## 2 Method  
### 3-D FDTD Model  
Using the 3-D FDTD method, a two-step approach [41] is employed to calculate the LIVs on an overhead distribution line: first, the lightning EM fields are calculated over mountainous terrain, and then EM fields along the wire are used to compute the LIVs by using the Agrawal coupling model. As shown in Figure 1, the simulation space is 2900 \(\times\) 2500 \(\times\) 1500 m\({}^{3}\), which is divided into \(\Delta x\)\(\times\)\(\Delta y\)\(\times\)\(\Delta z\) = 5 \(\times\) 5 \(\times\) 5 m\({}^{3}\) cells, and surrounded by the second-order Mur absorbing boundary condition [42], the time step is set to 8.33 ns. Three different lightning strike locations (SL) are considered in the analysis: SL1 represents the side point at a distance \(d\) from the mountain center, SL2 is located at the mountain top, and SL3 represents the location of the lighting strike behind the mountain at a distance \(d\) from the mountain center. Additionally, two configurations of distribution lines arranged above the flat lossy ground are considered as shown in Figure 1a,b: For Configuration A, the close end of the line is located at distance \(d\) from the mountain center, with the line oriented parallel to the \(x\)-axis. For Configuration B, the center of the line is placed at distance \(d\) from the mountain center, and the distribution line is aligned parallel to the \(y\)-axis. The overhead line is modeled as 1000 m long and 10 m high above flat ground, with a DC resistance of 0.5 \(\Omega\)/km, which represents an approximate distribution line of 12 kV.  
Figure 1: Geometry of Gaussian-shaped terrain developed in our 3-D FDTD model. Side view and top view for two different line configurations: (**a**) Configuration A and (**b**) Configuration B.  
The whole working space can be divided into three regions: the first region represents the air (with a conductivity of 0 S/m and a relative permittivity of 1), the second region represents the ground (with a soil thickness of 100 m, the soil conductivity \(\sigma_{g}\) and the relative permittivity \(\varepsilon_{r}\) are set to 0.001 S/m and 10), and the last region represents the mountainous terrain, with parameters consistent with those of the ground soil. In this paper, the geometry of terrain is represented by a Gaussian-shaped mountain in a 3-D Cartesian coordinate system, as expressed by a standard normal distribution function:  
\[z(x,y)= H_{m}\cdot exp(\frac{(x-Xd)^{2}+(y-Yd)^{2}}{-2\rho^{2}}) \tag{1}\]  
where \(H_{m}\) represents the height of the mountain, \((Xd,Yd)\) denote the center coordinate of the mountain's projection onto the \(x\)-\(y\) plane, the width of the mountain can be adjusted by \(\rho\) and \(H_{m}\).  
To simulate the curved surface of the Gaussian-shaped terrain, the conformal FDTD technique [43; 44] is adopted in this study. As shown in Figure 2, for a Yee cell containing two different media, the dielectric parameters (conductivity and relative permittivity) corresponding to the electric field nodes are modified based on the proportion of each material, which can be expressed as follows:  
\[\begin{cases}\sigma_{x}^{eff}=\frac{\frac{1}{\
u_{0}\sigma_{h}+(\Delta x-l_{x })\sigma_{a}}}{\Delta x},\epsilon_{x}^{eff}=\frac{l_{x}\sigma_{h}+(\Delta x-l_ {x})\varepsilon_{a}}{\Delta x}\\ \sigma_{y}^{eff}=\frac{l_{y}\sigma_{h}+(\Delta y-l_{y})\sigma_{a}}{\Delta y},\epsilon_{y}^{eff}=\frac{l_{x}\sigma_{h}+(\Delta y-l_{y})\sigma_{a}}{\Delta y} \\ \sigma_{z}^{eff}=\frac{l_{z}\sigma_{h}+(\Delta z-l_{z})\sigma_{a}}{\Delta z}, \epsilon_{z}^{eff}=\frac{l_{z}\sigma_{h}+(\Delta z-l_{z})\sigma_{a}}{\Delta z} \end{cases} \tag{2}\]