The increasing penetration of electric vehicles over the coming decades,
taken together with the high cost to upgrade local distribution networks
and consumer demand for home charging, suggest that managing congestion on
low voltage networks will be a crucial component of the electric vehicle
revolution and the move away from fossil fuels in transportation. Here, we
model the max-flow and proportional fairness protocols for the control of
congestion caused by a fleet of vehicles charging on two real-world
distribution networks. We show that the system undergoes a continuous phase
transition to a congested state as a function of the rate of vehicles
plugging to the network to charge. We focus on the order parameter and its
fluctuations close to the phase transition, and show that the critical
point depends on the choice of congestion protocol. Finally, we analyse the
inequality in the charging times as the vehicle arrival rate increases, and
show that charging times are considerably more equitable in proportional
fairness than in max-flow.