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

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.

• The paper develops a data-driven framework that identifies dynamic droop coefficients from terminal measurements and certifies small-signal frequency stability without requiring internal inverter models.
• The results show that GFM units can satisfy decentralized stability and performance conditions, while conventional SRF-PLL-based GFL droop violates them at higher frequencies because its gain grows and phase approaches 180°.
• The framework converts stability, droop, damping, and inertia requirements into Bode and Nyquist specifications while accounting for dynamic transmission lines and model uncertainty, though its guarantees depend on decoupling and uniform line R/X assumptions.

Overview

This paper develops a data-enabled, decentralized framework for certifying small-signal frequency stability and frequency-control performance of heterogeneous power systems comprising grid-forming (GFM) inverter-based resources (IBRs), grid-following (GFL) IBRs, and synchronous generators (SGs). The central modeling construct is the dynamic droop coefficient $m_{p,n}(s)$, a complex-valued transfer function from active power injections to bus frequency deviations that generalizes the classical steady-state droop constant to the full subsynchronous frequency range. Because it is recovered purely from input-output data at the unit's point of interconnection, the framework requires no knowledge of internal control structure or hardware topology — a property the authors position as directly relevant to interoperability certification and grid-code development.

The work addresses a well-recognized scalability gap: eigenvalue-based and impedance-based small-signal methods require models of the entire interconnected system, whereas the proposed conditions can be verified unit-by-unit against a network abstraction parameterized by only a few quantities ($R/X$ ratio, minimum line inductance $\ell_{\min}$, maximum node degree $e_{\max}$).

Dynamic droop model

Each unit at bus $n$ is represented by the $2\times 2$ transfer matrix

$$\begin{bmatrix}\Delta \omega_n \ \Delta V_n\end{bmatrix} = -M_n(s)\begin{bmatrix}\Delta p_n \ \Delta q_n\end{bmatrix},$$

whose diagonal entries $m_{p,n}(s)$ and $m_{q,n}(s)$ are the dynamic droop coefficients capturing the $P$-$R/X$0 and $R/X$1-$R/X$2 relationships. The steady-state droop constants are recovered as $R/X$3. The phase of $R/X$4 carries a direct stability interpretation: phase within $R/X$5 corresponds to dissipative (positively damped) response over an oscillation cycle, while phase excursions beyond $R/X$6 imply negative damping on average.

The network is modeled with second-order line dynamics whose normalized transfer function $R/X$7 reduces to the DC power flow at low frequencies but captures circuit resonances near nominal frequency. Higher-order line effects (distributed-parameter behavior, frequency-dependent parameters) are handled via additive norm-bounded uncertainty $R/X$8, calibrated from EMT simulations of lumped-, distributed-, and frequency-dependent-parameter line models. The simulations show that the second-order model is adequate for short lines at high voltage, while long lines and extra-high voltage require the uncertainty weight to cover resonances appearing at lower perturbation frequencies.

Two simplifying assumptions underpin the analysis: decoupling of active and reactive power, and a uniform resistance-inductance ratio across all lines. Both are standard for transmission-level studies but are nonetheless restrictive assumptions on which the certificates depend.

Data-driven recovery

The dynamic droop model is identified by connecting the unit under test to a controlled AC voltage source that injects sinusoidal magnitude and frequency perturbations at discrete frequencies $R/X$9. Terminal measurements of voltage phasor, current, and derived power are Fourier-analyzed, and the model is obtained pointwise as $\ell_{\min}$0. Only externally measurable signals (phase angle, voltage magnitude, power) are used, distinguishing this approach from earlier GFM characterization efforts that relied on internal control signals.

Illustrative results from EMT simulations are informative:

Unit	Behavior of $\ell_{\min}$1	Phase behavior
SG (6th-order, governor + AVR + PSS)	DC gain matches 0.05 pu droop; rises then falls with governor/inertia dynamics; rises again past ~2 Hz	Largely dissipative ($\ell_{\min}$2)
GFM droop/dVOC with inner loops	Desired roll-off maintained up to ~60 Hz	Negative damping beyond ~30 Hz, mitigated by low gain
GFM droop without inner loops	Gain rises beyond ~5 Hz (LC filter dominance)	Remains dissipative
GFL SRF-PLL	Increasing gain beyond ~2 Hz (reduced damping)	Negative damping beyond ~20 Hz; incorrect sign (~180°) near 19 Hz

Notably, the GFL unit with delay exhibits sign-inverted droop around 19 Hz, consistent with reported 18–20 Hz oscillation events on the Kaua'i island system (Dong et al., 2023). This is a strong empirical claim linking the proposed metric to documented instability incidents.

Decentralized stability certificate

The main theoretical result extends the scale-free synthesis conditions of Pates and Mallada to networks with dynamic line models. Under the assumption that all poles of $\ell_{\min}$3 lie in the open left half-plane, the interconnected system is asymptotically stable if there exists $\ell_{\min}$4 such that, for every bus,

$\ell_{\min}$5

for all frequencies, where $\ell_{\min}$6 quantifies electrical coupling strength (bounded by $\ell_{\min}$7) and $\ell_{\min}$8 is the unit rating. Geometrically, the Nyquist plot of each unit's interconnection transfer function must lie in a rotated half-plane through $\ell_{\min}$9.

The parameter $e_{\max}$0 trades off gain versus phase requirements. Two limiting cases yield interpretable Bode-plot specifications:

• Low-gain case ($e_{\max}$1): strict phase bounds at low frequency, mixed gain/phase bounds in a mid-band, and a pure gain bound $e_{\max}$2 at high frequency, requiring the unit to dampen the line resonance peak.
• Passive case ($e_{\max}$3): dissipativity is required throughout; the unit must actively compensate the $e_{\max}$4 phase drop of the line dynamics near resonance by increasing its own droop phase while reducing gain.

When line-dynamics uncertainty is included as circles superimposed on the Nyquist plot, a sharp asymmetry emerges: the GFM unit remains certifiable, whereas the GFL unit violates the condition. The mechanism is structural — because $e_{\max}$5 for the GFL unit grows with frequency at roughly the rate of $e_{\max}$6, the uncertainty radius does not shrink at high frequency, leaving negligible robustness margin. This result substantiates the paper's claim that GFM units are significantly less sensitive to unmodeled transmission dynamics than GFL units.

Performance specifications

Beyond stability, the framework translates ancillary-service functions into verifiable Bode-plot constraints: steady-state droop (gain and phase within tolerance of $e_{\max}$7 up to $e_{\max}$8), transient damping (bounded gain with dissipative phase), and an inertia-like specification requiring one-decade gain roll-off per decade of frequency beyond a cut-off $e_{\max}$9, matching the VSM transfer function $n$0 with $n$1. For typical parameters these performance specifications dominate the low-gain stability conditions at low frequency, so compliance with performance implies compliance with the corresponding stability bounds in that band.

Delineating grid-forming from grid-following

A notable analytical result is Proposition 1: for SRF-PLL-based GFL frequency droop, the relative degree of $n$2 is two, so $n$3 and $n$4 as $n$5 for all choices of PLL gains and filter time constants. This formally contradicts both the low-gain and passive stability conditions and the inertia performance specification — a strong claim that no tuning of conventional SRF-PLL droop can satisfy the proposed certificates, with violations occurring within the practically relevant frequency range due to finite PLL bandwidth. Full-order GFL models exhibit still larger violations than the reduced-order analysis suggests.

The numerical examples further show that network circuit dynamics materially affect certification: with a quasi-steady-state line model the GFL interconnection appears passive, but the dynamic line model reveals reduced margins, corroborating prior observations that line dynamics can improve apparent GFL stability margins while GFM units face challenges on low-impedance networks. Contour studies quantify the minimum stabilizing network inductance, showing it scales with transient droop capability $n$6, decreases with inertia $n$7 and resistance ratio $n$8, and is dominated by the transient-droop band rather than the inertial band — consistent with prior results for homogeneous GFM networks.

Limitations and open questions

The authors are explicit about several restrictions. The stability theorem relies on the decoupled active/reactive power assumption and uniform $n$9 ratio; cross-coupling terms $2\times 2$0, $2\times 2$1 appear in the model but not in the certificate. The conditions are sufficient, not necessary — a non-compliant unit may still operate stably in a specific system, though without scalable a-priori guarantees. The framework addresses only small-signal frequency stability; large-signal behavior, limiters, and saturation are outside its scope. Voltage stability conditions are not developed, and validation is confined to EMT simulation — recovery of dynamic droop coefficients from hardware experiments remains open. Finally, the choice of $2\times 2$2, the uncertainty weight calibration, and the treatment of multi-frequency or wideband excitation during identification are left to engineering judgment.

Conclusion

This paper contributes a coherent chain from data-driven input-output identification through decentralized Nyquist-domain stability certification to Bode-domain performance specifications, all expressed in terms of a single interpretable object — the dynamic droop coefficient. Its principal strengths are scalability (unit-level verification against a coarse network abstraction), technology agnosticism, and formal results delineating GFM from GFL capabilities, including the impossibility result for SRF-PLL droop and the robustness asymmetry under line-model uncertainty. The framework offers a concrete mathematical basis for grid-forming certification criteria, contingent on the stated modeling assumptions and extension to hardware validation and voltage stability.