# Methods Improving Energy Efficiency of Photovoltaic Systems Operating under Partial Shading
## 3 Efficiency Estimation of PV Installation
### PV Installation with Individual Matching Converters and Micro-Inverters
The operating point of a PV installation with an individual matching converter and the value of the partial shading loss can be calculated from the system:  
Figure 6: Current-voltage characteristic of a group of solar cells using an individual matching converter. \[\left\{ \begin{array}{c}P_{loss}=P-(P_{sh}-P_{IMC});\\ P_{IMC}=\frac{p}{i}\big{(}U_{IMC}^{in}I_{IMC}^{in}\eta_{IMC}\big{)}_{i}=I\sum \limits_{i=1}^{p}U_{IMC}\,i;\\ U_{IMC}=\frac{U_{IMC}^{in}I_{IMC}^{in}}{I}\eta_{IMC};\\ I_{IMC}^{in}=I_{ph}-I_{0}\bigg{[}exp\bigg{(}\frac{q(U_{IMC}^{in}+I_{IMC}^{in} \cdot R_{s}}{n_{in}(\text{AKT})}\bigg{)}-1\bigg{]};\\ \frac{q\big{(}U_{IMC}^{in}+I_{IMC}^{in}R_{s}\big{)}}{n_{in}(\text{AKT})}-ln \bigg{[}\frac{I_{ph}-I_{0}}{i\big{(}1+\frac{q(U_{IMC}^{in}+I_{IMC}^{in}R_{s})}{ n_{in}(\text{AKT})}\big{)}}\bigg{]}=0;\\ I_{ph}=\Big{[}I_{sc(stc)}+K_{i}(T-T_{stc})\Big{]}\frac{\beta_{in}}{\beta_{stc}},\end{array} \right. \tag{17}\]  
where \(n_{in}\) is the number of solar cells at the input of an individual matching converter; \(\beta_{in}\) is the intensity of solar radiation, the least illuminated solar cell at the input of an individual matching converter (W/m\({}^{2}\)); \(I=I_{IMC}\), the current in the solar cell array and at the output of the individual matching converter (A); \(K_{i}\) is the temperature coefficient of the short-circuit current of the solar cell (A/K); \(T_{stc}\) is the temperature under standard conditions (298 K); \(I_{sc(stc)}\) is the short-circuit current of the solar cell under standard conditions (A). If a group of solar cells in an array that has uneven illumination transmits electrical energy through individual matching converters, and the rest of the normally illuminated group does not have such, then the characteristic of such an array will be formed by the sum of the characteristics of each group with respect to voltage:  
\[U(I)=U_{il}(I)+U_{IMC}(I), \tag{18}\]  
where \(U_{il}(I)\) is the characteristic of a normally illuminated solar cell array; \(U_{IMC}(I)\) is the characteristic of an individual matching converter. Figure 6 shows the characteristic of such an array for the case of when a group of solar cells with installed individual matching converters \(U_{IMC}(I)\) has shading. Due to the fact that an individual matching converter selects electrical energy at the point of maximum power from the group of solar cells connected to its input, at different output current values the point of maximum power of the array will be determined by the current value at the point of maximum power of the entire PV installation that transmits energy without using converters (\(I_{(mpp)}=I_{IMC}=I_{il(mpp)}\)). Hence, it follows that the operating mode of such an installation can be described by the system of equations:  
\[\left\{ \begin{array}{c}P_{(MPP)}=I_{(MPP)}U_{(MPP)};\\ I_{(MPP)}=I_{ph}-I_{0}\bigg{[}exp\bigg{(}\frac{q(U_{(MPP)}+I_{(MPP)}R_{s})}{(n -n_{in})(\text{AKT})}\bigg{)}-1\bigg{]};\\ U_{(MPP)}=U_{il(MPP)}+U_{IMC};\\ \frac{q\big{(}U_{il(MPP)}+IR_{s}\big{)}}{(n-n_{in})(\text{AKT})}-ln\Bigg{[} \frac{I_{ph}-I_{0}}{i\big{(}1+\frac{q(U_{il(MPP)}+IR_{s})}{(n-n_{in})(\text{AKT })}\bigg{)}}\bigg{]}=0;\\ U_{IMC}=\frac{U_{IMC}^{in}I_{IMC}^{in}}{I_{IMC}^{in}}\eta_{IMC};\\ I_{IMC}^{in}=I_{ph}-I_{0}\bigg{[}exp\bigg{(}\frac{q\big{(}U_{IMC}^{in}+I_{IMC}^ {in}\cdot R_{s}\big{)}}{n_{in}(\text{AKT})}\bigg{)}-1\bigg{]}.\end{array} \right. \tag{19}\]  
Figure 7 shows the electrical characteristics obtained using a mathematical model and as a result of an experiment in which two KV-260M PV modules and Tigo \"Module Maximizer\" commercially available matching converters were used. The main disadvantages of individual matching converters are:  
* a significant increase in the cost of the installation (the cost of the ICP is about 15-20% of the cost of the module);
* reduction of the voltage value of the entire array of PV modules at the point of maximum power, which, as in the case of shunt diodes, significantly affects the energy efficiency during parallel switching of arrays.