Abstract: Non-intrusive load monitoring (NILM), as an important means to achieve fine-grained perception of the smart grid users, helps to achieve demand response, high efficiency 'source-grid-load' interaction and optimal energy use, strongly promoting the realization of the "3060 goal". The high-quality measurement is the foundation of the data-driven NILM. However, the data acquisition device failures, the channel congestion and delay may lead to data loss, especially the seriously continuous loss, resulting in a decrease in the accuracy of the non-intrusive load monitoring and disaggregation and affecting some advanced applications such as the user profiling and the demand response. To tackle this problem, a new data recovery approach for the missing NILM measurements based on the regularized low-rank tensor completion is proposed. Breaking through the limitations of the traditional single-dimensional data processing, this approach constructs a third-order observation tensor across the NILM multi-dimensional measurements, and formulates a regularized low-rank tensor completion so as to exploit the internal time series correlation of the measurements and the electrical correlation between the electrical parameters. Considering the calculation of the singular value decomposition of large factor matrices at each iteration usually takes too much time, the calculation of the trace norm based on the Canonical Polyadic factor matrix decomposition is used to reduce the calculation amount and save the calculation time, and mathematically proves the equivalence of the transformation. Finally, the promising experimental results on the NILM public dataset iAWE show the proposed method is able to improve the accuracy of the data recovery, and have a good completion effect under the conditions of having high missing rates or continuous losses. What's more, through the non-intrusive load disaggregation experiments, the proposed method is able to effectively improve the disaggregation accuracy, having a good practical significance for improving the fine-grained perception ability of a smart grid.