Electric mobility is expected to play a key role in the decarbonisation of the energy system. Continued development of battery electric vehicles is fundamental to achieving major reductions in the consumption of fossil fuels and of CO2 emissions in the transport sector. Hydrogen can become an important complementary synthetic fuel providing electric vehicles with longer ranges. However, the environmental benefit of electric vehicles is significant only if their additional electricity consumption is covered by power production from renewable energy sources. Analysing the implications of different scenarios of electric vehicles and renewable power generation considering their spatial and temporal characteristics, we investigate possible effects of electric mobility on the future power system in Germany and Europe. The time horizon of the scenario study is 2050. The approach is based on power system modelling that includes interchange of electricity between European regions, which allows assessing long-term structural effects in energy systems with over 80% of renewable power generation. The study exhibits strong potential of controlled charging and flexible hydrogen production infrastructure to avoid peak demand increases and to reduce the curtailment of renewable power resulting in reduced system operation, generation, and network expansion costs. A charging strategy that is optimised from a systems perspective avoids in our scenarios 3.5 to 4.5 GW of the residual peak load in Germany and leads to efficiency gains of 10% of the electricity demand of plug-in electric vehicles compared with uncontrolled loading.