The integration of photovoltaic (PV) power generation into the electrical grid is a cornerstone of the global transition towards sustainable energy. At the heart of this integration lies the grid-tied inverter, a critical power electronic interface responsible for converting the direct current (DC) produced by PV panels into alternating current (AC) suitable for grid injection. However, the inherent intermittency and stochastic nature of solar irradiance and ambient temperature introduce significant power fluctuations. These fluctuations manifest as substantial disturbances on the DC-link voltage of the inverter. Maintaining the stability and quality of this DC-link voltage is paramount, as it directly influences the performance, efficiency, and reliability of the entire power conversion chain and, consequently, the stability of the local grid segment. Therefore, developing advanced control strategies that can effectively reject these disturbances and ensure robust voltage regulation is a persistent and vital research challenge in power electronics and renewable energy systems.
Traditional control approaches for the outer voltage loop of a grid-tied inverter, such as Proportional-Integral (PI) control, are widely adopted due to their simplicity. Yet, they often exhibit limitations in handling rapid and large disturbances, leading to undesirable overshoot, slow recovery, and potential instability under highly variable conditions. Other strategies like sliding mode control, adaptive control, and repetitive control have been explored to address these issues, but they may introduce complexities related to chattering, parameter tuning, or computational burden. In recent years, Active Disturbance Rejection Control (ADRC) has emerged as a powerful alternative. Its core philosophy is to estimate and compensate for the total disturbance—encompassing model uncertainties, parameter variations, and external perturbations—in real-time, thereby simplifying the controller design and enhancing robustness. Linear ADRC (LADRC), a simplified version, has gained traction for its easier implementation and tuning. This work focuses on the critical task of DC-link voltage regulation and proposes a refined, first-order Modified Linear Active Disturbance Rejection Control (M-LADRC) strategy specifically designed for the outer voltage loop of a three-phase LCL-filter-based grid-tied inverter.
1. Mathematical Modeling of the LCL-Type Grid-Tied Inverter System
The topology of a three-phase, two-level PV inverter system with an LCL output filter is considered standard for medium to high-power applications due to its superior harmonic attenuation capabilities. The system comprises a PV array, a DC-DC boost converter (for Maximum Power Point Tracking – MPPT), a DC-link capacitor, a three-phase voltage source inverter (VSI), and an LCL filter connecting to the utility grid.
To derive a control-oriented model, key variables are defined: $U_{dc}$ is the DC-link voltage, $C_2$ is the DC-link capacitance, $i_o$ is the output current from the boost stage, and $i_{in}$ is the input current to the inverter bridge. Applying Kirchhoff’s current law at the DC-link node yields the fundamental dynamic equation:
$$ C_2 \frac{dU_{dc}}{dt} = i_o – i_{in} $$
For control system design, it is advantageous to express this model in the synchronous rotating (dq) reference frame, which transforms AC quantities into DC values, simplifying the controller design. Using Park’s transformation synchronized with the grid voltage vector, the dynamics of the LCL filter can be represented. However, for the bandwidth of the voltage control loop (which is typically much lower than the current control loop), the LCL filter can often be approximated by an equivalent L-filter in the low-frequency range, significantly simplifying the model without sacrificing accuracy for the outer loop design. The resulting dq-axis model of the inverter system is:
$$
\begin{aligned}
L_{eq} \frac{di_d}{dt} &= u_d – \omega L_{eq} i_q – u_{sd} \\[5pt]
L_{eq} \frac{di_q}{dt} &= u_q + \omega L_{eq} i_d – u_{sq} \\[5pt]
C_2 \frac{dU_{dc}}{dt} &= i_o – i_{in}
\end{aligned}
$$
where $L_{eq}$ is the equivalent inductance, $i_d$ and $i_q$ are the grid currents in the dq-frame, $u_d$ and $u_q$ are the inverter output voltages, $u_{sd}$ and $u_{sq}$ are the grid voltages, and $\omega$ is the grid angular frequency. The power balance between the DC and AC sides, assuming ideal conversion and a steady DC-link voltage near its reference $U_{dc}^{ref}$, can be stated as $P_{dc} \approx P_{ac}$. This leads to:
$$ U_{dc}^{ref} i_{in} \approx \frac{3}{2} (u_{sd} i_d + u_{sq} i_q) $$
Typically, by aligning the d-axis with the grid voltage vector ($u_{sd}=U_1$, $u_{sq}=0$), the expression simplifies. Substituting this into the DC-link dynamic equation provides a direct relationship between the DC-link voltage and the d-axis current reference $i_{d}^{ref}$, which is the primary control variable for voltage regulation:
$$ \frac{dU_{dc}}{dt} = \frac{3 U_1}{2 C_2 U_{dc}^{ref}} i_{d}^{ref} – \frac{1}{C_2} i_o $$
This equation reveals the first-order nature of the DC-link voltage dynamics with respect to the control input $i_{d}^{ref}$, making it suitable for a first-order controller design. The term $(-i_o / C_2)$ aggregates the effect of input power variation from the PV side and acts as a primary disturbance to the voltage loop.
2. Design of the Modified Linear Active Disturbance Rejection Controller (M-LADRC)
The standard first-order LADRC framework consists of two main components: a Linear Extended State Observer (LESO) and a Linear State Error Feedback (LSEF) control law. The innovation proposed here modifies both components to enhance performance.
Consider a generalized first-order plant affected by disturbance:
$$ \dot{y} = b_0 u + f $$
where $y$ is the output (DC-link voltage $U_{dc}$), $u$ is the control input (d-axis current reference $i_{d}^{ref}$), $b_0$ is a known approximate gain, and $f$ represents the “total disturbance,” which includes unknown internal dynamics and external disturbances (like $ -i_o / C_2 $).
2.1 Modified Linear Extended State Observer (M-LESO)
The objective of the LESO is to estimate both the system state $y$ and the total disturbance $f$ in real-time. A modified LESO structure with enhanced estimation dynamics is proposed. It is defined by the following set of equations:
$$
\begin{aligned}
e_1(t) &= Z_1 – y \\
\dot{Z}_1 &= Z_2 – \beta_1 e_1(t) + b_0 u \\
\dot{Z}_2 &= -\beta_2 e_1(t) – \beta_3 \dot{e}_1(t)
\end{aligned}
$$
Here, $Z_1$ is the estimate of the output $y$, and $Z_2$ is the estimate of the total disturbance $f$. The terms $\beta_1$, $\beta_2$, and $\beta_3$ are observer gains to be tuned. The inclusion of the derivative of the observation error $\dot{e}_1(t)$ is a key modification that can improve the observer’s transient response and noise handling. Using the pole placement method, all observer poles are placed at $-\omega_o$ (the observer bandwidth) for simplicity and ease of tuning. This yields the gain relationships:
$$ \beta_2 = \omega_o^2, \quad \beta_1 + \beta_3 = 2\omega_o $$
A symmetric choice of $\beta_1 = \beta_3 = \omega_o$ is adopted, simplifying parameterization to a single bandwidth parameter $\omega_o$.
2.2 Modified Control Law with Derivative Feedforward
The Linear State Error Feedback (LSEF) law generates the preliminary control signal $u_0$ based on the tracking error between the reference $v$ ($U_{dc}^{ref}$) and the estimated state $Z_1$. The proposed modification introduces a derivative feedforward term from the reference signal to improve tracking agility. The final control signal $u$ is then derived by compensating for the estimated disturbance:
$$
\begin{aligned}
u_0 &= k_p (v – Z_1) \\
u &= \frac{ \dot{v} + u_0 – Z_2 }{b_0}
\end{aligned}
$$
Here, $k_p$ is the proportional gain, often set as the controller bandwidth $\omega_c$ (i.e., $k_p = \omega_c$). The term $\dot{v}$ is the derivative of the reference signal. For a step reference (which is common for DC-link voltage control), $\dot{v}=0$, but this structure provides general theoretical benefits. The complete structure of the proposed M-LADRC for the grid-tied inverter voltage loop is illustrated in the block diagram below and integrated into the overall system control architecture.
The overall control structure for the grid-tied inverter employs a cascaded control scheme. The proposed M-LADRC serves as the outer loop controller, generating the d-axis current reference $i_{d}^{ref}$ to regulate $U_{dc}$. This reference is then fed to the inner current control loop, which typically uses PI controllers in the dq-frame to dictate the inverter modulation signals. A Phase-Locked Loop (PLL) synchronizes the control with the grid voltage. The integration ensures that the grid-tied inverter maintains a stable DC bus while injecting high-quality current into the grid.
3. Performance Analysis of the M-LADRC Voltage Controller
To theoretically validate the advantages of the proposed M-LADRC, a comprehensive analysis of its disturbance rejection, tracking error, and stability properties is conducted.
3.1 Disturbance Rejection Capability
The effectiveness of any ADRC-based scheme hinges on its ability to estimate and cancel disturbances. The transfer function from the total disturbance $f$ to the output $y$ under closed-loop M-LADRC control is derived. After algebraic manipulation and Laplace transformation, this disturbance transfer function $G_{df}(s)$ is found to be:
$$ G_{df}(s) = \frac{Y(s)}{F(s)} = \frac{s^2 + (\omega_o + \omega_c)s + \omega_o \omega_c}{(s + \omega_o)^2 (s + \omega_c)} $$
For comparison, the disturbance transfer function for the traditional first-order LADRC (without the derivative term in the observer and feedforward) is:
$$ G_{df, traditional}(s) = \frac{s}{(s + \omega_o)^2 (s + \omega_c)} \cdot (s + \omega_o + \omega_c) $$
A Bode plot analysis of these two transfer functions reveals the superior disturbance rejection of the M-LADRC in the low to medium frequency range, where most of the disturbance energy from irradiance changes resides. The proposed structure typically shows lower gain in this critical region, meaning the output is less affected by the disturbance $f$.
Table 1: Comparative Analysis of Disturbance Rejection Characteristics
| Frequency Range | Traditional LADRC | Modified LADRC (Proposed) | Implication |
|---|---|---|---|
| Low Frequency ($\omega << \omega_c$) | Higher gain magnitude | Lower gain magnitude | Better suppression of slow-varying disturbances (e.g., gradual irradiance change). |
| Crossover Frequency | Standard attenuation slope | Steeper attenuation slope | Faster roll-off, improving rejection near the bandwidth. |
| Parameter Sensitivity | Sensitive to $\omega_o$ tuning | More robust with symmetric tuning ($\beta_1=\beta_3$) | Easier and more reliable controller commissioning for the grid-tied inverter. |
3.2 Tracking Error and State Estimation Error
The steady-state performance is evaluated by analyzing the estimation errors for both the state ($E_1 = Z_1 – y$) and the disturbance ($E_2 = Z_2 – f$). Using the final value theorem, the steady-state errors for step changes in the output and disturbance are computed:
$$ \lim_{s \to 0} s E_1(s) = 0, \quad \lim_{s \to 0} s E_2(s) = 0 $$
This confirms that the modified LESO can perfectly estimate both the state and the constant (or slowly varying) disturbance in steady state, leading to zero steady-state tracking error for the DC-link voltage. The inclusion of the error derivative term $\dot{e}_1(t)$ helps reduce the peak estimation error during transients, contributing to lower overshoot.
3.3 Closed-Loop Stability Analysis
The closed-loop transfer function from the voltage reference $U_{dc}^{ref}(s)$ to the actual voltage $U_{dc}(s)$ is derived as:
$$ G_{cl}(s) = \frac{(s+\omega_c)(s+\omega_o)^2}{s^3 + (\omega_o+\omega_c)s^2 + (\omega_o^2+2\omega_o\omega_c)s + \omega_o^2\omega_c} $$
Stability is guaranteed if all poles of $G_{cl}(s)$ have negative real parts. Since $\omega_o > 0$ and $\omega_c > 0$, all coefficients of the characteristic polynomial are positive. Applying the Routh-Hurwitz stability criterion provides the necessary and sufficient condition for stability. For a third-order polynomial $a_0s^3 + a_1s^2 + a_2s + a_3$, the system is stable if all coefficients are positive and $a_1a_2 > a_0a_3$. Substituting the coefficients from $G_{cl}(s)$:
$$
\begin{aligned}
a_1a_2 &= (\omega_o+\omega_c)(\omega_o^2+2\omega_o\omega_c) \\
&= \omega_o^3 + 3\omega_o^2\omega_c + 2\omega_o\omega_c^2 \\
a_0a_3 &= 1 \cdot \omega_o^2\omega_c = \omega_o^2\omega_c
\end{aligned}
$$
Clearly, $a_1a_2 – a_0a_3 = \omega_o^3 + 3\omega_o^2\omega_c + 2\omega_o\omega_c^2 – \omega_o^2\omega_c > 0$ for all $\omega_o, \omega_c > 0$. Therefore, the closed-loop system employing the proposed M-LADRC is unconditionally stable for any positive values of the observer and controller bandwidths, a highly desirable property for the grid-tied inverter application.
4. Simulation Verification and Results
To validate the theoretical analysis and demonstrate the practical efficacy of the proposed M-LADRC strategy, a detailed simulation model of a three-phase LCL-type grid-tied inverter was built in MATLAB/Simulink. The system parameters are listed in the table below.
Table 2: Simulation Parameters of the Grid-Tied Inverter System
| Parameter | Symbol | Value |
|---|---|---|
| DC-Link Voltage Reference | $U_{dc}^{ref}$ | 600 V |
| DC-Link Capacitance | $C_2$ | 2200 µF |
| Grid Voltage (Phase-to-Phase RMS) | $U_{grid}$ | 380 V |
| Grid Frequency | $f$ | 50 Hz |
| Inverter-Side Inductor | $L_1$ | 3.0 mH |
| Grid-Side Inductor | $L_2$ | 1.5 mH |
| LCL Filter Capacitor | $C_f$ | 20 µF |
| Switching Frequency | $f_{sw}$ | 10 kHz |
| Observer Bandwidth (M-LADRC) | $\omega_o$ | 2000 rad/s |
| Controller Bandwidth (M-LADRC) | $\omega_c$ | 1000 rad/s |
The performance of the proposed M-LADRC is compared against a traditional LADRC under identical tuning parameters ($\omega_o$, $\omega_c$, $b_0$) to ensure a fair comparison. Two challenging dynamic scenarios were tested to emulate real-world operational conditions for a PV-based grid-tied inverter.
Scenario 1: Step Increase in Solar Irradiance. This test simulates a sudden clearing of clouds, causing a rapid increase in PV output power. The equivalent disturbance $i_o$ steps up, challenging the DC-link controller’s ability to maintain voltage stability.
Scenario 2: Step Decrease in Solar Irradiance. This test simulates a sudden cloud cover, causing a rapid drop in PV power. The controller must prevent a large voltage dip and recover quickly.
The key metrics for comparison are the DC-link voltage overshoot/undershoot (in percentage) and the settling time. Furthermore, the quality of the injected grid current is assessed by calculating the Total Harmonic Distortion (THD).
Table 3: Comparative Simulation Results for DC-Link Voltage Performance
| Test Scenario | Control Strategy | Voltage Overshoot/Undershoot | Settling Time (to within 2%) | Grid Current THD |
|---|---|---|---|---|
| Irradiance Step Increase | Traditional LADRC | 2.77 % | 28 ms | 3.8 % |
| Proposed M-LADRC | 1.67 % | 25 ms | 2.9 % | |
| Irradiance Step Decrease | Traditional LADRC | 2.16 % (dip) | 30 ms | 4.1 % |
| Proposed M-LADRC | 1.39 % (dip) | 26 ms | 3.3 % |
The simulation results unequivocally demonstrate the superiority of the proposed M-LADRC strategy. In both transient scenarios, it significantly reduces the voltage deviation (overshoot and undershoot) by approximately 1.1% and 0.77% respectively, compared to the traditional LADRC. This is a direct consequence of the improved disturbance estimation and the derivative feedforward action, which allows for a more aggressive yet stable rejection of the power disturbance. The settling time is also marginally improved. Crucially, the quality of the power injected into the grid remains high, with current THD consistently below 5% and lower than that achieved with the traditional method, meeting standard grid codes (e.g., IEEE 1547). This confirms that the enhanced voltage stability provided by the outer M-LADRC loop creates a more favorable operating condition for the inner current loop of the grid-tied inverter.
5. Conclusion
This work has addressed the critical challenge of robust DC-link voltage regulation in photovoltaic grid-tied inverters subjected to unpredictable environmental disturbances. A Modified Linear Active Disturbance Rejection Control (M-LADRC) strategy was proposed, featuring an enhanced Linear Extended State Observer (LESO) structure and a control law incorporating reference derivative feedforward. Detailed mathematical modeling of the LCL-type inverter system established the foundation for the first-order controller design. A thorough theoretical analysis proved the proposed controller’s superior disturbance rejection in the low-frequency band, zero steady-state tracking error, and guaranteed closed-loop stability.
Comprehensive numerical simulations validated the theoretical claims. Under stringent tests of sudden irradiance changes—common stress scenarios for a PV grid-tied inverter—the M-LADRC strategy demonstrated markedly improved transient performance over traditional LADRC. It achieved a significant reduction in DC-link voltage overshoot and undershoot by 30-40%, faster settling, and concurrently maintained grid current quality with a THD below 5%. The simplified, symmetric tuning procedure (based on two bandwidth parameters $\omega_o$ and $\omega_c$) enhances its practicality. These attributes make the proposed M-LADRC a highly effective, robust, and practically implementable solution for improving the resilience and power quality of grid-connected photovoltaic systems, offering substantial reference value for engineering applications in renewable energy integration.