## 2 Improved Virtual Impedance Control Principle  
This paper mainly studies a two-level topology structure. The analysis in this paper can be also applicable to an MMC system if the dc bus voltage ripples are insignificant [12]. A two-level VSC-HVDC system used in AC grid interconnection is depicted in Figure 1. Figure 1 shows a DC voltage-controlled converter and a power-controlled converter on the left and right sides, respectively. The two converters have an identical structure. \(R_{n1}+jX_{n1}\) and \(R_{n2}+jX_{n2}\) are the equivalent impedance of the AC system, \(R_{c1}+j\omega_{1}L_{1}\) and \(R_{c2}+j\omega_{2}L_{c2}\) are the impedance of the filter reactor, \(C_{f1}\) and \(C_{f2}\) are the filter capacitors, \(m_{1}\) and \(m_{2}\) are the modulations of the converter station, \(\dot{u}_{s1}\) and \(\dot{u}_{s2}\) are the AC voltage at point of common coupling, \(\dot{u}_{g1}\) and \(\dot{u}_{g2}\) are the AC voltage of the AC system, \(\dot{i}_{s1}\) and \(\dot{i}_{s2}\) are the AC current flowing through the filter reactor, and \(\dot{i}_{g1}\) and \(\dot{i}_{g2}\) are the AC current flowing through the AC system. The DC cable is a \(\pi\) type, with an equivalent resistance of \(R_{d}\), an equivalent inductance of \(L_{d}\), and an equivalent capacitance of \(C_{d}\). The modeling and control of the VSC-HVDC system are presented in a synchronous rotating frame (SRF). The transformation of the three-phase quantity from stationary reference frame to the SRF is based on the amplitude-invariant Park transformation, with the d-axis aligned with the voltage vector \(u_{s}\) and q-axis leading the d-axis by \(90^{\circ}\). The grid voltage defines the system's \(dq\) reference. A phase-locked loop (PLL) defines the controller \(dq\) reference. The system reference is aligned with the PLL reference in a steady state. When a small disturbance occurs, the system reference is no longer aligned with the PLL reference. The relationship between the system reference and the PLL reference under a small disturbance is shown in Figure 2.  
Here, subscript \(dq\) represent the components of the physical quantity in the SRF. Subscript 0 represents the value of the physical quantity at the static working point, subscript _ref_ represents the given value of physical quantity, and subscript \(\Delta\) represents the small disturbance components of the physical quantity. Superscript \(c\) represents the components of physical quantity in the PLL reference, and superscript \(s\) represents the components in the system reference. The simulation model of VSC-HVDC in Figure 1 is built on a MATLAB/Simulink. The system parameters are shown in the following table. The vector current control [13] is adopted in the two converters, and the control is shown in Figure 3. For symmetry of the VSC-HVDC system, subscripts 1,2 of the system parameters in Table 1 and the control diagram in Figure 3 are omitted. In the model, controller parameters are per unit. The base angular grid frequency is 50 Hz, the base grid voltage is 110 kV, the base system capacity is 500 MW, and the base DC voltage is 250 kV. According to the parameters in Table 1, the system's closed-loop bandwidth with a DC voltage controller is 65 Hz, and the system's open-loop phase margin is 58 degrees. The system's closed-loop bandwidth with an active power controller is 8 Hz, and the system's open-loop phase margin is 150 degrees. The system's closed-loop bandwidth with the current controller is 180 Hz, and the system's open-loop phase margin is 85 degrees. Figure 1: VSC–HVDC used in AC grid interconnection. Figure 2: System and phase-locked loop (PLL) references. The power flow direction from the power-controlled converter to the DC voltage-controlled converter is set to positive. Figure 4 shows the resulting time-domain responses of the DC voltage of the VSC-HVDC system. At 2 s, the active power instruction value steps from 500 MW to \(-\)500 MW, and the length of the DC cable is 50 km.