The fast-response feature from a superconducting magnetic energy storage (SMES) device is favored for suppressing instantaneous voltage and power fluctuations, but the SMES coil is much more expensive than a conventional battery energy storage device. In order to improve the energy utilization rate and reduce the energy storage cost under multiple-line power distribution conditions, this paper investigates a new interline DC dynamic voltage restorer (IDC-DVR) scheme with one SMES coil shared among multiple compensating circuits. In this new concept, an improved current-voltage (I/V) chopper assembly, which has a series of input/output power ports, is introduced to connect the single SMES coil with multiple power lines, and thereby satisfy the independent energy exchange requirements of any line to be compensated. Specifically, if two or more power lines have simultaneous compensating demands, the SMES coil can be selectively controlled to compensate the preferable line according to the priority order of the line. The feasibility of the proposed scheme is technically verified to maintain the transient voltage stability in multiple-line voltage swell and sag cases caused by either output voltage fluctuations from external power sources or power demand fluctuations from local sensitive loads. The simulation results provide a technical basis to develop a cost-effective SMES-based IDC-DVR for use in various DC distribution networks.