Modeling of the Maximum Entropy Problem as an Optimal Control Problem and its Application to pdf Estimation of Electricity Price

This paper proposes a novel two step modeling and analysis on the continuous random variable of electricity price. at the first step, the continuous optimal control theory is used to model and solve the maximum entropy problem for a continuous random variable. the maximum entropy principle provides a method to obtain least-biased probability density function (pdf) estimation. in this paper, to find a closed form solution for the maximum entropy problem with any number of moment constraints, the entropy is considered as a functional measure and the moment constraints are considered as the state equations. therefore, the pdf estimation problem can be reformulated as the optimal control problem. at the second step, the proposed unbiased pdf estimator is used to estimate the pdf of electricity price. moreover, the statistical indices and the distributional characteristics of electricity price are analyzed at each load level. the simulation results on the electricity price data of new england, ontario and nord pool electricity markets show the efficiency of the proposed pdf estimator. in addition, the obtained results show that by decreasing the load, the statistical and distributional characteristics of the electricity price inclined toward the statistical properties of the normal distribution.