#### Coupled and decoupled impedance models compared in power electronics systems

##### Atle Rygg, Marta Molinas, Chen Zhang, Xu Cai

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 $\epsilon$ 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.

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