Optimal Scheduling of Storage Batteries and Power Generators Based on Interval Prediction of Photovoltaics—Monotonicity Analysis for State of Charge

Abstract

This letter focuses on a day-ahead scheduling problem on thermal power plants and storage batteries based on the information of the interval prediction of photovoltaic (PV) power and demand. To realize an optimal operation, there are three kinds of decision variables to plan, which correspond to a generating power schedule, a charge/discharge (C/D) power schedule, and a state of charge (SOC) schedule. Our purpose is to obtain the exact range of each optimal schedule for any possible PV/demand profiles of the interval prediction. In the previous work, we proposed a method to find the exact ranges of the optimal generation power schedule and the optimal C/D power schedule by showing the sign patterns of the Jacobians with respect to an uncertain renewable parameter. However, the range of the optimal SOC schedule is not derived so far. If the exact range is found, we can offer a guideline for a necessary and sufficient capacity of storage batteries at each time step. To get the range, we show that the sign pattern of the Jacobian for SOC is invariant, more specifically, its sign pattern has negative elements in the lower triangle and positive elements in others. The key to analyzing the sign pattern is to utilize particular properties of an ${M}$ -matrix and a diagonally dominant matrix. As a result, we clarify the sign pattern mathematically, which enables to effectively calculate the exact range of the optimal SOC schedule. This letter completes the monotonicity analysis of all the decision variables in the optimal scheduling problem.