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This paper investigates the feasibility of damping multi-mode oscillations by a single FACTS device with multiple damping controllers in a multi-machine power system. The nonlinear dynamic model and the linearized Phillips- Heffron model of a power system installed with a MUPFC (Multi-terminal UPFC) are derived. While MUPFC is a type of FACTS device as an extended UPFC (Unified Power Flow Controller). Based on modal analysis, controllability indices are used to decide respective efficient control loops for weakly damped modes. Simulation result indicates that the multiple auxiliary damping controllers of a single FACTS determined by the proposed method can successfully suppress the multi-mode oscillations in the power system.