An accurate and consistent arc model under specific scenes is built. It is important to identify a fault arc in a timely and reliable fashion by studying the current signal characteristics of arc small current grounding and processing it based on the measurable electrical volume signal. A detection method for a fault arc in a small current grounding system is proposed. By establishing the fault arc model, the fault arc is accurately identified based on variational mode decomposition and a convolution neural network. First, the improved "cybernetic" arc model is adopted, and a typical 10 kV distribution network simulation model and the grounding "cybernetic" arc model are built on the PSCAD software platform. Secondly, the variational mode decomposition algorithm is used to process the electrical signal in the fault state, and the Intrinsic Mode Function (IMF) of current signals of four groups is obtained. Then, the first set of IMF (IMF1) containing the signal fundamental frequency components is extracted as the input to the Convolutional Neural Network (CNN). Finally, CNN is used to identify the characteristics of IMF1 and correctly identify the normal and arc fault situations. The experimental and simulation results show that the VMD-CNN identification method improves the accuracy of identifying the original current signal and accurately detects the fault arc. This work is supported by the Key Project of National Natural Science Foundation of China (No. 52077001).