# Long-Term Forecasting of Electrical Loads in Kuwait Using Prophet and Holt-Winters Models
## 3 Results and Discussion
### Model Accuracy  
Various accuracy and statistical performance metrics were used in this study, which include MAPE, MAE, RMSE, CVRMSE, and the coefficient of determination R\({}^{2}\). MAE demonstrates the gap between the estimated value and the real value using the absolute error. RMSE assesses the instability of model response regarding variance and sensitivity to large errors. CVRMSE normalizes the forecasted error and provides a unitless metric that evaluates the variability of the errors between real and predicted values. The advantage of using these statistical indicators is to explore various aspects of the presented models. Table 1 presents the calculated values of five statistical performance metrics (MAPE, MAE, RMSE, CVRMSE, and R\({}^{2}\)) for both the Prophet and Holt-Winters models using the equations aforementioned in the methodology section. MAPE is one of the most used tools for measuring the accuracy of models [35]. MAPE is proposed in [4] as a reference range for evaluating the performance of energy consumption forecasting at different horizons. Highly accurate models are considered when the MAPE is less than 10%, as benchmarked in [36]. The highly accurate models were divided into levels into four sublevels for each planning horizon. For example, the long-term forecasting is divided into these sublevels: I (\(\leq\)1.2%), II (1.2-2.8%), III (2.8-4.6%), and IV (4.6-10%) [4]. Accurate values of MAPE were achieved by the [PERSON] and Holt-Winters methods with superiority by the Prophet model. The values of MAPE were 1.75% and 4.17% for Prophet and Holt-Winters models, respectively. Both values indicate an accurate prediction of the values of both models with relatively low MAPE values. Based on the sublevels of accuracy mentioned above and \begin{table}
\begin{tabular}{c c c} \hline \hline
**Criteria** & **Prophet** & **Holt-Winters** \\ \hline MAPE & 1.75\% & 4.17\% \\ MAE & 147.89 & 343.33 \\ RMSE & 205.64 & 475.76 \\ CVRMSE & 7.61\% & 17.59\% \\ R\({}^{2}\) & 0.9942 & 0.9694 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The calculated values of different performance statistical metrics for both models. Figure 6: Enlarged forecast portion for both models along with the 99% confidence interval for the years 2020–2030. in [4], the Prophet model falls within sublevel II, whereas the Holt-Winters model falls within sublevel III accuracy. Figure 7 depicts the coefficient of determination, R\({}^{2}\), and illustrates the variations between the forecasted and real data values for both the Prophet and Holt-Winters models. Both models achieved high values of the coefficient of determination R\({}^{2}\) equal to 0.9942 and 0.9694 for the Prophet and Holt-Winters models, respectively. The MAE of the Prophet model is reported as 147.89, whereas the value of MAE for Holt-Winters model is 343.33, which is approximately more than the double of the Prophet model MAE. In addition, the Prophet model results in a better RMSE value than the Holt-Winters model with a value of 205.64. Moreover, the CVRMSE percentage of the Prophet model is less than the Holt-Winters model with values of 7.61% and 17.59%, respectively. It is clear that the Prophet model has a superior fitting to the real data and has better performance and accuracy than the Holt-Winters model. From the five different statistical metrics, both the Prophet and Holt-Winters models show high accuracy with the superiority of the Prophet model over the Holt-Winters model. Another measure to assess the accuracy of the models is to use the simulated historical forecast (SHF) [22; 37], as depicted in Figure 8. The Prophet model forecasts have a considerably lower prediction error than the Holt-Winters model. The Prophet model's MAPE values are limited within the range between 1.3 and 2.1 approximately, and there is no clear trend of MAPE with the forecast window. The MAPE values of the Holt-Winters model across the different forecasting periods were generally good but higher than the Prophet model.