Invariance of single diode equation and its application

The electrical power available from a photovoltaic (PV) module can be simulated with the well-known and widely used [1–4] single diode equation. Five parameters must be known in the equation to determine the current-voltage relationship (I–V curve) for the module. These parameters can be obtained by fitting the experimental I–V curve at a fixed temperature and solar radiation for a small but representative sub-cell of the PV module. This is true because the single diode equation is invariant with respect to an arbitrary linear transformation in the I–V plane and there is a simple linear transformation between I–V curve of the module and I–V curve of its representative sub-cell. Of course, the five parameters are the functions of this transformation. Furthermore, the single diode equation for an array of the same PV modules can be represented as a linear transformation from the single module in I–V plane. The inverse problem to derive the coefficients of the matrix of this transformation can be defined from only two measurements: I–V curve measurement for the module and I–V curve measurement for the array of the modules. The information about the matrix leads to the evaluation of the additional array´s series and shunt resistances which are essential for the real array due to the presence of connecting wires, inverter, etc. Other application of the invariance is the simple tool for the PV array power optimization. The key element of the single diode model is the effective fitting algorithm which yields the five parameters from a large number of experimental points at I–V curve.