This paper proposes a robust controller to improve power system stability and mitigate subsynchronous interaction (SSI) between doubly-fed induction generator (DFIG)-based wind farms and series compensated transmission lines. A robust stability analysis is first carried out to show the impact of uncertainties on the SSI phenomenon. The uncertainties are mainly due to the changes in the power system impedance (e.g., transmission line outages) and the variations of wind farm operating conditions. Then, using the l-synthesis technique, a robust SSI damping controller is designed and augmented to the DFIG control system to effectively damp the SSI oscillations. The output signals of the supplementary controller are dynamically limited to avoid saturating the converters and to provide DFIG with the desired fault-ride-through (FRT) operation during power system faults. The proposed controller is designed for a realistic test system with multiple series capacitor compensated lines. The frequency of the unstable SSI mode varies over a wide range due to the changes in power system topologies and wind farm operating conditions. The performance of the proposed controller is verified through electromagnetic transient (EMT) simulations using a detailed wind farm model. Simulation results also confirm the grid compliant operation of the DFIG.