[Paper Review] Modularized Bilinear Koopman Operator for Modeling and Predicting Transients of Microgrids

Xinyuan Jiang, Yan Li+1 more|arXiv (Cornell University)|May 6, 2022

Model Reduction and Neural Networks4 citations

TL;DR

This paper proposes Modularized Bilinear Koopman Form (M-KBF), a scalable data-driven method that models microgrid transient dynamics by decomposing the system into subsystems, applying Koopman bilinear forms via Extended Dynamic Mode Decomposition (EDMD) with eigenvalue-based truncation, and achieving fast, accurate predictions—demonstrated with real-time capable performance and plug-and-play modularity for online control applications.

ABSTRACT

Modularized Koopman Bilinear Form (M-KBF) is presented to model and predict the transient dynamics of microgrids in the presence of disturbances. As a scalable data-driven approach, M-KBF divides the identification and prediction of the high-dimensional nonlinear system into the individual study of subsystems; and thus, alleviating the difficulty of intensively handling high volume data and overcoming the curse of dimensionality. For each subsystem, Koopman bilinear form is applied to efficiently identify its model by developing eigenfunctions via the extended dynamic mode decomposition method with an eigenvalue-based order truncation. Extensive tests show that M-KBF can provide accurate transient dynamics prediction for the nonlinear microgrids and verify the plug-and-play modeling and prediction function, which offers a potent tool for identifying high-dimensional systems. The modularity feature of M-KBF enables the provision of fast and precise prediction for the microgrid operation and control, paving the way towards online applications.

Motivation & Objective

• Address the challenge of modeling and predicting transient dynamics in microgrids with high-dimensional, nonlinear behavior due to distributed energy resources (DERs) and reduced inertia.
• Overcome the limitations of local linear models (lack of extrapolation) and global nonlinear models (high computational cost) in transient prediction.
• Develop a scalable, modular framework that enables efficient, accurate, and fast prediction of microgrid transients using only operating data.
• Enable plug-and-play modeling and prediction for microgrid subsystems, supporting online control applications.
• Ensure robustness to reference frame ambiguity in DQ transformations through data preprocessing and observable function design.

Proposed method

• Decompose the microgrid into modular subsystems (e.g., V/f-controlled DERs, PQ-controlled DERs) to reduce dimensionality and enable parallel processing.
• Apply the Koopman bilinear form (KBF) to each subsystem to represent nonlinear dynamics as a finite-dimensional linear system in a lifted state space using eigenfunctions.
• Use Extended Dynamic Mode Decomposition (EDMD) with eigenvalue-based order truncation to identify Koopman eigenfunctions and eigenvalues from transient data.
• Construct observable functions as nonlinear combinations of state variables (e.g., voltages, currents, angles) to enable accurate lifting of the dynamics.
• Integrate individual subsystem models into a global microgrid model using a modular framework that preserves accuracy and scalability.
• Preprocess training data by rotating reference frames to reduce dependency on global DQ frame orientation and improve model invariance.

Figure 1: A typical microgrid test system

Experimental results

Research questions

• RQ1 Can a modular, data-driven Koopman-based framework accurately predict transient dynamics in microgrids with high-dimensional, nonlinear behavior?
• RQ2 How does the M-KBF framework handle the curse of dimensionality and scalability in large microgrid systems?
• RQ3 To what extent does the model remain accurate under large disturbances (e.g., ±100% load or generation changes) and reference frame shifts?
• RQ4 Can the M-KBF model achieve real-time or faster-than-real-time prediction performance suitable for online control?
• RQ5 How robust is the model to reference frame ambiguity, and can it be made invariant through proper observable function selection?

Key findings

• M-KBF achieves accurate transient predictions for microgrids under ±100% disturbances, with average voltage prediction errors below 0.01 pu across all nodes.
• The method enables faster-than-real-time prediction: 20-second predictions take only 5.0064 seconds on a 2.9 GHz PC, supporting online control applications.
• The modular design allows plug-and-play integration of subsystem models, enabling scalable and reusable modeling for evolving microgrid configurations.
• The model maintains accuracy within ±0.04 radian for PLL phase shifts in the V/f model and ±0.05 radian in the PQ model, with potential for improvement via invariant observable functions.
• EDMD with eigenvalue-based truncation effectively identifies a minimal, accurate set of Koopman eigenfunctions, reducing computational cost while preserving predictive fidelity.
• Preprocessing training data with random DQ frame shifts significantly reduces model dependency on reference frame orientation, enhancing robustness.

Figure 2: Prediction of Vf-controlled DER under disturbances in the $100\%$ range