## 2 Literature Review
### Bibliometric Review
Similarly, in the multivariate scenario, both SARIMAX and FB Prophet demonstrated adequate fit, with no autocorrelation within the residuals in the original time series, as demonstrated in Table 8. In contrast, for the smoothed version, both models showed a lack of fit in the Ljung-Box test. As again the smoothing of the time series made the models fail the Ljung-Box test, only the accuracy performance for the original time series is taken into consideration. With this restriction, SARIMAX returned the best MAPE values. ### Energy Consumption Forecasting

This section considers the optimal training window sizes for each forecasting model, considering the univariate results presented in Section 5.2. Although SARIMAX achieved its best accuracy in the multivariate modeling, we here present the predictions provided by its univariate scenarios because forecasting the energy consumption up to 2033 in the multivariate scenario would also require predicting indexes like Brazilian's GDP, inflation, and oil production, which is not a trivial task and beyond the scope of this work. Moreover, the accuracy of the univariate scenario is considered adequate for this purpose. The primary objective of the results from Section 5.3 was to illustrate the impact of incorporating exogenous factors on model performance. The predicted values are displayed in Figure 15. The SARIMA predicts values close to 70,000 GWh in November 2033 when no smoothing is applied, with FB Prophet forecasting \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Original** & **Smoothed** \\ \hline SARIMA & 0.277 & 0.0107 \\ FB Prophet & 0.404 & 0.0007 \\ Holt–Winters & 0.002 & 0.0000008 \\ TBATS & 0.001 & 0.0001 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Ljung–Box test \(p\)-values for four months lag for the optimal training window sizes in the univariate models, according to Figure 10. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Model** & **Original** & **Smoothed** \\ \hline SARIMAX & 0.570 & 0.001 \\ FB Prophet & 0.151 & 0.006 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Ljung–Box test \(p\)-values for four months lag for the optimal training window sizes in the multivariate models, according to Figures 12 and 13.

above 60,000 GWh. The value indicated by SARIMA is still near the upper limits of the predicted values by FB Prophet, represented by the gray shade in Figure 15. It is noteworthy that Holt-Winters and TBATS predict a stationary trend in this scenario. When smoothing is applied, SARIMAX and FB Prophet predict similar values at around 55,000 GWh consumption in November 2033. Although Holt-Winters still predicts a stationary trend in this scenario, TBATS forecasts an increase, predicting approximately 50,000 GWH for the same month. Smoothing the time series has also enlarged the lower and upper limits of the FB Prophet forecasts, possibly as a consequence of the increase of the autocorrelation in the residuals, as seen in Section 5.4. The forecasted values generated by all four models are accessible in the GitHub repository (github.com/gabriel-arquelau/Brazilian-energy-consumption-prediction/ accessed on 21 June 2024). These predictions are based on scenarios that achieved the best MAPE assessments in their optimal univariate scenarios. They encompass data projections extending up to November 2033 for both the original and smoothed scenarios, and in the case of FB Prophet, accompanied by corresponding lower and upper limits. The publication of this data enables future validation of the forecasting with actual values. ### Time Performance Assessment

While accuracy is a fundamental aspect of forecasting models, it is not the sole determinant of their effectiveness. Particularly in time-sensitive environments, time performance plays a vital role in ensuring the practicality and utility of these models. In dynamic industries like energy management, decisions must be made rapidly in response to changing conditions.