## 3 Problem Description
### Influence of PV on Distribution System  
Analyzing Figure 5, the specific structure is shown in Figure 6. From the substation to the end of the feeder line, they are numbered according to the load nodes along the line in order of 0, 1, 2, 3, \(\ldots\), \(n\). The impedance of each branch is \(Z_{i}=R_{i}+X_{i}\), and the load size and power supply size connected to each load node are \(P_{i}+jQ_{i}\), \(P_{\text{PV}i}+jQ_{\text{PV}i}\), respectively.  
When the PV is not connected to the distribution network,  
Figure 5: Simplified topology and the model: (**a**) simplified distribution diagram; (**b**) simplified circuit: branch I; (**c**) system simulation model.  
Figure 6: Feedline diagram with PV.  
The current of each branch is as follows:  
\[I_{l}=\sum\limits_{i=l}^{n}\frac{P_{i}+jQ_{i}}{U_{i}}\;l=1,2,3,\cdots,n \tag{2}\]  
The voltage drop due to line impedance in each branch is as follows:  
\[\Delta U=I_{l}Z_{l}\;l=1,2,3,\cdots,n \tag{3}\]  
The voltage of each node is as follows:  
\[U_{i}=U_{0}-\sum\limits_{l=1}^{n}\Delta U_{l}\;l=1,2,3,\cdots,n \tag{4}\]  
When the PV is connected to the distribution network node \(k\),  
Assuming that distributed PV are connected to node \(k\), the current of each branch becomes the following:  
\[I^{\prime}_{l}=\left\{\begin{array}{ll}\sum\limits_{i=l}^{n}\frac{P_{i}+jQ_{ i}}{U_{i}}-I_{\text{PV}}\;l\leq k\\ \sum\limits_{i=l}^{n}\frac{P_{i}+jQ_{i}}{U_{i}}\;\;\;\;\;\;l>k\end{array}\right. \tag{5}\]  
The voltage loss due to line impedance in each branch becomes the following:  
\[\Delta U^{\prime}_{l}=I^{\prime}_{l}Z_{l}=\left\{\begin{array}{ll}Z_{l}( \sum\limits_{i=l}^{n}I_{li}-I_{pv})&l\leq k\\ Z_{l}\sum\limits_{i=l}^{n}I_{li}&l>k\end{array}\right. \tag{6}\]  
\(I_{li}=\frac{P_{i}+jQ_{i}}{U_{i}}\) is the equivalent current of the load; \(I_{\text{PV}}=\frac{P_{\text{PV}}+jQ_{\text{PV}}k}{U_{k}}\) is the equivalent current of distributed PV.  
\[U_{i}=U_{0}-\sum\limits_{i=l}^{n}\Delta U^{\prime}_{l}\;l=1,2,3,\cdots,n \tag{7}\]  
By comparing Equations (4) and (7), it can be seen that after connecting PV, the voltage drop of each branch before node \(k\) decreases by \(Z_{l}I_{\text{PV}}\), and the voltage of the connection point and previous nodes is raised due to the connection of PV. Although nodes after \(k\) are not directly affected by \(I_{\text{PV}}\), since the voltage of nodes before \(k\) is raised, nodes after \(k\) are also passively raised by \(U_{i}\) (\(i>k\)). According to Equation (6), when \(l>k\), the voltage drop on the line decreases compared to before the PV is connected. Qualitative analysis shows that the node voltage after \(k\) is also raised. From the above analysis, it can be concluded that the integration of distributed PV can help improve the voltage quality of feeders [26].  
When the connected PV capacity increases, the \(I_{\text{PV}}\) will also increase accordingly. From Equations (6) and (7), it can be seen that the current and voltage drop of each branch before node \(k\) will decrease accordingly, causing the node voltage to be raised and the voltage quality to be improved. But when the capacity of PV is large enough, i.e., \(I_{\text{PV}}>I_{li}\), the power-reversal phenomenon will occur [27].