RECURSIVE LEAST SQUARES HARMONIC IDENTIFICATION IN ACTIVE POWER FILTERS A. El Zawawi, K. H. Youef, and O. A. Sebakhy Department of Electrical Engineering, Alexandria Univerity, Alexandria 21544, Egypt.P.O. BOX 21544. Tel: 203 5925557 Fax: 203 5921853. Email: omarebakhy @ hotmail.com and karim_hy24 @ hotmail.com) Abtract: A new leat quare Harmonic identification technique i developed. Thi technique i ued baically to etimate the fundamental component of the drawn by nonlinear load in order to calculate the reference of the active power filter on line. The effectivene of the propoed technique i teted under a udden change of the nonlinear load. The performance of the propoed technique i compared with the conventional harmonic extraction technique baed on the ynchronou reference frame method. Simulation reult are given for both the propoed and the conventional technique. Keyword: Harmonic compenation, Parameter etimation, Sytem identification, Active power filter. 1. INTRODUCTION Due to the advancement in power electronic technology and due to the intenive ue of power electronic equipment in all branche of indutry, the harmonic pollution ha increaed everely in electrical network. Thi affect the operation of motor, control equipment and protection device. Harmonic alo caue reonance and they increae neutral and ytem loe (Singh, et al., 1999). Conventionally paive (L-C) filter were ued to olve thi problem but they are characterized by fixed compenation, large ize and complex deign. They can alo caue reonance in the power ytem and their performance i affected by aging of the paive component or change of power frequency (Moran, et al., 1995). A a reult, the active power filter ha become the bet way for harmonic and reactive power compenation. A three phae hunt active power filter i hown in Fig. 1. where: ia, ib and ic are the three phae upply, ila, ilb and ilc are the three phae load, ica, icb and icc are the three phae filter, va, vb and vc are the three phae voltage and 1 : 6 are the inverter witche. va vb vc L L L ia ib ic ica L L L ila ilb ilc icb icc 1 3 5 2 4 6 Fig. 1- Three phae hunt active power filter The load can be expreed in the form: il( t) = ao a1 in ωt a3 in 3 ωt LL am in Mωt b1 co ωt b3 co3 ωt LL bm co Mωt (1) where M i the order of the maximum harmonic of interet. The role of the active power filter i to upply the harmonic and reactive power component of the load o that the ource upplie only the fundamental component in-phae with the upply voltage ( a1 in ωt ). Many technique have been developed for eparating the R L C
fundamental and harmonic component of the load and for calculating reference for active power filter, for example: the p-q theory, the ynchronou reference frame theory (Soare, et al., 2000) and the correlation technique (Harmelen and Enlin, 1993; Jung, et al., 2000). The reference can be etimated uing capacitor voltage regulation technique (Jain, et al., 2002; Chatterjee, et al., 1999;Torrey and Al-Zamel, 1995). There i alo the fat Fourier tranform (FFT) technique which perform the harmonic extraction in the frequency domain. It i found that the capacitor voltage regulation and the p-q technique are not efficient under non-inuoidal upply voltage condition. On the other hand, the Fat Fourier Tranform and the correlation technique method are dependent on the ampling frequency and require extenive calculation. They alo require toring of ample during one period at any time. The ynchronou reference frame method require le calculation burden and it i le dependent on upply voltage harmonic but it performance i affected by the filter ued to extract the reference. Recurive parameter identification technique can be ued to etimate the fundamental and harmonic component of the load in order to etimate the reference of active power filter. Thi paper applie the leat quare identification technique to generate the reference for an active power filter. Simulation reult for the propoed technique are given and compared with the reult of the ynchronou reference frame technique. 2. THE SYNCHRONOUS REFERENCE FRAME THEORY In thi method the nonlinear load il a,il b and il c are tranformed into the ynchronou reference frame baed on Park tranformation a follow: ila ilα 2 1 1/ 2 1/ 2 = (2). ilb ilβ 3 0 3 / 2 3 / 2 ilc va vα 2 1 1/ 2 1/ 2 = (3). vb vβ 3 0 3 / 2 3 / 2 vc il d co θ in θ ilα vβ θ = il =., tan 1 (4) q in θ co θ ilβ vα Intantaneou active and reactive load can be decompoed into average and ocillatory term: ~ ~ il d = Il d i l d and il q = Il q i l q.under balanced and inuoidal upply voltage condition, the firt harmonic of poitive equence i tranformed into a dc quantity and thi contitute the average component (Soare, et al.,2000). All higher order harmonic including the firt harmonic of negative equence are tranformed to non-dc quantitie and o contitute the ocillatory component. ila ilb ilc Vdc* Vdc θ LPF abc dq ild ilq LPF LPF PI 0 icd * dq icq* Fig. 2- Reference generation uing ynchronou frame method Extracting the average uing a low pa filter (LPF) and ubtracting it from the active and reactive component the ocillatory component can be obtained and the required reference compenating are then calculated a follow: ~ ic α 1 vα vβ i ld (5) = ~. icβ 2 2 β α α v v v v i lq β ila 2 1 1/ 2 1/ 2 T ilα (6) ilb =. 3 0 3 / 2 3 / 2 ilβ ilc A Butterworth type filter i choen in order to obtain magnitude and phae characteritic a cloe a poible to an ideal filter ince it magnitude repone i maximally flat in the pa-band (Soare, et al., 2000). The filter conidered i a fourthorder filter with a cut-off frequency 25 Hz. It hould be noted that the average component, being dc quantitie, do not uffer from phae hift and thi i the reaon for not extracting the ocillatory component directly uing high pa filter. The block diagram of harmonic extraction and reference generation i hown in Fig. 2. A proportional-integral (PI) controller i ued for the dc voltage regulation of the inverter and to control the active power flow for compenating the inverter loe. 3.THE RECURSIVE HARMONIC IDENTIFICATION TECHNIQUE. θ abc ica * icb* icc * Equation (1) can be written in the tandard form: yk il ( tk) = u k a (7) Where t k i the ampling time, u k = [ 1,in ωtk, K,in Mωtk, co ωtk, K, co Mωtk] i the regreor vector, and a = [ ao, a1, K, am, b1, K, bm ] i the unknown parameter vector to be etimated. In the teady-tate cae, the off-line etimate of the parameter a that minimize the weighted um of the quare of the error, namely J = E WE (8) where E = Y Ua, Y = [ y 1, K y N],
W = diag { λ N i, i = 1, K, N}, 0 < λ <1i a forgetting factor to peed up the etimation (uually λ 0.95-0.99) and U = [ u1, L, u N ] i given by: a ˆ U 1 = ( WU ) U WY (9) In the literature, there are many variant of the recurive leat quare algorithm (RLS) to etimate the unknown parameter vector a (Goodwin and Sin, 1984). We hall preent here the mot baic algorithm, correponding to the above mentioned off-line etimate. The recurive algorithm take the form: aˆ k = aˆ k 1 Pk u k ( y k u k aˆ k 1) (10) P = k 1u k uk Pk 1 1 P P (11) k k 1 λ uk Pk 1u k λ where P k i the covariance matrix. It can be hown that if the initial condition are: aˆ(0) = 0 P(0) = ρ I where I i the identity matrix and ρ i a very large contant, then at teady tate, the on-line etimate of aˆ k (10) tend to the off line etimation â a k ( Södertröm and Stoica, 1989). Thi technique can be ued for the on-line etimation of the fundamental component which i then ubtracted from the load to obtain the reference of the active power filter a hown in Fig. 3. A proportional-integral (PI) controller i ued for the dc voltage regulation of the inverter and to control the active power flow in the inverter. The etimated fundamental component i then ubtracted from the load to obtain the reference for the active power filter. Load Sampling harmonic Recurive uing etimation DSP Fundamenta component l Vdc * PI Vdc V/Vmax Reference filter Fig.3 Reference generation uing recurive identification 4. DESIGN OF THE PI CONTROLLER A PI controller i ued to regulate the capacitor voltage by adding an active component I cp to the reference. The deign of the PI controller i done by linearizing the DC voltage control ytem around the teady tate operating point (Soare, et al., 2000; Chatterjee, et al., 1999). Uing the average power balance principle, the open loop tranfer function model of the compenator about a particular operating pointv dco, i obtained a: V V G dc ( ) 3 1 ( ) = = = * I cp ( ) CV dco where; 3V A = CV dco A (12) V i the r.m. phae voltage V dco i the required teady tate capacitor voltage and C i the capacitance at the DC ide. Fig. 4 how the cloed loop DC voltage control ytem including the PI controller with a proportional and integral gain of k P and k I repectively and a firt-order low-pa filter F() with a gain of k I / k P and a pole at = k I / k P to eliminate the influence of the zero introduced by the PI controller (Soare, et al.,2000). V dc* k F() I k P G() V dc Fig.4- The cloed loop control ytem The overall cloed loop tranfer function i given by: V dc ( ) Ak = I (13) V * dc ( ) 2 Ak P Ak I Equation (13) repreent a econd order ytem and the deign of PI controller gain i baed on the choen damping ratio ζ and undamped natural frequencyω n. Thi dc voltage controller will give a zero teady tate error due to the introduction of a pole at = 0 by the PI controller, making a type 1 ytem. When the DC voltage control i done in the d-q ynchronou reference frame, the obtained value for k p and k I hould be multiplied by 3 due to tranformation. 5. THE CURRENT CONTROLLER The obtained reference for each * phaei c and it actual i c are compared uing a two level hyterei controller with hyterei band ± 0.25 A a hown in Fig. 5. The lower witch of each leg i ued to increae the (if the actual i le than the reference ) and the upper witch i ued to decreae the (if the actual i greater than the reference ). Reference Actual Error 1 0 h 0 h Fig. 5 - Hyterei control for one inverter leg Upper witch Lower witch
Fig. 6 Simulation reult of the conventional technique. Fig. 7 Simulation reult of the propoed technique
. Fig. 8. The etimated fundamental component.. Fig. 9. Spectrum of upply uing both technique. 6. SIMULATION RESULTS The ytem parameter are choen a follow: Supply rm phae voltage: 50 V at 50 Hz. Filter inductance and capacitance: L = 2.2 mh., C = 2000µF. Dc voltage operating point = 175 V. The PI controller i deigned to have ζ = 2 / 2 and ω n = 314 rad/ec uing a linearization method around the teady tate operating point and thi give: k P = 1. 796, k I = 398. 88 for the cae of the conventional controller and k P = 1. 036 k I = 223.29 for the new technique. The nonlinear load i a three phae bridge rectifier with a contant DC of 20 A which i uddenly decreaed to 10 A at t = 0.15 ec. Fig. 6 how the performance of the filter uing the conventional technique and Fig. 7, 8 how the performance of the filter uing the propoed technique. Fig.9 how the upply pectrum uing both technique. From Fig. 6, 7 and 9 it i obviou that the performance of the propoed technique i better than that of the conventional technique. The harmonic content of the upply uing the propoed technique i better than the one obtained uing the conventional technique. The reaon i that in the ynchronou reference frame, harmonic frequencie are reduced due to tranformation and they may pa through the low pa filter which i practically not ideal. Fig. 8 how the etimated fundamental component. The etimation proce ha reached teady tate in nearly half cycle time and it ha tracked the change in load effectively within the ame time period. Although both technique ue the ame DC voltage regulator, but the DC voltage profile in the conventional technique i better. Thi i due to the delay aociated with the etimation proce becaue of the relatively low ampling frequency (4 khz) and o, the DC voltage repone of the propoed technique will be much better at higher ampling frequencie. 7.CONCLUSION A new leat quare harmonic identification technique ha been propoed and it wa applied to generate the reference of a three phae active power filter. Simulation reult of the propoed technique have been given and they are found better than thoe of the ynchronou
reference frame technique. The on line recurive parameter etimation technique ue only two conecutive ample at any intant for the harmonic extraction proce and it i not affected by the harmonic in the upply voltage. REFERENCES Chatterjee, K., B. G. Fernande and G. K. Dubey, (1999) "An intantaneou reactive volt-ampere compenator and harmonic uppreor ytem," IEEE Tranaction on Power Electronic, Vol. 14, No.2, pp.381-392. Goodwin, G. C., and K.S. Sin, (1984) "Adaptive filtering prediction and control," Printice-Hall, Inc., New Jery, U.S.A. Harmelen, G. L. and J. H. R. Enlin, (1993), "Realtime dynamic control of dynamic power filter in upplie with high contamination," IEEE Tranaction on Power Electronic, Vol. 8, No.3,, pp.301-308. Jain, S. K., P. Agrawal and H. O. Gupta, (2002), "Fuzzy logic controlled hunt active power filter for power quality improvement," IEE Proc.-Electr. Power Appl., Vol. 149, No.5, pp.317-328. Jung, Y. G., Y. C. Lim and S. H. Yang, (2000), "Single-phae active power filter baed on three-dimenional coordinate," IEE Proc.-Electr. Power Appl., Vol. 147, No.6, pp.572-578. [Moran, L. A., J. W. Dixon and R. R. Wallace, (1995), "A three phae active power filter operating with fixed witching frequency for reactive power and harmonic compenation," IEEE Tranaction on Indutrial Electronic, Vol. 42, No.4, pp.402-408. Singh, B., K. Al-Haddad and A. Chandra, (1999), "A review of active filter for power quality improvement," IEEE Tranaction on Indutrial Electronic, Vol. 46, No.5, pp.960-967. Soare, V., P. Verdelho and G. D. Marque, (2000), "An intantaneou active and reactive component method for active filter," IEEE Tranaction on Power Electronic, Vol. 15, No.4, pp.660-669. Södertröm, T., and P. G. Stocia, (1989), "Sytem identification," Prentice-Hall,. Torrey, D. A., and A. M. Al-Zamel, (1995), "Single-phae active power filter for multiple nonlinear load," IEEE Tranaction on Power Electronic, Vol. 10, No.3, pp. 263-272..