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Small-signal stability of power systems with voltage droop

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Pith's one-line read A lossless grid with V-q droop at every bus is small-signal stable if each bus has a stable local response, a positive-definite response matrix, and a droop coefficient α_n large enough relative to its neighbors.

desk verdict A solid, genuinely new decentralized stability certificate, but the δ=0 semi-stability claim has a proof gap that needs fixing. read the letter →

arxiv 2411.10832 v2 pith:CMTELHZF submitted 2024-11-16 math.OC cs.SYeess.SYmath.DS

Jakob Niehues , Robin Delabays , Anna Büttner , Frank Hellmann This is my paper · ORCID
classification math.OCcs.SYeess.SYmath.DS MSC 93D0593D0993C80
keywords small-signalstabilitygrid-formingconvertersV-qdroopcontrolcomplexfrequencydecentralizedconditionssmallphasetheoreminverter-dominatedpowergridsheterogeneousbusdynamics
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

This paper tries to establish that small-signal stability of an inverter-based power grid can be certified locally, bus by bus, without knowing the rest of the grid, provided each bus implements an exact V-q droop and the lines are lossless (or have a homogeneous R/X ratio). The central claim is Proposition 1: stability follows from per-bus checks on 2x2 transfer matrices that map reactive and active power deviations to amplitude-velocity and frequency deviations, plus a lower bound on the droop coefficient α_n that depends only on neighboring voltages and phase differences. What makes the claim matter is that it is technology-neutral—no assumptions about internal inverter control details and no dynamical homogeneity across buses—so a heterogeneous mix of grid-forming devices is covered by one simple condition set. Numerical experiments on the IEEE 14-bus system show the predicted thresholds track the actual stability boundary closely and can single out one misconfigured inverter that triggers a slow voltage collapse.

What carries the argument

The machinery has three parts. (1) Complex frequency $\eta_n = \dot v_n/v_n = \varrho_n + j\omega_n$ and shifted reactive power $\hat q_n = q_n + \alpha_n V_n$, which let node dynamics be written as a phase-invariant 2x2 transfer matrix $T_n(s)$ mapping $(\Delta\hat q_n, \Delta p_n)$ to $(\varrho_n, \omega_n)$; choosing $\alpha$ to eliminate $V_n$ as a state keeps $T_n$ well behaved at small $s$. (2) The generalized small-phase theorem of [23], extended in Proposition 3 to block-structured loops $H = \bigoplus_n T_n$ and $G = B^\dagger\bigoplus_e T_e B$, so global phase conditions reduce to per-node and per-edge phase conditions. (3) An edge-wise decomposition of the network response $J^{\mathrm{net}}$ into 4x4 Hermitian blocks $J_e$ whose positive semidefiniteness is equivalent to $|\varphi_n-\varphi_m|<\pi/2$ and $\alpha_n \ge 2\sum_m \tilde Y_{nm} V_m \cos(\varphi_n-\varphi_m)$; the factor $1/s$ makes each edge transfer semi-stable with phase $-\pi/2$, so the loop phase sum stays inside $(-\pi,0)$.

What would settle it

Take the lossless IEEE 14-bus grid with generalized droop (8)-(9), choose parameters satisfying (3), (4), and (5) for every bus, and compute the eigenvalues of the full linearization; if any eigenvalue lies in the open right half-plane, Proposition 1 is false. A minimal version of the same test scans the two-bus lossless system over phase differences $|\varphi_1-\varphi_2|<\pi/2$ while enforcing (3)-(5) and checks the linearized dynamics by eigenvalue or time-domain computation.

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Extended reading notes

Core claim

In the coordinates of complex frequency and complex power, the closed loop between grid-forming nodes and transmission lines splits into per-node and per-edge blocks, and the global small-signal stability problem becomes a set of local phase conditions. Each node is a 2x2 transfer function $T_n(s)$ from $(\Delta\hat q_n, \Delta p_n)$ to $(\varrho_n, \omega_n)$ where $\hat q_n = q_n + \alpha_n V_n$ is the V-q-droop-shifted reactive power, and the whole grid is stable when every $T_n$ is internally stable and strictly accretive—conditions (3) and (4)—and when $\alpha_n$ satisfies $\alpha_n \ge 2\sum_m \tilde Y_{nm} V_m \cos(\varphi_n-\varphi_m)$ (condition (5)). The result is explicitly sufficient; the proof wires the small-phase theorem of [23] through a block-structured version (Proposition 3), with the network response decomposed into per-edge contributions whose phases are all $-\pi/2$. Along the way the paper shows that several established models—generalized droop laws, the third-order droop inverter model of [5], virtual synchronous machines, quadratic droop, reactive current control—fall into this framework, and that lossy lines with constant R/X are handled by rotating the inputs by $O(R/X)$. The simulations indicate the sufficient conditions are close to exact in the tested regime, with a predicted $\alpha_n$ that matches the numerical stability threshold almost perfectly except near zero.

Load-bearing premise

Every bus must react to a change in voltage amplitude exactly in proportion to its reaction to a change in reactive power, using the same coefficient $\alpha_n$, so that the voltage amplitude disappears as an independent state variable.

Editorial extensions

If this is right

  • A grid operator can certify stability bus by bus from each inverter's terminal transfer function and the voltages and phase differences on adjacent lines; no global model, no homogeneity assumption, and no knowledge of other buses' control laws is needed.
  • The conditions translate into design rules: the diagonal couplings $T^{\varrho\hat q}_n$ and $T^{\omega p}_n$ must be positive and dominate the off-diagonal crosstalk terms, quantifying how much frequency-voltage cross-coupling is tolerable.
  • The lower bound (5) on $\alpha_n$ is a new design constraint; it can be negative, meaning local grid conditions can make even a misconfigured droop tolerable, and it pins down the critical reactive-droop gain observed numerically.
  • For lossy lines with homogeneous R/X, the same theorem applies after rotating the control inputs by $O(R/X)$; the effective droop becomes one on $\hat q$ and $\hat p = p + \alpha V R/X$.
  • Established models—generalized droop, third-order droop inverters, virtual synchronous machines, quadratic droop, reactive current control—fall into the framework, recovering and improving previous matrix-inequality stability conditions with simpler decentralized expressions.

Reading between the lines

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

  • Beyond the paper, because the relevant transfer functions can be measured at the terminal, Proposition 1 could be turned into an acceptance test in the field: record $T_n$ by probing active and reactive power set-point changes, check (3)-(4), and verify (5) from local phasor measurements, without needing a vendor model.
  • The main obstacle the paper names—exact V-q droop—suggests a concrete next step: if a device's voltage response is not exactly proportional to its reactive-power response, the dynamics of $V_n$ make the Hermitian part indefinite at small $s$; adding gain information, as the companion work does for adaptive networks, is a plausible route to cover dVOC and lossy conventional models.
  • One could probe whether the sufficient conditions are nearly necessary in larger systems by initiating small perturbations near the certified boundary and recording the actual stability margin; the 14-bus simulations suggest the gap is small except when $\alpha_n$ is near zero.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. This paper derives sufficient conditions for small-signal stability of lossless and constant R/X power grids with heterogeneous grid-forming inverters that implement a V-q droop. The main result, Proposition 1, certifies stability from local nodal transfer matrices T_n(s) (conditions (3)-(4)) and a nodal lower bound on the droop coefficient α_n involving only neighboring voltage magnitudes and phase differences (condition (5)). The proof recasts the grid as a feedback interconnection of nodal and edge transfer matrices and applies a block-structured version of the generalized small-phase theorem. The conditions are applied to generalized droop and third-order converter models, and validated by numerical linearization of the IEEE 14-bus system, showing tightness of the α bound and identification of a misconfigured inverter.

Significance. If correct, the main theorem is a significant methodological advance: it gives a decentralized, model-agnostic stability certificate for a broad class of grid-forming controls, with conditions that are directly interpretable as design rules (positive diagonal couplings, bounded cross-coupling, sufficient voltage droop). The proof is largely self-contained, including a block-structured generalization of the small-phase theorem, and the numerical results support the claim that the sufficient conditions are not overly conservative. The authors are also explicit about the key limitation: the exact V-q droop assumption (Appendix C, Assumption II) excludes models with independent voltage-amplitude dynamics.

major comments (1)
  1. [Section III-B (third-order models)] The claim that semi-stability at δ_n=0 follows from stability for arbitrarily small δ_n>0 and continuity of the eigenvalues of the Jacobian is not a valid inference. Continuity of eigenvalues only implies Re λ ≤ 0 for the limiting matrix; it does not rule out a Jordan block associated with the eigenvalue at 0. For example, A(δ)=[[-δ,1],[0,-δ]] is Hurwitz for δ>0 but converges to [[0,1],[0,0]], which is not Lyapunov stable. Since the δ=0 case recovers the droop-controlled inverter model of [5], the paper should either prove directly that the zero eigenvalue (uniform phase mode) is simple and that no Jordan block occurs for the full system (13)-(15) at δ=0, or revise the claim to indicate that Proposition 1 applies only for δ>0 and that the δ=0 case requires separate treatment (e.g., via the gain-based analysis of [24]).
minor comments (5)
  1. [Eq. (5) and Eq. (79)] The quantity \tilde Y_nm is used in condition (5) and in the definition of α_theory_n in (79) but is never defined in the main text. Please define it explicitly (presumably |Y_nm| = -L_nm for a lossless grid).
  2. [Proposition 1 statement] The notation 'for all s∈[0,∞]' is confusing; since s is the Laplace variable, it should be made explicit that s runs over the imaginary axis s=jω with ω∈[0,∞].
  3. [Appendix D, Lemma 6] The proof of Lemma 6 is delegated to a 'straightforward calculation'; given that this lemma underpins the edge-wise decomposition and the bound (5), please include the calculation or a more detailed sketch (e.g., the Schur complement steps) so the result is independently verifiable.
  4. [Section IV, Figure 1 caption] The caption lists 'CV_p = 1, CV_q = 1, CV_q = 0.5, Cω_p = 0.5'; the second CV_q is likely a typo for Cω_q. Please correct.
  5. [Section V, Eqs. (21)-(23)] In Eq. (22), O(R/X) is defined and then used as O in Eq. (23); please use consistent notation. Also, 'O cos κ is a rotation matrix' would be clearer as 'O(R/X) cos κ is a rotation matrix'.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: stability conditions are derived from the stated V-q droop assumption via an external small-phase theorem and independently benchmarked in simulation; self-citations to the authors' complex-frequency framework are not load-bearing for the main result.

full rationale

The paper's central claim (Proposition 1) is a sufficient condition proved in Appendix D from the stated V-q droop model class, using Chen et al.'s Generalized Small Phase Theorem (Theorem 2, an external result) and the paper's own block-structure version (Proposition 3), which is proved in Appendix E. The conditions (3)-(5) are derived algebraic consequences of sectoriality of the nodal transfer matrices and the edgewise decomposition of the network response (Lemmas 4-7); they are not definitions of stability nor fitted parameters. In Section IV, the theoretical bound alpha_theory_n from (5) is tested against independently computed critical values alpha_crit_n of the full generalized droop model (8)-(9), so the simulation is an external benchmark rather than a prediction forced by construction. The self-citations to [20], [21], [22], and [24] supply the complex-frequency notation and a companion gain-based treatment, but the main proof reproduces the needed algebraic steps and the load-bearing phase theorem is external; these self-citations are therefore not circular. The paper also openly acknowledges the exact V-q droop assumption and the limitations for dVOC and pass-through-free models in Section VI. One correctness caveat, not a circularity: Section III-B argues that delta_n=0 semi-stability follows from stability at small delta_n>0 by continuity of eigenvalues, but eigenvalue continuity does not exclude a nilpotent Jordan block at the origin; the additional citation to [24] does not turn this into an input-output circularity. Overall, no step reduces the prediction to its own inputs, so the circularity score is low.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on a small set of modeling assumptions: exact V-q droop, no explicit phase dependence, grid-forming nodes, lossless or constant-R/X lines, and small phase-angle differences. None of these are hidden fitting parameters; they define the model class that the theorem covers. The proof also imports the external small-phase theorem, which is clearly cited. The edge-wise decomposition variables α'_nm are proof devices and do not enter the final sufficient bound.

assumptions (5)
  • domain assumption Nodal dynamics do not depend explicitly on phase angle φ_n, and near the operating point they depend on q_n and V_n only through the combination q_n + α_n V_n.
    Assumptions I and II in Appendix C. This is required for the linearized nodal response (55)-(59) and for the Hermitian structure of the network response J_net in (71).
  • domain assumption All nodes are grid-forming, so the operating-point complex frequency is zero: ω_n° = ρ_n° = 0.
    Stated in Section II as the choice of co-rotating frame; needed so that θ_n and θ_n are valid state variables and the linearization around the operating point is well posed.
  • standard math The Generalized Small Phase Theorem of Chen et al. is valid, together with the block-structure extension given as Proposition 3.
    Theorem 2 in Appendix B is cited from [23]; Proposition 3 is proved in Appendix E using properties of the numerical range. The main proof applies this theorem to the nodal and edge transfer matrices.
  • domain assumption Transmission lines are lossless (Y is a Laplacian) or have a constant R/X ratio on every line.
    Used throughout: L = -jY is assumed real symmetric positive semidefinite in Appendix C, and the lossy case is treated by the rotation matrix O in Section V.
  • domain assumption The voltage phase-angle difference across every connected line satisfies |φ_n° - φ_m°| < π/2.
    Included in Proposition 1 and required in Lemma 6 for the edge matrices J_e to be positive semidefinite, which makes the network response semi-sectorial.

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Pith. "Pith review of Small-signal stability of power systems with voltage droop." pith.science (2026). https://pith.science/paper/CMTELHZF

@misc{pith2026241110832,
  author       = {Pith},
  title        = {Pith review of: Small-signal stability of power systems with voltage droop},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CMTELHZF}},
  note         = {Machine review of arXiv:2411.10832}
}
abstract

The stability of inverter-dominated power grids remains an active area of research. This paper presents novel sufficient conditions for ensuring small-signal stability in lossless and constant $R/X$ grids with highly heterogeneous mixes of grid-forming inverters that implement an adapted $V$-$q$ droop control. The proposed conditions can be evaluated in the neighborhood of each bus without information on the rest of the grid. Apart from the presence of $V$-$q$ droop, no additional assumptions are made regarding the inverter control strategies, nor is dynamical homogeneity across the system assumed. The analysis is enabled by recasting the node dynamics in terms of complex frequency and power, resulting in transfer functions that directly capture the small-signal frequency and amplitude responses to active and reactive power imbalances. These transfer functions are directly aligned with typical design considerations in grid-forming control. Building on an adapted small-phase theorem and viewing the system as a closed feedback loop between nodes and lines, the derived stability conditions also yield new insights when applied to established inverter control designs. We demonstrate in simulations that our conditions are not overly conservative and can identify individual inverters that are misconfigured and cause instability.

Figures

Figures reproduced from arXiv: 2411.10832 by the authors.

Figure 1
Figure 1. Numerical small-signal stability of the IEEE 14- [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. illustrates example trajectories for a stable system where all nodes are at the theoretical α value, and an unstable system where one node violates the theoretical stability guarantee. We observe that the violation leads to a slow voltage collapse within the system. As only one node violates our theoretical bound in this system, our bounds successfully pinpoint the origin of instability in this case. Finally, we als… view at source ↗
Figure 3
Figure 3. Numerical small-signal stability of the IEEE 14- [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Block diagram representation of the system [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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Cited by 1 Pith paper

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Reference graph

Works this paper leans on

39 extracted references · 29 canonical work pages · cited by 1 Pith paper

  1. [5]

    Conditions for stability of droop-controlled inverter-based microgrids,

    J. Schiffer, R. Ortega, A. Astolfi, J. Raisch, and T. Sezi, “Conditions for stability of droop-controlled inverter-based microgrids,” Automatica, vol. 50, no. 10, pp. 2457–2469, Oct. 2014. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S0005109814003100

    work page 2014

  2. [24]

    Phase and gain stability for adaptive dynamical networks,

    N. Kastendiek, J. Niehues, R. Delabays, T. Gross, and F. Hellmann, “Phase and gain stability for adaptive dynamical networks,” Chaos: An Interdisciplinary Journal of Nonlinear Science, vol. 35, no. 5, p. 053142, May 2025. [Online]. Available: https://doi.org/10.1063/5.0249706

    work page doi:10.1063/5.0249706 2025

  3. [1]

    A Structure Preserving Model for PowerSystemStabilityAnalysis,

    A. Bergen and D. Hill, “A Structure Preserving Model for PowerSystemStabilityAnalysis,” IEEE Transactions on Power Apparatus and Systems , vol. PAS-100, no. 1, pp. 25–35, Jan. 1981,conferenceName:IEEETransactionsonPowerApparatus and Systems

    work page 1981

  4. [2]

    Machowski, J

    J. Machowski, J. Bialek, and J. Bumby,Power System Dynam- ics. Stability and Control . John Wiley & Sons, Jan. 2012

    work page 2012

  5. [3]

    Synchronization in complex oscillator networks and smart grids,

    F. Dörfler, M. Chertkov, and F. Bullo, “Synchronization in complex oscillator networks and smart grids,” Proceedings of the National Academy of Sciences , vol. 110, no. 6, pp. 2005–2010, Feb. 2013, publisher: Proceedings of the National Academy of Sciences. [Online]. Available: https: //www.pnas.org/doi/full/10.1073/pnas.1212134110

    doi:10.1073/pnas.1212134110 2005

  6. [4]

    Distributed Stability Conditions for Power Systems With Heterogeneous Nonlinear Bus Dynamics,

    P. Yang, F. Liu, Z. Wang, and C. Shen, “Distributed Stability Conditions for Power Systems With Heterogeneous Nonlinear Bus Dynamics,” IEEE Transactions on Power Systems , vol. 35, no. 3, pp. 2313–2324, May 2020. [Online]. Available: https://ieeexplore.ieee.org/abstract/document/8890862

    work page arXiv 2020

  7. [6]

    Collective nonlinear dynamics and self-organization in decentralized power grids,

    D. Witthaut, F. Hellmann, J. Kurths, S. Kettemann, H. Meyer- Ortmanns, and M. Timme, “Collective nonlinear dynamics and self-organization in decentralized power grids,” Reviews of Modern Physics , vol. 94, no. 1, p. 015005, Feb. 2022, publisher: American Physical Society. [Online]. Available: https://link.aps.org/doi/10.1103/RevModPhys.94.015005

    work page doi:10.1103/revmodphys.94.015005 2022

  8. [7]

    Stability BoundsofDroop-ControlledInvertersinPowerGridNetworks,

    P. C. Böttcher, L. R. Gorjão, and D. Witthaut, “Stability BoundsofDroop-ControlledInvertersinPowerGridNetworks,” IEEE Access, vol. 11, pp. 119947–119958, 2023, conference Name: IEEE Access. [Online]. Available: https://ieeexplore. ieee.org/document/10267988/?arnumber=10267988 12

    work page arXiv 2023

Show all 39 references
  1. [8]

    Mixed Gain/Phase Robustness Criterion for Structured Perturbations With an Application to Power System Stability,

    L. Woolcock and R. Schmid, “Mixed Gain/Phase Robustness Criterion for Structured Perturbations With an Application to Power System Stability,” IEEE Control Systems Letters , vol. 7, pp. 3193–3198, 2023, conference Name: IEEE Control Systems Letters. [Online]. Available: https:...

    2023

  2. [9]

    ScalableDesignofHeterogeneous Networks,

    R.PatesandG.Vinnicombe,“ScalableDesignofHeterogeneous Networks,” IEEE Transactions on Automatic Control , vol. 62, no. 5, pp. 2318–2333, May 2017, conference Name: IEEE Transactions on Automatic Control. [Online]. Available: https: //ieeexplore.ieee.org/document/7583731/?arnum...

    2017

  3. [10]

    Bregman storage functions for microgrid control,

    C. D. Persis and N. Monshizadeh, “Bregman storage functions for microgrid control,” Aug. 2016, arXiv:1510.05811 [math]. [Online]. Available: http://arxiv.org/abs/1510.05811

    2016 arXiv

  4. [11]

    Gain and Phase: Decentralized Stability Conditions for Power Electronics-Dominated Power Systems,

    L. Huang, D. Wang, X. Wang, H. Xin, P. Ju, K. H. Johansson, and F. Dörfler, “Gain and Phase: Decentralized Stability Conditions for Power Electronics-Dominated Power Systems,” Jan. 2024, arXiv:2309.08037. [Online]. Available: http://arxiv.org/abs/2309.08037

    2024 arXiv

  5. [12]

    Christensen, G

    P. Christensen, G. K. Andersen, M. Seidel, S. Bolik, S. Engelken, T. Knueppel, A. Krontiris, K. Wuerflinger, T. Bülo, J. Jahn, M. Ndreko, S. Salehi, I. Theologitis, B. Weise, H. Urdal, A. Egea Alvarez, A. J. Roscoe, and J. Fortmann, High Penetration of Power Electronic Interfa...

    2020

  6. [13]

    Testing Characteristics of Grid Forming Converters Part III: Inertial Behaviour

    A. Dyśko, A. Egea, Q. Hong, A. Khan, P. Ernst, R. Singer, and Ise, “Testing Characteristics of Grid Forming Converters Part III: Inertial Behaviour.”

  7. [14]

    Generalized Multivariable Grid-Forming Control Design for Power Converters,

    M. Chen, D. Zhou, A. Tayyebi, E. Prieto-Araujo, F. Dörfler, and F. Blaabjerg, “Generalized Multivariable Grid-Forming Control Design for Power Converters,” IEEE Transactions on Smart Grid, vol. 13, no. 4, pp. 2873–2885, Jul. 2022, conference Name: IEEE Transactions on Smart Gr...

    2022

  8. [15]

    Grid-Forming and Spatially Distributed Control Design of Dynamic Virtual Power Plants,

    V. Häberle, A. Tayyebi, X. He, E. Prieto-Araujo, and F. Dörfler, “Grid-Forming and Spatially Distributed Control Design of Dynamic Virtual Power Plants,”IEEE Transactions on Smart Grid, vol. 15, no. 2, pp. 1761–1777, Mar. 2024, conference Name: IEEE Transactions on Smart Grid....

    2024

  9. [16]

    Quantitative Stability Conditions forGrid-FormingConverters With Complex Droop Control,

    X. He, L. Huang, I. Subotić, V. Häberle, and F. Dörfler, “Quantitative Stability Conditions forGrid-FormingConverters With Complex Droop Control,”IEEE Transactions on Power Electronics, vol. 39, no. 9, pp. 10834–10852, Sep. 2024. [Online]. Available: https://ieeexplore.ieee.or...

    2024

  10. [17]

    A survey on modeling of microgrids—From fundamental physics to phasors and voltage sources,

    J. Schiffer, D. Zonetti, R. Ortega, A. M. Stanković, T. Sezi, and J. Raisch, “A survey on modeling of microgrids—From fundamental physics to phasors and voltage sources,” Automatica, vol. 74, pp. 135–150, Dec. 2016. [Online]. Available: https://www.sciencedirect.com/science/ar...

    2016

  11. [18]

    Self- organized synchronization and voltage stability in networks of synchronous machines,

    K. Schmietendorf, J. Peinke, R. Friedrich, and O. Kamps, “Self- organized synchronization and voltage stability in networks of synchronous machines,”The European Physical Journal Special Topics, vol. 223, no. 12, pp. 2577–2592, Oct. 2014. [Online]. Available: https://doi.org/1...

    2014 doi

  12. [19]

    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. [Online]. Available: https://ieeexplore.ieee.org/abstract/ document/9524491?casa_token=d7Q69Fh7p2sAAAAA: ZxtL-Bmd...

    2022

  13. [20]

    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. [Online]. Available: https://link.aps.org/doi/ 10.1103/PRXEnergy.1.013008

    2022 doi

  14. [21]

    Complex Couplings – A universal, adaptive and bilinear formulation of power grid dynamics,

    A. Büttner and F. Hellmann, “Complex Couplings – A universal, adaptive and bilinear formulation of power grid dynamics,” PRX Energy, vol. 3, no. 1, p. 013005, Feb. 2024, arXiv:2308.15285 [nlin]. [Online]. Available: http://arxiv.org/ abs/2308.15285

    2024 arXiv

  15. [22]

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

    A. Büttner, H. Würfel, S. Liemann, J. Schiffer, and F. Hellmann, “Complex-Phase, Data-Driven Identification of Grid-Forming Inverter Dynamics,” Sep. 2024, arXiv:2409.17132 [eess]. [Online]. Available: http://arxiv.org/abs/2409.17132

    2024 arXiv

  16. [23]

    A Phase Theory of Multi-Input Multi-Output Linear Time-Invariant Systems,

    W. Chen, D. Wang, S. Z. Khong, and L. Qiu, “A Phase Theory of Multi-Input Multi-Output Linear Time-Invariant Systems,” SIAM Journal on Control and Optimization , vol. 62, no. 2, pp. 1235–1260, Apr. 2024. [Online]. Available: https://epubs.siam.org/doi/10.1137/22M148968X

    2024 doi

  17. [25]

    Adaptive Dynamical Networks,

    R. Berner, T. Gross, C. Kuehn, J. Kurths, and S. Yanchuk, “Adaptive Dynamical Networks,” Apr. 2023, arXiv:2304.05652 [nlin]. [Online]. Available: http://arxiv.org/abs/2304.05652

    2023 arXiv

  18. [26]

    Scalable robust stability for nonsymmetric heterogeneous networks,

    I. C. Lestas and G. Vinnicombe, “Scalable robust stability for nonsymmetric heterogeneous networks,” Automatica, vol. 43, no. 4, pp. 714–723, Apr. 2007. [Online]. Available: https:// www.sciencedirect.com/science/article/pii/S0005109806004523

    2007

  19. [27]

    Passivity and Decentralized Stability Conditions for Grid-Forming Converters,

    X. He and F. Dörfler, “Passivity and Decentralized Stability Conditions for Grid-Forming Converters,” Oct. 2023, arXiv:2310.09935 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2310.09935

    2023 arXiv

  20. [28]

    Decentralized Parametric Stability Certificates for Grid- Forming Converter Control,

    V. Häberle, X. He, L. Huang, F. Dörfler, and S. Low, “Decentralized Parametric Stability Certificates for Grid- Forming Converter Control,” Jun. 2025, arXiv:2503.05403 [eess]. [Online]. Available: http://arxiv.org/abs/2503.05403

    2025 arXiv

  21. [29]

    Transient Angle Stability of Virtual Synchronous Generators Using Lyapunov’s Direct Method,

    Z. Shuai, C. Shen, X. Liu, Z. Li, and Z. J. Shen, “Transient Angle Stability of Virtual Synchronous Generators Using Lyapunov’s Direct Method,” IEEE Transactions on Smart Grid, vol. 10, no. 4, pp. 4648–4661, Jul. 2019, conference Name: IEEE Transactions on Smart Grid. [Online]...

    2019

  22. [30]

    Voltage Stabilization in Microgrids via Quadratic Droop Control,

    J. W. Simpson-Porco, F. Dörfler, and F. Bullo, “Voltage Stabilization in Microgrids via Quadratic Droop Control,” IEEE Transactions on Automatic Control , vol. 62, no. 3, pp. 1239–1253, Mar. 2017, conference Name: IEEE Transactions on Automatic Control. [Online]. Available: ht...

    2017

  23. [31]

    Stabilizing Large-Scale Electric Power Grids with Adaptive Inertia,

    J. Fritzsch and P. Jacquod, “Stabilizing Large-Scale Electric Power Grids with Adaptive Inertia,” PRX Energy , vol. 3, no. 3, p. 033003, Aug. 2024, publisher: American Physical Society. [Online]. Available: https://link.aps.org/doi/10.1103/ PRXEnergy.3.033003

    2024

  24. [32]

    Dynamic stability of electric power grids: Tracking the interplay of the network structure, transmission losses, and voltage dynamics,

    P. C. Böttcher, D. Witthaut, and L. Rydin Gorjão, “Dynamic stability of electric power grids: Tracking the interplay of the network structure, transmission losses, and voltage dynamics,” Chaos: An Interdisciplinary Journal of Nonlinear Science , vol. 32, no. 5, p. 053117, May ...

    2022 doi

  25. [33]

    Dynamic Ancillary Services: From Grid Codes to Transfer Function-Based Converter Control,

    V. Häberle, L. Huang, X. He, E. Prieto-Araujo, and F. Dörfler, “Dynamic Ancillary Services: From Grid Codes to Transfer Function-Based Converter Control,” Oct. 2023, arXiv:2310.01552 [cs, eess]. [Online]. Available: http://arxiv.org/abs/2310.01552

    2023 arXiv

  26. [34]

    Taxonomy of Power Converter Control Schemes Based on the Complex Frequency Concept,

    D. Moutevelis, J. Roldán-Pérez, M. Prodanovic, and F. Milano, “Taxonomy of Power Converter Control Schemes Based on the Complex Frequency Concept,” IEEE Transactions on Power Systems, vol. 39, no. 1, pp. 1996–2009, Jan. 2024, conference Name: IEEE Transactions on Power Systems...

    1996

  27. [35]

    Dispatchable Virtual Oscillator Control for Decentralized Inverter-dominated Power Systems: Analysis and Experiments,

    G.-S. Seo, M. Colombino, I. Subotic, B. Johnson, D. Groß, and F. Dörfler, “Dispatchable Virtual Oscillator Control for Decentralized Inverter-dominated Power Systems: Analysis and Experiments,” in 2019 IEEE Applied Power Electronics Con- ference and Exposition (APEC), Mar. 201...

    2019

  28. [36]

    The Effect of Transmission-Line Dynamics on Grid-Forming Dis- patchable Virtual Oscillator Control,

    D. Groß, M. Colombino, J.-S. Brouillon, and F. Dörfler, “The Effect of Transmission-Line Dynamics on Grid-Forming Dis- patchable Virtual Oscillator Control,” IEEE Transactions on Control of Network Systems , vol. 6, no. 3, pp. 1148–1160, Sep. 2019, conference Name: IEEE Transa...

    2019

  29. [37]

    R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, 1st ed. Cambridge University Press, Apr. 1991. 13 [Online]. Available: https://www.cambridge.org/core/product/ identifier/9780511840371/type/book

    1991

  30. [38]

    On the phases of a complex matrix,

    D. Wang, W. Chen, S. Z. Khong, and L. Qiu, “On the phases of a complex matrix,”Linear Algebra and its Applications , vol. 593, pp. 152–179, May 2020. [Online]. Available: https://www. sciencedirect.com/science/article/pii/S0024379520300458

    2020

  31. [39]

    On the phases of a semi-sectorial matrix and the essential phase of a Laplacian,

    D. Wang, X. Mao, W. Chen, and L. Qiu, “On the phases of a semi-sectorial matrix and the essential phase of a Laplacian,” Linear Algebra and its Applications , vol. 676, pp. 441–458, Nov. 2023. [Online]. Available: https://www.sciencedirect.com/ science/article/pii/S0024379523002744

    2023

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.