In recent years, the integration of renewable energy sources has gained significant attention, with photovoltaic (PV) systems becoming a key focus. Three-phase inverters play a critical role in converting DC power from PV arrays to AC power for grid connection. However, resonance issues in these three-phase inverter systems can lead to instability, current distortion, and voltage fluctuations, threatening grid reliability. Existing methods, such as phase-locked control, often suffer from negative resistance characteristics in equivalent loops, resulting in grid instability and poor resonance suppression. To address this, we propose a resonance suppression method based on virtual resistance for three-phase photovoltaic grid-connected inverters. This approach amplifies grid impedance to analyze resonance characteristics, adjusts parameters via virtual resistance to prevent frequency deviations, and terminates circulating currents on the grid side, effectively mitigating resonance. Our experimental results demonstrate lower resonance currents and a gain resonance margin of 2–3 dB, indicating superior performance suitable for practical applications.
Resonance in three-phase inverter systems typically arises from imbalances in PV grid-connected system parameters, time-varying grid impedance, and improper inverter design. For instance, when multiple inverters are connected in parallel, the amplification of grid impedance can lead to multiple resonance peaks, increasing current harmonics. The resonance frequency for a grid-connected system can be expressed as:
$$ f_g = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2 + N L_g}{L_1 (L_2 + n L_g) C}} $$
Here, \( f_g \) represents the resonance frequency, \( L_1 \) and \( L_2 \) denote the DC-side inductance and common-mode inductance, respectively, \( L_g \) is the grid-side inductance, \( N \) is the number of parallel inverters, \( n \) is the resonance point, and \( C \) is the DC-side capacitance. As the number of three-phase inverter units increases, resonance points shift toward lower frequencies, often stabilizing around 1.5 kHz. This behavior underscores the need for a method that adapts to impedance variations, which we achieve through virtual resistance.
Virtual resistance serves as an abstract concept to model equivalent impedance, enabling precise analysis of resonance characteristics in three-phase inverter systems. By incorporating virtual resistance, we can simulate the effects of additional damping without physical components, thus avoiding issues like power loss or overheating. The root locus of the virtual resistance system illustrates how changes in resistance coefficients affect stability. For example, as the virtual resistance coefficient increases from 0.01 to 2, the root locus remains on the left side of the imaginary axis, indicating system stability. Let \( \lambda_1, \lambda_2, \lambda_3 \) represent conjugate characteristic roots; as virtual resistance increases, their imaginary parts decrease, enhancing system damping. However, excessive virtual resistance can push these roots toward the imaginary axis, compromising stability. Therefore, we set virtual resistance at an intermediate position to balance damping and stability. The equivalent impedance must satisfy:
$$ \frac{R_i + R_{vi}}{R_j + R_{vj}} = \frac{f_g \lambda_i k_{pi}}{\lambda_j k_{pj}} $$
In this equation, \( R_i \) and \( R_j \) are equivalent resistances, \( R_{vi} \) and \( R_{vj} \) are virtual resistances added in resonance regions, \( \lambda_i \) and \( \lambda_j \) are conjugate characteristic coefficients, and \( k_{pi} \) and \( k_{pj} \) are virtual resistance coefficients. The virtual resistance value \( R_r \) is derived as:
$$ R_r = k_{pi} \frac{P}{U} \frac{R_i + R_{vi}}{R_j + R_{vj}} $$
where \( P \) is the output active power of the three-phase inverter, and \( U \) is the output voltage. This formulation allows for compensation control in the inverter, combining with active damping to adjust resonance suppression parameters. By introducing negative feedback in resonance regions, the frequency response is given by:
$$ \frac{C(j\omega_r)}{R_r(j\omega_r)} = R_r \frac{R_i + R_{vi}}{R_j + R_{vj}} \frac{G(j\omega_r)}{1 + G(j\omega_r) H(j\omega_r)} $$
Here, \( j \) denotes the resonance region, \( \omega_r \) is the resonance frequency, \( G(j\omega_r) \) is the real part of the frequency response with virtual resistance, and \( H(j\omega_r) \) is the imaginary part. When \( G(j\omega_r) \) is large at \( \omega_r \), \( H(j\omega_r) \) acts as negative feedback to reduce the response. The system remains stable if \( H(j\omega_r) G(j\omega_r) > -1 \), enabling effective resonance suppression through adjustments to \( H(j\omega_r) \) and \( G(j\omega_r) \). This approach ensures that the three-phase inverter maintains stability under varying grid conditions.
To further suppress resonance circulating currents, we leverage virtual resistance to terminate current loops on the grid side. In an ideal three-phase inverter scenario, without considering grid impedance or capacitor charging/discharging, the inverter switch transient states can be modeled as:
$$
\begin{aligned}
e_a(t) &= E_m \sin(\omega_r t) \cdot R_r \frac{C(j\omega_r)}{R_r(j\omega_r)} \\
e_b(t) &= E_m \sin\left(\omega_r t – \frac{2\pi}{3}\right) \cdot R_r \frac{C(j\omega_r)}{R_r(j\omega_r)} \\
e_c(t) &= E_m \sin\left(\omega_r t + \frac{2\pi}{3}\right) \cdot R_r \frac{C(j\omega_r)}{R_r(j\omega_r)}
\end{aligned}
$$
where \( e_a(t), e_b(t), e_c(t) \) are the resonance circulating current amplitudes for phases a, b, and c, respectively, \( E_m \) is the grid phase voltage amplitude, and \( t \) is the time of resonance occurrence. These values serve as ideal references for decoupling three-phase voltages into active and reactive components, allowing independent control to reduce the amplitude and frequency of harmonic circulating currents. Additionally, damping resistors are added at the output to absorb harmonic energy, and parallel operation of multiple three-phase inverter units disperses circulating currents. By interleaving carrier angles, current peaks are distributed across grid angle intervals, minimizing resonance impact without compromising grid stability.
We conducted simulation experiments to validate the proposed method for three-phase photovoltaic grid-connected inverters. The setup used Multisim software on a Windows platform, replicating circuit schematics and hardware description languages. The inverter DC side was supplied by an independent DC source and a PV simulator, with multiple inverters operating independently to ensure data validity. Key parameters included an inverter rated power of 20.0 kW, base frequency of 50.0 Hz, inverter-side inductance of 0.09 mH, grid-side inductance of 90.0 μH, AC common-mode inductance of 1.5 mH, AC common-mode capacitance of 3.3 μF, DC capacitance of 1.2 μF, and DC common-mode capacitance of 10.0 nF. Multiple inverters shared identical states, simulating PV grid connection, and an active damper was connected at the common bus to facilitate resonance suppression tests. The simulation structure involved inductors L1, L2, L3 of 10 mH, 2.5 mH, and 1.0 mH, respectively, a voltage source \( U_g \), and a capacitor C. During grid connection, the rated line voltage was maintained at 400 V, switching frequency at 16 kHz, filter capacitance varying between 1.0–3.0 μF, DC common-mode inductance at 1.8 mH, DC common-mode capacitance at 4.7 nF, and AC common-mode capacitance at 10 μF. Two inverters were operated simultaneously, with carrier angles controlled to analyze output currents under interleaved conditions. Fast Fourier Transform (FFT) was used to assess phase characteristics in resonance frequency ranges, determining resonance states.
The results of harmonic suppression performance are summarized in the table below, comparing our method with two existing approaches: Method A (based on improved active disturbance rejection control) and Method B (based on capacitor current feedback with active damping). Resonance current and gain resonance margin were key metrics, with lower currents and higher margins indicating better suppression.
| Method | Resonance Current Range (A) | Gain Resonance Margin (dB) |
|---|---|---|
| Proposed Virtual Resistance Method | 15–30 | 2–3 |
| Method A | 60–90 | 0.5–1.5 |
| Method B | 30–75 | 1–2 |
As shown, our method achieves resonance currents of 15–30 A, significantly lower than Method A (60–90 A) and Method B (30–75 A). This demonstrates effective resonance suppression in the three-phase inverter system. Moreover, the gain resonance margin for our approach ranges from 2 to 3 dB, higher than the 0.5–1.5 dB for Method A and 1–2 dB for Method B, indicating greater stability and better resonance control. These results highlight the superiority of the virtual resistance-based method in maintaining three-phase inverter performance under resonant conditions.
Further analysis of the frequency response in the three-phase inverter system reveals how virtual resistance parameters influence resonance behavior. The stability condition \( H(j\omega_r) G(j\omega_r) > -1 \) ensures that the system does not enter unstable regions. For instance, with virtual resistance coefficients optimized, the resonance peaks in the impedance spectrum are flattened, reducing the risk of harmonic amplification. The following equation models the adjusted output current \( I_{out} \) in terms of virtual resistance and grid impedance \( Z_g \):
$$ I_{out} = \frac{V_{inv}}{Z_g + R_r} $$
where \( V_{inv} \) is the inverter output voltage. This simplifies current control and minimizes distortions. Additionally, the use of parallel three-phase inverter units allows for current sharing, which can be expressed as:
$$ I_{total} = \sum_{k=1}^{N} I_k $$
where \( I_k \) is the current from the k-th inverter, and \( N \) is the number of units. By interleaving carriers, the peak current is reduced, enhancing overall system reliability.
In conclusion, the virtual resistance-based method for resonance suppression in three-phase photovoltaic grid-connected inverters effectively addresses limitations of existing approaches. By analyzing resonance characteristics through impedance amplification, adjusting parameters via virtual resistance, and suppressing circulating currents, this method ensures stable operation of the three-phase inverter under varying grid conditions. Simulation results confirm lower resonance currents and higher gain margins, making it suitable for real-world applications. Future work could focus on optimizing virtual resistance algorithms for larger-scale systems and integrating adaptive control to handle nonlinearities in three-phase inverter networks.