## 4 Methodology  
As mentioned earlier, SCADA measurements are commonly available from the beginning operation of ADNs to perform DSSE calculations and to make them observable [74]. The installation of PMU technology has gradually been developed as parts of grid modernization projects move from traditional DNs to smart DNs [75]. Thus, DSSE results will be improved by incorporating PMU measurements as well as SCADA measurements [67]. Unfortunately, SCADA measurements typically have a slower refresh rate compared with the newly installed PMU measurements [76]. With smart DNs, it is important to perform DSSE calculations as frequently as possible in order to understand the dynamic state of the system [77]. Rather than only performing SEs as each new SCADA measurement arrives, we want to perform DSSE calculations with every new PMU measurement. This requires a method for interpolating SCADA measurements so they can be incorporated into the DSSE algorithm. In this paper, we assume that we start with an ADN with SCADA measurements. After a period of time, the ADN is updated to include PMUs as part of a grid modernization procedure. Figures 2 and 3 illustrate how we trained a regression model to predict future SCADA values prior to and after installing PMUs. Regression models are a category of supervised machine learning, where the corresponding output (\(y\)) at each input (\(x\)) is available [78]. In the training phase, a relationship between \(x\) and \(y\) is determined and expressed as \(f\); i.e., \(y=f(x)\). A regression model was chosen in this study, since there were enough data points to train a model. Note that the regression model was only used for DSSE calculations after PMUs were installed in order to produce interpolated SCADA measurements that could be used to perform DSSE calculations more frequently. In this work, we pre-trained a SCADA prediction model prior to PMU installation and regularly updated the model after PMU installation. Prior to PMU installation, DSSE calculations based on WLS method were performed using SCADA and pseudo-measurements. A multi-output regression model was trained based on these DSSE results to predict an interpolated SCADA measurement, as shown in Figure 2. The ground truth used for the interpolated SCADA measurement was obtained using backward-forward power flow calculations. This procedure was repeated for every SCADA measurement that arrived prior to the PMU installation. A different training procedure was performed after adding the PMUs to the ADN. In the updated procedure, SCADA measurements were predicted using the most recent DSSE calculations, which in turn depended on the most recent PMU measurement and the most recent SCADA measurement or prediction. As shown in Figure 3, a training model was updated every time real SCADA measurements were provided in order to improve model performance and make it robust against dynamic changes of ADNs. For instance, when SCADA and PMU measurements were refreshed at \(t_{1}\), a training model was updated from \(M_{0}\) to \(M_{1}\). In order to evaluate the DSSE result, the Mean Absolute Error (MAE) and Root Mean Square Error (RMSE) are calculated using (12) and (13), respectively:  
\[\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}\left|\frac{\hat{x}_{i}-x_{i}}{x_{i}}\right| \tag{12}\]  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{N}(\hat{x}_{i}-x_{i})^{2}}{N}}. \tag{13}\]  
Figure 3: DSSE calculations and updating a training model after PMU installation. Figure 2: A multi-output regression model to predict SCADA measurements. For the figure, we assume that the PMUs are installed in the DN at time step N\({}_{1}\), so the procedure is performed through time step N\({}_{1}\)-5. In the above Equations, \(\hat{x}\) is the estimated value, \(x\) is the actual value, and \(n\) refers to the dataset size. The interface between MATLAB (2022) and Anaconda Python was used to train and update a model using the DSSE results obtained from the SCADA and PMU measurements.