The integration of distributed renewable energy sources into the power grid relies critically on the performance of the grid-tied inverter, the essential power electronic interface. Among various filter topologies, the LCL filter is predominantly adopted for grid-tied inverter applications due to its superior high-frequency harmonic attenuation capability compared to a simple L filter, allowing for smaller inductor sizes and reduced cost. However, the inherent third-order dynamics of the LCL filter introduce a resonant peak that can destabilize the system. Furthermore, the operation of a grid-tied inverter is perpetually challenged by complex grid conditions, including voltage sags, swells, background harmonics, and frequency deviations. These factors collectively degrade the quality of the injected current, leading to higher total harmonic distortion (THD), poor dynamic response, and potential instability.
Traditional control methods, such as Proportional-Integral (PI) regulators within a synchronous reference frame, often struggle with these challenges. While passive damping by adding resistors in series with the filter capacitor can suppress resonance, it incurs significant power losses. Active damping methods, which reshape the control loop through virtual impedance or state feedback, avoid these losses but may require additional sensors or precise system parameters, increasing complexity and cost. The quest for a control strategy that offers high-performance current tracking, inherent resonance damping, and robust disturbance rejection without relying on an exact system model or complex damping networks remains a central focus in grid-tied inverter research.
Active Disturbance Rejection Control (ADRC) presents a compelling paradigm shift. Its core philosophy is to treat all discrepancies between a simplified model and the actual plant—including unmodeled dynamics (like LCL resonance), parameter variations, and external disturbances—as a aggregated “total disturbance.” This disturbance is estimated in real-time by an Extended State Observer (ESO) and actively canceled in the control law. The Linear ADRC (LADRC) variant simplifies parameter tuning using bandwidth concepts. For a grid-tied inverter with an LCL filter, this means the resonant peak, grid voltage harmonics, and model inaccuracies can all be considered part of the total disturbance to be estimated and compensated, theoretically eliminating the need for separate active damping loops. However, the performance of a fixed-parameter LADRC is intrinsically linked to its observer and controller bandwidths. In a dynamic environment, fixed parameters may not yield optimal performance across all operating conditions.
This article explores an enhanced control strategy that synergizes the robustness of LADRC with the adaptability of fuzzy logic. We begin by establishing the mathematical model of a three-phase LCL-type grid-tied inverter and justify its model reduction for controller design. Subsequently, we delve into the design of an improved Linear Extended State Observer (LESO) with enhanced estimation dynamics. A fuzzy logic system is then integrated to dynamically adjust the key parameters of the LADRC, forming the Improved Fuzzy-LADRC (Fuzzy Linear Active Disturbance Rejection Control) strategy. This strategy is applied to both the current control loop and the phase-locked loop (PLL), which is crucial for synchronization. The system’s stability and disturbance rejection capabilities are rigorously analyzed using Lyapunov’s method and frequency-domain analysis. Finally, comprehensive simulations under various grid disturbance scenarios are conducted to validate the superior performance of the proposed approach compared to conventional methods.
Mathematical Modeling of the LCL Grid-Tied Inverter System
The topology of a three-phase two-level voltage source grid-tied inverter with an LCL filter is considered. The system comprises a DC-link voltage source \(U_{dc}\), a three-phase inverter bridge, the LCL filter (with inverter-side inductance \(L_1\), filter capacitance \(C\), and grid-side inductance \(L_2\)), and the grid connection point.
Applying Kirchhoff’s laws, the dynamic equations in the three-phase (abc) stationary frame are derived as:
$$ L_1 \frac{d i_{1x}}{dt} = u_x – u_{cx} $$
$$ L_2 \frac{d i_{2x}}{dt} = u_{cx} – u_{gx} $$
$$ C \frac{d u_{cx}}{dt} = i_{1x} – i_{2x} $$
where \(x = a, b, c\); \(i_{1x}\) and \(i_{2x}\) are the inverter-side and grid-side currents, respectively; \(u_x\) is the inverter output voltage; \(u_{cx}\) is the capacitor voltage; and \(u_{gx}\) is the grid voltage.
From these equations, the transfer function from the inverter output voltage \(u_x(s)\) to the grid current \(i_{2x}(s)\) is obtained as:
$$ G_{LCL}(s) = \frac{i_{2x}(s)}{u_x(s)} = \frac{1}{L_1 L_2 C s^3 + (L_1 + L_2)s} = \frac{1}{L_1 L_2 C} \cdot \frac{1}{s(s^2 + \omega_{res}^2)} $$
where the resonant frequency \(\omega_{res}\) is given by:
$$ \omega_{res} = \sqrt{\frac{L_1 + L_2}{L_1 L_2 C}} $$
This transfer function clearly shows the resonant peak at \(\omega_{res}\), which poses a stability challenge.
A fundamental principle of ADRC is that it does not require a precise model. For controller design, a simplified, reduced-order model is sufficient, as the observer will estimate and compensate for the modeling error. Using a first-order Padé approximation or simply considering the dominant low-frequency behavior, the LCL model can be approximated as:
$$ G_{LCL, approx}(s) = \frac{i_{2x}(s)}{u_x(s)} \approx \frac{1}{(L_1 + L_2)s} $$
This simplified model represents an ideal integrator. The discrepancy between this simplified model and the actual third-order LCL plant, along with all other disturbances, is lumped into the “total disturbance” to be handled by the ADRC framework. This significantly simplifies the controller design process for the grid-tied inverter.
Design of the Improved Linear Active Disturbance Rejection Controller
The core of LADRC for a first-order system is the Linear Extended State Observer (LESO) and the Linear State Error Feedback (LSEF) law. Consider the reduced-order model of the grid-tied inverter current loop:
$$ \frac{d i_2}{dt} = b_0 u + f $$
where \(i_2\) is the grid current (the controlled output \(y\)), \(u\) is the control input (inverter voltage reference), \(b_0 = 1/(L_1+L_2)\) is the approximate system gain, and \(f\) is the total disturbance encompassing model inaccuracies, resonance, grid voltage disturbances, and parameter variations.
We define the state variables: \(x_1 = i_2 = y\) and \(x_2 = f\). Assuming the derivative of the disturbance \(\dot{f} = h\) is bounded, the state-space representation is:
$$ \begin{cases} \dot{x}_1 = x_2 + b_0 u \\ \dot{x}_2 = h \end{cases} $$
The conventional LESO for this system is:
$$ \begin{cases} \dot{z}_1 = z_2 – \beta_1 (z_1 – y) + b_0 u \\ \dot{z}_2 = -\beta_2 (z_1 – y) \end{cases} $$
where \(z_1\) and \(z_2\) are the estimates of \(x_1\) (output) and \(x_2\) (total disturbance), respectively, and \(\beta_1, \beta_2\) are observer gains.
To enhance the disturbance estimation and rejection capability, we propose an improvement by modifying the error correction mechanism in the disturbance estimation channel. The improved LESO is designed as:
$$ \begin{cases} \dot{z}_1 = z_2 – \beta_1 (z_1 – y) + b_0 u \\ \dot{z}_2 = -\beta_2 [(\dot{z}_1 – \dot{y}) + \beta_1 (z_1 – y)] \end{cases} $$
This structure effectively increases the observer’s sensitivity to the rate of change of the estimation error, leading to faster and more accurate disturbance estimation for the grid-tied inverter system.
The control law (LSEF) is then formulated as:
$$ u_0 = k_p (r – z_1) $$
$$ u = \frac{u_0 – z_2}{b_0} $$
where \(r\) is the reference current, \(k_p\) is the controller gain, and \(u\) is the final control signal. The term \(-z_2/b_0\) actively cancels the estimated total disturbance.
Parameter tuning is simplified using the bandwidth parameterization method. For the improved LESO, we set \(\beta_1 = 2\omega_o\) and \(\beta_2 = \omega_o^2\), where \(\omega_o\) is the observer bandwidth. For the LSEF, we set \(k_p = \omega_c\), where \(\omega_c\) is the controller bandwidth. A common rule of thumb is \(\omega_o \approx 3 \sim 10 \times \omega_c\) to ensure the observer dynamics are sufficiently faster than the controller dynamics. The simplified control structure effectively turns the challenging control of an LCL-type grid-tied inverter into the control of a simple integrator, with all complexities handled by the disturbance observer.
Integration of Fuzzy Logic for Adaptive Parameter Tuning
While the bandwidth-based LADRC offers simplified tuning, its performance in the face of wide-ranging and dynamic disturbances in a grid-tied inverter application can be suboptimal with fixed parameters \(\omega_c\) and \(\omega_o\). A high fixed \(\omega_o\) provides fast disturbance estimation but may increase noise sensitivity. A high fixed \(\omega_c\) provides fast tracking but may reduce stability margins. To achieve optimal performance across different operating regimes (e.g., steady-state, step changes, under severe harmonics), we integrate a fuzzy logic controller to adaptively adjust these parameters.
The error \(e = r – y\) and its derivative \(\dot{e}\) are chosen as the inputs to the fuzzy inference system. The outputs are the adjustment values \(\Delta \omega_c\) and \(\Delta \omega_o\). The final adaptive parameters are:
$$ \omega_c = \omega_{c0} + \Delta \omega_c, \quad \omega_o = \omega_{o0} + \Delta \omega_o $$
where \(\omega_{c0}\) and \(\omega_{o0}\) are the carefully chosen nominal values.
The fuzzy sets for both inputs and outputs are defined as: Negative Big (NB), Negative Medium (NM), Negative Small (NS), Zero (ZE), Positive Small (PS), Positive Medium (PM), Positive Big (PB). Triangular membership functions are used. The fuzzy rule base is designed based on the following heuristic principles:
- When both \(|e|\) and \(|\dot{e}|\) are large, the system is far from the reference and moving quickly. Both \(\omega_c\) and \(\omega_o\) should be increased significantly to accelerate response and disturbance rejection.
- When \(|e|\) is large but \(|\dot{e}|\) is small or negative, the system is far from equilibrium but not moving quickly towards it. Increase \(\omega_c\) to improve tracking and moderately increase \(\omega_o\) to enhance disturbance estimation.
- When \(|e|\) is small, the system is near the desired state. To avoid overshoot and suppress noise, reduce \(\omega_c\) and \(\omega_o\), especially if \(\dot{e}\) is also small.
- When \(|e|\) is small but \(|\dot{e}|\) is large, a potential overshoot is indicated. Reduce \(\omega_c\) to dampen the response and adjust \(\omega_o\) appropriately.
A subset of the fuzzy rule table for \(\Delta \omega_c\) is illustrated below:
| e \ $\dot{e}$ | NB | NM | NS | ZE | PS | PM | PB |
|---|---|---|---|---|---|---|---|
| NB | PB | PB | PM | PM | PS | ZE | ZE |
| NM | PB | PB | PM | PS | ZE | ZE | NS |
| NS | PM | PM | PS | PS | ZE | NS | NM |
| ZE | PM | PS | PS | ZE | NS | NM | NM |
| PS | PS | PS | ZE | NS | NM | NM | NB |
| PM | ZE | ZE | NS | NM | NM | NB | NB |
| PB | ZE | NS | NM | NM | NB | NB | NB |
This fuzzy-adaptive mechanism empowers the LADRC to self-optimize its parameters in real-time, ensuring that the grid-tied inverter maintains excellent performance whether the grid is clean or severely distorted.
Stability and Robustness Analysis
The stability of the closed-loop system with the improved Fuzzy-LADRC is paramount. We analyze it in two parts: the stability of the improved LESO and the stability of the overall control loop.
1. Stability of the Improved LESO:
Let the estimation errors be \(\tilde{x}_1 = x_1 – z_1\) and \(\tilde{x}_2 = x_2 – z_2\). From the improved LESO dynamics and the plant model, the error dynamics can be derived as:
$$ \begin{cases} \dot{\tilde{x}}_1 = \tilde{x}_2 – \beta_1 \tilde{x}_1 \\ \dot{\tilde{x}}_2 = \delta(t) – \beta_2 (\dot{\tilde{x}}_1 + \beta_1 \tilde{x}_1) \end{cases} $$
where \(\delta(t) = h – \dot{f}_{actual}\) represents the estimation error of the disturbance derivative and is assumed bounded. By defining scaled errors \(\varepsilon_1 = \tilde{x}_1\) and \(\varepsilon_2 = \tilde{x}_2 / \omega_o\), the error dynamics can be transformed into a standard singular perturbation form. Using Lyapunov analysis, it can be shown that for a sufficiently large observer bandwidth \(\omega_o\), the estimation errors \(\tilde{x}_1\) and \(\tilde{x}_2\) are bounded and converge to a small region around zero whose size is inversely proportional to \(\omega_o\). This proves the observer’s stability and convergence.
2. Stability of the Overall Closed-Loop System:
With a stable LESO providing accurate estimates, the control law \(u = (\omega_c (r – z_1) – z_2)/b_0\) effectively reduces the plant to approximately: \(\dot{y} \approx \omega_c (r – y)\). The closed-loop transfer function from reference \(r\) to output \(y\) approximates:
$$ \frac{Y(s)}{R(s)} \approx \frac{\omega_c}{s + \omega_c} $$
which is a stable first-order system for \(\omega_c > 0\). The fuzzy logic adaptation varies \(\omega_c\) and \(\omega_o\) within positive bounds, ensuring the stability conditions for both the observer and the controller are maintained during adaptation.
3. Frequency-Domain Robustness:
The disturbance rejection capability can be visualized in the frequency domain. The sensitivity function and the complementary sensitivity function for the proposed controller show that the improved observer structure provides better attenuation of the LCL resonant peak compared to the conventional LESO. The integration of fuzzy logic does not alter the fundamental structure but optimizes the bandwidths, leading to an adaptive trade-off between fast response (high bandwidth) and robustness/noise rejection (lower bandwidth). This ensures the grid-tied inverter remains stable and provides high-quality current under a wide range of grid impedances and disturbances.
Simulation Results and Performance Evaluation
To validate the proposed Improved Fuzzy-LADRC strategy, a detailed simulation model of a three-phase grid-tied inverter with an LCL filter is built. The system parameters are listed in the table below:
| Parameter | Symbol | Value |
|---|---|---|
| DC-Link Voltage | \(U_{dc}\) | 800 V |
| Grid Voltage (phase-to-phase RMS) | \(U_g\) | 380 V |
| Inverter-side Inductance | \(L_1\) | 2.0 mH |
| Grid-side Inductance | \(L_2\) | 0.5 mH |
| Filter Capacitance | \(C\) | 4.7 µF |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
| Nominal Grid Frequency | \(f\) | 50 Hz |
| Rated Grid Current (peak) | \(I_{rated}\) | \(\sim\)100 A |
The proposed strategy is compared against three benchmarks: 1) Conventional PI control with capacitor-current-feedback active damping (PI+AD), 2) Standard first-order LADRC with fixed parameters, and 3) The Improved LADRC (without fuzzy adaptation). The following key performance indicators (KPIs) are evaluated: Total Harmonic Distortion (THD) of the grid current, dynamic response time and overshoot during step changes, and steady-state error. The tests are conducted under three challenging scenarios.
Scenario 1: Current Reference Step Change
At t = 0.25s, the current reference is stepped from 64A (peak) to 96A (peak). The performance metrics are summarized below:
| Control Strategy | Settling Time (ms) | Overshoot (%) | Steady-state THD after step (%) |
|---|---|---|---|
| PI + Active Damping | ~35 | 12.5 | 2.68 |
| Standard LADRC | ~15 | 6.3 | 2.41 |
| Improved LADRC | ~8 | 2.8 | 1.28 |
| Improved Fuzzy-LADRC | ~4 | 1.2 | 0.72 |
The results show that the Improved Fuzzy-LADRC strategy achieves the fastest settling time, the smallest overshoot, and the lowest steady-state harmonic distortion. The fuzzy adaptation allows the controller to aggressively increase its bandwidth during the transient for a fast response and then reduce it in steady-state for optimal filtering, outperforming all fixed-parameter controllers.
Scenario 2: Grid Voltage Sag
At t = 0.25s, the grid voltage experiences a 50% sag (from 380V to 190V line-to-line RMS) for 100ms. This is a severe disturbance that tests the controller’s disturbance rejection capability.
| Control Strategy | Max Current Deviation (A) | Recovery Time (ms) | THD during sag (%) |
|---|---|---|---|
| PI + Active Damping | 45 | 72 | 8.5 |
| Standard LADRC | 28 | 28 | 4.1 |
| Improved LADRC | 15 | 8 | 2.0 |
| Improved Fuzzy-LADRC | 8 | <5 | 1.1 |
The Improved Fuzzy-LADRC demonstrates exceptional robustness. The enhanced LESO provides a faster and more accurate estimate of the disturbance caused by the voltage sag, and the adaptive gains allow for swift compensation. The current waveform shows minimal distortion and a very rapid return to the reference value, which is critical for low-voltage ride-through (LVRT) compliance in modern grid-tied inverter standards.
Scenario 3: Grid with Background Harmonics
The grid voltage is contaminated with 5th (5%), 7th (4%), 11th (3%), and 13th (2%) order harmonics. This tests the controller’s ability to reject periodic disturbances and inject a clean sinusoidal current.
| Control Strategy | Grid Current THD (%) | Tracking Error (RMS, A) |
|---|---|---|
| PI + Active Damping | 4.75 | 3.8 |
| Standard LADRC | 3.10 | 2.1 |
| Improved LADRC | 2.65 | 1.5 |
| Improved Fuzzy-LADRC | 1.96 | 0.9 |
Once again, the proposed strategy yields the best performance. The THD is significantly reduced below the 5% limit stipulated by standards like IEEE 1547, and the fundamental current tracking is most accurate. The fuzzy system likely adjusts the observer bandwidth to optimally tune the disturbance estimation for the specific harmonic frequencies present, showcasing its adaptive superiority for the grid-tied inverter operating in a polluted grid environment.
Conclusion and Future Perspectives
This article has presented a comprehensive Improved Fuzzy-Linear Active Disturbance Rejection Control strategy for LCL-type grid-tied inverters. The core innovation lies in the synergistic combination of an enhanced Linear Extended State Observer (LESO) for faster and more robust total disturbance estimation and a fuzzy logic system for the online adaptation of the controller’s critical bandwidth parameters. This approach addresses the fundamental challenges of resonance damping, dynamic performance, and robustness against grid disturbances in a unified framework.
The key advantages demonstrated are:
- Superior Performance: The strategy achieves lower grid current THD, faster dynamic response with minimal overshoot, and stronger rejection of grid voltage sags and harmonics compared to conventional PI with active damping and standard fixed-gain LADRC.
- Structural Simplicity: It inherently damps the LCL resonance without requiring additional active damping loops or virtual impedance calculations, simplifying the control system design for the grid-tied inverter.
- Robustness and Adaptability: The fuzzy adaptation mechanism allows the controller to self-optimize its performance across different operating conditions, making it highly suitable for the unpredictable environment of modern power grids with high penetration of renewables.
The stability of the proposed system was substantiated through Lyapunov analysis for the observer and frequency-domain reasoning for the overall loop.
Future research directions could explore the integration of more advanced intelligent algorithms, such as neural networks or reinforcement learning, with the ADRC framework to automate the design of the fuzzy rule base and membership functions. Furthermore, the practical implementation aspects, including digital delay compensation, parameter sensitivity analysis, and experimental validation on a hardware prototype of a grid-tied inverter, are essential next steps to transition this promising control strategy from simulation to real-world application.