  This paper proposes a distributed framework for vehicle grid integration
(VGI) taking into account the communication and physical networks. To this end,
we model the electric vehicle (EV) behaviour that includes time of departure,
time of arrival, state of charge, required energy, and its objectives, e.g.,
avoid battery degradation. Next, we formulate the centralised day ahead
distribution market (DADM) which explicitly represents the physical system,
supports unbalanced three phase networks with delta and wye connections, and
incorporates the charging needs of EVs. The solution of the centralised market
requires knowledge of EV information in terms of desired energy, departure and
arrival times that EV owners are reluctant in providing. Moreover, the
computational effort required to solve the DADM in cases of numerous EVs is
very intensive. As such, we propose a distributed solution of the DADM clearing
mechanism over a time-varying communication network. We illustrate the proposed
VGI framework through the 13-bus, 33- bus, and 141-bus distribution feeders.
