Optimal Sizing of Photovoltaic Generation in Radial Distribution Systems Using Lagrange Multipliers

The integration of renewable distributed generation into distribution systems has been studied comprehensively, due to the potential benefits, such as the reduction of energy losses and mitigation of the environmental impacts resulting from power generation. The problem of minimizing energy losses in distribution systems and the methods used for optimal integration of the renewable distributed generation have been the subject of recent studies. The present study proposes an analytical method which addresses the problem of sizing the nominal power of photovoltaic generation, connected to the nodes of a radial distribution feeder. The goal of this method is to minimize the total energy losses during the daily insolation period, with an optimization constraint consisting in the energy flow in the slack bus, conditioned to the energetic independence of the feeder. The sizing is achieved from the photovoltaic generation capacity and load factors, calculated in time intervals defined in the typical production curve of a photovoltaic unit connected to the distribution system. The analytical method has its foundations on Lagrange multipliers and relies on the Gauss-Jacobi method to make the resulting equation system solution feasible. This optimization method was evaluated on the IEEE 37-bus test system, from which the scenarios of generation integration were considered. The obtained results display the optimal sizing as well as the energy losses related to additional power and the location of the photovoltaic generation in distributed generation integration scenarios.