Balancing energy demand and supply will become an even greater challenge considering the ongoing transition from traditional fuel to electric vehicles (EV). The management of this task will heavily depend on the pace of the adoption of light-duty EVs. Electric vehicles have seen their market share increase worldwide; the same is happening in Portugal, partly because the government has kept incentives for consumers to purchase EVs, despite the COVID-19 pandemic. The consequent shift to EVs entails various challenges for the distribution network, including coping with the expected growing demand for power. This article addresses this concern by presenting a case study of an area comprising 20 municipalities in Northern Portugal, for which battery electric vehicles (BEV) sales and their impact on distribution networks are estimated within the 2030 horizon. The power required from the grid is estimated under three BEV sales growth deterministic scenarios based on a daily consumption rate resulting from the combination of long- and short-distance routes. A Monte Carlo computational simulation is run to account for uncertainty under severe EV sales growth. The analysis is carried out considering three popular BEV models in Portugal, namely the Nissan Leaf, Tesla Model 3, and Renault Zoe. Their impacts on the available power of the distribution network are calculated for peak and off-peak hours. The results suggest that the current power grid capacity will not cope with demand increases as early as 2026. The modeling approach could be replicated in other regions with adjusted parameters.