## 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  
Tropical cyclones primarily cause abnormalities in transmission lines by damaging conductors (single-span lines) or towers. This section describes the mechanism of abnormal effects of extreme meteorological factors on transmission lines by constructing a probability model for tower collapse and a probability model for conductor breakage in single-span lines. According to the Palmgren-Miner linear fatigue damage criterion [26], the cumulative fatigue damage is obtained by summing the fatigue damage over several preceding time periods. Let \(C_{i,j}\) be the single fatigue damage for tower \(i\) during time period \(j\), which can be expressed as follows:  
\[C_{i,j}= \begin{cases}0&v_{i,j}\in[0,v_{\text{cr}})\\ ae^{h\
u_{ij}^{2}}&v_{i,j}\in[v_{\text{cr}},v_{\text{m}})\\ 1&v_{i,j}\in[v_{\text{m}},\infty)\end{cases}  \tag{8}\]In the equation, \(v_{i,j}\) represents the extreme wind speed perceived by tower \(i\) during time period \(j\); \(a\) and \(b\) are the fatigue damage coefficients; \(v_{\text{cr}}\) denotes the critical wind speed that causes fatigue in the tower; and \(v_{\text{m}}\) represents the ultimate wind speed corresponding to the damage that leads to breakage caused by a tropical cyclone in a single time period. In practice, the collapse of a tower is related to the cumulative fatigue damage sustained throughout the entire period of a tropical cyclone's impact. The probability \(\mathit{Prob}_{\text{to},j}^{i}\) that tower \(i\) will not collapse during time period \(j\) can be expressed as follows:  
\[\mathit{Prob}_{\text{to},j}^{i}=e^{-\frac{C_{ij}}{1-\left(\sum_{k=1}^{j-1}C_ {ijk}\right)-C_{ij}}\Delta t} \tag{9}\]  
In this equation, \(\Delta t\) denotes the duration of each time period. The probability \(\mathit{Prob}_{\text{to},j}^{i}\) that a tower remains intact during the first \(t\) periods is the product of the non-collapse probabilities from period 1 to period \(t\):  
\[\mathit{Prob}_{\text{to},j}^{i}=\mathit{Prob}_{\text{to},1}^{i}\cdot\mathit{ Prob}_{\text{to},2}^{i}\cdot\cdot\mathit{Prob}_{\text{to},j}^{i} \tag{10}\]  
Based on Equations (9) and (10), the probability \(\mathit{Prob}_{\text{to},j}^{i}\) that tower \(i\) does not collapse in the first \(t\) periods is as follows:  
\[\mathit{Prob}_{\text{to},j}^{i}=e^{\left[\sum_{j=1}^{t}(-\frac{C_{ij}}{1- \left(\sum_{k=1}^{j-1}C_{ijk}\right)-C_{ij}}\Delta t)\right]} \tag{11}\]  
The probability \(\mathit{Prob}_{\text{to},j}^{i}\) that tower \(i\) collapses during the first \(t\) periods is as follows:  
\[\mathit{Prob}_{\text{to},j}^{i}=1-\mathit{Prob}_{\text{to},j}^{i} \tag{12}\]  
Tropical cyclones induce short-term faults in single-span lines through wind pressure [27]. Let \(v_{i,j}\) represent the wind speed perceived by single-span line \(i\) during period \(j\). The wind pressure \(W_{\text{A}i,j}\) experienced by the line can be expressed as follows:  
\[W_{\text{A}i,j}=v_{i,j}^{2}\alpha\mu_{z}\mu_{sc}\beta_{c}\text{sin}^{2}\theta /1600 \tag{13}\]  
In the equation, \(\alpha\), \(\beta_{c}\), \(\mu_{z}\), and \(\mu_{sc}\) represent the coefficients for wind pressure non-uniformity, wind load, wind pressure height variation, and line type, respectively. \(\theta\) denotes the angle between the wind direction and the conductor. The failure rate \(\lambda_{i,j}\) for single-span line \(i\) during period \(j\) can be expressed as follows:  
\[\lambda_{i,j}= \begin{cases}0,&W_{\text{A}i,j}<W_{\text{i}\max}^{\text{normal}} \\ \frac{(1-\lambda_{i1})W_{\text{A}i,j}+\lambda_{\text{A}1}W_{\text{i}\max}-W_{ \text{i}\max}^{\text{normal}}}{W_{\text{i}\max}-W_{\text{i}\max}^{\text{normal }}},&W_{\text{i}\max}^{\text{normal}}\leq W_{\text{A}i,j}\leq W_{\text{i}\max} \\ 1,&W_{\text{A}i,j}>W_{\text{i}\max}\end{cases}