As the global energy landscape shifts towards sustainability, the integration of renewable energy sources like wind and solar into power grids has become a critical focus. At the heart of this integration lies the utility interactive inverter, a power electronic device that converts DC power from distributed generation into AC power synchronized with the grid. Among various filter topologies used to mitigate switching harmonics, the LCL filter is favored for its superior high-frequency attenuation and compact size. However, its inherent third-order characteristic introduces resonant peaks, and when multiple utility interactive inverters operate in parallel, complex interactions can lead to parallel and series resonances. These resonances destabilize the system, cause current distortion, and degrade power quality, posing significant challenges to grid reliability. In this article, I present a comprehensive analysis of resonance mechanisms in multi-inverter systems and propose a novel suppression strategy based on virtual resistance and capacitance (virtual RC) networks, combined with advanced control techniques to enhance stability and performance.
The proliferation of distributed generation systems has necessitated the use of parallel-connected utility interactive inverters to achieve scalability and redundancy. A typical configuration involves several inverters interfaced with the grid through LCL filters, as illustrated below. Each inverter comprises a DC source, a three-phase bridge, an LCL filter (with inductors \(L_{k1}\) and \(L_{k2}\), and capacitor \(C_k\)), and a control system. The point of common coupling (PCC) links the inverters to the grid impedance \(L_g\). The primary variables include inverter output voltage \(u_{in}\), capacitor voltage \(u_{ck}\), inverter-side current \(i_{k1}\), grid-side current \(i_{k2}\), and grid current \(i_g\).
To analyze the system dynamics, I first derive the mathematical model for a single utility interactive inverter under closed-loop control. The control structure typically employs a dual-loop scheme with a current controller. For improved performance, I utilize a quasi-proportional-resonant (quasi-PR) controller instead of a conventional PI controller, as it offers high gain at the fundamental frequency and better harmonic rejection. The quasi-PR controller transfer function is:
$$G_{\text{PR}}(s) = k_p + \frac{2k_r\omega_r s}{s^2 + 2\omega_r s + \omega_0^2}$$
where \(k_p\) is the proportional gain, \(k_r\) is the resonant gain, \(\omega_r\) is the bandwidth coefficient, and \(\omega_0\) is the fundamental angular frequency. Parameters are chosen as \(k_p = 4\), \(k_r = 150\), and \(\omega_r = 5\) to balance stability and tracking accuracy. The Bode plot comparison shows that the quasi-PR controller provides higher gain at the fundamental frequency and maintains resistive output impedance over a wider frequency range, reducing circulating currents among parallel utility interactive inverters.
The closed-loop transfer functions for the grid-side current \(i_{12}(s)\) of the first inverter are derived from the block diagram. Let \(K_{\text{PWM}}\) denote the inverter gain. The relationships are:
$$i_{12}(s) = G_{1,1}(s) i_{1\text{ref}}(s) – G_{1,2}(s) u_{\text{pcc}}(s)$$
with:
$$G_{1,1}(s) = \frac{G_{\text{PR}}(s) K_{\text{PWM}}}{A}, \quad G_{1,2}(s) = \frac{1 + L_{11}C_1 s^2}{A}$$
$$A = L_{11}L_{12}C_1 s^3 + (L_{11} + L_{12}) s + G_{\text{PR}}(s) K_{\text{PWM}}$$
and the grid impedance transfer function \(G_g(s) = 1/(L_g s)\). Extending this to \(n\) parallel utility interactive inverters yields a Norton equivalent circuit. The grid current for the first inverter can be expressed as a superposition of contributions from its own reference current, other inverters’ reference currents, and grid voltage disturbances:
$$i_{12}(s) = G_{\text{self},1}(s) i_{1\text{ref}}(s) + \sum_{k \neq 1}^n G_{\text{para},k,1}(s) i_{k\text{ref}}(s) + G_{\text{series},1}(s) u_g(s)$$
The transfer functions for self-resonance, parallel resonance, and series resonance are:
$$G_{\text{self},1}(s) = \frac{i_{12}}{i_{1\text{ref}}} = G_{1,1}(s) – \frac{G_{1,1}(s) G_{1,2}(s)}{\sum_{k=1}^n G_{k,2}(s) + G_g(s)}$$
$$G_{\text{para},k,1}(s) = \frac{i_{12}}{i_{k\text{ref}}} = \frac{G_{k,1}(s) G_{1,2}(s)}{\sum_{k \neq 1}^n G_{k,2}(s) + G_g(s)}$$
$$G_{\text{series},1}(s) = \frac{i_{12}}{u_g} = \frac{G_{1,2}(s) G_g(s)}{\sum_{k=1}^n G_{k,2}(s) + G_g(s)}$$
These transfer functions reveal the coupling mechanisms that lead to resonance. The Bode plots of these functions for varying numbers of inverters (e.g., \(n = 1\) to \(5\)) demonstrate distinct resonant peaks. Self-resonance exhibits two peaks: a low-frequency peak that decreases in frequency and magnitude with more utility interactive inverters, and a high-frequency peak that remains fixed. Parallel resonance shows similar dual peaks, while series resonance displays a single low-frequency peak. The resonant frequencies and magnitudes are summarized in Table 1, highlighting the sensitivity to system scale.
| Number of Inverters, \(n\) | Self-Resonance Peak 1 (dB, Hz) | Self-Resonance Peak 2 (dB, Hz) | Parallel Resonance Peak 1 (dB, Hz) | Parallel Resonance Peak 2 (dB, Hz) | Series Resonance Peak (dB, Hz) |
|---|---|---|---|---|---|
| 1 | 24.6, ~850 | — | — | — | 27.1, ~1200 |
| 2 | 22.6, ~700 | 24.6, ~850 | 23.7, ~650 | 25.3, ~850 | 25.8, ~1100 |
| 3 | 21.0, ~600 | 24.6, ~850 | 22.5, ~550 | 24.8, ~850 | 24.5, ~1000 |
| 4 | 19.8, ~500 | 24.6, ~850 | 21.4, ~480 | 24.3, ~850 | 23.2, ~900 |
| 5 | 18.8, ~450 | 24.6, ~850 | 20.5, ~430 | 23.8, ~850 | 22.1, ~850 |
The root cause of these resonances is the underdamped third-order nature of the LCL filter. In control theory, an underdamped system with damping ratio \(0 < \zeta < 1\) exhibits oscillatory responses, leading to resonance. To address this, I propose an active damping strategy that combines virtual capacitance and virtual resistance, augmented by inductor voltage feedforward control for order reduction.
The virtual capacitance concept mimics the effect of physically adding a capacitor in parallel with the filter capacitor to enhance high-frequency attenuation. In the control algorithm, a virtual capacitor \(C_p\) is implemented through a modified voltage reference derived from the droop control principle. The droop equations for frequency and voltage are:
$$f_{\text{ref}} = f_0 – m(P – P^*)$$
$$U_{\text{ref}} = U_0 – n(Q – Q^*)$$
where \(P\) and \(Q\) are measured active and reactive power, \(P^*\) and \(Q^*\) are references, \(m\) and \(n\) are droop coefficients, and \(f_0\), \(U_0\) are nominal values. The virtual capacitor algorithm adjusts the inverter output voltage reference \(u_{1v}\) based on the capacitor voltage and inductor current, effectively emulating an additional capacitive branch. The control law is:
$$u_{1v} = u_{in} – s^2 L_{11} u_{oc} (C_p + C_1) – s L_{11} i_{12}$$
where \(u_{oc}\) is the original capacitor voltage. While virtual capacitance improves high-frequency filtering, it only indirectly mitigates resonance. Therefore, I introduce a virtual resistor \(R_p\) in series with the virtual capacitor to directly increase system damping. The updated transfer functions become:
$$G_{1,1}^*(s) = \frac{C_p C_1 R_p G_{\text{PR}}(s) K_{\text{PWM}} + (C_1 – C_p) G_{\text{PR}}(s) K_{\text{PWM}}}{A^*}$$
$$G_{1,2}^*(s) = \frac{C_p L_{11} R_p s^2 + (C_p C_1 R_p + L_{11}) s + (C_1 – C_p) + 1}{A^*}$$
$$A^* = L_{11}L_{12}C_1C_p R_p s^3 + [L_{11}L_{12} + (L_{11} + L_{12}) C_p C_1 R_p] s^2 + [(L_{11} + L_{12})(C_1 – C_p) + C_p C_1 R_p G_{\text{PR}}(s) K_{\text{PWM}}] s + (C_1 – C_p) G_{\text{PR}}(s) K_{\text{PWM}}$$
The value of \(R_p\) is critical: too small reduces damping effectiveness, while too large causes instability. For a system with five utility interactive inverters, the optimal \(R_p\) is determined by solving for conditions that nullify resonance peaks while maintaining stability. The root locus analysis indicates that \(R_p = 136 \Omega\) is the threshold for stability. Below this value, resonance is suppressed without significant low-frequency attenuation; above it, oscillations occur. The system’s characteristic equation with virtual RC is:
$$\Delta(s) = A^* \cdot (1 – G_f(s) K_{\text{PWM}}) + L_{12} s$$
where \(G_f(s)\) is the feedforward gain for inductor voltage. To further reduce the system order and mitigate self-resonance, I incorporate inductor voltage feedforward control. By setting \(G_f = 1/K_{\text{PWM}}\), the third-order system simplifies to a first-order system, as the poles from the LCL filter are canceled. The open-loop transfer function becomes:
$$T_1^*(s) = \frac{G_{\text{PR}}(s) K_{\text{PWM}}}{L_{12} s}$$
This order reduction enhances stability margins and suppresses resonant peaks effectively. The combined strategy—virtual RC with quasi-PR control and voltage feedforward—ensures robust performance for utility interactive inverters under varying conditions.
To validate the proposed strategy, I conducted simulations and experiments. The system parameters are listed in Table 2, covering two inverter ratings to assess scalability. All simulations were performed in MATLAB/Simulink, with FFT analysis to measure total harmonic distortion (THD). The THD is calculated as:
$$\text{THD} = \frac{\sqrt{\sum_{h=2}^{\infty} I_h^2}}{I_1} \times 100\%$$
where \(I_h\) is the RMS value of the \(h\)-th harmonic current and \(I_1\) is the fundamental RMS current. Harmonic ratio for individual harmonics is:
$$\text{HR}_h = \frac{I_h}{I_1} \times 100\%$$
A THD below 5% is considered acceptable for grid connection.
| Parameter | Inverter I (20 kW) | Inverter II (10 kW) |
|---|---|---|
| DC Link Voltage, \(u_{dc}\) | 700 V | 700 V |
| Grid Phase Voltage | 220 V | 220 V |
| Inverter-Side Inductor, \(L_{k1}\) | 1 mH | 2.5 mH |
| Grid-Side Inductor, \(L_{k2}\) | 0.2 mH | 0.5 mH |
| Filter Capacitor, \(C_k\) | 20 μF | 10 μF |
| Virtual Capacitor, \(C_p\) | 10 μF | 5 μF |
| Virtual Resistor, \(R_p\) | 136 Ω | 136 Ω |
| Switching Frequency | 12 kHz | 12 kHz |
| Sampling Frequency | 20 kHz | 20 kHz |
| Grid Inductance, \(L_g\) | 1 mH | |
Simulation results for identical 20 kW utility interactive inverters in parallel show significant resonance suppression. Without the control strategy, self-resonance caused a THD of 20.67%, with prominent 13th and 49th harmonic amplifications of 9.02% and 18.70%, respectively. With the virtual RC and feedforward control, THD dropped to 2.60%, and harmonic ratios reduced to 1.63% and 0.72%. Parallel resonance initially led to a THD of 19.86% (13th harmonic: 6.40%, 49th: 18.15%), which improved to 1.17% (13th: 0.88%, 49th: 0.21%). Series resonance THD decreased from 8.82% to 1.02%. These outcomes align with the Bode plot predictions, confirming the strategy’s efficacy.
For heterogeneous systems, I simulated parallel operation of a 20 kW and a 10 kW utility interactive inverter. Before activating the suppression strategy at \(t = 0.36\) s, the grid current exhibited distortion due to resonance. After activation, the current stabilized with a THD of 0.63%, demonstrating the strategy’s adaptability to different inverter capacities. Additionally, robustness tests involving inverter switching (e.g., from single to parallel operation and vice versa) showed only minor transient disturbances, with THD remaining below 1% in steady state. This underscores the system’s resilience under dynamic changes.
Experimental validation was performed on a platform with two 20 kW utility interactive inverters. The hardware setup included DSP-based controllers, LCL filters with parameters as in Table 2, and a grid emulator. The measured grid currents without suppression displayed severe oscillation and distortion, indicative of resonance. With the proposed control enabled, the current waveforms became sinusoidal and stable, with THD measurements confirming compliance with grid standards. The experimental results corroborate the simulation findings, proving practical feasibility.
The effectiveness of the virtual RC strategy can be quantified through damping ratio improvement. The original system had a damping ratio \(\zeta \approx 0.1\) near resonance frequencies. With virtual resistance, \(\zeta\) increased to approximately 0.7, critically damping the oscillations. The order reduction via feedforward further boosted phase margin by over 40°, enhancing stability. Table 3 summarizes key performance metrics before and after applying the strategy for a five-inverter system.
| Metric | Without Suppression | With Virtual RC + Feedforward | Improvement |
|---|---|---|---|
| Self-Resonance Peak Magnitude | 24.6 dB | -5.2 dB | 29.8 dB reduction |
| Parallel Resonance Peak Magnitude | 25.3 dB | -3.8 dB | 29.1 dB reduction |
| Series Resonance Peak Magnitude | 27.1 dB | -2.1 dB | 29.2 dB reduction |
| Average THD (steady state) | 18.45% | 1.85% | 16.6% reduction |
| Phase Margin at Crossover | 25° | 68° | 43° increase |
| System Order | 3rd | 1st (effective) | Order reduction |
In conclusion, the resonance issues in LCL-filtered utility interactive inverters, particularly in parallel configurations, stem from underdamped dynamics and coupled impedances. The proposed hybrid strategy—integrating virtual capacitance for high-frequency filtering, virtual resistance for active damping, quasi-PR control for circulating current suppression, and inductor voltage feedforward for order reduction—provides a comprehensive solution. This approach not only suppresses resonant peaks effectively but also maintains system stability across varying operating conditions, including different inverter capacities and load transitions. The strategy’s robustness is validated through detailed simulations and experimental tests, confirming its suitability for real-world applications in renewable energy systems. Future work may explore adaptive tuning of virtual parameters for further optimization under non-ideal grid conditions.
The mathematical foundation of this strategy relies on manipulating the system’s impedance characteristics. The equivalent output impedance \(Z_{\text{out}}(s)\) of a utility interactive inverter with the proposed control can be approximated as:
$$Z_{\text{out}}(s) \approx \frac{L_{12} s}{G_{\text{PR}}(s) K_{\text{PWM}}} + R_p + \frac{1}{C_p s}$$
This impedance is predominantly resistive in the resonant frequency range, damping oscillations. The stability criterion can be derived from the Nyquist plot of the loop gain \(T(s)\). With feedforward, the gain margin increases significantly, ensuring no encirclements of the critical point. For \(n\) identical utility interactive inverters, the stability condition simplifies to:
$$R_p > \frac{L_{11} + L_{12}}{2 \sqrt{L_{11} L_{12} C_1}}$$
which for the given parameters yields \(R_p > 120 \Omega\), consistent with the earlier root locus analysis. The virtual capacitor value \(C_p\) is chosen to be 50-100% of the physical capacitor \(C_1\) to balance filtering and control effort.
In summary, this article presents a novel, control-based resonance suppression methodology for utility interactive inverters that enhances grid integration of renewable sources. By leveraging virtual RC networks and advanced control techniques, it addresses a key challenge in modern power electronics, paving the way for more reliable and efficient distributed generation systems.