## 2 Literature Review
### Bibliometric Review
As the features of the time series, such as the Hurst exponent, vary with time, each model is trained using an expanding training window that incrementally encompasses 36 months of historical data up to the complete training dataset. Moreover, to mitigate the influence of noise inherent in the residual component, the evaluation is conducted on both the original time series and on a smoothed version, with a smoothing parameter \(\alpha=0.3\). Figure 9 summarizes these forecasting scenarios. ### Univariate Models Accuracy Evaluation

The validity and performance of these time series forecasting models must be rigorously assessed to ensure their suitability for real-world applications. This section presents \begin{table}
\begin{tabular}{l l l c} \hline \hline
**Model** & **Library (Version)** & **Parameters** & **Reference Section** \\ \hline \multirow{3}{*}{SARIMA} & \multirow{3}{*}{statsmodels (0.14.2)} & Seasonality = 12 & Section 4.3 \\  & & Autoregressive = 1 & Section 4.2 \\  & & Differences = 1 & Section 4.1 \\  & & Moving Average = 1 & Section 4.2 \\ \hline FB Prophet & prophet (1.1.5) & Automatic & \\ \hline Holt–Winters & statsmodels (0.14.2) & Seasonality = 12 & Section 4.3 \\  & & Seasonality component = additive & Section 4.3.1 \\ \hline TBATS & TBATS (1.1.3) & Seasonality = 12 & Section 4.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters selected for each model. Figure 9: Conditions of the training and testing of the models. the test results of the time series univariate forecasting models SARIMA, FB Prophet, Holt-Winters, and TBATS for energy consumption, comparing them on their Mean Absolute Percentage Error (MAPE). As presented in Figure 10, FB Prophet's model achieved the best accuracy, with 0.71% MAPE for the smoothed energy usage time series, with a 108-month training window. Furthermore, upon comparing Figure 10,b, it becomes evident that smoothing the time series enhanced the accuracy of the models. Additionally, in most scenarios, the utilization of a 36-month training window yielded the poorest MAPE values, suggesting an insufficient amount of data for robust model performance. For the original time series, the optimal training window sizes for each model were 72 months for SARIMA (1.38% MAPE) and for FB Prophet (1.91%); 108 months for Holt-Winters (3.85%); and 215 months for TBATS (4.17%). For the smoothed version, the optimal sizes were 180 months for SARIMA (1.44%), 108 months for FB Prophet (0.71%, the lowest MAPE overall); 72 months for Holt-Winters (2.53%); and 215 months for TBATS (1.44%). The predictions for the validation period for each optimal model are further compared with additional accuracy metrics in Table 5, which presents additional accuracy metrics for these models. Both FB Prophet and SARIMA returned negative MPE values, meaning that their predicted values are greater than the actual values. In contrast, Holt-Winters and TBATS resulted in positive MPE values, as their estimated values are lower than the actual ones. The TBATS model also obtained a MAPE equal to MPE, meaning that the predicted values never cross over the actual values. These conclusions are supported by Equation (2) and visually represented in Figure 11, which compares these models, in these same scenarios, for both the original and the smoothed time series. It is noteworthy that all these models achieved Highly Accurate Prediction, according to [51]. These models are also used to forecast the energy consumption in Brazil for up to November 2033, in Section 5.5. The MAPE assessment shows that FB Prophet achieved the best accuracy among the models assessed for energy forecasting in Brazil. Our findings are in accordance with [17; 18]. Figure 10: Mean absolute percentage error (MAPE—%) for the univariate models in different scenarios. ### Multivariate Models Accuracy Evaluation

Similar to the results presented in Section 5.2, this section evaluates the accuracy metrics for the forecasting models.