This study proposes a robust sub-synchronous damping controller based on fractional-order sliding mode control (FOSMC) method to mitigate sub-synchronous control interaction (SSCI) induced by DFIG-based wind park connected to series-compensated transmissions. Firstly, the nonlinearity of DFIG is cancelled through feedback linearisation. Secondly, FOSMC is designed and applied to the control loop of the DFIG rotor-side converter. During the design process, the extra degree of freedom, provided by the fractional operator, is employed to achieve fast SSCI-damping. Thirdly, a genetic algorithm is used for parameter optimisation with the aim of making system eigenvalues approach the left plane. The effectiveness of the proposed damping controller is evaluated based on the adapted IEEE first benchmark model, non-aggregated wind system model with a realistic configuration, and experimental platform. Simulation and experiment results demonstrate the superior damping performance of FOSMC under different wind speeds, compensation levels, and the number of online wind turbines. Moreover, FOSMC also shows the robustness under parameter uncertainty, control parameter perturbation, and symmetric (three-phase short circuit) fault.