In the vision of the future smart grid, the communication is often featured by wide range, massive connect, and low latency, which poses new requirements on the reachable distance and spectral efficiency of wireless communication. In this regard, this paper studies ergodic capacity enhancement by applying successive relay (SR) technology to a non-orthogonal multiple access (NOMA) based Coordinated Direct and Relay transmission (CDRT) system, where a base station (BS) communicates with a near user (NU) directly while communicating with a far user (FU) with the help of a group of relays. We design a novel three-phase CDRT strategy based on SR technology to overcome the half-duplex (HD) constrain without introducing additional noise. The proposed strategy can improve the spectral efficiency while expanding the communication coverage, which to some extent improves the communication quality of the edge users of the smart grid and reduces the communication delay. To analyze the performance of the proposed three-phase CDRT strategy, an exact and closed-form expression for ergodic capacity of the NU, the FU, and the whole system is derived. Finally, the numerical and simulation results validate the analysis results and show that the proposed strategy can improve the ergodic capacity of FU without reducing the capacity scaling of NU.