With the widespread deployment of smart meters, power grids have accumulated vast amounts of raw electricity consumption data. However, data loss remains a challenge due to the complex operational environments of data acquisition equipment. This study addresses the problem of incomplete electricity consumption data by accounting for the influence of Gaussian noise and proposing a robust completion method. First, a electricity consumption data matrix is constructed by reorganizing the sequences of individual users, and the ideal electricity data matrix is approximated using nonnegative matrix factorization (NMF). Second, both the Frobenius norm and the nuclear norm are employed to regularize the Gaussian noise and promote low-rank characteristics of the ideal matrix, thereby formulating an optimization model. Finally, within a block coordinate descent framework, the EM algorithm and a direct updating method are applied alternately to update the matrix factors derived from NMF, enabling accurate and complete data reconstruction. Simulation and experimental results validate the proposed algorithm’s effectiveness and accuracy.