User classification is an important method for energy consumption analysis, and the wide application of smart meters provides a large number of available data for user classification. To improve the accuracy of user classification and the extraction ability of energy consumption features, this study proposes a graph convolutional network (GCN) of self-learned edge weights for user classification. It converts the original energy consumption data into a graph through a special initialization layer with attention mechanisms and extracts energy consumption features from the generated graph. Then, the proposed network outputs the user classes according to the learning features of the graph. Through comparative experiments on a real energy consumption dataset, it is proven that the feature extraction of the proposed method is more intuitive and clear, and the classification performance of the proposed method is better than the existing methods.