A Rational Canonical Grey Gompertz Forecasting Model Based on the Hausdorff Fractal Derivative

Accurately forecasting carbon emission trends in China’s power sector is of great significance for achieving the “dual-carbon” goals, optimizing the energy structure, and formulating low-carbon development strategies. This paper aims to develop a forecasting method capable of effectively characterizing the long-term evolutionary patterns and short-term dynamic features of carbon emissions in China’s power sector. However, existing models struggle to simultaneously describe the nonlinear S-shaped growth trend, periodic fluctuations, and long-term memory effects inherent in carbon emission time series. To address these issues, this paper proposes a rational canonical form grey Gompertz forecasting model based on the Hausdorff fractional derivative. By introducing a rational canonical form matrix structure, this model enhances the capability of the grey Gompertz model to represent multi-scale periodic information, and incorporates the Hausdorff fractional derivative to characterize the non-local dynamic features during the time-series evolution, thereby improving the model’s adaptability to complex nonlinear carbon emission sequences. Taking the quarterly and semi-annual carbon emission data of China’s power sector as the research object, simulation and forecasting experiments under various time scales and sample settings were conducted to verify the effectiveness of the new model, and a comparative analysis was performed against traditional grey models, fractional-order grey models, and statistical forecasting models. The results indicate that the model possesses certain advantages in structural representation and dynamic memory mechanisms, exhibiting high forecasting accuracy and stability across different time scales. The full-sample mean absolute percentage errors (MAPEs) were all below 3%, and the MAPEs of the optimal schemes for quarterly and semi-annual data reached 0.8353% and 0.0491%, respectively. Finally, the model was utilized to effectively forecast the carbon emissions of China’s power sector for the 2026–2027 period.