Discovering Unknown Inverter Governing Equations via Physics-Informed Sparse Machine Learning

1. Problem Statement

The rapid integration of Inverter-Based Resources (IBRs) has transformed power systems, shifting dynamics from physical synchronous machines to software-controlled power electronics. A critical challenge arises because the control logic of Grid-Forming (GFM) and Grid-Following (GFL) inverters is often proprietary and closed-source ("black boxes").

• The Modeling Gap: Traditional stability analysis relies on explicit state-space equations. However, existing identification methods fail to provide these:
  • Impedance Identification: Assumes local linearity and fails to capture global nonlinear dynamics or large-signal behaviors (e.g., transient stability).
  • Symbolic Regression: Struggles with high-dimensional, multi-time-scale systems and assumes a predefined function library is sufficient, often failing in complex control architectures.
  • Pure Neural Networks (NNs): While expressive, they lack physical consistency, often producing models that fit data but violate physical laws (e.g., incorrect eigenvalues), rendering them unsuitable for rigorous stability analysis.
• Goal: To discover explicit, interpretable, and physically consistent governing equations for unknown inverter control systems directly from external measurement data.

2. Methodology: The PISML Framework

The authors propose Physics-Informed Sparse Machine Learning (PISML), a hybrid framework that balances the interpretability of symbolic regression with the representational power of neural networks.

A. Hybrid Neural-Symbolic Architecture

The framework decomposes the inverter dynamics into two components:

1. Sparse Symbolic Backbone ($f_{sparse}$): Captures dominant physical laws (e.g., Kirchhoff's laws, RLC dynamics) and standard control structures (e.g., PI regulators, power calculations). It uses a Domain-Specific Physics-Informed Library containing linear terms, integral states, and bilinear power terms ($p=vdid+vqiq$).
2. Neural Residual Branch ($N(x, u)$): Modeled as a continuous-time Neural ODE (using a Multilayer Perceptron). This branch learns the "residual" dynamics—complex, unmodeled nonlinearities, proprietary control logic, and high-frequency behaviors that the symbolic backbone cannot capture.

• Total Dynamics: $\dot{x}(t) = f_{sparse}(x, u) + N(x, u; \theta_{NN})$

B. Multi-Scale Physics-Informed Training

To prevent the neural network from "overwriting" the physical backbone (a common issue in joint training), PISML employs a Jacobian-regularized training mechanism:

• Trajectory Loss ($L_{traj}$): Minimizes the error between predicted and measured state trajectories.
• Jacobian Consistency Loss ($L_{pert}$): Enforces agreement between the model's linearized small-signal response (Jacobian matrix) and empirical perturbation data. This ensures the learned model preserves correct local stability characteristics (eigenvalues).
• Sparsity Penalty ($L_{sparse}$): Uses $\ell_1$ regularization to force the symbolic backbone to select only the most critical physical terms, discarding redundant ones.

C. Symbolic Explicit Equation Discovery (Distillation)

The final step transforms the "grey-box" hybrid model into a "white-box" explicit equation:

1. The trained Neural Residual is used as a high-fidelity, noise-free predictor to generate synthetic data.
2. Sparse Symbolic Regression is performed on this synthetic data using an extended library (including trigonometric and high-order terms) to extract a closed-form mathematical expression for the residual.
3. The explicit residual is merged with the symbolic backbone to yield the final governing equation.

3. Key Contributions

1. Hybrid Architecture: A novel co-design of a sparse symbolic backbone and a neural residual ODE, decoupling known physical laws from unknown nonlinearities.
2. Multi-Scale Consistency: Introduction of a physics-informed training strategy that simultaneously ensures accurate large-signal trajectory reconstruction and physically valid small-signal linearization (correct Jacobian structure).
3. Fidelity-Preserving Compression: A two-stage process that converts a heavy, opaque neural network into a compact, explicit symbolic equation, restoring analytical tractability for stability analysis.

4. Experimental Results

The framework was validated on a high-fidelity Hardware-in-the-Loop (HIL) platform using a Grid-Forming inverter with unknown control logic.

• Trajectory Reconstruction:
  • PISML reduced trajectory reconstruction error by over 340 times compared to standard SINDy and Mod-SINDy baselines.
  • It demonstrated superior Out-of-Distribution (OOD) generalization, accurately tracking dynamics under severe voltage sags (0.4 p.u.) and control saturation where purely symbolic or purely neural methods failed.
• Small-Signal Stability:
  • Unlike pure Neural ODEs or unconstrained PISML, the PISML-Phy (with Jacobian constraints) correctly identified the system's eigenvalues, keeping them strictly in the left-half plane (stable) and matching the ground truth.
  • The Wasserstein distance (measuring spectral discrepancy) was reduced by a factor of 340 compared to baselines.
• Data Efficiency:
  • PISML achieved high accuracy with only 12 training trajectories, whereas pure Neural ODEs required ~64 trajectories to match similar performance.
• Model Compression:
  • The final distilled model reduced the parameter count from ~5,000+ (in the neural network) to **<50 coefficients**, achieving a compression ratio of >250x while maintaining high fidelity.
• System-Level Integration:
  • The discovered explicit equations were successfully integrated into a 3-bus microgrid simulation. The hybrid simulation (PISML model + known analytical inverters) accurately reproduced transient load-sharing dynamics and allowed for rigorous root-locus stability analysis, which is impossible with black-box models.

5. Significance

This work bridges the critical gap between data-driven identification and rigorous power system engineering.

• Restores Analytical Tractability: It converts structurally inaccessible "black box" devices into explicit mathematical models, enabling eigenvalue-based stability analysis and damping assessment.
• Operational Safety: Grid operators can now mathematically assess the stability impact of integrating unknown proprietary inverters before physical connection, mitigating risks of instability.
• Computational Efficiency: The distilled symbolic models are lightweight enough for embedded deployment, unlike heavy neural networks.
• Unified Pathway: It provides a generalizable methodology for discovering governing equations in complex, nonlinear, multi-time-scale systems beyond just inverters.

In conclusion, PISML successfully reconciles the conflict between identifying complex unmodeled nonlinearities and preserving physical consistency, offering a robust solution for the stability analysis of modern inverter-dominated power grids.