## 5 Simulation Results
### Method Comparison  
When choosing NSGA-II as the comparison algorithm, the proposed OPP methods based on NSGA-III and NSGA-II are individually simulated on the IEEE 33-bus system. Table 1 shows the comparison results between different methods. Compared with other algorithms, the proposed algorithm can achieve more ANOBC and NMR. Compared with the OPP method based on NSGA-II, the proposed OPP method based on NSGA-III can obtain a better solution with a higher NMR and a higher ANOBC. In addition, it can be seen from Table 1 that NSGA-III takes the shortest time because it has a faster convergence speed than NSGA-II. Therefore, compared with NSGA-II, NSGA-III is more suitable for optimizing the proposed multi-objective model of OPP. According to the measurement data from before and after the installation of the D-PMU, we carry out 1000 state estimation simulations. Pseudo measurements, SCADA and D-PMU accuracy have been derived from Refs. [10; 31]. The relative error is calculated according to the true value of power flow and the result of state estimation. The average estimation errors in the voltage amplitude and phase angle of each bus are effectively reduced by implementing the D-PMU placement scheme, which is shown in Figures 6 and 7. This is because the D-PMU has higher accuracy than traditional measurements and the ability to directly measure the phase angle. The state estimation results without D-PMUs are the state estimation results of replacing D-PMUs with SCADA measurements. In order to better reflect the advantages of the proposed method, several indexes considered in this paper are compared with the method from Refs. [32; 33] in IEEE 33- and 69-bus systems. Tables 2-4 show the comparison results. Appendix A Figure A2 shows the topology of the IEEE 69-bus system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Num. of**} & \multirow{2}{*}{**ANOBC**} & \multirow{2}{*}{**NMR**} & \multirow{2}{*}{
\begin{tabular}{c} **Execution** \\ **Time/s** \\ \end{tabular} } \\  & **D-PMUs** & & & \\ \hline NSGA-III & 11 & 30.64 & 36.5 & 2.3503 \\ NSGA-II & 11 & 30.36 & 35.5 & 2.6363 \\ NSGA-II (Only & 11 & 28.68 & 34.5 & 2.3728 \\ consider num) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of results based on different methods. Figure 5: Results of D-PMU placement for IEEE 33-bus system. Figure 6: Comparison of average errors in voltage amplitude. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**IEEE 33-Bus System**} & \multicolumn{2}{c}{**IEEE 69-Bus System**} \\ \cline{2-5}  & **Num** & **D-PMU Locations** & **Num** & **D-PMU Locations** \\ \hline Proposed Method & 11 & 2, 3, 9, 11, 14, 17, 20, 24, 26, 29, 32 & 24 & 2, 3, 8, 13, 16, 19, 23, 26, 30, 34, 37, 40, 43, 45, 49, 51, 53, 56, 59, 60, 63, 64, 66, 68 \\ \hline Ref. [32] & 14 & 2, 4, 6, 8, 11, 13, 15, 17, 21, 23, 24, 27, 29, 32 & 27 & 1, 4, 5, 8, 9, 12, 15, 18, 20, 23, 26, 29, 32, 34, 37, 40, 42, 45, 49, 52, 53, 55, 58, 61, 64, 66, 69 \\ \hline Ref. [33] & 11 & 2, 6, 8, 11, 15, 17, 21, 24, 28, 29, 32 & 26 & 1, 4, 8, 14, 17, 19, 21, 24, 27, 28, 31, 34, 37, 39, 42, 45, 49, 51, 54, 56, 59, 61, 64, 66, 68, 69 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of the minimum number of D-PMUs for complete observability. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**IEEE 33-Bus System**} \\ \cline{2-5}  & **NMR** & \multicolumn{2}{c}{**Observability of Topology**} \\ \cline{3-5}  & **Topology 1** & **Topology 2** \\ \hline Proposed Method & 36.5 & Yes & Yes & 30.64 \\ \hline Ref. [32] & 45.0 & Yes & Yes & 31.64 \\ \hline Ref. [33] & 37.5 & Yes & No & 29.54 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of multiple indicators of IEEE 33-bus system. Figure 7: Comparison of average errors in voltage phase angle. According to Table 2, the proposed method uses fewer D-PMUs than other methods.