In the context of rapid global development of renewable energy, the grid connected inverter has become a critical component in distributed power generation systems. Its performance directly impacts power quality, making it essential to enhance control strategies and reduce harmonic content in grid-injected currents. Compared to L-filters, LCL filters offer smaller inductance and superior high-frequency attenuation, which is beneficial for lowering current harmonics. This paper presents a constant frequency hysteresis control method for LCL-type three-level grid connected inverters. We derive the relationship between bridge-side and grid-side currents, enabling indirect control of grid-side currents through direct control of bridge-side currents. The reference voltage hexagon is divided into 12 small parallelogram regions to achieve decoupled control of phase-to-phase error currents. By adjusting the hysteresis width dynamically based on previous switching cycles, constant frequency operation is maintained. Additionally, an extended state observer is utilized for differential tracking to determine reference voltage vector sectors and estimate bridge-side reference currents. Simulation and semi-physical simulation results validate the effectiveness of the proposed control strategy.
The grid connected inverter is a pivotal element in modern power systems, especially with the integration of solar and wind energy. Its ability to inject clean current into the grid is paramount. Traditional control methods like proportional-resonant (PR) control, model predictive control, and repetitive control have limitations, such as complexity in multi-harmonic suppression, dependency on accurate models, or slow response times. Hysteresis control, known for its fast response and high precision, is widely used but suffers from variable switching frequency, complicating filter design and causing thermal issues in switches. Our research addresses these challenges by proposing a constant frequency hysteresis control for L-level grid connected inverters with LCL filters, ensuring improved performance and reduced total harmonic distortion (THD).
The three-level grid connected inverter topology, as illustrated, consists of a DC voltage source, switching devices, LCL filter components (including inductors L1 and L2, capacitor C, and damping resistor R), and grid connections. The mathematical model of the inverter is fundamental to our control design. The bridge-side voltage $u_{j}$ (where $j = ab, bc, ca$ for phase-to-phase quantities) is related to the switching states $S_a$, $S_b$, and $S_c$, which can take values of 1, 0, or -1. The output voltage is given by:
$$u_{j} = S_{j} \cdot U_{dc} / 2$$
where $U_{dc}$ is the DC-link voltage. The dynamics of the bridge-side current $i_{1j}$ and grid-side current $i_{2j}$ are described by:
$$L_1 \frac{di_{1j}}{dt} = u_{j} – u_{cj}$$
$$L_2 \frac{di_{2j}}{dt} = u_{cj} – u_{gj} – u_0$$
$$C \frac{du_{cj}}{dt} = i_{1j} – i_{2j}$$
Here, $u_{cj}$ is the capacitor voltage, $u_{gj}$ is the grid voltage, and $u_0$ is the neutral point voltage. For control purposes, we focus on phase-to-phase quantities to decouple the three phases. The bridge-side phase-to-phase error current $\Delta i_{1j}$ is defined as the difference between the actual bridge-side current and its reference $i_{1j}^*$. By manipulating the switching states, we can control $\Delta i_{1j}$ independently. For instance, when $S_b = -1$, the error current $\Delta i_{1ab}$ can be controlled solely by switch $S_a$, as shown in the derived relation:
$$L_1 \frac{d\Delta i_{1ab}}{dt} = u_{ab}^* – (S_a – S_b) \cdot U_{dc} / 2$$
This allows decoupled control by dividing the reference voltage vector space into regions. The reference voltage hexagon is segmented into 12 small parallelogram regions based on the magnitudes of phase-to-phase reference voltages $u_{ab}^*$, $u_{bc}^*$, and $u_{ca}^*$. Each region corresponds to a specific control strategy for the switches to regulate the error currents. The table below summarizes the control strategies for each region, where the switch states are determined by the hysteresis comparators for $\Delta i_{1ab}$, $\Delta i_{1bc}$, and $\Delta i_{1ca}$.
| Region | Control Strategy for Switch States | Controlled Error Currents |
|---|---|---|
| Q1 | $S_a$ controls $\Delta i_{1ab}$, $S_c$ controls $\Delta i_{1bc}$ | $\Delta i_{1ab}$, $\Delta i_{1bc}$ |
| Q2 | $S_a$ controls $\Delta i_{1ab}$, $S_b$ controls $\Delta i_{1bc}$ | $\Delta i_{1ab}$, $\Delta i_{1bc}$ |
| Q3 | $S_b$ controls $\Delta i_{1ab}$, $S_c$ controls $\Delta i_{1bc}$ | $\Delta i_{1ab}$, $\Delta i_{1bc}$ |
| Q4 | $S_a$ controls $\Delta i_{1ca}$, $S_b$ controls $\Delta i_{1bc}$ | $\Delta i_{1ca}$, $\Delta i_{1bc}$ |
| Q5 | $S_a$ controls $\Delta i_{1ca}$, $S_c$ controls $\Delta i_{1bc}$ | $\Delta i_{1ca}$, $\Delta i_{1bc}$ |
| Q6 | $S_b$ controls $\Delta i_{1ca}$, $S_c$ controls $\Delta i_{1bc}$ | $\Delta i_{1ca}$, $\Delta i_{1bc}$ |
| Q7 | $S_a$ controls $\Delta i_{1ab}$, $S_c$ controls $\Delta i_{1ca}$ | $\Delta i_{1ab}$, $\Delta i_{1ca}$ |
| Q8 | $S_a$ controls $\Delta i_{1ab}$, $S_b$ controls $\Delta i_{1ca}$ | $\Delta i_{1ab}$, $\Delta i_{1ca}$ |
| Q9 | $S_b$ controls $\Delta i_{1ab}$, $S_c$ controls $\Delta i_{1ca}$ | $\Delta i_{1ab}$, $\Delta i_{1ca}$ |
| Q10 | $S_a$ controls $\Delta i_{1bc}$, $S_b$ controls $\Delta i_{1ca}$ | $\Delta i_{1bc}$, $\Delta i_{1ca}$ |
| Q11 | $S_a$ controls $\Delta i_{1bc}$, $S_c$ controls $\Delta i_{1ca}$ | $\Delta i_{1bc}$, $\Delta i_{1ca}$ |
| Q12 | $S_b$ controls $\Delta i_{1bc}$, $S_c$ controls $\Delta i_{1ca}$ | $\Delta i_{1bc}$, $\Delta i_{1ca}$ |
To achieve constant switching frequency, the hysteresis width $H$ is adjusted in real-time. Based on the previous switching cycle, with times $T_1$ for the switch in state 0 and $T_2$ for state 1, the error current slopes are estimated. For the next cycle, the hysteresis width $H_{next}$ is calculated to maintain a fixed period $T_{sw}$ (e.g., corresponding to 25.6 kHz). The formula is derived from the geometry of the hysteresis loop:
$$\frac{H + H_{next}}{2H/T_1} + \frac{H + H_{next}}{2H/T_2} = T_{sw}$$
Solving for $H_{next}$ ensures that the switching frequency remains constant, addressing one of the key drawbacks of traditional hysteresis control in grid connected inverters.
The grid-side current $i_{2j}$ is indirectly controlled through the bridge-side current. The relationship between bridge-side and grid-side currents is given by:
$$i_{1j} = i_{2j} + L_1 C \frac{d^2 i_{2j}}{dt^2} + C \frac{du_{cj}}{dt}$$
By generating a reference bridge-side current $i_{1j}^*$ from the grid-side reference $i_{2j}^*$ and grid voltage $u_{gj}$, we can force $i_{2j}$ to track its reference. This involves computing derivatives, which is where the linear extended state observer (LESO) comes into play. The LESO is a fourth-order system used for differential tracking without direct differentiation, which is noisy and impractical. The state-space representation is:
$$\dot{z} = A z + B u$$
$$y = C z$$
For our application, the LESO estimates the first, second, and third derivatives of the input signal. With parameters set via pole placement, e.g., $\beta_1 = 4\omega_0$, $\beta_2 = 6\omega_0^2$, $\beta_3 = 4\omega_0^3$, $\beta_4 = \omega_0^4$ and bandwidth $\omega_0 = 10000$ rad/s, the LESO acts as a robust differentiator. It tracks the derivatives of grid voltage and reference currents, enabling accurate calculation of bridge-side reference voltages and sector determination. The reference voltage vector $u^*$ is derived from:
$$u_{j}^* = L_1 C L_2 \frac{d^3 i_{2j}^*}{dt^3} + L_1 C \frac{d^2 u_{gj}}{dt^2} + (L_1 + L_2) \frac{di_{2j}^*}{dt} + u_{gj}$$
Using LESO, we obtain the necessary derivatives for this computation, ensuring precise sector identification for the hysteresis control logic. This enhances the performance of the grid connected inverter by reducing harmonic distortion and improving dynamic response.
Simulation studies were conducted using MATLAB/Simulink to validate the proposed constant frequency hysteresis control for the LCL-type three-level grid connected inverter. The system parameters include: $U_{dc} = 800$ V, grid voltage $u_{g}$ RMS of 220 V, output current $i_{2}$ RMS of 100 A, $L_1 = 0.16$ mH, $C = 300 \mu$F, $R = 1 \Omega$, $L_2 = 0.04$ mH, and switching frequency of 25.6 kHz. Comparative results with traditional PI control show significant improvements. The THD of grid current with PI control was 1.68%, whereas with our constant frequency hysteresis control, it reduced to 0.47%. The dynamic response was also faster, as evidenced by step changes in reference current. The following equation summarizes the current error dynamics under the proposed control:
$$\Delta i_{1j}(t) = \int \left( \frac{u_{j}^* – (S_j) \cdot U_{dc}/2}{L_1} \right) dt$$
Moreover, the switching cycles were maintained consistently around 39 $\mu$s, confirming constant frequency operation. The sector judgment results over a grid period demonstrated proper decoupling, with error currents bounded within the hysteresis width. In regions where direct control is not possible, error currents are indirectly regulated through other phases, but their ripple remains at switching frequency and is attenuated by the LCL filter, minimizing grid current harmonics.
Semi-physical simulation experiments were performed using a Typhoon HIL404 platform with an Artix-7 FPGA-based control board. The same parameters as in software simulation were applied. The grid connected inverter model ran in real-time, with control signals generated by our algorithm. FFT analysis of grid currents revealed that with PI control, THD was 1.94%, with notable low-order harmonics. In contrast, the constant frequency hysteresis control achieved a THD of 0.82%, with harmonic amplitudes mostly below 0.5%. The switching ripple at 25.6 kHz was effectively filtered by the LCL filter, showcasing the superiority of our method for grid connected inverter applications. The table below compares key performance metrics:
| Control Method | THD of Grid Current | Switching Frequency Stability | Dynamic Response Time |
|---|---|---|---|
| Traditional PI Control | 1.68% (simulation), 1.94% (semi-physical) | Fixed (PWM-based) | Slower |
| Proposed Constant Frequency Hysteresis Control | 0.47% (simulation), 0.82% (semi-physical) | Constant at 25.6 kHz | Faster |
The mathematical foundation of our control strategy is further solidified by considering the damping resistor $R$ in the LCL filter. Although initially neglected for simplicity, its effect can be incorporated to improve accuracy. The modified bridge-side current equation becomes:
$$L_1 \frac{di_{1j}}{dt} = -R i_{1j} + u_{j} – u_{cj}$$
However, since $R$ is small, its impact on control performance is minimal. The key innovation lies in the decoupled phase-to-phase control and constant frequency adjustment. The hysteresis width adaptation algorithm ensures that even under varying grid conditions, the grid connected inverter maintains optimal performance. For instance, during grid voltage sags or swells, the LESO quickly tracks changes, updating reference currents and voltages to sustain low harmonic distortion.
In terms of implementation, the control algorithm can be embedded in digital signal processors or FPGAs for real-time operation. The computational burden is manageable, as the LESO requires only a few state updates per switching cycle. The sector determination logic, based on reference voltage magnitudes, is straightforward and can be implemented using comparators. This makes the proposed method practical for industrial grid connected inverter systems, where reliability and efficiency are crucial.
Future work could explore adaptive tuning of the LESO bandwidth or integration with other advanced control techniques, such as model predictive control, to further enhance the grid connected inverter’s robustness against parameter variations. Additionally, the method could be extended to multi-level inverters beyond three levels, catering to higher power applications. The constant frequency hysteresis control presented here offers a viable solution for modern renewable energy systems, contributing to cleaner power generation and stable grid integration.
In conclusion, we have developed and validated a constant frequency hysteresis control strategy for LCL-type three-level grid connected inverters. By decoupling phase-to-phase error currents and dynamically adjusting hysteresis width, we achieve fixed switching frequency, reducing filter design challenges and thermal stress. The use of an extended state observer enables accurate differential tracking for sector judgment and reference current estimation. Simulation and semi-physical results demonstrate significant reductions in current harmonic distortion and improved dynamic response compared to traditional PI control. This research underscores the potential of advanced hysteresis techniques in enhancing the performance of grid connected inverters, paving the way for more efficient and reliable power conversion in renewable energy systems.