Pauli Decomposition of Impedance Matrices for Understanding the Root Cause of Instabilities in Grid-Connected Power Electronic Converters

TL;DR

This work introduces a Pauli decomposition of dq impedance matrices to assign physical meaning to impedance terms and facilitate root-cause analysis of instabilities in grid-connected power electronic converters. By representing impedance as a quaternion-like object, the method provides a compact, interpretable link between impedance components and stability criteria such as the minor-loop Nyquist function L(s), eigenvalues, and passivity. A case study on a weak-grid benchmark demonstrates how specific components (notably PLL-related terms) drive instability and how PLL bandwidth tuning can stabilize the system. The approach is validated with EMT simulations and supported by public code (pauliCode), offering a practical tool for diagnosing and mitigating grid-connected converter instabilities.

Abstract

The impedance criterion has emerged as an alternative way to stability assessment of grid-connected power electronic converters. However, the lack of physical meaning of impedance and admittance matrices hinders the ability to understand the root cause of instabilities. To address this issue, this paper proposes the application of Pauli decomposition to the impedance matrices and the minor loop of grid-connected power electronic converters. The application of this methodology simplifies establishing the link between impedance matrix terms and closed-loop stability properties. Moreover, Pauli decomposition transforms impedance matrices in a quaternion-like form that is helpful to assess the root cause of instabilities. The theoretical contributions are validated using a case study consisting of a power electronic converter connected to a weak grid that has been previously analysed in the literature using existing techniques.

Figures (8)

• Figure 1: (a) Classical interpretation of the impedance matrix. (b) Pauli decomposed impedance (c) actions of the Pauli matrices on a vector.
• Figure 2: Electrical and control diagram of a power converter connected to the grid.
• Figure 3: Admittance model of the converter.
• Figure 4: Logarithmic Nyquist plot of $L(s)$, for the unstable case.
• Figure 5: Analysis of the components of the converter admittance.