## 1 Introduction  
The interaction between lightning strikes and electrical power distribution networks can cause significant disturbances, with lightning-induced voltages (LIVs) being one of the most critical threats to grid reliability and safety. LIVs can significantly exceed the rated voltage of systems, and such high voltages may cause insulation breakdown, leading to equipment short circuits or failures [1]. Accurate assessment of LIVs caused by indirect lightning strikes on distribution lines is a crucial aspect of lightning protection system design [2, 3]. Numerical calculation is a crucial component in the LIV research. A commonly adopted approach is the two-step method. The first step is to calculate the lightning electromagnetic (EM) fields [4]. There are numerous numerical methods available, such as the method of moments (MOMs), the finite element method (FEM) [5, 6], and the finite-difference time-domain (FDTD) method [7, 8, 9, 10]. The second step is to calculate the LIVs [11], using commonly used field-to-line coupling models, including the [PERSON], [PERSON], and [PERSON] coupling model [12, 13, 14, 15]. Nowadays, numerous studies have discussed the factorsinfluencing LIVs, including soil conductivities [16; 17; 18; 19], terrain topography [20; 21; 22; 23], lightning strikes [24; 25; 26], and distribution line configurations [27; 28]. Particularly an increasing number of studies focus on the influence of terrain on lightning EM fields and LIVs [29; 30; 31; 32; 33; 34; 35]. [PERSON] et al. [29] first used a cylindrical 2-D FDTD model to calculate the LIVs along a distribution line and analyze LIVs on other irregular terrain such as V-shaped and inverted V-shaped mountains [30]. [PERSON] et al. [31] evaluated the effect of the oblique cone-shaped mountain by using the 3-D FDTD method. [PERSON] et al. [32] further explored the impacts of the tower placed on a trapezoidal mountain on EM fields at different observation points by using the 3-D FDTD method. [PERSON] et al. [33] investigated the effects of lightning currents with different peaks and waveforms on solar photovoltaic (PV) power stations, in order to estimate the transient lightning that could potentially damage PV modules, inverters, and transformers. Subsequently, [PERSON] et al. [34] further examined the influence of lightning strike locations and cable length on a hybrid solar PV-battery energy storage system. [PERSON] et al. [35] investigate the characteristics of LIVs in large-scale PV over complex terrain such as lake terrain and mountainous terrain by using the 2-D FDTD method. However, the existing studies mentioned above [29; 30; 31; 32; 33; 34; 35] have adopted idealized mountain shapes including cone-shape, V-shape, and trapezoidal-shaped mountain, the real-world mountainous terrains are more complex and irregular, with geometries that can be more accurately represented by Gaussian surface. [PERSON] [36] investigated the propagation characteristics of EM waves along flat and curved surfaces; the results indicated that when EM waves propagate along the Gaussian surface and sinusoidal surface, the presence of local convex regions of the surface always leads to the enhancement of wave amplitude. [PERSON] et al. [37] further investigate the influence of irregular terrain on the propagation of EM waves; it is found that a sinusoidal obstacle has a significant impact on the reflection of EM waves, especially in regions with high curvature, leading to field enhancement in local regions. Existing studies predominantly focus on the impact of lightning strikes at the top of the mountain or another tall structure on the induced voltages of distribution lines. [PERSON] et al. [38] investigated of influence of lightning position on LIVs in 10 kV overhead distribution lines using a 2-D FDTD model. However, the range of strike positions that can be considered in a 2-D configuration is inherently limited, thereby limiting the effect of complex terrain features on LIVs.