An important scenario for smart grids which encompass distributed electrical networks is given by the simultaneous presence of aggregators and individual consumers. In this work, an aggregator is seen as an entity (a coalition) which is able to manage jointly the energy demand of a large group of consumers or users. More precisely, the demand consists in charging an electrical vehicle (EV) battery. The way the EVs user charge their batteries matters since it strongly impacts the network, especially the distribution network costs (e.g., in terms of Joule losses or transformer ageing). Since the charging policy is chosen by the users or the aggregators, the charging problem is naturally distributed. It turns out that one of the tools suited to tackle this heterogenous scenario has been introduced only recently namely, through the notion of composite games. This paper exploits for the first time in the literature of smart grids the notion of composite game and equilibrium. By assuming a rectangular charging profile for an EV, a composite equilibrium analysis is conducted, followed by a detailed analysis of a case study which assumes three possible charging periods or time-slots. Both the provided analytical and numerical results allow one to better understand the relationship between the size (which is a measure) of the coalition and the network sum-cost. In particular, a social dilemma, a situation where everybody prefers unilaterally defecting to cooperating, while the consequence is the worst for all, is exhibited.