In recent years, the increasing penetration of renewable energy sources has led to a rise in power electronic devices, which, while enhancing grid controllability and flexibility, have introduced new power quality challenges such as supraharmonics. As a key interface for new energy integration, the grid-connected inverter is a typical source of supraharmonics, and understanding its generation and propagation mechanisms is crucial for ensuring grid stability. In this paper, I delve into the fundamental aspects of supraharmonic emissions from grid-connected inverters, focusing on the sinusoidal pulse width modulation (SPWM) technique, and explore how various factors influence these emissions. I also develop models to analyze the propagation characteristics in multi-inverter systems, providing insights that can aid in mitigating supraharmonic-related issues.
Supraharmonics, typically defined in the frequency range of 2 to 150 kHz, can cause adverse effects such as communication interference, maloperation of protective devices, and excessive heating in power electronic switches. The grid-connected inverter, being ubiquitous in solar and wind power systems, is a primary contributor to these emissions. My analysis begins with the generation mechanism under SPWM control, where I derive mathematical expressions to characterize the supraharmonic voltage sources inherent in inverter outputs. I then examine the impact of parameters like DC-link voltage and modulation index on these sources. Following that, I establish circuit models for supraharmonic propagation, particularly in parallel configurations of multiple grid-connected inverters, and derive expressions for emission currents. Finally, I validate my theoretical findings through simulations, demonstrating the practical implications of my research.
The topology of a three-phase LCL-type grid-connected inverter is commonly used in renewable energy systems. It consists of an inverter bridge, an LCL filter for harmonic attenuation, and a grid connection. Under SPWM control, the inverter generates switching harmonics that extend into the supraharmonic range. The output phase voltage from the inverter bridge can be expressed using double Fourier series analysis. For a phase-a voltage, the expression is:
$$ u_{aN}(t) = \frac{M_r U_{dc}}{2} \sin(\omega_o t – \phi_o) + \frac{2U_{dc}}{\pi} \left( \sum_{a=1,3,\ldots}^{\infty} \sum_{b=0, \pm2, \pm4,\ldots}^{\infty} \frac{4J_b(a M_r \pi / 2)}{3a} \sin(a\pi/2) \sin^2(b\pi/3) \cos(a(\omega_{sw}t – \phi_{sw}) + b(\omega_o t – \phi_o)) + \sum_{a=2,4,\ldots}^{\infty} \sum_{b=\pm1, \pm3,\ldots}^{\infty} \frac{4J_b(a M_r \pi / 2)}{3a} \cos(a\pi/2) \sin^2(b\pi/3) \sin(a(\omega_{sw}t – \phi_{sw}) + b(\omega_o t – \phi_o)) \right) $$
where \( M_r \) is the modulation index, \( U_{dc} \) is the DC-link voltage, \( \omega_o \) is the fundamental angular frequency, \( \phi_o \) is the initial phase of the modulation wave, \( \omega_{sw} \) is the switching angular frequency, \( \phi_{sw} \) is the initial phase of the carrier wave, and \( J_b(x) \) is the Bessel function of the first kind. This equation reveals that supraharmonics primarily appear at integer multiples of the switching frequency, i.e., at frequencies \( a f_{sw} \pm b f_o \), where \( f_{sw} \) is the switching frequency and \( f_o \) is the fundamental frequency. The grid-connected inverter thus acts as a voltage source for these supraharmonics, with their amplitudes influenced by \( U_{dc} \) and \( M_r \).
To quantify the impact of DC-link voltage, I consider typical ranges for grid-connected inverters in photovoltaic systems. The modulation index is related to the DC-link voltage and the output fundamental voltage by:
$$ M_r = \frac{2\sqrt{2} u_{aN0}}{U_{dc}} $$
where \( u_{aN0} \) is the RMS value of the fundamental phase voltage. Variations in \( U_{dc} \) affect \( M_r \), and both parameters jointly determine supraharmonic amplitudes. For instance, at the first sideband around the switching frequency, the harmonic voltage amplitude \( V_{1-1} \) is given by:
$$ V_{1-1} = \frac{2U_{dc}}{\pi} J_2(M_r \pi / 2) $$
and at the second sideband, \( V_{2-1} \) is:
$$ V_{2-1} = \frac{U_{dc}}{\pi} J_1(M_r \pi) $$
Using Bessel function plots, I analyze how changes in \( U_{dc} \) and \( M_r \) influence these amplitudes. For a given output voltage, as \( U_{dc} \) increases, \( M_r \) decreases. In the range of 622 V to 1000 V, the second-order Bessel function \( J_2(x) \) increases with \( x \), while the first-order Bessel function \( J_1(x) \) decreases. Consequently, \( V_{1-1} \) tends to decrease with increasing \( U_{dc} \), whereas \( V_{2-1} \) increases. This highlights the nonlinear relationship between DC-link voltage and supraharmonic emissions in grid-connected inverters.
The capacity of the grid-connected inverter also plays a significant role. The output power affects the fundamental voltage drop across the filter and grid impedances, thereby altering \( M_r \). From the phasor diagram of the inverter, the fundamental output voltage can be expressed as:
$$ u_{aN0} = I_o j\omega_o (L_1 + L_2 + L_g) + I_o R_g + V_{aN} $$
where \( I_o \) is the grid current, \( L_1 \) and \( L_2 \) are filter inductors, \( L_g \) and \( R_g \) are grid impedance components, and \( V_{aN} \) is the grid phase voltage. Using the cosine law, \( u_{aN0} \) can be related to active power \( P \) and reactive power \( Q \):
$$ u_{aN0}^2 = \frac{P^2 + Q^2}{9V_{aN}^2} (Z^2 + R_g \cos \phi + Z R_g \sin 2\phi) + \frac{2}{3}(P R_g + Q Z) + V_{aN}^2 $$
where \( Z = \omega_o (L_1 + L_2 + L_g) \) and \( \phi \) is the power factor angle. Substituting into the modulation index formula shows that \( M_r \) increases with higher output power, especially reactive power, leading to changes in supraharmonic amplitudes. This underscores the importance of inverter operating conditions in supraharmonic generation.
Moving to propagation mechanisms, I consider multiple grid-connected inverters operating in parallel. Each inverter is modeled as a supraharmonic voltage source \( V_{SH} \) with an LCL filter. Using two-port network theory, the filter can be represented by a transmission matrix \( T_{LCL} \):
$$ T_{LCL} = \begin{bmatrix} A_{LCL} & B_{LCL} \\ C_{LCL} & D_{LCL} \end{bmatrix} = \begin{bmatrix} 1 – \omega^2 L_1 C & j\omega (L_1 + L_2) – j\omega^3 L_1 L_2 C \\ j\omega C & 1 – \omega^2 L_2 C \end{bmatrix} $$
For two inverters in parallel, the primary emission current \( I_{pri-1} \) from inverter 1 and the secondary emission current \( I_{sec-1} \) due to inverter 2 can be derived. The total emission current from inverter 1 is:
$$ I_{em-1} = I_{pri-1} – I_{sec-1} = \frac{V_{SH1}}{W_1} \left( \frac{1}{Z_{GRID}} + \frac{1}{Z_{LCL2}} \right) – \frac{V_{SH2}}{W_2} \times \frac{1}{Z_{LCL1}} $$
where \( Z_{GRID} = R_g + j\omega L_g \), \( Z_{LCL1} = B_{LCL1} / A_{LCL1} \), and \( W_1 = A_{LCL1} + B_{LCL1} \left( \frac{1}{Z_{GRID}} + \frac{1}{Z_{LCL2}} \right) \). The current injected into the grid is:
$$ I_{grid} = \left( \frac{V_{SH1}}{W_1} + \frac{V_{SH2}}{W_2} \right) \frac{1}{Z_{GRID}} $$
These expressions show that emission strengths depend on inverter parameters and grid impedance. For instance, the primary emission strength \( K_{pri-1} \) is proportional to \( 1/W_1 \left( 1/Z_{GRID} + 1/Z_{LCL2} \right) \), and the secondary emission strength \( K_{sec-1} \) is proportional to \( 1/(W_2 Z_{LCL1}) \). In grid-connected inverters, larger capacity units tend to have smaller filter impedance magnitudes, increasing both primary and secondary emission strengths.
To illustrate the impact of inverter capacity, I analyze filter parameters designed based on grid current. For an LCL filter, the inverter-side inductance \( L_1 \) is limited by:
$$ L_1 \leq \frac{V_g}{\omega_o I_o} \times 5\% $$
the capacitor \( C \) by:
$$ C \leq \frac{P_o}{\omega_o V_g^2} \times 5\% $$
and the grid-side inductance \( L_2 \) by:
$$ L_2 \geq \frac{1}{L_1 C \omega_h^2 – 1} \left( L_1 + \frac{u_{aN}(j\omega_h)}{0.3\% \omega_h I_o} \right) $$
where \( \omega_h \) is the angular frequency at maximum harmonic content. As grid current increases, \( L_1 \) and \( L_2 \) decrease, and \( C \) increases, leading to lower filter impedance magnitudes. This enhances emission strengths for higher-capacity grid-connected inverters.
Extending to n inverters in parallel, the total emission current for inverter 1 becomes:
$$ I_{em-1} = \frac{V_{SH1}}{W_1} \left( \frac{1}{Z_{GRID}} + \sum_{i=2}^n \frac{1}{Z_{LCLi}} \right) – \sum_{i=2}^n \frac{V_{SHi}}{W_i} \times \frac{1}{Z_{LCL1}} $$
and the grid current is:
$$ I_{grid} = \sum_{i=1}^n \frac{V_{SHi}}{W_i} \times \frac{1}{Z_{GRID}} $$
where \( W_i = A_{LCLi} + B_{LCLi} \left( \frac{1}{Z_{GRID}} + \sum_{j \neq i} \frac{1}{Z_{LCLj}} \right) \). For identical grid-connected inverters, these simplify to:
$$ I_{em-1} = \frac{V_{SH1}}{A_{LCL1} Z_{GRID} + B_{LCL1} [1 + (n-1) Z_{GRID} / Z_{LCL1}]} $$
and
$$ I_{grid} = \frac{n V_{SH1}}{n A_{LCL1} Z_{GRID} + B_{LCL1}} $$
This indicates that as the number of inverters increases, the emission current per grid-connected inverter decreases, while the total grid current increases, albeit at a diminishing rate. Higher grid impedance suppresses grid current injection.
To validate my analysis, I conduct simulations in Simulink for a three-phase LCL-type grid-connected inverter. Parameters are set as follows: grid voltage \( V_g = 220 \, \text{V} \) (phase), grid frequency \( f_o = 50 \, \text{Hz} \), switching frequency \( f_{sw} = 10 \, \text{kHz} \), filter inductors \( L_1 = 1 \, \text{mH} \), \( L_2 = 0.4 \, \text{mH} \), capacitor \( C = 15 \, \mu\text{F} \), grid resistance \( R_g = 0.02 \, \Omega \), grid inductance \( L_g = 1 \, \text{mH} \), and grid current \( I_o = 30 \, \text{A} \). I vary the DC-link voltage from 700 V to 1000 V and analyze the inverter bridge voltage spectrum using FFT. The results are summarized in Table 1.
| DC-Link Voltage (V) | Fundamental Voltage (V) | THD (%) | Voltage at 9.9 kHz (V) | Voltage at 19.95 kHz (V) |
|---|---|---|---|---|
| 700 | 311.6 | 59.14 | 91.62 | 91.68 |
| 800 | 311.6 | 70.28 | 83.13 | 130.24 |
| 900 | 311.5 | 80.76 | 75.85 | 160.69 |
| 1000 | 311.7 | 89.60 | 69.66 | 184.17 |
As predicted, the harmonic voltage at 9.9 kHz (near the switching frequency) decreases with increasing \( U_{dc} \), while at 19.95 kHz (near twice the switching frequency) it increases. This aligns with the Bessel function analysis, confirming the influence of DC-link voltage on supraharmonics in grid-connected inverters.
Next, I examine the effect of inverter power by varying active and reactive power outputs while keeping \( U_{dc} = 800 \, \text{V} \). Results are shown in Table 2.
| Power (kVA) | Fundamental Voltage (V) | THD (%) | Voltage at 9.9 kHz (V) | Voltage at 19.95 kHz (V) |
|---|---|---|---|---|
| 19.8 + j0 | 311.6 | 59.14 | 83.13 | 130.24 |
| 19.8 + j4.6 | 315.9 | 69.07 | 85.33 | 127.70 |
| 39.6 + j0 | 312.2 | 70.20 | 85.38 | 130.10 |
| 39.6 + j13.2 | 318.4 | 68.42 | 86.46 | 127.04 |
Reactive power injection increases the fundamental voltage and modulation index, leading to higher supraharmonic voltages at the switching frequency sideband but lower at twice the switching frequency. This verifies that power output, especially reactive power, significantly affects supraharmonic emissions from grid-connected inverters.
For propagation studies, I simulate multiple grid-connected inverters in parallel. First, I consider two identical inverters with different carrier phase shifts. The grid current \( I_{grid} \) at supraharmonic frequencies varies with phase difference, as shown in Figure 1. When carriers are in phase, supraharmonic currents add constructively, increasing grid injection. This emphasizes the need for phase diversity in multi-inverter systems to mitigate supraharmonic buildup.
Then, I simulate four inverters with two capacity levels: 40 A and 10 A grid currents. Filter parameters are designed accordingly: for 10 A, \( L_1 = 3.5 \, \text{mH} \), \( L_2 = 1.17 \, \text{mH} \), \( C = 7.2 \, \mu\text{F} \); for 40 A, \( L_1 = 0.87 \, \text{mH} \), \( L_2 = 0.3 \, \text{mH} \), \( C = 30 \, \mu\text{F} \). Three scenarios are tested: one 40 A with three 10 A inverters, two 40 A with two 10 A, and three 40 A with one 10 A. The supraharmonic currents at switching frequency sidebands are measured. In all cases, larger capacity grid-connected inverters exhibit higher emission currents, and as more high-capacity units are added, the grid current increases. This supports my theoretical finding that inverter capacity directly influences supraharmonic propagation strength.
Finally, I vary the number of identical grid-connected inverters from 1 to 5, each with \( I_o = 30 \, \text{A} \). The emission current per inverter and the total grid current are plotted against the number of units. As derived, the per-inverter emission decreases with more units, while the grid current increases but at a reducing rate. This has practical implications: in dense installations of grid-connected inverters, supraharmonic currents can accumulate, potentially exceeding limits if not managed.
In conclusion, my analysis provides a comprehensive understanding of supraharmonic generation and propagation in grid-connected inverters. Key findings include: the supraharmonic voltage source nature of inverters under SPWM control, the significant effects of DC-link voltage and modulation index on emission amplitudes, and the role of inverter capacity in propagation characteristics. For multiple grid-connected inverters, larger capacity units emit more supraharmonics, and increasing the number of units raises grid injection while reducing per-unit emissions. These insights can guide the design and operation of grid-connected inverter systems to suppress supraharmonics, such as by optimizing DC-link voltage, controlling power factor, and implementing phase-shifted carriers. Future work could explore non-ideal conditions, such as grid background harmonics and inverter control interactions, to further refine models. Overall, addressing supraharmonic issues is essential for the reliable integration of renewable energy sources, and grid-connected inverters remain a focal point in this endeavor.
Throughout this paper, I have emphasized the importance of grid-connected inverters in modern power systems. Their supraharmonic emissions are a critical aspect of power quality, and my research aims to contribute to safer and more efficient grid operations. By combining theoretical derivations with simulation validations, I hope this work aids engineers and researchers in developing effective mitigation strategies for supraharmonic disturbances in grid-connected inverter applications.