## 3 Methodology
### Component Swapping in the Original System  
In complex equipment, component redundancy is widely used to improve system reliability, such as parallel redundancy, load-sharing redundancy, standby redundancy and \(k\)-out-of-\(n\) redundancy. Once systems can not function due to the failures of critical components, shutdowns of entire systems will bring huge economic losses to enterprises. To make failed systems return to work quickly, failed components at a more critical position can be replaced with other functioning components at a less critical position in the original system. As an example, Figure 4 depicts a four-component system of two types. Components 1 and 2 are of the same Type 1, and components 3 and 4 are of the same Type 2. Component 1 is in a more important position than other components, and failure of component 1 will cause the failure of the entire system. Therefore, the system reliability will be improved a lot if component 1 fails and can be replaced by component 2 or components of the same type in the original system. \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**No.** & **Units** & **Type** & **MTBF (\(\times\)10\({}^{4}\) h)** & **Distribution** \\ \hline
1, 2 & Encoder of pitch motor & Standby & 56.869 & Exponential \\
3, 4 & Speed encoder & Standby & 38.625 & Exponential \\
5, 6 & Temperature sensor & Standby & 25.696 & Exponential \\
7/8 & Grid voltage/Battery & Standby & 35.215/21.651 & Exponential \\
9, 10, 11 & Limit switch & 2/3 & 180.855 & Weibull \\
12, 13, 14 & Charging circuit & 2/3 & 18.087 & Exponential \\
15, 16, 17 & Back power & 2/3 & 17.52 & Exponential \\
18/19 & PLC/Safety controller & Parallel & 25.835 & Exponential \\
20, 21 & Signal monitoring & Parallel & 3.5339 & Exponential \\
22 & Speed sensor of the rotor & Parallel & 16.459 & Exponential \\
23 & Speed sensor of the generator & Parallel & 10.512 & Exponential \\
24, 25 & Hydraulic system & Load sharing & 3.5 & Weibull \\
26, 27 & Nacelle axial-flow fan & Load sharing & 17.52 & Weibull \\
28 & Temp. sensor of gearbox & Series & 48.48 & Exponential \\
29/30 & Cooling fan/Cooling pump & Load sharing & 108.405/61.946 & Weibull \\
31 & Temp. sensor of control system & Series & 10.512 & Exponential \\
32, 34 & Generator & 4.5499 & Weibull \\
33, 35 & Frequency convertor & Standby & 0.601 & Exponential \\
36, 37, 38 & Anemometer & Standby & 16.637 & Exponential \\
39, 40, 41 & Wind vane & Standby & 4.2 & Exponential \\
42, 43 & Temp. sensor of nacelle & Parallel & 28.293 & Exponential \\
44 & Temp. sensor of genreator & Series & 48.18 & Exponential \\
45, 46 & Cooling fan of generator & & 86.724 & Weibull \\
47 & Cooling pump of generator & Load sharing & 61.946 & Weibull \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of basic events of fault tree. Survival signature-based component swapping is the possibility to replace a failed component by another component in the system which has not yet failed [11; 19]. For a coherent system that consists of \(m\) units of \(K\geq 2\) types, with \(m_{k}\) units of type \(k\in\{1,2,\cdots,K\}\) and \(\sum_{k=1}^{K}m_{k}=m\). Let \(\Phi(l)\) (\(l=1,2,\cdots,m\)) denote the probability that the system functions. If we assume that there are exactly \(l\) units functioning, then the remaining \((m-l)\) units do not function. \(\underline{x}^{k}=(x_{1}^{k},x_{2}^{k},\cdots,x_{m_{k}}^{k})\in\{0,1\}^{m_{k}}\) is the state vector of units of type \(k\), with \(x_{i}^{k}=1\) if the \(k\)th units of type \(k\) function and \(x_{i}^{k}=0\) if not. The state vector \(\underline{x}=(\underline{x}^{1},\underline{x}^{2},\cdots,\underline{x}^{K}) \in\{0,1\}^{m}\) is introduced to represent the states of units of the type \(k\) (\(\sum_{i=1}^{m_{k}}x_{i}^{k}=l^{k}\)). The system's survival function is represented by \(\Phi(l_{1},l_{2},\cdots,l_{K})\) that means the probability that the system functions in the condition that exactly \(l_{k}\) of type \(k\) units function, for \(l_{k}=0,1,\cdots,m_{k}\).