## 1 Introduction  
### The Solar Cell/Panel Equivalent Circuit Models  
The performance of photovoltaic systems (solar cell/panels), that is, the output current/voltage curve (\(I\)-\(V\) curve), is usually studied using an equivalent circuit model. This equivalent circuit consists of a current source with one or two diodes connected in parallel, and up to two resistors, one connected in parallel and the other one in series, to take into account energy losses in this model. Based on these electronic components, four basic configurations are normally used when studying photovoltaic systems (see Figure 1):  
(1) The 1-diode model, whose equation to relate the output current, \(I\), to the output voltage, \(V\), is:  
\[I=I_{pv}-I_{D1}=I_{pv}-I_{0}\left[\exp\left(\frac{V}{NaV_{T}}\right)-1\right], \tag{1}\]  
where \(I_{pv}\) is the photocurrent delivered by the constant current source, \(I_{0}\) is the reverse saturation current corresponding to the diode, \(V_{T}\) is the thermal voltage (\(V_{T}=kT/q\), where \(k\) is the Boltzmann constant, \(T\) the temperature expressed in kelvin, and \(q\) is the electron charge), \(a\) is the ideality factor that takes into account the deviation of the diodes from the Shockley diffusion theory, and \(N\) is the number of series-connected cells in the photovoltaic system to be analyzed (obviously, \(N=1\) in case of a single cell). (2) The 1-diode/1-resistor model, whose main equation is:\[I=I_{pv}-I_{0}\bigg{[}\exp\bigg{(}\frac{V+IR_{s}}{NaV_{T}}\bigg{)}-1\bigg{]}, \tag{2}\]  
where \(R_{s}\) is the series resistor. (3) The 1-diode/2-resistor model, whose main equation is:  
\[I=I_{pv}-I_{0}\bigg{[}\exp\bigg{(}\frac{V+IR_{s}}{NaV_{T}}\bigg{)}-1\bigg{]}- \frac{V+IR_{s}}{R_{sh}}, \tag{3}\]  
where \(R_{sh}\) is the shunt resistor. (4) And finally, the 2-diode/2-resistor model, whose main equation is:  
\[I=I_{pv}-I_{01}\bigg{[}\exp\bigg{(}\frac{V+IR_{s}}{Na_{1}V_{T}}\bigg{)}-1\bigg{]} -I_{02}\bigg{[}\exp\bigg{(}\frac{V+IR_{s}}{Na_{2}V_{T}}\bigg{)}-1\bigg{]}- \frac{V+IR_{s}}{R_{sh}}, \tag{4}\]  
Obviously, the parameters of these equations need to be estimated to adapt the corresponding model to the real performance of the solar cell/panel behavior. This parameter extraction can be carried out based on analytical or computational methods. A quite complete review of the different methods for fitting the equivalent circuit to the solar cell/panel behavior can be found in recent works by [PERSON] and [PERSON] [1] and [PERSON] et al. [2]. In Figures 2 and 3, the well-known experimental data (i.e., the \(I\)-\(V\) curve) of a R.T.C. France silicon cell (La Radiotechnique Compelec, Paris, France) and a Photowatt PWP201 solar panel (Photowatt, Bourgoin-Jallieu, France) from the work by [PERSON] et al. [4] is plotted (the data from the work by [PERSON] et al. have been used as reference in a very large number of scientific works related to photovoltaic performance, according to google scholar it has been cited more than 150 times since its publication--20 times in 2017--), together with five curves corresponding to Equations (1)-(4). See in Table 1 the characteristic data from these experimental curves, including the inverse of the slope of the \(I\)-\(V\) curves at the short-circuit and open-circuit points, respectively \(R_{sh0}\) and \(R_{sh0}\). The extracted parameters from these equations are included in Tables 2 and 3. The extraction was carried out analytically, based on procedures already described in previous works [3; 5; 6; 7], using different conditions to extract the parameters in each case:  
1. 1-diode model (indicated as 1D in Figures 2 and 3, and Tables 2 and 3): the three characteristic points from the testing data (short circuit current, \(I_{sc}\), open circuit voltage, \(V_{oc}\), and current and voltage levels at maximum power point, \(I_{mp}\), \(V_{mp}\)), are used as the three conditions needed to  
Figure 1: Four different equivalent circuit models: (**a**) 1-diode; (**b**) 1-diode/1-resistor; (**c**) 1-diode/2-resistor; (**d**) 2-diode/2-resistor [3].