## 1 Introduction  
With the development of power electronic devices, VSC-HVDC systems have been widely applied to AC grid interconnection because of their independent decoupling control of active and reactive power [1; 2; 3]. Recently, a large number of studies on modeling, control, and stability analysis of VSC-HVDC system have been published [4; 5; 6; 7; 8; 9; 10]. Previous studies have shown that the interaction between converters or between the converter and the grid influences the stability of a system. DC- side oscillation is a problem in VSC-HVDC and has been reported in a real project [11]. When DC side oscillation occurs, a DC system will not work and will impact on the power system. Therefore, the DC-side stability of VSC-HVDC should be evaluated before connecting it to the main grid. In VSC-HVDC systems applied to AC grid interconnection, active power often needs bidirectional transmission [12]. However, studies show that the maximum transmission power of the DC-voltage-controlled converter to power-controlled converter is less than that in the opposite power flow direction [13]. An Impedance-based approach can be adopted to analyze the influence of VSC-HVDC systems with different directions of transmission power on stability [13]. The impedance stability criterion was proposed in [14] and used in grid-connected inverters [15]. It was applied as a stability criterion in a VSC-HVDC system [12; 16; 17; 18; 19]. The impedance model of two-terminal VSC-HVDCs was built in [12], and the cause of DC current resonance was analyzed with Nyquist stability criteria. Different subsystems were selected to analyze the DC-side stability of the VSC-HVDC system with a transfer function method in [16]. The influence of overhead transmission lines, DC cables, and the DC-side filter on system stability was investigated in [17]. The different performances of the lumped parameter and distributed parameter circuit models in stability analysis were discussed in [18]. It was found that the distributed parameter circuit model is more accurate in stability analysis. In the condition of VSC-HVDC for AC Grid interconnection, the maximum transmission capacities of the different power flow directions are different. Thus, a control strategy is required to improve the transmission capacity from the DC-voltage-controlled converter to the power-controlled converter to improve resource utilization efficiency. To increase the maximum transmission power, the DC side oscillation must be suppressed. The suppression methods of DC side oscillation can be classified into passive methods [20; 21] and active methods [22; 23; 24; 25; 26; 27; 28; 29]. Passive methods suppress resonance by introducing a passive damper branch into the circuit to remodel the impedance of the source converter or load converter in a cascaded system. Active methods suppress resonance by introducing voltage and current feedback control in a controller to improve the impedance of the source converter or load converter. Virtual impedance is widely used in control systems as an active damping control method [24; 25; 26; 27; 28; 29]. It can be introduced to suppress DC-side oscillation [24; 25], to limit output current for voltage controlled inverters during overloads or faults [26; 27], to improve the stability of a grid-connected inverter by change its input admittance [28], and to enhance the small-signal stability of a modular multilevel converter (MMC) based DC grid [29]. Virtual impedance in the DC voltage control loop can suppress the DC-side oscillation of a VSC-HVDC transmission system and improve its stability margin and the transmission capacity of the system [24; 25]. However, virtual impedance control leads to steady-state errors in the DC output voltage of DC-voltage-controlled converters, due to different operating points [25]. An improved virtual impedance control strategy is proposed in this paper. To design an appropriate control strategy, a stability analysis is required.