## 2 Modeling of Photovoltaic Systems  
### The 1-D/2-R Equivalent Circuit Model  
The most widely used method of modeling the performance of a solar cell/panel (based on its _I-V_ curve, where \(I\) is the output current and \(V\) the output voltage) is an equivalent circuit based on one current source, one diode, and two resistors (series and shunt resistors), as shown in Figure 1. The equation that describes the performance of this model is as follows [57; 64; 65; 66]:  
\[I=I_{pv}-I_{0}\left[\exp\left(\frac{V+IR_{s}}{naV_{T}}\right)-1\right]-\frac{V+ IR_{s}}{R_{sh}} \tag{1}\]  
where \(I_{pv}\) is the photocurrent, \(I_{0}\) is the reverse saturation current of the diode, \(R_{sh}\) is the resistance of the shunt resistor, \(R_{s}\) is the resistance of the series-connected resistor, and \(V_{T}\) is the thermal voltage:  
\[V_{T}=\frac{\kappa T}{q} \tag{2}\]  
In the above equation, \(\kappa=1.38064852\cdot 10^{-23}\) m\({}^{2}\) kg s\({}^{-2}\) K\({}^{-1}\) is the Boltzmann constant, \(T\) is the temperature expressed in K, and \(q=1.60217662\cdot 10^{-19}\) C is the electron charge. In Equation (1), \(a\) is the ideality factor of the diode (in principle, the effect of temperature on this parameter can be left aside [64]), and \(n\) is the number of series-connected cells in the solar panel (obviously, \(n=1\) when studying the performance of a single cell). Figure 1: (**a**) _I-V_ curves for three different solar cells; (**b**) 1-D/2-R equivalent circuit model for analyzing the behavior of a solar cell/panel. The first problem of modeling a photovoltaic device with the 1-D/2-R equivalent circuit model lies in extracting the five parameters of the model: \(I_{pp}\), \(I_{0}\), \(a\), \(R_{s}\), and \(R_{slt}\). Depending on the available information, which might be either:  
* The \(I\)-\(V\) curve with an enough large number of points, or;
* The three characteristic points of the \(I\)-\(V\) curve (short circuit output current, \(I_{sc}\), open circuit output voltage, \(V_{oc}\), and output current and voltage levels at the Maximum Power Point (MPP), \(I_{mp}\) and \(V_{mp}\),  
It is possible to extract these five parameters either by numerically fitting Equation (1) to the \(I\)-\(V\) curve, or by using the three characteristic points, which in practice represent four conditions [67]: the first three of which force the equation to match the points (\(0\), \(I_{sc}\)), (\(V_{mp}\), \(I_{mp}\)) and (\(V_{oc}\), \(0\)), and the fourth is related to the maximum power condition:  
\[-\frac{\partial I}{\partial V}\bigg{|}_{(V_{mp},I_{mp})}=\frac{I_{mp}}{V_{mp}}. \tag{3}\]  
Modeling the performance of photovoltaic systems with the equivalent circuit described here can be very advantageous, as these circuits somehow preserve the physical processes of a pair formed of a current source and a p-n junction [68]. The effect of the series-connected resistor, \(R_{s}\), is mainly associated with power losses in the solder bonds and interconnections between cells, whereas the effect of the shunt resistor, \(R_{sh}\), is associated with current leakages across the p-n junction [69; 70]. In recent years, several reviews of the different procedures and techniques for the parameter extraction problem related to the 1-D/2-R model have been published [20; 21; 23; 24; 67; 71; 72; 73]. The present work focuses on an analytical approach based on the use of information from the characteristic points to calculate the five parameters in Equation (1).