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

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.

• The paper introduces PowerSINDy, a sparse-regression pipeline that uses time-dependent and Fourier-augmented libraries to identify nonlinear deterministic dynamics from noisy power-grid frequency measurements.
• LASSO delivers the best accuracy, reaching a stable RMSE of 0.0101 on the Continental Europe grid with cubic features, while STLSQ provides greater simulation stability and SR3 performs least reliably at higher model complexity.
• The identified models highlight nonlinear state interactions such as θω and θω², but cubic models often diverge and Gaussian filtering omits stochastic forcing, limiting immediate operational and physical interpretation.

PowerSINDy is a framework for sparse identification of governing equations from empirical power grid frequency measurements, extending the Sparse Identification of Nonlinear Dynamics (SINDy) method with time-dependent and Fourier-augmented candidate libraries. The paper applies the framework to 1-second-resolution frequency recordings from two structurally distinct synchronous systems—the Continental Europe (CE) grid (~800 GW capacity, 50 Hz) and the South Korea (SK) grid (~144 GW, 60 Hz)—and systematically benchmarks three sparsity-promoting optimizers: Sequentially Thresholded Least Squares (STLSQ), LASSO, and Sparse Relaxed Regularized Regression (SR3). The central contribution is less a new regression algorithm than a validated pipeline for moving SINDy from synthetic benchmarks to noisy, real-world, non-autonomous power system data (2605.02602).

Motivation and problem setting

The conventional model of aggregate grid frequency is the linear stochastic swing equation, in which the rate of change of the frequency deviation $\omega$ is governed by damping $c_\omega$, stiffness $c_\theta$ from secondary control, a power imbalance input $\Delta P(t)$, and Gaussian noise. The authors argue this linearity assumption is increasingly inadequate as renewable penetration introduces intermittency, reduced inertia, and nonlinear interactions. Prior work demonstrated a proof-of-concept application of SINDy to empirical frequency data but did not examine how optimizer choice or library complexity affects accuracy and stability—gaps this paper addresses directly.

Two properties of frequency data complicate identification: oscillatory components are not strictly periodic, so polynomial bases alone are insufficient; and the system is not autonomous, since load curves and dispatch schedules impose deterministic time dependence. PowerSINDy responds by augmenting the candidate library with an explicit time variable $T$, quadratic and cubic polynomials ($p2$, $p3$), and first-order Fourier terms ($p2f1$), giving features such as $\sin(\theta)$, $\cos(\omega)$, $c_\omega$0, and cross terms like $c_\omega$1. Higher-order combinations (e.g., $c_\omega$2) were excluded after preliminary tests indicated overfitting—a restriction the authors acknowledge limits expressiveness but keeps the search tractable.

Preprocessing

Raw trajectories are filtered with a Gaussian kernel whose bandwidth $c_\omega$3 is itself optimized: the authors sweep $c_\omega$4, run SINDy on each filtered series, simulate forward, and select the bandwidth minimizing RMSE against the empirical signal. The optimum, $c_\omega$5 s, yields an RMSE of 0.0453, with performance stable for $c_\omega$6 and degrading sharply beyond $c_\omega$7 due to oversmoothing. A key modeling assumption is stated here: because filtering removes stochastic fluctuations, the identified dynamics are treated as deterministic. This is a substantive concession—the stochastic component of the swing equation is deliberately excluded rather than modeled—and all reported accuracies should be read as applying to the filtered, deterministic component only.

Optimizer benchmarking results

The headline quantitative comparison uses mean stable RMSE (RMSE averaged only over trajectory chunks whose simulated model did not diverge) together with a stability fraction. Three findings stand out.

First, LASSO achieves the lowest stable RMSE across nearly all configurations. The single best result is CE under the cubic library, at 0.0101, which also beats the Euler–Maruyama baseline of 0.0282 (CE) and 0.0342 (SK). Notably, cubic models outperform quadratic ones on accuracy despite lower stability: LASSO-$c_\omega$8 reaches stability fractions of only 0.586 (CE) and 0.525 (SK), versus roughly 0.83–0.84 for $c_\omega$9. This is a clear accuracy–robustness trade-off embedded in the paper's strongest numerical claim.

Second, STLSQ offers the best stability: fractions exceed 0.79 in the $c_\theta$0 setting on both grids, with only modest RMSE penalty relative to LASSO. The authors position STLSQ as the most reliable choice when consistent, non-divergent behavior matters more than minimal error.

Third, SR3 is the least reliable optimizer here. Its $c_\theta$1 variant performs reasonably on $c_\theta$2 (CE RMSE 0.0156) but degrades sharply on cubic models for SK, where stability falls below 0.52 and convergence required a larger relaxation parameter ($c_\theta$3); lower values failed to converge within iteration limits. A striking negative result is that SR3's $c_\theta$4 and $c_\theta$5 penalties produce numerically indistinguishable RMSEs and stability fractions in every tested configuration, suggesting the two penalties are effectively equivalent in this regime—an outcome that runs counter to the theoretical distinctions typically motivating $c_\theta$6 regularization.

Model structure and interpretability

Feature-count analysis reinforces the optimizer ranking. LASSO produces the sparsest models throughout: approximately 5.9 active terms for $c_\theta$7, about 8 for $c_\theta$8, and around 10 for $c_\theta$9, whereas SR3 retains up to 14 active terms at cubic order. In the LASSO-$\Delta P(t)$0 model, the nonlinear interaction terms $\Delta P(t)$1 and $\Delta P(t)$2 carry the largest coefficients, indicating that state couplings beyond the linear swing equation genuinely contribute to the observed dynamics rather than serving as overfitting artifacts. This is arguably the paper's most interpretable physical finding, though it is reported without a statistical significance analysis.

Hyperparameter sensitivity studies in the appendix reveal a consistent pattern across all three optimizers: RMSE exhibits a broad plateau over several orders of magnitude of the sparsity parameter, while stability improves monotonically with stronger regularization. The selected operating points lie within these plateaus, supporting robustness of the reported results to small parameter perturbations.

Limitations and open questions

Several limitations are conceded explicitly. The determinism assumption following Gaussian filtering means the framework does not identify the stochastic forcing that is central to physically grounded swing-equation models. Hyperparameters were tuned via grid search on the CE dataset only, then transferred to SK; the paper does not test whether per-grid tuning would materially change the SK results. Stability fractions below 0.6 for all optimizers at cubic order mean that a substantial fraction of identified models diverge, and the paper does not diagnose why higher-order terms induce divergence beyond noting faster trajectory blow-up. Whether the identified $\Delta P(t)$3 and $\Delta P(t)$4 terms generalize across seasons, grid events, or additional synchronous areas remains unexamined, as does extension to time-varying parameters appropriate for renewable-dominated grids—all left open by the authors themselves.

Conclusion

PowerSINDy demonstrates that SINDy-style sparse regression, suitably extended with time and Fourier features, can recover accurate and parsimonious deterministic models from real grid frequency data, with LASSO providing the best accuracy–sparsity balance and STLSQ the best reliability. The framework's practical value for operational settings depends on resolving the open questions above, particularly the low stability of high-order models and the treatment of stochastic forcing.