## 5 Results
### Case Study II
The following assumptions were made about the system **after PMU installation**:  
1. Two PMUs were placed at buses 1 and 22 [80], and two more PVs were added to a DN at buses 38 and 64 with maximum generations of 20.83 kW and 16.67 kW, respectively, at \(t=4130\) to meet the new demands of the DNs in grid modernization development. 2. To model the uncertainty of the PMU measurements, a Gaussian error with \(3\sigma=1\%\) and \(3\sigma=1\)_crad_ was added to the voltage and branch current magnitudes, as well as the voltage and branch current phase angles, respectively [81]. 3. Since a PMU was installed in the slack bus, the phase angle at bus 1 was included in the state vector and is defined as: \(x=[\delta_{1},\cdots,\delta_{N},V_{1},\cdots,V_{N}]\). 4. We assumed that the PMU measurements were updated at each iteration, i.e., \(\tau_{SCADA}=5\tau_{PMU}\), and, at each iteration, the SCADA measurements were predicted from a training model. 5. The training model was updated once or twice per day using the DSSE results obtained from the SCADA measurements and PMU measurements during that time. The PMU measurement resolution was 20 measurements per day, so the model was updated every 10 (twice daily updates) or 20 (once daily updates) PMU measurements. 6. In order to verify the proposed method performance, the DSSE results were compared when the SCADA measurements were replaced using a sample-and-hold technique. 7. The estimated value from WLS calculations and the actual value from the power flow calculations were compared using MAE and RMSE criteria and shown in Table 2.  
This case study was based on actual data, and it had more severe dynamic behaviors. When the model was updated daily, the results were improved, but they were still close to those obtained using the sample-and-hold technique. In order to improve the results, the training model was updated twice a day to become robust against the dynamic behaviors of a DN. As it is clear from Table 2, the MAE and RMSE values for the state variables (voltage magnitude and phase angle) were improved using the proposed method when the model was updated twice a day compared to the sample-and-hold technique. In Figure 6, the estimated and actual values from the WLS and power flow calculations at bus 20 for 145 consecutive samples and for all buses at a given time are shown. As it is clear from a Figure 6, the estimated values from the WLS method using SCADA measurements from the proposed method and PMU measurements correctly follow the actual values, which were calculated from the power flow calculations. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Voltage Magnitude (_p.u._)} & \multicolumn{3}{c}{**Phase Angle (_rad_)**} \\ \cline{2-7}  & \multicolumn{3}{c}{Proposed Method} & \multicolumn{3}{c}{Proposed Method} \\ \cline{2-7} Criteria & Daily Update & \multirow{2}{*}{Twice Daily Update} & \multirow{2}{*}{Sample-and-hold technique} & Daily Update & \multirow{2}{*}{Twice Daily Update} & \multirow{2}{*}{Sample-and-hold technique} \\ \cline{1-1} \cline{5-7} MAE & 0.0031 & 0.0023 & & & & \\ \hline RMSE & \(4.8\times 10^{-5}\) & \(3.7\times 10^{-5}\) & \(5.2\times 10^{-5}\) & \(9.1\times 10^{-5}\) & \(8.5\times 10^{-5}\) & \(9.8\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: DSSE results of voltage magnitudes and phase angles after grid modernization when the training model was updated every 10 or 20 time steps (twice or once per day).