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Recent events have shown that doubly-fed induction generator (DFIG) wind turbines close to series capacitors are prone to sub-synchronous oscillations (SSO). This paper analyzes these sub-synchronous interactions (SSI) and a countermeasure to enable a reliable and secure integration of DFIG-based wind farms into series-compensated transmission systems. A case study based on a real power system is considered in the present work. It has several transmission lines (many of them series compensated), various conventional and wind power plants, and a distributed load. This system allows to study the SSI among various wind farms and multiple sub-synchronous modes, unlike previous studies which considered a single DFIG and one series-compensated line (i.e., one sub-synchronous mode). An analytical investigation and development of fundamental concepts of the SSO are also discussed. A supplementary damping control (SDC) is designed to enhance the operating range and to damp the SSO of these series-compensated wind farms. A detailed analysis is carried out using eigenvalues and non-linear time-domain simulations to evaluate the performance and robustness of the SDC. Besides the parameters usually studied, such as wind speed, series compensation level, and control gains, the effects of conventional synchronous generation, type of load model, and system demand level on the SSI are also analyzed.