Frequency-Domain Compliance Assessment of Grid-Forming Devices

Ambuj Gupta; Balarko Chaudhuri; Mark O'Malley; Muhammad Sharjeel Javaid

arxiv: 2605.02131 · v1 · submitted 2026-05-04 · 📡 eess.SY · cs.SY

Frequency-Domain Compliance Assessment of Grid-Forming Devices

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

Pith reviewed 2026-05-09 16:46 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords grid-forming invertersvoltage source behind impedancefrequency-domain complianceBode plotsJacobian elementssmall-signal stabilitypower system transients

0 comments

The pith

Frequency-domain Bode plot minima provide an equivalent compliance test for grid-forming inverters acting as voltage sources behind impedance.

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

Grid-forming inverters are required to supply voltage stiffness to the grid by behaving like a voltage source behind an impedance during fast disturbances. Existing compliance checks measure active and reactive power responses to sudden phase or magnitude steps, yet these tests suffer from measurement inaccuracies when transients last less than a cycle. The paper introduces a frequency-domain criterion that sets minimum levels on the Bode plots of the Jacobian elements P(s)/θ(s) and Q(s)/V(s) over a defined frequency band. It demonstrates that satisfying these levels produces the same pass-fail outcome as the time-domain rules and applies the test to standard inverter models while tracing effects on small-signal stability of a 39-bus test system.

Core claim

The voltage-source-behind-impedance characteristic of grid-forming inverters can be verified by requiring the frequency-domain Jacobian elements P(s)/θ(s) and Q(s)/V(s) to remain above specified minimum magnitudes in their Bode plots across the sub-transient frequency range. This frequency-domain rule is shown to be equivalent to the conventional time-domain criterion that evaluates power or current responses to step changes in terminal voltage phase and magnitude. The method is implemented on generic grid-forming inverter models in electromagnetic transient simulation and is used to illustrate how compliance status influences small-signal stability margins in the IEEE 39-bus bulk-power test

What carries the argument

The minimum expected Bode plot magnitudes of the frequency-domain Jacobian elements P(s)/θ(s) and Q(s)/V(s) across the chosen frequency band, which enforce the required stiffness response.

If this is right

Compliance assessment can be performed with frequency-response data that avoids the measurement challenges of sub-cycle transients.
System operators obtain a concrete specification for the expected stiffness behavior that can be checked in simulation or hardware.
The proven equivalence permits existing time-domain standards to be translated directly into frequency-domain requirements.
Small-signal stability studies of large grids can account for individual inverter compliance status without additional time-domain simulations.
Model validation in tools such as PSCAD can incorporate the frequency-domain checks as a standard pass-fail gate.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The frequency-domain formulation may combine naturally with other small-signal analysis methods already expressed in the same domain.
The exact frequency band and minimum levels could be tuned to local grid strength or specific operating points.
Phasor-based monitoring equipment might estimate the same transfer functions in real time to track ongoing compliance.

Load-bearing premise

The chosen frequency range and Bode plot minima in the Jacobian elements are assumed to fully and equivalently represent the voltage-source-behind-impedance behavior for practical grid-forming inverters under all relevant conditions.

What would settle it

A grid-forming inverter that passes the time-domain step-response compliance test but violates one or more of the proposed minimum Bode plot levels (or the reverse) would show the two criteria are not equivalent.

Figures

Figures reproduced from arXiv: 2605.02131 by Ambuj Gupta, Balarko Chaudhuri, Mark O'Malley, Muhammad Sharjeel Javaid.

**Figure 1.** Figure 1: Simulation results demonstrating the effect of view at source ↗

**Figure 2.** Figure 2: Block diagram representing the small-signal dy view at source ↗

**Figure 3.** Figure 3: Effect of varying damping ratio (ζLP F ) on a secondorder LPF with ωn,LP F = 1: (a). step response, and (b). rise time. For a standard second-order underdamped LPF, the widely accepted empirical formula to calculate the rise time (tr,LP F ) is given in (15) [15]. Thus, for the LPF boundary case with ζLP F = 1, the natural frequency ωn,LP F can be calculated from the specified maximum rise time using (15).… view at source ↗

**Figure 4.** Figure 4: Fitting accuracy for the simulated decay time view at source ↗

**Figure 5.** Figure 5: Minimum P(s)/θ(s) magnitude Bode compliance envelope derived from AEMO time-domain criteria view at source ↗

**Figure 6.** Figure 6: Active-power step response of a critically compli view at source ↗

**Figure 8.** Figure 8: GFMI frequency-domain Jacobian scan test setup. view at source ↗

**Figure 10.** Figure 10: Change in reactive power response of compliant view at source ↗

**Figure 9.** Figure 9: ERCOT Q(s)/V (s) compliance assessment of two NLR GFMI designs; one satisfies and one violates the minimum magnitude criterion. To verify the validity of the proposed frequency-domain compliance criteria, the test-setup voltage magnitude (|VT est|) is stepped from 1 p.u. to 0.97 p.u.. The resulting change in reactive power responses of the GFMIs is illustrated in view at source ↗

**Figure 11.** Figure 11: A modified IEEE 39-bus test system [18], with all machines replaced by IBRs and a GFMI at Bus 36. See Table II for the type and location of all IBRs. loop (PLL)-driven oscillations and small-signal stability issues in weak grids [19]. In contrast, a GFMI, owing to its voltage-source or VSBI characteristics, can enhance grid strength and help damp weak-grid oscillations. Thus, to highlight the impact of GF… view at source ↗

**Figure 13.** Figure 13: Critical eigenvalues of the modified IEEE 39-bus view at source ↗

**Figure 12.** Figure 12: ERCOT’s Q(s)/V (s) frequency domain compliance assessment for GFMI-A and GFMI-B for analysing small-signal stability issues in the modified IEEE-39 Bus system. System-level modal analysis of Case-A (with GFMI-A at Bus 36) and Case-B (with GFMI-B at Bus 36) is presented next view at source ↗

**Figure 16.** Figure 16: Observability Index Obs|v| λi of the critical 4.8 Hz oscillatory mode λi in voltage magnitude of non-IBR buses. (a). Case-A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 38 (b). Case-B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 39 38 Low High20logObs |v| λi view at source ↗

**Figure 15.** Figure 15: It is observed that the only generator exhibiting voltage source characteristics in the system, i.e., the GFMI at Bus 36, has the highest participation in the critical 4.8 Hz oscillatory mode, λi . This corroborates the fact that replacing the stronger GFMI-A in Case A with the weaker GFMI-B in Case B leads to instability of the mode and, consequently, the system. Moreover, in Case-B, the GFLs at buses 35… view at source ↗

**Figure 17.** Figure 17: Voltage magnitude observability heatmap indi view at source ↗

read the original abstract

Grid-ForMing Inverters (GFMIs) are expected to provide voltage stiffness to the grid. Explicitly, system operators (SOs) and regulators expect GFMIs to behave like a "voltage source behind impedance (VSBI)" in the (sub)-transient time frame. SOs assess this VSBI characteristic of GFMIs during compliance by defining a pass-fail time-domain criterion. This is done by evaluating the GFMIs' active (or reactive) power/current response to step changes in voltage phase (and magnitude) at its terminals. However, this approach is prone to errors due to poorly defined measurement specifications for very fast (less than a cycle) transients. To address this, this work proposes a compliance criterion for the VSBI characteristic of GFMIs in the frequency domain based on elements of the frequency-domain Jacobian. The compliance criterion is defined in terms of the minimum expected P(s)/\theta(s) and Q(s)/V(s) Bode plot characteristics across a specific frequency range. The equivalence between the time-domain and frequency-domain criteria is established. The proposed method is demonstrated by assessing the compliance of generic NLR (formerly NREL) GFMI models in PSCAD. Furthermore, the impact of GFMI compliance on the small-signal stability of the IEEE 39-bus bulk-power system is demonstrated.

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.

Desk Editor's Note private letter to a colleague

The paper shifts VSBI compliance checks for grid-forming inverters from time-domain step tests to frequency-domain Bode minima on the Jacobian's P/theta and Q/V responses, with demonstrations on models and a test system.

read the letter

The core idea is to replace error-prone time-domain step tests for very fast transients with a frequency-domain criterion: minimum expected gains in the Bode plots of the frequency-domain Jacobian elements over a chosen band. They establish equivalence between the two and apply it to generic NLR models in PSCAD, then show downstream effects on small-signal stability in the IEEE 39-bus system. That is the actual new piece and the part that works cleanly for the linearised case they consider. It directly tackles a measurement headache that system operators face as inverter shares grow. The demonstrations are concrete enough to show the method can be implemented and that compliance choices matter at system level. The citation pattern is straightforward and builds on existing Jacobian and frequency-response work without obvious holes. One soft spot is the linear small-signal foundation. The stress-test concern about nonlinear elements (current limiting, saturation, PLL behavior) is real for practical GFMIs; a step can excite trajectories that the Bode thresholds do not capture. The paper validates on its chosen models, but the gap between linear prediction and full nonlinear response is not explored in depth, so the equivalence claim is narrower than it first appears. The specific frequency range is also a tunable choice that needs clear robustness checks. This is for engineers and regulators who set or apply inverter compliance rules and for stability analysts working with high inverter penetration. It is solid enough on its own terms to deserve peer review, with the main referee questions being how far the linear criterion travels to real hardware and whether the frequency band holds across device types.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a frequency-domain compliance criterion for the voltage source behind impedance (VSBI) characteristic of Grid-Forming Inverters (GFMIs), defined via the minimum gains of the P(s)/θ(s) and Q(s)/V(s) elements of the frequency-domain Jacobian over a chosen frequency band. It claims this criterion is equivalent to the conventional time-domain pass/fail test based on active/reactive power or current responses to terminal voltage phase and magnitude steps. The equivalence is established analytically, the method is demonstrated on generic NLR GFMI models in PSCAD, and the effect of compliance on small-signal stability is illustrated for the IEEE 39-bus system.

Significance. If the claimed equivalence is shown to hold under realistic operating conditions, the frequency-domain formulation would provide a practical alternative to time-domain compliance testing that avoids the documented difficulties of sub-cycle transient measurements. The linkage to small-signal stability of a standard test system further indicates potential utility for system operators. The use of publicly referenced generic models supports reproducibility.

major comments (2)

[§4] §4 (Equivalence derivation): The equivalence between the time-domain step-response criterion and the Bode-plot minima is derived under small-signal linearization of the Jacobian. However, practical GFMIs incorporate nonlinear elements (current limiting, saturation, PLL dynamics, virtual-impedance switching) that a step change can excite; the manuscript does not demonstrate that the linear Bode thresholds remain predictive when these nonlinearities are active, which directly affects the central claim that the frequency-domain criterion fully captures VSBI compliance in the sub-transient regime.
[§5] §5 (PSCAD demonstration): The compliance assessment is performed on generic NLR models, yet no quantitative comparison is provided between the frequency-domain predictions and the actual nonlinear time-domain step responses under conditions that trigger current limiting or saturation. Without such a side-by-side error analysis or falsification test, it remains unclear whether the proposed minima thresholds correctly classify compliance when the linear assumption is violated.

minor comments (2)

[Abstract] The abstract states that equivalence is established but supplies no explicit frequency range or threshold values; including these numerical details would improve immediate readability.
[§5] Figure captions in the PSCAD results section could explicitly state the operating point and whether current limiting was engaged during the depicted transients.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive review of our manuscript. The comments raise important points about the scope of the linear equivalence and the need for validation under nonlinear conditions. We address each major comment below and outline the revisions we will make.

read point-by-point responses

Referee: [§4] §4 (Equivalence derivation): The equivalence between the time-domain step-response criterion and the Bode-plot minima is derived under small-signal linearization of the Jacobian. However, practical GFMIs incorporate nonlinear elements (current limiting, saturation, PLL dynamics, virtual-impedance switching) that a step change can excite; the manuscript does not demonstrate that the linear Bode thresholds remain predictive when these nonlinearities are active, which directly affects the central claim that the frequency-domain criterion fully captures VSBI compliance in the sub-transient regime.

Authors: The analytical derivation in §4 is performed under the small-signal linearization of the Jacobian, as is standard for frequency-domain analysis of this type. Conventional time-domain compliance tests likewise rely on small step perturbations precisely to remain within the linear operating region and avoid exciting nonlinearities such as current limiting or saturation. The proposed frequency-domain criterion is therefore intended as an equivalent assessment tool for the same linear VSBI characteristic that the time-domain tests target. We acknowledge that the manuscript does not provide explicit verification of the Bode minima when nonlinear elements become active. In the revised manuscript we will add a new subsection in §4 that explicitly states the small-signal scope of the equivalence, clarifies that the criterion does not replace time-domain testing for large-signal events, and discusses the conditions under which the linear thresholds are expected to remain predictive. revision: yes
Referee: [§5] §5 (PSCAD demonstration): The compliance assessment is performed on generic NLR models, yet no quantitative comparison is provided between the frequency-domain predictions and the actual nonlinear time-domain step responses under conditions that trigger current limiting or saturation. Without such a side-by-side error analysis or falsification test, it remains unclear whether the proposed minima thresholds correctly classify compliance when the linear assumption is violated.

Authors: The demonstrations in §5 employ the publicly available generic NLR GFMI models with step changes sized to keep the responses within the linear regime, consistent with the analytical equivalence derived earlier. No side-by-side quantitative comparison under current-limiting or saturation conditions was included, because the section focuses on validating the frequency-domain formulation against the linear time-domain criterion. We agree that such a comparison would strengthen the presentation. In the revised manuscript we will add new PSCAD results that activate current limiting, present the corresponding time-domain responses, and tabulate the classification outcomes (pass/fail) from both the frequency-domain minima and the nonlinear time-domain tests, including any observed discrepancies. revision: yes

Circularity Check

0 steps flagged

No circularity: frequency-domain criterion derived independently via linear systems equivalence

full rationale

The paper defines the time-domain VSBI compliance criterion from explicit step-response measurements and then derives the frequency-domain version directly from the small-signal Jacobian transfer functions P(s)/θ(s) and Q(s)/V(s). Equivalence is shown by standard Fourier/Laplace relationships between step inputs and Bode characteristics under linearised assumptions, without any fitted parameters, self-referential definitions, or load-bearing self-citations. The central claim therefore rests on independent mathematical structure rather than reducing to its own inputs by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the approach relies on standard small-signal assumptions in power systems and the definition of VSBI behavior as the target compliance model.

free parameters (1)

specific frequency range
The range over which minimum Bode plot characteristics are evaluated is part of the criterion definition but not quantified in the abstract.

axioms (2)

domain assumption Small-signal linearization applies to the sub-transient transients of interest.
Frequency-domain Jacobian analysis assumes small perturbations around an operating point.
domain assumption GFMIs are expected to exhibit VSBI behavior in the sub-transient time frame.
The compliance target is defined by system operators as voltage source behind impedance behavior.

pith-pipeline@v0.9.0 · 5547 in / 1324 out tokens · 49840 ms · 2026-05-09T16:46:37.985735+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages · 1 internal anchor

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

A. Gupta, B. Chaudhuri, and M. O’Malley, “Equivalent circuit modeling of grid-forming inverters in (sub)-transient time- frame,” arXiv preprint arXiv:2602.12202, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

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work page 2023

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Voluntary spec- ification for grid-forming inverters ,

Australian Energy Market Operator (AEMO), “ Voluntary spec- ification for grid-forming inverters ,” AEMO, Tech. Rep., May 2023

work page 2023

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Advanced Grid Support Energy Storage Resource (AGS-ESR) Functional Specification and Test Requirements for the ERCOT Grid, Version 1.0,

Electric Reliability Council of Texas (ERCOT), “ Advanced Grid Support Energy Storage Resource (AGS-ESR) Functional Specification and Test Requirements for the ERCOT Grid, Version 1.0,” Sep. 2024

work page 2024

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Grid- forming battery energy storage capabilities, performance, and simulation test requirements proposal ,

Midcontinent Independent System Operator (MISO), “ Grid- forming battery energy storage capabilities, performance, and simulation test requirements proposal ,” MISO, Tech. Rep., Jul. 2024

work page 2024

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Grid forming capability of power park modules: First interim report on technical requirements ,

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IEEE Power and Energy Society, IEEE Standard for Intercon- nection and Interoperability of Inverter-Based Resources (IBRs) Interconnecting with Associated Transmission Electric Power Systems, IEEE Std. IEEE Std 2800-2022, Apr. 2022

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work page 2019

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Testing the Per- formance of Grid-Forming Resources ,

Energy Systems Integration Group (ESIG), “ Testing the Per- formance of Grid-Forming Resources ,” ESIG, Tech. Rep., Sep. 2025

work page 2025

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A testing framework for grid-forming resources,

S. Shah, W. Yan, P. Koralewicz, V. Gevorgian, D. Ramasubra- manian, R. Wallen, A. Hoke, B. Kroposki, and B. Mather, “A testing framework for grid-forming resources,” in 2023 IEEE Power & Energy Society General Meeting (PESGM), 2023, pp. 1–5

work page 2023

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Methodology for analysing the essential voltage source characteristics of power systems components,

E. A vdiaj, F. J. Cifuentes Garcia, and J. Beerten, “Methodology for analysing the essential voltage source characteristics of power systems components,” TechRxiv Preprint, Jan. 2025

work page 2025

[13] [13]

Open- source PSCAD grid-following and grid-forming inverters and a benchmark for zero-inertia power system simulations,

R. W. Kenyon, A. Sajadi, A. Hoke, and B.-M. Hodge, “Open- source PSCAD grid-following and grid-forming inverters and a benchmark for zero-inertia power system simulations,” in Proc. IEEE Kansas Power and Energy Conf. (KPEC), Apr. 2021, pp. 1–6

work page 2021

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An overview of grid- forming inverter technologies and the readiness of power systems worldwide to deploy the technology ,

Electric Power Research Institute (EPRI), “ An overview of grid- forming inverter technologies and the readiness of power systems worldwide to deploy the technology ,” EPRI, Tech. Rep., Dec. 2024

work page 2024

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N. S. Nise, Control Systems Engineering, 7th ed. Hoboken, NJ: John Wiley & Sons, 2015

work page 2015

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A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 3rd ed. Upper Saddle River, NJ, USA: Prentice Hall, 2009

work page 2009

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UNIFI’s Grid-Forming (GFM) Inverter Reference Design ,

D. Chatterjee, J. Triemstra, C. Macías, W. Cai, K. Tatkare, R. Mallik, and B. Johnson, “ UNIFI’s Grid-Forming (GFM) Inverter Reference Design ,” The University of Texas at Austin, Technical Report NREL/TP-5D00-92994, July 2025

work page 2025

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Benchmark Systems for Small-Signal Stability Analysis and Control ,

IEEE PES Task Force on Benchmark Systems for Stability Con- trols, “ Benchmark Systems for Small-Signal Stability Analysis and Control ,” IEEE Power & Energy Society, Tech. Rep. PES- TR18, Aug. 2015

work page 2015

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Diagnosis and Mitigation of Observed Oscillations in IBR-Dominant Power Systems: A Practical Guide ,

Energy Systems Integration Group (ESIG), “ Diagnosis and Mitigation of Observed Oscillations in IBR-Dominant Power Systems: A Practical Guide ,” ESIG, Technical Report, Aug. 2024

work page 2024

[20] [20]

EMT— RMS modeling trade-off for IBR-driven sub-synchronous oscil- lations,

M. S. Javaid, B. Chaudhuri, F. Teng, and Z. Akhtar, “EMT— RMS modeling trade-off for IBR-driven sub-synchronous oscil- lations,” IEEE Transactions on Power Systems, 2025

work page 2025