## 3 Preliminary Work
### Algorithm to Calculate Mileage Matrix
We chose the two end nodes where the two hydro stations were to show the voltage change after the network optimization. The results are shown in Figure 8. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Branch**} & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** \\ \hline \multirow{2}{*}{\(PM\)} & \(G_{1}\) & 0.3 & 2.2 & 2.1 & 3.0 & 36.8 & 36.4 & 37.4 & 39.7 \\  & \(G_{1}\) & 53.6 & 49.6 & 45.3 & 41.2 & 13.5 & 12.1 & 10.2 & 8.0 \\ \multirow{2}{*}{Division} & 10.81 & 3.05 & 4.69 & 4.30 & 0.67 & 0.80 & 0.88 & 0.94 \\  & Maximal & 53.6 & 49.6 & 45.3 & 41.2 & 36.8 & 36.4 & 37.4 & 39.7 \\ \multirow{2}{*}{Ratio} & 0.93 & 0.86 & 0.79 & 0.72 & 0.64 & 0.63 & 0.65 & 0.69 \\ \hline \multicolumn{1}{c}{**Branch**} & **10** & **11** & **12** & **13** & **14** & **15** & **16** & **17** \\ \hline \multirow{2}{*}{\(PM\)} & \(G_{1}\) & 44.3 & 42.0 & 22.1 & 23.4 & 23.7 & 27.0 & 40.9 & 46.1 \\  & \(G_{1}\) & 5.6 & 2.5 & 23.5 & 20.2 & 16.8 & 13.5 & 10.0 & 5.2 \\ \multirow{2}{*}{Division} & 0.95 & 0.96 & 1.95 & 2.07 & 2.31 & 2.33 & 1.83 & 1.82 \\ \multirow{2}{*}{Maximal} & 44.3 & 42.0 & 23.5 & 23.4 & 23.7 & 27.0 & 40.9 & 46.1 \\  & Ratio & 0.77 & 0.73 & 0.40 & 0.41 & 0.41 & 0.47 & 0.71 & 0.80 \\ \hline \hline \end{tabular}

* Division is the average power mileage of the second graph over that of the first one. If it is larger than 1, Equation (5) is satisfied. Ratio is the maximal power mileage value for 2 subgraphs over the original graph’s power mileage. The unit of the power mileage in the table is MW-km or MVar-km.

\end{table} 
Table 5: Result of first iteration for the power grid in Chongqing. Figure 7: The resulting two power grids generated by network optimization algorithm. Figure 8 shows a great improvement in the voltage condition occurs in both node 11 and node 17. In the original power grid, the voltage at node 17 in May transcended the upper limit for the saving operation, which did not occur in the new network. The voltage dip in the winter was greatly alleviated. The large amount of surplus energy made the problem very difficult to solve. Energy at a magnitude of 100 MWh needed to be absorbed, which is a very challenging problem. However, with the help of our network optimization algorithm, the addition of a signal transmission line greatly improved the voltage profile of the grid. ### IEEE 123-bus Case

The classical IEEE 123-bus system was used to verify our algorithm [24; 25; 26]. We modified the case from a 3-phase unbalanced system to a balanced system and added 6 RESs to the grid [4]. Unlike the case in Chongqing, which had 12 time points that denote 12 months in a year, this case had data only for a single time point. With so many nodes and different lengths of transmission line between each two nodes, this case' situation is much more complicated. The topology of the case is shown in Figure 9, and the position and the nominal power of the 7 added power stations is shown in Table 6.

Figure 8: Voltage profile of two selected nodes. Figure 9: Topology of the IEEE 123-bus power grid. When \(\tau\) equals 0.8464, the network optimization algorithm runs in the 123-bus case. The algorithm also iterates three times, and the power mileage changes from 50.4 to 21.5, where the second value is the maximum power mileage of the sub-graphs and the unit is MW\(\cdot\)km or MVar-km. The resulting power grid is shown in Figure 10.

As we assumed, the sub-graphs with large hydro stations were separated from the main grid, and the resulting voltage profile for each node is shown in Figure 11. The voltage quality is much better than the original power grid in every node, and new grids were able to operate in the saving voltage range. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Node** & **P (kW)** & **Q (kVar)** \\ \hline
32 & 640 & 320 \\ \hline
38 & 1610 & 805 \\ \hline
45 & 1520 & 760 \\ \hline
72 & 180 & 0 \\ \hline
102 & 80 & 0 \\ \hline
109 & 920 & 560 \\ \hline
112 & 2020 & 910 \\ \hline \hline \end{tabular}
\end{table}