A new combined moment invariants is proposed to describe the features of a hydro-generator unit′s axis orbit based on Hu moment invariants and affine moment invariants and also act as the input characteristics of back propagation (BP) neural network. In order to overcome the problems of a slow convergence rate and a tendency to fall into a local minimum in the BP neural network algorithm, improved particle swarm optimization (PSO) is applied to train the weights of the BP neural network, so that the BP has enhanced convergence and precision. Quantitative feature parameters of the axis orbit shapes are solved with the optimization method to help operation personnel provide guidelines for the fault location of a hydro-generator unit. The experimental simulation and application results show that the axis orbit identification method with combined moments is better than that by the Hu moment invariants or affine moment invariants individually. The improved PSO-BP method has a faster rate of convergence and higher recognition precision than the classical BP during the identification of the axis orbit. The proposed method of axis orbit identification has been successfully applied to the imbalanced fault diagnosis of the hydro-generator unit.