Several European countries have launched programs to increase the market penetration of battery electric vehicles (BEV). Similarly, politicians in Switzerland have targeted a 15% BEV share of new car registrations by 2022. As each electric car increases the power demand, new challenges are posed to the operation of existing distribution grid infrastructure. Here, a new bottom-up physical approach is presented that couples agent-based traffic simulations through an unsteady vehicle power consumption model with distribution grid power flow simulations. The impacts on hourly powerline loads from charging a car fleet with an 8.5% BEV share are quantified in the real distribution grid for the canton of Zurich. The grid is composed of 12,000 buses and 9,800 powerlines, providing power to 398,000 individual customers. Results indicate that the risk of overloaded powerlines is highest in low-level distribution grids. In our most critical future scenario, with simultaneous 8.5% BEV charging at 8 pm with 11 kW, peak line loads reach up to 132% of rated capacity. Hence, in a potential energy transition towards a decarbonised future, each individual distribution grid could face critical loads at specific temporal and spatial bottlenecks. Thus, grids should be assessed individually to limit uncertainties and risks of critical power system situations.