## 4 Results and Discussion  
Before presenting the results of the tests, we briefly outline the expected effects of temperature on the PV cell performance from the theoretical point of view. The increase in temperature leads to a decrease in the band gap energy, \(E_{\mathrm{g}}\), of a semiconductor [3]  
\[E_{\mathrm{g}}(T)=E_{\mathrm{g}}(0)-\frac{\kappa T^{2}}{T+\beta} \tag{2}\]  
Here \(E_{\mathrm{g}}(T)\) is the band gap of a semiconductor at temperature \(T\), \(\alpha\) and \(\beta\) are material constants, and \(E_{\mathrm{g}}(0)\) is the band gap value at zero Kelvin. The decrease in the band gap leads to a slight increase in the short-circuit current, \(I_{\mathrm{SC}}\), owing to the enhanced absorption and photocurrent. However, the heating results in a more pronounced detrimental effect on the open-circuit voltage, \(V_{\mathrm{OC}}\), defined as the maximum voltage when the PV cell is open and zero current is flowing. The open-circuit voltage is given as [3]  
\[V_{\mathrm{OC}}=\frac{kT}{q}\ln\biggl{(}\frac{I_{\mathrm{SC}}}{I_{0}}+1\biggr{)} \tag{3}\]  
where \(I_{0}\) is the reverse saturation current, \(q\) is electron charge, and \(k\) is the Boltzmann constant. Many parameters in equations governing PV cell performance depend on temperature, including band gap, diffusion coefficients, built-in voltage, intrinsic carrier concentration, and depletion region width. It is challenging to disentangle the effect of temperature on each of them term by term. However, computational studies for different material systems predict a linear decrease in the open-circuit voltage and the fill factor (FF), slightly offset by the linear increase in the short-circuit current. The result of these trends is a linear decrease in the power conversion efficiency, \(\eta\), which determines how much of the input solar energy is converted into electricity  
\[\eta=\frac{P_{\mathrm{mp}}}{P_{\mathrm{in}}}=\frac{FF\cdot I_{\mathrm{SC}}V_{ \mathrm{OC}}}{P_{\mathrm{in}}} \tag{4}\]  
Here, the maximum power output, \(P_{\mathrm{mp}}\), corresponds to the point in the current-voltage characteristic where the current is at maximum, \(I_{\mathrm{mp}}\), and voltage, \(V_{\mathrm{mp}}\), is maximum. These parameters define the fill factor: \(FF=I_{\mathrm{mp}}V_{\mathrm{mp}}/\left(I_{\mathrm{SC}}V_{\mathrm{OC}}\right)=P _{\mathrm{mp}}/\left(I_{\mathrm{SC}}V_{\mathrm{OC}}\right)\). For crystalline Si, FF is in the range  
Figure 4: Schematic of the testing procedures for solar cells attached to the heat sink. The lenses allowed to test the performance under concentrated light conditions corresponding to up to 2000 suns. of 0.7 to 0.85, and the efficiency ranges from 13% to 16%. According to the theory, the maximum power output for crystalline solar cell decreases by 0.4-0.5% per every 1 \({}^{\circ}\)C increase in the cell temperature [58; 59; 60; 61]. Based on these theory considerations, we focus on investigation of the temperature effects on open-circuit voltage, \(V_{\text{OC}}\), and then calculate the change in the efficiency, \(\eta\). The first experimental runs have been conducted for a single junction large area polycrystalline Si solar panel in order to better elucidate the effects of solar cell panel overheating. No special heat sink or TIM have been used. Figure 5 shows the temperature effect on \(V_{\text{OC}}\) of the panel placed under the solar simulator. In the first 10 min we see a decrease in the open circuit voltage, \(V_{\text{OC}}\), as the temperature of the solar cell panel increases from 25 to 55 \({}^{\circ}\)C. The 30 \({}^{\circ}\)C temperature rise results in the \(V_{\text{OC}}\) change of about 12%. At the next step, the solar simulator is shut off for the next 10 min leaving the solar cell panel without illumination. The non-zero \(V_{\text{OC}}\) in time period from 10 to 20 min is the panel output due to ambient light in the laboratory. As the solar cell temperature decreases to RT (\(T\) = 25 \({}^{\circ}\)C) the open-circuit voltage gradually increases.