A rapid increase in the number of electric vehicles is expected in coming years, driven by government incentives and falling battery prices. Charging these vehicles will add significant load to the electricity network, and it is important to understand the impact this will have on both the transmission and distribution level systems, and how smart charging can alleviate it. Here we analyse the effects that charging a large electric vehicle fleet would have on the power network, taking into account the spatial heterogeneity of vehicle use, electricity demand, and network structure. A conditional probability based method is used to model uncontrolled charging demand, and convex optimisation is used to model smart charging. Stochasticity is captured using Monte Carlo simulations. It is shown that for Great Britain’s power system, smart charging can simultaneously eliminate the need for additional generation infrastructure required with 100% electric vehicle adoption, while also reducing the percentage of distribution networks which would require reinforcement from 28% to 9%. Discussion is included as to how far these results can be extended to other power systems.