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arxiv: 2510.26953 · v3 · pith:C7HQKF7Inew · submitted 2025-10-30 · 📡 eess.SY · cs.SY

Quantifying Grid-Forming Behavior: Bridging Device-Level Dynamics and System-Level Strength

Kehao Zhuang , Huanhai Xin , Verena H\"aberle , Xiuqiang He , Linbin Huang , Florian D\"orfler This is my paper

Pith reviewed 2026-05-22 11:59 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords grid-forming convertersforming indexsystem strengthbus strengthvoltage stiffnesssmall-signal analysispower system stabilityconverter placement
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0 comments X

The pith

A Forming Index measures converter sensitivity to grid voltage changes and formally proves that grid-forming units increase multi-bus system strength.

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

The paper defines a Forming Index at the device level that quantifies how strongly a converter reacts to grid voltage fluctuations, giving a single number for its grid-forming ability instead of comparing control architectures one by one. At the system level it introduces a strength measure that tracks how much voltage and phase angle shift at many buses when current or power is disturbed, then extends the idea to grid strength and bus strength to flag weak spots. A formal proof connects the two scales by showing that converters with high Forming Index raise the overall system strength. The result supplies one set of numbers that can guide converter design, decide where to install them, and judge stability in grids full of power electronics.

Core claim

At the device level the Forming Index is introduced to quantify a converter's grid-forming ability by its sensitivity to grid voltage fluctuations. At the system level a quantitative measure of system strength is defined that captures the stiffness of voltage and phase-angle responses across multiple buses to current or power disturbances; this measure is further specialized into grid strength and bus strength to locate weak areas. The paper then proves that grid-forming converters raise system strength, thereby linking device behavior directly to system-level performance.

What carries the argument

The Forming Index, a metric of converter sensitivity to grid voltage variations, which is shown to raise the multi-bus system strength defined by voltage and phase responses to disturbances.

If this is right

  • Different converter controls can be ranked by Forming Index without enumerating every architecture.
  • Bus strength and grid strength identify weak locations for targeted reinforcement.
  • Optimal placement of grid-forming units follows directly from maximizing the system strength metric.
  • Stability studies gain a single quantitative link between individual converter behavior and whole-system response.

Where Pith is reading between the lines

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

  • The same indices could support real-time monitoring that adjusts converter settings when bus strength drops.
  • Extension to larger disturbances would test whether the small-signal link still holds during faults.
  • Comparison with existing strength measures such as short-circuit ratio could show where the new metric adds information.

Load-bearing premise

Small-signal analysis at the device level is sufficient to quantify grid-forming behavior via sensitivity to voltage fluctuations and to connect it to system-level strength.

What would settle it

Compute the Forming Index and system strength in a laboratory multi-bus testbed, then add or remove grid-forming converters and check whether the predicted rise in voltage stiffness actually appears in measured responses.

Figures

Figures reproduced from arXiv: 2510.26953 by Florian D\"orfler, Huanhai Xin, Kehao Zhuang, Linbin Huang, Verena H\"aberle, Xiuqiang He.

Figure 1
Figure 1. Figure 1: The diagram of a single converter connected to the grid. [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The common control strategies. (a) PLL-based converter with reactive [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: The equivalent circuit of a single converter system [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: The control diagram of a single converter system. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: The F Is of PLL-PQ. (a) ωPLL = 10−50Hz. (b) Lg = 0.1−0.5pu [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The F Is of PLL-PV. (a) ωPLL = 10−50Hz. (b) Lg = 0.1−0.5pu. transient to transient time scales, a GFM converter is expected to maintain F I < 1 over the frequency range from a few Hz up to approximately a few hundred Hz, with F I ≈ 0 desirable in the tens-of-Hz range. Then, we examine four control architectures recognized as GFM, with their F I results shown in Figs. 7 - 10. 1) DC gain and Roll-off behavio… view at source ↗
Figure 7
Figure 7. Figure 7: The F Is of VSG. (a) J = 1 − 5pu, D = 50pu. (b) D = 30 − 70pu. (c) Lv = 0.05 − 0.25pu. (d) Lg = 0.1 − 0.5pu [PITH_FULL_IMAGE:figures/full_fig_p004_7.png] view at source ↗
Figure 10
Figure 10. Figure 10: The F Is of PLL-GFM. (a) Yv = 2 − 10pu. (b) Lg = 0.1 − 0.5pu. TABLE II DIFFERENT METRICS FOR QUALIFYING VOLTAGE SOURCE BEHAVIOR Metrics FI IN [13] FS [12] Definition σ [Sv(jω)] σ [Zde(jω)] [PITH_FULL_IMAGE:figures/full_fig_p005_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The control diagram of a power system. B. The concept of system strength A necessary requirement for a stable power system is to ensure that the multi-bus voltage vectors remain within safe limits under a disturbance, i.e., the system has stiff voltages at all buses. To formalize these requirements, CIGRE [21] and AEMO [22], have introduced the concept of system strength. It is defined as “the ability of … view at source ↗
Figure 12
Figure 12. Figure 12: System structure diagram. in [PITH_FULL_IMAGE:figures/full_fig_p007_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Different control of converter at bus 38 in the IEEE 39-bus system: [PITH_FULL_IMAGE:figures/full_fig_p008_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Time domain response waveforms. V. CASE RESULTS In order to demonstrate the utility of the proposed indices, the IEEE 39 bus system in [16] is used for validation. We connect PLL-PQ converters to bus {30 ∼ 37} and treat bus {39} as an infinite bus. An additional converter with different control strategies in [PITH_FULL_IMAGE:figures/full_fig_p008_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The bus strength of the IEEE 39-bus system. (a) without additional converter. (b) a VSG connected at bus 38. (b) a VSG connected at bus 34. [PITH_FULL_IMAGE:figures/full_fig_p009_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: VSG connected at diffenrent bus in the IEEE 39-bus system: grid [PITH_FULL_IMAGE:figures/full_fig_p009_16.png] view at source ↗
read the original abstract

Grid-forming (GFM) technology is widely regarded as a promising solution for future power systems dominated by power electronics. However, a precise method for quantifying GFM converter behavior and a universally accepted GFM definition remain elusive. Moreover, the impact of GFM on system stability is not precisely quantified, creating a significant disconnect between device and system levels. To address these gaps from a small-signal perspective, at the device level, we introduce a novel metric, the Forming Index (FI) to quantify a converter's response to grid voltage fluctuations. Rather than enumerating various control architectures, the FI provides a metric for the converter's GFM ability by quantifying its sensitivity to grid variations. At the system level, we propose a new quantitative measure of system strength that captures the multi-bus voltage stiffness, which quantifies the voltage and phase angle responses of multiple buses to current or power disturbances. We further extend and define this concept to grid strength and bus strength to identify weak areas within the system. Finally, we bridge the device and system levels by formally proving that GFM converters enhance system strength. Our proposed framework provides a unified benchmark for GFM converter design, optimal placement, and system stability assessment.

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

1 major / 2 minor

Summary. The paper introduces the Forming Index (FI) as a device-level metric to quantify grid-forming (GFM) converter behavior via sensitivity to grid voltage fluctuations under small-signal assumptions. At the system level, it defines a multi-bus voltage stiffness measure of system strength (along with extensions to grid strength and bus strength) that quantifies voltage and angle responses to current/power disturbances. The central claim is a formal proof that GFM converters enhance system strength, thereby bridging device-level dynamics to system-level strength.

Significance. If the proof and metrics are valid, the work supplies quantitative tools for GFM assessment and system-strength evaluation that could support converter design and weak-bus identification. The explicit bridging via a formal small-signal derivation is a potential strength, though its generality depends on the unaddressed limits of the linearization.

major comments (1)
  1. [Bridging device and system levels (proof section)] The formal proof that GFM converters enhance system strength (described in the abstract and the bridging section) is constructed entirely from small-signal linear sensitivities linking the device-level FI to the multi-bus stiffness measure. No explicit bounds on the linear regime are derived, and the manuscript contains no comparison of the stiffness metric against nonlinear time-domain responses under large disturbances, which is the regime where grid-strength quantification is most critical.
minor comments (2)
  1. [Device-level metric definition] Clarify the precise definitions and units of the Forming Index (FI) and the multi-bus stiffness measure to avoid potential ambiguity in how sensitivity to voltage fluctuations is normalized.
  2. [System-level measures] The extension from multi-bus stiffness to 'grid strength' and 'bus strength' for identifying weak areas would benefit from an explicit algorithmic procedure or pseudocode for practical application.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address the major comment below, clarifying the scope of our small-signal analysis while incorporating revisions to better acknowledge its limitations.

read point-by-point responses

  1. Referee: The formal proof that GFM converters enhance system strength (described in the abstract and the bridging section) is constructed entirely from small-signal linear sensitivities linking the device-level FI to the multi-bus stiffness measure. No explicit bounds on the linear regime are derived, and the manuscript contains no comparison of the stiffness metric against nonlinear time-domain responses under large disturbances, which is the regime where grid-strength quantification is most critical.

    Authors: We agree that our formal proof relies on small-signal linear sensitivities, consistent with the manuscript's explicit framing 'from a small-signal perspective' in the abstract and introduction. This approach enables a rigorous mathematical bridge between the device-level Forming Index and the multi-bus voltage stiffness metric. We acknowledge the absence of explicit bounds on the linear regime and the lack of nonlinear time-domain comparisons. In the revised manuscript, we have added a dedicated paragraph in the bridging section discussing the assumptions underlying the linearization and the conditions under which the small-signal approximation holds. We have also included a forward-looking statement in the conclusions noting that validation against large-signal nonlinear responses represents an important avenue for future work, as developing such comparisons would require an extended large-signal framework outside the current scope. revision: partial

Circularity Check

0 steps flagged

No significant circularity; metrics and proof defined independently from small-signal models

full rationale

The paper introduces the Forming Index (FI) at device level via sensitivity to grid voltage fluctuations and a separate multi-bus stiffness metric for system strength based on voltage/angle responses to disturbances. The formal proof bridges them by showing GFM enhancement without reducing either quantity to the other by definition or via fitted parameters. No self-citation chains, uniqueness theorems, or ansatzes are invoked as load-bearing in the abstract or described derivation. The small-signal assumption is stated explicitly rather than smuggled in, and the framework remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

Central claims rest on the small-signal linearization being representative of GFM dynamics and on the new indices providing meaningful quantification without additional empirical grounding shown in the abstract.

axioms (1)
  • domain assumption Small-signal approximation accurately captures the relevant dynamics of grid-forming converters and multi-bus voltage responses.
    Explicitly stated as the perspective for both device-level FI and system-level strength analysis.
invented entities (2)
  • Forming Index (FI) no independent evidence
    purpose: Quantify converter GFM ability via sensitivity to grid voltage fluctuations.
    Newly proposed metric at device level.
  • Multi-bus voltage stiffness measure of system strength no independent evidence
    purpose: Quantify voltage and phase angle responses across multiple buses to current or power disturbances.
    New quantitative measure proposed for system, grid, and bus strength.

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Reference graph

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