## 2 Materials and Methods
### Model of the Injection System
#### 2.1.1 Solar Panel Model  
A solar panel is made up of photovoltaic cells in series and in parallel. A model of the equivalent circuit can be seen in Figure 2. Its components are a set of classic pn-junction diodes, which, through the photoelectric effect, convert solar radiation into electrical energy, a current source, and resistive elements that emulate the losses that occur in a real environment [16].  
The source \(I_{ph}\) is the photogenerated current at a fixed value of solar radiation, the shunt resistance (\(R_{sh}\)) represents the current loss due to leakage, and the series resistance (\(R_{s}\)) is used to represent the voltage drop at the output. Moreover, \(N_{s}\) is the number of cells in series, and \(N_{p}\) is the number of modules in parallel.  
Figure 1: General diagram of the system.  
Figure 2: Equivalent circuit of the solar panel for \(N_{s}\) photovoltaic cells in series and \(N_{p}\) modules in parallel [17].  
The diode current can be written as:  
\[I_{D}=I_{s}\Big{[}e^{\alpha\big{(}V_{pv}+R_{s}i_{pv}\big{)}}-1\Big{]}, \tag{1}\]  
where  
\[\alpha=\frac{q}{AkT_{C}}, \tag{2}\]  
where, \(k=1.3807\times 10^{-23}\) JK\({}^{-1}\) is the constant of Boltzmann, \(q=1.6022\times 10^{-19}\) C is the charge of the electron, \(T_{C}\) is the operating temperature of the cell in Kelvin, \(A\) is the ideality factor of a p-n junction whose value is between 1 and 5, \(I_{s}\) is the saturation current, and \(V_{pv}\) is the output voltage of the panel [17]. By applying [PERSON]'s law and using Equation (1) in the equivalent circuit (Figure 2), the output current (\(i_{pv}\)) generated by a photovoltaic cell is given by:  
\[i_{pv}=I_{ph}-I_{s}\Big{[}e^{\alpha\big{(}V_{pv}+R_{s}i_{pv}\big{)}}-1\Big{]}- \frac{V_{pv}+R_{s}i_{pv}}{R_{sh}}. \tag{3}\]  
The photogenerated current \(I_{ph}\) depends on the solar irradiance and the cell temperature present in the measurement. It is described by:  
\[I_{ph}=\frac{G}{G_{ref}}\Big{[}I_{sc}+k_{i}\Big{(}T_{C}-T_{ref}\Big{)}\Big{]}, \tag{4}\]  
where \(I_{sc}\) is the short-circuit current, \(G\) is the solar irradiance, \(G_{ref}=1000\) W/m\({}^{2}\) is the solar irradiance under standard conditions, \(k_{i}\) is the short-circuit current temperature coefficient, and \(T_{ref}\) is the working temperature of the cell under normal conditions (298 \({}^{\circ}\)K).  
The saturation current of the cell \(I_{s}\) changes with temperature [17] according to:  
\[I_{s}=I_{RS}\Bigg{(}\frac{T_{C}}{T_{ref}}\Bigg{)}^{3}\,e^{\frac{qE_{S}}{Ak} \Big{(}\frac{1}{T_{ref}}-\frac{1}{T_{C}}\Big{)}}, \tag{5}\]  
where \(E_{g}\) is the energy of the semiconductor in its forbidden band or bandgap, and \(I_{RS}\) is the inverse saturation current of the cell at the reference temperature and solar irradiance, which is given by:  
\[I_{RS}=\frac{I_{sc}}{v_{oc}}. \tag{6}\]  
Finally, for \(N_{s}\) photovoltaic cells in series and for \(N_{p}\) modules in parallel, the output current \(i_{pv}\) of the panel can be written as:  
\[i_{pv}=N_{p}I_{ph}-N_{p}I_{s}\Big{[}e^{\alpha\big{(}\frac{V_{pv}}{N_{s}}+ \frac{R_{s}i_{pv}}{N_{p}}\big{)}}-1\Big{]}-\frac{N_{p}}{R_{sh}}\bigg{(}\frac{ V_{pv}}{N_{s}}+\frac{R_{s}i_{pv}}{N_{p}}\bigg{)}. \tag{7}\]