Frequency Characteristic Analysis of the VSC-HVDC DC Oscillating Power under AC Sub-Synchronous Disturbances

Abstract

The DC-side response characteristics of the VSC-HVDC transmission system under AC-side sub-synchronous disturbances may lead to the propagation of AC/DC oscillations, and then there may be non-fundamental frequency oscillation on the DC-side. To clarify the interactive influence mechanism and propagation evolution law of the AC and DC side sub-synchronous oscillations in the VSC-HVDC transmission system, the frequency characteristics of the DC-side oscillation power are investigated by considering the effect of converter control links. First, the AC-side voltage and current frequency responses under single/multiple SSO components are analyzed. Secondly, the frequency characteristics of the DC-side power after the injection of single/multiple SSO components are investigated. Meanwhile, the effect of system control parameters on the sub-synchronous responses of the DC-side power is investigated. Finally, the theoretical analysis is verified with PSCAD/EMTDC simulations. The results show that under the influence of the AC-side SSO components, the DC-side of VSC-HVDC faces the risk of multi-frequency oscillations; when there are multiple SSO components on the AC-side, there is a coupling between the components, and the system control parameters affect the amplitude of the DC-side oscillation power.

1. Introduction

The VSC-HVDC transmission system has the advantages of flexible control ability, low harmonic component, power supply for passive system, etc., which has been rapidly developed in the field of large-capacity long-distance transmission [1,2,3]. However, due to the large number of power electronic devices connected to the power systems, a series of SSO issues have arisen.

The SSO phenomenon was observed during the trial operation of the Shanghai Nanhui wind power via the VSC-HVDC project [4], and the SSO phenomenon occurred when wind turbines were connected to the weak AC grid at Xinjiang Hami region in 2015 [5], thus threatening the system stable operation. These studies attribute the SSOs to the increasing number of power electronic devices in the power systems due to the access to wind power. The interactions of power electronic devices/power grids are prone to cause the SSO phenomenon on the AC- side. However, few studies have focused on the propagation of SSOs on the AC and DC sides of the wind power via the VSC-HVDC system, which may bring new risks and pitfalls for the VSC-HVDC system’s stable operation. Therefore, it is urgent to study the response characteristics and mechanism of the AC and DC sides of the wind power via the VSC-HVDC system when SSO occurs.

The SSO analysis methods for power electronic power systems are mainly time domain analysis [6,7], eigenvalue analysis [8,9,10,11], impedance analysis [12,13,14,15], the complex torque coefficient analysis method [16,17], and the transfer function analysis method [18]. The literature [6] investigated the SSO characteristics triggered by a static reactive power compensator in a weak grid using the time domain analysis method. The effect of phase-locked loop (PLL) on the VSC-HVDC characteristics is investigated with the eigenvalue method in [11], but the eigenvalue transfer function is relatively complex and requires extensive calculations. The literature [14] develops an impedance model for the PLL-based VSCs that can be used to assess system-level converter-grid compatibility and power quality. The literature [15] analyzes the interaction of synchronous generators on the AC- side based on the impedance analysis method, and the effect of controller parameters on the interaction is analyzed. The literature [16] analyzed the SSO risks before and after the connection of the VSC-HVDC with the complex torque coefficient analysis method, and a hybrid damping control strategy for the VSC-HVDC was proposed. The above literature has carried out in-depth research on the response mechanism and influencing factors of SSOs. However, the phenomenon of the system generating multiple frequency components under SSOs has not been analyzed. Meanwhile, most focus has been put on the response characteristics of wind turbines or weak grids under sub-synchronous disturbances, but the multi-frequency response analysis of the VSC-HVDC system under AC sub-synchronous disturbances is not addressed. The literature [18] analyzed the current responses of the wind turbine to the SSO component, but the frequency component characteristics of the VSC-HVDC SSO oscillating power are not addressed.

Currently, the AC-side or DC-side oscillation characteristics of wind power via the VSC-HVDC system are generally analyzed separately [19,20,21,22,23,24]. The literature [19] shows that in a doubly-fed wind turbine via a VSC-HVDC system, the SSO current component appears on the VSC-HVDC AC- side due to the wind turbine cutover. The literature [20] shows that there is a risk of SSO on the AC- side of the VSC-HVDC system when multiple direct-drive turbines are connected to the VSC-HVDC. The literature [21] investigated the effect of the AC system and PLL on the DC-side current oscillation characteristics. The literature [22] proposed a method to suppress the DC-side oscillation of the VSC-HVDC system based on a superconducting magnetic energy storage device. The literature [23] established the DC-side impedance model of the island VSC-HVDC system with the impedance analysis method, and the DC-side SSO mechanism was revealed. The literature [24] analyzed the DC-side SSO mechanism of the VSC-HVDC system from the perspectives of its impedance characteristics, switching frequency characteristics, and system control damping. Although the above literature considers the oscillation mechanism of the AC or DC side of the system, respectively, the possible interaction between the AC and DC networks is ignored [25], which may bring certain errors in the system stability analysis. Meanwhile, the AC–DC coupled SSO problem may cause serious hazards such as wind turbine off-grid and DC transmission line power dipping [26]. For the response characteristics under the SSO disturbances, existing studies mainly focused on the frequency response characteristics under a single SSO disturbance component [27,28]. The nonlinear controllers proposed in [29] have a process of weight factor adjustment, which is complicated. The PI controller used in this paper is simple and easy to implement. The dq control structures are usually associated with PI controllers, and they perform well in controlling DC variables [30]. Moreover, the use of cross-coupling terms combined with voltage feedforward in this paper also improves the performance of the PI controller [31]. The literature [32,33] points out that forced oscillations will be induced when there is a continuous small periodic disturbance in the system and the frequency of the disturbance is close to the intrinsic frequency of the system. In this paper, there is no disturbance frequency close to the SSO component within the VSC-HVDC system, so the oscillation case discussed in this paper is not a forced oscillation.

In summary, the SSO response mechanism and frequency component analysis of wind power via the VSC-HVDC system are seldom involved at present. Therefore, this paper firstly derives the response of AC-side voltage and current of the VSC-HVDC system under the AC sub-synchronous disturbance. Secondly, the frequency response characteristics of the VSC-HVDC under single/multiple SSO disturbance components are analyzed by considering the role of converter inner-loop control. Finally, the effect of system control parameters on the DC-side power is analyzed.

2. VSC-HVDC System Structure

The structure of a typical VSC-HVDC system is shown in Figure 1, which mainly includes AC and DC circuit parts. The AC and DC circuit sections mainly include the AC power source, SSO disturbance source, grid inductor, VSC, and DC line. The rectifier station of the VSC-HVDC system adopts constant active power control, and the inverter station adopts constant DC voltage control. Figure 2 shows the current control loop of the VSC. The inner loop current control outputs u_dref and u_qref correspond to the VSC output voltage’s d and q axis components, respectively.

Figure 3 shows the PLL control loop, including the PARK conversion and the q-axis voltage control, where ω₀ and θ_pll are the synchronous angular velocity of the AC grid and the synchronous phase angle measured by the PLL, respectively.

3. DC Power Frequency Characteristics under AC-Side SSO Perturbation

When SSO occurs in the AC-side of the VSC-HVDC system, the response of the AC-side voltage and current is firstly analyzed by considering the VSC inner-loop control, and then the expression of DC-side power is obtained. This section investigates the DC-side power response characteristics based on the system shown in Figure 1. The three-phase fundamental frequency current and SSO current, when the SSO current i_{ac1_ss} is injected into the AC-side point of common coupling (PCC), are

$$i_{\mathrm{abc}\_0}=\left[\begin{array}{l}{I}_{\mathrm{f}0}\cos ({\omega }_{0}t+{\varphi }_{\mathrm{f}0})\hfill \\ I_{\mathrm{f}0}\cos ({\omega }_{0}t+{\varphi }_{\mathrm{f}0}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \\ I_{\mathrm{f}0}\cos ({\omega }_{0}t+{\varphi }_{\mathrm{f}0}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \end{array}\right]$$

(1)

$$i_{\mathrm{ac}1\_\mathrm{ss}}=\left[\begin{array}{l}{I}_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\varphi }_{\mathrm{s}})\hfill \\ I_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\varphi }_{\mathrm{s}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \\ I_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\varphi }_{\mathrm{s}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \end{array}\right]$$

(2)

Noting the three-phase current at the PCC of the rectifier AC system as i_{s_abc}, i_{s_abc} is then

$$i_{\mathrm{s}\_\mathrm{abc}}=i_{\mathrm{abc}\_0}+i_{\mathrm{ac}1\_\mathrm{ss}}$$

(3)

System three-phase current disturbances lead to three-phase voltage fluctuations, and then the three-phase fundamental frequency voltage and SSO voltage can be expressed as

$$u_{\mathrm{abc}\_0}=\left[\begin{array}{l}{U}_{\mathrm{f}0}\cos ({\omega }_{0}t+{\psi }_{\mathrm{f}0})\hfill \\ U_{\mathrm{f}0}\cos ({\omega }_{0}t+{\psi }_{\mathrm{f}0}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \\ U_{\mathrm{f}0}\cos ({\omega }_{0}t+{\psi }_{\mathrm{f}0}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \end{array}\right]$$

(4)

$$u_{\mathrm{ac}1\_\mathrm{ss}}=\left[\begin{array}{l}{U}_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\psi }_{\mathrm{s}})\hfill \\ U_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\psi }_{\mathrm{s}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \\ U_{\mathrm{ac}1}\cos ({\omega }_{\mathrm{ac}1}t+{\psi }_{\mathrm{s}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \end{array}\right]$$

(5)

Noting the three-phase voltage at the PCC of the rectifier AC system as u_{s_abc}, u_{s_abc} is then

$$u_{\mathrm{s}\_\mathrm{abc}}=u_{\mathrm{abc}\_0}+u_{\mathrm{ac}1\_\mathrm{ss}}$$

(6)

In (1)–(6), I_f0 and ϕ_f0 are the amplitude and initial phase of the fundamental frequency current, respectively; I_ac1 and ϕ_s are the amplitude and initial phase of the injected SSO current i_{ac1_ss}, respectively; U_f0 and Ψ_f0 are the amplitude and initial phase of the fundamental frequency voltage, respectively; U_ac1 and Ψ_s are the amplitude and initial phase of the SSO voltage u_{ac1_ss}, respectively; and ω₀ and ω_ac1 are the fundamental frequency and the SSO frequency.

3.1. PLL Response Characteristics to SSO Disturbance

The SSO voltage u_{ac1_ss} leads to the change in PLL control angle compared to the pre-disturbance period. The change in the PLL control angle affects the response characteristics of the DC-side power. Therefore, the response of the PLL under the SSO perturbation is analyzed in the following. As shown in Figure 3, the PLL control angle is

$$\left\{\begin{array}{l}{\theta }_{\mathrm{pll}}={\displaystyle \frac{\omega }{s}}\\ \omega ={\omega }_0+(K_{\mathrm{p}\_\mathrm{pll}}+{\displaystyle \frac{K_{\mathrm{i}\_\mathrm{pll}}}{s}})u_{\mathrm{sq}}\end{array}\right.$$

(7)

where ω is the synchronous coordinate rotation angular velocity of PLL output, K_{p_pll} and K_{i_pll} are PLL proportional and integral coefficients, respectively, and u_sq is the q-axis voltage tracked by PLL.

When SSO does not occur, the q-axis voltage satisfies u_sq0 = 0 and the steady state value of the PLL control angle can be expressed as θ_a = ω₀t. When the SSO voltage exists on the AC-side, the fluctuation in the d and q-axis voltage can be expressed as

$$\left[\begin{array}{c}{u}_{\mathrm{sd}1}\\ u_{\mathrm{sq}1}\end{array}\right]=P({\theta }_{\mathrm{a}})\left[\begin{array}{c}{u}_{\mathrm{s}\_\mathrm{a}}\\ u_{\mathrm{s}\_\mathrm{b}}\\ u_{\mathrm{s}\_\mathrm{c}}\end{array}\right]$$

(8)

where P(θ_a) is the PARK transformation matrix with the specific expression.

$$P({\theta }_{\mathrm{a}})={\displaystyle \frac{2}{3}}{\left[\begin{array}{cc}\cos {\theta }_{\mathrm{a}}& -\sin {\theta }_{\mathrm{a}}\\ \cos ({\theta }_{\mathrm{a}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})& -\sin ({\theta }_{\mathrm{a}}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\\ \cos ({\theta }_{\mathrm{a}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})& -\sin ({\theta }_{\mathrm{a}}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\end{array}\right]}^{\mathrm{T}}$$

(9)

Substituting (9) into (8) gives the q-axis voltage as

$$u_{\mathrm{sq}1}=U_{\mathrm{ac}1}\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\psi }_1]$$

(10)

It can be known from (7) that the PLL control angle will contain a component with frequency (ω₀ − ω_ac1) under the effect of u_sq1, which can be expressed as

$$\Delta {\theta }_{\mathrm{e}}=M_1\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{M}1}]$$

(11)

where M₁ is the amplitude of the output phase angle variation and ϕ_M1 is the initial phase of the output phase angle variation.

When the SSO component exists in the AC-side of the VSC-HVDC rectifier system, a perturbation Δθ_e will appear in the PLL control angle, which makes the PLL output phase angle change from θ_a to ${\theta }_{1 \mathrm{p}\mathrm{l}\mathrm{l}}$, and the expression of ${\theta }_{1 \mathrm{p}\mathrm{l}\mathrm{l}}$ is

$${\theta }_{\mathrm{pll}}^1={\theta }_{\mathrm{a}}+\Delta {\theta }_{\mathrm{e}}$$

(12)

As shown in Figure 3, there is a feedback loop in the PLL control system, and the PLL control angle while considering the perturbation amount affects the PARK transformation of u_{s_abc}. Then, a new d/q-axis component appears, and the AC-side d/q-axis current i_sdq and the voltage u_sdq can be expressed as

$$\left\{\begin{array}{l}{i}_{\mathrm{sdq}}=P({\theta }_{\mathrm{pll}}^1)i_{\mathrm{s}\_\mathrm{abc}}\\ u_{\mathrm{sdq}}=P({\theta }_{\mathrm{pll}}^1)u_{\mathrm{s}\_\mathrm{abc}}\end{array}\right.$$

(13)

where u_sdq denotes the d and q-axis voltage at PCC and i_sdq is the d and q-axis current of the rectifier AC-side.

Since the perturbation angle variation Δθ_e is small, the trigonometric expression of the PLL output phase ${\theta }_{1 \mathrm{p}\mathrm{l}\mathrm{l}}$ can be dealt with a first-order Taylor expansion, which can be expressed as [28]

$$\left[\begin{array}{c}\sin {\theta }_{\mathrm{pll}}^1\\ \cos {\theta }_{\mathrm{pll}}^1\end{array}\right]=\left[\begin{array}{cc}\sin ({\omega }_{0}t)& \cos ({\omega }_{0}t)\end{array}\right]\left[\begin{array}{cc}1& -\Delta {\theta }_{\mathrm{e}1}\\ \Delta {\theta }_{\mathrm{e}1}& 1\end{array}\right]$$

(14)

Substituting (3), (9), and (12) into (13) yields specific expressions for the feedthrough single SSO component and the AC-side d and q axis current components i_sd1 and i_sq1, as follows:

$$\left\{\begin{array}{l}{i}_{\mathrm{sd}1}=I_{\mathrm{d}1}+I_{\mathrm{d}2}\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{d}2}]+\\ I_{\mathrm{d}3}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{d}3}]\\ i_{\mathrm{sq}1}=I_{\mathrm{q}1}+I_{\mathrm{q}2}\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{q}2}]+\\ I_{\mathrm{q}3}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{q}3}]\end{array}\right.$$

(15)

where the SSO current i_{s_abc} with frequency ω_ac1 mainly generates two components with frequencies (ω₀ − ω_ac1) and 2(ω₀ − ω_ac1) after abc/dq coordinate transformation. The amplitude of the 2(ω₀ − ω_ac1) frequency component is smaller than that of the (ω₀ − ω_ac1) frequency component. I_d1 and I_q1 are the steady-state values of the AC-side d/q-axis currents; I_d2 and I_q2 are the amplitudes of the (ω₀ − ω_ac1) frequency components; and I_d3 and I_q3 are the amplitudes of the 2(ω₀ − ω_ac1) frequency components.

Considering the effect of PLL disturbances, substituting (6), (9), and (12) into (13) yields the expressions of the AC-side voltage d/q-axis components u_sd2 and u_sq2, as shown as

$$\left\{\begin{array}{l}{u}_{\mathrm{sd}2}=u_{\mathrm{sd}0}+u_{\mathrm{sds}}\\ u_{\mathrm{sq}2}=u_{\mathrm{sq}0}+u_{\mathrm{sqs}}\end{array}\right.$$

(16)

From (1)–(6), it can be seen that the fundamental frequency voltage and SSO voltage in the three-phase voltage u_{s_abc} are consistent with the three-phase current i_{s_abc}. Therefore, the frequency response components of the AC-side d/q-axis voltages after the PARK transformation are consistent with the AC-side d/q-axis currents. The steady-state values of the AC-side d/q-axis voltages are u_sd0 and u_sq0, and the voltages u_sds and u_sqs generated under the sub-synchronous current perturbation on the AC-side mainly contain the components of frequencies (ω₀ − ω_ac1) and 2(ω₀ − ω_ac1).

3.2. Response Characteristics of DC Power to Single SSO Component

When the SSO component exists in the AC-side of the wind power via the VSC-HVDC system, it can be known from [34] that the voltage outer-loop control link has a larger time constant and responds more slowly to the SSO disturbance. Therefore, this section analyzes the dynamic responses of the inner-loop current control under SSO disturbances. It can be known from Figure 2 that the d/q axis components of the converter output voltage are

$$\left\{\begin{array}{l}{u}_{\mathrm{td}}=(i_{\mathrm{sdref}}-i_{\mathrm{sd}})(K_{\mathrm{pd}}+{\displaystyle \frac{K_{\mathrm{id}}}{s}})-{\omega }_{0}L{i}_{\mathrm{sq}}+u_{\mathrm{sd}}\\ u_{\mathrm{tq}}=(i_{\mathrm{sqref}}-i_{\mathrm{sq}})(K_{\mathrm{pq}}+{\displaystyle \frac{K_{\mathrm{iq}}}{s}})+{\omega }_{0}L{i}_{\mathrm{sd}}+u_{\mathrm{sq}}\end{array}\right.$$

(17)

where K_pd, K_pq, K_id, and K_iq represent the proportional and integral coefficients of the d/q axis inner loop PI controllers, respectively; the three-phase current i_{s_abc} at the PCC of the rectifier AC system contains the SSO current component i_{ac1_ss}; and the d/q-axis current components under the effect of PLL and PARK transformation are i_sd1 and i_sq1. Therefore, u_td and u_tq also contain SSO components.

From the equal power on both sides of the AC and DC, it can be concluded that the AC-side power and DC-side power expressions are [35]

$$\left\{\begin{array}{l}{P}_{\mathrm{ac}}={\displaystyle \frac{3}{2}}(i_{\mathrm{sd}}{u}_{\mathrm{td}}+i_{\mathrm{sq}}{u}_{\mathrm{tq}})\\ P_{\mathrm{dc}}=P_{\mathrm{ac}}\end{array}\right.$$

(18)

where P_ac denotes the AC-side power of the VSC-HVDC rectifier system and P_dc denotes the DC-side power of the VSC-HVDC system. Substituting (15) and (17) into (18), the expression for the power on the DC-side after feeding the single SSO component can be obtained, as follows:

$$\begin{array}{l}{P}_{\mathrm{dc}1}=A_0+A_1\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\phi }_{\mathrm{A}1}]+\\ A_2\cos [2({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\phi }_{\mathrm{A}2}]+\\ A_3\cos [3({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\phi }_{\mathrm{A}3}]+\\ A_4\cos [4({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\phi }_{\mathrm{A}4}]\end{array}$$

(19)

where φ_A1, φ_A2, φ_A3, and φ_A4 denote the phase angles of the doubled to quadrupled frequency components in the (ω₀ − ω_ac1) reference. A₀, A₁, A₂, A₃, and A₄ represent the steady-state amplitude and the amplitude of the doubled to quadrupled frequency components in the (ω₀ − ω_ac1) reference, respectively. From (19), there are five terms in the DC-side power response components. A₀ is the active power of the basic frequency, which is independent of oscillations; the component with frequency (ω₀ − ω_ac1) is generated under the effect of the power frequency component and PLL perturbation component; the component with frequency 2(ω₀ − ω_ac1) is generated under the effect of the power frequency component, PLL perturbation component, and SSO component; and the components with frequencies 3(ω₀ − ω_ac1) and 4(ω₀ − ω_ac1) are generated under the effect of SSO component and PLL perturbation component.

The d/q-axis current response characteristics and DC-side power response characteristics under a single AC-side SSO disturbance are shown in

Table 1. The frequency response characteristics of the DC power under the AC sub-synchronous disturbance.

AC-Side Current Frequency Component	DC-Side Power Frequency Component
ω0 − ωac1	ω0 − ωac1 2ω0 − 2ωac1
2ω0 − 2ωac1	3ω0 − 3ωac1 4ω0 − 4ωac1

. It can be seen from Table 1 that the d/q-axis current components contain one and two times the frequency components in the (ω₀ − ω_ac1) reference and the DC-side power contains one to four times the frequency components in the (ω₀ − ω_ac1) reference.

3.3. Response Characteristics of DC Power to Multiple SSO Components

To obtain the multi-frequency response components of the DC-side power when the AC-side of the VSC-HVDC system is subjected to multiple AC-side SSO components, this section derives the multi-frequency response characteristics of the DC-side power by feeding two AC-side SSO components as an example.

As shown in Figure 1, the second SSO current i_{ac2_ss} is superimposed based on the first SSO current, i_{ac1_ss}, which is fed at the PCC point of the AC-side, and the expression of the three-phase current i_{1 s_abc} on the AC-side after superposition is

$$i_{\mathrm{s}\_\mathrm{abc}}^1=i_{\mathrm{s}\_\mathrm{abc}}+i_{\mathrm{ac}2\_\mathrm{ss}}$$

(20)

The expression for i_{ac2_ss} in (20) is

$$i_{\mathrm{ac}2\_\mathrm{ss}}=\left[\begin{array}{l}{I}_{\mathrm{ac}2}\cos ({\omega }_{\mathrm{ac}2}t+{\varphi }_{\mathrm{s}2})\hfill \\ I_{\mathrm{ac}2}\cos ({\omega }_{\mathrm{ac}2}t+{\varphi }_{\mathrm{s}2}-{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \\ I_{\mathrm{ac}2}\cos ({\omega }_{\mathrm{ac}2}t+{\varphi }_{\mathrm{s}2}+{\displaystyle \frac{2\mathsf{\pi }}{3}})\hfill \end{array}\right]$$

(21)

where I_ac2 and ϕ_s2 are the amplitude and initial phase of the SSO current i_{ac2_ss}, respectively; ω_ac2 is the frequency of the SSO current i_{ac2_ss}.

The voltage fluctuation caused by i_{ac2_ss} is ω_ac2L_ti_{ac2_ss} and the voltage fluctuation caused by the SSO current i_{ac2_ss} is denoted as u_{ac2_ss}. The three-phase voltage at the PCC point shown in Figure 1 after the superposition of u_{ac2_ss} is denoted as u_{1 s_abc} and the expression of u_{1 s_abc} is

$$u_{\mathrm{s}\_\mathrm{abc}}^1=u_{\mathrm{s}\_\mathrm{abc}}+u_{\mathrm{ac}2\_\mathrm{ss}}$$

(22)

When the SSO component i_{ac2_ss} is fed, the PLL phase angle will have a change in Δθ_e1, and then the PLL phase angle variation changes from Δθ_e to Δθ_e2 = Δθ_e + Δθ_e1. The PLL output phase angle becomes θ_vp = θ_a + Δθ_e2. Substituting θ_vp into (9) yields P(θ_vp), and then the combination of P(θ_vp), (20), and (13) gives the expressions of the AC d/q axis currents i_sd2 and i_sq2 when the two SSO components are fed. The expressions for the AC-side d/q-axis currents i_sd2 and i_sq2 are

$$\left\{\begin{array}{l}{i}_{\mathrm{sd}2}=C_0+C_{\mathrm{a}1}\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{a}1}]+\\ C_{\mathrm{a}2}\cos [({\omega }_0-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{a}2}]+\\ C_{\mathrm{a}3}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{a}3}]+\\ C_{\mathrm{a}4}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{a}4}]+\\ C_{\mathrm{a}5}\cos [(2{\omega }_0-{\omega }_{\mathrm{ac}1}-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{a}5}]+\\ C_{\mathrm{a}6}\cos [({\omega }_{\mathrm{ac}1}-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{a}6}]\\ i_{\mathrm{sq}2}=D_0+D_{\mathrm{b}1}\cos [({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{b}1}]+\\ D_{\mathrm{b}2}\cos [({\omega }_0-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{b}2}]+\\ D_{\mathrm{b}3}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\varphi }_{\mathrm{b}3}]+\\ D_{\mathrm{b}4}\cos [2({\omega }_0-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{b}4}]+\\ D_{\mathrm{b}5}\cos [(2{\omega }_0-{\omega }_{\mathrm{ac}1}-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{b}5}]+\\ D_{\mathrm{b}6}\cos [({\omega }_{\mathrm{ac}1}-{\omega }_{\mathrm{ac}2})t+{\varphi }_{\mathrm{b}6}]\end{array}\right.$$

(23)

It can be seen from (23) that the d/q-axis current components contain not only the components with frequencies (ω₀ − ω_ac1) and 2(ω₀ − ω_ac1) resulting from feeding the SSO component i_{ac1_ss} alone and the components with frequencies (ω₀ − ω_ac2) and 2(ω₀ − ω_ac2) resulting from feeding the SSO component i_{ac2_ss} alone but also the components with frequencies (2ω₀ − ω_ac1 − ω_ac2) and (ω_ac1 − ω_ac2) resulting from the interaction between i_{ac1_ss} and i_{ac2_ss}.

Based on the analysis of (16) in Section 3.1, the d- and q-axis components u_sd2 and u_sq2 of u_{1 s_abc} are consistent with the frequency components of the currents i_sd2 and i_sq2 in (23). The d- and q-axis current and voltage equations under the effect of SSO components i_{ac1_ss} and i_{ac2_ss} are calculated according to (15), (17), and (18). The d- and q-axis voltages and currents considering the SSO quantities are substituted into (18) to derive the DC-side power expression when the SSO components i_{ac1_ss} and i_{ac2_ss} are fed simultaneously, as shown in (24). The multi-frequency response characteristics of the DC power under multiple AC sub-synchronous disturbances are listed in

Table 2. The multi-frequency response characteristics of the DC power under the multiple AC sub-synchronous disturbances.

AC-Side Current Frequency Component	DC-Side Power Frequency Component
ω0 − ωac1 2ω0 − 2ωac1	ω0 − ωac1 2ω0 − 2ωac1 3ω0 − 3ωac1 4ω0 − 4ωac1
ω0 − ωac2 2ω0 − 2ωac2	ω0 − ωac2 2ω0 − 2ωac2 3ω0 − 3ωac2 4ω0 − 4ωac2
ωac1 − ωac2	ωac1 − ωac2 2ωac1 − 2ωac2 ω0 − 2ωac2 + ωac1 ω0 − 2ωac1 + ωac2 2ω0 − ωac1 − ωac2 2ω0 − 3ωac1 + ωac2
2ω0 − ωac1 − ωac2	2ω0 + ωac1 − 3ωac2 3ω0 − ωac1 − 2ωac2 3ω0 − 2ωac1 − ωac2 4ω0 − 2ωac1 − 2ωac2 4ω0 − 3ωac1 − ωac2 4ω0 − ωac1 − 3ωac2

.

$$\begin{array}{l}{P}_{\mathrm{dc}2}=B_0+B_1{\displaystyle \sum _{m=1}^4\cos [m({\omega }_0-{\omega }_{\mathrm{ac}1})t+{\phi }_{\mathrm{cm}}]}+\\ B_2{\displaystyle \sum _{n=1}^4\cos [n({\omega }_0-{\omega }_{\mathrm{ac}2})t+{\phi }_{\mathrm{dn}}]}+\\ B_{\mathrm{u}}{\displaystyle \sum _{u=1}^2\cos [u({\omega }_{\mathrm{ac}1}-{\omega }_{\mathrm{ac}2})t+{\phi }_{\mathrm{eu}}]}+\\ B_{\mathrm{x}}{\displaystyle \sum _{\begin{array}{l}{\alpha }_1=1\\ {\alpha }_1\ne 2\end{array}}^3{\displaystyle \sum _{\left|\beta \right|,\left|\gamma \right|=1}^2\cos [({\alpha }_1{\omega }_0-{\beta }_1{\omega }_{\mathrm{ac}1}-{\gamma }_1{\omega }_{\mathrm{ac}2})t+{\phi }_{\mathrm{x}}]}}+\\ B_{\mathrm{y}}{\displaystyle \sum _{\begin{array}{l}{\alpha }_2=2\\ {\alpha }_2\ne 3\end{array}}^4{\displaystyle \sum _{\left|\beta \right|,\left|\gamma \right|=1}^3\cos [({\alpha }_2{\omega }_0-{\beta }_2{\omega }_{\mathrm{ac}1}-{\gamma }_2{\omega }_{\mathrm{ac}2})t+{\phi }_{\mathrm{y}}]}}\end{array}$$

(24)

It can be seen from Table 2 that m(ω₀ − ω_ac1) and n(ω₀ − ω_ac2) components generated by the simultaneous action of SSO components i_{ac1_ss} and i_{ac2_ss} are the same as the frequency response components of the DC-side power when the SSO components i_{ac1_ss} and i_{ac2_ss} are fed individually; u(ω_ac1 − ω_ac2) and αω₀ − βω_ac1 − γω_ac2 (α = β + γ) frequency components are the frequency response components of the DC-side power under the coupling effect of the two SSO components i_{ac1_ss} and i_{ac2_ss}.

4. Simulation Verification

To verify the frequency response characteristics of the DC-side power under single/multiple AC-side SSO disturbances in Section 3.2 and Section 3.3, the system simulation model of Figure 1 is built in PSCAD/EMTDC. The quantitative selection principles of the system parameters are

(1)

The transmission power range of the VSC-HVDC system is 10–400 MW and the regulation range of the operating power is 10~100%. The upper limit of DC is 1.5 kA and the range of DC voltage level is (±10~±200) kV or 350 kV;

(2)

The selection principle of filter: considering the harmonic current flowing into the filter from other harmonic sources and considering the effects of system voltage imbalance, filter detuning, and harmonic impedance.

(3)

Capacitor selection principles: capacitor energy storage needs to be less than or equal to 5 ms, and it should be able to stabilize the voltage and weaken the coupling effect of the converter station.

(4)

Transformer selection principles: its rated capacity is decided by the converter capacity, the valve side voltage, and the converter outlet voltage. It should be designed to prevent the zero sequence current from transferring between the AC and DC systems.

Quantitative selection principles of controller parameters

(1)

Selection of proportional coefficient: a small proportional coefficient can be selected and then gradually increases to observe the system responses and adjusted to the appropriate value;

(2)

Selection of integral coefficient: it can be adjusted through simulations and is designed to eliminate the steady-state error and maintain stability.

According to the above principles, the system’s main parameters can be designed, as shown in

Table 3. System main parameters.

Parameters	Value
AC-side Voltage/kV	110
Filter Inductor Lt/H	0.01
Shunt capacitor Cs/μF	5
Kp_pll, Ki_pll	20, 50
D-axis inner loop proportional and integral parameters	5.28, 10,000
Q-axis inner loop proportional and integral parameters	5.28, 10,000
Active power reference Pref/MW	300
Reactive power reference Qref/Mvar	0

. SSO currents with frequencies of 40 Hz and 35 Hz are injected at the rectifier PCC point. On this basis, the corresponding frequency response characteristics of the DC-side power are observed.

4.1. D-Axis Current Response Characteristics to Single/Multiple SSO Components

The d-axis current response results are shown in Figure 4. It can be seen from Figure 4a that when the SSO component with a frequency of 35 Hz is injected into the PCC, SSO components with frequencies of 15 Hz (ω₀ − ω_ac1) and 30 Hz (2ω₀ − 2ω_ac1) appear in the d-axis current. In Figure 4b, when SSO components with frequencies of 20 and 30 Hz are simultaneously injected into the PCC, there are SSO currents with frequencies of 10 Hz (ω_ac1 − ω_ac2), 20 Hz (ω₀ − ω_ac2), 30 Hz (ω₀ − ω_ac1), 40 Hz (2ω₀ − 2ω_ac2), 50 Hz (2ω₀ − ω_ac1 − ω_ac2), and 60 Hz (2ω₀ − 2ω_ac1). The d-axis current response characteristics in Figure 4 verify the AC-side current frequency response components in Table 1 and Table 2.

4.2. Verification of Single AC-Side SSO Disturbance with DC-Side Power Response

The frequency response results of DC-side power and corresponding spectrum analysis are shown in Figure 5 and Figure 6. It can be seen from Figure 5 that the oscillation frequency (1/t₁ = 10 Hz) of the wind power via the VSC-HVDC system in Figure 5a is consistent with the oscillation frequency of 10 Hz in Figure 5b. It can be known from Figure 5b that when the AC-side SSO current with the frequency of 40 Hz is injected at the PCC, the sub-synchronous frequency components with the frequencies of 10 Hz (ω₀ − ω_ac1), 20 Hz (2ω₀ − 2ω_ac1), 30 Hz (3ω₀ − 3ω_ac1), and 40 Hz (4ω₀ − 4ω_ac1) appear in the DC-side power. It can be seen from Figure 6 that the oscillation frequency (1/t₂ = 15 Hz) in Figure 6a is consistent with the oscillation frequency of 15 Hz in Figure 6b. In Figure 6b, when the AC-side SSO current with the frequency of 35 Hz is injected at the PCC, there are super-synchronous DC oscillation components with a frequency of 60 Hz (4ω₀ − 4ω_ac1) and multiple SSO components with the frequencies of 15 Hz (ω₀ − ω_ac1), 30 Hz (2ω₀ − 2ω_ac1), and 45 Hz (3ω₀ − 3ω_ac1).

Therefore, when the SSO component is added to the AC-side PCC point of the wind power via the VSC-HVDC system, the DC-side power response components contain the sub/super-synchronous oscillation powers. Each frequency component of the DC-side power present in the simulation results of Figure 5 and Figure 6 is consistent with the theoretical analysis of Table 1 in Section 3.2.

4.3. Verification of Multiple AC-Side SSO Disturbance with DC-Side Power Response

To verify the DC-side power response characteristics to multiple AC-side SSO disturbance components, this section takes two AC-side SSO components acting at the same time as an example. Simultaneously feeding sub-synchronous disturbance components with frequencies of 20 Hz and 30 Hz on the VSC-HVDC AC-side, the multiple frequency response characteristics of the DC-side power are shown in Figure 7.

It can be seen from Figure 7 that the oscillation frequency of the VSC-HVDC system (1/t₃ = 10 Hz) in Figure 7a is consistent with the oscillation frequency of 10 Hz in Figure 7b. Meanwhile, it can be known from Figure 7b that when the SSO components with the frequencies of 20 Hz and 30 Hz are injected at the same time, there are 10 Hz (ω₀ − 2ω_ac2 + ω_ac1/ω_ac1 − ω_ac2), 20 Hz (ω₀ − ω_ac2/2ω_ac1 − 2ω_ac2), 30 Hz (ω₀ − ω_ac1), 40 Hz (2ω₀ − 2ω_ac2), 50 Hz (2ω₀ − ω_ac1 − ω_ac2), 60 Hz (3ω₀ − 3ω_ac2), 70 Hz (3ω₀ − ω_ac1 − 2ω_ac2), 80 Hz (4ω₀ − 4ω_ac2), 90 Hz (4ω₀ − 3ω_ac2 − ω_ac1), 100 Hz (4ω₀ − 2ω_ac1 − 2ω_ac2), 110 Hz (4ω₀ − 3ω_ac1 − ω_ac2), and 120 Hz (4ω₀ − 4ω_ac1) frequency components. Each frequency component of the DC−side output power in Figure 7 is consistent with the theoretical analysis of Table 2 in Section 3.3.

5. Influence of Control Parameters on the DC-Side Oscillation Power

5.1. PLL Controller Parameter

The effect of PLL proportional coefficient K_{p_pll} and integral coefficient K_{i_pll} on the VSC-HVDC DC-side oscillating power is shown in Figure 8 and

Table 4. The impact of K_{i_pll} on the DC-side oscillation power.

PLL Integral Coefficient (Ki_pll)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
25	1.01255	0.0244788	4.92673 × 10−4	9.53981 × 10−6
40	1.01257	0.0244808	4.92621 × 10−4	9.52625 × 10−6
50	1.01258	0.0244814	4.92623 × 10−4	9.52372 × 10−6
100	1.01260	0.0244836	4.92747 × 10−4	9.52804 × 10−6

when the SSO component of 35 Hz is injected at the rectifier AC-side. It can be seen from Figure 8 that the amplitude of the oscillation component in the DC-side power spectrum analysis is the largest when K_{p_pll} = 2. With the increase in K_{p_pll}, the oscillation frequency nearly does not change. The amplitude of each oscillation component in the DC-side power spectrum analysis gradually decreases, and thus the oscillation risk decreases. It can be seen from Table 4 that the amplitude of each oscillation component in the DC-side power varies little with the change in K_{i_pll}, so the influence of K_{i_pll} on the DC-side oscillation power is small.

5.2. Inner Loop Current Controller Parameter

The effect of d-axis inner-loop current control proportional and integral parameters on the VSC-HVDC DC-side oscillating power is shown in Figure 9 and

Table 5. The impact of K_pd on the DC-side oscillation power.

d Axis Current Inner Loop Proportional Coefficient (Kpd)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
5.08	1.01237	0.0244709	4.92113 × 10−4	9.50216 × 10−6
5.18	1.01247	0.0244761	4.92368 × 10−4	9.51295 × 10−6
5.28	1.01258	0.0244814	4.92623 × 10−4	9.52372 × 10−6
5.38	1.01237	0.0244709	4.92113 × 10−4	9.50216 × 10−6

when the SSO component of 35 Hz is injected at the rectifier AC-side. It can be seen from Figure 9 that as the d-axis integral coefficient K_id increases, the amplitude of each DC-side frequency component in the spectral analysis decreases, and thus the oscillation risk decreases. It can be seen from Table 5 that the d-axis inner loop current control proportional parameter has a small impact on the DC-side oscillation power.

Meanwhile, it can be seen from

Table 6. The impact of K_pq on the DC-side oscillation power.

q Axis Current Inner Loop Proportional Coefficient (Kpq)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
1.5	1.01256	0.0244803	4.92793 × 10−4	9.55026 × 10−6
2.0	1.01257	0.0244807	4.92748 × 10−4	9.5422 × 10−6
2.5	1.01258	0.0244814	4.92623 × 10−4	9.52372 × 10−6
3.0	1.01258	0.0244816	4.92639 × 10−4	9.52469 × 10−6

and

Table 7. The impact of K_iq on the DC-side oscillation power.

q Axis Current Inner Loop Integral Coefficient (Kiq)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
1000	1.0139	0.0247099	5.0705 × 10−4	1.00151 × 10−5
5000	1.01264	0.0244927	4.93534 × 10−4	9.58107 × 10−6
10,000	1.01258	0.0244814	4.92623 × 10−4	9.52372 × 10−6
20,000	1.01254	0.0244763	4.92551 × 10−4	9.54171 × 10−6

that after the change in the current inner-loop q-axis PI parameters, there is no change in each oscillation frequency point on the DC-side, and the amplitude of each oscillation component of the DC-side power changes slightly. Therefore, the effect of the current inner-loop q-axis PI parameters on the DC-side oscillating power amplitude is small.

5.3. Filter Inductor Parameter

When the filter inductor parameter L_t is 0.005 H, 0.01 H, 0.011 H, and 0.013 H with the injection of the 35 Hz SSO component at the AC-side rectifier, the oscillating power of the VSC-HVDC DC-side is shown in Figure 10. It can be seen from Figure 10 that the amplitude of the oscillation component in the DC-side power spectrum analysis is the smallest when L_t is 0.005 H. With the increase in L_t, there is no change in each oscillation frequency point, but the amplitude of each oscillation component in the DC-side power spectrum analysis gradually increases, and the oscillation risk increases.

5.4. Other Parameters

When the SSO component of 35 Hz is injected at the rectifier AC-side, the DC-side oscillation power of the VSC-HVDC is shown in

Table 8. The impact of L_f on the DC-side oscillation power.

AC Grid Inductance (Lf)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
0.004 H	0.3990	0.0037	2.829 × 10−5	1.985 × 10−7
0.007 H	0.7027	0.0117	1.593 × 10−4	2.050 × 10−6

with the AC-grid inductance L_f varying between 0.004 H and 0.007 H. The DC-side oscillating power of the VSC-HVDC is shown in

Table 9. The impact of P_ref on the DC-side oscillation power.

Active Power Reference Value (Pref)	DC-Side Power Amplitude (15 Hz)/MW	DC-Side Power Amplitude (30 Hz)/MW	DC-Side Power Amplitude (45 Hz)/MW	DC-Side Power Amplitude (60 Hz)/MW
100 MW	0.0667	0.0335	0.0225	0.017
200 MW	0.1291	0.0647	0.0434	0.0329

when the reference value of the active power P_ref varies between 100 MW and 200 MW. It can be seen from Table 8 that the change in the AC-grid inductance L_f does not influence the frequency points of each oscillation component on the DC- side. However, Table 8 shows that the amplitude of each oscillation component of the DC-side power increases, and the risk of oscillation increases. It can be seen from Table 9 that with the increase in active power reference P_ref, there is no change in the frequency point of each oscillation component on the DC- side. However, Table 9 shows that the risk of oscillation increases with the increase in P_ref, as the amplitude of each oscillation component of the DC-side power increases.

6. Conclusions

To reveal the oscillation propagation mechanism in the AC and DC sides of the VSC-HVDC system, the frequency response characteristics of the VSC-HVDC DC-side oscillation power under the AC SSO perturbations are investigated. The main conclusions are summarized as follows:

(1)

When the SSO component with frequency ω_ac1 exists on the AC-side of the VSC-HVDC system, the perturbation frequency component is generated through the effect of the PLL, and then the perturbation frequency component acts directly on the current inner loop. Finally, the DC-side oscillation power exhibits multi-frequency oscillation characteristics, containing one to four times the frequency component under the (ω₀-ω_ac1) reference;

(2)

When the SSO components of ω_ac1 and ω_ac2 exist simultaneously on the AC-side of the VSC-HVDC system, the m(ω₀ − ω_ac1), n(ω₀ − ω_ac2), and αω₀ − βω_ac1 − γω_ac2 (α = β + γ) DC-side oscillation power frequency components under the effect of the two sub-synchronous components appear. This indicates that different SSO components are coupled with each other, which affects the multi-frequency response characteristics of the DC-side oscillation power;

(3)

The increase in the PLL proportional coefficient and d-axis current inner-loop integral coefficient leads to the decrease in the amplitude of each DC-side oscillation power frequency component, and then the oscillation risk decreases. Meanwhile, with the increase in the filtering inductance, AC grid inductance, and active power, the amplitude of each DC-side oscillation power frequency component increases, and then the oscillation risk increases.