As an energy exchange interface between renewable energy generation units and the grid, grid-connected inverters play a critical role in converting DC power into high-quality AC power for injection into the grid. These systems are essential components of modern power systems, especially with the rapid growth of distributed generation. Among various filter topologies, the LCL filter is widely adopted in three-phase inverter outputs due to its superior high-frequency harmonic suppression capabilities. However, the open-loop transfer function of an LCL filter lacks damping terms, leading to resonant peaks that can cause instability in three-phase inverter operations. This paper addresses this issue by proposing an active damping control strategy based on inverter-side current feedback. Compared to traditional capacitive current active damping methods, the proposed approach maintains consistent fundamental frequency gain and cutoff frequency even under grid impedance variations. Furthermore, from a damping perspective, it offers improved stability margins and faster response times. To further mitigate the impact of grid voltage distortions on grid-connected current, grid current proportional feedback is incorporated, forming a double-loop structure that enhances output impedance. The inverter current loop eliminates resonant peaks, while the grid current loop reduces harmonic components. Extensive simulations validate the effectiveness and feasibility of the proposed control strategy for three-phase inverters.
The increasing penetration of distributed generation, particularly from solar and wind sources, has heightened the importance of reliable grid-connected inverters. As of recent statistics, wind and photovoltaic capacity has reached significant levels, underscoring the need for robust power electronic interfaces. In this context, three-phase inverters serve as the backbone for integrating renewable energy into the grid. The LCL filter, while effective at attenuating high-frequency harmonics, introduces resonance issues that can destabilize the system. Traditional damping methods, such as passive damping with resistors or active damping using capacitor current feedback, face limitations like reduced efficiency, sensitivity to grid impedance changes, and practical implementation challenges. This research introduces an innovative active damping technique utilizing inverter-side current feedback, which not only suppresses resonance but also enhances system robustness in weak grid conditions. The double-loop feedback mechanism further augments output impedance, reducing grid-induced harmonic distortions. Through mathematical modeling, parameter design, and simulation, this paper demonstrates the superior performance of the proposed strategy for three-phase inverters.
The mathematical model of an LCL-type grid-connected three-phase inverter begins with the equivalent circuit representation. The inverter bridge output voltage, denoted as \( v_{\text{inv}} \), is modeled as a voltage source. The LCL filter comprises an inverter-side inductor \( L_1 \), a filter capacitor \( C \), and a grid-side inductor \( L_2 \). The grid impedance \( L_g \) is considered to account for weak grid conditions. The open-loop transfer function from the inverter output voltage to the grid current \( i_2 \) is derived using Mason’s gain formula:
$$ G_{\text{LCL}}(s) = \frac{i_2(s)}{v_{\text{inv}}(s)} = \frac{1}{s^3 L_1 L_2 C + s(L_1 + L_2)} $$
This transfer function exhibits a resonant peak due to the absence of damping terms. To introduce damping, active feedback methods are employed. Among various options, inverter-side current feedback is selected for its advantages. The control diagram with inverter-side current proportional feedback, with gain \( k_f \), modifies the transfer function to:
$$ G_{\text{LCL}}(s) = \frac{1}{s^3 L_1 L_2 C + s^2 L_2 C k_f k_{\text{pwm}} + s(L_1 + L_2) + k_f k_{\text{pwm}}} $$
Here, \( k_{\text{pwm}} \) represents the ratio of the DC input voltage to the triangular carrier amplitude. Compared to capacitive current feedback, which only adds an \( s^2 \) term, the inverter-side current feedback introduces an additional constant term \( k_f k_{\text{pwm}} \), enhancing damping by acting as a virtual resistor. This results in better resonance suppression and stability for three-phase inverters.
In digital control systems, delays due to computation and zero-order hold effects must be considered. The total control delay \( G_d(s) \) is modeled as \( e^{-1.5 s T_s} \), where \( T_s \) is the sampling period. Incorporating this delay, the system loop gain becomes:
$$ T_A(s) = \frac{k_{\text{pwm}} e^{-1.5 s T_s} G_i(s)}{s^3 L_1 (L_2 + L_g) C + s^2 (L_2 + L_g) C k_f k_{\text{pwm}} e^{-1.5 s T_s} + s(L_1 + L_2 + L_g) + k_{\text{pwm}} k_f e^{-1.5 s T_s}} $$
where \( G_i(s) \) is the current controller transfer function. This expression highlights that the proposed method maintains fundamental frequency gain and cutoff frequency despite grid impedance \( L_g \) variations, unlike traditional approaches. For instance, when grid impedance increases, capacitive current damping reduces low-frequency gain, whereas inverter-side current damping preserves performance, making it more suitable for three-phase inverters in weak grids.
To ensure stability, parameter design is critical. The damping feedback coefficient \( k_f \) must be bounded to avoid instability. Using the Routh-Hurwitz criterion in the discrete domain via the w-transform, the stability conditions are derived. The characteristic equation in the w-domain is:
$$ a_0 w^4 + a_1 w^3 + a_2 w^2 + a_3 w + a_4 = 0 $$
where the coefficients are functions of \( k_f \), \( L_1 \), \( L_2 \), \( C \), and \( k_{\text{pwm}} \). For stability, all coefficients must be positive, and the first column of the Routh array must have no sign changes. The maximum allowable \( k_f \) is given by:
$$ k_f < \frac{L_1 + L_2}{k_{\text{pwm}} T_s + 2 \frac{L_2}{L_1} k_{\text{pwm}} \sin(\omega_r T_s)} \cdot \frac{1}{1 + \cos(\omega_r T_s)} $$
For the system parameters listed in Table 1, \( k_f \) is chosen as 0.08 to balance damping effectiveness and stability. The proportional-integral (PI) controller parameters \( k_p \) and \( k_i \) are designed to achieve a phase margin greater than 45° and a gain margin over 3 dB. The gain margin \( GM \) and phase margin \( PM \) are expressed as:
$$ GM = 20 \log \left( \frac{L_2 k_f}{L_1 k_p} \right) $$
$$ PM = 180^\circ + \arctan \left( \frac{2\pi L_1 (f_r^2 – f_c^2) + f_c k_f k_{\text{pwm}} – \frac{k_{\text{pwm}} k_f}{(2\pi)^2 f_c L_2 C} \sin(3\pi f_c T_s)}{f_c k_f k_{\text{pwm}} – \frac{k_{\text{pwm}} k_f}{(2\pi)^2 f_c L_2 C} \cos(3\pi f_c T_s)} \right) – 3\pi f_c T_s – \arctan \left( \frac{k_i}{2\pi f_c k_p} \right) $$
After optimization, \( k_p = 0.045 \) and \( k_i = 150 \) are selected to meet these criteria while ensuring adequate steady-state error performance and dynamic response for three-phase inverters.
| Parameter | Value |
|---|---|
| DC source voltage \( v_{\text{in}} \) | 400 V |
| AC grid voltage \( v_g \) | 180 V |
| Filter capacitor \( C \) | 10 μF |
| Inverter-side inductor \( L_1 \) | 4.0 mH |
| Grid-side inductor \( L_2 \) | 2.0 mH |
| Grid impedance \( L_g \) | 2.0 mH |
| Switching frequency \( f_s \) | 10 kHz |
| Current feedback coefficient \( k_f \) | 0.08 |
| Proportional coefficient \( k_p \) | 0.045 |
| Integral coefficient \( k_i \) | 150 |
Robustness analysis evaluates the system’s performance under parameter variations. Bode plots are generated for changes in \( k_f \), \( L_1 \), \( L_2 \), and \( C \). For example, when \( k_f \) varies from 0.07 to 0.09 (a 28% change), the gain margin changes from 5.27 dB to 5.81 dB, and the phase margin from 55.4° to 68.9°. Similarly, variations in \( L_1 \) (3.5 mH to 4.5 mH), \( L_2 \) (1.6 mH to 2.4 mH), and \( C \) (7 μF to 13 μF) show that gain and phase margins remain within acceptable limits, demonstrating the robustness of the proposed control strategy for three-phase inverters.
Comparative analysis with traditional capacitive current damping reveals significant advantages. Under weak grid conditions with increasing grid impedance, the inverter-side current damping maintains higher phase margins. For instance, at \( L_g = 8 \) mH, the phase margin is 48.5°, whereas capacitive current damping drops to 35.8°, below the 45° requirement. Additionally, the root mean square (RMS) of grid current shows lower oscillations with the proposed method, enhancing stability for three-phase inverters.
To further reduce harmonic distortions caused by grid voltage imperfections, a double-loop feedback structure is introduced. The grid current proportional feedback, with gain \( k_o \), increases the output impedance, thereby attenuating grid-induced harmonics. The output impedance \( Z_0(s) \) without grid current feedback is:
$$ Z_0(s) = \frac{s^3 L_1 L_2 C + s^2 L_2 C k_f k_{\text{pwm}} G_d(s) + s(L_1 + L_2) + k_{\text{pwm}} G_d(s) G_i(s) + k_f k_{\text{pwm}} G_d(s)}{s^2 L_1 C + s C k_f k_{\text{pwm}} G_d(s) + 1} $$
With grid current feedback, the corrected output impedance \( Z_0′(s) \) becomes:
$$ Z_0′(s) = Z_0(s) + Z_1(s) $$
$$ Z_1(s) = k_o \frac{s^2 L_1 C G_d(s)}{s^2 L_1 C + s C k_f k_{\text{pwm}} G_d(s) + 1} $$
This addition effectively series-connects a resistive component, boosting output impedance across frequencies. The design of \( k_o \) must ensure that the fundamental frequency impedance is not significantly altered, with \( k_o < 15 \) based on constraints. Bode plots of output impedance before and after correction confirm the enhancement, particularly in the low-frequency range where harmonics are prominent.
Simulation studies validate the harmonic suppression capability. Under grid voltage distortions with 3rd, 5th, 7th, 9th, 11th, and 13th harmonics, the total harmonic distortion (THD) of grid current is measured. As shown in Table 2, for grid impedances of 6 mH and 8 mH, the double-loop feedback reduces THD from 3.43% to 2.30% and from 5.44% to 2.69%, respectively. Waveforms demonstrate smoother current with reduced harmonic components, affirming the effectiveness of the double-loop strategy for three-phase inverters.
| Grid Impedance \( L_g \) (mH) | THD Before Correction (%) | THD After Correction (%) |
|---|---|---|
| 6 | 3.43 | 2.30 |
| 8 | 5.44 | 2.69 |
Experimental verification using a three-phase inverter prototype confirms the theoretical and simulation results. The system maintains stable grid-connected current waveforms even as grid impedance increases to 8.55 mH, demonstrating the practical applicability of the proposed control strategy.
In conclusion, this paper presents an active damped grid-connected control strategy for three-phase inverters using double-loop feedback. The inverter-side current damping effectively suppresses resonance while maintaining performance under grid impedance variations. The addition of grid current feedback enhances output impedance, reducing harmonic distortions. Stability analysis, parameter design, and simulations prove the strategy’s robustness and effectiveness. This approach offers a reliable solution for three-phase inverters in modern power systems, facilitating the integration of renewable energy sources.