1. Introduction

2. Methodology

2.3. Forecasting Models

The development of forecasting models holds paramount importance in anticipating and adapting to the dynamic demand profiles inherent in fuel consumption, particularly for natural gas (NG) in electricity generation [16]. In this research endeavor, we deployed four distinct forecasting methodologies to project fuel consumption for electricity generation in the United States for the upcoming years 2023 and 2024. The models underwent a rigorous training phase spanning an 8-year period from January 2015 to December 2022, followed by meticulous validation using test data from January 2023 to August 2023. To comprehensively assess their performance, we conducted a detailed comparative analysis, evaluating the forecasting models based on their error metrics. The effectiveness of each model was further scrutinized through a meticulous comparison of forecasted data against actual consumption figures.

2.3.1. Forecasting Models for NG Consumption

Figure 10 serves as a visual representation of the forecasted NG consumption for the years 2023 and 2024, employing diverse forecasting methodologies. The benchmark methods, STL decomposition, ETS method, and ARIMA method each contribute to a multifaceted analytical approach, offering a nuanced understanding of the varied strategies employed in forecasting. This rigorous methodology adheres to industry standards and contributes valuable insights to the realm of energy planning and policy formulation.

2.3.1.1. Forecasting using the Benchmark Methods

Benchmark approaches constitute foundational methodologies in time series forecasting, characterized by their simplicity and practicality. One such method is the Mean approach, which employs the average of historical observations to project future values. In contrast, the Naïve approach relies solely on the latest observation for forecasting. Extending the Naïve concept, the Drift method extrapolates future values by establishing a linear trend between the initial and final data points. Additionally, the Seasonal Naïve (SNaïve) method projects future values based on the last observed data point from the corresponding season in the preceding year. These benchmark methods serve as crucial reference points in fuel consumption forecasting, providing intuitive approaches for comparison with more sophisticated models [17]. Notably, among the illustrated benchmark methods (refer to Figure 10a), the Seasonal Naive (SNaïve) technique distinguishes itself by adeptly capturing the majority of fluctuations in fuel consumption.

2.3.1.2. Forecasting using the STL Decomposition Methods.

Decomposition techniques constitute a crucial element in the nuanced analysis of time series data, facilitating the discernment and isolation of pivotal components essential for unveiling trends, seasonal fluctuations, and cyclical patterns. One such sophisticated method is STL, denoting Seasonal and Trend decomposition using Loess. STL employs an intricate additive decomposition approach by iteratively applying the Loess smoother, which entails locally weighted polynomial regression at each data point.

This method stands out for its robustness in handling outliers and its adaptability in addressing seasonal time series characterized by frequencies exceeding one. Unlike methodologies confined to specific temporal resolutions, such as monthly or quarterly intervals, STL demonstrates versatility by accommodating a broader spectrum of seasonal patterns, rendering it particularly well-suited for precise fuel consumption forecasting [18,19]. The forecasting outcome following STL decomposition is illustrated in Figure 10b, attesting to its efficacy in enhancing predictive accuracy.

2.3.1.3. Forecasting using the ETS Methods

The ETS (Error, Trend, Seasonality) forecasting methodology, renowned for its efficacy in time series prediction, is instrumental in modeling and anticipating fuel consumption patterns. This approach dissects time series data into three fundamental components: error, trend, and seasonality. The error term encapsulates stochastic fluctuations, the trend encapsulates long-term directional movements, and seasonality captures recurring patterns at fixed intervals. Diverse ETS model variations, such as ETS(AAA), ETS(AAN), and ETS(MAM), tailor to distinct time series characteristics by combining different error, trend, and seasonality components. Its adaptability to varied time series patterns and capacity to unveil insights into future trends make ETS models pervasive in forecasting applications. Figure 10c illustrates the forecasted natural gas (NG) consumption data using the ETS method. The ETS decomposition plot in Figure 11, derived from the ETS (M, N, M) model—signifying multiplicative error, no trend, and multiplicative seasonality—was selected based on the minimal values of AIC, AICc, and BIC determined through the ETS () function in R. The estimated parameters for exponential smoothing include α = 0.583 and Υ = 0.0001, with a calculated σ2 of 0.0024. Notably, α at 0.583 denotes a moderate emphasis on recent observations in forecasting, and the lower σ2 value suggests heightened stability and predictability. These parameters assume a pivotal role in governing the rate of change for error, trend, and seasonality components, offering flexibility through α and Υ in adjusting level and trend, respectively.

2.3.1.4. Forecasting using the ARIMA Methods

The ARIMA (AutoRegressive Integrated Moving Average) forecasting methodology stands as a widely embraced approach for time series prediction, notably effective in forecasting natural gas (NG) consumption for electricity generation. By seamlessly incorporating autoregressive (AR) and moving average (MA) components, along differencing to attain temporal stationarity, this model proves adept at capturing nuanced patterns inherent in time series data. The pivotal parameters of the ARIMA model encompass the autoregressive component (p), vital for discerning correlations with antecedent values, the integrated component (d) determining the order of differencing requisite for achieving stationarity, and the moving average component (q), effective in accounting for correlations with prior prediction errors. The optimal values for ‘p,’ ‘d,’ and ‘q’ are calculated through optimization techniques such as grid search. Post-training on historical data, the ARIMA model proficiently extrapolates future NG consumption values, adeptly accommodating both transient oscillations and enduring trends within the temporal dataset. This renders ARIMA an invaluable tool for precise and holistic forecasting within the realm of energy consumption. The forecasted natural gas (NG) consumption for electricity generation in 2023 and 2024, derived from the ARIMA forecasting method, is visually presented in Figure 10d, thereby showcasing the model’s predictive prowess in delineating future trends.

2.3.2. Forecasting Models for Coal Consumption

In the prediction of coal consumption for electricity generation in the USA, we applied various forecasting methods, including the benchmark method, STL decomposition, ETS method, and ARIMA method, as outlined in Section 5. The coal consumption values from 2015 to 2022 were used as input data for predicting the values for the years 2023 and 2024. Figure 13 visually represents the forecasted coal consumption values obtained through different methods.

Figure 13. Forecasting coal consumption for the year 2023 and 2024 using different forecasting methods.

Figure 13. Forecasting coal consumption for the year 2023 and 2024 using different forecasting methods.

2.2.3. Forecasting Models for Petroleum Coke Consumption

The forecasting process for petroleum coke consumption in the years 2023 and 2024 followed the same methodology discussed in Section 5. Similar forecasting methods were applied to determine the most effective approach. Figure 15 visually presents the forecasted values for petroleum coke consumption obtained through the benchmark, STL decomposition, ETS, and ARIMA methods.

Figure 15. Forecasting petroleum coke consumption for the year 2023 and 2024.

Figure 15. Forecasting petroleum coke consumption for the year 2023 and 2024.

2.3.4. Forecasting Models for Petroleum Liquids Consumption

Applying similar forecasting methods discussed in Section 5, Figure 17 visually shows the forecasted values for petroleum liquid consumption obtained through the benchmark, STL decomposition, ETS, and ARIMA methods.

Figure 17. Forecasting petroleum liquid consumption for the year 2023 and 2024.

Figure 17. Forecasting petroleum liquid consumption for the year 2023 and 2024.

2.4. Model Comparison in Terms of Errors for the Energy Streams

2.4.1. Natural Gas

In assessing the accuracy of various forecasting methodologies for NG consumption, a comprehensive evaluation was conducted by partitioning the dataset into distinct training set (year 2015-2022) and test set (year 2023-2024). The performance of seven forecasting models was evaluated using six different accuracy measures, as demonstrated in Table 1 and Table 2. The measures, designed to explain biases and precision across different models, encompass metrics such as the ME for bias estimation, MAE for precision measurement, and RMSE as an indicator of precision that penalizes larger errors. These three metrics operate on a scale-dependent basis. Conversely, the MPE and MAPE are articulated in percentage terms, offering a more conducive platform for comparative analysis across diverse consumption levels. Remarkably, the ETS model emerged as the optimal performer, showing the lowest RMSE values in both the training (39237.5) and test (20687) datasets. The consistent excellence of the ETS model is evident in its ability to fit historical data and forecast future values. This makes it the chosen model for natural gas consumption forecasting in both training and test scenarios.

Moreover, from the residual plot of the ETS model as shown in Figure 12, a normal distribution of residuals appears in the histogram plot, signifying a normal distribution of errors. The ACF plot, which identifies any remaining patterns in the residuals, illustrates that only a limited number of lines cross the blue dashed line, indicating statistical significance. This suggests that the model has successfully captured the majority of autocorrelation in the data. Moreover, random fluctuation in the residuals plot indicates that the model is capturing the underlying patterns in the data. In summary, the ETS model is a good fit for forecasting NG consumption as the residuals display randomness, normality, and lack of systematic patterns.

Figure 12. Residual analysis of ETS forecasting method.

Figure 12. Residual analysis of ETS forecasting method.

2.4.2. Coal

Analyzing the performance on both training and test datasets reveals distinct characteristics of the models. While the ETS model boasts a lower RMSE of 3831.801 on the training data (Table 3), the STLF model demonstrates superior generalization on the test data with the lowest RMSE of 5936.203, compared to the ETS model’s RMSE of 7288.344 (Table 4).

This difference in performance suggests that the STLF model outperforms the ETS model when forecasting coal consumption, particularly in terms of its ability to generalize well to unseen data. The choice to prioritize the STLF model is supported by its superior performance on the test data, highlighting its potential for accurate and reliable coal consumption predictions in real-world scenarios.

From Figure 14, The STLF model for coal consumption forecasting is robust, with residuals displaying normality, minimal significant autocorrelation in the ACF plot, and random fluctuation. Overall, the model effectively captures underlying data patterns.

Figure 14. Residual analysis of STLF forecasting method.

Figure 14. Residual analysis of STLF forecasting method.

2.4.3. Coak

From Table 5, the ETS model exhibits the lowest RMSE of 44.60 on the training data, indicating a good fit to the training set. However, the situation changes when assessing the test data, where the snaive model achieves the lowest RMSE of 99.49, outperforming the ETS model with an RMSE of 134.36 (Table 6).

Based on the visual analysis as depicted in Figure 16, the snaïve model for petroleum coke consumption forecasting demonstrates robustness. The residuals exhibit a normal distribution in the histogram, minimal significant autocorrelation in the ACF plot, and random fluctuation.

2.4.4. Petroleum Liquid

From Table 7 and Table 8 ARIMA model exhibits the lowest RMSE on the training data at 1199.579, indicating a robust fit to historical observations. However, on the test data, the mean method surpasses the ARIMA model, achieving the lowest RMSE of 287.34, in contrast to ARIMA’s RMSE of 424.909. Despite the mean method’s superior performance on the test data, we prioritize the ARIMA model for its capability to capture underlying complex patterns, which is particularly advantageous in forecasting petroleum liquids consumption.

The visual analysis as in Figure 18 suggests the robustness of the ARIMA model for petroleum liquids consumption forecasting. Residuals exhibit a normal distribution in the histogram, minimal significant autocorrelation in the ACF plot, and random fluctuation.

3. Result and Discussion

4. Conclusion

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