Analysis of Influence of Grid-Following and Grid-Forming Static Var Generators on High-Frequency Resonance in Doubly Fed Induction Generator-Based Wind Farms

Abstract

In Doubly Fed Induction Generator (DFIG)-based wind farms with Static Var Generators (SVGs), high-frequency resonance will be more like to occur when an unloaded cable is put into operation, which will threaten the stable operation of the wind farm. To address this issue, the influence of power outer loops on the impedance of grid-connected inverters is considered. Based on harmonic linearization, theoretical models for the sequence impedances of DFIGs, Grid-following (GFL) SVGs, and Grid-forming (GFM) SVGs are established. The correctness of the three models is verified by impedance scanning using the frequency sweep method. Through a comparative analysis of these sequence impedances, it is found that unlike the GFM SVG (which exhibits inductive impedance), the GFL SVG exhibits capacitive impedance in the high-frequency band, which leads to negative damping characteristics in the high-frequency band for the wind farm system with the grid-following SVG; thereby, the risk of high-frequency resonance also increases accordingly. On the contrary, GFM control adopted by SVGs can effectively eliminate the negative damping region in the high-frequency band for wind farms to suppress high-frequency resonance. Meanwhile, for grid-forming SVGs, the parameter variations in power synchronous loops have no significant impact on the suppressing effect of high-frequency resonance for wind farms. Finally, an electromagnetic simulation model for a DFIG-based wind farm system with an SVG is established using the StarSim-HIL (hardware-in-the-loop) experiment platform, and the simulation results validate the correctness of the theoretical analysis.

1. Introduction

By the end of 2023, the cumulative capacity of global wind power generation reached 837 GW. In areas with abundant wind resources, wind power development follows a trend in clustered grid connection [1]. With the large-scale grid connection of wind turbines, SVGs, and other power electronic devices, the supporting role of the power grid on wind farms is gradually decreasing, and the risk of high-frequency resonance in wind farms is increasing, which may threaten the stable operation of the system [2,3].

The converters of wind turbines and SVGs in wind farms are both high-speed power electronic devices. In recent years, the system has experienced high-frequency resonance due to converter-to-converter interaction and converter-to-grid interaction. For example, in 2008, the Saihanba Wind Farm in Inner Mongolia encountered a resonance occurrence of twentieth harmonics, leading to the frequent tripping and shutdowns of wind turbines [4]. Similarly, near-kilohertz resonances were observed at the Borwin1 offshore wind farm in Germany’s North Sea in 2013 [5]. Furthermore, in 2021, the Wudalai Wind Farm in North Hebei Province experienced high-order harmonic resonances around 2500 Hz, and several SVGs tripped due to the excessive number of shutting locks [6].

The current discussion on the high-frequency resonance problem in wind farms mainly focuses on the analysis and research of grid-connected systems with GFL converters. Previous studies have shown that the main factors causing high-frequency resonance in wind farm systems include the following: filtering capacitors of grid-side converters (GSCs) in wind turbines [7], transmission cable length [8,9], the number of wind turbines [10,11], the control parameters and control delay of GFL converters [7,12,13,14,15], etc. In order to suppress harmonic injection into the power grid, LC filters are installed in the GSC of wind turbines. However, these filtering capacitors may not only cause interaction resonance at around 1650 Hz between the GSC and rotor-side converter (RSC) but they may also lead to a 1000–2000 Hz converter-to-grid resonance [7]. Research has found that with the increase in the length of transmission cables, the resonant frequency of the system will decrease [8], and the increase in the number of grid-connected wind turbines will also lead to a decrease in the high-frequency resonant frequency [10]. In the literature [7,11], a slightly larger parameter in the current inner loop of DFIG converters can improve the phase margin of DFIG impedance, while the increase in the control parameter of GFL SVGs will reduce system stability in the high-frequency band. Additionally, the literature [12,13,14,15] has highlighted that sampling and control delay associated with GFL SVGs and GSCs can shift the first negative damping region of equivalent impedance to the left and increase the risk of high-frequency resonance in interconnected systems. However, these investigations did not consider the influence of the power outer loop of DFIGs and GFL SVGs on wind farm system stability. Recent research indicates that the power outer loop significantly affects impedance characteristics within the 500~1500 Hz band. When accounting for the power outer loop, there is an increase in the converters’ impedance phase angle and expansion of the negative damping region, leading to deterioration in impedance characteristics [16].

Numerous scholars have proposed resonance suppression methods based on active damping technology, such as virtual impedance [9,13,17], additional damping [18], and other approaches, to address the high-frequency resonance issue in wind farms. However, these measures are primarily effective for resonances within a fixed frequency band and have certain limitations in their applicability. Because of the enhanced robustness in weak power grids, GFM converters have gained significant attention [19,20,21]. Compared to conventional GFL converters, GFM converters are generally inductive and have significantly smaller equivalent impedance, thereby offering improved stability under weak grid conditions [22]. Consequently, some researchers have suggested adopting GFM control for wind turbine converters to enhance the stability of wind farm systems. The authors of refs. [23,24] changed GFL PMSGs and GFL DFIGs into GFM control and analyzed the system stability of GFM PMSGs and DFIGs when they were connected to weak power grids by impedance analysis. It was found that grid-connected wind turbines showed positive damping characteristics regardless of the operating conditions of wind turbines, thus reducing the risk of resonance occurrence. However, in the practical field, the types of wind turbines in large-scale cluster wind farms are diverse, and the electrical parameters and control structures of different wind turbines also vary. Therefore, the GFM control of wind turbine converters poses significant technical challenges. Simultaneously, GFM control relies on calculating the active power output to track the phase angle of grid voltage. Nevertheless, wind power fluctuations will directly lead to the asynchronous tracking of phase angles among converters of different wind turbines, potentially resulting in adverse interactions and subsequently having a negative impact on the stable operation of the system.

Based on the aforementioned analysis, it can be inferred that the current research on high-frequency resonance in wind farms utilizing GFL control has overlooked the modeling of power outer loops for converters. However, when employing an SVG with constant power factor control mode, the influence of the power outer loop on system stability cannot be disregarded. Directly transforming the wind turbines’ GSC from GFL control into GFM control not only causes unknown stability issues but also increases reformation workload and technical complexity. Relatively speaking, GFM SVGs are more conducive to the stable operation of wind farms connected to weak grids [25]. Considering this, China NR Electric Co., Ltd. has added supercapacitors and GFM control reformations to SVGs, enhancing the frequency- and voltage-supporting capability for wind power systems. In May 2024, the 50,000 kW GFM SVG trial operation of the Bamian Wind Farm in Tongyu County, Jilin, China was successful, laying the foundation for the control reformation and promotion application of GFM SVGs are subsequent renewable energy stations. In view of the above, the main contributions of this paper are as follows:

• Based on harmonic linearization theory, a high-frequency sequence impedance model is established for a DFIG-based wind farm with SVG, taking into account the power factor control outer loop. Based on the theoretical impedance model, the variation law of high-frequency impedance in the DFIG-based wind farm with GFL SVG is analyzed to reveal the occurrence mechanism of high-frequency resonance after an unloaded cable is put into operation.
• Without additional investment, the wind farm’s self-equipped GFL SVG is transformed into GFM control, and a high-frequency sequence impedance model for GFM SVG is established. For GFM and GFL SVG, a comparative analysis is conducted on the impedance characteristics to investigate GFM SVG’s impact on the high-frequency impedance characteristics of the wind farm. The proposed method presents a novel perspective for suppressing high-frequency resonance in the wind farm system.

The subsequent sections of this paper are structured as follows: Section 2 introduces the structure and resonance issues of DFIG-based wind farms. In Section 3, a sequence impedance model is established for aggregated DFIG, followed by impedance scanning verification. Section 4 presents the establishment of impedance models for GFL SVG and GFM SVG, along with a comparative analysis of their impedance characteristics. Based on these theoretical models, Section 5 explores the high-frequency resonance mechanism in the wind power system and highlights the negative impact of the feedforward loop in GFL SVG on system stability. Furthermore, Section 6 outlines the advantages offered by GFM SVG in suppressing high-frequency resonances in the wind farm. Section 7 verifies the correctness of theoretical analysis based on the StarSim-HIL platform, and finally, Section 8 summarizes the paper.

2. Structure of DFIG-Based Wind Farm and Resonance Issues

The topology of a DFIG-based wind farm is illustrated in Figure 1, where multiple DFIGs are aggregated into a single DFIG [8]. Each individual DFIG comprises an RSC, a DFIG, a GSC, and an LC filter. The RSC and GSC are interconnected through a DC bus with parallel capacitors. Under the voltage outer loop control of GSC, the DC voltage is maintained at V_dc. L_D and C_D represent the filtering inductance and capacitance of the GSC, respectively. In this wind farm system, power from wind turbines 1–10 flows through a 690 V/35 kV generator transformer and converges to the 35 kV bus bar. Other wind turbines are also connected to the 35 kV bus through cable lines No.2-n, and the SVG is incorporated into the 35 kV bus through filtered inductance L_s.

However, with changes in operating conditions, wind farm systems containing SVGs are at risk of high-frequency resonance. For example, in the Hami region of Xinjiang, the wind farm experienced high-frequency resonances at 2200 Hz and 3100 Hz when unloaded cables were switched on at the convergence station. To address such issues, this paper analyzes the influence mechanism of SVG control mode and unloaded cables as well as suppression methods on high-frequency resonance in wind fields from an impedance perspective by establishing a sequence impedance model for the wind farm.

3. Modeling of Sequence Impedance for DFIG

When a DFIG is connected to the grid, its output impedance is actually the equivalent parallel impedance between the RSC and the GSC. Due to the decoupling control of both converters, they do not affect each other during modeling [26]. Therefore, they can be independently modeled, and the output impedance of DFIG can be obtained as

$$Z_{DFIG}=Z_{RSC+G}//Z_{GSC+L_D}//Z_{C_D}$$

(1)

3.1. RSC Sequence Impedance

The equivalent circuit of the stator side of DFIG and the control structure of the RSC are depicted in Figure 2. In Figure 2a, a single-phase equivalent circuit is presented for DFIG’s stator side, with parameters from the rotor side transformed stator side. In Figure 2a, $v^{rc\prime }$ and $i^{rc\prime }$ represent the converted output voltage and current for phase A on DFIG’s rotor side, respectively, while slip is the slip ratio, denoted as slip = (ω_s − ω_r)/ω_s. R_s denotes the stator resistance, R_r denotes the rotor resistance, L_σs represents leakage inductance for DFIG’s stator whereas L_σr denotes leakage inductance for its rotor, and L_m refers to excitation inductance.

The harmonic linearization method is used to model the sequence impedance of each component in the wind farm. Assuming small positive and negative sequence voltage disturbances are injected at PCC because the positive and negative sequence disturbance components enter the wind turbine from the stator side, it is necessary to convert them to the stator side when establishing the impedance model of the turbine side and the RSC. According to Figure 2a, considering the circuit relationship between the RSC and induction motor, the total impedance on the stator side is

$$Z_{RSC+G}=R_s+s{L}_s-{\displaystyle \frac{{\left(s{L}_m\right)}^2}{\frac{R_r}{slip\_s}+s{L}_r+\frac{Z_{RSC}}{slip\_s}}}$$

(2)

where $L_s=L_{\sigma s}+L_m$, $L_r=L_{\sigma r}+L_m$, slip_s is defined as the slip in the frequency domain, $slip\_s=(s-j2\pi f_r)/s$.

According to Figure 2b, it can be seen that when a disturbance is injected, the rotor current component undergoes frequency shift due to slip ratio. Therefore, the expression for the A-phase response current at the rotor and A-phase stator voltage in the time domain is as follows:

$$\left\{\begin{array}{l}{i}_a^{rc}(t)=I_1^{rc}\cos [2\pi (f_1-f_r)t+{\phi }_{i1}^{rc}]+I_p^{rc}\cos [2\pi (f_p-f_r)t+{\phi }_{ip}^{rc}]\\ v_a^d(t)=V_1^d\cos [2\pi (f_1-f_r)t+{\phi }_{v1}^d]+V_p^d\cos [2\pi (f_p-f_r)t+{\phi }_{vp}^d]\end{array}\right.$$

(3)

Among them, $I_1^{rc}$ and $I_p^{rc}$ represent the fundamental current and positive sequence disturbance current of RSC output, while $V_1^d$ and $V_p^d$ represent the fundamental voltage and positive sequence disturbance voltage of the wind turbine. ${\phi }_{i1}^{rc}$, ${\phi }_{ip}^{rc}$, ${\phi }_{v1,}^d$ and ${\phi }_{vp}^d$ are the corresponding initial vectors, and f_r represents the rotor rotation frequency.

Both RSC and GSC use a phase-locked loop (PLL) to realize the synchronization with stator voltage. When small disturbances of positive and negative sequence voltages are injected at the PCC, the PLL is affected by these disturbances, resulting in a minor disturbance component in its output phase angle. The Park transformation introduces this disturbance into the current inner loop, thereby affecting the converters’ impedance. Therefore, when modeling sequence impedances for converters, harmonic linearization modeling should be applied to the PLL. The existing research has already provided a derivation process for harmonic linearization of PLL [27], so it can provide the relationship between PLL output error and grid voltage harmonic disturbance:

$$\Delta {\theta }_{PLL}[f]=-j{T}_{PLL}(s){\mathit{V}}_{\mathit{p}}^{\mathit{d}},f=f_p-f_1$$

(4)

$$T_{PLL}(s)={\displaystyle \frac{H_{PLL}(s)}{1+V_1{H}_{PLL}(s)}}$$

(5)

The transfer function of the PLL is given as $H_{PLL}(s)=(k_{pp}+k_{pi}/s)$. In the equation, ${\mathit{V}}_{\mathit{p}}^{\mathit{d}}=0.5V_p^d{e}^{j{\phi }_{vp}^d}$.

Based on (3) and the frequency domain convolution rule, the input current of the inner loop of the RSC in the complex frequency domain is

$$\begin{array}{l}{i}_d^{rc}\left[f\right]=\left\{\begin{array}{ll}{I}_1^{rc}\cos {\phi }_{i1}^{rc},& \hfill f=0\\ {\mathit{I}}_{\mathit{p}}^{\mathit{rc}}-j\sin {\phi }_{i1}^{rc}{T}_{PLL}(s){\mathit{V}}_{\mathit{p}}^{\mathit{d}},& \hfill f=f_p-f_1\end{array}\right.\\ i_q^{rc}\left[f\right]=\left\{\begin{array}{ll}{I}_1^{rc}\sin {\phi }_{i1}^{rc},& \hfill f=0\\ -j{\mathit{I}}_{\mathit{p}}^{\mathit{rc}}+j\cos {\phi }_{i1}^{rc}{T}_{PLL}(s){\mathit{V}}_{\mathit{p}}^{\mathit{d}},& \hfill f=f_p-f_1\end{array}\right.\end{array}$$

(6)

The frequency domain expression of the dq-axis voltage is

$$\begin{array}{l}{v}_d^{rc}[f]=\left\{\begin{array}{ll}{V}_1^d,& \hfill f=0\\ {\mathit{V}}_{\mathit{p}}^{\mathit{d}},& \hfill f=f_p-f_1\end{array}\right.\\ v_q^{rc}[f]=\left\{\begin{array}{ll}0,& \hfill f=0\\ -j{\mathit{V}}_{\mathit{p}}^{\mathit{d}}-V_1^d{T}_{PLL}{\mathit{V}}_{\mathit{p}}^{\mathit{d}},& \hfill f=f_p-f_1\end{array}\right.\end{array}$$

(7)

where ${\mathit{I}}_{\mathit{p}}^{\mathit{rc}}=0.5I_p^{rc}{e}^{j{\phi }_{i1}^{rc}}$. According to the control structure in Figure 2b, the expression of the modulated wave $v_{md}^{rc}$ and $v_{mq}^{rc}$ in the frequency domain is

$$\left\{\begin{array}{l}{v}_{md}^{rc}\left[f\right]=-i_d^{rc}\left[f\right]H_i^{rc}(s)-i_q^{rc}\left[f\right]K_{dq}^{rc}+v_d^{rc}[f]\\ v_{mq}^{rc}\left[f\right]=-i_q^{rc}\left[f\right]H_i^{rc}(s)+i_d^{rc}\left[f\right]K_{dq}^{rc}+v_q^{rc}[f]\end{array}\right.$$

(8)

where $H_i^{rc}(s)=k_p^{rc}+k_i^{rc}/s$ represents the current inner loop of RSC, and $K_{dq}^{rc}$ is the rotor cross-coupling coefficient. Based on the established PLL model, (2), and (8), the final expression of positive sequence impedance can be derived between the induction motor and RSC.

$$Z_{RSC+G}=R_s+s{L}_s-{\displaystyle \frac{{(s{L}_m)}^2}{(R_r/slip\_s)+s{L}_r+\frac{H_i^{rc}(s-j2\pi f_1)-j{K}_{dq}^{rc}}{slip\_s(1+0.5(M_r+{\mathit{I}}_{\mathbf{1}}^{\mathit{rc}}(H_i^{rc}(s-j2\pi f_1)-j{K}_{dq}^{rc})+j{\mathit{V}}_{\mathbf{1}}^{\mathit{d}})T_{PLL}(s-j2\pi f_1))}}}$$

(9)

where M_r = ($V_1^d$ + ${\mathit{I}}_{\mathbf{1}}^{\mathit{r}\mathit{c}}$ (R_s + jωL_s − (jωL_m)²)/jωL_r)))/$V_{dc}^d$K_m represents the steady-state value of RSC voltage without harmonic interference.

3.2. GSC Sequence Impedance

For GSC, the power outer loop adopts constant power factor control, where the power factor of the wind turbine generator is set as the default value. The control structure is shown in Figure 3. Due to the large capacitance of DC capacitors and the narrow bandwidth of DC voltage outer loop control, the influence of the DC voltage outer loop is ignored during the modeling process [28].

According to Figure 3, the relationship between the stator-side voltage of the wind turbine and the valve-side voltage of the GSC is obtained as

$$v_{abc}^{gc}=v_{abc}^d+L_{gc}{\displaystyle \frac{d{i}_{abc}^{gc}}{dt}}$$

(10)

For GSC, the power outer loop of constant power factor is modeled as shown in Figure 3. The reactive power reference $Q_{ref}^d$ of the DFIG is obtained by using the instantaneous active power P^d and power factor λ^d at the DFIG outlet and then subtracted from the instantaneous reactive power Q^d. Through the reactive power PI link, the reactive current reference $I_{qref}^{gc}$ of the GSC is obtained as

$$\left\{\begin{array}{l}{I}_{qref}^{gc}=H_q^{gc}(s)(Q^d-P^{d}a)\\ H_q^{gc}(s)=k_{qp}^{gc}+{\displaystyle \frac{k_{qi}^{gc}}{s}}\end{array}\right.$$

(11)

where a = tan(arccos(λ)), $H_q^{gc}(s)$ is the GSC reactive outer loop.

According to the theory of instantaneous power, the frequency domain expression of instantaneous active and reactive power of DFIG can be obtained as

$$\begin{array}{l}{Q}^d[f]=j{\displaystyle \frac{3}{2}}{V}_1^d{\mathit{I}}_{\mathit{p}}^{\mathit{d}}-j{\displaystyle \frac{3}{2}}{I}_1^d{e}^{-j{\phi }_{ip}^d}{\mathit{V}}_{\mathit{p}}^{\mathit{d}},f=f_p-f_1\\ P^d[f]={\displaystyle \frac{3}{2}}{V}_1^d{\mathit{I}}_{\mathit{p}}^{\mathit{d}}+{\displaystyle \frac{3}{2}}{I}_1^d{e}^{-j{\phi }_{ip}^d}{\mathit{V}}_{\mathit{p}}^{\mathit{d}},f=f_p-f_1\end{array}$$

(12)

In the frequency domain, $I_{qref}^{gc}$ is derived by substituting (11) into (10).

$$I_{qref}^{gc}[f]={\displaystyle \frac{3}{2}}{H}_q^{gc}((j-a)V_1^d{\mathit{I}}_{\mathit{p}}^{\mathit{d}}+(-j-a)I_1^d{e}^{-j{\phi }_{ip}^d}{\mathit{V}}_{\mathit{p}}^{\mathit{d}}),f=f_p-f_1$$

(13)

Due to the fact that the power calculation of the GSC power outer loop is based on the voltage and current of DFIG, and the impedance of the GSC is calculated based on the output current of the GSC, the wind turbine outlet current in $I_{qref}^{gc}$ should be converted into the GSC outlet current. According to Figure 1’s topology diagram of the wind turbine and Kirchhoff’s law, the positive sequence current of DFIG can be converted into

$${\mathit{I}}_{\mathit{p}}^{\mathit{d}}={\mathit{I}}_{\mathit{p}}^{\mathit{gc}}+{\mathit{I}}_{\mathit{p}}^{\mathit{rc}}={\mathit{I}}_{\mathit{p}}^{\mathit{gc}}+(-\frac{{\mathit{V}}_{\mathit{p}}^{\mathit{d}}}{Z_{G+RSC}})$$

(14)

Equation (14) is substituted into (12) to obtain the frequency domain expression of $I_{qref}^{gc}$ after unifying the current:

$$I_{qref}^{gc}[f]={\displaystyle \frac{3}{2}}{H}_q^{gc}((j-a)V_1^d{\mathit{I}}_{\mathit{p}}^{\mathit{gc}}+(-\frac{{\mathit{V}}_{\mathit{p}}^{\mathit{d}}}{Z_{G+RSC}})+(-j-a)I_1^d{e}^{-j{\phi }_{ip}^d}{\mathit{V}}_{\mathit{p}}^{\mathit{d}}),f=f_p-f_1$$

(15)

The GSC and RSC share a PLL, and the expressions for the current and voltage output of the GSC in the dq-axis frequency domain are similar to those on the rotor side. According to the control structure shown in Figure 3, the expression of the modulated wave in the frequency domain is

$$\left\{\begin{array}{l}{v}_{md}^{gc}\left[f\right]=-i_d^{gc}\left[f\right]\cdot H_i^{gc}(s)-i_q^{gc}\left[f\right]\cdot K_{dq}^{gc}+v_d^{gc}[f]\\ v_{mq}^{gc}\left[f\right]=(i_{qref}^{gc}[f]-i_q^{gc}\left[f\right])\cdot H_i^{gc}(s)+i_d^{gc}\left[f\right]\cdot K_{dq}^{gc}+v_q^{gc}[f]\end{array}\right.$$

(16)

where $H_i^{gc}(s)=k_{ip}^{gc}+k_{ii}^{gc}/s$ is the GSC current inner loop, and $K_{dq}^{gc}$ is the GSC cross-decoupling coefficient.

By performing the Park inverse transform on $v_{md}^{gc}$ and $v_{mq}^{gc}$, the frequency domain expression of $v_{mabc}^{gc}$ can be obtained. Combining with (9), considering the influence of the power outer loop and output filter inductance, the expression for the positive sequence impedance of the GSC is obtained as

$$Z_{GSC+L_{gc}}={\displaystyle \frac{s{L}_{gc}+K_m{V}_{dc}(H_i^{gc}(s^{*})-j{K}_{dq}^{gc}+0.75V_1^d{H}_i^{gc}({s^{*}}_1)H_q^{gc}(s^{*})(1+aj)}{1-0.5T_{PLL}(s^{*})K_m{V}_{dc}^d((M_g+{\mathit{I}}_{\mathbf{1}}^{\mathit{gc}}(H_i^{gc}({s^{*}}_1)-j{K}_{dq}^{gc})-V_1^d)+\frac{0.75(1+aj)V_1^d{H}_i^{gc}(s^{*})H_q^{gc}(s^{*})}{Z_{RSC+G}}+0.75(1-aj)H_i^{gc}(s^{*})H_q^{gc}(s^{*}){\mathit{I}}_{\mathbf{1}}^{{\mathit{d}}^{*}}+1)}}$$

(17)

Among (17), s* = sj2πf₁, M_g = ($V_1^d$ + jωL_gc${\mathit{I}}_{\mathbf{1}}^{\mathit{g}\mathit{c}}$)/($V_{dc}^d$K_m) is the steady-state voltage value of the GSC without disturbance, ${\mathit{I}}_{\mathbf{1}}^{\mathit{g}\mathit{c}}$ is the fundamental component of GSC output current, and ${\mathit{I}}_{\mathbf{1}}^{\mathit{d}\mathbf{*}}$ is the conjugate component of fundamental current of the DFIG.

The establishment of the sequential impedance model for DFIG is based on the voltage and current at the low-voltage side of the transformer. Therefore, it is necessary to convert the DFIG’s impedance to the high-voltage side and consider the influence of the transformer. A derivation of a transformer model has been presented in [29]. Based on this transformer model, this paper provides an expression for converting the positive sequence impedance of DFIG to the high-voltage side:

$$Z_W(s)=K^2{Z}_{DFIG}(s)+Z_T$$

(18)

K is the transformer turns ratio, Z_T is the transformer impedance converted to the 35 kV side, and Z_W is the DFIG impedance converted to the 35 kV side.

3.3. Verification of Impedance Model

To verify the correctness of the theoretical modeling of the DFIG sequential impedance, the impedance scanning method is utilized based on electromagnetic simulation. The DFIG theoretical modeling and impedance scanning results are shown in Figure 4. As shown in Figure 4, the theoretical modeling results are basically consistent with the actual impedance scanning results, indicating that the theoretical model established is correct.

4. Modeling of Sequence Impedance for SVG of Two Control Types

4.1. Sequence Impedance of GFL SVG

The topology of the SVG converter configured in the wind farm is mostly a cascaded H-bridge structure, corresponding to the GFL control as shown in Figure 5. Compared to the current inner loop, the bandwidth of the DC voltage control outer loop is lower, and the time scale is larger, which mainly affects the low-frequency characteristics of the SVG. Therefore, we ignore the influence of the DC voltage outer loop during modeling.

According to Figure 5, the relationship between PCC voltage and SVG valve-side voltage can be obtained as follows:

$$v_{abc}^{pcc}=v_{abc}^{sg}+L_{sg}{\displaystyle \frac{d{i}_{abc}^{sg}}{dt}}$$

(19)

where $v_{abc}^{sg}$ represents the three-phase voltage at the SVG valve side, and $i_{abc}^{sg}$ represents the three-phase output current of SVG.

Since SVG has a similar control method to the GSC, its outer loop also utilizes constant power factor control. Therefore, the specific derivation process is not provided here. The frequency domain expression for the instantaneous power of the grid is

$$\begin{array}{l}{Q}^g[f]=j{\displaystyle \frac{3}{2}}{V}_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{g}}-j{\displaystyle \frac{3}{2}}{I}_1^g{e}^{-j{\phi }_{ip}^g}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}},f=f_p-f_1\\ P^g[f]={\displaystyle \frac{3}{2}}{V}_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{g}}+{\displaystyle \frac{3}{2}}{I}_1^g{e}^{-j{\phi }_{ip}^g}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}},f=f_p-f_1\end{array}$$

(20)

The reference value of the q-axis current in the SVG current inner loop is

$$I_{qref}^{sg}[f]={\displaystyle \frac{3}{2}}{H}_q^{sg}((j-a^g)V_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{g}}+(-j-a^g)I_1^g{e}^{-j{\phi }_{ip}^g}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}),f=f_p-f_1$$

(21)

where a^g = tan(arccos(λ^g)), λ^g represents the power factor at the grid side.

According to the derivation process of GSC impedance, it is necessary to replace the PCC current in $I_{qref}^{sg}$ with the SVG output current:

$${\mathit{I}}_{\mathit{p}}^{\mathit{g}}={\mathit{I}}_{\mathit{p}}^{\mathit{sg}}+{\mathit{I}}_{\mathit{p}}^{\mathit{W}}={\mathit{I}}_{\mathit{p}}^{\mathit{sg}}+(-\frac{{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}}{Z_W})$$

(22)

The expressions of modulated waves $v_{md}^{sg}$ and $v_{mq}^{sg}$ in the frequency domain for SVG are

$$\left\{\begin{array}{l}{v}_{md}^{sg}\left[f\right]=-i_d^{sg}\left[f\right]\cdot H_i^{sg}(s)-i_q^{sg}\left[f\right]\cdot K_{dq}^{sg}+K_f{v}_d^{sg}[f]\\ v_{mq}^{sg}\left[f\right]=(i_{qref}^{sg}[f]-i_q^{sg}\left[f\right])\cdot H_i^{sg}(s)+i_d^{sg}\left[f\right]\cdot K_{dq}^{sg}+K_f{v}_q^{sg}[f]\end{array}\right.$$

(23)

where $H_q^{sg}$ represents the PI controller for the SVG power outer loop, $H_i^{sg}$ represents the PI controller for the current inner loop, and K_f represents the voltage feedforward coefficient.

By employing the inverse park transformation and combining (19) and (23), the expression for the positive sequence impedance of the GFL SVG can be obtained as follows:

$$Z_{SVG}={\displaystyle \frac{s{L}_{sg}+K_m{V}_{dc}^{sg}(H_i^{sg}(s^{*})-j{K}_{dq}^{sg}+0.75V_1^{pcc}{H}_i^{sg}(s^{*})H_q^{sg}({s^{*}}_1)(1+a^{g}j))}{1-0.5T_{PLL}(s^{*})K_m{V}_{dc}^{sg}((M_{sg}+{\mathit{I}}_{\mathbf{1}}^{\mathit{sg}}(H_i^{sg}(s^{*})-j{K}_{dq}^{sg})-K_f{V}_1^{pcc})+\frac{0.75(1+a^{g}j)V_1^{pcc}{H}_i^{sg}(s^{*})H_q^{sg}(s^{*})}{Z_W}+0.75(1-a^{g}j)H_i^{sg}(s^{*})H_q^{sg}(s^{*}){\mathit{I}}_{\mathbf{1}}^{\mathit{g}*}+K_f)}}$$

(24)

4.2. Sequence Impedance of GFM SVG

Existing GFM control methods include VF control, virtual synchronous machine (VSM) control, and virtual oscillator control [19]. This section presents the impedance modeling of a GFM SVG using VSM control.

By simulating the mechanical and electromagnetic characteristics of a synchronous generator, the GFM SVG exhibits inertial and damping characteristics similar to a synchronous generator. The control structure is illustrated in Figure 6. The DC voltage outer loop in Figure 6 generates the active power reference after being processed by a PI controller. Similar to the modeling process described in Section 4.1, the influence of the DC voltage outer loop is neglected. Synchronization with grid voltage is achieved by simulating a synchronous machine. In terms of reactive power voltage control, constant power factor control is still used to obtain the SVG reactive power reference and then the phase angle $\theta$ and amplitude E_m of the virtual internal potential is obtained as (25). Then, the three-phase modulation wave is obtained.

$$\left\{\begin{array}{l}\theta (s)={\displaystyle \frac{(P_{ref}^{sg}-P^{sg})}{{\omega }_n}}N(s)\\ E_m(s)=(Q^g-Q_{ref}^g)H_q^{sg}(s)+V_{nom}\end{array}\right.$$

(25)

where N(s) = 1/(Js² + Ds), J represents virtual rotational inertia, D represents active damping coefficient, and $H_q^{sg}$(s) represents reactive outer loop PI controller.

The frequency domain expression for SVG output active and reactive power is the same as (20). By substituting (20) into (25) and ignoring the small signal component of the quadratic term, the expression for the virtual internal potential phase angle θ and amplitude E_m in the frequency domain is obtained as follows:

$$\begin{array}{l}\theta [f]=\left\{\begin{array}{l}(P_{ref}^{sg}-{\displaystyle \frac{3}{2}}{V}_1^{pcc}{I}_1^{sg}\cos ({\phi }_{i1}^{sg})){\displaystyle \frac{1}{{\omega }_n}}N(s),f=0\\ {\displaystyle \frac{(\frac{3}{2}{I}_1^{sg}{e}^{-j{\phi }_{ip}^{sg}}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}-\frac{3}{2}{V}_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{sg}})}{{\omega }_n}}N(s),f=f_p-f_1\end{array}\right.\\ E_m[f]=\left\{\begin{array}{l}-{\displaystyle \frac{3}{2}}{V}_1^{pcc}{I}_1^{sg}\sin ({\phi }_{i1}^{sg})+V_{nom},f=0\\ {\displaystyle \frac{3}{2}}{H}_q((j-a^g)V_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{g}}+(-j-a^g)\\ I_1^g{e}^{-j{\phi }_{ip}^g}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}),f=f_p-f_1\end{array}\right.\end{array}$$

(26)

Small signal voltage disturbances introduce phase angle perturbations $\Delta \theta$ and voltage magnitude perturbations $\Delta E_m$. The frequency domain expressions for $\Delta \theta$ and $\Delta E_m$ can be obtained from (26):

$$\left\{\begin{array}{l}\Delta \theta [f]=(({\displaystyle \frac{3}{2}}{I}_1^{sg}{e}^{-j{\phi }_{ip}^{sg}}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}-{\displaystyle \frac{3}{2}}{V}_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{sg}})N(s))/{\omega }_n,f=f_p-f_1\\ \Delta E_m[f]={\displaystyle \frac{3}{2}}{H}_q((j-a^g)V_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{g}}+(-j-a^g)I_1^g{e}^{-j{\phi }_{ip}^g}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}),f=f_p-f_1\end{array}\right.$$

(27)

In the subsequent modeling process, as mentioned earlier, the PCC current should be replaced by the SVG output current. The small signal modulation of phase A voltage is simplified as

$$\Delta e_a=\Delta E_m\cos \theta -E_m\Delta \theta \sin \theta$$

(28)

Substituting (26) and (27) into (28), we obtain:

$$\begin{array}{l}\Delta e_a[f]=0.75((j-a^g)V_1^{pcc}({\mathit{I}}_{\mathit{p}}^{\mathit{sg}}-{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}}/Z_W)-(j+a^g)I_1^g{e}^{-j{\phi }_{ig}}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}})e^{j{\phi }_{vir}}{H}_q^{sg}(s-j2\pi f_1)-\\ 0.75j{E}_m{e}^{j{\phi }_{vir}}N(s-j2\pi f_1)(V_1^{pcc}{\mathit{I}}_{\mathit{p}}^{\mathit{sg}}+I_1^{sg}{e}^{-j{\phi }_{isg}}{\mathit{V}}_{\mathit{p}}^{\mathit{pcc}})/{\omega }_n,f=f_p\end{array}$$

(29)

where ${\phi }_{vir}=\arcsin (P_{ref}^{sg}{\omega }_n{L}_{sg}/E_m{V}_1^{pcc})$ represents the power angle of the GFM SVG.

After substituting (29) into (19), the expression for the positive sequence impedance of the GFM SVG can be obtained:

$$Z_{SVG}={\displaystyle \frac{s{L}_{sg}-K_m{V}_{dc}^{sg}0.75e^{j{\phi }_{vir}}{V}_1^{pcc}((j-a^g)H_q^{sg}(s-j2\pi f_1)-\frac{j{E}_{m}N(s-j2\pi f_1)}{{\omega }_n})}{1+K_m{V}_{dc}^{sg}0.75e^{j{\phi }_{vir}}((\frac{(j-a^g)V_1^{pcc}}{Z_W}+(j+a^g)I_1^g{e}^{-j{\phi }_{ig}})H_q^{sg}(s-j2\pi f_1)+\frac{j{E}_m{I}_1^{sg}{e}^{-j{\phi }_{isg}}N(s-j2\pi f_1)}{{\omega }_n})}}$$

(30)

4.3. Verification of Impedance Model

To verify the accuracy of two SVG impedance models, an impedance scanning method was employed to scan the electromagnetic simulation model of the system. For GFM control, according to (31), J = 118 and D = 35,431 were obtained based on the system parameters in

Table 1. SVG parameters.

Parameter	Value
Filter inductance (mH)	14
Grid-connected voltage (kV)	35
Proportional/integral coefficient of the current controller	15/300
Proportional/integral coefficient of power controller	0.4/10
Virtual inertia	118
Damping coefficient	35,431
GFM SVG proportional/integral coefficient of power controller	0.1/1

.

$$\left\{\begin{array}{c}J={\displaystyle \frac{3E_0{U}_G}{X{\omega }_0{\omega }_n^2}}\\ D=2\zeta \sqrt{{\displaystyle \frac{3E_0{U}_{G}J}{X{\omega }_0}}}\end{array}\right.$$

(31)

where X represents SVG outlet inductance while $\zeta$ represents the damping ratio of the power synchronous loop. A comparison between theoretical modeling and frequency scanning results is plotted in Figure 7. Analysis of Figure 7 reveals that both theoretical impedance models for GFL and GFM SVGs match well with actual impedance scanning results. The GFM SVG mainly exhibits an inductive impedance at high-frequency bands, which is consistent with grid impedance characteristics; whereas the GFL SVG exhibits a capacitive impedance at high-frequency bands and is susceptible to resonance when interacting with an inductive grid.

The GFM SVG adopts power synchronous control, which manifests externally as a voltage source characteristic and possesses active voltage support capability. Due to its high-frequency impedance characteristics being inductive, it reduces the equivalent impedance of the grid and enhances systems’ robustness, making it more suitable for application in high-penetration new energy generation systems.

5. Analysis of the Influence of GFL SVG on High-Frequency Resonance in Wind Farm

5.1. Analysis of High-Frequency Resonance Mechanism in Wind Farm

When analyzing stability issues using the impedance analysis method, the power grid is usually equivalent to an ideal voltage source in series with impedance, and DFIG and SVG are Norton equivalent. According to the system’s structure shown in Figure 1, the impedance equivalent circuit of the wind farm is shown in Figure 8, where Z_g represents the grid impedance, Z_line represents the cables’ impedance, Z_s represents SVG’s impedance, V_g represents grid voltage, I_w represents the equivalent current source on the wind farm side, and I_sg represents the equivalent current source on the SVG side.

In Figure 8, the output current I of the DFIG-based wind farm can be expressed as

$$I(s)=\left[I_{sg}(s)+I_W(s)-{\displaystyle \frac{V_g(s)}{Z_o(s)}}\right]{\displaystyle \frac{1}{1+Z_g(s)/Z_o(s)}}$$

(32)

Z_o is the system impedance of the wind farm:

$$Z_o(s)=Z_W(s)//Z_{SVG}(s)//Z_{line}(s)$$

(33)

According to (30), the stability of Z_g(s)/Z_o(s) can represent the stability of the DFIG-based wind farm system. When there is a frequency f_c at which an intersection occurs between the grid impedance and the DFIG-based wind farm’s impedance, the phase margin at this intersection can be represented as

$${\delta }_{PM}=180-(\arg (Z_g(j2\pi f_c))-\arg (Z_o(j2\pi f_c)))$$

(34)

Based on the above equation, analyze whether the impedance of the grid-connected inverter system satisfies the Nyquist stability criterion, observe the phase margin characteristics at frequency intersection points, and analyze whether there is a risk of resonance in the system.

Combining (18), (24), (33) and Table 1,

Table 2. Parameters of wind transformer and cable line.

Parameter	Value
Primary side resistance (Ω)	6.3477
Secondary side resistance (Ω)	0.17653
Primary side leakage inductance (H)	0.0008223
Secondary side leakage inductance (H))	2.09 × 10−5
Cable length (km)	5
Cable inductance (mH/km)	0.297
Cable capacitance (μF/km)	0.14

and

Table 3. Parameters of the DFIG grid-connected system.

Parameter	Value
wind speed (m/s)	6
DFIG number	10
Rated voltage (V)	690
Rated power of a single DFIG (MW)	1.5
Excitation mutual inductance (mH)	2.6
Stator leakage inductance (mH)	0.16367
Rotor leakage inductance (mH)	0.14549
GSC filter inductance (mH)	0.272
Rotor resistance (mΩ)	4.6
Stator resistance (mΩ)	6.6
DC bus capacitor voltage (V)	1150

, the BODE diagram of the impedance Z_o(s) for the DFIG-based wind farm is shown in Figure 9, where Z_g represents the grid impedance as 0.017 H and SVG adopts GFL control. By analyzing Figure 9, it can be seen that when unloaded cables are not connected, there is no intersection between the DFIG-based wind farm and grid impedance, and the phase margin remains positive, which indicates no risk of resonance. However, when unloaded cables are connected, there is an intersection between the DFIG-based wind farm and grid impedance at 1158 Hz with a phase margin of −3°. It indicates that the wind farm exhibits negative damping characteristics around 1158 Hz and is at risk of resonance.

5.2. The Influence of Voltage Feedforward in GFL SVG on System Stability

Grid voltage feedforward can achieve fast regulation as the load changes, reduce the distortion of SVG output current caused by grid voltage distortion and minimize the impact current during startup. However, introducing a voltage feedforward loop will affect the stability of the converter [30], especially in the frequency band above 200 Hz, where the voltage feedforward coefficient has a significant impact on the high-frequency damping characteristics of the system [31].

Figure 10 shows the impedance characteristics of wind farms with SVG voltage feedforward control. Figure 10 reveals that when the feedforward coefficient K_f is 1, the voltage feedforward loop provides a path for disturbance propagation, resulting in a negative damping region at the high-frequency band and reducing the phase margin of the system. It increases the risk of resonance. As the feedforward coefficient decreases, the DFIG-based wind farm system gradually improves the impedance phase margin and reduces the negative damping region. Consequently, there is a significant decrease in risk for system resonance.

6. The Suppression Effect of GFM SVG on High-Frequency Resonance in Wind Farm

6.1. Resonance Suppression Mechanism of GFM SVG for Wind Farm

According to the analysis in Section 5.2, although reducing the voltage feedforward coefficient of GFL SVG can decrease the risk of system resonance, it also reduces the response speed of dynamic reactive power compensation. Therefore, the engineering practicality of this method needs to be discussed. Compared to GFL control, the converter adopts GFM control, which not only avoids the impact of PLL on system operation [19] but also has the ability to actively provide voltage support, making it more suitable for weak grids. Therefore, this section will further analyze the suppression mechanism of GFM SVG on high-frequency resonance in the wind farm.

Based on the sequential impedance model of GFM SVG established in Section 4.2, combined with (18) and (33), the BODE diagram of the impedance of the wind farm with GFM SVG is plotted in Figure 11. In Figure 11, there are two intersections between the wind farm impedance and the grid impedance after the unloaded cable is switched on, and the intersection frequency is 681 Hz and 1680 Hz, respectively. Near 681 Hz, the impedance of wind power with GFM SVG is inductive, and will not resonate with the grid. The intersection point at 1680 Hz has a phase margin of 2°, while the wind farm with GFL SVG exhibits negative damping characteristics at 1158 Hz, which indicates that the SVG with GFM control can inhibit the high-frequency resonance to a certain extent.

From the analysis in Section 5.2, it can be seen that the voltage feedforward link of GFL SVG introduces voltage disturbances into the control loop, thereby increasing the negative damping region of the system. In contrast, with the adoption of GFM control by SVG, its modulating waveform is calculated from the output of both the power synchronization loop and reactive voltage loop, omitting the grid voltage feedforward link, and then eliminating its negative impact on the system stability. At the same time, compared to the GFL control, the GFM control makes the equivalent output impedance of SVG basically inductive and consistent with the grid impedance characteristics, which increases the phase margin of the high-frequency equivalent impedance of the wind farm, and thus reduces the resonant risk.

6.2. Influence of SVG Control Parameters on the System Stability

The essence of the GFM SVG is to simulate the mechanical and electromagnetic parts of the synchronous generator, enabling the SVG to possess the inertial and damping characteristics of a synchronous generator. In actual field conditions, the virtual inertia J and the virtual damping coefficient D may be adjusted according to changes in wind turbine and wind speed, so this section will analyze the impact of the power synchronous loop parameter on the system stability. Figure 12 and Figure 13 show the system impedance BODE diagrams when the inertia constant J and damping coefficient D are changed, respectively.

By analyzing Figure 12 and Figure 13, it can be seen that when J decreases, the system will obtain a larger phase margin, and a larger D will result in a smoother impedance curve. Therefore, when designing the parameters of GFM SVG, both the impedance characteristics and dynamic reactive power compensation effect should be comprehensively considered within a reasonable band. It can be found that although changes in J and D will affect the impedance characteristics of the system, these occur in the frequency band within 100 Hz and have no effect on the impedance characteristics of the high-frequency band concerned in this paper.

7. Experimental Verification

Based on the StarSim-HIL MT 3200 experimental platform (the experimental details can be found in the Appendix A), the DFIG-based wind farm electromagnetic model is constructed as shown in Figure 1 (main parameters are shown in Table 1, Table 2 and Table 3). The SVG adopts the GFL and the GFM control, respectively, to compensate reactive power of the wind farm, and an unloaded cable is switched on at t₁.

7.1. GFL SVG

The SVG adopts the GFL control, and the wind farm operates stably before t₁. After the unloaded cable is switched on at t₁, high-frequency resonance occurs. The experimental results are shown in Figure 14. From Figure 14, it can be observed that the unloaded cable results in harmonic distortion of both the 35 kV bus voltage and current. The current flowing into the grid is analyzed by Fourier decomposition as shown in Figure 15. In Figure 15, after resonance occurs at t₁, 1160 Hz harmonics appear in the grid-side current, which is basically consistent with the resonant frequency of 1158 Hz of the DFIG-based wind farm in the theoretical analysis of Section 5.1. However, due to the asymmetry of the dq axis controller, there is a mirror frequency effect in the control system, causing the system to couple a mirror frequency with a difference of 100 Hz, which is 1060 Hz, on the basis of the original resonant frequency [32].

7.2. GFM SVG

Based on the experimental results in Figure 14, the SVG switches from GFL control to GFM control at t₂. The experimental results shown in Figure 16, when GFM control is enabled, the 35 kV bus voltage and the grid-side current return to normal after a transition period of approximately 0.5 s, and the resonance phenomenon of the DFIG-based wind farm disappears. The Fourier decomposition of the grid-side current is further performed in Figure 17. The THD of the grid-side current decreases from 116.56% during resonance to 1.11% in stability, which is consistent with the conclusion in Section 6. In Section 6.1, theoretical analysis indicates that GFM SVG is conducive to improving the impedance phase margin of wind farms and suppressing high-frequency resonance.

Based on the theoretical analysis in Section 6.2, the parameter change in the power synchronization loop of GFM SVG does not affect the impedance characteristics of the system in the high-frequency band. To prove its correctness, the virtual inertia coefficient J and the virtual damping coefficient D are, respectively, changed to repeat the above experiment. The dynamic effect of reactive power compensation is shown in Figure 18 and Figure 19. By analyzing Figure 18 and Figure 19, it can be seen that as J decreases, there is a gradual increase in the fluctuation of reactive power output by SVG, while a decrease in D prolongs the adjustment time for reactive power by GFM SVG. However, both changes do not affect the suppression effect of resonance, but only affect the dynamic effect of reactive power compensation, which is consistent with the above theoretical analysis results.

8. Conclusions

In this paper, the harmonic linearization method is adopted to establish the sequence impedance model of a DFIG-based wind farm with SVG, and the influence of SVG’s GFL and GFM control on the operation stability of wind farm is studied from the perspective of impedance. Through theoretical analysis and simulation verification, the following conclusions are obtained:

• The voltage feedforward loop of GFL SVG directly affects the impedance characteristics of the wind farm system. With an increase in the feedforward coefficient, the impedance of the system changes from positive damping characteristics to negative damping characteristics in the high-frequency band, increasing the resonant risk of the system.
• When the GFL SVG adopts a fixed power factor control mode, under the dual effects of cable-to-ground capacitance and capacitive impedance of the GFL SVG in the high-frequency band, the unloaded cable in the wind farm will further reduce the high-frequency impedance phase margin of the system, resulting in the lack of phase margin of the system, which ultimately triggering high-frequency resonance.
• Due to the omission of voltage feedforward in GFM SVG, it is evaded that resonance signal injection into the control system caused by voltage feedforward, so that the overall impedance of the system shows positive damping characteristics in the high-frequency band, and the resonance of the wind farm is suppressed. At the same time, within a reasonable band, changes in parameters in the power synchronization loop in the GFM only affect dynamic reactive power compensation of the SVG and do not affect its suppression of high-frequency resonance in the wind farm.