## 5 Simulation Results
### The Influence of Topology Number  
The topology generation method used in this paper has been applied to the IEEE 33-bus system. According to the operation data of the distribution network in Refs. [25; 26], the generated reconfigured topologies are obtained by a Monte Carlo simulation that has been run 2000 times. We select the topologies with a proportion of occurrence over 0.08, and we finally derive 11 topologies, as shown in Table 5. Method 2 is used to realize network observability without considering accidental faults. Method 3 is used to realize complete observability under \(N\)-1 D-PMU outages. According to the proposed method, the optimal compromise solution with a maximum cost of 20% compared with method 2 is selected from the Pareto solution set. The variation relation between the three objective functions and the number of topologies is as shown in Figures 8-10.  
According to Figure 8, after the number of topologies increases to eight, the number of D-PMUs that must be placed does not increase. The reason is that, when the number of D-PMUs is high, the system has high measurement redundancy, which increases the reliability of system observation. According to Figures 8-10, the proposed method can increase the ANOBC to near the average number of methods 2 and 3 at the cost of increasing the D-PMU number by 20%. The NMR is also increased by about 30% compared with Method 2. Although the more popular Method 3 can maximize the ANOBC, it increases the number about twofold compared with Method 2. When the number of topologies  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{4}{c}{**IEEE 69-Bus System**} \\ \cline{2-5}  & \multirow{2}{*}{**NMR**} & \multicolumn{2}{c}{**Observability of Topology**} & \\ \cline{3-5}  & & **1** & **2** & **ANOBC** \\ \hline Proposed Method & 77.5 & Yes & Yes & 62.63 \\ \hline Ref. [32] & 81.0 & Yes & Yes & 65.68 \\ \hline Ref. [33] & 79.0 & Yes & No & 65.07 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison of multiple indicators of IEEE 69-bus system.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Topology**} & \multicolumn{2}{c}{**IEEE 33-Bus System**} \\ \cline{2-3}  & **Closed** & **Open** \\ \hline
1 & no & no \\ \hline
2 & s33, s34, s35 & s7, s9, s14 \\ \hline
3 & s35, s37 & s7, s25 \\ \hline
4 & s34, s36 & s13, s17 \\ \hline
5 & s33, s37 & s5, s26 \\ \hline
6 & s36, s37 & s16, s25 \\ \hline
7 & s36, s37 & s24, s31 \\ \hline
8 & s33, s34, s35 & s6, s11, s12 \\ \hline
9 & s34, s36 & s14, s15 \\ \hline
10 & s35, s37 & s2, s3 \\ \hline
11 & s34, s36, s37 & s12, s23, s27 \\ \hline \hline \end{tabular}
\end{table}
Table 5: The topological set of IEEE 33-bus system used.  
reaches seven, Method 3 ensures that almost every bus is equipped with a D-PMU, which is extremely uneconomical. Therefore, it can be concluded that, compared with other methods, the proposed method can greatly improve the ANOBC and NMR with a small increase in the number of D-PMUs. There are two main reasons. On one hand, the greater that the number of D-PMU configurations is, the stronger that the ability of the system to maintain observability is in cases of topology change and component failure. On the other hand, reasonable D-PMU placement can reduce the demand for measurement quantity.  
Figure 8: Relationship between \(N_{\text{D-PMU}}\) and the number of topologies.  
Figure 10: Relationship between NMR and the number of topologies.  
Figure 9: Relationship between ANOBC and the number of topologies.