Low Swtchng Frequency Actve Harmonc Elmnaton n Multlevel Converters wth Unequal DC Voltages Zhong Du,, Leon M. Tolbert, John N. Chasson, Hu L The Unversty of Tennessee Electrcal and Computer Engneerng Knoxvlle, TN 7996-00 E-mal: zdu@utk.edu, tolbert@utk.edu, chasson@utk.edu Center for Advanced Power Systems Florda State Unversty Tallahassee, FL 0 E-mal: zdu@caps.fsu.edu, hl@caps.fsu.edu Abstract Ths paper presents an actve harmonc elmnaton modulaton control method for the cascaded H-brdges multlevel converter when suppled by unequal DC sources. Frst, the multlevel converter s decoupled nto ndvdual unpolar converters, and the low order harmoncs (such as the th, 7th, th, and th) are elmnated by usng resultant theory whle at the same tme the mnmum specfed harmonc dstorton (HD) of a combnaton of unpolar converters for multlevel converter control s found. Next, the swtchng angle sets correspondng to the mnmum harmonc dstorton are used as ntal guesses wth the Newton clmbng method to elmnate the specfed hgher order harmoncs. If the solutons are not avalable for some modulaton ndces, the unpolar swtchng scheme s used to elmnate hgh order harmoncs and the actve harmonc elmnaton method s used to elmnate low order harmoncs. Ths method has lower swtchng frequency than that of the prevously proposed actve harmonc elmnaton method. Keywords Multlevel converter, harmonc elmnaton. I. INTRODUCTION The multlevel converter s a promsng technology for renewable/dstrbuted energy applcatons of power electroncs because of ts easy connecton wth renewable/dstrbuted energy modules such as fuel cells, solar panels, and wnd turbnes. Other benefts also nclude ts low electromagnetc nterference (EMI) and hgh effcency wth low swtchng frequency control methods []. There are four knds of control methods for multlevel converters. They are the selectve harmonc elmnaton method, space vector control method, tradtonal PWM control method, and space vector PWM method [-]. The space vector control method, the space vector PWM method, and the tradtonal PWM control method wth carrer phase shftng requre equal DC voltages for the multlevel converters. Control of multlevel converters wth unequal DC voltages s much more complcated [-6]. To address the ssue of elmnatng hgher order harmoncs at low modulaton ndces, the actve harmonc elmnaton method has been proposed [7]. The actve harmonc elmnaton method wth unequal DC sources uses a unpolar swtchng scheme n whch the swtch angles are determned usng elmnaton theory to remove low-order harmoncs. Then, the specfcally chosen hgh-order harmoncs (e.g., the odd non trplen harmoncs) are elmnated by usng an addtonal swtchng angle (one for each hgher harmonc) to generate the negatve of the harmonc to cancel t [8]. The actve harmonc elmnaton n [7-8] has a dsadvantage n that t uses a hgh swtchng frequency to elmnate hgher order harmoncs. Ths paper proposes a modfcaton to the prevously proposed harmonc elmnaton method n [7-8] that s equally effectve at removng harmoncs but uses a lower swtchng frequency. Frst, the multlevel converter s decoupled nto ndvdual unpolar converters and the loworder harmoncs (such as the th, 7th, th, and th) are elmnated by usng resultant theory whle at the same tme the mnmum specfed harmonc dstorton (HD) of a combnaton of unpolar converters for multlevel converter control s found. Next, the swtchng angle sets correspondng to the mnmum harmonc dstorton are used as ntal guesses for the Newton clmbng method to elmnate the specfed hgh-order harmoncs. However, for some modulaton ndces, no solutons exst. Under such stuatons, the actve harmonc elmnaton method can use the unpolar swtchng scheme to elmnate hgh-order harmoncs, and use the actve harmonc method to elmnate low-order harmoncs to decrease the requred swtchng frequency. Compared to the actve harmonc elmnaton method proposed for H-brdge multlevel converters prevously [8], ths method has lower swtchng frequency. Therefore, t s referred as to the optmzed harmonc elmnaton method. A -level example and a 7-level example are gven n ths paper. Matlab smulatons and experments are employed to valdate the proposed method. The expermental results show that the method can effectvely elmnate the specfc harmoncs, and the output voltage waveforms have low total harmonc dstorton (THD) as expected n theory. II. UNIPOLAR SWITCHING SCHEME AND ITS SOLUTION Based on harmonc elmnaton theory [9-], the control of the snusodal wave generaton s to choose a seres of swtchng angles to synthesze a desred snusodal voltage waveform. A typcal -angle unpolar swtchng output s shown n Fg.. The Fourer seres expanson of the output voltage waveform shown n Fg. s V dc V( t) = [( nθ ) ( nθ ) ( nθ ) ( nθ ) ( nθ )]sn( nωt ) n=,,... nπ ()
Fg.. Fve-angle unpolar swtchng output. Fg.. Swtchng angle solutons to -angle unpolar swtchng scheme vs. m. Ideally, gven a desred fundamental voltage V, one wants to determne the swtchng angles θ, θ, θ, θ, and θ so that V(ωt)=V (ωt), and specfc hgher harmoncs of V n (ωt) are equal to zero. For a three-phase applcaton, the trplen harmoncs n each phase need not be canceled as they automatcally cancel n the lne-to-lne voltages. Here, the th, 7th, th, and th order harmoncs are chosen to be removed. That s, the swtchng angles must satsfy the followng equatons: ( θ ) ( θ ) ( θ ) ( θ ) ( θ ) = m ( θ ) ( θ ) ( θ ) ( θ ) ( θ ( 7θ ) ( 7θ ) ( 7θ ) ( 7θ ) ( 7θ ( θ ) ( θ ) ( θ ) ( θ ) ( θ ( θ ) ( θ ) ( θ ) ( θ ) ( θ Here, m s defned as the modulaton ndex as and the THD s computed as () m = πv /(V ), () THD = dc 9 V =,7,,, V. () The resultant method descrbed n [] s used to compute the solutons to (), and these are shown n Fg.. Fg. shows the THD correspondng to these solutons. From the swtchng angle solutons shown n Fg., t can be derved that the solutons exst n a range of the modulaton ndces from 0 to 0.9. Some modulaton ndces have a sngle soluton, and there are multple soluton sets for other modulaton ndces. Fg. shows that dfferent soluton sets have dfferent THD values. Another feature s the THD s very hgh (> 0%) for the low modulaton ndex range (m < 0.). Fg.. THD vs. m for swtchng angles shown n Fg.. III. PROPOSED OPTIMIZED HARMONIC ELIMINATION METHOD Cascaded H-brdge multlevel converters can be vewed as unpolar converters connected n seres; the total modulaton ndex for a multlevel converter when decoupled control s appled s m = s = c k m. () Here, s the nomnal DC voltage, s the th DC voltage, c (c {-, 0, }) s called the combnaton coeffcent, and k = / s called the unbalance coeffcent. If c =, level produces a postve voltage; f c, the level s bypassed; and f c = -, the level produces a negatve voltage. For each -angle unpolar converter, the resdual hgher harmonc contents also can be computed by (). The normalzed harmonc magntudes are computed as V n = [( nθ) ( nθ ) ( nθ ) ( nθ ) ( nθ )]. (6) nπ
H =k S, S, - S, S, V out H S, S, =k S, S, - - Fg.. Normalzed 7th, 9th, and rd harmonc magntudes vs. m. For example, Fg. shows the normalzed 7th, 9th, and rd harmonc magntudes vs. m. It can be seen that they are postve for some modulaton ndces and are negatve for other modulaton ndces. Thus, t s possble to obtan a combnaton of several unpolar converters for each modulaton ndex m of the multlevel converter wth the lowest harmonc dstorton for the specfed harmoncs. A. Fve-level case The multlevel crcut topology for the -level case s shown n Fg.. There are two H-brdges, H and H, and two unequal DC sources, = k and = k. If we choose to have a -angle unpolar output for each H-brdge level, then there are a total of 0 swtchng angles for the -level case. Therefore, one degree of freedom s used for fundamental control and any 9 harmoncs could then be elmnated. The modulaton ndex for ths -level nverter s m = c k m. (7) = In (7), m and k are gven so that c, m are chosen to satsfy (7). Next, the resultant method s then used to solve the system () twce; once for brdge H wth m = m n (), and then for brdge H wth m = m n (). The soluton set for m = m s denoted as θ, θ, θ, θ, θ whle the soluton set for m = m s denoted as θ, θ, θ, θ, θ. Resultant theory has elmnated the lower order harmoncs ( th, 7 th, th, th ). However, to elmnate harmoncs up to the 9th, dfferent angles need to be chosen. The ntal swtchng angle guess s found by checkng whch combnaton of the soluton sets found wth the resultant theory has the lowest harmonc dstorton for the remanng harmoncs to be elmnated, whch n ths example are the 7th, 9th, rd, th, and 9th: 7 9 V9 HD = V, (8) where, V n s the nth harmonc whch can be expressed as V n Fg.. Fve-level multlevel converter topology. = ckv [( nθ ) ( nθ ) dc nπ. (9) = ( nθ) ( nθ) ( nθ)] The swtchng angle sets correspondng to the lowest harmonc dstorton (HD mn ) are used as ntal guesses for the Newton Clmbng technque to solve (0) to control the fundamental and to elmnate the odd, non-trplen harmoncs (th, 7th, th, th, 7th, 9th, rd, th, and 9th): ( θj) ( θ j ) j kc ( ) = m j kc ( ) ( θ j ) j kc ( ) 0 = ( θ j ) j kc ( ) 9 0 =. (0) As a numercal example, an unequal case (k =0.7, k =) s computed usng ths method. The swtchng angles are shown n Fg. 6. The swtchng angles shown n Fg. 6 are for brdge H, and the swtchng angles shown n Fg. 6 are for brdge H. A varaton of ths method s to elmnate the st harmonc (for nstance) and not the 7th harmonc (for nstance). The 7th harmonc could then be elmnated usng the actve harmonc elmnaton method [7-8]. The harmonc dstorton s computed up to st by 7 9 9 V HD = V. () The swtchng angle sets correspondng to the lowest harmonc dstorton (HD mn ) could be used as ntal guesses for the Newton clmbng method to solve () to elmnate the odd, non-trplen harmoncs th, th, th, 7th, 9th, rd, th, 9th and st.
Fg. 6. Swtchng angles for -level multlevel converter to elmnate harmoncs below the 9th (k.7, k =.0), swtchng angles for brdge H ; swtchng angles for brdge H. ( θj) ( θ j ) ( θ j ) j kc ( ) = m j kc ( ) j kc ( ) ( θ j ) j kc ( ) 0 = () Note that the 7th s not elmnated n (). Here, the 7th harmonc, not the th harmonc, s chosen to be removed by the actve harmonc elmnaton method because cancelng the th harmonc wll generate a new th harmonc. The swtchng angles are shown n Fg. 7. Smlarly, the swtchng angles shown n Fg. 7 are for brdge H, and the swtchng angles shown n Fg. 7 are for brdge H. If there are no solutons avalable for some modulaton ndces, the unpolar swtchng scheme can be used to elmnate hgh-order harmoncs and the actve harmonc elmnaton method can be used to elmnate low-order harmoncs to decrease the swtchng frequency [7]. Fg. 7. Swtchng angles for -level multlevel converter to elmnate harmoncs below the st (k.7, k =.0), swtchng angles for brdge H ; swtchng angles for brdge H. For example, the equatons to elmnate hgh-order harmoncs up to the st usng the -angle unpolar swtchng scheme are: ( θ ) ( θ ) ( θ ) ( θ ) ( θ ) = m ( 9θ ) ( 9θ ) ( 9θ ) ( 9θ ) ( 9θ ( θ ) ( θ ) ( θ ) ( θ ) ( θ ) = ( 9θ ) ( 9θ ) ( 9θ ) ( 9θ ) ( 9θ ) = ( θ ) ( θ ) ( θ ) ( θ ) ( θ 0 0 () These equatons are solved by the Newton clmbng method []. The ntal guesses for the Newton clmbng method are obtaned from the solutons to elmnate the th, 7th, th, and th harmoncs usng the resultant method. Fg. 8 shows the solutons. The solutons can be used to obtan the lowest THD combnaton and elmnate the specfed harmoncs. Next, actve harmonc elmnaton s used where the magntudes and phases of the resdual lower order harmoncs (th, 7th, th, th, 7th, and th) are computed, generated, and subtracted from the orgnal voltage waveform to elmnate them. In ths method, the addtonal number of swtchngs requred to elmnate the th, 7th, th, th, 7th, and th harmoncs s bounded by N n. () sw n {,7,,,7,}
Fg. 8. Swtchng angles for -angle unpolar swtchng scheme to elmnate the 9th, rd, 9th and st harmoncs vs. m. The upper bound of the addtonal number of swtchngs for ths example s 78 by (). It s about one half of the actve harmonc elmnaton method n theory [8]. Thus, ths method can also decrease the requred number of swtchngs. B. Seven-level case The multlevel crcut topology for the 7-level case s shown n Fg. 9. There are three H-brdges, H, H, and H, and three unequal DC sources, = k, = k, and = k. If we choose to have a -angle unpolar output for each level, then there are a total of swtchng angles for the 7- level case ( H-brdges); therefore, any harmoncs could be elmnated (one degree of freedom s used for fundamental frequency ampltude control). The modulaton ndex for ths 7-level nverter s m = c k m () = H S, S, =k - S, S, (c) Fg. 0. Swtchng angles for 7-level multlevel converter to elmnate harmoncs below the rd (k.9, k.9, k =.0), swtchng angles for brdge H ; swtchng angles for brdge H ; (c) swtchng angles for brdge H. H S, S, =k - S, S, H S, =k S, S, S, Fg. 9. Seven-level multlevel converter topology. V out - - Just as n the prevous secton, resultant theory can be used to elmnate the lower order harmoncs (th, 7th, th, th). However, to elmnate harmoncs up to the rd, dfferent angles need to be chosen. The ntal swtchng angle guess s found by checkng whch combnaton of the soluton sets found wth the resultant theory has the lowest harmonc dstorton for the remanng harmoncs to be elmnated, whch n ths example are the odd, nontrplen harmoncs from the 7th through the rd: HD = V 7 9 7 9 (6)
The swtchng angle sets correspondng to the lowest harmonc dstorton (HD mn ) are used as ntal guesses for the Newton Clmbng technque to solve (7) to control the fundamental and to elmnate the odd, non-trplen harmoncs (th, 7th, th, th, 7th, 9th, rd, th, 9th, st, th, 7th, st, and rd): ( θj) ( θ j ) j kc ( ) = m j kc ( ) ( θ j ) j kc ( ) 0 = (7) As an example, an unequal case (k.9, k.9, k = ) s computed. The swtchng angles are shown n Fg. 0. The swtchng angles shown n Fg. 0 are for brdge H, the swtchng angles shown n Fg. 0 are for brdge H, and the swtchng angles shown n Fg. 0(c) are for brdge H. IV. SIMULATION AND EXPERIMENT Smulaton has been used to valdate the proposed algorthm. Fg. shows voltage waveforms of a -level smulaton case wth k.7, k =.0, = 8 V, m =.080, THD = 9.9% to elmnate harmoncs up to the 9th. From the normalzed FFT analyss shown n Fg., t can be derved that all the harmoncs up to 9th are zero. Ths confrmed the computaton results. A prototype three-phase -level cascaded H-brdge multlevel converter has been bult usng 60 V, 70 A MOSFETs as the swtchng devces to mplement the algorthm usng a feld programmable gate array (FPGA) controller wth 8 µs control resoluton. The -level smulaton case s chosen to mplement wth the multlevel converter. Fg. shows the expermental lnelne voltage, and Fg. shows ts correspondng normalzed lne-lne voltage FFT analyss. Fg. shows that the harmoncs have been elmnated up to 9th. The expermental THD s 7.%, and t corresponds very well wth the theoretcal computaton of 9.% and smulaton result of 9.9%. A 7-level case wth wth k.9, k.9, k =.0, = 8 V, and m =.66 s also chosen to mplement wth the multlevel converter to valdate the proposed algorthm. Fg. shows the expermental lne-lne voltage, and Fg. shows ts correspondng normalzed lne-lne voltage FFT analyss. Fg. shows that the harmoncs have been elmnated up to rd. The expermental THD s 6.%, and t corresponds very well wth the theoretcal computaton of 6.87% and smulaton result of 6.89%. Fg.. Smulaton of a -level case (k.7, k =, = 8 V, m =.080, THD = 9.9%) to elmnate harmoncs up to 9th, voltage waveform; normalzed FFT analyss of lne-lne voltage. Fg.. Experment of -level case (k.7, k =, = 8 V, m =.080, THD = 7.%) to elmnate harmoncs up to 9th, lne-lne voltage; normalzed FFT analyss of lne-lne voltage.
ACKNOWLEDGMENTS We would lke to thank the Natonal Scence Foundaton for partally supportng ths work through contract NSF ECS- 00988. We would also lke to thank Oak Rdge Natonal Laboratory for partally supportng ths work through UT/Battelle contract No. 0007. REFERENCES Fg.. Expermental voltages for 7-level case (k.9, k.9, k =, = 8 V, m =.66, THD = 6.%) to elmnate harmoncs up to rd, lne-lne voltage; Normalzed FFT analyss of lne-lne voltage V. CONCLUSIONS Ths paper presents an actve harmonc elmnaton modulaton control method for a cascaded H-brdges multlevel converter wth unequal DC sources. Ths method has lower swtchng frequency than the earler proposed method [8]. Frst, the multlevel converter s decoupled nto ndvdual unpolar converters and the low order harmoncs (such as the th, 7th, th and th) are elmnated by usng elmnaton theory whle at the same tme the mnmum specfed harmonc dstorton (HD) of a combnaton of unpolar converters for multlevel converter control s found. Next, the swtchng angle sets correspondng to the mnmum harmonc dstorton are used as ntal guesses wth the Newton clmbng method to elmnate the specfed hgher order harmoncs. If the solutons are not avalable for some modulaton ndces, the unpolar swtchng scheme s used to elmnate hgh order harmoncs and the actve harmonc elmnaton method s used to elmnate low order harmoncs. The experments valdated that the proposed method can elmnate the specfed harmoncs as expected. [] J. S. La, F. Z. Peng, Multlevel converters A new breed of power converters, IEEE Transactons on Industry Applcatons, vol., no., May./June 996, pp. 09-7. [] J. Rodríguez, J. La, F. Peng, Multlevel nverters: a survey of topologes, controls and applcatons, IEEE Transactons on Industry Applcatons, vol. 9, no., Aug. 00, pp. 7-78. [] G. Carrara, S. Gardella, M. Marcheson, R. Salutar, G. Scutto, A new multlevel PWM method: a theoretcal analyss, IEEE Transactons on Power Electroncs, vol. 7, no., July 99, pp. 97-0. [] J. K. Stenke, Control strategy for a three phase AC tracton drve wth a -Level GTO PWM nverter, IEEE PESC, 988, pp. -8. [] L. M. Tolbert, J. N. Chasson, Z. Du, and K. J. McKenze, Elmnaton of harmoncs n a multlevel converter wth non equal DC sources, IEEE Transactons on Industry Applcatons, vol., no., Jan./Feb. 00, pp. 7-8. [6] L. M. Tolbert, F. Z. Peng, T. Cunnyngham, J. N. Chasson, Charge balance control schemes for multlevel converter n hybrd electrc vehcles, IEEE Transactons on Industral Electroncs, vol. 9, no., October 00, pp. 08-06. [7] Z. Du, L. M. Tolbert, J. N. Chasson, Harmonc elmnaton for multlevel converter wth programmed PWM method, IEEE Industry Applcatons Socety Annual Meetng, October -7, 00, Seattle, Washngton, pp. 0-. [8] Z. Du, L. M. Tolbert, J. N. Chasson, Actve Harmonc Elmnaton n Multlevel Converters Usng FPGA Control, IEEE COMPEL, August -8, 00, Urbana-Champagn, Illnos, pp. 7-. [9] H. S. Patel, R. G. Hoft, Generalzed harmonc elmnaton and voltage control n thyrstor nverters: Part I harmonc elmnaton, IEEE Transactons on Industry Applcatons, vol. 9, May/June 97. pp. 0-7. [0] H. S. Patel, R. G. Hoft, Generalzed harmonc elmnaton and voltage control n thyrstor nverters: Part II voltage control technque, IEEE Transactons on Industry Applcatons, vol. 0, Sept./Oct. 97, pp. 666-67. [] P. N. Enjet, P. D. Zogas, J. F. Lndsay, Programmed PWM technques to elmnate harmoncs: A crtcal evaluaton IEEE Transactons on Industry Applcatons, vol. 6, no., March/Aprl. 990. pp. 0 6. [] J. N. Chasson, L. M. Tolbert, K. J. McKenze, Z. Du, Control of a multlevel converter usng resultant theory, IEEE Transactons on Control Systems Technology, vol., no., May 00, pp. -. [] Z. Du, L. M. Tolbert, J. N. Chasson, Modulaton extenson control for multlevel converters usng trplen harmonc njecton wth low swtchng frequency, IEEE Appled Power Electroncs Conference, March 6-0, 00, Austn, Texas, pp. 9-. [] C. K. Duffey, R. P. Stratford, Update of harmonc standard IEEE-9: IEEE recommended practces and requrements for harmonc control n electrc power systems, IEEE Transactons on Industry Applcatons, vol., no. 6, Nov./Dec. 989, pp. 0-0.