# Methods Improving Energy Efficiency of Photovoltaic Systems Operating under Partial Shading
## 2 Materials and Methods  
The proposed scientific results are based on a set of analytical and experimental research methods, including mathematical modeling, the \"light equivalent\" method, and methods of mathematical processing of the experimental results. The works [12; 13; 14; 15; 16; 17; 18] are the theoretical and methodological basis of this research. For a theoretical analysis of the processes occurring in PV installations with various switching options for solar cells operating in conditions of uneven illumination, a mathematical model of a solar cell was compiled, with an accuracy sufficient for engineering calculations--not exceeding the design tolerance. Thus, the initial mathematical model describing the operation of the solar cell model is as follows [19; 20; 21]. \[I=I_{ph}-I_{0}\left[\exp(\frac{q(U+IR_{s}}{\Lambda kT})-1\right]-\frac{U+IR_{s} }{R_{sh}}, \tag{1}\]  
where \(I\) is theoutput current of the solar cell, (A); \(I_{ph}\) is the photocurrent of the solar cell, (A); \(I_{0}\) is the reverse saturation current of the solar cell, (A); \(q\) is the electron charge, (1.602:\(10^{-19}\) C); \(U\) is the voltage at the output of the solar cell, (V); \(k\) is the Boltzmann constant (1.381\(\cdot 10^{-23}\) J/K); \(T\) is the temperature of the solar cell, (K); \(R_{s}\) is the series resistance of the solar cell, (\(\Omega\)); \(R_{sh}\) is the parallel resistance of the solar element, (\(\Omega\)); A is the ideality factor, depending on the thickness of the p-n-junction and the material, takes values for silicon PEC from 1.2 to 5 (Table 1). The photocurrent (\(I_{ph}\)) can be calculated from the short circuit current under various lighting and temperature conditions by dividing the current, assuming that during a short \begin{table}
\begin{tabular}{c c} \hline
**Material** & **Coefficient A** \\ \hline Si–monocrystalline & 1.2 \\ Si–polycrystalline & 1.3 \\ a-Si:H–amorphous hydrogenated & 1.8 \\ a-Si:H double & 3.3 \\ a-Si:H triple & 5 \\ \hline \end{tabular}
\end{table} 
Table 1: Ideality factor of the solar cell from material [22; 23; 24]. circuit the current flowing through the diode is negligible and almost all of the generated current flows into the load. According to this:  
\[I_{ph}=\left[I_{sc(stc)}+K_{i}(T-T_{stc})\right]\frac{\beta}{\beta_{stc}}\frac{R _{sh}+R_{s}}{R_{sh}}, \tag{2}\]  
where \(K_{i}\) is the temperature coefficient of the short-circuit current of the solar cell, (A/K); \(\beta\) is the intensity of solar radiation (W/m\({}^{2}\)); \(\beta_{stc}\) is the intensity of solar radiation at standard test conditions (W/m\({}^{2}\)); \(I_{sc(stc)}\) is the short-circuit current in standard test conditions (A); \(T_{stc}\) is the temperature at standard test conditions (298 K). The reverse saturation current \(I_{0}\) depends significantly on the temperature of the solar cell, while the change in the photocurrent is not so great. As a result of an increase in temperature, the height of the potential barrier decreases, and the concentration of minority charge carriers grows exponentially, which leads to an increase in the photocurrent. The amount of reverse current is also influenced by the structure and material of the solar cell. Thus, the reverse saturation current can be calculated from Equation (1) solved for the open circuit point:  
\[I_{0}=\frac{I_{ph}}{\exp\left[\frac{q[U_{sc}+K_{u}(T-T_{stc})]}{\Delta kT} \right]-1}, \tag{3}\]  
where \(K_{u}\) is the temperature coefficient of open circuit voltage (V/K); \(U_{oc}\) is the open circuit voltage (V). To assess the energy efficiency of PV installations, an information-measuring complex was proposed as a tool. The block diagram of the complex is shown in Figure 1. It allows automatic load regulation with simultaneous measurement, registration, and graphical display of voltage and current values. In addition, this complex allows you to measure and record the temperature of the environment and the surface of solar cells by the pyrometric method, as well as the intensity of solar radiation.