Harmonic Stability Analysis of Microgrids with Converter-Interfaced Distributed Energy Resources, Part II: Case Studies

TL;DR

This work extends the Harmonic Stability Assessment (HSA) framework from Part I to Converter-Interfaced Distributed Energy Resources (CIDERs) and a CIGRÉ LV microgrid. By leveraging Harmonic State-Space (HSS) models in Linear Time-Periodic (LTP) form, it highlights cross-harmonic coupling and demonstrates that harmonic instability can be detected only when time-periodic effects are included, not by conventional LTI analysis. The study develops and applies methods for (i) maximum-harmonic truncation impact, (ii) eigenvalue classification into CDI/CDV/DI, and (iii) sensitivity analyses of control parameters, validating findings with Time-Domain Simulations (TDS) in Simulink. A representative microgrid case shows harmonic instability arising from system-wide interactions, underscoring the practical value of HSA for robust design and tuning of CIDERs in distribution networks. The results affirm the framework's generic and modular nature for analyzing a broad class of CIDERs and grid configurations under harmonic distortion.

Abstract

In Part I of this paper a method for the Harmonic Stability Assessment (HSA) of power systems with a high share of Converter-Interfaced Distributed Energy Resources (CIDERs) was proposed. Specifically, the Harmonic State-Space (HSS) model of a generic power system is derived through combination of the components HSS models. The HSS models of CIDERs and grid are based on Linear Time-Periodic (LTP) models, capable of representing the coupling between different harmonics. In Part II, the HSA of a grid-forming, and two grid-following CIDERs (i.e., ex- and including the DC-side modelling) is performed. More precisely, the classification of the eigenvalues, the impact of the maximum harmonic order on the locations of the eigenvalues, and the sensitivity curves of the eigenvalues w.r.t. to control parameters are provided. These analyses allow to study the physical meaning and origin of the CIDERs eigenvalues. Additionally, the HSA is performed for a representative example system derived from the CIGRE low-voltage benchmark system. A case of harmonic instability is identified through the system eigenvalues, and validated with Time-Domain Simulations (TDS) in Simulink. It is demonstrated that, as opposed to stability analyses based on Linear Time-Invariant (LTI) models, the HSA is suitable for the detection of harmonic instability.

Figures (13)

• Figure 1: Block diagram of a generic CIDER. The internal response is consists of the power hardware $\pi$, control software $\kappa$ and coordinate transformations $\tau_{\cdot|\cdot}$. Together with the reference calculation $\rho$ and additional transformations (e.g., accounting for changes in circuit configurations) it builds the CIDER response at the point of connection to the grid $\gamma$.
• Figure 2: Test setup for the validation of the individual CIDER models. The resource is represented by a detailed state-space model, and the power system by a TE(see \ref{['tab:TE:parameters', 'tab:TE:harmonics']}).
• Figure 3: Analysis of the impact of the maximum harmonic order on the location of the LTP eigenvalues for the grid-following CIDER that models the DC side.
• Figure 4: Analysis of the impact of $h_{max}$ on the location of the LTP compared to the LTI eigenvalues for the three CIDERs. The CIDERs being analysed are the grid-forming and grid-following CIDERs that only model the AC-side characteristics, and the grid-following CIDER that includes the DC-side modelling.
• Figure 5: Classification of the eigenvalues of the grid-forming CIDER. \ref{['fig:hsa-rsc:classification:frm']}: Eigenvalues on the left-hand side and eigenvector matrix $\boldsymbol{\mathcal{V}}$ on the right-hand side. The dark grey line in the plot of the eigenvector matrix indicates the separation between the states of the power hardware and control software. The individual states are further partitioned w.r.t. their harmonic order by the dotted light grey lines. \ref{['fig:hsa:schema:transformation']}: Mapping of the harmonic sequences for a transformation from ABC coordinates to DQZ components and vice versa. \ref{['fig:hsa:frm:classification:Vdetail']}: Representation of the entries of $\boldsymbol{\mathcal{V}}$ by triplets (ABC) and pairs (DQ) in the sequence domain.