Interaction Analysis among PMSG-Based Wind Turbines Based on Self- and En-Stabilizing Functions

Abstract

Sub-synchronous oscillation is one of the most challenging issues in modern power systems with a high proportion of wind power. Interactions among the wind turbines are key factors causing the oscillations. This paper aims to propose a method to quantify the interactions among N permanent-magnet-synchronous-generator (PMSG)-based wind turbines by transfer functions and analyze the influence of the interactions on the system stability. We derived different terms of the self- and en-stabilizing functions to describe different interactions caused by different wind turbines analytically. Meanwhile, the parameter sensitivity of this method is proposed to analyze the impact of the system and control parameters on the interactions and the system stability. Examples and simulations are given to show the effectiveness of this method.

1. Introduction

There is no doubt that wind power has become one of the most-important sources in modern power systems. Sub-synchronous oscillation is a challenging issue in systems with a high proportion of wind power [1,2,3]. Since the dynamic stability performance of such systems is determined by the characteristics of the wind turbines (WTs) and the interactions among the devices, the interactions related to WTs have attracted extensive attention from researchers.

Theoretical research has been conducted to investigate the mechanism of interactions related to WTs. The interaction between WTs and the synchronous generator was the earliest to be discussed by researchers [4,5]. The net damping method is used in the analysis. WTs are modeled as an equivalent impedance. The reached results show that a WT may cause negative electrical damping in the frequency region where it acts as a negative resistance.

With the increase of the proportion of wind power, the interaction between the series-compensation and WTs [6,7,8,9] and the interaction between high-voltage direct current (HVDC) system and WTs [10,11,12] are analyzed. Eigenvalue-based analysis was applied to the system with WTs and series-compensation and the system with WTs and HVDC in [8,13], respectively. The researchers found that the grid-side converters of permanent-magnet-synchronous-generator (PMSG)-based WTs and the voltage source converter (VSC) of HVDC play important roles in the dominant mode and stability. The impedance analysis method is another popular frequency domain method to analyze the interactions. X. Xie et al. established the impedance matrix model of a system with multiple doubly fed induction-generator (DFIG)-based WTs and series-compensation in [6]. They also analyzed the interactions caused by multiple PMSG-based WTs and series-compensation in [7]. The researchers concluded that the controllers reinforce the oscillations, and the increasing series-compensation level may cause the oscillations. The interaction between WTs and modular multilevel converter (MMC)-HVDC and the interaction between WTs and diode-rectifier-based HVDC were also analyzed by impedance analysis in [10,11,12].

In practice, the interactions among WTs are also very important since there exist hundreds of WTs in a wind farm. Several papers have concentrate on this problem [14,15,16,17,18]. Among them, the open-loop mode analysis method was an effective method to analyze the interactions among WTs in [14,15,16]. A system connected with multiple WTs is divided into two subsystems. When the modes of the two subsystems are close to each other, the risk of instability increases. Although some regularities about the interactions can be obtained from the above analysis, the interactions could not be quantified by transfer functions in those methods. This leads to the result that one cannot analyze how the interactions affect the stability of the system directly. Moreover, the relationship between the parameters, interactions, and stability are not clear enough, so that the influence of the parameters on the system stability caused by different interactions cannot be analyzed.

Recently, the concepts of self-stabilization and en-stabilization were proposed to quantify the interactions among devices in [19,20]. An interaction quantification method of DC grids based on these concepts was presented in [21]. However, the transfer functions based on these concepts in an AC system were not defined. The self- and en-stabilizing coefficients in DC grids, which refer to the transfer functions between the active power and the DC voltage, make no sense for an AC system because DC voltage is not the interface of a device in AC systems.

To address the above problems, this paper proposes a method based on the self- and en-stabilizing functions to analyze the interactions among PMSG-based WTs. The contributions of this paper are as follows: (1) The self-stabilizing functions and en-stabilizing functions of AC systems are defined to quantify the interactions by transfer functions. (2) The expressions of different terms of the self- and en-stabilizing functions are presented to quantify different interactions among N PMSG-based WTs. (3) The parameter sensitivity of the self- and en-stabilizing functions is proposed, so that the influence of the parameters on the system stability through different interactions can be analyzed. The proposed method in this paper can explain the mechanism of the sub-synchronous oscillation more deeply from the perspective of the interactions. It can also provide new ideas for the setting of the parameters in a system with multiple WTs.

The rest of this article is as follows. In Section 2, the basic control and the small-signal model of a PMSG-based WT is introduced. Section 3 proposes the analytic quantification method of interactions among N WTs. In Section 4, examples and simulation results are given to show the effectiveness of this method. Finally, conclusions are drawn in Section 5.

2. Basic Control Scheme and the Small-Signal Model of a PMSG-Based WT

This section first introduces the basic control scheme of a PMSG-based WT, then presents the small-signal model of a weak AC system connected with N WTs based on the motion equation concept.

Figure 1 shows the typical control scheme of the machine-side and grid-side VSCs in a PMSG-based WT. Typical vector control was utilized in the machine-side and grid-side VSCs. Suppression methods, such as sliding mode control [22,23], were not considered. In a machine-side VSC, when the synchronous machine chain is orientated, the active power to the AC grid is controlled by the q-axis current of the synchronous machine, while the reactive power is controlled by the d-axis current. To be specific, when the WT is running in maximum power point tracking (MPPT) mode, the reference value of the q-axis current on this side is regulated by the rotor speed control (RSC), and the d-axis current is usually set to zero. The inner current control (ICC) regulates the AC voltage in the machine-side Vs. For the grid-side VSC, vector control and the phase-locked loop (PLL) are applied. The reference value of the d-axis in this side is manipulated by DC voltage control (DVC). The reference value of the q-axis current is controlled by AC terminal voltage control (TVC). PI controllers are used by RSC, DVC, PLL, ICC, and TVC. In addition, the bandwidths of the DVC, TVC, and PLL is about 10 Hz, the bandwidth of the RSC is about 1 Hz, and the bandwidth of the ICC is about 100 Hz.

Next, the small-signal model of a weak AC system connected with N WTs for the interaction analysis in the DVC timescale is presented. According to the control strategy, the bandwidths of the DVC and TVC are designed much higher than the RSC, but much lower than the ICC. Since this paper considers the interactions in sub-synchronous oscillation, which is in the DVC timescale [1], the machine-side VSC and its dynamics can be neglected. This means that the active power from the machine-side to the grid-side is considered to be constant. With these assumptions, a small-signal model of a single grid-side VSC based on the motion equation concept in the DVC timescale was established in [1]. When considering a weak AC system connected with N WTs as depicted in Figure 2, the small-signal model of such a system is shown in Figure 3. $M_{VSC1}\left(s\right)\sim M_{VSCN}\left(s\right)$ and $D_{VSC1}\left(s\right)\sim D_{VSCN}\left(s\right)$ represent the equivalent inertia and damping of the grid-side VSC of WT $1\sim N$. $G_{EQ1}\left(s\right)\sim G_{EQN}\left(s\right)$ represent the transfer functions of TVC. The specific expressions of the transfer functions presented by (32), (33), and (35) in [1] are shown in Appendix A. Note that the accuracy of this model was verified by eigenvalue analysis in [1]. Therefore, the model can be used to analyze the interactions among PMSG-based WTs in the DVC timescale.

Then, denote

$$\begin{array}{cc}\hfill \Delta {\mathit{P}}_{out}& ={[\Delta P_{out1}\Delta P_{out2}\cdots \Delta P_{outN}]}^T,\hfill \\ \hfill \Delta \mathit{Q}& ={[\Delta Q_1\Delta Q_2\cdots \Delta Q_N]}^T,\hfill \\ \hfill \Delta \mathit{\theta }& ={[\Delta {\theta }_1\Delta {\theta }_2\cdots \Delta {\theta }_N]}^T,\hfill \\ \hfill \Delta \mathit{E}& ={[\Delta E_1\Delta E_2\cdots \Delta E_N]}^T.\hfill \end{array}$$

The active and reactive power flow in AC networks of the model in Figure 2 can be expressed as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\Delta {\mathit{P}}_{out}\\ \Delta \mathit{Q}\end{array}\right]=\left[\begin{array}{cc}{\mathit{K}}_{P\theta }& {\mathit{K}}_{PE}\\ {\mathit{K}}_{Q\theta }& {\mathit{K}}_{QE}\end{array}\right]\left[\begin{array}{c}\Delta \mathit{\theta }\\ \Delta \mathit{E}\end{array}\right].\end{array}$$

(1)

Each component of the matrix can be computed as

$$\begin{array}{cc}\hfill K_{P\theta ij}& =\left\{\begin{array}{cc}{E}_i{E}_j{B}_{ij}cos({\theta }_i-{\theta }_j),\hfill & i\ne j\hfill \\ E_i^2{B}_{ii}-E_i{\displaystyle \sum _{j=1}^N}{E}_j{B}_{ij}cos({\theta }_i-{\theta }_j)-E_i{U}_g{B}_{ig}cos({\theta }_i-{\theta }_g)),\hfill & i=j\hfill \end{array}\right.,\hfill \\ \hfill K_{PEij}& =\left\{\begin{array}{cc}-E_j{B}_{ij}sin({\theta }_i-{\theta }_j),\hfill & i\ne j\hfill \\ -E_i{\displaystyle \sum _{j=1}^N}{E}_j{B}_{ij}sin({\theta }_i-{\theta }_j)-E_i{U}_g{B}_{ig}sin({\theta }_i-{\theta }_g),\hfill & i=j\hfill \end{array}\right.,\hfill \\ \hfill K_{Q\theta ij}& =\left\{\begin{array}{cc}{E}_i{E}_j{B}_{ij}sin({\theta }_i-{\theta }_j),\hfill & i\ne j\hfill \\ -E_i{\displaystyle \sum _{j=1}^N}{E}_j{B}_{ij}sin({\theta }_i-{\theta }_j)-E_i{U}_g{B}_{ig}sin({\theta }_i-{\theta }_g),\hfill & i=j\hfill \end{array}\right.,\hfill \\ \hfill K_{QEij}& =\left\{\begin{array}{cc}{E}_j{B}_{ij}cos({\theta }_i-{\theta }_j),\hfill & i\ne j\hfill \\ E_i{B}_{ii}+{\displaystyle \sum _{j=1}^N}{E}_j{B}_{ij}cos({\theta }_i-{\theta }_j)+U_g{B}_{ig}cos({\theta }_i-{\theta }_g),\hfill & i=j\hfill \end{array}\right..\hfill \end{array}$$

where i, j represent the i-th and j-th WT, and $i,j=1,2,\cdots ,N$. Note that $B_{ij}$ and $B_{ig}$ refer to the susceptance of the admittance of the system. The conductance of the AC lines is ignored, that is $Y_{ij}=j{B}_{ij}$ and $Y_{ig}=j{B}_{ig}$.

Next, we quantify the interactions on the basis of this model.

3. Analytic Quantification Method of the Interactions among N WTs

This section presents a method to quantify the interactions among WTs based on the self- and en-stabilizing functions. Firstly, the self- and en-stabilizing functions are defined. Secondly, a method to calculate the self- and en-stabilizing functions of an AC system with N WTs is proposed. Thirdly, the stability criterion of this method is presented. Finally, we propose the parameter sensitivity of this method to study the influence of the control parameters on the interactions among different WTs.

3.1. The Definition of Self- and En-Stabilizing Functions

In this subsection, the authors define the self- and en-stabilizing functions in AC systems as the basis of the proposed method.

The concept of self- and en-stabilization was first proposed in [19], in which the process of the focused device to stabilize itself is called self-stabilization, while the process in other devices to stabilize the focused device is called the other devices’ en-stabilization. Based on this concept, when focusing on the i-th WT, the active power output of it to the AC network can be expressed as

$$\begin{array}{cc}\hfill \Delta P_{outi}& =\sum _{j=1}^N(K_{P\theta ij}\Delta {\theta }_j+K_{PEij}\Delta E_j).\hfill \end{array}$$

(2)

As mentioned above, the dynamics of the active power input from the machine-side VSC is zero. Then, the unbalanced active power can be written as

$$\begin{array}{cc}\hfill \Delta P_{dci}& =0-\Delta P_{outi}=-\sum _{j=1}^N(K_{P\theta ij}\Delta {\theta }_j+K_{PEij}\Delta E_j).\hfill \end{array}$$

(3)

According to (3), denote the active power affected by the grid-side VSC i as

$$\begin{array}{c}\hfill \Delta P_{Si}=-(K_{P\theta ii}\Delta {\theta }_i+K_{PEii}\Delta E_i).\end{array}$$

(4)

Denote the active power affected by other grid-side VSCs as

$$\begin{array}{c}\hfill \Delta P_{Ei}=-\sum _{j=1,j\ne i}^N(K_{P\theta ij}\Delta {\theta }_j+K_{PEij}\Delta E_j).\end{array}$$

(5)

Based on the small-signal model in Figure 3, $\Delta {\theta }_{i,j}$ and $\Delta E_{i,j}$ can be replaced by $\Delta {\omega }_i$. Then, the unbalanced active power can be further expressed as

$$\begin{array}{c}\hfill \Delta P_{dci}=\Delta P_{Si}+\Delta P_{Ei}=G_{Si}\left(s\right)\Delta {\omega }_i+G_{Si}\left(s\right)\Delta {\omega }_i.\end{array}$$

(6)

Define the self-stabilizing function as the transfer function between $\Delta P_{Si}$ and $\Delta {\omega }_i$, that is $G_{Si}\left(s\right)=\Delta P_{Si}/\Delta {\omega }_i$. Define the en-stabilizing function as the transfer function between $\Delta P_{Ei}$ and $\Delta {\omega }_i$, that is $G_{Ei}\left(s\right)=\Delta P_{Ei}/\Delta {\omega }_i$.

Obviously, the en-stabilizing function $G_{Ei}\left(s\right)$ must be very complicated because it reflects the dynamics from all WTs except i. Thus, $G_{Ei}\left(s\right)$ can be decomposed into different terms to reflect different interactions caused by different WTs. According to the quantity of WTs participating in the interactions, $G_{Ei}\left(s\right)$ can be decomposed into $N-1$ types of terms, including single en-stabilizing functions, double en-stabilizing functions, triple en-stabilizing functions, and so on. The specific process to obtain the self- and en-stabilizing functions is derived in the next subsection.

3.2. Calculation of the Self- and En-Stabilizing Functions

Now, we show how to calculate the different terms of the self- and en-stabilizing functions of a system with N WTs.

3.2.1. MIMO System Represented by Transfer Function Matrix

For a convenient expression, we use the transfer function matrix to describe the multi-input and multi-output (MIMO) system in Figure 3.

The relationship between the angular frequency and the phase of the internal voltage can be expressed as

$$\begin{array}{c}\hfill \Delta \mathit{\theta }=diag\left[\begin{array}{c}\frac{1+s{D}_{VSC1}\left(s\right)}{s}\cdots \frac{1+s{D}_{VSCN}\left(s\right)}{s}\end{array}\right]\Delta \mathit{\omega }\triangleq {\mathsf{\Gamma }}_1\left(s\right)\Delta \mathit{\omega }.\end{array}$$

(7)

By direct computation, it has $\Delta \mathit{\omega }={\mathsf{\Gamma }}_1^{-1}\left(s\right)\Delta \mathit{\theta }$. The relationship between the active power output and the phase of the internal voltage can be written as

$$\begin{array}{c}\hfill \Delta \mathit{\theta }=diag\left[\begin{array}{c}-G_1\left(s\right)\cdots -G_N\left(s\right)\end{array}\right]\Delta {\mathit{P}}_{out}\triangleq {\mathsf{\Gamma }}_2\left(s\right)\Delta {\mathit{P}}_{out}.\end{array}$$

(8)

where $G_i\left(s\right)=(1+s{D}_{VSCi}\left(s\right))/\left(s^2{M}_{VSCi}\left(s\right)\right)$. The relationship between the reactive power and the magnitude of the internal voltage is

$$\begin{array}{c}\hfill \Delta \mathit{E}=diag\left[\begin{array}{c}{G}_{EQ1}\left(s\right)\cdots G_{EQN}\left(s\right)\end{array}\right]\Delta \mathit{Q}\triangleq {\mathsf{\Gamma }}_3\left(s\right)\Delta \mathit{Q}.\end{array}$$

(9)

Then, the model of the WTs without AC networks in Figure 3 can be represented as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\Delta \mathit{\omega }\\ \Delta \mathit{E}\end{array}\right]=\left[\begin{array}{cc}{\mathsf{\Gamma }}_1^{-1}\left(s\right)& \\ & 1\end{array}\right]\left[\begin{array}{c}\Delta \mathit{\theta }\\ \Delta \mathit{E}\end{array}\right]=\left[\begin{array}{cc}{\mathsf{\Gamma }}_1^{-1}\left(s\right){\mathsf{\Gamma }}_2\left(s\right)& \\ & {\mathsf{\Gamma }}_3\left(s\right)\end{array}\right]\left[\begin{array}{c}\Delta {\mathit{P}}_{out}\\ \Delta \mathit{Q}\end{array}\right].\end{array}$$

(10)

Substituting (7) into (1), the power flow in the AC network can be computed as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\Delta {\mathit{P}}_{out}\\ \Delta \mathit{Q}\end{array}\right]=\left[\begin{array}{cc}{\mathit{K}}_{P\theta }{\mathsf{\Gamma }}_1\left(s\right)& {\mathit{K}}_{PE}\\ {\mathit{K}}_{Q\theta }{\mathsf{\Gamma }}_1\left(s\right)& {\mathit{K}}_{QE}\end{array}\right]\left[\begin{array}{c}\Delta \mathit{\omega }\\ \Delta \mathit{E}\end{array}\right].\end{array}$$

(11)

Then, the model in Figure 3 can be converted into the MIMO system model represented by the transfer function matrix in Figure 4a.

Since the self- and en-stabilizing functions are related to the transfer function between the angular frequency and the unbalanced active power, the dynamics of the reactive power and the magnitude of internal voltage need to be converted into the dynamics of the active power and the phase of internal voltage. Expanding (1), the active power output and the reactive power can be written as

$$\begin{array}{c}\hfill \Delta {\mathit{P}}_{out}={\mathit{K}}_{P\theta }\Delta \mathit{\theta }+{\mathit{K}}_{PE}\Delta \mathit{E},\end{array}$$

(12)

$$\begin{array}{c}\hfill \Delta \mathit{Q}={\mathit{K}}_{Q\theta }\Delta \mathit{\theta }+{\mathit{K}}_{QE}\Delta \mathit{E}.\end{array}$$

(13)

Substituting (9) into (13), it has

$$\begin{array}{c}\hfill \Delta \mathit{Q}={(\mathit{I}-{\mathit{K}}_{QE}{\mathsf{\Gamma }}_3\left(s\right))}^{-1}{\mathit{K}}_{Q\theta }\Delta \mathit{\theta }.\end{array}$$

(14)

Then, substituting (14) into (9), the relationship between the magnitude and the phase of the internal voltage can be expressed as

$$\begin{array}{c}\hfill \Delta \mathit{E}={\mathsf{\Gamma }}_3\left(s\right){(\mathit{I}-{\mathit{K}}_{QE}{\mathsf{\Gamma }}_3\left(s\right))}^{-1}{\mathit{K}}_{Q\theta }\Delta \mathit{\theta }.\end{array}$$

(15)

Combining (7), (12), and (15), the relationship between the active power output and the angular frequency of the internal voltage can be written as

$$\begin{array}{c}\hfill \Delta {\mathit{P}}_{out}=\mathit{F}\left(s\right)\Delta \mathit{\omega }.\end{array}$$

(16)

where

$$\begin{array}{c}\hfill \mathit{F}\left(s\right)=({\mathit{K}}_{P\theta }+{\mathsf{\Gamma }}_3\left(s\right){(\mathit{I}-{\mathit{K}}_{QE}{\mathsf{\Gamma }}_3\left(s\right))}^{-1}{\mathit{K}}_{Q\theta }{\mathit{K}}_{QE}){\mathsf{\Gamma }}_1\left(s\right)\end{array}$$

Thus, the model in Figure 4a can be converted into the model in Figure 4b, in which ${\mathsf{\Gamma }}_1^{-1}\left(s\right){\mathsf{\Gamma }}_2\left(s\right)$ is the feedforward channel and F(s) is the feedback channel.

3.2.2. The Self-Stabilizing Function and the Overall En-Stabilizing Function

We are now ready to compute the self-stabilizing function and the overall en-stabilizing function.

Suppose WT 1 is focused. Consider the feedback channel of the model in Figure 4b. By expanding (16), the active power output of WT 1 and the WTs except 1 can be expressed by (17) and (18) as

$$\begin{array}{cc}\hfill \Delta P_{out1}& =F_{11}\left(s\right)\Delta {\omega }_1+\left[\begin{array}{c}{F}_{12}\left(s\right)\cdots F_{1N}\left(s\right)\end{array}\right]\left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right],\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \left[\begin{array}{c}\Delta P_{out2}\\ ⋮\\ \Delta P_{outN}\end{array}\right]& =\left[\begin{array}{c}{F}_{21}\left(s\right)\cdots F_{N1}\left(s\right)\end{array}\right]\Delta {\omega }_1\hfill \\ \hfill & +\left[\begin{array}{ccc}{F}_{22}\left(s\right)& \cdots & F_{2N}\left(s\right)\\ ⋮& \ddots & ⋮\\ F_{N2}\left(s\right)& \cdots & F_{NN}\left(s\right)\end{array}\right]\left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right].\hfill \end{array}$$

(18)

According to (10) and the model in Figure 3, the dynamic characteristics of WT $2\sim N$ can be written as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right]=\left[\begin{array}{ccc}-\frac{1}{s{M}_{VSC2}\left(s\right)}& & \\ & \ddots & \\ & & -\frac{1}{s{M}_{VSCN}\left(s\right)}\end{array}\right]\left[\begin{array}{c}\Delta P_{out2}\\ ⋮\\ \Delta P_{outN}\end{array}\right].\end{array}$$

(19)

Inverting the matrix in (19) and substituting the result into (18), we have

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{F}_{21}\left(s\right)\\ ⋮\\ F_{N1}\left(s\right)\end{array}\right]\Delta {\omega }_1& ={\left[\begin{array}{ccc}-\frac{1}{s{M}_{VSC2}\left(s\right)}& & \\ & \ddots & \\ & & -\frac{1}{s{M}_{VSCN}\left(s\right)}\end{array}\right]}^{-1}\left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right]\hfill \\ \hfill & -\left[\begin{array}{ccc}{F}_{22}\left(s\right)& \cdots & F_{2N}\left(s\right)\\ ⋮& \ddots & ⋮\\ F_{N2}\left(s\right)& \cdots & F_{NN}\left(s\right)\end{array}\right]\left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right].\hfill \end{array}$$

(20)

According to (20), the relationship between the angular frequency in WT 1 and the angular frequency in the other WTs can be written as

$$\begin{array}{c}\hfill \left[\begin{array}{c}\Delta {\omega }_2\\ ⋮\\ \Delta {\omega }_N\end{array}\right]={\mathit{L}}^{-1}\left(s\right)\left[\begin{array}{c}{F}_{21}\left(s\right)\\ ⋮\\ F_{N1}\left(s\right)\end{array}\right]\Delta {\omega }_1\end{array}$$

(21)

where each element of L(s) is

$$L_{ij}\left(s\right)=\left\{\begin{array}{cc}-s{M}_{VSCi}\left(s\right)-F_{ii}\left(s\right)\hfill & i=j\hfill \\ -F_{ij}\left(s\right)\hfill & i\ne j\hfill \end{array}\right.,i,j\in \{2,3,\cdots ,N\}.$$

Since $\Delta P_{dc1}=-\Delta P_{out1}$, the relationship between the unbalanced active power and the angular frequency can be calculated by substituting (21) into (17), that is

$$\begin{array}{cc}\hfill \Delta P_{dc1}& =-F_{11}\left(s\right)\Delta {\omega }_1-\left[\begin{array}{c}{F}_{12}\left(s\right)\cdots F_{1N}\left(s\right)\end{array}\right]{\mathit{L}}^{-1}\left(s\right)\left[\begin{array}{c}{F}_{21}\left(s\right)\\ ⋮\\ F_{N1}\left(s\right)\end{array}\right]\Delta {\omega }_1\hfill \end{array}$$

(22)

According to the definitions of the self- and en-stabilizing functions in (6), the self-stabilizing function $G_{S1}\left(s\right)=-F_{11}\left(s\right)$, and the overall en-stabilizing function is

$$\begin{array}{c}\hfill G_{E1}\left(s\right)=-\left[\begin{array}{c}{F}_{12}\left(s\right)\cdots F_{1N}\left(s\right)\end{array}\right]{\mathit{L}}^{-1}\left(s\right)\left[\begin{array}{c}{F}_{21}\left(s\right)\\ ⋮\\ F_{N1}\left(s\right)\end{array}\right].\end{array}$$

3.2.3. Different Paths of En-Stabilizing Functions

Next, the en-stabilizing functions can be decomposed into different terms to quantify the interactions among different WTs. According to the quantity of WTs participating in the interactions, the paths of interactions have N-1 types, which are calculated as follows:

(1) Single en-stabilizing function caused by WT j: Considering the effect of WT j on WT 1, denote the single en-stabilizing function as $G_{E1}^j\left(s\right)$, which can be computed as

$$\begin{array}{c}\hfill G_{E1}^j\left(s\right)=\frac{F_{j1}\left(s\right)F_{1j}\left(s\right)}{-s{M}_{VSCj}\left(s\right)-F_{jj}\left(s\right)}.\end{array}$$

(23)

(2) Double en-stabilizing function caused by WT j,k: When considering the effect of WT j,k on WT 1, double en-stabilizing function $G_{E1}^{jk}\left(s\right)$ can be calculated as

$$\begin{array}{cc}\hfill G_{E1}^{jk}\left(s\right)& =\left[\begin{array}{c}{F}_{1j}\left(s\right)F_{1k}\left(s\right)\end{array}\right]{\left[\begin{array}{cc}{L}_{jj}\left(s\right)& L_{jk}\left(s\right)\\ L_{kj}\left(s\right)& L_{kk}\left(s\right)\end{array}\right]}^{-1}\left[\begin{array}{c}{F}_{j1}\left(s\right)\\ F_{k1}\left(s\right)\end{array}\right]-G_{E1}^j\left(s\right)-G_{E1}^k\left(s\right).\hfill \end{array}$$

(24)

(3) En-stabilizing function caused by WTs 2∼N: When all the other WTs interact with WT 1, the en-stabilizing functions caused by WT 2∼N can be expressed as

$$\begin{array}{cc}\hfill G_{E1}^U\left(s\right)& =-\left[\begin{array}{c}{F}_{12}\left(s\right)\cdots F_{1N}\left(s\right)\end{array}\right]{\mathit{L}}^{-1}\left(s\right)\left[\begin{array}{c}{F}_{21}\left(s\right)\\ ⋮\\ F_{N1}\left(s\right)\end{array}\right]-\sum _{j=2}^N{G}_{E1}^j\left(s\right)\hfill \\ \hfill & -\sum _{j=2,k=3,j<k}^N{G}_{E1}^{jk}\left(s\right)-\sum _{j=2}^{N-1}{G}_{E1}^{J_p}\left(s\right),\hfill \end{array}$$

(25)

where $U=\{2,3,\cdots ,N\},J_p\subset U,J_p\subset R^{1(N-2)},p\in \{1,2,\cdots ,N-1\}$.

Then, the small-signal model with quantification of the interactions can be depicted by Figure 5.

3.3. Stability Criterion

In this subsection, the stability criterion is presented.

According to Figure 5, the closed-loop transfer function of the whole system can be written as

$$\begin{array}{c}\hfill G\left(s\right)=\frac{1/\left(s{M}_{VSC1}\left(s\right)\right)}{1-(G_{S1}\left(s\right)+G_{E1}\left(s\right))/\left(s{M}_{VSC1}\left(s\right)\right)}.\end{array}$$

(26)

Let the denominator of $G\left(s\right)$ in (26) equal zero. The poles of the system can be computed by

$$\begin{array}{c}\hfill s=(G_{S1}\left(s\right)+G_{E1}\left(s\right))/M_{VSC1}\left(s\right).\end{array}$$

(27)

By substituting $s=\sigma +j{\omega }_d$ into (27), (27) can be written as

$$\begin{array}{c}\hfill \sigma +j{\omega }_d=Re\left[\frac{G_{S1}(\sigma +j{\omega }_d)+G_{E1}(\sigma +j{\omega }_d)}{M_{VSC1}(\sigma +j{\omega }_d)}\right]+jIm\left[\frac{G_{S1}(\sigma +j{\omega }_d)+G_{E1}(\sigma +j{\omega }_d)}{M_{VSC1}(\sigma +j{\omega }_d)}\right].\end{array}$$

(28)

Since $\sigma \ll {\omega }_d$ in weak damping systems, (28) can be written as

$$\begin{array}{c}\hfill \sigma =Re\left[\frac{G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)}{M_{VSC1}\left(j{\omega }_d\right)}\right],{\omega }_d=Im\left[\frac{G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)}{M_{VSC1}\left(j{\omega }_d\right)}\right].\end{array}$$

(29)

According to the expression of $M_{VSC1}\left(s\right)$, we can see that $M_{VSC1}\left(j{\omega }_d\right)$ is a real number because it only contains the even power of s. Once one knows the sign of $M_{VSC1}\left(j{\omega }_d\right)$, the stability of the system can be justified according to the sign of $Re[G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)]$. Then, the criterion of the proposed method can be expressed as follows. (i) When $M_{VSC1}\left(j{\omega }_d\right)>0$, if the imaginary curve crosses 0 from negative to positive and $Re[G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)]<0$, the system is stable with positive damping. The smaller the $Re[G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)]<0$, the better the stability performance is. Otherwise, the system is unstable. (ii) When $M_{VSC1}\left(j{\omega }_d\right)<0$, if the imaginary curve crosses 0 from negative to positive and $Re[G_{S1}\left(j{\omega }_d\right)+G_{E1}\left(j{\omega }_d\right)]>0$, the system is stable with positive damping. Otherwise, the system is unstable. On the basis of the stability criterion, how different interactions among WTs affect the stability of the system can be evaluated.

3.4. Parameter Sensitivity of the Self- and En-Stabilizing Functions

In order to analyze the influence of the control parameters on the stability of the system, the parameter sensitivity of the self- and en-stabilizing functions is proposed in this subsection.

Denote the small variation of a certain parameter as $\Delta \beta$. Denote any paths of self- and en-stabilizing functions as $G_f\left(s\right)$. Since the real part of the pole is only related to the real part of $Re\left[G_f\left(s\right)\right]$, the sensitivity parameter of the self- and en-stabilizing functions is defined as

$$\begin{array}{c}\hfill S_{\beta }=Re\left[\underset{\Delta \beta \to 0}{lim}\frac{\Delta G_f\left(s\right)}{\Delta \beta }\right]=Re\left[\frac{\partial G_f\left(s\right)}{\partial \beta }\right].\end{array}$$

(30)

The absolute value of the numerical result is related to the different impacts of the parameters on the system stability through different interactions. The larger the absolute value, the stronger the influence of the parameter on the stability through this path of interaction is. In addition, the sign of $S_{\beta }$ determines the trend of the stability. If $S_{\beta }<0$, the system is more stable when the parameter decreases. If $S_{\beta }>0$, the system becomes more stable when the parameter increases.

Based on the proposed method, one can quantify the interactions among WTs by transfer functions first. Then, how different interactions affect the poles of the system can be evaluated. Finally, the effect of different parameters on the system stability through different interactions can be analyzed utilizing the proposed parameter sensitivity. We will use examples and simulations to show the effectiveness of this method in the next section.

4. Simulation and Examples

In this section, specific examples are given to demonstrate the effectiveness and advantages of the proposed method. First, we analyzed the interactions among WTs in an example and compared the results with the eigenvalue-based analysis. Second, how the parameters affect the system stability through different interactions was analyzed. Let us consider a weak AC system connected with five WTs. The system and control parameters are shown in Appendix B.

4.1. Comparison with Eigenvalue-Based Analysis

Consider the weak AC system connected with five WTs.

Table 1. Eigenvalues of the model except for real roots.

$\mathit{\lambda }$ of Detailed Model		Poles of the Proposed Model		States with Participation Factors $\ge 0.05$
${\lambda }_{1,2}$	$2.119\pm j53.782$	${\lambda }_{1,2}$	$2.417\pm j54.755$	$\Delta U_{dc1}$, $\Delta i_{d1}$, $\Delta U_{dc3}$, $\Delta i_{d3}$
${\lambda }_{3,4}$	$-1.240\pm j97.894$	${\lambda }_{3,4}$	$-1.235\pm j100.0$	$\Delta U_{dc2}$, $\Delta i_{d2}$, $\Delta U_{dc3}$, $\Delta i_{d3}$
${\lambda }_{5,6}$	$-3.039\pm j56.174$	${\lambda }_{5,6}$	$-2.914\pm j55.166$	$\Delta U_{dc1}$, $\Delta i_{d1}$, $\Delta U_{dc2}$, $\Delta i_{d2}$, $\Delta U_{dc3}$, $\Delta i_{d3}$
${\lambda }_{7,8}$	$-3.142\pm j57.048$	${\lambda }_{7,8}$	$-3.04\pm j55.476$	$\Delta U_{dc2}$, $\Delta i_{d2}$, $\Delta U_{dc4}$, $\Delta i_{d4}$, $\Delta U_{dc5}$, $\Delta i_{d5}$
${\lambda }_{9,10}$	$-3.350\pm j51.447$	${\lambda }_{9,10}$	$-3.005\pm j50.659$	$\Delta U_{dc4}$, $\Delta i_{d4}$, $\Delta U_{dc5}$, $\Delta i_{d5}$
${\lambda }_{11,12}$	$-22.993\pm j34.792$	${\lambda }_{11,12}$	$-24.743\pm j35.074$	$\Delta {\theta }_{pll1}$, $\Delta {\theta }_{pll2}$, $\Delta {\omega }_{pll2}$, $\Delta i_{q1}$
${\lambda }_{13,14}$	$-23.178\pm j36.016$	${\lambda }_{13,14}$	$-24.86\pm j37.024$	$\Delta {\theta }_{pll1}$, $\Delta {\theta }_{pll2}$, $\Delta {\omega }_{pll2}$, $\Delta i_{q1}$
${\lambda }_{15,16}$	$-35.038\pm j30.841$	${\lambda }_{15,16}$	$-34.498\pm j28.604$	$\Delta {\theta }_{pll3}$, $\Delta {\theta }_{pll4}$, $\Delta {\theta }_{pll5}$, $\Delta {\omega }_{pll3}$, $\Delta {\omega }_{pll4}$, $\Delta {\omega }_{pll5}$
${\lambda }_{17,18}$	$-35.652\pm j28.794$	${\lambda }_{17,18}$	$-34.959\pm j27.848$	$\Delta {\theta }_{pll3}$, $\Delta {\theta }_{pll4}$, $\Delta {\theta }_{pll5}$, $\Delta {\omega }_{pll3}$, $\Delta {\omega }_{pll4}$, $\Delta {\omega }_{pll5}$
${\lambda }_{19,20}$	$-35.625\pm j28.794$	${\lambda }_{19,20}$	$-34.959\pm j27.848$	$\Delta {\theta }_{pll3}$, $\Delta {\theta }_{pll4}$, $\Delta {\theta }_{pll5}$, $\Delta {\omega }_{pll3}$, $\Delta {\omega }_{pll4}$, $\Delta {\omega }_{pll5}$

lists the eigenvalues in the DVC timescale (except for the real roots) and participation factors. The eigenvalues of both the nonlinear model and the proposed model with the quantification of the interactions are presented. It can be seen from Table 1 that the unstable eigenvalues ${\lambda }_{1,2}$ are dominated by the DC voltage, the d-axis current, the angular frequency, and the output angle of the PLL of WT 3. However, one could not know which paths of interactions lead to the oscillation and how different parameters influence the interactions and further affect the system stability.

Next, we use our proposed method to analyze the interactions among the five WTs. Suppose WT 1 is focused. The real parts of the self-/en-stabilizing functions through different interactions are shown in

Table 2. Frequency characteristics of the real part through different paths.

Interaction Path	Value	Interaction Path	Value
$G_{S1}\left(s\right)$	14	$G_{E1}^{34}\left(s\right)$	$-64$
$G_{E1}^2\left(s\right)$	42	$G_{E1}^{35}\left(s\right)$	$-66$
$G_{E1}^3\left(s\right)$	$-45$	$G_{E1}^{45}\left(s\right)$	44
$G_{E1}^4\left(s\right)$	44	$G_{E1}^{234}\left(s\right)$	$-34$
$G_{E1}^5\left(s\right)$	45	$G_{E1}^{245}\left(s\right)$	35
$G_{E1}^{23}\left(s\right)$	$-51$	$G_{E1}^{235}\left(s\right)$	$-35$
$G_{E1}^{24}\left(s\right)$	48	$G_{E1}^{345}\left(s\right)$	$-40$
$G_{E1}^{25}\left(s\right)$	51	$G_{E1}^{2345}\left(s\right)$	$-27$

. The oscillation frequency is ${\omega }_{d1}=54.8$ rad/s. At the same time, Figure 6 shows that $M_{VSC1}<0$. From Table 2, it can be seen that $Re\left[G_{S1}\right]$ > 0, which means WT 1 is stable. $Re\left[G_{E1}^2\left(s\right)\right]$, $Re\left[G_{E1}^4\left(s\right)\right]$, $Re\left[G_{E1}^5\left(s\right)\right]$, $Re\left[G_{E1}^{24}\left(s\right)\right]$, $Re\left[G_{E1}^{25}\left(s\right)\right]$, $Re\left[G_{E1}^{45}\left(s\right)\right]$, and $Re\left[G_{E1}^{245}\left(s\right)\right]$ are positive, which means these interactions produce positive damping to make the system stable. The rest of the paths of interaction are negative, which means they produce negative damping and lead to the oscillation. In particular, $Re\left[G_{E1}^{34}\left(s\right)\right],Re\left[G_{E1}^{35}\left(s\right)\right]$ have the largest negative value to make the system unstable.

Beyond this, one can even justify the stability of the system with fewer WTs on the basis of Table 2. It can be seen that $Re\left[G_{E1}^3\left(s\right)\right]$, $Re\left[G_{E1}^{23}\left(s\right)\right]$, $Re\left[G_{E1}^{34}\left(s\right)\right]$, $Re\left[G_{E1}^{35}\left(s\right)\right]$, $Re\left[G_{E1}^{234}\left(s\right)\right]$, $Re\left[G_{E1}^{235}\left(s\right)\right]$, $Re\left[G_{E1}^{345}\left(s\right)\right]$, and $Re\left[G_{E1}^{2345}\left(s\right)\right]$ are negative. Thus, if the weak AC system only includes WT 1, WT 2, WT 4, and WT 5, the system would be stable. The time domain simulation in Figure 7 verifies the analysis. From Table 2, one can also see that $Re\left[G_{E1}^5\left(s\right)\right]+Re\left[G_{E1}^{25}\left(s\right)\right]+Re\left[G_{E1}^{35}\left(s\right)\right]+Re\left[G_{E1}^{45}\left(s\right)\right]+Re\left[G_{E1}^{235}\left(s\right)\right]+Re\left[G_{E1}^{245}\left(s\right)\right]$$+Re\left[G_{E1}^{345}\left(s\right)\right]$$+Re\left[G_{E1}^{2345}\left(s\right)\right]>0$. This indicates that, compared with the system with WT 1 ∼WT 5, the system with WT 1∼WT 4 would have a worse stability performance. The time domain simulation in Figure 7 also verifies the analysis. Other similar results of the system with fewer WTs can be obtained according to Table 2 as well.

4.2. Effect of System and Control Parameters

In this subsection, we analyze how the system and control parameters affect the stability through different interactions. The DVC, PLL, and steady-state value of active power in WT 3 are discussed.

First, consider the control parameters of DC voltage control in WT 3. According to the analytic expressions of different paths of en-stabilizing functions, notice that $k_{P13}$ and $k_{I13}$ exist in $G_{E1}^3\left(s\right)$, $G_{E1}^{23}\left(s\right)$, $G_{E1}^{34}\left(s\right)$, $G_{E1}^{35}\left(s\right)$, $G_{E1}^{234}\left(s\right)$, $G_{E1}^{235}\left(s\right)$, $G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$.

Table 3. The sensitivity of different paths of the en-stabilizing functions to the parameters of DC voltage.

$\mathit{Controller}$ $\mathit{Parameters}$	${\mathit{G}}_{\mathit{E}1}^3\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{23}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{34}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{35}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{234}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{235}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{2345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}\left(\mathit{s}\right)$
$S_{kP13}$	$0.0472$	$0.0598$	$0.0451$	$0.0451$	$0.0897$	$0.0742$	$0.0832$	$0.1251$	$0.5694$
$S_{kI13}$	$-0.0031$	$-0.0018$	$-0.0061$	$-0.0061$	$-0.0098$	$-0.0083$	$-0.0087$	$-0.0139$	$-0.0578$

lists the sensitivity of $k_{P13}$ and $k_{I13}$ with respect to different terms of en-stabilizing functions. The results indicate that $G_{E1}^{234}\left(s\right),G_{E1}^{235}\left(s\right),G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$ are more sensitive to the variation of $k_{P13}$. Since the sensitivity with respect to all the interaction paths is positive, the larger the $k_{P13}$, the more stable the system is. The analysis results can be verified by time domain simulation, as depicted in Figure 8. On the other hand, $G_{E1}^{2345}\left(s\right)$ is the most-sensitive path to the variation of $k_{I13}$. Since $S_{KI13}$ with respect to the overall $G_{E1}\left(s\right)$ is negative, the system is more stable with the decrease of $k_{I13}$. The time domain simulation verifies the analysis, as shown in Figure 9.

Then, we analyzed how the control parameters of PLL in WT3 affect the stability through different paths of interaction. According to the analytic expressions of different paths of en-stabilizing functions, it can be seen that $k_{P23}$ and $k_{I23}$ exist in $G_{E1}^3\left(s\right)$, $G_{E1}^{23}\left(s\right)$, $G_{E1}^{34}\left(s\right)$, $G_{E1}^{35}\left(s\right)$, $G_{E1}^{234}\left(s\right)$, $G_{E1}^{235}\left(s\right)$, $G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$.

Table 4. The sensitivity of different paths of the en-stabilizing functions to the parameters of PLL.

$\mathit{Controller}$ $\mathit{Parameters}$	${\mathit{G}}_{\mathit{E}1}^3\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{23}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{34}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{35}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{234}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{235}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{2345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}\left(\mathit{s}\right)$
$S_{kP23}$	$0.3879$	$0.3126$	$0.2816$	$0.2816$	$0.5649$	$0.4893$	$0.5142$	$0.7892$	$3.6213$
$S_{kI23}$	$-0.0045$	$-0.0036$	$-0.0072$	$-0.0072$	$-0.0099$	$-0.0101$	$-0.0093$	$-0.0117$	$-0.0635$

lists the sensitivity of $k_{P23}$ and $k_{I23}$ with respect to the different terms of the en-stabilizing functions. The results show that all the paths are sensitive to $k_{P23}$, especially $G_{E1}^{234}\left(s\right),G_{E1}^{235}\left(s\right),G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$. Since $S_{KP23}$ with respect to the overall $G_{E1}\left(s\right)$ is positive, the larger the $k_{P23}$, the more stable the system is. The analysis results can be verified by the time domain simulation, as depicted in Figure 10. At the same time, $G_{E1}^{234}\left(s\right),G_{E1}^{235}\left(s\right),G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$ are more sensitive to the variation of $k_{I23}$. Since $S_{KI23}$ with respect to the overall $G_{E1}\left(s\right)$ is negative, the system is more stable with the decrease of $k_{I23}$. The time domain simulation in Figure 11 verifies the results.

Next, we discuss how the steady-state value of the active power in WT3 influences the stability through different interactions. According to the analytic expressions of different paths of en-stabilizing functions, it can be found that $P_{30}$ exists in $G_{E1}^3\left(s\right)$, $G_{E1}^{23}\left(s\right)$, $G_{E1}^{34}\left(s\right)$, $G_{E1}^{35}\left(s\right)$, $G_{E1}^{234}\left(s\right)$, $G_{E1}^{235}\left(s\right)$, $G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$.

Table 5. The sensitivity of different paths of the en-stabilizing functions to the parameters of $P_{30}$.

$\mathit{Controller}$ $\mathit{Parameters}$	${\mathit{G}}_{\mathit{E}1}^3\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{23}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{34}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{35}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{234}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{235}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}^{2345}\left(\mathit{s}\right)$	${\mathit{G}}_{\mathit{E}1}\left(\mathit{s}\right)$
$S_{P30}$	$-1.4392$	$-2.7491$	$-2.8865$	$-2.9458$	$-4.0179$	$-3.9953$	$-4.1182$	$-5.6209$	$-27.7729$

lists the sensitivity of $P_{30}$ with respect to the different terms of the en-stabilizing functions. The results show that all the paths are sensitive to $P_{30}$, especially $G_{E1}^{234}\left(s\right),G_{E1}^{235}\left(s\right),G_{E1}^{345}\left(s\right)$, and $G_{E1}^{2345}\left(s\right)$. Since $S_{P30}$ with respect to $G_{E1}\left(s\right)$ is negative, the system will be more stable with the decrease of $P_{30}$. The analysis results can be verified by the time domain simulation, as depicted in Figure 12.

5. Conclusions

This paper proposed an interaction quantification method based on the self- and en-stabilizing functions to analyze the interactions among N PMSG-based WTs. The general expressions of the self- and en-stabilizing functions through different paths were derived. These transfer functions can represent the interactions among different WTs directly and analytically. Next, the relation between the system stability and the self- and en-stabilizing functions was demonstrated. Based on this, how different interactions influence the system stability can be analyzed. Moreover, the parameter sensitivity of this method was proposed to present the relationship among the interactions, parameters, and the stability. We analyzed how the parameters of the DVC and PLL in different VSCs impact the system stability through different interactions. The results were verified by simulations. This paper explains the mechanism of the oscillations from the perspective of interactions. It also provides new ideas for the setting of the parameters in a system with multiple WTs. In the near future, we will study the interacts among PMSG-based WTs and other electronic devices (such as LCC-HVDC, DFIG-based WT, etc.) through the coupling of both AC and DC networks.