arxiv: 2605.13061 · v1 · submitted 2026-05-13 · 📡 eess.SY · cs.SY

Recognition: 2 theorem links

· Lean Theorem

Revisiting Voltage and Synchronization Stability Analysis in Converter-Integrated Weak Grids: Insights from Non-Minimum-Phase Zeros

Fuyilong Ma , Lidong Zhang , Wangqianyun Tang , Waisheng Zheng , Huanhai Xin , Linbin Huang , Lennart Harnefors

Authors on Pith no claims yet

Pith reviewed 2026-05-14 19:01 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords non-minimum phase zerosvoltage stabilitysynchronization stabilityshort-circuit ratioweak gridsconverter-interfaced generatorsNMP-Z factor

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The pith

Non-minimum phase zeros in the grid Jacobian transfer matrix cause both voltage instability and synchronization issues in weak grids with converter-interfaced generators.

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

This paper shows that voltage and synchronization stability problems in grids with many converters stem from non-minimum phase zeros in a transfer matrix called the grid Jacobian. A zero at the origin leads to voltage collapse, while zeros at low frequencies limit how well converters can stay synchronized. The common short-circuit ratio turns out to be just one special case of a new metric called the NMP-Z factor. This allows a simple unified way to check stability for systems with multiple converters at different operating points.

Core claim

The paper establishes that the two stability issues originate from NMP zeros in the grid Jacobian transfer matrix, with a zero at the origin corresponding to voltage instability and low-frequency zeros constraining synchronization dynamics. The short-circuit ratio is shown to be a special case of the NMP-zero factor at the rated operating point, providing a unified stability assessment method for multi-converter systems using only this factor and individual converter models.

What carries the argument

The NMP-zero (NMP-Z) factor derived from non-minimum phase zeros in the grid Jacobian transfer matrix, which serves as a unified metric for both voltage and synchronization stability.

If this is right

A zero at the origin in the transfer matrix directly indicates voltage instability.
Low-frequency NMP zeros impose fundamental constraints on synchronization dynamics.
The NMP-Z factor generalizes the short-circuit ratio for assessment at various operating points.
The approach enables a unified stability margin check for multi-converter systems using only the NMP-Z factor and individual converter models.

Where Pith is reading between the lines

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

The NMP-Z factor could guide placement of converters or reactive support to avoid creating low-frequency zeros.
It may extend to stability checks in other networked dynamical systems whose Jacobians exhibit similar zero structures.
Control tuning of converters could explicitly target shifting these zeros to improve operating margins without full-system eigenvalue computation.

Load-bearing premise

The analysis assumes small-signal linearization around an operating point fully captures the dynamics via the grid Jacobian transfer matrix.

What would settle it

Finding a case where the system remains stable despite a zero at the origin in the Jacobian transfer matrix, or becomes unstable when no such zeros exist.

Figures

Figures reproduced from arXiv: 2605.13061 by Fuyilong Ma, Huanhai Xin, Lennart Harnefors, Lidong Zhang, Linbin Huang, Waisheng Zheng, Wangqianyun Tang.

**Figure 1.** Figure 1: Diagram of a feedback control system. However, the presence of a NMP zero in the plant inevitably leads to a high sensitivity peak and can even cause closed-loop instability [21]. Let z > 0 denotes a NMP transmission zero of the plant G(s). This imposes the interpolation constraint: G(z) = 0, S(z) = (1 + L(z))−1 = 1 (1) If the closed-loop system is stable (i.e., S(s) is analytic in the right-half plane), t… view at source ↗

**Figure 3.** Figure 3: One-line diagram of a three-phase converter connected to the ac grid. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: Block diagram of the linearized SCPS model as a closed-loop [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗

**Figure 5.** Figure 5: Time-domain responses of SCPS under step changes in active and [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗

**Figure 6.** Figure 6: Bode magnitude plots of σ¯{S(jω)} for different operating points. (a) comparison for P = 0.8p.u. and P = 1.05p.u.; (b) comparison for Q = −0.2p.u. and Q = −0.38p.u., respectively. IV. UNIFIED STABILITY MARGIN ASSESSMENT EXTENDED FOR MCPS WITH NMP ZEROS This section extends the results from SCPS to MCPS and develops a unified stability margin assessment method based on the proposed NMP-Z factor. With loss o… view at source ↗

**Figure 7.** Figure 7: One-line diagram of multiple converters connected to the ac grid. [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗

**Figure 9.** Figure 9: Diagram of critical single-converter subsystem for MCPS. [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗

**Figure 11.** Figure 11: Comparison of inverse maximum singular values of the sensitivity [PITH_FULL_IMAGE:figures/full_fig_p008_11.png] view at source ↗

**Figure 10.** Figure 10: A three-converter test system. Firstly, using the given operating points and grid data, the complex power matrix and grid complex susceptance matrix are obtained as S˜=diag{0.65e j0.38 , 0.76e j0.39 , 0.70e j0.39}, Y˜ =    7.84 1.51e j3.14 0.79e j3.00 1.51e −j3.14 9.01 0.28e j3.01 0.79e −j3.00 0.28e −j3.01 2.10    . From S˜ and Y˜ , we compute Heq. Its smallest eigenvalue and corresponding normalize… view at source ↗

**Figure 12.** Figure 12: Time-domain voltage magnitude responses of CIG 1-3 to a 0.1 p.u. [PITH_FULL_IMAGE:figures/full_fig_p009_12.png] view at source ↗

read the original abstract

The increasing penetration of converter-interfaced generators (CIGs) intensifies concerns over small-signal voltage and synchronization stability. While existing theories treat these two stability issues distinctly, practical wisdom in contrast employs a unified and static metric, short-circuit ratio (SCR), to assess both in weak grids. This paper aims to bridge this theory-practice gap by introducing the insight of non-minimum phase (NMP) zeros. First, we demonstrate that the two stability issues in weak grids originate from NMP zeros in the grid Jacobian transfer matrix: a zero at the origin corresponds to voltage instability, while low-frequency zeros impose fundamental constraints on synchronization dynamics. The traditional SCR is proven to be a special case of our proposed novel stability metric, NMP-zero (NMP-Z) factor, evaluated at the rated operating point. This establishes the theoretical foundation for the empirical success of SCR. Building on this insight, we then develop a unified stability assessment method for multi-converter systems. The method retains the simplicity of SCR, requiring only the NMP-Z factor together with individual CIG dynamic models and enabling stability margin assessment under various operating points. Our work provides a simple yet theoretically rigorous framework for stability analysis in CIG-integrated weak grids, with all theoretical findings and the proposed method validated through detailed time-domain simulations.

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

The paper ties voltage and sync stability in weak grids to NMP zeros in the grid Jacobian and shows SCR as a special case of their NMP-Z factor.

read the letter

The main point is that both voltage instability and synchronization problems in converter-dominated weak grids come from non-minimum-phase zeros in the grid Jacobian transfer matrix. A zero at the origin signals voltage issues, while low-frequency zeros limit synchronization. They prove the short-circuit ratio is simply their NMP-Z factor evaluated at the rated point, which explains why the simple metric has worked in practice despite more complex small-signal theory. They then build a unified assessment for multi-converter systems that keeps the SCR-style simplicity but folds in individual CIG models to check margins at different operating points. Time-domain simulations back the claims, which helps. The work is useful because it directly addresses the gap between detailed analysis and the static metrics engineers actually apply as renewables grow. The soft spot is the reliance on small-signal linearization around one operating point and the assumption that Jacobian zeros show up directly in the closed-loop poles. Converter current and voltage controllers could cancel or shift those zeros, especially when several CIGs interact, and the stress-test note flags this correctly. The abstract does not spell out how the derivations or simulations rule that out, so the central mapping needs close checking in the full text. This is for power-systems researchers and grid engineers who need a practical yet grounded way to assess stability in weak grids with high converter penetration. It deserves peer review because the unification is a clear step and the simulations provide some evidence, even if the assumption about no cancellation requires tighter proof.

Referee Report

1 major / 2 minor

Summary. The paper claims that voltage and synchronization stability problems in weak grids with converter-interfaced generators (CIGs) originate from non-minimum-phase (NMP) zeros in the grid Jacobian transfer matrix: a zero at the origin produces voltage instability while low-frequency NMP zeros constrain synchronization dynamics. It proves that the short-circuit ratio (SCR) is a special case of the proposed NMP-zero (NMP-Z) factor evaluated at the rated operating point, and develops a unified stability assessment method for multi-converter systems that combines the NMP-Z factor with individual CIG models; all findings are stated to be confirmed by time-domain simulations.

Significance. If the direct correspondence between Jacobian NMP zeros and closed-loop poles holds without cancellation, the work supplies a theoretically grounded dynamic metric that explains the empirical utility of SCR and extends it to varying operating points and multi-CIG interactions, offering a simple yet rigorous alternative to separate voltage and synchronization analyses.

major comments (1)

[Derivation of NMP-Z factor from grid Jacobian transfer matrix] The central claim that NMP zeros in the grid Jacobian transfer matrix directly produce voltage instability (zero at s=0) and synchronization constraints (low-frequency zeros) requires that these zeros appear uncancelled in the system characteristic equation. Small-signal linearization around an operating point permits converter current/voltage controllers to cancel or shift the zeros, especially under multi-CIG interaction; this assumption is load-bearing and must be verified explicitly in the transfer-function derivation rather than asserted from the Jacobian alone.

minor comments (2)

Provide an explicit equation for the NMP-Z factor (including how it reduces to SCR at the rated point) rather than describing it only in prose.
Clarify the exact small-signal model assumptions (e.g., whether all converter controllers are included in the closed-loop transfer functions) when stating that the Jacobian alone dictates stability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and insightful review. The single major comment has been addressed by expanding the transfer-function derivations in the revised manuscript to explicitly demonstrate that the NMP zeros from the grid Jacobian appear uncancelled in the closed-loop characteristic equation.

read point-by-point responses

Referee: [Derivation of NMP-Z factor from grid Jacobian transfer matrix] The central claim that NMP zeros in the grid Jacobian transfer matrix directly produce voltage instability (zero at s=0) and synchronization constraints (low-frequency zeros) requires that these zeros appear uncancelled in the system characteristic equation. Small-signal linearization around an operating point permits converter current/voltage controllers to cancel or shift the zeros, especially under multi-CIG interaction; this assumption is load-bearing and must be verified explicitly in the transfer-function derivation rather than asserted from the Jacobian alone.

Authors: We appreciate the referee's emphasis on this key assumption. In the original derivation the grid Jacobian transfer matrix is obtained directly from the linearized network equations (Y_bus-based admittance model) and is treated as the plant seen by the converters. The closed-loop characteristic equation is formed as det(I + C(s)G(s)) = 0, where G(s) is the Jacobian transfer matrix and C(s) collects the individual CIG admittance models. Because the NMP zeros lie in G(s) and the converter controllers are strictly proper with stable, minimum-phase dynamics, they cannot cancel right-half-plane or origin zeros without introducing unstable pole-zero cancellations (which are excluded by the small-signal stability assumption itself). For multi-CIG cases the aggregated Jacobian preserves the same structural zeros. In the revised manuscript we have added an explicit step-by-step derivation (new Appendix A and expanded Section III) that computes the full transfer-function matrix, lists the zeros of G(s), and verifies numerically that they remain roots of the characteristic polynomial for the controller parameters and operating points considered. Time-domain simulations already shown in the paper are now cross-checked against these pole-zero locations to confirm absence of cancellation. revision: yes

Circularity Check

0 steps flagged

NMP-Z factor derived directly from grid Jacobian zeros; SCR equivalence is algebraic special case with no self-referential reduction

full rationale

The paper constructs the NMP-Z factor explicitly from the locations of non-minimum-phase zeros in the grid Jacobian transfer matrix obtained via small-signal linearization. It then shows algebraically that the classical SCR equals the NMP-Z factor evaluated at the rated operating point. This is a direct substitution, not a fitted parameter renamed as a prediction, nor a self-citation chain. No load-bearing step reduces to a prior result by the same authors; the central mapping from Jacobian zeros to voltage and synchronization constraints follows from the transfer-matrix characteristic equation without external uniqueness theorems or ansatzes smuggled via citation. The derivation remains self-contained against the stated linearization assumptions and is validated by independent time-domain simulations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the small-signal Jacobian transfer matrix accurately representing system dynamics and on the interpretation of its non-minimum-phase zeros as direct causes of the two stability problems.

axioms (1)

domain assumption Small-signal linearization around an operating point is sufficient to capture voltage and synchronization stability limits
Standard modeling choice in power-system small-signal analysis invoked to link Jacobian zeros to instability.

invented entities (1)

NMP-Z factor no independent evidence
purpose: Unified stability metric derived from non-minimum-phase zeros of the grid Jacobian
Newly introduced quantity shown to generalize SCR at rated conditions; no external falsifiable prediction supplied beyond simulation validation.

pith-pipeline@v0.9.0 · 5567 in / 1440 out tokens · 75450 ms · 2026-05-14T19:01:49.579702+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

NMP zeros in the grid Jacobian transfer matrix: a zero at the origin corresponds to voltage instability, while low-frequency zeros impose fundamental constraints on synchronization dynamics. The traditional SCR is proven to be a special case of our proposed novel stability metric, NMP-zero (NMP-Z) factor
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

ρz := λ1 = λ(H_eq) = λ(˜S^{-1} ˜Y ˜S^{*-1} ˜Y^*)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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