Sub-Synchronous Oscillations (SSOs) pose significant challenges to both conventional and modern power grids. For years, the power industry has relied on Fixed Series Compensators (FSCs) for long-distance AC power transmission, but heavily compensated lines are a major cause for SSOs. Most Thyristor Controlled Series Compensator (TCSC) applications use Sub-Synchronous Damping Controllers (SSDCs) to avoid SSR. The TCSC inherently provides some level of damping in the sub synchronous frequency range even without SSDCs. However, the mechanism behind the inherent damping capability of the TCSC and its application of it to mitigate SSR is not well established. Furthermore, there remains uncertainty regarding the extent to which the inherent damping capability can be relied upon to mitigate SSOs and the need for SSDCs. This thesis investigates the role of TCSC in damping SSOs in power systems, mainly considering its inherent damping nature, while accommodating SSDCs when required. A simplified Dynamic Phasor (DP) small signal model of the TCSC is proposed for SSO studies, which accurately represent its inherent damping capability in the sub-synchronous frequency range. The proposed model is used to evaluate the inherent damping capability of the TCSC to mitigate Induction Generator Effect (IGE) and Torsional Interactions (TIs) in conventional power systems and Wind Sub-Synchronous Controller Interactions (W-SSCIs) in systems with Type 3 Wind power plants, through DP based small signal stability analysis. The study reveals that proper selection of TCSC parameters and operation close to its highly non-linear range improves its inherent damping capability. The inherent damping capability of the TCSC is adequate to avoid IGE and W-SSCIs even at low levels of TCSC. However, to mitigate TIs in Conventional Turbine Generator (CTG) systems with low mechanical damping, high levels of TCSC and operation near highly non-linear region are necessary. Consequently, the utilization of SSDCs becomes imperative to ensure effective mitigation of TIs. In networks with multiple series compensated lines and CTGs, there can be many network resonances which can excite IGE and TIs, and replacement of all FSCs with TCSCs is not a feasible solution. This thesis proposes a methodology to utilize TCSCs to address SSO problems in such networks.