## 5 MTTF Analysis
\[MTTF^{(2)}=\frac{1+\frac{2\widetilde{\lambda}^{(2)}}{\widetilde{\lambda}^{(1) }}\widetilde{P}_{c}}{2\widetilde{\lambda}^{(2)}} \tag{14}\]  
\[MTTF^{(1)}=\frac{1}{2\widetilde{\lambda}^{(1)}} \tag{15}\]  
The fuzzy memberships of the switches are used to extract the fuzzy curves of the redundant-switch configuration. According to the obtained triangular functions, the MTTF functions of the converters can be extracted through Figures 6-8. By referring to Equation (9) and (10), and Figure 5a, the fuzzy curve of the failure rate for each single switch in the parallel-switch and standby-switch configurations are obtained. The results are shown in Figure 6a,b (for standby and parallel switches in the single-phase mode, respectively), Figure 7a,b (for standby and parallel switches in the two-phase mode, respectively), and Figure 8a,b (for standby and parallel switches in the single-phase mode respectively). Then, by considering the probability of the successful operation of the fault managing system (Pc) the fuzzy curves of the standby and parallel configurations are individually obtained. The results are shown in Figure 6c,d (for standby and parallel configurations in the single-phase mode, respectively), Figure 7c,d (for standby and parallel configurations in the two-phase mode, respectively), and Figure 8c,d (for standby and parallel configuration in the single-phase mode, respectively). Finally, the fuzzy curves of the MTTF of different modes are extracted. The results are shown in Figure 6e (for single-phase mode), Figure 7e (for two-phase mode), and Figure 8e (for three-phase mode). Figure 6: Fuzzy curves related to single-phase converter (**a**) failure rate of a switch in the parallel-switch configuration, (**b**) failure rate of a switch rate in the standby-switch configuration, (**c**) failure rate of parallel-switch configuration, (**d**) failure rate of standby-switch configuration, and (**e**) Mean Time to Failure (MTTF) of single-phase converter with parallel-switch and standby-switch modules. Figure 7: Fuzzy curves related to two-phase converter (**a**) failure rate of a switch in parallel-switch configuration, (**b**) failure rate of a switch in standby-switch configuration, (**c**) failure rate of parallel-switch configuration, (**d**) failure rate of standby-switch configuration, and (**e**) MTTF of two-phase converter with parallel-switch and standby-switch modules. The described fuzzy curves of the MTTFs show the probability of the healthy life of the converter equipped with the mentioned redundant structure. According to the results from fuzzy analysis, the standby-switch-based converters offer higher reliability compared to the parallel-switch-based converters. The MTTF of the standby-switch-based converter is in a narrower range than that of the parallel-switch-based ones. For example, for the two-phase converter, the probabilities of high and low MTTF for the standby-switch-based converter are small values. The medium MTTF value is more probable for the standby-switch-based converters than that for the parallel-switch-based ones. As seen in fuzzy MTTF curves, it is probable that under certain conditions the MTTF of an \(n\)-phase converter is lower than that of a (\(n\)-1)-phase converter. To sum up, employing a proper fault-managing strategy makes it possible to prolong the MTTF of an \(n\)-phase converter in a way that it exceeds the MTTF of an (\(n\)\(-\)1)-phase converter. Mapping the fuzzy numbers to a real number is the final step in fuzzy-based reliability evaluation. The defuzzified central (\(Defvalue\)) of triangular fuzzy numbers \(\widetilde{R}=(a,b,c)\) can be determined by Equation (16). The results of the defuzzified values of the considered converters are provided in Table 2.  
\[Defvalue=\frac{c^{2}+bc-a^{2}-ab}{3(c-a)} \tag{16}\]