## 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
 \tag{14}\]  
The probability of a break occurring in single-span line \(i\) during the first \(j\) periods can be expressed as follows:  
\[\mathit{Prob}_{\text{ulc},j}^{i}=1-\text{e}^{-\sum_{j=1}^{t}\lambda_{ij} \Delta t} \tag{15}\]Let transmission line \(i\) consist of \(n\) towers and \(n\)-\(I\) single-span lines connected in a series. Based on Equations (12) and (15), the failure probability of transmission line \(i\) during the first \(j\) periods can be expressed as follows:  
\[Prob^{i}_{\text{lic},j}=1-\prod_{k=1}^{n}(1-Prob^{k}_{\text{toc},j})\prod_{m=1}^ {n-1}(1-Prob^{m}_{\text{ulc},j}) \tag{16}\]  
The Abnormal Evolution Process of the Power Grid Under Extreme Meteorological Conditions During Tropical Cyclones  
Based on Section 3.1, the time-varying extreme weather conditions caused by a tropical cyclone not only lead to time-varying abnormal outputs \(P_{\text{wi},j}\) in wind farm \(i\), but also cause transmission line \(i\) to experience line faults with a specific probability \(Prob^{i}_{\text{lic},j}\) at different periods. Suppose the affected wind farms are numbered from \(1\) to \(N\), and the affected transmission lines are numbered from \(1\) to \(L\). The tropical cyclone's passage is divided into \(T\) periods. The abnormal grid evolution process, considering only extreme meteorological factors, can be described as a binary tuple \(\mathbf{AE}=\left(\mathbf{P}_{\text{w}},\mathbf{Prob}_{\text{lic}}\right)\), where element \(\mathbf{P}_{\text{w}}\) represents the spatiotemporal matrix of abnormal outputs from \(N\) wind farms:  
\[\mathbf{P}_{\text{w}}= \begin{bmatrix}P_{\text{w}1,1}&P_{\text{w}1,2}&\cdots&P_{ \text{w}1,T}\\ P_{\text{w}2,1}&P_{\text{w}2,2}&\cdots&P_{\text{w}2,T}\\ \vdots&\vdots&\vdots&\vdots\\ P_{\text{w}N,1}&P_{\text{w}N,2}&\cdots&P_{\text{w}N,T}\end{bmatrix}  \tag{17}\]  
In this equation, \(P_{\text{wi},j}\) denotes the abnormal output of wind farm \(i\) during time period \(j\). The element \(\mathbf{Prob}_{\text{lic},c}\) represents the spatiotemporal matrix of failure probabilities:  
\[\mathbf{Prob}_{\text{lic}}= \begin{bmatrix}Prob^{1}_{\text{lic},1}&Prob^{1}_{\text{ lic},2}&\cdots&Prob^{1}_{\text{lic},T}\\ Prob^{2}_{\text{lic},1}&Prob^{2}_{\text{lic},2}&\cdots&Prob^{2}_{\text{lic},T}\\ \vdots&\vdots&\vdots&\vdots\\ Prob^{L}_{\text{lic},1}&Prob^{L}_{\text{lic},2}&\cdots&Prob^{L}_{\text{lic},T} \end{bmatrix}  \tag{18}\]  
In this equation, \(Prob^{l}_{\text{lic},j}\) represents the probability of a fault occurring on branch \(l\) during the first \(j\) periods. Since Equation (18) does not capture the actual operational state of the branch, it is necessary to determine the spatiotemporal matrix of equipment failure states through sampling:  
\[\mathbf{S}_{\text{lic}}= \begin{bmatrix}S^{1}_{\text{lic},1}&S^{1}_{\text{lic},2}& \cdots&S^{1}_{\text{lic},T}\\ S^{1}_{\text{lic},2}&S^{2}_{\text{lic},2}&\cdots&S^{2}_{\text{lic},T}\\ \vdots&\vdots&\vdots&\vdots\\ S^{L}_{\text{lic},1}&S^{L}_{\text{lic},2}&\cdots&S^{L}_{\text{lic},T}\end{bmatrix}  \tag{19}\]  
In this equation, \(S^{l}_{\text{lic},j}\) represents the operational state of branch \(l\) at time period \(j\), where \(0\) indicates the branch is closed, and \(1\) indicates the branch is open. The Abnormal Evolution Mechanism of the Power Grid Considering the Coupling Effects of Meteorological Factors and Power Flows  
The abnormal evolution process described by Equations (17)-(19) considers only the abnormal impacts of extreme meteorological factors on power grid equipment across different time periods. However, branch faults triggered by extreme weather may lead to overloads and further branch faults, or even cascading failures, due to power flow redistribution. As time passes, the alternating coupling of time-varying extreme weather conditions and the post-fault power flow redistribution further complicates the propagation paths of abnormalities within the grid's branches.