The proliferation of solar energy has made the photovoltaic (PV) grid-connected inverter a critical component in modern power systems. The core control objectives for a three-phase solar inverter are rapid dynamic response, zero steady-state error, strong robustness, and low harmonic distortion in its output voltage and current. While various control strategies like Proportional-Integral (PI) and repetitive control have been proposed, their performance can be significantly degraded by inherent non-idealities within the inverter system. Among these, the dead-time effect, introduced deliberately to prevent shoot-through faults in the switching bridge, is a primary source of voltage waveform distortion and reduced conversion efficiency. Furthermore, external grid disturbances, load variations, and internal parameter uncertainties pose additional challenges to achieving high-quality power injection. This article presents a comprehensive control methodology for a solar inverter that explicitly addresses the dead-time effect alongside other disturbances through an integrated approach combining a robust state observer and a non-singular terminal sliding mode controller (NTSMC).
The dead-time is a small interval inserted between the turn-off of one switch and the turn-on of its complementary switch in the same inverter leg. This essential protection measure, combined with the intrinsic turn-on/off delays of semiconductor devices, causes a deviation between the ideal gate signal and the actual output voltage. This deviation, or dead-time effect, manifests as voltage errors that are dependent on the polarity of the output phase current. Over a fundamental cycle, these errors introduce low-order harmonics, primarily the 5th and 7th, distorting the output waveform, increasing total harmonic distortion (THD), and causing torque pulsations in motor loads. Therefore, any high-fidelity model and advanced control design for a solar inverter must incorporate this phenomenon.
We begin by analyzing a standard two-level, three-phase voltage source inverter (VSI) topology used for PV grid connection. The system comprises a DC-link capacitor, six insulated-gate bipolar transistors (IGBTs) with anti-parallel diodes, an LCL filter (inverter-side inductor, filter capacitor, and grid-side inductor), and the grid connection point. Parasitic resistances of the inductors and capacitor are also considered. The primary control goal is to force the inverter output currents to precisely track their sinusoidal references, which are in phase with the grid voltages, thereby ensuring unity power factor operation.
To model the solar inverter with dead-time, we employ the state-space averaging method over a switching period $T_s$. Taking phase-A as an example, the ideal output voltage $U_{AN’}$ (point A relative to the DC-link midpoint N’) is given by the switching function $S_A$:
$$ U_{AN’} = \frac{U_{dc}}{2}(2S_A – 1) $$
where $S_A = 1$ when the upper switch is on, and $S_A = 0$ when the lower switch is on. The average value over $T_s$ is:
$$ \langle U_{AN’} \rangle_{T_s} = \frac{U_{dc}}{2}(2\langle S_A \rangle_{T_s} – 1) $$
Considering the dead-time $t_d$, switch turn-on delay $t_{on}$, and turn-off delay $t_{off}$, the actual average duty cycle becomes $\langle S_A’ \rangle_{T_s} = D \pm D’$, where $D$ is the ideal duty cycle from the modulator and $D’ = (t_d + t_{on} – t_{off})/T_s$. The sign depends on the direction of the phase-A current $i_A$. The average output voltage, incorporating this effect, is derived as:
$$ \langle U_{AN} \rangle_{T_s} = \frac{U_{dc}}{2U_{tri}} U_m – \text{sgn}(i_A) U_{dc} D’ $$
where $U_m$ is the modulation wave amplitude, $U_{tri}$ is the carrier wave amplitude, and $\text{sgn}(\cdot)$ is the signum function. The term $-\text{sgn}(i_A) U_{dc} D’$ represents the dead-time distortion voltage $\Delta U_{DT}$. Generalizing for all three phases and applying Kirchhoff’s laws, the full state-space model of the three-phase solar inverter in the stationary frame is derived. For control design, we consider a single-phase equivalent and define the state vector $\mathbf{x}(t) = [i, i_g, v_c]^T$, representing the inverter-side current, grid-side current, and capacitor voltage, respectively.
The system model can be expressed in a standard state-space form that lumps the dead-time effect, parameter variations (in $L$, $C$, $R$), and external disturbances into a compounded disturbance term $d(t)$:
$$ \dot{\mathbf{x}} = \mathbf{A} \mathbf{x} + \mathbf{B} \mathbf{u} + \mathbf{G} $$
$$ y = \mathbf{F} \mathbf{x} $$
where $\mathbf{A}$, $\mathbf{B}$ are system matrices, $\mathbf{u}=[U_{dc}, U_g, \Delta U]^T$ is the input vector, $\mathbf{G}=[0, 0, d(t)]^T$, and $y$ is the system output. The compounded disturbance $d(t)$ is defined as $d(t) = \beta_1 i + \beta_2 i_g + \beta_3 v_c + h(t)$, where $\beta_i$ represent uncertainties from component tolerances and $h(t)$ encapsulates unmodeled dynamics and external disturbances.
| Symbol | Description |
|---|---|
| $U_{dc}$ | DC-link input voltage |
| $i, i_g$ | Inverter-side and grid-side current |
| $v_c$ | Filter capacitor voltage |
| $L, R$ | Filter inductance and parasitic resistance |
| $C, R_c$ | Filter capacitance and parasitic resistance |
| $t_d$ | Dead-time |
| $d(t)$ | Compounded lumped disturbance |
The core of the proposed control strategy is a Proportional-Integral (PI) based Extended State Observer (ESO). Its purpose is to actively estimate the compounded lumped disturbance $d(t)$ in real-time, which includes the effects specific to the solar inverter’s non-idealities. The observer dynamics are designed as:
$$ \dot{\hat{\mathbf{x}}} = \mathbf{A} \hat{\mathbf{x}} + \mathbf{B} \mathbf{u} + \hat{\mathbf{G}} + \mathbf{M} (\mathbf{x} – \hat{\mathbf{x}}) $$
$$ \hat{y} = \mathbf{F} \hat{\mathbf{x}} $$
where $\hat{\mathbf{x}}$ is the estimated state, $\hat{\mathbf{G}}=[0, 0, \hat{d}(t)]^T$ is the estimated disturbance vector, and $\mathbf{M}$ is the observer gain matrix designed to ensure stability. The disturbance estimate $\hat{d}(t)$ is generated through a PI mechanism on the estimation error:
$$ \hat{d} = \mathbf{L}_1 \mathbf{H} + \int_0^t \mathbf{L}_2 \mathbf{H} d\tau + \hat{d}(0) $$
Here, $\mathbf{H}$ is a generalized error signal derived from the observer, and $\mathbf{L}_1$, $\mathbf{L}_2$ are diagonal gain matrices. Using Lyapunov stability theory, it can be proven that if the transfer function $\mathbf{G}(s) = \mathbf{E}[s\mathbf{I} – (\mathbf{A} – \mathbf{M})]^{-1}$ is strictly positive real and certain passivity conditions hold, the estimation errors $\tilde{\mathbf{x}} = \mathbf{x} – \hat{\mathbf{x}}$ and $\tilde{d} = d – \hat{d}$ converge to zero asymptotically. This accurate estimation is crucial for the subsequent controller to achieve precise compensation.
With the disturbance estimate $\hat{d}$ available, we design a Non-Singular Terminal Sliding Mode Controller (NTSMC) to achieve finite-time convergence and high tracking accuracy. Define the tracking error vector $\boldsymbol{\sigma} = \mathbf{x}_{ref} – \mathbf{x}$, where $\mathbf{x}_{ref}$ is the reference state vector. We construct the following sliding surface vectors:
$$ \mathbf{J}_1 = \theta_1 \boldsymbol{\sigma} + \theta_2 \int_0^t \boldsymbol{\sigma} d\tau $$
$$ \mathbf{J}_2 = \dot{\mathbf{J}}_1 = \theta_1 \dot{\boldsymbol{\sigma}} + \theta_2 \boldsymbol{\sigma} $$
where $\theta_1, \theta_2 > 0$ are design parameters. The non-singular terminal sliding surface is defined as:
$$ \mathbf{z} = \mathbf{J}_1 + \boldsymbol{\delta} \mathbf{J}_2^{p/q} $$
Here, $\boldsymbol{\delta} = \text{diag}(\delta_1, \delta_2) > 0$, and $p$ and $q$ are positive odd integers satisfying $1 < p/q < 2$. This structure avoids the singularity problem present in conventional terminal sliding mode control. The control law is synthesized as:
$$ \mathbf{e} = \frac{1}{b} ( \mathbf{e}_1 – \mathbf{e}_2 + \hat{d} ) $$
$$ \mathbf{e}_1 = \mathbf{A} \mathbf{x} – \frac{\theta_2}{\theta_1} \boldsymbol{\sigma} $$
$$ \mathbf{e}_2 = \int_0^t k_1 \text{sat}\left(\frac{\mathbf{z}}{\alpha}\right) d\tau + \int_0^t \frac{q}{\theta_1 p} \boldsymbol{\delta}^{-1} \mathbf{J}_2^{2-p/q} d\tau + \int_0^t k_2 \mathbf{z} d\tau $$
In the control law, $b$, $k_1$, $k_2 > 0$ are controller gains, and $\text{sat}(\cdot)$ is a saturation function replacing the signum function to mitigate chattering, with $\alpha$ being the boundary layer thickness. The stability of the closed-loop solar inverter system is rigorously proven using Lyapunov’s direct method. Consider the Lyapunov function candidate $V = \frac{1}{2} \mathbf{z}^T \mathbf{z}$. Its time derivative, after substituting the controller and the disturbance estimation error, becomes:
$$ \dot{V} = \mathbf{z}^T \dot{\mathbf{z}} \leq -\frac{\theta_1 p}{q} \min_i \{\delta_i J_{2i}^{p/q-1} \} \left[ k_1 \| \mathbf{z} \| + k_2 \| \mathbf{z} \|^2 \right] \leq 0 $$
Since $1 < p/q < 2$, we have $J_{2i}^{p/q-1} > 0$. The inequality $\dot{V} \leq 0$ guarantees that the system trajectories will reach the sliding manifold $\mathbf{z} = 0$ in finite time. Once on the manifold, it follows that $\mathbf{J}_1 = \mathbf{J}_2 = 0$, which implies $\boldsymbol{\sigma} = \dot{\boldsymbol{\sigma}} = 0$. Therefore, the output of the solar inverter tracks its reference precisely, and the system is globally asymptotically stable despite the presence of dead-time and other disturbances.
| Parameter | Value / Description | Role |
|---|---|---|
| $p, q$ | 13, 11 (odd integers, 1 < p/q < 2) | NTSMC convergence dynamics |
| $\delta_1, \delta_2$ | 0.001 | Sliding surface coefficients |
| $k_1, k_2$ | 100, $8 \times 10^3$ | NTSMC gains for robustness |
| $\theta_1, \theta_2$ | 12 | Sliding surface integration weights |
| $\mathbf{L}_1, \mathbf{L}_2$ | Diagonal matrices | ESO PI gains for disturbance estimation |
| $\alpha$ | Boundary layer thickness | Chattering reduction |
The proposed integrated control strategy for the solar inverter was validated through detailed simulations using the PSIM software. The system parameters were set as follows: DC input voltage $U_{dc} = 360 \text{V}$, AC output voltage $220 \text{V} \text{ rms}$, switching frequency $f_{sw} = 10 \text{kHz}$, nominal load $R_L = 44 \Omega$, filter inductance $L = 4 \pm 2 \text{mH}$, filter capacitance $C = 28.2 \pm 13 \mu\text{F}$. The controller parameters are listed in Table 2.
Under a pure resistive nominal load startup, the inverter’s voltage and current reached steady-state within one fundamental cycle (20 ms) with virtually no overshoot or oscillation. The output voltage total harmonic distortion (THD) was measured at a remarkably low 0.024%, demonstrating the excellent steady-state performance of the controlled solar inverter.
To test robustness against load disturbances, the load resistance was abruptly changed from 34 Ω to 15 Ω at t=0.1s, and back to 34 Ω at t=0.2s. The results showed that the load voltage remained stable and sinusoidal during these transitions, with minimal deviation. The THD was maintained at 0.025%. For a nonlinear load (a single-phase diode bridge rectifier with an RC load), the solar inverter controller successfully maintained a stable sinusoidal output voltage with a THD of 0.034%, showcasing its capability to handle difficult load conditions.
A key test involved enabling the dead-time effect (with $t_d=3\mu s$, $t_{on}=1\mu s$, $t_{off}=2\mu s$) in the simulation. Initially, without compensation, the output voltage amplitude was reduced to approximately 210V. Upon activation of the proposed controller at t=0.02s, which incorporates the disturbance observer estimating the dead-time effect within $d(t)$, the output voltage swiftly recovered to its nominal 220V amplitude by t=0.025s. This confirms that the integrated observer-controller structure effectively compensates for the dead-time-induced voltage drop and distortion in the solar inverter.
Finally, the system’s resilience to input DC voltage variations was tested. The DC-link voltage $U_{dc}$ was stepped from 360V to 380V and back. The AC output voltage and current showed exceptional immunity to these input disturbances, holding constant amplitude and waveform quality. This feature is particularly beneficial for PV applications, as it relaxes the requirement for an extremely stiff DC-link, potentially allowing for smaller DC-link capacitors.
| Test Condition | Key Performance Indicator | Result |
|---|---|---|
| Nominal Startup | Settling Time, Voltage THD | < 20 ms, 0.024% |
| Load Step (34Ω ↔ 15Ω) | Voltage Regulation, THD | Excellent, 0.025% |
| Nonlinear Load | Voltage THD | 0.034% |
| Dead-Time Enabled | Voltage Recovery & Compensation | Full compensation achieved in ~5 ms |
| DC Input Step (360V ↔ 380V) | Output Voltage Stability | Highly stable, negligible impact |
This article has presented a robust integrated control scheme for grid-connected solar inverters. The methodology explicitly addresses two major practical challenges: the dead-time effect inherent in the inverter’s switching action and the broader category of compounded disturbances including parameter uncertainties and external perturbations. The solution is bifurcated into a sophisticated estimation stage and a robust control stage. First, a Proportional-Integral Extended State Observer (PI-ESO) is designed to actively estimate the aggregated lumped disturbance in real-time. Second, a Non-Singular Terminal Sliding Mode Controller (NTSMC) utilizes these estimates to construct a control law that ensures finite-time convergence and precise reference tracking. The formal stability of the entire closed-loop system is guaranteed via Lyapunov analysis.
The simulation results comprehensively validate the approach, demonstrating that the controlled solar inverter achieves excellent dynamic response, negligible steady-state error, very low output THD, and strong robustness against load changes, input voltage variations, and the detrimental dead-time effect. The integrated use of the disturbance observer makes the sliding mode controller highly effective without requiring excessively large switching gains, thereby reducing chattering. This control strategy significantly enhances the power quality delivered by the photovoltaic generation system, contributing to the stability and reliability of the utility grid. Future work may focus on the experimental validation of this controller on a hardware prototype and its extension to manage power flow in unbalanced grid conditions or under grid faults. The principles established here are fundamental to advancing the control and performance of modern solar inverter technology.