# Decision Optimization for Power Grid Operating Conditions with High- and Low-Voltage Parallel Loops
## 5 Case Study
### New England 39-Bus System  
The New England 39-bus system is a pure AC system, which is shown in Figure 3. Many parallel loops would emerge if a higher-voltage power grid is connected to this system via node 5, 16 and 26. With node 5, 16 and 26 as hub substations, this power grid can be divided into three basic partitions at most after switching off several specific lines.  
According to the complex network theory in Section 2, edge betweenness of each branch should be calculated before switching off any line. The calculation results are shown in Table 1.  
Figure 3: New England 39-bus system.  
In the original power grid network, the maximum edge betweenness is 2.0273 of line 3-4, which is between bus 3 and bus 4. According to GN algorithm, line 3-4 should be switched off firstly. No new partition emerges after line 3-4 is switched off. Then, the edge betweenness of the modified network without the line 3-4 is calculated. The calculation results are shown in Table 1 as well. In this round, the maximum edge betweenness is 2.8089 of line 14-15. No new partition emerges after line 14-15 is switched off. During the third round, line 1-39 has the largest edge betweenness, with a value of 4.9039. After line 1-39 is switched off, the original network would be divided into two partitions. Since there are still two hub substations in the same partition, the line switching process should be continued until all of the hub substations are divided into different partitions. It should be noted that the edge betweenness of some specific branches in a subsequent line switching process might be larger than their corresponding values in the previous process, which is due to the change of network connectivity and the branch number along with the line switching process.  
The basic partitions of the New England 39-bus system is shown in Figure 4 After three branches are switched off, the network is divided into two partitions. And three partitions would emerge with four branches switched off.  
There are three basic partitions illustrated in Figure 4, which can be combined to generate four different opening schemes in total according to Combinatorics and actual neighboring relationships.  
\begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Branch**} & \multicolumn{4}{c}{**Edge Betweenness**} \\ \cline{2-5}  & **First Partition** & **Second Partition** & **Third Partition** & **Forth Partition** \\ \hline bus1–bus2 & 1.2581 & 1.6293 & 4.0836 & 0.3300 \\ bus1–bus39 & 1.1759 & 1.9265 & 4.9039 & switched off \\ bus2–bus3 & 1.4398 & 0.7426 & 2.0915 & 0.3789 \\ bus2–bus25 & 0.6875 & 0.4990 & 0.7762 & 0.1663 \\ bus3–bus4 & 2.0273 & switched off & switched off & switched off \\ bus3–bus18 & 0.7607 & 0.6539 & 1.7082 & 0.3870 \\  
Because the number of generated opening schemes is quite small, it is not necessary to filter them using the filtering index. The filtering index (only weighted modularity indicator is considered in this example with \(\lambda_{2}\) = 0) and priority index of these four generated opening schemes (No. 1-4) as well as two experiential schemes (No. 5-6), are shown in Table 2.  
It is obvious in Table 2 that the filtering index and priority index of the first four opening schemes generated by proposed method are overall larger than two experiential schemes. For New England 39-bus system, which is a pure AC power system, a larger filtering index indicates a better network structure after specific lines are switched off. And a larger priority index indicates better operational characteristics if the opening scheme is employed. By comprehensively considering the filtering index and priority index, scheme 1 should be selected as the optimal opening scheme. Based on the analysis above, it is easy to recognize the effectiveness of the proposed method to find optimal opening schemes, as well as its superiority over experiential method. The simulation results show that the proposed opening method can successfully be applied in a pure AC power system.