Aiming at solving the predicting problems for the power function series, the grey Euler model(GEM(1,1)) with grey action optimization is proposed by introducing the nonhomogeneous Euler equation. The modeling mechanism, parameter estimation and the time response function are studied. The initial point of GEM(1,1) is optimized with the minimum of the square sum of the error as the target. And further studies on three basic forms and three optimized forms of the GEM(1,1) model are also discussed. Researches show that the OSGEM(1,1) model can fit the series of power function characteristics well, which expands the system of gray prediction theory and the application scope of gray prediction theory. Finally, in the case studies, the natural gas consumption of China and the fitting of the power function series data are adopted to test the effectiveness of this model, and the good results obtained show the effectiveness and practicality of the proposed optimization model.