Geometric properties of the single-diode photovoltaic model and a new very simple method for parameters extraction

One of the most important models to predict the electrical behavior of a photovoltaic (PV) module is the so-called single-diode model. This model is derived from the electrical equivalent circuit formed by a current source in parallel with one diode, a shunt resistor and a series resistor. The model equation depends on five parameters, if these parameters are obtained, it has been tested that this model fits accurately the real behavior of the PV module under a minimum of illumination. Nevertheless, the extraction of the parameters is quite difficult since none of the variables in the model equation can be expressed in explicit form. This fact also implies the difficulty of knowing the real properties of the current as an implicit function of the voltage in this model. Knowing these properties deeply will involve a more suitable use of the model and better understanding of the behavior of the photovoltaic module. The first goal of this paper is the rigorous mathematical study of the model. In particular, it is demonstrated that the model equation actually defines the current as an implicit function of the voltage which is indefinitely differentiable along the real line. We will provide the most significant geometric properties of the current function by means of the study of the first and the second derivatives functions which are also implicitly given. The second goal of the paper is, given real data of voltage and current measured from a PV module, to show how the parameters of the model equation can be extracted in a very simple way, giving rise to an estimated curve which fits accurately the real one. The proposed new analytical method is as good, for instance, as the well-known analytical five point method but significantly simpler.