pith. sign in

arxiv: 2604.21234 · v1 · submitted 2026-04-23 · 📡 eess.SY · cs.SY

A Dynamic Phasor Framework for Analysis of IBR-Induced SSOs in Multi-Machine Systems

Fiaz Hossain , Nilanjan Ray Chaudhuri , Constantino M. Lagoa This is my paper

Pith reviewed 2026-05-09 21:34 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords dynamic phasorsubsynchronous oscillationsinverter-based resourcesgrid-following inverterseigenanalysisdamping controlmulti-machine systemsunbalanced conditions
0
0 comments X

The pith

A dynamic phasor framework renders mixed IBR and synchronous generator models linear and time-invariant so eigen decomposition can identify and damp subsynchronous oscillation modes.

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

The paper proposes a generalized dynamic phasor framework that analyzes subsynchronous oscillations induced by grid-following inverter-based resources connected to multi-machine power systems under both balanced and unbalanced conditions. Grid-following IBRs are represented in dq-frame dynamic phasors while synchronous generators and the dynamic transmission network use pnz-frame dynamic phasors, producing an overall linear time-invariant system model. This property allows eigen decomposition for root-cause analysis of the oscillation modes and for designing decentralized damping controllers based on local IBR signals. The same model also quantifies how data-center loads can excite the modes and shows their locational dependence. Validation occurs on a modified IEEE two-area system where two synchronous generators are replaced by IBRs, with results cross-checked against detailed EMTDC/PSCAD simulations.

Core claim

By transforming the system into dynamic phasors that keep the models linear and time-invariant, the framework enables eigen decomposition of the resulting state-space representation. This decomposition directly reveals the subsynchronous oscillation modes, their participation factors, and the effect of control parameters, even when the system experiences unbalanced faults or additional DC loads.

What carries the argument

Dynamic phasor transformation applied separately to dq-frames for GFL IBRs and pnz-frames for SGs plus dynamic network, producing a linear time-invariant state-space model amenable to eigen decomposition.

If this is right

  • Eigen decomposition supplies participation factors and root-cause information for each SSO mode.
  • Robust decentralized H-infinity controllers can be synthesized directly from local IBR signals to damp the modes.
  • The same linear model supports analysis of unbalanced faults and quantifies how DC load location affects mode excitation.
  • Results on the modified IEEE two-area benchmark confirm the approach reproduces EMT behavior for both time- and frequency-domain responses.

Where Pith is reading between the lines

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

  • The framework could be used to derive stability margins or placement guidelines for new IBRs and loads before they are installed.
  • Because the model remains LTI under unbalanced conditions, it might support real-time monitoring or adaptive retuning of damping controllers as operating points drift.
  • Extension to larger networks would allow systematic comparison of different inverter control structures for their SSO susceptibility.

Load-bearing premise

The chosen dq-frame models for GFL IBRs and pnz-frame models for SGs plus the dynamic network accurately capture the interactions that produce the observed SSOs, and linearization around the operating point remains valid for the phenomena of interest.

What would settle it

If the frequencies, damping ratios, or mode shapes obtained from eigen decomposition of the dynamic-phasor state-space model differ materially from those measured in the corresponding EMTDC/PSCAD time-domain simulations, the framework's predictive accuracy would be refuted.

Figures

Figures reproduced from arXiv: 2604.21234 by Constantino M. Lagoa, Fiaz Hossain, Nilanjan Ray Chaudhuri.

Figure 2
Figure 2. Figure 2: Modified IEEE two-area benchmark system with 50% IBR penetration [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: Proposed genralized DP-based modeling framework capable of [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: Block diagram representation of the models of (a) PLL, (b) outer [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of tie line power flow between DP and EMT models [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Frequency response of the transfer functions from perturbation ( [PITH_FULL_IMAGE:figures/full_fig_p004_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Performance of decentralized controller for damping SSO mode [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Magnitude responses of the transfer functions from DC load input [PITH_FULL_IMAGE:figures/full_fig_p005_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: (a) DC load active power and SSO mode excitation due to DC load [PITH_FULL_IMAGE:figures/full_fig_p005_8.png] view at source ↗
read the original abstract

We propose a generalized dynamic phasor (DP) framework to analyze inverter-based resources (IBRs) connected to multi-machine systems under balanced and unbalanced conditions. It captures subsynchronous oscillations (SSOs) induced by grid-following (GFL) IBRs. The linearizability and time invariance of the framework enables us to perform eigen decomposition, which is a powerful tool for root-cause analysis of the SSO modes and damping controller design. The same framework also enables analysis of excitation of the SSO modes in presence of data center (DC) loads. The GFL IBRs are modeled in their respective $dq$-frame DPs and the detailed model of synchronous generators (SGs) along with dynamic transmission network models are represented in $pnz$-frame DPs. Several case studies are performed on the modified IEEE two-area benchmark system, where $2$ SGs are replaced by GFL IBRs and validated with EMTDC/PSCAD simulations. First, time- and frequency-domain analyses of the SSO mode are presented followed by the design of a robust decentralized $\mathcal{H}_\infty$ damping controller based on local signals of the GFL IBRs. Second, the dynamic behavior of the system following an unbalanced fault is demonstrated that is damped by the proposed damping controller. Finally, excitation of the SSO mode in presence of DC load is exhibited and its locational impact is analytically quantified.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a generalized dynamic phasor (DP) framework for analyzing inverter-based resource (IBR)-induced subsynchronous oscillations (SSOs) in multi-machine systems under balanced and unbalanced conditions. Grid-following (GFL) IBRs are modeled in dq-frame DPs while synchronous generators (SGs) and the dynamic transmission network use pnz-frame DPs. The authors claim the resulting model is linear and time-invariant, enabling eigen decomposition for root-cause analysis of SSO modes and design of a robust decentralized H-infinity damping controller using local IBR signals. Case studies on a modified IEEE two-area benchmark system (with two SGs replaced by GFL IBRs) include time/frequency-domain SSO analysis, unbalanced fault response, and DC-load excitation of SSO modes; results are validated against EMTDC/PSCAD simulations.

Significance. If the mixed-frame DP model is strictly linear and time-invariant, the work provides a useful extension of established dynamic-phasor techniques to mixed dq-pnz settings for SSO root-cause analysis and H-infinity controller synthesis in IBR-dominated systems. The EMTDC/PSCAD validation and explicit treatment of unbalanced faults plus DC-load effects are concrete strengths that would support practical applicability in stability studies of renewable-integrated grids.

major comments (2)
  1. [DP modeling and interfacing section] The central claim that the combined dq-pnz DP model is strictly time-invariant (enabling exact eigen decomposition) requires explicit derivation showing that interface transformations remove all time-dependent cross-coupling terms. Under unbalanced conditions the pnz components of the network and SGs introduce frequency shifts and admittance-matrix interactions whose time dependence is eliminated only if the DP frequency shift is applied uniformly and the reference frame is chosen consistently; any residual explicit time variation would make the state matrix time-varying and undermine the eigenanalysis and H-infinity design. This point is load-bearing for the abstract's strongest claim and must be demonstrated with the state-space equations in the modeling section.
  2. [Case studies and validation section] Validation against EMTDC/PSCAD is stated for the SSO mode eigenvalues, time responses, and controller performance, but no quantitative error metrics (e.g., eigenvalue deviation, RMS time-domain error), data-exclusion criteria, or operating-point linearization validity ranges are reported. Without these, the support for the claim that the framework accurately captures IBR-SG interactions and that the H-infinity controller damps the modes cannot be fully assessed.
minor comments (2)
  1. [Abstract and Introduction] The abstract and introduction could more explicitly contrast the mixed dq-pnz DP approach with prior single-frame DP or EMT-based SSO studies to clarify novelty.
  2. [Notation and modeling equations] Notation for the pnz-frame variables and the interface transformation matrices should be defined once with a consistent symbol table to improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and positive assessment of the manuscript's significance. We address each major comment below and will revise the paper to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [DP modeling and interfacing section] The central claim that the combined dq-pnz DP model is strictly time-invariant (enabling exact eigen decomposition) requires explicit derivation showing that interface transformations remove all time-dependent cross-coupling terms. Under unbalanced conditions the pnz components of the network and SGs introduce frequency shifts and admittance-matrix interactions whose time dependence is eliminated only if the DP frequency shift is applied uniformly and the reference frame is chosen consistently; any residual explicit time variation would make the state matrix time-varying and undermine the eigenanalysis and H-infinity design. This point is load-bearing for the abstract's strongest claim and must be demonstrated with the state-space equations in the modeling section.

    Authors: We acknowledge the need for an explicit derivation to confirm time-invariance. While the manuscript describes the dq-frame modeling of GFL IBRs and pnz-frame modeling of SGs and the network along with their interfacing, the full state-space equations demonstrating elimination of time-dependent terms were not presented in detail. In the revised manuscript, we will expand the modeling section to include these equations, showing that the uniform DP frequency shift applied across pnz components, combined with consistent reference-frame alignment at the interfaces, cancels all explicit time variations arising from frequency shifts and admittance interactions under both balanced and unbalanced conditions. This will rigorously establish the linearity and time-invariance of the overall model. revision: yes

  2. Referee: [Case studies and validation section] Validation against EMTDC/PSCAD is stated for the SSO mode eigenvalues, time responses, and controller performance, but no quantitative error metrics (e.g., eigenvalue deviation, RMS time-domain error), data-exclusion criteria, or operating-point linearization validity ranges are reported. Without these, the support for the claim that the framework accurately captures IBR-SG interactions and that the H-infinity controller damps the modes cannot be fully assessed.

    Authors: We agree that quantitative metrics are necessary for a complete assessment. In the revised manuscript, we will add explicit error metrics, including eigenvalue deviations between the DP model and EMTDC/PSCAD results for the SSO modes, RMS errors for the time-domain responses, and a clear statement of the operating-point ranges over which the linearization is valid. We will also document the data-exclusion criteria applied in the comparisons. These additions will strengthen the validation of the framework's accuracy for IBR-SG interactions and the damping controller performance. revision: yes

Circularity Check

0 steps flagged

No circularity: LTI property follows from standard DP transformations, not by construction or self-reference

full rationale

The paper constructs the mixed dq/pnz DP model by applying established dynamic-phasor frequency-shift transformations separately to GFL IBRs (dq) and SGs/network (pnz), then interfaces the states. The resulting claim of overall linear time-invariance is presented as a direct algebraic consequence of those transformations (constant-coefficient state matrix after uniform shift), not as a fitted quantity or a redefinition of the SSO modes themselves. Eigen decomposition is then applied as a standard tool to the derived LTI system; no prediction is shown to equal its own input data or a self-cited uniqueness theorem. Validation against independent EMTDC/PSCAD runs further separates the modeling step from the analysis step. No load-bearing self-citation, ansatz smuggling, or renaming of known results occurs in the core derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Information is limited to the abstract; the framework rests on standard small-signal linearization and frame-specific modeling assumptions common to power-system dynamics.

axioms (2)
  • domain assumption The system dynamics can be linearized around a steady-state operating point while preserving the essential SSO modes.
    Required for eigen decomposition as stated.
  • domain assumption Dynamic phasor representations in dq and pnz frames accurately reproduce the interaction between GFL IBRs and synchronous generators.
    Core modeling choice enabling the unified framework.

pith-pipeline@v0.9.0 · 5568 in / 1386 out tokens · 39327 ms · 2026-05-09T21:34:34.235072+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Limiting the Impact of AI Data Centers on Fatigue Life of Thermal Turbine Generators in the Grid: A Frequency-Domain Approach

    eess.SY 2026-05 unverdicted novelty 5.0

    A three-step frequency-domain method using mechanical first-principles models, load-flow interaction factors, and optimization sets safe limits on AI data center load variations to protect synchronous generator fatigue life.

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · cited by 1 Pith paper

  1. [1]

    Real-world subsynchronous oscillation events in power grids with high penetrations of inverter-based resources,

    Y . Chenget al., “Real-world subsynchronous oscillation events in power grids with high penetrations of inverter-based resources,”IEEE Trans. on Power Syst., vol. 38, no. 1, pp. 316–330, 2023

    work page 2023

  2. [2]

    Characteristics and risks of emerging large loads,

    NERC, “Characteristics and risks of emerging large loads,” North American Electric Reliability Corporation, White Paper, 2025

    work page 2025

  3. [3]

    Generalized averaging method for power conversion circuits,

    S. Sanders, J. Noworolski, X. Liu, and G. Verghese, “Generalized averaging method for power conversion circuits,”IEEE Trans. on Power Electronics, vol. 6, no. 2, pp. 251–259, 1991

    work page 1991

  4. [4]

    Analysis of asymmetrical faults in power systems using dynamic phasors,

    A. Stankovic and T. Aydin, “Analysis of asymmetrical faults in power systems using dynamic phasors,”IEEE Trans. on Power Syst., vol. 15, no. 3, pp. 1062–1068, 2000

    work page 2000

  5. [5]

    Dynamic phasor analysis of SSR mitigation schemes based on passive phase imbalance,

    M. C. Chudasama and A. M. Kulkarni, “Dynamic phasor analysis of SSR mitigation schemes based on passive phase imbalance,”IEEE Trans. on Power Syst., vol. 26, no. 3, pp. 1668–1676, 2011

    work page 2011

  6. [6]

    A dynamic phasor framework for analysis of grid-forming converter connected to series-compensated line,

    F. Hossain and N. R. Chaudhuri, “A dynamic phasor framework for analysis of grid-forming converter connected to series-compensated line,” presented in IEEE PES General Meeting 2025. [Online]. Available: https://arxiv.org/abs/2505.17414

    work page arXiv 2025

  7. [7]

    Analysis and application of quasi-static and dynamic phasor calculus for stability assessment of integrated power electric and electronic systems,

    J. Vega-Herrera, C. Rahmann, F. Valencia, and K. Strunz, “Analysis and application of quasi-static and dynamic phasor calculus for stability assessment of integrated power electric and electronic systems,”IEEE Trans. on Power Syst., vol. 36, no. 3, pp. 1750–1760, 2021

    work page 2021

  8. [8]

    Dynamic phasor modeling of multi-generator variable frequency electrical power systems,

    T. Yanget al., “Dynamic phasor modeling of multi-generator variable frequency electrical power systems,”IEEE Trans. on Power Syst., vol. 31, no. 1, pp. 563–571, 2016

    work page 2016

  9. [9]

    Simulation of power system dynamics using dynamic phasor models,

    T. Demiray, “Simulation of power system dynamics using dynamic phasor models,” Ph.D. dissertation, ETH Zurich, 2008

    work page 2008

  10. [10]

    Application of dynamic phasor models for the analysis of power systems with phase imbalance,

    M. Chudasama, “Application of dynamic phasor models for the analysis of power systems with phase imbalance,” Ph.D. dissertation, Indian Institute of Technology Bombay, 2012

    work page 2012

  11. [11]

    IEEE recommended practice for excitation system models for power system stability studies (IEEE Std. 421.5-2005),

    IEEE Power & Energy Society, “IEEE recommended practice for excitation system models for power system stability studies (IEEE Std. 421.5-2005),” IEEE, Tech. Rep., Apr. 2005

    work page 2005

  12. [12]

    Pal and B

    B. Pal and B. Chaudhuri,Robust control in power systems. Springer, 2005. [13]MATLAB version 9.14.0.2286388 (R2023a), The MathWorks, Inc., Nat- ick, Massachusetts, 2023. [14]PSCAD/EMTDC, Manitoba Hydro International Ltd., Winnipeg, MB, Canada, 2024, version 5.0.2

    work page 2005

  13. [13]

    DecentralizedH ∞ controller design: A matrix inequality approach using a homotopy method,

    G. Zhai, M. Ikeda, and Y . Fujisaki, “DecentralizedH ∞ controller design: A matrix inequality approach using a homotopy method,”IF AC Proceedings V olumes, vol. 31, no. 20, pp. 151–156, 1998

    work page 1998