Predicting power grid frequency dynamics with invertible Koopman-based architectures

1. Problem Statement

The stability of modern power grids is increasingly threatened by the integration of inverter-based renewable energy sources, which lack inherent inertia. This makes system frequency stability a critical challenge. While Koopman operator-based modeling offers a powerful data-driven method to approximate nonlinear system dynamics by lifting them into a linear space, traditional implementations often rely on autoencoders that are non-invertible.

• The Core Issue: Non-invertible mappings can lead to information loss during the transformation from physical state space to latent (observable) space. This distortion degrades the accuracy of the learned spectral quantities (eigenvalues, eigenfunctions) and results in significant prediction errors.
• The Constraint: While Invertible Neural Networks (INNs) guarantee bijective (lossless) mappings, they face two specific challenges in this context:
  1. Expressiveness: They are often less expressive than standard neural networks.
  2. Dimensionality: INNs require input and output dimensions to be equal, whereas Koopman theory often requires "dimension lifting" (embedding low-dimensional nonlinear dynamics into higher-dimensional linear spaces) to achieve linearity.

2. Methodology

The authors propose a framework combining Invertible Neural Networks (INNs) with the Koopman operator to model power system frequency dynamics. The study evaluates various architectural strategies and hybrid approaches.

A. Architectural Strategies Evaluated

The paper categorizes and tests several INN architectures:

1. Coupling Flow INNs (CF-INNs): Based on coupling layers (e.g., Additive, Affine). These partition input vectors and transform subsets conditionally. Variants tested include NICE, RealNVP, Glow, HINT, and an All-In-One model.
2. Residual Flows: Including i-RevNet/RevNet and iResNet. These use residual connections ($x + F(x)$) but face challenges with Lipschitz constraints and numerical inversion accuracy.
3. ODE-based Approaches: Neural ODEs, which treat layers as continuous transformations. The authors note these are theoretically less suitable for the discrete-time mappings required by Koopman operators ($t \to t+1$).

B. Hybrid Architecture (Augmented Invertible Koopman Autoencoder - AIKAE)

To address the dimensionality and expressiveness limitations of pure INNs, the authors implement a hybrid approach inspired by Frion et al. [7]:

• Structure: The input is processed in parallel by:
  1. An Invertible Network ($i$): Preserves bijectivity for state reconstruction.
  2. A Non-Invertible Extension Network ($a$): Generates additional features (dimension lifting) without invertibility constraints.
• Process: The outputs ($z_i$ and $z_a$) are concatenated. The Koopman operator evolves this high-dimensional vector. Only the invertible part ($\hat{z}_i$) is used to reconstruct the future state, ensuring the overall model remains invertible.
• Extension Types Tested:
  • CNN-based: Extracts local spatial/temporal structures.
  • Kernel-based (RBF): Maps inputs to high-dimensional spaces to capture similarity structures.
  • Deep Residual (Multitimescale): Uses SiLU activations and skip connections to capture long-term dependencies and multi-scale frequency components.

C. Experimental Setup

• Datasets: Two benchmark systems simulated using the ANDES package:
  • IEEE-14 bus: Faster-decaying dynamics.
  • WECC-179 bus: Weaker damping, significant inter-area oscillations.
• Scenarios: Short-circuit faults (0.1s duration) applied at various buses.
• Metrics: Relative Root Mean Squared Error (RRMSE) for trajectory prediction.
• Loss Functions: Joint optimization of prediction loss and latent-space consistency (Koopman loss).

3. Key Contributions

1. Systematic Architectural Comparison: The first comprehensive analysis of different INN classes (Coupling, Residual, ODE) specifically for power system frequency dynamics within a Koopman framework.
2. Hybrid Synergy Analysis: Demonstrates that combining invertible and non-invertible layers is not universally beneficial; the success depends heavily on the specific pairing of the base INN and the extension type.
3. Ablation Study on Feature Augmentation: By replacing the invertible network with an identity function, the authors isolated the contribution of the extension networks, effectively comparing them against Extended Dynamic Mode Decomposition (EDMD) with learned observables.
4. Practical Guidelines: Provides actionable insights for selecting INN architectures based on the specific dynamics of the power grid (e.g., damping characteristics).

4. Key Results

• Base INN Performance:
  • CF-INNs (Coupling Flows) consistently outperformed other architectures. Specifically, Affine Coupling models (All-In-One, Glow, RealNVP) showed the best generalization and lowest RRMSE.
  • NICE (Additive coupling) performed well on training data but struggled with generalization on complex datasets (WECC) due to limited expressiveness.
  • ODE-based and iResNet models performed poorly. ODEs produced overly smooth mappings that missed detailed dynamics, while iResNet suffered from numerical inversion errors and Lipschitz constraints.
• Impact of Hybrid Extensions:
  • Strong Base Models (CF-INNs): Generally did not benefit from extensions. In some cases (e.g., Glow), extensions slightly degraded performance on unseen data.
  • Weak Base Models (ODE, iResNet): Showed significant improvement (up to 5–10% RRMSE reduction) when paired with extensions, particularly the Multitimescale (Residual) extension.
  • Best Overall Performance: The All-In-One model combined with the Multitimescale extension achieved the best results on the challenging WECC test set (19.57% RRMSE), outperforming the base model alone.
• Ablation Findings:
  • When the invertible network was removed (Identity Map + Extension), only the Multitimescale extension significantly improved performance over the baseline EDMD.
  • CNN and Kernel extensions often degraded performance in the ablation study, suggesting they introduced features unsuitable for the linear Koopman approximation in this context.
• Generalization Anomaly: Some models performed better on test trajectories than training data, attributed to the varying difficulty of specific fault scenarios in the dataset.

5. Significance and Conclusion

This study establishes that Architectural Choice is paramount in Koopman-based power system modeling.

• Optimal Strategy: For high-fidelity frequency prediction, Affine Coupling INNs are the superior choice due to their structural stability and ability to preserve invertibility without information loss.
• Role of Extensions: Non-invertible extensions are a double-edged sword. They are highly effective at "rescuing" weaker architectures (like ODEs) by providing necessary feature richness, but they can hinder already robust architectures (like CF-INNs) if the added features introduce noise or overfitting.
• Practical Impact: The proposed hybrid approach allows for accurate, model-free prediction of grid frequency dynamics. This is crucial for real-time stability monitoring and control in grids with high renewable penetration, where traditional physics-based models may be too slow or unavailable.

The paper concludes that there is no "one-size-fits-all" solution; the optimal architecture depends on the specific dynamics of the grid and the trade-off between structural stability and feature expressiveness.