Study on the Resonance Characteristics and Active Damping Suppression Strategies of Multi-Inverter Grid-Connected Systems Under Weak Grid Conditions

Abstract

When a multi-inverter grid-connected system is influenced by the parasitic parameters of LCL-type inverters and the impedance of the connected system’s lines, its resonance characteristics become more complex and difficult to predict. For LCL-type multi-inverter grid-connected systems, a mathematical model that considers the effects of parasitic parameters and line impedance has been established, leading to the derivation of the system’s Norton equivalent circuit and a general expression for the inverter output current. The resonance characteristics of multi-inverter grid-connected systems composed of inverters with the same and different parameters were analyzed under the influence of parasitic parameters and line impedance. To suppress the resonance in multi-inverter grid-connected systems and address the issue of traditional PI control not meeting grid requirements for LCL-type grid-connected inverters, a strategy combining superhelical sliding mode control with active damping was adopted. To verify the practical performance of the adopted resonance suppression strategy in complex environments, a grid-connected system model containing two LCL-type inverters was constructed using the MATLAB/Simulink software platform, followed by simulation analysis. The simulation results strongly confirm the feasibility and effectiveness of the adopted resonance suppression strategy, considering the effects of parasitic parameters and line impedance. This strategy demonstrates significant suppression effects in addressing resonance issues caused by parasitic parameters and line impedance, effectively enhancing the quality of grid-connected electrical energy.

1. Introduction

Given the trend towards large-scale photovoltaic (PV) power generation, PV power plants are increasingly adopting the technology of multi-inverter parallel grid-connection systems [1]. However, as the scale of grid-connected PV systems expands, the issue of parallel resonance has become more prominent. Specifically, harmonics at certain frequencies are amplified in the grid-connected current, degrading the overall power quality [2]. In weak grid environments [3], the integration of grid-connected inverters [4] causes significant changes in grid impedance characteristics. These changes interact with the output impedance of inverters, resulting in grid-connected series resonance. More critically, when grid-connected inverters are introduced, the increased grid impedance directly reduces the phase margin of the inverter system, severely affecting grid stability and operational safety. In some cases, these changes could lead to equipment failures, potentially triggering grid collapse and causing catastrophic consequences [5].

Given these potential risks, it is of significant theoretical and practical importance to conduct an in-depth study on the resonance characteristics of multi-inverter grid-connected PV systems and develop corresponding suppression strategies.

The core conditions for the generation of resonance in grid-connected systems are twofold: first, the presence of a resonance network with matched impedance parameters [6]; second, the existence of a harmonic source capable of generating the resonant frequency within the system [7]. To effectively suppress resonance phenomena, the primary strategy is to prevent harmonic sources from entering the resonance network. Additionally, reshaping the output impedance of the inverter by adjusting network impedance parameters can change the resonance conditions of the system. There are currently two methods for reshaping inverter output impedance: passive damping and active damping.

The passive damping method enhances damping by directly adding resistors in series or parallel within the system. Although simple to implement, this method inevitably results in additional power loss [8]. Active damping, on the other hand, suppresses system resonance by selecting appropriate feedback variables combined with a well-designed control strategy. This approach not only avoids extra power loss but is also applicable to a wide range of scenarios [9]. The following sections will delve into the research status and development of both passive and active damping strategies for suppressing resonance in inverter-based grid-connected systems.

2. Literature Review

Ma et al. (2023) [8] introduced a passive damping solution based on a parallel resistor-capacitor circuit, offering a comprehensive design guide applicable to various scenarios involving current control on the inverter or grid side, with or without the use of active damping methods. This parallel resistor-capacitor scheme enabled the designed grid-connected inverter to achieve passive output admittance across all frequencies, effectively preventing potential harmonic oscillations near the Nyquist frequency. Jo et al. (2019) [10] proposed three criteria for determining passive damping resistors, considering system delay through an equivalent model and deriving the applicable range for the damping resistor to suppress filter resonance. Wang et al. (2024) [11] improved the grid voltage feedforward control strategy, increasing the system’s phase margin and enhancing its adaptability to grid impedance variations.

Xie et al. (2020) [12] constructed an admittance model for an LCL-type inverter with grid-side current control and active damping using capacitor voltage feedforward. They found that designing the anti-resonance frequency of the inverter-side inductance and filter capacitance within a specific range could achieve full-frequency passive output admittance through a straightforward active damping scheme with capacitor voltage feedforward. Ali Azghandi et al. (2021) [13] proposed a fractional-order active damping control method with more adjustable parameters. They also developed a design strategy for a fractional-order damper that accounts for digital control delays. The proposed active damper reduced passive filter resonance and maintained power quality. Zhang et al. (2017) [14] investigated current control strategies for high-power grid-connected inverters using LCL filters. They explored a multi-sampling active scheme, which effectively reduced control delays and improved active performance. Additionally, they studied multi-sampling control without additional damping. By feeding back the inverter-side current, the system could achieve single-current loop control based on multiple frequency domains, simplifying the control system and enhancing system stability with better dynamic performance. Zhao et al. (2022) [15] developed a unified impedance model for LCL-type grid-connected inverters with passive capacitor current feedback and active damping. This model was suitable for both inverter current control and grid current control, facilitating passive stability assessments and controller parameter design. Through passivity analysis, they found that when the anti-resonance frequency of the LCL filter falls within a specific range, the proposed design criteria could enable full-frequency passive admittance output from the inverter. This approach, using observers and passive output admittance across all frequencies, not only eliminated the need for additional current sensors for active damping but also allowed the inverter to operate stably in the grid regardless of grid impedance. Ma et al. (2021) [16] proposed an innovative control scheme based on the active disturbance rejection control (ADRC) principle, aimed at enhancing system robustness and stability. They provided a thorough root locus analysis to detail the scientific method for system parameter design, offering strong theoretical support for practical applications. This control strategy achieved direct control of grid-injected current for LCL-type grid-connected inverters by detecting only the grid-injected current. The effects of digital control delays and grid uncertainties on system stability were also carefully analyzed.

Akhavan et al. (2018) [17] explored the design of control systems for multi-parallel grid-connected inverters using active damping. They modeled inverters with different characteristics in a weak grid as a multi-variable system, considering the coupling effects of wide-ranging grid impedance changes. To address shortcomings in conventional grid voltage feedforward control, they proposed an improved feedforward control method. Guo et al. (2018) [18] designed a dual-layer active damping framework using function rotation to simultaneously suppress multiple resonance peaks. This approach involved embedding active damping functions into the designated distributed energy interface inverter in the first layer, while other inverters did not participate in damping. In the second layer, interface inverters were selected to suppress multiple resonance peaks, with the real-time monitoring and coordination of each distributed energy’s operational state.

Fang et al. (2021) [19] proposed an equivalent allocation of grid impedance and improved the system’s stability against grid impedance variations by combining a phase lead compensator with capacitive current feedback. Y et al. (2023) [20] addressed system instability by compensating for output admittance while ensuring the system’s capability to attenuate low-order harmonics. Zhou et al. (2022) [21] investigated the stability of a system with multiple single-phase inverters connected to the grid and employed a dual second-order filter to suppress system resonance. Akhavan et al. (2024) [22] studied the stability of a multi-inverter grid-connected system under asymmetric conditions. Zeng et al. (2023) [23] analyzed multi-inverter systems by defining a contribution factor to identify the inverter causing system instability and then removing it to restore stable operation. Li et al. (2023) [24] tackled the instability caused by grid impedance fluctuations and plug-and-play inverter connections using upper-level D partitioning and lower-level adaptive control, demonstrating its effectiveness. In another study, Li et al. (2023) [25] addressed the nonlinear oscillations in grid voltage and current caused by grid impedance fluctuations and plug-and-play inverters, employing a nonlinear observer and an adaptive dynamic coordination loop to validate their approach. Zhang et al. (2023) [26] improved the stability margin of multi-inverter parallel systems under weak grids by using a modified weighted average current and parallel virtual conductance at the point of common coupling. Khajeh et al. (2022) [27] proposed a solution to the need for individual feedforward control of each inverter in a parallel-connected multi-inverter system by introducing an inverter with virtual negative conductance, verifying its feasibility through simulations and experiments.

In summary, most current research on multi-inverter grid-connected systems primarily focuses on the parallel connection of multiple inverters at a common point, with limited discussion on the impact of line impedance and parasitic parameters. There is a lack of in-depth analysis and research on this issue. Given the significant influence of parasitic parameters and line impedance on resonance in multi-inverter grid-connected systems, conducting a thorough and systematic investigation into this complex problem is of critical importance.

3. Modeling and Resonance Characterization of Multi-Converter Systems with LCL Topology

3.1. System Modeling Considering Parasitic Parameters

The circuit diagram of the LCL-type inverter multi-machine grid-connected system, considering parasitic parameters and line impedance, is shown in Figure 1.

The variables are defined as follows: u_dc represents the DC-side voltage; L_1i and L_2i are the inverter-side and grid-side inductances, respectively; C_i denotes the filter capacitor; C_L1i is the parasitic capacitance of the inverter-side inductor; R_1i is the parasitic resistance of the inverter-side inductor; L_ci represents the parasitic inductance of the filter capacitor; R_ci is the parasitic resistance of the filter capacitor; C_L2i is the parasitic capacitance of the grid-side inductor; R_2i is the parasitic resistance of the grid-side inductor; R_fi and L_fi represent the line resistance and line inductance, respectively; u_g denotes the grid voltage; L_g represents the grid inductance; and i_gi is the grid-connected current output by each inverter, where i = 1,2,…,n.

The grid-connected inverter system adopts a control method based on grid-side output current feedback. The equivalent control block diagram is shown in Figure 2. In this diagram, K_PWM represents the equivalent gain of the inverter; i_ref(s) is the reference current; G_c(s) is the current controller; Z₁(s) = (s²L₁C_L1R₁ + sL₁ + R₁)/(s²L₁C_L1 + 1) represents the inverter-side impedance, including parasitic capacitance and resistance; Z₂(s) = (s²L₂C_L2R₂ + sL₂ + R₂)/(s²L₂C_L2 + 1) is the grid-side impedance, which also includes parasitic capacitance and resistance; and Z₃(s) = (s²L_c + sCR_c + 1)/sC is the filter-side impedance, accounting for parasitic inductance and resistance.

The control block diagram in Figure 2 is simplified to obtain the reduced control structure, as shown in Figure 3.

Based on Figure 3, an analysis of the single inverter grid-connected system yields the expression for i_g(s) as follows:

$$i_{\mathrm{g}}\left(s\right)=G_{\mathrm{i}}{i}_{\mathrm{ref}}\left(s\right)-Y_{\mathrm{g}}{u}_{\mathrm{PCC}}\left(s\right)$$

(1)

where

$$\left\{\begin{array}{l}{G}_{\mathrm{i}}={\displaystyle \frac{G_1{G}_2}{1+G_1{G}_2}}\\ Y_{\mathrm{g}}={\displaystyle \frac{G_2}{1+G_1{G}_2}}\end{array}\right.$$

(2)

$$\left\{\begin{array}{l}{G}_1={\displaystyle \frac{G_{\mathrm{c}}{K}_{\mathrm{PWM}}{\mathrm{Z}}_3}{Z_1+Z_3}}\\ G_2={\displaystyle \frac{Z_1+Z_3}{Z_1{Z}_2+\left(Z_1+Z_2\right)Z_3}}\end{array}\right.$$

(3)

As shown in Figure 3, the transfer function T(s) is:

$$T\left(s\right)={\displaystyle \frac{G_{\mathrm{c}}{K}_{\mathrm{PWM}}{Z}_3}{Z_1{Z}_2+\left(Z_1+Z_2\right)Z_3}}$$

(4)

Based on the derivation from Equation (1), a Norton equivalent model for the single inverter grid connection is constructed, as shown in Figure 4. This model employs an AC current source in parallel with an equivalent admittance, providing an intuitive representation of the equivalent circuit structure on the inverter side, while the grid side is represented by an AC current source in series with equivalent impedance.

Typically, each inverter is controlled independently. Therefore, when n inverters are connected in parallel at a common point, a multi-inverter grid-connected system is formed, resulting in the Norton equivalent circuit diagram of the multi-inverter grid-connected system, as shown in Figure 5.

According to Figure 5 and the superposition theorem, the expression for i_g(s) is obtained as follows:

$$i_{\mathrm{g}}\left(s\right)=i_{21}\left(s\right)+i_{22}\left(s\right)+\cdots +i_{2\mathrm{n}}\left(s\right)$$

(5)

The point of common coupling (PCC) voltage u_PCC(s) is given by:

$$u_{\mathrm{pcc}}\left(s\right)={\displaystyle \frac{G_{\mathrm{i}1}s{L}_{\mathrm{g}}}{1+Y_{\mathrm{g}1}s{L}_{\mathrm{g}}}}{i}_{\mathrm{ref}1}+{\displaystyle \sum _{k=2}^n\frac{G_{\mathrm{ik}}s{L}_{\mathrm{g}}}{1+Y_{\mathrm{gk}}s{L}_{\mathrm{g}}}}{i}_{\mathrm{refk}}+{\displaystyle \sum _{k=1}^n\frac{1}{1+Y_{\mathrm{gk}}s{L}_{\mathrm{g}}}{u}_{\mathrm{g}}}$$

(6)

where i_ref1 is the reference current of the first inverter, and i_refk represents the reference currents of the other inverters.

Thus, substituting Equation (6) into Equation (1), the output current expression for the first inverter is obtained as Equations (7) and (8).

$$i_{21}\left(s\right)=A\left(s\right)i_{\mathrm{ref}1}\left(s\right)-B\left(s\right)i_{\mathrm{refk}}\left(s\right)-C\left(s\right)u_{\mathrm{g}}\left(s\right)$$

(7)

$$\left\{\begin{array}{l}A\left(s\right)=G_{\mathrm{i}1}\left(s\right)-{\displaystyle \frac{G_{\mathrm{i}1}\left(s\right)Y_{\mathrm{g}}\left(s\right)s{L}_{\mathrm{g}}}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{gk}}\left(s\right)s{L}_{\mathrm{g}}}}}\\ B\left(s\right)={\displaystyle \sum _{k=2}^n\frac{G_{\mathrm{ik}}\left(s\right)s{L}_{\mathrm{g}}}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{gk}}\left(s\right)s{L}_{\mathrm{g}}}}}\\ C\left(s\right)={\displaystyle \frac{Y_{\mathrm{g}}\left(s\right)}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{gk}}\left(s\right)s{L}_{\mathrm{g}}}}}\end{array}\right.$$

(8)

From the derivation of these equations, it can be observed that the grid-connected output current of the first inverter is the result of multiple excitation sources. These sources include the inverter’s own excitation, the excitation from other inverters, and the excitation from the grid voltage. The influence of each source on the output current is characterized by different impact factors: A(s) represents the influence of the inverter itself, B(s) represents the influence of other inverters, and C(s) reflects the influence of the grid voltage. Therefore, by thoroughly analyzing the specific characteristics of A(s), B(s), and C(s), the resonance characteristics of multi-inverter grid-connected systems can be effectively evaluated.

3.2. Analysis of System Resonance Characteristics Considering Parasitic Parameters

The influence coefficient of the inverter’s own excitation source already incorporates the effects of other inverter excitation sources and the grid voltage excitation source at both low and high-frequency resonance points. Therefore, this study focuses on analyzing the influence coefficients A(s) and A₁(s) of the inverter’s own excitation source.

For the inverters with the same parameters, two A-type inverters were selected, with their main technical specifications shown in

Table 1. Control strategy parameter settings.

Parameters	Value
Inverter-side inductance, L1	3 mH
Integral coefficient, Ki	2200
Filter capacitor, C	10 μF
Inverter power, Po	30 kW
Grid-side inductance, L2	1 mH
Grid voltage, ug	311 V
Proportional coefficient, KP	0.35
Grid reactance, Lg	0.5 mH
Switching frequency, fsw	10 kHz
DC-side voltage, udc	800 V

. For inverters with different parameters, one A-type inverter and one B-type inverter were selected. The main technical specifications for the B-type inverter are presented in

Table 2. Control strategy parameter settings.

Parameters	Value
Inverter-side inductance, L1	6 mH
Integral coefficient, Ki	4000
Filter capacitor, C	15 μF
Inverter power, Po	30 kW
Grid-side inductance, L2	3 mH
Grid voltage, ug	311 V
Proportional coefficient, KP	0.6
Grid reactance, Lg	0.5 mH
Switching frequency, fsw	10 kHz
DC-side voltage, udc	800 V

.

3.2.1. Analysis of System Resonance Characteristics with Two Identical Parameter Inverters

This section analyzes the effects of parasitic parameters on the inverter-side inductance, grid-side inductance, and filter capacitance of LCL-type inverters. These are categorized into six components: the parasitic resistance and capacitance of the inverter-side inductance, the parasitic resistance and capacitance of the grid-side inductance, and the parasitic resistance and inductance of the filter capacitance.

As shown in Figure 6, the two resonant frequency points remain unchanged at 506 Hz and 568 Hz as the parasitic resistance of the inverter-side inductance increases. However, the peak values of the resonant spikes decrease with the increase in parasitic resistance. This indicates that the parasitic resistance of the inverter-side inductance has a damping effect on the resonant peaks.

As shown in Figure 7, the two resonant frequency points remain unchanged at 506 Hz and 568 Hz as the parasitic capacitance of the inverter-side inductance increases. However, the peak values of the resonant spikes decrease. This indicates that the parasitic capacitance of the inverter-side inductance has a certain damping effect on the system’s resonance.

Figure 8 shows that, even with an increase in the parasitic resistance of the grid-side inductance, the two resonant frequency points remain constant at 506 Hz and 568 Hz, without any shift. However, it is noteworthy that as the parasitic resistance of the grid-side inductance increases, the peak values of the resonant spikes significantly decrease. This indicates that the parasitic resistance of the grid-side inductance has a damping effect on the resonant peaks.

As shown in Figure 9, the two resonant frequency points remain unchanged at 506 Hz and 568 Hz as the parasitic capacitance of the grid-side inductance increases, and the peak values of the resonant spikes also show no variation. This indicates that the parasitic capacitance of the grid-side inductance has almost no effect on the system’s resonance.

As shown in Figure 10, the two resonant frequency points remain constant at 506 Hz and 568 Hz as the parasitic resistance of the filter capacitor increases. However, the peak values of the resonant spikes decrease with the increase in parasitic resistance. This indicates that the parasitic resistance of the filter capacitor has a damping effect on the resonant peaks.

As shown in Figure 11, with the increase in the parasitic inductance of the filter capacitor, both resonant frequency points shift. A reduction in the parasitic inductance of the filter capacitor causes the two resonant frequency points to shift towards higher frequencies, while an increase in parasitic inductance causes them to shift towards lower frequencies. Specifically, when the parasitic inductance is 50 nH, 100 nH, and 150 nH, the system’s first resonant frequency points are 716 Hz, 506 Hz, and 413 Hz, respectively, and the second resonant frequency points are 774 Hz, 568 Hz, and 471 Hz, respectively. Additionally, the peak values of the resonant spikes gradually increase. This indicates that an increase in the parasitic inductance of the filter capacitor exacerbates the resonant spikes of the LCL filter.

3.2.2. Analysis of System Resonance Characteristics with Two Different Parameter Inverters

The system’s amplitude–frequency characteristics with varying R1 are shown in Figure 12.

As shown in Figure 12, the two resonant frequency points remain unchanged at 506 Hz and 558 Hz as the parasitic resistance of the inverter-side inductance increases. However, the peak values of the resonant spikes decrease with the increase in parasitic resistance. This indicates that the parasitic resistance of the inverter-side inductance has a damping effect on the resonant peaks.

As shown in Figure 13, the two resonant frequency points remain unchanged at 506 Hz and 558 Hz as the parasitic capacitance of the inverter-side inductance increases, and the peak values of the resonant spikes also show little to no variation. This indicates that the parasitic capacitance of the inverter-side inductance has almost no effect on the system’s resonance.

As shown in Figure 14, the two resonant frequency points remain constant at 506 Hz and 558 Hz as the parasitic resistance of the grid-side inductance increases. However, the peak values of the resonant spikes decrease with the increase in parasitic resistance. This indicates that the parasitic resistance of the grid-side inductance has a damping effect on the resonant peaks.

As shown in Figure 15, the two resonant frequency points remain unchanged at 506 Hz and 558 Hz as the parasitic capacitance of the grid-side inductance increases, and the peak values of the resonant spikes also show no variation. This indicates that the parasitic capacitance of the grid-side inductance has almost no effect on the system’s resonance.

Figure 16 shows that, as the parasitic resistance of the filter capacitor increases, the two resonance frequency points remain unchanged at 506 Hz and 558 Hz, respectively. However, the peak values of the resonance spikes decrease with an increase in the parasitic resistance of the filter capacitor. This indicates that the parasitic resistance of the filter capacitor exerts a certain suppressive effect on the resonance spikes.

From Figure 17, it can be observed that both resonant frequency points shift toward lower frequencies as the parasitic inductance of the filter capacitor increases. Specifically, when the parasitic inductances are 50 nH, 100 nH, and 150 nH, the first resonant frequency points of the system are 716 Hz, 506 Hz, and 413 Hz, respectively. The second resonant frequency points are 764 Hz, 558 Hz, and 461 Hz, respectively, and the peak values of the resonance spikes gradually increase. This indicates that the increase in parasitic inductance of the filter capacitor intensifies the resonance peaks of the LCL-type filter.

3.3. System Modeling Considering Line Impedance

Considering the effect of line impedance, Figure 18 illustrates the Norton equivalent model corresponding to a single inverter.

An in-depth analysis was conducted on the characteristics of a grid-connected system with a single inverter as shown in Figure 18, including the line impedance. The mathematical expression for the output current i_g (s) of the single inverter, considering line impedance, is derived as follows:

$$i_{\mathrm{g}}\left(s\right)=G_{\mathrm{t}}{i}_{\mathrm{ref}}\left(s\right)-Y_{\mathrm{t}}{u}_{\mathrm{PCC}}\left(s\right)$$

(9)

where

$$\left\{\begin{array}{c}{G}_{\mathrm{t}}={\displaystyle \frac{G_{\mathrm{i}}}{1+{\mathrm{Z}}_{\mathrm{f}}{Y}_{\mathrm{g}}}}\\ Y_{\mathrm{t}}={\displaystyle \frac{Y_{\mathrm{g}}}{1+{\mathrm{Z}}_{\mathrm{f}}{Y}_{\mathrm{g}}}}\end{array}\right.$$

(10)

By paralleling multiple single inverter models, the multi-inverter model is obtained. Then, by connecting this multi-inverter parallel model to the point of common coupling (PCC), the Norton equivalent circuit for multiple inverters with line impedance is derived, as shown in Figure 19.

According to Figure 19 and the superposition theorem, the grid-connected current i_g(s) is given by

$$i_{\mathrm{g}}\left(s\right)=i_{21}\left(s\right)+i_{22}\left(s\right)+\cdots +i_{2\mathrm{n}}\left(s\right)$$

(11)

The PCC voltage u_PCC(s) is expressed as

$$u_{\mathrm{pcc}}\left(s\right)={\displaystyle \frac{G_{\mathrm{t}1}s{L}_{\mathrm{g}}}{1+Y_{\mathrm{t}1}s{L}_{\mathrm{g}}}}{i}_{\mathrm{ref}1}+{\displaystyle \sum _{k=2}^n\frac{G_{\mathrm{tk}}s{L}_{\mathrm{g}}}{1+Y_{\mathrm{tk}}s{L}_{\mathrm{g}}}}{i}_{\mathrm{refk}}+{\displaystyle \sum _{k=1}^n\frac{1}{1+Y_{\mathrm{tk}}s{L}_{\mathrm{g}}}{u}_{\mathrm{g}}}$$

(12)

Therefore, for the first inverter, the general expression for its output current is

$$i_{21}\left(s\right)=A_1\left(s\right)i_{\mathrm{ref}1}\left(s\right)-B_1\left(s\right)i_{\mathrm{refk}}\left(s\right)-C_1\left(s\right)u_{\mathrm{g}}\left(s\right)$$

(13)

where

$$\left\{\begin{array}{l}{A}_1\left(s\right)=G_{\mathrm{t}1}\left(s\right)-{\displaystyle \frac{G_{\mathrm{t}1}\left(s\right)Y_{\mathrm{t}}\left(s\right)s{L}_{\mathrm{g}}}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{tk}}\left(s\right)s{L}_{\mathrm{g}}}}}\\ B_1\left(s\right)={\displaystyle \sum _{k=2}^n\frac{G_{\mathrm{tk}}\left(s\right)s{L}_{\mathrm{g}}}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{tk}}\left(s\right)s{L}_{\mathrm{g}}}}}\\ C_1\left(s\right)={\displaystyle \frac{Y_{\mathrm{t}}\left(s\right)}{1+{\displaystyle \sum _{k=1}^n{Y}_{\mathrm{tk}}\left(s\right)s{L}_{\mathrm{g}}}}}\end{array}\right.$$

(14)

3.4. Analysis of System Resonance Characteristics Considering Line Impedance

Analyzing the impact of line impedance on the system’s resonant characteristics is crucial due to the differing positions of various types of inverters connected to the system.

3.4.1. Analysis of System Resonance Characteristics with Two Identical Parameter Inverters

The system’s amplitude–frequency characteristics with varying R_f are shown in Figure 20.

By analyzing the data in Figure 20, it is evident that even with an increase in line resistance, the two resonant frequency points of the system remain stable at 1671 Hz and 5176 Hz, without significant shifts. However, the peak value of the first resonance spike decreases as the line resistance increases. This suggests that line resistance has a suppressive effect on the first resonance spike.

From Figure 21, it can be observed that the first resonant frequency point shifts towards lower frequencies as the line inductance increases, specifically at 1671 Hz, 1395 Hz, and 1268 Hz. Additionally, the resonance spike decreases with the increase in line inductance. The second resonant frequency point remains unchanged at 5176 Hz, and the peak value of the resonance spike also shows no variation. This indicates that line inductance causes the first resonant frequency point to shift toward lower frequencies and has a suppressive effect on the first resonance spike.

3.4.2. Analysis of System Resonance Characteristics with Two Different Parameter Inverters

The system’s amplitude–frequency characteristics with varying R_f are shown in Figure 22.

By examining the data in Figure 22, it is clear that even with an increase in line resistance, the two resonant frequency points of the system remain stable at 1187 Hz and 2431 Hz. However, the peak value of the first resonance spike decreases as the line resistance increases. This suggests that line resistance has a suppressive effect on the first resonance spike.

From Figure 23, it can be seen that the first resonant frequency point shifts toward lower frequencies as the line inductance increases, specifically at 1186 Hz, 978 Hz, and 874 Hz. Additionally, the resonance spike decreases with the increase in line inductance. The second resonant frequency point remains unchanged at 2431 Hz, and the peak value of the resonance spike also remains unaffected. This indicates that line inductance causes the first resonant frequency point to shift toward lower frequencies and has a suppressive effect on the first resonance spike.

4. Active Damping Strategy Design and Simulation Results

Based on the detailed resonance analysis in Section 3, it can be seen that in a multi-inverter grid-connected system, LCL-type inverters are not only influenced by their own excitations, other inverters’ excitations, and grid harmonics but are also inevitably affected by parasitic parameters and line impedance. Therefore, selecting and implementing appropriate resonance suppression strategies is crucial to ensuring the stable operation of multi-inverter grid-connected systems. This section focuses on control strategies for grid-connected inverters, in particular addressing resonance issues that arise under the conditions of parasitic parameters and line impedance in multi-inverter systems. Given the limited performance of the traditional PI control when handling such issues, we adopted an improved active damping suppression strategy. To verify the effectiveness and broad applicability of this strategy, a corresponding simulation model was built in the MATLAB/Simulink 2018b environment. The circuit topology and a control structure diagram of the system using the active damping suppression strategy are shown in Figure 24. Simulink model design revolves around the core components of power electronic devices. The main circuit model provides a detailed description of the electrical characteristics of the inverter, filter, DC power supply, and power grid. The parameters of the inverter and filter are based on the settings shown in Table 1 and Table 2. The control logic includes the implementation of active damping strategies, involving simulation of feedback loops and digital signal processing. The signal processing module is responsible for signal acquisition, conversion, and processing to ensure the accurate implementation of control strategies.

4.1. Active Damping Strategy Design

Sliding mode control is an outstanding nonlinear control strategy that is widely favored in engineering practice. However, it also has certain limitations. When the system reaches the sliding surface, its motion trajectory does not strictly follow the surface but frequently oscillates back and forth across it, a phenomenon known as “chattering.” Since chattering cannot be completely eliminated theoretically, only suppressed to a certain extent, this section introduces the super-twisting algorithm as an optimization method to effectively mitigate chattering, thereby improving the system’s dynamic performance.

Super-twisting sliding mode control is a higher-order strategy that incorporates the essence of the super-twisting algorithm. Compared to traditional exponential reaching laws, the innovation of the super-twisting algorithm lies in its embedding of the switching term sgn(s) into higher-order derivatives, allowing for continuous output of the control signal, which effectively suppresses chattering. Its mathematical expression is

$$\left\{\begin{array}{l}{u}_{\mathrm{st}}=u_{\mathrm{s}1}+u_{\mathrm{s}2}\\ u_{\mathrm{s}1}=\alpha {\left|s\right|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}\left(s\right)\\ u_{\mathrm{s}2}=\beta \mathrm{sgn}\left(s\right)\end{array}\right.$$

(15)

From Equation (15), it can be concluded that the super-twisting control (STC) strategy consists of two parts. u_s1 is introduced to transform sgn(s) into a continuous function, thereby enhancing the smoothness of the system. u_s2 employs a clever mechanism to transfer sgn(s) from u to ū and process it through integration, which not only maintains the continuity of the final output control signal but also significantly reduces chattering phenomena in the system, thereby improving overall control performance. The structural diagram is depicted in Figure 25.

To ensure the power quality of the grid-connected current, the STC strategy is applied to the current loop. This current loop first collects the three-phase grid-connected currents i_a, i_b, and i_c and transforms them into d and q axis components i_d and i_q using the Park transformation. Next, i_d and i_q are compared with preset reference values, where i*_{ref_iq} is set to 0 to eliminate the reactive current component. Subsequently, the STC processes the obtained difference, and after applying the inverse Park and inverse Clark transformations—while considering the impact of filter capacitor current—the voltage signals on the α and β axes, u_α and u_β, are finally calculated. Using the SVPWM technique, the corresponding trigger signals are then generated based on the computed u_α and u_β to achieve precise control of the switching devices.

Define the current tracking error as

$$\left\{\begin{array}{l}{e}_1=i_{\mathrm{d}}^{*}-i_{\mathrm{d}}\hfill \\ e_2=i_{\mathrm{q}}^{*}-i_{\mathrm{q}}\hfill \end{array}\right.$$

(16)

The design of the integral sliding surface is

$$\left\{\begin{array}{l}{s}_1=e_1+{\lambda }_1{\displaystyle {\int }_0^t{e}_1}\mathrm{d}t=0\\ s_2=e_2+{\lambda }_1{\displaystyle {\int }_0^t{e}_2}\mathrm{d}t=0\end{array}\right.$$

(17)

in which λ is the control gain and is greater than zero.

The initial value for the integral is chosen as

$$\left\{\begin{array}{l}{I}_1=-{\displaystyle \frac{1}{{\lambda }_1}}{e}_1(0)\hfill \\ I_2=-{\displaystyle \frac{1}{{\lambda }_1}}{e}_2(0)\hfill \end{array}\right.$$

(18)

The design of the control law integrates the concepts of equivalent control strategy and the super-twisting algorithm, leading to the expressions for S_d and S_q in Equations (2)–(12):

$$\left\{\begin{array}{l}{\mathrm{S}}_{\mathrm{d}}=u_{\mathrm{eq}1}+u_{\mathrm{st}1}\hfill \\ {\mathrm{S}}_{\mathrm{q}}=u_{\mathrm{eq}2}+u_{\mathrm{st}2}\hfill \end{array}\right.$$

(19)

Differentiating Equation (17) yields

$$\left\{\begin{array}{l}{\dot{s}}_1={\dot{e}}_1+{\lambda }_1{e}_1=-{\dot{i}}_{\mathrm{d}}+{\lambda }_1{e}_1={\displaystyle \frac{R}{L}}{i}_{\mathrm{d}}-\omega i_{\mathrm{q}}-{\displaystyle \frac{1}{L}}{\displaystyle \frac{u_{\mathrm{dc}}}{2}}{S}_{\mathrm{d}}+{\displaystyle \frac{1}{L}}{u}_{\mathrm{gd}}+{\lambda }_1{e}_1\\ {\dot{s}}_2={\dot{e}}_2+{\lambda }_1{e}_2=-{\dot{i}}_{\mathrm{q}}+{\lambda }_1{e}_1={\displaystyle \frac{R}{L}}{i}_{\mathrm{q}}-\omega i_{\mathrm{d}}-{\displaystyle \frac{1}{L}}{\displaystyle \frac{u_{\mathrm{dc}}}{2}}{S}_{\mathrm{q}}+{\displaystyle \frac{1}{L}}{u}_{\mathrm{gq}}+{\lambda }_1{e}_2\end{array}\right.$$

(20)

When setting s = 0, the equivalent control law is obtained as

$$\left\{\begin{array}{l}{u}_{\mathrm{e}\mathrm{q}1}={\displaystyle \frac{2R{i}_{\mathrm{d}}-2\omega L{i}_{\mathrm{q}}+2u_{\mathrm{g}\mathrm{d}}+2{\lambda }_1{L}_1{e}_1}{u_{\mathrm{d}\mathrm{c}}}}\hfill \\ u_{\mathrm{e}\mathrm{q}2}={\displaystyle \frac{2R{i}_{\mathrm{q}}+2\omega L{i}_{\mathrm{d}}+2u_{\mathrm{g}\mathrm{q}}+2{\lambda }_1{L}_1{e}_2}{u_{\mathrm{d}\mathrm{c}}}}\hfill \end{array}\right.$$

(21)

By combining Equations (20) and (21), the control equation for the current loop is derived as

$$\left\{\begin{array}{l}{S}_{\mathrm{d}}={\displaystyle \frac{2R{i}_{\mathrm{d}}-2\omega L{i}_{\mathrm{q}}+2u_{\mathrm{g}\mathrm{d}}+2{\lambda }_{1}L{e}_1}{u_{dc}}}+\alpha |s_1{|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}(s_1)+\beta {\displaystyle {\int }_0^t\mathrm{sgn}}(s_1)\mathrm{d}t\\ S_{\mathrm{q}}={\displaystyle \frac{2R{i}_{\mathrm{q}}+2\omega L{i}_{\mathrm{d}}+2u_{\mathrm{g}\mathrm{q}}+2{\lambda }_{1}L{e}_2}{u_{\mathrm{d}\mathrm{c}}}}+\alpha |s_2{|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}(s_2)+\beta {\displaystyle {\int }_0^t\mathrm{sgn}}(s_2)\mathrm{d}t\end{array}\right.$$

(22)

Next, we perform a stability analysis of the sliding mode controller, defining the Lyapunov function as V = 0.5s^Ts. Its derivative is obtained as follows:

$$\begin{array}{cc}\hfill V& =s_1{\dot{s}}_1+s_2{\dot{s}}_2\hfill \\ & =s_1\left({\dot{e}}_1+{\lambda }_1{e}_1\right)+s_2\left({\dot{e}}_2+{\lambda }_1{e}_2\right)\hfill \\ & =s_1\left({\displaystyle \frac{R}{L}}{i}_{\mathrm{d}}-\omega i_{\mathrm{q}}-{\displaystyle \frac{1}{L}}{\displaystyle \frac{u_{ \mathrm{d}\mathrm{c}}}{2}}{S}_{\mathrm{d}}+{\displaystyle \frac{1}{L}}{u}_{\mathrm{g}\mathrm{d}}+{\lambda }_1{e}_1\right)\hfill \\ & +s_2\left({\displaystyle \frac{R}{L}}{i}_{\mathrm{q}}+\omega i_{\mathrm{d}}-{\displaystyle \frac{1}{L}}{\displaystyle \frac{u_{ \mathrm{d}\mathrm{c}}}{2}}{S}_{\mathrm{q}}+{\displaystyle \frac{1}{L}}{u}_{\mathrm{g}\mathrm{q}}+{\lambda }_l{e}_2\right)\hfill \\ & =s_1\left[-{\displaystyle \frac{u_{\mathrm{d}\mathrm{c}}}{2L}}\left(\alpha {\left|s_1\right|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}\left(s_1\right)+\beta {\displaystyle {\int }_0^t\mathrm{sgn}}\left(s_1\right)\mathrm{d}t)\right.\right]\hfill \\ & +s_2\left[-{\displaystyle \frac{u_{\mathrm{d}\mathrm{c}}}{2L}}(\alpha |s_2{|}^{{\displaystyle \frac{1}{2}}}\mathrm{sgn}(s_2)+\beta {\displaystyle {\int }_0^t\mathrm{sgn}}(s_2)\mathrm{d}t)\right]\hfill \end{array}$$

(23)

where α and β are positive values, indicating that the system remains stable when sliding mode control is employed for current control.

4.2. Simulation Results

To thoroughly investigate the practical effectiveness of the active damping resonance suppression strategy, we constructed a system model for LCL-type inverters in a multi-inverter grid-connected scenario within the MATLAB/Simulink simulation environment. In this system, the specific parameters for Class A and Class B inverters are set according to Table 1 and Table 2, ensuring the accuracy and reliability of the simulation experiments.

Building on the resonance analysis in Section 3.2 and Section 3.4, it was determined that the parasitic parameters of the filter capacitors and line impedance are key factors influencing the system’s resonance frequency points. Subsequently, the resonance suppression effects in multi-inverter systems under different grid connection scenarios are simulated and analyzed for three cases: (1) the system without resonance suppression strategies, (2) with PI control, and (3) with improved sliding mode control, specifically focusing on (a) inverter systems with varying parameters under parasitic parameters of the filter capacitors, and (b) inverter systems with varying parameters under line impedance conditions.

To investigate the effects of the resonance suppression strategy in depth, a grid-connected system model comprising parallel A-type and B-type inverters was constructed. In this model, the value of the parasitic inductance R_c is set to 1 Ω, while the parasitic inductance L_c is set to 50 nH. Subsequently, a series of comparative experiments was conducted to analyze the changes in system performance before and after the implementation of the resonance suppression strategy.

(1)

Without Resonance Suppression Strategy

Figure 26 illustrates the grid-connected voltage waveform and its frequency spectrum analysis without the implementation of the resonance suppression strategy. It is evident from the figure that the system exhibits resonance phenomena, with a total harmonic distortion (THD) rate reaching 26.16%. This indicates that significant resonance issues arise under these conditions, adversely affecting the stability and performance of the system.

(2)

Using the PI Control-Based Active Damping Resonance Suppression Strategy

Figure 27 presents the grid-connected voltage waveform and its frequency spectrum analysis after applying the PI control-based active damping resonance suppression strategy. Although this strategy introduces virtual impedance to generate some damping effects, the system’s resonance is not fully suppressed, and significant voltage spikes remain in the waveform. Additionally, while the THD has decreased to 14.24% with a change rate of 45.57%, it still does not meet the required performance standards for grid connection.

(3)

Using the Improved Active Damping Resonance Suppression Strategy

Figure 28 displays the grid-connected voltage waveform and frequency spectrum analysis after implementing the improved active damping resonance suppression strategy. It is clear from the figure that this improved strategy effectively suppresses the resonance phenomena in the system, significantly enhancing the quality of the grid-connected voltage waveform. Notably, the THD has dramatically reduced to 0.87% with a change rate of 96.67%, indicating the significant effectiveness of this strategy in improving system stability and performance.

A grid-connected system model with one A-type inverter and one B-type inverter was constructed, with the line resistance R_f set to 1 Ω and the line inductance L_f set to 1 mH. Comparative experiments will be conducted before and after the implementation of the suppression strategy.

(1)

Without Resonance Suppression Strategy

Figure 29 shows the grid-connected voltage waveform and its frequency spectrum analysis without the resonance suppression strategy. It is evident from the figure that the system exhibits resonance phenomena, with a THD rate of 19.27%, indicating significant resonance issues that adversely affect system stability and performance.

(2)

Using the PI Control-Based Active Damping Resonance Suppression Strategy

Figure 30 presents the grid-connected voltage waveform and its frequency spectrum analysis after applying the PI control-based active damping resonance suppression strategy. Despite the introduction of virtual impedance to create some damping effects, the system’s resonance is not fully suppressed, and the voltage waveform still exhibits considerable spike phenomena. Furthermore, the THD has decreased to 11.04% with a change rate of 42.71%, yet it still fails to meet the performance standards required for grid connection.

(3)

Using the Improved Active Damping Resonance Suppression Strategy

Figure 31 shows the grid-connected voltage waveform and frequency spectrum analysis when the improved active damping resonance suppression strategy is implemented. It is clear from the figure that this improved strategy effectively suppresses the resonance phenomena in the system, significantly enhancing the quality of the grid-connected voltage waveform. In particular, the THD has been substantially reduced to 0.34% with a change rate of 98.24%, demonstrating the considerable effectiveness of this strategy in enhancing system stability and performance.

This article addresses the resonant issues of LCL-type inverter multi-machine grid systems under the influence of parasitic inductance in the filter capacitor and line impedance, adopting an optimized active damping resonance suppression method. Given the limitations of traditional PI control in LCL-type grid-connected inverters, which fail to meet system performance requirements, this study introduces a new strategy based on second-order super-spiral sliding mode control. This strategy ensures the continuity of control output through discontinuous terms in the integral approximation law. To validate the effectiveness of the proposed method, a model of the LCL-type inverter multi-machine grid system, considering parasitic inductance in the filter capacitor and line impedance, was constructed. Simulation results indicate that the improved active damping resonance strategy significantly suppresses system resonance and enhances performance.

5. Conclusions

In light of the urgent pursuit of large-scale development in photovoltaic generation, photovoltaic power plants adopt multi-machine parallel grid connection modes. However, as the scale of grid connection continues to expand, the resonance issues within the multi-machine grid system of photovoltaic inverters have become increasingly prominent, severely impacting power quality and the stable operation of the grid. Therefore, an in-depth study of the resonant characteristics and suppression strategies of LCL-type photovoltaic inverter multi-machine grid systems is particularly important. This paper successfully establishes an equivalent model for the multi-machine grid system, thoroughly analyzes the key factors affecting system resonance characteristics, and comprehensively explores the resonant characteristics of the inverter multi-machine grid system under different grid connection scenarios. Based on these studies, an improved active damping suppression strategy is adopted. The main research contents of this paper are summarized as follows:

(1) An equivalent model of the LCL-type inverter multi-machine grid system is established, taking into account parasitic parameters and line impedance, and a general expression for the inverter output current is derived. The influence of parasitic capacitance and resistance in the inverter-side inductance, grid-side inductance, and filter capacitor, as well as line impedance on the resonance characteristics, is analyzed.

(2) To address the issue of LCL-type grid-connected inverters failing to meet grid connection requirements under PI control systems, this paper adopts an improved active damping suppression strategy by combining super-spiral sliding mode control with active damping. Experimental results indicate that the proposed strategy effectively suppresses resonance phenomena in multi-machine grid systems, significantly improving the quality of grid-connected voltage, even when considering the impacts of parasitic parameters and line impedance.

(3) In the MATLAB/Simulink software environment, a corresponding simulation model was constructed based on the adopted improved active damping resonance suppression strategy, followed by detailed simulation analysis and validation. Through comparative analysis, it was successfully demonstrated that the application of this improved strategy effectively suppresses resonance phenomena in multi-machine grid systems when considering the comprehensive effects of parasitic parameters and line impedance.

(4) Although the adopted active damping control strategy is effective, it still requires further exploration and refinement. Specifically, the current strategy does not comprehensively address power quality issues under adverse grid conditions, such as voltage imbalance and the presence of harmonics.