A Frequency Adaptive Scheme Based on Newton Structure of PRRC for LCL-Type Inverter Connected with Weak Grid

Repetitive control (RC) is gradually used in inverters tied with weak grid. To achieve the zero steady-state error tracking of inverter current and compensate the harmonic distortion caused by frequency fluctuation, a frequency adaptive (FA) control scheme for LCL-type inverter connected with weak grid is proposed. This scheme adopts a proportional resonance (PR) controller in parallel with RC (PRRC) to overcome the disadvantages caused by RC inherent one-cycle time delay. A fractional delay (FD) filter based on the Newton structure is proposed to approximate the fraction item of fs/f, where fs and f are sample frequency and grid frequency, respectively. The structure of the proposed FD filter is relatively simple; moreover, coefficients of the filter maintain constant so as not to need online tuning even when grid frequency fluctuates, which decreases the computational burden considerably. The feasibility and effectiveness of the proposed FA control scheme, named as Newton-FAPRRC, are all verified by the simulation and experimental results.


Introduction
With the growing increase of distributed generation (DG) penetration [1,2], the impedance of conventional ideal grid can no longer be ignored and the grid presents the characteristics of weak grid [3,4]. Under the weak grid condition, the current harmonics of grid-connected inverter will be further amplified and grid frequency fluctuation will be more serious than ideal grid, which may even lead to the problems of system stability. To improve the power quality, control schemes of grid-connected inverter have been investigated extensively and many controllers are developed, such as proportional integral (PI) controller [5,6], proportional resonant (PR) controller [6,7], deadbeat (DB) controller [8], RC [9][10][11], etc.
The PI control is widely used for inverters due to its relatively easy implementation [6]. However, limited by the control gain, PI control cannot achieve zero steady-state error tracking for periodic signals in α, β coordinate [12]. To improve control accuracy, a common method is to transfer state variable equations to dq coordinate. However, mutual coupling of the d-axis and q-axis equations will significantly increase the complexity of controller design [13]. PR control can achieve zero steady-state error tracking for periodic signals of a specific frequency (resonant frequency of the PR controller) in the α, β coordinate due to its infinite gain at the resonant frequency. However, the disadvantage of PR is that it is difficult to suppress the periodic harmonics that are not of the resonance frequency of PR [7,14]. Multi-PR control is proposed, i.e., several resonant controllers are parallel connected and every resonant controller responds for the suppression of a harmonic at a specific resonance frequency [15]. Obviously, this scheme is complex because the control effect depends on the large number of resonant controllers. Moreover, if any harmonic with unexpected resonant frequency while having relatively large magnitude appears in the system, the control accuracy will be greatly deteriorated. DB control is a type of predictive

Configuration of LCL-Type Grid-Connected Inverter System
The configuration and control scheme of investigated inverter are shown in Figure 1. C is the filter capacitor of LCL filter; L1 and L2 are the filter inductance of LCL filter; Zg is the grid impedance; iL, ig and ic are the output current of inverter, grid-connected current and capacitor current respectively; vc, vpcc and vg are the capacitor voltage, the voltage at the point of common coupling and grid voltage, respectively; The reference current iref of grid connected inverter is calculated from command Iref. GCFAD is the abbreviation of grid current feedback active damping, which is used to suppress the inherent resonance peak of LCL filter. Figure 2 shows the control structure diagram of grid-connected inverter, which adopts a control scheme of PR in parallel with RC (PRRC). Gpr and Grc are the transfer functions of PR control and RC, respectively; vin is the output voltage of inverter; H is the transfer function of GCFAD; kpwm is the gain of inverter bridges; Gd is the time delay of the system, which includes times of PWM transfer and program calculation. The expression of Gd is shown in (1) [1]; Tc is the sampling period. (1 ) ( ) C is the filter capacitor of LCL filter; L 1 and L 2 are the filter inductance of LCL filter; Z g is the grid impedance; i L , i g and i c are the output current of inverter, gridconnected current and capacitor current respectively; v c , v pcc and v g are the capacitor voltage, the voltage at the point of common coupling and grid voltage, respectively; The reference current i ref of grid connected inverter is calculated from command I ref . GCFAD is the abbreviation of grid current feedback active damping, which is used to suppress the inherent resonance peak of LCL filter. Figure 2 shows the control structure diagram of grid-connected inverter, which adopts a control scheme of PR in parallel with RC (PRRC). G pr and G rc are the transfer functions of PR control and RC, respectively; v in is the output voltage of inverter; H is the transfer function of GCFAD; k pwm is the gain of inverter bridges; G d is the time delay of the system, which includes times of PWM transfer and program calculation. The expression of G d is shown in (1) [1]; T c is the sampling period.

Configuration of LCL-Type Grid-Connected Inverter System
The configuration and control scheme of investigated inverter are shown in Figure 1. C is the filter capacitor of LCL filter; L1 and L2 are the filter inductance of LCL filter; Zg is the grid impedance; iL, ig and ic are the output current of inverter, grid-connected current and capacitor current respectively; vc, vpcc and vg are the capacitor voltage, the voltage at the point of common coupling and grid voltage, respectively; The reference current iref of grid connected inverter is calculated from command Iref. GCFAD is the abbreviation of grid current feedback active damping, which is used to suppress the inherent resonance peak of LCL filter. Figure 2 shows the control structure diagram of grid-connected inverter, which adopts a control scheme of PR in parallel with RC (PRRC). Gpr and Grc are the transfer functions of PR control and RC, respectively; vin is the output voltage of inverter; H is the transfer function of GCFAD; kpwm is the gain of inverter bridges; Gd is the time delay of the system, which includes times of PWM transfer and program calculation. The expression of Gd is shown in (1) [1]; Tc is the sampling period.

Proposed PRRC
Actually, the adopted GCFAD is a high pass filter [24] and its one order expression is: where kc is the gain and ωc is the cut-off frequency of the high pass filter. According to Figure 3, the transfer function of LCL filter in s domain can be obtained. Actually, the adopted GCFAD is a high pass filter [24] and its one order expression is: where k c is the gain and ω c is the cut-off frequency of the high pass filter. According to Figure 3, the transfer function of LCL filter in s domain can be obtained. Figure 3. Control structure diagram of grid-connected LCL inverter in z domain.
Applying a bilinear transformation to (1), (2) and (3), the transfer function of the inner closed loop in z domain is: Figure 3 is the control structure diagram of grid-connected LCL inverter in z domain, in which Q(z) is a constant introduced in the feedback branch to improve the stability of RC [25,26].
( ) c G z is the compensator of RC, which is used to compensate the amplitude and phase of the equivalent control object of RC, i.e., the transfer function of ig to yrc. The transfer functions of RC and PR in z-domain are as follows: In (5), kp and ki are the proportional gain and integral gain of PR; ωi is the bandwidth of PR; ω0 is the grid angular frequency, and 0 =2 f   .

FARC Based on Newton Structure
For weak grids, f often fluctuates severely and N contains an integer term i N and a fractional term d. Therefore, the time delay z −N can be expressed as [27]: In this paper, a FD filter based on the Newton structure is proposed to approximate fractional delay item d z  , and frequency adaptation of RC (FARC) can be expected. The relationship of d and d will be introduced in Section 3.2. and corresponding The proposed scheme is shown in Figure 4. Applying a bilinear transformation to (1), (2) and (3), the transfer function of the inner closed loop in z domain is: Figure 3 is the control structure diagram of grid-connected LCL inverter in z domain, in which Q(z) is a constant introduced in the feedback branch to improve the stability of RC [25,26]. G c (z) is the compensator of RC, which is used to compensate the amplitude and phase of the equivalent control object of RC, i.e., the transfer function of i g to y rc .
The transfer functions of RC and PR in z-domain are as follows: In (5), k p and k i are the proportional gain and integral gain of PR; ω i is the bandwidth of PR; ω 0 is the grid angular frequency, and ω 0 = 2π f .

FARC Based on Newton Structure
For weak grids, f often fluctuates severely and N contains an integer term N i and a fractional term d. Therefore, the time delay z −N can be expressed as [27]: In this paper, a FD filter based on the Newton structure is proposed to approximate fractional delay item z −d , and frequency adaptation of RC (FARC) can be expected. The relationship ofd and d will be introduced in Section 3.2. and corresponding N i = N −d. The proposed scheme is shown in Figure 4.
Applying a bilinear transformation to (1), (2) and (3), the transfer function of the inner closed loop in z domain is: Figure 3 is the control structure diagram of grid-connected LCL inverter in z domain, in which Q(z) is a constant introduced in the feedback branch to improve the stability of RC [25,26].
( ) c G z is the compensator of RC, which is used to compensate the amplitude and phase of the equivalent control object of RC, i.e., the transfer function of ig to yrc. The transfer functions of RC and PR in z-domain are as follows: In (5), kp and ki are the proportional gain and integral gain of PR; ωi is the bandwidth of PR; ω0 is the grid angular frequency, and 0 =2 f   .

FARC Based on Newton Structure
For weak grids, f often fluctuates severely and N contains an integer term i N and a fractional term d. Therefore, the time delay z −N can be expressed as [27]: In this paper, a FD filter based on the Newton structure is proposed to approximate fractional delay item d z  , and frequency adaptation of RC (FARC) can be expected. The relationship of d and d will be introduced in Section 3.2. and corresponding The proposed scheme is shown in Figure 4.   It should be noted that there is also a z −N in the forward branch of RC, but it will only affect the phase of the system slightly due to high sampling frequency in this paper. Therefore, no FD compensation is made for it.

Transformation from Farrow to Newton
In order to realize the implementation of various interpolation methods in Newton structure and simplify the structure of FD filter, a Newton structure FD filter is derived from a Farrow structure, which is shown in Figure 4. The expression of Farrow structure [28] is represented as (7).
In (7), T ; M is the number of the sub-filters contained in Farrow FD filter; (N ' -1) is the order of sub-filter; and C is the coefficient matrix of Farrow FD filter. The expression of Newton structure [28] can be represented as (8).
In (8), ; andĈ is the coefficient matrix of Newton FD filter.
The transformation from Farrow structure to Newton structure can be realized by (9).
where T d and T z are the transformation matrices that transform D and Z intoD andẐ, respectively.D = T d D, T d can be calculated as: In (10), T d is the matrix with each row i containing the coefficients of the polynomial (d + M−1 2 ) i−1 , and elements of the matrix T d are calculated by (11): The elements in the matrix T d are the Stirling numbers of the first kind S Transformation matrix can be expressed as follows:

Three-Order Newton FD Filter
Because the third-order filter has the best trade-off between structure complexity and approximate effect [29,30], M, N ' are set the value of 4. The coefficient matrix C spline of the third-order filter with Farrow structure is calculated by using spline interpolation and is shown as (13) [29]: The coefficient matrix of the third-order FD filter based on Newton structure can be obtained as (14).
The structure of third-order Newton FD filter can be obtained as Figure 5.
The coefficient matrix of the third-order FD filter based on Newton structure can be obtained as (14).
The structure of third-order Newton FD filter can be obtained as Figure 5. Figure 5. Structure of third-order Newton FD filter.
According to Figure 5, the proposed Newton FD filter has the characteristics of constant coefficients and relatively simple structure, which significantly reduces the computational burden and implementation complexity [31].

Performance Analysis of Newton-FAPRRC
To verify the performance of proposed Newton-FAPRRC, the amplitude frequency characteristics of conventional PRRC (CPRRC) and Newton-FAPRRC are both shown in Figure 6.  According to Figure 5, the proposed Newton FD filter has the characteristics of constant coefficients and relatively simple structure, which significantly reduces the computational burden and implementation complexity [31].

Performance Analysis of Newton-FAPRRC
To verify the performance of proposed Newton-FAPRRC, the amplitude frequency characteristics of conventional PRRC (CPRRC) and Newton-FAPRRC are both shown in Figure 6.
The coefficient matrix of the third-order FD filter based on Newton structure can be obtained as (14).
The structure of third-order Newton FD filter can be obtained as Figure 5. Figure 5. Structure of third-order Newton FD filter.
According to Figure 5, the proposed Newton FD filter has the characteristics of constant coefficients and relatively simple structure, which significantly reduces the computational burden and implementation complexity [31].

Performance Analysis of Newton-FAPRRC
To verify the performance of proposed Newton-FAPRRC, the amplitude frequency characteristics of conventional PRRC (CPRRC) and Newton-FAPRRC are both shown in Figure 6.  Newton-FAPRRC is 31 dB at 7th harmonic frequency (f = 344.4 Hz), while the amplitude of CPRRC is 27 dB. If grid frequency is 50.8 Hz, the amplitude of CPRRC is considerably reduced to 18 dB while Newton-FAPRRC keeps unchanged. The same conclusions also can be drawn for other harmonics having different frequencies. It is obviously that the tracking effect of Newton-FAPRRC is much better than CPRRC due to its higher amplitude when grid frequency fluctuates severely.

Parameters of PRRC
From Figure 3, the closed-loop transfer function of the system can be derived as follows: According to (15), the characteristic equation of the closed-loop system can be derived as follows: P(z) is the control object of RC, the expression is as follows: In order to keep the system stable, N roots of the characteristic (16) should be inside the unit circle. However, N is a relatively large figure factually (generally, the f s is 10 kHz and N is 200), and it is difficult to judge the system stability by conventional stability criterion. According to (16) and replacing z with e jwTs , a sufficient condition for system stability is adopted by using the small gain theorem, which is shown in (18).

Parameters of PR and GCFAD
Parameters of investigated LCL inverter and controller (as shown in Figure 1) are shown in Table 1, where the parameters of PR and GCFAD control are determined according to the design principles of [24] and [32]. Q(z) in RC is adopted intentionally to improve system stability. It can be a constant less than 1, and the smaller value of Q(z) means a better stability and a worse zero-error characteristic of RC. Therefore, a trade-off is made and Q(z) is set as 0.98 in this paper.

Parameters of Gc
Gc is the compensator of RC. The expression is as follows: where k r is the gain compensation of the system; z m is the phase lead compensation of the system; and S(z) is a filter to accelerate decay rate of high frequency band of the system.

Filter S(z)
In order to accelerate the attenuation of the high-frequency band of P(z), a fourth-order Butterworth filter is designed by the Filter Designer of MATLAB, and the cut-off frequency of filter is 1 kHz. The transfer function is as follows: Figure 7 shows that with the introducing of Butterworth filter, the gain of P(z) in high-frequency band will be further decreased, which means a more efficient suppression is achieved of harmonics with frequency higher than 1000 kHz. However, the phase lag is more serious at the same time.

Setting of Q(z)
Q(z) in RC is adopted intentionally to improve system stability. It can be a constant less than 1, and the smaller value of Q(z) means a better stability and a worse zero-error characteristic of RC. Therefore, a trade-off is made and Q(z) is set as 0.98 in this paper.

Parameters of Gc
Gc is the compensator of RC. The expression is as follows: where kr is the gain compensation of the system; z m is the phase lead compensation of the system; and S(z) is a filter to accelerate decay rate of high frequency band of the system.

Filter S(z)
In order to accelerate the attenuation of the high-frequency band of P(z), a fourthorder Butterworth filter is designed by the Filter Designer of MATLAB, and the cut-off frequency of filter is 1 kHz. The transfer function is as follows:  Figure 7 shows that with the introducing of Butterworth filter, the gain of P(z) in high-frequency band will be further decreased, which means a more efficient suppression is achieved of harmonics with frequency higher than 1000 kHz. However, the phase lag is more serious at the same time.

Phase Compensator z m
In order to compensate the phase lag induced by P(z) and S(z), a z m is introduced in G c (z). Figure 8 shows the phase frequency characteristic of z m S(z)P(z) with different m values. It can be seen that when m equals 9, the system phase is closer to 0 • and the compensation effect is the optimal, therefore, m is set to 9 in this paper.

The Gain k r
The gain k r is usually taken as a constant less than 1 to adjust the compensation intensity of RC. According to (17), in the whole process of ω increasing from zero frequency to Nyquist frequency (1/2 sampling frequency), the stability of the control system can be guaranteed if the trajectory of G(e jωT s ) does not exceed the unit circle. Figure 9 shows the trajectory of G F (e jωT s ) when k r changes from 0.2 to 0.8. In order to compensate the phase lag induced by P(z) and S(z), a z m is introduced in Gc(z). Figure 8 shows the phase frequency characteristic of z m S(z)P(z) with different m values. It can be seen that when m equals 9, the system phase is closer to 0° and the compensation effect is the optimal, therefore, m is set to 9 in this paper.

The Gain kr
The gain kr is usually taken as a constant less than 1 to adjust the compensation intensity of RC. According to (17), in the whole process of ω increasing from zero frequency to Nyquist frequency (1/2 sampling frequency), the stability of the control system can be guaranteed if the trajectory of ( ) s j T G e  does not exceed the unit circle. Figure 9 shows the trajectory of ( ) In general, kr is smaller than 1 [33] and because the value of kr is inversely proportional to the system stability and is directly proportional to the compensation accuracy, kr is taken as 0.6 in this paper.

Stability Analysis of FAPRRC
The feedback branch of Figure 4 can be written as . Thus, when the proposed Newton FD filter is adopted, the transfer function of Figure 4 GFARC can be expressed by (22):

The Gain kr
The gain kr is usually taken as a constant less than 1 to adjust the compensation intensity of RC. According to (17), in the whole process of ω increasing from zero frequency to Nyquist frequency (1/2 sampling frequency), the stability of the control system can be guaranteed if the trajectory of ( ) s j T G e  does not exceed the unit circle. Figure 9 shows the trajectory of ( ) In general, kr is smaller than 1 [33] and because the value of kr is inversely proportional to the system stability and is directly proportional to the compensation accuracy, kr is taken as 0.6 in this paper.

Stability Analysis of FAPRRC
The feedback branch of Figure 4 can be written as . Thus, when the proposed Newton FD filter is adopted, the transfer function of Figure 4 GFARC can be expressed by (22): In general, k r is smaller than 1 [33] and because the value of k r is inversely proportional to the system stability and is directly proportional to the compensation accuracy, k r is taken as 0.6 in this paper.

Stability Analysis of FAPRRC
The feedback branch of Figure 4 can be written as z −N i z −d Q(z). Thus, when the proposed Newton FD filter is adopted, the transfer function of Figure 4 G FARC can be expressed by (22): Based on (22), the Q(z) in (18) adds a coefficient z −N i z −d . Replace z with e jwTs and combine it with (18) to obtain the stability condition of FAPRRC: Sinced is very small comparing to N and N i is close to N, according to (19), (24) can be obtained: According to (23) and (24), the stability conditions of FAPRRC and PRRC are the same, i.e., the trajectory of G F (e jωT s ) is inside the unit circle.
In the following analysis, the resistance of weak grid is neglected because it benefits to system stability. Therefore, the grid impedance can be represented by a inductances L g and the grid-side inductance L 2 of LCL filter is (L 2 + L g ). Presently, power systems with short-circuit ratio (SCR) less than 10 are often regarded as weak grids [34]. In this paper, SCR is 10 corresponds to L g is 2.5 mH. Figure 10a-d shows the trajectory of G F (e jωT s ) when L g is 0 mH, 1 mH, 3 mH and 5 mH respectively. It can be seen that even when L g is as large as 5 mH, which corresponds to SCR = 5, the trajectory of G F (e jωT s ) is still inside the unit circle and the system keeps stable.  (24) According to (23) and (24), the stability conditions of FAPRRC and PRRC are the same, i.e., the trajectory of ( ) s j T F G e  is inside the unit circle.
In the following analysis, the resistance of weak grid is neglected because it benefits to system stability. Therefore, the grid impedance can be represented by a inductances Lg and the grid-side inductance L2 of LCL filter is (L2 + Lg). Presently, power systems with short-circuit ratio (SCR) less than 10 are often regarded as weak grids [34]. In this paper, SCR is 10 corresponds to Lg is 2.5 mH. Figure 10a-d shows the trajectory of ( ) s j T F G e  when Lg is 0 mH, 1 mH, 3 mH and 5 mH respectively. It can be seen that even when Lg is as large as 5 mH, which corresponds to SCR = 5, the trajectory of

Simulation and Experiment
In order to verify the effectiveness of the proposed Newton-FAPRRC scheme, the simulation was carried out under Matlab/Simulink and the corresponding experimental platform was also built in our laboratory.

Simulation
In order to verify the effectiveness of the proposed Newton-FAPRRC scheme for LCLtype inverter tied with weak grid, simulations of three cases were carried out: (1) adaptability of Newton-FAPRRC under weak grid condition; (2) verification of Newton-FAPRRC dynamic performance; (3) verification of Newton-FAPRRC frequency adaptability.

Adaptability of Newton FAPRRC under Weak Grid
Simulation was carried out when L g is 0, 3 and 5 mH, respectively, and parameters used are listed as in Table 1 of the Section 4. Figures 11-13 indicate that harmonic of the voltage at the PCC v pcc increased significantly with the increasing of L g and its THD rises from 0.83% at 3 mH to 1.22% at 5 mH. The grid current i g has the opposite trend with v pcc . When grid inductance was 0 mH, 3 mH and 5 mH, the THD of i g was 1.43%, 1.19% and 1.15%, respectively, which means that a larger L g has a better filter effect of grid-current harmonics. Figures 13-15 also show the current tracking error ε = i re f − i g ; it can be seen that ε decreases when the Lg increases. The grid current ig has the opposite trend with vpcc. When grid inductance was 0 mH, 3 mH and 5 mH, the THD of ig was 1.43%, 1.19% and 1.15%, respectively, which means that a larger Lg has a better filter effect of grid-current harmonics. Figures 13-15 Figure 14 shows the dynamic response of the system when the grid current reference signal iref is set to jump from 14 A to 10.5 A at 0.4 s. It can be seen that even with the large jump of the command, the proposed FAPRRC can accurately track the reference signal and the transition time is as short as about 5 ms. According to Figures 15-17, for CPRRC, the THD reaches 2.82% and the maximum ε approaches 1.5 A. The steady-state responses of Farrow-FAPRRC and Newton-FOPRRC are similar. The THD of ig_c is 1.15% and 1.16%, respectively. At the same time, the ε is greatly reduced and the maximum ε is less than 1 A. Figures 15-17 indicate that FAPRRC still has high tracking accuracy even when grid frequency fluctuates, and the quality of ig is significantly improved compared to CPRRC.   Figure 14 shows the dynamic response of the system when the grid current reference signal iref is set to jump from 14 A to 10.5 A at 0.4 s. It can be seen that even with the large jump of the command, the proposed FAPRRC can accurately track the reference signal and the transition time is as short as about 5 ms. According to Figures 15-17, for CPRRC, the THD reaches 2.82% and the maximum ε approaches 1.5 A. The steady-state responses of Farrow-FAPRRC and Newton-FOPRRC are similar. The THD of ig_c is 1.15% and 1.16%, respectively. At the same time, the ε is greatly reduced and the maximum ε is less than 1 A. Figures 15-17 indicate that FAPRRC still has high tracking accuracy even when grid frequency fluctuates, and the quality of ig is significantly improved compared to CPRRC.  More detailed simulation results at different grid frequencies are given in Table 2. It should be noted that although both Newton and Farrow structure filters have the advantages of constant coefficients and high quality of ig, the proposed Newton FD filter has simpler constructure and is easier to realize, which significantly reduces the computational burden [31] compared to the Farrow FD filter which compromises M sub-filters.

Experiment
A downscale experiment was held to verify above research. Figure 18 is the experimental platform of three-phase grid-connected LCL inverter. In the experiment, vpcc and ig were reduced to one-third of the simulated value, and other parameters remained unchanged.   According to Figures 15-17, for CPRRC, the THD reaches 2.82% and the maximum ε approaches 1.5 A. The steady-state responses of Farrow-FAPRRC and Newton-FOPRRC are similar. The THD of i g_c is 1.15% and 1.16%, respectively. At the same time, the ε is greatly reduced and the maximum ε is less than 1 A. Figures 15-17 indicate that FAPRRC still has high tracking accuracy even when grid frequency fluctuates, and the quality of i g is significantly improved compared to CPRRC.
More detailed simulation results at different grid frequencies are given in Table 2. It should be noted that although both Newton and Farrow structure filters have the advantages of constant coefficients and high quality of i g , the proposed Newton FD filter has simpler constructure and is easier to realize, which significantly reduces the computational burden [31] compared to the Farrow FD filter which compromises M sub-filters. Table 2. The THD of grid-current i g with different grid frequencies.

Experiment
A downscale experiment was held to verify above research. Figure 18 is the experimental platform of three-phase grid-connected LCL inverter. In the experiment, v pcc and i g were reduced to one-third of the simulated value, and other parameters remained unchanged. More detailed simulation results at different grid frequencies are given in Table 2. It should be noted that although both Newton and Farrow structure filters have the advantages of constant coefficients and high quality of ig, the proposed Newton FD filter has simpler constructure and is easier to realize, which significantly reduces the computational burden [31] compared to the Farrow FD filter which compromises M sub-filters.

Experiment
A downscale experiment was held to verify above research. Figure 18 is the experimental platform of three-phase grid-connected LCL inverter. In the experiment, vpcc and ig were reduced to one-third of the simulated value, and other parameters remained unchanged.  Similar to simulation, the experiment was also carried out with three scenes: (1) adaptability of FAPRRC under weak grid condition; (2) verification of Newton-FAPRRC dynamic performance; (3) verification of Newton-FAPRRC frequency adaptability.

Adaptability of Newton FAPRRC under Weak Grid Condition
In experiments, L g was also set to 0 mH, 3 mH and 5 mH respectively. The corresponding steady-state responses, including voltage at PCC of phase C, v pcc_c and i g , are shown in Figures 19-21      It can be seen that with the increase of L g , the harmonic of v pcc_c also increases. The THD of v pcc_c is 0.72%, 0.95% and 1.17% when L g is 0 mH, 3 mH and 5 mH, respectively. However, the quality of i g has a certain improvement. Figure 21 shows that even when L g is as large as 5 mH, the system is still stable and the THD i g_c is 1.92%. It is obvious that the proposed control scheme has a relatively large stable range under weak grid condition. It can be seen that with the increase of Lg, the harmonic of vpcc_c also increases. The THD of vpcc_c is 0.72%, 0.95% and 1.17% when Lg is 0 mH, 3 mH and 5 mH, respectively. However, the quality of ig has a certain improvement. Figure 21 shows that even when Lg is as large as 5 mH, the system is still stable and the THD ig_c is 1.92%. It is obvious that the proposed control scheme has a relatively large stable range under weak grid condition.

Verification of Newton-FAPRRC Dynamic Performance
In the experiment, the reference value of grid-current iref was operated to jump from 4.6 A to 3.5 A at time t and the transient rate achieves as 24%. The dynamic response is shown in Figure 22. It can be concluded that ig can track the reference value iref effectively and appears perfect dynamic performance.
3.5A t Figure 22. Experimental results of the dynamic response with iref mutation.

Verification of Newton-FAPRRC Frequency Adaptability
Because the grid frequency cannot be adjusted in our lab, the variable grid frequency was simulated by changing the sampling frequency in experiment. To correspond to grid frequency 50.8, the sampling frequency was set to 9920 Hz. It should be noted that the little mismatch between simulated and measured results for harmonic spectrums and THD was caused by the influence of the solver and the harmonic spectrum acquisition algorithm of Matlab/Simulink. In fact, Matlab can give a more detailed harmonic spectrum,

Verification of Newton-FAPRRC Dynamic Performance
In the experiment, the reference value of grid-current i ref was operated to jump from 4.6 A to 3.5 A at time t and the transient rate achieves as 24%. The dynamic response is shown in Figure 22. It can be concluded that i g can track the reference value i ref effectively and appears perfect dynamic performance.  It can be seen that with the increase of Lg, the harmonic of vpcc_c also increases. The THD of vpcc_c is 0.72%, 0.95% and 1.17% when Lg is 0 mH, 3 mH and 5 mH, respectively. However, the quality of ig has a certain improvement. Figure 21 shows that even when Lg is as large as 5 mH, the system is still stable and the THD ig_c is 1.92%. It is obvious that the proposed control scheme has a relatively large stable range under weak grid condition.

Verification of Newton-FAPRRC Dynamic Performance
In the experiment, the reference value of grid-current iref was operated to jump from 4.6 A to 3.5 A at time t and the transient rate achieves as 24%. The dynamic response is shown in Figure 22. It can be concluded that ig can track the reference value iref effectively and appears perfect dynamic performance. t Figure 22. Experimental results of the dynamic response with iref mutation.

Verification of Newton-FAPRRC Frequency Adaptability
Because the grid frequency cannot be adjusted in our lab, the variable grid frequency was simulated by changing the sampling frequency in experiment. To correspond to grid frequency 50.8, the sampling frequency was set to 9920 Hz. It should be noted that the little mismatch between simulated and measured results for harmonic spectrums and THD was caused by the influence of the solver and the harmonic spectrum acquisition algorithm of Matlab/Simulink. In fact, Matlab can give a more detailed harmonic spectrum,

Verification of Newton-FAPRRC Frequency Adaptability
Because the grid frequency cannot be adjusted in our lab, the variable grid frequency was simulated by changing the sampling frequency in experiment. To correspond to grid frequency 50.8, the sampling frequency was set to 9920 Hz. It should be noted that the little mismatch between simulated and measured results for harmonic spectrums and THD was caused by the influence of the solver and the harmonic spectrum acquisition algorithm of Matlab/Simulink. In fact, Matlab can give a more detailed harmonic spectrum, while the oscilloscope of our laboratory superimposes many non-integer harmonic components into the closest integer harmonic components.
Experimental results of the CPRRC, Farrow-FAPRRC and Newton-FAPRR (L g = 0) are shown in the Figures 23-25 respectively. Figures 23a, 24a and 25a shows the steady state responses of i g ; Figures 23b, 24b and 25b and Figures 23c, 24c and 25c show the harmonic spectrum of i g_c and current tracking error ε.
while the oscilloscope of our laboratory superimposes many non-integer harmonic components into the closest integer harmonic components.
Experimental results of the CPRRC, Farrow-FAPRRC and Newton-FAPRR (Lg = 0) are shown in the Figures 23-25 respectively. Figures 23-25a shows the steady state responses of ig; Figure

Figures 23-25
indicate that compared with the other two FAPRRC control schemes, the harmonics of CPRRC is significantly larger and the distortion is also more serious. The THD of ig_c achieves 6.34% when CPRRC is adopted. From Figure 23c, Figure 24c, Figure  25c, ε of Newton and Farrow-FAPRRC is as small as about 1 A, but for CPRRC, it can reach about 2 A. Figures 24 and 25 show that Newton-FAPRRC and Farrow-FAPRRC have similar control effect and the THD of ig is 2.41% and 2.57%, respectively. while the oscilloscope of our laboratory superimposes many non-integer harmonic components into the closest integer harmonic components.
Experimental results of the CPRRC, Farrow-FAPRRC and Newton-FAPRR (Lg = 0) are shown in the Figures 23-25 respectively. Figures 23-25a shows the steady state responses of ig; Figure

Figures 23-25
indicate that compared with the other two FAPRRC control schemes, the harmonics of CPRRC is significantly larger and the distortion is also more serious. The THD of ig_c achieves 6.34% when CPRRC is adopted. From Figure 23c, Figure 24c, Figure  25c, ε of Newton and Farrow-FAPRRC is as small as about 1 A, but for CPRRC, it can reach about 2 A. Figures 24 and 25 show that Newton-FAPRRC and Farrow-FAPRRC have similar control effect and the THD of ig is 2.41% and 2.57%, respectively. while the oscilloscope of our laboratory superimposes many non-integer harmonic components into the closest integer harmonic components.
Experimental results of the CPRRC, Farrow-FAPRRC and Newton-FAPRR (Lg = 0) are shown in the Figures 23-25 respectively. Figures 23-25a shows the steady state responses of ig; Figure [23][24][25] indicate that compared with the other two FAPRRC control schemes, the harmonics of CPRRC is significantly larger and the distortion is also more serious. The THD of ig_c achieves 6.34% when CPRRC is adopted. From Figure 23c, Figure 24c, Figure  25c, ε of Newton and Farrow-FAPRRC is as small as about 1 A, but for CPRRC, it can reach about 2 A. Figures 24 and 25 show that Newton-FAPRRC and Farrow-FAPRRC have similar control effect and the THD of ig is 2.41% and 2.57%, respectively.   [23][24][25] indicate that compared with the other two FAPRRC control schemes, the harmonics of CPRRC is significantly larger and the distortion is also more serious. The THD of i g_c achieves 6.34% when CPRRC is adopted. From Figures 23c, 24c and 25c, ε of Newton and Farrow-FAPRRC is as small as about 1 A, but for CPRRC, it can reach about 2 A. Figures 24 and 25 show that Newton-FAPRRC and Farrow-FAPRRC have similar control effect and the THD of i g is 2.41% and 2.57%, respectively.
It should be noted that although Newton-FAPRRC and Farrow-FAPRRC appear as similar control effects, the former has a relatively simple structure and is easy to realize in implementation.

Conclusions
In order to suppress the grid current harmonics of LCL-type inverter under weak grid condition, a frequency adaptive scheme based on Newton Structure of PRRC is proposed. Comparing to FD filter based on Lagrange interpolation and Farrow structure, Newton FD filter has the advantages of simple structure and constant coefficients even when the grid frequency fluctuates, and is applied to RC to approximate the fractional delay item z -d to improve frequency adaptability. In this paper, the detail derivation of the general Newton structure and the parameters setting of proposed FAPRRC are given. At the same time, the stability of the control system under weak grid is analyzed. The simulation and experimental results indicate that the proposed Newton-FAPRRC can improve grid current quality and appears strong adaptability under weak grid condition.