With the gradual depletion of traditional fossil fuels and the increasing severity of environmental pollution, renewable energy generation systems such as solar and wind power have developed rapidly. The on-grid inverter, as a critical interface equipment between renewable energy sources and the public grid, plays a key role in the stability of grid-connected systems. The control of switching devices in on-grid inverters often employs pulse width modulation (PWM) technology, leading to abundant high-frequency switching harmonics in the output current. Typically, L-type or LCL-type filters are connected to the grid to ensure grid current quality. Compared to L-type on-grid inverters, LCL-type on-grid inverters offer advantages such as strong harmonic attenuation capability, low cost, and small size. However, LCL-type filters have inherent resonance peaks, which may trigger high-frequency oscillations and other stability issues in grid-connected systems.
To suppress the high-frequency resonance of LCL-type filters, the most direct method is to connect a resistor in series or parallel with the filter capacitor, known as passive damping. This method is simple and reliable but increases power losses and degrades filtering performance. Active damping, on the other hand, simulates virtual resistance through state variable feedback, achieving suppression effects comparable to passive damping without adding passive components. Among these, capacitor current feedback active damping is widely used due to its simplicity and effectiveness. However, digital control in on-grid inverter systems introduces control delays that have damping effects. Therefore, for on-grid inverters with grid-side current control, stability under single current-loop control can be ensured by designing the resonant frequency of the LCL filter to lie between one-sixth of the sampling frequency and half the sampling frequency, even without external damping. Nonetheless, variations in grid inductive impedance can cause shifts in the system’s resonant frequency. When the resonant frequency equals or falls below one-sixth of the sampling frequency, the system may become unstable. Thus, even when the resonant frequency is within the stable range, appropriate external damping is needed to enhance the system’s robustness to grid impedance.
The presence of control delays causes capacitor current feedback active damping to behave as a frequency-dependent virtual impedance. Typically, the capacitor current feedback coefficient is designed based on the resonant frequency. However, in practical systems, wide variations in grid impedance affect the resonant frequency, making the selection of the capacitor current feedback coefficient challenging. Previous studies have designed optimal capacitor current feedback coefficients based on amplitude margin requirements under different grid impedance conditions, but the design process is complex. Other works have introduced the concept of damping coefficients and designed passive resistors and capacitor current feedback coefficients for hybrid damping under known grid impedance conditions. Adaptive control strategies based on hybrid damping have also been proposed, but online measurement of grid impedance increases control difficulty and system complexity. These studies largely rely on loop gain analysis for system stability, ignoring the nonlinear characteristics of the system and depending on grid impedance information, which has limitations.
For interactive systems between on-grid inverters and the grid, impedance analysis methods establish impedance/admittance models for both and analyze system stability based on impedance stability criteria. To ensure that stability analysis does not depend on grid impedance information, parameter design and improved control based on passivity theory have gained widespread attention. Some studies have proposed delay compensation strategies to enhance the passivity of the output admittance of on-grid inverters and designed capacitor current feedback coefficients and current controllers. Others have introduced voltage feedforward control on top of delay compensation strategies, achieving full-frequency passivity of the output admittance of on-grid inverters. However, these studies focus on on-grid inverters controlled in stationary frames. Research has established dq-frame admittance models for LCL-type on-grid inverters, but the dq admittance is a 2×2 matrix, requiring separate study of passivity for four sub-admittances, significantly increasing analysis and design difficulty.
In summary, this paper focuses on three-phase LCL-type on-grid inverters controlled in the dq frame. Using the harmonic linearization method, a unified sequence admittance model for the on-grid inverter is established. Combined with passivity theory, the effects of different damping methods on high-frequency admittance characteristics and system stability are analyzed in detail. Based on the design of the optimal active damping coefficient, a hybrid damping method combining optimal active damping and passive damping is proposed. This method improves the system’s stability margin at one-sixth of the sampling frequency and enhances robustness to grid impedance and filter parameter variations. Finally, simulations based on MATLAB/Simulink verify the correctness of the theory and the effectiveness of the proposed hybrid damping method.
The structure of a three-phase LCL-type on-grid inverter is shown in the figure above. It includes the main circuit topology, grid current control structure, and SRF-PLL control structure. Key parameters include: inverter-side inductance \(L_1\) with equivalent resistance \(R_1\), grid-side inductance \(L_2\) with equivalent resistance \(R_2\), filter capacitor \(C\), damping resistor \(R_d\), grid impedance \(Z_g\), DC voltage source \(U_{dc}\), PCC voltages \(u_a, u_b, u_c\), grid voltage \(u_g\), inverter-side inductor currents \(i_{1a}, i_{1b}, i_{1c}\), capacitor currents \(i_{Ca}, i_{Cb}, i_{Cc}\), and grid currents \(i_{2a}, i_{2b}, i_{2c}\). The control system uses a PI controller \(H_{PI}(s) = K_p + K_i/s\) for current loop, capacitor current feedback coefficient \(K_C\), and equivalent delay functions \(G_i(s) = e^{-1.5sT_s}\) for current sampling and \(G_u(s)\) for voltage sampling. The SRF-PLL has a PI controller \(G_{PI}(s) = K_{p,PLL} + K_{i,PLL}/s\) and open-loop transfer function \(H_{PLL}(s) = G_{PI}(s)/s\).
To establish the sequence admittance model, symmetrical positive-sequence and negative-sequence voltage disturbances are injected at the PCC. The A-phase PCC voltage in time and frequency domains is given by:
$$ u_a(t) = U_1 \cos(2\pi f_1 t) + U_p \cos(2\pi f_p t + \phi_{up}) + U_n \cos(2\pi f_n t + \phi_{un}) $$
$$ U_a[f] = \begin{cases}
U_1, & f = \pm f_1 \\
U_p, & f = \pm f_p \\
U_n, & f = \pm f_n
\end{cases} $$
where \(f_1\), \(f_p\), and \(f_n\) are frequencies of fundamental, positive-sequence disturbance, and negative-sequence disturbance voltages, respectively; \(U_1\), \(U_p\), \(U_n\) are their amplitudes; and \(\phi_{up}\), \(\phi_{un}\) are initial phases. Similarly, the A-phase grid current is expressed as:
$$ i_{2a}(t) = I_1 \cos(2\pi f_1 t + \phi_{i1}) + I_p \cos(2\pi f_p t + \phi_{ip}) + I_n \cos(2\pi f_n t + \phi_{in}) $$
$$ I_{2a}[f] = \begin{cases}
I_1, & f = \pm f_1 \\
I_p, & f = \pm f_p \\
I_n, & f = \pm f_n
\end{cases} $$
From the main circuit, the relationships between capacitor current, grid current, and PCC voltage in the frequency domain are derived. For fundamental frequency \(f_1\):
$$ I_{C1} = \frac{(sL_2 + R_2)Cs}{1 + R_d C s} I_1 + \frac{Cs}{1 + R_d C s} U_1 $$
For positive-sequence disturbance frequency \(f_p\):
$$ I_{Cp} = \frac{(sL_2 + R_2)Cs}{1 + R_d C s} I_p + \frac{Cs}{1 + R_d C s} U_p $$
For negative-sequence disturbance frequency \(f_n\):
$$ I_{Cn} = \frac{(sL_2 + R_2)Cs}{1 + R_d C s} I_n + \frac{Cs}{1 + R_d C s} U_n $$
The inverter output voltage, grid current, and PCC voltage relationship is:
$$ K_m U_{dc} M_a[f] = \left[1 + \frac{(sL_1 + R_1)Cs}{1 + R_d C s}\right] U_a[f] + \left[(sL_1 + R_1) + (sL_2 + R_2) + \frac{(sL_1 + R_1)(sL_2 + R_2)Cs}{1 + R_d C s}\right] I_{2a}[f] $$
where \(K_m\) is the inverter modulation gain, and \(M_a[f]\) is the frequency-domain form of the A-phase modulation signal, which is a function of \(U_a[f]\) and \(I_{2a}[f]\) related to the control loop.
For the SRF-PLL modeling, without voltage disturbance, the phase-locked angle \(\theta_{PLL} = \theta_1 = 2\pi f_1 t\). The Park transformation matrix from abc to dq coordinates is:
$$ T(\theta_1) = \frac{2}{3} \begin{bmatrix}
\cos \theta_1 & \cos(\theta_1 – 2\pi/3) & \cos(\theta_1 + 2\pi/3) \\
-\sin \theta_1 & -\sin(\theta_1 – 2\pi/3) & -\sin(\theta_1 + 2\pi/3)
\end{bmatrix} $$
With voltage disturbances, a small phase angle disturbance \(\Delta \theta\) occurs, so \(\theta_{PLL} = \theta_1 + \Delta \theta\). The Park transformation matrix becomes:
$$ T(\theta_{PLL}) = \begin{bmatrix}
\cos \Delta \theta & \sin \Delta \theta \\
-\sin \Delta \theta & \cos \Delta \theta
\end{bmatrix} T(\theta_1) $$
Since \(\Delta \theta\) is small, \(\sin \Delta \theta \approx \Delta \theta\) and \(\cos \Delta \theta \approx 1\). The relationship between \(\Delta \theta\) and voltage disturbances is derived as:
$$ \Delta \theta[f] = \begin{cases}
\mp j T_{PLL}(s) G_u(s \pm j2\pi f_1) U_p, & f = \pm(f_p – f_1) \\
\pm j T_{PLL}(s) G_u(s \mp j2\pi f_1) U_n, & f = \pm(f_n + f_1)
\end{cases} $$
where \(T_{PLL}(s) = \frac{H_{PLL}(s)}{1 + U_1 H_{PLL}(s)}\).
For current control and modulation, the three-phase grid currents and capacitor currents undergo Park transformation into the dq-frame control loop. The d-axis and q-axis grid currents in frequency domain are:
$$ I_d[f] = \begin{cases}
I_1 \cos \phi_{i1}, & f = 0 \\
\mp j I_1 \sin \phi_{i1} T_{PLL}(s) G_u(s \pm j2\pi f_1) U_p + G_i(s \pm j2\pi f_1) I_p, & f = \pm(f_p – f_1) \\
\pm j I_1 \sin \phi_{i1} T_{PLL}(s) G_u(s \mp j2\pi f_1) U_n + G_i(s \mp j2\pi f_1) I_n, & f = \pm(f_n + f_1)
\end{cases} $$
$$ I_q[f] = \begin{cases}
I_1 \sin \phi_{i1}, & f = 0 \\
\pm j I_1 \cos \phi_{i1} T_{PLL}(s) G_u(s \pm j2\pi f_1) U_p \mp j G_i(s \pm j2\pi f_1) I_p, & f = \pm(f_p – f_1) \\
\mp j I_1 \cos \phi_{i1} T_{PLL}(s) G_u(s \mp j2\pi f_1) U_n \pm j G_i(s \mp j2\pi f_1) I_n, & f = \pm(f_n + f_1)
\end{cases} $$
Similarly, the d-axis and q-axis capacitor currents are:
$$ I_{Cd}[f] = \begin{cases}
I_{C1} \cos \phi_{C1}, & f = 0 \\
\mp j I_{C1} \sin \phi_{C1} T_{PLL}(s) G_u(s \pm j2\pi f_1) U_p + G_i(s \pm j2\pi f_1) I_{Cp}, & f = \pm(f_p – f_1) \\
\pm j I_{C1} \sin \phi_{C1} T_{PLL}(s) G_u(s \mp j2\pi f_1) U_n + G_i(s \mp j2\pi f_1) I_{Cn}, & f = \pm(f_n + f_1)
\end{cases} $$
$$ I_{Cq}[f] = \begin{cases}
I_{C1} \sin \phi_{C1}, & f = 0 \\
\pm j I_{C1} \cos \phi_{C1} T_{PLL}(s) G_u(s \pm j2\pi f_1) U_p \mp j G_i(s \pm j2\pi f_1) I_{Cp}, & f = \pm(f_p – f_1) \\
\mp j I_{C1} \cos \phi_{C1} T_{PLL}(s) G_u(s \mp j2\pi f_1) U_n \pm j G_i(s \mp j2\pi f_1) I_{Cn}, & f = \pm(f_n + f_1)
\end{cases} $$
After current control, the dq-frame modulation signals are:
$$ M_d[f] = -H_{PI}(s) I_d[f] – K_C I_{Cd}[f] $$
$$ M_q[f] = -H_{PI}(s) I_q[f] – K_C I_{Cq}[f] $$
Through inverse Park transform, the A-phase modulation signal is:
$$ M_a[f] = \cos \theta_{PLL}[f] \ast M_d[f] – \sin \theta_{PLL}[f] \ast M_q[f] $$
where \(\ast\) denotes convolution. Substituting into the main circuit equation yields the positive-sequence and negative-sequence admittance models for the three-phase LCL-type on-grid inverter:
$$ Y_p(s) = -\frac{I_p}{U_p} = \frac{1 + \frac{(sL_1 + R_1)sC}{1 + R_d C s} + \frac{K_m U_{dc} G_u(s) K_C C s}{1 + R_d C s} – K_m U_{dc} T_{PLL}(s – j\omega_1) G_u(s) \left\{ H_i(s – j\omega_1) I_1 + K_C I_{C1} + M_1 \right\}}{s(L_1 + L_2) + R_1 + R_2 + \frac{(sL_1 + R_1)(sL_2 + R_2)Cs}{1 + R_d C s} + K_m U_{dc} G_i(s) \left[ H_i(s – j\omega_1) + K_C \frac{(sL_2 + R_2)Cs}{1 + R_d C s} \right]} $$
$$ Y_n(s) = -\frac{I_n}{U_n} = \frac{1 + \frac{(sL_1 + R_1)Cs}{1 + R_d C s} + \frac{K_m U_{dc} G_u(s) K_C C s}{1 + R_d C s} – K_m U_{dc} T_{PLL}(s + j\omega_1) G_u(s) \left\{ H_i(s + j\omega_1) I_1^* + K_C I_{C1}^* + M_1^* \right\}}{s(L_1 + L_2) + R_1 + R_2 + \frac{(sL_1 + R_1)(sL_2 + R_2)Cs}{1 + R_d C s} + K_m U_{dc} G_i(s) \left[ H_i(s + j\omega_1) + K_C \frac{(sL_2 + R_2)Cs}{1 + R_d C s} \right]} $$
where \(\omega_1 = 2\pi f_1\), \(I_1^*\) is the conjugate of \(I_1\), \(I_{C1}^*\) is the conjugate of \(I_{C1}\), \(M_1\) is the steady-state operating point, and \(M_1^*\) is its conjugate. This model includes both passive and active damping and can be reduced to L-type on-grid inverter systems, offering generality and practicality. Compared to dq admittance models, it has only two sub-admittances, simplifying analysis and design.
The sequence admittance model is verified using frequency scanning in MATLAB/Simulink. Simulation parameters are listed in Table 1.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| DC-side voltage \(U_{dc}\) | 700 V | Switching frequency \(f_{sw}\) | 10 kHz |
| Grid phase voltage amplitude \(U_1\) | 311 V | Sampling frequency \(f_s\) | 10 kHz |
| Grid frequency \(f_1\) | 50 Hz | PLL proportional gain \(K_{p,PLL}\) | 1.388 |
| Rated active power \(P\) | 20 kW | PLL integral gain \(K_{i,PLL}\) | 299.67 |
| Inverter-side inductance \(L_1\) | 2.4 mH | Current controller proportional gain \(K_p\) | 10 |
| Inductor \(L_1\) resistance \(R_1\) | 0.01 Ω | Current controller integral gain \(K_i\) | 1668 |
| Grid-side inductance \(L_2\) | 0.6 mH | Modulation gain \(K_m\) | 1/700 |
| Inductor \(L_2\) resistance \(R_2\) | 0.01 Ω | Damping resistor \(R_d\) | 1 Ω |
| Filter capacitor \(C\) | 10 μF | Capacitor current feedback coefficient \(K_C\) | 6.2 |
The comparison between theoretical model curves and simulation scanning results shows good agreement, validating the correctness of the sequence admittance model. This provides a model foundation and theoretical basis for subsequent system stability analysis.
Passivity theory, originating from electrical network theory, has been introduced into power electronics stability analysis. It offers a sufficient but not necessary stability criterion: if the real part of the on-grid inverter port impedance/admittance is non-negative in a certain frequency band, then the on-grid inverter system satisfies passivity in that band and is judged stable. The equivalent condition for passivity is that the phase of the on-grid inverter port impedance/admittance lies within \([-90^\circ, 90^\circ]\) in that frequency band. The passivity criterion only focuses on the inverter’s own impedance, not relying on grid impedance information, reducing the difficulty of system stability assessment. If a single inverter satisfies full-frequency passivity, it can achieve plug-and-play requirements for the grid. From a system-level perspective, if each inverter is passive, the multi-inverter grid-connected system is also passive and stable. Therefore, enhancing the passivity of on-grid inverters can be seen as improving system stability and robustness to grid impedance.
Damping methods primarily suppress high-frequency resonance peaks of LCL filters. This paper focuses on the passivity and stability of the system in the high-frequency band. From the admittance curves, the positive- and negative-sequence admittances of the on-grid inverter almost coincide in the high-frequency band, while the negative-sequence admittance has stronger passivity in the mid-low frequency band. Thus, only the passivity of the positive-sequence admittance is analyzed. PLL dynamics and current controller integral gain \(K_i\) mainly affect the admittance characteristics in the mid-low frequency band. In this analysis, they are neglected, i.e., in the positive-sequence admittance, \(T_{PLL}(s – j\omega_1) = 0\) and \(H_{PI}(s – j\omega_1) = K_p\). Additionally, inductor resistances have some damping effect, beneficial for system stability. Considering the worst case, let \(R_1 = R_2 = 0\), and the grid impedance is considered as pure inductance \(L_g\). Under these simplifications, the positive-sequence admittance becomes:
$$ Y_{ps}(s) = \left[1 + \frac{L_1 C s^2}{1 + R_d C s} + \frac{K_m U_{dc} G_u(s) K_C s}{1 + R_d C s}\right] \left[sL_1 + sL_2 + \frac{L_1 L_2 C s^3}{1 + R_d C s} + K_m U_{dc} G_i(s) \left(K_p + \frac{K_C L_2 C s^2}{1 + R_d C s}\right)\right]^{-1} $$
First, consider the case with no external damping, where \(R_d = 0\) and \(K_C = 0\). Based on the basic parameters in Table 1, the positive-sequence admittance characteristics of the on-grid inverter are analyzed. The admittance phase has a non-passive region in the frequency band [1027 Hz, 1667 Hz], where the phase exceeds \(90^\circ\). The grid admittance \(Y_g\), considered as pure inductance \(L_g\), always has a phase of \(-90^\circ\). The phase margin of the grid-connected system is given by:
$$ \phi_{PM} = 180^\circ – \left[ \angle Y_{ps}(2\pi f_x) – \angle Y_g(2\pi f_x) \right] $$
where \(f_x\) is the intersection frequency of the magnitude-frequency characteristics of grid admittance and positive-sequence admittance. If \(f_x\) falls within the non-passive band, the phase margin \(\phi_{PM} < 0^\circ\), and the system becomes unstable. The resonant frequency of the system is calculated as:
$$ f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} = 2297 \text{ Hz} $$
Since \(f_r > f_s/6\) (1667 Hz), with \(L_g = 0\) and no external damping, the on-grid inverter with only grid-side current loop control can ensure stability solely through the damping effect of control delay. However, when \(L_g = 0.5\) mH, the intersection frequency is 1543 Hz, lying within the non-passive band, leading to instability.
Second, consider only passive damping, where \(K_C = 0\). With different damping resistors \(R_d\), the positive-sequence admittance characteristics are analyzed. As \(R_d\) increases, the high-frequency resonance peak gradually decreases, and the passive region frequency band of the on-grid inverter positive-sequence admittance gradually expands, improving system phase margin. This indicates enhanced robustness and stability to grid impedance.
Third, consider only active damping, where \(R_d = 0\). With different capacitor current feedback coefficients \(K_C\), the positive-sequence admittance characteristics are analyzed. As \(K_C\) increases, the passive region frequency band first expands and then shrinks, meaning system robustness and stability first strengthen and then weaken.
Based on the analysis, when using only active damping, there exists an optimal active damping coefficient that maximizes the passive region frequency band of the on-grid inverter positive-sequence admittance, thereby maximizing system robustness and stability to grid impedance. Using Euler’s formula, the delay functions are expressed as:
$$ G_u(j\omega) = G_i(j\omega) = \cos \alpha – j \sin \alpha $$
where \(\alpha = 1.5 T_s \omega = 3\pi T_s f\). Substituting \(R_d = 0\), \(s = j\omega\), and the delay expressions into the simplified admittance formula, the real part of \(Y_{ps}\) is derived as:
$$ \text{Re}(Y_{ps}) = \frac{A_1 A_2 + B_1 B_2}{A_2^2 + B_2^2} $$
with:
$$ A_1 = 1 – L_1 C \omega^2 + K_C C \omega \sin \alpha $$
$$ B_1 = K_C C \omega \cos \alpha $$
$$ A_2 = (K_p – K_C L_2 C \omega^2) \cos \alpha $$
$$ B_2 = \omega(L_1 + L_2) – L_1 L_2 C \omega^3 – (K_p – K_C L_2 C \omega^2) \sin \alpha $$
The numerator of the real part is:
$$ A_1 A_2 + B_1 B_2 = \cos \alpha \left[ L_1 C (K_C – K_p) \omega^2 + K_p \right] $$
This is the product of two expressions: \(y_1 = \cos \alpha = \cos(1.5 T_s \omega)\) and \(y_2 = L_1 C (K_C – K_p) \omega^2 + K_p\). For \(K_C \geq K_p\), \(y_2 \geq K_p > 0\), so the sign of \(\text{Re}(Y_{ps})\) matches \(y_1\): positive for \(0 < \omega < \omega_s/6\) and negative for \(\omega_s/6 < \omega < \omega_s/2\), where \(\omega_s = 2\pi f_s\). Thus, according to the passivity criterion, for \(K_C \geq K_p\), the system satisfies passivity in \(0 < \omega < \omega_s/6\) but not in \(\omega_s/6 < \omega < \omega_s/2\). For \(K_C < K_p\), \(y_2\) is a quadratic function opening downward, with a positive root at:
$$ \omega_x = \sqrt{\frac{K_p}{L_1 C (K_p – K_C)}} $$
To ensure \(\text{Re}(Y_{ps})\) is non-negative in \((0, \omega_s/2)\), the sign intervals of \(y_1\) and \(y_2\) must align, i.e., \(\omega_x = \omega_s/6\). Then, the on-grid inverter system maintains passivity throughout \((0, \omega_s/2)\), and the selected active damping coefficient is optimal. The optimal active damping coefficient \(K_{Co}\) is:
$$ K_{Co} = K_p \left(1 – \frac{36}{L_1 C \omega_s^2}\right) $$
Using parameters from Table 1, \(K_{Co} = 6.2\). Comparing cases with \(K_C < K_{Co}\), \(K_C = K_{Co}\), and \(K_C > K_{Co}\), when \(K_C = K_{Co}\), the phase-frequency characteristic curve of the positive-sequence admittance is almost tangent to the \(90^\circ\) line at \(f_s/6\), with no non-passive region in the high-frequency band. For \(K_C < K_{Co}\) or \(K_C > K_{Co}\), the phase exceeds \(90^\circ\) near \(f_s/6\), failing to satisfy passivity, confirming the uniqueness of the optimal active damping coefficient.
For single damping methods, larger passive damping resistors enhance system robustness and stability but increase power losses. Typically, the damping resistor is chosen as one-third of the capacitive reactance at the resonant frequency, i.e., the optimal value \(R_{do}\) is:
$$ R_{do} = \frac{1}{6\pi f_r C} $$
In active damping, the optimal coefficient ensures full passivity in the high-frequency band, meaning the on-grid inverter system remains stable under any grid impedance variation. However, with optimal active damping, the phase margin at \(f_s/6\) is very small, nearly zero, making the system prone to instability. Moreover, \(K_{Co}\) depends on filter parameters \(L_1\) and \(C\), and in practice, fluctuations in these parameters make accurate acquisition of \(K_{Co}\) difficult, hindering true passivity achievement.
Addressing issues with single damping methods, this paper proposes a hybrid damping method combining optimal active damping with passive damping, leveraging the advantages of both and compensating for their shortcomings. Based on the optimal active damping coefficient design, a small passive damping resistor is added, ensuring low power loss while improving the phase margin at \(f_s/6\), thereby enhancing system robustness and stability to grid impedance and filter parameters. For single damping methods, with LCL filter parameters from Table 1, \(R_{do} = 2.4\ \Omega\) and \(K_{Co} = 6.2\). For the hybrid damping method, with \(K_C = K_{Co}\), add a damping resistor of \(R_{do}/4 = 0.6\ \Omega\). Comparisons show that compared to single optimal active damping, the hybrid method achieves high-frequency passivity while effectively increasing the phase margin at \(f_s/6\), ensuring robustness to grid impedance and enhancing high-frequency stability. Compared to single passive damping, the hybrid method achieves similar phase margins in the high-frequency band but with only one-fourth the resistor loss.
To verify the theoretical analysis and effectiveness of the proposed hybrid damping method, a system model is built in MATLAB/Simulink with parameters from Table 1. Simulation results for three-phase grid currents under varying grid impedance \(L_g\) for no external damping, single passive damping (\(R_d = R_{do}\)), and single active damping (\(K_C = K_{Co}\)) are examined. When \(L_g = 0\) mH, all three methods yield sinusoidal and smooth grid currents, indicating stable operation. When \(L_g = 0.5\) mH, no external damping causes significant harmonic oscillation near 1543 Hz, making the system unstable; single passive damping and optimal active damping maintain stable grid currents, as \(R_d = R_{do}\) and \(K_C = K_{Co}\) satisfy high-frequency passivity, ensuring robustness to grid impedance changes. These results align with theoretical analysis.
With optimal active damping and \(L_g = 0.86\) mH, the intersection frequency of grid admittance and positive-sequence admittance magnitude-frequency characteristics is approximately \(f_s/6\), yielding a very small phase margin. When the filter capacitor \(C\) degrades by 20%, optimal active damping leads to distorted grid currents with a total harmonic distortion (THD) of 20.01% and resonance at 1880 Hz, causing high-frequency oscillation and poor grid current quality. In contrast, with the hybrid damping method (\(K_C = K_{Co}\), \(R_d = R_{do}/4\)), even with \(C\) degraded by 20%, grid currents remain sinusoidal and smooth after a minor initial distortion, with THD only 0.47%, demonstrating stable operation. Compared to optimal active damping, the hybrid method ensures high-frequency stability and robustness to grid impedance while maintaining robustness to filter parameter variations. Compared to single passive damping, it reduces resistor loss by 75%.
Further demonstrating the superiority of the hybrid damping method, under \(L_g = 0.86\) mH and with filter parameters \(L_1\) and \(C\) both fluctuating by -15%, optimal active damping causes divergent grid currents and instability, whereas the hybrid method restores stability within one fundamental cycle, keeping the system stable. This confirms that the proposed hybrid damping method, combining optimal active damping and passive damping, enhances system robustness to grid impedance and LCL filter parameters without relying on grid impedance information, offering a simple and practical design approach.
In conclusion, for dq-frame controlled three-phase LCL-type on-grid inverters, high-frequency stability is addressed through sequence admittance modeling and passivity theory analysis. The effects of different damping methods on system passivity and stability are compared. A hybrid damping method combining optimal active damping with passive damping is proposed. Under grid impedance variations and filter parameter fluctuations, system robustness and passivity stability are analyzed, with simulations verifying the method’s effectiveness. Key findings are:
- With single passive damping, larger resistors expand the passive region and phase margin, strengthening system robustness and stability to grid impedance.
- With single active damping, increasing the capacitor current feedback coefficient first enlarges then shrinks the passive region, with an optimal coefficient ensuring full high-frequency passivity. However, the phase margin at \(f_s/6\) remains small, and system instability may occur under filter parameter fluctuations.
- Compared to single damping methods, the proposed hybrid damping method increases the high-frequency phase margin at \(f_s/6\), ensuring high-frequency stability under wide grid impedance variations and filter parameter fluctuations, with advantages of simple design, low power loss, and strong robustness.
Future work will focus on mid-low frequency stability issues for LCL-type on-grid inverter systems based on sequence admittance models and passivity theory. The on-grid inverter, as a critical component in renewable energy integration, requires continuous research to enhance performance and reliability in modern power systems.