Improvement of Frequency Support for a DFIG Using a Virtual Synchronous Generator Strategy at Large Power Angles

Abstract

The frequency regulation rate and operation stability of a doubly fed induction generator (DFIG) based on a virtual synchronous generator (VSG) strategy decreases under large-power-angle conditions, which reduces the grid frequency support capacity. This paper proposes the compound adaptive parameter (CAP) and coordinated primary frequency regulation (CPFR) strategies to improve the grid frequency support capacity in terms of multiple dimensions of the transient properties, operation condition range, and regulation duration. Mathematical and small signal models of the DFIG-VSG system are constructed. The effect of large-power-angle conditions on the transient properties under grid frequency perturbations is analyzed based on these models, and the CAP strategy for excitation control and virtual damping is formulated. The constraints of the rotor kinetic energy and the load increase capacity of the grid-side converter are analyzed, and the CPFR strategy is formulated based on this. Finally, the effectiveness of the proposed strategies is verified via simulations of single-machine and wind farm scenarios under grid frequency perturbation.

1. Introduction

Because of the increase in the penetration of large-scale wind power in power systems, the grid structure has gradually exhibited renewable energy and power electronic features, thereby reducing the contribution of synchronous generators to power generation [1]. A synchronous generator system has a high rotating kinetic energy and large boiler energy storage; therefore, reducing its contribution weakens the frequency support capacity of the grid and affects the safety of grid operation [2,3].

Improving the grid frequency support capability of wind generators is key to solving this problem, and there are two main existing research areas. The first research area focuses on the method used to achieve grid frequency support. The second research area focuses on energy reserve control for primary frequency regulation (PFR) [4].

Previously, the implementation methods of frequency inertia and frequency regulation based on the current source type have been proposed. The study reported in [5] adjusted the output power reference value based on the frequency change rate to simulate the frequency inertia characteristics. The study in [6] used frequency droop control to adjust the output power reference value to simulate primary frequency regulation (PFR). The study in [7] combined these two methods. These methods use proportional or differential control to regulate the active power reference value to support the grid frequency. Power control typically relies on a voltage or magnetic chain direction based on a phase-locked loop to detect the grid phase. However, the phase-locked loop may lead to negative system damping, resulting in resonance [8].

The studies in [9,10,11] introduced the virtual synchronous generator (VSG) control method, which simulates synchronous generators using synchronous generator rotor motion and excitation equations. The VSG strategy enables the device to operate as a voltage source, supports automatic synchronization with the grid frequency, and does not require the use of phase-locked loops. The VSG strategies in [9,10] were implemented for grid-connected inverters in permanent magnet wind generator and photovoltaic scenarios. The capacitor in the inverter DC bus decouples the VSG from the power source. The study in [11] developed a VSG strategy for a doubly fed induction generator (DFIG) based on rotor excitation control. Therefore, the control of a DFIG under the VSG strategy (DFIG-VSG) is close to that of a synchronous generator. The study in [12] reported an independent flexible link (IFL) strategy for DFIG-VSG, which makes up for the deficiency of the fixed parameters of the VSG strategy and improves the system frequency stability. In the DFIG-VSG, a power transfer model is usually constructed based on the excitation voltage, grid voltage, and stator inductance [13]. The power, grid voltage, and stator inductance determine the power angle for the transfer model. In actual operation scenarios, the power angle is generally large, which leads to active and reactive power coupling. However, the effect on the system properties and corresponding improved strategies at large power angles have not been investigated.

The energy reserve methods of a DFIG for primary frequency regulation are divided into two categories: load shedding operation and additional energy storage devices. The study in [14] developed an operation method with an increased pitch angle, and the energy for PFR was obtained by reducing the pitch angle. The study in [15] developed an improved maximum power point tracking (MPPT) method with increased rotor velocity to realize power reserve control, which has a higher probability density and lower mechanical losses than the pitch angle reserve method. In both methods, the wind turbine was not operated in the MPPT mode, which reduced the economic efficiency of the generator unit.

The studies in [16,17] developed the use of parallel energy storage devices at the AC bus to achieve PFR in cooperation with DFIG. This topology requires an additional set of inverters and step-up transformers, which is complex and costly and more suitable for centralized wind power scenarios. The study in [18] developed a scheme to connect lithium-ion supercapacitors directly to the DC bus, whereas the studies in [19,20,21] formulated a scheme to connect supercapacitors to the DC bus through a DC–DC converter. These schemes are simple and suitable for centralized and decentralized wind power scenarios. However, they use the remaining capacity of the grid-side converter (GSC) to provide frequency regulation power; therefore, the GSC limits the PFR capacity. In terms of the control strategy, the study in [18] proposed a synthetic inertia control (SIC) strategy with DC voltage control to simulate inertia. The study in [19] used differential control to achieve frequency inertia and a linear active disturbance rejection control strategy to improve the DC voltage stability. The study in [20] developed a strategy that uses rotor kinetic energy to realize frequency inertia and energy storage to realize PFR. The study in [21] reported a layered frequency regulation strategy to improve wind energy capture, which dynamically assigned the inertia control task to the DFIG DC capacitor and rotor speed inertia. However, these strategies must fully consider the GSC capacity limitation in PFR operation.

This study aims to improve the grid frequency support capability for DFIG-VSG under large-power-angle conditions. The main contributions of this study are summarized below.

(1)

This study analyzed the factors that lead to large-power-angle conditions and their effects on the frequency response of a DFIG-VSG. The mechanism of this effect was examined from the perspective of the transfer function by constructing a small-signal model.

(2)

A composite adaptive parameter (CAP) strategy, including excitation parameter adaptation and damping parameter adaptation, is proposed to improve the transient frequency response properties of a DFIG-VSG. This strategy enables the adaptation of the control parameters to the power angle state, thereby improving the power response rate and reducing transient disturbances and oscillations.

(3)

A coordinated primary frequency regulation (CPFR) strategy is proposed based on the complementary relationship between the rotor kinetic energy and the GSC surplus capacity to solve the problem of GSC capacity limitation on frequency regulation power. This strategy improves the frequency regulation capacity of a DFIG-VSG while avoiding GSC overload.

The rest of this study is organized as follows. Section 2 presents the structure of the DFIG-VSG system and the causes of large-power-angle conditions. Section 3 constructs a small-signal model and analyzes the effect of large-power-angles on the transient properties. Section 4 develops the CAP and CPFR strategies and introduces the theory. Case studies and simulation results are presented in Section 5. Section 6 concludes this study.

2. DFIG-VSG Wind Power System and Large-Power-Angle Operating Condition

2.1. Mathematical Model and Control Strategy of a DFIG-VSG Wind Power System

The stator of a DFIG connects to the grid and the rotor to the excitation controller, which is similar to those of a SG. This structure is the basis of simulating the mechanical and electrical properties of a SG on a DFIG. The rotor-side converter (RSC) control strategy based on the models of a DFIG and a SG is critical to achieving the VSG operation method for a DFIG.

The stator follows the generator convention In a DFIG model, and the rotor follows the motor convention. The equivalent circuit in a dq synchronously rotating reference frame is shown in Figure 1a.

The magnetic linkage equation is:

$$\left[\begin{array}{c}{\mathsf{\Psi }}_{\mathrm{s}}\\ {\mathsf{\Psi }}_{\mathrm{r}}\end{array}\right]=\left[\begin{array}{cc}{\mathrm{L}}_{\mathrm{ls}}+{\mathrm{L}}_{\mathrm{m}}& {\mathrm{L}}_{\mathrm{m}}\\ {\mathrm{L}}_{\mathrm{m}}& {\mathrm{L}}_{\mathrm{lr}}+{\mathrm{L}}_{\mathrm{m}}\end{array}\right]\left[\begin{array}{c}-{\mathrm{I}}_{\mathrm{s}}\\ {\mathrm{I}}_{\mathrm{r}}\end{array}\right]$$

(1)

here, ${\mathsf{\Psi }}_{\mathrm{s}}$ and ${\mathsf{\Psi }}_{\mathrm{r}}$ are the stator and rotor magnetic linkage, respectively; I_s and I_r are the stator and rotor current, respectively; L_ls and L_lr are the stator and rotor leaky inductance, respectively; L_m is the mutual inductance.

The voltage equation is given by the magnetic linkage derivative calculation.

$$\left[\begin{array}{c}{\mathrm{U}}_{\mathrm{s}}\\ {\mathrm{U}}_{\mathrm{r}}\end{array}\right]=p\left[\begin{array}{c}{\mathsf{\Psi }}_{\mathrm{s}}\\ {\mathsf{\Psi }}_{\mathrm{r}}\end{array}\right]+\left[\begin{array}{cc}{\mathrm{j}\mathsf{\omega }}_{\mathrm{s}}& 0\\ 0& {\mathrm{j}(\mathsf{\omega }}_{\mathrm{s}}-{\mathsf{\omega }}_{\mathrm{r}})\end{array}\right]\left[\begin{array}{c}{\mathsf{\Psi }}_{\mathrm{s}}\\ {\mathsf{\Psi }}_{\mathrm{r}}\end{array}\right]+\left[\begin{array}{cc}{\mathrm{R}}_{\mathrm{s}}& 0\\ 0& {\mathrm{R}}_{\mathrm{r}}\end{array}\right]\left[\begin{array}{c}-{\mathrm{I}}_{\mathrm{s}}\\ {\mathrm{I}}_{\mathrm{r}}\end{array}\right]$$

(2)

here, U_s and U_r are the stator and rotor voltage, respectively; p is derivative calculation; ω_s and ω_r are the power grid and rotor speeds, respectively; R_s and R_r are the stator and rotor resistance, respectively.

The excitation voltage E is defined as

$$\mathrm{E}={\mathrm{U}}_{\mathrm{s}}{|}_{{\mathrm{I}}_{\mathrm{s}}=0}={\mathrm{pL}}_{\mathrm{m}}{\mathrm{I}}_{\mathrm{r}}+{\mathrm{j}\mathsf{\omega }}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{m}}{\mathrm{I}}_{\mathrm{r}}$$

(3)

Therefore, the relation between the excitation voltage and the stator voltage is

$$\mathrm{E}={\mathrm{U}}_{\mathrm{s}}+{\mathrm{pL}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{s}}+{\mathrm{j}\mathsf{\omega }}_{\mathrm{s}}{\mathrm{L}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{s}}+{\mathrm{R}}_{\mathrm{s}}{\mathrm{I}}_{\mathrm{s}}$$

(4)

here, ${\mathrm{L}}_{\mathrm{s}}={\mathrm{L}}_{\mathrm{ls}}+{\mathrm{L}}_{\mathrm{m}}$ is the stator inductance.

The vector relation between the excitation voltage and grid voltage is shown in Figure 1b. From the figure, the d-axis is in the orientation of the excitation voltage; the rotation velocity of E is ω_v, and that of U_s is ω_s; the power angle σ is between vector E and U_s.

The stator active/reactive powers are formulated based on a power transfer model.

$$\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{s}}=\frac{{\mathrm{U}}_{\mathrm{s}}}{\sqrt{{\mathrm{R}}_{\mathrm{s}}^2+{\mathrm{X}}_{\mathrm{s}}^2}}{\left(\mathrm{Ecos}\right(\mathsf{\theta }}_{\mathrm{z}}-\mathsf{\sigma })-{\mathrm{U}}_{\mathrm{s}}{\cos (\mathsf{\theta }}_{\mathrm{z}}\left)\right)\\ {\mathrm{Q}}_{\mathrm{s}}=\frac{{\mathrm{U}}_{\mathrm{s}}}{\sqrt{{\mathrm{R}}_{\mathrm{s}}^2+{\mathrm{X}}_{\mathrm{s}}^2}}{\left(\mathrm{Esin}\right(\mathsf{\theta }}_{\mathrm{z}}-\mathsf{\sigma })-{\mathrm{U}}_{\mathrm{s}}{\sin (\mathsf{\theta }}_{\mathrm{z}}\left)\right)\end{array}\right.$$

(5)

here, θ_z is the impedance angle. Because R_s is far smaller than X_s, θ_z approximates 90°, and Equation (5) is simplified as

$$\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{s}}=\frac{{\mathrm{EU}}_{\mathrm{s}}\sin \left(\mathsf{\sigma }\right)}{{\mathrm{X}}_{\mathrm{s}}} \hfill \\ {\mathrm{Q}}_{\mathrm{s}}=\frac{{\mathrm{EU}}_{\mathrm{s}}\cos \left(\mathsf{\sigma }\right)-{\mathrm{U}}_{\mathrm{s}}^2}{{\mathrm{X}}_{\mathrm{s}}} \hfill \end{array}\right.$$

(6)

Ulteriorly, when the power angle is small enough, sin(σ) = σ and cos(σ) = 1. The active power P_s is adjusted by power angle σ and reactive power Q_s by excitation voltage E.

The rotor motion equation in a SG is used to control the power angle as

$$\left\{\begin{array}{c}{\mathsf{\omega }}_{\mathrm{v}}=\int \left(\frac{{\mathrm{n}}_{\mathrm{p}}{(\mathrm{P}}_{\mathrm{m}}-{\mathrm{P}}_{\mathrm{g}})}{{\mathsf{\omega }}_{\mathrm{v}}}-{\mathrm{D}}_{\mathrm{p}}{(\mathsf{\omega }}_{\mathrm{v}}-{\mathsf{\omega }}_{\mathrm{s}})+{\mathrm{k}}_{\mathrm{f}}{(\mathsf{\omega }}_{\mathrm{v}}-{\mathsf{\omega }}_0)\right)\times \frac{1}{{\mathrm{J}}_{\mathrm{v}}}\mathrm{dt}\\ \mathsf{\sigma }=\int {(\mathsf{\omega }}_{\mathrm{v}}-{\mathsf{\omega }}_{\mathrm{s}})\mathrm{dt}\hfill \end{array}\right.$$

(7)

here, P_m is the input mechanical power; P_g is the DFIG unit power; k_f is the frequency regulation coefficient; ω₀ is the rated velocity; D_p is the virtual damping coefficient; J_v is the virtual moment of inertia.

The excitation PI controller is used to regulate the excitation voltage as

$${\mathrm{E}}^{*}={(\mathrm{k}}_{\mathrm{p}}+{\mathrm{k}}_{\mathrm{i}}ʃ{\left)\right(\mathrm{Q}}_{\mathrm{s}}^{*}-{\mathrm{k}}_{\mathrm{u}}{(\mathrm{U}}_{\mathrm{s}}-{\mathrm{U}}_0)-{\mathrm{Q}}_{\mathrm{s}})$$

(8)

here, E^* is the reference value of the excitation voltage; k_p and k_i are the PI controller parameters; Q_s^* is the reference value of the stator reactive power; k_u is the coefficient for voltage regulation; U₀ is the rated stator voltage.

The modulation voltage U_r* for the RSC is formulated based on inner-loop current control and the rotor voltage equation in Equation (2).

$$\left\{\begin{array}{c}{\mathrm{U}}_{\mathrm{rd}}^{*}={(\mathrm{k}}_{\mathrm{p}1}+{\mathrm{k}}_{\mathrm{i}1}\int {\left)\right(\mathrm{I}}_{\mathrm{rd}}^{*}-{\mathrm{I}}_{\mathrm{rd}})-{(\mathsf{\omega }}_{\mathrm{v}}-{\mathrm{n}}_{\mathrm{p}}{\mathsf{\omega }}_{\mathrm{m}}{\left)\right(\mathrm{L}}_{\mathrm{m}}{\mathrm{I}}_{\mathrm{sq}}+{\mathrm{L}}_{\mathrm{r}}{\mathrm{I}}_{\mathrm{rq}})\\ {\mathrm{U}}_{\mathrm{rq}}^{*}={(\mathrm{k}}_{\mathrm{p}1}+{\mathrm{k}}_{\mathrm{i}1}\int {\left)\right(\mathrm{I}}_{\mathrm{rq}}^{*}-{\mathrm{I}}_{\mathrm{rq}})+{(\mathsf{\omega }}_{\mathrm{v}}-{\mathrm{n}}_{\mathrm{p}}{\mathsf{\omega }}_{\mathsf{\Omega }}{\left)\right(\mathrm{L}}_{\mathrm{m}}{\mathrm{I}}_{\mathrm{sd}}+{\mathrm{L}}_{\mathrm{r}}{\mathrm{I}}_{\mathrm{rd}})\end{array}\right.$$

(9)

here, I_r^* is the reference value for current control, and it is calculated as

$$\left\{\begin{array}{c}{\mathrm{I}}_{\mathrm{rq}}^{*}=-\frac{{\mathrm{E}}^{*}}{{\mathsf{\omega }}_{\mathrm{v}}{\mathrm{L}}_{\mathrm{m}}}\\ {\mathrm{I}}_{\mathrm{rd}}^{*}=0 \hfill \end{array}\right.$$

(10)

The grid-side converter (GSC) usually operates in the power outer-loop and current inner-loop mode to maintain the DC bus voltage stability [22]. A supercapacitor energy storage system (ESS) is connected to the DC bus through a DC–DC converter, decoupling the power between the rotor and GSC.

Additionally, the improved strategies reported in Section IV cooperate with the conventional VSG strategy. The CAP strategy can adjust the parameters of the excitation PI controller and virtual damping; the CPFR strategy can adjust the operation modes of the VSG, GSC, and ESS. Figure 2 shows the system topology and overall control diagram.

2.2. Cause and Effect of The Large-Power-Angle Conditions

From Equation (6), the power angle and excitation voltage under a stable state are

$$\left\{\begin{array}{c}\mathsf{\sigma }=\mathrm{atan}\left(\frac{{\mathrm{P}}_{\mathrm{s}}}{{\mathrm{Q}}_{\mathrm{s}}+{\mathrm{U}}_{\mathrm{s}}^2{/\mathrm{X}}_{\mathrm{s}}}\right)\\ \mathrm{E}=\frac{\sqrt{{{(\mathrm{P}}_{\mathrm{s}}{\mathrm{X}}_{\mathrm{s}})}^2+{{(\mathrm{Q}}_{\mathrm{s}}{\mathrm{X}}_{\mathrm{s}}+{\mathrm{U}}_{\mathrm{s}}^2)}^2}}{{\mathrm{U}}_{\mathrm{s}}}\end{array}\right.$$

(11)

In practical application scenarios, the rated power of a DFIG is at megawatt level, stator inductance at millihenry level, and stator voltage at kilovolt level. Then the power angle calculated from Equation (11) is large, and the previous approximation for sine and cosine functions does not hold.

The active power regulation per unit power angle is defined to quantify the effect of large-power-angle conditions on frequency regulation capability.

$${\left.\frac{\mathrm{\Delta }{\mathrm{P}}_{\mathrm{s}}}{\mathrm{\Delta }\mathsf{\sigma }}\right|}_{{\mathrm{E}}_0}=\frac{{\mathrm{E}}_0{\mathrm{U}}_{\mathrm{s}}\cos \left(\mathsf{\sigma }\right)}{{\mathrm{X}}_{\mathrm{s}}}$$

(12)

The active power regulation per unit power angle for different power-angle conditions is shown in Figure 3.

The power regulation is significant and stable when the power angle is small. In contrast, the power regulation is tiny and nonlinear at large power angles. When grid frequency perturbations occur, the small active power regulation leads to a slower active power increase rate and larger power angle fluctuation, which weakens the grid frequency support capacity and operation stability of a DFIG-VSG.

3. Small Signal Model of DFIG-VSG and Transient Response to Frequency Perturbation

The transfer function for the DFIG-VSG can reveal the theory of the transient response to the grid frequency. The small-signal model of a DFIG-VSG is constructed, considering the presence of nonlinear links in the system under large-power-angle conditions [23,24].

3.1. Small-Signal Model of a DFIG-VSG

The small-signal expression for the power equation in Equation (6) at the steady-state operating point (E₀, σ₀) is

$$\left\{\begin{array}{c}\mathrm{\Delta }{\mathrm{P}}_{\mathrm{s}}={\mathrm{k}}_0{\mathrm{k}}_1\mathrm{\Delta }\mathrm{E}+{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{k}}_2\mathrm{\Delta }\mathsf{\sigma }\\ \mathrm{\Delta }{\mathrm{Q}}_{\mathrm{s}}={\mathrm{k}}_0{\mathrm{k}}_2\mathrm{\Delta }\mathrm{E}-{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{k}}_1\mathrm{\Delta }\mathsf{\sigma }\end{array}\right.$$

(13)

here, ${\mathrm{k}}_0=U_{\mathrm{s}}{/X}_{\mathrm{s}}$, ${\mathrm{k}}_1={\sin (\sigma }_0)$, ${\mathrm{k}}_2{=\cos (\sigma }_0)$. The rotor motion equation and excitation control are linear, so their small-signal expressions are consistent with Equations (8) and (9).

Because the time constant of the current inner-loop control is much smaller than that of the outer loop, the transfer function of the inner loop approximates to 1. The small-signal model of a DFIG-VSG is construed in Figure 4.

From Figure 4, the grid frequency perturbation Δω_g results in the power angle perturbation Δσ, which affects both the active and reactive power feedback. The rotor excitation control adjusts the excitation voltage ΔE according to the reactive power feedback, which further affects the active power feedback. Therefore, the model can be analyzed in two ways.

In terms of the rotor excitation control, the transfer function of the power angle Δσ to the excitation voltage ΔE is

$${\mathrm{G}}_{\mathsf{\sigma }\mathrm{E}}\left(\mathrm{s}\right)=\frac{\mathrm{\Delta }\mathrm{E}\left(\mathrm{s}\right)}{\mathrm{\Delta }\mathsf{\sigma }\left(\mathrm{s}\right)}=\frac{{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{k}}_1{(\mathrm{k}}_{\mathrm{p}}\mathrm{s}+{\mathrm{k}}_{\mathrm{i}})}{(1+{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{p}})\mathrm{s}+{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{i}}}=\frac{{\mathrm{E}}_0{\mathrm{k}}_1}{{\mathrm{k}}_2}(\frac{\mathrm{As}+1}{\mathrm{Bs}+1})$$

(14)

Here,

$$\left\{\begin{array}{c}\mathrm{A}=\frac{{\mathrm{k}}_{\mathrm{p}}}{{\mathrm{k}}_{\mathrm{i}}}\hfill \\ \mathrm{B}=\frac{{1+\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{p}}}{{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{i}}}\end{array}\right.$$

(15)

Because

$$\frac{\mathrm{A}}{\mathrm{B}}=\frac{{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{p}}}{1+{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{p}}}<1$$

(16)

the transfer function G_σE(s) corresponds to a lag system. Putting Equation (15) into Equation (13), the expression of the active power feedback is calculated as

$$\mathrm{\Delta }{\mathrm{P}}_{\mathrm{s}}=\left(\frac{{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{k}}_1^2}{{\mathrm{k}}_2}\left(\frac{\mathrm{As}+1}{\mathrm{Bs}+1}\right)+{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{k}}_2\right)\mathrm{\Delta }\mathsf{\sigma }$$

(17)

In terms of the whole model, the closed-loop transfer function of the grid frequency perturbation Δω_g to power angle Δσ is

$${\mathrm{G}}_{\mathsf{\omega }\mathsf{\sigma }}\left(\mathrm{s}\right)=\frac{\mathrm{\Delta }\mathsf{\sigma }\left(\mathrm{s}\right)}{\mathrm{\Delta }{\mathsf{\omega }}_{\mathrm{g}}\left(\mathrm{s}\right)}=-\frac{1}{{\mathrm{s}}^2+\frac{{\mathrm{D}}_{\mathrm{p}}}{{\mathrm{J}}_{\mathrm{v}}}\mathrm{s}+\frac{{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{n}}_{\mathrm{p}}}{{\mathsf{\omega }}_{\mathrm{e}}}\left({\mathrm{k}}_2+\frac{{\mathrm{k}}_1^2}{{\mathrm{k}}_2}\left(\frac{\mathrm{As}+1}{\mathrm{Bs}+1}\right)\right)}$$

(18)

3.2. Analysis of Transient Properties of a DFIG-VSG under Grid Frequency Perturbations

The root locus and damping ratio of the transfer function G_ωσ(s) during the power angle from 5 to 85° are plotted in Figure 5a,b. The operating state of the DFIG-VSG at small power angles can be a benchmark for comparing operating states at different power angles. The figures show that the variation of the transient properties with the power angle has two phases.

When the power angle is small, k₁ is small, and k₂ is large. Because the coefficient $\frac{k_1^2}{k_2}$ of $\frac{\mathit{As}+1}{\mathit{Bs}+1}$ in Equation (18) is close to 0, this model is approximately a second-order system. As the power angle increases, only the constant term in the characteristic equation of G_ωσ(s) decreases. As a result, the real part of the root remains unchanged, while the absolute value of the imaginary part decreases. The root locus in Figure 5a is consistent with this. The decrease in the imaginary part leads to a slower response rate; the increase in the damping ratio in Figure 5b proves the change.

When the power angle increases to a certain degree, the coefficient $\frac{k_1^2}{k_2}$ cannot be taken as 0, and the lag component $\frac{\mathit{As}+1}{\mathit{Bs}+1}$ lifts the degree of the characteristic equation to cubic. The changes in the roots of the characteristic equation turn complicated. From Figure 5a, the poles move towards the imaginary axis as the power angle increases. The decrease in the absolute value of the real part leads to a large time constant of the system, which may result in oscillations in the system; the decrease in the damping ratio in Figure 5b proves this.

Analysis of the transfer function, root locus, and damping ratio reveals that the active power regulation rate and system stability of the DFIG-VSG are decreased under larger-power-angle conditions. The lag component is the leading cause of the deteriorated transient properties.

4. Compound Adaptive Parameter Strategy and Coordinated Primary Frequency Regulation Strategy for a DFIG-VSG

The CAP strategy for the PI coefficient of the rotor excitation controller and virtual damping coefficient can improve the transient properties of a DFIG-VSG at large power angles. The CPFR strategy coordinates the primary frequency regulation power of the stator VSG, GSC, and ESS to extend the range and duration of frequency regulation. The CAP and CPFR strategies improve the grid frequency support capacity of a DFIG-VSG under larger-power-angle conditions.

4.1. Adaptive Parameters for a Rotor Excitation Controller

From Section 3.2, reducing the effect of the lag component $\frac{\mathit{As}+1}{\mathit{Bs}+1}$ on the transfer function G_ωσ(s) can reduce the order of the system and make the transient response easy to control.

Reducing the time constant of the lag component and making it much smaller than that of the rotor motion system, the lag component can be considered as a proportional component at the time scale of the rotor motion system time constant. The adaptive parameters for the rotor excitation controller are developed in two steps based on this theory.

First, design an adaptive strategy for k_p and k_i to eliminate the effect of power angle variation on the time constant τ_l of the lag component.

The formula of the adaptive parameters is

$$\left\{\begin{array}{c}{\mathrm{k}}_{\mathrm{p}}=\frac{1}{{\mathrm{k}}_2}{\mathrm{k}}_{\mathrm{p}0}\\ {\mathrm{k}}_{\mathrm{i}}=\frac{1}{{\mathrm{k}}_2}{\mathrm{k}}_{\mathrm{i}0}\end{array}\right.$$

(19)

here, k_p0 and k_i0 are the initial parameters for the rotor excitation PI controller.

The time constant τ_l is

$${\mathsf{\tau }}_{\mathrm{l}}=\frac{1}{\mathrm{B}}=\frac{{1+\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{p}}}{{\mathrm{k}}_0{\mathrm{k}}_2{\mathrm{k}}_{\mathrm{i}}}=\frac{{1+\mathrm{k}}_0{\mathrm{k}}_{\mathrm{p}0}}{{\mathrm{k}}_0{\mathrm{k}}_{\mathrm{i}0}}$$

(20)

The time constant is decoupled with the power angle.

Second, optimize k_p0 and k_i0 to achieve the approximation of the lag component to a proportional component.

The constraint of the parameters is

$${\mathsf{\tau }}_{\mathrm{i}}\ll {\mathsf{\tau }}_{\mathrm{l}}\ll {\mathsf{\tau }}_{\mathsf{\sigma }}$$

(21)

here, τ_i is the time constant of the current inner-loop control; τ_σ is the time constant of the rotor motion system. The initial parameters are optimized as Equation (21), and the system’s physical parameters determine the time constants τ_i and τ_σ.

Under this strategy, the transient response of the lag component is neglected, and the steady state is taken as the response. Therefore, the transfer function of the lag component approximates 1, and the transfer function in Equation (18) can be formulated as

$${\mathrm{G}}_{\mathsf{\omega }\mathsf{\sigma }}\left(\mathrm{s}\right)\approx -\frac{1}{{\mathrm{s}}^2+\frac{{\mathrm{D}}_{\mathrm{p}}}{{\mathrm{J}}_{\mathrm{v}}}\mathrm{s}+\frac{{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{n}}_{\mathrm{p}}}{{\mathsf{\omega }}_{\mathrm{e}}{\mathrm{k}}_2}}$$

(22)

The real part of the roots of the characteristic equation remains constant at different power angles. The time constant of the system remains stable.

The root locus and damping ratio for this strategy are plotted in Figure 5c,d. Compared with Figure 5a,b, the poles avoid moving towards the imaginary axis, and the damping ratio improves.

4.2. Adaptive Parameters for Virtual Damping

The order of the system’s transfer function approximates to second-order under the adaptive parameter strategy for the excitation controller, and the damping ratio of the system is

$$\mathsf{\xi }=\frac{{\mathrm{D}}_{\mathrm{p}}}{2}\sqrt{\frac{{\mathsf{\omega }}_{\mathrm{e}}{\mathrm{k}}_2}{{\mathrm{J}}_{\mathrm{v}}{\mathrm{E}}_0{\mathrm{k}}_0{\mathrm{n}}_{\mathrm{p}}}}$$

(23)

Considering the effect of the changes in E₀ and k₂ on the damping ratio with different power angles, the adaptive virtual damping strategy is formulated as follows.

$${\mathrm{D}}_{\mathrm{p}}={\mathrm{D}}_{\mathrm{p}0}\sqrt{\frac{{\mathrm{E}}_0}{{\mathrm{k}}_2}}$$

(24)

here, D_p0 is the initial damping ratio.

The root locus and damping ratio for the compound adaptive parameter strategy are plotted in Figure 5e,f. The absolute values of the real and imaginary parts of the poles increase proportionally with the power angle, improving the response rate of the system; the damping ratio remains stable, which improves the system stability.

4.3. Analysis of Power Angle Properties under the CAP Strategy

The lag component $\frac{\mathit{As}+1}{\mathit{Bs}+1}$ approximates 1 under the CAP strategy. Then, the transfer function G_σE(s) in Equation (14) is simplified as

$${\mathrm{G}}_{\mathsf{\sigma }\mathrm{E}}\left(\mathrm{s}\right)=\frac{{\mathrm{E}}_0{\mathrm{k}}_1}{{\mathrm{k}}_2}$$

(25)

Plug $\mathsf{\Delta }E=G\mathrm{E}\mathsf{\sigma }\left(\mathrm{s}\right)\mathsf{\Delta }\sigma$ into the reactive power formula in Equation (13), and it turns $\mathsf{\Delta }\mathrm{Qs}=0$. Therefore, the stator reactive power Q_s steadily keeps at the rated value Q_n under frequency perturbation. In this condition, the power equation in Equation (7) is simplified as

$$\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{s}}=(\frac{{\mathrm{U}}_{\mathrm{s}}^2}{{\mathrm{X}}_{\mathrm{s}}}+{\mathrm{Q}}_{\mathrm{n}}\left)\tan \right(\mathsf{\sigma })\\ {\mathrm{Q}}_{\mathrm{s}}={\mathrm{Q}}_{\mathrm{n}}\hfill \end{array}\right.$$

(26)

The relations between the excitation voltage E, stator active power P_s and power angle σ for CAP strategy are shown in Figure 6.

Here, L₁ corresponds to the function P_s(E, σ), its projection L₂ to the function P_s(σ) and its projection L₃ to the function E(σ). L₂ shows that the stator active power varies with the power angle as a tangent function.

The power angle properties improve in two aspects with the CAP strategy.

• The amount and rate of the power regulation with a tangential power-angle property are larger and faster than those with a sinusoidal one.
• The reactive power remains robust when the power angle changes with the grid frequency perturbation.

4.4. Coordinated Primary Frequency Regulation Strategy for a DFIG-VSG

The CAP strategy improves the transient properties of a DFIG-VSG at grid frequency perturbations. To further improve the frequency support capability during the perturbations, the CPFR strategy is developed.

The rotor kinetic energy is the primary power reserve of a DFIG-VSG in frequency regulation. At high wind speeds, the rotor kinetic energy is abundant, and the stator and GSC power are almost at their rated values. The GSC power increases with the stator power during the frequency regulation operation, which causes the GSC to overload. At low wind speeds, the rotor velocity is nearly at its minimum, and little additional kinetic energy is released for frequency regulation. The analysis of the restriction is critical for developing the CPFR strategy.

The wind turbine control strategy determines the relation between wind speed and rotor velocity [13]. The stator and rotor active power is

$$\left\{\begin{array}{c}{\mathrm{P}}_{\mathrm{s}}=\frac{1}{1-{\mathrm{s}}_{\mathsf{\omega }}}{\mathrm{P}}_{\mathrm{g}}\\ {\mathrm{P}}_{\mathrm{r}}=-\frac{{\mathrm{s}}_{\mathsf{\omega }}}{1-{\mathrm{s}}_{\mathsf{\omega }}}{\mathrm{P}}_{\mathrm{g}}\end{array}\right.$$

(27)

here, s_ω is the slip rate; P_g is the generator unit power.

The capacity of the GSC for load-increase S_gsc is defined as

$${\mathrm{S}}_{\mathrm{gsc}}={\mathrm{P}}_{\mathrm{gscN}}-{\mathrm{P}}_{\mathrm{r}}$$

(28)

here, S_gscN is the GSC rated capacity.

The effective rotor kinetic energy for frequency regulation is

$${\mathrm{E}}_{\mathrm{k}}=0{.5\mathrm{J}}_{\mathrm{V}}{(\mathsf{\omega }}_{\mathrm{r}}^2-{\mathsf{\omega }}_{\min }^2)$$

(29)

From Equations (27)–(29), the relations between S_gsc, E_k, and rotor velocity are shown in Figure 7.

From Figure 7, the frequency regulation power is constrained by the GSC capacity for loading increase at high wind speeds and the rotor kinetic energy at low wind speeds. To remove these constraints, the CPFR strategy based on complementary properties is designed in Figure 8.

The CPFR strategy assigns operating methods for the stator VSG, GSC, and ES based on wind speed conditions and stages in frequency perturbation.

When grid frequency perturbations occur, the CPFR strategy assigns operating modes by different wind speeds. At high wind speeds, the stator VSG operates in both frequency inertial and PFR modes, GSC switches to constant power control, and ESS is used to maintain the DC bus voltage stability. At low wind speeds, the stator VSG operates in frequency inertial mode, GSC in PFR mode, and ESS in DC voltage control mode. The CPFR strategy eliminates the overload of the GSC at high wind speeds and remedies the insufficient rotor kinetic energy at low wind speeds.

When the grid frequency is restored, the stator VSG is in velocity recovery mode, GSC is in compensation control for stator load-shedding, and ESS remains in DC voltage control. The GSC compensation control reduces power loss during rotor velocity recovery.

When the grid frequency is normal, the GSC switches to DC voltage control, and the ESS is in state of charge (SOC) control.

5. Simulation Verification

To verify the improvement in the grid frequency support capacity for large-power-angle conditions using CAP and CPFR strategies, we constructed a single DFIG-VSG model and an integrated wind farm connected to an IEEE 9-node grid in MATLAB/Simulink 2020b. We formulated contrasting scenarios with different wind speeds and control strategies. The DFIG-VSG operation scenarios in Section 5.1 and Section 5.2 verify the validity and feasibility of the CAP strategy, and scenarios in Section 5.3 and Section 5.4 verify those of the CPFR strategy. Furthermore, grid-connected wind farm scenarios in Section 5.5 and Section 5.6 verify the improvement in the frequency support capability with the proposed strategy.

Table 1. Key Parameters of the DFIG-VSG System.

Parameter and Designation	Value
Rated voltage Us	950 V
Rated frequency fs	50 Hz
Generator unit rated power Pg	3.4 MW
Stator and rotor leakage inductance Lls/Llr	0.06 mH/0.07 mH
Mutual inductance Lm	1.2 mH
Converter DC voltage udc	1600 V
Supercapacitor unit capacitance and voltage	50 F/1100 V
Initial SOC of the supercapacitor unit	78.8%
Number of pole pairs np	2
Rated rotor speed rω	1782 r/min
Mechanical rotational inertia JΩ	750 Kg·m2
Virtual rotational inertia JV	150 Kg·m2
Frequency regulation coefficient kf	0.43 MW/0.25 Hz
Rotor excitation PI coefficients kp0 / ki0	0.002/0.012
Virtual damping coefficient Dp0	83

. lists the parameters of the system.

5.1. Inertial Operation of a DFIG-VSG at a Low Wind Speed

The active power and power angle are low at low wind speeds. Therefore, the operation at low wind speeds demonstrates the properties of a DFIG-VSG at low power angles. In this scenario, the grid frequency drops by 0.5 Hz from 0.5 to 1.5 s, and the primary frequency regulation is disabled. Figure 9 shows the resulting waveforms for the FPC and APC strategies.

The grid frequency f_s in (a) dropped to 49.5 Hz at 0.5 s; The accumulation of positive frequency difference increased the power angle σ in (b). The stator active power P_s in (d) increased with the power angle, and the maximum increase value was 0.42 MW. The excitation voltage E in (c) was adjusted with σ to maintain the stator reactive power Q_s in (e) stability. When the grid frequency recovered at 1.5 s, the variation rules for these variables were opposite to those at 0.5 s.

At low power angles, the transient properties of the DFIG-VSG system are approximately the same for both the fixed parameter control (FPC) and CAP strategies.

5.2. Inertial Operation of a DFIG-VSG at the Rated Wind Speed

This scenario demonstrates the operation properties of a DFIG-VSG at large power angles. The conditions are the same as those in Section 5.1, except for the wind speed. Figure 10 shows the resulting waves for the FPC and CAP strategies.

The variation rules of the variables are similar to those in Section 5.1. However, the transient processes are different. Compared with the transient properties of the FPC strategy, the advantages of the CAP strategy are twofold. First, the system responds quicker to grid frequency perturbation. The virtual synchronous frequency f_v in (a) traced the grid frequency quicker, and the stator power P_s in (d) reached the peak 70 ms in advance. Secondly, the transient fluctuations of the system are smaller. The power angle increment in (b) is reduced by 3.5°. Transient fluctuations in the power angle, excitation voltage, active power, and reactive power were eliminated.

At large power angles, the CAP strategy achieves a faster active power response and lower transient fluctuations, which improves the transient properties of the DFIG-VSG during grid frequency participation.

5.3. Grid Frequency Regulation Operation of a DFIG-VSG at a Low Wind Speed

The rotor velocity approaches its minimum at low wind speeds. Therefore, the DFIG-VSG cannot support the grid frequency without an energy storage device. In this scenario, the grid frequency drops by 0.5 Hz from 1 to 16 s, and the CAP strategy is enabled. Figure 11 shows the resulting waves with and without the CPFR strategy.

The primary frequency regulation of the stator VSG is disabled in this scenario with or without the CPFR strategy. Thus, the power angle σ in (b), excitation voltage E in (c), and stator active power P_s in (d) were similar to those in Section 5.1.

The GSC operation is a feature of the CPFR strategy. The GSC active power P_gsc in (e) increased to 0.18 MW from −0.22 MW during the grid frequency drop. The power of ESS P_ess in (f) was 0.4 MW, which balanced with the active power increase of the GSC. The SOC of supercapacitors in (i) gradually decreased with frequency regulation until the grid frequency was restored.

When the CPFR strategy was implemented, the maximum increase of the generator unit power P_g in (g) was 0.9 MW, and the primary frequency regulation power of the unit was 0.4 MW. The CPFR strategy improved the grid frequency support capacity of a DFIG-VSG at low wind speeds.

5.4. Grid Frequency Regulation Operation of a DFIG-VSG at the Rated Wind Speed

The conditions are the same as those in Section 5.3, except for the wind speed. Figure 12 shows the resulting waves with and without the CPFR strategy.

The primary frequency regulation of the stator VSG was enabled in this scenario, with and without the CPFR strategy. Therefore, the power angle σ in (b), excitation voltage E in (c), stator active power P_s in (d), and rotor velocity n_r in (h) were the same in these contrasting waveforms.

The GSC operation is a feature of the CPFR strategy. When a frequency sag occurred, the GSC active power P_gsc in (e) was kept at 0.6 MW and overload was avoided. As the rotor kinetic energy was released, the rotor velocity and rotor active power decreased. The energy storage device supported P_gsc to remain stable. When the grid frequency was restored, rotor velocity control resulted in a reduction in P_s. The GSC compensated for part of the power reduction, which reduced the power reduction of the generator unit. The rotor velocity n_r reduced to 1360 r/min with the frequency regulation operation, and the SOC of the ESS in (i) decreased by 46.5%.

When the CPFR strategy was implemented, the generator unit power P_g in (g) steadily increased by 0.4 MW for frequency regulation during grid frequency sag, and the minimum value of P_g increased by 1 MW during rotor velocity recovery. The CPFR strategy eliminates the overload in the GSC and power reduction of the generator unit in frequency regulation, which improves the grid frequency support capacity at high wind speeds.

5.5. Frequency Regulation Operation of a Wind Farm at a Low Wind Speed

To validate the effectiveness of the proposed strategies in a grid-connected wind farm scenario, we constructed a wind farm integrated with 90 wind generator units connected to an IEEE 9-node power grid; the topology is shown in Figure 13. The rated power of SG G₁ and G₂ is 500 MW. G₁ operates for grid frequency and voltage regulation, and its frequency regulation coefficient is 20 (50 MW/0.25 Hz). G₂ operates at the rated state. The loads L₁, L₂, and L₃ are 200 MW, 200 MW, and 400 MW, respectively. The perturbation load L_d is 60 MW.

At the low wind speed, the active power of the wind farm is 54 MW, and the wind power penetration rate is 6%. The perturbation load starts at 6 s. Figure 14 shows the resulting waves with and without the proposed strategies.

Compared to the conventional strategy, the wind farm active power P_wind in (b) increased by 30 MW at maximum in the proposed CAP and CPFR strategies, accounting for 57% of the wind farm power before the frequency failure. The lowest grid frequency in (a) increased from 49.76 Hz to 49.87 Hz, and the frequency drop was reduced by 50%. Charts (c–g) show the operation waveforms of a DFIG-VSG unit. The increased power in the wind farm is from energy storage devices.

The CAP and CPFR strategies improve the grid frequency support capacity of a wind farm at low wind speeds and low wind power penetration.

5.6. Frequency Regulation Operation of a Wind Farm at the Rated Wind Speed

At the rated wind speed, the active power of the wind farm is 310 MW, and the wind power penetration rate is 34%. The perturbation load starts at 6 s. Figure 15 shows the resulting waves with and without the proposed strategies.

Compared to the conventional strategy, the frequency regulation power in (b) was higher and more stable in the proposed CAP and CPFR strategies, and the minimum power of the wind farm improved by 30 MW during rotor velocity recovery. Therefore, the grid frequency in (a) recovered more quickly, and the secondary grid frequency drop was smaller. Charts (c–g) show the operation waveforms of a DFIG unit.

The CAP and CPFR strategies improve the grid frequency support capacity of wind farms at high wind speeds and high wind power penetration.

6. Conclusions

This study aims to improve the grid frequency support capacity of a DFIG-VSG at large power angles. Based on the analysis of the impact of large power angles on the grid frequency support, the CAP and CPFR strategies are proposed, corresponding to frequency inertial response and primary frequency regulation, separately. Simulations for single generator and wind farm scenarios under the proposed and conventional strategies validate the effectiveness. The main findings of this study are summarized below.

(1)

The impact of the lag link in the rotor excitation controller on the stator active power of a DFIG-VSG increases at large power angles, which turns the power-angle properties into a nonlinear form and changes the transfer function with the power angle. These changes reduce the active power response rate and system stability, thus reducing the grid frequency support capacity.

(2)

The CAP strategy adjusts the rotor excitation controller and virtual damping parameters based on the power angle, eliminating the transfer function change. As a result, the frequency inertia response of the DFIG-VSG maintains a stable and optimized state at different power angles.

(3)

The CPFR strategy dynamically allocates the GSC power to assist the DFIG-VSG in frequency regulation. It extends the operating range for primary frequency regulation and improves the stability of the regulation power. Moreover, the proposed strategy can also constrain the power drop during rotor velocity recovery and overload of the GSC at the start of frequency perturbation.

The proposed strategy is suitable for improving the grid frequency support capability of a DFIG in both single generator and centralized wind farm scenarios. The impact of the proposed strategies on the voltage disturbance response and analysis of the economy need further study.