# Reliability Modelling and Evaluation for LTD System Based on Load-Sharing Model
## 3 Load-Sharing Based Voltage Redistribution Mechanism for Modules in Each Level  
According to the LTD system feature, modules in each level are fault dependent because of the existence of the isolating mechanism. Initially, modules in the same level are loaded with the same voltage according to a pre-specified work coefficient. If any module fails and currently the total number of failure modules is lower than the pre-specified values \(k\), the voltage load will be reassigned to the remaining normal modules, which will cause a higher failure probability. The level failure can be regarded as a process of state migration comprising a set of states and a set of transitions [38]. Figure 2 illustrates a failure state migration road. Transitions here are source-action-target triples, which represent the influence of the module failure. There are many failure state migration roads actually corresponding to different failure sequences.  
The equal rule is the most commonly considered load-sharing rule. However, the system structure should have effects on the load redistribution process. The closer the normal module is to the failed one, the more likely it will be affected. Thus, it is more realistic to combine the structure factor into the load-sharing rule, such as local and monotone load-sharing rules [31]. Corresponding to the migration process, we assume the voltage of module \(p\) at state \(x-1\) as \(V_{p}(x-1)\). If the failure of module \(q\) causes the level to migrate to state \(x\) from state \(x-1\), the voltage variation of module \(p\) at state \(x\) can be represented as:  
\[\Delta V_{p}(x)=\begin{cases}0&p\in S_{1}(x)\\ f\big{(}V_{q}(x-1),\ D_{pq}\big{)}&p\in S_{0}(x)\end{cases} \tag{9}\]  
where \(S_{0}(x)\) is the set of the normal module at state \(x\), \(S_{1}(x)\) is the set of the failure module at state \(x\), and \(D_{pq}\) is the distance factor related to module \(p\) and module \(q\) in order to reflect the structure factor. The function \(f\big{(}V_{q}(x-1),\ D_{pq}\big{)}\) illustrates that the voltage variation is affected by both the voltage of the failed module and the distance factor. Constraint of the normalization process can be given as:  
\[\sum\Delta V_{p}(x)=\delta\cdot V_{q}(x-1) \tag{10}\]  
where \(0<\delta\leq 1\) is an adjustment factor to restrict that the sum of all the voltage variations does not exceed the original voltage of module \(q\). Then the voltage of module \(p\) at state \(x\) can be expressed as:  
\[V_{p}(x)=\begin{cases}0&p\in S_{1}(x)\\ V_{p}(x-1)+f\big{(}V_{q}(x-1),\ D_{pq}\big{)}&p\in S_{0}(x)\end{cases} \tag{11}\]  
Additionally, the function \(f\big{(}V_{q}(x-1),\ D_{pq}\big{)}\) can be adjusted according to different rules. Meanwhile, the case \(f\big{(}V_{q}(x-1),\ D_{pq}\big{)}\equiv 0\) can cover the traditional reliability analysis process where the load-sharing mechanism is not considered.  
Based on the existing discussion of the type of load-sharing rules [30, 31, 32, 39], in this part, we consider four typical rules corresponding to equal, linear, exponential, and local-equal relationship in detail to reflect common function types, that is linear, non-linear, and segment. The proposed rules cover the generally accepted load-sharing rules.  
Figure 2: A failure state migration road.