We analyze the impacts of a hypothetical fleet of plug-in electric vehicles on the imperfectly competitive German electricity market with a game-theoretic model. Electric vehicles bring both additional demand and additional storage capacity to the market. We determine their effects on prices, welfare, and electricity generation for various cases with different players being in charge of vehicle operations. We find that vehicle loading increases generator profits, but decreases consumer surplus. If excess vehicle batteries can be used for storage, welfare results are reversed: generating firms suffer from the price-smoothing effect of additional storage, whereas consumers benefit despite increasing overall demand. Results however depend on the player being in charge of storage operations, and on battery degradation costs. Strategic players tend to underutilize the storage capacity of the vehicle fleet, which may have negative welfare implications. In contrast, we find a small market power mitigating effect of electric vehicle recharging on oligopolistic generators. Overall, electric vehicles are unlikely to be a relevant source of market power in Germany.