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arxiv: 2510.20544 · v2 · submitted 2025-10-23 · 📡 eess.SY · cs.SY

Decentralized Small Gain and Phase Stability Conditions for Grid-Forming Converters: Limitations and Extensions

Diego Cifelli , Adolfo Anta This is my paper

Pith reviewed 2026-05-18 04:49 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords decentralized stabilitygrid-forming converterssmall-gainsmall-phaseloop shapingsectorialitypower system stabilitystability certification
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The pith

Loop shaping transformations resolve non-sectoriality at low frequencies for decentralized stability analysis of grid-forming converters.

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

The paper establishes that loop shaping transformations can reformulate converter and network models in new coordinate frames to satisfy the sectoriality condition required by mixed gain-phase stability criteria. This is necessary because grid-forming converters typically violate sectoriality at low frequencies, limiting the use of decentralized small-gain and small-phase conditions. If successful, the transformations reduce conservativeness and enable scalable stability certification without needing exhaustive full-system simulations. The approach is illustrated first with an infinite bus system and then with the IEEE 14-bus network to demonstrate practicality.

Core claim

The central discovery is that by applying loop shaping transformations, the models of the converter and the network are rewritten in alternative coordinates where the sectoriality assumption holds, even though it does not in the original coordinates at low frequencies. These transformations maintain the stability properties of the original interconnection, allowing the application of decentralized small-gain and small-phase conditions that were previously inapplicable to grid-forming converters.

What carries the argument

loop shaping transformations applied to converter and network models to create alternative coordinate frames where sectoriality holds

If this is right

  • Decentralized stability criteria can now be used for grid-forming converters that previously failed the sectoriality test at low frequencies.
  • The stability analysis becomes less conservative while remaining decentralized and scalable.
  • The method works for both simple infinite bus setups and larger networks such as the IEEE 14 bus system.
  • This provides a practical way to certify stability in power grids with increasing shares of converter-based resources.

Where Pith is reading between the lines

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

  • The transformations could potentially be generalized to other converter control strategies beyond those analyzed in the paper.
  • Similar coordinate changes might help in stability analysis of interconnected systems in other engineering domains.
  • Future work could test the method on real-time hardware-in-the-loop simulations for dynamic grid conditions.
  • Integrating these certificates with existing power system tools could accelerate adoption for grid planning.

Load-bearing premise

The loop shaping transformations preserve the stability properties of the original converter and network interconnection.

What would settle it

Observing whether a grid-forming converter connected to an infinite bus remains stable exactly when the transformed small-phase condition is satisfied but the original one is not, particularly at low frequencies.

Figures

Figures reproduced from arXiv: 2510.20544 by Adolfo Anta, Diego Cifelli.

Figure 1
Figure 1. Figure 1: a. The following theorem gives a sufficient condition for the stability of the resulting closed-loop system [14]. H1(s) H2(s) w1 w2 u1 y1 u2 − y2 (a) Generic Feedback System YC (s) Y −1 net (s) w1 w2 u1 y1 u2 − y2 (b) Admittance Feedback [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Mixed small gain–phase condition for a GFM converter. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Small-signal model of a generic converter with syn [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: Small phase condition for a GFM converter in Power [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Small-phase bounds without and with virtual admit [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Small phase condition of a GFM converter with [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: GFM Infinite Bus - Stable vs Unstable. 10−3 10−2 10−1 100 101 102 103 104 105 −50 0 50 Frequency (Hz) Gain (dB) J˜C,2 J˜C,3 J˜C,6 J˜C,8 J˜ net 10−3 10−2 10−1 100 101 102 103 104 105 -2π -π 0 π 2π Frequency (Hz) Phase (rad) J˜C,2 J˜C,3 J˜C,6 J˜C,8 J˜−1 net [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: IEEE14 Bus system to all converters for the transformation in (17). This shows that, even when the exact virtual admittance is unknown, it is still possible to use a reasonable estimate to obtain adequate phase bounds and thus an effective stability validation with the proposed approach. We intentionally excluded GFL converters. Although the concepts presented are general and applicable to different contro… view at source ↗
Figure 10
Figure 10. Figure 10: GFM control Structure [15] Y. Gu, Y. Li, Y. Zhu, and T. C. Green, “Impedance-Based Whole-System Modeling for a Composite Grid via Embedding of Frame Dynamics,” IEEE Transactions on Power Systems, vol. 36, no. 1, pp. 336–345, Jan. 2021. [16] K. Dey and A. M. Kulkarni, “Analysis of the Passivity Characteristics of Synchronous Generators and Converter-Interfaced Systems for Grid Interaction Studies,” Interna… view at source ↗
read the original abstract

The increasing share of converter based resources in power systems calls for scalable methods to analyse stability without relying on exhaustive system wide simulations. Decentralized small gain and small-phase criteria have recently been proposed for this purpose, but their applicability to grid forming converters is severely limited by the sectoriality assumption, which is not typically satisfied at low frequencies. This work revisits and extends mixed gain phase conditions by introducing loop shaping transformations that reformulate converter and network models in alternative coordinate frames. The proposed approach resolves intrinsic non sectoriality at low frequencies and reduces conservativeness, thereby improving the applicability of decentralized stability certification. Analytical results are illustrated using an infinite bus system first and then extended to the IEEE 14 bus network, demonstrating the practicality and scalability of the method. These findings provide a pathway toward less conservative and more widely applicable decentralized stability certificates in power grids.

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

2 major / 2 minor

Summary. The paper claims that decentralized small-gain and small-phase stability criteria for grid-forming converters are limited by the sectoriality assumption at low frequencies, and proposes loop-shaping transformations that reformulate converter and network models in alternative coordinates to resolve non-sectoriality, reduce conservativeness, and enable scalable decentralized stability certificates. The approach is illustrated analytically on an infinite-bus system and then on the IEEE 14-bus network.

Significance. If the transformations are shown to preserve closed-loop stability equivalence without introducing unstable hidden dynamics, the work would meaningfully extend the applicability of decentralized small-gain/phase methods to realistic grid-forming converter models, providing a scalable alternative to exhaustive system-wide simulations in high-renewable power systems.

major comments (2)
  1. [Abstract and Section on loop-shaping transformations] The central claim requires that the proposed loop-shaping transformations preserve the stability properties of the original converter-network interconnection. However, the manuscript does not explicitly demonstrate that the frequency-dependent coordinate changes are proper, invertible, introduce no unstable hidden modes, and leave the number of right-half-plane poles and Nyquist encirclements unchanged (see the abstract and the application to the infinite-bus and IEEE 14-bus cases).
  2. [IEEE 14-bus case study] The soundness of the stability conclusions on the test systems rests on the transformations making the sectoriality condition hold while leaving the original interconnection dynamics equivalent; without explicit verification of this equivalence (e.g., via pole-zero analysis or Nyquist plots before/after transformation), the reduction in conservativeness cannot be confirmed as arising solely from the reformulation rather than from altered dynamics.
minor comments (2)
  1. [Abstract] The abstract states that analytical results are illustrated on the infinite-bus system and extended to the IEEE 14-bus network, but quantitative metrics (e.g., stability margins or conservativeness reduction percentages) are not reported, making it difficult to assess the practical improvement.
  2. [Method section] Notation for the transformed coordinates and the specific definitions of the loop-shaping filters should be introduced earlier and used consistently to improve readability of the derivations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough and constructive review. The comments highlight important aspects of rigor regarding stability equivalence under the proposed transformations. We address each major comment below and will incorporate clarifications and additional verification in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract and Section on loop-shaping transformations] The central claim requires that the proposed loop-shaping transformations preserve the stability properties of the original converter-network interconnection. However, the manuscript does not explicitly demonstrate that the frequency-dependent coordinate changes are proper, invertible, introduce no unstable hidden modes, and leave the number of right-half-plane poles and Nyquist encirclements unchanged (see the abstract and the application to the infinite-bus and IEEE 14-bus cases).

    Authors: We agree that an explicit demonstration is required to support the central claim. The loop-shaping transformations are constructed as biproper, stable, and minimum-phase mappings to ensure invertibility without introducing unstable hidden dynamics. In the revised manuscript we will add a formal proposition in the loop-shaping section proving that the transformations preserve the number of right-half-plane poles and the net number of Nyquist encirclements of the original interconnection. This will be supported by explicit pole-zero analysis for the infinite-bus example. revision: yes

  2. Referee: [IEEE 14-bus case study] The soundness of the stability conclusions on the test systems rests on the transformations making the sectoriality condition hold while leaving the original interconnection dynamics equivalent; without explicit verification of this equivalence (e.g., via pole-zero analysis or Nyquist plots before/after transformation), the reduction in conservativeness cannot be confirmed as arising solely from the reformulation rather than from altered dynamics.

    Authors: We accept this point. To confirm that the reported reduction in conservativeness arises from the reformulation and not from changed dynamics, the revised IEEE 14-bus section will include side-by-side Nyquist plots and pole-location comparisons of the interconnection before and after the loop-shaping transformations. These plots will verify that the closed-loop stability properties remain equivalent while the sectoriality condition is satisfied in the transformed coordinates. revision: yes

Circularity Check

0 steps flagged

No circularity: transformations and stability claims grounded in independent test cases

full rationale

The paper introduces loop-shaping coordinate transformations to address non-sectoriality in grid-forming converter models and applies decentralized small-gain/phase criteria in the new coordinates. The abstract and described results rely on explicit analytical derivations followed by validation on an infinite-bus system and the IEEE 14-bus network. No equations or claims reduce a prediction to a fitted parameter by construction, no load-bearing premise rests solely on self-citation, and no uniqueness theorem is imported from prior author work. The stability-preservation property of the transformations is presented as a modeling step whose consequences are checked against the concrete test systems rather than assumed tautologically. This constitutes a self-contained derivation chain against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the domain assumption that sectoriality fails at low frequencies for typical grid-forming converter models and on the ad-hoc proposal that loop shaping transformations can be chosen to restore it without changing stability conclusions.

axioms (2)
  • domain assumption Sectoriality assumption is not typically satisfied at low frequencies for grid-forming converters.
    Explicitly stated in the abstract as the core limitation of existing decentralized criteria.
  • ad hoc to paper Loop shaping transformations can be applied to reformulate models while preserving interconnection stability properties.
    This is the key premise introduced by the paper to enable the extension.

pith-pipeline@v0.9.0 · 5677 in / 1395 out tokens · 44877 ms · 2026-05-18T04:49:34.070752+00:00 · methodology

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Reference graph

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