  In this paper we study theoretical properties of inverter-based microgrids
controlled via primary and secondary loops. Stability of these microgrids has
been the subject of a number of recent studies. Conventional approaches based
on standard hierarchical control rely on time-scale separation between primary
and secondary control loops to show local stability of equilibria. In this
paper we show that (i) frequency regulation can be ensured without assuming
time-scale separation and, (ii) ultimate boundedness of the trajectories
starting inside a region of the state space can be guaranteed under a condition
on the inverters power injection errors. The trajectory ultimate bound can be
computed by simple iterations of a nonlinear mapping and provides a certificate
of the overall performance of the controlled microgrid.
