In this paper, a comprehensive model in Hamiltonian form of a Microgrid (MG) composed of heterogeneous components, i.e. wind turbine generator, battery storage and local loads together with their power conversion units, is developed. The proposed model analytically captures the energy conversion capabilities of different sustainable energy sources. Based on this model description, novel primary (nonlinear PI) and secondary controllers (receding horizon) are proposed that ensure boundedness of the currents injected by each energy source and optimal power management operation of the entire MG. Furthermore, closed-loop stability analysis is rigorously proven for both primary and secondary control loops taking into account the accurate Hamiltonian description of the whole MG that includes the energy conversion characteristics. Detailed simulation results of the entire MG connected to a weak grid and operating in islanded mode are provided to validate the proposed model, the control design and the stability analysis under various scenarios.