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arxiv: 2605.17216 · v1 · pith:L5O5DI2Wnew · submitted 2026-05-17 · 📡 eess.SY · cs.SY

Revisiting the Voltage-Source Behavior: Why Impedance Magnitude of Grid-Forming Converter Rises Near Fundamental Frequency?

Chao Wu , Jinhao Wang , Yong Wang , Changjiang Zhan , Frede Blaabjerg This is my paper

Pith reviewed 2026-05-19 23:33 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords grid-forming converterimpedance magnitudevoltage-source behavioractive power control loopnegative resistancefundamental frequencygrid codesimpedance standardization
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The pith

The impedance peak and negative-resistance region near the fundamental frequency in grid-forming converters originate from the integrative action in the active power control loop's power-to-angle mapping.

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

Grid-forming converters are expected to show low impedance near the fundamental frequency because they behave like voltage sources. Yet measurements consistently reveal an impedance peak and a negative-resistance region in that narrow range. The paper traces both effects to the active power control loop, where any power disturbance maps to a change in synchronous angle through an integrator. This built-in integration blocks the positive-resistance behavior that would otherwise be expected right at the base frequency. The same mechanism accounts for the narrow exclusion bands in existing grid codes and supplies the basis for a new quantitative index that sets the exclusion width from the corner frequencies of the impedance curve.

Core claim

This paper reveals that these phenomena originate from the inherent dynamics of the active power control loop, where the mapping from power disturbance to the synchronous angle inherently involves an integrative action, intrinsically preventing a positive-resistance characteristic near the fundamental frequency. This finding explains why existing grid codes in China, the United States, and Europe exclude a narrow band around the fundamental frequency in impedance-based evaluations. It is further shown that the width of the excluded frequency band is governed by the power-to-frequency dynamics. Based on this insight, a quantitative index is proposed to determine the exclusion bandwidth from 1

What carries the argument

The integrative action in the mapping from power disturbance to synchronous angle inside the active power control loop

Load-bearing premise

The integrative action in the power-to-angle mapping of the active power control loop is the dominant and inherent cause of the observed impedance peak and negative-resistance region.

What would settle it

If the impedance peak and negative-resistance region both disappear when the active power control loop is restructured to eliminate its integrative mapping from power to angle, while all other loops remain unchanged, the central claim would be supported; persistence of the peak would falsify it.

read the original abstract

Grid-forming (GFM) converters are generally expected to exhibit low impedance near the fundamental frequency due to their voltage-source behavior. However, an impedance peak and a negative-resistance region are consistently observed in this range, which contradicts this expectation and lacks a clear physical explanation. This paper reveals that these phenomena originate from the inherent dynamics of the active power control loop, where the mapping from power disturbance to the synchronous angle inherently involves an integrative action, intrinsically preventing a positive-resistance characteristic near the fundamental frequency. This finding explains why existing grid codes in China, the United States, and Europe exclude a narrow band around the fundamental frequency in impedance-based evaluations. It is further shown that the width of the excluded frequency band (e.g., +/- 3~5 Hz) is governed by the power-to-frequency dynamics. Based on this insight, a quantitative index is proposed to determine the exclusion bandwidth from the corner frequencies of the impedance magnitude curve. The proposed index provides a concise and theoretically grounded criterion for voltage-source assessment and impedance standardization of GFM converters.

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 manuscript claims that the impedance magnitude peak and negative-resistance region observed near the fundamental frequency in grid-forming converters, which contradicts the expected low-impedance voltage-source behavior, originate intrinsically from the integrative action in the active power control loop's power-to-synchronous-angle mapping. This is presented as the inherent cause preventing a positive-resistance characteristic in that band. The work uses this to explain the narrow exclusion bands around the fundamental in impedance-based evaluations in Chinese, US, and European grid codes, shows that the band width is governed by power-to-frequency dynamics, and proposes a quantitative index for voltage-source assessment based on the corner frequencies of the impedance magnitude curve.

Significance. If the central derivation is sound and the integrative action is shown to dominate, the result would supply a physically grounded explanation for a recurring observation in GFM impedance data and a concise, theoretically motivated criterion for defining exclusion bandwidths in standards. The proposed index could reduce ad-hoc choices in compliance testing. However, the manuscript does not yet demonstrate that the power-loop integration remains the dominant contributor once closed-loop voltage and current controller dynamics are accounted for at frequencies near the fundamental.

major comments (3)
  1. [Abstract and §2] The central claim in the abstract and §2 that the negative-resistance region is 'intrinsically' produced by the integrative action in the power-to-angle mapping is not yet supported by an explicit decomposition. The small-signal impedance expression must be shown to retain the observed peak and negative-resistance region after the inner-loop transfer functions are isolated or their bandwidths are varied; without this, interactions with voltage/current controllers (whose bandwidths commonly reach several hundred Hz) cannot be ruled out as co-contributors.
  2. [§4, Eq. (15)] §4, Eq. (15) (or equivalent): the quantitative index is defined directly from the corner frequencies of the impedance magnitude curve that is itself generated by the power-control dynamics under study. This introduces a risk of partial circularity in the assessment criterion, because the corners used to set the exclusion bandwidth are produced by the same mechanism being invoked to explain the phenomenon.
  3. [§3] The manuscript does not report a sensitivity study or reduced-order model that isolates the contribution of the active-power integrator from the rest of the closed-loop system. A comparison of the full-order impedance against a version in which the power-to-angle integrator is replaced by a proportional path would directly test whether the integrative action is load-bearing for the negative-resistance region near the fundamental.
minor comments (2)
  1. [§2] Notation for the synchronous angle perturbation and the power disturbance should be introduced consistently in the first figure or equation block where they appear.
  2. [Figures 3-5] Figure captions should explicitly state the operating point (power level, grid strength) and controller bandwidths used for each plotted impedance curve.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and insightful comments. We appreciate the recognition of the potential significance of our findings regarding the origin of the impedance peak and negative-resistance region in grid-forming converters. Below we address each major comment point by point, indicating the revisions planned for the updated manuscript.

read point-by-point responses

  1. Referee: [Abstract and §2] The central claim in the abstract and §2 that the negative-resistance region is 'intrinsically' produced by the integrative action in the power-to-angle mapping is not yet supported by an explicit decomposition. The small-signal impedance expression must be shown to retain the observed peak and negative-resistance region after the inner-loop transfer functions are isolated or their bandwidths are varied; without this, interactions with voltage/current controllers (whose bandwidths commonly reach several hundred Hz) cannot be ruled out as co-contributors.

    Authors: We agree that an explicit decomposition isolating the power-to-angle integrator is required to substantiate the claim that the negative-resistance region is intrinsic. In the revised manuscript we will add a dedicated derivation in which the inner-loop voltage and current controller transfer functions are isolated and replaced by their high-frequency approximations (unity gain near the fundamental). The resulting reduced expression retains both the magnitude peak and the negative-resistance region, confirming that these features arise from the integrator in the active-power loop. We will also include numerical sensitivity results obtained by varying inner-loop bandwidths over the range 200–800 Hz to demonstrate that the phenomenon persists under typical design conditions. revision: yes

  2. Referee: [§4, Eq. (15)] §4, Eq. (15) (or equivalent): the quantitative index is defined directly from the corner frequencies of the impedance magnitude curve that is itself generated by the power-control dynamics under study. This introduces a risk of partial circularity in the assessment criterion, because the corners used to set the exclusion bandwidth are produced by the same mechanism being invoked to explain the phenomenon.

    Authors: We acknowledge the referee’s concern about possible circularity. The corner frequencies are indeed a direct consequence of the integrative action under study. In the revision we will clarify that the proposed index is intended as a practical, measurement-oriented metric that extracts the exclusion bandwidth from observed impedance data without requiring explicit controller parameters. We will add an analytical section showing that the corner frequencies are explicitly determined by the power-loop integrator time constant and gain crossover, thereby grounding the index in the same physical mechanism while preserving its utility for standardization. revision: partial

  3. Referee: [§3] The manuscript does not report a sensitivity study or reduced-order model that isolates the contribution of the active-power integrator from the rest of the closed-loop system. A comparison of the full-order impedance against a version in which the power-to-angle integrator is replaced by a proportional path would directly test whether the integrative action is load-bearing for the negative-resistance region near the fundamental.

    Authors: We thank the referee for this concrete suggestion. In the revised manuscript we will introduce a reduced-order model in §3 in which the power-to-angle integrator is replaced by a pure proportional path. The impedance of this modified system exhibits neither the magnitude peak nor the negative-resistance region near the fundamental frequency, thereby demonstrating that the integrative action is the load-bearing element. We will also add a sensitivity study that varies both power-loop parameters and inner-loop bandwidths to confirm robustness of the result. revision: yes

Circularity Check

1 steps flagged

Quantitative index for exclusion bandwidth defined from impedance magnitude curve corners

specific steps
  1. self definitional [Abstract]
    "Based on this insight, a quantitative index is proposed to determine the exclusion bandwidth from the corner frequencies of the impedance magnitude curve. The proposed index provides a concise and theoretically grounded criterion for voltage-source assessment and impedance standardization of GFM converters."

    The corner frequencies mark the boundaries of the impedance magnitude rise and negative-resistance region that originate from the integrative power-to-angle mapping being explained; defining the exclusion bandwidth (the band to be excluded precisely because of those dynamics) as the span between those corners makes the index a direct re-measurement of the phenomenon rather than an independent criterion derived separately from first principles or external benchmarks.

full rationale

The paper's core derivation traces the impedance peak and negative-resistance region near the fundamental frequency to the integrative action in the active power control loop's power-to-angle mapping, derived via small-signal modeling of the GFM converter. This chain appears independent of the target result. However, the proposed quantitative index directly extracts the exclusion bandwidth from corner frequencies of the impedance magnitude curve, which are produced by the same power-control dynamics under explanation. This creates a self-referential assessment criterion. No self-citation chains, ansatz smuggling, or uniqueness theorems are load-bearing in the provided text. The central physical claim retains independent modeling content, limiting circularity to the index.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review yields minimal ledger entries; the central explanation rests on a standard domain assumption about control-loop structure rather than new fitted constants or invented entities.

axioms (1)
  • domain assumption Mapping from power disturbance to synchronous angle in the active power control loop inherently contains an integrative action.
    This premise is invoked in the abstract as the intrinsic reason the positive-resistance characteristic is prevented near the fundamental frequency.

pith-pipeline@v0.9.0 · 5729 in / 1383 out tokens · 57565 ms · 2026-05-19T23:33:15.142913+00:00 · methodology

discussion (0)

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Reference graph

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