[Paper Review] Coupled and decoupled impedance models compared in power electronics systems

Atle Rygg, Marta Molinas+2 more|arXiv (Cornell University)|Oct 17, 2016

Microgrid Control and Optimization7 references3 citations

TL;DR

This paper compares coupled (matrix) and decoupled impedance models for power electronic systems, introducing the concept of Mirror Frequency Coupling (MFC) to assess accuracy. It defines decoupled and semi-decoupled models and proposes a norm ε to quantify coupling error, concluding that decoupled models in the sequence domain offer near-equal accuracy to semi-decoupled models but with significantly lower complexity.

ABSTRACT

This paper provides a comparative analysis of impedance models for power electronic converters and systems for the purpose of stability investigations. Such models can be divided into either decoupled models or matrix models. A decoupled impedance model is highly appealing since the Single-Input-Single-Output (SISO) structure makes the analysis and result interpretation very simple. On the other hand, matrix impedance models are more accurate, and in some cases necessary. Previous works have applied various approximations to obtain decoupled models, and both the dq- and sequence domains have been used. This paper introduces the terms decoupled and semi-decoupled impedance models in order to have a clear classification of the available approximations. The accuracy of 4 decoupled impedance models are discussed based on the concept of Mirror Frequency Coupling (MFC). By definition the decoupled models based on sequence domain impedances will be exact for systems without MFC. In the general case, they are expected to be more accurate than the decoupled dq-impedance models. The paper defines a norm $ε$ to measure the degree of coupling in the impedance matrices. This norm equals the error in the eigenvalue loci between the matrix and semi-decoupled models. This can also be viewed as the error in the semi-decoupled Nyquist plot. An example case study consisting of a grid-connected VSC with current controller and PLL is used to compare the different methods. It is found that decoupled and semi-decoupled models in the dq-domain are only applicable in grids with very low X/R-ratio. Furthermore, it is concluded that the decoupled model in the sequence domain gives close to equal results as the semi-decoupled model.

Motivation & Objective

• To systematically classify and compare decoupled, semi-decoupled, and exact impedance models in power electronic systems.
• To address the ambiguity in existing literature regarding the definitions of decoupled and semi-decoupled models.
• To introduce a quantitative measure, ε, to assess the error introduced by decoupling approximations in impedance models.
• To evaluate the accuracy of different impedance modeling approaches under varying grid conditions, particularly regarding X/R ratio.
• To provide practical recommendations on when to use decoupled, semi-decoupled, or exact matrix models for stability analysis.

Proposed method

• Defines three model types: decoupled (initially neglecting coupling), semi-decoupled (captures full coupling but neglects it in final analysis), and exact (full 2x2 matrix models).
• Introduces the norm ε to quantify the error between semi-decoupled and exact models, defined as the deviation in eigenvalue loci or Nyquist plots.
• Applies the modified sequence domain transformation to relate dq-domain impedance matrices to sequence-domain matrices via a unitary transformation matrix A_Z.
• Uses a grid-connected VSC with current control and PLL as a case study to compare model performance across different grid X/R ratios.
• Employs perturbation injection signals in both dq and sequence domains to extract impedance matrices from simulation.
• Analyzes stability using eigenvalue loci and Nyquist plots to compare the accuracy of different modeling approaches.

Experimental results

Research questions

• RQ1 How do decoupled and semi-decoupled impedance models compare in accuracy to exact matrix models in power electronic systems?
• RQ2 What is the role of Mirror Frequency Coupling (MFC) in degrading the accuracy of decoupled models in the dq-domain?
• RQ3 In which grid conditions do decoupled models in the dq-domain remain valid, and when do they fail?
• RQ4 How does the choice of impedance domain (dq vs. sequence) affect the accuracy and practicality of decoupled modeling?
• RQ5 Can the norm ε reliably predict when matrix models are necessary for accurate stability analysis?

Key findings

• Decoupled and semi-decoupled models in the dq-domain are only accurate for grids with very low X/R ratios (e.g., X/R ≈ 0.1), failing for typical inductive grids.
• The decoupled model in the sequence domain provides near-identical accuracy to the semi-decoupled model, as confirmed by eigenvalue locus and Nyquist plot comparisons.
• The norm ε effectively quantifies the error introduced by decoupling, with higher values indicating stronger coupling effects requiring matrix models.
• For systems without Mirror Frequency Coupling (MFC), decoupled sequence-domain models are exact by definition.
• Semi-decoupled models offer no significant accuracy improvement over decoupled models but require substantially more complex measurement or simulation procedures.
• The paper recommends using decoupled models in the sequence domain over semi-decoupled or dq-domain models due to superior balance of accuracy and simplicity.