## 2 Materials and Methods  
The system analyzed in this study corresponds to the configuration of IEEE 14, 30, and 57-bus systems, as illustrated in Figure 1a-c. The IEEE 14 system is composed of 1 slack bus, 4 generation buses, 9 load buses, and 20 transmission lines. Meanwhile, the IEEE 30 system has 1 slack bus, 5 generation buses, 24 load buses, and 41 transmission lines. Finally, the IEEE 57 system consists of 1 slack bus, 6 generation buses, 50 load buses, and 80 transmission lines. Figure 1: Systems studied: (**a**) IEEE 14 buses; (**b**) IEEE 30 buses; (**c**) IEEE 57 buses. For the IEEE 14-bus system, 213 samples were utilized, while for the IEEE 30-bus system, 184 samples were used, and for the IEEE 57-bus system, 202 samples were employed, resulting in a total of 599 samples. These samples were acquired according to the parameterization continuation power flow (PCPF) method described in [2; 11] and were used for training and validation of ANN. Each sample is composed of five pieces of data: three input data for the ANN, which include the loading factor (\(\lambda\)) and the real and reactive powers generated at the slack bus (P\({}_{\text{g}}^{\text{slack}}\) and Q\({}_{\text{g}}^{\text{slack}}\)), and two output data representing the loss total real (\(Pa\)) and reactive (\(Pr\)) power of the system. The ANN used is an MLP (Multi-Layer Perceptron) with a backpropagation learning algorithm with three layers: input with 3 neurons, hidden with 10 neurons, and 2 neurons in the output layer, as shown in Figure 2. The software used to obtain the results was Matlab(r), version 2024 [18]. Continuation power flow (CPF), unlike experimental random classification, is characterized by a sequence of nonlinear equations. Its resolution is achieved by varying the load factor parameter, resulting in different values for the real (\(Pa\)) and reactive (\(Pr\)) power losses of the system in each case. For this type of dataset, MLP networks prove to be well-suited. Figure 3 illustrates the flowchart of the MLP network used to estimate the total real (\(Pa\)) and reactive (\(Pr\)) power losses of electrical power systems. Figure 3: Flowchart depicting the MLP utilized in this study. Figure 2: ANN used in this work. The activation functions used were the hyperbolic tangent for the hidden layer (1) and linear for the output layer (2). \[f(u)=\frac{(1-e^{-\text{tu}})}{(1+e^{-\text{tu}})} \tag{1}\]  
where,  
\(f(u)=\text{hyperbolic tangent activation function}\);  
\(t=\text{estimate of the parameter that determines the slope of the curve;}\)  
\(u=\text{function activation potential}\). \[f(u)=u \tag{2}\]  
The mean square error (MSE) vector of the ANN is calculated according to Equation (3), where \(\mathbf{Y_{ob}}\) and \(\mathbf{Y_{des}}\) represent the obtained and desired outputs of the ANN, respectively. The more similar these outputs are to each other, the smaller the error, indicating a more accurate weight adjustment. \[\text{MSE}=\frac{1}{\text{P}}\sum_{i=1}^{n}(\mathbf{Y_{ob}}-\mathbf{Y_{des}}) ^{2} \tag{3}\]  
where,  
\(\text{P}=\text{number of data}\);  
\(\mathbf{Y_{des}}=\text{desired values (target via experiment)}\);  
\(\mathbf{Y_{ob}}=\text{values obtained via ANN}\). As the MSE, the Mean Absolute Percentage Error (MAPE) was also applied to the performance of the models used in Equation (4). The Mean Absolute Percentage Error (MAPE) is one of the most commonly used Key Performance Indicators to measure forecast accuracy. The results are presented in Table 1.  
\[\text{MAPE}=\frac{1}{\text{P}}\,\sum_{i=1}^{n}\biggl{(}\frac{\mathbf{Y_{ob}}- \mathbf{Y_{des}}}{\mathbf{Y_{des}}}\biggr{)} \tag{4}\]  
A test used in the literature is the Kruskal-Wallis test, which is a nonparametric version of classical one-way ANOVA, and an extension of the Wilcoxon rank sum test to two groups or more. It compares the medians of the groups of data in x to determine if the samples come from the same population (or, equivalently, from different populations with the same distribution) [18].