Analytic Solution to the Photovoltaic Maximum Power Point Problem

Photovoltaic (PV) power has been successfully used for over five decades. Whether in dc or ac form, photovoltaic cells provide power for systems in many applications on earth and space. Its principles of operation are therefore well understood, and circuit equivalents have been developed that accurately model the nonlinear relationship between the current and voltage of a photovoltaic cell. With the improved efficiencies of power electronics converters, it is now possible to operate photovoltaic system about its maximum power point (MPP) in order to improve the overall system efficiency. Hitherto, this problem has been tackled using tracking (MPPT) algorithms that iteratively find the point of maximum power and respond to changes in solar irradiance accordingly. A mathematical manipulation that uses the mean value theorem is presented here that provides the analytic solution of a point in a close neighborhood of the MPP. It is thoroughly proved that this point is enclosed in a ball of small radius that also contains the MPP and therefore can practically be considered as the MPP. Since the solution is analytic, no iterative schemes are necessary, and only a periodic measurement is required to adjust to changes in solar irradiance. A circuit is implemented that shows the validity of the theory and the accuracy of the solution.