# Optimization of Active Distribution Network Operation with SOP Considering Reverse Power Flow
## 3 Actual Maximum Transfer Capacity of Reverse Power Flow in the Distribution Network
Additionally, the input \(K^{*}_{t}\) in Step 2 can be reduced to optimize the operational cost of the distribution network. Figure 6: Real power reverse flow transmission capacity distribution map of the distribution network. \begin{table}
\begin{tabular}{c l} \hline \hline
**Step** & **Methodology** \\ \hline \hline  & Input the data for the distribution network, including the internal impedance of the OLTC \(Z_{T}^{*}\), the maximum transformation ratio \(K_{max}^{*}\), the rated capacity \(S_{N}^{*}\), the initial reactive power compensation on the low-voltage side, and the impedance from the equivalent DG node to the OLTC’s low-voltage side \(Z_{DG}^{*}\). \\  & Input the initial value of \(S_{2,re,t}^{*}\), designating this initial value as the rated capacity of the OLTC, \(S_{N}^{*}\). Set the transformation ratio for the OTLC as \(K_{max}^{*}\). \\  & Substitute the input data from Step 2 into Equation (7) to calculate the value of \(\eta_{t}\). \\  & Evaluate whether \(b_{t}\) falls within the permissible range. If it does not, revert to Step 1 and decrease the initial value of \(S_{2,re,t}^{*}\) accordingly until \(b_{t}\) remains within the allowable limits. The current value of \(S_{2,re,t}^{*}\) represents the maximum transmission capacity \(S_{2,re,max}^{*}\) of the distribution network, considering only the OLTC transformation ratio. \\  & Input the values of \(b_{t}\), \(K_{max}^{*}\), and \(S_{2,re,t}^{*}\) into Equations (4) and (20) to calculate the magnitude of \(U_{3,t}^{*}\). \\  & Determine if \(U_{3,t}^{*}\) exceeds the limit. If it does, decrease \(S_{2,re,t}^{*}\) and return to Step 2. \(U_{3,t}^{*}\) is the highest voltage point in the distribution network. If \(U_{3,t}^{*}\) is within limits, other nodes will also be within limits. \\  & Substitute the calculated \(S_{2,re,t}^{*}\) from Step 5 into Equation (19) to determine the required reactive compensation capacity \(Q_{C2,n,t}^{*}\). \\  & To obtain \(S_{2,re,t}^{*}\) represents the maximum transmission capacity \(S_{2,re,max}^{*}\) of the distribution network under reverse flow, taking all factors into account, as well as the required reactive power compensation capacity, producing the output results. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The specific computational procedure for \(S_{2,re,t,max}^{*}\). Figure 7: Calculation procedure for maximum transmission capacity of reverse power flow in distribution networks. Joint Operation Optimization Model for Multi-Distribution Networks with SOP Considering Reverse Power Flow  
Since the maximum value of \(S_{2,re,t,max}^{*}\) in the distribution network is only about 50% of the OLTC's rated capacity, it limits further development. To address this issue, this paper employs a soft open point (SOP) to flexibly interconnect multiple distribution grids. This enables sharing of backward power transfer capacities among sub-distribution networks and enhances the grid's capacity to handle reverse flow under high penetration of the DG. Figure 8 illustrates the SOP interconnection schematic, where the SOP interlinks two distribution networks at an equivalent power source node. When the reverse power flow in distribution network 1 exceeds its maximum transferable capacity, the SOP diverts the excess power to distribution network 2, which then transmits it to the upstream grid, thus reducing occurrences of wind and solar energy curtailment.