Harmonic Stability Analysis of Microgrids with Converter-Interfaced Distributed Energy Resources, Part I: Modelling and Theoretical Foundations

TL;DR

This work develops a modular Harmonic State-Space (HSS) framework for Harmonic Stability Assessment (HSA) in power systems with a high share of Converter-Interfaced Distributed Energy Resources (CIDERs). By deriving HSS models for individual CIDERs and the grid, and combining them into a closed-loop power-system HSS, the method captures frequency coupling across harmonics via Fourier-Toeplitz representations. It introduces operators for eigenvalue sensitivity and a taxonomy of eigenvalues (CDI, CDV, DI), along with discussion of spurious modes due to harmonic truncation, setting the stage for Part II’s detailed analysis and validation on a realistic test system. The approach aims to provide a universal, scalable tool for identifying harmonic instabilities and guiding control-design decisions in modern low-voltage distribution networks with substantial CIDER penetration.

Abstract

This paper proposes a method for the Harmonic Stability Assessment (HSA) of power systems with a high share of Converter-Interfaced Distributed Energy Resources (CIDERs). To this end, the Harmonic State-Space (HSS) model of a generic power system is formulated by combining the HSS models of the resources and the grid in closed-loop configuration. The HSS model of the resources is obtained from the Linear Time Periodic (LTP) models of the CIDER components transformed to frequency domain using Fourier theory and Toeplitz matrices. Notably, the HSS of a CIDER is capable of representing the coupling between harmonic frequencies in detail. The HSS model of the grid is derived from the dynamic equations of the individual branch and shunt elements. The system matrix of the HSS models on power-system or resource level is employed for eigenvalue analysis in the context of HSA. A sensitivity analysis of the eigenvalue loci w.r.t. changes in model parameters, and a classification of eigenvalues into control-design variant, control-design invariant, and design invariant eigenvalues is proposed. A case of harmonic instability is identified by the HSA and validated via Time-Domain Simulations (TDS) in Simulink.

Figures (5)

• Figure 1: Illustration of the effect of frequency coupling in a model. In- and output signals are depicted through their Fourier coefficients, assuming a maximum harmonic order of two. The model behaviour w.r.t. coupling of frequencies is illustrated by rectangular blocks. The positive and negative spectrum of the signal are taken into account.
• Figure 2: Block diagram of the generic model of CIDERs in harmonic domain. The internal response is the closed-loop configuration between power hardware $\pi$ and control software $\kappa$. The reference calculation is represented by $\mathbf{\hat{R}}(\cdot,\cdot)$ and signals are subject to transformation matrices $\mathbf{\hat{T}}$ that represent changes of coordinate frames between power hardware and control software and circuit configurations between power hardware and grid $\gamma$.
• Figure 3: Lumped-element model of the grid shown at nodes with a grid-forming resource (\ref{['fig:Grid:model:forming']}) and a grid-following resource (\ref{['fig:Grid:model:following']}). The electrical quantity controlled by the resource is highlighted in red.
• Figure 4: Block diagram of the generic power system.
• Figure 5: Example of a mode of eigenvalues for an LTI model, and a corresponding HSS model with and without frequency coupling (FC) effect.

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Definition 3
• Definition 4