The widespread integration of renewable energy sources has made the utility interactive inverter a critical interface between distributed generation systems and the AC grid. Its performance directly impacts grid stability and power quality. LCL filters are predominantly used at the output of utility interactive inverters due to their superior high-frequency harmonic attenuation capabilities compared to simple L filters. However, the proliferation of power electronic devices has led to increasing grid impedance, creating a “weak grid” environment characterized by significant background voltage harmonics. These harmonics can induce substantial distortion in the grid-injected current of the utility interactive inverter, compromising compliance with power quality standards such as IEEE 1547 and GB/T 37408.
Among various harmonic suppression strategies, grid voltage feedforward is a widely adopted technique due to its simplicity and effectiveness in mitigating disturbances caused by grid voltage harmonics without altering the system’s closed-loop stability margins. The ideal full feedforward function can, in theory, completely cancel the influence of grid voltage distortion on the output current. However, its practical implementation faces challenges, particularly when the Point of Common Coupling (PCC) voltage is not directly accessible, as is often the case when the utility interactive inverter is connected through a step-up transformer. In such configurations, the transformer’s secondary-side leakage inductance can serve as the grid-side filter inductor, making the capacitor voltage a readily available internal state variable that correlates with the PCC voltage.
This leads to the concept of Capacitor Voltage Full Feedback (CVFF). By feeding back a processed version of the capacitor voltage, the utility interactive inverter can emulate the harmonic rejection effect of grid voltage feedforward. The ideal CVFF function, derived from the principle of disturbance rejection, comprises three terms: a proportional term, a first-derivative term, and a second-derivative term. In practice, the derivative terms, especially the second-derivative, are challenging to implement and can negatively interact with the system’s current control loop, potentially introducing instability. Existing CVFF schemes often simplify the ideal function, typically by omitting the derivative term that cancels with active damping feedback or by approximating complex digital delay terms as unity. While these simplifications ensure implementability and basic stability, they come at a significant cost: a severe degradation in the suppression capability against medium and high-order harmonics (e.g., above the 13th harmonic). In weak grids with rich harmonic spectra, this shortcoming becomes particularly pronounced, as the simplified CVFF may even amplify certain high-frequency harmonics.
This paper addresses this critical limitation. We identify the core contradiction: achieving strong high-frequency harmonic suppression requires the CVFF function to closely match its ideal form, but this often compromises system stability, especially under weak grid conditions with wide impedance variations. To resolve this, we propose a novel Adaptive Capacitor Voltage Full Feedback (A-CVFF) method for LCL-type utility interactive inverters. The proposed method introduces a phase compensation link to correct the mid-to-high frequency phase deviation caused by neglecting digital control delays. Furthermore, it incorporates a virtual impedance correction mechanism by introducing an adjustable impedance coefficient into the second-derivative feedback path. Most importantly, an adaptive adjustment module is designed to dynamically regulate this impedance coefficient based on the real-time harmonic content of the grid current. This allows the utility interactive inverter to autonomously balance the trade-off between high-order harmonic attenuation and system stability margin, ensuring robust performance across varying grid conditions. The proposed A-CVFF method significantly enhances the harmonic immunity of the utility interactive inverter, making it highly suitable for modern weak grid applications.
Analysis of Conventional CVFF Limitations and Digital Delay Impact
The control structure of a three-phase LCL-type utility interactive inverter with conventional capacitor voltage feedback in the stationary (αβ) frame is considered. The system includes the inverter bridge, an LCL filter (with inverter-side inductance $L_1$, filter capacitance $C$, and grid-side inductance $L_2$ which includes transformer leakage and grid impedance $L_g$), and a digital controller. The current control loop typically employs a Proportional-Integral (PI) regulator $G_{PI}(s)$ and active damping via capacitor current feedback with gain $K_c$. The total digital control delay $G_d(s)$, encompassing computation and Pulse-Width Modulation (PWM) hold effects, is modeled as $G_d(s) = e^{-1.5T_s s}$, where $T_s$ is the sampling period.
When the PCC voltage is not measured, the capacitor voltage $u_c$ is fed back. The ideal full-feedforward function from the capacitor voltage to cancel grid voltage disturbance is derived from the system’s open-loop transfer function. The simplified control block diagram of the utility interactive inverter with CVFF is shown below, where $G_{x1}(s)$ and $G_{x2}(s)$ represent the forward path and the plant transfer function from the modulator output to the grid current, respectively.
$$ G_{x1}(s) = \frac{G_{PI}(s) G_d(s)}{s^2 L_1 C + s C K_c G_d(s) + 1} $$
$$ G_{x2}(s) = \frac{s^2 L_1 C + s C K_c G_d(s) + 1}{s^3 L_1 L_2 C + s^2 L_2 C K_c G_d(s) + s(L_1 + L_2)} $$
The ideal CVFF function $G_{cf,ideal}(s)$ is then given by:
$$ G_{cf,ideal}(s) = \frac{G_{PI}(s)}{G_{x1}(s)} = \frac{1}{G_d(s)} + s C K_c + \frac{s^2 L_1 C}{G_d(s)} $$
This function has three components. The term $s C K_c$ corresponds to the existing capacitor current feedback for active damping. If this path is already present, it can be omitted from the CVFF implementation, simplifying it to:
$$ G’_{cf,ideal}(s) = \frac{1}{G_d(s)} + \frac{s^2 L_1 C}{G_d(s)} $$
The major implementation difficulty lies in the term $1/G_d(s) = e^{1.5T_s s}$, which is a non-causal, phase-advancing element. Conventional CVFF schemes approximate $G_d(s) \approx 1$, leading to the practically implemented feedback function:
$$ G_{cf,conv}(s) = 1 + s^2 L_1 C $$
This approximation introduces a significant phase error between the ideal and implemented feedback functions. The digital delay $G_d(s)$ introduces a phase lag of $-1.5 \omega T_s$ radians. By neglecting it ($G_d(s)=1$), the conventional CVFF function $G_{cf,conv}(s)$ lacks this lag, making its phase response lead the ideal one. The actual phase deviation $\Delta \phi(\omega)$ is:
$$ \Delta \phi(\omega) = \arg(G_{cf,conv}(j\omega)) – \arg(G’_{cf,ideal}(j\omega)) $$
Since $\arg(G’_{cf,ideal}(j\omega)) = \arg(e^{1.5 j \omega T_s} (1 – \omega^2 L_1 C)) = 1.5 \omega T_s + \arg(1 – \omega^2 L_1 C)$, and $\arg(G_{cf,conv}(j\omega)) = \arg(1 – \omega^2 L_1 C)$, we get:
$$ \Delta \phi(\omega) = -1.5 \omega T_s $$
This linear phase lag increases with frequency. For a 20 kHz sampling frequency ($T_s = 50 \mu s$), the phase lag reaches approximately $27^\circ$ at 1 kHz. This mismatch severely undermines the harmonic cancellation performance at medium and high frequencies. The following table summarizes the phase characteristics.
| Feedback Function | Mathematical Form | Key Limitation |
|---|---|---|
| Ideal CVFF | $G’_{cf,ideal}(s)= e^{1.5T_s s}(1+s^2 L_1 C)$ | Non-causal, unimplementable |
| Conventional CVFF | $G_{cf,conv}(s)= 1+s^2 L_1 C$ | Large phase error at high frequency, poor high-order harmonic suppression |
Furthermore, the second-derivative term $s^2 L_1 C$ interacts with the system dynamics. From an active damping perspective, the CVFF introduces equivalent virtual impedances in parallel with the filter capacitor. The stability of the utility interactive inverter depends on the resistive part of these equivalent impedances. The conventional CVFF can introduce negative resistance near the LCL resonance frequency, especially when the grid impedance $L_g$ is large, pushing the system towards instability or requiring very conservative controller tuning that limits bandwidth. This creates the fundamental conflict: using a more “ideal” feedback for better harmonic suppression risks stability, while a simplified, stable feedback offers inadequate high-frequency performance.
Proposed Adaptive Capacitor Voltage Full Feedback (A-CVFF) Method
The proposed A-CVFF method for the utility interactive inverter is designed to overcome the aforementioned limitations through three key innovations: 1) Phase compensation for the digital delay, 2) Virtual impedance correction with an adjustable coefficient, and 3) An adaptive mechanism to dynamically optimize this coefficient.
Phase Compensation Strategy
Instead of approximating $G_d(s)$ as 1, we propose to approximate its inverse $1/G_d(s) = e^{1.5T_s s}$ using a first-order lead-like function that is physically realizable. A suitable approximation is a first-order high-pass filter with a carefully chosen time constant to match the phase lead characteristics within the frequency range of interest (typically below $f_s/6$). We use a first-order inertial element in a positive feedback path to create the required phase compensation. The compensated feedback function $G_{cp}(s)$ becomes:
$$ G_{cp}(s) = G_{IE}(s) + \frac{G_{IE}(s) s^2 L_1 C}{R_h} $$
Here, $G_{IE}(s)$ is the phase compensation link designed to mimic the phase response of $e^{1.5T_s s}$ at lower frequencies, and $R_h$ is the newly introduced impedance correction coefficient (to be discussed next). A effective choice for $G_{IE}(s)$ is:
$$ G_{IE}(s) = \frac{1}{1.5 T_s s + 1} $$
This first-order transfer function provides a phase response that partially compensates for the lag introduced by neglecting $G_d(s)$. While it does not perfectly match the ideal advance across all frequencies, it significantly reduces the phase error in the mid-frequency range (e.g., a few hundred Hz to a few kHz), which is crucial for suppressing typical harmonic orders (e.g., 11th, 13th, 17th, 23rd). The modified control block diagram of the utility interactive inverter with the proposed phase-compensated feedback is integrated into the system.
Virtual Impedance Correction and Stability Analysis
The term $\frac{G_{IE}(s) s^2 L_1 C}{R_h}$ represents the second-derivative feedback path. The coefficient $R_h$ (with units of $\Omega$) is not merely a gain but acts as a crucial virtual impedance correction parameter. To understand its role, we analyze the equivalent virtual impedance $Z_{eq}(s)$ introduced by the A-CVFF loop across the filter capacitor. By moving the feedback summation point, the A-CVFF can be shown to equivalently place two virtual impedances, $Z_P(s)$ and $Z_{D2}(s)$, in parallel with the capacitor $C$.
$$ Z_P(s) = -\frac{s L_1}{G_d(s) G_{IE}(s)} $$
$$ Z_{D2}(s) = -\frac{R_h}{s C G_d(s) G_{IE}(s)} $$
The stability of the current control loop in the utility interactive inverter is strongly influenced by the real (resistive) parts of $Z_P(s)$ and $Z_{D2}(s)$. For the system to be stable, the equivalent parallel resistance must be positive around the LCL resonant frequency to provide adequate damping. Substituting $s = j\omega$ and analyzing the real parts $R_P(\omega)$ and $R_{D2}(\omega)$ reveals that their signs depend on frequency and $R_h$. A critical condition for stability is that the parallel combination $R_P(\omega) // R_{D2}(\omega)$ remains positive for all frequencies below the Nyquist frequency. Through detailed derivation, a lower bound for $R_h$ is found to ensure this condition:
$$ R_h > 4\pi^2 f_{max}^2 L_1 C $$
where $f_{max}$ is the highest frequency of concern for stability, typically taken as $f_s/3$. This gives a practical design rule:
$$ R_{h,min} = \frac{4\pi^2 f_s^2 L_1 C}{9} $$
Selecting $R_h$ greater than $R_{h,min}$ guarantees that the A-CVFF scheme does not introduce negative resistance that could destabilize the utility interactive inverter. However, as will be shown next, the value of $R_h$ also directly impacts the harmonic suppression performance.
Adaptive Adjustment of the Impedance Coefficient $R_h$
While a large $R_h$ ensures good stability margin, it degrades the harmonic suppression capability of the A-CVFF, particularly at higher frequencies. The transfer function from grid voltage disturbance $u_g(s)$ to grid current $i_g(s)$ with A-CVFF is:
$$ i_g(s) = … – \frac{G_{x2}(s)G_{PI}(s) – G_{cp}(s)T_A(s)}{[1+T_A(s)]G_{PI}(s) – sL_2 G_{cp}(s)T_A(s)} u_g(s) $$
where $T_A(s)=G_{x1}(s)G_{x2}(s)$ is the loop gain. The magnitude of this disturbance rejection term $|i_g(j\omega)/u_g(j\omega)|$ indicates the susceptibility to grid voltage harmonics. Analysis shows that as $R_h$ increases, this magnitude increases at medium and high frequencies, meaning weaker harmonic suppression.
To resolve the stability-performance trade-off dynamically, we propose an adaptive module that adjusts $R_h$ in real-time based on the measured harmonic content of the grid current. The block diagram of this module is as follows:
- Harmonic Extraction: The measured grid current $i_g$ is passed through a notch filter bank $G_{NA}(s)$ to remove the fundamental component (50/60 Hz) and dominant low-order harmonics (e.g., 5th, 7th). The output $i_{g,h}$ contains the higher-order harmonic content.
$$ G_{NA}(s) = \prod_{h \in \{1,5,7\}} \frac{s^2 + (h\omega_0)^2}{s^2 + h\omega_0 s / Q + (h\omega_0)^2} $$
where $\omega_0$ is the fundamental frequency and $Q$ is the quality factor. - Harmonic Energy Calculation: The extracted harmonic current $i_{g,h}$ (in αβ coordinates) is used to compute its squared RMS value. For each axis (α, β), the signal is squared and passed through a low-pass filter (LPF) $G_{LPF}(s)$ with a cutoff of, e.g., 50 Hz, to obtain the average square.
$$ I_{h,\alpha}^2 = \text{LPF}(i_{g,h,\alpha}^2), \quad I_{h,\beta}^2 = \text{LPF}(i_{g,h,\beta}^2) $$
The total harmonic energy indicator is $I_h^2 = I_{h,\alpha}^2 + I_{h,\beta}^2$. - PI Regulation: This indicator $I_h^2$ is compared to a predefined threshold $I_{lim}^2$, which corresponds to the maximum allowable high-order harmonic distortion (e.g., corresponding to a 2% THD contribution from harmonics above the 7th). The error $e = I_{lim}^2 – I_h^2$ is fed into a PI regulator $G_{PA}(s)$.
$$ G_{PA}(s) = K_{PA} + \frac{K_{IA}}{s} $$ - Dynamic Coefficient Generation: The output of the PI regulator is the impedance coefficient $R_h(t)$. A limiter is applied to ensure $R_h$ always stays within a safe range: $[R_{h,min}, R_{h,max}]$. The lower limit $R_{h,min}$ ensures stability as derived. The upper limit $R_{h,max}$ prevents excessive gain reduction that might make the feedback ineffective.
The adaptive loop works as follows: If the high-order harmonic content $I_h^2$ exceeds the threshold $I_{lim}^2$, the error becomes negative, causing the PI regulator to decrease $R_h$. A smaller $R_h$ strengthens the second-derivative feedback action, improving high-frequency harmonic suppression at the potential cost of reduced stability margin. Conversely, if harmonics are well suppressed ($I_h^2 < I_{lim}^2$), the PI regulator increases $R_h$, prioritizing a larger stability margin. Thus, the utility interactive inverter autonomously finds an operating point where harmonic distortion is just at the acceptable limit, maximizing the robustness against grid impedance variations.
The parameters for the key components of the A-CVFF scheme for a typical utility interactive inverter are determined as follows:
| Component | Parameter Determination Method | Typical Value / Range |
|---|---|---|
| Phase Compensator $G_{IE}(s)$ | Time constant set to match $1.5T_s$ to compensate for dominant delay. | $\tau = 1.5T_s$ (e.g., 75 $\mu$s for $f_s=20$ kHz) |
| Impedance Coef. Lower Limit $R_{h,min}$ | $R_{h,min} = \frac{4\pi^2 f_s^2 L_1 C}{9}$ | Calculated from $L_1$, $C$, $f_s$ (e.g., 8.5 $\Omega$) |
| Impedance Coef. Upper Limit $R_{h,max}$ | Set to 2-5 times $R_{h,min}$, or based on the minimum required harmonic suppression at highest frequency of concern. | e.g., 40 $\Omega$ |
| Harmonic Threshold $I_{lim}$ | Based on power quality standard for high-order harmonics (e.g., limit for harmonics > 15th). Can be set as a fraction of rated current $I_N$. | $I_{lim} = (0.01 \text{ to } 0.02) \cdot I_N$ |
| Adaptive PI Regulator $G_{PA}(s)$ | Tuned for slow adaptation to avoid interference with current control dynamics. Bandwidth < 10 Hz. | $K_{PA}$ small, $K_{IA}$ to achieve settling in ~100 ms. |
Experimental Verification
A 10 kVA three-phase LCL-type utility interactive inverter prototype was built to validate the proposed A-CVFF method. The system parameters are listed in the table below. A programmable AC source was used to emulate a weak grid with background voltage harmonics. A series inductor ($L_g = 3$ mH) was added to simulate high grid impedance. The proposed control algorithm was implemented on a TMS320F28379D digital signal processor.
| Parameter | Symbol | Value |
|---|---|---|
| DC Link Voltage | $U_{dc}$ | 700 V |
| Grid Voltage (Phase) | $U_g$ | 220 Vrms |
| Rated Power | $S_N$ | 10 kVA |
| Switching/Sampling Frequency | $f_{sw}$, $f_s$ | 20 kHz |
| Inverter-side Inductor | $L_1$ | 0.6 mH |
| Filter Capacitor | $C$ | 8 $\mu$F |
| Total Grid-side Inductor | $L_2$ ($L_{2inv}+L_g$) | 0.4 mH + 3 mH |
| Current PI Controller | $K_P$, $K_I$ | 12, 1000 rad/s |
| Cap. Current Feedback Gain | $K_c$ | 0.6 |
| Phase Compensator | $G_{IE}(s)$ | $1/(7.5e-5 s + 1)$ |
| Impedance Coef. Limits | $[R_{h,min}, R_{h,max}]$ | [8.5 $\Omega$, 40 $\Omega$] |
| Harmonic Threshold | $I_{lim}$ | 2% of $I_N$ ($\approx$0.43 A) |
The grid voltage was programmed with significant background harmonics at the 5th (3%), 7th (3%), 11th (2%), 13th (2%), 17th (2%), 23rd (1%), and 31st (1%) orders to create a challenging distortion environment for the utility interactive inverter.
Case 1: No CVFF. The grid current waveform exhibited clear distortion due to the polluted grid voltage. The Total Harmonic Distortion (THD) of the grid current was measured at 9.12%, with prominent low-order harmonics.
Case 2: Conventional CVFF ($G_{cf,conv}(s)=1+s^2L_1C$). The low-order harmonic distortion (5th, 7th) was reduced. However, the high-order harmonic content (17th, 23rd, 31st) was visibly higher than in Case 1. The current THD improved to 7.65%, but the high-frequency spectrum was worsened, confirming the amplification effect predicted by the analysis.
Case 3: Proposed A-CVFF with Fixed $R_h=17 \Omega$ (Adaptive module OFF). With the phase compensation active, the high-order harmonic amplification was mitigated compared to Case 2. The current THD reduced to 5.64%. However, the harmonic levels, especially the 17th and above, were still not optimally suppressed, as the fixed $R_h$ was chosen conservatively for stability.
Case 4: Proposed A-CVFF with Adaptive $R_h$ (Adaptive module ON). Initially, the system operated as in Case 3. Upon enabling the adaptive module, it detected that the high-order harmonic energy $I_h^2$ exceeded the set threshold $I_{lim}^2$. The PI regulator dynamically decreased $R_h$ from 17 $\Omega$ and settled around 12 $\Omega$. This stronger feedback action further suppressed the high-frequency harmonics. The grid current waveform became significantly cleaner. A detailed spectral analysis showed a drastic reduction in the 17th, 23rd, and 31st harmonic components. The final grid current THD was reduced to 2.81%, well within standard limits. The utility interactive inverter maintained stable operation throughout the adaptation process.
The following table summarizes the key performance metrics from the experimental validation, demonstrating the effectiveness of the A-CVFF method for the utility interactive inverter.
| Control Scheme | Key Observation | Grid Current THD | High-order Harmonic Trend (17th+) | Stability |
|---|---|---|---|---|
| No CVFF | Severe distortion from grid harmonics. | 9.12% | Present | Stable |
| Conventional CVFF | Low-order harmonics reduced, but high-order harmonics amplified. | 7.65% | Amplified | Stable |
| A-CVFF (Fixed $R_h$) | Phase compensation helps; high-order harmonics still elevated. | 5.64% | Moderately Suppressed | Stable |
| A-CVFF (Adaptive $R_h$) | Optimal balance; harmonics effectively suppressed to meet limit. | 2.81% | Strongly Suppressed | Stable |
Conclusion
This paper has presented a comprehensive Adaptive Capacitor Voltage Full Feedback (A-CVFF) method for enhancing the performance of LCL-type utility interactive inverters in weak and distorted grid environments. The method systematically addresses the core limitations of existing CVFF schemes: phase error due to unmodeled digital delays and the inherent conflict between high-frequency harmonic suppression and system stability.
The proposed solution incorporates a first-order phase compensation link to better approximate the ideal feedback function’s behavior in the medium-frequency range. More importantly, it introduces a virtual impedance correction coefficient $R_h$ within the second-derivative feedback path. The theoretical derivation of a lower bound for $R_h$ ensures that the utility interactive inverter maintains stability under weak grid conditions. The cornerstone of the A-CVFF is the adaptive adjustment module, which dynamically regulates $R_h$ based on the real-time high-order harmonic content of the grid current. This allows the utility interactive inverter to autonomously navigate the trade-off, minimizing current distortion while preserving adequate stability margin.
Experimental results on a 10 kVA prototype validate the theory and demonstrate the superior performance of the A-CVFF method. Compared to conventional CVFF, the proposed method not only prevents the amplification of high-order harmonics but actively suppresses them, reducing the overall grid current THD from over 7.6% to below 3% under severe grid voltage distortion. The utility interactive inverter equipped with A-CVFF exhibits strong robustness against grid impedance variation and background harmonic pollution, making it a highly viable control strategy for ensuring high power quality and reliable operation of distributed generation systems in modern electrical grids.