Harmonic Suppression Strategy of LCL Grid-Connected PV Inverter Based on Adaptive QPR_PC Control

Abstract

To reduce the influence of voltage harmonics on the grid current, a control strategy based on adaptive quasi-proportional phase compensated resonance (QPR_PC) is proposed. Firstly, the LCL grid-connected photovoltaic inverter system model is established, and the stability performance of the three-level inverter system under double closed-loop control is analyzed using the output impedance model of the inverter. Then, a QPR regulator for the zero steady-state error tracking of AC signals is studied. To solve the problem that the system has poor robustness against frequency changes when the traditional QPR regulator is used in the static coordinate system, this paper improves the traditional QPR regulator to optimize the response characteristics of the closed-loop system. Based on the QPR regulator, a phase margin compensation structure is introduced to form QPR_PC control. Then adaptive frequency design is added to ensure good control even when the power grid frequency drifts, which is the control strategy proposed in this paper. Verification shows that the proposed method improves the phase margin of output impedance at a specific frequency, restrains the interference of the 3rd, 5th, and 7th harmonics of grid voltage, and improves the dynamic performance of the system and the quality of grid-connected current. Finally, the simulation results show that the total harmonic distortion rate of grid-connected current is reduced by 1.03% after adopting this strategy, which verifies the effectiveness and correctness of the proposed method.

1. Introduction

The continuous expansion of distributed generation systems, multiple transformers, and long transmission lines is used for interconnection and connection systems [1]. Therefore, in the modern distributed generation system of the public power grid, the inverter cannot be ignored, and its structure and control strategy will directly affect the power quality of the grid [2]. In photovoltaic power generation systems, many non-linear devices are usually connected at the point of common coupling (PCC) of the grid-connected inverter to ensure living needs. After the harmonic current generated by these devices passes through the line impedance, the grid voltage at PCC contains more harmonics, eventually leading to the distortion of the grid-connected current and the instability of the power system [3,4]. Renewable energy is fed to the power grid by a grid-connected inverter, which constitutes a local microgrid. Therefore, grid-connected inverter technology is one of the key technologies for renewable energy utilization. Grid-connected inverters generally use current source control under pulse-width modulation(PWM) to inject large amounts of high-frequency current into the grid and affect the grid’s quality [5,6]. To filter out the high-frequency current, a filter inductor is usually added between the inverter and the grid. Many scholars have proposed an LCL filter [7]. Compared with the traditional L-type filter, an LCL filter provides a high-frequency bypass through the capacitor branch, thus greatly attenuating the high-frequency component of the current flowing into the grid, which is widely used in high-power equipment.

The three-phase sinusoidal input is still a sinusoidal quantity in a two-phase stationary coordinate system and, therefore, cannot be controlled using a PI regulator [8]. Although static error can be reduced using PI plus power grid voltage feed-forward control, static error still cannot be completely eliminated. In addition, due to the low-frequency harmonics in the inverter output, the PI regulator cannot eliminate such harmonics [9]. Quasi-proportional resonance (QPR) control proposed in some literature can provide infinite gain at a specified frequency, thus achieving static error-free control at a specific frequency [10,11]. If the resonance control is added at 3, 5, 7, etc., harmonics of these frequencies can be eliminated in a targeted manner, and the quality of current injected into the grid can be significantly improved [12].

To improve the stability and robustness of the system to grid-connected harmonics, several control strategies for grid-connected inverters are proposed. A notch-like digital filter-based single-loop active damping method is proposed in [13] to make the LCL inverter property parameters and characteristic grid changes robust. In [14], the authors demonstrated the existing admittance coupling among multiple parallel inverters and obtained the multiple parallel grid-connected inverter’s admittance by considering the influence of PLL with the admittance coupling factor. In [15], the authors studied a harmonic voltage distortion damping method to parallel-connected LCL-type inverters in islanded operations, which includes a P + Repetitive controller and the parallel virtual-admittance control. The performance of controllers with different phase compensation is analyzed and compared in [16]. This paper reveals that the traditional PI and HQR controllers with P-GVF are unsuitable for parallel applications. In the low-frequency band, the output impedance of the inverter is passivated by WP-GVF and HQR-PC controllers improved by the traditional control method. In the high-frequency band, the system has the minimum phase characteristic after using PQR-PC for delay compensation. Multiple harmonic quasi-resonant (HQR) controllers are used in [17] to suppress harmonic current. This proves that the maximum order of a plug-in HQR controller is limited by the control bandwidth, which is significantly reduced due to grid impedance. To reduce the computational number and complexity, the control parameter design method based on the dual-current loop active damping scheme is proposed in [18]. The harmonic suppression method is proposed in [19] for quasi-proportional resonant of multiple parallel inverters and voltage external trap control. Focus on a traditional inverter, an improved active damping inhibition strategy based on PIR current controller is proposed in [20], which can effectively inhibit system resonance, though this strategy makes it difficult to accurately track command current and makes no static difference in the tracking of harmonic components.

Given the limitations in the above literature, the stability analysis of output impedance under double closed-loop control and the correlation between grid-connected voltage and grid-connected current are described in detail in this paper, thus affecting the stability of an LCL grid-connected inverter cluster system. The QPR_PC current control strategy with phase compensation is adopted to improve the output impedance phase margin at a specific subfrequency and to track the AC component of a specific frequency. The influence of network voltage on network current is suppressed quickly. This scheme ensures the excellent quality of the system and improves the stability of the LCL grid-connected inverter. Finally, the effectiveness of the proposed strategy is verified by simulation and experimental analysis.

2. Mathematical Model of LCL Photovoltaic Grid-Connected Inverter

In this section, the relationship between the ratio of power grid impedance and output impedance is found after a comprehensive analysis of the output impedance model of the LCL grid-connected inverter based on double closed-loop control.

2.1. System Model

The cluster system model of LCL grid-connected photovoltaic inverters studied in this paper is shown in Figure 1, where C₁,C₂ are the support capacitors of the DC side; PVi is the photovoltaic array, where #i = 1, 2, 3,…,n; three-phase state variable u_g, i₁, are grid voltage, inverter side current, grid current; Z_g = sL_g is the grid perceptual impedance; L₁, L₂, C is LCL filter side inductance, network side inductance, and filter capacitor. I* is grid current [21,22,23].

The parasitic resistance and inductance of the filter capacitor are ignored [24]; the quantitative relationship between variables of the LCL filter in Figure 1 is:

$$\{\begin{array}{l}{L}_1\frac{d{i}_{L_1}}{dt}=u_{i\mathrm{n}v}-u_c\\ C\frac{d{u}_c}{dt}=i_L-i_s\\ L_2\frac{d{i}_{L_2}}{dt}=u_c-u_{pcc}\end{array}$$

(1)

2.2. Inverter Output Impedance Model

According to Equation (1), the equivalent model of inverter #i in the continuous domain is shown in the dotted line part in Figure 2a. Figure 2a is the application of capacitive current i₁ and grid-connected current i_L2 feedback double closed-loop control block diagram; where kd is the active damping proportional coefficient, PWM amplification gain k_PWM = 1, i_ref is a grid-connected current reference value, U_PCC is the grid-connected point voltage, G_c is the current loop control function, and G_d is the delay function introduced in the digital control process, namely:

$$G_d(s)=\frac{1}{T_S}\frac{1-e^{-T_{s}s}}{s}{e}^{-T_{s}s}$$

(2)

where Ts is the sampling and switching period.

The open-loop transfer function of the system is:

$$G_o=G_c{k}_{pwm}{G}_d/\left[s^{3}C{L}_1{L}_2+s^2{k}_{pwm}{G}_d{k}_d\left(L_2+L_g\right)C+s\left(L_1+L_2+L_g\right)\right]$$

(3)

Equivalent transformation of Figure 2a yielded Figure 2b. Among them:

$$\begin{gathered} G_{s1}=k_{pwm}{G}_d{G}_c/[1+sC{k}_{pwm}{k}_d{G}_d+s^2{L}_{1}C] \\ \begin{array}{l}{G}_{s2}=[1+sC{k}_{pwm}{k}_d{G}_d+s^2{L}_{1}C]/[s(L_1+L_2+L_g)\\ +s^2(L_2+L_g)C{k}_{pwm}{k}_d{G}_d+s^3{L}_1(L_2+L_g)C]\end{array} \end{gathered}$$

The mesh current i_g of the #i inverter can be derived from Figure 2b:

$$i_g=G_0{i}_{ref}-Y_0{u}_{pcc}$$

(4)

In Equation (4), G₀ is the closed-loop transfer function of the current reference value to the inverter output current:

$$G_0={\frac{i_g}{i_{ref}}|}_{u_{pcc}=0}=\frac{G_{s1}{G}_{s2}}{1+G_{s1}{G}_{s2}}$$

(5)

Y₀ is a closed-loop transmission function of the grid voltage to the inverter output current:

$$Y_0={\frac{i_g}{u_{pcc}}|}_{i_{ref}=0}=\frac{G_{s2}}{1+G_{s1}{G}_{s2}}$$

(6)

It can be seen as an equivalent output guide of the inverter. Then the current control inverter can be described using a Norton model; namely, the current source is parallel with the output conductor Y₀ [25]. A Norton model of the n-parallel inverter system can be built from the perspective of each sub-frequency converter, as shown in Figure 3a.

2.3. Stability Analysis

According to Figure 3a, the line current i_gi of the #i inverter is known:

$$i_{\mathrm{gi}}=i_{c\mathrm{i}}-u_{pcc}{Y}_0$$

(7)

and

$$u_{pcc}=\frac{i_{c\mathrm{i}}+{\displaystyle \sum _{\mathrm{j}\ne \mathrm{i}}^n{i}_{c\mathrm{j}}+u_g{Y}_g}}{n{Y}_0+Y_g}$$

(8)

According to the literature [26,27], when all inverters have the same working conditions and the same parameters and n parallel inverters are taken as the whole, the circulating current is approximately 0. Therefore, a simplified equivalent impedance model of N parallel inverters can be established from the perspective of PCC, as shown in Figure 3b. The injection current i_gi of the #i inverter, which can be derived:

$$i_{g\mathrm{i}}=\frac{Z_0}{Z_0+n{Z}_g}{\displaystyle \sum _{\mathrm{i}=0,1\dots }^n{i}_{c\mathrm{i}}}-n\frac{u_g}{Z_0+n{Z}_g}$$

(9)

In Equation (9), the quality of the grid current of the LCL grid-connected inverter system under double closed-loop control is related to nZ_g/Z₀ and ug. When gate impedance Z_g ≠ 0, in order to ensure system stability, the phase margin θ_PM between Z_g and Z₀ satisfying Nyquist criterion should be greater than zero. Therefore, the subsequent research mainly focuses on ensuring the amplitude margin and phase margin of the output impedance.

3. QPR_PC Control Strategy

3.1. The QPR Control Policy

According to the analysis in the previous section, to maintain good system stability, effective control strategies can be adopted to improve the amplitude margin and phase margin of the output impedance Z₀. The output impedance of the inverter under double closed-loop control is expressed by Equation (6):

$$\begin{array}{l}{Z}_0=[k_{pwm}{G}_d{G}_c+s(L_1+L_2+L_g)+s^2(L_2+L_g)C{k}_{pwm}{k}_d{G}_d\\ +s^3{L}_1(L_2+L_g)C]/[1+sC{k}_{pwm}{k}_d{G}_d+s^2{L}_{1}C]\end{array}$$

(10)

In (10), the amplitude margin and phase margin of output impedance Z₀ can be adjusted by designing the current controller G_c. The QPR regulator is widely used due to its good performance and low cost in a static coordinate system for fundamental wave AC tracking. Therefore, this section mainly analyzes the performance of the QPR regulator applied in double closed-loop knowable conditions.

The typical QPR controller transfer function expression is:

$$G_{QPR}=k_p+\frac{2k_r{\omega }_{c}s}{s^2+2{\omega }_{c}s+{\omega }_0{}^2}$$

(11)

where k_p is the proportional gain and k_r is the resonant gain; ω₀ = 2πf₀ is base wave angular frequency; ω_c is resonance term bandwidth, which increasing ω_c can improve the adaptability of QPR controller to power grid voltage frequency.

The gain at ω₀ is:

$$G_{QPR}(j{\omega }_0)=\sqrt{k_p{}^2+{(\frac{k_r{\omega }_c{\omega }_0}{{\omega }_0{}^2+{\omega }_c{\omega }_0})}^2}$$

(12)

According to (12), the QPR controller has infinite gain at ω0 to realize AC signal control without static error and small steady-state error. The Bode diagram corresponding to the PR controller is shown in Figure 4. It can be seen that the gain of the proportional resonance controller does tend to infinity at the set resonant frequency. If the control system in Figure 4 adopts the QPR controller, the closed-loop transfer function of the system tends to 1 at the resonant frequency; that is, the input reference signal is tracked theoretically without error. However, without improvement, the QPR controller inevitably has its defects; that is, when the grid voltage frequency deviates from the fundamental frequency, the QPR in the mains frequency controller gain falls sharply, no longer going to infinity, so that the tracking grid voltage frequency sine to time unable to realize zero steady-state error, and the resulting system control effect is poor. If the parameters kp and kr are set improperly, the controller gain at the reference signal frequency becomes negative. The controller will suppress the reference signal instead, which deteriorates the control performance of the whole system. Therefore, the influence of the voltage and frequency fluctuation of the power grid on the controller must be considered.

In the industrial field, the frequency of commercial power often fluctuates, especially in harsh environments of electricity consumption, such as a local weak power grid system on the edge of the edge power grid, a system powered by a diesel generator, an independent wind power system, or places where electricity is in short supply. When the frequency fluctuation, the control effect of the resonance controller will be affected, especially the resonance controller working at three, five, and seven times of fundamental wave frequency resonance point, the influence is very obvious. Therefore, to effectively improve the anti-power grid frequency disturbance ability of the PR controller, reduce its sensitivity to power grid frequency fluctuation so that it can still maintain a good control effect when the power grid frequency drift has a strong practical guiding significance.

3.2. QPR_PC Control Strategy

To solve the problem of a system subject to frequency changes when the traditional QPR regulator is used in the static coordinate system, this paper improves the traditional QPR regulator to optimize the response characteristics of the closed-loop system. To effectively suppress harmonic components at a specific frequency and enable the current controller to output high-quality power grid current even under power grid voltage disturbance, a QPR regulator with phase compensation is proposed, as shown in Figure 5. Compared to the traditional QPR regulator, the QPR_PC regulator maintains the high-gain characteristic in the fundamental frequency, not only processing the current reference signal through the proportion compensation and harmonic compensation but also adding the phase margin compensation, which improves the performance of the grid current tracking under the grid voltage disturbance.

As shown in Figure 4, the QPR expression at the base frequency is:

$$G_{QPR}=k_p+k_r\frac{{\omega }_{c}s}{s^2+{\omega }_{c}s+{\omega }_0{}^2}$$

(13)

The phase margin compensation expression at a particular harmonic frequency is:

$$G_{PC}=\sum _{n=3,5,7}{k}_r\frac{\sqrt{1-{\lambda }_n^2}s-n{\lambda }_n{\omega }_0}{s^2+2{\omega }_{c}s+{\left(n{\omega }_0\right)}^2}$$

(14)

In Equation (14), the relationship between the phase angle lag quantity θ_n, λ_n, and θ_n to determine the harmonic frequency is:

$${\theta }_n=\arctan ({\lambda }_n/\sqrt{1-{\lambda }_n{}^2})$$

(15)

In the actual system operation process, the system parameters will change with the change of external factors, which will have an important impact on the system’s performance, especially the LCL filter parameters. Due to the increase in temperature and current, the inductance, capacitance, and resistance values will change greatly. Therefore, it is necessary to analyze the influence of filter parameters on the closed-loop poles of the system to ensure the robustness of the system during operation. The LCL filter and system parameters obtained in this paper are shown in

Table 1. Paramenters of the grid-connected inverters system.

Parameter	Numerical Value
DC bus voltage, Udc	800 V
Power rating, P	5 kW
Inverter side inductance, L1	0.78 mH
Net side electrical feeling, L2	0.45 mH
Filter capacitor, C	6.8 μF
Power grid voltage, Ug	220 V
Frequency of sampling, fs	15 kHz
Resonance frequency, fc	1 kHz
Power grid voltage frequency, f	50 Hz

.

According to the principle of control theory, to make the system stable, the system’s roots must be distributed in the left half plane of the coordinate system. Secondly, the system parameters should be designed with a certain margin so that the system can still work stably when the parameters of the LCL filter change. Thirdly, to ensure that the system can effectively suppress the resonant link of the LCL filter and improve the quality of the output current, the poles generated by the LCL filter should have enough damping to attenuate the oscillation, and should be kept away from the virtual axis to weaken the influence on the system. Since the QPR controller only affects its resonant frequency, at the system crossover frequency ω_i, the G_c can be approximately equal to the k_p, binding Table 1 parameters available:

$$k_p=2\pi (L_1+L_2+L_g)f_{\mathrm{c}}$$

(16)

In this paper, k_p is set to 9.2. The harmonic suppression structure with the 3rd, 5th, and 7th harmonics phase compensation is shown in Figure 5.

The resonance bandwidth ω_c = 2πf_c = 6.3. For the system to have sufficient gain at nω₀, it should satisfy:

$$20\lg (\frac{k_r}{{\omega }_c})>40dB$$

(17)

Here, the k_r value is 600.

The ideal phase angle θ_n* between the modified QPR controller and λ_n is:

$${\lambda }_n=\frac{{\omega }_c{k}_p\tan {\theta }_n^{*}+\sqrt{k_r^2{\mathrm{t}\mathrm{a}\mathrm{n}}^2{\theta }_n^{*}-{\left({\omega }_c{k}_p\mathrm{t}\mathrm{a}\mathrm{n}{\theta }_n^{*}\right)}^2+k_r^2}}{k_r\left(1+{\mathrm{t}\mathrm{a}\mathrm{n}}^2{\theta }_n^{*}\right)}$$

(18)

To stabilize the system under the control of the QPR_PC above −45°at nω₀, λ₃ = 0.454, λ₅ = 0.643, and λ₇ = 0.666 are selected; the parameters of the QPR_PC controller are shown in the table below.

The bode diagram corresponding to the QPR controller can be drawn from

Table 2. The QPR_PC controller parameters.

Parameter	Numerical Value
Proportional gain, kp	9.2
Harmonic gain, kr	600
Tuning bandwidth, ωc	6.3
λ3	0.454
λ5	0.643
λ7	0.766

parameters, combining Equations (11), (13) and (14).

Figure 6 shows the bode diagram corresponding to the QPR controller. It can be seen that the QPR_PC controller does not change the general trend of the original QPR controller; that is, it maintains the original stability margin and turn frequency. Compared with the latter, the system can maintain a larger gain at a specific frequency, nω₀, and effectively suppress the influence of power grid voltage disturbance on the system. This solves the problem of the poor robustness of the system against frequency changes when using the traditional QPR regulator in a static coordinate system to optimize the response characteristics of the closed-loop system. It provides a large gain for the system to realize the station-free tracking of the ac component with the frequency nω₀. At the same time, according to the control theory, the larger the open-loop gain of the system, the stronger the ability to resist the disturbance in the ring. Therefore, the system has a strong resistance to the interference of the nω₀ frequency.

According to Equations (11), (13) and (14) and combining the parameters of Table 1 and Table 2, a Potter diagram of the output impedance Z₀ under QPR control and QPR_PC control is shown in Figure 7.

Figure 7 illustrates the output impedance Z₀ under the control of QPR. Obviously, it has an amplitude of high modulus over 40 dB in the low-frequency band, which favors the suppression of the harmonic. However, the phase of Z₀ is similar to the −90° line, requiring more inverters in parallel, which may lead to system instability. In contrast, the QPR_PC control proposed in this paper can not only keep the output impedance Z₀ at high amplitude but also improve the phase margin of Z₀, in particular at specific frequencies, due to the addition of the phase margin compensation link; the harmonic of the power grid is well suppressed, effectively improving the system’s stability.

In conclusion, the QPR_PC control strategy of band phase margin compensation proposed in this paper can not only reduce system cost but also track the base wave component of the grid current under the condition of grid voltage disturbance, effectively inhibit the main low harmonic, improve the quality of grid-connected current, and then improve the stability of the grid-connected system. The poor robustness of the system against the frequency change is solved when the traditional QPR regulator is used in the static coordinate system, and the response characteristics of the closed-loop system are optimized.

3.3. Adaptive Frequency Design

The cutoff frequency ω_c is mainly used to restrain the power grid frequency fluctuation and the adverse influence of the precision limit on the control effect in the digital realization of the transfer function. The value of ω_c does not affect the gain of the QPR controller at the resonant frequency nω₀. When the power grid frequency fluctuation is small, ω_c can be taken to a smaller value, and the frequency signal in a smaller bandwidth range can have a high enough steady-state gain by adjusting k_r. However, the allowable frequency deviation of China’s power grid for small-capacity systems is ±0.5 Hz, and the harmonic frequency fluctuation range is an integer multiple of this. When the ω_c value is small, the actual harmonic frequency can be outside the high-gain band range of the controller, so it is difficult to achieve zero steady-state error control of the harmonic signal.

By increasing the cutoff frequency ω_c or resonant control gain k_r, the gain of the control system in the frequency fluctuation range can be improved, but the system’s stability will be reduced.

In the actual operation, the power grid frequency does not fluctuate dramatically, and the harmonic frequency is always an integer multiple of the fundamental frequency. To realize the robust adaptive of the control system to the frequency fluctuation, a digital phase-locked loop (DPLL) is adopted in this paper to detect the power frequency period T₀(=1/f₀) of the system in real-time, and the analog-to-digital conversion chip is controlled to make the sampling number N in unit power frequency period T₀ fixed. In this case, the sampling period T_s = 1/(Nf₀), Zn can be expressed as:

$$Z_n=\frac{n{\omega }_0}{\tan (n\pi /N)}$$

(19)

According to Equation (19), the corresponding difference equation of the QPR controller with harmonic number n is:

$$y_n(k)=k_r{Y}_n(u_n(k)-u_n(k-2))-X_{1n}{y}_n(k-1)X_{2n}{y}_n(k-2)$$

(20)

where y_n(k) is the output of the QPR controller with harmonic number n; u_n(k) is the input to the corresponding QPR controller. And:

$$Y_n=\frac{2{\omega }_c{Z}_n}{Z_n{}^2+2{\omega }_c{Z}_n+{\omega }_n{}^2}$$

(21)

In Equation (21), n is the given harmonic number, n is a constant, and the fluctuation of system frequency leads to the change of ω_c/ω₀ value. Since the value of ω_c only affects the controller’s bandwidth, allowing a certain adjustment range, and the fluctuation of power grid frequency during actual operation is small, ω_c/ω₀ can be set as a constant.

$$k={\left.\frac{{\omega }_c}{{\omega }_0}\right|}_{{\omega }_0=2\pi f_e}$$

(22)

where f_e is the rated frequency of the power grid, and f_e = 50 Hz. nω₀ is always based on the resonant frequency point of the difference Equation (20) of the QPR controller with this coefficient, thus ensuring the robustness of the control system against frequency fluctuation. In the specific implementation, W_C only needs to restrain the adverse influence of precision limitation in the digital implementation of the transfer function on the control effect, so the constant k should be a smaller value.

The Potter diagram of output impedance Z₀ based on adaptive QPR_PC control is shown in Figure 8. It can be seen from the figure that the application of adaptive QPR_PC control can greatly improve the amplitude and phase of output impedance Z₀, improve the anti-interference of the system, and ensure the stability of the system. Because the resonant angular frequency is very stable, even during the process of power grid voltage distortion or mutation, adaptive QPR_PC resonance output can remain stable. The integral output fluctuations in the power grid voltage mutation moment, though not completely, is a dc component. The rotation transformation will generate a harmonic under the static coordinate system and reduce the system’s stability. The adaptive QPR_PC controller has stronger power grid adaptability. The harmonic output content of adaptive QPR_PC is very small, so the output current waveform is very sinusoidal.

To reduce the burden on the QPR regulator, voltage feed-forward upcc is also introduced into the system control, as shown in Figure 1. The voltage feed-forward upcc is directly proportional to the grid voltage and inversely proportional to the DC side voltage. The modulation wave is output by the feed-forward quantity upcc and the current loop PR regulator. Feed-forward reduces the burden of the PR regulator in the modulation process, so the adjustment speed of DC bus fluctuation can be improved, and its response speed can be increased by adding feed-forward.

Where G_f is the proportional feed-forward coefficient of power grid voltage, and when G_d is ignored, the angular frequency of the denominator D_Z0 is:

$$f_{cor}=\frac{1}{2\pi }\sqrt{\frac{1-G_f}{L_{1}C}}$$

(23)

To avoid a 90° phase lag caused by G_f = 1, thus affecting the passive characteristics of Z₀, the f_cor is pushed away from the inverter bandwidth f_bw to alleviate the phase delay. According to Equation (23), the following equation can be obtained:

$$G_f<1-L_{1}C{(2\pi f_{bw})}^2$$

(24)

When G_f = 0.7, it can guarantee type (8) was set up without significantly reduced |Z₀|, improving the stability of the low-frequency band. This section may be divided by subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

4. Simulation and Experimental Verification

4.1. Incisive

To verify the effectiveness of the proposed method, the type T three-level grid-connected inverter model was built in the simulation software Matlab/Simulink for analysis, and the parameters are shown in Table 1 and Table 2. The PV power side voltage Udc and the grid voltage Ug are provided by the DC constant voltage source and the AC voltage source, respectively, in which the Ug contains the 3rd, 5th, and 7th harmonics; the grid impedance Zg is simulated by a series of inductance between the PCC point and the Ug.

Figure 9a,b are the reference current i_ref and grid current i_g waveforms during QPR control and adaptive QPR_PC control strategy. We can see that the grid-connected current contains many of the 3rd, 5th, and 7th harmonics and the phase lag. Under the adaptive QPR_PC control, the grid current can also track the reference current excellently, and the current quality is significantly improved.

Figure 10 shows the parallel current THD analysis using the QPR control strategy and the adaptive QPR_PC control strategy proposed. The harmonic distortion rate of the grid-connected current has changed from 1.97% to 0.94%, reduced by 1.03%, which shows that the control strategy can effectively reduce the grid-connected current harmonic, improve the current quality, and verify the effectiveness of the control strategy proposed in this paper.

From the simulation results, the proposed control strategy can effectively reduce the grid-connected current harmonic and improve the current quality.

4.2. Validation

To verify the effectiveness of the control strategy of a three-phase LCL grid-connected inverter, a three-phase LCL grid-connected inverter is built in this paper for experiments. The parameters of the experimental platform are the same as those in the previous table. In the experiment, a programmable AC source is used to simulate the distorted network conditions. Figure 11 shows the experimental waveform of grid-connected current under the traditional grid voltage feed-forward control and the proposed coordinated control strategy.

By comparing Figure 11a,b, it can be seen that the current is seriously distorted when the traditional power grid voltage feed-forward control is adopted, while the current is perfect when the proposed coordinated control strategy is adopted. Therefore, it can be seen that the proposed coordinated control strategy can effectively suppress the harmonic content, ensure the system’s stability, and improve the power quality, which verifies the conclusions mentioned above.

5. Conclusions

In LCL grid-connected photovoltaic inverter systems, the factors affecting stability include grid-connected impedance and grid-connected voltage. To improve the stability of the grid-connected inverter, a control strategy based on adaptive QPR_PC was proposed in a static coordinate system to solve the problem of multi-frequency component interference during compensation, and phase margin compensation was introduced based on QPR control. Simulation results show that compared with classical QPR control, the proposed control strategy can effectively suppress the adverse effects of the system and effectively track the current without any difference. Through simulation and experimental verification, the harmonic distortion rate of the connection current is reduced by 1.03%, and the waveform of the grid-connected current has better quality, indicating that the system can resist the interference of the 3rd, 5th, and 7th harmonics. LCL grid-connected inverter cluster system runs stably and can inject high-quality current. There are still some theoretical and practical problems in the study of this manuscript. Experiments based on four parallel grid-connected inverters further verify the effectiveness of the proposed control method.