# Impedance-Based Stability Analysis of Grid-Connected Inverters under the Unbalanced Grid Condition
## 1 Introduction  
In recent years, the penetration level of generated renewable energy such as wind and solar power into the power system has increased [1]. In grid-connected devices for renewable energy generation, inverters are widely applied [2; 3; 4]. Photovoltaic power stations and wind farms are usually located in remote places, and electricity generated by renewable energy sources needs to be transmitted over long distances [5]. For long-distance transmission lines, the line impedance and transformer impedance cannot be ignored, and various faults may occur throughout the line, which make the grid extremely sensitive to vulnerability and unbalance [6; 7; 8]. Due to the complex interaction between inverters and the weak and unbalanced power grid, there are many oscillation events caused by inverters in the power system [9; 10]. Hence, the stability analysis of a grid-connected inverter system is significant for improving the utilization rate of renewable energy and maintaining the stable operation of the whole power system [11; 12]. Stability analysis methods for inverter systems are mainly categorized into time-domain-based and frequency-domain-based analysis methods [13; 14; 15; 16; 17]. The state-space-model-based time-domain analysis method has been investigated for a long time and the techniques are very mature [13]. However, due to the large number of black-box and grey-box devices accessed by modern power systems, the time-domain approach is not valid or efficient [14]. Moreover, the topology of modern power systems is rapidly changing, leading to inefficient analysis using time-domain methods. In recent years, impedance-model-based frequency-domain analysis methods have attracted researchers and scholars due to their great advantages in modeling, measurement, and scalability [15]. Impedance models can be categorized into sequence impedance models and \(dq\)-axis impedance models [16; 17]. Compared with the latter, the physical meaning of the sequence impedance model is clearand the sequence impedance model is highly scalable, so it is more suitable for modern power systems. Therefore, the sequence impedance-based modeling method and stability analysis method are adopted in this paper. A weak and unbalanced grid condition will result in an inaccurate phase being obtained by the phase-locked loop (PLL), and further cause oscillations in the PLL-based grid-following inverter system [18; 19]. In addition, it has been found in recent years that PLL-induced oscillations exhibit frequency-coupling characteristics. The frequencies of the oscillation harmonic components are mirror-coupled about the fundamental frequency [20]. In addition to the PLL, the grid impedance and other \(dq\)-axis control asymmetries have a great influence on frequency-coupling oscillations [21; 22]. The traditional impedance models consider the inverter as a single-input single-output (SISO) system with positive and negative sequence impedance decoupling. Obviously, the traditional impedance model cannot accurately describe the frequency coupling characteristics of the inverter. Frequency-coupled impedance modeling is significant for analyzing PLL-based grid-following inverters. Using the multi-harmonic linearization method, a detailed frequency-coupled impedance model of the voltage source converter has been established [23]. Various frequency-coupling factors such as PLL, outer DC voltage loop, and inner current loop are considered in [23]. The PLL is the most significant factor in the frequency-coupling effect. Many investigations have improved the implementation and structure of the PLL to achieve suppression of the frequency-coupling effect and enhance system stability [24; 25; 26]. Compared with the traditional synchronous reference frame phase-locked loop (SRF-PLL) and its improvement, the dual second-order generalized integrator PLL (DSOGI-PLL) is widely used under the grid condition where three-phase unbalance and complex system harmonics exist.