arxiv: 2605.05107 · v1 · submitted 2026-05-06 · 📡 eess.SY · cs.SY

Recognition: unknown

Input-Output Specifications and Dynamic Droop Coefficients: Stability and Performance Conditions for Grid-Forming IBRs

Jennifer T. Bui , Dominic Gro{\ss}

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Pith reviewed 2026-05-08 15:55 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords grid-forming invertersdynamic droop coefficientsinput-output specificationsfrequency stabilityinverter-based resourcessmall-signal analysisancillary servicesgrid certification

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The pith

Dynamic droop coefficients from input-output data yield scalable, unit-level stability conditions for grid-forming IBRs.

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

The paper extends the idea of steady-state droop to dynamic droop coefficients that capture the small-signal frequency-power response of both inverter-based resources and synchronous generators. These coefficients are measured directly from terminal input-output data at the point of interconnection, without any need for internal control details. When combined with a lightweight dynamic model of the transmission network that includes line uncertainty, the approach produces frequency stability conditions that can be checked locally at each unit using only a few network parameters. The conditions are translated into practical bounds on the Bode plot of the dynamic droop coefficient, turning basic frequency control services into verifiable requirements and offering a concrete way to assess whether an IBR qualifies as grid-forming.

Core claim

By defining dynamic droop coefficients that describe the small-signal dynamics of IBRs and SGs, the work shows that frequency stability conditions can be obtained from input-output data collected at the unit's point of interconnection. These coefficients, when paired with a lightweight dynamic transmission network model that accounts for line dynamics uncertainty, produce highly scalable stability criteria verifiable at the unit level given only a few key network parameters. The resulting specifications map to Bode-plot bounds for two broad classes of IBR responses, directly translating frequency control ancillary services into testable requirements and supplying insights for certifying grid

What carries the argument

The dynamic droop coefficient, which encodes the small-signal relationship between frequency and active power at the unit terminals and is extracted from input-output data to derive stability bounds.

Load-bearing premise

Input-output data collected at the point of interconnection fully captures the small-signal dynamics needed for stability, and the lightweight network model sufficiently accounts for line dynamics uncertainty.

What would settle it

A full-system simulation in which an IBR's measured dynamic droop coefficient satisfies the derived Bode-plot bounds yet the closed-loop system becomes unstable under the proposed network model, or the reverse case where the bounds are violated but stability holds.

Figures

Figures reproduced from arXiv: 2605.05107 by Dominic Gro{\ss}, Jennifer T. Bui.

**Figure 2.** Figure 2: The gain and phase shift of the frequency deviation view at source ↗

**Figure 1.** Figure 1: Abstraction of an original network (top) through an analytical model view at source ↗

**Figure 3.** Figure 3: Change in instantaneous active power injection view at source ↗

**Figure 5.** Figure 5: Small-signal frequency dynamics of the multi-IBR/multi-SG system. view at source ↗

**Figure 6.** Figure 6: Two-bus system and signal processing for capturing frequency view at source ↗

**Figure 4.** Figure 4: Normalized dynamic line model µ(j2πfp) ( ) for ρ = 0.0817 pu, weight |W(j2πfp)| ( ), and maximum gain of µ∆(j2πfp) ( ). Gω,p(s)Ψ−1 BGp,θ(s)BT 1 s δp(s) ∆p(s) − ∆θ(s) ∆ω(s) view at source ↗

**Figure 7.** Figure 7: Recovered dynamic droop coefficients for GFM droop without inner view at source ↗

**Figure 8.** Figure 8: Recovered dynamics for transmission lines based on lumped-parameter ( view at source ↗

**Figure 9.** Figure 9: Rearranged small-signal frequency dynamics for a system with view at source ↗

**Figure 10.** Figure 10: illustrates the stability condition defined by Theorem 1 along with the bus dynamics of a GFM and GFL IBR, −1 0 1 2 3 4 5 6 7 −2 −1 0 1 γ¯ ψnωp view at source ↗

**Figure 12.** Figure 12: Illustration of the restrictions imposed on the dynamic droop view at source ↗

**Figure 13.** Figure 13: Illustration of the restrictions imposed on the dynamic droop coeffi view at source ↗

**Figure 15.** Figure 15: Minimum network inductance ℓmin in the frequency range of fα < fp ≤ fc, resulting from the settings of fα = 0.6 Hz, ψ = 1, emax,n = 2, ϵl = 0.1µ0, and ϵd = 0.1mp,0. 0.0005 0.0005 0.0005 001 . 0 001 . 0 0.001 0.001 0015 . 0 0015 . 0 0.0015 0.0015 002 . 0 002 . 0 0.002 0.002 003 . 0 0.003 0.02 0.04 0.06 0.08 0.1 0.1 0.2 0.3 0.4 mp [pu] H [sec] 1 2 3 ·10−3 view at source ↗

**Figure 16.** Figure 16: Minimum network inductance ℓmin,n for fc < fp ≤ f ′′, where the unit is required to demonstrate an inertial response. The results are based on fα = 0.6 Hz, ψ = 1, emax,n = 2, ϵl = 0.1µ0, ϵd = 0.1mp,0, and ρ = 0.1 view at source ↗

**Figure 17.** Figure 17: illustrates the dynamic droop coefficients for three IBR controls obtained in Sec. IV-B. It can be seen that both GFM IBRs meet all three performance specifications, while the GFL IBR only meets the steady-state and transient droop specifications. Notably, the GFL unit does not provide significant damping for high-frequency disturbances. Instead, the GFL response results in negative damping, since the pha… view at source ↗

**Figure 18.** Figure 18: Bode plot of hn(j2πfp) for a GFM IBR with inner-loops ( ), GFM IBR without inner-loops ( ), and GFL IBR ( ), with (solid) and without (dashed) network circuit dynamics. C. Impact of line model on certifying stability Finally, we briefly illustrate how our framework can be used to explain prior results that show that considering network circuit dynamics can significantly impact stability of networks of GFM… view at source ↗

read the original abstract

This paper proposes dynamic stability and performance conditions for grid-connected inverter-based resources (IBRs). To this end, we extend the notion of steady-state droop coefficients to dynamic droop coefficients to capture the small-signal dynamics of IBRs and synchronous generators (SGs). Notably, the dynamic droop coefficients can be obtained from input-output data collected at the unit's (e.g., IBR or SG) point of interconnection without requiring prior knowledge of IBR internals or controls structure. To obtain frequency stability conditions, this IBR model is combined with a lightweight dynamic transmission network model that accounts for uncertainty of line dynamics. The resulting stability conditions are highly scalable and, given a few key network parameters, can be verified at the unit level. To make the conditions practical and offer intuitive and illustrative interpretations, we map the frequency stability conditions to bounds on the Bode plot of the dynamic droop coefficient for two broad types of IBR responses. Moreover, our specifications on the dynamic droop coefficient (i) translate basic frequency control ancillary services into verifiable requirements, and (ii) provide insights into the much-debated question of how to certify an IBR as grid-forming (GFM). The results are illustrated using dynamic droop coefficients obtained using detailed simulations of GFM and GFL IBRs as well as SGs.

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

Dynamic droop coefficients pulled from POI input-output data give a scalable path to unit-level stability checks for IBRs, though the lightweight network model needs closer validation on uncertainty bounds.

read the letter

The key point is that this paper turns steady-state droop into a dynamic version derived directly from input-output measurements at the point of interconnection. That step lets them skip internal IBR control details and still produce frequency stability conditions that check out at the unit level using only a handful of network parameters plus a simplified transmission model that folds in line dynamics uncertainty. They then translate those conditions into Bode plot bounds for two common IBR response types, which also maps onto basic frequency ancillary services and offers some concrete guidance on what counts as grid-forming behavior. The simulations with GFM, GFL, and synchronous generator models illustrate the approach in a concrete way. This framing is new enough to stand out from standard droop work and the data-driven extraction plus local verification angle is genuinely practical for large-scale systems. The soft spot sits in the network model itself. The abstract claims it accounts for line dynamics uncertainty, but without explicit error bounds or tests against full-order meshed networks with multiple IBRs, it is not yet clear whether the reduced model preserves all relevant coupling modes or keeps the uncertainty set tight enough for the unit-level checks to remain reliable. The POI data collection step also rests on the assumption that measurements capture the closed-loop dynamics without significant network feedback contamination, which may require more careful experimental design than is shown here. This paper is for power systems researchers and engineers who work on IBR grid integration, stability certification, and ancillary service specifications. Readers who need concrete, verifiable requirements rather than purely theoretical small-signal analysis will get the most from it. It deserves a serious referee because the core method is coherent, the simulations provide initial evidence, and the practical mappings address real industry questions, even if the network abstraction will need tightening in revision.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes dynamic stability and performance conditions for grid-connected inverter-based resources (IBRs) by extending steady-state droop coefficients to dynamic droop coefficients that capture small-signal dynamics of IBRs and synchronous generators (SGs). These coefficients are obtained from input-output data at the point of interconnection (POI) without requiring knowledge of internal controls. The IBR model is combined with a lightweight dynamic transmission network model that accounts for line dynamics uncertainty to derive scalable frequency stability conditions verifiable at the unit level using only a few key network parameters. The conditions are mapped to Bode-plot bounds for two broad types of IBR responses, translating ancillary service requirements into verifiable specifications and offering insights on certifying IBRs as grid-forming (GFM). Results are illustrated via simulations of GFM and GFL IBRs as well as SGs.

Significance. If the central claims hold, the work would be significant for providing a practical, data-driven pathway to stability certification and GFM classification that avoids detailed internal IBR models and full-system simulations. The scalability of unit-level verification and the explicit mapping of stability conditions to Bode bounds and ancillary services address real needs in high-IBR grids. The input-output extraction approach and simulation-based illustrations are strengths that enhance applicability for manufacturers and grid operators.

major comments (3)

[Derivation of stability conditions using the lightweight network model] The section deriving the stability conditions from the lightweight dynamic transmission network model: the claim that this reduced-order model with uncertainty bounds on line dynamics is sufficient for unit-level verification to imply global small-signal stability requires explicit justification. It is unclear whether the abstraction preserves all relevant coupling modes (e.g., in meshed topologies or multi-IBR cases) or whether the uncertainty set is tight enough that local Bode-plot bounds on the dynamic droop coefficient guarantee stability of the full system; if higher-order dynamics or operating-point variations are omitted, the unit-level check could certify stability when the interconnected system is not.
[Input-output data collection and dynamic droop coefficient extraction] The section on extraction of dynamic droop coefficients from POI input-output data: the assumption that measurements at the POI fully capture the closed-loop small-signal map without contamination from network feedback during data collection is load-bearing for the model-free claim. Without explicit excitation protocols, error bounds, or validation against detailed models showing that network effects are negligible or accounted for, the extracted coefficients may not reliably represent the unit's standalone dynamics for subsequent stability analysis.
[Bode-plot bounds and ancillary service mapping] The section mapping stability conditions to Bode-plot bounds and ancillary services: while the translation to verifiable requirements for two IBR response types is conceptually useful, the paper must demonstrate (via the underlying transfer-function derivations) that the proposed bounds are both necessary and sufficient for the claimed frequency stability under the network uncertainty model; otherwise the practical certification insights for GFM IBRs rest on unverified equivalences.

minor comments (2)

The abstract references simulations but provides no quantitative metrics (e.g., stability margins, error norms, or comparison to full-order models); adding such numbers would strengthen the illustration of the claims.
Notation for the dynamic droop coefficient and its Bode representation should be introduced with a clear definition early in the manuscript to aid readability for readers outside the immediate subfield.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed feedback on our manuscript. The comments identify important areas for clarification and strengthening, particularly around the justification of the modeling assumptions and derivations. We address each major comment point-by-point below and indicate the revisions we will make in the next version of the manuscript.

read point-by-point responses

Referee: The section deriving the stability conditions from the lightweight dynamic transmission network model: the claim that this reduced-order model with uncertainty bounds on line dynamics is sufficient for unit-level verification to imply global small-signal stability requires explicit justification. It is unclear whether the abstraction preserves all relevant coupling modes (e.g., in meshed topologies or multi-IBR cases) or whether the uncertainty set is tight enough that local Bode-plot bounds on the dynamic droop coefficient guarantee stability of the full system; if higher-order dynamics or operating-point variations are omitted, the unit-level check could certify stability when the interconnected system is not.

Authors: We agree that the link between the lightweight network model and global stability requires more explicit justification. The model uses bounded uncertainty on line dynamics to derive conservative unit-level conditions via a robust small-gain argument applied to the interconnection at the POI. This is intended to ensure that satisfaction of the local Bode bounds implies stability of the full system under the modeled uncertainties, including equivalent representations for meshed networks. However, to address the concern directly, we will revise the derivation section to include a formal theorem statement with a proof outline, an expanded discussion of how the uncertainty set covers multi-IBR and meshed cases, and explicit acknowledgment of limitations regarding omitted higher-order dynamics or operating-point variations. revision: yes
Referee: The section on extraction of dynamic droop coefficients from POI input-output data: the assumption that measurements at the POI fully capture the closed-loop small-signal map without contamination from network feedback during data collection is load-bearing for the model-free claim. Without explicit excitation protocols, error bounds, or validation against detailed models showing that network effects are negligible or accounted for, the extracted coefficients may not reliably represent the unit's standalone dynamics for subsequent stability analysis.

Authors: This is a fair point regarding the practical applicability of the model-free extraction. The manuscript presents the dynamic droop coefficients as identifiable from POI measurements, with the implicit assumption that data collection can be performed under conditions where network feedback is either negligible (e.g., stiff grid or controlled test setup) or can be de-embedded. To strengthen this, we will add a new subsection detailing the excitation protocols used (including specific input signals and measurement considerations), quantitative error bounds on the identified coefficients, and validation comparisons against detailed internal IBR models to demonstrate when network contamination remains negligible. This will better support the claim that the extracted coefficients reliably represent standalone unit dynamics. revision: yes
Referee: The section mapping stability conditions to Bode-plot bounds and ancillary services: while the translation to verifiable requirements for two IBR response types is conceptually useful, the paper must demonstrate (via the underlying transfer-function derivations) that the proposed bounds are both necessary and sufficient for the claimed frequency stability under the network uncertainty model; otherwise the practical certification insights for GFM IBRs rest on unverified equivalences.

Authors: We acknowledge the need to make the necessity and sufficiency of the Bode bounds fully explicit through the derivations. In the paper, the bounds are obtained by substituting the dynamic droop coefficient into the closed-loop characteristic equation under the lightweight network model and applying frequency-domain stability criteria (e.g., Nyquist/Bode), yielding equivalent conditions for the two response types. To clarify this, we will expand the mapping section with additional step-by-step transfer-function algebra showing the direct equivalence between the stability inequalities and the proposed magnitude/phase bounds. This will also reinforce the ancillary service translations and GFM certification insights as direct consequences of the derived conditions. revision: yes

Circularity Check

0 steps flagged

No significant circularity: data-driven extraction and independent network model keep derivation self-contained

full rationale

The paper derives dynamic droop coefficients directly from external input-output measurements collected at the POI, without reference to internal IBR models or controls. These coefficients are then combined with a separate lightweight dynamic transmission network model (accounting for line dynamics uncertainty) to produce unit-level stability conditions. No step reduces a claimed prediction or stability bound to a fitted parameter by construction, nor does any load-bearing premise rely on self-citation chains or imported uniqueness theorems. The approach is explicitly measurement-based and externally verifiable using a few network parameters, rendering the derivation chain independent of its own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the validity of small-signal linearization, the sufficiency of point-of-interconnection measurements to capture relevant dynamics, and the adequacy of a lightweight network model with uncertainty; no explicit free parameters or new entities are stated in the abstract.

axioms (2)

domain assumption Small-signal linearization is valid for deriving frequency stability conditions
Implicit in extending droop coefficients to dynamic versions and combining with network model.
domain assumption Lightweight dynamic transmission network model captures line dynamics uncertainty sufficiently for unit-level verification
Used to obtain scalable stability conditions from the IBR model.

pith-pipeline@v0.9.0 · 5548 in / 1550 out tokens · 126373 ms · 2026-05-08T15:55:17.746759+00:00 · methodology

discussion (0)

Reference graph

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