State-Space Representation Model of Phase-Lock Loop Systems for Stability Analysis of Grid-connected Converters

The paper presents the description, analysis, and linearization in space-state form of three distinct phase-lock loop (PLL) systems, which are aimed for grid-following converters in a microgrid. PLL systems play an important role to assure proper phase angle and frequency tracking, which are fundamental to allow the proper operation of grid-connected converters and droop-operation of multiple converters in a microgrid. Due to its non-linear nature, accurate linearization methods must be used to include the dynamics of the PLL into stability analysis of an individual converter or of a cluster of grid-connected converters. The capability of analyzing dynamics by the means of reduced subsystems brings advantages of space-state form linearization when used for complex and non-linear systems such as the PLL. The accuracy of the described space-state linearized model is verified. Simulation results show that the error between the non-linear model and the space-state representation linearized model is below 2.5 %. The modelling presented is a good candidate to be used in analysis of local stability, when concerning an individual grid-following converter, as well for global stability, when concerning a cluster of converters within a microgrid.