## 5 Results  
### Case Study I  
The effectiveness of the proposed method was evaluated on the modified IEEE 33 bus distribution network as one of the case studies (shown in Figure 4). The actual values of a DN were calculated based on power flow calculations. The following assumptions were made about the system **prior to PMU installation**:  
1. Measurements were randomly calculated based on their probability density function for each Monte Carlo trial. 2. Gaussian distribution, with \(3\sigma=50\%\) of the nominal value, was considered for the power injection on the buses [79]. 3. We assumed that one Photovoltaic (PV) panel was placed on bus 8 [80] with a maximum generation of 104 kW (The PV was modeled as a negative load), and three SCADA devices were located at buses 3, 8, and 21. Voltage magnitudes, active and reactive power flows, and injections were considered as real measurements (\(|V_{|s},|V|_{|g},|V|_{21}\)), \((P_{3-4}^{flow},\,P_{3-23}^{flow},P_{8-9}^{flow},\,P_{21-22}^{flow})\), \((Q_{3-4}^{flow},Q_{3-23}^{flow},Q_{8-9}^{flow},\,Q_{21-22}^{flow})\), \((P_{3}^{inj},P_{8}^{inj},P_{21}^{inj})\), and \((Q_{3}^{inj},Q_{8}^{inj},Q_{21}^{inj})\) from SCADA measurements to perform DSSE calculations. 4. The pseudo-measurements of the active and reactive power injections and flows were generated to make the system observable and to perform WLS calculations. 5. 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 active and reactive power flow and injection measurements [44]. 6. 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. 7. SCADA measurements were refreshed every five iterations, and DSSE calculations were performed using SCADA and pseudo-measurements every five iterations, con  
Figure 4: IEEE standard 33-bus distribution system before and after grid modernization. Prior to modernization, there was a PV on bus 8 and SCADA units on buses 3, 8, and 21. After modernization, an additional PV was installed on bus 29, and PMU devices were installed on buses 1, 16, and 31.  
cluding at \(N_{MC}=5995\). After this, Monte Carlo trial and PMUs were added to the system. 8. The DSSE results would be used for training a model to interpolate the SCADA measurements, as discussed in Section 4. It should be noted that the inputs were corrupted by random Gaussian error, with \(3\sigma=2\%\) and \(3\sigma=1\)_crad_ for the voltage magnitude and phase angle values, respectively, before feeding to a training model. This initial model would be used to predict the SCADA measurements after the installation of the PMUs to enable the DSSE calculations to be performed more frequently. 9. WLS calculations were performed using MATLAB, and the iteration process was stopped when the minimum difference between two iterations was \(\varepsilon=10^{-6}\). The following assumptions were made about the system after PMU **installation**:
1. Three PMUs were placed at buses \(1\), \(16\), and \(31\)[80], and one more PV with a maximum generation of \(80\) kW was added to a DN at bus \(29\) at \(N_{MC}=6000\) 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, so the full state vector is defined as: \(x=[\delta_{1},\cdots,\delta_{N},V_{1},\cdots,V_{N}]\).