In my research on renewable energy integration, I have extensively studied the challenges posed by weak grid conditions, particularly the harmonic resonance issues in multi-utility interactive inverter grid-connected systems. The increasing penetration of distributed generation sources, such as solar and wind, has led to the widespread deployment of utility interactive inverters, which convert DC power to AC and synchronize with the grid. However, in weak grids characterized by high grid impedance, the interaction among multiple utility interactive inverters and between inverters and the grid can lead to destabilizing resonances. This not only degrades power quality but also threatens system stability. Therefore, I propose a comprehensive global resonance suppression strategy tailored for multi-utility interactive inverter systems operating in weak grid environments. This strategy combines improved grid-current feedback active damping with enhanced grid-voltage feedforward control, effectively mitigating self-resonance, parallel resonance, and series resonance. Throughout this article, I will detail the modeling, analysis, and validation of this approach, emphasizing the critical role of utility interactive inverters in modern power systems.
To begin, I analyze the resonance mechanisms in multi-utility interactive inverter grid-connected systems. Each utility interactive inverter typically employs an LCL filter to attenuate switching harmonics, but this introduces inherent resonant peaks. In a weak grid, the grid impedance becomes significant and variable, creating complex coupling effects. When multiple utility interactive inverters are connected in parallel, their output impedances interact with the grid impedance, leading to additional resonance bands that can excite harmonic oscillations. My modeling approach is based on the Norton equivalent circuit, which allows for a systematic analysis of these interactions. For a system with n utility interactive inverters, the grid-connected current of the first inverter, denoted as \(i_{g1}(s)\), can be expressed as a function of reference currents, coupling currents from other inverters, and grid voltage disturbances. The transfer functions are derived as follows:
$$ i_{g1}(s) = R_1(s) I_{ref1}(s) – \sum_{k=2}^{n} P_{1k}(s) I_{refk}(s) – S_1(s) U_g(s) $$
where \(R_1(s)\) is the self-excitation transfer function, \(P_{1k}(s)\) represents the parallel coupling transfer function between inverters, and \(S_1(s)\) is the grid-voltage excitation transfer function. These functions depend on the LCL filter parameters (\(L_1\), \(L_2\), \(C_f\)), controller gains, and grid impedance \(L_g\). For instance, the self-excitation transfer function for a utility interactive inverter can be approximated as:
$$ R_1(s) = \frac{G_{pi}(s) K_{PWM} Z_c}{Z_{L1} Z_{L2} + (Z_{L1} + Z_{L2}) Z_c + G_{pi}(s) K_{PWM} Z_c} $$
with \(Z_{L1} = sL_1\), \(Z_{L2} = sL_2\), \(Z_c = 1/(sC_f)\), and \(G_{pi}(s) = K_p + K_i/s\) as the PI controller. The grid impedance \(L_g\) introduces additional terms in the denominator, affecting resonance frequencies. My analysis shows that as the number of utility interactive inverters increases, the system exhibits a fixed resonance band and a shifting resonance band, with the latter moving to lower frequencies as grid impedance rises. This behavior is summarized in Table 1, which compares resonance characteristics for different numbers of utility interactive inverters under weak grid conditions.
| Number of Utility Interactive Inverters | Fixed Resonance Band (Hz) | Shifting Resonance Band (Hz) | Dominant Coupling Effect |
|---|---|---|---|
| 1 | ~950 | N/A | Self-resonance |
| 2 | ~950 | ~600-800 | Parallel resonance |
| 3 | ~950 | ~400-600 | Parallel and series resonance |
| ≥4 | ~950 | <400 | Strong grid coupling |
The resonance peaks can lead to high total harmonic distortion (THD) in grid currents, exceeding standards like IEEE 1547. To address this, I propose a global resonance suppression strategy that enhances damping without additional hardware costs. This strategy is particularly relevant for utility interactive inverters, which must maintain stability while feeding power into weak grids. The core idea is to implement virtual impedance through active damping and compensate for grid impedance variations via feedforward control.
First, I focus on the improved grid-current feedback active damping method. Traditional active damping often requires capacitor current feedback, necessitating extra sensors. To simplify, I use grid-current feedback with a virtual resistor \(R_v\) placed in parallel with the filter capacitor. However, direct differentiation of grid current can amplify noise. Therefore, I propose a modified active damping controller \(G_{ad}(s)\) that incorporates a second-order low-pass filter to approximate the derivative effect while reducing noise sensitivity. The controller is defined as:
$$ G_{ad}(s) = \frac{L_1 L_2 \omega_n^2 s}{R_v (s^2 + 2\zeta \omega_n s + \omega_n^2)} $$
where \(\omega_n\) is the cutoff frequency (set 4-5 times the LCL resonant frequency for effective damping), and \(\zeta = 0.707\) for optimal noise rejection. This controller emulates a virtual impedance \(R_v\) (e.g., 10 Ω) across the capacitor, increasing system damping and suppressing resonance peaks from both self and parallel interactions among utility interactive inverters. The effectiveness is evident in the modified open-loop transfer function, where the resonance spike is flattened without compromising phase margin.
Second, I address the series resonance between utility interactive inverters and the grid using an improved grid-voltage feedforward strategy. In weak grids, the grid impedance \(L_g\) causes a positive feedback loop when traditional proportional feedforward is applied, as the PCC voltage \(U_{pcc}(s)\) includes a term \(i_g(s) s L_g\). This exacerbates instability. To counteract this, I measure \(L_g\) via non-characteristic harmonic injection methods and subtract the impedance-induced voltage drop from the feedforward path. The feedforward control law becomes:
$$ U_{ff}(s) = U_g(s) – i_g(s) s L_g $$
This eliminates the positive feedback component, enhancing stability. The overall control diagram for a utility interactive inverter integrates both strategies, as shown in the inserted figure below. The combination ensures robust performance across varying grid conditions.
To validate my approach, I conducted simulations and experiments on a system with three utility interactive inverters. The parameters are listed in Table 2, reflecting typical values for grid-connected applications. Each utility interactive inverter operated under identical conditions, with a weak grid simulated by adding series inductance at the PCC.
| Parameter | Value | Description |
|---|---|---|
| Grid Voltage \(U_g\) | 220 V (RMS) | Nominal AC voltage |
| DC-Link Voltage \(V_{dc}\) | 700 V | Input to utility interactive inverter |
| Switching Frequency \(f_s\) | 12.8 kHz | PWM frequency for utility interactive inverter |
| LCL Filter: \(L_1\) | 350 μH | Inverter-side inductance |
| LCL Filter: \(L_2\) | 50 μH | Grid-side inductance |
| LCL Filter: \(C_f\) | 10 μF | Filter capacitance |
| Grid Impedance \(L_g\) | 0.2-2 mH | Variable weak grid condition |
| PI Controller: \(K_p\) | 2 | Proportional gain for utility interactive inverter |
| PI Controller: \(K_i\) | 5 | Integral gain for utility interactive inverter |
| Virtual Resistor \(R_v\) | 10 Ω | Active damping parameter |
Simulation results demonstrated significant improvement. Without the global resonance suppression strategy, the grid current THD reached 6.86%, with visible distortion due to resonances at multiple frequencies. After applying my strategy, the THD dropped to 1.76%, well within the 5% limit often required by grid codes. The Bode plots of the open-loop transfer function confirmed resonance suppression, showing increased phase margin (above 45°) and gain margin (above 6 dB) across a wide frequency range, even as \(L_g\) varied from 0.2 to 2 mH. The key resonance frequencies, calculated using the formula for LCL resonance \(f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C_f}}\), were effectively damped. For my parameters, \(f_r \approx 950\) Hz, and the active damping controller reduced the peak by over 20 dB.
Experimental verification on a hardware platform with three utility interactive inverters further supported these findings. The grid current waveforms transitioned from highly distorted to sinusoidal after implementing the strategy. Dynamic tests, such as sudden disconnection of one utility interactive inverter, showed fast recovery with minimal transients, indicating robust stability. The measured impedance at PCC was approximately 40 μH, and with added series inductance, the weak grid condition was emulated. The global resonance suppression strategy maintained current quality under these perturbations, underscoring its practicality for real-world applications involving multiple utility interactive inverters.
The mathematical foundation of my strategy can be summarized through key equations. The overall transfer function from reference current to grid current for a utility interactive inverter with both active damping and feedforward is:
$$ \frac{i_g(s)}{I_{ref}(s)} = \frac{G_{pi}(s) K_{PWM}}{s^3 L_1 L_2 C_f + s^2 (L_1 + L_2 + L_g) C_f K + s (L_1 + L_2 + L_g) + K} $$
where \(K = G_{pi}(s) K_{PWM} + G_{ad}(s) R_v\). This equation highlights how the virtual impedance and feedforward terms reshape the denominator, adding damping coefficients that suppress resonances. To quantify the damping improvement, I define a damping ratio \(\zeta_{sys}\) for the dominant resonance mode:
$$ \zeta_{sys} = \frac{R_v C_f \omega_n}{2} \cdot \frac{1}{\sqrt{1 + \frac{L_g}{L_1 + L_2}}} $$
With my parameters, \(\zeta_{sys}\) increased from 0.1 (underdamped) to 0.7 (near critically damped), ensuring stable operation. Additionally, the sensitivity of the system to grid impedance variations is reduced, as shown by the sensitivity function \(S(s)\):
$$ S(s) = \frac{1}{1 + L(s)} $$
where \(L(s)\) is the loop gain. With my strategy, \(S(s)\) remains below 0 dB at resonance frequencies, minimizing harmonic amplification.
In conclusion, my global resonance suppression strategy offers a viable solution for multi-utility interactive inverter grid-connected systems in weak grids. By integrating improved grid-current feedback active damping and grid-voltage feedforward, it addresses self, parallel, and series resonances without additional sensors, making it cost-effective for widespread deployment of utility interactive inverters. The strategy enhances damping, maintains power quality, and ensures stability under dynamic conditions. Future work could explore adaptive tuning of virtual impedance based on real-time grid impedance measurements, further optimizing performance for utility interactive inverters in evolving grid environments. This research underscores the importance of advanced control techniques in enabling reliable integration of renewable energy sources through utility interactive inverters, contributing to a more resilient and sustainable power system.
To further elaborate on the technical aspects, I present Table 3, which compares my strategy with conventional methods for resonance suppression in utility interactive inverters. This highlights the advantages in terms of stability, cost, and implementation complexity.
| Method | Key Principle | Hardware Cost | Stability in Weak Grid | Suitability for Multi-Utility Interactive Inverters |
|---|---|---|---|---|
| Capacitor Current Feedback | Active damping via sensor | High (extra sensor) | Moderate | Limited due to coupling |
| Grid Current Differential Feedback | Derivative-based damping | Low | Poor (noise-sensitive) | Moderate |
| Notch Filter Control | Frequency-selective suppression | Low | Poor (ignores grid impedance) | Limited |
| My Global Strategy | Virtual impedance + feedforward | Low | High | Excellent |
The efficacy of my approach is also reflected in the harmonic spectrum analysis. Before suppression, prominent harmonics at multiples of the resonance frequencies (e.g., 950 Hz, 600 Hz) were observed. After suppression, these harmonics were attenuated by over 15 dB, as calculated using the formula for harmonic reduction \(H_{red} = 20 \log_{10} \left( \frac{|G_{after}(j\omega)|}{|G_{before}(j\omega)|} \right)\), where \(G\) is the transfer function at harmonic frequency \(\omega\). For instance, at 950 Hz, \(H_{red} \approx -25\) dB, demonstrating effective damping.
In summary, the global resonance suppression strategy I propose is a significant advancement for utility interactive inverter systems. It leverages control theory to mitigate inherent challenges in weak grids, ensuring that utility interactive inverters can operate reliably in large-scale deployments. As the power grid evolves with more distributed generation, such strategies will be crucial for maintaining stability and power quality, making utility interactive inverters key enablers of the energy transition.