A Stability Control Method to Maintain Synchronization Stability of Wind Generation under Weak Grid

Abstract

When wind generation systems operate under weak grid conditions, synchronization stability issues may arise, restricting the wind farms’ power transfer capacity. This paper aims to address these challenges on the grid side. Firstly, a clear exposition of the coupling mechanism between the grid-connected inverters (GCI) of wind generations and the weak grid is provided. Then, an equivalent parallel compensation method integrated into the PLL to enhance synchronization stability is proposed. The method changes the reference of the PLL and equivalently parallels the virtual resistance with the grid impedance, which alters the strength of the grid. It reshapes the inverter qq-axis impedance at the impedance level. And the proper design of the virtual resistance will enhance the system’s stability without compromising the dynamic performance of PLL. In addition, the proposed method is robust to the parameter changes of the grid-connected system and the grid impedance measurement error. Experimental results are presented to validate the effectiveness of the compensation method.

1. Introduction

The development of wind generation has gained significant momentum as a solution to environmental pollution in recent years. Given the decentralized nature of wind resources, power plants are often situated in remote locales. And it relies on GCI to access the conventional grid.

However, employing PLL as synchronization units in wind generation systems can potentially introduce synchronization instability problems, particularly under weak grid conditions [1]. The sub-synchronous resonance (SSR) was observed in the Xinjiang Type-IV wind generation plant [2].

The impedance-based stability criterion combined with the generalized Nyquist criterion (GNC) is extensively adopted to address the synchronization instability issues [3]. The dq-frame impedance model of GCI is established in [4], and the qq-axis negative resistance is considered as the fundamental factor contributing to synchronization instability. Ref. [5] analyzes the PLL feedforward effect on the control of GCI and concludes that the PLL current feedforward effect has a more adverse impact on system stability. Ref. [6] simplifies the MIMO impedance model to the SISO model to assess the impacts on the synchronization stability of PLL, current control, voltage feedforward, and DC voltage controller. Additionally, ref. [7] derives a SISO impedance model related to the synchronization loop, with PLL loop shaping employed to enhance stability. In [8], a d-axis current error is feedforward to current control to damp the resonance based on the steady-state relationship.

The stability improvement methods are proposed for both the motor side [9] and the grid side. On the grid side, the widely adopted method is a feedforward method to reshape the impedance characteristic [10].

Since the PLL is pivotal for synchronization, several studies have focused on modifying the PLL [11,12]. Recent efforts have predominantly focused on changing the PLL input. Ref. [13] enhances the passivity of a GCI by multiplying the output impedance phase regulator of normalized magnitude with the phase-locked loop inputs. Ref. [14] introduces a compensation frame to achieve current control by phase lag compensation, which is employed in PLL. By modifying the PLL input, ref. [15] introduces a series resistance in the grid impedance to increase damping in the studied system. However, this method entails a complex impedance model. Ref. [16] analogous to the back electromotive force observer in DC motor control, the input ofthe PLL is readjusted by state estimation.

Some of the literature achieves oscillation suppression by employing novel control methods. The literature [17] points out that the grid-forming control has more stability under weak grids compared to the grid-following control, but it will be a risk of instability under strong grids. Ref. [18] controls the GCI without adopting PLL implemented by the voltage-modulated direct power control. And it can also be utilized in weak grids by employing a band-pass filter [19]. The adaptive dual-mode control is suggested for oscillation suppression, i.e., current source mode under a weak grid and voltage source mode under a strong grid [20]. Ref. [21] proposes a grid voltage feedforward control strategy based on multi-objective objective constraint to broaden the adaptation range of the system to the grid impedance.

In addition, ref. [22] analyzes the stability range of PLL parameters under weak grid, finding suitable parameters by iterative adjustment method to enhance the power transfer capability. Ref. [23] concludes that the guideline that the PLL bandwidth should not exceed one-tenth of the current loop may not be applicable under weak grid. Ref. [24] proposes a control parameter design method considering the fluctuation of the grid impedance to improve the dynamic stability under weak grid. The supplementary damping controllers, such as TCSC, have also been adopted to damp the SSR of wind generations [25].

However, stability enhancement methods based on optimizing system parameters sacrifice the bandwidth of the PLL or current control, increasing the coupling between the current control and PLL. The supplementary damping controllers are limited in their promotion by their high cost and complex operation control modes. Therefore, optimizing the grid-side or motor-side control system for the studied system is more appropriate.

But the current study of the coupling mechanism of a grid-connected system is still incomplete, resulting in the existing stability control strategies mainly relying on feedforward compensation based on the q-axis voltage. And the compensation term is complicated. When the grid impedance changes and the parameter changes during the actual operation of the system, it brings new challenges for the stability control methods. Hence, there is a pressing need to introduce novel stability control methods derived from the coupling mechanism of grid-connected systems.

This article proposes an equivalent parallel resistance compensation method on the grid side employed in the PLL, which is based on comprehending the root of instability. The highlights are summarized as below:

(a)

A clear exposition of the coupling mechanism among the PLL, weak grid, and current control is provided, laying the groundwork for forthcoming stability enhancement techniques.

(b)

An equivalent parallel resistance compensation method integrated into PLL is proposed to improve synchronization stability. It reshapes the qq-axis impedance and will not decrease the dynamic performance of PLL with a proper design of virtual resistance.

(c)

The compensation method demonstrates robustness against system parameter variations and grid impedance measurement errors.

The subsequent sections are structured as follows: Section 2 derives the investigated system’s impedance model and analyzes the coupling mechanism between the PLL, weak grid, and current control. Section 3 presents the proposed equivalent parallel resistance compensation method, including its design and performance analysis. Section 4 conducts the experimental verification. Section 5 concludes and outlines future work.

2. Impedance Model and Stability Analysis of the Grid-Connected System

2.1. Description and Model of System

Because the dynamic of the DC capacitor is slow, the motor side and the grid side are decoupled in type-IV wind generation. Thus, the rotor side is represented by DC-source. The topology is shown in Figure 1. Filter inductor L_f is utilized to filter the GCI output voltage V_iabc to PCC voltage V_abc. By controlling the dq-component ($i_d^c,i_q^c$) of the current I_abc and synchronizing with the voltage of PCC, the grid-following control based on PLL can be achieved.

Table 1. System parameters.

Parameter	Values
DC-link voltage Vdc	400 V
Grid voltage RMS value Vg	130 V
Rated power Pn	2 kW
Filter inductors Lf	3 mH
Filter capacitors Cf	20 μF
Damping resistors Rf	10 Ω
Grid inductors Lg	9/16 mH
Switching frequency fsw	10 kHz
Sampling frequency fs	10 kHz
Proportional gain of Gi kpi	5.24
Integral gain of Gi kii	1370
Proportional gain of Gpll kppll	4.2
Integral gain of Gpll kipll	384

lists the introduction and the value of system parameters. SCR is usually used to define the grid strength, and the equation of SCR can be expressed as:

$$SCR={\displaystyle \frac{3V_g^2}{Z_{g}S}}$$

(1)

where V_g is the grid voltage magnitude, Z_g is the grid impedance, and S is the rated capacity of the grid-connected system. When SCR > 3, it is considered a strong grid; 2 < SCR < 3, it is considered a weak grid; and SCR < 2, it is considered a very weak grid.

The input of the PLL is V_abc rather than V_gabc under the weak grid case. This leads to distinct system frames and control frames. The dq-component of variables in the system and control frame are equal in steady state, namely ${\mathit{x}}_{dq}^s={\mathit{x}}_{dq}^c$, where the control frame and the system frame are designated with “c” and ”s”.

Figure 2a illustrates the control block of the GCI with PLL effects. The relationship between the current ∆i_dq and modulated voltage ∆e_dq in different frames is as follows [8]:

$$\Delta {\mathit{i}}_{dq}^c=\Delta {\mathit{i}}_{dq}^s+\underset{G_{pll}^i(s)}{\underbrace{\left[\begin{array}{cc}0& -I_{qref}\\ 0& I_{dref}\end{array}\right]}}{H}_{pll}(s)\Delta {\mathit{v}}_{dq}^s$$

(2)

$$\Delta {\mathit{e}}_{dq}^c=\Delta {\mathit{e}}_{dq}^s+\underset{G_{pll}^v(s)}{\underbrace{\left[\begin{array}{cc}0& -V_{iq}\\ 0& V_{id}\end{array}\right]}}{H}_{pll}(s)\Delta {\mathit{v}}_{dq}^s$$

(3)

${\mathit{G}}_{pll}^i(s)$ and ${\mathit{G}}_{pll}^v(s)$ represent the PLL feedforward effects on current and modulated voltage. I_dqref and V_idq denote the steady-state values [8]. H_pll(s) describes the relationship from the $\Delta v_q^s$ to the ∆θ:

$$H_{pll}(s)={\displaystyle \frac{G_{pll}(s)}{s+V_d^s{G}_{pll}(s)}}$$

(4)

$$G_{pll}(s)=k_{ppll}+k_{ipll}\cdot s^{-1}$$

(5)

G_i(s) represents the matrix of current control, G_dec(s) is the matrix of decoupling term, and I is the unit matrix.

$${\mathit{G}}_i(s)=Gi(s)\cdot \mathit{I}=\left(k_{pi}+k_{ii}\cdot s^{-1}\right)\cdot \mathit{I}$$

(6)

$${\mathit{G}}_{dec}(s)=\left[\begin{array}{cc}0& -\omega L_f\\ \omega L_f& 0\end{array}\right]$$

(7)

The impedance matrix Z_f corresponds to the inverter side filter inductance L_f and is expressed as:

$${\mathit{Z}}_f=\left[\begin{array}{cc}\mathrm{s}{L}_f& -\omega L_f\\ \omega L_f& \mathrm{s}{L}_f\end{array}\right]$$

(8)

The time delay matrix G_del(s) accounts for the digital control delay, with T_del = 1.5 T_s. It is defined as:

$${\mathit{G}}_{del}(s)=\left({\displaystyle \frac{1-0.5T_{del}\cdot s}{1+0.5T_{del}\cdot s}}\right)\cdot \mathit{I}$$

(9)

Figure 2a can be further converted to Figure 2b, where ${\mathit{G}}_{pll}^m(s)$ can be given by:

$${\mathit{G}}_{pll}^m={\mathit{G}}_{del}(({\mathit{G}}_i-{\mathit{G}}_{dec}){\mathit{G}}_{pll}^i-{\mathit{G}}_{pll}^v)$$

(10)

Considering that the capacitor branch has a more significant influence on the high-frequency dynamics, while the oscillation of the studied system primarily occurs at frequencies less than 200 Hz, the filter capacitor branch is simplified. Consequently, the expression for the inverter impedance Z_out can be obtained as:

$${\mathit{Z}}_{out}=\left[\begin{array}{cc}{Z}_{out,dd}& Z_{out,dd}\\ Z_{out,qd}& Z_{out,qq}\end{array}\right]=({\mathit{Z}}_f+({\mathit{G}}_i-{\mathit{G}}_{dec}){\mathit{G}}_{del})\cdot {(\mathit{I}+{\mathit{G}}_{pll}^m{H}_{pll})}^{-1}$$

(11)

The grid impedance, denoted as Z_g, can be obtained as follows:

$${\mathit{Z}}_g=\left[\begin{array}{cc}s{L}_g& -\omega L_g\\ \omega L_g& s{L}_g\end{array}\right]$$

(12)

2.2. Stability and the Coupling Mechanism Analysis

Based on (8), the Z_out can be presented in a more intuitive way:

$${\mathit{Z}}_{out}={(\underset{{{\mathit{Z}}_{plant}}^{-1}}{\underbrace{I\cdot {({\mathit{Z}}_f+({\mathit{G}}_i-{\mathit{G}}_{dec}){\mathit{G}}_{del})}^{-1}}}+\underset{{{\mathit{Z}}_{pll}^m}^{-1}}{\underbrace{{\mathit{G}}_{pll}^m{H}_{pll}\cdot {({\mathit{Z}}_f+({\mathit{G}}_i-{\mathit{G}}_{dec}){\mathit{G}}_{del})}^{-1}}})}^{-1}$$

(13)

Z_out denotes the impedance of the GCI incorporating the PLL feedforward effect, while Z_plant refers to the GCI impedance when the PLL feedforward path is not taken into account. ${\mathit{Z}}_{pll}^m(s)$ represents the impedance formulated by the feedforward effect.

Figure 3 shows the GNC plot of the ${{\mathit{L}}_p\left(s\right)={\mathit{Z}}_g\cdot \mathit{Z}}_{plant}^{-1}$ when L_g = 16 mH and the ${{\mathit{L}}_o(s)={\mathit{Z}}_g\cdot \mathit{Z}}_{out}^{-1}$ when L_g = 9 mH, L_g = 16 mH. The eigen-traces λ_1p, λ_2p correspond to L_p(s), and the eigen-traces λ_1o, λ_2o correspond to L_o(s).

As depicted in Figure 3, when the PLL effect is not taken into account, and L_g = 16 mH, the λ_1p, λ_2p do not intersect (−1, j0), where the stable operation can be maintained. When taking into account the PLL effect, it is evident that the system remains stable when L_g = 9 mH, whereas instability arises when L_g = 16 mH. Therefore, the system’s stability is primarily influenced by both the PLL and the grid impedance. The coupling between the weak grid and the PLL will be explored in the following.

The Bode plot of the GCI impedance in qq-axis when considering PLL effects and not considering PLL effects is shown in Figure 4. It can be deduced that Z_out,qq exhibits capacitive impedance characteristics in qq-axis due to the PLL effect, which may couple with a weak grid and further contribute to instability. Thus, the PLL affects the stability of the studied system via current control by the feedforward effect.

Further exploration of the coupling between the weak grid and PLL will be conducted subsequently. The steady-state relationship between the ${\mathit{v}}_{dq}^s$, ${\mathit{v}}_{gdq}^s$ and Z_g can be formulated as:

$${\mathit{v}}_{dq}^s={\mathit{v}}_{gdq}^s+{\mathit{Z}}_g\cdot {\mathit{i}}_{dq}^s$$

(14)

Linearizing Formula (14) and assuming negligible variation in the grid voltage ${\mathit{v}}_{gdq}^s$ at steady state:

$$\Delta {\mathit{v}}_{dq}^s={\mathit{Z}}_g\cdot \Delta {\mathit{i}}_{dq}^s$$

(15)

Based on Formula (15), the small signal model of the PLL can be represented in Figure 5a. This depiction clearly reveals the interconnection between the Z_g and PLL. PLL further feedforwards the current control, resulting in the formulation of the qq-axis capacitive impedance area.

Figure 5b further provides an intuitive representation of the coupling, where G denotes the coupling term, which can be mathematically obtained as:

$$\mathit{G}={\mathit{G}}_{pll}^m{H}_{pll}{\mathit{Z}}_g$$

(16)

3. Analysis and Design of Equivalent Parallel Compensation Method

3.1. Analysis of the Equivalent Parallel Compensation Method

When a virtual resistance is parallel with Z_g, it leads to a modification in the reference of the PLL, consequently changing the coupling method. It results in a reduction in the coupling degree between the qq-axis capacitive impedance of the GCI and the inductive weak grid. Meanwhile, the damping of the studied system can be improved by the virtual resistance.

To achieve the stability control of the grid-connected system, an equivalent parallel compensation method integrated into PLL is proposed. The Z_g in complex space is expressed as follows [26]:

$${\mathit{Z}}_g=(s+j{\omega }_1)L_g$$

(17)

Assuming the virtual resistance R_v is parallel with Z_g, the expression for the grid impedance Z_g,com after parallel connection can be expressed as:

$${\mathit{Z}}_{g,com}=\underset{G_{vr,dq}}{\underbrace{{\displaystyle \frac{R_v}{(s+j{\omega }_1)L_{mg}+R_v}}}}(s+j{\omega }_1)L_g$$

(18)

where L_mg is the grid impedance value measured by the impedance measuring device, G can be expressed as another expression:

$$\mathit{G}(s)={\displaystyle \frac{{\mathit{G}}_{pll}^m(s)}{H_{pll}(s)}}\cdot \underset{H_{pll,com}}{\underbrace{G_{vr,dq}\cdot H_{pll}(s)}}\cdot (s+j{\omega }_1)\cdot L_g$$

(19)

The specific implementation of the virtual resistance will be discussed below. Due to the difficulty of paralleling the resistance with the grid impedance in practical applications, it can modify H_pll to H_pll,com, namely modify the input of the PLL as shown in Formula (18). Where G_vr,dq in dq domain can be converted to G_vr,αβ in αβ domain:

$$G_{vr,\alpha \beta }(s)=e^{-j\theta }{G}_{vr,dq}(s+j\omega )e^{j\theta }={\displaystyle \frac{R_v}{s{L}_{mg}+R_v}}$$

(20)

According to Formula (19), when multiplying the transfer function G_vr,αβ with the PLL input in the αβ domain, the transfer function of H_pll becomes H_pll,com, which equivalently parallels the virtual resistance R_v with the grid impedance. The equivalent parallel resistance compensation method alters the coupling method in the studied system. Adopting the equivalent parallel resistance compensation method, the coupling degree will decrease, which theoretically improves the damping of the studied system.

Expanding on the above analysis, the modified PLL structure is shown in Figure 6. Here, G_vr,αβ = G_vr,αβ·I, and G_bpf = G_bpf·I. G_bpf is the band-pass filter aimed at preserving high-frequency characteristics and suppressing additional harmonics, expressed as:

$$G_{bpf}(s)={\displaystyle \frac{2\xi {\omega }_{1}s}{s^2+2\xi {\omega }_{1}s+{{\omega }_1}^2}}$$

(21)

where ξ = 0.707, ω₁ denotes the grid angular frequency. The introduction of G_vr,αβ changes the phase reference of PLL input, consequently altering the stable operational point. This leads to an undesirable reactive power. Therefore, the dq transform angle should be modified, formulated as:

$${\mathit{T}}_{\alpha \beta ,dq}=e^{j{\theta }_{pll,com}}=\left[\begin{array}{cc}\cos ({\theta }_{pll,com})& -\sin ({\theta }_{pll,com})\\ \sin ({\theta }_{pll,com})& \cos ({\theta }_{pll,com})\end{array}\right]$$

(22)

where θ_pll,com represents the corrected dq transform angle. Since the phase shift of the G_bpf is 0 at the fundamental frequency, θ_pll,com can be formulated as:

$${\theta }_{pll,com}=-\angle G_{vr,\alpha \beta }+{\theta }_{pll}$$

(23)

Consequently, the impedance of GCI with the proposed compensation method can be modified as follows:

$${\mathit{Z}}_{out,com}=({\mathit{Z}}_f+({\mathit{G}}_i+{\mathit{G}}_{com}){\mathit{G}}_{del})\cdot {(\mathit{I}+{\mathit{G}}_{pll}^m{G}_{vr,dq}{H}_{pll})}^{-1}$$

(24)

3.2. Dynamic Performance of Compensation Method

The compensation method is utilized in the PLL without impacting the current tracking dynamic performance. The impact of the compensation method on PLL dynamic performance is assessed through MATLAB R2021b/Simulink.

The dynamic performance of the frequency tracking for both the conventional PLL and the improved PLL with different values of R_v (R_v = 4 and R_v = 1) is illustrated in Figure 7. The improved PLL exhibits reduced undershoot compared to the conventional PLL. Furthermore, as R_v decreases, the undershoot is further minimized. Therefore, the improved PLL effectively mitigates the overshoot and undershoot observed in the conventional PLL.

The dynamic response time is slightly changed with the improved PLL when R_v = 4. But it will further decrease with improved PLL when R_v = 1. Thus, the R_v will not be designed too small for a better dynamic response time of the PLL.

Thus, with a well-designed R_v, the proposed compensation method effectively maintains the stability while preserving minimal impact on the dynamic performance of the frequency tracking. For the selection of R_v, if R_v is too high, the coupling degree cannot be decreased. Conversely, if R_v is set too low, it may adversely affect the dynamic performance of the PLL.

3.3. Design and Stability Analysis

The stable range of the R_v is (0, 6) by employing the GNC to ${{\mathit{L}}_c\left(s\right)={\mathit{Z}}_g\cdot \mathit{Z}}_{out,com}^{-1}$. In this study, R_v is chosen as 2, as it offers appropriate damping while preserving the dynamic performance of the PLL. Figure 8 illustrates the GNC plot for L_g = 16 mH and R_v = 2. The GNC plot clearly indicates that (−1, j0) lies outside the λ_1oc, λ_2oc, indicating the stable operation of the interconnection system.

Figure 9 showcases the Bode plots of the GCI output impedance with and without compensation. Because the magnitude of the dq-axis impedance and the qd-axis impedance are relatively small, which can be omitted. The primary influence of the compensation method is located on the qq-axis impedance. With the implementation of the compensation method, the positive resistance area of qq-axis impedance expands. This signifies a reduction in the coupling degree between the PLL and the weak grid.

3.4. Robustness Analysis

The different operation modes of the grid-connected system may further decrease the stability. Factors such as the introduction of reactive power, increased PLL bandwidth, and grid impedance variations can destabilize the system. The errors of the estimated grid impedance L_mg of G_vr,dq also influence the effect of the compensation method. Therefore, it is imperative to conduct further analysis on the robustness of the proposed compensation method.

To validate the robustness of the equivalent parallel resistance compensation method, a series of test cases listed in

Table 2. Test cases.

Cases	Lmg (mH)	Lg (mH)	PLL Parameters	dq-Axis Current (A)
I	16	16	kppll = 4.2, kipll = 384	Idref = 10, Iqref = 2
II	16	16	kppll = 4.2, kipll = 484	Idref = 10, Iqref = 0
III	16	17.5	kppll = 4.2, kipll = 384	Idref = 10, Iqref = 0
IV	4.8 (−70% Lg)	16	kppll = 4.2, kipll = 384	Idref = 10, Iqref = 0
V	41.6 (+160% Lg)	16	kppll = 4.2, kipll = 384	Idref = 10, Iqref = 0

were investigated. Depicted in Figure 10, the grid-connected system exhibited instability under cases I, II, and III. However, with the equivalent parallel resistance compensation method, stability was achieved in the grid-connected system for all three cases. It demonstrates the robustness of the equivalent parallel resistance compensation in accommodating reactive power introduction, fluctuations in PLL bandwidth, and grid impedance.

The proposed compensation method relies on obtaining grid impedance information. However, in practice, the unavoidable error of the grid impedance measurement device will lead to a deviation between the L_mg and the L_g. Therefore, it is necessary to analyze whether the equivalent parallel resistance compensation can achieve stable control when there are errors in grid impedance measurement.

The impacts of grid impedance estimation value L_mg, as revealed by Equation (19), on the compensation method can be inferred. Case IV and case V demonstrate that the compensation method can maintain stability even when the error in grid impedance measurement reaches −70% L_g and +160 L_g.

Compared with [27], the proposed compensation method is highly tolerated for grid impedance measurement errors. In addition, compared with the series resistance compensation method in [15], the impedance model of the proposed compensation method is not complicated.

4. Experiment and Simulation Verification

To verify the proposed equivalent parallel compensation method, a 2 kW grid-connected system is built up, employing a PM50CLA20 inverter made by Mitsubishi in Tokyo, Japan, and YXSPACE-SP2000 made by YANXU in Nanjing, China. Additionally, IT6535C serves as the DC power source, which is made by ITCH in Nanjing, China. And GCT33060 functions as the grid simulator, which is made by HANSUN in Shanghai, China. The weak grid is realized by connecting the inductor between the converter and grid simulator. The experimental setup is depicted in Figure 11.

Figure 12 shows the waveforms with different L_g. It can be observed from Figure 12b that when the SCR is switched from 4.6 to 2.59, the oscillations will occur. Notably, a positive-sequence frequency of 111.25 Hz and a negative-sequence frequency of 11.25 Hz are observed due to the mirror frequency effect, as illustrated in Figure 12c.

Figure 13 showcases the waveforms with the compensation method when R_v = 1, R_v = 2, R_v = 4. As depicted in Figure 13, the utilization of the compensation method reduces voltage and current distortion. Consequently, the studied system demonstrates stable operation with the utilization of the compensation method. Additionally, Figure 14 presents the fast Fourier transform (FFT) analysis of the current, indicating the damping of oscillation frequencies in all three cases.

Figure 15 exhibits the waveforms with the compensation method for R_v = 2 under case I. The parameter variations of the studied system alter the oscillation frequency. From Figure 15a, when the compensation is not utilized, the investigated system experiences resonance. When the compensation method is adopted, the resonance can be mitigated. But the reference of PLL will be changed. Thus, there is a transient current overshoot and reactive power before the phase lock process, as depicted in Figure 15c. Figure 16 shows the damping process of i_a, which is based on discrete Fourier transformation (DFT). The oscillation frequency of 16 Hz and 116 Hz will be attenuated after the adoption of the compensation method.

Figure 17 and Figure 18 present the waveforms when R_v = 2 under case II and case III. It can be deduced that the compensation method effectively dampens the resonance even when the oscillation frequency has shifted, indicating the robustness of the method against the introduction of reactive power, fluctuations in PLL bandwidth, and grid impedance. These experimental results provide further evidence of the effectiveness and robustness of the compensation method.

Figure 19 illustrates the waveforms of when R_v = 2 under case IV and case V. Given Formula (23), the error in grid impedance measurement does not affect the effectiveness of the phase correction. In addition, the error of grid impedance measurement can be up to −70% and +160%, demonstrating the robustness of the compensation method to errors in grid impedance measurement, as verified experimentally.

Moreover, the waveforms of when the frequency of the grid changes 0.5 Hz and −0.5 Hz and R_v = 2 are shown in Figure 20, which is conducted in MATLAB/Simulink. The system can maintain stability in both cases, which indicates the proposed compensation method can deal with the change of grid frequency.

5. Conclusions and Future Work

This paper provides a clear exposition of the coupling mechanism in the grid-connected system. Based on this coupling mechanism, an equivalent parallel compensation method is proposed by altering the structure of PLL. This method introduces a virtual resistance in parallel with the grid impedance, which achieves the decouple of the studied system. It reshapes the inverter qq-axis impedance at the impedance level. By carefully designing the virtual resistance, the PLL dynamic performance can be maintained. Additionally, the method also has robustness to the parameter variations, such as the introduction of reactive power, the fluctuations of the PLL bandwidth, and grid impedance. And the error of grid impedance measurement can be up to −70% L_g and +160% L_g. Experimental waveforms further validate the effectiveness of this method.

The outlook of future work on this topic can be directed towards the following aspects:

(1)

The instability mechanism of the grid-connected system can be deeply analyzed, which enables us to propose a more effective stability control method.

(2)

The effectiveness and implementation of the equivalent parallel resistance compensation method in scenarios such as multiple inverters should be explored.