Electric vehicles (EV) pose a major challenge to the local power distribution network, particularly the low-voltage (LV) network that serves residential demand. Autonomous electric vehicles (AEVs) may reduce the need to expand the LV network capacity to meet residential charging needs by shifting charging to public charging stations connected to the medium-voltage (MV) network. To study this question, we develop a model that links EV users’ charging and travel choices with the distribution system operators’ (DSOs) decisions on discriminatory grid network fees. According to our results, LV expansion costs from induced travel and charging demands with AEVs fully offset the LV grid cost reductions from a substantial shift to public charging. Therefore, switching to AEVs is not a way to lower overall network costs. We further study different regulation schemes for DSO fees, including Ramsey pricing. Regulation enhances welfare, though it increases network costs relative to monopoly fee discrimination. The results are robust to changes in in-vehicle time use, density of charging stations, battery size, and wholesale electricity prices.