This article analysis, through simulation, the impact that local transactions and the penetration of EVs have on the distribution network and the costs and incomes of local market participants. An auction-based competitive market and mathematical model are provided to simulate end-user transactions with EVs on the low-voltage grid. The proposed framework is validated in a scenario considering 55 end-users (some with PV generation and EVs) and six combined heat and power generators trading energy in the IEEE European Low Voltage Test Feeder system. Furthermore, a distribution system operator is considered by defining network constraints (voltage limits and lines overloading) that ensure the correct operation of the distribution grid.