# Methods Improving Energy Efficiency of Photovoltaic Systems Operating under Partial Shading
## 3 Efficiency Estimation of PV Installation  
### PV Installation with Shunt Diodes  
The use of shunt diodes eliminates the influence of shaded modules on the power generation of other (unshaded) modules located in the same series circuit. Without their use, the current of the entire array of series-connected modules does not exceed the current of the least illuminated photocell [25]. This causes energy dissipation on the shaded element (the so-called \"hot spot effect\") with subsequent damage [26; 27]. In this regard, the bypass diodes also perform the protective function of the module, protecting it from damage [28; 29; 30]. Shunt diodes are located directly on the output terminals of the modules. They are connected in such a way that they conduct current in the forward direction (I\({}_{\text{sh1}}\), I\({}_{\text{sh2}}\)) when reverse bias voltage (U\({}_{\text{b1}}\), U\({}_{\text{b2}}\)) is applied to shaded panels (Figure 2). Reverse bias voltage occurs if the current flowing in the serial chain I creates a voltage drop in the shaded area greater than the voltage generated by the group of shaded panels themselves. In this case, the only loss is the power dissipated in the diodes themselves, due to the voltage drop across them. Due to the fact that shunt diodes, when uneven illumination occurs, disconnect (shunt) a group of solar cells that has come under a reverse bias voltage (shaded group) that is not able to provide a load current, the current-voltage characteristic of a PV installation is formed by individual characteristics of each such group (Figure 3). When shunting groups of solar cells with diodes, it is necessary to take into account the voltage drop across them, which is determined by the ratio:  
\[U_{d}=\sum_{i=1}^{p}U_{di}, \tag{4}\]  
where \(p\) is the number of solar cell groups shunted by diodes; \(U_{d}\) is the shunt diode voltage drop (V). The number of solar cell groups shunted by diodes can be calculated from the ratio:  
\[p=\frac{n_{m}-n_{x}}{z}, \tag{5}\]  
Figure 2: Switching on shunt diodes with PV modules. where \(n_{m}\) is the total number of solar cells in the array; \(n_{x}\) is the number of solar cells in a group not shunted by a diode; \(z\) is the number of solar cells per shunt diode. Thus, the current-voltage characteristic U(I) of such an array can be described by the following system of equations:  
\[\left\{ \begin{array}{c}U_{1}(I)=\sum\limits_{i=1}^{n_{1}}U_{i}(I)-U_{d}p= \frac{n_{1}(AkT)}{q}ln\left(\frac{I_{ph}\;1}{I_{0}}-\frac{I}{I_{0}}+1\right)-n _{1}IR_{s}-U_{d}\left(\frac{n_{m}-n_{1}}{z}\right)\text{, at }I>I_{max2}\text{,}\\ U_{2}(I)=\sum\limits_{i=1}^{n_{2}}U_{i}(I)-U_{d}p=\frac{n_{2}(AkT)}{q}ln\left( \frac{I_{ph}\;2}{I_{0}}-\frac{I}{I_{0}}+1\right)-n_{2}IR_{s}-U_{d}\left(\frac{ n_{m}-n_{2}}{z}\right)\text{, at }I_{max2}\geq I>I_{max3}\text{,}\\ \hskip 142.26378 pt\ldots\;\ldots\;\ldots\\ U_{m}(I)=\sum\
olimits_{i=1}^{n_{m}}U_{i}(I)=\frac{n_{m}(AkT)}{q}ln\left( \frac{I_{ph}\;m}{I_{0}}-\frac{I}{I_{0}}+1\right)-n_{m}IR_{s}\text{, \ at }I\leq I_{max\;m}\text{,}\end{array} \right. \tag{6}\]  
where \(m\) is the number of groups with elements with different lighting; \(n_{1}\), \(n_{2}\), \(\ldots\), \(n_{m}\) are the number of elements in the corresponding group; \(I_{max1}\), \(I_{max2}\), \(\ldots\), \(I_{maxm}\) are the maximum current of the corresponding group, (A); \(I_{ph1}\), \(I_{ph2}\), \(\ldots\), \(I_{ph\;m}\) are the photocurrent of the group with the corresponding value of the intensity of solar radiation \(\beta_{1}\), \(\beta_{2}\), \(\ldots\), \(\beta_{m}\), (A). Obviously, the characteristic of each group of solar cells in such an installation has its own point of maximum power (MPP), which is the point of local maximum (LMP) general characteristics U(I).