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

Abstract

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 f_s/f, where f_s 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.

1. 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 $\alpha ,\beta$ 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 $\alpha ,\beta$ 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 control and has a fast response. However, it must be noticed that this scheme depends on the highly accuracy of system model [16]. In the weak grid condition, the grid impedance cannot be ignored and may fluctuate, which imposes critical requirements on impedance identification for the DB controller design.

At present, RC, which is based on the internal model principle, gains more attention due to its advantages such as high tracking accuracy of periodic signals, easy digital realization and parameters setting [17,18,19], etc. However, the inherent one-cycle time delay causes relatively poor dynamic characteristic of RC. Usually, to achieve fast dynamic response, RC is in parallel with other controllers [10,11], which is called as a composite control strategy. The composite control scheme of proportional control in parallel with RC (PRC) can effectively improve the dynamic response of the system, and the structure is relatively simple. However, the PRC scheme has the disadvantage of large steady-state tracking error [10]. A control scheme of PI in parallel with RC (PIRC) is proposed in [11] to improve the dynamic and steady-state performance of the system. However, as mentioned before, PI control only has zero steady-state error in the dq coordinate, which indicates that a complex decoupling transformation is indispensable to employ PIRC [11].

In the applying of RC, its digital form of $\frac{z^{-N}}{1-z^{-N}}$ is usually adopted, where N = f_s/f, f_s and f are sample frequency and grid frequency respectively. RC has high gains at fundamental frequency and integer multiple harmonic frequencies when N is an integer, which means that RC can achieve the zero steady-state error tracking for periodic signals with these frequencies. However, when the grid frequency fluctuates, which is a common scene under weak grid condition, N is mainly a fractional when f_s keeps constant. However, due to the limitation of the digital realization of the controller, N can only be an integer, therefore, the high tracking accuracy RC cannot be achieved. In order to solve this problem, [20] proposes a variable sampling frequency control scheme to keep N as an integer. However, the variable f_s will affect other controllers in the system and also increase the complexity of real-time implementation. In order to avoid the disadvantages caused by the change of sampling frequency, a fractional delay filter based on Lagrange interpolation is proposed to approximate the FD item z^−d [21,22], where d is the fractional part of N. This scheme makes the resonant frequencies of RC track the fundamental and harmonic frequencies of the power grid by tuning the coefficients of the filter on-line. However, if the grid frequency fluctuates, all the coefficients of the FD filter will have to be recalculated and retuned, which greatly increases the computational burden. In [23], a FD filter based on Farrow structure is proposed, which effectively avoids the disadvantage as online calculation and reset of filter coefficients at the cost of the structure complexity.

To suppress the grid current harmonics of LCL-type inverter tied with weak grid and improve system stability, a novel frequency-adaptive of PRRC (FAPRRC) control scheme is proposed. Compared with PRC and PIRC, the PR in parallel with RC scheme is chosen due to its advantages of the control accuracy in the dynamic response process and the small control error due to the infinite gain of PR at a specific frequency [14]. FD filter based on the Newton structure is applied to approximate the FD. Comparing to FD filter based on Lagrange interpolation and Farrow structure, the proposed Newton FD filter has simple structure; moreover, its coefficients do not need to be tuned on-line even when frequency fluctuates severely. Both mathematical analysis and experimental results show that the proposed Newton-FAPRRC significantly improves the accuracy and stability of the LCL-type inverter tied with weak grid. The rest of this paper is organized as follows: Section 2 describes the structure of grid-connected inverter system and PRRC scheme. Section 3 presents the FD filter with the Newton structure. Section 4 gives the Newton-FAPRRC parameters setting. Section 5 analyzes the stability of Newton-FAPRRC in weak grid condition. Simulation and experimental results are shown in Section 6 to verify the effectiveness of the proposed scheme.

2. PRRC Control Scheme

2.1. 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; Z_g is the grid impedance; i_L, i_g and i_c are the output current of inverter, grid-connected 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.

2.2. Proposed PRRC

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.

$$G_d(s)=\frac{e^{-T_{c}s}(1-e^{-T_{c}s})}{T_{c}s}$$

(1)

Actually, the adopted GCFAD is a high pass filter [24] and its one order expression is:

$$H(s)=-\frac{k_{c}s}{s+{\omega }_c}$$

(2)

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.

$$G_{LCL}(s)=\frac{1}{L_1{L}_{2}C{s}^3+(L_1+L_2)s}$$

(3)

Applying a bilinear transformation to (1), (2) and (3), the transfer function of the inner closed loop in z domain is:

$$P_0(z)=\frac{K_{pwm}{G}_d(z)G_{LCL}(z)}{1+K_{pwm}{G}_d(z)G_{LCL}(z)H(z)}$$

(4)

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:

$$\{\begin{array}{c}{G}_{rc}(z)=\frac{z^{-N}}{1-z^{-N}Q(z)}{G}_c(z)\begin{array}{ccc}\begin{array}{ccc}\begin{array}{cc}\begin{array}{cccc}\begin{array}{ccc}\begin{array}{ccc}& & \end{array}& & \end{array}& & & \end{array}& \end{array}& & \end{array}& & \end{array}\\ G_{pr}(z)=k_p+\frac{4k_i{\omega }_i{T}_c(z^2-1)}{(4+4{\omega }_i{T}_c+{\omega }_0^2{T}_c^2)z^2+(2{\omega }_0^2{T}_c^2-8)z+(4-4{\omega }_i{T}_c+{\omega }_0^2{T}_c^2)}\end{array}$$

(5)

In (5), k_p and k_i are the proportional gain and integral gain of PR; ω_i is the bandwidth of PR; ω₀ is the grid angular frequency, and ${\omega }_0=2\pi f$.

3. Frequency Adaptation of RC

3.1. 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]:

$$z^{-N}=z^{-N_i}{z}^{-d}$$

(6)

In this paper, a FD filter based on the Newton structure is proposed to approximate fractional delay item $z^{-\widehat{d}}$, and frequency adaptation of RC (FARC) can be expected. The relationship of $\widehat{d}$ and d will be introduced in Section 3.2. and corresponding ${N^{\prime }}_i=N-\widehat{d}$. 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.

3.2. 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).

$$H_{Farrow}(d,z)=D^{T}CZ$$

(7)

In (7), $D={\left[\begin{array}{cccc}1& d& d^2\cdot \cdot \cdot & d^{M-1}\end{array}\right]}^T$; $Z={\left[\begin{array}{cccc}1& z^{-1}& z^{-2}\cdot \cdot \cdot & z^{-N^{\prime }+1}\end{array}\right]}^T$; 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).

$$H_{Newton}(\widehat{d},z)={\widehat{D}}^T\widehat{C}\widehat{Z}$$

(8)

In (8), $\widehat{D}={\left[\begin{array}{cccc}1& \widehat{d}& \widehat{d}(\widehat{d}-1)\cdot \cdot \cdot & {\displaystyle \prod _{i=0}^{M-2}(\widehat{d}-i)}\end{array}\right]}^T$; $\widehat{d}=\overline{d}+\frac{M-1}{2},\widehat{d}\in [1,2]$; $\overline{d}=d-0.5$; $\widehat{Z}={\left[\begin{array}{cccc}1& 1-z^{-1}& {(1-z^{-1})}^2\cdot \cdot \cdot & {(1-z^{-1})}^{N^{\prime }-1}\end{array}\right]}^T$; and $\widehat{C}$ is the coefficient matrix of Newton FD filter.

The transformation from Farrow structure to Newton structure can be realized by (9).

$$\begin{array}{c}{H}_{Farrow}(d,z)=D^{T}CZ=D^T(T_d^T{T}_d^{-T})C(T_z^{-1}{T}_z)Z={(T_{d}D)}^T(T_d^{-T}C{T}_z^{-1})(T_{z}Z)\\ ={\widehat{D}}^T\widehat{C}\widehat{Z}=H_{Newton}(\widehat{d},z)\end{array}$$

(9)

where $T_d$ and $T_z$ are the transformation matrices that transform D and Z into $\widehat{D}$ and $\widehat{Z}$, respectively. $\widehat{D}=T_{d}D$, $\widehat{C}=T_d^{-T}C{T}_z^{-1}$, $\widehat{Z}=T_{z}Z$, $T_d^{-T}={(T_d^T)}^{-1}$. Transformation matrix $T_d$ can be calculated as:

$$T_d={T_d}^{″}{T_d}^{\prime }$$

(10)

In (10), ${T_d}^{\prime }$ is the matrix with each row i containing the coefficients of the polynomial ${(\overline{d}+\frac{M-1}{2})}^{i-1}$, and elements of the matrix ${T_d}^{\prime }$ are calculated by (11):

$${T_d}^{\prime }\left[i,j\right]=\left(\begin{array}{c}i-1\\ j-1\end{array}\right){\left(\frac{M-1}{2}\right)}^{i-j}$$

(11)

The elements in the matrix ${T_d}^{″}$ are the Stirling numbers of the first kind $S_j^{(i)}$, i.e., ${T_d}^{″}\left[i,j\right]=S_j^{(i)}$.

Transformation matrix can be expressed as follows:

$$T_z[i,j]=\left(\begin{array}{c}i-1\\ j-1\end{array}\right){\left(-1\right)}^{j+1}$$

(12)

3.3. 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]:

$$C_{spline}=\left[\begin{array}{cccc}1& 23& 23& 1\\ -6& -30& 30& 6\\ 12& -12& -12& 12\\ -8& 24& -24& 8\end{array}\right]$$

(13)

The coefficient matrix of the third-order FD filter based on Newton structure can be obtained as (14).

$$\widehat{C}=T_d^{-T}{C}_{spline}{T}_z^{-1}={({T_d}^{″}{T_d}^{\prime })}^{-T}{C}_{spline}{T}_z^{-1}=\left[\begin{array}{cccc}1& 0& 1/6& 1/6\\ 0& -1& 0& -1/6\\ 0& 0& 1/2& 0\\ 0& 0& 0& -1/6\end{array}\right]$$

(14)

The structure of third-order Newton FD filter can be obtained as Figure 5.

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].

3.4. 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.

In this paper, M = 4, so $\widehat{d}=d+1$, ${N^{\prime }}_i=N-\widehat{d}=N_i-1$ in Figure 4. Taking 7th harmonic as an example, if grid frequency is 49.2 Hz, $N=f_s/f=203.25$, z^−N can be expressed as $z^{-N}=z^{-202}{z}^{-1.25},\widehat{d}=1.25,{{\mathrm{N}}^{\prime }}_i=202$. It can be seen that the amplitude of 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.

4. Parameters of PRRC

From Figure 3, the closed-loop transfer function of the system can be derived as follows:

$$\frac{i_g}{i_{ref}}=\frac{\left[G_{pr}(z)+G_{rc}(z)\right]P_0(z)}{1+\left[G_{pr}(z)+G_{rc}(z)\right]P_0(z)}=\frac{\left[G_{pr}(z)+\frac{z^{-N}}{1-z^{-N}Q(z)}{G}_c(z)\right]P_0(z)}{1+\left[G_{pr}(z)+\frac{z^{-N}}{1-z^{-N}Q(z)}{G}_c(z)\right]P_0(z)}=\frac{\left\{G_{pr}(z)-z^{-N}\left[G_{pr}(z)Q(z)-G_c(z)\right]\right\}\frac{P_0(z)}{1+G_{pr}(z)P_0(z)}}{1-z^{-N}\left[Q(z)-G_c(z)\frac{P_0(z)}{1+G_{pr}(z)P_0(z)}\right]}$$

(15)

According to (15), the characteristic equation of the closed-loop system can be derived as follows:

$$G^{\prime }(z)=1-z^{-N}\left[Q(z)-k_r{z}^{m}S(z)P(z)\right]$$

(16)

P(z) is the control object of RC, the expression is as follows:

$$P(z)=\frac{P_0(z)}{1+G_{pr}(z)P_0(z)}$$

(17)

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).

$$\left|G(e^{j\omega T_s})\right|=\left|e^{-j\omega N{T}_s}\left[Q(e^{j\omega T_s})-k_r{e}^{j\omega m{T}_s}S(e^{j\omega T_s})P(e^{j\omega T_s})\right]\right|<1,\omega \in [0,\pi /T_s]$$

(18)

It can be verified that:

$$\left|e^{-j\omega N{T}_s}\right|=1$$

(19)

4.1. Parameters of PR and GCFAD

Parameters of investigated LCL inverter and controller (as shown in Figure 1) are shown in

Table 1. Parameters of LCL-type inverter and controller.

Parameter	Symbol	Value
Rated power	P	9.1 kW
Grid voltage	vg	110 V
Grid-side inductor	L2	1 mH
Inverter-side inductor	L1	3 mH
Capacitor of LCL-filter	C	10 uF
DC voltage	VDC	200 V
Frequency	f	50 Hz
Sampling frequency	fs	10 kHz
Switching frequency	fsw	10 kHz
Feedback coefficient of GCFAD	kc	30
Cut-off frequency of GCFAD	ωc	12,165
Proportional gain of PR	kp	5
Integral gain of PR	ki	2500
Bandwidth coefficient of PR	ωi	3.14

, where the parameters of PR and GCFAD control are determined according to the design principles of [24] and [32].

4.2. 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.

4.3. Parameters of Gc

Gc is the compensator of RC. The expression is as follows:

$$G_c(z)=k_r{z}^{m}S(z)$$

(20)

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.

4.3.1. 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:

$$S(z)=\frac{0.00482z^4+0.0193z^3+0.02895z^2+0.0193z+0.00482}{z^4-2.36951z^3+2.314z^2-1.05467z+0.18738}$$

(21)

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.

4.3.2. 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^mS(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.

4.3.3. 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 $G(e^{j\omega T_s})$ does not exceed the unit circle. Figure 9 shows the trajectory of $G_F(e^{j\omega T_s})$ when k_r changes from 0.2 to 0.8.

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.

5. Stability Analysis of FAPRRC

The feedback branch of Figure 4 can be written as $z^{-N_i^{\prime }}{z}^{-\widehat{d}}Q(z)$. Thus, when the proposed Newton FD filter is adopted, the transfer function of Figure 4 G_FARC can be expressed by (22):

$$G_{FARC}(z)=\frac{z^{-N}}{1-z^{-N_i^{\prime }}{z}^{-\widehat{d}}Q(z)}{G}_c(z)=\frac{z^{-N}}{1-z^{-N_i^{\prime }}{z}^{-\widehat{d}}Q(z)}{k}_r{z}^{m}S(z)$$

(22)

Based on (22), the Q(z) in (18) adds a coefficient $z^{-N_i^{\prime }}{z}^{-\widehat{d}}$. Replace z with e^jwTs and combine it with (18) to obtain the stability condition of FAPRRC:

$$\left|G_F(e^{j\omega T_s})\right|=|e^{-j\omega [{N^{\prime }}_i+\widehat{d}]T_s}Q(e^{j\omega T_s})-k_r{e}^{-j\omega (N-m)T_s}S(e^{j\omega T_s})P(e^{j\omega T_s})|<1,\omega \in [0,\pi /T_s]$$

(23)

Since $\widehat{d}$ is very small comparing to N and ${N^{\prime }}_i$ is close to N, according to (19), (24) can be obtained:

$$\{\begin{array}{c}\left|e^{-j\omega [{N^{\prime }}_i+\widehat{d}]T_s}\right|\stackrel{}{\to }\left|e^{-j\omega N{T}_s}\right|=1\\ \left|e^{-j\omega [N-m]T_s}\right|\stackrel{}{\to }\left|e^{j\omega m{T}_s}\right|\begin{array}{cc}& \end{array}\end{array}$$

(24)

According to (23) and (24), the stability conditions of FAPRRC and PRRC are the same, i.e., the trajectory of $G_F(e^{j\omega T_s})$ 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 L₂ of LCL filter is (L₂ + 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 $G_F(e^{j\omega T_s})$ 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 $G_F(e^{j\omega T_s})$ is still inside the unit circle and the system keeps stable.

6. 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.

6.1. Simulation

In order to verify the effectiveness of the proposed Newton-FAPRRC scheme for LCL-type 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.

6.1.1. 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.

Figure 11, Figure 12 and Figure 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. Figure 13, Figure 14 and Figure 15 also show the current tracking error $\epsilon =i_{ref}-i_g$; it can be seen that $\epsilon$ decreases when the Lg increases.

6.1.2. Newton FAPRRC Dynamic Performance

Figure 14 shows the dynamic response of the system when the grid current reference signal i_ref 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.

6.1.3. Verification of Newton-FAPRRC Frequency Adaptability

To verify the effectiveness of the proposed Newton-FAPRRC, frequency adaptabilities of CPRRC, Farrow-FAPRRC and Newton-FAPRRC are compared. Figure 15, Figure 16 and Figure 17 shows the simulation results of the three schemes when grid frequency is 50.8 Hz, where Figure 15, Figure 16 and Figure 17a shows the steady-state responses of i_g, Figure 15, Figure 16 and Figure 17b shows the current tracking error $\epsilon$ and Figure 15, Figure 16 and Figure 17c shows the harmonic spectrum of grid current phase C i_{g_c}.

According to Figure 15, Figure 16 and Figure 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. Figure 15, Figure 16 and Figure 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. The THD of grid-current i_g with different grid frequencies.

Frequency/Hz	CPRRC			Farrow			Newton
	ia	ib	ic	ia	ib	ic	ia	ib	ic
49.2	2.73%	2.89%	2.77%	1.15%	1.18%	1.21%	1.11%	1.17%	1.19%
49.6	1.87%	1.95%	1.85%	1.07%	1.06%	1.13%	1.08%	1.11%	1.06%
50	1.35%	1.37%	1.33%	1.35%	1.35%	1.38%	1.34%	1.35%	1.36%
50.4	1.88%	1.96%	1.78%	1.17%	1.20%	1.22%	1.13%	1.19%	1.20%
50.8	2.82%	3.00%	2.83%	1.15%	1.14%	1.18%	1.16%	1.18%	1.16%

. 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.

6.2. 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.

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.

6.2.1. 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 Figure 19, Figure 20 and Figure 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.

6.2.2. 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.

6.2.3. 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 Figure 23, Figure 24 and Figure 25 respectively. Figure 23a, Figure 24a and Figure 25a shows the steady state responses of i_g; Figure 23b, Figure 24b and Figure 25b and Figure 23c, Figure 24c and Figure 25c show the harmonic spectrum of i_{g_c} and current tracking error ε.

Figure 23, Figure 24 and Figure 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 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. Figure 24 and Figure 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.

7. 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.