Data-Driven Discovery of Multiscale Power System Oscillation Governing Equations Using SINDy-SENDAI

Alberto Berizzi; Andrea Pomarico; Giorgio Maria Giannuzzi; J. Nathan Kutz; Liyao Mars Gao; Salvatore Tessitore; Yuxuan Bao

Data-Driven Discovery of Multiscale Power System Oscillation Governing Equations Using SINDy-SENDAI

T0 review · 4 major / 0 minor · reviewed 2026-07-12 · grok-4.5

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Pith's one-line read A hierarchical deep model separates power-grid sensor frequencies and recovers explicit equations for the dominant electromechanical oscillations, beating Hankel-DMD on both synthetic and real European data.

desk verdict Solid engineering of SINDy inside SENDAI for PMU streams that beats Hankel-DMD on reconstruction and modal recovery; the multi-minute autonomous forecast claim is overstated by the paper’s own figures. read the letter →

arxiv 2607.03485 v1 pith:75LCT7DA submitted 2026-07-03 eess.SY cs.SY

Andrea Pomarico , Yuxuan Bao , Liyao Mars Gao , Salvatore Tessitore , Giorgio Maria Giannuzzi , Alberto Berizzi , J. Nathan Kutz This is my paper · ORCID

classification eess.SYcs.SY

keywords electromechanicaloscillationsSINDySENDAIwide-areameasurementsystemsmodalanalysispowersystemstabilitydata-drivendynamicsPMU

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

Power grids must keep electromechanical oscillations under control, yet modern inverter-based resources make those oscillations harder to see and interpret from noisy, sparse PMU streams. This paper introduces SINDy-SENDAI, a two-pathway network that first peels low-frequency dynamics into a low-dimensional latent space and then discovers a sparse ordinary differential equation there, while a stack of high-frequency correction layers reconstructs the residual fast content. Because the latent equation is explicit, classical linear algebra immediately yields frequencies, damping ratios and mode shapes, and the same equation can be integrated forward for short-term forecasts. On the classic Kundur two-area system the recovered modal parameters match conventional modal analysis more closely than Hankel-DMD; the same architecture also reconstructs and forecasts the 2016 Iberian inter-area event and ambient Italian measurements. The claim is that an interpretable, multiscale dynamical model can be learned directly from wide-area data and is accurate enough for practical stability assessment.

What carries the argument

SINDy-SENDAI: a hierarchical architecture whose low-frequency pathway (GRU encoder + shallow decoder + SINDy regularization) isolates a sparse latent ODE while successive high-frequency peel layers correct residuals under a temporal-frequency sparsity penalty.

What would settle it

On a fresh multi-hour PMU archive that includes a known change of operating point or a forced oscillation, retrain once and then compare autonomous multi-minute forecasts against measured frequencies; systematic growth of error or loss of the known inter-area mode would falsify the claim that the latent ODE remains predictive without re-initialization.

Watch

Extended reading notes

Core claim

SINDy-SENDAI consistently outperforms Hankel-DMD at recovering electromechanical modal frequency and damping, and the four-dimensional linear ODE identified in its low-frequency latent space is already rich enough to reconstruct full-state PMU trajectories and to produce usable one-minute forecasts on both simulated and real European events.

Load-bearing premise

That a single four-dimensional linear ODE fitted to one short training window continues to describe the grid's dominant modes for multi-minute autonomous forecasts even when operating conditions slowly change.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The usable advance here is a concrete pipeline: SINDy-regularized latent dynamics inside the SENDAI low-frequency path, plus temporal-frequency peeling tuned for one-dimensional PMU series. That combination is not in the prior SHRED/SENDAI/SINDy papers, and they show it works on real European data.

What they do well is clear. On the Kundur two-area system they recover the known inter-area mode against classical modal analysis with small relative error and beat Hankel-DMD on both frequency and damping. On the 2016 Iberian event and 2021 Italian ambient data they reconstruct the full sensor set from sparse inputs, recover the expected East–Central–West mode shapes, and again outperform the industrial Hankel-DMD baseline. The latent ODEs are linear and four-dimensional, so eigendecomposition immediately gives frequency, damping, and mode shapes that map back through the decoder. Code is public; the limitations section is honest about manual hyper-parameters, confidential data, and the inability to separate forced oscillations.

The soft spot is the forecasting half of the abstract claim. The paper itself shows open-loop integration of the fixed latent ODE already diverging after roughly thirty seconds (Figs. 6 and 11) and states that re-initialization or full retraining is required once conditions drift. High-frequency peels cannot be integrated open-loop at all, and the mean-frequency component is deliberately omitted. Reconstruction and modal-parameter recovery remain solid; multi-minute autonomous prediction does not. That is a real but bounded over-claim, not a collapse of the rest of the work.

This is for people who care about interpretable, deployable monitoring tools for TSOs under rising IBR penetration. The math is standard, the citation pattern is appropriate (self-cites are to the building-block methods), and the evidence is stronger than most pure ML-for-power papers. I would send it to referees; they will ask for tighter language on forecast horizon and perhaps an automated hyper-parameter study, but the core contribution is real enough to deserve that scrutiny.

Referee Report

4 major / 0 minor

Summary. The manuscript proposes SINDy-SENDAI, a hierarchical multiscale architecture that combines SENDAI-style low/high-frequency residual peeling with a SINDy-regularized latent dynamical model (primarily linear) on the low-frequency pathway. The LF pathway (GRU encoder + shallow decoder + SINDy) is intended to isolate electromechanical oscillations, recover parsimonious latent ODEs, and support modal analysis, mode-shape reconstruction, and short-horizon autonomous forecasting; sequential HF peel layers refine residual high-frequency content. Validation is performed on the two-area Kundur system against classical modal analysis, and on two European PMU datasets (2016 Iberian inter-area event; 2021 southern Italian ambient operation), with Hankel-DMD as the main industrial baseline. The authors report improved modal-parameter accuracy on Kundur, competitive frequency estimates and interpretable mode shapes on real events, and improved reconstruction when HF peels are included.

Significance. If the claims hold under realistic TSO operating conditions, the work would be a meaningful step beyond black-box ML and purely signal-processing modal estimators: it couples multiscale reconstruction from sparse PMUs with explicit latent ODEs that operators can eigendecompose for frequency, damping, and mode shapes. Strengths include a clear Kundur comparison against modal analysis with quantitative relative errors (Table I), real-event reconstructions and mode shapes consistent with known European geography, open code, and an explicit limitations section on forced oscillations and hyperparameter tuning. The combination of hierarchical spectral peeling with latent SINDy is a concrete, deployable-oriented contribution for WAMS oscillation monitoring, provided the forecasting and robustness claims are scoped to the evidence.

major comments (4)

Abstract and §II.A.3 / §III.B–C (Figs. 6, 11; Eqs. 3, 13, 18–19): The central claim that the learned latent dynamics are “sufficiently informative to accurately … predict the behavior of the full system” is not supported at the multi-minute horizon suggested for practical use. Only the LF pathway is integrated open-loop; HF peels cannot be forecasted; the mean-frequency component is omitted by design; and the paper’s own forecasts diverge after roughly 30 s, with the authors stating that re-initialization or full retraining is required when conditions drift. Please either (i) restrict the abstract/conclusions to short-horizon LF forecasting with a quantified valid horizon and error growth metrics, or (ii) add experiments that demonstrate stable multi-minute prediction under re-initialization/online update protocols that a control room could actually run.
§III opening configuration paragraph and the deployment claim of “stable performance under diverse operating conditions” (Abstract, §I, §V): All free parameters (d_z=4, M=10 peels, lag L=40 s, λ_SINDy, sparsity thresholds, λ_sp/λ_mag/β, linear library only) are fixed by “engineering judgment,” with no sensitivity study, ablation on d_z/M/L, or cross-event transfer of the same hyperparameters. Because robustness is listed as a design requirement for TSO deployment, at least a limited sensitivity or leave-one-event-out check is load-bearing; otherwise the claim should be narrowed to the three reported operating regimes.
Tables II–III and the “consistently outperforms Hankel-DMD” claim (Abstract, §I.B, §III): On Kundur (Table I) the comparison is fair and favorable. On the real events there is no ground-truth damping, and the Iberian damping estimates differ substantially (hDMD −0.528% vs SINDy-SENDAI −1.920%) while frequencies agree. Reconstruction RMSE gains (Figs. 5, 10) do not by themselves establish superior modal identification. Please separate reconstruction metrics from modal-parameter claims, state clearly what “outperforms” means on real data, and, if possible, add an independent check (e.g., Prony/ESPRIT on the same windows, or consistency across sliding windows) for damping.
§II.A.1 and Eqs. (14), (18), (19): The latent SINDy model is constrained a priori to be linear and four-dimensional, then used for classical eigendecomposition. The manuscript does not show that a nonlinear library is unnecessary, nor that d_z=4 is minimal/sufficient across events (ambient Italian data may involve weaker excitation and different modal content). A brief comparison—linear vs sparse nonlinear library, or d_z∈{2,4,6}—would substantiate that the recovered modes are not an artifact of the linear 4-D ansatz.

Circularity Check

1 steps flagged · score 2.0 of 10

Minor self-citations supply architectural building blocks (SENDAI/SHRED/SINDy-SHRED) from overlapping authors, but the load-bearing empirical claims rest on independent external benchmarks (classical modal analysis and Hankel-DMD) and are not forced by construction.

self citation load bearing [§I (Bibliography Review) and §II (SINDy-SENDAI Algorithm), citations [9], [33], [34], [36]]
"The Sparse-measurement, EfficieNt Data AssImilation (SENDAI) model [9] is specifically designed to address the robust, multiscale reconstruction problem. ... The LF backbone of SENDAI builds upon the SHallow REcurrent Decoder (SHRED) architecture [33], [34] ... we augment the LF pathway by embedding a SINDy module [10], [36] directly within the latent space."

SENDAI, SHRED and SINDy-SHRED are prior works whose author lists overlap with the present paper (Kutz, Gao, Pomarico). They supply the hierarchical peeling and latent-SINDy machinery that the method is built upon. The circularity is only mild: these citations justify the architectural choice, not the numerical performance claims, which are independently checked against classical modal analysis and Hankel-DMD.

full rationale

The paper's derivation chain is a standard data-driven pipeline: sparse PMU windows are encoded by a GRU into a 4-D latent trajectory, a linear SINDy model is identified post-hoc by sequential thresholding on that trajectory (Eqs. 3, 6–8, 14, 18, 19), the LF decoder reconstructs the full state, and HF peel layers correct residuals. Modal frequencies/damping are then read off by eigendecomposition of the discovered A matrix and compared to classical modal analysis (Kundur ground truth, Table I) and to Hankel-DMD run in the original measurement space (Tables II–III). Reconstruction RMSE and open-loop forecasts are likewise evaluated on held-out real PMU windows. None of these numerical claims reduce by definition or by fit to their own inputs; the latent ODEs are discovered from data rather than imposed, and the architecture citations ([9], [33], [34], [36]) merely supply reusable components. The only mild circularity is ordinary co-author self-citation of those components; it is not load-bearing for the strongest claim of outperforming hDMD or matching MA. Forecasting drift (Figs. 6, 11) is an honesty issue, not a circularity issue. Score 2 reflects the presence of those self-citations without elevating them to a forced result.

Assumptions & free parameters 5 free parameters · 3 assumptions · 1 invented entities

The central empirical claims rest on a modest set of architectural free parameters chosen by hand, standard dynamical-systems assumptions (Takens, linear modal analysis), and the newly assembled SINDy-SENDAI pipeline itself. No exotic physical entities are postulated.

free parameters (5)

latent dimension d_z = 4
Fixed at 4 for all experiments; controls capacity of the discovered ODE and is not cross-validated.
number of HF peel layers M = 10
Set to 10; each layer adds residual capacity and is trained sequentially.
lag length L = 40 s
40 s history window for the GRU encoder; chosen by engineering judgment.
SINDy regularization weight λ_SINDy and sparsity threshold
Warmed up during training then used for post-hoc sequential thresholding; values not reported numerically.
HF sparsity and magnitude weights λ_sp, λ_mag, β
Control spectral concentration of each peel layer; set by hand and warmed up.

assumptions (3)

standard math Takens’ embedding theorem guarantees that a sufficiently long lag of a few sensors reconstructs the attractor of the full state.
Invoked to justify the SHRED-style encoder (§II).
domain assumption Dominant electromechanical modes lie in a low-dimensional linear subspace that can be recovered by sparse regression on latent trajectories.
Underpins the linear SINDy model and subsequent eigendecomposition for frequency/damping (§II.A.1, §II.A.4).
ad hoc to paper Hierarchical residual peeling cleanly separates spectral bands without destructive mode interference.
Core design claim of SENDAI extended to 1-D PMU series; not independently proved for power-system spectra.

invented entities (1)

SINDy-SENDAI architecture
purpose: Joint multiscale reconstruction and sparse latent ODE discovery from sparse PMU streams.
The paper’s central methodological object; assembled from prior SINDy, SHRED and SENDAI pieces but not previously published as a unit.

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Pith. "Pith review of Data-Driven Discovery of Multiscale Power System Oscillation Governing Equations Using SINDy-SENDAI." pith.science (2026). https://pith.science/paper/75LCT7DA

@misc{pith2026260703485,
  author       = {Pith},
  title        = {Pith review of: Data-Driven Discovery of Multiscale Power System Oscillation Governing Equations Using SINDy-SENDAI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/75LCT7DA}},
  note         = {Machine review of arXiv:2607.03485}
}

read the original abstract

Monitoring electromechanical oscillations is crucial for maintaining the stability of modern power systems, particularly in the presence of increasing penetrations of inverter-based resources (IBRs), which introduce new dynamic behaviors. In this work, we propose a hierarchical multiscale framework based on the SINDy-SENDAI algorithm to characterize the transient dynamics captured by wide-area measurements. The proposed deep learning architecture robustly separates low- and high-frequency components embedded in sensor data and incorporates a Sparse Identification of Nonlinear Dynamical Systems (SINDy) module in the latent space to identify parsimonious governing equations. In contrast to conventional deep learning approaches that often produce black-box models with limited interpretability, the proposed framework learns an explicit dynamical representation, enabling physical interpretation, stability assessment, and forecasting of electromechanical oscillations. Given the societal importance of modern power systems, the proposed approach is specifically designed to satisfy key requirements for practical deployment, namely robustness, interpretability, and stable performance under diverse operating conditions. The framework is first validated on the two-area Kundur test system using conventional modal analysis as ground truth and subsequently demonstrated on two real-world datasets: the 2016 Iberian oscillatory event and the 2021 ambient measurements from the southern Italian power grid. The results show that SINDy-SENDAI consistently outperforms the state-of-the-art Hankel-DMD method and that the learned latent dynamics are sufficiently informative to accurately reconstruct and predict the behavior of the full system in the original state space.

Figures

Figures reproduced from arXiv: 2607.03485 by the authors.

**Figure 3.** PMU measurements of the oscillatory event on December 1 [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 4.** Reconstruction of the LF pathway and the combined LF+HF pathways across [PITH_FULL_IMAGE:figures/full_fig_p007_4.png] view at source ↗

Figures from the paper (7 more)

**Figure 5.** Figure 5: RMSE [Hz] and Gain [%] between LF and LF+HF for all the PMUs [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: One-minute forecast of the LF pathway across [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]

**Figure 7.** Figure 7: Mode shapes for the oscillatory event in December 2016. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]

**Figure 8.** Figure 8: PMU measurements of the normal grid operation in March 2021. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

**Figure 9.** Figure 9: Visualization of the ground truth, the LF pathway, and the LF+HF pathways obtained with the SINDy-SENDAI approach for the measurement [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]

**Figure 10.** Figure 10: RMSE [Hz] and Gain [%] between LF and LF+HF for all the PMUs [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: One-minute forecast of the LF pathway for [PITH_FULL_IMAGE:figures/full_fig_p010_11.png]

Discussion (0). Continue with ORCID to comment.

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

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