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

TL;DR

The paper addresses the ambiguity in where a grid-forming inverter's internal voltage source (IVS) resides and how to quantify its effective impedance. It introduces a frequency-domain method that fits the admittance $Y_{qd}$ of a black-box GFM model to a Thevenin equivalent with impedance $Z_{th}=R_{th}+j\,2\pi f L_{th}$ across the 5–100 Hz band, effectively yielding $Z_{Eff}=R_{Eff}+jX_{Eff}$. The method is validated on ideal voltage sources, a synchronous machine, and NREL PSCAD GFM models, showing accurate replication of sub-transient responses and static voltage-stability limits. This approach enables rapid assessment of GFM-grid interactions and compliance with voltage-support criteria without resorting to computationally intensive EMT simulations.

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 NREL, we demonstrate that the GFM's equivalent impedance model captures the sub-transient response and static voltage stability limit reasonably accurately.

Figures (13)

• Figure 1: (a) A generic GFM plant connected to POI, (b) Ideal voltage source behind a series impedance ($Z_{Id}$) at POI.
• Figure 2: Incremental voltage change at ST and VCP of GFM.
• Figure 4: Varying $Z_{Filter}$: (a) Case III - Idealistic GFM, (b) Case IV - Realistic GFM.
• Figure 5: Equivalent GFM plant (a) without virtual impedance, (b) with virtual $Z_{Control}$.
• Figure 6: $Y_{qd}$ admittance plot for IDVS with proposed default values at HV terminals: $X_{Eff} = 0.48$ p.u., $X_{Eff}/R_{Eff} = 10$