In order to damp subsynchronous oscillation（SSO） completely and reliably， it is necessary to study the active suppressing strategies of TCSC for SSO， which can provide additional guarantee besides the damping effect of TCSC's nature characters. Based on the analysis of the principles of attenuating SSO， a TCSC supplementary damping control for SSO is proposed as an active suppression approach. Using phase compensation principle， the proposed method extracts the subsynchronous components through measuring the remote or local signals and provides supplementary electromagnetic torque for electrical damping when integrated with existing TCSC controls. Considering both damping effect and controller reliability， controls based on remote shaft speed deviation are used as the main and those based on local electrical quantities are treated as standby. A test system is derived from a real system for research purpose. Through small signal stability analysis and time domain simulation， the oscillation variation is compared between cases with and without the TCSC supplementary damping control for SSO. The effect of different control inputs and the standby control is also studied. The results show that based on this control principle， the electrical damping increases using whether remote or local input， and the effect of remote shaft speed deviation is better than that of local active power deviation.