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

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1 energie Article A Repetitive Control Scheme Aimed at Compenating 6k + 1 Harmonic for a Three-Phae Hybrid Active Filter Zhaoxu Luo 1, Mei Su 1, Jian Yang 1, *, Yao Sun 1, Xiaochao Hou 1 and Joep M. Guerrero 2 1 School of Information Science and Engineering, Central South Univerity, Changha , China; luozhaoxu@cu.edu.cn (Z.L.); umeicu@cu.edu.cn (M.S.); yaouncu@gmail.com (Y.S.); houxc10@cu.edu.cn (X.H.) 2 Department of Energy Technology, Aalborg Univerity, Aalborg Eat DK-9220, Denmark; joz@et.aau.dk * Correpondence: jian.yang@cu.edu.cn; Tel.: Academic Editor: Ying-Yi Hong Received: 4 Augut 2016; Accepted: 23 September 2016; Publihed: 29 September 2016 Abtract: The traditional repetitive controller ha relatively wore tability and poor tranient performance becaue it generate infinite gain at all integer multiple of fundamental frequency, and it control action i potponed by one fundamental period (T 0 ). To improve e diadvantage, many repetitive controller with reduced delay time have been propoed, which can electively compenate odd harmonic or 6k ± 1 harmonic with delay time reduced to T 0 /2 and T 0 /3, repectively. To furr tudy in thi area, thi paper propoe an improved repetitive cheme implemented in a tationary reference frame, which only compenate 6k + 1 harmonic (e.g., 5, +7, 11, +13) in three-phae ytem and reduce time delay to T 0 /6. Thu compared with earlier reduced delay time repetitive controller, robutne and tranient performance i furr improved, wate of control effort i reduced, and poibility of amplifying and even injecting any harmonic noie into ytem i avoided to a great extent. Moreover, propoed repetitive cheme i ued in control of a three-phae hybrid active power filter. The experimental reult validate effectivene of propoed repetitive control cheme. Keyword: repetitive control; hybrid active power filter; power quality; harmonic compenation 1. Introduction Recently, due to widepread application of ditributed generation, adjutable peed drive, uncontrolled AC/DC rectifier, and or nonlinear load, harmonic pollution in power ytem i becoming increaingly more eriou. The paive power filter (PPF) and active power filter (APF) are two common olution applied to mitigate e harmonic [1,2]. PPF have advantage of low-cot and high-efficiency. However, y alo have ome inherent drawback. Their compenation characteritic are trongly influenced by upply impedance and y are highly uceptible to erie and parallel reonance with upply and load impedance. The APF, which are baed on power electronic, can overcome above drawback of PPF [3,4]. Additionally, APF are more flexible and efficient compared with PPF. However, pure APF uually require a PWM inverter with a large kilovolt ampere (KVA) rating. Thu, y do not contitute a cot-effective harmonic filtering olution for nonlinear load above kw [5,6]. To addre thi iue, hybrid active power filter (HAPF) have been developed, which are compoed of a mall rated APF and a PPF in different configuration. Among variou viable hybrid active filter topologie, parallel hybrid active filter preent a cot-effective olution for harmonic filtering and reactive power compenation of high power nonlinear indutrial load, due to mall rating of active filter (2% 3%) of load KVA rating [6]. Thu, y have attracted increaing attention [7 11]. Energie 2016, 9, 787; doi: /en

2 Energie 2016, 9, of 17 Among variou control trategie, repetitive control, a a kind of control method baed on internal model principle, can accurately track periodic ignal or reject periodic interference. Hence it i widely ued in harmonic compenation cheme for active filter [12 15]. The traditional repetitive control technique can generate infinite gain at all integer multiple of fundamental frequency, including odd, even harmonic, and dc component. However, in mot cae, introduction of high gain for all frequencie i not neceary, a it could wate control effort and reduce ytem robutne without improving ytem performance, even amplify irrelevant ignal, and reinject ditortion to ytem [16]. Moreover, control action of traditional repetitive controller i potponed by one fundamental period (T 0 ), hence tranient performance i poorer. To improve drawback above, and conidering fact that even harmonic component do not regularly appear in power ytem, literature reference [16,17] propoe a repetitive control cheme aimed at compenating only odd harmonic. Thi ue a negative feedback array intead of uual poitive feedback in traditional repetitive controller. Meanwhile, delay time of control action i reduced to T 0 /2. A a equence, control performance i improved and convergence rate i enhanced. Furrmore, among odd harmonic, group of 6k ± 1 (k = 0, ±1, ±2 ) harmonic component in electrical indutry are dominant due to wide ue of uncontrolled rectifier and ix-pule converter. Thu, many improved repetitive control cheme aiming at compenating 6k ± 1 harmonic have been developed [18 20]. For intance, in [18] a repetitive control cheme baed on feedback array of two delay line plu a feed forward path i preented, which can only compenate 6k ± 1 harmonic and reduce delay time to T 0 /3; In [19], author propoe a 6k ± 1 repetitive control cheme in re-phae ynchronou reference frame (SRF). It ha an advantage of T 0 /6 delay time. However, it need complex coordinate tranformation and much more calculation in both poitive-rotating and negative-rotating SRF. Conidering three-phae power ytem, harmonic of ame frequency can be decompoed into poitive equence, negative equence, and zero equence. Generally peaking, a normal balanced three-phae ytem mainly contain 6k + 1 harmonic (uch a 5, +7, 11, +13), and rarely contain 6k 1 harmonic (uch a +5, 7, +11, 13). For thi reaon, thi paper propoe a repetitive control cheme aimed at compenating 6k + 1 harmonic implemented in a three-phae tationary reference frame with T 0 /6 delay time in order that tranient performance i furr improved. The 6k + 1 repetitive controller i expreed with complex-vector notation, o that dual-input/dual-output control ytem (in αβ reference frame) can be implified into one ingle-input/ingle-output ytem. Meanwhile, general deign method of Lk + M repetitive controller i alo introduced, with which a repetitive controller aimed at compenating Lk + M harmonic can be eaily deduced. Moreover, taking tranformerle parallel hybrid active filter a controlled object, a harmonic compenating control ytem baed on propoed 6k + 1 repetitive control cheme i preented. Finally, experimental reult validate effectivene of 6k + 1 repetitive control cheme. 2. Sytem Structure and Mamatical Modeling of HAPF 2.1. Topological Structure Analyi The topology of tranformerle parallel hybrid active filter i hown in Figure 1. It conit of an LC paive filter and a three-phae voltage ource inverter (VSI). The purpoe of intalling LC filter are: (1) to provide reactive power compenating and aborb ome harmonic; (2) to utain fundamental voltage at point of common coupling (PCC). Additionally, active filter (VSI) i reponible for improving filtering characteritic of paive filter and avoiding undeirable reonance with grid. To minimize it own KVA rating, VSI doe not participate in reactive compenation, and grid voltage i almot fully dropped on capacitor in LC filter. Thu fundamental voltage utained by VSI i mall uch that dc bu voltage rating of VSI can be et very low, KVA rating and power loe are n reduced greatly. Due to preence of VSI, it i not neceary that LC filter i accurately tuned at a certain harmonic frequency. The deign

3 Energie 2016, 9, of 17 objective of LC filter i to offer lowet poible impedance path for injecting harmonic current, on Energie premie 2016, 9, 787 of enuring reactive power compenating. 3 of 17 i a i b i c v a v b v c i la i lb i lc C v Ca v Cb v Cc i c i b i a LR, Sc v c Sb v b S a v a C dc idc + v dc - Sc Sb S a Figure Topology of of parallel hybrid hybrid active active power power filter. filter Mamatical Mamatical Modeling Modeling According According to to Figure Figure 1, 1, mamatical mamatical model model in in tate-pace tate pace repreentation repreentation for for ytem ytem i i formulated formulated a a L di a dt di = v a + v Ca Ri a + v a a L L di b va Ca Ria va dt dt = v b + v Cb b + v b (1) L di c dt di = v c + v Cc Ri c + v c b L vb vcb Rib vb (1) dt C dv Ca dt = i a dic L Cv dv Cb c dt = v i Cc b Ric v (2) c dt C dv Cc dt = i c dv dvca C dc dc = i C ia dt dc = (S a i a + S b i b + S a i a ) (3) dt where S a, S b, and S c are witching function defined dvcb by C ib (2) { dt 1, (when S S dv x = x on, S Cc x o f f ) C i (x = a, b, c) (4) 0, (when S c x off, dt S x on) 3. Sytem Control dvdc Cdc idc ( Si a a Si b b Si a a ) (3) According to (1) and (2), it can dt be inferred that tate equation of output current iwhere x (x = a, S b, a, S c) b, i and a econd-order S c are witching differential function equation. defined by If output current control i implemented in a dq ynchronou reference frame, it need to ample and feedback AC capacitor voltage v Cx (x = a, b, c) to achieve decoupling 1, control ( when Sbetween xon, Sx off ) d-axi and q-axi. Therefore, in thi paper, Sx ( xa, b, c) (4) output current control i implemented 0, ( when in Sx off, αβ tationary Sx ) frame, which ha advantage of no requirement for complex decoupling control and AC capacitor voltage ampling. The overall ytem control diagram i hown in Figure 2, it i mainly compoed of dc-link voltage control and harmonic 3. Sytem Control current tacking control. In thi figure, Bpf () i a band-pa filter to extract fundamental frequency component According of to input (1) and ignal, (2), and it it can expreion be inferred i given that by tate equation of output current ix ( x abc,, ) i a econd order differential equation. If output current control i implemented in γω a dq ynchronou reference frame, Bp it f need () = 0 to ample and feedback AC capacitor voltage vcx ( x abc,, ) 2 + γω 0 + ω 2 (5) 0 to achieve decoupling control between d axi and q axi. Therefore, in thi paper, where output ω 0 icurrent gridcontrol frequency; i implemented γ i control in coefficient tationary of paband frame, width which and ha γ > 0. advantage of no requirement for complex decoupling control and AC capacitor voltage ampling. The overall ytem control diagram i hown in Figure 2, it i mainly compoed of dc link voltage control and harmonic current tacking control. In thi figure, Bpf() i a band pa filter to extract fundamental frequency component of input ignal, and it expreion i given by

4 Bpf () (5) Energie 2016, 9, of 17 where 0 i grid frequency; i control coefficient of paband width and 0. * V dc 0 V fq d T q j e dq/ v f ( ) v( ) S a S a S c i labc ( ) v dc i abc l( ) 1 Bpf ( ) ih( ) vh( ) i( abc) i abc ( ) 1 Bpf ( ) ih( ) Figure Figure Overall Overall block block diagram diagram of of ytem ytem control. control DC Link Voltage Stabilization Method 3.1. DC-Link Voltage Stabilization Method Auming that VSI doe not provide reactive power compenation for load and only Auming that VSI doe not provide reactive power compenation for load and only aborb active power from grid to maintain it power lo n according to power aborb active power from grid to maintain it power lo n according to power conervation conervation principle, we have principle, we have P in Cdcv dv P in = C dc v dc dc dc + Plo (6) dt lo (6) dv P in i active power aborbed from grid, dv C dc where P in i active power aborbed from grid, C dc v dc dc dcv dc i power of dc link dt i dt power of dc-link capacitor, and capacitor, P lo iand power P lo i lo power of inverter. lo of It inverter. can be inferred It can be from inferred (6) that from if P lo (6) that i regarded if P a lo i a diturbance, v regarded a a diturbance, dc could be controlled by adjuting P v dc could be controlled in. In dq ynchronou reference frame by adjuting P in. In dq ynchronou (grid voltage orientation), active power P in and reactive power Q in aborbed by VSI can be given by reference frame (grid voltage orientation), active power P in and reactive power Q in aborbed by VSI can be given by P in = V dv f q X Q in = V V2 f d dv V dv f d +V 2 (7) f q fq X Pin X where X denote fundamental frequency impedance 2 of 2LC filter; V (7) Vfd VdVfd V d i d-axi component of fq grid voltage (V q = 0); V f q, V f q are Qd-axi in and q-axi component of VSI output fundamental voltage, repectively. X where It X can denote be inferred fundamental from (7) thatfrequency P in i onlyimpedance regulated by of V f q LC. Generally, filter; V V f q d i i mall. d axi Thu, component Q in 0 could be achieved when V of grid voltage ( Vq 0 f q i et to 0. Thu, dc-link voltage regulator can be deigned a (8), ); V fq, V fq are d axi and q axi component of VSI output and correponding bode diagram i hown in Figure 2. fundamental voltage, repectively. { It can be inferred from (7) that V P f in q = i (k only p + kregulated i /)(Vdc by v dc) V fq. Generally, V fq i mall. Thu, Qin 0 could be achieved when V V fq i f d et = 0 (8) to 0. Thu, dc link voltage regulator can be deigned where a (8), and V correponding bode diagram i hown in Figure 2. dc i rated value of dc-link voltage; k p, k i are parameter of PI controller. Vfq ( kp ki )( Vdc vdc) 3.2. Harmonic Current Tracking Control (8) V fd 0 Harmonic current tracking control i important part of ytem control, which contribute directly * where V to dc i performance rated value of harmonic of dc link compenating. voltage; k The p, k block i are diagram parameter of current of control PI controller. i hown in Figure 2. Conidering cae that three-phae load current mainly contain 6k + 1 harmonic, thi paper preent a 6k + 1 repetitive control cheme to compenate e harmonic. The detailed oretical derivation, analyi, and deign of propoed 6k + 1 repetitive controller i given in next ection.

5 Energie 2016, 9, of Repetitive Control Scheme (6k + 1) 4.1. Internal Model of 6k + 1 Repetitive Controller Firtly, internal model of well-known traditional repetitive controller i given by RC t () = e T 0 1 e T 0 (9) where e T 0 i periodic time delay unit, and T 0 i fundamental period, i.e., T 0 = 2π/ω 0. By etting denominator 1 e T 0 in (9) equal to zero, following can be obtained pk = jkω 0 (k = 0, ±1, ±2,..., ± ) (10) where pk i pole of (9). Seen from (10), it i clear that traditional repetitive controller ha an infinite number of pole located at jkω 0, which i reaon that traditional repetitive controller ha reonant peak at every integral multiple of fundamental frequency ω 0. In order to make a repetitive controller with pole only located at a elected group of harmonic frequencie, a new internal model need to be tructured. Aume order h of e harmonic meet rule: h = Lk + M (11) where L, M are interger, and L i not equal to zero. Then, pole of new internal model hould be located at pk = j(lk + M)ω 0 (k = 0, ±1, ±2,..., ± ) (12) Moreover, to enhance frequency electivity, an infinite number of zero of new internal model that are located in midpoint between two conecutive pole are introduced a zk = j(lk + M + 0.5L)ω 0 (k = 0, ±1, ±2,..., ± ) (13) Thee zero bring anor benefit of allowing bigger gain with improved performance. In order to atify (12) and (13), general internal model for Lk + M harmonic can be tructured a RC g () = 1 e ( jω0(m+0.5l)) L T 0 1 e ( jω 0 M) L T 0 (14) After ubtitution of L = 6 and M = 1 in (14), 6k + 1 internal model i given by RC() = 1 + ej π 3 e T0 6 1 e j π 3 e T 0 6 (15) Comparing (15) with traditional internal model given by (9), it can be found that delay time of 6k + 1 internal model i reduced to T 0 /6, which mean a much fater dynamic repone. What i more, it hould be noted that 6k + 1 internal model i expreed uing complex-vector notation, a it contain complex coefficient e j π 3. A a conequence, input ignal of RC() i required to be a complex vector. The block of propoed 6k + 1 internal model i hown in Figure 3. Auming that fundamental frequency f 0 = 50 Hz, i.e., T 0 = 0.02, bode plot of 6k + 1 repetitive controller internal model i hown in Figure 4. A expected, amplitude-frequency repone curve how that 6k + 1 internal model ha reonant peak that are located at frequency multiple 6k + 1 of 50 Hz (50, 250, 350, 550, 650 Hz...), and ha notche that are located at frequency multiple 6k + 4 of 50 Hz ( 100, 200, 400, 500 Hz...). The phae-frequency repone curve how phae hift i bound between 90 and 90, and zero at peak and notche.

6 repetitive controller internal model i hown in Figure 4. A expected, amplitude frequency Auming that fundamental frequency f0 = 50 Hz, i.e., T0 = 0.02, bode plot of 6k + 1 repone repetitive curve controller how that internal 6k model + 1 internal i hown model in Figure ha reonant 4. A expected, peak that are amplitude frequency located at multiple repone 6k curve + 1 of how 50 Hz that (50, 250, 6k 350, + 1 internal 550, 650 model Hz...), ha and reonant ha notche peak that are located at at frequency multiple 6k 6k of 1 of Hz Hz ( 100, (50, 250, 200, 350, 400, 550, Hz...). Hz...), The and phae frequency notche that repone are located curve at frequency how phae Energie multiple hift 2016, i 9, 6k 787 bound + 4 of between 50 Hz ( 100, , and 400, 90, 500 and Hz...). zero at The phae frequency peak and notche. repone curve how 6 of 17 phae hift i bound between 90 and 90, and zero at peak and notche. e p j p j3 3 e() e() T 6-6 e - 0 T0 y () (a) (a) ea e() a () y a () T e - T0 e T 0 6 e - T0 6 e - ee() b () () b y b () (b) (b) Figure 3. Block diagram of 6k repetitive controller internal model: (a) Complex vector notation; Figure Block diagram of of 6k 6k repetitive controller internal model: (a) (a) Complex vector Complex-vectornotation; (b) Scalar notation. (b) Scalar notation. 1 2 Figure 4. Bode plot of propoed repetitive controller internal model. Figure 4. Bode plot of propoed repetitive controller internal model. Figure 4. Bode plot of propoed repetitive controller internal model Fractional Delay Compenation In a practical application, implementation of repetitive control cheme i uually performed in digital form. Uing tranformation z = e T, (15) can be dicretized and it expreion in dicrete time domain i given by RC(z) z=e T = 1 + ej π 3 z N0 6 1 e j π 3 z N 0 6 (16)

7 Energie 2016, 9, of 17 where T i ampling period, and N 0 = T 0 /T ( number of ample per fundamental period). In mot cae, ampling frequency f ( f = 1/T ) i a fixed rate (e.g., 10 khz, 12.8 khz, 20 khz), and grid frequency detected by PLL i variable in a certain range (e.g., Hz). Thu, N 0 /6 i uually non-integer. Let z N 0 6 = z (D+d) = z D z d (17) where D and d are integral and fractional part of N 0 /6, repectively. In a common implementation, z N 0/6 i approximately treated a z D and performed by reerving D memory location, with fractional order part z d neglected. However, thi will caue reonant peak to deviate from harmonic frequencie. A a conequence, harmonic compenation performance could be degraded. To addre thi problem, fractional delay (FD) filter have been ued a approximation of z d. The magnitude-frequency and phae-frequency characteritic of z d can be given by { z d = 1 z d = dωt (18) Thu, it require that FD filter hould have a unit gain and linear phae in low-middle frequencie, and achieve a high attenuation rate in high frequencie to enhance ytem tability. In condition of z 1 1 < 1 (i.e., π/(3t ) < ω < π/(3t )), with ue of Taylor expanion, z d can be expreed a z d = (1 + z 1 1) d = 1 + d(z 1 1) d(d 1)... (d n + 1) (z 1 1) n (19) n! Specifically, chooing firt-order Taylor expanion of z d a a FD filter, that i Fd(z) = 1 d + dz 1 (20) Figure 5 how bode plot of Fd(z), with T = µ, d = 0.2, 0.5, and 0.8, repectively. It can be een that Fd(z) ha a low-pa filter nature. At low frequencie, Fd(z) ha a good linear phae approximating to ideal value. However, main diadvantage of Fd(z) are that cutoff frequency i too high (greater than 3000 Hz), and it change with value of d. Only when d = 0.5, Energie doe Fd(z) 2016, 9, achieve 787 lowet cutoff frequency and bet linear phae. 8 of db ideal value ideal value d 0.8 d 0.2,0.8 d 0.5 d 0.5 d 0.2 ideal value Frequency (Hz) Figure 5. Bode plot of Fd(z). To overcome above iue, thi paper preent a FD filter Qz ( ) by cacading Fd(z) with a zero phae digital low pa filter, i.e., Qz ( ) FdzM ( ) ( z) (21)

8 d Frequency (Hz) Energie 2016, 9, 787 Figure 5. Bode plot of Fd(z). 8 of 17 To overcome above iue, thi paper preent a FD filter Qz ( ) by cacading Fd(z) with a zero phae To overcome digital low pa abovefilter, iue, i.e., thi paper preent a FD filter Q(z) by cacading Fd(z) with a zero-phae digital low-pa filter, i.e., Qz ( ) FdzM ( ) ( z) (21) where M ( z ) i zero phae digital Q(z) low pa = Fd(z)M(z) filter ued to lower cut off frequency (21) and where increae M(z) i attenuation zero-phae rate at digital high frequencie. low-pa filter It ued expreion to lower i given cut-off a frequency and increae attenuation rate at high frequencie. It expreion i given 1 M( z) ( a za a z ) n a (22) where a M(z) = (a 1 z + a 0 + a 1 z 1 ) n (22) 0, a1 0 and a0 2a1 1; n i order of filter. D Although Qz () i non caual, time delay term z make it applicable. After fractional where a 0, a 1 > 0 and a 0 + 2a 1 = 1; n i order of filter. delay compenation, (16) hould be revied a Although Q(z) i non-caual, time delay term z D make it applicable. After fractional delay compenation, (16) hould be revied a j 3 D 1 e Q( z) z RC( z) RC(z) = 1 + ej π 3 Q(z)z D (23) j 1 1 e j π 3 D e Q( z) z (23) 3 Q(z)z D 4.3. Deign of 6k + 1 Repetitive Controller Figure 6 how block diagram of harmonic current tracking control. Thi Thipaper adopt a aplug in plug-inrepetitive controller tructure in in control loop, where PI controller i ued to enhance tability and improve dynamic repone, and repetitive controller i ued to eliminate teady-tate teady tate error. * i h e Gf ( z) P() z k P() z c v h i h Figure 6. Block diagram of harmonic current tracking control. Figure 6. Block diagram of harmonic current tracking control. In Figure 6, P(z) i plant of current control. According to (1) and (2), it expreion in continuou domain can be obtained a P() = 1 L + R + 1/C Obviouly, P() i a econd-order ytem. To modify characteritic of P(), a method of output current tatu feedback i ued. According to Figure 6, modified plant expreion i given by (24) P () = k cp() 1 + k c P() = 1 L k c + (k c+r) k c + k 1 c C (25) Equation (25) reveal that P () can be viewed a R become 1 (k c R), while L, C become 1/k c and k c time ir original value in P(), repectively. The bode plot of P() and P () are hown in Figure 7, with L = 3 mh, C = 90 µf, R = 0.1 Ω. A een, P () ha cancelled reonant peak appearing in P(), and preent characteritic of a band-pa filter. The paband width depend on value of k c. A larger k c lead to a bigger paband width and maller phae lead/lag.

9 Equation (25) reveal that P () can be viewed a R become 1 ( k c R), while L, C become 1 k c and k c time ir original value in P(), repectively. The bode plot of P() and P () are hown in Figure 7, with L 3 mh, C 90 F, R 0.1. A een, P () ha cancelled reonant peak appearing in P(), and preent characteritic of a band pa filter. The paband width depend Energie 2016, 9, of 17 on value of k c. A larger k c lead to a bigger paband width and maller phae lead/lag. P() P() with k c 3 P() with k c 6 P() P() with k c 3 P() with k c 6 Figure 7. Bode plot of P() and P ().. * In Figure 7, 7, without repetitive controller, tracking error error e between e between reference reference i i h and h output and output i h i i h i 1 e 0 (z) = e 1 + PI(z)P 1 * 0() z (z) i 1 PI( z) P( z) i h (26) h (26) where PI(z) hould be deigned to guarantee tability of e 0 (z). where With PI ( z ) propoed hould be 6k deigned + 1 repetitive to guarantee controller, tability tracking of error e () z e 0. can be written a With propoed 6k + 1 repetitive controller, 1 e(z) = e tracking error e can be written a 0 (z) 1+H(z)G f (z)r(z) 1 e( = z) 1 2 e (1 e 0(z) 0( j π 3 Q(z)z D ) (27) 1 (1 H(z)G 1 HzG ( ) f (z)) ( zrz ) (1+e ( ) j π 3 Q(z)z D )/2 j 1 (1 3 D e Q( z) z ) e0( z) 2 j P (z) 3 f where G f (z) i compenation function, and D 1 H(z) (1- H= ( z) G ( z))(1 e Q( z) z ) PI(z)P (z) where Gf () z i compenation function, and By mall gain orem, a ufficient condition for enuring (27) table can be given a f (27) (28) (1 H(z)G f (z)) (1 + e j π 3 Q(z)z D )/2 < 1 (29) Clearly, (1 + e j π 3 Q(z)z D )/2 1 i true. To make (29) true, it only need 1 H(z)G f (z) < 1 to be atified. Thu, G f (z) can be choen a G f () = 1 H() 1 τ + 1 (30) where G f () and H() are function of G f (z) and H(z) in Laplace domain, repectively; 1/(τ + 1) i a low-pa filter. Moreover, on premie of ytem tability, it can be derived that numerator of (27) ha uch a teady-tate relationhip: 1 e j π 3 Q(e j(6k+1)ω 0 T )e j(6k+1)ω 0DT = 0 (31)

10 f 1( 1) i a low pa filter. Moreover, on premie of ytem tability, it can be derived that numerator of (27) ha uch a teady tate relationhip: Energie 2016, 9, of 17 j 3 j(6k1) 0T j(6k1) 0DT Equation (31) indicate that 6k + 1 repetitive control cheme can eliminate teady-tate teady tate error of 6k + 1 harmonic tracking in D + d TT (i.e., T0/6), T 0 which mean propoed repetitive control cheme could have a much fater tranient tate repone than traditional one. 5. Experimental Reult To validate correctne and effectivene of of propoed 6k 6k repetitive control cheme, a prototype a of of a three phae a three-phae parallel parallel hybrid hybrid APF APF wa wa built built in inlaboratory, laboratory, which which i hown i hown in Figure in 8. Figure The control 8. The control ytem ytem i realized i realized by a combination by a combination of digital of digital ignal ignal proceor proceor TMS320F28335 and field and programmable field gate gate array array FPGA FPGA EP2C8T144C8N. The The power power witche ue uethree Infineon IGBT module and drive circuit ue M57962L driver chip. The non linear non-linear load ued in experiment i a three-phae three phae diode rectifier bridge with reitive load. The overall experimental parameter are given in Table 1. f 1 e Q( e ) e 0 (31) Figure 8. Photograph of prototype. Figure 8. Photograph of prototype. Table 1. Experimental parameter. Parameter Symbol Value Unit grid phae voltage v 60 V (rm) grid frequency f 50 Hz inductance L 3 mh capacitor C 90 µf dc bu voltage V dc 80 V load reitance R L 6.6 Ω witching period T µ 5.1. Controller Parameter In implementation of experiment, parameter of controller are given a follow. (1) Dc-link voltage PI controller: k p1 = 2, k i1 = 5. (2) In harmonic current tacking loop: a. The zero-phae low-pa filter M(z) i given a M(z) = (0.25z z 1 ) 2 ;

11 Energie 2016, 9, of 17 b. The number of delay ample i 42, and FD filter i given a c. Output current tate feedback gain k c = 3; Fd(z) = 1 d + dz 1 = z 1 (32) d. PI controller in plug-in repetitive controller: k p2 = 1, k i2 = 1. e. The compenation function G f (z) i given a 5.2. LC Filter Parameter G f (z) = z z z z 2 (33) A non-linear load ued in thi paper i a three-phae diode rectifier bridge with reitive load, 5th, 7th, 11th, 13th harmonic current are dominant in load current. Auming that load harmonic current are fully compenated by hybrid APF, voltage drop acro LC filter by injected compenating harmonic current i v h = ilh m(ω ml 1 ω m>1 C m ) (Ilh m m>1 ωm L 1 ) (34) where i m lh i m-th order harmonic component of load current, and Im lh i amplitude of im lh. For hybrid APF, deign objective of LC filter i to offer a lowet poible impedance path for injecting harmonic current, in or word, to minimize voltage drop v h. Thu, dc-link voltage rating of VSI can be minimized. Then, an optimization function can be given a f min = (Ih m m=5,7,11,13 ω ml 1 ω m C ) = I f m=5,7,11,13 ω m C (HD m ω m L 1 ω m C ) (35) where HD m i individual m-th order harmonic ditortion rate, and I l f fundamental component in load current. The capacitor C in LC filter can be choen by rule a follow: i amplitude of Q C = 3ω 1 CV 2 (36) where Q C i reactive power demanded by load, ω 1 i grid frequency, V i grid voltage amplitude. Aume capacitor C ha been determined, uch a C = 90 uf. According to Figure 11b, it can be een that HD 5 = 22.4%, HD 7 = 8%, HD 11 = 5.7% and HD 13 = 2.6%. Subtituting above parameter into (32), optimal inductor L can be determined a L = 2.8 mh. So we can chooe L = 3 mh for hybrid APF experimental prototype without lo of much performance, and reonant frequency of LC filter i 306 Hz Experimental Reult Figure 9 how dynamic behavior of dc-link capacitor voltage in tart-up proce. To avoid inruh current caued by capacitor, erie-reitance oft-tart mode wa ued in experiment. Specifically, when v dc V et = 60 V, IGBT are turned off, capacitor are charged up with mall current due to erie-reitance; when v dc > V et, erie-reitance i bypaed and n PWM pule will be activated. v dc reache etting value 80 V in teady tate, which verifie correctne of dc voltage control trategy.

12 avoid inruh current caued by capacitor, erie reitance oft tart mode wa ued in experiment. Specifically, when v V 60 V, IGBT are turned off, capacitor are dc et charged up with mall current due to erie reitance; when vdc Vet, erie reitance i bypaed and n PWM pule will be activated. v reache etting value 80 V in teady dc Energie 2016, 9, of 17 tate, which verifie correctne of dc voltage control trategy. v dc V et Figure 9. Start up Start-up proce of dc link dc-link voltage. To validate tatic and dynamic performance of of propoed 6k + 1 repetitive control cheme, related experimental reult are are hown hown in in Figure Figure For For ake ake of implicity, of implicity, only only a-phae a waveform phae waveform are diplayed. are diplayed. Figure 10 how harmonic compenation reult when a nonlinear load i diconnected (i ( i la = 00). A A een, i a = iai, a and, andi a i a i almot reactive power current provided by LC filter. The waveform of i a i i a i inuoidal with with le le ditortion, which indicate that propoed repetitive control control cheme cheme can can well well uppre uppre undeired undeired harmonic harmonic component. component. Figure Figure and and how how teady-tate teady tate harmonic harmonic compenation compenation reult reult with with and and without without fractional fractional delay delay compenation compenation when when nonlinear nonlinear load load i i connected, connected, repectively. repectively. In Figure In Figure 11, 11, total harmonic ditortion (THD) of ource current i total harmonic ditortion (THD) of ource current i a i a i reduced to 3.8% from 24.8% (THD of reduced to 3.8% from 24.8% (THD of load current), and ditortion ratio of 5th, 7th, 11th, and 13th harmonic in i a are reduced to load 2.3%, current), 1.3%, and 1.6%, and ditortion 1.2%, repectively. ratio of 5th, 7th, A 11th, a comparion, and 13th harmonic THD in of ia are reduced to 2.3%, i 4.9% in Figure 12. Thee 1.3%, 1.6%, comparion and 1.2%, experimental repectively. reult A demontrate a comparion, good THD taticof performance i a i 4.9% ofin Figure 6k repetitive Thee controller Energie comparion 2016, and 9, experimental 787 effectivene reult of demontrate fractional delay good compenation. tatic performance of 6k + 1 repetitive 13 of 17 controller and effectivene of fractional delay compenation. v a i a i a i la Figure 10. Harmonic compenation reult with nonlinear load diconnected. i a i a i la

13 Energie 2016, 9, of 17 Figure 10. Harmonic compenation reult with nonlinear load diconnected. i a i a i la (a) (b) Figure 11. Figure Harmonic 11. Harmonic compenation reult with fractional delay (FD) (FD) compenation: (a) Source (a) (L1), Source (L1), compenating compenating (L2), load (L2), (L3) load (L3) current waveform; (b) Harmonic ditortion rate graph. rate graph.

14 Energie 2016, 9, of 17 Energie 2016, 9, of 17 i a i a i la (a) (b) Figure Figure Harmonic compenation reult without FD compenation: (a) (a) Source Source (L1), (L1), compenating (L2), load (L3) current waveform; (b) Harmonic ditortion rate graph. (L2), load (L3) current waveform; (b) Harmonic ditortion rate graph. Alo, to highlight effectivene of 6k + 1 repetitive control cheme, harmonic Alo, to highlight effectivene of 6k 1 repetitive control cheme, harmonic compenation reult by only LC filter are hown in Figure 13. A een, ource current compenation reult by only LC filter are hown in Figure 13. A een, ource current i till i till highly ditorted after compenation of LC filter, with a THD of 15.9%. The main reaon highly areditorted that reonant after frequency compenation of LCof filter i LC not filter, preciely with tuned a THD at aof domain 15.9%. harmonic The main frequency, reaon are that and reonant performance frequency of of LC filter LC eriouly filter i depend not preciely on tuned internal at reitance a domain ofharmonic grid ource. frequency, and performance of LC filter eriouly depend on internal reitance of grid ource.

15 Energie 2016, 9, of 17 Energie 2016, 9, of 17 i a i a i la (a) (b) Figure 13. Harmonic compenation reult by only LC filter: (a) Source (L1), compenating (L2), load load (L3) (L3) current current waveform; waveform; (b) (b) Harmonic Harmonic ditortion ditortion rate rate graph. graph. To verify dynamic performance of 6k + 1 repetitive control cheme, Figure 14 how comparion experimental reult of propoed and traditional repetitive control cheme in tranient proce. A een, before time t 1, t 1, harmonic compenation function function i not i not enabled, enabled, i a i i only a i reactive power current provided by LC filter with inuoidal waveform, and i only reactive power current provided by LC filter with inuoidal waveform, a i ditorted and i by a i load harmonic. At time t 1, harmonic compenation function i enabled. The 6k + 1 repetitive ditorted by load harmonic. At time t 1, harmonic compenation function i enabled. The 6k control cheme can take effect after T 0 /6 time, and eliminate teady-tate error of harmonic tracking quickly. + 1 repetitive A a comparion, control cheme traditional can take effect repetitive after control T0/6 time, cheme and take eliminate effect after teady tate T 0 time, and error need of everal harmonic T 0 tracking period toquickly. eliminate A a comparion, teady-tate error. traditional The experimental repetitive reult control demontrate cheme take that effect after T0 time, and need everal T0 6k + 1 repetitive control cheme ha period a muchto better eliminate dynamic teady tate performance error. thanthe experimental traditional repetitive reult control demontrate cheme, that which 6k i conitent + 1 repetitive with control oretical cheme ha analyi. a much better dynamic performance than traditional repetitive control cheme, which i conitent with oretical analyi.

16 Energie 2016, 9, of 17 Energie 2016, 9, of 17 i a T0 6 i a i la t 1 (a) i a T 0 i a i la t 1 (b) Figure 14. Dynamic performance comparion: (a) 6k + 1 repetitive control cheme; (b) Traditional repetitive repetitive control control cheme. cheme. 6. Concluion In thi thi paper, paper, a 6k a + 6k 1 repetitive + 1 repetitive control control cheme for cheme HAPFfor i propoed, HAPF i which propoed, aim at which compenating aim compenating 6k + 1 harmonic 6k + in 1 harmonic three-phaein power three phae ytem. power Theytem. internalthe model internal of model 6k + of 1 repetitive 6k + 1 controller repetitive controller i contructed i contructed general on mamatical general mamatical principle principle of a traditional of a traditional repetitive controller, repetitive and controller, expreed and uing expreed complex-vector uing complex vector notation. Annotation. FD compenating An FD compenating method for method 6k + 1 repetitive for 6k controller + 1 repetitive i alo controller preented. i Through alo preented. oretical Through analyi oretical and experiment, analyi and it wa experiment, demontrated it that wa demontrated 6k + 1 repetitive that control 6k + 1 cheme repetitive cancontrol achievecheme a fat tranient can achieve repone a fat with tranient a delay repone time ofwith T 0 /6, a and delay a good time of performance T0/6, and a for good compenating performance orfor uppreing compenating 6k or + uppreing 1 harmonic. Furrmore, 6k + 1 harmonic. due to Furrmore, above feature, due to 6k above + 1 repetitive feature, control 6k + cheme 1 repetitive i alocontrol uitable cheme to be ued i alo inuitable current to orbe voltage ued control in current foror or voltage three-phae control for grid-connected or three phae inverter. grid connected inverter. Acknowledgment: Thi work wa upported by National Natural Science Foundation of China under Acknowledgment: Thi work wa upported by National Natural Science Foundation of China under Grant , National High-tech R&D Program of China under Grant 2015AA050604, Project of Innovation-driven Grant , Plan National in Central High tech South Univerity. R&D Program of China under Grant 2015AA050604, Project of Innovation driven Plan in Central South Univerity.

17 Energie 2016, 9, of 17 Author Contribution: Zhaoxu Luo conceived main idea and wrote manucript with guidance from Mei Su, Jian Yang and Yao Sun; Xiaochao Hou perforemed experment. Joep M. Guerrero reviewed work and given helpful improvement uggetion. Conflict of Interet: The author declare no conflict of interet. Reference 1. Singh, B.; Al-Haddad, K.; Chandra, A. A review of active filter for power quality improvement. IEEE Tran. Ind. Electron. 1999, 46, [CroRef] 2. Peng, F.Z. Application iue of active power filter. IEEE Ind. Appl. Mag. 1998, 4, [CroRef] 3. Trinh, Q.N.; Lee, H.H. An advanced current control trategy for three-phae hunt active power filter. IEEE Tran. Ind. Electron. 2013, 60, [CroRef] 4. Cao, W.; Liu, K.; Ji, Y.; Wang, Y.; Zhao, J. Deign of a four-branch lcl-type grid-connecting interface for a three-phae, four-leg active power filter. Energie 2015, 8, [CroRef] 5. Bhattacharya, S.; Divan, D.M. Active filter olution for utility interface of indutrial load. In Proceeding of 1996 International Conference on Power Electronic, Drive and Energy Sytem for Indutrial Growth, New Delhi, India, 8 11 January 1996; pp Bhattacharya, S.; Cheng, P.T.; Divan, D.M. Hybrid olution for improving paive filter performance in high power application. IEEE Tran. Ind. Appl. 1997, 33, [CroRef] 7. Inzunza, R.; Akagi, H. A 6.6-kV tranformerle hunt hybrid active filter for intallation on a power ditribution ytem. IEEE Tran. Power Electron. 2005, 20, [CroRef] 8. Lee, T.L.; Wang, Y.C.; Li, J.C. Hybrid active filter with variable conductance for harmonic reonance uppreion in indutrial power ytem. IEEE Tran. Ind. Electron. 2015, 62, [CroRef] 9. Luo, Z.X.; Su, M.; Sun, Y.; Zhang, W.; Lin, Z.L. Analyi and control of a reduced witch hybrid active power filter. IET Power Electron. 2016, 9, [CroRef] 10. Luo, A.; Xu, X.Y.; Fang, H.H. Feedback-feedforward PI-type iterative learning control trategy for hybrid active power filter with injection circuit. IEEE Tran. Ind. Electron. 2010, 57, [CroRef] 11. Deng, Y.P.; Tong, X.Q.; Jia, H. A bidirectional control principle of active tuned hybrid power filter baed on active reactor uing active technique. IEEE Tran. Ind. Inform. 2015, 11, [CroRef] 12. Zou, Z.X.; Zhou, K.L.; Wang, Z. Frequency-adaptive fractional-order repetitive control of hunt active power filter. IEEE Tran. Ind. Electron. 2015, 62, [CroRef] 13. Sun, J.J.; Gong, J.W.; Chen, B.F. Analyi and deign of repetitive controller baed on regeneration pectrum and enitivity function in active power filter ytem. IET Power Electron. 2014, 7, [CroRef] 14. Miret, J.; Catilla, M.; Mata, J. Selective harmonic-compenation control for ingle-phae active power filter with high harmonic rejection. IEEE Tran. Ind. Electron. 2009, 56, [CroRef] 15. Grino, R.; Cardoner, R.; Catelló, C.R. Digital repetitive control of a three-phae four-wire hunt active filter. IEEE Tran. Ind. Electron. 2007, 54, [CroRef] 16. Catelló, C.R.; Grino, R.; Foa, E. Odd-harmonic digital repetitive control of a ingle-phae current active filter. IEEE Tran. Power Electron. 2004, 19, [CroRef] 17. Ecobar, G.; Martinez, P.R.; Ramo, L.J. A Negative feedback repetitive control cheme for harmonic compenation. IEEE Tran. Ind. Electron. 2006, 53, [CroRef] 18. Ecobar, G.; Hernandez-Brione, P.G.; Martinez, P.R. A repetitive-baed controller for compenation of 6l ± 1 harmonic component. IEEE Tran. Ind. Electron. 2008, 55, [CroRef] 19. Chen, D.; Zhang, J.M.; Qian, Z.M. Reearch on fat tranient and 6n ± 1 harmonic uppreing repetitive control cheme for three-phae grid-connected inverter. IET Power Electron. 2013, 6, [CroRef] 20. Ecobar, G.; Gomez, H.M.; Valdez-Fernandez, A.A. Implementation of a 6n ± 1 repetitive controller ubject to fractional delay. IEEE Tran. Ind. Electron. 2015, 62, [CroRef] 2016 by author; licenee MDPI, Bael, Switzerland. Thi article i an open acce article ditributed under term and condition of Creative Common Attribution (CC-BY) licene (