In my research on robust control of grid-connected multilevel inverters, I encountered persistent challenges such as large output voltage fluctuations and high harmonic distortion. To overcome these issues, I proposed a novel approach based on harmonic elimination SHEPWM (Selective Harmonic Elimination Pulse Width Modulation). This method significantly improves the robustness of the inverter system. Among the various types of solar inverters, multilevel inverters are widely adopted due to their low harmonic content and reduced switching stress. However, when they are connected to the grid, uncertainties such as grid impedance variations and load disturbances degrade performance. My work focuses on developing a robust control framework that leverages the SHEPWM mathematical model to eliminate specific harmonics and enhance system stability.
I first established the mathematical model for the SHEPWM method from both unipolar and bipolar perspectives. The output voltage waveform of the grid-connected multilevel inverter can be expressed using Fourier series expansion:
$$
u(t) = \sum_{i=1}^{\infty} \left[ \alpha_i \cos(i\omega t) + \beta_i \sin(i\omega t) \right]
$$
where $\alpha_i$ and $\beta_i$ are Fourier coefficients. For a unipolar waveform shown in Figure 1 (which I describe here without an image reference), the waveform exhibits odd symmetry about $\pi/2$ and point symmetry about $(\pi,0)$. This implies that the Fourier coefficients simplify to:
$$
\alpha_i = \frac{U}{i\pi} \sum_{n=1}^{N} \sin(i\phi_n)
$$
$$
\beta_i = 0, \quad i=1,2,3,\dots
$$
Here, $U$ is the voltage amplitude, $N$ is the carrier ratio, and $\phi_n$ are the switching angles for the $n$-th harmonic. To eliminate low-order harmonics, I set the amplitudes of the selected harmonics to zero, leading to the unipolar SHEPWM model:
$$
\phi_n = \frac{U}{i\pi} \sum_{n=3,5,7,\dots}^{N} \sin(i\alpha_i) = q
$$
where $q$ represents the fundamental amplitude. The switching angles must satisfy:
$$
0 \le \phi_1 \le \phi_2 \le \dots \le \phi_n \le \frac{\pi}{2}
$$
For three-phase systems, triplen harmonics (multiples of 3) cancel in line voltages, so the model reduces to:
$$
\phi_n = q, \quad n=3,5,7,\dots
$$
Similarly, I derived the bipolar SHEPWM model:
$$
\phi_n = q, \quad n=5,7,11,\dots
$$
The switching angles are determined by solving these nonlinear algebraic equations using Wu’s method. Wu’s method transforms the polynomial set $GS$ into a characteristic set $VS$. The zero set relation is:
$$
\mathrm{Heros}(GS) = \mathrm{Heros}(VS / \phi_n) + \mathrm{Heros}(GS, \epsilon)
$$
Since $\mathrm{Heros}(GS, \epsilon)$ is empty, I only need to solve $\mathrm{Heros}(VS / \phi_n)$. Using Mathematica, I obtained the optimal switching angles $\phi_n$ that eliminate specific harmonics, thereby improving inverter robustness.
Experimental Setup and Results
To validate my method, I built a simulation model in MATLAB/Simulink with parameters listed in Table 1. The inverter output voltage was set to 220 V, grid voltage to 380 V, and a 20 kΩ resistor was used as a load. A TMS320F28335 chip acted as the robust controller, and grid inductance was added to simulate real grid conditions.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| Inverter output voltage | 220 V | Fundamental angular frequency | 314 rad/s |
| Sampling frequency | 1000 Hz | Damping coefficient | 0.411 |
| Resonant frequency | 50 Hz | Virtual reactance | 0.1 Ω |
| Grid voltage | 380 V | Virtual resistance | 20 kΩ |
I compared my method with two mainstream approaches from literature [1] and [2]. The output voltage waveforms captured by an oscilloscope (RFHARF-A4F4) are described here. Note that the actual waveforms are not shown in this text; instead, I insert the relevant image link for illustration:
Under my proposed method, the grid-connected multilevel inverter output voltage exhibited very small ripple and high stability. In contrast, the methods from literature [1] and [2] produced larger voltage fluctuations, indicating inferior robustness.
Harmonic Distortion Analysis
I further evaluated the total harmonic distortion (THD) for each method. The results are summarized in Table 2 and Figure 5 (described here without an image reference). My method achieved a THD below 1%, while the other two methods had THD values exceeding 2%. This demonstrates that the SHEPWM-based robust control effectively suppresses harmonics and improves system stability.
| Control Method | THD (%) |
|---|---|
| Proposed SHEPWM method | 0.85 |
| Literature [1] method | 2.31 |
| Literature [2] method | 2.15 |
The mathematical relationship between THD and harmonic amplitudes can be expressed as:
$$
\text{THD} = \frac{\sqrt{\sum_{n=2}^{\infty} U_n^2}}{U_1} \times 100\%
$$
where $U_n$ is the RMS value of the $n$-th harmonic. My method minimizes $U_n$ for the targeted low-order harmonics, resulting in a very low THD.
Conclusion
In this work, I successfully developed a robust control strategy for grid-connected multilevel inverters using harmonic elimination SHEPWM. The mathematical model was derived for both unipolar and bipolar cases, and Wu’s method efficiently solved the switching angles. Experimental results confirm that the output voltage waveform is smooth with THD under 1%, significantly outperforming conventional methods. This approach is highly effective for various types of solar inverters used in renewable energy systems, enhancing their resilience to grid uncertainties and load disturbances. The proposed methodology provides a practical solution for improving power quality and system robustness in grid-connected applications.