In the realm of renewable energy systems, solar power stands out due to its cleanliness and sustainability. The integration of photovoltaic (PV) systems into the electrical grid relies heavily on the performance of grid tied inverters. As a key interface, the grid tied inverter must ensure efficient power conversion, high-quality output, and robust operation under varying grid conditions. This article delves into an advanced control strategy for grid tied inverters, combining fractional-order sliding mode control with two-degree-of-freedom internal model control to address limitations in traditional methods.
The widespread adoption of grid tied inverters in applications such as DC microgrids, islanded power systems, and urban rail transit underscores their importance. However, the nonlinear nature of PV systems poses significant control challenges. Conventional approaches like PI control often struggle with parameter tuning and dynamic response, while sliding mode control, though robust, introduces chattering that degrades performance. Moreover, one-degree-of-freedom internal model control cannot independently optimize reference tracking and disturbance rejection. To overcome these issues, this work proposes a novel control framework that leverages fractional calculus and two-degree-of-freedom structures to enhance the dynamic quality and robustness of grid tied inverters.
The topology of a three-phase PWM grid tied inverter is foundational to understanding its control. The system comprises a PV array, a DC-link capacitor, and an inverter bridge connected to the grid via filtering inductors. The mathematical model in the dq reference frame captures the dynamics essential for control design. For a grid tied inverter, the voltage and current relationships are expressed as:
$$ u_{kd} = L \frac{di_d}{dt} + R i_d + e_d – \omega L i_q $$
$$ u_{kq} = L \frac{di_q}{dt} + R i_q + e_q + \omega L i_d $$
$$ C \frac{du_{dc}}{dt} = i_{pv} – i_{in} $$
where \( u_{kd} \) and \( u_{kq} \) are the dq-axis inverter output voltages, \( i_d \) and \( i_q \) are the grid currents, \( e_d \) and \( e_q \) are the grid voltages, \( L \) and \( R \) are the filter inductance and resistance, \( \omega \) is the grid frequency, \( C \) is the DC-link capacitance, \( u_{dc} \) is the DC voltage, \( i_{pv} \) is the PV output current, and \( i_{in} \) is the inverter input current. The power balance, assuming negligible losses, is given by \( P_{dc} = u_{dc} i_{in} = \frac{3}{2}(e_d i_d + e_q i_q) \). For unity power factor operation with \( i_q = 0 \), the DC voltage dynamics simplify to:
$$ \frac{du_{dc}}{dt} = \frac{1}{C} \left( i_{pv} – \frac{3 e_d i_d}{2 u_{dc}} \right) $$
This model forms the basis for designing both outer voltage and inner current loops in a grid tied inverter. The outer loop regulates the DC voltage to ensure stable power transfer, while the inner loop controls the grid currents for precise power injection. Traditional control methods often rely on integer-order sliding mode control, which can cause chattering, and one-degree-of-freedom internal model control, which lacks independent tuning for tracking and disturbance rejection. The proposed method integrates fractional-order sliding mode for the voltage loop and two-degree-of-freedom internal model control for the current loop, offering a synergistic improvement.
Fractional-order calculus extends the concepts of differentiation and integration to non-integer orders, providing additional degrees of freedom for control design. In the context of a grid tied inverter, a fractional-order sliding surface is defined for the voltage error \( e_1 = u_{dc}^* – u_{dc} \), where \( u_{dc}^* \) is the reference DC voltage. The sliding surface \( s \) is given by:
$$ s = e_1 + c_1 D^{-\alpha} e_1 $$
Here, \( D^{-\alpha} \) denotes the fractional integral operator of order \( \alpha \) (with \( 0 < \alpha < 1 \)), and \( c_1 \) is a positive tuning parameter. The derivative of \( s \) is:
$$ \dot{s} = \dot{e}_1 + c_1 D^{1-\alpha} e_1 $$
Using the voltage dynamics and an exponential reaching law \( \dot{s} = -\epsilon \text{sgn}(s) – k s \) (with \( \epsilon > 0 \), \( k > 0 \)), the reference current for the inner loop, \( i_d^* \), is derived as:
$$ i_d^* = \frac{2 u_{dc}}{3 e_d} \left[ i_{pv} – C \left( \epsilon \text{sgn}(s) + k s + c_1 D^{1-\alpha} e_1 \right) \right] $$
This fractional-order approach reduces chattering by smoothing the control action, a critical advantage for grid tied inverters where high-frequency switching can introduce harmonics. The parameter selection for the fractional-order sliding mode control is summarized in the table below, which contrasts it with integer-order methods.
| Control Parameter | Fractional-Order Value | Integer-Order Value | Effect on Grid Tied Inverter |
|---|---|---|---|
| Sliding Order \( \alpha \) | 0.3 | 1 (implicit) | Reduces chattering, improves transient response |
| Gain \( c_1 \) | 47 | 50 | Adjusts convergence speed |
| Reaching Law \( \epsilon \) | 5.3 | 6.0 | Controls disturbance rejection |
| Reaching Law \( k \) | 0.025 | 0.03 | Influences steady-state error |
For the current inner loop, the two-degree-of-freedom internal model control offers independent optimization of setpoint tracking and disturbance rejection. The system model \( G(s) \) for the current dynamics, considering the dq-axis cross-coupling, is:
$$ G(s) = \begin{bmatrix} \frac{1}{R + sL} & -\frac{\omega L}{R + sL} \\ \frac{\omega L}{R + sL} & \frac{1}{R + sL} \end{bmatrix} $$
In practice, the grid tied inverter’s current loop can be approximated as a first-order system due to the high PWM switching frequency (e.g., 10 kHz). The internal model controllers are designed as \( C_{\text{IMC}}^1(s) \) for tracking and \( C_{\text{IMC}}^2(s) \) for disturbance compensation. With a model \( \hat{G}(s) \) and a low-pass filter \( L_1(s) = \frac{1}{\alpha_1 s + 1} \), the controllers are:
$$ C_{\text{IMC}}^2(s) = \hat{G}^{-1}(s) L_1(s) = \begin{bmatrix} \frac{\hat{R} + s\hat{L}}{\alpha_1 s + 1} & \hat{\omega}\hat{L} \\ -\hat{\omega}\hat{L} & \frac{\hat{R} + s\hat{L}}{\alpha_1 s + 1} \end{bmatrix} $$
$$ C_{\text{IMC}}^1(s) = \frac{\alpha_2 s + 1}{\alpha_1 s + 1} $$
where \( \hat{R} \), \( \hat{L} \), and \( \hat{\omega} \) are estimated parameters, and \( \alpha_1 \), \( \alpha_2 \) are tuning filters. The overall control law for the grid tied inverter currents yields:
$$ Y(s) = \left[ C_{\text{IMC}}^1(s) R(s) + C_{\text{IMC}}^2(s) D(s) \right] G(s) $$
with \( Y(s) \) as the output current, \( R(s) \) as the reference, and \( D(s) \) as disturbances. This structure decouples the dq-axis currents and enhances robustness against grid voltage variations. The following table outlines key parameters for the internal model control in a typical grid tied inverter setup.
| System Parameter | Symbol | Value | Role in Grid Tied Inverter |
|---|---|---|---|
| Output Power | \( P_e \) | 6 kW | Rated capacity for grid connection |
| DC Voltage | \( U_{dc} \) | 700 V | Input to inverter stage |
| Grid Voltage | \( e \) | 380 V (RMS) | Grid synchronization target |
| Filter Inductance | \( L \) | 4 mH | Attenuates switching harmonics |
| Filter Resistance | \( R \) | 0.2 Ω | Accounts for losses |
| Switching Frequency | \( f \) | 10 kHz | PWM resolution for grid tied inverter |
| IMC Filter \( \alpha_1 \) | \( \alpha_1 \) | 0.6 | Speeds up disturbance rejection |
| IMC Filter \( \alpha_2 \) | \( \alpha_2 \) | 3 × 10² | Tunes tracking performance |
Simulation studies validate the superiority of the proposed fractional-order sliding mode two-degree-of-freedom internal model control for grid tied inverters. Comparative tests against integer-order sliding mode with one-degree-of-freedom internal model control were conducted under two scenarios: grid-following performance and grid voltage sag/swell disturbances. The grid tied inverter was modeled in a simulation environment with parameters as listed above. For the grid-following test, the inverter was tasked to synchronize with a 380 V, 50 Hz grid while maintaining unity power factor.
The results demonstrated that the proposed control strategy achieves faster settling with minimal overshoot. The three-phase grid currents exhibited reduced ripple, and the total harmonic distortion (THD) was significantly lower. Specifically, the THD for the grid tied inverter with the new method was 1.03%, compared to 1.99% for the traditional method—a reduction of 0.96%. This improvement is crucial for grid tied inverters, as it ensures compliance with power quality standards like IEEE 1547. The current waveforms can be described mathematically by Fourier series, where the harmonic content is minimized through the control action:
$$ i_a(t) = \sum_{n=1}^{\infty} I_n \sin(n\omega t + \phi_n) $$
with \( I_n \) decreasing rapidly for \( n > 1 \) under the proposed control. The reduction in low-order harmonics (e.g., 3rd, 5th) is particularly notable, as shown in the harmonic spectrum analysis.
In the disturbance test, the grid voltage was increased from 380 V to 400 V at 0.3 seconds to simulate a voltage swell. The grid tied inverter with the proposed method maintained stable power output with negligible transients, whereas the traditional control showed pronounced oscillations in both active and reactive power. The active power \( P \) and reactive power \( Q \) are given by:
$$ P = \frac{3}{2} (e_d i_d + e_q i_q) $$
$$ Q = \frac{3}{2} (e_q i_d – e_d i_q) $$
Under the proposed control, \( P \) remained constant at 6 kW, and \( Q \) stayed near zero, confirming effective decoupling and disturbance rejection. The settling time for the grid tied inverter to regain steady-state was merely 0.02 seconds, highlighting the enhanced robustness. This performance is quantified in the table below, summarizing key metrics from the simulations.
| Performance Metric | Proposed Control | Traditional Control | Improvement |
|---|---|---|---|
| THD of Grid Current | 1.03% | 1.99% | 48% reduction |
| Settling Time (grid follow) | 0.05 s | 0.10 s | 50% faster |
| Overshoot in DC Voltage | < 2% | 5% | Reduced chattering |
| Power Ripple During Swell | 0.1 kW | 0.5 kW | 80% suppression |
| Reactive Power Deviation | 0.05 kvar | 0.3 kvar | Better grid support |
The efficacy of the fractional-order sliding mode aspect can be further analyzed through stability criteria. Using Lyapunov theory, the sliding surface dynamics ensure convergence if \( \dot{V} = s \dot{s} \leq 0 \). For the grid tied inverter, with the chosen reaching law, this condition holds globally, guaranteeing robustness against parameter variations in the PV array or grid impedance. The fractional-order operator introduces memory effects, which smooth the control signal and reduce high-frequency switching noise—a common issue in grid tied inverters that can lead to electromagnetic interference.
Moreover, the two-degree-of-freedom internal model control provides inherent decoupling of the dq-axis currents. The transfer function from reference to output, assuming perfect model matching (\( \hat{G}(s) = G(s) \)), is:
$$ T_r(s) = C_{\text{IMC}}^1(s) G(s) $$
$$ T_d(s) = [I – C_{\text{IMC}}^2(s) G(s)] G(s) $$
where \( T_r(s) \) governs tracking and \( T_d(s) \) governs disturbance response. By tuning \( \alpha_1 \) and \( \alpha_2 \) independently, the grid tied inverter can achieve fast tracking without amplifying grid disturbances. This is particularly beneficial in weak grids where voltage fluctuations are common. The parameter selection can be optimized using pole placement or frequency-domain methods, ensuring that the bandwidth of the grid tied inverter aligns with grid requirements.
In practical implementations, the control algorithm can be deployed on digital signal processors (DSPs) or field-programmable gate arrays (FPGAs) for real-time operation. The computational overhead for fractional-order calculus is manageable with approximate methods like Oustaloup’s recursive filter, which discretizes the fractional operator into integer-order transfer functions. For a grid tied inverter with a 10 kHz switching frequency, the sampling time can be set to 100 μs, allowing sufficient margin for control computations. The overall control structure for the grid tied inverter integrates the voltage and current loops, as depicted in the block diagram earlier.
Future work could explore adaptive versions of this control strategy to handle aging effects in PV modules or grid impedance changes. Additionally, integrating maximum power point tracking (MPPT) algorithms with the proposed control could further optimize energy harvest for grid tied inverters. The scalability of the method to multi-inverter systems or microgrids also warrants investigation, as grid tied inverters are increasingly deployed in distributed generation networks.
In conclusion, the fusion of fractional-order sliding mode and two-degree-of-freedom internal model control presents a significant advancement for grid tied inverters. By mitigating chattering, enhancing tracking accuracy, and improving disturbance rejection, this approach elevates the robustness and power quality of PV integration systems. Simulation results confirm notable gains in THD reduction, settling time, and power stability during grid anomalies. As renewable penetration grows, such advanced control strategies will be pivotal in ensuring reliable and efficient grid tied inverter operation, contributing to a sustainable energy future.