The Frequency Spectrum Analysis of Wideband Oscillations of Grid-Connected Voltage Source Converter

Abstract

With the widespread application of voltage source converters (VSCs) in power systems with high renewable resource penetration, the problem of wideband oscillations caused by interactions between power converters and the power grid has drawn great attention. This paper presents a new methodology for simplifying the mathematical derivations of the impedance model of a VSC. Using the new simplified model, the stability and spectrum characteristics of the performance of the VSC can be clearly analyzed. The proposed second-order d-q impedance model of a grid-connected VSC in the presence of a PLL not only mathematically demonstrates that wideband oscillatory modes always occur in conjugate pairs, but also delivers a clear vision of the physical essence of these wideband oscillations. A classic modal analysis is executed in this paper to demonstrate the stability and frequency spectral characteristics of wideband oscillations in a VSC integrated system. The impact of the control parameters of the VSC on wideband oscillatory stability is thoroughly analyzed by modal analysis using ‘DIgSILENT PowerFactory’ as well as hardware-in-the-loop experiments. This methodology can be considered a powerful tool for the control design of VSCs in the modern power system.

1. Introduction

Voltage source converters (VSCs) are the dominant conversion equipment for renewable power grid connections. In a power system with high renewable resource penetration, the performance of VSC-interfaced generation plays a key role in system stability. The results of the current research show that the overall performance of VSC-interfaced generation is dominated by the control system. Improperly set control parameters can cause unstable interactions between power electronic device systems, including low-frequency oscillations [1], sub-synchronous oscillations, super-synchronous oscillations [2,3,4], and even wideband oscillations with high frequencies [5,6,7]. Therefore, studying the proper control parameter settings for VSC-interfaced equipment is crucial for ensuring system stability and analyzing the wideband frequency spectrum characteristics of the power system.

At present, modal analysis and impedance analysis are the well-known and widely used methods for the study of VSC stability [8]. Modal analysis based on linearized state-space modeling is an effective method that provides valuable information about the dynamics of the system, i.e., the frequency and damping factor of the dominant oscillatory modes. Reference [9] uses the modal analysis method for analyzing the sub-synchronous oscillation problem caused by resonance between a wind farm and series compensation capacitors. However, this method highly depends on the complete modeling of the system. Hence, for complex systems with high penetration of converter-interfaced generations (CIGs), the problem of dimension disaster creates sizeable problems for solving the linear characteristic equation [10].

The impedance method provides a clear physical observation of the electric system and is widely used in engineering. Two impedance models have been developed: the sequence impedance model and the dq-axis impedance model. References [11,12] apply the classical Nyquist criterion on the system’s sequence impedance model for system stability analysis. However, the sequence impedance method is built on the three-phase symmetrical system [13], and a small disturbance from the PLL could impact the accuracy of the sequence impedance method [14].

The dq-axis impedance method using Park transformation converts the three-phase rotating component of the system into a component that is stationary relative to the dq-axis [15]. This method delivers a clearer vision of the coupling between the components in the system [16]. Figure 1 presents a block diagram of a typical VSC control system with the coupling terms of the dq-axis system. These coupling terms pose significant challenges to a mathematical solution that much research has been dedicated to solving. Reference [17] transforms a typical VSC test system to a composite equivalent current loop to offset the coupling terms. Reference [18] applies the Nyquist criterion to the d-axis and q-axis impedances separately and proposes that the system is stable when the corresponding criteria are satisfied; additionally, this conclusion is not supported in Reference [19]. According to Reference [19], when using the Nyquist criterion for a second-order transfer function, analyzing individual transfer functions separately to determine system stability lacks theoretical evidence.

References [20,21] propose robust control to improve the performance of LCL-filter-based grid-connected VSC. The LCL filter is widely used for filtering switching harmonics, but LCL-filters might cause the resonance phenomenon. References [22,23] present that increasing the grid-side inductance will lead to a decrease in the system short-circuit ratio (SCR) and take a negative effect for the system stability. References [24,25] discuss the impact of PI control parameters in the PLL on the oscillatory stability. However, there are few papers that clearly and systematically analyze how the control parameters influence the frequency and damping of the oscillatory modes. References [26,27] comprehensively analyse the impact of line inductance for system stability based on the Nyquist criterion. This paper presents a new methodology for simplifying the impedance model by shifting the observation of the output of the VSC control system so that the coupling terms do not exist in the mathematical derivations for impedance modeling. Using the new simplified model, the stability and the spectrum characteristics of the performance of the VSC can be clearly analyzed. This paper is structured as follows: Section 2 and Section 3 present the derivation of the proposed second-order dq impedance model and the stability criterion of a VSC; Section 4 presents the impact of the control parameters on the wideband oscillatory stability of a VSC using modal analysis and nonlinear simulation; and Section 5 demonstrates the impact of key parameters of the VSC controller on oscillations using hardware-in-the-loop (HIL) simulation. In the end, Section 6 concludes the paper. Figure 2 gives the flow chart of the research methodology.

2. Linearized Impedance Model of a VSC Grid-Connected Inverter System

2.1. Impedance Modeling of a VSC

Figure 3 presents the structure of the test system for the grid-connected VSC, which consists of the main circuit and control schemes. The VSC control consists of the d-axis control and the q-axis control: the d-axis control ensures the DC voltage of the capacitor remains at a constant value, and the q-axis control regulates the q-axis current feeding into the power grid at the expected value. As seen in Figure 3, U_dc represents the DC voltage on the VSC side, and I_dc represents the DC current on the DC side. $U_d$ and $U_q$ represent the voltage at the terminal of the VSC. $U_{d0}^c$ and $U_{q0}^c$ are the initial values at the PCC under steady-state conditions. L_f represents the filtering inductance, and L_g represents the equivalent impedance of the external grid. $U^c$ and $U^g$ represent the voltage at the PCC and the voltage on the grid side, respectively. θ is the phase angle of the voltage at the PCC provided by the PLL.

Based on the VSC control equations at each stage in the VSC grid-connected structure shown in Figure 3, the second-order impedance model of the VSC can be obtained by taking the output voltage and input current of the VSC in the dq-axis rotating coordinate system as the output and input variables, respectively.

The dynamics of the DC capacitor of a VSC can be described by the following equation:

$$C\frac{d{U}_{dc}}{dt}{U}_{dc}=P_m-P_e$$

(1)

where P_m is the power injected into the DC capacitor and P_e is the active power output of the VSC. Under steady-state conditions, the small-signal expression for Equation (1) is obtained as follows:

$$C\frac{d{U}_{dc}}{dt}∆U_{dc}=-∆P_e$$

(2)

$$P_e=1.5\left(U_d^c{I}_d+U_q^c{I}_q\right)$$

(3)

where $U_d^c$, $U_q^c$, I_d, and I_q are the electric components at the PCC in the rotating coordinate system. When a small disturbance occurs, by letting

$$\left\{\begin{array}{c}{U}_q^c=U_{q0}^c+∆U_q^c\\ U_d^c=U_{d0}^c+∆U_d^c\\ I_d=I_{d0}+∆I_d\\ I_q=I_{q0}+∆I_q\end{array}\right.$$

(4)

a new equation is obtained:

$$∆P_e=1.5\left(∆U_d^c{I}_{d0}+∆I_d{U}_{d0}^c+∆U_q^c{I}_{q0}+U_{q0}^c∆I_q\right)$$

(5)

$U_{d0}^c$, $U_{q0}^c$, I_d0, and I_q0 are the initial values.

Here, I_q0 and I_qref are set to zero; hence,

$$∆P_e=1.5\left(∆U_d^c{I}_{d0}+∆I_d{U}_{d0}^c+U_{q0}^c∆I_q\right)$$

(6)

Taking the Laplace transform of Equation (2) and combining it with Equation (6), Equation (7) is obtained:

$$∆U_{dc}=-\frac{3}{2}\times \frac{∆U_d^c{I}_{d0}+∆I_d{U}_{d0}^c+U_{q0}^c∆I_q}{sC{U}_{dc0}}$$

(7)

As shown in Figure 3, the dynamics of I_qref are described by

$$∆I_{dref}=H_1∆U_{dc}$$

(8)

Since $U_{d0}^c$ and $U_{q0}^c$ are constants, we have

$$\left\{\begin{array}{c}∆U_d^c=∆U_{dref}=H_2\left(∆I_d-∆I_{dref}\right)\\ ∆U_q^c=∆U_{qref}=H_3∆I_q\end{array}\right.$$

(9)

where $H_1=K_{p1}+\frac{K_{i1}}{S},H_2=K_{p2}+\frac{K_{i2}}{S} \mathrm{and} H_3=K_{p3}+\frac{K_{i3}}{S}$.

Combining Equations (7)–(9), the following equation is obtained:

$$\left[\begin{array}{c}∆U_d^c\\ ∆U_q^c\end{array}\right]=Z\left(s\right)\left[\begin{array}{c}∆I_d\\ ∆I_q\end{array}\right]$$

(10)

where Z(s) is a 2D impedance matrix, as follows:

$$Z\left(s\right)=\left[\begin{array}{cc}\frac{2H_{2}sC{U}_{dc0}+3H_1{H}_2{U}_{d0}^c}{2sC{U}_{dc0}-3H_1{H}_2{I}_{d0}}& 3H_1{H}_2{U}_{q0}^c\\ 0& H_3\end{array}\right]$$

(11)

2.2. The Influence of PLL on VSC Impedance Modeling

Figure 4 presents the control block diagram of a typical PLL. When the PLL is integrated into the control loop of the VSC, the impedance matrix shown in (11) is restructured.

Assuming a symmetric three-phase sinusoidal voltage with an amplitude of $U_0$, the time-domain expression is given as follows when the grid operates at a steady-state frequency of f₀:

$$\left[\begin{array}{c}{u}_a\left(t\right)\\ u_b\left(t\right)\\ u_c\left(t\right)\end{array}\right]=U_0\left[\begin{array}{c}\mathit{cos}\left(2\pi f_{0}t+{\phi }_0\right)\\ \mathit{cos}\left(2\pi f_{0}t+{\phi }_0-\frac{2}{3}\pi \right)\\ \mathit{cos}\left(2\pi f_{0}t+{\phi }_0+\frac{2}{3}\pi \right)\end{array}\right]$$

(12)

Performing Clark transformation on Equation (12) gives

$$\left[\begin{array}{c}{u}_{\alpha }\left(t\right)\\ u_{\beta }\left(t\right)\end{array} \right]=T_{\alpha \beta }\left[\begin{array}{c}{u}_a\left(t\right)\\ u_b\left(t\right)\\ u_c\left(t\right)\end{array}\right]=U_0\left[\begin{array}{c}\mathit{cos}\left(2\pi f_{0}t+{\phi }_0\right)\\ \mathit{sin}\left(2\pi f_{0}t+{\phi }_0\right)\end{array}\right]$$

(13)

Then, the Clark transformation matrix is

$$T_{\alpha \beta }=\frac{2}{3}\left[\begin{array}{ccc}1& -\frac{1}{2}& -\frac{1}{2}\\ 0& \frac{\sqrt{3}}{2}& -\frac{\sqrt{3}}{2}\end{array} \right]$$

(14)

According to Euler’s formula, in the time-domain, the voltage can be expressed in exponential form in a two-phase stationary coordinate system:

$$u_{\alpha \beta }\left(t\right)=u_{\alpha }\left(t\right)+j{u}_{\beta }\left(t\right)=U_0{e}^{j2\pi f_{0}t}{e}^{j{\phi }_{0}t}$$

(15)

Taking the Park transformation of (15), expression (16) is obtained, and u_dq is the term defined in the rotating d-q reference frame system:

$$u_{dq}\left(t\right)=u_d\left(t\right)+j{u}_q\left(t\right)=T_{dq}{u}_{\alpha \beta }\left(t\right)$$

(16)

Here, the Park transformation matrix is

$$T_{dq}=\left[\begin{array}{cc}\mathit{cos}\left(2\pi ft\right)& \mathit{sin}\left(2\pi ft\right)\\ -\mathit{sin}\left(2\pi ft\right)& \mathit{cos}\left(2\pi ft\right)\end{array}\right]$$

(17)

According to Euler’s formula, the exponential form of T_dq is obtained on the complex plane:

$$T_{dq}=e^{-j2\pi ft}=\mathit{cos}\left(2\pi ft\right)-j\mathit{sin}\left(2\pi ft\right)$$

(18)

f and θ (θ = 2$\pi ft$) represent PLL output frequency and angle. Then, substituting Equation (18) into Equation (16), we have

$$u_{dq}\left(t\right)=u_d\left(t\right)+j{u}_q\left(t\right)=U_0{e}^{j2\pi (f_0-f)t}{e}^{j{\phi }_{0}t}$$

(19)

Now, a small perturbation with frequency f_p (w_p = 2πf_p) is applied to Equation (12), and setting ${\phi }_0$ = 0, expression (20) is obtained:

$$\left[\begin{array}{c}{u}_a\left(t\right)\\ u_b\left(t\right)\\ u_c\left(t\right)\end{array}\right]=U_0\left[\begin{array}{c}\mathit{cos}\left(2\pi f_{0}t\right)\\ \mathit{cos}\left(2\pi f_{0}t-\frac{2}{3}\pi \right)\\ \mathit{cos}\left(2\pi f_{0}t+\frac{2}{3}\pi \right)\end{array}\right]+U_p\left[\begin{array}{c}\mathit{cos}\left(2\pi f_{p}t+{\phi }_p\right)\\ \mathit{cos}\left(2\pi f_{p}t+{\phi }_p-\frac{2}{3}\pi \right)\\ \mathit{cos}\left(2\pi f_{p}t+{\phi }_p+\frac{2}{3}\pi \right)\end{array}\right]$$

(20)

Correspondingly, (21) and (22) represent the form of (20) in the stationary coordinate system and the rotating coordinate system, respectively:

$$u_{\alpha \beta p}\left(t\right)=U_0{e}^{j2\pi f_{0}t}+U_p{e}^{j2\pi f_{p}t}{e}^{j{\phi }_{p}t}$$

(21)

$$u_{dqp}\left(t\right)=T_{dq}{u}_{\alpha \beta p}\left(t\right)=U_0{e}^{j2\pi (f_0-f)t}+U_p{e}^{j2\pi (f_p-f)t}{e}^{j{\phi }_{p}t}$$

(22)

The perturbation error frequency $∆f$ and phase angle $∆\theta$ of the system are represented by

$$∆f=f-f_0$$

(23)

$$∆\theta =2\pi t∆f$$

(24)

Then, using Euler’s formula and the equivalent infinitesimal, we can obtain

$$e^{-j∆\theta }=\mathit{cos}\left(∆\theta \right)-j\mathit{sin}\left(∆\theta \right)=1-j∆\theta$$

(25)

By substituting Equations (23)–(25) into Equation (22) and neglecting the high-order infinitesimal in $u_{dqp}\left(t\right)$, Equation (26) is obtained:

$$u_{dqp}\left(t\right)=(U_0+U_p{e}^{j2\pi (f_p-f_0)t})e^{-j2\pi ∆ft}=U_0-j{U}_0∆\theta +U_p{e}^{j2\pi (f_p-f_0)t}{e}^{j{\phi }_{p}t}$$

(26)

As seen from Equation (26), when the system is subjected to a small disturbance, the system will produce two disturbance components caused by the phase-locked loop. The first component is defined as $u_{p1}\left(t\right)$, and the second component is defined as $u_{p2}\left(t\right)$.

$$u_{p1}\left(t\right)=-j{U}_0∆\theta$$

(27)

$$u_{p2}\left(t\right)=U_p{e}^{j2\pi (f_p-f_0)t}{e}^{j{\phi }_{p}t}$$

(28)

$u_{p1}\left(t\right)$ is the mapped disturbance voltage on the q-axis caused by the phase angle disturbance. $u_{p2}\left(t\right)$ is the complex space voltage component caused by the frequency disturbance, which can be further decomposed into d-axis and q-axis components:

$$U_{p2}\left(s\right)=∆U_d+∆U_q$$

(29)

$$\left\{\begin{array}{c}∆U_{ds}=∆U_d\\ ∆U_{qs}=∆U_q+U_{p1}\left(s\right)=∆U_q-U_0∆\theta \end{array}\right.$$

(30)

where $∆U_{ds}$ and $∆U_{qs}$ are the voltage disturbance components in the rotating coordinate system in the presence of the phase-locked loop.

As seen from Figure 4:

$$∆\theta =\frac{H_{pll}\left(s\right)}{S}∆U_{qs}$$

(31)

$$H_{pll}\left(s\right)=K_p+\frac{K_i}{s}$$

(32)

By combining Equations (30) and (31), Equations (33) and (34) are obtained:

$$∆\theta =G_{pll}\left(s\right)∆U_q$$

(33)

$$G_{pll}\left(s\right)=\frac{H_{pll}\left(s\right)/s}{U_0{H}_{pll}\left(s\right)/s+1}$$

(34)

Similar to the voltage expressions, the linearized model of the perturbation current considering the dynamics of the PLL is obtained:

$$\left\{\begin{array}{c}∆I_{ds}=∆I_d\\ ∆I_{qs}=∆I_q-I_0∆\theta \end{array}\right.$$

(35)

Then, combining Equations (10), (11), (30), and (35), the impedance model of the VSC considering the dynamics of the PLL is obtained:

$$\left[\begin{array}{c}∆U_{ds}\\ ∆U_{qs}\end{array}\right]=Z_{PLL}\left(s\right)\left[\begin{array}{c}∆I_{ds}\\ ∆I_{qs}\end{array}\right]$$

(36)

$$Z_{PLL}\left(s\right)=\left[\begin{array}{cc}\frac{2H_{2}sC{U}_{dc0}+3H_1{H}_2{U}_{d0}^c}{2sC{U}_{dc0}-3H_1{H}_2{I}_{d0}}& 3H_1{H}_2{U}_{q0}^c\\ 0& \frac{s{H}_3}{s+H_{PLL}\left(U_0-I_0{H}_3\right)}\end{array}\right]$$

(37)

Transforming Equations (36) and (37) yields the admittance model, where $Y_{PLL}$ and $Z_{PLL}$ are inverse matrices of each other.

$$\left[\begin{array}{c}∆I_{ds}\\ ∆I_{qs}\end{array}\right]=Y_{PLL}\left(s\right)\left[\begin{array}{c}∆U_{ds}\\ ∆U_{qs}\end{array}\right]$$

(38)

$$Y_{PLL}\left(s\right)=Z_{PLL}^{-1}\left(s\right)$$

(39)

2.3. The Impedance Model of the External Power Grid

By taking the PCC in Figure 3 as the input, we can obtain the voltage equation on the grid side:

$$\left[\begin{array}{c}{U}_d^c\\ U_q^c\end{array}\right]=\left[\begin{array}{cc}S{L}_g& -\omega L_g\\ \omega L_g& S{L}_g\end{array}\right]\left[\begin{array}{c}∆I_{ds}\\ ∆I_{qs}\end{array}\right]+\left[\begin{array}{c}{U}_d^g\\ U_q^g\end{array}\right]$$

(40)

By letting the voltage source $U_g$ to zero and combining it with Equation (40), the impedance model of the external power grid, $Z_{grid}$, is

$$Z_{grid}\left(s\right)=\left[\begin{array}{cc}S{L}_g& -\omega L_g\\ \omega L_g& S{L}_g\end{array}\right]$$

(41)

2.4. Symmetry Analysis of Wideband Oscillatory Spectra

Taking $u_{\alpha \beta p}\left(t\right)$ in Equation (21) as an example, we can analyze the characteristics of the wideband oscillation spectrum. By ignoring the fundamental frequency component $U_0{e}^{j2\pi f_{0}t}$ in $u_{\alpha \beta p}\left(t\right)$ and setting $\omega =2\pi f,{\omega }_0=2\pi f_{0,}{\omega }_p=2\pi f_{p,}$ and ${\phi }_p=0$, we can obtain

$$u_{\alpha \beta p}^{~}\left(t\right)=U_p{e}^{j{\omega }_{p}t}$$

(42)

$$T_{dq}=e^{-j2\pi ft}=e^{-j\omega t}$$

(43)

$$u_{dqp}^{~}\left(t\right)=T_{dq}{u}_{\alpha \beta p}^{~}\left(t\right)=U_p{e}^{j({\omega }_p-\omega )t}$$

(44)

According to Euler’s formula, the time-domain voltage $u_{dqp}^{~}\left(t\right)$ can be expressed in trigonometric form in the two-phase rotating coordinate system:

$$u_{dqp}^{~}\left(t\right)=U_p\left[\mathit{cos}\left({\omega }_{p}t-\omega t\right)+j\mathit{sin}\left({\omega }_{p}t-\omega t\right)\right]$$

(45)

$$\left\{\begin{array}{c}{u}_d^{~}\left(t\right)=U_p\mathit{cos}\left({\omega }_{p}t-\omega t\right)\\ u_q^{~}\left(t\right)=U_p\mathit{sin}\left({\omega }_{p}t-\omega t\right)\end{array}\right.$$

(46)

$$\begin{gathered} i_{dqp}^{~}\left(t\right)=i_d^{~}\left(t\right)+j{i}_q^{~}\left(t\right)=Y_{dd}{u}_d^{~}\left(t\right)+(Y_{dq}+j{Y}_{qq})u_q^{~}\left(t\right)=U_p[Y_{dd}\mathit{cos}\left({\omega }_{p}t- \\ \omega t\right)+(Y_{dq}+j{Y}_{qq})\mathit{sin}\left({\omega }_{p}t-\omega t\right)] \end{gathered}$$

(47)

The output variable of the perturbation current in the stationary coordinate system is

$$\begin{array}{lll}{i}_{\alpha \beta p}^{~}\left(t\right)=T_{dq}^{-1}{i}_{dqp}^{~}\left(t\right)& =e^{j\omega t}{i}_{dqp}^{~}\left(t\right)& \\ & =\left[\mathit{cos}\left(2\pi ft\right)+j\mathit{sin}\left(2\pi ft\right)\right]U_p[Y_{dd}\mathit{cos}\left({\omega }_{p}t-\omega t\right)+(Y_{dq}\\ & +j{Y}_{qq})\mathit{sin}\left({\omega }_{p}t-\omega t\right)]\end{array}$$

(48)

Using the prosthaphaeresis on Equation (48), we obtain

$$\begin{gathered} i_{\alpha \beta p}^{~}\left(t\right)=\left(\frac{Y_{dd}+Y_{qq}}{2}-j\frac{Y_{dq}}{2}\right)U_p\left[\mathit{cos}\left({\omega }_{p}t\right)+j\mathit{sin}\left({\omega }_{p}t\right)\right]+\left(\frac{Y_{dd}-Y_{qq}}{2} \\ +j\frac{Y_{dq}}{2}\right)U_p\left[\mathit{cos}\left(2\omega t-{\omega }_{p}t\right)+j\mathit{sin}\left(2\omega t-{\omega }_{p}t\right)\right] \end{gathered}$$

(49)

As seen in Equation (49), when a small perturbation with frequency f_p (${\omega }_p$ = 2πf_p) exists, a corresponding component with frequency of $2\omega -{\omega }_p$ emerges during the coordinate transformation. Similarly, if the frequency of perturbation is $2\omega -{\omega }_p$, the output of the system contains two components with frequencies ${\omega }_p$ and $2\omega -{\omega }_p$. Therefore, the oscillatory components not only emerge in pairs, but are also symmetrical, with the fundamental frequency as the center. The magnitude of each component depends on the magnitude of the positive-sequence admittance and negative-sequence admittance. When the frequency of perturbation ${\omega }_p$ is negative, the output amplitude of the negative-sequence oscillatory component coincides with some positive-sequence oscillatory components (oscillatory frequency is more than 0). Therefore, it is difficult to identify the pairs of oscillatory components in the frequency spectrum of wideband oscillatory signals.

3. Stability Criterion and Frequency Spectrum Symmetry Verification for a VSC

3.1. Stability Criterion for a Grid-Connected VSC

Using circuit theory on the system presented in Figure 3, a second-order nodal voltage equation in the dq rotating coordinate system is derived at the PCC:

$$U_{PCC}^{d/q}=\frac{\frac{U_g^{d/q}}{Z_{grid}}+E\times \left[\begin{array}{c}∆I_d\\ ∆I_q\end{array}\right]}{\frac{E}{Z_{PLL}}+\frac{E}{Z_{grid}}}=\frac{U_g^{d/q}+Z_{grid}\times \left[\begin{array}{c}∆I_d\\ ∆I_q\end{array}\right]}{\frac{Z_{grid}}{Z_{PLL}}+E}$$

(50)

In the above equation, E represents the second-order identity matrix. L represents the sum of $\frac{Z_{grid}}{Z_{PLL}}$ and E. When the determinant of matrix L approaches zero, the system impedance exhibits the “negative resistance” effect, which can cause the voltage at the PCC to oscillate continuously and lead to oscillations with wideband frequency. The characteristic equation for the grid-connected VSC is

$$\frac{Z_{grid}}{Z_{PLL}}+E=0$$

(51)

Due to the non-commutative property of matrix multiplication, $\frac{Z_{grid}}{Z_{PLL}}$ could take on two forms: $Z_{grid}\times Y_{PLL}$ or $Y_{PLL}\times Z_{grid}$. Substituting either of the forms into the characteristic equation for the system, a simplified equation is obtained:

$$det\left[Z_{PLL}+Z_{grid}\right]=0$$

(52)

Equation (52) is consistent with the conclusions of references [26,27]. Reference [27] uses a reversible transformation matrix T to simplify and analyze Equation (52) and obtains the sequence impedance stability criterion:

$$det\left[T{Z}_{PLL}{T}^{-1}+T{Z}_{grid}{T}^{-1}\right]=0$$

(53)

$$T=\frac{1}{\sqrt{2}}\left[\begin{array}{cc}1& i\\ 1& -i\end{array}\right]$$

(54)

In Equation (53), $T{Z}_{grid}{T}^{-1}$ is a second-order diagonal matrix, and the diagonal elements represent the positive and negative sequence impedances of the power grid, respectively.

When analyzing the impedance stability of grid-connected VSCs, different impedance models correspond to different stability criteria. Although the emphasis of each criterion is different, their physical meaning is ultimately the same. From the perspective of basic control theory, the main purpose of stability criteria is to simplify complex systems and analyze the stability from different perspectives. As long as the mathematical derivation is rigorous, the conclusion of the system’s stability is consistent.

The roots of the characteristic Equation (51) or (52) correspond to the closed-loop poles of the system, and the characteristic roots can be expressed by Equation (55):

$$\lambda =\sigma +i\omega$$

(55)

Here, the real part of the characteristic roots represents the oscillatory damping, and the imaginary part represents the oscillatory frequency.

3.2. Verification of the System Impedance Criterion and the Symmetry of the Wideband Oscillation Frequency Spectrum

To verify the effectiveness of the characteristic Equation (51) or (52), a typical grid-connected VSC test system was constructed using “DIgSILENT Powfactory”, as shown in Figure 5. This system consists of an ideal DC current source $I_{dc}$, a DC-side capacitor C, a grid-side VSC, a filter inductance L_f, a transformer, an equivalent impedance of the external grid $L_g$, and the external grid (an ideal voltage source ${\mathrm{U}}_{\mathrm{g}}$). The control structure of the VSC is given in Figure 3, and the detailed parameters of the test system are given in

Table 1. Parameter table for the grid-connected VSC test system.

Parameters	Numeric Value
Proportional gain of the outer loop $K_{p1}$	0.01
Integral gain of the outer loop $K_{i1}$	10
Proportional gain of the inner loop d-axis $K_{p2}$	0.01
Integral gain of inner d-axis loop $K_{i2}$	10
Proportional gain of the inner loop q-axis $K_{p3}$	5
Integral gain of inner q-axis loop $K_{i3}$	5
Phase-locked loop proportional parameter $K_p$	31.62
Phase-locked loop integral parameter $K_i$	500
DC-side capacitance C/mF	3
Grid-side inductance $L_g/\mathrm{mH}$	31.83099
Rated power of VSC/MW	5
DC-side voltage $U_{dc0}$/KV	1.5
The filter inductor Lf of VSC/mH	15

.

Under steady-state conditions, a classic modal analysis was performed to find the system’s oscillatory modes. Six eigenvalues were found in the system, as shown in

Table 2. The eigenvalues in the grid-connected VSC test system.

No. of Oscillatory Mode	Real Parts of the Eigenvalues (1/s)	Imaginary Part of Eigenvalues (Hz)
1, 2	11.360662	±66.6547
3, 4	−14.761669	±31.4314
5	−43.963275	0
6	−1.221247	0

. The layouts of the eigenvalues that are associated with the oscillatory modes are presented in a complex plane (see Figure 6).

The presence of a pair of positive real part eigenvalues indicates the existence of unstable oscillatory modes in the system. The nonlinear time-domain was used to demonstrate the results of the eigenvalue analysis. Figure 7 presents the active power at the PCC, and Figure 8 presents the results of FFT analysis of the oscillatory signal. As confirmed by the FFT results, there is an oscillatory mode of 68 Hz in the system, which is consistent with the eigenvalue analysis. Figure 9 and Figure 10 present the instantaneous signals of voltage and current and the results of the corresponding FFT analysis.

As seen from the simulation results in Figure 9 and Figure 10, there are two dominant oscillatory modes in the test system, i.e., (1) 118 Hz (50 Hz + 68 Hz) and (2) 18 Hz (50 Hz − 68 Hz). The oscillation frequency of 68 Hz is symmetrical to the fundamental frequency of 50 Hz, which is consistent with the analysis in Section 2.4, confirming the accuracy of the impedance model and criteria.

4. The Sensitivity of the Spectrum to Control Parameters

The control parameters of a VSC and the grid-side equivalent impedance directly affect the spectral characteristics of the wideband oscillations. In this section, the sensitivity of the spectrum to some key parameters is analyzed.

4.1. The influence of K_p1 and K_p2

Eigenvalue analysis was again applied to the test system, setting K_p = 10, K_i = 50, and K_i1 = K_i2 = 25 and keeping other parameters consistent with Table 1. After setting K_p2 = 0.025 and increasing K_p1 from 0.01 to 0.027, Figure 11 shows the system’s oscillatory modes for different K_p1 values. Similarly, after setting K_p1 = 0.025 and increasing K_p2 from 0.01 to 0.027, Figure 12 presents the system’s oscillatory modes for different K_p2 values. As K_p1 and K_p2 increase, the oscillatory modes 1 and 2 gradually move to the left, then enter the left half-plane, and the other oscillation modes hardly move. As seen from Figure 11 and Figure 12, the damping of the dominant oscillatory modes is sensitive to K_p1 and K_p2, whereas the frequency of the dominant oscillatory modes is relatively insensitive.

4.2. The Influence of K_i1 and K_i2

First, the settings of K_p = 10 and K_i = 50 were applied, keeping other parameters consistent with Table 1. Then, the setting of K_i2 = 200 was applied and K_i1 was increased from 10 to 200; Figure 13 shows the system’s oscillatory modes for different K_i1 values. Similarly, after setting K_i1 = 200 and increasing K_i2 from 10 to 200, Figure 14 presents the system’s oscillatory modes for different K_i2 values. As K_i1 and K_i2 increase, the damping and oscillatory frequency of modes 1 and 2 gradually increase, mode 5 gradually moves to the left, and the other oscillation modes hardly change.

4.3. The influence of K_p3 and K_i3

After setting K_p = 10, K_i = 50, and K_i1 = K_i2 = 20 and keeping other parameters consistent with Table 1, then setting K_i3 = 5 and increasing K_p3 from 5 to 200, Figure 15 shows the system’s oscillatory modes for different K_p3. Similar to occurrence with K_i1 and K_i2, the oscillatory frequency and damping of modes 1 and 2 increase, mode 4 gradually moves to the left, and the other oscillation modes hardly change. K_p3 has a significant impact on both the oscillatory damping and frequency of modes 1 and 2.

After setting K_p = 10, K_i = 50, and K_i1 = K_i2 = 20 and keeping other parameters consistent with Table 1, then setting K_p3 = 5 and increasing K_i3 from 5 to 200, Figure 16 shows the system’s oscillatory modes for different K_i3 values. As seen from Figure 16, oscillatory modes 1, 2, 3, and 4 are insensitive to K_i3. As K_i3 increases, mode 5 and mode 6 become complex eigenvalues and occur in conjugate pairs.

4.4. The Influence of the Coefficients of the PLL

In this section, the influence of the coefficients of the control loop of the PLL, i.e., K_p and K_i, is analyzed. For a three-phase PLL (SRF-PLL) based on dq synchronous rotating coordinate transformation, the following equation was used:

$$\left\{\begin{array}{c}{K}_P=2\beta {\omega }_n\\ K_i={\omega }_n^2\end{array}\right.$$

(56)

Here, ω_n represents the natural frequency of the PLL, and β represents the damping coefficient, which is taken as $1/\sqrt{2}$ in this paper. A basic structure of the PLL was used, with the settings of K_p and K_i obtained from the current research and engineering experience.

After setting K_i1 = K_i2 = 18 and K_i3 = 150 and keeping other parameters consistent with Table 1, then changing ω_n from 18.4 to 42.4, Figure 17 presents the system’s oscillatory modes for different K_p and K_i values. As K_p and K_i increase, oscillatory modes 1 and 2 move to the left, then enter the left half-plane, whereas oscillatory modes 3 and 4 move gradually to the right and enter the right half-plane. The oscillatory modes 5 and 6 hardly change as K_p and K_i increase.

4.5. The Influence of L_f and L_g

For the grid-connected VSC system in Figure 5, L_f and L_g are in series connection between the VSC inverter and the grid. Therefore, the values of L_f and L_g have the same impact on the stability of the system. We applied the settings of K_p = 10 and K_i = 50, keeping other parameters consistent with Table 1.

After setting L_g = 31.83 mH and increasing L_f from 9 mH to 22.50 mH, Figure 18 shows the system’s oscillatory modes for different L_f values. Similarly, after setting L_f = 15 mH and increasing L_g from 6.37 mH to 127.32 mH, Figure 19 presents the system’s oscillatory modes for different L_g values. As L_f and L_g increase, the damping of modes 1 and 2 increases and the oscillatory frequency of modes 1 and 2 decreases. There are no significant changes in the other oscillatory modes.

5. Hardware-in-the-Loop (HIL) Simulations

To further investigate the impact of control parameters of VSC on the oscillations, a HIL simulation platform is set up. The HIL simulation platform mainly consists of a PC, RT-LAB simulation machine (OP5700) and a hardware of the VSC controller. Here, the controller has a sampling and control period of 4 kHz, and the frequency of the Space Vector Pulse Width Modulation (SVPWM) is 4 kHz. The hardware and the schematic diagram simulation platform are given in Figure 20.

At steady state, the active power output of the VSC is 1.9 MW. The FFT analysis results of the active power at the PCC are shown in Figure 21, which indicates that the system has no oscillatory components. The key control parameters of VSC are given in

Table 3. The key control parameters of the VSC.

Parameters	Numeric Value
$K_{p1}$	0.3
$K_{i1}$	1
$K_{p2}$	0.3
$K_{i2}$	1
$K_{p3}$	0.3
$K_{i3}$	1
$K_p$	4
$K_i$	8
C/mF	14.66
$L_g/\mathrm{mH}$	156
Lf$/\mathrm{mH}$	6

.

Then, the key control parameters of the VSC are individually modified. The active power at the PCC and its FFT analysis results are shown in Figure 22, Figure 23, Figure 24, Figure 25, Figure 26, Figure 27. The HIL simulation results are consistent with the previous simulation results obtained from “DIgSILENT Powfactory”.

6. Conclusions

In this paper, we developed a second-order impedance model of the grid-connected VSC that mathematically demonstrates the symmetry of the frequency spectrum of wideband oscillations in the VSC grid-connected system, and we also explained the physical essence of the characteristic equation of this system. At the conclusion, the effects of the various parameters on the frequency spectrum were analyzed. The main conclusions are as follows:

(1)

The influence of the d-axis control loop

The proportional gains of the d-axis control loop have a relatively small impact on the frequency of the dominant oscillatory modes, mainly affecting the damping factor. As the proportional gains increase, system stability is improved. The integral gains of the d-axis control loop have impacts on both the frequency and damping factor of the dominant oscillatory modes. When larger integral gains are used, higher oscillation frequencies and damping factors are obtained.

(2)

The influence of the q-axis control loop

The proportional gain of the q-axis control loop has a significant impact on the system’s dominant oscillatory modes. As the proportional gain increases, both the oscillatory frequency and damping factor increase, which make the system more unstable. The q-axis integral gain has a relatively small impact on the dominant oscillatory modes, whereas new oscillatory modes appear as the integral gain increases.

(3)

The influence of the PLL control loop

The proportional gain and integral gain of the PLL have a large influence on the oscillatory modes. When the proportional gain and integral gain increase, one pair of oscillatory modes moves to the left, whereas other pairs of oscillatory modes move to the right of the plane.

(4)

The influence of the filter inductor and the external grid

As L_f and L_g increase, the oscillatory frequency decreases and the damping factor increases. L_f represents a simplified filter inductance, which depends on the design of the filter circuit, whereas L_g represents the equivalent impedance of the external grid, which depend on the configuration and operational modes of the external grid.