As wind power penetration continues to increase, the sub-synchronous control interaction (SSCI) problem caused by the interaction between doubly fed induction generators (DFIGs) and series-compensated transmission lines has become increasingly prominent, posing a serious threat to power system stability. To address this problem, this research proposes a centralized sub-synchronous oscillation damping controller (CSODC) for wind farms. First, a DFIG impedance model was constructed based on multi-operating-point impedance scanning and a Taylor series expansion, achieving impedance prediction with an error of less than 2% under various power conditions. Subsequently, a CSODC comprising a sub-synchronous damping calculator (SSDC) and a power electronic converter is designed. By optimizing feedback signals, phase shift angles, gain parameters, and filter parameters, dynamic adjustment of controllable impedance in the sub-synchronous frequency band is achieved. Frequency-domain impedance analysis demonstrates that the CSODC significantly enhances the system’s equivalent resistance, reversing it from negative to positive at the resonance frequency point. Time-domain simulations validated the CSODC’s effectiveness in scenarios involving series capacitor switching and wind speed disturbances, demonstrating rapid sub-synchronous current decay. The results confirm that the proposed strategy effectively suppresses sub-synchronous oscillations across multiple scenarios, offering an economical and efficient solution to stability challenges in high-penetration renewable energy grids.