  We compare stability regions for different power flow models in the process
of charging electric vehicles (EVs) by considering their random arrivals, their
stochastic demand for energy at charging stations, and the characteristics of
the electricity distribution network. We assume the distribution network is a
line with charging stations located on it. We consider the Distflow and the
Linearized Distflow models and we assume that EVs have an exponential charging
requirement, that voltage drops on the distribution network stay under control
and that the number of charging stations $N$ goes to infinity. We investigate
the stability of utility-optimizing power allocations in large distribution
networks for both power flow models by controlling the arrival rate of EVs to
charging stations. For both power flow models, we show that to obtain
stability, the maximum feasible arrival rate, i.e. stability region of vehicles
is decaying as $1/N^2$, and the difference between those arrival rates is up to
constants, which we compare explicitly.
