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We present an application of the theory of Arnold diffusion to interconnected power systems. Using a Hamiltonian formulation, we show that Arnold diffusion arises on certain energy levels of the swing equations model. The occurrence of Arnold diffusion entails complex nonperiodic dynamics and erratic transfer of energy between the subsystems. Conditions under which Arnold diffusion exists in the dynamics of the swing equations are found by using the vector-Melnikov method. These conditions become analytically explicit in the case when some of the subsystems undergo relatively small oscillations. Perturbation and parameter regions are found for which Arnold diffusion occurs. These regions allow for a class of interesting systems from the point of view of power systems engineering.