Complex-Valued State Matrices for Simple Representation of Large Autonomous Microgrids Supplied by $PQ$ and

This paper focuses on representing the state space model of a microgrid in which power regulated (PQ ) and voltage/frequency regulated (Vf) generation units share a distribution system. The generation units considered in this paper are inverter interfaced. This introduces some interesting modeling problems which are treated in the paper, such as the decoupled cascaded control schemes or the non-negligible grid dynamics. A modeling approach is proposed based on four defined complex vectors. These vectors allow for complex-valued system matrices to be formed in a quite automated way. Moreover, a convenient partition of the system matrices is proposed, which in turn allows fast and easy modifications. Additionally, a multivariable methodology is proposed to simultaneously find the control system gains in an optimal sense. A 69-bus radial system, supplied by 20 generation units, is used to demonstrate how the proposal is of easy implementation to conduct small-signal stability analyses.