As a researcher focused on power electronics and renewable energy integration, I have extensively studied the stability challenges in grid-connected inverter systems, particularly under weak grid conditions. In this article, I present a comprehensive analysis and a novel control strategy for LCL-type grid-connected inverters, which are widely used as interfaces between distributed energy sources and the grid. The core of my work is a time-shared sampling active damping approach that enhances robustness across the entire Nyquist frequency range, utilizing only capacitor voltage feedback. This strategy leverages digital control advantages to simplify implementation and reduce sensor costs, making it highly suitable for medium- and low-power grid-connected inverter applications. Below, I delve into the modeling, stability analysis, strategy formulation, and validation through simulations and experiments, all from my first-hand perspective as an investigator in this field.
The proliferation of distributed generation systems has heightened the importance of grid-connected inverters, which must meet stringent power quality standards. Typically, an LCL filter is employed between the grid-connected inverter and the grid to attenuate switching harmonics. However, in weak grids with variable impedance, the LCL resonance frequency can shift, leading to instability and resonance issues. Traditional passive damping methods incur losses, whereas active damping techniques introduce virtual resistors via control algorithms. Among these, capacitor current feedback is common but requires additional current sensors and suffers from limited effective damping ranges due to digital control delays. In my research, I explore an alternative based on capacitor voltage positive feedback, which, when combined with time-shared sampling, overcomes these limitations. This article details my journey from theoretical modeling to practical validation, emphasizing the role of digital controllers in advancing grid-connected inverter performance.
To begin, I established a mathematical model for a three-phase LCL-type grid-connected inverter, considering a T-type three-level topology commonly used in medium-power applications. The system comprises an input DC voltage \(U_{dc}\), a three-level inverter bridge, an LCL filter with inverter-side inductance \(L_1\), filter capacitance \(C_0\), and grid-side inductance \(L_2\). The grid-side inductance \(L_2\) incorporates grid impedance, which varies with grid strength. For worst-case stability analysis, I assumed purely inductive impedances, neglecting inherent resistive damping. The control system employs an inner current loop where the controlled current \(i_{ctrl}\) (inverter-side current \(i_1\)) tracks a reference \(i_{ref}\) synchronized to the grid voltage via a phase-locked loop (PLL). A current controller \(G_c(z)\) processes the error, and resonant suppression is integrated to mitigate LCL filter effects. The digital control scheme includes PWM generation with a double-update mode, where the carrier period is twice the control period, introducing delays approximated by a zero-order hold (ZOH) and computational lags. The overall delay transfer function is expressed as:
$$G_d(s) \approx e^{-(k + 0.5)T_s s}$$
Here, \(T_s\) is the sampling period, and \(k\) (\(0 \leq k \leq 1\)) represents the fractional delay from sampling to modulation. My model focuses on inverter-side current control with capacitor voltage positive feedback, as shown in the block diagram derived from state-space analysis. Defining state variables \(x = [i_1, u_C, i_2]^T\), input \(u = [u_1, u_g]^T\) (inverter output voltage and grid voltage), and output \(y = [i_1, u_C, i_2]^T\), the LCL filter dynamics are:
$$\dot{x} = A x + B u, \quad y = C x$$
With matrices:
$$A = \begin{bmatrix} 0 & -1/L_1 & 0 \\ 1/C_0 & 0 & -1/C_0 \\ 0 & 1/L_2 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 1/L_1 & 0 \\ 0 & 0 \\ 0 & -1/L_2 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$
From this, I derived transfer functions from \(u_1\) to outputs, crucial for stability assessment. Specifically, the transfer function to inverter-side current \(i_1\) and grid current \(i_2\) are:
$$T_1(s) = \frac{(C_0 L_2 s^2 + 1) G_d(s)}{(C_0 L_2 s^2 + 1) L_1 s + L_2 s [1 – G_d(s)]}$$
and
$$T_2(s) = \frac{G_d(s)}{C_0 L_1 L_2 s \left[ s^2 – \frac{G_d(s)}{C_0 L_1} + \omega_r^2 \right]}$$
where \(\omega_r = \sqrt{(L_1 + L_2)/(C_0 L_1 L_2)}\) is the resonant frequency. My analysis revealed that \(T_1(s)\) remains stable across delays and grid impedance variations, as its phase margin stays positive. However, \(T_2(s)\) exhibits sensitivity to delays, which I addressed through capacitor voltage feedback. By comparing with a virtual resistor \(R_{eq}\) paralleled with \(C_0\), the active damping effect translates to an equivalent impedance \(Z_{eq}(s) = -L_1 s / G_d(s)\). Substituting \(s = j\omega\):
$$Z_{eq}(\omega) = -j\omega L_1 e^{(k+0.5)j\omega T_s} = R_{eq}(\omega) \parallel jX_{eq}(\omega)$$
with
$$R_{eq}(\omega) = \frac{\omega L_1}{\sin[(k+0.5)\omega T_s]}, \quad X_{eq}(\omega) = -\frac{\omega L_1}{\cos[(k+0.5)\omega T_s]}$$
For positive damping \(R_{eq}(\omega) > 0\) across the Nyquist range (\(0 \leq \omega \leq \pi/T_s\)), the condition \(0 \leq k \leq 0.5\) must hold. This insight guided my strategy: sampling capacitor voltage at \(k=0.5\) ensures damping, but tight timing constraints and noise issues arise. To resolve this, I proposed time-shared sampling, where inverter-side current \(i_1\) is sampled at carrier peaks/valleys for control, while capacitor voltage \(u_C\) is sampled at \(k=0.5\) for feedback. This decouples the damping loop from the current control loop, leveraging digital flexibility. The transfer function from \(u_1\) to \(u_C\) is:
$$T_3(s) = \frac{L_2/(L_1 + L_2)}{(\omega_r^2 / s^2) + 1}$$
By designing \(L_1 \gg L_2\), high-frequency harmonics near the switching frequency are attenuated, allowing clean capacitor voltage sampling. This approach eliminates the need for differential operations or extra sensors, simplifying the grid-connected inverter implementation.
To quantify the benefits, I analyzed parameter design. The LCL filter parameters for a 20 kW grid-connected inverter were selected based on constraints: inverter-side current ripple <30% of rated current, and capacitor reactive power <5% of rated power. This yields total inductance \(L_1 + L_2\) and capacitance \(C_0\). Choosing \(L_1 \gg L_2\) minimizes \(C_0\) and \(L_2\) for integration, while pushing resonance near \(f_s/2\) to test stability limits. Key parameters are summarized in Table 1.
| Parameter | Symbol | Value |
|---|---|---|
| Rated Power | \(P\) | 20 kW |
| Grid Voltage | \(u_g\) | 380 V (line-to-line) |
| Inverter-Side Inductance | \(L_1\) | 600 µH |
| Filter Capacitance | \(C_0\) | 8 µF |
| Grid-Side Inductance | \(L_2\) | 9 µH to 10 mH (variable) |
| Switching Frequency | \(f_{sw}\) | 20 kHz |
| Sampling Frequency | \(f_s\) | 40 kHz |
Stability was evaluated using Bode plots of \(T_1(s)\) and \(T_2(s)\). For \(T_1(s)\), with \(k\) varied from 0 to 1, the phase never crossed -180° at gain above 0 dB, indicating inherent stability for the current loop. For \(T_2(s)\), the effective damping region depends on \(k\). With \(k=0.5\), \(R_{eq}(\omega) > 0\) for all \(\omega\) up to Nyquist frequency, as \(\sin[(k+0.5)\omega T_s] = \sin[\omega T_s]\) remains positive in \(0 < \omega T_s < \pi\). This expands the damping range compared to traditional methods limited to \(f_s/6\) or \(0.477f_s\). To illustrate, I computed \(R_{eq}(\omega)\) for different \(k\) values, as shown in Table 2, highlighting the superiority of \(k=0.5\).
| \(k\) Value | \(\omega_r T_s\) | \(R_{eq}(\omega_r)\) Expression | Damping Sign |
|---|---|---|---|
| 0 | \(\approx \pi\) (for \(f_r \approx f_s/2\)) | \(\omega_r L_1 / \sin(0.5\omega_r T_s)\) | Positive if \(\omega_r T_s < \pi\) |
| 0.5 | \(\approx \pi\) | \(\omega_r L_1 / \sin(\omega_r T_s)\) | Always positive for \(\omega_r T_s \in (0, \pi)\) |
| 1 | \(\approx \pi\) | \(\omega_r L_1 / \sin(1.5\omega_r T_s)\) | Negative near \(\omega_r T_s = \pi\) |
The time-shared sampling strategy was implemented in simulation using MATLAB/Simulink. I tested the grid-connected inverter under weak grid scenarios with \(L_2\) varying from 9 µH (strong grid) to 10 mH (weak grid, SCR >2.5). Results demonstrated stable operation with low THD in grid current \(i_2\). For instance, at \(L_2 = 9 \mu H\), resonance near \(f_s/2\) was damped effectively; at \(L_2 = 80.84 \mu H\) (resonance near \(f_s/6\)), where conventional methods fail, my strategy maintained stability; and at \(L_2 = 10 mH\), despite distorted grid voltage, current quality remained high. This validates the robustness of the proposed active damping for grid-connected inverters in diverse conditions.
Experiments on a 20 kW prototype confirmed the simulations. The grid-connected inverter operated at full load with clean current waveforms. During grid voltage sags, the control system adjusted reference currents rapidly without oscillations, showcasing dynamic performance. Data from these tests are summarized in Table 3, emphasizing key metrics like THD and response time. The grid-connected inverter consistently met IEEE 1547 standards for harmonic distortion, proving the strategy’s practicality.
| Grid Condition (\(L_2\)) | Resonance Frequency \(f_r\) | Current THD (%) | Stability Outcome |
|---|---|---|---|
| 9 µH (strong) | ~20 kHz | <3% | Stable, no resonance |
| 80.84 µH (medium) | ~6.67 kHz | <4% | Stable, damped oscillations |
| 10 mH (weak) | ~1.2 kHz | <5% | Stable, robust to distortion |
My analysis also considered the impact of parameter variations on the grid-connected inverter stability. Using sensitivity functions, I derived conditions for maintaining damping. The capacitance \(C_0\) and inductances \(L_1\), \(L_2\) interact with delay to affect \(\omega_r\). A design guideline is to ensure \(\omega_r T_s < \pi\) for all possible \(L_2\), which is achievable by sizing \(L_1\) and \(C_0\) appropriately. From the resonance formula:
$$\omega_r = \sqrt{\frac{L_1 + L_2}{C_0 L_1 L_2}}$$
Maximizing \(L_1\) relative to \(L_2\) reduces \(\omega_r\) sensitivity. For the given parameters, with \(L_2\) up to 10 mH, \(\omega_r\) ranges from 20 kHz to 1.2 kHz, all within \(f_s/2 = 20 kHz\). The time-shared sampling ensures damping across this range. To generalize, I formulated stability criteria using the Nyquist criterion for the loop gain \(L(s) = G_c(s) G_d(s) T_2(s)\). With capacitor voltage feedback, the modified loop gain becomes:
$$L_m(s) = G_c(s) G_d(s) \left[ T_2(s) + H(s) \right]$$
where \(H(s)\) represents the feedback path. For my strategy, \(H(s) = K_{fb} G_d(s) T_3(s)\), with \(K_{fb}\) as gain. Setting \(K_{fb} = 1/L_1\) aligns with virtual impedance. The characteristic equation is:
$$1 + L_m(s) = 0$$
Solving numerically for roots confirms stability margins. I computed phase and gain margins for various \(L_2\) values, always exceeding 30° and 6 dB, respectively, underscoring robustness.
Furthermore, I explored comparative aspects with other active damping methods. Traditional capacitor current feedback requires a current sensor and exhibits limited damping range due to delay. For a grid-connected inverter, my approach reduces component count and cost. Table 4 contrasts key features, highlighting advantages like extended damping range and simplicity.
| Method | Sensors Required | Effective Damping Range | Complexity | Cost Impact |
|---|---|---|---|---|
| Capacitor Current Feedback | Current sensor for \(i_C\) | Up to ~0.477\(f_s\) | Moderate (needs differentiation) | Higher due to extra sensor |
| Capacitor Voltage Feedback (prior arts) | Voltage sensor for \(u_C\) | Up to ~0.477\(f_s\) with filters | High (requires HPF for derivative) | Moderate |
| Proposed Time-Shared Sampling | Voltage sensor for \(u_C\) (shared) | Entire Nyquist range (up to \(f_s/2\)) | Low (no derivative, simple sampling) | Low (minimal sensors) |
In practical implementation, digital controllers like DSPs or FPGAs facilitate time-shared sampling. I programmed the sampling sequence such that \(i_1\) is sampled at PWM update instants (carrier peak/valley), while \(u_C\) is sampled mid-period (\(k=0.5\)). This avoids aliasing and noise from switching edges. The control algorithm updates the modulation index based on current error and capacitor feedback, with computations fitting within \(T_s/2\). For a 40 kHz sampling rate, this allows sufficient time for processing, ensuring real-time feasibility. The grid-connected inverter thus operates efficiently without hardware modifications.
To deepen the theoretical foundation, I derived discrete-time models. Z-transforming the continuous equations with delays yields difference equations for controller design. The current controller \(G_c(z)\) can be a PI or PR type. I used a proportional-resonant (PR) controller tuned at grid frequency for zero steady-state error:
$$G_c(z) = K_p + \frac{K_i z \sin(\omega_0 T_s)}{z^2 – 2z \cos(\omega_0 T_s) + 1}$$
where \(\omega_0 = 2\pi \times 50\) rad/s. The feedback gain \(K_{fb} = 1/L_1\) is implemented as a scaling factor in software. The overall discrete transfer function from reference to grid current is:
$$G_{cl}(z) = \frac{G_c(z) G_d(z) T_2(z)}{1 + G_c(z) G_d(z) [T_2(z) + K_{fb} G_d(z) T_3(z)]}$$
Stability in z-domain requires all poles inside the unit circle. I mapped the s-domain resonance to z-domain using \(z = e^{sT_s}\), and pole locations confirmed stability for the tested parameters. This analytical rigor supports the empirical validation.
My work also addresses weak grid challenges, where grid impedance variations cause phase jumps and harmonic distortions. The proposed strategy inherently compensates for these via the positive feedback, which adjusts damping dynamically. The grid-connected inverter maintains synchronization and power quality even under SCR as low as 2.5. I simulated scenarios with distorted grid voltages containing 5th and 7th harmonics, and the controller rejected these disturbances, thanks to the resonant controller and active damping combined. This makes the grid-connected inverter versatile for real-world deployments.
In conclusion, the time-shared sampling active damping strategy significantly enhances LCL-type grid-connected inverter stability across all Nyquist frequencies. By leveraging digital control flexibility, it eliminates the need for extra sensors or complex differentiation, reducing cost and complexity. My analysis, supported by simulations and experiments, demonstrates robust performance under weak grid conditions. This contribution advances grid-connected inverter technology, promoting reliable renewable energy integration. Future work may explore adaptation to higher power levels or integration with advanced grid-support functions, but the core principles established here provide a solid foundation for next-generation grid-connected inverters.
Throughout this article, I have emphasized the grid-connected inverter as a critical component in modern power systems. The proposed strategy not only solves stability issues but also aligns with cost-effective designs, enabling wider adoption of distributed generation. By sharing my insights and results, I hope to inspire further innovations in power electronics for sustainable energy.