Frequency-Adaptive Fractional-Order Repetitive Control ... - IEEE Xplore

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 3, MARCH 2015

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Frequency-Adaptive Fractional-Order Repetitive Control of Shunt Active Power Filters Zhi-Xiang Zou, Student Member, IEEE, Keliang Zhou, Senior Member, IEEE, Zheng Wang, Member, IEEE, and Ming Cheng, Senior Member, IEEE Abstract—Repetitive control (RC), which can achieve zero steady-state error tracking of any periodic signal with known integer period, offers active power filters (APFs) a promising accurate current control scheme to compensate the harmonic distortion caused by nonlinear loads. However, classical RC cannot exactly compensate periodic signals of variable frequency and would lead to significant performance degradation of APFs. In this paper, a fractional-order RC (FORC) strategy at a fixed sampling rate is proposed to deal with any periodic signal of variable frequency, where a Lagrange-interpolation-based fractional delay (FD) filter is used to approximate the factional delay items. The synthesis and analysis of FORC systems are also presented. The proposed FORC offers fast online tuning of the FD and the fast update of the coefficients, and then provides APFs with a simple but very accurate realtime frequency-adaptive control solution to the elimination of harmonic distortions under grid frequency variations. A case study on a single-phase shunt APF is conducted. Experimental results are provided to demonstrate the validity of the proposed FORC. Index Terms—Active power filter (APF), fractional order, frequency variation, repetitive control (RC).

I. I NTRODUCTION

T

HE rapidly growing use of nonlinear loads causes considerable power quality degradation in power distribution networks, such as power harmonic distortions, and resonance problems. Active power filters (APFs) that operate as controllable power sources and offer fast response to dynamic load changes are widely used to cancel power harmonics produced by nonlinear loads [1], [2]. Due to its simplicity and effectiveness, a shunt APF is the most popular tool to compensate load current harmonics. Numerous current control schemes have been proposed for shunt APFs, such as hysteresis control, proportional–integral (PI) control, proportional resonant (PR) control, and deadbeat (DB) control [3]–[6]. However, high switching stress or inaccuManuscript received July 31, 2013; revised December 9, 2013, June 1, 2014, and August 4, 2014; accepted September 29, 2014. Date of publication October 16, 2014; date of current version February 6, 2015. This work was supported in part by the National Key Basic Research Program of China (973 program) under Grant 2013CB035603 and in part by the National Natural Science Foundation of China (No. 50977013, No. 51007008, and No. 51137001). Z.-X. Zou, Z. Wang, and M. Cheng are with the School of Electrical Engineering, Southeast University, Nanjing 210096, China (e-mail: [email protected]; [email protected]; [email protected]). K. Zhou is with the School of Engineering, University of Glasgow, Glasgow, G12 8LT, U.K. (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2363442

rate elimination of the load harmonic currents may occur due to random switching frequency for hysteresis control, sensitivity to inaccurate parameters for DB control, and failure in dealing with complex harmonic currents for both PI control and PR control. Multiple resonant control can achieve zero steadystate error tracking of sinusoidal signals at selected harmonic frequencies [5]–[7]. However, a large number of paralleled resonant controllers might cause heavy parallel computation burden and tuning complexity. Based on the internal model principle, repetitive control (RC) [8]–[15] can achieve zero steady-state error tracking of any periodic signal with a known period due to the introduction of high gains at interested harmonic frequencies. It offers promising accurate current control solutions for shunt APFs to compensate harmonics. In practical applications, most modern controllers are usually implemented in their digital forms. The conventional RC (CRC) controller in its digital form of z −N /(1 − z −N ) can track any periodic reference signal with an integer period of N = fs /f , where f is the fundamental frequency of the reference signal, fs is the sampling rate, and N is the order of RC (i.e., the number of samples per period). Since the grid frequency is usually variable in a certain range (e.g., 49 Hz ∼ 51 Hz) in practice, the order of RC would be often fractional in the case of a fixed sampling rate. Since only z −N with integer N can be implemented in practice, CRC is sensitive to grid frequency variations and cannot exactly compensate fractional period signals. Ensuring the integer period of N is always the same in the presence of grid frequency variations, and variable sampling rate approach enables RC to reject harmonics completely [16]–[18]. However, a variable sampling rate will significantly increase the real-time implementation complexity of the control systems, such as online controller redesign. To address this issue, a frequency-adaptive fractional-order RC (FORC) strategy at a fixed sampling rate is proposed to track or eliminate any periodic signal of variable frequency in this paper. The item z −N with a fractional delay (FD) element number N will be well approximated by a Lagrangeinterpolation-based finite-impulse-response (FIR) filter with integer order. The proposed FIR filter only needs a small number of multiplications and additions for coefficient update, and is well suited to fast online tuning of the FD. The proposed FORC will enable shunt APFs to always precisely cancel the load harmonic current under variable grid frequency. The rest of this paper is organized as follows: 1) Section II presents the FORC strategy with a fixed sampling rate and the design criteria of FORC systems; 2) Section III describe the shunt APF configuration and the design of the proposed control; 3) Section IV demonstrates some experiment results to verify

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Fig. 1.


Plug-in RC system.

the expected performance of the proposed FORC strategy; and 4) Section V gives the conclusions of this paper. II. F REQUENCY-A DAPTIVE FORC A. CRC Fig. 1 shows the typical closed-loop control system with a plug-in digital CRC, where R(z) is the reference input, Y (z) is the output, E(z) = R(z)−Y (z) is the tracking error, D(z) is the disturbance, Gp (z) is the plant, Gc (z) is the conventional feedback controller, Gr (z) is a feedforward plug-in CRC, kr is the RC gain, Ur (z) is the output of the CRC, Gf (z) is a phase lead compensation filter to stabilize the overall closed-loop system, and Q(z) = a1 z+a0 +a1 z −1 with 2a1 +a0 = 1 is a lowpass filter (LPF) to enhance the robustness of the system [8]. The transfer function of the plug-in CRC shown in Fig. 1 can be written as z −N Q(z) Ur (z) Gr (z) = = kr Gf (z) (1) E(z) 1 − z −N Q(z) where N = f s/f ∈ N with f being the fundamental frequency of reference signal R(z) and/or disturbance D(z), fs being the sampling rate, and N being the order of RC; the poles of Gr (z) are located around 2mπf , with m = 0, 1, 2, . . . , M (M = N/2 for even N and M = (N − 1)/2 for odd N ). It is clearly seen that the amplitudes of Gr (z) at frequencies 2mπf approach infinity if Q(z) = 1. Therefore, CRC provides zero steady-state error tracking of all harmonic components below the Nyquist frequency if Q(z) = 1 and its order N are integers [8]. B. FORC However, in practical electrical systems, the grid frequency f is time-varying within a certain range. Therefore, the order N (i.e., the ratio of the sampling rate to the grid frequency) would often be fractional with a fixed sampling rate fs . CRC even with the order N being the nearest integer value cannot exactly track fractional period signals because high control gains shift away from interested harmonic frequencies. According to the FD filters design method [19], [20], FD z −N can be well approximated by FD filters with integer orders. Assuming that z −N = z −Ni −F with Ni = int[N ] being the integer part of N and F = N − Ni (0 ≤ F < 1) being the fractional part of N , the FD z −F can be approximated by a Lagrange interpolation polynomial FIR filter as follows [21], [22]: n z −F ≈ Ak z −k (2) k=0

where k = 0, 1, . . . , n, and the coefficient Ak can be obtained as n F −i , k, i = 0, 1, . . . , n. (3) Ak = k−i i=0

Fig. 2.

Block diagram of the proposed frequency-adaptive FORC.

Substituting (2) and (3) into (1), a FORC will be obtained as Ur (z) = kr Gfr (z) = E(z)

z −Ni 1−

n

Ak z −k Q(z)

k=0 n z −Ni

Ak z −k Q(z)

Gf (z). (4)

k=0

The FORC of (4) will become the CRC of (1) when F = 0. It provides a general approach to tracking or elimination of any periodic signal with arbitrary fundamental frequency. Its block diagram is shown in Fig. 2, and Gr (z) in Fig. 1 can be replaced by FORC Gfr (z). Note that both Ni and F changes very slowly in grid-tied converter control applications. According to the properties of Lagrange interpolation method, the approximation remainder term of FD can be derived as

Rn = z

−F

−

n k=0

ξ −F −n Ak z

−k

=

n−1 i=0

(−F −i) n

(n+1)!

(F −i)

(5)

i=0

where ξ ∈ [Tk , Tk+1 ] with Tk and Tk+1 being the kth and k + 1th sampling instants, respectively. With an increase in the degree n, a more accurate approximation can be acquired. Fig. 3 shows the magnitude responses of the Lagrangeinterpolation-based FD filter of (2) with the order n = 1 and n = 3 for various fractional F from 0 to 0.9. It is seen that the FD filter of (2) with order n = 3 give an excellent approximation of FD z −F at low frequencies within bandwidth of 75% Nyquist frequency, whereas the bandwidth of 50% Nyquist frequency for the FD filter of order n = 1. Within its passband of the FD filter of (2), the magnitude of proposed FORC is close to one; then, the proposed FORC can exactly track fractional period signals. Moreover, Lagrange interpolation is one of the easiest ways to design a FD filter to approximate a given FD [19]–[22]. The coefficient of (3) for the FD filter only consumes a small number of additions and multiplications for a fast online update of the coefficients. Therefore, Lagrange-interpolationbased FIR FD filter not only can be easily implemented in real-time applications but can also offer high approximation accuracy in most cases. Therefore, FIR FD-filter-based FORC offers an attractive method for the real-time control of highswitching-frequency converters. Before the plug-in FORC is employed, the transfer function of the closed-loop system can be written as

i=k

Specifically, if n = 1 in (2), a linear interpolation polynomial z −F ≈ (1 − F ) + F z −1 will be obtained here.

H(z) =

Gc (z)Gp (z) z−d B + (z −1)B −(z −1 ) Y (z) = = (6) R(z) 1+Gc (z)Gp (z) A(z −1)

ZOU et al.: FREQUENCY-ADAPTIVE FORC OF SHUNT APFs

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Fig. 4. load.

Single-phase shunt APF connected to the grid with nonlinear

2) The roots of 1−z−Ni nk=0 Akz−kQ(z)(1−krGf (z)H(z)) = 0 are inside the unit circle, and n −1 |(1 − kr Gf (z)H(z))| < |Q(z)|−1 Ak z −k . (9) k=0

Fig. 3. Frequency responses of Lagrange-interpolation-based FD filters. (a) n = 1. (b) n = 3.

where d ∈ R is the known delay steps of the system; B + (z −1 ) and B − (z −1 ) are the cancelable and uncancelable parts of the numerator, respectively; and A(z −1 ) = 0 is the system characteristic equation. In order to achieve zero-phase compensation [23], [24], the compensating filter Gf (z) can be chosen as Gf (z) =

z d A(z −1 )B − (z −1 ) B + (z −1 )b

(7)

where b ≥ B − (z −1 )2 . The delay steps d can be determined by experiments in practical applications. With the plug-in FORC, the error transfer function of the overall system can be derived as follows: E(z) R(z) − D(z) −1

=

−Ni

n

−k

(1 + Gc (z)Gp (z)) Ak z 1−z k=0 n . −N −k i 1−z Ak z Q(z) (1 − kr Gf (z)H(z))

(8)

k=0

Therefore, from (8), the closed-loop FORC system is asymptotically stable if the following two conditions hold. 1) The roots of 1+Gc (z)Gp (z) = 0 are inside the unit circle.

The above stability criteria for the FORC systems are fully compatible to that of a CRC system [8], [23]. The stability condition 1 is the same as that for CRC systems; within the passband of the FD filter of (2), i.e., | nk=0 Ak z −k | → 1, the stability criterion of (9) for FORC systems is almost equivalent to that for CRC systems. Furthermore, assuming that the bandwidth of the proposed FD filter of (2) is larger than the bandwidth of the LPF Q(z) in practical applications, the FORC of (4) would be almost the same as that of the CRC of (2) due to |Q(z)|| nk=0 Ak z −k | → 1. When Gf (z) of (7) is applied to achieve zero-phase compensation, the stability range for the FORC gain will be 0 < kr < 2. It is clear that the synthesis of FORC systems can be almost the same as that of well-known CRC systems [8], [23]. III. FORC OF S INGLE -P HASE S HUNT APF Fig. 4 shows the typical configuration of a single-phase pulsewidth-modulated (PWM) inverter-based APF circuit, which is connected to the grid with the nonlinear diode rectifier load. While keeping dc bus voltage vdc constant at setting level, the shunt APF inverter is regulated to accurately generate the current ic to compensate the harmonic components of the load current iL , and to achieve sinusoidal grid current ig with a unity power factor. Fig. 5 shows the diagram of the proposed control scheme for the shunt APF shown in Fig. 4. The proposed control scheme includes two control loops: a PI-controller-based outer voltage loop for forcing dc bus voltage vdc to track the reference voltage vdcref and a DB-plus-proposed FORC-controller-based inner current loop for forcing current ic to track reference current iref , which would be the inverse of harmonic components and reactive power components of the load current iL . Since the bandwidth of the inner current loop is usually much larger than that of the outer voltage loop, the reference current signal iref can be treated as constant during the regulation of compensation current ic . Therefore, an outer voltage loop controller and an inner current loop controller can be designed independently.

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C. FORC-Based Current Control

Fig. 5.

Fig. 6.

The sampled-data form of (10) can be expressed as Ts (vg (k) − vi (k)) (17) ic (k + 1) = ic (k) + L where k and k+1 represent the kth and k+1th sampling intervals, respectively, and Ts = 1/fs is the sampling period. Let ic (k+1) = iref (k); a DB current controller is obtained as follows: L vi (k) = vg (k) − (iref (k) − ic (k)) (18) Ts Vi (k) = uc (k)vdc (19)

Dual-loop control scheme for the single-phase APF.

Small-signal model of the outer voltage loop.

A. Modeling of Shunt APF The dynamics of the shunt APF in Fig. 4 can be written as 1 (10) i˙ c = (vg − vi ) L 1 (11) v˙ dc = ic S(t) C where vdc , vg , and vi are the dc bus voltage, the grid voltage and the output voltage of the inverter, respectively; ic is the compensation current; L and C are the filter inductor and dc bus capacitor, respectively; and S(t) denotes the switching function with value being 1 or −1. B. PI-Based Voltage Control Due to the power exchange between the grid and the APF, the fluctuations of dc bus voltage Δvdc can be written as 1 Δvdc = C

t k+1

1 S(t)ic dt = C

tk

t k+1

S(t) tk

icn sin(nωt) dt (12)

n=1

where icn is the nth-order frequency component of ic . The fluctuations of dc bus voltage will lead to unexpected harmonics in the reference current iref and in the compensation current ic hereafter. An LFP is usually used to alleviate the impact from dc bus voltage fluctuations. A PI controller is usually to force the dc bus voltage vdc to track its reference value vdcref as follows:

2

2 2 2 dt. (13) vdcref − vdcf u = kp vdcref − vdcf + ki The output of PI controller of (13) is used to regulate the magnitude of current ic to have a unity grid power factor, i.e., igref = uvg .

(14)

The small-signal model of the outer voltage loop shown in Fig. 6 can be derived as follows [25]: ki 2 2 2 2 · vˆdcref (s) − vˆdcf vˆdc (s) = kp + (s) (15) · s Cs ωc 2 2 (s) = vˆdc (s) (16) vˇdcf s + ωc where ωc is the cutoff frequency of the LPF.

where vi (k) is the output voltage of the inverter, −1 ≤ uc (k) ≤ 1 is the output of the current controller of (17), iref = iL − igref is as shown in Fig. 5. Since ic (k + 1) = iref (k), the transfer function from iref to ic is H(z) = z −1 , i.e., a DB current control loop is achieved. To ensure exact current harmonics compensation in the presence of parameter and frequency uncertainties, a frequencyadaptive FORC controller Gfr (z) of (4) is plugged into the DBcontrolled current loop, where Gf (z) = z d+1 is a linear-phase lead filter with lead step d being determined by experiments [23]. For the FORC of (4), n = 3 is chosen to be the Lagrange polynomial degree. Hence, the corresponding FD will be z −N = A0 z −Ni +A1 z −Ni −1 +A2 z −Ni −2 +A3 z −Ni −3 . (20) D. PLL As shown in Fig. 2, for the online update of the Lagrange coefficients Ak of the proposed FORC, a phase-locked loop (PLL) provides the real-time measurement of grid frequency f . A frequency-locked loop (FLL) technique [26] is employed in this scenario. The FLL is not sensitive to grid disturbances and harmonics, and can be operated within 47 ∼ 52 Hz [26]. The transient response of FLL with a settling time of 50 ms here is much slower than that of a current loop and is much faster than the maximum rate of grid frequency, which is subject to utility codes and generator inertia. For example, in the case of sampling frequency of 5 kHz and the change grid frequency from 50 to 50 ± 1 Hz, a large fundamental signal period N will change from 100 to about 100 ± 2. If the rate of change of grid frequency is 1 Hz/s (a typical maximum rate value), it takes 500 ms to increase or decrease the integer part Ni by 1, and the fractional period for FORC would change ±4 × 10−4 over one sampling cycle. Online updating coefficients Ak will correspondingly change only a very little bit and can be treated as constants in stability analysis in (8) and (9). This will lead to a sufficiently slow change of the gains at interested harmonic frequencies. Hence, for the FORC-controlled shunt APF with a sufficiently large fundamental period N , the upper bound for system robustness to grid frequency disturbances is actually determined by the operation frequency range of PLL, e.g., 47 ∼ 52 Hz in this case. IV. E XPERIMENTAL VALIDATION A. Experimental Setup To validate above theoretical analysis, a dSPACE 1104-based single-phase shunt system has been built in our laboratory. The


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TABLE I S YSTEM PARAMETERS

Fig. 7.

Magnitude response of the FD filters and the LPF Q(z).

grid voltage is generated by a programmable ac power source. The system parameter values are listed in Table I. The phase-lead compensation filter Gf (z) = z 2 is employed. The bandwidth for LPF Q(z) = 0.1z + 0.8 + 0.1z −1 and the FD filter of (20) are about 1630 and 1875 Hz, respectively, and enable FORC to efficiently compensate up to the 30th-order current harmonics. As two examples, when the grid frequency changes from 50 to 50 ± 0.2 Hz in the case of a fixed sampling frequency of 5 kHz, i.e., the period N will change from integer N = 100 to fractional N = 100 ± 0.4. The magnitude-frequency responses of Q(z) and FD filters of z −99.6 and z −100.4 are plotted in Fig. 7. Bode diagrams for CRC (N = 100) and FORC with FD filters of z −99.6 and z −100.4 are shown in Fig. 8. It can be noted that, both CRC and FORC have expected large gains at their harmonic frequencies of 50n and (50 ± 0.2)n Hz (n = 1, 2, . . .), respectively. Moreover, the phase shifts at these interested frequencies are zero. Fig. 8 indicates that, if fundamental frequency changes from 50 to 50 ± 0.2 Hz, CRC cannot exactly compensate harmonic frequencies of (50 ± 0.2)n Hz (n = 1, 2, . . .) due to its low gains at these frequencies. That means CRC is sensitive to grid frequency variations. With listed parameters, the DB controller of (18) for the shunt APF can be given as follows: uc (k) = 0.008333vg (k) − 0.2083 (iref (k) − ic (k)) .

(21)

Assuming that Lt = L + ΔL is the actual inductance of the filter inductor with ΔL being the uncertainty inductance and L being the nominal inductance, the transfer function of the DB-controlled current loop without plug-in FORC will change from H(z) = z −1 to Ht (z) =

L ic (z) = . iref (z) Lt z − ΔL

Fig. 8. Bode diagrams of CRC, FORC with N = 100.4 and 99.6 within (a) low frequency band and (b) neighboring frequency band around seventh-order harmonics.

(22)

Fig. 9. Robustness analysis. (a) Pole map of Ht (z). (b) Amplitude of stability criteria of (8).

Fig. 9(a) shows the pole map of Ht (z) with the parameter uncertainty. Fig. 9(b) shows the amplitude of the stability criteria of (8) for the DB-plus-plug-in-FORC-controlled current loop. It is seen that, if inductance uncertainty ΔL ≥ −0.5L and kr < 1.4, DB-plus-FORC-controlled current loop is stable.

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Fig. 11. Steady-state responses at 50 Hz with a CRC-controlled APF. (a) Grid voltage vg and grid current ig . (b) Harmonic spectrum of compensated ig .

Fig. 10. Steady-state responses at 50 Hz without an APF. (a) Grid voltage vg and grid current ig . (b) Harmonic spectrum of vg . (c) Harmonic spectrum of ig .

B. Experimental Results Fig. 10 shows the steady-state responses of the system without an APF under grid frequency of 50 Hz: the grid voltage vg and the grid current ig , the harmonic spectrum of vg , and the harmonic spectrum of ig . Fig. 11 shows the steady-state responses of the system with a CRC-controlled APF (i.e., FORC with F = 0) under grid frequency of 50 Hz : grid voltage vg and the grid current ig , and the harmonic spectrum of ig , where the period of N = 100. Figs. 10 and 11 indicate that the CRC-controlled APF efficiently reduce the total harmonic distortion (THD) of the grid current ig from 44.15% to 3.719% under integer period grid voltage vg with THD of 2.41%. Fig. 12 shows the steady-state responses of the system with a CRC-controlled APF under grid frequency of 49.8 Hz: grid voltage vg and the grid current ig , the current tracking error of the compensation current ic , and the harmonic spectrum of ig . Fig. 13 shows the steady-state responses of the system with a FORC-controlled APF under grid frequency of 49.8 Hz: grid voltage vg and the grid current ig , the current tracking error of the compensation current ic , and the harmonic spectrum of ig , where N = 100.4. Figs. 12 and 13 indicate that, in

Fig. 12. Steady-state responses at 49.8 Hz with a CRC-controlled APF. (a) Grid voltage vg and grid current ig . (b) Compensation current ic , reference current iref , and current tracking error. (c) Harmonic spectrum of ig .


Fig. 13. Steady-state responses at 49.8 Hz with a FORC-controlled APF. (a) Grid voltage vg and grid current ig . (b) Compensation current ic , reference current iref , and current tracking error. (c) Harmonic spectrum of ig .

the case of fractional period grid voltage, the grid current ig compensated by the CRC-controlled APF contains considerable THD(= 7.179%), and FORC enables the APF to provide much better quality sinusoidal grid current with THD = 2.987%. Figs. 12(b) and 13(b) indicate that FORC offers much better control accuracy in the case of fractional period grid voltage. Fig. 14 shows the steady-state responses of the system with a CRC-controlled APF under grid frequency of 50.2 Hz: grid voltage vg and the grid current ig , the current tracking error of the compensation current ic , and the harmonic spectrum of ig . Fig. 15 shows the steady-state responses of the system with a FORC-controlled APF under grid frequency of 50.2 Hz: grid voltage vg and the grid current ig , the current tracking error of the compensation current ic , and the harmonic spectrum of ig , where N = 99.6. Figs. 14 and 15 indicate that, in the case of fractional period grid voltage, the grid current ig compensated by a CRC-controlled APF contains considerable THD(= 8.749%), and FORC enables the APF to provide much better quality sinusoidal grid current with THD = 2.795%. Figs. 14(b) and 15(b) indicate that FORC offers

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Fig. 14. Steady-state responses at 50.2 Hz with a CRC-controlled APF. (a) Grid voltage vg and grid current ig . (b) Compensation current ic , reference current iref , and current tracking error. (c) Harmonic spectrum of ig .

much better control accuracy in the case of fractional period grid voltage. Table II lists the experimental THD data of steady-state compensated grid current ig under grid frequency from 49.5 to 50.5 Hz. The THD data in Table II indicates that, in the cases of various grid frequencies, the FORC-controlled APF offers very good current harmonics compensation capacity, whereas the CRC-controlled APF fails to provide satisfactory current harmonics compensation capacity. Generally speaking, Figs. 10–15 and Table II show that CRC is sensitive to grid frequency variations, whereas FORC is not. Fig. 16 shows the transient responses of the system with a FORC-controlled APF to step changes of grid frequency between 49.5 and 50.5 Hz: grid voltage vg and the grid current ig , and the PLL detected grid frequency f . Fig. 16 indicates it takes FLL-based PLL about 50 ms (i.e., the settling time) to detect the step changes of grid frequency, and the grid current ig compensated by a FORC-controlled APF is immune to the step grid frequency in these cases. Therefore, Fig. 16 implies that,

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Fig. 16. Responses to step changes of grid frequency. (a) 49.5 Hz → 50.5 Hz. (b) 50.5 Hz → 49.5 Hz.

Fig. 15. Steady-state responses at 50.2 Hz with a CRC-controlled APF. (a) Grid voltage vg and grid current ig . (b) Compensation current ic , reference current iref , and current tracking error. (c) Harmonic spectrum of ig . TABLE II P ERFORMANCE U NDER D IFFERENT F UNDAMENTAL F REQUENCIES

for the FORC-controlled shunt APF with a sufficiently large fundamental period N , the upper bound for system robustness to grid frequency disturbances is actually determined by the operation frequency range of PLL since the tuning rate of the proposed FORC is much faster than the transient response of PLL.

Fig. 17. Responses to step load changes at 49.8-Hz fundamental frequency. (a) R 15 Ω → 30 Ω, (b) R 30 Ω → 15 Ω.

Fig. 17 shows the transient responses to step load changes between rectifier resistor R = 15Ω and R = 30Ω: dc bus voltage vdc , the load current iL , and the grid current ig . It is seen that


the FORC-controlled APF maintains a sinusoidal line current even during the step load change. The recovery time of dc bus voltage in both two cases is about 250 ms. The dynamic responses of FORC are promising.

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[15] [16]

V. C ONCLUSION This paper has proposed a frequency-adaptive FORC scheme with a fixed sampling rate to track or eliminate any periodic signal with variable frequency. Using Lagrange-interpolationbased FD filter to approximate the FD items in RC, the proposed FORC offers fast online tuning of the FD and the fast update of the coefficients. It provides APFs with a simple but very accurate real-time frequency-adaptive control solution to harmonics distortions compensation under grid frequency variations. The stability criteria of FORC systems are given, which are compatible with those of CRC systems. A study case on a FORC-based single-phase shunt APF is done. Experiment results show the effectiveness of the proposed FORC strategy. Furthermore, the Lagrange-interpolation-based FORC can be used in extensive applications, such as the feeding currents control of grid-connected converters [10], [11], [27], programmable ac power supply [28], active noise cancelation, and so on. R EFERENCES [1] H. Akagi, “New trends in active filters for power conditioning,” IEEE Trans. Ind. Appl., vol. 32, no. 6, pp. 1312–1322, Nov./Dec. 1996. [2] H. Akagi, “Active harmonic filters,” Proc. IEEE, vol. 93, no. 12, pp. 2128– 2141, Dec. 2005. [3] Y. Han et al., “Robust deadbeat control scheme for a hybrid APF with resetting filter and ADALINE-based harmonic estimation algorithm,” IEEE Trans. Ind. Electron., vol. 58, no. 9, pp. 3893–3904, Sep. 2011. [4] M. Angulo, D. A. Ruiz-Caballero, J. Lago, M. L. Heldwein, and S. A. Mussa, “Active power filter control strategy with implicit closedloop current control and resonant controller,” IEEE Trans. Ind. Electron., vol. 60, no. 7, pp. 2721–2730, Jul. 2013. [5] P. Mattavelli and F. P. Marafao, “Repetitive-based control for selective harmonic compensation in active power filters,” IEEE Trans. Ind. Electron., vol. 51, no. 5, pp. 1018–1024, Oct. 2004. [6] L. R. Limongi, R. Bojoi, G. Griva, and A. Tenconi, “Digital currentcontrol schemes,” IEEE Ind. Electron. Mag., vol. 3, no. 1, pp. 20–31, Mar. 2009. [7] A. D. Le Roux, H. Mouton, and H. Akagi, “DFT-based repetitive control of a series active filter integrated with a 12-pulse diode rectifier,” IEEE Trans. Power Electron., vol. 24, no. 6, pp. 1515–1521, Jun. 2009. [8] K. Zhou et al., “Zero-phase odd-harmonic repetitive controller for a single-phase PWM inverter,” IEEE Trans. Power Electron., vol. 21, no. 1, pp. 193–201, Jan. 2006. [9] P. C. Loh, Y. Tang, F. Blaabjerg, and P. Wang, “Mixed-frame and stationary-frame repetitive control schemes for compensating typical load and grid harmonics,” IET Power Electron., vol. 4, no. 2, pp. 218–226, Feb. 2011. [10] D. Chen, J. Zhang, and Z. Qian, “An improved repetitive control scheme for grid-connected inverter with frequency-adaptive capability,” IEEE Trans. Ind. Electron., vol. 60, no. 2, pp. 814–823, Feb. 2013. [11] M. Rashed, C. Klumpner, and G. Asher, “Repetitive and resonant control for a single-phase grid-connected hybrid cascaded multilevel converter,” IEEE Trans. Power Electron., vol. 28, no. 5, pp. 2224–2234, May 2013. [12] B. Zhang, K. Zhou, and D. Wang, “Multirate repetitive control for PWM DC/AC converters,” IEEE Trans. Ind. Electron., vol. 61, no. 6, pp. 2883– 2890, Jun. 2014. [13] G. Escobar, P. Mattavelli, M. Hernandez-Gomez, and P. R. MartinezRodriguez, “Filters with linear-phase properties for repetitive feedback,” IEEE Trans. Ind. Electron., vol. 61, no. 1, pp. 405–413, Jan. 2014. [14] Z. Zou, Z. Wang, and M. Cheng, “Design and analysis of operating strategies for a generalized voltage-source power supply based on inter-

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Zhi-Xiang Zou (S’12) received the B.Eng. degree from Southeast University, Nanjing, China, in 2007, where he currently working toward the Ph.D. degree. From 2007 to 2008, he was with the State Grid Electric Power Research Institute, Nanjing, where he was engaged in relay protection design projects. He is the author or coauthor of more than 20 technical papers published in refereed journals and conference proceedings. His research interests include smart transformers, distributed generation, power quality, and analysis and control of converters.

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Keliang Zhou (M’04–SM’08) received the B.Sc. degree from Huazhong University of Science and Technology, Wuhan, China, in 1992, the M.Eng. degree from Wuhan Transportation University (now Wuhan University of Technology), Wuhan, in 1995, and the Ph.D. degree from Nanyang Technological University, Singapore, in 2002. From 2003 to 2006, he was a Research Fellow with Nanyang Technological University, and with Delft University of Technology, Delft, The Netherlands. From 2006 to 2014, he was with Southeast University, Nanjing, China, and the University of Canterbury, Christchurch, New Zealand. He is currently a Senior Lecturer with the School of Engineering, University of Glasgow, Glasgow, U.K. He is the author or coauthor of more than 100 technical papers and several granted patents in relevant areas. His research interests include power electronics and electric drives, renewable energy generation, control theory and applications, and microgrid technology.

Zheng Wang (S’05–M’09) received the B.Eng. and M.Eng. degrees from Southeast University, Nanjing, China, in 2000 and 2003, respectively, and the Ph.D. degree from The University of Hong Kong, Pokfulam, Hong Kong, in 2008. From 2008 to 2009, he was a Postdoctoral Fellow with Ryerson University, Toronto, ON, Canada. He is currently an Associate Professor with the School of Electrical Engineering, Southeast University. He is the author or coauthor of several technical papers and industrial reports. His research interests include power converters, electric drives, renewable energy, and distributed generation.

Ming Cheng (M’01–SM’02) received the B.Sc. and M.Sc. degrees from Southeast University, Nanjing, China, in 1982 and 1987, respectively, and the Ph.D. degree from The University of Hong Kong, Pokfulam, Hong Kong, in 2001. Since 1987, he has been with Southeast University, where he is currently a Professor with the School of Electrical Engineering and the Director of the Research Center for Wind Power Generation. As a Visiting Professor, he worked with Wisconsin Electric Machines and Power Electronic Consortium, The University of Wisconsin-Madison, Madison, WI, USA, from January to April 2011, and with the Department of Energy Technology, Aalborg University, Aalborg, Denmark, from June to July 2012. He is the author or coauthor of over 280 technical papers and five books, and is a holder of 50 patents. His main research interests include electrical machines, motor drives for electric vehicles, and renewable energy generation. Dr. Cheng has served as a Chair and as an organizing committee member for many international conferences. He received The Institution of Engineering and Technology (IET) Premium Award in Electric Power Application, the GM Automotive Innovative Talent Award for China University, and the Society of Automotive Engineers Environmental Excellence in Transportation Award–Education, Training and Public Awareness. He is a Fellow of The IET.