## 3 LE Identification Method Considering the Fluctuation of Wind Power Output
### Calculation Method of APEV Considering the Output Fluctuation of the Wind Farm
#### 3.3.2 Generalized Sensitivity When the Faulty Branch Is Removed  
The conventional sensitivity GSDF reflects the APCV of the branch caused by the change value in active power output at bus node \(i\) when the power grid topology is unchanged. After removing the faulty branch, the power grid's topology changes. Each bus node's sensitivity to the remaining branch will also change. Therefore, it is necessary to extend the conventional sensitivity to the case of branch removal, which is called generalized sensitivity. The generalized sensitivity of each bus node to the line is derived below based on the LERPF and GSDF obtained above. 1. Scenarios where the active power of a single bus node output changes When branch \(l\) is removed, the active power of branch \(k\) will change from \(\mathrm{P}_{k}^{0}\) to \(\mathrm{P}_{k}^{l}\), and \[\mathrm{P}_{k}^{l}=\mathrm{P}_{k}^{0}+d_{k-l}\mathrm{P}_{l}^{0}\] (12) Considering that the active power generated by large-scale wind farms has huge volatility, it is assumed that the active power generated by the wind farms connected to bus node \(i\) changes after branch \(l\) is removed. Then, the active power transmitted by branch \(k\) after branch \(l\) is removed will become as follows, \[\mathrm{P}_{k}^{l,i}=\mathrm{P}_{k}^{i}+d_{k-l}\mathrm{P}_{l}^{i}\] (13) where the superscripts '\(l\)' and '\(i\)' represent the removed branch \(l\) and the bus node \(i\) containing the wind farm, respectively. \(\mathrm{P}_{k}^{l,i}\) is the APEV of branch \(k\) under the condition that the active power generated by node \(i\) changes after branch \(l\) is removed; According to Equation (11), Equation (14) can be obtained. \[ \begin{cases}\mathrm{P}_{k}^{i}=\mathrm{P}_{k}^{0}+G_{k-i}\mathrm{\Delta G}_{ i}\\ \mathrm{P}_{l}^{i}=\mathrm{P}_{l}^{0}+G_{l-i}\mathrm{\Delta G}_{i}\end{cases}  \tag{14}\]  
Substitute Equation (14) into Equation (13); then Equation (15) is obtained as follows. \[ \begin{array}{ll}\mathrm{P}_{k}^{l,i}&=(\mathrm{P}_{k}^{0}+G_{k-i}\mathrm{ \Delta G}_{i})+d_{k-l}(\mathrm{P}_{l}^{0}+G_{l-i}\mathrm{\Delta G}_{i})\\ &=(\mathrm{P}_{k}^{0}+d_{k-l}\mathrm{P}_{l}^{0})+(G_{k-i}+d_{k-l}G_{l-i}) \mathrm{\Delta G}_{i}\\ &=\mathrm{P}_{k}^{l}+G_{k-i}^{l}\mathrm{\Delta G}_{i}\end{array}  \tag{15}\]  
where \(\mathrm{P}_{k}^{l}\) is the APEV of branch \(k\) after branch \(l\) is removed without considering the change value of the active power generated by the wind farms; \(G_{k-i}^{l}\mathrm{\Delta G}_{i}\) is the APCV of branch \(k\) caused by the change value of active power generated by bus node \(i\) after branch \(l\) is removed. According to Equation (15), when the bus node output at node \(i\) changes before and after branch \(l\) is removed, the active power on branch \(k\) consists of two parts:  
* \(\mathrm{P}_{k}^{l}\): the APEV of branch \(k\) after branch \(l\) is removed when the active power generated by the wind farms is constant; that is, the APEV obtained by the conventional method. * \(\mathrm{G}_{k-i}^{l}\mathrm{\Delta G}_{i}\): the APCV of branch \(k\) caused by the \(\mathrm{\Delta G}_{i}\) after branch \(l\) is removed. Define \(\mathrm{G}_{k-i}^{l}\) as generalized sensitivity. According to Equation (15), \[G_{k-i}^{l}=G_{k-i}+d_{k-l}G_{l-i}\] (16)It can be seen from Equation (16) that the generalized sensitivity is related to the LERPF and GSDF and can reflect the change in active power caused by the change in bus node output before and after the branch is removed. From Section 3.3.1, for any branch in the subgraph, the nodes outside the subgraph and the breakpoints of the subgraph have the same GSDF for the branch. It can be seen from Equation (16) that after branch \(l\) is removed, for branch \(k\) in the subgraph, \(d_{k-l}\) is a fixed value. Therefore, the nodes outside the subgraph and the breakpoints of the subgraph also have the same generalized sensitivity to the branch. 2. Scenarios where the active power of multi-node output changes.