{"id":"841510db-42f7-4a64-9726-4c4c3ef57bec","arxiv_id":"2506.22054","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Linearizing grid-forming inverter dynamics in complex phase coordinates gives a time-invariant, phase-independent linear model that remains valid under frequency and phase drifts.","lead":"This paper argues that modeling grid-forming inverters using the complex phase, the logarithm of the voltage phasor, yields linearized dynamics that stay valid during phase drifts and do not depend on an arbitrary reference phase. The authors derive this property and point to companion papers where it enables data-driven inverter identification and small-signal stability conditions.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq.28's phase-independence is the crux: it holds only for phase-symmetric devices, and the paper neither states this as a limitation nor bounds it, so the LTI-valid-through-drift claim is conditional.","rationale":"The derivation in Sections II-III is internally consistent: the complex-phase change of variables turns the problematic finite rotation into a translation, and the zero second column of Jη is exactly the statement that translations in φ are a symmetry of both the full and the linearized device dynamics. The reader's weakest_assumption correctly identifies eq.28 as the load-bearing premise, and I agree that it is contentful: it is satisfied by the usual power-based droop and virtual-synchronous-machine controls, but it is not guaranteed for controls that use an absolute phase reference. That assumption is not stated as a limitation in the discussion, and the time-varying-ω sentence overclaims because the matrices of eq.29 depend on P◦,Q◦,σ◦, not on φ or ω directly. Since no internal inconsistency or algebraic error was found, the concern does not warrant rejection or a stronger verdict than the reader's CONDITIONAL; it supports keeping CONDITIONAL, with the request that the authors state the phase-symmetry scope explicitly and qualify the ω◦ sentence. The secondary observation that Sections IV-V rely on companion papers reinforces the conditional framing but does not change it.","tokens_in":9020,"tokens_out":23400,"duration_ms":288861,"concrete_test":"Simulate the H-W normal form (35)-(36) with an added symmetry-breaking term, e.g., replace η by η0(σ,P,Q)+ε sin(φ0+δφ), and estimate the transfer matrix T(s) in eq.37 at two operating phases φ0=0 and φ0=π/2 with identical P,Q,V. If the estimated Jη,Dη or T(s) differ for any ε>0, the necessity of eq.28 for the LTI-independence claim is demonstrated; as ε→0 the difference must vanish. For a hardware check, rerun the Section IV islanding experiment with the inverter's PLL reference held to a fixed external frame rather than free-running: if the identified model no longer tracks the post-drift dq-voltage transient, the phase-symmetry assumption is violated in practice.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central result is eq.29, an LTI system whose matrices Jη and Dη are independent of the operating phase φ◦ and frequency ω◦. This rests entirely on eq.28, η(σ,φ,P,Q)=η(σ,P,Q). For a memoryless voltage source the derivation from (11) is correct: a rotation-equivariant complex frequency can depend only on the rotation invariants V, P, and Q. The load-bearing premise is that the device, including internal control states, is phase-symmetric. Any grid-forming implementation with an absolute phase reference, e.g., a PLL locked to an external frame or a GPS-synchronized phase, violates eq.28; the second column of Jη in eq.29 becomes nonzero and the claimed invariance under phase shifts fails for that device. The paper does not list this as a limitation in Section VI. A second, less central overreach is the sentence 'ω◦ can vary with time, but the linearization remains unchanged': eq.29's coefficients depend on P◦, Q◦, and σ◦, so this is true only for a pure phase drift with those quantities held fixed; a frequency variation accompanied by a power-flow change (the physically generic case) changes the coefficient matrices. Sections IV and V summarize applications from companion papers, so the evidence for the full 'enables robust system identification and stability analysis' claim is not self-contained.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9243,"tokens_out":3949,"duration_ms":40420,"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":[{"comment":"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.","section":"Section III, after Eq. (29)"},{"comment":"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.","section":"Sections IV and V"},{"comment":"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].","section":"Sections IV and V"}],"minor_comments":[{"comment":"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.","section":"Section II, Eq. (11)"},{"comment":"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.","section":"Fig. 1 caption"},{"comment":"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.","section":"References [3], [18]"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is essentially a methods paper whose two application sections rest on the authors' own prior work, [3] and [18]. This raises a scope question: if the journal expects a self-contained contribution, the dependence on companion manuscripts needs to be made explicit and perhaps reduced. The central derivation is sound within the phase-symmetry assumption, but the paper currently understates that assumption and overstates the time-varying-frequency claim; both should be corrected in revision."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nNone 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.","headline":"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.","tokens_in":9809,"tokens_out":1288,"would_cite":true,"duration_ms":15072,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["complex frequency","complex phase","grid-forming inverters","phase-invariant linearization","linear-time-invariant systems","normal form","system identification","small-signal stability"],"falsifier":"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.","tokens_in":8780,"feed_emoji":"⚡","tokens_out":9930,"duration_ms":97212,"temperature":0.7,"pith_summary":"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.","feed_headline":"Complex phase keeps linear grid models valid through phase drifts","feed_subtitle":"In inverter-heavy grids, power mismatches shift phases; this linearization stays valid through those shifts.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the complex frequency concept used to define the complex phase variable.","marker":"[2]"},{"why":"supplies the PHIL system identification pipeline and the islanding experiment with phase drift.","marker":"[3]"},{"why":"introduces the complex phase formulation of voltage used to build the coordinates.","marker":"[12]"},{"why":"provides the normal form assumption that device response depends only on rotation-invariant errors, the premise behind equation (28).","marker":"[13]"},{"why":"provides the Park/dq transformation baseline that the paper contrasts with, showing the conventional phase-dependence problem.","marker":"[14]"},{"why":"provides the small-signal stability conditions (40)-(42) that the complex phase linearization enables.","marker":"[18]"},{"why":"supplies the small-phase theorem variant used to prove the interconnected stability result.","marker":"[19]"}],"fun_headline_variants":["Complex phase makes grid linearization immune to phase drift","Complex frequency formulation yields drift-invariant grid models","Phase-independent linearization for inverter-based grids","LTI form for grid inverters via complex phase"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Complex phase makes grid linearization immune to phase drift","Complex frequency formulation yields drift-invariant grid models","Phase-independent linearization for inverter-based grids","LTI form for grid inverters via complex phase"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000768,"raw_usage":{"total_tokens":3354,"prompt_tokens":844,"completion_tokens":2510,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":460,"completion_tokens_details":{"reasoning_tokens":2450}},"tokens_in":460,"tokens_out":2510,"duration_ms":18193,"temperature":1.0,"reasoning_tokens":2450,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T22:12:33.314825+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Complex Frequency,","cited_arxiv_id":null,"evidence_quote":"supplies the complex frequency concept used to define the complex phase variable."},{"cited_title":"Complex Couplings - A Universal, Adaptive, and Bilinear Formulation of Power Grid Dynamics,","cited_arxiv_id":null,"evidence_quote":"introduces the complex phase formulation of voltage used to build the coordinates."},{"cited_title":"Normal Form for Grid-Forming Power Grid Actors,","cited_arxiv_id":null,"evidence_quote":"provides the normal form assumption that device response depends only on rotation-invariant errors, the premise behind equation (28)."},{"cited_title":"Power System Dynamics. Stability and Control,","cited_arxiv_id":null,"evidence_quote":"provides the Park/dq transformation baseline that the paper contrasts with, showing the conventional phase-dependence problem."},{"cited_title":"Small-signal stability of power systems with voltage droop","cited_arxiv_id":"2411.10832","evidence_quote":"provides the small-signal stability conditions (40)-(42) that the complex phase linearization enables."},{"cited_title":"A Phase Theory of Multi-Input Multi-Output Linear Time-Invariant Systems,","cited_arxiv_id":null,"evidence_quote":"supplies the small-phase theorem variant used to prove the interconnected stability result."}],"review_version":1}