Stability Region of Grid-Forming Wind Turbine with Variable Parameters Using Bialternate Sum Matrix Approach

Abstract

Although the stability regions of wind turbines in the islanding mode have been widely researched, small-signal modeling of grid-forming wind turbines (GFWTs) in the islanding mode has yet to be explored. In addition, the state-space matrix of the wind turbine system has yet to be fully represented. Therefore, this paper proposes a small-signal modeling of GFWT and a method for identifying the stabilization region of a system with variable parameters. First, small-signal modeling of a GFWT based on virtual synchronous generator control is developed. Second, the bialternate sum matrix approach is used to determine the system stabilization region. The system matrix with multiple variable parameters is first decomposed into the sum of several matrices in this paper. Furthermore, the rotor-side generator control is simplified. It can reduce the dimensionality of the system matrix model. Finally, the simulation shows that the proposed method for determining the stabilization region of the variable system is accurate.

1. Introduction

Currently, more and more wind turbines are connected to microgrids to effectively utilize wind resources [1,2]. Wind turbines with traditional grid-forming control are no longer suitable for microgrids. Unlike traditional grid-following control, grid-forming control does not require the involvement of phase-locked loops to achieve grid synchronization. In the microgrid system of islanded wind turbines, the application of grid-forming control is more promising. In the past, grid-forming control was applied to voltage converters (VSCs). The advantages of grid-forming control are beginning to be realized as the voltage and frequency fluctuations in the internal systems of the expanding renewable energy generation become more frequent. R. H. Lasseter [3] proposes a VSC based on P-f and Q-V control which can provide voltage and frequency support to the grid. Wu [4] demonstrates that the VSC with grid-forming control is more stable under a network with low short circuit ratio. Along with the hot trend of renewable energy generation and microgrids, there is a strong interest in wind turbines operating in the islanding mode, coupled with the fact that the traditional grid-following control is no longer suitable for islanded wind turbines, which provides the possibility of applying grid-forming control strategies to islanded wind turbine systems. The application of grid-forming control to wind turbines is also attractive, and the stability advantages of GFWT operating under weak grids have been revealed in the literature [5]. The transient characteristics of GFWT and the transient problem of fault conditions are investigated in literature [6,7]. Wang [6] focuses on the turbine air-gap flux attenuation and overcurrent problems, and the effects of these problems are reduced by establishing air-gap flux feedback and virtual impedance. When a sudden drop in grid voltage occurs, it is often difficult for a wind turbine system with grid-forming control to respond to it in a timely manner, and thus literature [7] focuses on the wind turbine power dynamic response problem. However, the parameter stabilization region problem of GFWT, especially the stabilization region problem of wind turbine systems with multiple variable parameters, has rarely been studied. Although the parameter stabilizing region problem of VSC and GFWT [8,9,10,11] have been well studied, the parameter stabilizing region problem of GFWT still needs further research. In addition, small-signal modeling of GFWT for detailed expression is rarely provided. To bridge this gap, an exact small-signal model of the GFWT in the islanding mode and the bialternate sum matrix approach is used to determine the stabilization region of the GFWT system containing variable parameters.

On the one hand, small-signal models based on state space have been widely studied [12,13,14]. However, estimating the stabilization region of the system is difficult when there are variable parameters in the model. Zhu [15] demonstrates the complexity of state space matrices for a complex system. Although many studies have focused on methods for quickly determining the stable regions of variable system parameters based on state-space matrices, they remain to be investigated. The process of determining the system stability boundary through eigenvalues and selecting the system stability region applies to low-dimensional systems in the literature [16,17]. This is currently the dominant approach for performing system stability analysis. However, this method does not apply to systems with multiple variable parameters, such as the wind turbine system presented in this article. Hao [18] enhances computational efficiency by simplifying the system and downgrading it to realize the estimation of the system stability region. However, the stability region of the system obtained based on this method is inaccurate. Also, the method is complicated in the selection of the retained matrix features. The parametric approximation method is widely used to estimate the stabilization region of parameters. Qiu [19] uses Kernel ridge regression to compute the stabilization region for voltage. Similarly, the method is used to calculate the stabilization region for voltage and power in literature [20]. The bifurcation theory is accessed in the literature [21,22] to calculate the critical points one by one, and finally the stability region of the variable parameter system is obtained. However, this computational method imposes a significant burden on the computation of nonlinear functions and implicit functions. A method based on Taylor expansion and inequality constraints has been proposed in the literature [23] for determining the effect of multiple sag coefficients on the stability of a system. However, the parameter transformations on the basis of this method to judge variable parameters are not bidirectional. By constructing a linearized model of the microgrid and obtaining the closed-loop transfer function of the system for the stability judgment of the system, it is extremely common to use the Ross criterion [24] and the Routh–Hurwitz criterion [25] to judge the stability of low-order systems. However, the global model of the system is difficult to obtain, and the closed-loop transfer function is not scalable. On the other hand, impedance modeling methods have been widely used. Mohammad Kazem Bakhshizadeh [26] and Qian [27] determine system stability by measuring the output impedance ratio at two different locations of the system and applying the Nyquist criterion. Ji [28] models the AC impedance of the MMC-HVDC system for offshore integrated wind power. However, impedance modeling is more often applied to analyze the steady state of the system as the frequency changes and its use to estimate the stability region of a system containing variable parameters is difficult. In this paper, on the basis of the small-signal model of GFWT containing variable parameters, the discrimination of the stable region of the system is realized by utilizing the bialternate sum matrix approach.

Therefore, on the basis of the model of GFWT containing variable parameters, the discrimination of the stable region of the system is realized through the utilization of the bialternate sum matrix approach. The specific main features and advantages are as follows:

(1)

The detailed small-signal model of the GFWT is finally obtained by detailed modeling of each part of the permanent magnet synchronous wind turbine.

(2)

In this paper, a method for determining the stable region of a wind turbine system based on the bialternate sum matrix approach is used. The method can quickly determine the stable region of the system under the situation of system multi-parameter changes. Compared to the traditional methods, the method proposed in this paper requires fewer iterations to obtain the stabilization region of the GFWT system.

In Section 2, the modeling of an islanded wind turbine system is introduced, and the small-signal model of the GFWT is established. In addition, the effects of the parameters on the frequency and power of the GFWT system are visualized through simulation results. In Section 3, the influence of variable parameters on system stability is visually represented by the change in eigenvalues of the system state matrix. In Section 4, the bialternate sum matrix approach is proposed. In Section 5, the system stability region of the GFWT with multiple variable parameters is analyzed on the basis of the bialternate sum matrix approach. In addition, eigenvalue methods have been applied to compute the stabilization region of the system. In Section 6, the system stabilization regions obtained based on the method proposed in this paper and the eigenvalue method are compared, and it is verified that the present method is effective and accurate. Also, the advantages of the proposed method in terms of iteration speed are visually represented. The article is summarized in Section 7.

2. The Model of the GFWT

In order to model the turbine as a whole, the turbine parts are first modeled (Figure 1).

2.1. The Model of the Generator and Rotor-Side Converter

The modeling objectives of a single inverter include a generator, a rotor-side field-oriented controller, a virtual synchronous-controlled grid-side converter, a transformer, a transmission line, and load. In this paper, only the modeling processes of the grid-side converter, transmission lines, transformers, and loads are described in detail. The presence of DC-side capacitance decouples the rotor-side and grid-side converters from each other. Their mutual influence is small. Simplifying the machine-side converter structure into a constant-value voltage source helps to reduce the state-space matrix dimension. The machine-side output active power P is as follows:

$$P=1.5\left(U_{od}{i}_{od}+U_{oq}{i}_{oq}\right)$$

(1)

where $u_{od}$ is the output voltage of the d-axis of the net-side converter and $u_{oq}$ is the output voltage of the q-axis of the whole system. $i_{od}$ is the output current of the d-axis of the whole system and $i_{oq}$ is the output current of the q-axis of the whole system. The small-signal state-space equation for the grid-side converter in the $dq$ axis is

$$\Delta P=1.5\left(\Delta U_{od}{i}_{od}+U_{od}\Delta i_{od}+\Delta U_{oq}{i}_{oq}+U_{oq}\Delta i_{oq}\right)$$

(2)

It is important to mention that there is no intricate connection between the stator and rotor sides, and only a straightforward transfer of energy occurs between them. The rotor side control strategy is as follows (Figure 2):

2.2. Modeling of P-f Control in Grid-Side Converter

The equations for the P-f loop in Figure 3 are as follows:

$$\left\{\begin{array}{c}\frac{d{\omega }_m}{dt}=\frac{1}{J}\left[\frac{P_r}{{\omega }_1}-\frac{P}{{\omega }_1}-D_p\left({\omega }_r-{\omega }_1\right)\right]\hfill \\ \frac{d{\varphi }_0}{dt}={\omega }_r\hfill \end{array}\right.$$

(3)

where ${\omega }_r$ is the voltage reference angular frequency, J is the rotational inertia, $P_r$ represents the grid-side voltage reference, $D_p$ denotes the damping coefficient and ${\theta }_0$ is the voltage phase reference.

Secondary frequency regulation is also incorporated in the network side control link to enhance the stability of the system. This helps the frequency to remain stable. The secondary frequency modulation formula is as follows:

$$P_r=P_s+k_p\left({\omega }_1-\omega \right)+k_i\left({\omega }_1-\omega \right)dt$$

(4)

The variables $k_p$ and $k_i$ represent the proportional and integral regulation coefficients for quadratic frequency modulation, while $P_s$ denotes the active power reference value obtained through maximum power tracking and $P_r$ represents the active power reference value. The discrepancy between $\omega$ and ${\omega }_r$ is negligible, allowing for the approximation of ${\omega }_r$ as $\omega$. The model of the active control section of the grid-side converter can be described as follows:

$$\Delta \dot{\omega }=-\frac{k_p+{\omega }_1{D}_p}{J{\omega }_1}\Delta \omega -\frac{\Delta P}{J{\omega }_1}-\frac{k_i\Delta \gamma }{J{\omega }_1}$$

(5)

where $\Delta \gamma$ is a newly defined variable, and it can be represented as follows:

$$\frac{d\gamma }{dt}={\omega }_1-\omega$$

(6)

2.3. Modeling of Closed-Loop Control in Grid-Side Converter

Among the grid-forming control strategies, virtual synchronous generator control (VSG) is a widely used means of grid-forming control. The electromagnetic characteristics of the virtual synchronous generator without abrupt changes in the input can be expressed as follows:

$$\left\{\begin{array}{c}{u}_{odabc}=E\left[sin({\theta }_0+\psi )\right]\hfill \\ E={\omega }_1{M}_{fc}{i}_{fc}\hfill \end{array}\right.$$

(7)

where $i_{fc}$ is the excitation current, $M_{fc}$ is the virtual excitation mutual inductance, and E is the grid-side converter output voltage amplitude. The VSG control simulates the voltage regulation function of the excitation system of the synchronous machine through the constant regulation process of the excitation current by the PI controller, thus providing virtual inertia and damping for the microgrid, which can be expressed as follows:

$$i_{fc}=\frac{k_{p}c}{M_{fc}}\left(u_{\mathrm{odref}}-u_{od}\right)+\frac{k_{i}c}{M_{fc}}\int \left(u_{\mathrm{odref}}-u_{od}\right)dt$$

(8)

In the given equation, $u_{odref}$ represents the reference value for the d-axis component of the output voltage from the grid-side converter, while $u_{od}$ denotes the d-axis component of the overall output voltage. Additionally, $k_{pc}$ and $k_{ic}$ are the control coefficients for the voltage control ratio and the integration link, respectively. Setting state variable $d{\zeta }_{od}/dt=u_{odref}-u_{od}$, the following equations can be obtained:

$$\Delta {\dot{\zeta }}_{od}=-u_{od}$$

(9)

Associating (8) and (9), the virtual excitation current can be expressed as

$$i_{fc}=\frac{k_{pc}}{M_{fc}}\left(u_{odref}-u_{od}\right)+\frac{k_{ic}}{M_{fc}}{\zeta }_{od}$$

(10)

where $M_{fc}$ is the virtual excitation mutual inductance and $i_{fc}$ is the excitation current converter that is modeled and linearized using the switching averaging method, and the reference output voltage from the converter in the d-q axis is

$$\left\{\begin{array}{cc}\hfill \Delta u_{od}^{*}& =M_{fc}{i}_{fc}\Delta \omega -{\omega }_{ref}{k}_{pc}\Delta u_{od}+k_{i}c{\omega }_0{\zeta }_{od}\hfill \\ \hfill \Delta u_{oq}^{*}& =0\hfill \end{array}\right.$$

(11)

According to the closed-loop control strategy in Figure 4, the current loop model can be expressed as

$$\left\{\begin{array}{c}{i}_{id}^{*}=i_{od}-{\omega }_{1}C{u}_{oq}+\left(k_{p3}+\frac{k_{i3}}{s}\right)\left(u_{od}^{*}-u_{od}\right)\hfill \\ i_{iq}^{*}=i_{oq}-{\omega }_{1}C{u}_{od}+\left(k_{p3}+\frac{k_{i3}}{s}\right)\left(u_{oq}^{*}-u_{oq}\right)\hfill \end{array}\right.$$

(12)

The voltage and current loops have a similar structure with no notable differences, and it can be expressed as

$$\left\{\begin{array}{c}{u}_{id}^{*}=u_{od}-{\omega }_{1}C{i}_{oq}+\left(k_{p4}+\frac{k_{i4}}{s}\right)\left(i_{id}^{*}-i_{id}\right)\hfill \\ u_{iq}^{*}=u_{oq}-{\omega }_{1}C{i}_{od}+\left(k_{p4}+\frac{k_{i4}}{s}\right)\left(i_{iq}^{*}-i_{iq}\right)\hfill \end{array}\right.$$

(13)

Four state variables are set up according to the four proportional-integral links present in the voltage-current closed-loop control, and they can be expressed as

$$\left\{\begin{array}{c}\frac{d{\zeta }_{1d}}{dt}=u_{od}^{*}-u_{od}\hfill \\ \frac{d{\zeta }_{1q}}{dt}=u_{oq}^{*}-u_{oq}\hfill \\ \frac{d{\zeta }_{2d}}{dt}=i_{id}^{*}-i_{id}\hfill \\ \frac{d{\zeta }_{2q}}{dt}=i_{iq}^{*}-i_{iq}\hfill \end{array}\right.$$

(14)

By using (12) and (13) and neglecting the power electronics losses, the model of the dual loop of the converter can be expressed as

$$\left\{\begin{array}{c}{i}_{id}^{*}=i_{od}-{\omega }_{1}C{u}_{oq}+k_{p3}\left(u_{od}^{*}-u_{od}\right)+k_{i3}{x}_{id}\hfill \\ i_{iq}^{*}=i_{oq}-{\omega }_{1}C{u}_{od}+k_{p3}\left(u_{oq}^{*}-u_{oq}\right)+k_{i3}{x}_{iq}\hfill \\ u_{id}^{*}=u_{od}-{\omega }_{1}L{i}_{iq}+k_{p4}\left(i_{id}^{*}-i_{id}\right)+k_{i4}{x}_{id}\hfill \\ u_{iq}^{*}=u_{oq}-{\omega }_{1}L{i}_{id}+k_{p4}\left(i_{iq}^{*}-i_{iq}\right)+k_{i4}{x}_{2d}\hfill \end{array}\right.$$

(15)

Ignoring power electronics losses, the closed-loop control model can be expressed as

$$\left\{\begin{array}{c}\Delta {\dot{\zeta }}_{1d}=M_f{i}_f\Delta \omega +k_i{\omega }_{ref}\Delta x_{0d}-\left(k_p{\omega }_{ref}+1\right)\Delta u_{od}\hfill \\ \Delta {\dot{\zeta }}_{1q}=-\Delta u_{oq}\hfill \\ \Delta {\dot{\zeta }}_{2d}=k_{p3}{M}_f{i}_f\Delta \omega +k_{p3}{k}_i{\omega }_{ref}\Delta x_{0d}+k_{i3}\Delta x_{1d}\hfill \\ -\Delta i_{id}-k_{p3}\left(k_p{\omega }_{ref}+1\right)\Delta u_{od}-{\omega }_{ref}C\Delta u_{oq}+\Delta i_{od}\hfill \\ \Delta {\dot{\zeta }}_{2q}=k_{i3}\Delta x_{1q}-\Delta i_{iq}+{\omega }_{ref}C\Delta u_{0d}+k_{p3}\Delta u_{oq}+\Delta i_{oq}\hfill \end{array}\right.$$

(16)

The dual loop structure can be seen in Figure 4. $d_{abc}$ is the drive signal of the controller. The loop control transforms the information generated by the P-f control link to generate a trigger signal. The voltage loop enhances the stability by controlling the $C_{dc}$ voltage through the proportional-integral link.

Likewise, the impacts of transformers and loads are considered by modeling them after equivalent transformations. P-f control and closed-loop control are the main control aspects of the GFWT proposed in this paper. A thorough representation of the grid-side converter link is provided in both the preceding and following sections.

2.4. Modeling of Lines, LC Filter and Load

The wind turbine transformer and load are modeled as shown in Figure 1. The leakage inductance of the transformer and load is considered and converted to the side near the filter. It is interesting to note that the parameters of the $LC$ filter and the load are denoted by $R_f$, $L_f$ and $C_f$, and the model can be expressed as follows:

$$\left\{\begin{array}{c}\frac{d{i}_{id}}{dt}=-\frac{R_f}{L_f}{i}_{id}+\frac{1}{L_f}{u}_{id}-\frac{1}{L_f}{u}_{od}+\omega i_{iq}\hfill \\ \frac{d{i}_{iq}}{dt}=-\frac{R_f}{L_f}{i}_{iq}+\frac{1}{L_f}{u}_{iq}-\frac{1}{L_f}{u}_{oq}-\omega i_{id}\hfill \\ \frac{d{u}_{od}}{dt}=\frac{1}{C_f}{i}_{id}-\frac{1}{C_f}{i}_{od}+\omega u_{oq}\hfill \\ \frac{d{u}_{oq}}{dt}=\frac{1}{C_f}{i}_{iq}-\frac{1}{C_f}{i}_{oq}-\omega u_{od}\hfill \end{array}\right.$$

(17)

However, the effect of $R_{load}$ on the stability of GFWT cannot be ignored. In order to reduce the complexity of the small-signal model of the system, the transformer and the load are normalized, and the sum of their impedances is denoted by $R_T$ and $L_T$. The transformer and load models can be described as follows:

$$\left\{\begin{array}{c}\frac{d{i}_{od}}{dt}=-\frac{R_T}{L_T}{i}_{od}+\frac{1}{L_T}{u}_{od}+\omega i_{oq}\hfill \\ \frac{d{i}_{oq}}{dt}=-\frac{R_T}{L_T}{i}_{oq}+\frac{1}{L_T}{u}_{oq}-\omega i_{od}\hfill \end{array}\right.$$

(18)

where $u_{id}$ and $u_{iq}$ are the d-axis and q-axis components of the output voltage of the grid-side converter, respectively, $u_{od}$ and $u_{oq}$ are the d-axis and q-axis components of the transformer voltage near the filter side, respectively. Based on (16) and (17), the small-signal model of the transformer and filter can be expressed as follows:

$$\left\{\begin{array}{c}\Delta i_{id}=-\frac{R_f}{L_f}\Delta i_{id}+\frac{1}{L_f}\Delta u_{id}-\frac{1}{L_f}\Delta u_{od}+{\omega }_0\Delta i_{iq}+i_{iq}\Delta \omega \hfill \\ \Delta i_{iq}=-\frac{R_f}{L_f}\Delta i_{iq}+\frac{1}{L_f}\Delta u_{iq}-\frac{1}{L_f}\Delta u_{oq}-{\omega }_0\Delta i_{id}-i_{id}\Delta \omega \hfill \\ \Delta u_{od}=\frac{1}{C_f}\Delta i_{id}-\frac{1}{C_f}\Delta i_{od}+{\omega }_0\Delta u_{oq}+u_{oq}\Delta \omega \hfill \\ \Delta u_{oq}=\frac{1}{C_f}\Delta i_{iq}-\frac{1}{C_f}\Delta i_{oq}-{\omega }_0\Delta u_{od}-u_{od}\Delta \omega \hfill \\ \Delta i_{od}=-\frac{R_T}{L_T}\Delta i_{od}+\frac{1}{L_T}\Delta u_{od}-\frac{1}{L_T}\Delta u_{pd}+{\omega }_0\Delta i_{oq}+i_{oq}\Delta \omega \hfill \\ \Delta i_{oq}=-\frac{R_T}{L_T}\Delta i_{oq}+\frac{1}{L_T}\Delta u_{oq}-\frac{1}{L_T}\Delta u_{pq}-{\omega }_0\Delta i_{od}-i_{od}\Delta \omega \hfill \end{array}\right.$$

(19)

2.5. Modeling of the Whole Wind Turbine System

Based on the small-signal models above, the small-signal model of the wind turbine system can be expressed as

$$\Delta \dot{x_w}=A_w\Delta x_w$$

(20)

where

$$\begin{array}{c}\Delta x_w=[\Delta \omega ;\Delta \zeta ;\Delta x_{od};\Delta x_{1d};\Delta x_{1q};\Delta x_{2d};\Delta x_{2q};\hfill \\ \Delta i_{id};\Delta i_{iq};\Delta u_{od};\Delta u_{oq};\Delta i_{od};\Delta i_{oq}];\end{array}$$

(21)

The detailed structure of the state space matrix $A_w$ is given in Appendix A.

2.6. Simulation of the Effect of Different Control Parameters on System Parameters

In order to verify the correctness of the system’s small-signal modeling and the dominant features following the trajectory analysis, this section builds a system simulation model under the Matlab/Simulink simulation environment, whose structural topology is shown in Figure 1. The initial wind speed of the turbine is set to be 11 m/s, which is abruptly increased to 13 m/s at t = 0.5 s. The initial parameters of the system are shown in

Table A1. Steady-state operating point of the system.

Parameters	Values	Parameters	Values	Parameters	Values
${\omega }_1$	314.1592	$k_i$	$\mathrm{100,000}$	$k_{i4}$	$0.1$
$L_f$	$0.003\mathrm{H}$	$k_{pc}$	$0.01$	$k_{p3}$	$0.3$
$R_f$	$0.0002\Omega$	$k_{ic}$	$0.2$	$k_{i3}$	$0.1$
$C_f$	$0.025\mathrm{F}$	$u_p$	$3.5\mathrm{kV}$	$S_N$	$1.1\mathrm{MW}$
$D_p$	600	$p_m$	76	J	3
$k_p$	$\mathrm{10,000}$	$k_{p4}$	7	$u_o$	$0.69\mathrm{kV}$

. In the following section, separate simulations are performed on the effects of virtual inertia, virtual damping, and integration coefficients on the stability of the system, respectively. The system is underdamped before and after the parameter change.

Figure 5 and Figure 6 show the simulated waveforms when J is varied. As the value of J increases, the appearance of the frequency peak is delayed sequentially. Meanwhile, the maximum deviation of the system frequency shows a tendency to decrease and then increase as J increases. A similar trend is observed for the power variation of the wind turbine.

Figure 7 and Figure 8 show the simulated waveforms when $D_p$ is varied. It can be seen that the maximum deviation of frequency decreases and the amplitude of system oscillation decreases when $D_p$ increases within a reasonable range. The power variation of the wind turbine has a similar trend.

From Figure 9 and Figure 10, it can be seen that as $k_{ic}$ increases, the system oscillation increases during the oscillation decay process after the system frequency reaches the peak; the time for the system to recover to the steady state also increases significantly. The power change trend is basically consistent with the frequency change trend.

3. Effect of Various Variable Parameters on System Stability

In fact, numerous parameters included in the small-signal model of the wind turbine affect the stability of the system, and changes in the stability of the system intuitively respond with eigenvalue shifts. To verify the accuracy of the method proposed in this paper for stability analysis, the stability of the system containing variable parameters was analyzed by applying the method of this paper and the eigenvalue method [29], respectively.

In Figure 11, the eigenvalue criterion is clearly expressed. A clear expression is shown. The eigenvalues exist symmetrically about the real axis, and the system is stable when all of the system matrix eigenvalues lie to the left of the imaginary axis. This is a prerequisite for stability analysis. However, a change in the variable parameters of the system causes a change in the eigenvalues of the system matrix. When the eigenvalues move in the positive direction of the real axis, the system is less stable. When there are positive values of the eigenvalues, the system is unstable. To simplify the trajectories of eigenvalues, the complex trajectories of eigenvalue changes were simplified with straight lines (Figure 12, Figure 13 and Figure 14). In order to clearly represent the effect of changes in eigenvalues on the stability of the system, the lines are separated into two colors. The blue line area represents that the system is stable under the current eigenvalue, and the red line area represents that the system is stable under the current eigenvalue.

The system is stable when it is at a static operating point. However, changes in specific variable parameters in the system can affect the stability of the system. In particular, when there are multiple variable parameters in the system, it is essential to analyze its stabilization region. However, eigenvalue methods make calculating stability with multiple variable parameters difficult. Therefore, the method based on the bialternate sum matrix approach to estimate the stability region of the system is proposed in the next chapter.

In this section, the effect of each variable parameter of the system on the eigenvalues is directly expressed.

4. Proposed Bialternate Sum Matrix Approach for Identifying Stability Regions

This paper uses the bialternate sum matrix approach to determine the stability intervals of variable parameters in a wind turbine system. Linear time-invariant systems with variable parameters without external inputs can be described by

$$\dot{\mathit{x}}=Ass(\mathit{p},\mathit{x},\dots ,{\mathit{x}}_{\mathit{n}})$$

(22)

where x is a fixed state vector, p is the parameter vector after nominalization. The p vector is also affected due to the presence of variable parameters in the system. For a small-signal model containing variable parameters, the model can be expressed as

$$\dot{\mathit{x}}=\left[A_{w0}\right]\mathit{x}+\left[E_w\right]\mathit{x}$$

(23)

Based on (23), a state space matrix with variable parameters is represented as the sum of the stabilization matrix $\left[A_{w0}\right]$ and the ingress matrix $\left[E_w\right]$, where matrix $\left[E_w\right]$ is related to the variation of variable parameters. To ensure the stability of $\dot{\mathit{x}}$ in (22), it is vital to restrict the matrix $\left[E_w\right]$. The matrix $\left[E_w\right]$ is determined by the variable parameters. Therefore, by imposing limits on the range of variation of variable parameters, we can ensure the stability of the system.

The authors present a methodology for evaluating the stabilization region of time-invariant systems in [30]. This method is described below. First, the matrix containing the variable parameters is decomposed into the sum of the deterministic and variable matrices. Then, based on the previous method, the range of variation of the variable parameters is restricted. The method is applied to analyze the stabilization region of the GFWT in the subsequent chapters. The following is an example of limiting the range of variation of system parameters using bialternate sum matrix approach.

For a system that is linear and where the variable parameters do not shadow each other, (23) can be extended to

$$\left[A_{wq}\right]=\left[A_{wq0}\right]+\sum _{i=1}^r{q}_i\left[A_{wqi}\right]$$

(24)

where r is the number of variable parameters and $q_i$ is a specific variable parameter. $A_{wqi}$ is the matrix corresponding to $q_i$, the position of the element corresponding to $\left[A_{wq0}\right]$ to the position of its variable parameter in $\left[A_{wqi}\right]$. The following is a brief explanation of the proposed methodology:

$$\dot{x}\left(t\right)=\left[\begin{array}{ccc}a11& a12& a13\\ a21& k_v& a23\\ a31& a32& a33\end{array}\right]\mathit{x}\left(t\right)$$

(25)

where a is the actual parameters of the system and $k_v$ is the variable parameter. The matrix obtained in this example based on the decomposition of (24) can be expressed as

$$\left[A_w\left(0\right)\right]=\left[\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& k_{v0}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right]$$

(26)

and

$$\left[A_w\left(1\right)\right]=\left[\begin{array}{ccc}0& 0& 0\\ 0& 1& 0\\ 0& 0& 0\end{array}\right]$$

(27)

The bialternation and matrix theory is extremely effective for determining the stabilization region of systems containing variable parameters. Important lemmas and conclusions based on this theory are presented here.

Theorem 1. 

A system with variable parameters described in (25) is stable if the following condition is satisfied:

$$\left|\Delta k_{ui}\right|=\left|k_{ui}-k_{ui0}\right|<{\lambda }_u$$

(28)

$$\begin{array}{cc}\hfill {\lambda }_u& =min\left({\lambda }_u,{\lambda }_u\right)\hfill \\ \hfill {\lambda }_{A_{w}0}& =\frac{1}{\epsilon \left\{{\sum }_{i=1}^l{\left[\left[A_{wi}\right]{\left[A_{w0}\right]}^{-1}\right]}_m\right\}}\hfill \\ \hfill {\lambda }_{A_B}& =\frac{1}{\epsilon \left\{{\sum }_{i=1}^l{\left[B\left[A_{wi}\right]{\left(B\left[A_{w0}\right]\right)}^{-1}\right]}_m\right\}}\hfill \end{array}$$

(29)

Variables $k_i$ and $k_{i0}$ represent the perturbation and nominal values of parameter $k_i$. Symbol $\epsilon \left[A_i\right]$ stands for the spectral radius of matrix $\left[A_i\right]$, while ${\left[A\right]}_m$ refers to the matrix containing the absolute values of the elements in $\left[A\right]$, and $B\left[A\right]$ is the binary summation matrix. If the matrix has a dimension of d, the dimension of the neighbor-sum matrix is $m=\frac{1}{2}\left[d(d-1)\right]$.

$$b_{\kappa \sigma ,\tau \upsilon }=\left\{\begin{array}{cc}-a_{\kappa \upsilon }\hfill & \tau =\sigma \hfill \\ a_{\kappa \tau }\hfill & \tau \ne \kappa ,\upsilon =\sigma \hfill \\ a_{\kappa \kappa }+a_{\sigma \sigma }\hfill & \tau =\kappa \bigcap \upsilon =\sigma \hfill \\ a_{\sigma \upsilon }\hfill & \tau =\kappa \bigcap \upsilon \ne \sigma \hfill \\ -a_{\sigma \kappa }\hfill & \upsilon =\kappa \hfill \\ 0\hfill & \mathit{else}\hfill \end{array}\right.$$

(30)

Based on (29), for a third-order square matrix, its double alternating sum matrix can be expressed as

$$B=\left[\begin{array}{ccc}{a}_{11}+a_{22}& a_{23}& -a_{13}\\ a_{21}& a_{11}+a_{33}& a_{12}\\ -a_{31}& a_{21}& a_{22}+a_{33}\end{array}\right]$$

(31)

$$k_{ui0}-\Delta k_{u}i<k_i<k_{ui0}+\Delta k_{ui}$$

(32)

5. Stability Region of the GFWT in Islanding Mode

In this section of the article, the stabilization region of the GFWT described in the second section is substantiated by applying the bialternate sum matrix approach. Based on the grid-forming wind small-signal model shown in (20), the stability in all scenarios is analyzed using method based on Theorem I and eigenvalue method (Figure 15, Figure 16, Figure 17, Figure 18, Figure 19 and Figure 20).

For the GFWT presented in Section 2, the effect of the variation of the wind turbine inertia J on the wind turbine system is extremely significant, which can also be described in the wind turbine. The effect of each variable parameter associated with the grid-side converter on the durable system stability is also evident. When analyzing the range of stability containing multiple variable parameters, the first step is to analyze the stability of the system containing multiple variable parameters which is a prerequisite for analyzing whether a system is small-signal. If the system is stabilized, matrix [A] can be decomposed, and in the system with two variable parameters, two additional matrices can be obtained, which correspond to the two variable parameters in the system. The determination of the stabilization region for a multi-parameter system is performed in steps. First, one of the two different variable parameters and their stability are determined. Then, Theorem 1 is applied to analyze the range of the stability around this stabilization point. Afterward, the judgment area is extended until the judgment of the whole stability region is realized.

In the following section, the stabilization region is analyzed by applying the bialternate sum matrix approach and the eigenvalue method. The variation of the integral parameter seriously affects the stability of the system, and the effects of J (Figure 15 and Figure 16), $D_p$(Figure 17 and Figure 18), and $R_T$ (Figure 19 and Figure 20) on the stability of the system are investigated using the two methods described above, respectively, taking into account the variation of the $k_{ic}$.

Compared to the stabilized region obtained by the eigenvalue method, the stabilized region obtained using the method proposed in this paper suffers from a small discrepancy in the estimation of the boundary region, though. However, its iteration speed method is highly advantageous. The next section discusses the above two methods in detail.

6. Discussion

The capability of the method proposed in this paper is demonstrated in this section. The eigenvalue method has been applied earlier in the literature [16] as a method to determine the stability of a system. The stability intervals of the system parameters obtained by calculating the eigenvalues of the system point by point are accurate. The accuracy of the proposed method in this paper is confirmed by comparing it with the stabilization region of the text parameters obtained by solving the traditional eigenvalue method. In the following, the stability regions derived from the two methods are compared separately. In particular, the blue shadow is the stability region of the exact parameters of the system obtained through the eigenvalue method, and the region consisting of irregular rectangles is the stabilized region obtained using the method proposed method.

The comparison shows that the stabilization regions of the system parameters obtained using the method proposed in this paper are extremely similar to those obtained by the traditional method. In addition, in this section, the stabilization regions of multiple sets of parameters are calculated using both methods as a way to verify the accuracy of the method proposed in this paper.

The stability regions obtained by the two methods are very similar, and the discrepancies can be ignored. However, it is difficult to ignore the difference in iteration speed between the two methods.

In addition, the impedance in the network has an extremely important effect on the stability of the GFWT system. Therefore, the effect of impedance cannot be ignored for GFWT systems. Similarly, the stabilization region of the system when the impedance of the system is changed is also portrayed by the proposed method in this paper along with the conventional method.

By way of comparison, the results are obvious. The stabilized regions obtained by the two methods are essentially the same, and their discrepancies are essentially negligible. However, the speed advantage of the method proposed in this paper is extremely obvious.

The method proposed in this paper is applied to find the stabilization region of a GFWT system containing several variable parameters. It is shown in Figure 21, Figure 22 and Figure 23. The rest of the parameters are kept constant during the process. By comparing the results obtained in two different ways, the correctness of the stabilization region obtained through the proposed method in this paper can be proven.

With roughly the same accuracy, the method proposed in this paper is highly advantageous in terms of speed, especially when dealing with higher-order state space matrices. The great advantage of the method proposed in this paper in terms of computational speed compared to the eigenvalue methods proposed in the literature [16] is visualized in the following by the number of iterations.

In Figure 24, the two methods show a huge difference in speed. The stabilized region obtained by the eigenvalue method with 25 point-by-point operations can be completed only once by applying the method proposed in this paper.

7. Conclusions

Although small-signal state-space models play a significant role in solving stability, the eigenvalue method has been widely used as a standard stability analysis method. However, facing a system containing variable parameters, it is difficult to use the eigenvalue method to calculate the computational effort to compute the stabilization region of the system. In the grid-forming wind turbine system proposed in this paper, even though the control part of the wind turbine is decoupled and the devices such as the load and the transformer are normalized, the final state-space matrix obtained remains in the 13th order. The amount of computation required to determine the stable region of the system using the eigenvalue method is extremely large. It would be impractical to apply the eigenvalue method to higher-order systems. Therefore, in this paper, a grid-forming small-signal state-space model is developed, and the stability intervals are obtained using the bialternate sum matrix approach. First, the small-signal model of the GFWT is established to retain the critical control aspects of the GFWT. In addition, the space matrix of this wind turbine system is decomposed, and the system stability region is determined using the bialternate sum matrix approach. Under the guarantee of accuracy, the stabilization region of the system obtained by the traditional eigenvalue method with 25 iterations, the method proposed in this paper needs only one iteration. In addition, it is feasible and effective to apply the bialternate sum matrix approach to the wind turbine system and calculate its parameter stabilization region. In addition, the method detailed in this paper offers a noteworthy advantage when dealing with high-order matrices requiring much computation. Eventually, this paper achieves reliable judgment of grid-forming turbine systems containing variable parameters.