The proliferation of renewable energy sources has placed the utility interactive inverter at the forefront of modern power systems, serving as the critical interface between distributed generation and the main grid. Among various filter topologies, the LCL filter is favored for its superior high-frequency harmonic attenuation and dynamic performance. However, this structure introduces a resonant peak that can threaten system stability, especially under weak grid conditions characterized by significant grid impedance. While inverter-side current feedback control inherently offers higher stability compared to grid-side feedback, it is still critically limited by digital control delays. These delays restrict the stable operating region to a ratio between the filter’s resonant frequency and the sampling frequency. Furthermore, a problematic harmonic amplification band exists near the resonant frequency, which can excite harmonic resonance when interacting with grid background harmonics. Traditional dual-loop control methods with active damping often struggle with bandwidth limitations and complex parameter design, failing to achieve an ideal stability boundary and robust performance. This paper addresses these challenges by proposing a novel Capacitor Voltage Feedback Active Damping (CVFAD) design method, which reconfigures system zeros to eliminate the detrimental reverse resonance peak and ensures robust operation across a wide range of grid impedances.
System Modeling and Stability Challenges
The analysis focuses on a single-phase LCL-type utility interactive inverter. The system’s plant, from the inverter output voltage \( u_{in} \) to the inverter-side inductor current \( i_1 \), neglecting parasitic resistances and grid voltage disturbances for stability analysis, is given by:
$$G_{in}(s) = \frac{i_1(s)}{u_{in}(s)} = \frac{1}{sL_1} \cdot \frac{s^2 + \omega_r^2}{s^2 + \omega_{res}^2}$$
where the conjugate zero frequency \( \omega_r \) and the resonant pole frequency \( \omega_{res} \) are defined as:
$$\omega_r = \frac{1}{\sqrt{L_T C}}, \quad \omega_{res} = \sqrt{\frac{L_1 + L_T}{L_1 L_T C}}$$
with \( L_T = L_2 + L_g \) representing the total grid-side inductance.
Digital control introduces a computational delay, typically modeled as a 1.5-sample delay \( G_d(s) \approx e^{-1.5sT_s} \). With a proportional-resonant (PR) current controller \( G_c(s) \) and pulse-width modulation gain \( k_{PWM} \), the open-loop transfer function for the basic inverter-side current feedback control is:
$$T_{open}(s) = G_c(s) k_{PWM} G_d(s) G_{in}(s)$$
Stability analysis of this loop reveals a critical constraint: for stability, the resonant frequency \( f_{res} \) must be less than one-sixth of the sampling frequency \( f_s \) (i.e., \( f_{res} < f_s/6 \)). This significantly limits the design of the LCL filter parameters for a given digital controller speed.
Even when stable (\( f_{res} < f_s/6 \)), a deeper issue persists. The utility interactive inverter’s output admittance \( Y_o(s) \), which determines its interaction with the grid voltage harmonics, exhibits a band of positive gain (magnitude > 0 dB) around the resonant frequency. This implies a negative incremental impedance behavior in that frequency range, leading to amplification of any grid voltage harmonics present at the point of common coupling (PCC). This harmonic amplification can severely degrade grid current quality and may precipitate harmonic resonance under weak grid conditions. The table below summarizes the core limitations of the basic single-loop control.
| Aspect | Limitation in Basic Inverter-Side Current Feedback |
|---|---|
| Stability Boundary | Stable only if \( f_{res} < f_s/6 \). Limits filter design. |
| Harmonic Interaction | Existence of a harmonic amplification band near \( f_{res} \). Poor robustness to grid voltage distortion. |
| Grid Impedance Variation | Performance and stability margins vary significantly with changing \( L_g \). |
Proposed Capacitor Voltage Feedback Active Damping (CVFAD) Method
To overcome these limitations, we propose a novel active damping strategy that feeds back the filter capacitor voltage. The core idea is to use this feedback to actively damp the LCL resonance and, through an additional feedback path, reconfigure the system’s open-loop zeros to cancel the problematic resonant poles. The block diagram of the proposed control structure is implemented within the utility interactive inverter’s digital controller.
The capacitor voltage feedback inner loop, with gain \( H_v \), is added directly. This modifies the system dynamics. Analyzing the inner loop stability in the discrete domain (z-domain) using the Routh-Hurwitz criterion in the w-plane provides the stability conditions for the proposed utility interactive inverter. The key finding is that the stable boundary frequency is extended. Specifically, with negative feedback gain (\( H_v < 0 \)), the system can remain stable for resonant frequencies up to \( f_s/3 \), doubling the allowable range compared to the basic single-loop case. The condition for stability is:
$$-\frac{L_1 + L_T}{k_{PWM} L_T} < H_v < 0 \quad \text{for} \quad 0 < f_{res} < f_s/3$$
This greatly relaxes the design constraints on the LCL filter for the utility interactive inverter.
However, while stabilizing the system, this inner loop alone does not fully address robustness. The open-loop transfer function \( T_{o-Hv}(s) \) still contains a pair of complex zeros that interact with the poles, creating a “reverse resonance” peak in the frequency response. This peak reduces phase margin and its frequency shifts with grid inductance \( L_g \), making the system sensitive to grid strength variations.
The pivotal step in our method is the introduction of an additional capacitor voltage feedback path, \( G_V(s) \), as shown in the final control block diagram. This path is designed specifically to cancel the undesirable zeros introduced by the main CVFAD loop. By choosing:
$$G_V(s) = \frac{1 + H_v k_{PWM} G_d(s)}{sL_1}$$
the open-loop transfer function of the utility interactive inverter simplifies remarkably to a first-order system:
$$T_{down}(s) = \frac{G_c(s) G_d(s) k_{PWM}}{sL_1}$$
This expression is completely independent of the total grid-side inductance \( L_T \). Consequently, the gain crossover frequency and phase margin of the current control loop become invariant to changes in grid strength, granting the utility interactive inverter exceptional robustness.
Parameter Design and Performance Characteristics
The parameter design for this utility interactive inverter control scheme is straightforward. The capacitor voltage feedback gain \( H_v \) is critical. Analysis of the output admittance \( Y_{HG-Mc}(s) \) reveals that to avoid any frequency band with positive gain (which would cause harmonic amplification), the optimal choice is:
$$H_v = -1/k_{PWM}$$
This value ensures the output admittance magnitude remains below 0 dB across all frequencies, granting the utility interactive inverter inherent immunity to grid voltage harmonics.
Since the addition of the \( G_V(s) \) path alters the effective current feedback signal, a magnitude correction factor \( M_c \) is required to ensure the reference current \( i_{ref} \) correctly commands the desired grid current amplitude \( I_m \). The correction factor is given by:
$$M_c = \frac{I_m}{I_m + (t_d / L_1) U_C}$$
where \( t_d = 1.5T_s \) is the digital delay time and \( U_C \) is the amplitude of the capacitor voltage. This correction is implemented digitally.
Furthermore, as the control uses capacitor voltage for phase-locked loop (PLL) synchronization and regulates the inverter-side current, a small phase shift exists between the grid voltage and current. To achieve unity power factor operation, this phase angle \( \gamma \) must be compensated within the PLL. The angle is calculated as:
$$\gamma = \arcsin\left(\frac{\omega_0 L_2 I_2}{U_C}\right) – \arctan\left(\frac{\omega_0 C U_C}{I_1 (1 – \omega_0^2 L_2 C)}\right)$$
where \( \omega_0 \) is the grid fundamental angular frequency. Compensating the PLL by this angle \( \gamma \) enables the utility interactive inverter to operate at unity power factor.
| Parameter | Design Equation / Value | Purpose |
|---|---|---|
| Capacitor Voltage Gain \( H_v \) | \( H_v = -1/k_{PWM} \) | Optimizes damping and eliminates harmonic amplification band. |
| Additional Path \( G_V(s) \) | \( G_V(s) = (1 + H_v k_{PWM} G_d(s)) / (sL_1) \) | Cancels resonant zeros, achieves grid-impedance-invariant open-loop response. |
| Magnitude Corrector \( M_c \) | \( M_c = I_m / (I_m + (t_d / L_1) U_C) \) | Corrects steady-state amplitude error from added path. |
| Power Factor Angle \( \gamma \) | \( \gamma = \arcsin(\omega_0 L_2 I_2 / U_C) – \arctan(\omega_0 C U_C / (I_1 (1 – \omega_0^2 L_2 C))) \) | Phase compensation for unity power factor operation. |
Theoretical Analysis of Robustness and Performance
The superiority of the proposed CVFAD method for the utility interactive inverter is evident in the analytical results. First, the stability boundary is expanded from \( f_{res} < f_s/6 \) to \( f_{res} < f_s/3 \), offering more flexibility in LCL filter design. Second, and most importantly, the open-loop transfer function after zero-pole cancellation is:
$$T_{down}(s) = \frac{G_c(s) e^{-1.5sT_s} k_{PWM}}{sL_1}$$
Its Bode plot is unaffected by \( L_T \). This means the phase margin (PM) and gain margin (GM) remain constant regardless of grid impedance variations. For example, if the controller is designed for a 45° PM at a nominal grid condition, it will maintain exactly 45° PM even when the grid inductance \( L_g \) varies from 0 to a large value representing a very weak grid.
Third, the output admittance is reshaped. With \( H_v = -1/k_{PWM} \), the magnitude of \( Y_o(s) \) is suppressed below 0 dB for all frequencies. Mathematically, this ensures the real part of the output impedance remains positive across the spectrum (i.e., \( \text{Re}\{Z_o(j\omega)\} > 0 \)), fulfilling the passivity criterion in a wide frequency range. This makes the utility interactive inverter inherently stable when connected to any grid impedance and highly resistant to harmonic distortion at the PCC.
The pole maps of the closed-loop system further confirm robustness. As \( L_g \) varies over a wide range (e.g., 0 to 12.73 mH, representing a short-circuit ratio down to 2), all closed-loop poles remain in the left-half plane, with no migration towards the instability boundary. This demonstrates the exceptional stability robustness of the proposed design for the utility interactive inverter in weak grids.
Simulation and Experimental Validation
The proposed control method was validated through detailed simulations and a 1.8 kW laboratory prototype of a utility interactive inverter. The system parameters are listed below.
| Parameter | Value |
|---|---|
| Rated Power \( P_{out} \) | 1.8 kW |
| DC-Link Voltage \( U_{dc} \) | 300 V |
| Grid Voltage (RMS) \( U_{grms} \) | 120 V |
| Inverter-side Inductor \( L_1 \) | 2 mH |
| Grid-side Inductor \( L_2 \) | 0.5 mH |
| Filter Capacitor \( C \) | 5 µF |
| Switching Frequency \( f_{sw} \) | 10 kHz |
| Sampling Frequency \( f_s \) | 20 kHz |
| PR Controller (\( k_p, k_r \)) | 0.052, 3.6 |
| Feedback Gain \( H_v \) | -1/300 |
Simulations compared the proposed CVFAD method against a traditional inductor current feedback active damping (ICFAD) method. Upon startup without damping, both systems exhibited resonant instability. When active damping was enabled at 0.05s, both stabilized. However, spectral analysis revealed a critical difference: the traditional ICFAD method still showed a pronounced harmonic amplification band around the resonant frequency, leaving the utility interactive inverter vulnerable to grid harmonics. In contrast, the proposed CVFAD method completely suppressed this band, resulting in a grid current total harmonic distortion (THD) well within IEEE Std 519-2022 limits.
To test robustness, simulations injected multiple background voltage harmonics (3rd, 5th, 7th, etc., up to 13th) at the PCC and varied the grid inductance \( L_g \) from 0 to 12.73 mH. The grid current waveform maintained high quality with low THD (consistently below 2.5%) across all conditions, demonstrating the method’s effectiveness in harmonic rejection and stability. The unity power factor correction was also verified, showing reactive power reduction to nearly zero after compensation.
Experimental results from the 1.8 kW utility interactive inverter prototype confirmed the simulation findings. The waveforms showed stable operation with grid current THD values between 1.83% and 2.37% as \( L_g \) was varied, proving the robustness of the hardware implementation. Dynamic performance tests showed the inverter could smoothly transition between half-load and full-load within half a grid cycle, demonstrating excellent transient response. These results collectively validate the theoretical analysis and practical feasibility of the proposed CVFAD design method for robust utility interactive inverters.
Conclusion
This paper has presented a comprehensive analysis and a novel solution for enhancing the stability and robustness of LCL-type utility interactive inverters using inverter-side current feedback. The proposed Capacitor Voltage Feedback Active Damping (CVFAD) method effectively addresses the dual challenges imposed by digital control delay and weak grid conditions. By employing a capacitor voltage feedback inner loop, the stable operating boundary is extended. Crucially, through the strategic addition of a complementary capacitor voltage feedback path, the system’s open-loop zeros are reconfigured to cancel the resonant poles. This yields a first-order open-loop response that is invariant to grid impedance variations, thereby eliminating the harmonic amplification band and the sensitivity of stability margins to grid strength. The parameter design is straightforward and relies directly on known circuit and controller parameters. Both simulation and experimental results on a 1.8 kW prototype confirm that the utility interactive inverter employing the proposed CVFAD method achieves superior performance: it maintains high stability and low grid current THD across a wide range of grid impedances, effectively suppresses background harmonic influences, and can operate at unity power factor with appropriate compensation. This method offers a reliable and practical control solution for the next generation of robust utility interactive inverters in weak and distorted grid environments.