# DC Cabling of Large-Scale Photovoltaic Power Plants
## 4 Impact of Temperature on Cable Losses
Without the established ratio of the factors \(\gamma\) and \(\theta\), Equation (39) cannot be applied to determine the change in current as a function of temperature at the maximum power point. In determining this relationship, it is necessary to start with the definition of temperature coefficients. The temperature coefficients of power, current, and voltage are the quotients of power change and temperature change, current change and temperature change, and voltage change and temperature change, respectively:  
\[k_{\rm p}=\frac{100}{P_{\rm T}}\frac{dP}{dT},\quad\gamma k_{\rm i}=\frac{100 }{I_{\rm T}}\frac{dI}{dT},\quad\theta k_{\rm u}=\frac{100}{U_{\rm T}}\frac{dU }{dT}, \tag{40}\]  
where the measuring unit of coefficients \(k_{\rm p}\), \(k_{\rm u}\), and \(k_{\rm i}\) is \([\%/^{\circ}\! {\rm C}]\), and \(P_{\rm T}\), \(U_{\rm T}\), and \(I_{\rm T}\) are values of power, voltage, and current, respectively, in the specific point. For the maximum power point, using Equation (40), the following applies [36]:  
\[k_{\rm p}=\frac{100}{P_{\rm T}}\frac{d}{dT}(U\cdot I)=\frac{100}{P_{\rm T}} \frac{dU}{dT}I+\frac{100}{P_{\rm T}}\frac{dI}{dT}U=\frac{\theta k_{\rm u}I}{I _{\rm T}}+\frac{\gamma k_{\rm i}U}{U_{\rm T}}. \tag{41}\]  
The coefficients from Equation (40) refer to any point of the UI characteristic curve and represent the percentage change in power, current, and voltage as a function of temperature in relation to the value of power, current, and voltage at that point. By normalizing these coefficients to the unit amount of power, current, and voltage, we obtain these coefficients for any point. This means that in Equation (41) we set \(P_{\rm T}=1\,\rm W\), \(U_{\rm T}=1\,\rm V\), and \(I_{\rm T}=1\,\rm A\). Now Equation (41) becomes  
\[U_{\text{O}}(T) = U_{\text{O,STD}}\cdot\left[1+\frac{k_{\text{u}}}{100}(T_{\text{ C}}-25\,^{\circ}\text{C})\right],\]0Equalizing Equations (39) and (42), and neglecting expressions with small values (product of \(k_{\rm u}\) and \(k_{\rm i}\)) the ratio of \(\gamma\) and \(\theta\) is obtained as follows:  
\[U_{\text{O}}(T) = U_{\text{O,STD}}\cdot\left[1+\frac{k_{\text{u}}}{100}(T_{\text{ C}}-25\,^{\circ}\text{C})\right],\]1  
\(U\) and \(I\) are replaced by \(U_{\rm PN}\) and \(I_{\rm PN}\) because the change in current as a function of temperature change at the maximum power point according to STC conditions is calculated. By including Equation (43) in Equation (39), \(\gamma\) is determined as follows:  
\[U_{\text{O}}(T) = U_{\text{O,STD}}\cdot\left[1+\frac{k_{\text{u}}}{100}(T_{\text{ C}}-25\,^{\circ}\text{C})\right],\]2  
The problem with the temperature coefficients of current and voltage was discussed in [37]; catalog data is provided only for short-circuit current and open-circuit voltages but not for other points on the current-voltage characteristic. It was found that some authors use the same coefficient for the whole characteristic, but that is not justified [38]. It is proposed to determine this coefficient by measuring in specific conditions. It is also evident that temperature coefficients change with insolation [36; 38]. For the following values of coefficients \(k_{\rm i}=0.05\,\%^{\circ}\mathrm{C}^{-1}\), \(k_{\rm u}=-0.29\,\%^{\circ}\mathrm{C}^{-1}\), \(k_{\rm p}=-0.37\,\%^{\circ}\mathrm{C}^{-1}\), and ambient temperature of \(35\,^{\circ}\mathrm{C}\), \(\gamma=1.707862\) is calculated. Figure 7 presents the change in characteristic values (\(I_{\rm SC}\), \(U_{\rm O}\), \(I_{\rm I}\), \(U_{\rm I}\)) for cell temperatures between \(25\,^{\circ}\mathrm{C}\) and \(71.25\,^{\circ}\mathrm{C}\) at \(1000\,\mathrm{W}\,\mathrm{/}\,\mathrm{m}^{2}\) of insolation. For a temperature of \(25\,^{\circ}\mathrm{C}\), the obtained values are as follows: \(I_{\rm SC}=10.25\,\mathrm{A}\), \(U_{\rm I}=53.9\,\mathrm{V}\), \(P_{\rm P}=440\,\mathrm{W}\), \(U_{\rm I}=44.56\,\mathrm{V}\), and \(I_{\rm I}=9.65\,\mathrm{A}\). These values correspond to the catalog data under STC conditions.