Uncertainty cost functions for solar photovoltaic generation, wind energy generation, and plug-in electric vehicles: mathematical expected value and verification by Monte Carlo simulation

Electrical power systems which incorporate solar or wind energy sources, or electric vehicles, must deal with the uncertainty about the availability of injected or demanded power. This creates uncertainty costs to be considered in stochastic economic dispatch models. The estimation of these costs is important for proper management of energy resources and accurate allocation of the amount of energy available for the system. In this paper, analytical formulas of uncertainty penalty costs are calculated, for solar and wind energy and for electric vehicles, through a mathematical expected value formulation. In order to get the proposed uncertainty cost functions, probability distribution functions (PDF) of the energy primary sources are considered, that is to say: log-normal distribution for solar irradiance PDF, Rayleigh distribution for wind speed PDF and normal distribution for loading and unloading behaviour PDF of electric vehicles. The analytical formulation is verified through Monte Carlo simulations.