A Repetitive Control Scheme Aimed at Compensating the 6k + 1 Harmonics for a Three-Phase Hybrid Active Filter

: The traditional repetitive controller has relatively worse stability and poor transient performance due to the facts that it generates infinite gain at all the integer multiples of the fundamental frequency, and its control action is postponed by one fundamental period (T 0 ). To improve these disadvantages, many repetitive controllers with reduced delay time have been proposed, which can selectively compensate the odd harmonics or 6k±1 harmonics with delay time reduced to T 0 /2 and T 0 /3,repectively. To further study in this area, this paper proposes an improved repetitive scheme implemented in stationary reference frame, which only compensates the 6k+1 harmonics (e.g. -5, +7, -11, +13) in three-phase systems and reduces the time delay to T 0 /6 . So compared with the earlier reduced delay time repetitive controllers, the robustness and transient performance is further improved, the waste of control effort is reduced, and the possibility of amplifying and even injecting any harmonic noises into system is avoided to the greatest extent. Moreover, the proposed repetitive scheme is used in the control of a three-phase hybrid active power filter. The experimental results validate the effectiveness of the proposed repetitive control scheme.


Introduction
Recently, due to the widespread applications of distributed generations, adjustable speed drives，uncontrolled AC/DC rectifiers, and other nonlinear loads, the harmonic pollution in power systems is getting more and more serious.The passive power filter (PPF) and active power filter (APF) are the two common solutions applied to mitigate these harmonics [1][2].PPFs have the advantages of low-cost and high-efficiency.However, they also have some inherent drawbacks.Their compensation characteristics are strongly influenced by supply impedance and they are highly susceptible to series and parallel resonances with the supply and load impedance.The APFs，which are based on the power electronics, can overcome above drawbacks of PPFs [3][4].Additionally, APFs are more flexible and efficient compared with PPFs.But, pure APFs usually require a PWM inverter with large kilovoltampere (KVA) rating.Thus, they do not constitute a cost-effective harmonic filtering solution for nonlinear loads above 500-1000 kW [5][6].To address this issue, hybrid active power filter (HAPF) have been developed, which are composed of small rated APF and PPF in different configurations.Among the various viable hybrid active filter topologies, parallel hybrid active filters present a cost-effective solution for harmonic filtering and reactive power compensation of high power nonlinear industrial loads, due to small rating of the active filter-2%-3% of load KVA rating [6].Thus, they have attracted increasing attention [7][8][9][10][11].
Among various control strategies, the repetitive control, as a kind of control method based on internal model principle, can accurately track the periodic signal or reject periodic interference.
Hence it is widely used in harmonic compensation scheme for active filters [12][13][14][15].The traditional repetitive control technique can generate infinite gains at all the integer multiples of the fundamental frequency, including the odd, even harmonics and dc component.However, in most cases, the introduction of high gain for all frequencies is not necessary, as it could waste control effort and reduce the system robustness without improving system performance, even amplify irrelevant signal and reinject distortions to systems [16].Moreover, the control action of traditional repetitive controller is postponed by one fundamental period (T0), hence the transient performance is poorer.To improve the drawbacks above, and considering the fact that even harmonic components do not regularly appear in power systems, literatures [16][17] propose a repetitive control scheme aiming at compensating only the odd harmonics.It uses a negative feedback array instead of the usual positive feedback in the traditional repetitive controller.Meanwhile, the delay time of control action is reduced to T0/2.As a sequence, the control performance is improved and the convergence rate is enhanced.Furthermore, among the odd harmonics, the group of ± ± ⋅ ⋅⋅ harmonic components in electric industry are dominated due to the wide use of uncontrolled rectifiers and six-pulse converters.Thus, many improved repetitive control schemes aiming at compensating 6 1 k ± harmonics have been developed [18-20].For instance, in [18] a repetitive control scheme based on the feedback array of two delay lines plus a feedforward path is presented, which can only compensate 6 1 k ± harmonics and reduce delay time to T0/3; In [19], the authors propose a 6 1 k ± repetitive control scheme in there-phase synchronous reference frame (SRF).It has an advantage of T0/6 delay time.However, it needs complex coordinate transformation and much more calculation in both positive-rotating and negative-rotating SRFs.
Considering in three-phase power systems, harmonic of the same frequency can be decomposed into positive sequence, negative sequence and zero sequence.Generally speaking, a normal balanced three-phase system mainly contains 6k+1 harmonics (such as -5, +7, -11, +13), and rarely contains 6k-1 harmonics (such as +5, -7, +11, -13).For this reason, this paper proposes a repetitive control scheme aiming at compensating the 6k+1 harmonics implemented in three-phase stationary reference frame with T0/6 delay time.So that the transient performance is further improved.The 6k+1 repetitive controller is expressed with complex-vector notation, so that the dual-input/dual-output control system (in the αβ reference frame) can be simplified into one single-input/single-output system.Meanwhile, the general design method of Lk+M repetitive controller is also introduced, with which a repetitive controller aiming at compensating Lk+M harmonics can be easily deduced.Moreover, taking the transformerless parallel hybrid active filter as controlled object, a harmonic compensating control system based on the proposed 6 1 k + repetitive control scheme is presented.Finally, the experimental results validate the effectiveness of the 6 1 k + repetitive control scheme.

Topological structure analysis
The topology of the transformerless parallel hybrid active filter is shown in Figure 1

Mathematical modeling
According to Figure 1, the mathematical model in state-space representation for the system is formulated as  to achieve decoupling control between d-axis and q-axis.Therefore, in this paper, the output current control is implemented in αβ stationary frame, which has the advantages of no need of complex decoupling control and AC capacitor voltage sampling.The overall system control diagram is shown in Figure 2, it is mainly composed of dc-link voltage control and harmonic current tacking control.In this figure, Bpf(s) is a band-pass filter to extract the fundamental frequency component of input signal, and its expression is given by where 0 ω is the grid frequency； γ is the control coefficient of passband width and

DC-link voltage stabilisation method
Assuming that the VSI doesn't provide reactive power compensation for the load and only absorbs active power from grid to maintain its power loss.According to the power conservation principle, there is where X denotes the fundamental frequency impedance of LC filter; sd V is the d-axis component of grid voltage ( V are the d-axis and q-axis component of VSI output fundamental voltage, respectively.It can be inferred from ( 7) that in P is only regulated by fq V .Generally, fq V is small.Thus, 0 in Q ≈ could be achieved when fq V is set to 0. Thus, the dc-link voltage regulator can be designed as ( 8), and the corresponding bode diagram is shown in Figure 2.

( )( )
where * dc V is the rated value of dc-link voltage; p k , i k are the parameters of PI controller.

Harmonic current tracking control
Harmonic current tracking control is the important part of system control, which contributes directly to the performance of harmonic compensating.The block diagram of current control is shown in Figure 2. Considering the case that the three-phase load current mainly contains 6k+1 harmonics, this paper presents a 6k+1 repetitive control scheme to compensate these harmonics.
The detail theoretical derivation, analysis and design of proposed 6k+1 repetitive controller is given in the next section.

internal model of 6k+1 repetitive controller
Firstly, the internal model of the wellknown traditional repetitive controller is given by where pk s is the pole of (9).Seen from (10), it is clear that the traditional repetitive controller has an infinite number of poles located at 0 jkω , which is the reason traditional repetitive controller has resonant peaks at every integral multiple of fundermental frequency 0 ω .
In order to make repetitive controller with poles only located at a selected group of harmonic frequencies, a new internal model is need to be structured.Assume the order h of these harmonics meets the rule: h Lk M = + (11) where L, M are intergers, and L is not equal to zero.
Then, the poles of the new internal model should be located at Moreover, to enhance the frequency selectivity, an infinite number of zeros of the new internal model that located in the midpoints between two consecutive poles are introduced as These zeros bring another benefit that allowing bigger gains with improved performance.
In order to satisfy (12)(13) , the general internal model for Lk M + harmonics can be structured as After the substitution of L=6 and M=1 in (14), the 6k+1 internal model is given by Comparing (15) with the traditional internal model given by ( 9), it can be found that the delay time of 6k+1 internal model is reduced to T0/6, which means a much faster dynamic response.what's more, it should be noted that the 6k+1 internal model is expressed using the complex-vector notation, as it contains the complex coefficient 3 j e π .As a consequence, the input signal of RC (s) is required to be a complex vector.The block of the proposed 6k+1 internal model is shown in Figure 3. Assuming that the fundermental frequcny f0 = 50 Hz, i.e., T0 = 0.02 s, the bode plot of the 6k+1 repetitive controller internal model is shown in Figure 4.As expected, the amplitude-frequency response curve shows that 6k+1 internal model has resonant peaks that located at frequency multiples 6k+1 of 50 Hz (50, -250, 350, -550, 650 Hz ...), and has notches that located at frequency multiples 6k+4 of 50 Hz (-100, 200, -400, 500 Hz ...).The phase-frequency response curve shows the phase shift is bounded between 90 and -90 degree, and zero at the peaks and notches.

Fractional delay compensation
In a practical application, the implementation of repetitive control scheme is usually performed in the digital form.Using the transformation where Ts is the sampling period, and 0 0 s

N T T =
(the number of samples per fundamental period ).
In most cases, the sampling frequncy s f ( 1 ) is a fixed rate(e.g. 10 kHz，12.8kHz, 20 kHz), and the grid frequecny detected by PLL is variabe in a certain range (e.g.49~51Hz).Thus, where D and d are the integral and fractional parts of 0 6 N , repectively.
In a common implementation, But, this will cause the resonant peaks to deviate from the harmonic frequencies.As a consequence, the harmonic comperseation performance could be degraded.
To address this problem, fractional delay (FD) filters have been usd as approximations of d z − .The magnitude-frequency and phase-frequency characteristics of d z − can be given by Thus, it requires that FD filters should have a unit gain and linear phase in the low-middle frequencies, and acheive a high attenuation rate in the high frequencies to enhance the system stablity.
In the condition of 1 ), with the use of the Taylor expansion, d z − can be expressed as Specifically, choose the first-order Taylor expansion of d z − as a FD filter, that is ( ) Fd z has the low-pass filter nature.In low frequecnies, ( ) Fd z has a well linear phase approximated to the ideal value.However, the main disadvantages of ( ) Fd z are that the cutoff frequecny is too high (greater than 3000 Hz), and it changes with the value of d.Only when d=0.5, ( ) Fd z achieves the lowest cutoff frequecny and best linear phase.To overcome above issue, this paper presents a FD filter ( ) Q z by cascading Fd(z) with a zero-phase digital low-pass filter, i.e.,

( ) ( ) ( )
where ( ) M z is the zero-phase digital low-pass filter used to low the cut-off frequency and Increase the attenuation rate in high frequecnies.Its expression is given as where 0 1 , 0 a a > and 0 1 2 1 a a + = ; n is the order of filter.Although ( ) Q z is noncausal, the time delay term D z − makes it applicable.After the fractional delay compseation, ( 16) should be revised as

Design of 6k+1 repetitive controller
Figure 6 shows the block diagram of the harmonic current tracking control.This paper adopts a plug-in repetive controller structure in the control loop，where the PI controller is used to enhance the stability and improve dynamic response, and the repetive controller is used to eliminate the steady-state error.

Figure 6. Block diagram of harmonic current tracking control
In Figure 6, P(z) is the plant of current control.According to (1-2), its expression in continuous domain can be obtained as Obviously, P(s) is a second-order system.To modify the characteristic of P(s), a method of output current status feedback is used.Accoding to Figure 6, the modified plant expression is given by In Figure 7, without the repetitive controller, the tracking error e between the reference where ( ) PI z should be designed to guarantee the stability of 0 ( ) z e .With the proposed 6k+1 repetitive controller, the tracking error e can be written as By the small gain theorem, the sufficient condition for ensuring (27) stable can be given as f G s and ( ) H s are the functions of ( ) f G z and ( ) H z in Laplace domain, respectively; 1 ( 1) s τ + is a low-pass filter.
Moreover, on the premise of system stability, it can be derived that the numerator of (27) has such a steady-state relationship: Equation (31) indicates that the 6k+1 repetitive control scheme can eliminate the steady-state error of 6k+1 harmonics tracking in D+d Ts (i.e., T0/6), which means the proposed repetitive control scheme could has a much faster transient state response than the traditional one.

Experimental results
To validate the correctness and effectiveness of the proposed 6k+1 repetitive control scheme, a prototype of three-phase parallel hybrid APF is built in lab, which is shown in Figure 8.The control system is realized by a combination of digital signal processor TMS320F28335 and field programmable gate array FPGA EP2C8T144C8N.The power switches use three Infineon IGBT modules and the drive circuit uses M57962L driver chips.The non-linear load used in the experiments is a three-phase diode rectifier bridge with resistive load.The overall experimental parameters are given in Table 1.2) In the harmonic current tacking loop: a.The zero-phase low-pass filter ( ) M z is given as

LC filter parmaters
As the non-linear load used in this paper is a three-phase diode rectifier bridge with resistive load，the 5 th , 7 th , 11 th , 13 th harmonic currents are dominated in load current.Assume that the load harmonic currents are fully compensated by the hybrid APF, the voltage drop across the LC filter by the injected compensating harmonic current is where m lh i is the m-th order harmonic component of load current, and m lh I is the amplitude of m lh i .
For the hybrid APF, the design objective of LC filter is to offer a lowest possible impedance path for injecting harmonic currents, in other words, to minimize the voltage drop h v .Thus, the dc-link voltage rating of VSI can be minimized.
Then, an optimization function can be given as The capacitor C in LC filter can be chose by the rule as follow: .Substituting the above parameters into (32), the optimal inductor L can be obtained as L=2.8 mH.So we choose L=3 mH for the hybrid APF experimental prototype without loss of much performance, and the resonant frequency of LC filter is 306 Hz.

Experimental results
Figure 9 shows the dynamic behaviour of dc-link capacitor voltage in start-up process.To avoid the inrush current caused by capacitors, the series-resistance soft-start mode is used in experiments.Specifically, when 60 V dc set v V ≤ = , the IGBTs are turned off, the capacitors are charged up with small current due to the series-resistance ; When dc set v V > , the series-resistance is bypassed and then the PWM pulses will be activated.dc v reaches the setting value 80 V in the steady state, which verifies the correctness of the dc voltage control strategy.Figure 10 shows the harmonic compensation results when nonlinear load is disconnected ( delay compensation when the nonlinear load is connected, respectively.In Figure 11, the total harmonic distortion (THD) of the source current sa i is reduced to 3.8% from 24.8% (THD of the load current), and the distortion ratio of 5 th , 7 th , 11 th and 13 th harmonics in sa i are reduced to 2.3%, 1.3%, 1.6% and 1.2%, respectively.As a contrast, the THD of sa i is 4.9% in Figure 12.These comparison experiment results demonstrate the good static performance of 6k+1 repetitive controller and effectiveness of the fractional delay compensation.Also, to highlight the effectiveness of the 6k+1 repetitive control scheme, the harmonic compensation results by only the LC filter is shown in Figure 13.As seen, the source current is still highly distorted after the compensation of the LC filter, with a THD of 15.9%.The main reasons are that the resonant frequency of LC filter is not precisely tuned at a domain harmonic frequency, and the performance of LC filter seriously depends on the internal resistance of grid source.To verify the dynamic performance of the 6k+1 repetitive control scheme, Figure 14 shows the comparison experimental results of the proposed and traditional repetitive control schemes in transient process.As seen, before the time 1 t , the harmonic compensation function is not enabled, a i is only the reactive power current provided by LC filter with sinusoidal waveform, and sa i is distorted by the load harmonics.At the time 1 t , the harmonic compensation function is enabled.
The 6k+1 repetitive control scheme can take effect after T0/6 time, and eliminate the steady-state error of harmonic tracking quickly.As a contrast, the traditional repetitive control scheme takes effect after T0 time, and needs several T0 periods to eliminate the steady-state error.The experimental results demonstrates that the 6k+1 repetitive control scheme has a much better dynamic performance than the traditional repetitive control scheme, which is consistent with theoretical analysis.

Conclusions
In this paper, a 6k+1 repetitive control scheme for HAPF is proposed, which aims at compensating the 6k+1 harmonics in three-phase power systems.The internal model of the 6k+1 repetitive controller is constructed by the general mathematical principles of traditional repetitive controller, and expressed using the complex-vector notation.A FD compensating method for 6k+1 repetitive controller is also presented.Through theoretical analysis and experiments, it is demonstrated that the 6k+1 repetitive control scheme can achieve a fast transient response with delay time of T0/6, and good performance for compensating or suppressing the 6k+1 harmonics.
Furthermore, due to the above features, the 6k+1 repetitive control scheme is also suitable to used in the current or voltage control for other three-phase grid-connected inverters.

Figure 1 .
Figure 1.Topology of parallel hybrid active power filter

Figure 2 .
Figure 2. Overall block diagram of system control the periodic time delay unit, and T0 is the fundamental period, i.e., equal to zero, it can be obtained that 0 ( 0, 1, 2, , ) pk s jk k

Figure 4 .
Figure 4. Bode plot of the proposed repetitive controller internal model.
) can be discretized and its expression in discrete time domain is given by performed by reserving D memory locations, with the fractional order part d z − neglected.

,
Figure 5 shows the bode plot of ( ) Fd z , with 78.125 s s T µ =

kFigure 7 .
Figure 7. Bode plots of P(s) and ( ) P s ′ the compensation function, and

Figure 8 .
Figure 8. Photograph of the prototype In the implementation of experiments, the parameters of controllers are given as follows.1)Dc-link voltage PI controller: number of delay sample is 42, and the FD filter is given as PI controller in the plug-in repetitive controller: individual m-th order harmonic distortion rate, and lf I is the amplitude of fundamental component in load current.

Q 1 ω
is the reactive power demanded by load, is the grid frequency, s V is the grid voltage amplitude.Assume the capacitor C has been determined, such as C=90uf.According to Fig.11(b)

Figure 9 .
Figure 9. Start-up process of dc-link voltage

Figure 10 .
Figure 10.Harmonic compensation results with nonlinear load disconnected

Figure 14 .
Figure 14.Dynamic performance comparison: (a) 6k+1 repetitive control scheme; (b) traditional repetitive control scheme is the power of the dc-link capacitor and loss P is the power loss of inverter.It can be inferred from (6) that if loss P is regarded as a disturbance, dc v could be controlled by adjusting in P .In the dq synchronous reference frame (grid voltage orientation), the active power in P and reactive power in Q absorbed by VSI can be given

Table 1 . Experimental parameters Parameters Symbol Value Unit grid
phase voltage