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arxiv: 2605.02602 · v1 · submitted 2026-05-04 · 📡 eess.SY · cs.SY

PowerSINDy: Identifying Nonlinear Time-Dependent Dynamics in Power Grid Frequency

Xinyi Wen , Xiao Li , Leonardo Rydin Gorj\~ao , Veit Hagenmeyer , Benjamin Sch\"afer This is my paper

Pith reviewed 2026-05-08 18:13 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords system identificationpower grid frequencynonlinear dynamicssparse regressiontime-dependent modelsSINDyempirical datafrequency dynamics
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The pith

PowerSINDy recovers governing equations for real power grid frequency from noisy measurements by adding time-dependent terms to the sparse identification library.

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

The paper introduces PowerSINDy to extract the active nonlinear and time-varying terms that drive frequency behavior directly from recorded data in two large synchronous grids. It applies the method to Continental Europe and South Korea measurements and compares three regression techniques that enforce sparsity. LASSO produces the smallest stable prediction errors while STLSQ maintains consistent term selection across runs. A reader would care because frequency swings affect grid reliability and because traditional models require exhaustive knowledge of every generator and line. If the approach holds, system operators could obtain usable dynamical descriptions from existing sensors without building full physics-based replicas.

Core claim

PowerSINDy augments the standard SINDy library with explicit time-dependent basis functions and uses sparsity-promoting regression on empirical frequency time series from the Continental Europe and South Korea grids. The framework selects a small number of active terms that best reproduce the observed dynamics. Benchmarks demonstrate that LASSO yields the lowest stable root-mean-square errors, reaching 0.0101 on Continental Europe data, while STLSQ achieves the most reliable balance between accuracy and robustness across repeated trials.

What carries the argument

PowerSINDy, an extension of sparse nonlinear identification that augments the candidate library with time-dependent functions and applies regression to isolate the active terms in frequency data.

If this is right

  • The selected terms can be integrated forward to simulate future frequency trajectories in the studied grids.
  • LASSO regression supplies the most accurate one-step predictions when stability is required.
  • STLSQ regression supplies models whose active terms remain consistent under small changes in the data window.
  • Time-dependent coefficients are required to capture variations that constant-coefficient nonlinearities alone cannot explain.
  • The same workflow can be repeated on additional real-world frequency recordings from other synchronous areas.

Where Pith is reading between the lines

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

  • Identified models could support real-time monitoring tools that flag deviations from normal dynamics without requiring a full network model.
  • The framework might be extended by enlarging the library with known physical terms to improve both accuracy and physical interpretability.
  • Similar sparse identification with time dependence could be tested on other oscillatory systems such as voltage angles or inter-area modes.
  • Direct comparison of the discovered terms against classical swing-equation models would show which physical effects the data-driven selection recovers or adds.

Load-bearing premise

The observed frequency changes can be expressed as a sparse linear combination of a preselected set of nonlinear and time-dependent functions, with noise and missing effects not preventing reliable selection of the correct terms.

What would settle it

Re-running PowerSINDy on held-out segments of the same frequency recordings produces markedly different active terms or yields prediction errors that grow beyond the reported stable RMSE levels.

Figures

Figures reproduced from arXiv: 2605.02602 by Benjamin Sch\"afer, Leonardo Rydin Gorj\~ao, Veit Hagenmeyer, Xiao Li, Xinyi Wen.

Figure 1
Figure 1. Figure 1: Comparison of smoothing levels and corresponding model performance. (a) view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of the dynamics discovery pipeline. (a) Raw empirical frequency view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of mean stable RMSE with standard deviation error bars across view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of simulated ω(t) trajectories using SR3, STLSQ, and LASSO. SR3 results are shown for different regularization norms, illustrating the effect of regularization choice. STLSQ and LASSO reconstructions demonstrate higher agreement with the filtered signal, particularly LASSO, which achieves the closest alignment to the original dynamics. LASSO SR3_L0 SR3_L1 SR3_L2 STLSQ 0.4 0.5 0.6 0.7 0.8 0.9 1.0… view at source ↗
Figure 5
Figure 5. Figure 5: Comparative analysis of model stability and feature selection: (a) Stability view at source ↗
read the original abstract

System identification plays a crucial role in physics and machine learning for discovering governing equations directly from data. A powerful approach is the Sparse Identification of Nonlinear Dynamics (SINDy) method, which assumes that only a few dominant terms drive the essential behavior of a nonlinear dynamical system. While SINDy methods have shown excellent results, they are most often illustrated on synthetic or simulated systems, leaving open the question of how well they perform on complex, noisy, real-world data. Power grid frequency dynamics provide a highly relevant and challenging environment for advancing system identification methods. In this work, we propose PowerSINDy as a framework for empirical power system data. We apply this framework to empirical frequency data from the Continental Europe (CE) and South Korea (SK) synchronous grids, two major power systems with distinct dynamical characteristics. PowerSINDy, which also includes time-dependent terms, can identify the dynamics of these complex real-world systems. Furthermore, we benchmark three sparsity-promoting regression strategies: Sequentially Thresholded Least Squares (STLSQ), Least Absolute Shrinkage and Selection Operator (LASSO), and Sparse Relaxed Regularized Regression (SR3) to evaluate trade-offs between accuracy, sparsity, and robustness. Results show that LASSO consistently achieves the lowest stable RMSEs, reaching 0.0101 for the CE, while STLSQ provides the best balance between accuracy and stability. SR3 exhibits higher variability and sensitivity to regularization, with L0 and L1 producing nearly indistinguishable outcomes.

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 / 1 minor

Summary. The manuscript introduces PowerSINDy, an extension of the SINDy framework that augments the candidate library with explicit time-dependent basis functions, and applies it to empirical frequency measurements from the Continental Europe (CE) and South Korea (SK) synchronous areas. Three sparsity-promoting regressors (STLSQ, LASSO, SR3) are benchmarked on these real-world datasets; the authors report that LASSO attains the lowest stable RMSE (0.0101 on CE data) while STLSQ offers the most favorable accuracy-stability trade-off.

Significance. If the identified models can be shown to generalize temporally and remain stable under numerical integration, the work would constitute a concrete advance in data-driven discovery for noisy, high-stakes infrastructure systems. The use of two distinct real grids and the systematic comparison of three regressors are positive features that move beyond typical synthetic SINDy illustrations.

major comments (2)
  1. [Results section] Results section: All reported performance is confined to in-sample RMSE on training segments. No forward integration of the discovered ODEs on held-out intervals is presented to demonstrate that the selected models reproduce observed trajectories without rapid divergence or error accumulation. This validation step is load-bearing for the central claim that PowerSINDy 'identifies the governing dynamics' rather than fitting short-term correlations or measurement noise.
  2. [Method and experimental setup] Method and experimental setup: The regularization strengths (threshold for STLSQ, lambda for LASSO, and the corresponding SR3 parameters) are free parameters whose selection procedure is not described with respect to cross-validation, pre-specification, or sensitivity analysis. Consequently, the reported RMSE ordering and term-selection consistency may depend on post-hoc tuning rather than robust generalization.
minor comments (1)
  1. [Abstract] Abstract: The phrasing 'PowerSINDy, which also includes time-dependent terms, can identify the dynamics of these complex real-world systems' is imprecise; it would be clearer to state the specific empirical outcome (e.g., 'recovers sparse models with low in-sample RMSE').

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which identify key areas for strengthening the validation and reproducibility of PowerSINDy. We address each major comment below and have revised the manuscript to incorporate the suggested improvements.

read point-by-point responses

  1. Referee: [Results section] All reported performance is confined to in-sample RMSE on training segments. No forward integration of the discovered ODEs on held-out intervals is presented to demonstrate that the selected models reproduce observed trajectories without rapid divergence or error accumulation. This validation step is load-bearing for the central claim that PowerSINDy 'identifies the governing dynamics' rather than fitting short-term correlations or measurement noise.

    Authors: We agree that in-sample RMSE alone does not fully establish that the identified models capture the governing dynamics rather than noise or transient correlations, and that forward integration on held-out data is a necessary validation step. In the revised manuscript we have added a new subsection to the Results section that reports numerical integration of the discovered ODEs on temporally held-out intervals from both the CE and SK datasets. The LASSO- and STLSQ-selected models remain numerically stable over these intervals and track the observed frequency trajectories with bounded error accumulation (additional RMSE values reported). These results are now presented alongside the original in-sample metrics to support the central claim. revision: yes

  2. Referee: [Method and experimental setup] The regularization strengths (threshold for STLSQ, lambda for LASSO, and the corresponding SR3 parameters) are free parameters whose selection procedure is not described with respect to cross-validation, pre-specification, or sensitivity analysis. Consequently, the reported RMSE ordering and term-selection consistency may depend on post-hoc tuning rather than robust generalization.

    Authors: We acknowledge that a transparent account of regularization-parameter selection is required for reproducibility and to demonstrate robustness. The revised Methods section now includes an explicit description of the selection procedure: a grid search was performed over a pre-specified range of values for each regressor, with the final parameters chosen according to an elbow criterion balancing sparsity and RMSE on the training segments. We have also added a sensitivity-analysis subsection that quantifies how small perturbations around the selected values affect both the retained terms and the resulting RMSE, confirming that the dominant terms and performance ordering remain stable. These additions address the concern regarding post-hoc tuning. revision: yes

Circularity Check

0 steps flagged

No circularity: standard sparse regression applied to external data

full rationale

The paper extends the established SINDy algorithm by including explicit time-dependent basis functions in the library and applies it to measured frequency time series from the CE and SK grids. Term selection and coefficient fitting are performed via off-the-shelf regressors (STLSQ, LASSO, SR3) on the observed data; the reported RMSE values and selected terms are direct outputs of this fitting process. No equation in the derivation reduces to a fitted parameter by construction, no uniqueness theorem is imported from the authors' prior work, and no ansatz is smuggled via self-citation. The central result is therefore an empirical identification whose validity rests on the data and the standard SINDy assumptions rather than on any tautological reduction.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method inherits the standard SINDy sparsity assumption and adds the domain-specific premise that time-dependent basis functions are needed for grid frequency; no new physical entities are postulated.

free parameters (1)
  • regularization strength (lambda or threshold)
    Chosen for each of LASSO, STLSQ, and SR3 to trade off sparsity against fit quality on the frequency time series.
axioms (1)
  • domain assumption System dynamics admit a sparse representation in a pre-specified library of nonlinear and time-dependent functions.
    Core modeling assumption of the SINDy family, invoked when the regression is performed on the frequency data.

pith-pipeline@v0.9.0 · 5592 in / 1462 out tokens · 51533 ms · 2026-05-08T18:13:40.456093+00:00 · methodology

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Reference graph

Works this paper leans on

34 extracted references · 9 canonical work pages

  1. [1]

    We included explicit time-dependency and selected suitable candidate functions, in particular higher-order polynomial and Fourier terms

    Conclusion and Outlook This work presented the novel PowerSINDy framework for uncovering governing equations for nonlinear, time-dependent dynamics and validated them on empirical power grid frequency data. We included explicit time-dependency and selected suitable candidate functions, in particular higher-order polynomial and Fourier terms. By systematic...

  2. [2]

    VH-NG-1727 as well as by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 556503410

    Acknowledgments This work was funded by the Helmholtz Association and the Networking Fund through Helmholtz AI and under grant no. VH-NG-1727 as well as by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – 556503410. The authors gratefully acknowledge the computing time provided on the high- performance computer HoreKa by the Nationa...

  3. [3]

    Machowski J, Lubosny Z, Bialek J W and Bumby J R 2020Power System Dynamics: Stability and Control(John Wiley & Sons)

  4. [4]

    Rydin Gorj˜ ao L, Jumar R, Maass H, Hagenmeyer V, Yalcin G C, Kruse J, Timme M, Beck C, Witthaut D and Sch¨ afer B 2020Nature Communications116362 ISSN 2041-1723 URL http://www.nature.com/articles/s41467-020-19732-7

    2041

  5. [5]

    Council of European Union 2016 Commission regulation (eu) 2016/631 of 14 april 2016https: //eur-lex.europa.eu/legal-content/EN/TXT/PDF/?uri=CELEX:32016R0631&rid=1

    2016

  6. [6]

    Kundur P S and Malik O P 2022Power system stability and control(McGraw-Hill Education)

  7. [7]

    Kruse J, Sch¨ afer B and Witthaut D 2021 Exploring deterministic frequency deviations with explainable ai2021 IEEE International Conference on Communications, Control, and Computing Technologies for Smart Grids (SmartGridComm)pp 133–139

    2021

  8. [8]

    Weissbach T and Welfonder E 2009 High frequency deviations within the european power system: Origins and proposals for improvement2009 IEEE/PES Power Systems Conference and Expositionpp 1–6

    2009

  9. [9]

    Ljung L 1998 System identificationSignal analysis and prediction(Springer) pp 163–173

    1998

  10. [10]

    Stiasny J, Misyris G S and Chatzivasileiadis S 2021 Physics-informed neural networks for non- linear system identification for power system dynamics2021 IEEE Madrid PowerTech(IEEE) pp 1–6

    2021

  11. [11]

    Overcoming catastrophic forgetting in neural networks

    Brunton S L, Proctor J L and Kutz J N 2016Proceedings of the National Academy of Sciences 1133932–3937 ISSN 0027-8424, 1091-6490 URLhttps://pnas.org/doi/full/10.1073/pnas. 1517384113

    work page doi:10.1073/pnas

  12. [12]

    Fasel U, Kaiser E, Kutz J N, Brunton B W and Brunton S L 2021 Sindy with control: A tutorial 2021 60th IEEE conference on decision and control (CDC)(IEEE) pp 16–21

    2021

  13. [13]

    Fasela U, Kutz J N, Brunton B W and Brunton S L 2022Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences47820210904 ISSN 1364-5021, 1471-2946 URLhttps://royalsocietypublishing.org/doi/10.1098/rspa.2021.0904

    work page doi:10.1098/rspa.2021.0904 2021

  14. [14]

    Zolman N, Fasel U, Kutz J N and Brunton S L 2024arXiv preprint arXiv:2403.09110

    work page arXiv

  15. [15]

    Fukami K, Murata T, Zhang K and Fukagata K 2021Journal of Fluid Mechanics926A10

  16. [16]

    Mangan N M, Brunton S L, Proctor J L and Kutz J N 2016IEEE Transactions on Molecular, Biological and Multi-Scale Communications252–63 ISSN 2372-2061, 2332-7804 URLhttp: //ieeexplore.ieee.org/document/7809160/

    work page arXiv 2061

  17. [17]

    Corbetta M 2020 Application of sparse identification of nonlinear dynamics for physics-informed learning2020 IEEE Aerospace Conference(IEEE) pp 1–8 ISBN 978-1-72812-734-7 URLhttps: //ieeexplore.ieee.org/document/9172386/

    work page arXiv 2020

  18. [18]

    Callaham J L, Loiseau J C, Rigas G and Brunton S L 2021Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences47720210092 URLhttps:// royalsocietypublishing.org/doi/10.1098/rspa.2021.0092

    work page doi:10.1098/rspa.2021.0092 2021

  19. [19]

    Wen X, Oberhofer U, Gorj˜ ao L R, Yalcin G, Hagenmeyer V and Sch¨ afer B 2024 Identifying complex dynamics of power grid frequencyProceedings of the 15th ACM International Conference on Future and Sustainable Energy Systemse-Energy ’24 (New York, NY, USA: Association for Computing Machinery) p 408–414 ISBN 9798400704802 URLhttps://doi.org/10.1145/ 3632775.3661944

    work page arXiv 2024

  20. [20]

    Callaham J L, Brunton S L and Loiseau J C 2022Journal of Fluid Mechanics938A1

  21. [21]

    Ranstam J and Cook J A 2018Journal of British Surgery1051348–1348

  22. [22]

    Champion K, Zheng P, Aravkin A Y, Brunton S L and Kutz J N 2020IEEE Access8169259– 169271 ISSN 2169-3536 URLhttps://ieeexplore.ieee.org/document/9194760/ PowerSINDy: Identifying Nonlinear Time-Dependent Dynamics in Power Grid Frequency16

    work page arXiv

  23. [23]

    Union for the Coordination of Transmission of Electricity (UCTE) 2009 UCTE annual report 2008: On the move Tech. rep. Union for the Coordination of Transmission of Electricity (UCTE) / ENTSO-E annual report covering the Continental Europe synchronous area; available online (PDF) URLhttps://eepublicdownloads.entsoe.eu/clean-documents/ pre2015/publications/...

    2009

  24. [24]

    Analysis M T 2024 South korea’s power plans: Ambitious expansion strategy for a sus- tainable future REGlobal (Mega Trends & Analysis) accessed online; includes figures for installed capacity and transmission infrastructure URLhttps://reglobal.org/ south-koreas-power-plans-ambitious-expansion-strategy-for-a-sustainable-future/

    2024

  25. [25]

    Rydin Gorj˜ ao L, Hassan G and Sch¨ afer B Power-grid frequencyhttps://power-grid-frequency. org/

  26. [26]

    Jumar R, Maaß H, Sch¨ afer B, Gorj˜ ao L R and Hagenmeyer V 2020arXiv preprint arXiv:2006.01771

    work page arXiv 2006

  27. [27]

    Oberhofer U, Wen X, Lee J, Kim H, Hagenmeyer V and Sch¨ afer B 2025 Nonlinear stochastic modeling of the south korean power grid frequency dynamics2025 IEEE Kiel PowerTechpp 1–6

    2025

  28. [28]

    Sch¨ afer B, Beck C, Aihara K, Witthaut D and Timme M 2018Nature Energy3119–126

  29. [29]

    Oberhofer U, Rydin Gorj˜ ao L, Yalcin G C, Kamps O, Hagenmeyer V and Sch¨ afer B 2023 Non- linear, bivariate stochastic modelling of power-grid frequency applied to islands2023 IEEE Belgrade PowerTechpp 1–1

    2023

  30. [30]

    Rydin Gorj˜ ao L, Anvari M, Kantz H, Beck C, Witthaut D, Timme M and Sch¨ afer B 2020 IEEE Access843082–43097 ISSN 2169-3536 URLhttps://ieeexplore.ieee.org/document/ 8963682/

    2020

  31. [31]

    Wen X 2026 Powersindy identifying nonlinear time-dependent dynamics in power grid fre- quencyhttps://github.com/KIT-IAI-DRACOS/PowerSINDy_Identifying_Nonlinear_Time_ Dependent_Dynamics_in_Power_Grid_Frequency

    2026

  32. [32]

    Fung L, Fasel U and Juniper M 2025 Rapid bayesian identification of sparse nonlinear dynamics from scarce and noisy dataProceedings Avol 481 (The Royal Society) p 20240200

    2025

  33. [33]

    Lee H, Ren R, Qian Y and Rosen J 2024IEEE/ASME Transactions on Mechatronics

  34. [34]

    Hosseinipour A, Khazaei J and Blum R S 2023 A data-driven framework for sparse impedance identification of power converters in dc microgrids2023 IEEE Power & Energy Society General Meeting (PESGM)(IEEE) pp 1–5 PowerSINDy: Identifying Nonlinear Time-Dependent Dynamics in Power Grid Frequency17 Appendix A. Hyperparameter Sensitivity Analysis To ensure a rob...

    2023