Aliasing Suppression Method for a Three-Phase Grid-Connected Photovoltaic Inverter Based on Multi-Sampling and Mean Filtering

: In order to reduce the sampling delay and improve bandwidth, sability margin, and the robustness of the active damping in LCL-filtered grid-connected inverters, real-time sampling provides a convenient method. However, aliasing is easily introduced in the control loop because of high-frequency switching harmonics, resulting in a rise in low-order harmonics. To address the challenge of aliasing under real-time sampling, a new method based on multisampling and mean filtering is proposed by combining the proposed harmonic detection and control methods in this paper. This method works by sampling and controlling the fundamental current and selective subharmonic currents separately. The fundamental current control loop without any additional filters maintains real-time sampling, while multisampling and mean filters are applied to the harmonic current control loop. The proposed control method can not only improve the dynamic response performance and control stability of the system but also effectively suppress the selective low-order current harmonics and the aliasing of sampling. Finally, the correctness of the proposed sampling scheme and control strategy is verified by the simulation based on R2018b, MathWorks, Natick, MA, USA and an experimental prototype of a three-phase grid-connected photovoltaic inverter rated at 230 V/50 Hz/40 kW.


Introduction
With the rapid development of new energy sources, a large number of power electronic devices are connected to the power grid, so power quality problems (especially gridconnected current harmonics) are becoming a concern for power systems [1].In the research related to digitally controlled inverters, some advanced control methods have been proposed to reduce the steady state error and improve the total harmonic distortion (THD) of the output current, such as deadbeat control [2], repetitive control [3], proportional resonant [4], state observer [5], hysteresis droop controller [6], and others [7][8][9][10][11][12][13].However, the high-frequency operation of the switching device interferes with the sampling of the signal, causing the sampled signal to carry high-frequency harmonic components.High-frequency switching harmonics are introduced into the control loop, resulting in lowfrequency aliasing in regular-sampling pulse-width modulation (PWM) [14].Due to sample aliasing, the grid-connected current of the photovoltaic (PV) inverter may be severely distorted [15].Therefore, the PV grid-connected inverter-based aliasing suppression needs to be further investigated, especially based on engineering applications, in order to improve the PV grid-connected inverter output power quality.
At present, research on aliasing suppression technology mainly focuses on the fields of signal measurement and digital control.The inevitable delay of the digital control inverter will change the phase-frequency characteristics and affect the stability of the Energies 2024, 17, 907 2 of 15 system.In [16], the influence of delay time on system stability is studied systematically, and the critical delay time to keep the system stable is analyzed.Although real-time sampling can effectively reduce the sampling delay and improve the robustness of the active damping control for grid-connected LCL-filtered inverters, this method is limited by the aliasing phenomenon in practical applications [17].Moreover, the phase angle of the aliased signal is closely related to the sampling delay time.In order to clarify the relationship between the aliasing phenomenon and the sampling time, the aliasing analysis model is established in [18].Several classes of methods have been proposed to suppress aliasing in the past.The first method is to suppress aliasing directly by controlling the sample delay time.In order to reveal the characteristic of the aliasing phenomenon in digitally controlled inverters, the effect of the sample delay mainly caused by the sample filter and the analog-to-digital (AD) transfer time on the alias signal is further researched in [19].The selective delay sampling method and complementary sampling method are proposed by controlling the sampling delay.However, the selective delay sampling method is not suitable for three-phase inverters.Moreover, since the method depends on the sampling delay information, which is different for different systems, the generality is weak.Another complementary sampling method also does not completely suppress lower harmonics.Second, multiple samplings in one period can make the low-order aliasing harmonics cancel themselves.Multiple sampling control provides an excellent option for reducing computational delays and pulse width modulation (PWM) delays [20,21].Low odd-order aliasing harmonics can be eliminated by multiple sampling within a control period, and the key is to control the phase difference between the two sampling instants [15].However, conventional multi-sampling methods introduce current ripples into the control loop and degrade the dynamic performance of the control system [22].In [18], this paper uses the double-sampling average current to reconstruct the multi-sampling current to remove most of the switching harmonics and suppress the aliasing phenomenon.But this method fails to completely remove high-frequency switching harmonics and has not been verified experimentally.Third, suppressing low-order harmonics through anti-aliasing filters is also a common approach.In [23], it is verified by simulation that the LCL-LC filter produces much lighter sampling aliasing than the LCL filter.However, this method can affect the sampling accuracy and require additional filtering components.In addition, digital filters are the most common method of filtering out switching harmonics, such as low-pass filters or moving average filters [24].A multiple sampling-based improved repetitive filter was proposed to remove high-frequency switching harmonics in [25].However, repetitive filtering reduces the advantage of multiple sampling and only removes integer switching harmonics, resulting in the presence of low-frequency harmonics in the control loop.A novel hybrid anti-aliasing filter based on a combination of notch and low-pass analog filters was proposed in [26].However, the effect of anti-aliasing filters in the control loop to eliminate switching harmonics is in contradiction with the phase lag, so the introduction of anti-aliasing filters in the control loop as mentioned above will reduce the stability margin and affect the transient response [27,28].Finally, a new harmonic detection and control method is proposed in this paper.Meanwhile, an aliasing suppression method based on oversampling and mean filtering is proposed in conjunction with the proposed harmonic detection and control method.The method works by sampling and controlling harmonic and fundamental currents independently.This method can prevent the collected harmonic current signal from being affected by the high-frequency signal aliasing, which can effectively suppress the low harmonics.At the same time, it can avoid the fundamental current signal delay and improve the stability of the system.
The rest of this article is organized as described further below.Section 2 analyzes the control structure of a three-phase grid-connected photovoltaic inverter, the sampling and aliasing phenomenon of grid-connected inverters based on real-time sampling in a digital control system, and the traditional harmonic extraction and control method.In Section 3, a new method based on multi-sampling and mean filtering is proposed by combining the proposed harmonic detection and control method and independently controlling the fundamental current and selective harmonic currents.Compared with the real-time sampling method, the MATLAB/Simulink-based simulation results are provided to verify the aliasing suppression capability of the proposed method in Section 4. In Section 5, the effectiveness of the proposed method is verified on a 40 KW three-phase grid-tied inverter.Section 6 gives the conclusion.

Aliasing in Digitally Controlled Inverters
The control structure diagram of the three-phase, three-wire grid-connected inverter is shown in Figure 1.The DC power is converted into AC power through the inverter, and then the AC power is transmitted to the grid through the LCL filter.The control system of the inverter consists of a bus voltage loop, a current loop, harmonic detection and control, and a phase-locked loop (PLL).
Energies 2024, 17, x FOR PEER REVIEW 3 of 1 proposed harmonic detection and control method and independently controlling the fun damental current and selective harmonic currents.Compared with the real-time sampling method, the MATLAB/Simulink-based simulation results are provided to verify the alias ing suppression capability of the proposed method in Section 4. In Section 5, the effective ness of the proposed method is verified on a 40 KW three-phase grid-tied inverter.Section 6 gives the conclusion.

Aliasing in Digitally Controlled Inverters
The control structure diagram of the three-phase, three-wire grid-connected inverte is shown in Figure 1.The DC power is converted into AC power through the inverter, and then the AC power is transmitted to the grid through the LCL filter.The control system o the inverter consists of a bus voltage loop, a current loop, harmonic detection and control and a phase-locked loop (PLL).Here, the bus voltage control loop ensures that the DC bus voltage is stable.In th process of harmonic detection and control, the controlled harmonic components in th grid current are extracted by the coordinate transformation method.Then the control re sults are sent into the current control loop for tracking control to suppress harmonics.Th sampled grid voltage is passed through the phase-locked loop to provide the phase angl for the current control loop.A phase-locked loop is used to keep the inverter output cur rent synchronized with the grid voltage.
For the control system, the higher the crossover frequency, the greater the effect o the delay on the phase margin.The delay reduces the anti-interference ability of the sys tem and, in severe cases, makes the system unstable.Therefore, delays should be mini mized in digital control systems.Inherent delays in digitally controlled inverters mainly include computation delays and PWM loading delays [29].The calculation delay is th time interval from the AD sampling instant to the modulation signal loading instant.Off set sampling time is a commonly used method to reduce the calculation delay so that th sampling instant is close to the loading instant.This sampling method will no longer b the standard carrier zero-crossing sampling or periodic sampling, which is called real time sampling, as shown in Figure 2.Although the real-time sampling method can greatly reduce the calculation delay of the system, this method causes the two sampling interval to not be fixed, which will lead to aliasing, affect the sampling accuracy, and deteriorat the power quality.Moreover, when the multiple sampling values are averaged, the stabil ity of the system will be affected.Thus, the significance of real-time sampling is lost.Here, the bus voltage control loop ensures that the DC bus voltage is stable.In the process of harmonic detection and control, the controlled harmonic components in the grid current are extracted by the coordinate transformation method.Then the control results are sent into the current control loop for tracking control to suppress harmonics.The sampled grid voltage is passed through the phase-locked loop to provide the phase angle for the current control loop.A phase-locked loop is used to keep the inverter output current synchronized with the grid voltage.
For the control system, the higher the crossover frequency, the greater the effect of the delay on the phase margin.The delay reduces the anti-interference ability of the system and, in severe cases, makes the system unstable.Therefore, delays should be minimized in digital control systems.Inherent delays in digitally controlled inverters mainly include computation delays and PWM loading delays [29].The calculation delay is the time interval from the AD sampling instant to the modulation signal loading instant.Offset sampling time is a commonly used method to reduce the calculation delay so that the sampling instant is close to the loading instant.This sampling method will no longer be the standard carrier zero-crossing sampling or periodic sampling, which is called real-time sampling, as shown in Figure 2.Although the real-time sampling method can greatly reduce the calculation delay of the system, this method causes the two sampling intervals to not be fixed, which will lead to aliasing, affect the sampling accuracy, and deteriorate the power quality.Moreover, when the multiple sampling values are averaged, the stability of the system will be affected.Thus, the significance of real-time sampling is lost.

Selective Harmonic Detection and Control
The extraction of the selective harmonic component adopts the methods of fractional detection and fractional compensation.In order to improve the steady-state and dynamic performance of the control system, separated low-order harmonic current is both transformed into DC components by respective multiple synchronous reference frames (MSRF) [30], which are controlled by proportion-integral (PI) regulators in order to implement zero steady-state error compensation.Based on the above method, the n-th harmonic can be extracted through the n-th synchronous reference frame (SRF).
In a three-phase, three-wire power system, the grid current usually contains various harmonic currents.Therefore, the three-phase currents with harmonics can be expressed as follows: The current is rotated to the stationary d-q coordinate system, and the following expression is obtained: Here, the selective expressions of each parameter are as follows:

Selective Harmonic Detection and Control
The extraction of the selective harmonic component adopts the methods of fractional detection and fractional compensation.In order to improve the steady-state and dynamic performance of the control system, separated low-order harmonic current is both transformed into DC components by respective multiple synchronous reference frames (MSRF) [30], which are controlled by proportion-integral (PI) regulators in order to implement zero steady-state error compensation.Based on the above method, the n-th harmonic can be extracted through the n-th synchronous reference frame (SRF).
In a three-phase, three-wire power system, the grid current usually contains various harmonic currents.Therefore, the three-phase currents with harmonics can be expressed as follows: The current is rotated to the stationary d-q coordinate system, and the following expression is obtained: Here, the selective expressions of each parameter are as follows: In Equation ( 5), k is an integer other than 0. For the three-phase system, the harmonic components with high content mainly include the 5th and 7th harmonics, and the harmonics above the 7th order are relatively small and are not considered.Therefore, when k = 1, the fundamental signal can be obtained: Energies 2024, 17, 907 5 of 15 It can be known from Equation ( 6) that the main components in the expressions of i d and i q are the DC component and the 6-fold frequency component, of which the 6-fold frequency component is the harmonic component.It can be known from Equation ( 6) that the main components in the expressions of i d and i q are the DC component and the 6-fold frequency component, of which the 6-fold frequency component is the harmonic component.Therefore, it is considered to adopt a nested coordinate system based on Formula (6).The coordinate systems of the two controllers are rotated by −6ω and 6ω, respectively, corresponding to the 5th negative sequence component and the 7th positive sequence component.Thus, single-harmonic extraction is achieved.
The extraction and control block diagram of a single harmonic is shown in Figure 3.The collected three-phase current is transformed into the two-phase rotating coordinate system through Formula (3) to obtain the i α and i β components.Then i α and i β are transformed to the d-q coordinate system by Formula (2) to obtain the i d and i q components of the selective harmonics.The i d and i q components are rotated at −6ω and 6ω, respectively, to obtain the DC components of the 6th wave with different phase sequences.Because this DC component has ripples, a low-pass filter (LPF) is used to filter out high-frequency components.Finally, the data after the PI regulator and C idq inverse transformation is sent to the corresponding control loop.
monics above the 7th order are relatively small and are not considered.Therefore, when k = 1, the fundamental signal can be obtained: cos( )+ cos (6 ) sin( )+ sin( 6) It can be known from Equation ( 6) that the main components in the expressions of id and iq are the DC component and the 6-fold frequency component, of which the 6-fold frequency component is the harmonic component.It can be known from Equation ( 6) that the main components in the expressions of id and iq are the DC component and the 6-fold frequency component, of which the 6-fold frequency component is the harmonic component.Therefore, it is considered to adopt a nested coordinate system based on Formula (6).The coordinate systems of the two controllers are rotated by −6ω and 6ω, respectively, corresponding to the 5th negative sequence component and the 7th positive sequence component.Thus, single-harmonic extraction is achieved.
The extraction and control block diagram of a single harmonic is shown in Figure 3.The collected three-phase current is transformed into the two-phase rotating coordinate system through Formula (3) to obtain the iα and iβ components.Then iα and iβ are transformed to the d-q coordinate system by Formula (2) to obtain the id and iq components of the selective harmonics.The id and iq components are rotated at −6ω and 6ω, respectively, to obtain the DC components of the 6th wave with different phase sequences.Because this DC component has ripples, a low-pass filter (LPF) is used to filter out high-frequency components.Finally, the data after the PI regulator and Cidq inverse transformation is sent to the corresponding control loop.Among them, the frequency domain form of the PI controller is as follows: Here, K p and K i are the proportional coefficient and integral coefficient of the PI controller of the selective harmonic in the rotating coordinate system.
The cutoff frequency of the LPF is set to 10 Hz.Therefore, the expression of the transfer function of the LPF can be written as follows: Energies 2024, 17, 907 6 of 15

Reconstruction of the Control Algorithm Based on Multi-Sampling and Mean Filtering
According to the Shannon sampling theorem [31], when the sampling frequency is less than the Nyquist frequency, the aliasing phenomenon will be inevitable in the sampling process.As can be seen from Section 2, when the switching frequency and the sampling frequency are both equal to 16 kHz, the sampling method is called realtime sampling.However, due to factors such as interrupt loading method, sampling signal timing, switching noise, and non-linearity of reactance, real-time sampling will cause signal aliasing, which affects low-frequency sampling signals and deteriorates loworder harmonic suppression.Therefore, the above control method fails to control the single harmonic.
The multi-sampling technique is used to sample the high signal to obtain all the sampled data.The high-frequency components are then filtered out by mean filtering [32].In this article, the sampling frequency of 128 kHz is used to sample the signal, and the sampling instantaneous value of each switching cycle can be obtained: where k is an even number greater than 0, T s is the switching period, I real is the sampling instantaneous value, and j is R, S, or T. Then average multiple instantaneous values sampled by each interrupt to implement mean filtering, and the following can be obtained: In order to analyze the influence of different multi-sampling values on the system, the transfer functions based on multi-sampling under different conditions are obtained when k is 4, 8, and 16, respectively.The Bode diagram of Formula (10) as shown in Figure 4 can be obtained when k is 4, 8, and 16, respectively.From the amplitude-frequency characteristics of Bode, it can be known that: 1.
Multi-sampling has a large attenuation ability for high-order harmonics, especially the switching frequency harmonics and their multiplier harmonics.Therefore, the influence of high-frequency components, switching frequency harmonics, and multiplied signals on sampling can be better avoided.

2.
The more times of multi-sampling, the lower the cutoff frequency of the filter.

3.
The greater the number of multi-sampling, the greater the delay of the system at half the switching frequency.
Combined with the Bode diagram shown in Figure 4, in order to filter out switching frequency harmonics as much as possible and avoid excessive multi-sampling occupying system resources, k is taken as 8.And subsequent simulations and experiments are based on this value.
In this article, the switching frequency is 16 kHz, and the crossover frequency is one tenth of the switching frequency.It can be seen from the phase-frequency characteristics in Figure 4 that the phase angle has a large lag near the crossover frequency due to the mean filter, which reduces the low phase margin of the system and affects the dynamic performance and stability of the grid-connected current.Therefore, it is not appropriate to use the average value for the inner current loop control, and the entire control system needs to be reconstructed.
A new method based on multi-sampling and mean filtering is reconstructed by independently controlling the fundamental current and the harmonic current in inner loop control.The control block diagram of harmonic control is shown in Figure 5.The selective harmonics are sampled based on the multi-sampling technique, and the sampled harmonic currents are averaged.Finally, an independent coordinate system is used to extract the selective harmonic, while the inner current loop control uses the previous sampling signal and control method, which remains unchanged.In this article, the switching frequency is 16 kHz, and the crossover frequency is one tenth of the switching frequency.It can be seen from the phase-frequency characteristics in Figure 4 that the phase angle has a large lag near the crossover frequency due to the mean filter, which reduces the low phase margin of the system and affects the dynamic performance and stability of the grid-connected current.Therefore, it is not appropriate to use the average value for the inner current loop control, and the entire control system needs to be reconstructed.
A new method based on multi-sampling and mean filtering is reconstructed by independently controlling the fundamental current and the harmonic current in inner loop control.The control block diagram of harmonic control is shown in Figure 5.The selective harmonics are sampled based on the multi-sampling technique, and the sampled harmonic currents are averaged.Finally, an independent coordinate system is used to extract the selective harmonic, while the inner current loop control uses the previous sampling signal and control method, which remains unchanged.In this article, the switching frequency is 16 kHz, and the crossover frequency is one tenth of the switching frequency.It can be seen from the phase-frequency characteristics in Figure 4 that the phase angle has a large lag near the crossover frequency due to the mean filter, which reduces the low phase margin of the system and affects the dynamic performance and stability of the grid-connected current.Therefore, it is not appropriate to use the average value for the inner current loop control, and the entire control system needs to be reconstructed.
A new method based on multi-sampling and mean filtering is reconstructed by independently controlling the fundamental current and the harmonic current in inner loop control.The control block diagram of harmonic control is shown in Figure 5.The selective harmonics are sampled based on the multi-sampling technique, and the sampled harmonic currents are averaged.Finally, an independent coordinate system is used to extract the selective harmonic, while the inner current loop control uses the previous sampling signal and control method, which remains unchanged.

Comparison of Different Harmonic and Aliasing Suppression Methods
Since the fundamental current control loop takes real-time samples and has no additional filters, the control delay is almost equal to zero.Table 1 gives a brief comparison of the different aliasing suppression methods.A large control delay time can degrade the steady and dynamic performance of the system, while the control delay in [16] is related to the duty ratio and the multisampling factor k (the ratio between the sampling frequency and switching frequency).The control delay will increase when the absolute value of the effective duty ratio is greater than 0.875, which will decrease the stability margin [16].The proposed control method and the method proposed in [25] have almost no delay, while other methods inevitably introduce delay.However, the method proposed in [25] has the highest computational complexity compared with other harmonic and aliasing suppression methods and cannot filter out all switching harmonics (SHs).On the other hand, the improved repetitive filter in [23] reduces the advantage of multiple sampling and only removes integer switching harmonics, resulting in the presence of low-frequency harmonics in the control loop.However, the proposed method in this paper can filter out all switching harmonics.Therefore, the analyses show that the proposed method can not only effectively filter out high-frequency harmonics and avoid aliasing but also does not introduce control delays to affect system stability.

Method Harmonic Suppression Delay Time Complexity of the Method
Method in [18] Cannot filter out all SHs T sw /4 or T sw /2 Low Method in [24] Can filter out all SHs T sw /2 Low Method in [25] Can cause high-frequency noise amplification T sw /4 Low Method in [27] Can filter out all main SHs ≈0 High Methods in this paper Can filter out all SHs ≈0 Medium

Simulation Results
In order to verify the abnormal sampling of low-order harmonics caused by sampling aliasing and to verify the proposed reconstruction algorithm based on multi-sampling and mean filtering, this article performs simulation verification on signal extraction based on the MATLAB/Simulink environment.The input signal consists of a fundamental wave with a peak value of 1 and a triangular wave with a frequency of 16 kHz, and the triangular wave component is 10%.The proposed algorithm is verified by different sampling methods.

Simulation Based on Real-Time Sampling
Since the interrupt loading is affected by the content of the interrupt calculation, other interrupts, and the interference of hardware sampling, real-time sampling will cause the time interval of the two sampling periods to be inconsistent.Here, the simulation experiments are carried out with variable sampling frequencies, and sampling is performed at sampling frequencies of 15.87 kHz, 16 kHz, and 16.13 kHz by switching sampling frequencies.Figure 6 is the fast Fourier transform (FFT) analysis result of data obtained by switching sampling frequencies.

Simulation Based on Multi-Sampling and Mean Filtering
Figure 7 shows the fast Fourier transform analysis results of data obtained by switching the sampling frequency at sampling frequencies of 126.96 kHz, 128 kHz, and 129.04 kHz.It can be seen from the simulation results that the low-order harmonic components in the sampling results are almost 0, the 5th harmonic component and the 7th harmonic component are both less than 0.01%, and the 16 kHz harmonic component is attenuated from 10% to less than 0.2%.Therefore, the closed-loop control algorithm based on this sampling method can theoretically completely attenuate the low-order harmonics to zero.Moreover, the sampling data obtained based on the above reconstruction algorithm will neither affect the sampling of the low-frequency signal nor the control of the low-fre- From the above simulation results, real-time sampling will cause low-frequency aliasing to affect the sampling of low-frequency harmonics, which in turn affects the control of low-frequency harmonics.There are more low-order harmonic components in the sampling results, of which the 5th harmonic component is 0.6% and the 7th harmonic component is 0.41%.

Simulation Based on Multi-Sampling and Mean Filtering
Figure 7 shows the fast Fourier transform analysis results of data obtained by switching the sampling frequency at sampling frequencies of 126.96 kHz, 128 kHz, and 129.04 kHz.It can be seen from the simulation results that the low-order harmonic components in the sampling results are almost 0, the 5th harmonic component and the 7th harmonic component are both less than 0.01%, and the 16 kHz harmonic component is attenuated from 10% to less than 0.2%.Therefore, the closed-loop control algorithm based on this sampling method can theoretically completely attenuate the low-order harmonics to zero.Moreover, the sampling data obtained based on the above reconstruction algorithm will neither affect the sampling of the low-frequency signal nor the control of the low-frequency signal.

Simulation Based on Multi-Sampling and Mean Filtering
Figure 7 shows the fast Fourier transform analysis results of data obtained by switching the sampling frequency at sampling frequencies of 126.96 kHz, 128 kHz, and 129.04 kHz.It can be seen from the simulation results that the low-order harmonic components in the sampling results are almost 0, the 5th harmonic component and the 7th harmonic component are both less than 0.01%, and the 16 kHz harmonic component is attenuated from 10% to less than 0.2%.Therefore, the closed-loop control algorithm based on this sampling method can theoretically completely attenuate the low-order harmonics to zero.Moreover, the sampling data obtained based on the above reconstruction algorithm will neither affect the sampling of the low-frequency signal nor the control of the low-frequency signal.

Experimental Verification
In order to verify the correctness of the proposed algorithm and the feasibility of the scheme, an experimental prototype of a three-phase grid-connected photovoltaic inverter with a rated power of 40 kW is built and tested, as depicted in Figure 8.A Chroma programmable regenerative grid simulator of model no.61860 is used to simulate the power grid.Six Chroma programmable photovoltaic simulation power supplies of model no.

Experimental Verification
In order to verify the correctness of the proposed algorithm and the feasibility of the scheme, an experimental prototype of a three-phase grid-connected photovoltaic inverter with a rated power of 40 kW is built and tested, as depicted in Figure 8.A Chroma programmable regenerative grid simulator of model no.61860 is used to simulate the power grid.Six Chroma programmable photovoltaic simulation power supplies of model no.62180H-1800S are used to simulate the photovoltaic array input.Experimental results are obtained from an experimental prototype with the detailed parameters as shown in Table 2.
Energies 2024, 17, x FOR PEER REVIEW 10 of 16 62180H-1800S are used to simulate the photovoltaic array input.Experimental results are obtained from an experimental prototype with the detailed parameters as shown in Table 2.

Experimental Results under Different Operating Conditions
Figure 9 shows the experimental results for 100% rated power without applying the harmonic suppression method.Figure 9a is the output current waveform, and Figure 9b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from Figure 9 that the 5th harmonic component, the 7th harmonic component, and the total harmonic distortion without the harmonic suppression method are 2.078%, 1.114%, and 2.539%, respectively.

Experimental Results under Different Operating Conditions
Figure 9 shows the experimental results for 100% rated power without applying the harmonic suppression method.Figure 9a is the output current waveform, and Figure 9b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from Figure 9 that the 5th harmonic component, the 7th harmonic component, and the total harmonic distortion without the harmonic suppression method are 2.078%, 1.114%, and 2.539%, respectively.Figure 10 shows the experimental results for 100% rated power with the reconstruction algorithm.Figure 10a is the output current waveform.Figure 10b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from the experimental results shown in Figure 10 that the reconstructed control algorithm can attenuate the 5th harmonic component to 0.041%, the 7th harmonic component to 0.046%, and the total harmonic distortion of the grid-connected current to 1.124%.Figure 10 shows the experimental results for 100% rated power with the reconstruction algorithm.Figure 10a is the output current waveform.Figure 10b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from the experimental results shown in Figure 10 that the reconstructed control algorithm can attenuate the 5th harmonic component to 0.041%, the 7th harmonic component to 0.046%, and the total harmonic distortion of the grid-connected current to 1.124%.Figure 11 shows the experimental results for 50% rated power without the harmonic suppression method.Figure 11a is the output current waveform, and Figure 11b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from Figure 11 that the 5th harmonic component, the 7th harmonic component, and the total harmonic distortion without the harmonic suppression method are 2.091%, 1.098%, and 2.857%, respectively.Figure 11 shows the experimental results for 50% rated power without the harmonic suppression method.Figure 11a is the output current waveform, and Figure 11b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from Figure 11 that the 5th harmonic component, the 7th harmonic component, and the total harmonic distortion without the harmonic suppression method are 2.091%, 1.098%, and 2.857%, respectively.Figure 11 shows the experimental results for 50% rated power without the harmonic suppression method.Figure 11a is the output current waveform, and Figure 11b is the FFT analysis result of the output current obtained by the WT3000 power analyzer.It can be seen from Figure 11 that the 5th harmonic component, the 7th harmonic component, and the total harmonic distortion without the harmonic suppression method are 2.091%, 1.098%, and 2.857%, respectively.Figure 12 shows the experimental results for 50% rated power with the reconstruction algorithm.It can be seen from the experimental results shown in Figure 12 that the reconstructed control algorithm can attenuate the 5th harmonic component to 0.235%, the 7th harmonic component to 0.118%, and the total harmonic distortion of the grid-connected current to 1.544%.From Figures 11 and 12, it can be seen that the proposed control method, can also effectively reduce the 5th and 7th harmonics and the total harmonic distortion at 50% rated power.Figure 12 shows the experimental results for 50% rated power with the reconstruction algorithm.It can be seen from the experimental results shown in Figure 12 that the reconstructed control algorithm can attenuate the 5th harmonic component to 0.235%, the 7th harmonic component to 0.118%, and the total harmonic distortion of the grid-connected current to 1.544%.From Figures 11 and 12, it can be seen that the proposed control method, can also effectively reduce the 5th and 7th harmonics and the total harmonic distortion at 50% rated power.

Experimental Results of Dynamic Characteristics
The dynamic experiment is tested as shown in Figure 13, when the output ac power changes from 40 kW to 30 kW and 30 kW to 35 kW, respectively.According to Chinese standard GB/T 37408-2019 [33], the inverter is able to operate continuously in the grid frequency range of 48.5 Hz to 50.5 Hz.The experimental results under different output power conditions are shown in Figure 14, when the grid frequency is abruptly changed from 50 Hz to 49 Hz and 50.5 Hz.From Figures 13 and 14, it can be seen that the proposed control method has good dynamic performance.

Experimental Results of Dynamic Characteristics
The dynamic experiment is tested as shown in Figure 13, when the output ac power changes from 40 kW to 30 kW and 30 kW to 35 kW, respectively.According to Chinese standard GB/T 37408-2019 [33], the inverter is able to operate continuously in the grid frequency range of 48.5 Hz to 50.5 Hz.The experimental results under different output power conditions are shown in Figure 14, when the grid frequency is abruptly changed from 50 Hz to 49 Hz and 50.5 Hz.From Figures 13 and 14, it can be seen that the proposed control method has good dynamic performance.
The dynamic experiment is tested as shown in Figure 13, when the output ac power changes from 40 kW to 30 kW and 30 kW to 35 kW, respectively.According to Chinese standard GB/T 37408-2019 [33], the inverter is able to operate continuously in the grid frequency range of 48.5 Hz to 50.5 Hz.The experimental results under different output power conditions are shown in Figure 14, when the grid frequency is abruptly changed from 50 Hz to 49 Hz and 50.5 Hz.From Figures 13 and 14, it can be seen that the proposed control method has good dynamic performance.Therefore, it can be seen from Figures 9-12 that the 5th and 7th harmonics are effectively suppressed, and the total harmonic distortion of the output current is greatly reduced.The experimental results show that the proposed method in this article can greatly suppress the selective harmonics and reduce the total harmonic distortion.On the other hand, it can be seen from Figures 13 and 14 that the proposed method has good dynamic performance under the conditions of output power perturbation and grid frequency perturbation.The experimental results are consistent with the simulation results, which verifies the correctness of the proposed sampling and control strategy.

Comparison of THD for Different Harmonic Suppression Methods
The performance of different harmonic suppression methods is shown in Table 3 at rated power.Some harmonic suppression methods are provided in [7][8][9][10][11][12][13], especially in [8,10], but the ability of aliasing suppression is not analyzed.The total harmonic distortion (THD), 5th harmonic, and 7th harmonic can be reduced by less than 0.1% by the harmonic and aliasing suppression method in [18], while the method has not been further experimentally verified.Both the proposed method in this paper and the method in [26] are able to reduce the 5th and 7th harmonics to less than 0.1%, and the proposed method in this paper has a lower THD.Therefore, the proposed method in this paper has better harmonic suppression capability.

Methods
5th Harmonic 7th Harmonic THD Method in [7] 0.481% 0.844% 1.690% Method in [8] <0.100% <0.100% 0.940% Therefore, it can be seen from Figures 9-12 that the 5th and 7th harmonics are effectively suppressed, and the total harmonic distortion of the output current is greatly reduced.The experimental results show that the proposed method in this article can greatly suppress the selective harmonics and reduce the total harmonic distortion.On the other hand, it can be seen from Figures 13 and 14 that the proposed method has good dynamic performance under the conditions of output power perturbation and grid frequency perturbation.The experimental results are consistent with the simulation results, which verifies the correctness of the proposed sampling scheme and control strategy.

Comparison of THD for Different Harmonic Suppression Methods
The performance of different harmonic suppression methods is shown in Table 3 at rated power.Some harmonic suppression methods are provided in [7][8][9][10][11][12][13], especially in [8,10], but the ability of aliasing suppression is not analyzed.The total harmonic distortion (THD), 5th harmonic, and 7th harmonic can be reduced by less than 0.1% by the harmonic and aliasing suppression method in [18], while the method has not been further experimentally verified.Both the proposed method in this paper and the method in [26] are able to reduce the 5th and 7th harmonics to less than 0.1%, and the proposed method in this paper has a lower THD.Therefore, the proposed method in this paper has better harmonic suppression capability.

Conclusions
This paper presents the control structure of a three-phase grid-connected photovoltaic inverter and sampling and aliasing in a digital control system.The traditional harmonic current frequency dividing control strategy for a three-phase grid-connected photovoltaic inverter based on multiple synchronous reference frames is derived.Then, an improved harmonic suppression method based on multisampling and mean filtering and a new harmonic detection and control method are proposed.The proposed control method achieves accurate sampling and control of the fundamental and harmonic currents by sampling and controlling the fundamental and harmonic currents independently, so that aliasing is not introduced into the control loop.Therefore, the control delay, the total harmonic distortion, and the low odd harmonic content are effectively reduced under the premise of guaranteeing the stability of the system.
Another contribution of this paper is that MATLAB/Simulink-based simulations and experiments verify that real-time sampling introduces aliasing in the control loop, resulting in an increase in the total harmonic distortion and low-order odd harmonic content.The proposed control method can almost reduce the selective low-order harmonic content to zero at rated power.First, the experimental results have shown that the proposed method can effectively suppress the selective low-order current harmonics, such as the 5th and 7th harmonics.The experimental results are consistent with the simulation results, which verifies the correctness of the proposed sampling scheme and control strategy.Therefore, the scheme can also be applied to the suppression of other harmonics, such as the 11th and 13th harmonics, with high practicality.Secondly, the experimental results have shown that the proposed method has good dynamic performance under the conditions of output power perturbation and grid frequency perturbation.On the other hand, the comparative analysis reveals that the proposed method has some advantages in harmonic and aliasing suppression and control delay.In conclusion, experimental results have shown the good harmonic suppression and anti-aliasing effects of the proposed method and the improvement of control performance.

Figure 2 .
Figure 2. Sampling and loading of a digitally controlled inverter.

Figure 2 .
Figure 2. Sampling and loading of a digitally controlled inverter.

Figure 3 .
Figure 3. Traditional 5th and 7th harmonic detection methods and control block diagram.

Figure 3 .
Figure 3. Traditional 5th and 7th harmonic detection methods and control block diagram.

Energies 2024 ,
17, x FOR PEER REVIEW 7 of 16 system resources, k is taken as 8.And subsequent simulations and experiments are based on this value.

Figure 4 .
Figure 4. Bode diagram of the transfer function based on multi-sampling and mean filtering.

Figure 5 .
Figure 5.The proposed control block diagram of the 5th and 7th harmonics.

Figure 4 .
Figure 4. Bode diagram of the transfer function based on multi-sampling and mean filtering.

Figure 4 .
Figure 4. Bode diagram of the transfer function based on multi-sampling and mean filtering.

Figure 5 .
Figure 5.The proposed control block diagram of the 5th and 7th harmonics.

Figure 5 .
Figure 5.The proposed control block diagram of the 5th and 7th harmonics.

Figure 6 .
Figure 6.FFT analysis result of data obtained by switching sampling frequency based on real-time sampling.

Figure 6 .
Figure 6.FFT analysis result of data obtained by switching sampling frequency based on real-time sampling.

Figure 6 .
Figure 6.FFT analysis result of data obtained by switching sampling frequency based on real-time sampling.

Figure 7 .
Figure 7. FFT analysis result of data obtained by switching sampling frequency based on the proposed method.

Figure 7 .
Figure 7. FFT analysis result of data obtained by switching sampling frequency based on the proposed method.

Figure 9 .Figure 9 .
Figure 9.The experimental results for 100% rated power without the harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 10 .
Figure 10.The experimental results for 100% rated power with the proposed harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 10 .
Figure 10.The experimental results for 100% rated power with the proposed harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 10 .
Figure 10.The experimental results for 100% rated power with the proposed harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 11 .
Figure 11.The experimental results for 50% rated power without the harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 11 .
Figure 11.The experimental results for 50% rated power without the harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 12 .
Figure 12.The experimental results for 50% rated power with the proposed harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 12 .
Figure 12.The experimental results for 50% rated power with the proposed harmonic suppression algorithm: (a) grid-connected current waveform; (b) the FFT analysis result of the output current.

Figure 13 .
Figure 13.Measured grid-injected current of the proposed system: (a) when the output ac power changes from 40 kW to 30 kW; (b) when the output ac power changes from 30 kW to 35 kW.

Figure 13 .Figure 14 .
Figure 13.Measured grid-injected current of the proposed system: (a) when the output ac power changes from 40 kW to 30 kW; (b) when the output ac power changes from 30 kW to 35 kW.Energies 2024, 17, x FOR PEER REVIEW 13 of 16

Figure 14 .
Figure 14.The FFT analysis results of the output current under different output power and grid frequencies: (a) 100% rated power and 49 Hz; (b) 100% rated power and 50.5 Hz; (c) 50% rated power and 49 Hz; (d) 50% rated power and 50.5 Hz.

Table 1 .
Comparison of different aliasing suppression methods.

Table 2 .
Main parameters of a grid-connected inverter.Symbol Quantity Value LA/LB/LC Inverter-side filter inductor 185 uH LgA/LgB/LgC Grid-side inductor 12 uH

Table 2 .
Main parameters of a grid-connected inverter.

Table 2 .
Main parameters of a grid-connected inverter.

Table 3 .
Comparison of THD for different harmonic suppression methods.

Table 3 .
Comparison of THD for different harmonic suppression methods.