# Computational Modeling of Passive and Active Cooling Methods to Improve PV Panels Efficiency
## 2 Modeling and Computational Details
### Description of the Physical Processes
#### 2.2.5 Photovoltaic Panel Equation  
To obtain the power differences in the modeled PV panels, from the temperature differences obtained with the passive and active cooling methods, the mathematical modeling of the PV panel was performed with the \(I\)-\(V\) relationship described in Equation (9) [22; 23].  
\[I(V)=\frac{I_{x}}{1-e^{-\frac{1}{b}}}\left[1-e^{\frac{V}{ BV_{x}}-\frac{1}{b}}\ \right] \tag{9}\]  
where \(V_{x}\) and \(I_{x}\) are the open-circuit voltage and the short-circuit current for dynamic values of irradiance and temperature and are defined by:  
\[V_{x}=s\frac{E_{i}}{E_{iN}}T_{CV}(T-T_{N})+s(V_{\max})-s(V_{\max}-V_{\min})e^ {\frac{E_{i}}{E_{iN}}L_{\max}}\frac{V_{\max}-V_{\infty}}{V_{\max}-V_{\min}} \tag{10}\]  
\[I_{x}=p\frac{E_{i}}{E_{iN}}[I_{sc}+T_{Ci}(T-T_{N})] \tag{11}\]  
where \(b\) is the constant characteristic of the PV panel; it has no units, and it is the only parameter that must be calculated in this model. \(s\): represents the number of panels in series. \(p\): represents the number of panels in parallel. \(E_{i}\): is the effective irradiation at which the panel operates. \(E_{iN}\): is an irradiation constant of 1000 W/m\({}^{2}\). \(T\): is the operatingtemperature at which the panel operates. \(T_{N}\): is a temperature constant of 25 \({}^{\circ}\)C. \(V_{oc}\): is the open-circuit voltage. \(I_{sc}\): is the short-circuit current. \(V_{\max}-V_{\min}\): are the open-circuit voltage values for irradiation levels less than 200 W/m\({}^{2}\) and greater than 1200 W/m\({}^{2}\) with an operating temperature of 25 \({}^{\circ}\)C. These values are around 85% of \(V_{oc}\) for \(V_{\min}\) and 103% of \(V_{oc}\) for the case of \(V_{\max}\). \(T_{Cl}\): is the temperature coefficient of current. \(T_{CV}\): is the temperature coefficient of voltage.  
Equations (9)-(11) were used for the modeling of the PV panel in Matlab for variable operating conditions in temperature and irradiation. For this, a monocrystalline 100 W PV panel was used (Table 1), with a tolerance of \(\pm\)3% and the following electrical characteristics for \(E_{i}=1000\) W/m\({}^{2}\) and \(T=25\)\({}^{\circ}\)C: Short-circuit current (\(I_{sc}\)) = 5.7 A. Open-circuit voltage (\(V_{oc}\)) = 22.53 V. Voltage at maximum power point (\(V_{\rm{pmax}}\)) = 18.75 V. Current at maximum power point (\(I_{\rm{pmax}}\)) = 5.34 A. Temperature coefficient of voltage (\(T_{CV}\)) = -0.35 V/\({}^{\circ}\)C. Temperature coefficient of current (\(T_{Cl}\)) = 0.05 A/\({}^{\circ}\)C. Maximum voltage (\(V_{\max}\)) = 23.21 V. Minimum voltage (\(V_{\min}\)) = 19.16 V.  
In addition to the parameters presented, it is necessary to find the value of b, which is the only parameter that must be calculated. The value of b is in the range of 0.01 to 0.18, therefore, the following approximation can be established [24]:  
\[1-e^{\frac{-1}{b}}\approx 1 \tag{12}\]  
For \(V_{x}=22.53\) V, \(I_{x}=5.7\)A, \(I=5.34\) A, and \(V=18.75\) V, the value of \(b\) found using Equation (9) is \(b=0.06073\).