In modern power systems, the integration of distributed generation, such as photovoltaic systems, has become increasingly prevalent. However, the geographical mismatch between energy resources and load centers often necessitates long-distance transmission, leading to the incorporation of extensive transmission lines and transformers. This results in increased grid impedance, characterizing the network as a weak grid. Under such conditions, grid impedance fluctuations can severely threaten the stability of utility interactive inverter systems. Additionally, LCL-type filters are widely used in utility interactive inverters due to their superior harmonic attenuation capabilities. Still, they suffer from inherent resonance issues, and parameter perturbations—such as those caused by aging in inductors and capacitors—can further degrade system performance and destabilize operation. Traditional resonance suppression methods, including active and passive damping, often face limitations like reduced bandwidth, increased sensor costs, and sensitivity to parameter variations. Therefore, developing robust control strategies that ensure stability and high power quality in weak grids is paramount.
In this work, we propose a composite control strategy based on super-twisting sliding mode theory for LCL-type utility interactive inverters operating in weak grids. Our approach leverages a super-twisting sliding mode observer to estimate key system states—such as capacitor voltage and inverter-side current—eliminating the need for additional sensors and reducing cost and noise contamination. Furthermore, we design a controller that combines a non-singular fast terminal sliding mode surface with an improved super-twisting algorithm to enhance dynamic response, minimize steady-state error, and bolster robustness against external disturbances and parameter uncertainties. Extensive simulations validate the effectiveness of our method, demonstrating strong performance under grid impedance variations and filter parameter perturbations.
To lay the groundwork, we first establish a mathematical model for the LCL-type utility interactive inverter. The system comprises a single-phase full-bridge inverter, an LCL filter, and the grid. Key variables include the DC-link voltage $U_{dc}$, the filter capacitor voltage $U_C$, the capacitor branch voltage $U_{CR}$, and the grid voltage $U_g$. The switching states are denoted by $S_1$ to $S_4$, while $i_1$, $i_C$, and $i_g$ represent the inverter-side current, capacitor current, and grid current, respectively. Applying Kirchhoff’s laws, the dynamic equations are derived as:
$$
\begin{aligned}
L_1 \frac{di_1}{dt} &= u U_{dc} – U_C – R_d i_1 + R_d i_g – d_1(t), \\
C \frac{dU_C}{dt} &= i_1 – i_g – d_2(t), \\
(L_2 + L_g) \frac{di_g}{dt} &= U_C + R_d i_1 – R_d i_g – U_g – d_3(t),
\end{aligned}
$$
where $u$ is the duty cycle (control input), $R_d$ is a passive damping resistor, $L_1$ and $L_2$ are the inverter-side and grid-side inductors, $C$ is the filter capacitor, $L_g$ is the grid impedance, and $d_1(t)$, $d_2(t)$, $d_3(t)$ represent lumped disturbances encompassing parameter uncertainties and external interferences. These disturbances are expressed as:
$$
\begin{aligned}
d_1(t) &= \Delta L_1 \frac{di_1}{dt} + \epsilon_1, \\
d_2(t) &= \Delta C \frac{dU_C}{dt} + \epsilon_2, \\
d_3(t) &= (\Delta L_2 + \Delta L_g) \frac{di_g}{dt} + \epsilon_3,
\end{aligned}
$$
with $\Delta L_1$, $\Delta C$, $\Delta L_2$, $\Delta L_g$ denoting deviations from nominal values, and $\epsilon_1$, $\epsilon_2$, $\epsilon_3$ being unmeasurable disturbances.
Defining the state vector $\mathbf{x} = [i_1, U_C, i_g]^T$, we formulate the state-space model:
$$
\dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} u + \mathbf{D} + \mathbf{d}, \quad y = \mathbf{C} \mathbf{x},
$$
where the matrices are given by:
$$
\mathbf{A} = \begin{bmatrix}
-\frac{R_d}{L_1} & -\frac{1}{L_1} & \frac{R_d}{L_1} \\
\frac{1}{C} & 0 & -\frac{1}{C} \\
\frac{R_d}{L_2 + L_g} & \frac{1}{L_2 + L_g} & -\frac{R_d}{L_2 + L_g}
\end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix}
\frac{U_{dc}}{L_1} \\ 0 \\ 0
\end{bmatrix}, \quad \mathbf{C} = \begin{bmatrix}
0 & 0 & 1
\end{bmatrix},
$$
$$
\mathbf{D} = \begin{bmatrix}
0 \\ 0 \\ -\frac{U_g}{L_2 + L_g}
\end{bmatrix}, \quad \mathbf{d} = \begin{bmatrix}
-\frac{d_1(t)}{L_1} \\ -\frac{d_2(t)}{C} \\ -\frac{d_3(t)}{L_2 + L_g}
\end{bmatrix}.
$$
The resonance angular frequency of the LCL filter is a critical factor affecting stability. It is derived from the characteristic equation and expressed as:
$$
\omega_r = \sqrt{\frac{L_1 + L_2 + L_g}{L_1 (L_2 + L_g) C}}.
$$
Parameter perturbations directly influence $\omega_r$, causing resonance shifts that can render conventional damping methods ineffective. To quantify this impact, we analyze the system’s frequency response under various parameter deviations. The following table summarizes nominal parameters and their variations used in our study:
| Parameter | Nominal Value | Variation Range |
|---|---|---|
| $L_1$ | 2.75 mH | ±50% |
| $L_2$ | 2.45 mH | ±20% |
| $C$ | 10 µF | ±50% |
| $L_g$ | 1 mH | 1–8 mH |
| $R_d$ | 0.8 Ω | Fixed |
| $U_{dc}$ | 700 V | Fixed |
| $U_g$ | 220 V (RMS) | Fixed |
The proposed control strategy consists of two main components: a super-twisting sliding mode observer for state estimation and a composite super-twisting sliding mode controller for current regulation. The observer is designed to reconstruct the capacitor voltage $U_C$, inverter-side current $i_1$, and grid current $i_g$ without direct measurement, thereby reducing sensor count and noise. The observer dynamics are given by:
$$
\dot{\hat{\mathbf{x}}} = \mathbf{A} \hat{\mathbf{x}} + \mathbf{B} u + \mathbf{D} + \mathbf{L} (y – \hat{y}) – \mathbf{v}, \quad \hat{y} = \mathbf{C} \hat{\mathbf{x}},
$$
where $\hat{\mathbf{x}} = [\hat{i}_1, \hat{U}_C, \hat{i}_g]^T$ are estimated states, $\mathbf{L} = [l_1, l_2, l_3]^T$ is the Luenberger gain matrix, and $\mathbf{v}$ is the super-twisting term defined as:
$$
\mathbf{v} = -k_1 |\mathbf{e}|^{1/2} \text{sgn}(\mathbf{e}) – \int k_2 \text{sgn}(\mathbf{e}) \, dt,
$$
with $\mathbf{e} = \mathbf{x} – \hat{\mathbf{x}}$ being the estimation error, and $k_1, k_2 > 0$ are gains chosen to ensure convergence. The stability of the observer is proven via Lyapunov analysis. Consider the Lyapunov function candidate $V = \boldsymbol{\zeta}^T \mathbf{Q} \boldsymbol{\zeta}$, where $\boldsymbol{\zeta} = [|\mathbf{e}|^{1/2} \text{sgn}(\mathbf{e}), \mathbf{e}, \boldsymbol{\phi}]^T$ and $\boldsymbol{\phi}$ is an auxiliary variable. Through detailed derivation, we show that $\dot{V} \leq -\mu \sqrt{V}$ for some $\mu > 0$, guaranteeing finite-time convergence of the estimation error to zero.
For the controller, we aim to track the reference grid current $i_{ref} = I_{ref} \sin(\omega t)$, where $I_{ref}$ is the amplitude and $\omega$ is the grid angular frequency. The tracking error is defined as $e_1 = i_{ref} – \hat{i}_g$, and we introduce augmented errors to incorporate capacitor voltage dynamics. A non-singular fast terminal sliding surface is designed to achieve fast convergence and avoid singularity issues:
$$
s = e_1 + \alpha \int e_1^{g/h} \, dt + \beta e_2^{p/q},
$$
where $e_2 = \dot{e}_1$, $\alpha, \beta > 0$, and $g, h, p, q$ are positive odd integers satisfying $1 < p/q < 2$ and $g/h > p/q$. This surface ensures that when $s = 0$, the error dynamics converge to zero in finite time. The control law is synthesized as the sum of an equivalent control $u_{eq}$ and a super-twisting switching control $u_{st}$:
$$
u = u_{eq} + u_{st}.
$$
The equivalent control is derived from the nominal system by setting $\dot{s} = 0$:
$$
u_{eq} = \frac{L_1 A_2}{A_1 R_d U_{dc} (k_4 + k_5 A_2)} \left[ \left(1 + \alpha \frac{g}{h} e_1^{g/h-1}\right) E(t) + A_1 k_4 \left( \dot{i}_{ref} – \frac{\dot{i}_g}{A_2} + \frac{R_d \dot{U}_{CR}}{L_1 A_2} \right) + A_1 k_5 \left( \dot{U}^*_{CR} – \frac{i_1 – i_g}{C} + \frac{R_d U_{CR}}{L_1} \right) + A_1 e_2 \right],
$$
where $A_1 = \beta \frac{p}{q} e_2^{p/q-1}$, $A_2 = L_2 + L_g$, $E(t) = e_1 + k_4 e_2 + k_5 e_3$ with $e_3 = U_C – \hat{U}_C$, and $k_4, k_5$ are positive constants. The switching control employs the super-twisting algorithm to reject disturbances:
$$
u_{st} = \frac{L_1 A_2}{A_1 R_d U_{dc} (k_4 + k_5 A_2)} \left[ \varepsilon_1 |s|^{1/2} \text{sgn}(s) + \int \varepsilon_2 \text{sgn}(s) \, dt \right],
$$
with $\varepsilon_1, \varepsilon_2 > 0$. The closed-loop stability is analyzed using Lyapunov theory. Consider the Lyapunov function $V_2 = 2 \varepsilon_2 |s| + \frac{1}{2} u_1^2$, where $u_1$ is an auxiliary variable. After rigorous manipulation, we obtain $\dot{V}_2 \leq -\gamma \sqrt{V_2}$ for some $\gamma > 0$, confirming finite-time convergence of $s$ to zero and thus robust tracking performance.
To validate our proposed strategy, we conduct simulations in MATLAB/Simulink, comparing it with two benchmark methods: a conventional sliding mode control with exponential reaching law and a quasi-proportional-resonant (QPR) control with a state observer. The system parameters are as listed in the table above. The utility interactive inverter is tested under various scenarios, including step changes in reference current, filter parameter perturbations, and grid impedance variations.
First, we evaluate dynamic tracking performance. The reference current $i_{ref}$ steps from 10 A to 20 A at 0.045 s. Our proposed controller exhibits superior response with minimal overshoot and settling time compared to the benchmarks. The tracking error, defined as $i_{err} = i_{ref} – i_g$, remains below 0.5 A in our method, while the others show larger deviations. This demonstrates the enhanced dynamic capability of the super-twisting sliding mode approach.
Next, the observer’s performance is assessed. The super-twisting sliding mode observer accurately estimates $i_1$, $U_C$, and $i_g$ with rapid convergence and negligible chattering, unlike the Luenberger observer and conventional sliding mode observer which exhibit slower response or steady-state oscillations. The estimation errors are summarized below:
| Observer Type | Max Error in $i_1$ (A) | Max Error in $U_C$ (V) | Max Error in $i_g$ (A) |
|---|---|---|---|
| Super-Twisting SMO | 0.15 | 5.2 | 0.12 |
| Conventional SMO | 0.45 | 12.8 | 0.38 |
| Luenberger Observer | 0.85 | 20.5 | 0.72 |
We then investigate robustness against filter parameter perturbations. At time intervals 0–0.025 s, 0.025–0.065 s, and 0.065–0.1 s, the values of $L_1$ and $C$ are varied to 50%, 100%, and 150% of their nominal values, respectively. Our controller maintains stable operation with grid current total harmonic distortion (THD) consistently around 0.5%, whereas the benchmark methods show THD up to 2.0% under the same conditions. The following table compares THD values under parameter perturbations:
| Control Strategy | THD at 50% $L_1$, 50% $C$ | THD at 100% $L_1$, 100% $C$ | THD at 150% $L_1$, 150% $C$ |
|---|---|---|---|
| Proposed Method | 0.52% | 0.48% | 0.51% |
| Conventional SMC | 0.91% | 0.87% | 0.93% |
| QPR Control | 2.15% | 1.98% | 2.22% |
Finally, the system’s behavior under weak grid conditions is examined by varying the grid impedance $L_g$ from 1 mH to 8 mH. Our strategy demonstrates strong adaptability, with grid current THD staying below 0.4% across all impedance values. In contrast, the conventional SMC and QPR control show THD degradation up to 1.3% and 2.5%, respectively. The robustness is attributed to the integral action in the sliding surface and the disturbance rejection capability of the super-twisting algorithm. The impedance variation results are condensed as:
| Grid Impedance $L_g$ (mH) | THD (Proposed Method) | THD (Conventional SMC) | THD (QPR Control) |
|---|---|---|---|
| 1 | 0.20% | 0.25% | 1.05% |
| 5 | 0.38% | 1.30% | 2.10% |
| 8 | 0.31% | 0.40% | 1.13% |
In conclusion, we have developed a robust control strategy for LCL-type utility interactive inverters in weak grids. The integration of a super-twisting sliding mode observer eliminates additional sensors and enhances estimation accuracy. The composite controller, blending non-singular fast terminal sliding mode and super-twisting algorithms, ensures fast dynamic response, low steady-state error, and high robustness against parameter perturbations and grid impedance fluctuations. Simulation results confirm that our method maintains grid current THD around 0.5% under diverse disturbances, outperforming conventional approaches. Future work may focus on experimental validation and extension to three-phase systems or microgrid applications. This research contributes to the reliable operation of utility interactive inverters in evolving power networks with high penetration of renewable energy.