This paper presents an improved group of energy functions reflecting generator damping effects for multi-machine power systems by using Center of Inertia (COI) formulation as an extension of the previous work. Since rotor angles at the Stable Equilibrium Point (SEP) of post-fault systems are generally calculated in COI, system transient energy can be found without assumption of infinite or slack bus, which is a crucial drawback of the absolute rotor angle frame approach. The developed energy functions have a structure preserving property with which it is very flexible to incorporate various models of power system components, especially various load and generator models. The proposed damping-reflected energy functions are applied to the Potential Energy Boundary Surface (PEBS) method, one of the direct methods. Numerical simulation of WSCC 9-bus shows that conservativeness of the PEBS method can be considerably reduced.