  We study the problem of phase optimization for electric-vehicle (EV)
charging. We formulate our problem as a non-convex mixed-integer programming
problem whose objective is to minimize the charging loss. Despite the hardness
of directly solving this non-convex problem, we solve a relaxation of the
original problem by proposing the PXA algorithm where "P", "X", and "A" stand
for three variable matrices in the formed phase optimization problems. We show
that under certain conditions, the solution is given by the PXA precisely
converges to the global optimum. In addition, using the idea of model
predictive control (MPC), we design the {PXA-MPC}, which is an online
implementation of the PXA. Compared to other empirical phase balancing
strategies, the PXA algorithm significantly improves the charging performance
by maximizing energy delivery, minimizing charging price, and assisting future
energy planning. The efficacy of our algorithm is demonstrated using data
collected from a real-world adaptive EV charging network (ACN).
