## 4 Reliability Analysis of Redundant Systems
According to the FT of redundant systems shown in Figures 2 and 3, the minimal cut set of subsystems can obtained as follows:  
T: \(\{E_{1},E_{2},E_{3},E_{4},E_{5},E_{6},E_{7},E_{8},E_{9},E_{10},E_{11}\}\)  
E1: \(\{E_{11},E_{12},E_{13},E_{14},E_{15},E_{16},E_{17}\}\)  
Therefore, the reliability of the redundant system of OWT can be obtained by  
\[R_{sys}^{r}(t)=R_{E_{11}}^{sb}(t)\cdot R_{E_{12}}^{sb}(t)\cdot R_{E_{14}}^{sb }(t)\cdot\prod_{i=15}^{17}R_{E_{i}}^{2/3}(t)\cdot\prod_{j=2}^{4}R_{E_{j}}^{ pair}(t)\cdot\prod_{k=5}^{6}R_{E_{k}}^{ls}(t)\cdot R_{E_{k}}^{sb}(t)\cdot R_{E_{ 9}}^{ser}(t)\cdot R_{RS}^{w}(t) \tag{29}\]  
Using the methods given above, the time-dependent reliability of redundant systems in Figure 3 can be obtained, and the reliability curves are shown in Figure 10. As it is shown in Figure 10, the dotted line and the solid line represent system reliability with and without considerations of component swapping, respectively. The system reliability considering component swapping is significantly larger than that of the system without swapping. To measure the increase of system reliability, a function \(\Delta R_{sys}(t)=R_{sys}^{r}(t)-R_{sys}(t)\) that is the difference of system reliability between two situations is defined. Figure 11 presents the absolute increase in system reliability at a given time \(t\). It can be seen from Figure 11 that the absolute increase of system reliability (\(\Delta R_{sys}(t)\)) is increasing with the increase of time \(t\) (\(T<8800\) h), and the absolute increase of system reliability will decrease when time \(t>8800\) h. Therefore, the component swapping makes the greatest contribution to the reliability  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c c} \hline \hline \(\mathbf{I_{1}}\) & \(\mathbf{I_{2}}\) & \(\mathbf{I_{3}}\) & \(\mathbf{\Phi_{0}}\) & \(\mathbf{\Phi_{1}^{w}}\) & \(\mathbf{\Phi_{2}^{w}}\) & \(\mathbf{\Phi_{3}^{w}}\) & \(\mathbf{\Phi_{4}^{w}}\) & \(\mathbf{I_{1}}\) & \(\mathbf{I_{2}}\) & \(\mathbf{I_improvement of redundant systems of offshore wind turbine at \(t=8800\) h, which means that the \"optimal\" performance will be obtained if swapping failed components is performed at time \(t=8800\) h.  
Besides, to examine the system reliability with different swapping components, three swapping cases are compared. In case 2, only Type 1 components are able to be swapped, in case 3, only Type 2 components can be swapped, and in case 4, only Type 3 components can be swapped. The system reliability curves are given in Figure 12. Clearly, Case 2 provides the best improvement of system reliability than others, which is mainly because Type 1 components are at critical locations in the original system, and the effects of component swapping of Case 3 and 4 on system reliability are the same. Because the levels of redundancy of Type 2 and 3 components are similar. Therefore, the positions of temperature sensors of Type 1 are more important than that of the cooling fan of Type 2 and the cooling pump of Type 3. The positions of temperature sensors should be paid more attention and be allocated more resources (inspections, spare components, etc.) than other components. Figure 11: Improvement of system reliability with component swapping. Figure 12: System reliability with different swapping strategies. Figure 10: System reliability with all possible swapping.