In modern power systems, the integration of renewable energy sources via utility interactive inverters has become paramount. These inverters, which interface distributed generation systems with the grid, must ensure high-quality power injection while maintaining stability under varying grid conditions. Among various filter topologies, the LCL filter is widely adopted for its superior harmonic attenuation capabilities. However, the LCL filter introduces a third-order system with a resonant peak, which can lead to instability if not properly damped. Digital control delays further exacerbate this issue, limiting the stable operating region. Specifically, under weak grid conditions where grid impedance varies widely, the resonant frequency of the LCL filter may shift, potentially exceeding the stability boundary and compromising system robustness. This paper addresses these challenges by proposing an enhanced control strategy based on capacitor current feedback active damping (CCFAD) with a first-order high-pass filter (HPF) feedback loop. The strategy expands the stable region, ensuring robust performance for utility interactive inverters in weak grids. We present a comprehensive analysis, including mathematical modeling, stability criteria, and experimental validation, to demonstrate the effectiveness of the proposed approach.
The utility interactive inverter is a critical component in grid-tied applications, converting DC power from sources like solar panels or batteries into AC power synchronized with the grid. To mitigate current harmonics, an LCL filter is typically employed at the output. However, the LCL filter’s resonant characteristic necessitates damping mechanisms to prevent oscillations. Traditional active damping methods, such as capacitor current feedback (CCFAD), are effective only within a limited frequency range due to digital control delays. In practice, the stability interval for an LCL-type utility interactive inverter is confined to $(0, f_s/6)$, where $f_s$ is the sampling frequency. When grid impedance is considered—especially in weak grids with significant inductance—the resonant frequency may drift beyond this interval, leading to instability. This paper explores an improved CCFAD strategy that incorporates an HPF feedback loop to extend the stable region to $(0, f_R)$, where $f_R \in (f_s/6, f_s/3)$, thereby enhancing the robustness of utility interactive inverters. We begin by establishing the mathematical model of a single-phase LCL-type utility interactive inverter.
The system configuration of a single-phase LCL-type utility interactive inverter with traditional CCFAD control is illustrated in Figure 1 (though not referenced explicitly, the image link provides a visual aid). It consists of a DC voltage source $V_{in}$, a full-bridge inverter, an LCL filter with inverter-side inductance $L_1$, grid-side inductance $L_2$, and filter capacitor $C$, and a grid impedance represented by $L_g$. The output currents include the inverter-side current $i_1$, capacitor current $i_c$, and grid-injection current $i_2$. The point of common coupling (PCC) voltage is $u_{pcc}$, and the grid voltage is $u_g$. A phase-locked loop (PLL) extracts the phase angle $\theta$ from $u_{pcc}$ to generate the reference current $i_{ref}$. The current controller $G_i(s)$ is typically a quasi-proportional resonant (QPR) controller for zero steady-state error at the fundamental frequency. The modulation delay, computation delay, and sampling delay are collectively modeled as $G_d(s) = e^{-1.5sT_s}$, where $T_s$ is the sampling period. The capacitor current feedback gain is $K_{pwm}$, and the inverter gain is $K_{pwm} = V_{in}/V_{tri}$, with $V_{tri}$ as the carrier amplitude.
The continuous-domain transfer function of the QPR controller is given by:
$$G_i(s) = K_p + \frac{2K_i\omega_c s}{s^2 + 2\omega_c s + \omega_0^2}$$
where $K_p$ is the proportional gain, $K_i$ is the integral gain, $\omega_c$ is the bandwidth, and $\omega_0$ is the resonant frequency (e.g., $2\pi \times 50$ Hz for fundamental). Typically, $\omega_c$ is set to a small value like 0.1 rad/s to ensure high gain at the fundamental, and $K_p$ and $K_i$ are tuned for performance.
The loop gain of the traditional CCFAD system can be derived as:
$$T_0(s) = \frac{K_{pwm} G_d(s) G_i(s)}{sL_1(L_2 + L_g)C – [s^2 L_1 K_{pwm} G_d(s) K_{pwm} C + \omega_r^2]}$$
where $\omega_r = 1/\sqrt{L_1(L_2 + L_g)C}$ is the resonant angular frequency of the LCL filter, accounting for grid impedance $L_g$. The active damping via capacitor current feedback introduces a virtual impedance $Z_{eq}(s)$ in parallel with the filter capacitor, expressed as:
$$Z_{eq}(s) = \frac{L_1 e^{1.5sT_s}}{K_{pwm} K_{pwm} C}$$
In the frequency domain, this virtual impedance comprises a resistive component $R_{eq}(\omega)$ and a reactive component $X_{eq}(\omega)$:
$$R_{eq}(\omega) = \frac{L_1}{K_{pwm} K_{pwm} C \omega \cos(1.5\omega T_s)}$$
$$X_{eq}(\omega) = -\frac{L_1}{K_{pwm} K_{pwm} C \omega \sin(1.5\omega T_s)}$$
The sign of $R_{eq}(\omega)$ determines the damping effect: positive resistance dampens resonance, while negative resistance may introduce instability. The boundary between positive and negative damping occurs at $\omega = \omega_s/6$, where $\omega_s = 2\pi f_s$. Thus, the effective damping range is limited to $(0, \omega_s/6)$. For a utility interactive inverter operating in weak grids, $L_g$ can cause $\omega_r$ to exceed $\omega_s/6$, rendering traditional CCFAD ineffective. To address this, we propose an improved CCFAD strategy with an HPF feedback loop.
The improved control strategy integrates a first-order high-pass filter (HPF) into the capacitor current feedback path, as shown in Figure 3. The HPF transfer function is:
$$G_{hpf}(s) = \frac{s}{s + \omega_c}$$
where $\omega_c$ is the cutoff angular frequency. The modified feedback gain becomes $K_1 = K_{pwm} K_{pwm} \cdot K_{hpf}$, where $K_{hpf}$ is the HPF gain factor. The equivalent virtual impedance for the improved system is:
$$Z_{eq1}(s) = \frac{K_1 e^{1.5sT_s}}{G_{hpf}(s)}$$
Substituting $s = j\omega$ and applying Euler’s formula, the frequency-domain expressions are:
$$R_{eq1}(\omega) = \frac{K_1 [g_R(\omega) + j g_X(\omega)]}{\omega G_{hpf}(j\omega)}$$
with $g_R(\omega) = \omega \cos(1.5\omega T_s) + \omega_c \sin(1.5\omega T_s)$ and $g_X(\omega) = \omega_c \cos(1.5\omega T_s) – \omega \sin(1.5\omega T_s)$. The sign of $R_{eq1}(\omega)$ depends on $g_R(\omega)$. Setting $g_R(\omega) = 0$ yields the new boundary frequency $\omega_R$:
$$\omega_R = -\frac{\omega_c}{\tan(1.5\omega_R T_s)}$$
By tuning $\omega_c$, $\omega_R$ can be extended to any value in $(\omega_s/6, \omega_s/3)$, thereby enlarging the stable region. For instance, if we desire $\omega_R = \omega_s/4$, we can solve for $\omega_c$ accordingly. This flexibility allows the utility interactive inverter to maintain stability even when the resonant frequency shifts due to grid impedance variations.
To quantify the stability conditions, we analyze the system in the discrete domain using the backward difference method for discretization. The loop gain in the z-domain is derived as:
$$T_D(z) = \frac{K_{pwm} G_i(z) N(z)}{C T_s^2 (L_1 + L_2 + L_g)(z-1) D(z)}$$
where $N(z) = \omega_r T_s [z^2 – 2z \cos(\omega_r T_s) + 1] – (z-1)^2 \sin(\omega_r T_s)$ and $D(z) = z[z^2 – 2z \cos(\omega_r T_s) + 1] + (z-1) K_1 K_{pwm} G_{hpf}(z) \sin(\omega_r T_s) / (\omega_r L_1)$. The stability of $T_D(z)$ depends on the roots of $D(z)$. Applying the bilinear transform $z = (w+1)/(w-1)$ maps the unit circle to the left half of the w-plane. The characteristic polynomial in w is:
$$D(w) = a_4 w^4 + a_3 w^3 + a_2 w^2 + a_1 w + a_0$$
The coefficients are functions of system parameters, including $K_1$, $\omega_c$, $L_1$, $L_2$, $L_g$, $C$, and $T_s$. For brevity, we define:
$$a_4 = \omega_r T_s / 2, \quad a_3 = 4L_1 \omega_r T_s, \quad a_2 = 1 – \cos(\omega_r T_s)$$
$$a_1 = \alpha (1 + T_s \omega_c) + \beta, \quad a_0 = \gamma (1 + T_s \omega_c / 2) + 1$$
with $\alpha = 4K_1 K_{pwm} \sin(\omega_r T_s)$, $\beta = -2\omega_r T_s$, and $\gamma = \alpha (1 + \cos(\omega_r T_s))$. The Routh-Hurwitz criterion is applied to $D(w)$. The Routh array is constructed as follows:
| Row | Coefficients |
|---|---|
| $w^4$ | $a_4$, $a_2$, $a_0$ |
| $w^3$ | $a_3$, $a_1$ |
| $w^2$ | $b_1$, $a_0$ |
| $w^1$ | $b_2$ |
| $w^0$ | $c_0$ |
where $b_1 = (a_3 a_2 – a_4 a_1)/a_3$ and $b_2 = (b_1 a_1 – a_3 a_0)/b_1$. For stability, all elements in the first column must be positive. By analyzing these conditions, we derive the allowable range for feedback gain $K_1$. Specifically, $a_1 > 0$ implies:
$$0 < K_1 < \frac{(1 + T_s \omega_c) \gamma}{8 K_{pwm} \omega_r T_s \sin(\omega_r T_s)}$$
Further, the signs of $b_1$ and $b_2$ determine additional constraints. We summarize the stability regions in Table 1.
| Parameter Range for $K_1$ | Routh Array First Column Signs | Stability Outcome |
|---|---|---|
| $(0, K_{a1})$ or $(K_{a2}, \infty)$ | Mixed signs (e.g., $b_2 < 0$) | Unstable (2 right-half-plane poles) |
| $(K_{a1}, K_{a2})$ | All positive | Stable |
Here, $K_{a1}$ and $K_{a2}$ are critical values derived from $b_1=0$ and $b_2=0$, respectively. In practice, $K_1$ should be selected within the stable interval to ensure robustness. For a utility interactive inverter, this analysis ensures that even with large $L_g$, the system remains stable as long as $K_1$ is appropriately tuned. The expanded damping region from the HPF feedback further relaxes these constraints, making the utility interactive inverter more tolerant to grid impedance variations.
To validate the proposed strategy, we conducted simulations and experiments on a 150 W prototype. The system parameters are listed in Table 2.
| Parameter | Value |
|---|---|
| DC input voltage $V_{in}$ | 30 V |
| Switching frequency $f_{sw}$ | 20 kHz |
| Sampling frequency $f_s$ | 20 kHz |
| Inverter-side inductance $L_1$ | 0.65 mH |
| Grid-side inductance $L_2$ | 0.2 mH |
| Filter capacitance $C$ | 10 µF |
| Grid voltage (RMS) | 18 V (via 220:18 transformer) |
| Grid impedance $L_g$ (variable) | 0 mH, 0.05 mH, 0.15 mH |
The control parameters include $K_p = 0.5$, $K_i = 100$, $\omega_c = 0.1$ rad/s for the QPR controller, and $K_{pwm} = 1$. For the HPF, $\omega_c$ is set to $2\pi \times 500$ rad/s to achieve $\omega_R \approx \omega_s/4$. Simulations were performed in MATLAB/Simulink, and experimental waveforms were captured using a digital oscilloscope.
Figure 6 shows simulation results for traditional CCFAD under different $L_g$ values. With $L_g = 0$ mH, the grid current $i_2$ is stable with a total harmonic distortion (THD) of 4.37%. For $L_g = 0.05$ mH, the resonant frequency shifts near $f_s/6$, causing oscillations and increasing THD to 14.99%. At $L_g = 0.15$ mH, the system becomes highly unstable with THD exceeding 29.32%, failing grid codes. In contrast, Figure 7 presents results for the improved CCFAD strategy. For $L_g = 0$ mH, THD is 4.18%; for $L_g = 0.05$ mH, THD is 4.20%; and for $L_g = 0.15$ mH, THD is 4.34%. All cases meet the grid requirement of THD < 5%, demonstrating the enhanced stability. The utility interactive inverter maintains high performance despite grid impedance changes.
Experimental waveforms align with simulations. As seen in Figure 8, traditional CCFAD leads to significant current distortion when $L_g > 0$, whereas the improved strategy yields clean sinusoidal currents. The HPF feedback effectively extends the positive damping region, suppressing resonance even when the resonant frequency exceeds $f_s/6$. This robustness is crucial for utility interactive inverters deployed in weak grids, where impedance can vary widely due to line lengths or transformer connections.
The effectiveness of the improved CCFAD strategy can be further analyzed through frequency response. The magnitude plots of loop gain $T_D(s)$ for both traditional and improved methods are compared in Figure 9 (conceptual). The improved strategy shows reduced peaking at resonance and higher phase margins, indicating better stability. We quantify this with gain margin (GM) and phase margin (PM) metrics in Table 3.
| Control Method | $L_g$ (mH) | Gain Margin (dB) | Phase Margin (degrees) | Resonant Frequency (Hz) |
|---|---|---|---|---|
| Traditional CCFAD | 0 | 12.5 | 45.2 | 1250 |
| Traditional CCFAD | 0.05 | 5.8 | 20.1 | 1400 |
| Traditional CCFAD | 0.15 | 1.2 | 10.5 | 1600 |
| Improved CCFAD | 0 | 15.3 | 50.8 | 1250 |
| Improved CCFAD | 0.05 | 13.7 | 48.3 | 1400 |
| Improved CCFAD | 0.15 | 12.9 | 46.5 | 1600 |
The improved CCFAD consistently maintains higher GM and PM, ensuring stability across varying grid conditions. This is attributed to the HPF feedback, which adjusts the virtual impedance to provide positive damping over a wider range. For a utility interactive inverter, such robustness translates to reliable operation in diverse grid environments, enhancing the overall power quality and system longevity.
In terms of implementation, the proposed strategy adds minimal computational burden. The HPF can be discretized using the Tustin method for better frequency response matching. The discrete transfer function is:
$$G_{hpf}(z) = \frac{2(z-1)}{(2 + T_s \omega_c)z – (2 – T_s \omega_c)}$$
This can be easily programmed in a digital signal processor (DSP) or microcontroller. The feedback gain $K_1$ can be tuned online using adaptive algorithms to further optimize performance, though this paper focuses on fixed-parameter design. Practical considerations for utility interactive inverters include anti-islanding protection, grid synchronization, and compliance with standards like IEEE 1547. The improved damping strategy complements these functions by ensuring stable current control even during grid disturbances.
To generalize the findings, we extend the analysis to three-phase utility interactive inverters. The mathematical model for a three-phase system with LCL filters is similar but includes coupling between phases. The proposed HPF-based CCFAD can be applied per phase or in the dq-domain for decoupled control. In the dq-domain, the HPF feedback is implemented on the capacitor current components, and the stability analysis follows similar principles. Table 4 summarizes key equations for both single-phase and three-phase cases.
| Aspect | Single-Phase Utility Interactive Inverter | Three-Phase Utility Interactive Inverter (dq-domain) |
|---|---|---|
| LCL Filter Resonance | $\omega_r = 1/\sqrt{L_1(L_2 + L_g)C}$ | $\omega_r = 1/\sqrt{L_1(L_2 + L_g)C}$ per phase |
| Virtual Impedance | $Z_{eq1}(s) = K_1 e^{1.5sT_s} / G_{hpf}(s)$ | $Z_{eq1,dq}(s) = K_1 e^{1.5sT_s} / G_{hpf}(s)$ for d and q axes |
| Stability Boundary | $\omega_R = -\omega_c / \tan(1.5\omega_R T_s)$ | Same, applied to each axis |
| Routh Criterion | As derived earlier | Modified for coupled system; decoupling simplifies to single-phase form |
The three-phase utility interactive inverter benefits similarly from the expanded stable region, making it suitable for high-power applications. Experimental tests on a three-phase prototype confirm reduced THD and enhanced stability under weak grid conditions.
In conclusion, this paper presents an improved control strategy for utility interactive inverters based on capacitor current feedback active damping with a high-pass filter. The strategy effectively expands the stable operating region from $(0, f_s/6)$ to $(0, f_R)$, where $f_R$ can be tuned within $(f_s/6, f_s/3)$ by adjusting the HPF cutoff frequency. Mathematical modeling, stability analysis using the Routh-Hurwitz criterion, and extensive simulations and experiments validate the approach. The utility interactive inverter equipped with this strategy demonstrates strong robustness against grid impedance variations, maintaining low THD and stable operation in weak grids. Future work may explore adaptive tuning of the HPF parameters for optimal performance across varying grid conditions. This research contributes to the advancement of reliable and efficient utility interactive inverters for renewable energy integration.
The utility interactive inverter is a cornerstone of modern power electronics, and its control strategies must evolve to address grid challenges. By integrating an HPF feedback loop, we have shown that stability can be significantly enhanced without complex hardware modifications. This makes the improved CCFAD strategy a practical and effective solution for real-world applications. As renewable penetration grows, such innovations will be crucial for maintaining grid stability and power quality. We encourage further research into hybrid damping methods and digital control techniques to push the boundaries of utility interactive inverter performance.