The increasing integration of renewable energy sources into the power grid has placed stringent demands on the performance and reliability of power conversion interfaces. The grid tied inverter serves as the critical link between distributed generation resources, such as photovoltaic arrays or wind turbines, and the utility grid. Its primary function is to convert direct current (DC) into high-quality alternating current (AC) that is synchronized with the grid voltage. The quality of the output current, characterized by low total harmonic distortion (THD), fast dynamic response, and robust operation under disturbances, is paramount for grid stability and compliance with standards.
While traditional two-level voltage source inverters (VSIs) are widely used, they suffer from significant drawbacks, including high switching losses, elevated electromagnetic interference (EMI), and substantial output current ripple, which necessitates large and costly output filters. In contrast, multilevel inverters, particularly the three-level topology, offer superior performance. The T-Type three-level inverter has emerged as a compelling solution, combining the benefits of multilevel output—such as reduced voltage stress on switches, lower output harmonic content, and smaller filter requirements—with a relatively simple structure. It achieves this by using a bidirectional switch connected to the neutral point of the DC-link capacitors, enabling output voltages of $+U_{dc}/2$, $0$, and $-U_{dc}/2$. The switching states for one phase leg are summarized below.
| Switch State | \(S_{a1}\) | \(S_{a2}\) | \(S_{a3}\) | \(S_{a4}\) | Output Voltage Level |
|---|---|---|---|---|---|
| P | ON | ON | OFF | OFF | \(+U_{dc}/2\) |
| O | OFF | ON | ON | OFF | 0 |
| N | OFF | OFF | ON | ON | \(-U_{dc}/2\) |
The advantages of the T-type three-level grid tied inverter are clear: lower switching losses compared to neutral-point-clamped (NPC) topologies, improved efficiency, and better output waveform quality. However, to fully exploit these hardware benefits, an advanced and robust control strategy is essential. The controller must ensure precise current tracking, rapid response to reference changes, and immunity to internal parameter variations and external grid disturbances. This article delves into the design of such a controller, proposing a novel hybrid approach that combines the strengths of sliding mode control (SMC) with the adaptive learning capability of Radial Basis Function (RBF) neural networks.
Mathematical Modeling of the T-Type Grid Tied Inverter
The foundation of any high-performance control system is an accurate mathematical model. For a three-phase T-type grid tied inverter connected to the grid through an L-filter, the dynamic equations in the stationary abc-frame can be derived using Kirchhoff’s voltage law (KVL).
$$L\frac{di_a}{dt} = \left(S_a – \frac{2(S_a + S_b + S_c)}{3}\right)\frac{U_{dc}}{2} – R i_a – e_a$$
$$L\frac{di_b}{dt} = \left(S_b – \frac{2(S_a + S_b + S_c)}{3}\right)\frac{U_{dc}}{2} – R i_b – e_b$$
$$L\frac{di_c}{dt} = \left(S_c – \frac{2(S_a + S_b + S_c)}{3}\right)\frac{U_{dc}}{2} – R i_c – e_c$$
Here, \(L\) and \(R\) represent the filter inductance and its parasitic resistance, \(i_{a,b,c}\) and \(e_{a,b,c}\) are the three-phase output currents and grid voltages, \(U_{dc}\) is the DC-link voltage, and \(S_{a,b,c}\) denote the switching functions for each phase leg, taking discrete values corresponding to the output levels (e.g., +1 for P, 0 for O, -1 for N).
To simplify the control design, the model is transformed from the three-phase stationary coordinate system (abc) to the synchronous rotating coordinate system (dq) using the Park transformation. This transformation aligns the reference frame with the grid voltage vector, converting the sinusoidal AC quantities into DC quantities, which are easier to regulate. The resulting state-space model in the dq-frame is:
$$L\frac{di_d}{dt} = L\omega i_q + \frac{U_{dc}}{2}S_d – R i_d – e_d$$
$$L\frac{di_q}{dt} = -L\omega i_d + \frac{U_{dc}}{2}S_q – R i_q – e_q$$
Where \(\omega\) is the grid angular frequency, \(i_d\) and \(i_q\) are the direct and quadrature-axis currents, \(e_d\) and \(e_q\) are the grid voltage components, and \(S_d\) and \(S_q\) are the control inputs in the dq-frame. This can be written in a compact state-space form:
$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{F}\mathbf{e}$$
with \(\mathbf{x} = [i_d, i_q]^T\), \(\mathbf{u} = [S_d, S_q]^T\), \(\mathbf{e} = [e_d, e_q]^T\), and matrices defined as:
$$\mathbf{A} = \begin{bmatrix} -R/L & \omega \\ -\omega & -R/L \end{bmatrix}, \quad \mathbf{B} = \begin{bmatrix} U_{dc}/(2L) & 0 \\ 0 & U_{dc}/(2L) \end{bmatrix}, \quad \mathbf{F} = \begin{bmatrix} -1/L & 0 \\ 0 & -1/L \end{bmatrix}$$.
In a practical grid tied inverter system, uncertainties are inevitable. These include parameter variations (e.g., inductance \(L\) and resistance \(R\) changing due to temperature or aging) and unmodeled dynamics. Furthermore, the grid voltage \(e\) may contain background harmonics and experience sags or swells. Therefore, a more realistic model incorporates a lumped disturbance term \(\mathbf{d}_t\):
$$\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u} + \mathbf{F}\mathbf{e} + \mathbf{d}_t$$
The primary challenge for the controller is to achieve precise current tracking despite the presence of this unknown disturbance \(\mathbf{d}_t\).
Limitations of Conventional Control and the Sliding Mode Control Approach
Proportional-Integral (PI) controllers in the dq-frame are the industry standard for current control in grid tied inverters due to their simplicity. However, their performance is highly dependent on accurate system parameters. Under parameter mismatches or disturbances, the dynamic response degrades, and steady-state error or increased harmonic distortion can occur. More advanced techniques like model predictive control (MPC) offer fast dynamics but are computationally intensive and sensitive to model accuracy.
Sliding Mode Control (SMC) is a robust nonlinear control technique renowned for its invariance to matched uncertainties once the system state is driven onto a predefined sliding surface. Its design involves two steps: 1) defining a sliding surface, and 2) formulating a control law that forces the system trajectory to reach and stay on that surface. For the current control of our grid tied inverter, we define the current tracking error as \(\mathbf{e}_c = \mathbf{i}^* – \mathbf{i} = [i_d^* – i_d, i_q^* – i_q]^T\). A linear sliding surface is chosen:
$$\mathbf{S} = \mathbf{C} \mathbf{e}_c$$
where \(\mathbf{C}\) is a positive definite diagonal matrix, often simply an identity matrix. The control objective is to make \(\mathbf{S} = \mathbf{0}\), which implies the tracking error is zero.
The control law is derived using the reachability condition. A common approach is to use the constant rate reaching law:
$$\dot{\mathbf{S}} = -\boldsymbol{\alpha}\, \text{sign}(\mathbf{S})$$
where \(\boldsymbol{\alpha}\) is a positive diagonal matrix and \(\text{sign}()\) is the signum function. Solving for the control input \(\mathbf{u}\) yields a basic SMC law:
$$\mathbf{u} = (\mathbf{C}\mathbf{B})^{-1}[\boldsymbol{\alpha}\,\text{sign}(\mathbf{S}) – \mathbf{C}\mathbf{A}\mathbf{x} – \mathbf{C}\mathbf{F}\mathbf{e} – \mathbf{C}\mathbf{d}_t]$$
The primary issue with this basic SMC is the “chattering” phenomenon—high-frequency oscillations of the control signal and system state around the sliding surface caused by the discontinuous \(\text{sign}()\) function. Chattering is highly undesirable in a grid tied inverter as it increases switching losses, excites unmodeled high-frequency dynamics, and degrades output waveform quality. A standard mitigation technique is to replace the \(\text{sign}()\) function with a smooth approximation like the saturation function \(\text{sat}(\mathbf{S}/\boldsymbol{\phi})\), where \(\boldsymbol{\phi}\) is the boundary layer thickness. While this reduces chattering, it also compromises the controller’s robustness within the boundary layer, as the invariance property is lost. Furthermore, this basic SMC still explicitly contains the disturbance term \(\mathbf{d}_t\), which is unknown. To truly enhance robustness, an adaptive mechanism to estimate and compensate for \(\mathbf{d}_t\) in real-time is required. This leads to the proposed integration with an RBF neural network.
Design of the RBF Neural Network Enhanced Sliding Mode Controller (RBF-SMC)
The proposed control strategy synergistically combines the robustness of SMC with the adaptive learning capability of an RBF neural network. The RBF network’s role is to online estimate the lumped uncertainty \(\mathbf{d}_t\), thereby allowing the SMC to operate with a smaller control gain and a thicker boundary layer, effectively suppressing chattering while maintaining strong disturbance rejection. The overall control structure for the current loop of the T-type grid tied inverter is a dual-loop system with an outer DC-link voltage loop (typically PI-controlled) providing the reference for the d-axis current \(i_d^*\) (active power), while the q-axis reference \(i_q^*\) is set to zero for unity power factor operation. The inner fast current loop employs the proposed RBF-SMC.
The RBF neural network is a three-layer feedforward network known for its simple structure, fast learning convergence, and ability to universally approximate any continuous function. For a multi-input, multi-output system like our grid tied inverter, we can design separate networks for the d and q axes, or a combined network with a vector output. The network input is chosen as the tracking error vector \(\mathbf{e}_c\). The output of the RBF network is the estimate of the disturbance, \(\hat{\mathbf{d}}_t\).
The Gaussian basis function for the \(j\)-th hidden node is:
$$h_j(\mathbf{x}) = \exp\left(-\frac{\|\mathbf{x} – \mathbf{c}_j\|^2}{2b_j^2}\right)$$
where \(\mathbf{x}\) is the input vector (e.g., \(\mathbf{e}_c\)), \(\mathbf{c}_j\) is the center vector, and \(b_j\) is the width of the \(j\)-th node. The network output is:
$$\hat{\mathbf{d}}_t = \mathbf{W}^T \mathbf{H}(\mathbf{x})$$
where \(\mathbf{H}(\mathbf{x}) = [h_1(\mathbf{x}), h_2(\mathbf{x}), …, h_m(\mathbf{x})]^T\) is the vector of radial basis functions, and \(\mathbf{W}\) is the weight matrix. The ideal weight matrix \(\mathbf{W}^*\) and an optimal RBF vector \(\mathbf{H}^*(\mathbf{x})\) exist such that the actual disturbance can be represented as:
$$\mathbf{d}_t = \mathbf{W}^{*T} \mathbf{H}^*(\mathbf{x}) + \boldsymbol{\varepsilon}$$
where \(\boldsymbol{\varepsilon}\) is a bounded approximation error, \(\|\boldsymbol{\varepsilon}\| \leq \varepsilon_N\).
We define the weight estimation error as \(\tilde{\mathbf{W}} = \mathbf{W}^* – \mathbf{W}\) and the disturbance estimation error as \(\tilde{\mathbf{d}}_t = \mathbf{d}_t – \hat{\mathbf{d}}_t\). The key is to develop an adaptive law to update the weights \(\mathbf{W}\) online to minimize this estimation error. We now reformulate the sliding mode control law using the disturbance estimate:
$$\mathbf{u} = (\mathbf{C}\mathbf{B})^{-1}[\boldsymbol{\alpha}\,\text{sat}(\mathbf{S}/\boldsymbol{\phi}) + \mathbf{K}\mathbf{S} – \mathbf{C}\mathbf{A}\mathbf{x} – \mathbf{C}\mathbf{F}\mathbf{e} – \hat{\mathbf{d}}_t]$$
Here, a proportional term \(\mathbf{K}\mathbf{S}\) (\(\mathbf{K}>0\)) is added to the reaching law to improve the convergence rate within the boundary layer. The term \(\boldsymbol{\alpha}\,\text{sat}(\mathbf{S}/\boldsymbol{\phi})\) handles the bounded approximation error \(\boldsymbol{\varepsilon}\).
The stability of the closed-loop system and the derivation of the adaptive law are rigorously analyzed using Lyapunov’s direct method. Consider the following Lyapunov function candidate:
$$V = \frac{1}{2}\mathbf{S}^T\mathbf{S} + \frac{1}{2\gamma}\text{tr}(\tilde{\mathbf{W}}^T\tilde{\mathbf{W}})$$
where \(\gamma > 0\) is the learning rate. Taking its time derivative and substituting the system dynamics and control law, we obtain:
$$\dot{V} = \mathbf{S}^T\dot{\mathbf{S}} + \frac{1}{\gamma}\text{tr}(\tilde{\mathbf{W}}^T\dot{\tilde{\mathbf{W}}}) = \mathbf{S}^T(-\boldsymbol{\alpha}\,\text{sat}(\mathbf{S}/\boldsymbol{\phi}) – \mathbf{K}\mathbf{S} – \tilde{\mathbf{d}}_t) + \frac{1}{\gamma}\text{tr}(\tilde{\mathbf{W}}^T\dot{\tilde{\mathbf{W}}})$$
Substituting \(\tilde{\mathbf{d}}_t = \tilde{\mathbf{W}}^T\mathbf{H}(\mathbf{x}) + \boldsymbol{\varepsilon}\), we get:
$$\dot{V} = -\mathbf{S}^T\mathbf{K}\mathbf{S} – \mathbf{S}^T\boldsymbol{\alpha}\,\text{sat}(\mathbf{S}/\boldsymbol{\phi}) + \mathbf{S}^T\boldsymbol{\varepsilon} + \text{tr}\left[\tilde{\mathbf{W}}^T \left( \frac{1}{\gamma}\dot{\tilde{\mathbf{W}}} – \mathbf{H}(\mathbf{x})\mathbf{S}^T \right) \right]$$
To ensure \(\dot{V} \leq 0\), we choose the weight adaptation law as:
$$\dot{\mathbf{W}} = \gamma \mathbf{H}(\mathbf{x}) \mathbf{S}^T$$
This cancels the last trace term. Furthermore, by selecting the gain \(\boldsymbol{\alpha}\) such that \(\alpha_i \geq \varepsilon_{N,i}\) for each axis, the combined effect of the saturation and error terms becomes non-positive. Thus, we have:
$$\dot{V} \leq -\mathbf{S}^T\mathbf{K}\mathbf{S} \leq 0$$
Since \(V > 0\) and \(\dot{V} \leq 0\), the Lyapunov function is non-increasing, proving that the sliding surface \(\mathbf{S}\) and the weight estimation error \(\tilde{\mathbf{W}}\) are bounded. According to Barbalat’s lemma, this implies that \(\mathbf{S} \to \mathbf{0}\) as \(t \to \infty\), guaranteeing asymptotic convergence of the current tracking error. The RBF neural network’s online adaptation allows the grid tied inverter controller to continuously learn and counteract uncertainties, leading to superior performance.
| Control Strategy | Key Advantages | Key Disadvantages | Robustness to Uncertainty |
|---|---|---|---|
| PI Control | Simple design, well-understood | Poor performance under parameter mismatch, slow response to disturbances | Low |
| Basic SMC | High robustness to matched uncertainties, fast dynamic response | Chattering phenomenon, requires known disturbance bounds | High (with chattering) |
| RBF-SMC (Proposed) | Adaptive disturbance estimation, strong robustness, minimized chattering, excellent dynamic/steady-state performance | More complex design, requires tuning of RBF parameters | Very High |
Simulation Analysis and Performance Evaluation
To validate the effectiveness of the proposed RBF-SMC strategy for the T-type three-level grid tied inverter, a detailed simulation model is built in MATLAB/Simulink. The system parameters are listed below. The performance is compared against a conventional SMC controller using a saturation function.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (phase, RMS) | \(e\) | 310 V |
| DC-Link Voltage | \(U_{dc}\) | 800 V |
| Grid Frequency | \(f\) | 50 Hz |
| Filter Inductance | \(L\) | 5 mH |
| Parasitic Resistance | \(R\) | 0.003 Ω |
| Switching Frequency | \(f_{sw}\) | 10 kHz |
1. Startup Transient Response: At time t=0s, the grid tied inverter is connected and commanded to inject a 30A (peak) sinusoidal current. The RBF-SMC controller demonstrates a significantly faster and smoother startup transient. The output current reaches its steady-state reference with negligible overshoot. In contrast, the conventional SMC exhibits a slower rise time and a noticeable current surge during startup before settling. This shows the RBF-SMC’s superior ability to manage the initial tracking error and system uncertainties right from the start.
2. Dynamic Response to Load Step Change: To test dynamic performance, the current reference is stepped down from 30A peak to 15A peak at t=0.05s. The conventional SMC-controlled grid tied inverter shows a visible transient oscillation and a slower settling time after the step. The proposed RBF-SMC controller, however, tracks the new reference almost instantaneously with a very smooth transition and no observable overshoot. The adaptive estimation of disturbances allows the controller to adjust the control effort precisely during the transient, showcasing excellent dynamic regulation.
3. Robustness Against Grid Disturbances and Harmonics: A severe test of robustness is performed by injecting 3rd and 5th order voltage harmonics (10% each) into the grid voltage model between t=0.05s and t=0.15s. This simulates a highly distorted grid condition. The conventional SMC controller struggles, resulting in a visibly distorted output current waveform from the grid tied inverter, with increased ripple and deviation from the sinusoidal shape. The RBF-SMC controller, thanks to its online disturbance estimation, successfully compensates for the grid harmonic disturbance. The output current remains largely sinusoidal and tightly locked to its reference. A Fourier analysis (FFT) of the steady-state current under this condition quantitatively confirms the superiority.
| Control Method | Fundamental Current (50 Hz) Magnitude [A] | Total Harmonic Distortion (THD) |
|---|---|---|
| Conventional SMC | 30.59 | 4.89% |
| Proposed RBF-SMC | 29.17 | 1.54% |
The THD results are decisive. The RBF-SMC controller reduces the current THD by approximately 68% compared to the conventional SMC under the same adverse grid conditions. This meets stringent grid codes (often requiring THD < 5%, with many standards recommending < 3%), demonstrating that the proposed controller can ensure high-quality power injection even in non-ideal grid environments.
Conclusion
This article has presented a comprehensive study on the application of an advanced RBF neural network-enhanced sliding mode control strategy for T-type three-level grid tied inverters. The inherent benefits of the T-type topology, such as reduced switching losses and improved output waveform quality, are fully leveraged by the sophisticated control algorithm. The proposed RBF-SMC effectively addresses the key challenges in grid tied inverter control: chattering mitigation in SMC and robust adaptation to unknown internal and external disturbances.
The mathematical foundation, including the dq-frame modeling and the integrated controller design with Lyapunov-based stability proof, ensures a theoretically sound approach. The RBF network’s role in online disturbance estimation is pivotal, allowing the sliding mode controller to operate with reduced aggressiveness, thereby minimizing chattering while maintaining and even enhancing its robustness. Simulation studies under various operational scenarios—startup, load transient, and harmonic grid distortion—conclusively demonstrate the superiority of the RBF-SMC over conventional SMC. The proposed controller delivers faster dynamic response, negligible overshoot, and most importantly, maintains exceptionally low output current THD under disturbed grid conditions, ensuring compliance with power quality standards.
Future work may involve the experimental validation of this control strategy on a hardware prototype, extension to inverters with LCL filters (requiring additional state variables and damping), and the exploration of other intelligent adaptive techniques for further performance optimization. The RBF-SMC strategy represents a significant step forward in the control of modern grid tied inverters, contributing to the stable and high-quality integration of renewable energy into the power grid.