Time-delayed fractional grey Bernoulli model with independent fractional orders for fossil energy consumption forecasting

Fossil fuels serve as the primary energy source in the global energy landscape. Through an in-depth understanding and forecasting of fossil fuel consumption, it becomes possible to address energy and environmental challenges more effectively. This study contributes to artificial intelligence by proposing a novel time-delayed fractional grey Bernoulli model with independent fractional orders, which leverages the structural properties of the Bernoulli equation, the cumulative nature of fractional orders, and the driving influence of time-delay term. The model parameters are optimized using the particle swarm optimization algorithm, enhancing its adaptability and accuracy. The engineering application of this model focuses on forecasting fossil fuel consumption, a critical challenge in energy and environmental engineering. Using real datasets from 2000 to 2022, the proposed model is applied to predict the consumption of natural gas, coal and oil in the Middle East and North America, alongside ten benchmark models. The results demonstrate the superior performance of the proposed model, achieving the mean absolute percentage errors of 2.632991%, 5.793513%, 5.432220% and 2.816756% across four case studies, significantly outperforming other models. These findings highlight the potential of the proposed model as a robust and reliable decision-support tool in energy engineering.