Most of the approaches developed to solve the supply restoration problem are based on the power flow models. Therefore, the practical implementation of these approaches requires a complete and accurate set of distribution network models, which are not available in many distribution networks. This paper proposes a new measurement-based approach for supply restoration which uses: (1) real-time current measurements gathered by the distribution system control center, and (2) topology of the distribution network, without requiring any other model information. The proposed approach formulates the supply restoration as the robust optimization problem in order to consider uncertainty involved with the used current measurements. The approach is applicable in distribution networks with distributed generators. The robustness and efficiency of the proposed methodology is demonstrated by calculating both a small test network and a 123-node network.