This paper investigates the sub-synchronous resonance (SSR) phenomenon in a series compensated doubly-fed induction generator (DFIG) based wind farms. A detailed linear state space model of a fixed-series compensated DFIG wind farm is presented for different operating point conditions. The model of the system includes a wind turbine aerodynamics, a 6th order induction generator, a 3rd order two-mass shaft system, a 4th order series compensated transmission line, an 8th order rotor and generator side converter controllers, and 1st order DC link model. The 22nd order system is modeled in Matlab/Simulink, and modal analysis is performed on the modeled DFIG wind farm, and eigenvalues of the system are calculated. The eigenvalue analysis and time-domain simulation results show that SSR can potentially occur in the fixed-series compensated DFIG-based wind farm even at realistic levels of series compensation. The fixed-series capacitor is replaced with a gate-controlled series capacitor (GCSC), which is a series flexible ac transmission system (FACTS) device composed of a pair of gate-commutated switches in parallel with a capacitor that enables one to control impedance and power flow of transmission lines, for series compensation and SSR damping. The root-locus analysis is employed to design a SSR damping controller (SSRDC) for the GCSC. It is shown than a well-designed SSRDC for the GCSC can alleviate the SSR in DFIG-based wind farms. The IEEE first benchmark model adapted with a DFIG-based wind farm is employed as a case study.