Two-stage stochastic optimization for cost-minimal charging of electric vehicles at public charging stations with photovoltaics

Electric vehicles are one promising technology towards an improved sustainable transportation sector, especially when charged with electricity from renewable energy sources. However, the fluctuating generation of renewable energy resources, as well as the changing driving patterns of electric vehicles, have the offset of an uncertain nature. This paper compares three approaches (heuristic, optimization, and stochastic programming) to schedule the charging process of three different electric vehicles fleets (commuters, opportunity, and commercial fleets) at a common charging infrastructure under uncertainty. In the setting of a car park case study, several technical restrictions are taken into consideration when the load shift potential of the electric vehicles fleets are evaluated in order to minimize charging costs or to maximize the utilization of generated electricity by local photovoltaic. The two-stage Stochastic Mixed Integer Optimization Problem is solved by a Latin Hypercube based Sample Average Approximation method. Uncertainties of electricity generation by the photovoltaic system are considered by three different forecasting options and the mobility characteristics of the three electric vehicles fleets are modeled with a non-parametric probability density function (Kernel Density Estimation). The differences in charging costs and utilization of electricity from photovoltaic when applying the three approaches are identified and discussed. The numerical results show the feasibility to charge different electric vehicle fleets in a car park according to different signals and taking thereby technical restrictions as well as uncertainties into consideration. An operator for facilitation of the charging control is needed to enable the load flexibilities of each electric vehicle fleet.