Equivalent Circuit Modeling of Grid-Forming Inverters in (Sub)-Transient Time-Frame

TL;DR

This paper introduces a systematic method to model grid-forming inverters as equivalent circuits in the sub-transient time frame, enabling better analysis of their stability and impedance characteristics.

Contribution

It proposes a frequency-domain admittance-based approach to quantify the effective impedance of black-box GFM inverters, validated with standard models and system simulations.

Findings

The equivalent impedance model accurately captures sub-transient responses.

Replacing GFM with the equivalent circuit reproduces stability characteristics.

The method provides a systematic way to analyze GFM inverter stability.

Abstract

The widely accepted definition of grid-forming (GFM) inverter states that it should behave as a (nearly) constant voltage source behind an impedance by maintaining a (nearly) constant internal voltage phasor in the sub-transient to transient time frame. Some system operators further mandate permissible ranges for this effective impedance. However, these specifications do not clearly define the location of the internal voltage source, and no systematic method exists to quantify its effective impedance for a black-box GFM model. To address this, we first compare the transient responses of an ideal voltage source and a GFM to show that an idealistic GFM maintains a (nearly) constant voltage across the filter capacitor, rather than at the inverter switches. Then we propose a systematic method to quantify the effective impedance of a GFM from its black-box model using frequency-domain admittance plots. Using standard PSCAD GFM models developed by NLR (formerly NREL), we demonstrate that the GFM's equivalent impedance model captures the sub-transient response and static voltage stability limit accurately. Further, replacing the GFM with the proposed equivalent circuit model in the modified IEEE-39 bus system is shown to reproduce the small-signal stability characteristics with reasonable accuracy.