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Power system normal form analysis has developed co-efficients and indices in modal coordinates to quantity nonlinear modal Interactions. We study the changes in the coefficients and indices when the power system equations are expressed in different coordinates or units and show that they can be normalized to be invariant to coordinate changes and thus intrinsic to the power system. The results are illustrated on a 4-generator System. An example shows that the coefficients and indices not only detect non-linear interactions but also can become very large near a strong resonance in the system linearization. [ABSTRACT FROM AUTHOR]

Highlights • Power flow modeling of hierarchically-controlled DG units for islanded AC microgrids. • The control includes a primary conventional droop control and a secondary control. • Comparisons against the solutions obtained with PSCAD and Simulink are presented. • The differences between the proposed model against the droop model are presented. • The solutions computed with the proposed models match with the time-domain models. Abstract This paper presents the power flow modeling of droop-controlled distributed generation units with secondary frequency and voltage restoration control for hierarchically controlled islanded microgrids. These models are incorporated in the conventional Newton-Raphson power flow method as a new bus, without the necessity of a slack bus, and include the gains of the control systems that influence the steady-state solution. Two case studies are addressed. In the first case study, comparisons of the proposed models against the steady-state solutions obtained with PSCAD and Simscape Power System of Simulink, where the closed-loop controls are explicitly modeled, are presented. In the second case study, the proposed method is contrasted against the droop-based approach. The results obtained exhibit low computing effort, reliability, and effectiveness of the proposed models since quadratic convergence behavior is maintained independently of the size and topology of the microgrid. Besides, it is demonstrated that hierarchical and droop controllers lead to different solutions, which confirms the necessity of including the hierarchical control in the power flow model. [ABSTRACT FROM AUTHOR]