*2.3. PV Plant Model and Control*

The dynamic model of a solar module can be formulated as a current source with series and shunt resistance to mimic the internal voltage drop and heating due to leakage current. The dynamic model is popularly described as shown in Figure 4. The dynamic model of solar module consists of the photo current or the light-generated current *Iph*, the dark current or reverse saturation current *Id*, and the leakage current *Ish*.

$$I\_{PV} = I\_{ph} - I\_d - I\_{sh} \,\tag{5}$$

**Figure 4.** Solar cell model.

The current of a solar module, as a function of voltage, is given as follows [13,20]:

$$I\_{\rm pl} = G \frac{I\_{\rm sc} \left[1 + K\_i (T - T\_n) \right]}{G\_{\rm n}} - I\_o \left(e^{V\_D/V\_l} - 1\right) - \frac{V\_D}{R\_{\rm sl}},\tag{6}$$

where *Isc* and *Io* are the short-circuit and the diode reverse saturation currents, respectively, *Vt* is the thermal voltage, *Rsh* is the parallel resistances, and *VD* is the diode voltage of the cell. *Ki* is the temperature coefficient of cell's short-circuit current, *T* is the cell's absolute temperature, *Tn* is the reference temperature, and *G* and *Gn* are the solar radiation at actual and nominal level, respectively. The diode reverse saturation current, *Io*, used in Equation (6) is given by

$$I\_o = \frac{I\_{ph} - V\_{oc}/R\_{sh}}{\varepsilon^{\frac{qV\_{sc}}{knT}} - 1}. \tag{7a}$$

An alternative and more accurate formulation of Equation (7a) can be given as follows [20]:

$$I\_o = \frac{I\_{\rm sc}}{e^{\left(\frac{V\_{\rm sc}}{V\_t}\right)} - 1} \left(\frac{T}{T\_n}\right)^3 e^{\left(\frac{E\_{\rm c}}{V\_t}\right) \left(\frac{1}{T\_n} - \frac{1}{T}\right)}\,\tag{7b}$$

where *Voc* is the cell's open circuit voltage, *Eg* is the band gap energy of the semiconductor used in the cell, and *Vt* is the thermal voltage, given by

$$V\_t = \frac{akT}{q},\tag{8}$$

where *<sup>q</sup>* is the electric charge (1.6 × <sup>10</sup>−<sup>19</sup> C), *<sup>a</sup>* is the cell idealizing factor (1.1), and *<sup>k</sup>* is the Boltzmann constant (1.38 × <sup>10</sup>−<sup>23</sup> J/K). The diode voltage *VD* can be expressed as

$$V\_D = V\_t + I\_{PV} R\_s.\tag{9}$$

To form the array, the generated current and voltage are multiplied by the number of modules in a string or in series and the number of such parallel strings, respectively. The three subsections below describe the control and implementation of array and inverter control. The parameter values of the above equations can be found in Table A4 (Appendix C).
