Small-Signal Converter Admittance in the $pn$-Frame: Systematic Derivation and Analysis of the Cross-Coupling Terms

In this paper, a methodology is presented to derive the small-signal admittance of a converter-based control system in the <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame. Such a methodology makes use of a set of derived equations that link the <inline-formula><tex-math notation="LaTeX">$qd$</tex-math></inline-formula>-frame small-signal admittance data to the corresponding <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame small-signal admittance terms. Compared to existing results providing the <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame small-signal impedance of a grid-connected converter, the presented technique allows a calculation of the <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame small-signal admittance (or impedance) when additional elements are present in the controller, as in many practical designs. The presented technique is therefore generic and provides a systematic way to calculate the <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame admittance regardless of the used converter control scheme. In this study, the method has been applied to assess how different parts of the controller affect the <inline-formula><tex-math notation="LaTeX">$pn$</tex-math></inline-formula>-frame admittance terms and, in particular, the cross-coupling that exists between its positive and negative sequence terms. Thereby, the results of the proposed methodology have been used to study the impact of the negative sequence current compensator on the stability performance of a grid-connected converter. Experimental tests using a real hardware setup are used to support the obtained results.