Analytical Prediction of Voltage Collapse in Current-Limited Grid-Forming Inverters

The limited overcurrent capability of grid-forming (GFM) inverters necessitates effective current limiting during large disturbances. This study presents an analytical framework to predict the boundary voltages at which a circular current limiter (CCL) is activated, assess the persistence of operating equilibrium as a saturated stable equilibrium point (satSEP), and pinpoint the voltage at which this equilibrium is lost. By formulating the CCL-based GFM inverter as a piecewise-smooth system comprising normal-control and current-limited modes, we interpret limiter activation as a boundary-equilibrium bifurcation (BEB). The paper introduces a continuation formulation that transitions to a reduced current-limited model post CCL activation, effectively circumventing rank deficiency linked to frozen integrator states. Additionally, by incorporating an equivalent circuit that includes the filter capacitor, closed-form expressions are derived for the upper and lower boundary voltages at which the CCL is activated. Results indicate that a positive lower-boundary slope signifies the persistence of a satSEP in current-limited mode, while a non-positive slope suggests a non-smooth fold and subsequent equilibrium loss at the BEB. Validation through power-angle analysis and time-domain simulations demonstrates that CCL activation can either instantaneously cause equilibrium loss or permit satSEP persistence until an eventual saddle-node loss.