# Reliability Modelling and Evaluation for LTD System Based on Load-Sharing Model
## 4 Numerical Simulation and Analysis
### Effect Analysis of Different Load-Sharing Rules on the Modules in a Level  
In Section 2, we have introduced the LTD system in which there are 30 modules in each level. As a typical load-sharing system, the failure probability of the remaining module rises with the increasing voltage load caused by the random failures. Although empirically the linear load-sharing rule may be inferred as the most appropriate rule, we cannot ignore the fact that the current data are too insufficient to determine specific load-sharing parameters. Therefore, we compare four typical rules to show the commonness and differences between them, which is meaningful to help to further identify the detailed rules. Figure 4 presents load redistribution results of different load-sharing rules in different failure cases. The initial working coefficient is set as 0.60. Moreover, solid lines are used to describe the working coefficients of the normal modules and dotted lines are the continuation of curves. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Working Coefficient (\%)** & **Failure Probability** & **Working Coefficient (\%)** & **Failure Probability** \\ \hline
35 & 0 & 70 & \(1.40\times 10^{-5}\) \\
40 & \(2.00\times 10^{-14}\) & 75 & \(1.10\times 10^{-4}\) \\
45 & \(3.10\times 10^{-12}\) & 80 & \(7.70\times 10^{-4}\) \\
50 & \(1.70\times 10^{-10}\) & 85 & \(4.40\times 10^{-3}\) \\
55 & \(5.30\times 10^{-9}\) & 90 & 0.022 \\
60 & \(1.00\times 10^{-7}\) & 95 & 0.098 \\
65 & \(1.40\times 10^{-6}\) & 100 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Relationship between working coefficients and failure probability for switch. Figure 3: Failure probability of the switch. Working coefficient curves in the condition that only one module fails are shown in Figure 4a,b. From those graphs, we can clearly see that the coefficient curve of equal rule keeps horizontal while the coefficient curve of local-equal rule is composed of several horizontal segments. As for the linear or exponential rules, the coefficient curve decreases with a special rate centered on the failed module. Furthermore, the impacts of the adjustment factor in different load-sharing rules are also reflected. As shown in Figure 4a,b, there is a trend that other rules can be degraded to the equal rule with the adjustment factor increasing. That is, the bigger the parameter \(b\), \(d\) or \(f\) in the corresponding rule, the smaller the impact of the distance between the normal module and the failed one. Additionally, comparing the coefficient increment sizes in Figure 4a with those in Figure 4b, it can be seen that the failure of the module around the structure edge tends to cause a larger coefficient change near it. Working coefficient curves in the condition that there are two failed modules are shown in Figure 4c,d. The results show that the coefficient curve segment between two failure sources is still subject to a linear relationship in the linear rule, while the same segment seems to be more complicated in the exponential or local-equal rule. The curve of the exponential rule shows a quite obvious fault centrality than the linear rule as before, where the centrality here is used to describe the phenomenon that the failure modules tend to occur within a small local range. The local-equal rule shows differences according to the failed modules. The modules in the overlapping part influenced by different failures will reach a rather high point with the local-equal assumption. Figure 4: Load redistribution results of different load-sharing rules under different failure cases. (**a**) single failure: module #1; (**b**) single failure: module #7; (**c**) cascading failures: module #15 after module #7; (**d**) cascading failures: module #23 after module #7. According to the discussion above, the exponential and local-equal rules show a more obvious fault centrality than the linear rule, while the equal rule assumes all the remaining modules to be the same. Also, the equal rule can be seen as a special case of other rules.