In recent years, the increasing penetration of distributed generation systems based on renewable energy sources has led to the emergence of weak grid characteristics in medium- and low-voltage distribution networks. The grid tied inverter serves as a critical interface between these distributed sources and the utility grid, and its stable and efficient operation is paramount. Weak grids are often characterized by non-ideal conditions such as grid voltage distortion, frequency fluctuations, and voltage sags, which pose significant challenges to the current control strategies of grid tied inverters. The total harmonic distortion (THD) of the output current is a key metric for assessing power quality, with standards like IEEE Std. 519-2022 requiring it to be below 5%. Therefore, developing advanced control methods that maintain high performance under weak grid conditions is essential.
Traditional plug-in repetitive controllers (PRCs) are widely used in grid tied inverters due to their ability to track periodic reference signals and suppress harmonic disturbances simultaneously. However, their performance degrades significantly when the grid frequency fluctuates, as the fixed resonant frequencies of the repetitive control internal model no longer align with the actual harmonic frequencies. This misalignment reduces the harmonic suppression capability. To address this, various approaches have been proposed, such as increasing the controller order or using multi-bandwidth repetitive control, but they often involve complex parameter design or insufficient robustness for large frequency variations. Feedforward repetitive control (FRC) offers a wider resonance bandwidth and higher open-loop gain, improving robustness against frequency variations. Nonetheless, under substantial grid frequency swings, even FRC’s bandwidth may be inadequate. A promising solution is the use of fractional-order repetitive control, which adapts the controller’s delay to match fractional periods caused by frequency deviations, thereby maintaining precise harmonic cancellation. This paper presents a novel fractional-order plug-in feedforward repetitive control (FOPFRC) strategy that leverages a Farrow-structure filter to approximate the fractional delay efficiently. The method enhances the robustness of grid tied inverters against grid frequency fluctuations while ensuring low computational burden, as the Farrow filter’s coefficients do not require online recalculations.
The core of our proposed FOPFRC method lies in modifying the internal model of the conventional PRC. The standard repetitive control internal model for a plug-in structure is given by:
$$ G_{RC}(z) = \frac{k_r z^{-N}}{1 – Q z^{-N}} $$
where \( N = f_s / f_g \) is the delay order, \( f_s \) is the sampling frequency, \( f_g \) is the grid fundamental frequency, \( k_r \) is the gain, and \( Q \) is a filter usually chosen as a constant less than 1 for stability. In the FRC structure, the internal model is altered to:
$$ G_{FRC}(z) = k_r \frac{1 + Q z^{-N}}{1 – Q z^{-N}} z^{-N} $$
This modification effectively doubles the open-loop gain and widens the resonance bandwidth around the harmonic frequencies. However, when \( f_g \) fluctuates, \( N \) becomes a non-integer. To handle this, we express \( N \) as:
$$ N = D + d $$
where \( D = \text{int}(N) \) is the integer part and \( d \) ( \( 0 \le d < 1 \) ) is the fractional part. The fractional delay \( z^{-d} \) must be approximated accurately. We employ a Farrow-structure variable fractional-delay filter, which is implemented as:
$$ z^{-d} \approx \sum_{i=0}^{W} L_i(z) d^i $$
Here, \( W \) is the order of the polynomial approximation, and \( L_i(z) \) are fixed-coefficient sub-filters derived based on Lagrangian interpolation. For our application, we select \( W = 3 \) to balance approximation accuracy and computational complexity. The sub-filter coefficients are predetermined and do not change with \( d \); only the value of \( d \) needs to be updated based on the real-time grid frequency estimated by a phase-locked loop (PLL). This significantly reduces the online computational load compared to methods that require recomputing filter coefficients. To ensure stability, a zero-phase low-pass filter \( G_L(z) \) is added to attenuate high-frequency gains introduced by the Farrow filter. The complete FOPFRC controller transfer function becomes:
$$ G_{FOPFRC}(z) = k_r \frac{1 + Q z^{-D-d} G_L(z)}{1 – Q z^{-D-d} G_L(z)} z^{-D-d+m} G_L(z) S(z) $$
where \( S(z) \) is a stability compensator (typically a low-pass filter), and \( z^m \) is a phase lead compensator to counteract phase lags in the plant and compensators.
The stability of the overall control system, which includes a PI controller in parallel with the FOPFRC, is analyzed using the small gain theorem. The system is stable if two conditions are met: first, the loop gain without the repetitive controller, \( 1 + G_{PI}(z)P(z) \), has all poles inside the unit circle; second, the repetitive control loop satisfies:
$$ |1 + G_{FOPFRC}(z) G_{PI}(z) P(z)| \neq 0 \quad \text{for} \quad z = e^{j\omega T_s} $$
Through derivation, this leads to constraints on the design parameters \( m \) and \( k_r \):
$$ \theta_s(\omega) + \theta_p(\omega) + m \omega T_s < 90^\circ $$
$$ k_r < \frac{\cos[\theta_s(\omega) + \theta_p(\omega) + m \omega T_s]}{N_s(\omega) N_p(\omega)} $$
where \( \theta_s, \theta_p \) are the phases and \( N_s, N_p \) the magnitudes of \( S(z) \) and the modified plant \( P_0(z) = G_{PI}(z)P(z)/(1+G_{PI}(z)P(z)) \), respectively. These inequalities guide the parameter selection to ensure robustness.
For implementation in a single-phase LCL-type three-level neutral-point-clamped (NPC) grid tied inverter, the plant model including computational and PWM delays is derived. The continuous-time transfer function from inverter voltage to grid current is:
$$ P(s) = \frac{T(s)}{L_1 L_2 C s^3 + L_2 C H_{ic} T(s) s^2 + (L_1 + L_2)s} $$
where \( T(s) = e^{-1.5 T_s s} \) represents a 1.5-sample delay, \( L_1, L_2, C \) are the LCL filter parameters, and \( H_{ic} \) is an active damping coefficient from capacitor current feedback. Discretizing with a zero-order hold gives \( P(z) \), which is used for design. The controller parameters are chosen as follows. The PI controller parameters are set to \( k_p = 16 \) and \( k_i = 5300 \) to ensure stable operation of the inner loop. The Farrow sub-filters for \( W=3 \) are calculated offline; their coefficients are summarized in Table 1.
| Sub-filter | Coefficient Polynomial |
|---|---|
| \( L_0(z) \) | 1 |
| \( L_1(z) \) | \(-1.833 + 3z^{-1} – 1.5z^{-2} + 0.3333z^{-3}\) |
| \( L_2(z) \) | \(1 – 2.5z^{-1} + 2z^{-2} – 0.5z^{-3}\) |
| \( L_3(z) \) | \(-0.1667 + 0.5z^{-1} – 0.5z^{-2} + 0.1667z^{-3}\) |
The stability compensator \( S(z) \) is a fourth-order Butterworth low-pass filter with a cutoff frequency of 1000 Hz to attenuate high-frequency noise. The phase lead compensator order is chosen as \( m = 8 \) to minimize phase lag in the low-frequency range. The internal model coefficient is set to \( Q = 0.95 \) to widen the resonance peaks slightly. The repetitive control gain \( k_r \) is selected as 0.44 based on the stability condition, ensuring the Nyquist plot of the sensitivity function remains inside the unit circle. The low-pass filter \( G_L(z) \) is:
$$ G_L(z) = \frac{z + 2 + z^{-1}}{4} $$
which provides zero-phase shift and attenuates frequencies above the Nyquist rate.
Experimental validation was conducted on a 3-kW LCL-type single-phase three-level NPC grid tied inverter prototype. The system parameters are listed in Table 2. A digital signal processor (TMS320F28335) was used to implement the control algorithm. A programmable grid simulator injected distorted grid voltage with 5.7% THD to emulate weak grid conditions. The performance of the proposed FOPFRC was compared with conventional PRC and integer-order PFRC under various scenarios: ideal grid frequency, grid frequency fluctuations, reference current steps, and grid voltage dips.
| Parameter | Value |
|---|---|
| DC-link voltage \( E_{dc} \) | 400 V |
| Grid voltage (RMS) \( u_g \) | 220 V |
| Nominal grid frequency \( f_g \) | 50 Hz |
| Sampling frequency \( f_s \) | 10 kHz |
| Inverter-side inductor \( L_1 \) | 4.4 mH |
| Grid-side inductor \( L_2 \) | 0.4 mH |
| Filter capacitor \( C \) | 4.7 µF |
| Active damping coefficient \( H_{ic} \) | 8 |
The experimental setup includes the inverter platform, a programmable DC source, a grid simulator, and a power quality analyzer. Under ideal 50 Hz grid conditions, the output current THD for different controllers is shown in Table 3. The FOPFRC achieves a THD of 1.0%, comparable to PFRC and better than PRC (1.2%) and PI alone (4.5%). This demonstrates the inherent harmonic suppression capability of repetitive control in grid tied inverters.
| Control Method | THD (%) |
|---|---|
| PI only | 4.5 |
| PRC | 1.2 |
| PFRC | 1.0 |
| FOPFRC | 1.0 |
When the grid frequency deviates to 49.6 Hz, the performance difference becomes evident. The PFRC’s THD rises to 2.0%, while the FOPFRC maintains a THD of 1.0%. Similarly, at 50.4 Hz, PFRC yields 2.3% THD versus 0.9% for FOPFRC. The current spectra show that FOPFRC effectively suppresses low-order harmonics (3rd, 5th, 7th, etc.) even under frequency shifts, thanks to the adaptive fractional delay. The dynamic response was tested by stepping the current reference amplitude from 5 A to 10 A. The FOPFRC settles within about 1.6 fundamental cycles (0.032 s) in all frequency scenarios, indicating fast transient performance. Moreover, during grid voltage dips (from 220 V to 200 V) with frequency fluctuations, the FOPFRC-controlled grid tied inverter maintains stable operation with minimal current distortion, showcasing its robustness.
To quantify robustness over a wider frequency range, we measured THD for frequencies from 49.5 Hz to 50.5 Hz. The results are summarized in Table 4. The FOPFRC consistently keeps THD below 1.1%, whereas PRC and PFRC exhibit significant degradation, with THD exceeding 2.5% at the extremes. This underscores the frequency-adaptive advantage of the fractional-order design in practical grid tied inverter applications.
| Grid Frequency (Hz) | PRC THD (%) | PFRC THD (%) | FOPFRC THD (%) |
|---|---|---|---|
| 49.5 | 3.3 | 2.5 | 1.1 |
| 49.6 | 2.8 | 2.0 | 1.0 |
| 49.8 | 2.0 | 1.5 | 1.0 |
| 50.0 | 1.2 | 1.0 | 1.0 |
| 50.2 | 1.8 | 1.4 | 0.9 |
| 50.4 | 2.5 | 2.3 | 0.9 |
| 50.5 | 3.0 | 2.6 | 1.0 |
The effectiveness of the FOPFRC can be further understood through its frequency response. The magnitude of the sensitivity function \( |S(j\omega)| \) demonstrates deep notches at the harmonic frequencies, which shift with grid frequency. The mathematical expression for the ideal internal model gain at harmonic frequencies is:
$$ |G_{internal}(e^{j 2\pi h f_g / f_s})| \approx \frac{k_r}{1 – Q} \quad \text{for integer } h $$
With fractional delay approximation, this gain remains high even when \( f_g \) varies, ensuring continuous harmonic attenuation. The Farrow filter’s approximation error is analyzed by evaluating its deviation from the ideal fractional delay. For \( W=3 \), the maximum magnitude error in the low-frequency range (below 2500 Hz) is less than 0.1 dB, which is negligible for harmonic suppression purposes. The phase error is also minimal, preserving the stability margins.
In terms of computational complexity, the FOPFRC requires additional operations for the Farrow filter compared to integer-order repetitive control. However, since the sub-filters are fixed FIR filters of order 3, the extra cost is modest: for each sample, it involves evaluating a cubic polynomial in \( d \) and filtering through four FIR filters. This is significantly cheaper than online redesign of fractional delay filters, making it suitable for real-time implementation in digital signal processors for grid tied inverters.
In conclusion, we have proposed and validated a fractional-order plug-in feedforward repetitive control strategy for single-phase grid tied inverters operating in weak grids. The method combines the wide resonance bandwidth of feedforward repetitive control with the frequency adaptability of a Farrow-structure fractional delay filter. Key contributions include: a modified internal model structure that enhances robustness against grid frequency variations; an efficient fractional delay implementation that avoids online coefficient recalculations; and a comprehensive stability analysis and parameter design guideline. Experimental results confirm that the FOPFRC maintains low output current THD (below 1.1%) under grid voltage distortion, frequency fluctuations up to ±0.5 Hz, and voltage dips, while exhibiting fast dynamic response. This makes it a highly suitable control solution for modern grid tied inverters facing increasingly challenging grid conditions. Future work may extend the approach to three-phase systems, incorporate adaptive tuning of parameters, and explore integration with other advanced grid-support functions.