Increasing the proportion of renewable energy sources, such as wind and solar, in the power system is a crucial pathway towards achieving the goals of carbon peak and carbon neutrality. The grid-connected inverter serves as the core unit of these renewable energy generation systems, making its performance paramount. To filter out the switching harmonics generated by Pulse-Width Modulation (PWM), L or LCL filters are connected at the output of the grid-connected inverter. The LCL filter offers superior switching harmonic attenuation capability and allows for a reduced filter volume, leading to its widespread adoption. However, it exhibits a resonance peak and a corresponding -180° phase jump at its resonant frequency. This frequency characteristic can easily lead to instability and oscillation in the grid-connected inverter.
To address this issue, damping must be applied to the resonance peak. To avoid introducing additional losses, active damping techniques are employed. This method involves feeding back the state variables of the LCL filter to virtually create a resistor, simulating the damping function of a physical resistor. Active damping can utilize feedback from a single state variable or combinations of different state variables, offering flexible implementation. Among these, capacitor-current feedback active damping has garnered significant attention due to its simplicity.
For capacitor-current proportional feedback active damping based on analog control, it is equivalent to connecting a virtual resistor in parallel with the filter capacitor. However, when digital control is employed, the inherent control delay causes this proportional feedback to no longer be equivalent to a pure virtual resistor, but rather a frequency-dependent virtual impedance. Considering a 1.5-sample digital control delay, the resistive part of the equivalent virtual impedance exhibits a watershed at \(f_s/6\) (where \(f_s\) is the sampling frequency), dividing the frequency range into positive and negative resistance regions. When the system’s resonance frequency falls within the negative resistance range, it creates a negative-damping effect. At this point, the system possesses a pair of open-loop unstable poles and exhibits non-minimum phase characteristics. Correspondingly, the stability condition involving the Gain Margin (GM) becomes complex. Particularly at \(1/6\) of the sampling frequency, the GM requirement cannot be satisfied, making it difficult to stabilize the grid-connected inverter. To mitigate or even eliminate the negative-damping effect, numerous studies have investigated capacitor-current feedback active damping to expand the positive-damping frequency range. For instance, using a first-order high-pass filter as the feedback function can expand the positive-damping range to \(0.28f_s\). Using a negative proportional-integral function can expand it to \(0.48f_s\). Typically, analyzing the positive-damping frequency range corresponding to a feedback function requires deriving the equivalent virtual resistance for different functions through block diagram equivalent transformations, followed by analyzing their frequency characteristics one by one. This process necessitates enumerating various capacitor-current feedback functions and analyzing them individually, which is labor-intensive and relatively complex. Moreover, recalculation is inevitable when the feedback function or its parameters change. Therefore, this paper proposes a general evaluation method suitable for different capacitor-current feedback functions to assess active damping characteristics. The main contributions are summarized as follows:
- The graphical method proposed in this paper can identify the positive-damping frequency range corresponding to different capacitor-current feedback functions. It can analyze their applicability and robustness against fluctuations in the LCL resonance frequency, providing guidance for selecting capacitor-current feedback functions.
- For a specific capacitor-current feedback function, using the proposed graphical method allows for an intuitive and rapid definition of its corresponding positive-damping frequency range. Selecting an appropriate feedback function and optimizing its parameters can expand the positive-damping range to cover the entire control frequency band, i.e., the Nyquist frequency \(f_s/2\), ensuring stable operation of the grid-connected inverter.
Compared to traditional methods, the graphical results obtained in this paper allow for an intuitive analysis of the virtual resistance frequency characteristics under various capacitor-current feedback functions, thereby defining the frequency interval where the feedback function achieves positive damping. This method conveniently facilitates the exploration of suitable feedback functions.
The main content of this paper includes: Firstly, establishing a dual-loop model with the active damping inner loop and the grid current outer loop, clarifying that ensuring the stability of the damping inner loop can prevent the emergence of the negative-damping effect. Deriving the phase restriction condition for the stability of the damping inner loop on the feedback function, obtaining a general phase constraint region. Utilizing this phase constraint region to intuitively define the positive-damping frequency interval for capacitor-current feedback active damping. Furthermore, proposing a capacitor-current feedback active damping strategy based on a phase-lag compensator, achieving strong robustness against LCL resonance frequency fluctuations. The correctness of the theoretical analysis is verified through a 6kW prototype.
Negative-Damping Effect of Capacitor-Current Feedback Active Damping
System Description and Modeling of LCL Grid-connected Inverter
The structure of a single-phase LCL grid-connected inverter is shown in the figure below. The LCL filter consists of the inverter-side inductor \(L_1\), the grid-side inductor \(L_2\), and the filter capacitor \(C\). \(V_{in}\) is the input DC voltage, \(v_{inv}\) is the bridge arm output voltage, and \(v_g\) is the grid voltage. Typically, \(L_g\) represents the grid impedance. In the control unit, the Phase-Locked Loop (PLL) obtains the phase \(\theta\) of the Point of Common Coupling (PCC) voltage \(v_{PCC}\). Its cosine value, \(\cos\theta\), is multiplied by the grid current reference amplitude \(I^*\) to obtain the grid current command \(i_{ref}\). The error signal between \(i_{ref}\) and the feedback grid current \(i_{L2}\) is sent to the current regulator \(G_i(s)\). Its output is subtracted from the capacitor-current feedback active damping signal, and the difference is sent to the PWM modulator to generate the control signals for the switches. \(G_{ad}(s)\) in the figure is the feedback function for the capacitor current.
The control block diagram corresponding to the system structure and its equivalent dual-loop model are shown below. \(G_d(s)\) represents the 1.5-sample digital control delay, expressed as:
$$G_d(s) = e^{-1.5sT_s}$$
where \(T_s=1/f_s\) is the sampling period, and \(f_s\) is the sampling frequency.
Equivalent Dual-Loop Model
Neglecting grid voltage \(v_g\) disturbance, the block diagram can be simplified into a dual-loop model. Here, \(G_{iL2}(s)\) and \(G_{iC}(s)\) are the transfer functions from the inverter bridge arm output voltage \(v_{inv}(s)\) to the grid current \(i_{L2}(s)\) and the capacitor current \(i_C(s)\), respectively. Their expressions are:
$$G_{iL2}(s) = \frac{i_{L2}(s)}{v_{inv}(s)} = \frac{1}{s(L_1 + L_2)} \cdot \frac{\omega_r^2}{s^2 + \omega_r^2}$$
$$G_{iC}(s) = \frac{i_C(s)}{v_{inv}(s)} = \frac{1}{L_1} \cdot \frac{s}{s^2 + \omega_r^2}$$
where \(\omega_r\) is the resonant frequency of the LCL filter, expressed as \(\omega_r = \sqrt{\frac{L_1+L_2+L_g}{L_1(L_2+L_g)C}}\).
From the dual-loop model, the inner-loop (active damping loop) gain \(T_{ad}(s)\), the inner-loop closed-loop transfer function \(\Phi_{ad}(s)\), and the outer-loop (grid current loop) gain \(T_i(s)\) can be derived:
$$T_{ad}(s) = G_d(s)G_{iC}(s)G_{ad}(s) = e^{-1.5sT_s} \cdot \frac{1}{L_1} \cdot \frac{s}{s^2 + \omega_r^2} \cdot G_{ad}(s)$$
$$\Phi_{ad}(s) = \frac{G_d(s)}{1 + T_{ad}(s)}$$
$$T_i(s) = G_i(s)\Phi_{ad}(s)G_{iL2}(s)$$
General Graphical Evaluation Method for Capacitor-Current Feedback Active Damping Characteristics
To achieve system stability, it is necessary to provide positive damping to the system by optimizing the feedback function. This section proposes a general graphical method for evaluating active damping characteristics. It intuitively defines the positive-damping frequency range corresponding to different feedback functions, avoiding repetitive and cumbersome derivations, and offers a new approach to discovering suitable feedback functions.
Analysis of Active Damping Characteristics
Research indicates that the negative-damping effect in capacitor-current feedback active damping introduces a pair of open-loop unstable poles into the outer-loop gain \(T_i(s)\). From the expressions for \(G_d(s)\) and \(G_{iL2}(s)\), it is known that neither has unstable poles. Therefore, according to \(T_i(s) = G_i(s)\Phi_{ad}(s)G_{iL2}(s)\), the open-loop unstable poles in \(T_i(s)\) must originate from the closed-loop transfer function \(\Phi_{ad}(s)\). In other words, instability in the inner-loop gain \(T_{ad}(s)\) leads to the negative-damping effect in capacitor-current feedback active damping. Thus, the positive-damping frequency interval corresponding to a feedback function \(G_{ad}(s)\) can be defined by judging the stability of the inner loop where it resides. This allows for the evaluation of whether the active damping exhibits positive or negative damping characteristics.
To analyze inner-loop stability, consider the case of capacitor-current proportional feedback active damping first, i.e., \(G_{ad}(s)=H_p\), where \(H_p\) is the proportional feedback coefficient. The Bode diagram of the inner-loop gain \(T_{ad}(s)\) can be plotted. The stability of the inner loop depends on whether it contains unstable closed-loop poles. According to the Nyquist stability criterion, the number of inner-loop closed-loop unstable poles \(Z\) is related to the number of inner-loop open-loop unstable poles \(P\), and the positive and negative crossover counts \(N_+\) and \(N_-\) by \(Z = P – 2(N_+ – N_-)\). The inner loop is stable when \(Z=0\). \(N_+\) and \(N_-\) refer to the crossings of the phase curve through \(-180°+360°·k\) (k is an integer) in the direction of increasing and decreasing phase, respectively, at frequencies where the magnitude curve is above 0dB. From \(T_{ad}(s)\), the open-loop unstable pole count is \(P=0\). Analysis shows that the condition for inner-loop stability is \(N_-=0\).
The analysis reveals that the phase of the inner-loop gain at the left limit of the resonance frequency, \(\theta_{T_{ad}}(f_{r\_l})\), is critical. The condition for inner-loop stability translates to a constraint on the phase of the capacitor-current feedback function, \(\theta_{ad}(f)\), at the resonance frequency \(f_r\):
$$ -90° + 540° \cdot f_r T_s + 360° \cdot k < \theta_{ad}(f_r) < 90° + 540° \cdot f_r T_s + 360° \cdot k $$
where \(k\) is an integer (…, -1, 0, 1, …).
Graphical Evaluation Method
The inequality above, for different values of \(k\), represents a series of phase constraint regions when plotted against frequency. By plotting the phase-frequency curve of any capacitor-current feedback function \(G_{ad}(s)\) on this graph, the frequency range where the curve falls within the shaded (constraint) regions can be directly observed. If the phase at the resonance frequency \(f_r\) lies within a constraint region, the inequality holds, meaning the inner loop can be stabilized (by adjusting the gain), and thus that frequency is within the positive-damping range for that feedback function. Conversely, if the phase at \(f_r\) is outside these regions, the inner loop is inherently unstable, leading to a negative-damping effect. This provides a universal graphical tool to evaluate and compare the positive-damping capability of different feedback functions for a grid-connected inverter.
The following table summarizes the positive-damping frequency ranges for common feedback functions derived using this graphical method, compared with results from traditional virtual impedance analysis.
| Feedback Function \(G_{ad}(s)\) | Positive-Damping Frequency Range (Graphical Method) | Positive-Damping Frequency Range (Traditional Analysis) | Notes |
|---|---|---|---|
| Proportional (\(H_p\)) | \( (0, f_s/6) \) | \( (0, f_s/6) \) | Consistent with literature. |
| First-order High-Pass Filter | \( (0, \approx 0.28f_s) \) | \( (0, \approx 0.28f_s) \) | Extends range compared to proportional. |
| Negative Proportional-Integral | \( (0, \approx 0.48f_s) \) | \( (0, \approx 0.48f_s) \) | Further extends the range. |
This method avoids the need for deriving the equivalent virtual impedance for each new feedback function, simplifying the analysis process significantly.
Application of the General Graphical Evaluation Method
As described, by plotting the phase-frequency curve of a capacitor-current feedback function, its corresponding positive-damping frequency range can be defined based on its relationship with the phase constraint regions. Benefiting from the intuitive graphical form, feedback functions with suitable positive-damping ranges can be selected.
For functions like the high-pass filter, the positive-damping range has an upper limit. If \(f_r\) fluctuates above this limit, a negative-damping effect occurs. To achieve strong robustness against \(f_r\) fluctuations, the introduced compensator should make the feedback function’s positive-damping range cover all possible \(f_r\).
The resonance frequency \(f_r\) generally has a minimum value \(f_{r\_min}\), corresponding to the maximum grid impedance \(L_{g\_max}\):
$$f_{r\_min} = \sqrt{ \frac{L_1 + L_2 + L_{g\_max}}{L_1(L_2 + L_{g\_max})C} }$$
Here, \(L_{g\_max}\) can be determined based on the short circuit ratio (SCR).
From the phase constraint graph, the region for \(k=1\) lies below 0°, suggesting the use of a phase-lag compensator. Therefore, this paper proposes a capacitor-current feedback active damping strategy based on a phase-lag compensator, as shown in the modified block diagram. The phase-lag compensator \(G_{plc}(s)\) is expressed as:
$$G_{plc}(s) = \frac{k_{plc}}{m e^{-sT_s} – 1}$$
where \(e^{-sT_s}\) represents a one-sample delay, \(0 < m < 1\), and \(k_{plc} > 0\).
The phase-frequency curve of \(G_{plc}(s)\) is plotted. Most of its phase curve falls within the constraint region for \(k=1\). Its positive-damping frequency range has a lower limit, which decreases as \(m\) increases. By selecting an appropriate \(m\), this lower limit can be set below the minimum possible \(f_r\) (\(f_{r\_min}\)), preventing \(f_r\) from falling outside the positive-damping range and thus achieving strong robustness against \(f_r\) fluctuations for the grid-connected inverter.
Parameter Design
The previous section proposed a capacitor-current feedback active damping strategy based on a phase-lag compensator. This section details its parameter design and provides a design example.
Parameter Design for the Phase-Lag Compensator
First, determine the minimum resonance frequency \(f_{r\_min}\) using the formula above. The inner-loop gain with \(G_{plc}(s)\) is:
$$T_{ad\_plc}(s) = G_d(s)G_{iC}(s)G_{plc}(s) = e^{-1.5sT_s} \cdot \frac{k_{plc}}{m e^{-sT_s} – 1} \cdot \frac{s}{s^2 + \omega_r^2} \cdot \frac{1}{L_1}$$
From this, the phase of \(T_{ad\_plc}(s)\) can be derived as \(\theta_{ad\_plc}(f)\). To ensure the inner loop is stable, the phase at \(f_r\) must satisfy the constraint condition (fall within the shaded region for \(k=1\)). This condition is used to select \(m\). After choosing \(m\), the gain \(k_{plc}\) is designed to ensure sufficient gain margins at the critical crossover frequencies (like \(f_s/2\) and another frequency \(f_b\)) in the inner loop, maintaining stability even with parameter variations.
Design Example and Robustness Comparison
Key parameters for a prototype single-phase LCL grid-connected inverter are given in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (RMS) | \(V_g\) | 220 V |
| Input DC Voltage | \(V_{in}\) | 360 V |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
| Sampling Frequency | \(f_s\) | 20 kHz |
| Inverter-side Inductor | \(L_1\) | 600 μH |
| Grid-side Inductor | \(L_2\) | 150 μH |
| Filter Capacitor | \(C\) | 5 μF |
| Nominal Resonance Frequency | \(f_r\) | ≈6.5 kHz |
Based on the parameters, \(f_{r\_min}\) is calculated to be approximately 3.2 kHz (for SCR=10, \(L_{g\_max}=2.6\) mH). To make the lower limit of the positive-damping range for \(G_{plc}(s)\) less than \(f_{r\_min}\) with some margin, \(m=0.9\) is chosen, corresponding to a lower limit of about 0.05\(f_s\) (1 kHz). The gain \(k_{plc}\) is then designed based on gain margin requirements, resulting in \(k_{plc}=4\).
The outer-loop grid current regulator uses a Proportional-Resonant (PR) controller:
$$G_i(s) = K_p + \frac{2K_r \omega_i s}{s^2 + 2\omega_i s + \omega_o^2}$$
where \(K_p\) and \(K_r\) are the proportional and resonant coefficients, \(\omega_o\) is the fundamental angular frequency, and \(\omega_i\) is the resonant cutoff angular frequency (typically \(\pi\) rad/s for ±1% grid frequency variation). Following standard design procedures with a target cutoff frequency of 800 Hz, the parameters are chosen as \(K_p=3.77\) and \(K_r=301.6\).
To demonstrate the strong robustness of the proposed phase-lag compensator based active damping against \(f_r\) fluctuations, it is compared with capacitor-current proportional feedback and high-pass filter feedback active damping. The proportional feedback uses the optimal gain \(H_{opt}=0.91\). The high-pass filter has a cutoff at \(f_{hpf}=0.5f_s=10\) kHz for maximum positive-damping range and a gain \(k_{hpf}=4\) (same as \(k_{plc}\)). Root locus analysis under varying grid impedance \(L_g\) and with ±30% variations in \(L_1\) and \(C\) shows that the phase-lag compensator maintains stability over the widest range of conditions, confirming its superior robustness.
Experimental Verification
To validate the theoretical analysis, a 6kW grid-connected inverter prototype was built according to the parameters in Table 1. The experimental platform consists of a programmable DC power supply, the single-phase LCL grid-connected inverter, a programmable AC power source (simulating the grid), and an oscilloscope. An external inductor is connected in series to simulate grid impedance \(L_g\).
Experimental results under nominal \(L_1\) and \(C\) are shown. The feedback function is switched between the phase-lag compensator, the high-pass filter, and the proportional feedback via a trigger signal. When \(L_g=0\) and \(L_g=2.6\)mH, the grid current remains stable with all three methods. However, when \(L_g=1.75\)mH, the system becomes unstable after switching to proportional feedback, while remaining stable with the phase-lag compensator and the high-pass filter. This aligns with the theoretical prediction that for this \(L_g\), \(f_r\) falls outside the positive-damping range of the proportional method but inside the ranges of the other two.
Further experiments were conducted with ±30% variations in \(L_1\) and \(C\) parameters. With a +30% increase and \(L_g=1.75\)mH, results are similar to the nominal case: stable with the phase-lag compensator and high-pass filter, unstable with proportional feedback. With a -30% decrease, the resonance frequency increases. When \(L_g=0\), the system becomes unstable after switching to the high-pass filter. When \(L_g=2.6\)mH, it becomes unstable after switching to proportional feedback. In both -30% variation cases, the grid current remains stable only with the phase-lag compensator active. These results match the root locus analysis and conclusively verify the strong robustness of the proposed phase-lag compensator based active damping strategy against parameter fluctuations in the grid-connected inverter.
Conclusion
This paper proposed a general graphical evaluation method for the active damping characteristics of capacitor-current feedback in LCL-type grid-connected inverters. First, an equivalent dual-loop model of the grid-connected inverter was established. By analyzing the stability of the active damping inner loop, a phase constraint region for the feedback function was derived. The relationship between the phase-frequency curve of the feedback function and this constraint region was used to define the positive-damping frequency interval, thereby evaluating the active damping characteristics. Using the proposed method, the characteristics of capacitor-current high-pass filter feedback active damping were assessed, yielding results consistent with traditional virtual impedance analysis while avoiding cumbersome calculations. Benefiting from the intuitiveness of the graphical method, a capacitor-current active damping strategy based on a phase-lag compensator was proposed. Its parameters were designed accordingly. Experimental results validated the strong robustness of the proposed active damping strategy and the correctness of the parameter design. The method proposed in this paper can also be extended and applied to three-phase LCL grid-connected inverters.