energies Article Coupling Influence on the dq Impedance Stability Analysis for the Three-Phase Grid-Connected Inverter Chuanyue Li 1 , Taoufik Qoria 1 , Frederic Colas 1 , Jun Liang 2 , Wenlong Ming 2 , Francois Gruson 1 and Xavier Guillaud 1, * 1 Laboratory of Electricl Engineering and Power Electronics (L2EP), 59046 Lille, France; Chuanyue.Li@outlook.com (C.L.); taoufik.qoria@ensam.eu (T.Q.); Frederic.COLAS@ENSAM.eu (F.C.); francois.gruson@ensam.eu (F.G.) 2 School of Engineering, Cardiff University, Cardiff CF24 3AA, UK; Liangj1@Cardiff.ac.uk (J.L.); MingW@cardiff.ac.uk (W.M.) * Correspondence: xavier.guillaud@centralelille.fr Received: 1 August 2019; Accepted: 23 September 2019; Published: 26 September 2019   Abstract: The dq impedance stability analysis for a grid-connected current-control inverter is based on the impedance ratio matrix. However, the coupled matrix brings difficulties in deriving its eigenvalues for the analysis based on the general Nyquist criterion. If the couplings are ignored for simplification, unacceptable errors will be present in the analysis. In this paper, the influence of the couplings on the dq impedance stability analysis is studied. To take the couplings into account simply, the determinant-based impedance stability analysis is used. The mechanism between the determinant of the impedance-ratio matrix and the inverter stability is unveiled. Compared to the eigenvalues-based analysis, only one determinant rather than two eigenvalue s-function is required for the stability analysis. One Nyquist plot or pole map can be applied to the determinant to check the right-half-plane poles. The accuracy of the determinant-based stability analysis is also checked by comparing with the state-space stability analysis method. For the stability analysis, the coupling influence on the current control, the phase-locked loop, and the grid impedance are studied. The errors can be 10% in the stability analysis if the couplings are ignored. Keywords: impedance stability analysis; VSC; small-signal modelling 1. Introduction The integration of renewable energy sources is normally assisted by power electronic converters due to its ability for asynchronous connection and fully-AC voltage control. The high demand for renewable energies requires more and more inverters to be connected to the grid. The interaction between the grid-connected inverter and the grid may cause instabilities [1]. The stability analysis for the grid-connected inverter is essential to ensure secure power transportation to the grid. Two stability analysis methods can be applied according to the small-signal linearization technology. The state-space stability analysis [2] is a mature and commonly-used method. However, a high order and a complex state matrix have to be built. Impedance stability analysis is achieved via the impedance ratio, which is determined via the equivalent impedance of the inverter and the grid impedance. The impedance ratio can also be drawn as the Bode plot for the frequency analysis. Both Norton-based [3] and Thevenin-based [4] equivalent impedances of the inverter can be derived in the impedance stability analysis. For a three-phase inverter controlled via the dq frame, the impedance ratio is normally derived in the dq frame, which is a 2 × 2 matrix. Both eigenvalues of the impedance-ratio matrix are required for the stability analysis via the generalized Nyquist criterion (GNC) [5]. The criterion is commonly used Energies 2019, 12, 3676; doi:10.3390/en12193676 www.mdpi.com/journal/energies