Microgrids are a special kind of power systems that offer an attractive alternative for sustainable power supply. However, their power electronics based operation still imposes several challenges from a control point of view. This feature usually has been simplified under two main conditions, namely: it is assumed the existence of power converters controllers that decouples the command of the amplitude and frequency of their output voltage and it is assumed that the network operates under a sinusoidal regime. In this paper, exploiting the Hamiltonian structure exhibited by the network, a distributed control scheme is developed for a Mesh Microgrid including the dynamic of the power converters and without assuming a steady state operation. This stabilization result is attained guaranteeing the dispatch of prescribed active and reactive power values.