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Scientific Reports volume 14, Article number: 22911 (2024 ) Cite this article hybrid power solutions
LCL filters are extensively utilized in Grid-connected inverters due to their exceptional capability in suppressing high-frequency harmonics. The active damping method is commonly employed to mitigate the resonance peak of the LCL filter. However, this control strategy induces a shift in the natural resonance point. To address this issue, a novel active damping control strategy based on the principle of equivalent transformation is proposed in this paper, which not only effectively suppresses the resonance peak but also avoids deviation from the natural resonance point. Finally, experiments are carried out on a three-phase LCL Grid-connected inverter, and the experimental results show that the control strategy has good steady-state performance, dynamic response, and robustness under both rigid and ultra-weak network conditions.
In the context of achieving carbon neutrality, the renewable energy sector, particularly wind and solar power, has seen significant advancement. Currently, the predominant method of harnessing renewable energy is distributed Grid-connected generation1. Grid-tied inverters play a crucial role in these systems, with L-type and LCL-type inverters distinguished by their filter configurations. Compared to L-type inverters, LCL-type inverters offer enhanced capabilities for suppressing high-frequency harmonics, making them extensively utilized in distributed Grid-connected generation. However, LCL filters, characterized as third-order undamped systems, exhibit resonance peaks at their resonant frequency, resulting in phase shifts up to -180° that can trigger oscillations within the Grid-tied system. Therefore, effectively mitigating these inherent resonance peaks in LCL Grid-tied inverters is imperative to ensure stable control of grid-connected currents2,3.
To mitigate resonance peaks, damping is typically increased using either passive damping (PD) or active damping (AD) methods. In PD methods, damping is achieved through series or parallel resistors. Although this method is simple, it introduces additional power losses due to increased damping resistance and also reduces the ability of the LCL filter to suppress high-frequency harmonics. Contrastingly, AD methods involve feedback of the filter’s voltage or current as state variables into the system, effectively simulating a virtual damping resistor to achieve resonance peak suppression4. Because this method does not need to increase the external passive components, there is no external power loss, so it is widely used in LCL inverter resonance peak suppression.
In5 proposed the concept of virtual impedance based on the idea of model equivalent transformation, and revealed the relationship between the active damping method and passive damping method based on state variable feedback. Literature6 proposes an active damping method using capacitive current PI positive feedback, which extends the effective damping interval to \({f}_{s}/2\) . Reference7 proposed that adding lead compensation control to the capacitor current feedback branch could extend the effective damping interval to \({f}_{s}/4\) . Reference8 used an all-pass filter to phase reshape the output impedance of the converter to ensure positive virtual impedance characteristics in the range of \((0,{f}_{s}/2)\) . These AD methods have successfully suppressed the resonance peak of the LCL inverter, but they do not consider the digital control delay will cause the natural resonance to shift, resulting in system stability decline.
To solve the problem of LCL resonance point offset caused by active damping control strategy, this paper will take capacitive current feedback (CCFB) as an example, based on the idea of virtual impedance equivalence principle, to analyze the reason for resonance point offset. At the same time, an improved active damping control strategy and controller design method are proposed. The control strategy not only provides effective damping for the LCL inverter but also successfully avoids the ground resonance point deviation caused by digital control. Finally, according to the proposed design method, experiments are carried out on the three-phase LCL Grid-connected inverter platform, and the experimental results are analyzed. The results show that the improved active damping strategy is feasible and correct.
Figure 1 depicts a voltage source inverter (VSI) interfaced with the grid through an LCL filter. The equivalent series resistances of inductance and capacitance parasitism are ignored to avoid being affected by PD. In the circuit, inverter side inductance \({L}_{1}\) , grid side inductance \({L}_{2}\) , and capacitor \(C\) constructs the LCL filter and \({L}_{g}\) represent the equivalent power grid inductance9.
The natural resonant frequency \({\omega }_{r}\) can be expressed as Eq. (1), where \({L}_{T}={L}_{2}+{L}_{g}\) . The PCC voltage is measured to realize phase synchronization of grid-connected current, and the current on the grid side is detected to adjust active power and martial power. \({H}_{1}\) is the feedback coefficient of AD control. Moreover, the PI controller \({G}_{c}\left(s\right)\) is used to track reference currents in three-phase systems with a synchronous rotating frame. When asymmetric regular sampling is adopted, the calculated delay is a sampling period \({T}_{s}\) , and the PWM modulation delay can be approximated as half a sampling period \({T}_{s}/2\) , so the total digital control delay \({G}_{d}\left(s\right)\) is modeled as Eq. (3). Regardless of capacitive current feedback, the transfer function of inverter voltage \({v}_{\text{inv}}\left(s\right)\) , inverter current \({i}_{{L}_{1}}\left(s\right)\) , capacitor current \({i}_{c}\left(s\right)\) , grid current \({i}_{{L}_{2}}\left(s\right)\) , capacitor voltage \({v}_{c}\left(s\right)\) and PCC voltage \({v}_{pcc}\left(s\right)\) can be obtained from Fig. 1, as shown in Eqs. (4)–(8).
Control structure diagram of AD system based on CCFB.
Figure 2 is a block diagram of the CCFB active damped transfer function. \({K}_{\text{PWM}}\) is the inverter gain, which can be approximated as Eq. (9), where \({U}_{tri}\) is the carrier amplitude10.
The block diagram depicted in Fig. 2b results from the repositioning of the active damped feedback node caused by the capacitive current illustrated in Fig. 2a. After this simplification, the transfer function of the capacitive current feedback type LCL inverter can be described as Eq. (10).
Control block diagrams of the LCL-type Grid-connected VSI with CCFB approaches. (a) Without simplification. (b) With simplification.
Similarly, the transfer functions of inverter side current feedback (ICFB), capacitor voltage feedback (CVFB), grid current feedback (GCFB), capacitor voltage feed-forward (CVFF), and common coupling point voltage feed-forward (PVFF) can be obtained successively, as shown in Table 111. \({G}_{\text{f}\text{b}}\) and \({G}_{\text{f}\text{f}}\) are respectively feedback coefficients and feedforward coefficients of AD method.
The circuit diagrams of two different passive damping methods are shown in Fig. 3. When the damping resistance is in parallel with the filter capacitor, the transfer function block diagram is shown in Fig. 4, and the transfer function from the inverter voltage to the grid current is shown in the Eq. (11). When the damping resistance is connected in series with the inductor on the inverter side, the transfer function and damping factor are shown in Eq. (12).
Based on the idea of virtual impedance equivalent transformation, the unified virtual impedance models of 6 common AD methods are obtained by defining the feedforward function \({G}_{\text{f}\text{f}}\) and feedback function \({G}_{\text{f}\text{b}}\) , Which are shown in Table 112,13.
Two different passive damping circuit diagrams. (a) The damping resistance is in parallel with the filter capacitor. (b) The damping resistance is connected in series with the inverter side inductor.
Control block diagram of LCL Grid-connected inverter with damping resistance and filter capacitance in parallel.
\(\zeta\) is the damping factor: \(\zeta =\frac{1}{2{R}_{d}C{\omega }_{r}}\) .
\(\zeta\) is the damping factor: \(\zeta =\frac{{R}_{d}}{2{L}_{1}{\omega }_{r}}.\)
According to Eqs. (10) and (11), the AD method increases the damping term in the transfer function of the system by introducing state variables. Without considering the digital delay condition (\({G}_{d}\left(s\right)=1\) ), the damping effect is equivalent to the parallel damping resistance and capacitance in PD method. However, due to the existence of digital control delay, the damping term of the CCFB transfer function in Eq. (10) is equivalent to not virtual resistance, but virtual impedance, as shown in Eq. (13). Upon substituting (\(s=j\omega\) ) into Eq. (13), Eqs. (14)–(16) are derived.
From Eq. (15), it can be seen that only when the frequency is in the range of \((0,{f}_{s}/6)\) , the resistance \({R}_{\text{eq}}\) equivalent to the virtual impedance will be greater than 0. That is, the damping introduced by the AD method in the frequency segment is in the effective region. As shown in Fig. 5, the virtual impedance is paralleled at both ends of the filter capacitor. The filter capacitance of the LCL filter itself is combined with the virtual reactance, which is expressed by \({C}_{\text{eq}}\) , as shown in Eqs. (17)–(18). \({L}_{1}\) , \({C}_{\text{eq}}\) , and \({L}_{2}\) constitute the actual LCL filter, and the resonant frequency is therefore changed, as shown in Eq. (19). Substitute Eq. (18) into Eq. (19) to get the Eq. (20).
Virtual impedance model of CCFB active damping method. (a) Without simplification. (b) With simplification.
When only a single state variable is used as the damping source, the natural resonant frequency of the system will shift due to the digital control delay, and the new resonant frequency is determined by several factors. In this paper, an active damping superposition control strategy is proposed to solve the problem of resonance point offset caused by virtual reactance by introducing CVFF based on the original CCFB. Figure 6 is the control block diagram of the active damped superposition system. \({G}_{x1}\) and \({G}_{x2}\) are respectively the transfer functions of the inverter voltage to the active damped feedback or feedforward state variables, such as Eqs. (4)–(8).
Then the open-loop gain is shown in Eq. (21).
CCFB and CVFF are introduced at the same time, and their control block diagram is shown in Fig. 7a. Figure 7b is obtained by moving its feedback back with the feedforward point. \({G}_{LCL}\) is the transfer function of LCL filter under active damped superposition control strategy, \({G}_{1}={G}_{2}={K}_{\text{PWM}}{G}_{d}\) . The gain of the whole system is shown in Eq. (22). The transfer function of the LCL filter under the active damping superposition control strategy can be obtained from Eq. (21) and Eq. (22), as shown in Eq. (23). Equation (24) is obtained by substituting Eqs. (5) and (7) into Eq. (24).
System control block diagram of active damping superposition.
CCFB and CVFF active damping superposition control block diagram. (a) Without simplification. (b) With simplification.
In Eq. (24), \({K}_{\text{PWM}}{G}_{d}{H}_{1}{L}_{T}C{s}^{2}\) is the active damping introduced by CCFB, and its virtual impedance is shown in Eqs. (13)–(16), denoted as \({R}_{eq1}\) and \({X}_{eq1}\) respectively. \(-{G}_{d}{K}_{\text{ff}}{L}_{T}s\) is the active damping introduced by CVFF, and its virtual impedance is denoted as \({R}_{eq2}\) and \({X}_{eq2}\) respectively, as shown in Eqs. (25)–(27).
The two groups of active damping are converted into virtual impedances in parallel at both ends of the filter capacitor, and the block diagram of their equivalent parallel impedances is shown in Fig. 8.
The virtual impedance after active damping superposition is Eqs. (28) and (29). By connecting Eqs. (11), (17), (18), (19), (20), (28), and (29), the damping factor Eq. (30) after superposition is obtained. In this case, the damping resistance and damping factor are jointly determined by CCFB and CVFF.
When the feedback coefficient of CCFB and the feedforward coefficient of CVFF meet Eq. (31), the corresponding virtual reactance \({X}_{eq1}\) and \({X}_{eq2}\) have parallel resonance. The impedance of \({X}_{eq1}\) in parallel with \({X}_{eq2}\) is infinite, which is equivalent to an open circuit. Therefore, by selecting a reasonable \({H}_{1}\) and \({K}_{\text{ff}}\) , the natural resonant frequency of the LCL filter can be guaranteed without deviation.
Table 2 shows the values of \({H}_{1}\) and \({K}_{\text{ff}}\) in different frequency ranges if Eq. (31) is satisfied and the damping resistance is greater than zero (Eq. (28) > 0). It can be seen from Table 2 that the \({K}_{\text{ff}}\) polarity does not change in the range \((0,{f}_{s}/3)\) . Therefore, it expands the effective damping region, when designing the parameters of the active damping superposition control strategy, the appropriate \({K}_{\text{ff}}\) can be determined first, and then the matching \({H}_{1}\) can be obtained through the adaptive principle. Therefore, the effective positive damping region of the new control strategy can be obtained as \((0,{f}_{s}/3)\) .
The parameters of the Grid-connected inverter in Table 3 are substituted into Eqs. (6), (10), and (24), and the Bode diagram of undamped control, CCFB control, and active damped superposition control is drawn, as shown in Fig. 9. The CCFB control makes the natural resonance point shift to the right. The active damping superposition control strategy is better than the CCFB control strategy in suppressing the resonant peak, and its resonant frequency is consistent with the natural resonant frequency of the LCL filter.
Bode diagram of LCL transfer function.
At the time of design, the phase margin (PM) of the closed-loop system is expected to be greater than 15° and the amplitude margin (GM) is greater than 3 dB to ensure that the system has ideal dynamic response performance and stability margin. Figure 10 shows the Bode diagram of the open-loop transfer function of the inverter system using the active damping superposition control strategy under different network obstacles. Both the phase margin and the amplitude margin meet the design requirements.
Bode diagram of LCL transfer function.
Based on the above analysis, the design process of the control strategy of active damping superposition can be summarized as shown in Fig. 11.
Step 1 (Design the LCL filter): Design the filter parameters according to the requirements of the inverter current ripple, the reactive power absorbed by the filter capacitor, and the harmonic standard injected into the grid. Specific parameter design can be referred to reference9,11,14.
Step 2 (Obtain appropriate \({H}_{1}\) and \({K}_{\text{ff}}\) ): Select the appropriate filter capacitor voltage feedback coefficient \({K}_{\text{ff}}\) by trial and error. At the beginning of the circuit operation, the grid impedance is obtained by injecting high-frequency pulse harmonics into the grid for a short time, and the natural resonant frequency of the LCL is calculated. According to Eq. (31), the capacitance current feedback coefficient \({H}_{1}\) is obtained by adaptive matching.
Step 3 (Check the effectiveness of damping): Substitute \({H}_{1}\) and\({K}_{\text{ff}}\) obtained in step 2 into Eqs. (28) and (30). If the equivalent virtual damping resistance is greater than 0 and the damping factor is greater than 0.1, the design is terminated. Otherwise, repeat step two until the condition is met.
Design process of active damping superposition control method.
To verify the correctness of the proposed control strategy, simulation experiments were carried out under different network impedances, as shown in Fig. 12. The distortion degree of PCC point voltage increases gradually with the increase of grid impedance. However, when the grid impedance is 0mH, 2mH, and 5mH, the THD is shown in Fig. 13, and is only 0.41%, 0.48%, and 0.50% respectively, which can still meet the grid-connection conditions. The experiment shows that the proposed control strategy still has good steady-state performance under weak network conditions.
Waveform of PCC voltage and grid-connected current under different grid impediments. (a) \({L}_{g}=0\text{mH}({f}_{r}<{f}_{s}/6)\) ; (b) \({L}_{g}=3\text{mH}({f}_{r}\approx {f}_{s}/6)\) ; (c) \({L}_{g}=5\text{mH}({f}_{r}>{f}_{s}/6)\) .
Fast Fourier transform analysis results of grid current Ig.
As shown in Fig. 14, during the period of 0–0.1 s, the inverter runs at 100% rated load, and its peak current on the grid side is 20 A; At 0.1s, the load suddenly drops to 50% of the rated load; At 0.2s, the load rises from 50% to the rated load. Although the PCC point voltage fluctuates obviously at the moment of load change, the voltage quickly returns to a stable state within half a cycle. And the grid-connected current can quickly track the load transformation, and reach a new equilibrium state in 1 cycle.
Waveform of PCC voltage and grid-connected current when the inverter load changes.
To verify the steady-state performance of the active damping superposition control strategy, RT-box1 and TI LaunchPad XL-F28069M were used to build a hardware-in-the-loop (HIL), the rated power of the experimental platform is 2100 W, the experimental platform is shown in Fig. 15.
The peak value of the grid-connected current is set to 20 A, and the grid-connected experiments are carried out under the grid impedance of 0mH, 2mH, and 5mH. The experimental results are shown in Fig. 16. Despite the increase in PCC point voltage distortion with higher grid impedance, no discernible oscillation occurs, and the system maintains stability. The three-phase grid current exhibits an almost ideal sinusoidal waveform, while the voltage phase can be accurately tracked, maintaining a stable peak at 20 A. The experimental results align with simulation outcomes, validating that the control strategy retains robust steady-state performance under weak grid conditions.
Waveform of phase A voltage at PCC and three-phase grid-connected current under different grid impediments. (a) \({L}_{g}=0\text{mH}({f}_{r}<{f}_{s}/6)\) ; (b) \({L}_{g}=3\text{mH}({f}_{r}\approx {f}_{s}/6)\) ; (c) \({L}_{g}=5\text{mH}({f}_{r}>{f}_{s}/6)\) .
To verify the anti-interference ability and dynamic performance of the proposed control strategy, the load variation experiment is carried out during the normal operation of the power grid, and the results are shown in Fig. 17. Figure 17a shows that at time t1, the inverter load drops from 100% of the rated value to 50%, that is, the command peak current drops from 20 A to 10 A. At this moment of change, both voltage and current respond quickly. In addition, both voltage and current enter a new steady state in less than one cycle. Figure 17b illustrates that at time t2, the load is restored from 50% to the rated value. In this process, the peak value of grid-connected current increases rapidly in a short time and exceeds 20 A, and there is a small overshoot, but the current remains stable at 20 A after a cycle. In the two dynamic experiments, the THD of the grid current is less than 5%, which can meet the grid-connected conditions. Therefore, the system has good anti-interference ability and dynamic response performance.
Waveform of phase A voltage at PCC and three-phase grid-connected current when Inverter load changes. Where \({L}_{g}=5\text{mH}({f}_{r}>{f}_{s}/6)\) . (a) The load was reduced from 100% value to 50% of the rated. (b) The load is increased from 50–100% of the rated value.
When the grid voltage contains 5, 7, 11 and 13 harmonics, the proposed control method is adopted to carry out grid-connected experiments, and the obtained waveform of A-phase voltage and three-phase grid-connected current at PCC is shown in Fig. 18. Obviously, the voltage waveform of the grid is distorted to a large extent, but the grid-connected current is less affected by the background harmonics of the grid. In this complex condition, the control scheme proposed in this paper can still make the grid-connected inverter system run stably.
Waveform of phase A voltage at PCC and three-phase grid-connected current when the grid voltage contains 5th, 7th, 11th, and 13th harmonics.
The frequency of the power grid was changed to 49.5 Hz and 50.5 Hz respectively, and the grid-connected experiment was conducted under this condition. The experimental results are shown in Fig. 19. As can be seen from Fig. 19, when the grid frequency varies within the range of 1 Hz, the proposed control strategy is adopted. Although the amplitude of the three-phase grid-connected current is slightly different and there is a slight distortion, the system remains stable and does not appear to collapse, and the current is still close to the standard sine wave, which can meet the requirements of grid-connected. The above two experiments show that the control method has good anti-disturbance capability.
Waveform of phase A voltage at PCC and three-phase grid-connected current under different grid frequencies. (a) The grid frequency is 49.5 Hz. (b) The grid frequency is 50.5 Hz.
The feedback and feedforward function is defined to solve the problem of natural resonance deviation of the LCL inverter caused by active damping, and the virtual impedance model of active damping is established. The control strategy of active damping superposition is proposed.
By setting up LCL three-phase Grid-connected inverter system, the experimental results show that the new control strategy can suppress the resonant peak, solve the problem of resonant point offset, and expand the effective damping region to \((0,{f}_{s}/3)\) . After adopting the control strategy of active damping superposition, the inverter system has good anti-interference ability, dynamic performance, and steady performance.
The data used to support the findings of this study are included in the article.
Blaabjerg, F. et al. Overview of control and grid synchronization for distributed power generation systems. IEEE Trans. Ind. Electron.53(5):1398–1409 (2006).
Hava, A., Lipo, T. & Erdman, W. Utility interface issues for line connected PWM voltage source converters: a comparative study. In Proceedings of IEEE Applied Power Electronics Conference.Osaka, Japan 125–13 (IEEE,1995).
Liserre, M., Teodorescu, R. & Blaabjerg, F. Stability of grid-connected PV inverters with large grid impedance variation. In Proceedings of IEEE Power Electronics Specialists Conference 4773–4779 (IEEE, 2004).
Liserre, M., Blaabjerg, F. & Hansen, S. Design and control of an LCL-filter-based three-phase active rectifier. IEEE Trans. Ind. Appl.41(5):1281–1291 (2005).
Dahono, P. et al. Damping of transient oscillations on the output LC filter of PWM inverters by using a virtual resistor. In Proceedings of IEEE International Conference on Power Electronics and Drive Systems 403–407 (IEEE, 2001).
He, Y. Y. et al. Capacitor-current proportional-integral positive feedback active damping for LCL-type grid-connected inverter to achieve high robustness against grid impedance variation. IEEE Trans. Power Electron.34(12), 12423–11243 (2019).
He, Y. et al. Hybrid active damping combining capacitor current feedback and point of common coupling voltage feedforward for LCL-type grid-connected inverter. IEEE Trans. Power Electron.36(2), 2373–2383 (2021).
Yao, W. L. et al. Phase reshaping via all-pass filters for robust LCL-filter active damping. IEEE Trans. Power Electron.35(3), 3114–3126 (2020).
Wei Chen, Z. et al. Active damp-ing control for LCL Filters with inverter-side cur-rent feedback only. IEEE Trans. Power Electron.37(9), 10065–10069 (2022).
Wang X., Ruan, X. & Liu, S. Control strategy for grid-connected Inverter to suppress current distortion effected by background harmonics in grid voltage. Proc. CSEE 31(06), 7–14 (2011).
Chen, W. et al. Unified active damping strategy based on generalized virtual impedance in LCL-type grid-connected inverter. IEEE Trans. Ind. Electron.70(8), 8129–8139 (2023).
Wang, Y., Wang, W. & Li, Y. PCC voltage feedback active damping control strategy for grid-connected converter with LCL filter. Electr. Drive47(02), 29–34 (2017).
Geng, Y., Qi, Y., Dong, W. et al. An active damping method with improved grid current feedback. Proc. CSEE38(18), 5557–5567 (2018).
Tang, W. & Ke, M. Critical damping ratio to ensure design efficiency and stability of LCL filters. IEEE Trans. Power Electron.36(01), 315–325 (2021).
This work was supported by the Science and Technology Project of Quanzhou City under Grant 2024QZC004R, and the Natural Science Foundation of Fujian Province under Grant 2024J01208.
School of Electronics and Electrical Engineering, Minnan University of Science and Technology, Quanzhou, 362700, Fujian, China
Dongdong Chen & Shengqi Zhang
College of Electrical Engineering and Automation, Fuzhou University, Fuzhou, 350108, Fujian, China
Dongdong Chen & Shengqi Zhang
School of Electrical Engineering, Xi'an Jiaotong University, Xi'an, 710049, Shaanxi, China
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Dongdong Chen: Dongdong Chen led the overall research design and methodology, including the conceptualization of the active damping control strategy. He also played a key role in drafting and revising the manuscript.Shengqi Zhang: Shengqi Zhang contributed significantly to the development of the theoretical framework and the mathematical modeling. He also carried out the simulations and data analysis, and provided critical feedback on the manuscript.Hongyi Lin: Hongyi Lin was responsible for the experimental setup and validation of the proposed control strategy. He also contributed to the interpretation of the results and assisted in finalizing the manuscript.
The authors declare no competing interests.
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Chen, D., Zhang, S. & Lin, H. An active damping control strategy for suppressing LCL resonant point migration for three-phase grid-tied inverter. Sci Rep 14, 22911 (2024). https://doi.org/10.1038/s41598-024-73897-5
DOI: https://doi.org/10.1038/s41598-024-73897-5
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