# Impedance-Based Stability Analysis of Grid-Connected Inverters under the Unbalanced Grid Condition
## 1 Introduction
The reason for this is that the DSOGI-PLL can accurately obtain the phase in the presence of three-phase unbalance and complex system harmonics [27]. However, the second-order generalized integrator (SOGI) of DSOGI-PLL cannot attenuate the low-frequency signal sufficiently, which can lead to low-frequency oscillations. Accurate small-signal models with frequency-adaptive DSOGI-PLL have been developed for DSOGI-PLL as well as PR current-controlled inverters [28]. However, studies on the stability of the inverter system under unbalanced grids, where DSOGI-PLL is most commonly used, are still lacking. Therefore, in this paper, the stability of DSOGI-PLL-based grid-connected inverter systems under a weak and unbalanced grid, on which few studies have been carried out until now, is investigated based on the impedance-based method. There have been many studies on the influence of the unbalanced grid impedance on the stability of grid-connected inverters [29; 30; 31; 32; 33; 34; 35]. The multi-input multi-output (MIMO) impedance model has been established for inverters under the unbalanced grid condition. Therefore, to simplify the stability analysis steps, the individual channel method [32] is adopted to extract the SISO subsystems from the established MIMO models [33]. Similarly, the MIMO system is decomposed into three SISO systems [34]. However, these studies neglect the frequency-coupling effect, which may lead to inaccurate analysis results. Therefore, for the inverter working under the unbalanced grid condition, the coupling of sequence impedance caused by both the unbalance and the frequency-coupling effect should be considered for modeling. In [35], the impedance model of the inverter is established by considering the frequency-coupling effect and the unbalance through the harmonic transfer matrix-based method. And it is pointed out in [35] that the fourth-order impedance model can accurately describe the impedance characteristics of the inverter under the unbalanced grid condition. Inspired by this study, for investigating the DSOGI-PLL-based inverter under the unbalanced grid condition, a fourth-order impedance model considering the modeling of DSOGI-PLL is required to be established. In summary, although there are many studies on the modeling and stability of inverters under unbalanced grid conditions, the modeling and stability analysis of DSOGI-PLL-based inverters under three-phase unbalanced conditions are not well developed. In order to remedy this deficiency, some investigations of the DSOGI-PLL-based inverter are carried out in this paper. For the stability analysis of higher-order MIMO systems, there are two common methods. One is to use the generalized Nyquist criterion (GNC) directly [29; 30]. The other is to transform the frequency-coupled impedance model into equivalent decoupled sequence impedances via matrix diagonalization, and then use the SISO stability criterion [35]. However, for the stability analysis of a fourth-order system, this diagonalization method is very complicated and not suitable. And there may be RHP poles in the equivalent decoupled sequence impedance model, which will lead to incorrect stability analysis results [36]. Therefore, in this paper, based on the GNC and matrix theory, a simplified stability analysis method is proposed. The main contributions of this paper are as follows  
1. The frequency-coupled impedance models of the DSOGI-PLL-based inverter under the balanced and unbalanced grid conditions are derived. This provides the theoretical basis for the analysis of the impedance characteristics and stability of the DSOGI-PLL-based inverter. 2. Based on the GNC and matrix theory, the stability analysis of high-order systems is transformed into the analysis of eigenvalues of two second-order matrices, which simplifies the stability analysis process of high-order systems. 3. The influences of pivotal parameters on the stability of the DSOGI-PLL-based grid-connected inverter system are comprehensively investigated.