Complex Phase Analysis of Power Grid Dynamics

Anna B\"uttner; Anne Riegler; Frank Hellmann; Jakob Niehues

REVIEW 3 major objections 4 minor 19 references

Complex Phase Analysis of Power Grid Dynamics

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read The paper claims that linearizing grid-forming inverter dynamics in complex phase and power variables yields a linear, time-invariant system whose coefficient matrices do not depend on the operating phase or frequency, so the linear model…

desk verdict Sound and genuinely new linearization result, but the phase-symmetry assumption behind Eq. 28 is a real limitation that the paper does not flag, and the applications are borrowed from companion papers. read the letter →

arxiv 2506.22054 v1 pith:6Z6VOU2I submitted 2025-06-27 eess.SY cs.SY

Jakob Niehues , Anna Büttner , Anne Riegler , Frank Hellmann This is my paper · ORCID

classification eess.SYcs.SY

keywords complexfrequencyphasegrid-forminginvertersphase-invariantlinearizationlinear-time-invariantsystemsnormalformsystemidentificationsmall-signalstability

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Power grids with many grid-forming inverters are usually analyzed by linearizing voltage and current dynamics around an operating point, but that linearization depends on an arbitrary reference phase and becomes invalid once the phase drifts. This paper shows that working in complex phase coordinates—the logarithm of the complex voltage, whose real part is log amplitude and imaginary part is phase—removes that dependence: the linearized system is time-invariant and identical for every reference phase and frequency. The consequence is practical: a small-signal model of an inverter remains valid through phase drifts caused by temporary power mismatches, which is exactly the situation in real islanding events. The paper demonstrates this with a hardware-in-the-loop identification experiment and derives small-signal stability conditions for heterogeneous inverter systems.

What carries the argument

The central object is the complex phase $\hat\Theta = \sigma + j\varphi = \ln(\hat v)$, the complex logarithm of the instantaneous voltage phasor, together with the complex frequency $\hat\eta = \dot{\Theta} = \rho + j\omega$ whose real part is the relative amplitude velocity and whose imaginary part is the instantaneous frequency. The load-bearing identity is $\eta(\sigma, \varphi, P, Q) = \eta(\sigma, P, Q)$: the complex frequency is independent of the imaginary part of the complex phase, a consequence of the rotational symmetry condition (11) applied to the normal form description of the device. This identity converts the rotating operating trajectory into a linear motion in the complex phase plane, and makes the linearized matrices $J_\eta$ and $D_\eta$ in equation (29) independent of the reference phase and frequency. The same machinery, with the normal form's input nonlinearity $e=(P-P_s, Q-Q_s, V-V_s)$, supports the Hammerstein-Wiener identification structure and the transfer-matrix stability analysis.

What would settle it

Using a detailed electromagnetic-transient model of a phase-symmetric droop-controlled inverter, compute the linearized matrices $J_\eta$ and $D_\eta$ in equation (29) at two operating points with identical power flow ($P,Q$) and voltage amplitude $\sigma$ but different absolute phase $\varphi_\circ$; any material difference between the two matrices would disprove the claimed phase-independence.

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Extended reading notes

Core claim

Restating the voltage dynamics in terms of the complex phase $\hat\Theta = \sigma + j\varphi = \ln(\hat v)$ and the complex frequency $\hat\eta = \dot{\hat v}/\hat v = \rho + j\omega$, the paper proves that the linearized dynamics take the LTI form of equation (29), with coefficient matrices $J_\eta$ and $D_\eta$ that do not depend on the operating phase $\varphi_\circ$ or the operating frequency $\omega_\circ$. This follows from the rotational symmetry condition in equation (11) together with the normal form assumption that a grid-forming device's complex frequency depends only on rotation-invariant error quantities (active power error, reactive power error, and voltage amplitude deviation). Because the phase is a linear direction in the complex phase plane, linearized phase shifts and nonlinear phase shifts coincide, so the linear model remains valid while the phase drifts. A phase shift leaves both the full and the linearized dynamics invariant, and different linearization points on the same voltage circle give the same linearized dynamics.

Load-bearing premise

The result rests on the assumption that a grid-forming inverter's response depends only on quantities that are unchanged by rotating the whole system—deviations of active power, reactive power, and voltage amplitude from their set points—and not on the absolute phase angle; if the control uses an absolute phase reference or an asymmetric phase-locked loop, the phase-independence and drift-validity of the linearization break down.

Editorial extensions

If this is right

A single LTI model identified at one operating point can be used to predict inverter behavior throughout a phase drift, without re-linearization, as long as the power flow operating point is restored.
Linear stability of an interconnected inverter-based grid can be certified from per-device transfer matrices that depend only on power flow and device dynamics, not on the arbitrary phase of the linearization point.
System identification in complex phase variables needs no co-rotating frame or precise frequency measurement, removing a major practical error source in lab and field experiments.
The stability conditions (40)-(42) reproduce known droop-based conditions as a special case and extend them to devices with arbitrarily complex internal states through their transfer functions.
The paper's future-work sketch indicates the stability results can be made robust in the $H_\infty$ sense and extended to grids with homogeneous losses, though that extension is not carried out here.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension: train the Hammerstein-Wiener normal form at one phase offset and measure its prediction error at another offset with identical power flow; if the complex-phase invariance holds, the error should be the same, and any difference would reveal hidden phase references in the control implementation.
The same phase-symmetry argument applies to any oscillator network whose dynamics are invariant under global phase rotation, not just power inverters; coupled oscillator models in biology or mechanics could inherit the same phase-drift robustness.
Because the transfer matrices are phase-independent, one could in principle identify device-level stability conditions online from measured data and use them to certify grid stability without relying on a known reference angle, which would ease real-time monitoring in grids with fluctuating frequency.
The paper leaves open the extension to devices without an exact V-Q droop and to inhomogeneous losses; a concrete next step would be to construct counterexamples where inequality (42) fails under inhomogeneous line losses, testing how much droop structure the stability theorem really needs.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The core result here is real. Linearizing grid-forming inverter dynamics in complex phase and power variables does give an LTI system, Eq. (29), whose matrices are independent of the operating phase, and the phase-shift invariance argument checks out. The dq-frame discussion in Section II is a clean setup for the problem, and the step through the normal form in Section III is a useful connection. This is a genuinely new way of presenting the linearization, even if it builds directly on the authors' own prior work. I also think the paper is honest about most of its scope: the system identification and stability results in Sections IV and V are explicitly summaries of [3] and [18], and the reader is not misled into thinking those are derived here. The math in the central section is, as far as I can tell, sound.

The soft spots are real. The crux is Eq. (28), the phase-independence of the complex frequency. It follows from the rotational symmetry condition plus the normal form assumption that the device response depends only on rotation-invariant errors. But that is a substantive assumption: any grid-forming implementation with an absolute phase reference (a PLL locked to an external frame, GPS synchronization, or any control that uses absolute angle) violates it, and then the second column of Jη in Eq. (29) is not zero and the invariance under phase drifts fails. The paper never lists this as a limitation in Section VI, and it should. Also, the sentence in Section III that "ω◦ can vary with time, but the linearization remains unchanged" is too strong. The coefficient matrices depend on P◦, Q◦, and σ◦; a frequency variation that is exactly a pure phase drift with those held fixed leaves the matrices unchanged, but any realistic frequency excursion comes with power-flow changes and the matrices move. This is a smaller overreach, but it is the kind of sentence a reader will quote, so it should be qualified.

None of this undermines the central derivation. The paper is best read as a focused theory note that consolidates a modeling framework, not as a self-contained treatment of system identification or stability analysis. For someone working on grid-forming inverter modeling, this is worth knowing and likely worth citing. I would send it to review, with the expectation that the authors add a clear statement of the phase-symmetry assumption and temper the time-varying frequency claim.

Referee Report

3 major / 4 minor

Summary. The paper argues that linearizing grid-forming inverter dynamics in complex phase and power variables yields an LTI system, Equation (29), whose matrices Jη and Dη are independent of the operating phase φ◦ and, in a stated sense, of the frequency ω◦. It further claims that in these coordinates linearized phase shifts and nonlinear phase shifts coincide, so the linear model remains valid during phase drifts. Section IV summarizes a companion system-identification result using a Hammerstein-Wiener normal form, and Section V summarizes a companion small-signal stability analysis based on transfer matrices derived from the complex phase formulation.

Significance. If the central claim holds, the complex phase provides an elegant coordinate choice that removes the arbitrary reference-phase dependence that plagues conventional dq-frame linearizations, and it would indeed be useful for system identification and stability analysis of inverter-based grids. The derivation in Section III is explicit and does not rely on fitted parameters. The main caveat is that the validity of the result is conditional on a phase-symmetry assumption that the paper does not carefully scope; this is a correctness-risk concern but not a circularity. The applications in Sections IV and V are summaries of the authors' own prior work and are not self-contained evidence.

major comments (3)

[Section III, after Eq. (29)] The independence η(σ, φ, P, Q) = η(σ, P, Q) is asserted with the phrase 'it can then be checked,' but it is a load-bearing assumption rather than a purely algebraic consequence for all devices. For a memoryless voltage source satisfying (11) it follows from rotation equivariance; for devices with internal states, however, it requires that the device dynamics—including internal control states—are phase-symmetric, which is exactly the normal-form assumption of [13]. A grid-forming inverter with an absolute phase reference (e.g., a PLL locked to an external frame or a GPS-synchronized phase) violates (28), and then the second column of Jη in (29) is nonzero. The paper does not state this as a limitation in Section VI. Please either prove (28) from the stated assumptions or explicitly scope the main theorem to phase-symmetric devices and add a limitation statement.
[Sections IV and V] The sentence 'ω◦ can vary with time, but the linearization remains unchanged' is an overstatement. As the text immediately acknowledges, Jη and Dη depend on P◦, Q◦, and σ◦. A time-varying ω◦ caused by a power mismatch is generically accompanied by changes in P◦, Q◦, and σ◦, so the coefficient matrices do change. The claim is valid only for a pure phase drift with P◦, Q◦, and σ◦ held fixed. This distinction is crucial for the paper's robustness claim and should be stated precisely.
[Sections IV and V] The applications sections are summaries of companion papers [3] and [18] rather than self-contained demonstrations. In particular, the abstract's claim that the approach 'enables robust system identification during realistic conditions' is supported only by reference to [3], and the stability conditions (40)–(42) are quoted from [18] without a derivation. If the paper's contribution is meant to be the theoretical linearization result, this should be stated clearly; if the applications are part of the contribution, the paper needs to include enough detail to verify the claimed connection between (29) and the models identified in [3] and analyzed in [18].

minor comments (4)

[Section II, Eq. (11)] The phrase 'linear-time-independent system' should be 'linear time-invariant system' (or 'linear-time-invariant'), which is the standard term used elsewhere in the paper.
[Fig. 1 caption] The derivation of (28) from (11) is not shown. Even if it is a short calculation, a displayed step or a reference to the property that P and Q are rotation invariants would help the reader verify the claim.
[References [3], [18]] The phrase 'both leave the linearization and the full model invariant' is ambiguous; it should specify that the invariant direction in the complex phase coordinates coincides with the direction of phase shifts for both the linearized and the nonlinear dynamics.
[General] References [3] and [18] are arXiv preprints; if any peer-reviewed versions exist, they should be cited instead or in addition, so that readers can locate the published results.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LTI complex-phase result follows by coordinate transformation from the stated equivariance condition, not from its own conclusion.

full rationale

The central derivation is self-contained and not circular. The paper defines complex frequency as eta = d/dt ln(v_hat) and complex phase as Theta = ln(v_hat) (eqs. 22-25), and assumes the voltage-source dynamics f satisfy the rotational equivariance condition R(-Phi) f(R(Phi) v, R(Phi) i) = f(v, i) (eq. 11). From this and the rotation-invariance of P, Q, and sigma, eq. (28) that eta is independent of phi follows by direct transformation, not by fitting or by importing a conclusion. The LTI system (29) then has vanishing phi-columns and coefficients depending only on P_o, Q_o, and sigma_o, so the claimed invariance under phase shifts is a mathematical consequence of the stated assumptions. The claim that linearized and nonlinear phase shifts coincide likewise follows because phi is a neutral coordinate of both the nonlinear and the linearized dynamics. No parameter is fitted to data in this paper; the identification results in Sec. IV and stability conditions in Sec. V are summaries of companion works [3], [13], and [18] and are not used to derive the LTI result. These self-citations are not load-bearing for the central derivation. A scope caveat exists: eq. (28) relies on the phase-symmetry assumption (11), which excludes devices with absolute phase references, but the paper states (11) explicitly; this is a limitation of applicability, not circularity. Similarly, the statement that omega_o can vary with time while the linearization remains unchanged is conditional on P_o, Q_o, and sigma_o remaining fixed; again, this is a precision issue, not a circular step.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on the phase-symmetry assumption of grid-forming device dynamics and the validity of the normal form representation from [13]. No free parameters are fitted in this paper, and no new entities are introduced. The stability conditions add further assumptions about V-Q droop and the network, but these belong to the companion paper [18].

assumptions (3)

domain assumption Phase invariance of grid-forming device dynamics: f satisfies R(−Φ)f(R(Φ)v, R(Φ)i) = f(v,i), which implies the complex frequency η is independent of the phase φ (eqs. 11 and 28).
The central LTI and phase-independent linearization (29) depends on this symmetry. It is inherited from the normal form framework [13], which assumes devices respond only to rotation-invariant errors (P, Q, V). Real inverters with explicit phase references or unsymmetrical PLLs may violate it.
domain assumption The dynamics of a grid-forming device can be fully described by complex phase Θ and complex frequency η with P and Q as inputs (eq. 27, normal form [13]).
Section III's derivation treats η(σ,P,Q) as the device model; the validity of this representation for arbitrary grid-forming converters is asserted via [13] and is not established in this paper.
domain assumption For the stability analysis, the system implements an exact linear V-Q droop (38) and the network is lossless with admittance Laplacian Y.
Conditions (40)-(42), reproduced from [18] in Section V, assume these properties. They are not derived or justified within this paper.

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Cite this review

Pith. "Pith review of Complex Phase Analysis of Power Grid Dynamics." pith.science (2026). https://pith.science/paper/6Z6VOU2I

@misc{pith2026250622054,
  author       = {Pith},
  title        = {Pith review of: Complex Phase Analysis of Power Grid Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6Z6VOU2I}},
  note         = {Machine review of arXiv:2506.22054}
}

read the original abstract

With an increasing share of renewable energy sources, accurate and efficient modeling of grid-forming inverters is becoming crucial for system stability. Linear methods are a powerful tool for understanding dynamics close to an operating point, but usually depend on the reference trajectory. Thus, small deviations can render linear models invalid over time, posing a significant challenge in practice, and complicating theoretical analysis. As a solution, we show that the complex phase offers a robust formulation independent of reference phases and frequencies, thus preserving invariance properties under linearization. This enables robust system identification during realistic conditions and opens the road to powerful stability analysis of inverter-based grids.

Figures

Figures reproduced from arXiv: 2506.22054 by the authors.

**Figure 2.** A single grid-forming inverter, modeled by the normal form, coupled [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** Four inverters in the power hardware in the loop lab. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Performance for the measured and predicted [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

19 extracted references · 18 canonical work pages

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Complex-Phase, Data-Driven Identification of Grid-Forming Inverter Dynamics,

A. B ¨uttner, H. W ¨urfel, S. Liemann, J. Schiffer, and F. Hellmann, “Complex-Phase, Data-Driven Identification of Grid-Forming Inverter Dynamics,” Sep. 2024, arXiv:2409.17132 [eess]

arXiv 2024

[2] [18]

Small-signal stability of power systems with voltage droop

J. Niehues, R. Delabays, and F. Hellmann, “Small-signal stability of power systems with voltage droop,” Nov. 2024, arXiv:2411.10832

work page Pith review arXiv 2024

[3] [13]

Normal Form for Grid-Forming Power Grid Actors,

R. Kogler, A. Plietzsch, P. Schultz, and F. Hellmann, “Normal Form for Grid-Forming Power Grid Actors,” PRX Energy , vol. 1, no. 1, p. 013008, Jun. 2022, publisher: American Physical Society

work page 2022

[4] [1]

Grid-Forming Inverters: Are They the Key for High Renewable Penetration?

J. Matevosyan, B. Badrzadeh, T. Prevost, E. Quitmann, D. Ramasubramanian, H. Urdal, S. Achilles, J. MacDowell, S. H. Huang, V . Vital, J. O’Sullivan, and R. Quint, “Grid-Forming Inverters: Are They the Key for High Renewable Penetration?” IEEE Power and Energy Magazine, vol. 17, no. 6, pp. 89–98, Nov. 2019

work page 2019

[5] [2]

Complex Frequency,

F. Milano, “Complex Frequency,” IEEE Transactions on Power Systems , vol. 37, no. 2, pp. 1230–1240, Mar. 2022, conference Name: IEEE Transactions on Power Systems

work page 2022

[6] [4]

Design of Virtual Impedance Control Loop using the Complex Frequency Approach,

D. Moutevelis, J. Rold ´an-P´erez, M. Prodanovic, and F. Milano, “Design of Virtual Impedance Control Loop using the Complex Frequency Approach,” in 2023 IEEE Belgrade PowerTech , Jun. 2023, pp. 1–6

work page 2023

[7] [5]

Enhancing Frequency Control Through Rate of Change of V oltage Feedback,

F. Milano, B. Alhanjari, and G. Tzounas, “Enhancing Frequency Control Through Rate of Change of V oltage Feedback,” IEEE Transactions on Power Systems , vol. 39, no. 1, pp. 2385–2388, Jan. 2024, conference Name: IEEE Transactions on Power Systems

work page 2024

[8] [6]

Improving voltage and frequency control of DERs through Dynamic Power Compensation,

R. Bernal and F. Milano, “Improving voltage and frequency control of DERs through Dynamic Power Compensation,” Electric Power Systems Research, vol. 235, p. 110768, Oct. 2024

work page 2024