A power network is a large-scale and highly nonlinear dynamical complex system with generators and loads interconnected in a network structure. The transient stability of the power system that we study here refers to its ability for bus angles to remain in synchronism. The usual view of a stable power system is in terms of being able to return from the postfault state to a system equilibrium after severe failures or faults occur. In fact, power systems experience instantaneous load and generation fluctuations even in the absence of system faults. This paper reframes the stability definition in terms of continuous synchronous behavior and looks at the stability of power systems as the capability of withstanding these various fluctuations as external disturbances. The new stability definition is characterized by the property that angles stay cohesive with each other and frequencies of generators stay bounded. The main objective is to establish a stability analysis method based on a class of new energy functions. An important stability lemma is proposed for nonlinear systems which later is used to derive the phase cohesiveness and frequency boundedness conditions. Motivated by the recent study of complex networks, coupled phase oscillators, and synchronization of power systems, the paper also derives a purely algebraic condition showing explicitly how the stability of the power network is related to the underlying network topology, system parameters and affected by the disturbances.