## Appendix A
If the heights of the two towers are different, higher towers are subject to greater stress at the distribution line interface, and the high difference angle \(\varphi\) should be considered. The projection distance \(l_{m}\) between the distribution line at the lowest point of the line and the top of the higher tower on the horizontal plane is:  
\[l_{m}\ =\ \frac{l}{2}\frac{\sigma_{n}}{\gamma}sin\varphi \tag{16}\]The internal stress \(\sigma_{0}\) of the distribution line at the connection of the higher tower is:  
\[\sigma_{0}\ =\ \sqrt{\sigma_{n}\ +\ \frac{l}{4}\frac{\sigma_{n}^{2}}{\gamma^{2}} tan^{2}\varphi} \tag{10}\]  
After the maximum stress of the distribution line is obtained, the function is used to compare the maximum stress \(\sigma_{0}\) with the distribution line stress upper limit \(\sigma_{s}\), and the probability of the distribution line disconnection event is obtained:  
\[P_{kj}^{t}(Y_{2})\ =\ \left\{ \begin{array}{ll}0&\sigma_{0}\ <\ 0\\ K_{ice}e^{\frac{\sigma_{0}}{\tau_{ic}}}&0\ \leq\ \sigma_{0}\ <\ \sigma_{s}\\ 1&\sigma_{s}\ \leq\ \sigma_{0}\end{array} \right. \tag{11}\]  
\[T_{ic}\ =\ \frac{\sigma_{s}}{ln\Big{(}\frac{1}{K_{ic}}\Big{)}} \tag{12}\]  
In the formula, \(\mu_{1}\) is the safety factor, which can be taken as \(\mu_{1}\) = 2.5. \(K_{ic}\) is a constant, which is selected according to statistical data and distribution line fragility. When the line is covered with ice, the tower may collapse due to ice accretion imbalance stress \(\Delta F\) on the distribution lines exceeding the design upper limit:  
\[\Delta F\ =\ \frac{\pi nD^{2}}{4}\sqrt{\sigma_{1}^{2}\ +\ \sigma_{2}^{2}\ -\ 2 \sigma_{1}\sigma_{2}cos\theta_{z}} \tag{13}\]  
The probability of the inverted tower event is:  
\[P_{kj}^{t}(Y_{3})\ =\ \left\{ \begin{array}{ll}0&\sigma_{0}\ <\ 0\\ K_{2}e^{\frac{\Delta F}{\sigma_{2}T}}&0\ \leq\ \sigma_{0}\ <\ \sigma_{s}\\ 1&\sigma_{s}\ \leq\ \sigma_{0}\end{array} \right. \tag{14}\]  
For distribution line dancing events, the vulnerability of the distribution line and the degree of asymmetry of the section increase with the ice accretion, so the \"thickness coefficient of ice accretion\" \(\omega\) is proposed. The formula is as follows:  
\[\omega\ =\ \frac{R}{40} \tag{15}\]  
In summary, the range of line dancing can be simplified to a function only related to ice thickness \(R\), wind speed \(V\), and angle \(\beta\):  
\[z\ =\ \widetilde{A}\sqrt{\Big{(}1\ -\ \widetilde{B}\omega_{d}\omega_{r}R \Big{)}}(V\ -\ 4)sin(\theta\ -\ 45^{\circ}) \tag{16}\]  
where \(\widetilde{A},\ \widetilde{B}\) are the coefficients related to distribution line length, radius, cross-sectional area, etc. The probability of the distribution line dancing events is:  
\[P_{kj}^{t}(Y_{4})\ =\ \begin{cases}0&z<z_{1}\\ \frac{z-z_{1}}{z_{2}-z_{1}}&z_{1}\leq z<z_{2}\\ 1&z\geq z_{2}\end{cases}  \tag{17}\]  
where \(z_{1}\), \(z_{2}\) mainly depend on the cross-sectional area, the tension level of the length distribution line. The relationship between the insulator flashover voltage \(U\) and the density of the insulator salt before ice accretion \(\rho_{SDD}\), altitude \(H\), and ice weight \(M\) is:  
\[U\ =\ C\rho_{SDD}^{-c}\sigma_{20^{\circ}C}^{-d}(1\ -\ \frac{H}{45.1})^{3,216}(1\ -\ \frac{M}{45.1})^{4,36} \tag{11}\]  
where \(C\) is the coefficient related to the insulator type and the quantity. \(e\) is the pollution characteristic index. \(\sigma_{20}\,\text{\textasciitifanc{\tiny{c}}}\) is the conductivity of the water at 20 \({}^{\circ}\)C. \(d\) is a characteristic index affected by the conductivity of water. During the icing weather, \(\rho_{SDD}\) and \(H\) are unchanged, then \(U\) is only related to the insulator ice weight \(M\):  
\[U\ =\ \overline{A}(1\ -\ \overline{B}W)^{4,36} \tag{12}\]  
where \(\overline{A},\overline{B}\) are the coefficients related to the salt density, elevation, insulator type, insulator quantity, and density of ice. The probability of insulator flashover event is:  
\[P_{kj}^{t}(Y_{5})\ =\ \begin{cases}0.01&U\leq U_{0}\\ K\ exp(\frac{U_{0}-U}{T})&U>U_{0}\end{cases}