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On the basis of the available gray models, a new fractional gray Bernoulli model (GFGBM (1,1,tα)) is proposed to predict the per capita primary energy consumption (PPEC) of major economies in the world. First, this paper introduces the modeling mechanism and characteristics of the GFGBM (1,1,tα). The new model can be converted to other gray models through parameter changes, so the new model has strong adaptability. Second, the predictive performance of the GFGBM (1,1,tα) is assessed by the four groups of PPEC. The optimal parameters of the model are solved by the moth flame optimization and gray wolf optimization algorithms, and the prediction results of the models are evaluated by two error metrics. The results show that the GFGBM (1,1,tα) is more feasible and effective than the other tested gray models. Third, the GFGBM (1,1,tα) is applied to forecast the PPEC of India, the world, the Organization for Economic Cooperation and Development (OECD) countries, and non-OECD countries over the next 5 years. The forecasting results indicate that the PPEC of the four economies will increase by 5.36 GJ, 42.09 GJ, 5.75 GJ, and 29.22 GJ, respectively, an increase of 51.53%, 55.61%, 3.22%, and 53.41%, respectively.

Estimating per Capita Primary Energy Consumption Using a Novel Fractional Gray Bernoulli Model