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arxiv: 2607.01688 · v1 · pith:4F75BS2Jnew · submitted 2026-07-02 · 📡 eess.SY · cs.SY

A Dynamic Phasor Framework for Analysis of Subsynchronous Oscillations in Multi-Machine Systems with IBRs and Large Loads

Fiaz Hossain , Nilanjan Ray Chaudhuri , Constantino M. Lagoa , Alok Sinha , ai Gopal Vennelaganti , Mohammed E. Nassar This is my paper

Pith reviewed 2026-07-03 08:05 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords dynamic phasorsubsynchronous oscillationsinverter-based resourcesmulti-machine systemseigen decompositiondamping controllersgrid-followinggrid-forming
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The pith

A mixed dq- and pnz-frame dynamic phasor model keeps large multi-machine systems linear and time-invariant so eigen decomposition can identify 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 introduces a generalized dynamic phasor framework for studying subsynchronous oscillations in multi-machine power systems that include inverter-based resources and large loads. It represents grid-following and grid-forming IBRs using dq-frame dynamic phasors while placing multi-mass synchronous generators, networks, and loads in pnz-frame dynamic phasors. This choice keeps the overall model linear and time-invariant, which supports eigen decomposition to trace the origins of oscillation modes and to design damping controllers. The same structure also permits faster time-domain simulation of big systems compared with full electromagnetic transient models. The approach is demonstrated on benchmark systems and a modified IEEE 68-bus case that includes IBR-induced modes and artificial intelligence data center loads.

Core claim

A generalized dynamic phasor framework is proposed in which GFL and GFM IBRs are modeled in dq-frame DPs while detailed multi-mass SGs, dynamic networks, and loads are represented in pnz-frame DPs. The linearizability and time-invariance of this mixed-frame model enable eigen decomposition for root-cause analysis of SSO modes and damping controller design, in addition to faster simulation of large-scale systems under balanced and unbalanced conditions. The framework is validated against EMTDC/PSCAD on the IEEE first benchmark model and modified IEEE 4-machine and 68-bus systems, with use cases including IBR SSO analysis, PSO-based decentralized control, GFM replacement, and AI DC load impact

What carries the argument

Mixed dq-frame and pnz-frame dynamic phasor models that maintain time-invariance for linear eigen analysis of subsynchronous oscillation modes.

If this is right

  • Root-cause analysis of SSO modes becomes possible through eigen decomposition.
  • Damping controllers can be designed directly from the linearized DP model.
  • Simulation of large-scale systems with IBRs and loads runs faster than EMT equivalents.
  • Analysis extends to unbalanced conditions and effects of control mode changes such as GFM versus GFL.
  • Primary frequency response impacts from large loads like AI data centers can be studied on multi-mass turbines.

Where Pith is reading between the lines

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

  • The framework may allow integration with real-time monitoring systems for online mode tracking.
  • Similar DP modeling could extend to other stability phenomena beyond subsynchronous oscillations.
  • Controller design via particle swarm optimization might generalize to other optimization methods for damping.
  • The pnz-frame choice for networks could simplify unbalanced fault studies in future applications.

Load-bearing premise

The chosen dq-frame and pnz-frame dynamic phasor models remain accurate enough representations of the underlying electromagnetic transient dynamics at the frequencies of interest, even under unbalanced conditions.

What would settle it

Running the DP model on the IEEE first benchmark model and comparing its predicted SSO mode frequencies and damping ratios against those from a full EMTDC/PSCAD simulation; large discrepancies would falsify the accuracy claim.

Figures

Figures reproduced from arXiv: 2607.01688 by ai Gopal Vennelaganti, Alok Sinha, Constantino M. Lagoa, Fiaz Hossain, Mohammed E. Nassar, Nilanjan Ray Chaudhuri.

Figure 1
Figure 1. Figure 1: Proposed genralized DP-based modeling framework capable of analyzing balanced/unbalance conditions. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 4
Figure 4. Figure 4: Block diagram representation of the GFL IBR: (a) PLL, (b) outer [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 3
Figure 3. Figure 3: Current limiting strategy in DP framework. [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: IEEE First benchmark model for SSR [14]. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of responses from DP model and EMT model of IEEE [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of tie line power flow between DP and EMT models [PITH_FULL_IMAGE:figures/full_fig_p007_10.png] view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of responses from DP and EMT model following a [PITH_FULL_IMAGE:figures/full_fig_p007_8.png] view at source ↗
Figure 12
Figure 12. Figure 12: Decentralized controller block diagram. Pc 2; p:u: 6.8 7 7.2 power output of GFL IBR2 NC WC fpll2; H z 59 60 61 PLL f requency of GFL IBR2 NC WC time, s 0 0.5 1 1.5 jv d q 2j; p:u: 0.96 0.98 1 1.02 1.04 voltage magnitude at GFL IBR2 terminal NC WC [PITH_FULL_IMAGE:figures/full_fig_p008_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Responses following a 2 cycles self-clearing L-G fault near bus 50 while one of the parallel tie lines between bus 18 and bus 49 are out for maintenance. (NC: no damping control, WC: with damping control) Pc 2; p:u: 6.99 7 7.01 power output of GFL IBR2 NC WC fpll2; H z 59.9 60 60.1 PLL f requency of GFL IBR2 NC WC time, s 0 1 2 3 4 5 6 7 jv d q 2j; p:u: 0.998 1 1.002 voltage magnitude at GFL IBR2 terminal… view at source ↗
Figure 16
Figure 16. Figure 16: Responses following a 2 cycles self-clearing LLL-G fault near bus 50 (GFL IBR2 is replaced with GFM IBR1). 1) Solution I: Decentralized damping controller design: As described in our previous work [19] the design problem of a low order, decentralized, fixed structure, and stable supplemen￾tary control for oscillation damping is non-convex. We have used a heuristic method called particle swarm optimization… view at source ↗
Figure 17
Figure 17. Figure 17: Primary frequency response following AI DC load ramp. [PITH_FULL_IMAGE:figures/full_fig_p009_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Time-domain response and corresponding heatmap for the considered [PITH_FULL_IMAGE:figures/full_fig_p009_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Boxplots of normalized shaft stresses and generator frequency with respect to their corresponding limits (maximum values considered) for 100 Monte [PITH_FULL_IMAGE:figures/full_fig_p010_19.png] view at source ↗
read the original abstract

Although the electromagnetic transient (EMT) framework can capture subsynchronous oscillations (SSOs), it faces scalability issues for large-scale systems. Thus motivated, we propose a generalized dynamic phasor (DP) framework to analyze SSOs in multi-machine systems with inverter-based resources (IBRs) and large loads such as artificial intelligence data centers (AI DCs) under balanced and unbalanced conditions. The grid-following (GFL) and grid-forming (GFM) IBRs are modeled in their respective $dq$-frame DPs. In contrast, the detailed model of multi-mass turbine driven synchronous generators (SGs) along with dynamic transmission network models and loads are represented in $pnz$-frame DPs. The linearizability and time-invariance of the framework enable us to perform eigen decomposition, which is a powerful tool for root-cause analysis of SSO modes and the design of damping controllers. In addition, the DP modeling approach facilitates faster simulation of large-scale systems. The generalized framework is validated with EMTDC/PSCAD simulations using the IEEE first benchmark model for subsynchronous resonance and the modified IEEE 4-machine system. Several use cases are presented on the modified IEEE 68-bus system with two GFL IBRs to show the applicability of the framework. First, time- and frequency-domain analyses of the IBR-induced SSO mode are presented. Then, two solutions are proposed to damp the poorly damped SSO mode: (a) a decentralized controller is designed using particle swarm optimization, and (b) the control of one GFL IBR is replaced by GFM control. Finally, the impact of AI DC load on primary frequency response of the system and the multi-mass turbines of the SGs are studied.

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 paper proposes a generalized dynamic phasor (DP) framework for subsynchronous oscillation (SSO) analysis in multi-machine systems with GFL/GFM IBRs and large loads (e.g., AI DCs). IBRs are modeled in dq-frame DPs while multi-mass SGs, dynamic networks, and loads use pnz-frame DPs. The framework is asserted to be linear and time-invariant, enabling eigen decomposition for root-cause analysis of SSO modes, damping controller design (via PSO or GFM replacement), and faster large-scale simulation. Validation against EMTDC/PSCAD is claimed on the IEEE first benchmark model for SSR and a modified 4-machine system, with use-case demonstrations on the modified 68-bus system including IBR-induced SSO time/frequency-domain analysis, damping solutions, and AI DC load effects on primary frequency response.

Significance. If the mixed dq/pnz DP models remain faithful to underlying EMT dynamics at subsynchronous frequencies (including under unbalanced conditions), the approach would provide a scalable linear model for modal analysis and controller synthesis in systems too large for full EMT simulation. The use of standard coordinate transformations without free parameters or circular fitting is a strength, and the 68-bus use cases illustrate practical applicability for IBR-induced modes and load impacts.

major comments (2)
  1. [Abstract and §4] Abstract and §4 (validation): the claim that the DP framework enables valid eigen decomposition for SSO modes rests on the unverified assumption that dq-frame IBR and pnz-frame SG/network/load models reproduce EMT frequencies and damping ratios with sufficient accuracy. No quantitative error bounds, frequency-dependent metrics, or explicit SSO eigenvalue comparisons (e.g., real/imaginary parts) between DP and PSCAD are reported for either benchmark system, particularly under unbalanced conditions.
  2. [§5] §5 (68-bus use cases): the reported IBR-induced SSO mode analysis, PSO-based decentralized controller, and GFM replacement results assume the overall DP system remains linear time-invariant and accurate at the relevant frequencies. Without error-bar reporting or unbalanced-condition checks on the pnz-frame network/load models, the extracted eigenvalues and designed damping performance lack demonstrated physical correspondence to EMT.
minor comments (2)
  1. [§2] §2 (modeling): the transition between dq and pnz frames at the point of common coupling should include an explicit statement of how the overall system matrix remains time-invariant when both frame types are present.
  2. [Figures in §4–§5] Figures in §4–§5: overlay DP and EMT time-domain waveforms and eigenvalue loci directly on the same plots to allow visual assessment of approximation quality.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the validation aspects of the manuscript. We address each major comment below and will incorporate quantitative comparisons in the revision.

read point-by-point responses

  1. Referee: [Abstract and §4] Abstract and §4 (validation): the claim that the DP framework enables valid eigen decomposition for SSO modes rests on the unverified assumption that dq-frame IBR and pnz-frame SG/network/load models reproduce EMT frequencies and damping ratios with sufficient accuracy. No quantitative error bounds, frequency-dependent metrics, or explicit SSO eigenvalue comparisons (e.g., real/imaginary parts) between DP and PSCAD are reported for either benchmark system, particularly under unbalanced conditions.

    Authors: We agree that explicit quantitative comparisons are not detailed in the current version. While §4 presents time-domain validation against PSCAD/EMTDC, direct SSO eigenvalue (real/imaginary) comparisons, error bounds, and unbalanced-condition checks are absent. In revision, we will add tables in §4 reporting these metrics for the IEEE first benchmark model and modified 4-machine system to confirm accuracy at subsynchronous frequencies. revision: yes

  2. Referee: [§5] §5 (68-bus use cases): the reported IBR-induced SSO mode analysis, PSO-based decentralized controller, and GFM replacement results assume the overall DP system remains linear time-invariant and accurate at the relevant frequencies. Without error-bar reporting or unbalanced-condition checks on the pnz-frame network/load models, the extracted eigenvalues and designed damping performance lack demonstrated physical correspondence to EMT.

    Authors: The LTI property derives directly from the DP modeling in §2–§3 without time-varying coefficients. We acknowledge the need for stronger linkage in §5. In the revision, we will include error metrics or bars for the 68-bus eigenvalues and add unbalanced network/load validation checks to better demonstrate physical correspondence to EMT for the reported modes and damping results. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper's core steps consist of applying standard dynamic phasor coordinate transformations (dq-frame for IBRs, pnz-frame for SGs/networks/loads) to obtain an LTI representation, followed by standard eigen decomposition for SSO analysis. These transformations and the resulting linearity/time-invariance are properties of the DP method itself, not derived from or fitted to the target SSO modes or controller designs. Validation occurs via external EMTDC/PSCAD comparison on IEEE benchmark systems rather than parameter fitting reused as prediction. No load-bearing self-citations, self-definitional loops, or ansatz smuggling are present in the provided text; the derivation chain remains independent of its outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the standard assumption that dynamic phasor representations preserve the essential dynamics of the EMT model within the frequency band of interest. No new free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Dynamic phasor models in dq and pnz frames accurately capture the subsynchronous dynamics of the underlying EMT system under balanced and unbalanced conditions.
    Invoked when the paper states that the linearizability and time-invariance enable eigen decomposition and faster simulation.

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discussion (0)

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Reference graph

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    SG: 1 Bus: 4 × × × ×

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    SG: 3 IBR: 3 Bus: 9 × × × ×

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    All other multi-mass parameters except damping are taken from

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