## 4 Simplified Methods for Rapid Calculation in Distribution Networks
### Validation of Model Equivalence Based on Node Voltage and System Losses  
For the proposed system simplification method, Figure 5 is the object for verification, comparing from two aspects: PV impact on node voltage and system loss. PV is installed at the bus TS and gradually increases the PV penetration rate, including the following:  
Case a. Not connected to PV power supply.  
Case b. Connected to a 36 kW PV power source with a power factor of 0.95 to the TS node.  
Case c. Connected to a 72 kW PV power source with a power factor of 0.95 to the TS node.  
Case d. Connected to a 108 kW PV power source with a power factor of 0.95 to the TS node.  
Case e. Connected to a 144 kW PV power source with a power factor of 0.95 to the TS node.  
Case f. Connected to a 160 kW PV power source with a power factor of 0.95 to the TS node.  
The data table and curve are as follows:  
The system simplification method proposed in this section can transform a complex network into a simple impedance model structure, with the error in the estimated value of node voltage stabilized within 0.4%. The data as shown in Table 5. For practical engineering purposes, this error falls within the allowable range. The efficiency benefits brought by the simplification method to engineering calculations are quite significant.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multicolumn{2}{c}{**PV Capacity (kw)**} & **0** & **36** & **72** & **108** & **144** & **160** \\ \hline \multirow{5}{*}{Bus voltage voltage (V)} & MS1 (Theory) & 400 & 400 & 400 & 400 & 400 & 400 \\  & MS1 (Simulation) & 400 & 400 & 400 & 400 & 400 & 400 \\  & MS2 (Theory) & 400 & 400 & 400 & 400 & 400 & 400 \\  & MS2 (Simulation) & 400 & 400 & 400 & 400 & 400 & 400 \\  & SS3 (Theory) & 396 & 400 & 403 & 406 & 410 & 411 \\  & SS3 (Simulation) & 397 & 401 & 402.3 & 405.6 & 409.8 & 411.3 \\  & TS (Theory) & 394.5 & 400.1 & 405.5 & 410.7 & 415.9 & 418.4 \\  & TS (Simulation) & 395 & 401 & 406 & 412 & 417 & 420 \\ \hline \multirow{2}{*}{\(P_{loss}\)} & Theory & 0.923 & 0.183 & 0.863 & 2.843 & 4.333 & 5.733 \\  & Simulation & 0.758 & 0.3844 & 1.0272 & 2.5932 & 5.1032 & 6.6332 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Node Voltage and \(P_{loss}\) Data Sheet.  
Observing Figure 14, at the end of the PV-integrated distribution network, as the PV penetration rate increases, the voltage changes at various nodes are shown in Figure 14a-d. When the PV capacity reaches 160 kw, node TS reaches the voltage upper limit of 1.05. According to the analysis in 3.1, the integration of PV reduces the demand for system power at the integration node, leading to a decrease in branch current before the node, thereby achieving the goal of improving node voltage and reducing line loss.  
Observing Figure 15, the total loss of the system exhibits a trend of high-low-high variation. This is due to the fact that when the PV capacity is relatively small, energy consumption is better, the branch current before the access node decreases, and the system loss decreases. As the PV capacity gradually increases, excess energy flows back, the branch current increases in reverse, and the loss increases accordingly. When the PV capacity increases from 36 kw to 160 kw, the loss of the system increases by 1725.6%. Therefore, calculating the balance between PV capacity and system loss should be an important step in planning costs.  
Figure 14: Simplify system theory/simulation voltage curve: (**a**) nodes MS1 and MS2; (**b**) nodes SS3; (**c**) nodes TS; (**d**) comparison of node voltage curves.