## 6 Simulation Results and Discussion
On the other hand, under the same number of PMUs, the state estimation error of the solution is reduced by network compression. Note that: the most economical solution is the solution that has the minimum number of PMUs in the Pareto set; the compromised solution is the solution selected by the TOPSIS from the Pareto set. In order to further analyze the influence of the compromised solution on the state estimation accuracy, 1000 state estimation simulations were carried out, the comparison of error in voltage amplitude estimation is shown in Figure 5, the comparison of error in voltage phase angle estimation is shown in Figure 6.  
As shown in Figures 5 and 6, the compromised solution obtained by the OPP method proposed in this paper can not only reduce not only the voltage amplitude estimation error but also the voltage phase angle estimation error of each bus effectively. In order to further analyze the effectiveness of searching the minimum number of PMUs required for the full observability, the simulation results of the proposed method in IEEE 33, 34 and 69 bus systems are compared with those of other methods in References [46; 47; 48], which are shown in Table 2. The topologies of the IEEE 34 bus system and IEEE 69 bus system are shown in Appendix A. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case** & **Optimal Solution** & **No. of PMUs** & **State Estimation Error** \\ \hline
1: Without Network & The Most Economical Solution & 11 & 6.64328588 \(\times\) 10\({}^{\circ}\)(–4) \\ Compression & The Compromised Solution & 12 & 6.46038299 \(\times\) 10\({}^{\circ}\)(–4) \\
2: With Network & The Most Economical Solution & 10 & 6.38764872 \(\times\) 10\({}^{\circ}\)(–4) \\ Compression & The Compromised Solution & 11 & 6.13058913 \(\times\) 10\({}^{\circ}\)(–4) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of two cases based on network compression. Figure 4: The Pareto set. As shown in Table 2, compared with the methods in References [46; 47; 48], the OPP method proposed in this paper can provide the optimal solution with the minimum number of PMUs required for the full observability of IEEE 33, 34 and 69 bus systems respectively. Furthermore, the proposed OPP method can also obtain the compromise optimal scheme which comprehensively considers the cost and state estimation error. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{2}{c}{**IEEE 33 Bus System**} & \multicolumn{2}{c}{**IEEE 34 Bus System**} & \multicolumn{2}{c}{**IEEE 69 Bus System**} \\ \cline{2-5}  & **No. of** & **PMU** & **No. of** & **PMU** & **No. of** & **PMU Locations** \\ \hline Proposed Method & & 2, 8, 11, 15, & & 2, 4, 6, 10, 12, & & 2, 3, 8, 11, 14, 17, 22, 26, 28, 33, \\ (The Most Economical Solution) & 10 & 17, 20, 24, 26, & 11 & 17, 21, 23, 28, & 22 & 34, 37, 41, 45, 49, 51, 56, 59, 62, \\  & & 30, 32 & 31, 33 & & 64, 66, 68 \\ \hline Proposed Method & & 2, 4, 8, 11, 15, & & 2, 4, 6, 10, 12, & 2, 3, 8, 11, 14, 17, 22, 26, 28, \\ (The Compromised Solution) & 11 & 17, 20, 24, 26, & 12 & 17, 20, 21, 23, & 23 & 33, 34, 37, 41, 45, 49, 51, 56, 59, \\  & & 30, 32 & 28, 31, 33 & & 62, 64, 66, 68 \\ \hline \multirow{2}{*}{Reference [46]} & & 2, 6, 8, 11, 15, & & 2, 5, 7, 10, 13, & 1, 4, 8, 14, 17, 19, 21, 24, 27, 28, \\  & & 17, 21, 24, 28, & 12 & 17, 21, 24, 28, & 26 & 31, 34, 37, 39, 42, 45, 49, 51, 54, \\  & & 29, 32 & 30, 31, 33 & & 56, 59, 61, 64, 66, 68, 69 \\ \hline \multirow{2}{*}{Reference [47]} & & 2, 4, 6, 8, 11, & & 2, 4, 8, 12, 14, & 1, 4, 5, 8, 9, 12, 15, 18, 20, 23, \\  & & 13, 15, 17, 21, & 16, 17, 21, 24, 27, & 27 & 26, 29, 32, 34, 37, 40, 42, 45, 49, \\  & & 23, 24, 27, 29, & 27, 28, 31, 33 & & 52, 53, 55, 58, 61, 64, 66, 69 \\ \hline \multirow{2}{*}{Reference [48]} & & 2, 5, 8, 11, 14, & & 2, 4, 8, 10, 12, & 2, 4, 7, 9, 14, 17, 20, 23, 26, 29, \\  & & 17, 21, 24, 27, & 13 & 17, 21, 23, 25, & 25 & 32, 34, 37, 40, 43, 45, 49, 51, 54, \\  & & 30, 32 & 28, 31, 33 & & 57, 58, 61, 64, 6