Negative Imaginary and Passivity Properties of Synchronous Machine Systems

Elizabeth L. Ratnam; Ian R. Petersen; Maryam Khodabakhshloo

arxiv: 2605.04796 · v1 · submitted 2026-05-06 · 📡 eess.SY · cs.SY

Negative Imaginary and Passivity Properties of Synchronous Machine Systems

Maryam Khodabakhshloo , Elizabeth L. Ratnam , Ian R. Petersen This is my paper

Pith reviewed 2026-05-08 16:56 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords synchronous machinespassivitynegative imaginary systemsdq-framedroop controlpower system stabilitydissipativityrenewable integration

0 comments

The pith

Synchronous machine systems in the nonlinear dq-frame are passive from current input to voltage output and negative imaginary from torque input to rotor angle output.

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

The paper establishes that synchronous machine systems modeled nonlinearly in the dq-frame exhibit passivity from current inputs to voltage outputs and a nonlinear negative imaginary property from torque inputs to rotor angle outputs. Shifting the equations around an equilibrium point yields explicit conditions under which both properties hold. The authors then show that these dissipativity traits are preserved with identical supply rates when the machine interconnects to a passive droop controller, which directly implies closed-loop stability. This matters for power systems because rising renewable penetration makes dynamics more nonlinear and demands stability certificates that survive controller interconnections.

Core claim

We prove that synchronous machine systems, modeled in the nonlinear dq-frame, possess fundamental dissipativity properties. Specifically, we show passivity from current input to voltage output and a nonlinear negative imaginary property from torque input to rotor angle output. For the nonlinear system shifted around an equilibrium point, we derive explicit conditions for both passivity and the NI property to hold. Finally, we demonstrate that interconnection with passive droop controllers preserves these dissipativity properties with identical supply rates, thereby ensuring closed-loop stability.

What carries the argument

The nonlinear dq-frame model of the synchronous machine, shifted around an equilibrium, which supplies the passivity and nonlinear negative imaginary supply rates used for stability proofs.

If this is right

Closed-loop stability follows directly when the machine is interconnected to any passive droop controller that uses the same supply rates.
Explicit algebraic conditions on machine parameters and equilibrium values certify both passivity and the nonlinear NI property.
The same dissipativity framework can be reused to certify stability of larger networks containing multiple synchronous machines and droop-controlled inverters.
Stability margins remain unchanged upon interconnection because the supply rates are identical before and after the connection.

Where Pith is reading between the lines

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

The same passivity and NI arguments may extend to averaged models of inverter-based resources that share a similar dq-frame structure.
Real-time grid simulators could be used to check whether the derived equilibrium conditions remain satisfied under measured renewable variability.
Controller synthesis tools that enforce passivity or NI properties could be applied directly to the shifted machine model to enlarge the region of stable operation.

Load-bearing premise

The nonlinear dq-frame equations accurately represent the machine and the equilibrium shift yields conditions that remain valid for the operating points of interest.

What would settle it

A numerical simulation or hardware test in which the supply-rate inequality for passivity or the nonlinear NI property is violated for the shifted model under the stated conditions.

Figures

Figures reproduced from arXiv: 2605.04796 by Elizabeth L. Ratnam, Ian R. Petersen, Maryam Khodabakhshloo.

**Figure 1.** Figure 1: SM connected to an impedance load via transmis view at source ↗

**Figure 2.** Figure 2: Feedback interconnection of the SM (P) and a torque controller (C). The plant includes both an electrical port (˜ue, y˜e) and a mechanical port (˜um, y˜m) system is passive under the condition (15) for the inputoutput pair (˜ue, y˜e). Next, consider a torque controller that is passive, such as a droop-based PI controller, defined by T˜m = kpω˜ + kiz, ˜ ˜˙z = ˜ω, kp, ki ≥ 0. (18) This PI controller admits … view at source ↗

read the original abstract

The recent rapid proliferation of renewable energy is fundamentally changing the dynamic operations of power systems, necessitating new approaches to assess stability for these highly nonlinear systems. In this paper, we prove that synchronous machine systems, modeled in the nonlinear dq-frame, possess fundamental dissipativity properties. Specifically, we show passivity from current input to voltage output and a nonlinear negative imaginary property from torque input to rotor angle output. For the nonlinear system shifted around an equilibrium point, we derive explicit conditions for both passivity and the NI property to hold. Finally, we demonstrate that interconnection with passive droop controllers preserves these dissipativity properties with identical supply rates, thereby ensuring closed-loop stability.

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 derives explicit conditions for passivity and nonlinear negative-imaginary behavior in the dq-frame synchronous machine after an equilibrium shift, plus preservation under passive droop interconnection, but the shift's effect on supply rates needs tighter verification.

read the letter

The core result is that the nonlinear dq-frame synchronous machine model is passive from current input to voltage output and satisfies a nonlinear negative-imaginary property from torque input to rotor angle output once shifted to an equilibrium, with explicit conditions given for both. The authors also show that connecting the machine to a passive droop controller keeps the same supply rates, which lets them conclude closed-loop stability by standard dissipativity arguments. This combination applied to the machine equations is the main new piece relative to existing power-system dissipativity work. They do a clean job of starting from the standard machine dynamics, performing the shift, and checking how the interconnection affects the supply rates without introducing extra terms. That kind of explicit preservation result is useful when you want to certify larger interconnections without re-deriving everything. The soft spot sits with the equilibrium shift itself. For a genuinely nonlinear system the coordinate change does not automatically carry the original storage function and supply rate over to the new variables in a neighborhood or globally; you need to confirm the inequality still holds after the transformation and that the dq-frame assumptions remain compatible. The abstract states explicit conditions, but if those conditions are only local or rest on unstated bounds on operating region or parameters, the claim that the properties survive with identical supply rates for the droop interconnection becomes narrower than it first appears. Readers working on dissipativity-based stability certificates for inverter-rich grids will find the explicit conditions and interconnection step practical. The paper is grounded enough in the equations to deserve a serious referee, mainly to check the derivations around the shift and any simulation support for the conditions. I would send it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript proves that synchronous machine systems modeled in the nonlinear dq-frame possess passivity from current input to voltage output and a nonlinear negative imaginary property from torque input to rotor angle output. For the system shifted around an equilibrium, explicit conditions are derived under which both properties hold; interconnection with passive droop controllers is shown to preserve the dissipativity properties with identical supply rates, thereby guaranteeing closed-loop stability.

Significance. If the central derivations hold, the work supplies a dissipativity framework for stability assessment of nonlinear power systems with synchronous machines and high renewable penetration. The explicit conditions after equilibrium shift and the preservation result under interconnection are potentially valuable for controller synthesis and analysis; the application of nonlinear NI and passivity concepts to this setting is a clear strength.

major comments (2)

[Shifted-system analysis and supply-rate definitions] The derivation of explicit conditions for passivity and the nonlinear NI property after the equilibrium shift (detailed in the section presenting the shifted model and supply rates) must verify that the storage functions continue to satisfy the dissipation inequalities in the new coordinates. For nonlinear systems the shift is not automatic; the paper should confirm that the supply rates remain valid in a neighborhood without additional restrictions on the operating region or machine parameters.
[Interconnection and closed-loop stability] The claim that interconnection with passive droop controllers preserves the original passivity and NI properties with identical supply rates (in the interconnection-stability section) requires an explicit check that the droop-controller terms do not reintroduce nonlinearities that alter the supply rates or violate the dissipation inequalities. The dq-frame modeling assumptions should also be shown to remain compatible after interconnection.

minor comments (2)

[Preliminaries] Clarify the precise definition of the nonlinear negative-imaginary supply rate and its relation to the standard linear NI definition; a short comparison paragraph would aid readability.
[Notation and modeling] Ensure all equilibrium-shift variables are consistently notated (e.g., use of subscript 'e' or '0') throughout the derivations and figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed review, which highlights key aspects of the nonlinear dissipativity analysis. We address each major comment below and will incorporate clarifications to strengthen the manuscript.

read point-by-point responses

Referee: The derivation of explicit conditions for passivity and the nonlinear NI property after the equilibrium shift (detailed in the section presenting the shifted model and supply rates) must verify that the storage functions continue to satisfy the dissipation inequalities in the new coordinates. For nonlinear systems the shift is not automatic; the paper should confirm that the supply rates remain valid in a neighborhood without additional restrictions on the operating region or machine parameters.

Authors: We agree that the equilibrium shift for nonlinear systems requires explicit verification of the dissipation inequalities. Our storage functions are selected to be positive definite around the equilibrium, and the explicit conditions derived in terms of machine parameters and the equilibrium point ensure the inequalities hold locally. These conditions inherently define the neighborhood of validity without imposing further restrictions on the operating region. To address the concern directly, we will add a clarifying statement in the revised version confirming that the supply rates remain valid under the stated conditions. revision: partial
Referee: The claim that interconnection with passive droop controllers preserves the original passivity and NI properties with identical supply rates (in the interconnection-stability section) requires an explicit check that the droop-controller terms do not reintroduce nonlinearities that alter the supply rates or violate the dissipation inequalities. The dq-frame modeling assumptions should also be shown to remain compatible after interconnection.

Authors: The interconnection result relies on the passivity of the droop controllers and the structure of the feedback interconnection, which is power-preserving. The droop terms are linear in the dq-frame and do not alter the original nonlinear supply rates or violate the dissipation inequalities; the combined system satisfies the sum of the individual supply rates. The dq-frame assumptions remain compatible because the controllers operate in the same reference frame as the machine model. We will add an explicit verification step or remark in the interconnection section to confirm these points without changing the core claims. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations follow from model equations and standard dissipativity definitions

full rationale

The paper states that passivity (current input to voltage output) and the nonlinear negative-imaginary property (torque input to rotor angle output) are shown for the nonlinear dq-frame synchronous machine model after an equilibrium shift, with explicit conditions derived and preservation shown under interconnection with passive droop controllers using identical supply rates. No quoted steps reduce a claimed prediction or property to a fitted parameter or self-referential definition by construction, nor do any load-bearing steps rely on self-citations whose content is unverified or imported as uniqueness theorems. The central claims rest on application of dissipativity theory to the given system dynamics, which is independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger reflects standard modeling choices typical for this domain rather than paper-specific inventions. The central claims rest on the dq-frame transformation being a valid representation and on the definitions of passivity and negative imaginary supply rates being applicable to the shifted nonlinear system.

axioms (2)

domain assumption The synchronous machine dynamics are accurately captured by the standard nonlinear dq-frame equations
Invoked implicitly when the paper states the model possesses the claimed properties.
domain assumption Equilibrium shift preserves the dissipativity structure with the stated supply rates
Required for the explicit conditions on the shifted system to hold.

pith-pipeline@v0.9.0 · 5412 in / 1348 out tokens · 40298 ms · 2026-05-08T16:56:52.429337+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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Yang, P., Liu, F., Wang, Z., and Shen, C. (2019). Dis- tributed stability conditions for power systems with het- erogeneous nonlinear bus dynamics.IEEE Transactions on Power Systems, 35(3), 2313–2324

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and Weiss, G

Zhong, Q.C. and Weiss, G. (2010). Synchronverters: Inverters that mimic synchronous generators.IEEE transactions on industrial electronics, 58(4), 1259–1267

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and Ohsawa, Y

Zhou, J. and Ohsawa, Y. (2008). Improved swing equation and its properties in synchronous generators.IEEE Transactions on Circuits and Systems I: Regular Papers, 56(1), 200–209

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Arghir, C., Jouini, T., and D¨ orfler, F. (2018). Grid- forming control for power converters based on matching of synchronous machines.Automatica, 95, 273–282

work page 2018

[2] [2]

Brogliato, B., Lozano, R., Maschke, B., Egeland, O., et al. (2007). Dissipative systems analysis and control.Theory and applications, 2, 2–5

work page 2007

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Ghallab, A.G., Mabrok, M.A., and Petersen, I.R. (2025). Negative imaginary systems theory for nonlinear sys- tems: A dissipativity approach.IEEE Transactions on Automatic Control

work page 2025

[4] [4]

(1999).Power system analysis

Grainger, J.J. (1999).Power system analysis. McGraw- Hill

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Kundur, P. (2007). Power system stability.Power system stability and control, 10(1), 7–1

work page 2007

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Stankovic, A., Taylor, C., et al. (2004). Definition and classification of power system stability ieee/cigre joint task force on stability terms and definitions.IEEE transactions on Power Systems, 19(3), 1387–1401

work page 2004

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(2020).Power system dynamics: stability and control

Machowski, J., Lubosny, Z., Bialek, J.W., and Bumby, J.R. (2020).Power system dynamics: stability and control. John Wiley & Sons

work page 2020

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Monshizadeh, P., Monshizadeh, N., De Persis, C., and van der Schaft, A. (2018). Output impedance diffusion into lossy power lines.IEEE Transactions on Power Systems, 34(3), 1659–1668

work page 2018

[9] [9]

and Ishizaki, T

Nishino, T. and Ishizaki, T. (2025). Equilibrium- independent passivity of power systems composed of park synchronous generator models. In2025 American Control Conference (ACC), 748–753. IEEE

work page 2025

[10] [10]

and Lanzon, A

Petersen, I.R. and Lanzon, A. (2010). Feedback control of negative-imaginary systems.IEEE Control Systems Magazine, 30(5), 54–72

work page 2010

[11] [11]

(2017).Power system dynamics and stability: with synchrophasor mea- surement and power system toolbox

Sauer, P.W., Pai, M.A., and Chow, J.H. (2017).Power system dynamics and stability: with synchrophasor mea- surement and power system toolbox. John Wiley & Sons

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Trip, S., B¨ urger, M., and De Persis, C. (2016). An internal model approach to (optimal) frequency regulation in power grids with time-varying voltages.Automatica, 64, 240–253

work page 2016

[13] [13]

Willems, J.C. (1972). Dissipative dynamical systems part i: General theory.Archive for rational mechanics and analysis, 45(5), 321–351

work page 1972

[14] [14]

Yang, P., Liu, F., Wang, Z., and Shen, C. (2019). Dis- tributed stability conditions for power systems with het- erogeneous nonlinear bus dynamics.IEEE Transactions on Power Systems, 35(3), 2313–2324

work page 2019

[15] [15]

and Weiss, G

Zhong, Q.C. and Weiss, G. (2010). Synchronverters: Inverters that mimic synchronous generators.IEEE transactions on industrial electronics, 58(4), 1259–1267

work page 2010

[16] [16]

and Ohsawa, Y

Zhou, J. and Ohsawa, Y. (2008). Improved swing equation and its properties in synchronous generators.IEEE Transactions on Circuits and Systems I: Regular Papers, 56(1), 200–209

work page 2008