## 1 Introduction
### Mechanisms of Abnormal Impacts of Extreme Meteorological Factors on Power Grid Equipment
#### 3.1.2 Mechanism of Abnormal Effects of Extreme Meteorological Factors on Transmission Lines
Considering the Abnormal Coupling of Extreme Meteorological Factors and Power Flow Redistribution in the Power Grid  
During the passage of a tropical cyclone in period \(t\), following the abnormal state evolution method of the power grid described in Section 3, the abnormal state of the grid at period \(t\) can be represented as follows:  
\[\mathbf{A}\mathbf{E}_{t}=\left\{\mathbf{P}_{\text{w},t},\mathbf{S}_{\text{lic},t}\right\} \tag{20}\]  
In the equation, \(\mathbf{P}_{\text{w},t}=\left[P_{\text{w}1,t},\cdots P_{\text{w}N,t}\right]\) represents the abnormal output vector of the wind farm in period \(t\), and \(\mathbf{S}_{\text{lic},t}=\left[S_{\text{lic},t}^{1},\cdots S_{\text{lic},t}^{L}\right]\) represents the operating state vector of the branch in period \(t\). Based on Equation (20), the power flow redistribution at period \(t\) can be analyzed:  
\[\mathbf{P}_{t,t}=\mathbf{F}(\mathbf{A}\mathbf{E}_{t},\mathbf{x}_{t}) \tag{21}\]  
In this equation, \(\mathbf{P}_{t,t}\) represents the transmission power vector of all branches in the system at period \(t\); \(\mathbf{F}(\cdot)\) is the branch transmission power function; \(\mathbf{x}_{t}\) is the system parameter vector. \(\mathbf{P}_{t,t}\) represents potential overloaded transmission power. The failure probability of the overloaded branch \(l\) (\(l\in\mathcal{N}_{\text{cl}}\)) is as follows:  
\[Prob_{t}^{l}= \begin{cases}1,&P_{\text{fl},t}\geq P_{\text{fl}}^{\text{max}}\\ \frac{P_{\text{fl},t}-P_{\text{fl}}^{\text{c}}}{P_{\text{fl}}^{\text{max}}-P_{ \text{fl}}^{\text{c}}},&P_{\text{fl}}^{\text{c}}<P_{\text{fl},t}<P_{\text{fl}} ^{\text{max}}\\ 0,&P_{\text{fl},t}\leq P_{\text{fl}}^{\text{c}}\end{cases}  \tag{22}\]  
In this equation, \(P_{\text{fl},t}\) represents the transmission power of overloaded branch \(l\); \(P_{\text{fl}}^{\text{max}}\) represents the transmission power limit of overloaded branch \(l\); and \(P_{\text{fl}}^{\text{c}}\) represents the rated transmission power of overloaded branch \(l\). By sampling based on Equation (22), the abnormal transition state of the branch under power flow redistribution can be obtained:  
\[\mathbf{S}_{\text{lic},t}^{\text{a}}=\left[S_{\text{lic},t}^{\text{la}},\cdots S_{ \text{lic},t}^{\text{la}}\right] \tag{23}\]  
Based on \(\mathbf{S}_{\text{lic},t}^{\text{a}}\) and the branch fault vector \(\mathbf{Prob}_{\text{lic},t}\) caused by extreme meteorological factors, the branch fault state at period \(t+1\) can be obtained:  
\[\mathbf{S}_{\text{lic},t+1}=\mathbf{G}\left(\mathbf{S}_{\text{lic},t}^{\text{a}},\mathbf{Prob }_{\text{lic},t+1}\right) \tag{24}\]  
In this equation, \(\mathbf{G}(\cdot)\) represents the following conditional function:  
\[\mathbf{S}_{\text{lic},t+1}^{l}= \begin{cases}S_{\text{lic},t^{\prime}}^{\text{la} }&\text{if}\quad S_{\text{lic},t}^{\text{la}}=0\\ Sa\left(Prob_{\text{lic},t+1}^{l}\right),&\text{if}\quad S_{\text{lic},t}^{ \text{la}}=1\end{cases}  \tag{25}\]  
In this equation, \(Sa(\cdot)\) represents the sampling function. If branch \(l\) is in a disconnected state (\(S_{\text{lic},t}^{\text{la}}\)) at period \(t\), it will remain disconnected at period \(t+1\). Otherwise, the state will be determined by sampling based on the fault probability at period \(t+1\). Finally, by incorporating the abnormal output vector of the wind power at period \(t+1\), the abnormal system state \(\mathbf{A}\mathbf{E}_{t+1}=\left\{\mathbf{P}_{\text{w},t+1},\mathbf{S}_{\text{lic},t+1}\right\}\) at period \(t+1\) is obtained. From Equations (20)-(25), it is evident that the system transitions from abnormal state \(\mathbf{A}\mathbf{E}_{t}\) to state \(\mathbf{A}\mathbf{E}_{t+1}\), influenced not only by extreme meteorological factors at period \(t+1\)but also by the transition state of the branches caused by power flow redistribution at period \(t\). The alternating coupling of extreme meteorological factors and power flow redistribution drives the system's abnormal evolution.