In order to further improve the fitting prediction accuracy of the grey prediction model with time power term,\ the FPDGM(1,1,N) model with grey action optimization is proposed by introducing the fractional order polynomial. The constructed model can generate a time response sequence closer to the general characteristics, so as to obtain a grey prediction model with higher fitting prediction accuracy. The modeling mechanism, parameter estimation and recursive time response formula of the model are studied, and the properties of the model under several special values of model parameters are discussed. The results show that the DGM(1,1) model, the NDGM(1,1) model and the DGM(1,1,N) model are all special forms of the FPDGM(1,1,N) model. Therefore, this model formally unifies the existing grey model with time power term and enlarges the application scope of grey prediction theory. Finally, experiments show that the proposed model has better fitting and prediction accuracy,\ which verifies the validity and applicability of the proposed model.