## 2 Materials and Methods
### Data Synthesis  
To validate the proposed model for estimating \(\mathrm{d}P_{n}\), the electrical flow issue was resolved for the circuit depicted in Figure 1, utilizing the pandapower (version 3.1.2) tool ([[https://www.pandapower.org/](https://www.pandapower.org/)]([https://www.pandapower.org/](https://www.pandapower.org/)), accessed on 14 June 2025). In the simulations, the MV/LV transformer was represented by the 0.25 MVA, 10/0.4 kV transformer model from the standard pandapower library. The scenarios of DL resistances (Figure 1) are provided in Table 2. Table 3 presents cases of the active power loads of the network. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{**Grid Case**} & \(R_{s}\) & \(R_{w1}\), & \(R_{w2}\), & \(R_{w3}\), & \(R_{w4}\), & \(R_{w5}\), & \(R_{w6}\), & \(R_{wsum}\), \\  & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) & \(\mathbf{m\Omega}\) \\ \hline
1 & 30 & 30 & 30 & 30 & 30 & 30 & 30 & 210 \\
2 & 30 & 15 & 15 & 15 & 30 & 30 & 30 & 165 \\
3 & 30 & 30 & 30 & 30 & 15 & 15 & 15 & 165 \\ \hline \end{tabular}
\end{table} 
Table 2: Resistances of distribution lines. Figure 1: Equivalent circuit representing the energy delivery route from sum meter to consumer MUT. The arrows and dots represent the other branches. The circles represent electricity meters. The synthesized data covers variety scenarios in order to assess the main effects of the branch operational modes on the method performance including different branch parameters, different loading patterns and uneven load distribution along the branch, different load types and different sizes of the consumer power step. Combination of the loads and wire resistances during the worst-case scenario, i.e., grid case 1 presented in Table 2, together with grid load case 8 presented in Table 3, causes 10% voltage drop in the power supply path. The other grid load cases were meant to consider the effect of unequal line lengths between consecutive loads. Grid load cases presented in Table 3 were utilized to assess the effect of different voltage drops and power losses in the branch. The arrangement of network load powers, smart meters metered power, and the consumer power step size of \(dP_{c}\) (= 0.3, 0.5, 0.7, 1.0, 1.5) kW, to cover the relevant power ranges of the conventional household appliances, alongside the power consumption preceding the power step of \(P_{c1}\) = (0, 0.3, 0.6, 0.9, 1.2) kW, is depicted in Figure 2, accompanied by the corresponding \(\mathrm{d}P_{n}\) for each configuration scenario. The network loads were simulated using the ZIP model, encompassing both constant impedance (ZZ) and constant power (ZP) coefficients, with \(\mathrm{ZP}=1-\mathrm{ZZ}\) within this model. Every load \(P_{Li}\), \(i\) = 1,\(N\), assumed the constant impedance coefficient as illustrated in Figure 2. The aggregate power consumption of the DG branch is  
\[P_{eqn}=\sum_{i=1}^{N}P_{Li}+\sum_{i=1}^{N}P_{wi}+\ P_{Rs} \tag{5}\]  
where \(P_{wi}\) and \(P_{Rs}\) represent the power losses within the _ith_ energy delivery branch and the \(R_{s}\) branch, respectively. The distribution of \(P_{Li}\), along with its metadata, is available at [[https://github.com/KTU-ANODETEL/Grid_of_6_Nodes_One_Phase_Dataset1](https://github.com/KTU-ANODETEL/Grid_of_6_Nodes_One_Phase_Dataset1)]([https://github.com/KTU-ANODETEL/Grid_of_6_Nodes_One_Phase_Dataset1](https://github.com/KTU-ANODETEL/Grid_of_6_Nodes_One_Phase_Dataset1)), accessed on 14 June 2025. This study initially focusses on the consumption of active power by consumer loads. Future plans include adapting the method to account for both active and reactive power components. As shown in Figure 2, the values of \(\mathrm{d}P_{n}\) can exhibit both positive and negative variations dependent upon the nature of loads and the magnitude of technical losses within the DG branches. This inherent complexity renders the prediction of \(\mathrm{d}P_{n}\) a challenging task.