**1. Introduction**

Transmission and Distribution Systems are highly affected and damaged by direct and indirect lightning events. Direct events occur when lightning directly strikes the line; such events are hazardous but rare and are typically studied and analyzed in Transmission System (TS). On the other hand, indirect events occur when lightning strikes the ground in the proximity of a power system; these events are much more frequent with respect to direct ones, but the overall voltage induced in the power system is usually much lower. For this reason, indirect events are not of interest for TS since the induced voltages are generally lower than the line Critical FlashOver voltage (CFO), but they are vital when dealing with Distribution Systems (DS), which are characterized by a low CFO.

Most works address lightning-induced voltages in DS model electric grounding as a constant value resistance *RLF* [1–16]. This parameter is associated with a low-frequency behavior, i.e., disregarding its electromagnetic dynamic. Therefore, this low-frequency grounding resistance cannot reproduce the reactive (inductive and capacitive) and electromagnetic wave propagation effects (attenuation and distortion), prominent in the highfrequency range related to the voltage and current wavefronts. Additionally, the determination of overvoltage on TS, due to direct lightning, is highly sensible on the electromagnetic modeling of the electrical grounding [17].

Given the above, this work presents an evaluation of the impact of grounding modeling on lightning-induced voltage. Thus, the main original contribution of this paper is to include, in the time domain type simulations, an equivalent electric circuit that reproduces the complete frequency response of grounding, with full inclusion of the aforementioned effects. The Hybrid Electromagnetic Model (HEM) is used to determine the wideband grounding frequency response *<sup>Z</sup>*(*ω*) [18,19]. To implement the *<sup>Z</sup>*(*ω*) in silico, the Vector Fitting (VF) technique is applied to generate an equivalent electric circuit that is easily inserted in EMT-type software [20,21]. In the following, the grounding circuit will be

implemented in the software developed in [22]. In this paper, as commonly proposed in the IEEE Standard [5], the coupling between the tower and the lightning channel and the coupling between the lightning channel and the grounding electrodes are neglected.

The results illustrate that the induced voltages, considering the grounding modeled via *RLF* are quite different from those results using *<sup>Z</sup>*(*ω*), with perceptual differences reaching values of around 25%. It is noticeable that the differences increase with the soil resistivity and with the point of occurrence of the lightning (lightning striking closer to the DS increase the perceptual differences) for both first and subsequent return strokes. The paper is organized as follows: Sections 2–4 show the lightning field-to-line coupling problem equations, the tower and the grounding modeling, respectively; while Sections 5 and 6 present the test cases and the results. Section 7 is dedicated to the conclusions.

### **2. Induced-Lightning Modeling**

The lightning-induced voltages occurring in a DS are here evaluated, recalling the procedure presented in [22,23]. This procedure is usually divided into two steps: (i) the ElectroMagnetic (EM) fields computation and (ii) the field-to-line coupling.

### *2.1. EM Fields Computation*

The EM fields are computed analytically considering the approach proposed in [24] and validated in [25]. The method requires as input the knowledge of the channel-base current, the return stroke height and the return stroke velocity. It can be applied both to perfect electric conductor ground and soil characterized by a finite conductivity. The only assumption required is the Transmission Line model for the attenuation of the current along the channel. The main advantage of this approach consists of the possibility of dealing with analytical formulas, which guarantee a fast solution and a low computational effort.
