Analytical Large-Signal Modeling of Inverter-based Microgrids with Koopman Operator Theory for Autonomous Control

TL;DR

This work addresses large-signal nonlinear dynamics in islanded inverter-based microgrids at EMT time scales by formulating an analytical Koopman-operator (KO) based linearization that preserves primary and zero-control dynamics. By carefully designing observable functions, the authors derive a finite-dimensional KO representation that enables standard linear control methods, demonstrated through an LQI-based voltage-control design and a lifting-to-recovery mapping back to the original MG. The KO model is validated on a three-DER MG, showing effective voltage restoration with robust performance under initial-condition variations and explicit model-error analysis. The approach avoids data-driven KO identification, offering explainability and generality to adapt to different MG structures and control objectives, while maintaining compatibility with advanced linear control techniques. Practically, this provides a scalable, transparent pathway to apply textbook controllers to complex EMT MGs without resorting to purely numerical KO estimation.

Abstract

The microgrid (MG) plays a crucial role in the energy transition, but its nonlinearity presents a significant challenge for large-signal power systems studies in the electromagnetic transient (EMT) time scale. In this paper, we develop a large-signal linear MG model that considers the detailed dynamics of the primary and zero-control levels based on the Koopman operator (KO) theory. Firstly, a set of observable functions is carefully designed to capture the nonlinear dynamics of the MG. The corresponding linear KO is then analytically derived based on these observables, resulting in the linear representation of the original nonlinear MG with observables as the new coordinate. The influence of external input on the system dynamics is also considered during the derivation, enabling control of the MG. We solve the voltage control problem using the traditional linear quadratic integrator (LQI) method to demonstrate that textbook linear control techniques can accurately control the original nonlinear MG via the developed KO linearized MG model. Our proposed KO linearization method is generic and can be easily extended for different control objectives and MG structures using our analytical derivation procedure. We validate the effectiveness of our methodology through various case studies.

Figures (9)

• Figure 1: Overall diagram of a nonlinear MG system model.
• Figure 2: Illustration of the KO theory. The upper row illustrates that a dynamic system can be measured by an infinite set of observable functions $\mathbf{g}$. The lower row explains that the KO, $\mathcal{K}$, describes the dynamical evolution of the observation of the state and input, $\mathbf{g}(\mathbf{x},\mathbf{u})$, in a linear manner.
• Figure 3: Diagram of the test MG system.
• Figure 4: Closed-loop MG control system based on the KO linearized model and LQI. The LQI gain is $\mathbf{K}=\mathbf{R}^{-1}\widetilde{\mathbf{B}}^{\top}\mathbf{P}$.
• Figure 5: Dynamic responses of DER output voltages of the test MG.