The global transition towards sustainable energy has placed photovoltaic (PV) systems at the forefront of power generation technologies. The solar inverter, acting as the critical interface between the PV panels and the utility grid, is responsible for converting direct current into grid-compliant alternating current. Its performance directly dictates the efficiency, reliability, and power quality of the entire PV installation. For high-power applications exceeding 1 MVA, minimizing switching losses is paramount for system efficiency and thermal management. This necessitates operating the solar inverter at very low switching frequencies, typically below 1 kHz.
However, low-frequency Pulse Width Modulation (PWM) inherently generates significant low-order harmonic currents. The conventional Selective Harmonic Elimination (SHE) modulation technique has been widely adopted for large-power converters to directly eliminate specific low-order harmonics from the inverter’s output waveform. While effective under ideal grid conditions, traditional SHE exhibits a critical limitation: it cannot mitigate harmonic distortion present in the grid voltage itself. When a solar inverter is connected to a weak or polluted grid, these background voltage harmonics interact with the inverter’s output filter, often an LC circuit. This interaction can lead to resonance amplification, severely degrading the quality of the injected grid current and potentially violating stringent grid codes.
This article addresses this fundamental challenge. We propose a modified SHE control strategy specifically designed for parallel current-source solar inverter systems operating at extremely low switching frequencies. The core innovation lies in transforming the inverter’s role from a passive harmonic eliminator to an active harmonic compensator. By strategically relaxing the waveform constraints of traditional SHE, we enable the inverter bridges to generate specific, controllable harmonic currents. These injected currents are designed to actively counteract and cancel the harmonic currents induced by the distorted grid voltage at the Point of Common Coupling (PCC), thereby significantly improving the overall grid current quality.
System Modeling and Harmonic Analysis
The foundation of the proposed control strategy is a clear understanding of the system’s harmonic interactions. We consider a topology employing two three-phase full-bridge current source inverters (CSIs) connected in parallel to a common LC filter and the grid. This architecture is common for scaling up power in solar inverter applications.
Let \( L_{d1} \) and \( L_{d2} \) represent the DC-link inductors for inverter bridge #1 and #2, respectively. The AC-side filter consists of an inductance \( L_s \) and a capacitance \( C_s \), with \( R_s \) representing the line resistance and \( R_c \) a damping resistor typically placed in series with the capacitor. The key currents are: \( i_{w1} \) and \( i_{w2} \), the PWM output currents from the two inverter bridges; \( i_w = i_{w1} + i_{w2} \), the total inverter output current; \( i_s \), the final grid-injected current; and \( i_c \), the filter capacitor current. The grid voltage is denoted as \( u_s \).
The harmonic performance at the PCC is governed by the superposition of two sources: the harmonic content of the grid voltage \( u_{sn} \) and the harmonic content of the total inverter PWM current \( i_{wn} \). We can analyze the \( n^{th} \) harmonic component using the equivalent circuit model. The transfer function from the grid voltage harmonic to the grid current harmonic \( i_{sn1} \) is given by:
$$ F_1(s) = \frac{i_{sn1}}{u_{sn}} = \frac{1}{\frac{1}{s C_s} + R_c + s L_s + R_s} \bigg|_{s=j n \omega} $$
The transfer function from the inverter output current harmonic to the resulting grid current harmonic \( i_{sn2} \) is:
$$ F_2(s) = \frac{i_{sn2}}{i_{wn}} = \frac{\frac{1}{s C_s} + R_c}{\frac{1}{s C_s} + R_c + s L_s + R_s} \bigg|_{s=j n \omega} $$
Applying superposition, the total \( n^{th} \) harmonic grid current is:
$$ i_{sn} = i_{sn2} – i_{sn1} = i_{wn} \cdot F_2(j n \omega) – u_{sn} \cdot F_1(j n \omega) $$
A Bode plot analysis of \( F_1 \) and \( F_2 \) for typical solar inverter parameters reveals a critical insight. The LC filter has a resonant frequency, often around 300 Hz for a 50 Hz system. At this resonance and nearby frequencies (notably the 5th (250 Hz) and 7th (350 Hz) harmonics), the gain of \( F_1 \) can be significantly high. This means low-order grid voltage harmonics are amplified by the filter, leading to substantial distortion in \( i_s \). Equation (3) is the cornerstone of our approach. It shows that if we can control \( i_{wn} \)—specifically, its amplitude and phase for a particular harmonic order \( n \)—we can produce a compensating current \( i_{sn2} \) that cancels out the undesirable current \( i_{sn1} \) caused by \( u_{sn} \). The goal is to achieve \( i_{sn} = 0 \).
Limitations of Traditional SHE and the Proposed Modification
Traditional SHE-PWM for current source solar inverters is defined by strict waveform symmetries to simplify the solving of the transcendental equations and to ensure device safe-commutation. A typical 7-pulse per quarter-wave pattern is characterized by three independent switching angles \( \alpha_1, \alpha_2, \alpha_3 \) within the interval \( (0, \pi/6) \). The constraints are:
- Half-wave symmetry: \( f(\omega t) = -f(\omega t + \pi) \).
- Quarter-wave symmetry: \( f(\omega t) = f(\pi – \omega t) \).
- Additional mirror symmetry at \( \pi/6 \) and \( 5\pi/6 \).
- No modulation in the middle \( \pi/3 \) interval of each half-cycle.
These constraints lead to the relationships: \( \alpha_i = \pi/3 – \alpha_{7-i} \) for \( i=4,5,6 \) and the switching instants \( \theta_i = 2\pi/3 + \alpha_i \). The Fourier series expansion of the normalized switching function \( H_m(\omega t) \) is:
$$ H_m(\omega t) = \sum_{n=1,5,7,11…}^{\infty} \left[ \frac{4}{n\pi} \sin\left(\frac{n\pi}{3}\right) \sum_{i=1}^{3} (-1)^{i+1} \cos(n \alpha_i) \right] \sin(n \omega t) $$
The critical observation is that due to quarter-wave symmetry, all cosine terms (\( b_n \) coefficients) vanish, resulting in harmonics that are purely in phase or out of phase with a fixed reference. The traditional SHE solver only controls the amplitude of selected harmonics (typically setting \( A_5 = 0, A_7 = 0 \)), with no ability to control their phase. Therefore, it cannot generate the specific harmonic current \( i_{wn} \) with the required phase to satisfy the compensation condition in Equation (3).
Our proposed modification strategically relaxes the constraints. We remove the requirement for quarter-wave symmetry (Constraint 2 and its derived mirror symmetry). The new 7-pulse waveform, while maintaining half-wave symmetry for safe operation, now has seven independent angles \( \alpha_1, \alpha_2, …, \alpha_7 \) within \( (0, \pi/3) \). The switching instants are still defined as \( \theta_i = 2\pi/3 + \alpha_i \). The Fourier coefficients for this modified waveform become:
$$ a_n = \frac{4}{n\pi} \sin\left(\frac{n\pi}{3}\right) \left\{ \sum_{i=1}^{7} (-1)^{i+1} \sin\left[n\left(\alpha_i + \frac{\pi}{3}\right)\right] \right\} $$
$$ b_n = \frac{4}{n\pi} \sin\left(\frac{n\pi}{3}\right) \left\{ \sum_{i=1}^{7} (-1)^{i+1} \cos\left[n\left(\alpha_i + \frac{\pi}{3}\right)\right] \right\} $$
for \( n = 1, 5, 7, 11, 13, … \). The amplitude \( A_n \) and phase \( \phi_n \) of the \( n^{th} \) harmonic in the normalized PWM waveform are:
$$ A_n = \sqrt{a_n^2 + b_n^2}, \quad \phi_n = \tan^{-1}(b_n / a_n) $$
This is the key enabler. By solving for the seven angles \( \alpha_1 … \alpha_7 \), we can now simultaneously control both the amplitude \( A_n \) and the phase \( \phi_n \) of selected harmonic components in the output of the solar inverter.
Control Strategy for Parallel Solar Inverter System
The relationship between the actual inverter bridge output current harmonic \( i_{wn} \) and the normalized harmonic component \( H_{mn} \) (with amplitude \( A_n \) and phase \( \phi_n \)) is:
$$ i_{wn} = I_{dc} \cdot H_{mn} = I_{dc} \cdot A_n \angle \phi_n $$
where \( I_{dc} \) is the DC-link current. Substituting into the compensation condition \( i_{sn}=0 \) from Equation (3), we derive the required reference for the normalized harmonic output of the solar inverter:
$$ H_{mn\_ref} = \frac{u_{sn}}{I_{dc} \cdot Z_c}, \quad \text{where } Z_c = \frac{1}{j n \omega C_s} + R_c $$
Therefore, the target amplitude \( A_{n\_ref} \) and phase \( \phi_{n\_ref} \) for the PWM modulation waveform are:
$$ A_{n\_ref} = \frac{|u_{sn}|}{I_{dc} \cdot |Z_c|} = \frac{|u_{sn}|}{I_{dc} \cdot \sqrt{R_c^2 + (1/(n \omega C_s))^2}} $$
$$ \phi_{n\_ref} = \angle u_{sn} – \angle Z_c = \phi_{u_{sn}} – \tan^{-1}\left( \frac{-1}{n \omega C_s R_c} \right) $$
The proposed architecture utilizes the two parallel solar inverter bridges for collaborative harmonic compensation. A practical and effective scheme is to assign each bridge the responsibility for compensating one dominant low-order harmonic. Typically, the 5th and 7th harmonics are the most prevalent in grid voltages. Thus, we configure:
Solar Inverter Bridge #1: Control its output to have a controllable 5th harmonic (\( A_{1\_5} = M_c, \phi_{1\_5} = \phi_c \)) while eliminating its native 7th harmonic (\( A_{1\_7} = 0 \)).
Solar Inverter Bridge #2: Control its output to have a controllable 7th harmonic (\( A_{2\_7} = M_c, \phi_{2\_7} = \phi_c \)) while eliminating its native 5th harmonic (\( A_{2\_5} = 0 \)).
For each bridge, we must also maintain an acceptable level of higher-order harmonics (e.g., 11th and 13th). This leads to the formulation of two distinct nonlinear equation systems to be solved for the switching angles \( \{\alpha_1, …, \alpha_7\} \).
For Bridge #1 (5th Harmonic Control):
$$ \begin{cases}
A_{1\_5}(\alpha_1, …, \alpha_7) = M_c \\
\phi_{1\_5}(\alpha_1, …, \alpha_7) = \phi_c \\
A_{1\_7}(\alpha_1, …, \alpha_7) = 0 \\
A_{1\_11} + A_{1\_13} \leq \text{TOL}
\end{cases} $$
For Bridge #2 (7th Harmonic Control):
$$ \begin{cases}
A_{2\_7}(\alpha_1, …, \alpha_7) = M_c \\
\phi_{2\_7}(\alpha_1, …, \alpha_7) = \phi_c \\
A_{2\_5}(\alpha_1, …, \alpha_7) = 0 \\
A_{2\_11} + A_{2\_13} \leq \text{TOL}
\end{cases} $$
Here, \( M_c \) is the commanded normalized harmonic amplitude (derived from Eq. (8)), \( \phi_c \) is the commanded phase (derived from Eq. (9)), and TOL is a small tolerance value. Given the complexity of solving these transcendental equations in real-time, an offline pre-calculation approach is adopted. Global search algorithms are used to pre-compute solutions for a wide range of \( (M_c, \phi_c) \) pairs, storing the results in a lookup table for each solar inverter bridge. The real-time controller measures \( u_{sn} \) (amplitude and phase), calculates \( (A_{n\_ref}, \phi_{n\_ref}) \), and fetches the corresponding set of switching angles from the lookup table.
Simulation Verification and Performance Analysis
To validate the proposed modified SHE strategy, a detailed simulation model of the parallel current-source solar inverter system was developed. The key parameters are summarized in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Grid Voltage (Line-to-Line) | \( U_s \) | 380 V |
| DC Link Current (per bridge) | \( I_{dc1}, I_{dc2} \) | 20 A |
| Fundamental Frequency | \( f \) | 50 Hz |
| Switching Frequency (Equivalent) | \( f_{sw} \) | 350 Hz |
| Filter Inductance | \( L_s \) | 4 mH |
| Filter Capacitance | \( C_s \) | 70 µF |
| Damping Resistor | \( R_c \) | 0.5 Ω |
| Line Resistor | \( R_s \) | 0.01 Ω |
A Sliding Discrete Fourier Transform (SDFT) is implemented for fast and accurate detection of the amplitude and phase of the 5th and 7th grid voltage harmonics in real-time.
Steady-State Performance under Distorted Grid
The grid voltage was deliberately distorted with 2% 5th harmonic (at 30°) and 2% 7th harmonic (at 40°). First, the conventional SHE method was applied, which only aims to eliminate these harmonics from the inverter’s own output. The results, shown in the table below, confirm its limitation.
| Method & Current | THD | 5th Harmonic (%) | 7th Harmonic (%) | Key Observation |
|---|---|---|---|---|
| Traditional SHE (Grid Current \( i_s \)) |
10.09% | 5.20% | 7.03% | Severe distortion due to amplified grid voltage harmonics. |
| Traditional SHE (Inverter Current \( i_w \)) |
– | 0.52% | 0.56% | Inverter output is clean, but grid current is polluted. |
| Modified SHE (Grid Current \( i_s \)) |
2.63% | 0.73% | 1.42% | Dramatic improvement. Active compensation is effective. |
| Modified SHE (Inverter Current \( i_w \)) |
– | Controlled | Controlled | Inverters deliberately inject compensating 5th & 7th harmonics. |
The modified SHE strategy demonstrates remarkable efficacy. The solar inverter bridges successfully generate the precise 5th and 7th harmonic currents required to cancel the effects of the grid voltage distortion. The grid current THD is reduced from over 10% to below 3%, and the individual 5th and 7th harmonic components are suppressed to very low levels.
Dynamic Response to Harmonic Transient
A critical test for any grid-connected controller is its dynamic response. A step change in grid distortion was simulated at t = 0.2s, introducing the same 2% 5th and 7th harmonics. The modified SHE controller’s performance was monitored. The system demonstrates a stable and relatively fast transient response. The grid current harmonics converge to their new steady-state compensated values within approximately two fundamental cycles (40 ms). The primary sources of this transient period are the inherent delay of the SDFT harmonic detection algorithm and the natural response time of the LC filter network. This performance is considered fully adequate for compensating slowly varying background grid harmonics, which are typical in power systems.
| Feature | Performance | Comment |
|---|---|---|
| Response Time | ~2 cycles (40 ms) | Governed by detection delay and filter dynamics. |
| Stability | Stable, no oscillation | The system smoothly transitions to the new operating point. |
| Practicality | Excellent | Suitable for real-world grid conditions with slowly varying harmonic distortion. |
Conclusion
This article has presented a novel and effective modified Selective Harmonic Elimination strategy tailored for parallel current-source solar inverter systems operating at very low switching frequencies. By analytically modeling the harmonic interaction at the PCC, we identified the inability of traditional SHE to compensate for grid voltage distortion as its fundamental shortcoming. The proposed solution involves a strategic relaxation of the PWM waveform constraints, enabling independent control over both the amplitude and phase of specific harmonic components in the solar inverter’s output current.
The core strength of the method lies in its collaborative use of multiple inverter bridges. By dedicating each bridge to the active compensation of a specific dominant harmonic (e.g., 5th and 7th), the overall system can effectively cancel multiple low-order harmonic currents at the PCC. The implementation via offline pre-calculated lookup tables makes the strategy computationally feasible for real-time control, even with complex transcendental equations.
Comprehensive simulation results validate the superior performance of the proposed method. Under steady-state distorted grid conditions, it reduces grid current THD by more than 70% compared to traditional SHE. It also exhibits stable and acceptably fast dynamic response to changes in grid harmonic levels. This enhanced harmonic compensation capability allows high-power solar inverter systems to maintain excellent power quality and strict compliance with grid codes, even when connected to non-ideal, harmonic-polluted grids, thereby improving the robustness and reliability of large-scale solar power integration.