  We study how high charging rate demands from electric vehicles (EVs) in a
power distribution grid may collectively cause its dynamic instability, and,
accordingly, how a price incentivization strategy can be used to steer
customers to settle for lesser charging rate demands so that these
instabilities can be avoided. We pose the problem as a joint optimization and
optimal control formulation. The optimization determines the optimal charging
setpoints for EVs to minimize the $\mathcal{H}_2$-norm of the transfer function
of the grid model, while the optimal control simultaneously develops a linear
quadratic regulator (LQR) based state-feedback control signal for the
battery-currents of those EVs to jointly minimize the risk of grid instability.
A subsequent algorithm is developed to determine how much customers may be
willing to sacrifice their intended charging rate demands in return for
financial incentives. Results are derived for both unidirectional and
bidirectional charging, and validated using numerical simulations of multiple
EV charging stations in the IEEE 33-bus power distribution model.
