in AM modelling within the RCM approach.