# A Hybrid State Estimator Based on SCADA and PMU Measurements for Medium Voltage Distribution System
## 4 Case Studies
### Comparison of State Estimation Results under the Bad Data Condition
To characterize the accuracy of state estimation, the absolute error for the estimated voltage amplitude of each bus is defined using Equation (17), and the relative error of the estimated voltage phase of each bus is defined using Equation (18). Because the values of the phase angle were close to zero, to avoid large fluctuation in the absolute error, the relative error was used to reflect the estimation error of the phase angles. \[r_{\text{absolute}}=\left|\hat{\mathbf{x}}-\mathbf{x}_{pf}\right|/\mathbf{x}_ {pf} \tag{17}\]  
\[r_{\text{relative}}=\left|\hat{\mathbf{x}}-\mathbf{x}_{pf}\right| \tag{18}\]  
From Figure 9a, the estimation errors of voltage amplitude obtained by the EXP were larger than that obtained by the improved IGGIII, especially in the vicinity of bad data. From Figure 9b, the estimation errors of phase angle in the vicinity of bad data obtained by the EXP were more than 1\({}^{\circ}\), while the improved IGGIII could maintain the estimation error to lower than 0.1\({}^{\circ}\), which highlights the better robustness of the improved IGGIII. The contrast results in Figures 8 and 9 also show that reasonable weights of measurements were very important during the iterations, especially under the bad data condition. Furthermore, the good estimation results obtained by the improved IGGIII indicated that the normalized residuals and adaptive robust thresholds could effectively improve the estimation accuracy under the bad condition. To sum up, the improved IGGIII could converge under the bad data condition with good robustness. When the bad data appeared, the proposed IIGGIII-LSE could also track the changes in the distribution system, and the dynamic estimation results are shown in Table 4, Figures 10 and 11. To emphasize the advantage of the proposed IIGGIII-LSE, the EXP combined with LSE (EXP-LSE) was used. For EXP-LSE, the estimation results of bus voltage and phase angle at bus 6 without PMU, were unsatisfying, as shown in Figure 10, because bus 6 was in the vicinity of bad data and there were no PMU measurements to update the inaccuracy states. The phase angle estimation results at bus 30 equipped with PMU were inaccurate due to the bad robustness of EXP at the SCADA data acquisition time. However, the estimation results could be corrected by the highly accurate PMU measurements, as shown in Figure 11b. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{**Time (s)**} & \multicolumn{4}{c}{**MAE of Bus Voltage Amplitude**} & \multicolumn{4}{c}{**The Relative Error of the Estimated Voltage Phase**} \\ \cline{2-9}  & \multicolumn{2}{c}{**Bus 6**} & \multicolumn{2}{c}{**Bus 30**} & \multicolumn{2}{c}{**Bus 6**} & \multicolumn{2}{c}{**Bus 30**} \\ \cline{2-9}  & **EXP-LSE** & **IIGGIII-LSE** & **EXP-LSE** & **IIGGIII-LSE** & **EXP-LSE** & **IIGGIII-LSE** & **EXP-LSE** & **IIGGIII-LSE** \\ \hline
20 & 0.00120 & 0.00032 & 0.00145 & 0.00066 & 0.85947 & 0.01101 & 0.63956 & 0.01316 \\
20.5 & 0.00317 & 0.00284 & 0.00154 & 0.00043 & 0.11463  
For the proposed IIGGIII-LSE, the dynamic estimation results under the bad data condition were as accurate as under the normal condition. While at the bus with the PMU installation, the proposed method could track the changes more accurately compared to the bus without the PMU installation, which indicated the effectiveness of using PMU measurements. To sum up, the robust state estimation was indispensable in the dynamic state estimation and the proposed IIGGIII-LSE performed well under the bad data condition. Besides, the estimation results would be less accurate if the estimated bus or area was far from the installation of the PMUs. The feasibility of the proposed IIGGIII-LSE was also verified by the actual medium voltage distribution system in the suburb of Guangzhou, China. The system included 62 buses and 108 real-time measurements, and the diagram is shown in Figure 12. Figure 12 shows the application results of the proposed method in this system.