In modern renewable energy systems, the grid-connected inverter plays a pivotal role in converting DC power from sources like solar panels or batteries into AC power that can be fed into the electrical grid. The performance of a grid-connected inverter directly impacts the stability, power quality, and efficiency of the entire power system. Among various filter topologies, the LCL filter has gained widespread adoption due to its superior high-frequency harmonic attenuation capabilities, compact size, and cost-effectiveness compared to simple L filters. However, the LCL filter introduces a third-order resonant system, which can lead to stability challenges such as resonance peaks and sensitivity to parameter variations. This necessitates advanced control strategies to ensure robust operation under uncertainties and disturbances. In this article, we present a comprehensive design and analysis of an active disturbance rejection control (ADRC) approach for LCL grid-connected inverters, focusing on enhancing current quality, robustness, and control performance through a linear extended state observer (LESO) and frequency-domain analysis.
The core challenge in controlling LCL grid-connected inverters lies in their inherent nonlinearity, strong coupling between d-q axis components, and susceptibility to external disturbances like grid voltage fluctuations. Traditional control methods, such as PI controllers combined with active damping techniques, often require additional sensors or complex compensation loops, increasing cost and complexity. Alternatively, methods like sliding mode control or repetitive control offer improved robustness but may involve intricate parameter tuning. Inspired by the ADRC framework, which treats all internal and external uncertainties as a “total disturbance” and compensates for it in real-time, we propose a third-order linear ADRC (LADRC) current controller for LCL grid-connected inverters. This approach leverages a fourth-order LESO to estimate and cancel disturbances, effectively decoupling the system and transforming it into a canonical integral series form. We delve into the mathematical modeling, controller design, performance analysis using frequency response methods, and stability proof via Lyapunov theory, supported by simulation results to validate the effectiveness.
To begin, we establish the mathematical model of a three-phase LCL grid-connected inverter in the synchronous rotating d-q reference frame. The inverter topology consists of a DC source, a PWM inverter bridge, an LCL filter (with inductors L1 and L2, capacitor C, and parasitic resistances R1 and R2), and the grid connection. By applying Kirchhoff’s laws and Park transformation, the dynamic equations can be derived. For instance, the voltage and current relationships are expressed as follows, where ω_g is the grid angular frequency, i_{1d}, i_{1q} are inverter-side currents, i_{2d}, i_{2q} are grid-side currents, u_d, u_q are inverter output voltages, u_{gd}, u_{gq} are grid voltages, and u_{cd}, u_{cq} are capacitor voltages.
The state-space representation highlights the coupling terms, making control design non-trivial. After neglecting small parasitic resistances for simplicity, we can derive a third-order differential equation for the grid-side current i_{2d} (similarly for i_{2q}), which serves as the controlled output:
$$ \frac{d^3 i_{2d}}{dt^3} = b_0 u_d + f_d(t) $$
Here, \( b_0 = \frac{1}{L_1 L_2 C} \) is a known gain, and \( f_d(t) \) represents the total disturbance encompassing model uncertainties, internal couplings (e.g., cross-coupling between d and q axes), parameter variations (e.g., changes in L1, L2, or C), and external disturbances (e.g., grid voltage harmonics or sags). For the q-axis, a similar equation holds: \( \frac{d^3 i_{2q}}{dt^3} = b_0 u_q + f_q(t) \). This formulation allows us to treat the grid-connected inverter as a perturbed third-order system, where the control objective is to regulate i_{2d} and i_{2q} to reference values (e.g., from a power or voltage controller) despite disturbances.
The ADRC philosophy involves estimating the total disturbance in real-time and compensating for it, thereby reducing the plant to a simple cascade of integrators. We design a fourth-order linear extended state observer (LESO) to estimate not only the states (grid-side current and its derivatives) but also the extended state representing the total disturbance. Defining the state vector as \( x_d = [i_{2d}, \dot{i}_{2d}, \ddot{i}_{2d}, f_d]^T \), the system can be rewritten in augmented state-space form:
$$ \dot{x}_d = A x_d + B u_d + E h(t) $$
$$ y = C x_d $$
with matrices:
$$ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{bmatrix}, \quad B = \begin{bmatrix} 0 \\ 0 \\ b_0 \\ 0 \end{bmatrix}, \quad E = \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix} $$
where \( h(t) = \dot{f}_d(t) \) is the derivative of the disturbance, assumed to be bounded. The LESO is constructed as:
$$ \dot{z}_d = A z_d + B u_d + L (y – \hat{y}) $$
$$ \hat{y} = C z_d $$
where \( z_d = [z_{d1}, z_{d2}, z_{d3}, z_{d4}]^T \) are the estimates of \( x_d \), and L is the observer gain vector. To simplify tuning, we parameterize L using a single bandwidth parameter \( \omega_o > 0 \) by placing all observer poles at \( -\omega_o \):
$$ L = [4\omega_o, 6\omega_o^2, 4\omega_o^3, \omega_o^4]^T $$
This ensures that the estimation error dynamics are stable and fast. The LESO provides accurate estimates of the grid-side current, its derivatives, and the total disturbance \( z_{d4} \approx f_d \). With these estimates, we design a linear state feedback control law combined with disturbance compensation:
$$ u_0 = k_{d1} (i_{2d}^* – z_{d1}) – k_{d2} z_{d2} – k_{d3} z_{d3} $$
$$ u_d = \frac{u_0 – z_{d4}}{b_0} $$
where \( i_{2d}^* \) is the reference current, and \( k_{d1}, k_{d2}, k_{d3} \) are controller gains. Similarly, for the q-axis, an identical structure is applied. The gains are chosen via pole placement to achieve a desired closed-loop bandwidth \( \omega_c \), typically set as \( [k_{d1}, k_{d2}, k_{d3}] = [\omega_c^3, 3\omega_c^2, 3\omega_c] \). Substituting the control law into the plant equation yields:
$$ \frac{d^3 i_{2d}}{dt^3} \approx u_0 $$
provided that \( z_{d4} \approx f_d \). Thus, the compensated system behaves like a triple-integrator, which is easier to control and inherently decoupled between d and q axes. This ADRC structure for the grid-connected inverter effectively handles uncertainties without requiring precise model knowledge.
To analyze the performance of the proposed LADRC for the grid-connected inverter, we employ frequency response methods. The tracking performance (from reference to output) and disturbance rejection (from grid voltage variations to output) can be evaluated through transfer functions derived from the closed-loop system. For instance, the closed-loop transfer function for the d-axis current control can be expressed as:
$$ G_{cl}(s) = \frac{i_{2d}(s)}{i_{2d}^*(s)} = \frac{R_1(s)}{b_0 N_1(s) P(s) + R_2(s)} $$
where \( P(s) = s^3 L_1 L_2 C + (L_1 + L_2)s \) represents the nominal plant, and \( R_1(s), R_2(s), N_1(s) \) are polynomials in s involving \( \omega_o \) and \( \omega_c \). Similarly, the disturbance transfer function from grid voltage \( u_{gd} \) to current \( i_{2d} \) is:
$$ G_{dist}(s) = \frac{i_{2d}(s)}{u_{gd}(s)} = -\frac{b_0 N_1(s) (s^2 L_1 C + 1)}{b_0 N_1(s) P(s) + R_2(s)} $$
By varying \( \omega_o \) and \( \omega_c \), we can assess the trade-offs between robustness, response speed, and noise sensitivity. A higher \( \omega_o \) improves disturbance estimation but may amplify high-frequency measurement noise, while a higher \( \omega_c \) speeds up response but could reduce stability margins. The following table summarizes key parameters and their effects on the grid-connected inverter control system:
| Parameter | Symbol | Typical Value | Effect on Performance |
|---|---|---|---|
| Observer Bandwidth | \( \omega_o \) | 12000 rad/s | Higher values improve disturbance rejection but increase noise sensitivity. |
| Controller Bandwidth | \( \omega_c \) | 4200 rad/s | Higher values speed up response but may reduce phase margin. |
| Inverter-side Inductance | \( L_1 \) | 3.2 mH | Affects filter dynamics and current ripple; variations are part of disturbance. |
| Grid-side Inductance | \( L_2 \) | 1.5 mH | Influences grid current quality and resonance frequency. |
| Filter Capacitance | \( C \) | 6.6 μF | Determines capacitive reactance and resonance; parameter shifts are compensated by ADRC. |
| Switching Frequency | \( f_s \) | 10 kHz | Sets the PWM resolution; harmonics are attenuated by the LCL filter. |
Stability is a critical aspect for any grid-connected inverter control system. We prove the asymptotic stability of the closed-loop system using Lyapunov theory. First, consider the estimation error of the LESO, defined as \( e = x_d – z_d \). By scaling the error as \( \epsilon_i = e_i / \omega_o^{i-1} \) for i=1,2,3,4, the error dynamics can be written as:
$$ \dot{\epsilon} = \omega_o A_0 \epsilon + B_0 \frac{h(t) – \hat{h}(t)}{\omega_o^3} $$
where \( A_0 \) is a Hurwitz matrix with all eigenvalues at -1, and \( B_0 = [0, 0, 0, 1]^T \). Choosing a Lyapunov function \( V(\epsilon) = \epsilon^T P \epsilon \) with P positive definite satisfying \( P A_0 + A_0^T P = -I \), we can show that \( \dot{V} < 0 \) for sufficiently large \( \omega_o \), ensuring \( \lim_{t \to \infty} e(t) = 0 \). Next, for the tracking error \( \eta_1 = i_{2d} – i_{2d}^* \), \( \eta_2 = \dot{i}_{2d} \), \( \eta_3 = \ddot{i}_{2d} \), the closed-loop dynamics reduce to:
$$ \dot{\eta} = K_1 \eta + K_2 e $$
where \( K_1 \) is a stable matrix due to pole placement, and \( K_2 \) involves controller gains. Since \( e(t) \) converges to zero, we have \( \lim_{t \to \infty} \eta(t) = 0 \), proving asymptotic tracking. This stability guarantee holds under bounded disturbances, making the ADRC-based grid-connected inverter robust to real-world uncertainties.
To validate the theoretical findings, we conduct simulations in MATLAB/Simulink for a 1 kW grid-connected inverter system with parameters as listed in the table above. The grid voltage is 220 V (RMS), DC link voltage is 700 V, and the control algorithms are implemented in discrete-time with a sampling frequency matching the switching frequency. We compare the proposed LADRC against a conventional PI controller with active damping to highlight improvements. The simulation scenarios include steady-state operation, step changes in current references, and grid voltage disturbances.
In steady-state, the grid-connected inverter with LADRC exhibits excellent current quality. The total harmonic distortion (THD) of the grid current is measured below 0.2%, significantly meeting standards like IEEE 1547. This is attributed to the effective suppression of switching harmonics by the LCL filter and the precise current regulation by ADRC. For dynamic performance, we test decoupling capability by applying step changes in d-axis reference current from 40 A to 80 A while keeping q-axis reference at zero. With LADRC, the q-axis current shows minimal coupling (less than 2.3 A deviation), whereas PI control causes larger cross-coupling effects. Similarly, step changes in q-axis reference are well-managed with fast settling time (under 5 ms) and low d-axis interference. These results demonstrate that the ADRC inherently decouples the d-q currents by treating coupling terms as part of the total disturbance.
Disturbance rejection is evaluated by subjecting the grid-connected inverter to sudden grid voltage sags and swells. For a 20% voltage sag at 0.1 s, the grid current with LADRC recovers within 2 ms with minimal overshoot (62 A peak), compared to 8 ms and higher overshoot (65 A) for PI control. Even under a severe 40% sag, LADRC maintains stability with bounded current transients, while PI control leads to larger oscillations. The disturbance estimation by LESO allows immediate compensation, showcasing the robustness of ADRC in maintaining power quality under grid faults. Furthermore, we test parameter robustness by varying L1 and L2 by ±20% from nominal values; the LADRC system maintains stable operation with negligible performance degradation, whereas PI control requires retuning.
In practical applications, grid-connected inverters are often integrated with energy storage systems to form hybrid power solutions, as illustrated in the image above. Such systems require high reliability and adaptability to fluctuating renewable generation and load demands. The ADRC approach, with its model-independent design and strong disturbance rejection, is well-suited for these scenarios. For instance, in a hybrid inverter with battery storage, the DC link voltage may vary due to charging/discharging cycles, and grid conditions can change abruptly. By treating these variations as disturbances, the ADRC controller can ensure seamless grid synchronization and current control without needing complex adaptive algorithms. This makes it a promising candidate for next-generation smart grid technologies.
From a design perspective, tuning the ADRC for a grid-connected inverter involves selecting \( \omega_o \) and \( \omega_c \) based on bandwidth requirements and noise considerations. A systematic approach is to set \( \omega_o \) about 3-5 times \( \omega_c \) to ensure accurate disturbance estimation, while \( \omega_c \) is chosen below one-tenth of the switching frequency to avoid aliasing. For the LCL filter, the resonance frequency \( f_r = \frac{1}{2\pi} \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} \) should be considered; typically, \( f_r \) is placed between the fundamental frequency (50/60 Hz) and half the switching frequency to avoid resonance issues. The ADRC naturally damps resonance through disturbance compensation, eliminating the need for additional active damping loops. However, in highly noisy environments, reducing \( \omega_o \) may be necessary, albeit at the cost of slower disturbance rejection.
To further quantify performance, we can derive key transfer functions and analyze their frequency responses. For example, the sensitivity function \( S(s) = \frac{1}{1 + G_{ol}(s)} \), where \( G_{ol}(s) \) is the open-loop transfer function, indicates robustness to model uncertainties. With ADRC, \( S(s) \) shows lower peaks at resonance frequencies compared to PI control, implying better stability margins. Additionally, the complementary sensitivity function \( T(s) = 1 – S(s) \) reflects tracking performance; ADRC provides faster roll-off at high frequencies, attenuating noise effectively. These characteristics are summarized in the following table comparing ADRC and PI control for the grid-connected inverter:
| Feature | ADRC with LESO | Conventional PI with Active Damping |
|---|---|---|
| Decoupling Capability | High (inherent via disturbance estimation) | Moderate (requires feedforward or decoupling networks) |
| Disturbance Rejection | Excellent (fast estimation and compensation) | Limited (dependent on integral action and damping loops) |
| Parameter Robustness | High (insensitive to LCL parameter variations) | Low (requires accurate model for damping design) |
| Implementation Complexity | Moderate (single observer and simple feedback) | High (multiple sensors and compensation circuits) |
| Tuning Effort | Low (two bandwidth parameters) | High (multiple gains and damping coefficients) |
| Stability Guarantee | Asymptotic per Lyapunov proof | Empirical or based on linear models |
In conclusion, the active disturbance rejection control strategy offers a robust and efficient solution for current control in LCL grid-connected inverters. By employing a linear extended state observer to estimate and cancel total disturbances, the system is transformed into a decoupled integral series form, simplifying control design and enhancing performance. Frequency-domain analysis confirms superior tracking and disturbance rejection, while Lyapunov stability theory provides a solid foundation for reliability. Simulation results validate that the ADRC approach improves grid current quality, reduces coupling effects, and maintains robustness under grid voltage variations and parameter uncertainties. For future work, real-time implementation on digital signal processors and experimental validation in microgrid setups could further demonstrate practicality. As renewable integration expands, advanced control methods like ADRC will be crucial for ensuring the stability and power quality of grid-connected inverter systems, paving the way for smarter and more resilient power networks.
Throughout this article, we have emphasized the importance of the grid-connected inverter as a key component in modern energy systems. The proposed ADRC method, with its emphasis on disturbance estimation and compensation, represents a significant step forward in addressing the challenges posed by LCL filters and grid interactions. By minimizing reliance on precise models and reducing sensor requirements, it offers a cost-effective and scalable solution for various applications, from residential solar inverters to large-scale wind farms. As technology evolves, integrating ADRC with other smart grid functionalities, such as fault detection and energy management, could unlock new potentials for efficient and sustainable power generation and distribution.