The application of an optimized fractional order accumulated grey model with variable parameters in the total energy consumption of Jiangsu Province and the consumption level of Chinese residents

Fractional order imply the idea of "in between", the grey model generated by fractional accumulation has better prediction and adaptability than that generated by first-order accumulation. General grey model of the differential equation of the left is a cumulative function derivative of time, in order to improve the adaptability of the model and prediction ability, general fractional order differential equation model is presented. In this paper, on the basis of the derivation of time $t$ extensions to the derivation of $t^u$, added a variable coefficient, and through the integral differential equation and tectonic background value. We establish an optimized fractional order cumulative grey model with variable parameters, i.e., optimized fractional order accumulated grey model (FOGM (1, 1)). By using the Particle swarm optimization (PSO) algorithm, we search for the order and variable parameters of the optimal fractional order. Then we apply the proposed model to predict the total energy consumption of Jiangsu province and the consumption level of Chinese residents. The results indicate that the proposed model has high fitting and prediction accuracy compared to other classical grey prediction models, such as grey model (GM (1, 1)), non-homogeneous grey model (NGM (1, 1)) and fractional order accumulated grey model (FGM (1, 1)). It also validates that the proposed model is a practical and promising model for forecasting the energy consumption as well as the consumption level of Chinese residents.