Subsynchronous resonance (SSR) caused by the interaction between thermal power generators and series-capacitor-compensated AC lines, has been a significant concern in China's power grids. An energy storage supplementary damping control (SDC) to suppress SSR is proposed in this paper. SDC enables local feedback, and utilizes the coupling between generators and power grid to enhance the overall damping level. Firstly, the target system is modelled and the mechanism and characteristics of its SSR problem are clarified through frequency and time-domain analyses. Then, the supplementary damping control strategy is established based on battery energy storage system (BESS). Its control parameters and adding positions are optimized, and the impact of SDC's control capacity on BESS's normal functions is analyzed. Finally, the effectiveness and feasibility of the proposed SDC are verified through electromagnetic transient simulations with a real-world project, i.e., the Shangdu wind-thermal-energy storage bundling system. The simulation results also show that the proposed SDC can effectively address the SSR issue with reduced control cost and equipment investment cost, serving as a more economical solution.