In photovoltaic (PV) systems, the control of grid-connected inverters is critical for ensuring stability and safety, particularly in three-phase configurations. Traditional control methods often fall short in performance, leading to high harmonic currents and significant overshoot in the three phase inverter. To address these issues, this paper proposes a control strategy based on the Space Vector Pulse Width Modulation (SVPWM) algorithm. The method involves deriving a mathematical model of the three phase inverter using Kirchhoff’s laws, determining the reference voltage sector through spatial vector analysis, and synthesizing the reference voltage using adjacent voltage vectors and zero vectors via SVPWM. Experimental results demonstrate that this approach reduces harmonic currents below 1A and overshoot below 0.1%, enabling precise control of the three phase inverter in grid-connected PV systems.
The intermittent and fluctuating nature of PV power output poses challenges to grid stability and power quality. Therefore, advanced control techniques for the three phase inverter are essential. SVPWM algorithm, known for its high efficiency and performance, leverages voltage space vectors to generate optimal output waveforms. This paper details the implementation of SVPWM for controlling a three phase inverter, focusing on mathematical modeling, sector identification, and modulation techniques.
Mathematical Model of the Three-Phase Inverter
To develop an accurate mathematical model of the three phase inverter, the following assumptions are made:
- The grid voltage is sinusoidal and free of harmonics.
- The grid-side filter inductance operates linearly without saturation.
- Power switches are ideal, with no parasitic resistance or dead time.
- Three-phase circuit parameters are symmetrical.
Based on these assumptions, the topology of the three phase grid-connected inverter is represented as follows:
In this topology, the upper and lower arms of the three phase inverter cannot conduct simultaneously. The switch function is defined as:
$$ S_k = \begin{cases} 1 & \text{if switch } k \text{ is on} \\ 0 & \text{if switch } k \text{ is off} \end{cases} $$
where \( k \) denotes the number of switches. Applying Kirchhoff’s voltage and current laws, the output current equation on the grid side, the output current equation on the inverter side, and the filter capacitor voltage equation are derived. The mathematical model of the three phase inverter in the stationary coordinate system is expressed as:
$$ \frac{d}{dt} \begin{bmatrix} I_a \\ I_b \\ I_c \end{bmatrix} = \frac{1}{L} \begin{bmatrix} V_a – U_a \\ V_b – U_b \\ V_c – U_c \end{bmatrix} – \frac{R}{L} \begin{bmatrix} I_a \\ I_b \\ I_c \end{bmatrix} $$
where \( I_a, I_b, I_c \) are the output currents, \( U_a, U_b, U_c \) are the grid voltages, \( V_a, V_b, V_c \) are the inverter output voltages, \( R \) is the filter resistance, and \( L \) is the filter inductance. This model describes the operational state of the three phase inverter during grid connection.
Reference Voltage Sector Identification
To implement SVPWM control, the sector of the reference voltage vector must be identified. The reference voltage vector magnitude is 1.5 times the grid voltage, and the three-phase sinusoidal voltages are given by:
$$ U_A = U_m \cos(\alpha), \quad U_B = U_m \cos(\alpha – 120^\circ), \quad U_C = U_m \cos(\alpha + 120^\circ) $$
where \( U_m \) is the voltage amplitude and \( \alpha \) is the phase angle. The horizontal distances between phases in the PV coordinate system are calculated as:
$$ X_{ab} = U_A – U_B, \quad X_{bc} = U_B – U_C, \quad X_{ac} = U_A – U_C $$
The sector judgment condition \( H \) is defined as:
$$ H = \text{sign}(X_{ab}) + 2 \cdot \text{sign}(X_{bc}) + 4 \cdot \text{sign}(X_{ac}) $$
where \( \text{sign}(x) \) returns 1 if \( x > 0 \), else 0. The reference voltage sector is determined based on \( H \), as shown in Table 1.
| H Value | Sector |
|---|---|
| 1 | I |
| 2 | II |
| 3 | III |
| 4 | IV |
| 5 | V |
| 6 | VI |
This sector identification is crucial for the subsequent SVPWM-based control of the three phase inverter.
Inverter Control Using SVPWM Algorithm
The SVPWM algorithm controls the three phase inverter by synthesizing a reference voltage vector using adjacent active voltage vectors and zero vectors. The six power switches in the three phase inverter result in eight switching states, corresponding to six active voltage vectors and two zero vectors, as summarized in Table 2.
| Switch State (S1-S6) | Voltage Vector | Magnitude |
|---|---|---|
| 100 | Y1 | \( \frac{2}{3} V_{dc} \) |
| 110 | Y2 | \( \frac{2}{3} V_{dc} \) |
| 010 | Y3 | \( \frac{2}{3} V_{dc} \) |
| 011 | Y4 | \( \frac{2}{3} V_{dc} \) |
| 001 | Y5 | \( \frac{2}{3} V_{dc} \) |
| 101 | Y6 | \( \frac{2}{3} V_{dc} \) |
| 000 | Y0 | 0 |
| 111 | Y7 | 0 |
When the reference voltage vector \( V_{\text{ref}} \) lies in a specific sector, it is synthesized using the two adjacent active vectors and a zero vector. The duty cycles \( t_1 \), \( t_2 \), and \( t_0 \) for these vectors must satisfy:
$$ t_1 + t_2 + t_0 = T_{\text{SVPWM}} $$
where \( T_{\text{SVPWM}} \) is the SVPWM control period. Using the principle of vector equivalence, the reference voltage is expressed as:
$$ V_{\text{ref}} = \frac{t_1}{T_{\text{SVPWM}}} V_1 + \frac{t_2}{T_{\text{SVPWM}}} V_2 + \frac{t_0}{T_{\text{SVPWM}}} V_0 $$
where \( V_1 \) and \( V_2 \) are the adjacent active vectors, and \( V_0 \) is the zero vector. Solving for the times:
$$ t_1 = \frac{\sqrt{3} T_{\text{SVPWM}} |V_{\text{ref}}|}{V_{dc}} \sin\left(\frac{\pi}{3} – \theta\right), \quad t_2 = \frac{\sqrt{3} T_{\text{SVPWM}} |V_{\text{ref}}|}{V_{dc}} \sin(\theta), \quad t_0 = T_{\text{SVPWM}} – t_1 – t_2 $$
where \( \theta \) is the angle of \( V_{\text{ref}} \) within the sector. This synthesis generates the reference voltage for each SVPWM cycle, which serves as the control input to the three phase inverter model, optimizing its performance.
Experimental Validation
To validate the proposed method, experiments were conducted on a three-phase PV power station. The three phase inverter parameters were set as follows: filter inductance 0.5 mH, filter capacitance 120 μF, DC-side voltage 400 V, grid frequency 45.54 Hz, switching frequency 85.25 Hz, rated power 5 MW, number of switching angles 8, and dead time 20 μs. The sampling frequency was 1.25 kHz. The SVPWM algorithm processed current and voltage signals to generate PWM signals for real-time control of the three phase inverter.
During experiments, the dynamic and steady-state performance of the three phase inverter were evaluated. Under varying conditions, such as sudden changes in irradiance, the SVPWM-controlled three phase inverter maintained synchronization with the grid within 0.5 s, with total harmonic distortion below 2% and power factor near 1. The efficiency reached 98.5% at rated power. Harmonic currents and overshoot were measured over multiple tests, as shown in Tables 3 and 4.
| Harmonic Order | Proposed Method | Traditional Method 1 | Traditional Method 2 |
|---|---|---|---|
| 1 | 0.15 | 0.45 | 0.35 |
| 3 | 0.22 | 0.67 | 0.52 |
| 5 | 0.18 | 0.58 | 0.44 |
| 7 | 0.12 | 0.41 | 0.31 |
| 9 | 0.09 | 0.33 | 0.25 |
| 11 | 0.07 | 0.28 | 0.21 |
| 13 | 0.05 | 0.24 | 0.18 |
| 15 | 0.04 | 0.21 | 0.16 |
| Test Group | Proposed Method | Traditional Method 1 | Traditional Method 2 |
|---|---|---|---|
| 1 | 0.05 | 0.25 | 0.18 |
| 2 | 0.06 | 0.27 | 0.20 |
| 3 | 0.04 | 0.23 | 0.17 |
| 4 | 0.05 | 0.26 | 0.19 |
| 5 | 0.07 | 0.29 | 0.22 |
| 6 | 0.05 | 0.24 | 0.18 |
| 7 | 0.06 | 0.28 | 0.21 |
| 8 | 0.04 | 0.22 | 0.16 |
The results indicate that the proposed SVPWM-based method keeps harmonic currents below 1A and overshoot below 0.1%, outperforming traditional methods. This demonstrates the efficacy of the SVPWM algorithm in controlling the three phase inverter for enhanced grid stability and power quality.
Conclusion
This paper presents a control method for the three phase photovoltaic grid-connected inverter using the SVPWM algorithm. By deriving a mathematical model, identifying reference voltage sectors, and synthesizing voltages with SVPWM, the method achieves precise control of the three phase inverter. Experimental validation confirms significant reductions in harmonic currents and overshoot, highlighting the algorithm’s superiority in improving the performance and reliability of grid-connected PV systems. The SVPWM-based approach offers a robust solution for managing the complexities of three phase inverter control in renewable energy applications.