## 5 Results
### Case Study II  
The effectiveness of the proposed method was also evaluated using the modified IEEE standard 69-bus distribution network (shown in Figure 5). The system is suitably adopted to include a mix of commercial and residential loads and DGs. A set of experimental data (available for a time period of one year), obtained from Open Energy Information (OpenEI) [82], was utilized with the simulation time step of \(1\) h. The hourly data of power generation of a photovoltaic system were computed and adopted based on the actual \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Voltage Magnitude (\(\mathbf{p.u.}\))** & \multicolumn{2}{c}{**Phase Angle (\(\mathbf{rad}\))**} \\ \hline Criteria & Proposed Method & Sample-and-hold technique & Proposed Method & Sample-and-hold technique \\ \hline MAE & \(8.2\times 10^{-4}\) & \(9.7\times 10^{-4}\) & \(0.0060\) & \(0.0071\) \\ \hline RMSE & \(1.02\times 10^{-5}\) & \(1.45\times 10^{-5}\) & \(7.9\times 10^{-5}\) & \(9.2\times 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: DSSE results of voltage magnitudes and phase angles after grid modernization. data for Bozeman, MT, USA [83, 84]. Because reactive power injection is not available, the reactive power injection at bus \(i\) with a random power factor is defined as [85]:  
\[Q_{i}(\text{t})=P_{i}(t)\frac{\sqrt{1-Pf_{i}^{2}(t)}}{Pf_{i}(t)}, \tag{14}\]  
where \(Pf_{i}(t)\sim Unif(0.85,0.95)\). The following assumptions were made about the system **prior to PMU installation**:  
1. We assumed that one Photovoltaic (PV) panel was placed on bus 20 [80] with a maximum generation of 15.33 kW (The PV was modeled as a negative load), and three SCADA devices were located at bused 6, 33, and 53. Voltage magnitudes, active and reactive power flows, and injections were considered as real measurements (\(|V|_{6,}V|_{33,}V|_{53}\)), (\(P_{6-7}^{flow}\), \(P_{33-34}^{flow}\), \(P_{53-54}^{flow}\)), (\(Q_{6-7}^{flow}\), \(Q_{33-34}^{flow}\), \(Q_{53-54}^{flow}\)), (\(P_{6}^{inj}\), \(P_{33}^{inj}\), \(P_{53}^{inj}\)), and (\(Q_{6}^{inj}\), \(Q_{33}^{inj}\), \(Q_{53}^{inj}\)) from SCADA measurements to perform DSSE calculations. 2. The pseudo-measurements of the active and reactive power injections and flows were generated to make the system observable and to perform WLS calculations. 3. The standard deviation was considered as 50% of the nominal value for pseudo-measurements, 2% of the voltage magnitude, and 3% of the actual value of the active and reactive power flow and injection measurements [44]. 4. The voltage magnitudes and phase angles for all the buses were considered as a state variable: \(x=[\delta_{2},\cdots,\delta_{N},V_{1},\cdots,V_{N}]\), where \(\delta_{N}\), \(V_{N}\) are the voltage phase angle and magnitude, respectively, and \(N\) is the number of buses. It was assumed that there were no measurement devices installed in the slack bus. As well, \(\delta_{1}=0\) and \(V_{1}=1\). The Jacobian matrix will be formed as \(H_{Traditional}\), which is shown in Section 3. 5. The SCADA measurements were refreshed every five iterations, so DSSE calculations were performed using SCADA and pseudo-measurements every five iterations, concluding at \(t=4125\). (After this time, PMUs were added to the system.) 6. The DSSE results would be used for training a model to interpolate the SCADA measurements, as discussed in Section 4. This initial model would be used to predict the SCADA measurements after the installation of the PMUs to enable DSSE calculations to be performed more frequently. 7. The WLS calculations were performed using MATLAB, and the iteration process was stopped when the minimum difference between two iterations was \(\varepsilon=10^{-6}\). Figure 5: IEEE standard 69-bus distribution system before and after grid modernization. Prior to modernization, there was a PV on bus 20 and SCADA units on buses 33 and 53. After modernization, two additional PVs were installed on bus 38 and 64, and PMU devices were installed on buses 1 and 22.