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arxiv: 2602.12202 · v2 · submitted 2026-02-12 · 📡 eess.SY · cs.SY

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

Ambuj Gupta , Balarko Chaudhuri , Mark O'Malley This is my paper

Pith reviewed 2026-05-16 02:28 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords grid-forming invertersequivalent circuit modelingsub-transient time-framefrequency-domain admittancesmall-signal stabilityvoltage stabilityIEEE-39 bus system
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The pith

Grid-forming inverters maintain constant voltage at the filter capacitor, enabling an equivalent impedance model from admittance plots.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows that an ideal grid-forming inverter keeps its voltage nearly constant across the filter capacitor in the sub-transient time frame, rather than at the switches. It introduces a method to calculate the effective impedance using frequency-domain admittance plots from black-box models. Validation with PSCAD simulations confirms that this equivalent circuit matches the sub-transient response and static voltage stability limits. When used in place of the full GFM model in a modified IEEE-39 bus system, it reproduces small-signal stability characteristics with reasonable accuracy. This provides a practical way for system studies without needing detailed internal models.

Core claim

The paper claims that the widely accepted definition of a grid-forming inverter as a constant voltage source behind an impedance is realized by maintaining constant voltage at the filter capacitor, and that frequency-domain admittance plots provide a systematic way to quantify this effective impedance for black-box models, allowing accurate representation of sub-transient dynamics and stability limits.

What carries the argument

The equivalent circuit model derived from frequency-domain admittance plots of the black-box GFM, positioning the constant voltage source behind the impedance at the filter capacitor.

Load-bearing premise

That an idealistic GFM maintains a nearly constant voltage across the filter capacitor and that admittance plots fully quantify the impedance for sub-transient behavior.

What would settle it

A simulation where the equivalent circuit model fails to match the detailed GFM response in sub-transient dynamics or stability metrics in a test system beyond the IEEE-39 bus.

Figures

Figures reproduced from arXiv: 2602.12202 by Ambuj Gupta, Balarko Chaudhuri, Mark O'Malley.

Figure 1
Figure 1. Figure 1: (a) A generic GFM plant connected to POI, (b) [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Incremental voltage change at ST and VCP of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 5
Figure 5. Figure 5: Equivalent GFM plant (a) without virtual [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: Varying ZF ilter: (a) Case III - Idealistic GFM, (b) Case IV - Realistic GFM. Therefore, a GFM (without any virtual impedance) can be represented as a (nearly) constant voltage source VGFM behind effective impedance ZGFM, as shown in [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: Yqd admittance plot for IDVS with proposed default values at HV terminals: XEf f = 0.48 p.u., XEf f /REf f = 10 B. Curve Fitting For a GFM to behave like a Thevenin source as specified by ENTSO-E, it must exhibit similar admittance charac￾teristics in the frequency range of interest. To achieve comparable time-domain behavior in the (sub)-transient time frame (0–10 cycles), the frequency range of interest … view at source ↗
Figure 7
Figure 7. Figure 7: Yqd admittance spectra for full vs equivalent models of (a). an IDVS and (b) a SM 2) GFM: In this subsection, the equivalent impedance model of a GFM model developed in PSCAD by NLR is obtained. The control parameters of this model, de￾scribed in [10], are adjusted within the usual bandwidths [19]. However, these models are treated as black-box representations at their POI. The combined collector system im… view at source ↗
Figure 8
Figure 8. Figure 8: Yqd admittance spectra for full vs equivalent models of NLR GFM Model in PSCAD Yqd admittance spectra for full vs equivalent models of the NLR GFM model are shown in [PITH_FULL_IMAGE:figures/full_fig_p006_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Yqd admittance spectra for full vs equivalent models of GFM with changing voltage control gains (V GFM F ull ) and the Thevenin equivalent (V GFM Eq ) is reduced by 5% from 1.0 p.u. to 0.95 p.u. Holding its IVS voltage constant, the GFM responds by increasing its reactive power output. Since the IVS voltage cannot be measured in a black-box model, ENTSO-E requires that compliance be demonstrated at the POI… view at source ↗
Figure 10
Figure 10. Figure 10: Reactive power response of full GFM model vs [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: A modified IEEE 39-bus test system [11], with all machines replaced by IBRs and a GFM at Bus 36. TABLE V: IBR configurations and dispatch scenario at IBR buses 30–39 in the modified IEEE 39-bus system. Bus 30 31 32 33 34 35 36 37 38 39 Type GFL GFL GFL GFL GFL GFL GFM GFL GFL GFL P (pu) 2.75 5.21 6.50 6.32 5.08 6.50 5.60 5.40 8.30 10.29 Q (pu) 1.88 1.57 1.69 0.94 1.59 4.72 2.73 0.32 0.64 3.56 are consider… view at source ↗
Figure 11
Figure 11. Figure 11: PV curve for full GFM model vs Thevenin equivalent 6) Application II - Small Signal Stability: Here we show that replacing the full GFM model with the pro￾posed equivalent circuit model can reproduce the small￾signal stability characteristics of the modified IEEE 39- bus system [11] with reasonable accuracy. The IEEE 39- bus system is modified by replacing all 10 synchronous generators with IBRs, as shown… view at source ↗
Figure 13
Figure 13. Figure 13: Critical modes of the modified IEEE 39-bus system for Case A (with full GFM model) and Case B (with equivalent GFM model). Hz mode has better damping for case B, while the ∼7.7 Hz mode has better damping for Case A. This indicates that the equivalent GFM model does not guarantee a strictly conservative estimate; however, it provides a reliable and accurate initial assessment and can serve as an effective … view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that grid-forming (GFM) inverters behave as a nearly constant voltage source behind an impedance in the sub-transient to transient timeframe, with the internal voltage located across the filter capacitor rather than the switches. It proposes extracting the effective impedance systematically from black-box frequency-domain admittance plots, validates the resulting equivalent circuit against PSCAD GFM models for sub-transient response and static voltage stability, and shows that substituting the equivalent circuit into a modified IEEE-39 bus system reproduces small-signal stability characteristics with reasonable accuracy.

Significance. If the validation holds, the work supplies a practical, black-box-compatible method to meet system-operator impedance specifications for GFM units and enables reduced-order modeling for stability studies without requiring full control details. The use of standard NLR PSCAD models and the IEEE-39 demonstration are concrete strengths that could improve reproducibility in the field.

major comments (3)
  1. [Abstract] The central validation for sub-transient response (Abstract) is performed only on the isolated NLR PSCAD GFM unit; no large-signal fault or disturbance waveforms are shown when the equivalent circuit replaces the GFM inside the IEEE-39 system. Because frequency-domain admittance extraction is inherently small-signal and linear, the claim that the same circuit “captures the sub-transient response accurately” under control saturation or current limiting remains unsupported.
  2. [Abstract] No quantitative error metrics (e.g., RMS voltage or current deviation, phase-error bounds) are reported for either the isolated-unit sub-transient comparisons or the IEEE-39 small-signal eigenvalue matches. The phrase “reasonable accuracy” is therefore impossible to assess against typical engineering tolerances.
  3. [Abstract] The assumption that an idealistic GFM maintains a nearly constant voltage across the filter capacitor (Abstract) is stated without a supporting derivation or sensitivity study showing how this location is identified from the admittance data versus alternative reference points (e.g., switch terminals).
minor comments (2)
  1. The manuscript should include a brief limitations paragraph discussing the small-signal nature of the admittance extraction and the operating-point range over which the equivalent circuit remains valid.
  2. Figure captions and axis labels for the frequency-domain admittance plots and time-domain waveforms should explicitly state the perturbation amplitude used for admittance extraction and the fault severity used for time-domain checks.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the scope and strengthen the presentation of our work. We address each major comment point by point below and have revised the manuscript to incorporate clarifications and additional quantitative details where feasible.

read point-by-point responses

  1. Referee: [Abstract] The central validation for sub-transient response (Abstract) is performed only on the isolated NLR PSCAD GFM unit; no large-signal fault or disturbance waveforms are shown when the equivalent circuit replaces the GFM inside the IEEE-39 system. Because frequency-domain admittance extraction is inherently small-signal and linear, the claim that the same circuit “captures the sub-transient response accurately” under control saturation or current limiting remains unsupported.

    Authors: We agree that the sub-transient validation is performed exclusively on the isolated GFM unit, as stated in the manuscript. The IEEE-39 study is limited to small-signal stability to demonstrate the utility of the reduced-order model in a multi-machine setting. The equivalent circuit is derived from small-signal admittance data and is not intended to represent behavior under control saturation or current limiting; the abstract does not claim such coverage. We will revise the abstract to explicitly note that sub-transient validation applies to the isolated unit under linear operating conditions and that the IEEE-39 results pertain to small-signal analysis. revision: yes

  2. Referee: [Abstract] No quantitative error metrics (e.g., RMS voltage or current deviation, phase-error bounds) are reported for either the isolated-unit sub-transient comparisons or the IEEE-39 small-signal eigenvalue matches. The phrase “reasonable accuracy” is therefore impossible to assess against typical engineering tolerances.

    Authors: We acknowledge that the absence of quantitative metrics makes the term “reasonable accuracy” difficult to evaluate. In the revised manuscript we will add RMS voltage and current deviation values for the isolated-unit waveform comparisons and report maximum eigenvalue deviation or damping ratio differences for the IEEE-39 small-signal results, allowing direct comparison with typical engineering tolerances. revision: yes

  3. Referee: [Abstract] The assumption that an idealistic GFM maintains a nearly constant voltage across the filter capacitor (Abstract) is stated without a supporting derivation or sensitivity study showing how this location is identified from the admittance data versus alternative reference points (e.g., switch terminals).

    Authors: The manuscript already contains a direct comparison of transient responses between an ideal voltage source and the GFM model that identifies the filter capacitor voltage as the appropriate internal reference. This is corroborated by the admittance plots, which align with a voltage-behind-impedance representation at that location. To address the request for a more explicit derivation and sensitivity analysis, we will add a brief subsection (or appendix) that formalizes the identification procedure and includes a sensitivity comparison of capacitor versus switch-terminal voltage under varying operating points. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper extracts effective impedance directly from frequency-domain admittance plots of the black-box GFM model and then validates the resulting equivalent circuit against time-domain sub-transient responses and small-signal stability metrics in the IEEE-39 system. This extraction is a direct computation from the model's own data rather than a self-definitional loop, a fitted parameter renamed as prediction, or a load-bearing self-citation. The central claim is an approximation whose accuracy is checked against independent simulation benchmarks, keeping the derivation self-contained without reduction to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on one domain assumption about voltage location inside the GFM; no free parameters or new entities are introduced because impedance is read out from existing admittance data.

axioms (1)
  • domain assumption An idealistic GFM maintains a nearly constant voltage across the filter capacitor in the sub-transient to transient time frame
    Stated explicitly after comparing transient responses of ideal voltage source and GFM.

pith-pipeline@v0.9.0 · 5520 in / 1223 out tokens · 74899 ms · 2026-05-16T02:28:41.845590+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Frequency-Domain Compliance Assessment of Grid-Forming Devices

    eess.SY 2026-05 unverdicted novelty 6.0

    A frequency-domain compliance criterion based on minimum expected P(s)/theta(s) and Q(s)/V(s) Bode plots is proposed for grid-forming inverters, shown equivalent to time-domain tests and demonstrated on models with st...

Reference graph

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