A nearly parametric solution to Selective Harmonic Elimination PWM Bao-Xin Shang 1, Shu-Gong Zhang 2, Na Lei 3 *, Jing-Yi Chen 4

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1 A early parametrc oluto to Selectve Harmoc Elmato PWM Bao-X Shag Shu-Gog Zhag a Le 3 * Jg-Y Che 4 Abtract Selectve Harmoc Elmato Pule Wdth Modulato (SHEPWM a mportat techque to olve PWM problem whch cotrol the output voltage of a verter va electg approprate wtchg agle. Baed o the Ratoal Uvarate Repreetato (RUR theory for olvg polyomal ytem the paper preet a algorthm to compute a early parametrc oluto to a SHEPWM problem. Whe the umber of wtchg agle fed a early parametrc oluto ca be codered a fucto of the modulato de m. So we ca adapt the ampltude of the output voltage wth the ame ource voltage by chagg the modulato de. Whe m gve a a pecfc value complete oluto to the SHEPWM problem ca be obtaed ealy ug uvarate polyomal olvg. Compared wth other method m codered a a ymbolc parameter for the frt tme ad th ca help avod totally retartg whe m chage. he average tme for computg complete oluto aocated to 46 modulato dee baed o a early parametrc oluto whe =5.84 o the algorthm practcal. hree group of wtchg agle aocated to =5 m=.75 mulated MALAB ad t verfe the algorthm correcte. Key word: SHEPWM Ratoal Uvarate Repreetato Parametrc oluto Polyomal ytem Ratoal Iterpolato I. IRODUCIO Multlevel verter have lot of applcato the feld of dutry []. Pule Wdth Modulato (PWM of great mportace medum-voltage hgh-power verter whch cotrol the output voltage of a verter through electg approprate wtchg agle. []-[] revew topologe ad modulato techque of multlevel verter. SHEPWM a wdely-ued method to olve PWM problem ad t ha advatage uch a low wtchg lo learly cotrollg fudametal voltage compoet ad pecfyg the order of elmated harmoc. he method frtly troduced [3] whch ue the ewto-rapho terato method to olve aocated olear equato. Abdul Moeed Amjad ad Zaal Salam ummarze a varety of method to olve SHEPWM problem [4]. Wth the umber of wtchg agle fed the dffculty of a SHEPWM problem le how to olve the aocated tracedetal ytem. here maly et three dffculte. he frt oe how to get a oluto gve a pecfc modulato de m. I th cae a group of good tal value mportat. Ital value electo techque are codered [5]-[7]. I addto a great umber of optmzato algorthm uch a geetc method bee method ad warm partcle method etc are ued [8]-[] ad they ca fd more tha oe group of wtchg agle ad allow rough tal value. he ecod oe how to get complete oluto gve a pecfc modulato de m. I 4 Joh Chao etc preet a method to get complete oluto baed o reultat theory ymbolc computato []. I 7 Qujg Wag etc mprove the reult ug ymmetrc polyomal theory [3]. he lat dffculty whe the modulato de m chage how to recompute related complete oluto qucly. If th come true output voltage ca be adapted more ealy through modfyg m wth ource voltage uchaged. o th ed all above method have to retart totally. o f th problem we ue the RUR method to olve parametrc zero-dmeoal polyomal ytem ad frtly coder utlzg algebrac terpolato techque to accelerate the computato. We mplemet the algorthm Maple 8. Whe m chage recomputato tart from a early parametrc oluto ad t oly volve lttle tme of uvarate polyomal olvg. More formato o theore ad mplemetato of RUR referred to [4]-[5]. he tructure of the paper a follow. I ecto II we decrbe how to traform the tracedetal equato aocated to SHEPWM problem to polyomal equato elemetary ymmetrc polyomal. We alo mplemet a fat covero algorthm Maple 8 to accelerate the traformg proce whch traform power um ymmetrc polyomal to polyomal where ' are elemetary ymmetrc polyomal '. Secto III cota theore o the RUR method ad we ue algebrac terpolato to accelerate the computato of a RUR of a parametrc zero-dmeoal polyomal ytem. h help u get a early parametrc oluto to a SHEPWM problem. Secto IV depct the tep to olve a SHEPWM problem wth the umber of wtchg agle fed. It maly cota two part. he frt part focue o preproceg ad get a early parametrc oluto to a SHEPWM problem whch the modulato de m codered a a parameter. he other part how how to get complete oluto whe m gve a a pecfc value. Secto V lt umerc epermet ad mulato reult. At lat the algorthm decrbed ecto IV llumated by a detaled Emal: hb@63.com. School of Mathematc Jl Uverty Chagchu 3 PR Cha; College of Scece ortheat Dal Uverty Jl 3 PR Cha School of Mathematc Jl Uverty Chagchu 3 PR Cha 3 Emal: lea@jlu.edu.c. School of Mathematc Jl Uverty Chagchu 3 PR Cha 4 Electrcal Egeerg College ortheat Dal Uverty Jl 3 PR Cha

2 eample whe =3 apped A. Apped B lt ome fact about ymmetrc polyomal. I apped C the proof of heorem gve. II. MODEL DESCRIPIO We maly coder half-wave ymmetry SHEPWM problem of three-level voltage verter. Coder the upolar wtchg cheme ad the bpolar cae mlar. Fg. how oe phae voltage waveform whe the umber of wtchg agle eve; Fg. how that whe odd. he am of SHEPWM problem to determe wtchg agle to elmate harmoc the output voltage. 4 ( co m 4 4 b ( co(6. 6 (6 ( If odd ued the lat epreo; f eve ued. b6m ( co(6m M (6M ote that co 3 ca be repreeted a a polyomal co. Set ( co the equato ( ca be coverted to ymmetrc polyomal equato []-[3]. Ad we deote them orderly by g ( m g ( ( Fg.. Oe phae voltage waveform ( eve. Fg.. Oe phae voltage waveform ( odd. Wthout lo of geeralty we ca aume the ampltude of U(. Apply Fourer aaly to U( ad get U( b. Due to the ymmetry property the Fourer coeffcet 4 / b U( d. 4 j Wheever eve or odd b ( co j j 3 5. We coder how to elmate 5 th 7 th th 3 th 7 th 9 th harmoc. Sce there are wtchg agle we 4 ca elmate electve harmoc. Set b m where m called the modulato de. m a meaure of the magtude of the fudametal output voltage. o elmate th harmoc et b he we get equato he reao why g (. ( co rather tha co are ued to get ymmetrc polyomal. ow the problem how to olve polyomal equato (. Coderg the ymmetrc property of the equato ( t ca be coverted to equato whch deote the elemetary ymmetrc polyomal [3]. Deote them orderly by f( m f (. (3 Both Mathematca 9 ad Maple 8 have buld- procedure to mplemet above covero proce. But for th problem both of them have effcecy problem. ote that polyomal equato ( are um of power um ymmetrc polyomal o the eplct covero formula from power um ymmetrc polyomal to elemetary ymmetrc polyomal much more effcet. We mplemet the covero algorthm Maple 8 ad t coume much le CPU tme ad memory tha buld- procedure Mathematca 9 ad Maple 8. For our problem there are two ma advatage to do the covero. he frt to reduce the umber of varable by. he ecod to reduce the umber of zero of polyomal equato to be olved gfcatly. hey wll be llumated by the detaled eample Apped A. et we wll gve a theorem baed o whch uele computato ca be reduced. If a moc uvarate polyomal of degree ha dtct real zero atfyg ( the the g of t coeffcet are determed. It ca be tated a follow. heorem Suppoe that (. Let r r r r. he 4d 4d 4d 3 4d 4 d. heorem ca help u avod uele computato ad t eplaed the ecod part of the algorthm ecto IV. A proof of heorem lted Apped C.

3 he followg ecto wll depct the tool ued to olve polyomal equato (3 whch cota m a a parameter. III. RAIOAL UIVARIAE REPRESEAIO FOR SOLVIG POLYOMIAL SYSEMS A. RUR for zero-dmeoal polyomal ytem Ratoal Uvarate Repreetato (RUR a effcet method for olvg zero-dmeoal polyomal ytem whch oly have fte zero. I 999 Rouller detal the method [4] ad gve a effcet way to compute uch a repreetato. I eece t cotruct a oe-to-oe map betwee zero of the orgal ytem ad thoe of a uvarate polyomal t (. he the zero of the orgal ytem ca be repreeted by thoe of t (. By ow th method provde cloer oluto to parametrc oe for a geeral zero-dmeoal polyomal tha ay other ymbolc or umerc method. More formato about RUR referred to [ ]. Let be the feld of comple umber ad the feld of ratoal umber. Let f f [ ] whch have fte zero. Let t be a lear combato of. t called a eparatg elemet f t ha dtct value at zero of f f. he the RUR of f f wth repect to t ( g ( g ( g ( t t t t where t( gt ( gt( gt( [ ]. he RUR uque up to t. For coveece we alo call t a RUR of f f. he the zero of f f ca be epreed a g t ( gt ( t (. gt( gt( Moreover there et a bjecto betwee the ratoal zero (real zero comple zero of t ( ad thoe of f f. I Maple 8 the bult- procedure RatoalUvarate- Repreetato ued to compute a RUR of a zero-dmeoal polyomal ytem. For our problem due to the chageablty of m drectly ug RatoalUvarateRepreetato mpoble. So we coder RUR of parametrc zero-dmeoal polyomal ytem. B. RUR for parametrc zero-dmeoal polyomal ytem Sce polyomal ytem the paper volve oly oe parameter we lmt the umber of parameter to ad deote t by u. Let [ u] be the et of all polyomal u wth coeffcet ( u the et of all ratoal fucto wth ther umerator ad deomator [ u ]. Let f f [ u][ ]. If f f ha oly fte zero the algebrac cloure of ( u the f f called a parametrc zero-dmeoal polyomal ytem wth u a parameter. he the RUR of f f wth repect to u t repreeted a ( g ( g ( g ( t u t u t u t u where t u t u t u t u ( g ( g ( g ( ( u[ ]. tu ( moc ad t a lear combato of whch a eparatg elemet of f f. he computg of eparatg elemet referred to [5 6]. For u f u atfe: ( ( g ( g ( g ( are well-defed. ( t u u u t u u u t u u u t u uu tu ( uu quare-free part of (. quare-free where ( deote the tu tu he gt ( gt ( t ( u u gt( gt( uu uu (4 the zero et of f f. More dcuo about uu u u RUR of parametrc zero-dmeoal polyomal ytem are referred to [5]. A far a I ow there o bult- procedure Maple 8 or Mathematca 9 to compute a RUR of a parametrc zero-dmeoal polyomal ytem. he algorthm [4 5] ca be ued to olve th problem but t computato volve ratoal fucto o t coume too much CPU tme ad RAM. o f th problem ote that the coeffcet wth repect to of polyomal RUR of parametrc zero-dmeoal polyomal ytem are ratoal fucto u o we ca ue algebrac terpolato theory to mprove the algorthm [5] ad the mproved algorthm compute RUR wth operato all over the feld of ratoal umber. We mplemet the mproved algorthm Maple 8. A RUR of equato (3 wth repect to m t called a early parametrc oluto to the SHEPWM problem whch oly related to the umber of wtchg agle. Whe m chage computato oly eed to retart from the early parametrc oluto ad th avod much uele computato. I the et ecto we how detal the tep to compute early parametrc oluto to SHEPWM problem ad how to get complete oluto baed o a early parametrc oluto. IV. SOLVIG SHEPWM PROBLEMS F the umber of wtchg agle ad coder the modulato de m a a parameter. he algorthm to olve SHEPWM problem plt to two part. he frt part maly clude preproceg ad t target to get a early parametrc oluto to a SHEPWM problem whch a RUR of equato (3. he ecod part to get complete oluto to equato ( baed o the early parametrc oluto whe m gve a a pecfc value. I th proce oly uvarate polyomal olvg volved. So f m chage we oly eed to retart the computato from the early parametrc oluto ad computato cot cut dow. he frt part of the algorthm decrbed a follow. ( Gve the umber of wtchg agle get

4 tracedetal equato (. ( Let ( co ad covert equato ( to polyomal equato. Deote them orderly by g m g g. he ug elemetary ymmetrc polyomal [ ] tralate g g g to polyomal equato ad we get f( m f( (5 f(. (3 Accordg to the phycal gfcace of m or m ca be codered a a parameter of the lat equato of (5. Ug the theory of RUR compute a RUR of equato (5 wth repect to m ad we get a early parametrc oluto to the SHEPWM problem ( g ( g ( g ( (6 t m t m t m t m Whe the modulato de m fed we ca ue formula (6 to get. A a reult of heorem oly 4d 4d 4d 3 4d 4 d ha a poblty to derve a effectve group of wtchg agle. For a fed modulato de m the ecod part of the algorthm depcted a follow. ( Subttute m m to formula (6 ad get ( g ( g ( g (. (7 ( m ( m ( m ( m t m t m t m t m ( ( Solve m ( tm ad deote t root by. (3 Subttute l to t m gtm tm gtm l g ( ( g ( ( ad deote the reult by S ' ' ' ' l' l' l; ' ' atfy d. ote that we '4d '4d '4d 3 '4d 4 abado the ' whoe g do t coform to the cocluo of heorem. (4 Accordg to Veta formula for S ' olve m ( the we ca get ' '. ( ' ( ' (5 If there et a permutato ( ' ( ' ( ' ( ' j j of ( ' ( ' j j atfyg ( we get ( ' a group of wtchg agle arcco(. Oly tep ( ad (4 volve uvarate polyomal olvg o the ecod part of the algorthm perform at mot l tme uvarate polyomal olvg. We gve a detaled eample Apped A to llumate how the algorthm wor. j V. UMERIC EXPERIMES AD SIMULAIO Baed o a early parametrc oluto wth 5 we compute complete oluto to SHEPWM problem aocated to m / 5 46 Maple 8. Ad t averagely coume.84 o a laptop wth a Itel Dual Core P84 (.6GHz ad 4G RAM. We draw the wtchg agle trajectore Fg. 3 whe 5. It cot of 35 group of wtchg agle aocated to m he pot wth the ame m ad color form a group of wtchg agle. Accordg to the epermet there et two group of wtchg agle whe m m.58 ad.786 m.98; three group of wtchg agle whe.479 m.487 ad.59 m.785 ; oly oe group of wtchg agle whe.488 m.55 ad.98 m.987; f m.988 o avalable wtchg agle are foud. Fg. 3. Swtchg agle trajectore whe 5. We buld a three-level eutral Pot Clamped verter ytem ug MALAB/Smul. he mulato crcut cofgurato how Fg. 4. he DC de ue VSC tructure wth parallel coecto of capactace voltage ource order to mata the put voltage a a cotat. Iputtg dfferet wtchg agle produce correpodg wtchg gal ad by th way we cotrol the output voltage of the verter. o verfy the effectvee of the algorthm o flter coected to the outlet. A three-phae retve load coected to AC de ad the load eutral pot coected to the groud. We meaure the voltage waveform at the verter et to verfy the effectvee of harmoc uppreo. Whe m.75 three group of wtchg agle: G = [ ] G = [ ] G 3 = [ ] are obtaed. We mulate the three group of wtchg agle baed o the above verter ytem. he mulato reult of G how Fg 5 Fg 6 that of G how Fg. 7 Fg. 8 ad that of G3 how Fg 9 Fg. A ca be ee from the Fg. 6 Fg. 8 ad Fg. the 5 th 7 th th 3 th harmoc ha bee elmated. h llutrate the valdty ad effectvee of th method.

5 Fg. 4. Smulato crcut cofgurato. Fg. 9. Phae to phae output voltage of G3. Fg. 5. Phae to phae output voltage of G. Fg.. Harmoc pectrum of phae to phae output voltage of G3. VI. COCLUSIO Fg. 6. Harmoc pectrum of phae to phae output voltage of G. I th paper baed o RUR for olvg polyomal ytem we preet a algorthm to compute complete oluto to SHEPWM problem. he algorthm ca get a early parametrc oluto to a SHEPWM problem wth the umber of wtchg agle fed ad the parameter the modulato de m. Gve a pecfc m oly ug uvarate polyomal olvg we ca obta the complete oluto to the SHEPWM problem. Whe m chage the algorthm eed t retart totally ad t peed up the olvg. Epermet how that t effectve to olve SHEPWM problem baed o a early parametrc oluto. APPEDIX Fg. 7. Phae to phae output voltage of G. Fg. 8. Harmoc pectrum of phae to phae output voltage of G. A. A detaled eample wth 3 he frt part of the algorthm ecto IV goe a follow. ( he tracedetal equato are 4 (co co co 3 m 4 b5 co5 co5 co5 3 (8 5 4 b7 co7 co7 co ( Let co co 3 co 3. ralate equato (8 to polyomal equato 3 ad get

6 3 g m g (5 6 ( g3 ( (3 Ug ymmetrc polyomal theory equato (9 are coverted to polyomal equato elemetary ymmetrc polyomal 3. he we ubttute m for f f 3 ad get f m f 6m 8m m 8m 3 8m 6m 83 5m f3 64m 448m m 448m 3 896m m 344m 3 448m 56m 56m 3 56m 448m m 563 7m683. ( Due to m equvalet to m. So actually equato ( have le varable tha equato (9. he umber of zero of equato (9 8 ad that of equato ( 3 ( the algebrac cloure of ( m. I fact due to the ymmetrc property of equato (9 the umber of t zero 3!=6 tme of that of ( becaue 3. (4 Compute the RUR of f f3wth repect to m ad a eparatg elemet t. We get t m( gt m( g ( tm g where ( tm m 448m m 394m 55m 8m m 4m m 456m m m 56m m 394m 55m 8m 55 gtm ( 4 79m 4m m 456m m m 8m m m m m m gtm ( m 448m m 394m 55m 8m m m m 456m m m 56m m 356m m m m gtm ( m 448m m 536m 86m 48m 98m 575 tm ( m m m 4m 4m 35 4 m m m m m m 4m 35 ( It a early parametrc oluto to the SHEPWM problem. For a fed modulato de m we ca get aocated wtchg agle va uvarate polyomal olvg. Ad the ecod part of the algorthm llumated by the followg. For eample let m. (For a pecfc m we alway retart the olvg proce here. ( Subttute m m to ( the we get 3 tm ( 47 4 (45 56 (47 8 gtm ( ( ( gtm ( 74 4 (45 8 (47 8 gtm ( (549 ( ( ( he three real root of m tm ( are (3 Subttute 3 to g tm 3 3 gtm g ( ( tm gtm ( the the three zero of equato ( got ( m ; 3 m ; 3 m (3 Accordg to heorem we eed 3 o 3 mut t geerate a group of avalable wtchg agle.

7 he root of are ad there doe t et a permutato of the three root uch that j j j o 3 3 mut t geerate a group of avalable wtchg agle ether. 3 he root of are whch atfy So they form a oluto of equato (9 the we 3. ca get the aocated three wtchg agle whch a oluto to equato (8. From above dcuo we ow that whe 3 m there oly et oe oluto to equato (8. B. oto ad Covero relato betwee ymmetrc polyomal Defto ([8] A polyomal f [ ] ymmetrc f f ( f ( for every poble permutato of the varable. Defto ([8] Gve varable the elemetary ymmetrc polyomal [ ] are defed by the followg formula. r r r. Defto 3 ([9] Gve varable the power um ymmetrc polyomal p p[ ] are defed by the followg formula. r r r. he covero relato from power um ymmetrc polyomal to elemetary ymmetrc polyomal ca be eplctly repreeted a follow. [9] p p 3 p p p m m( r r r! r pm (. r r! r! r! C. Proof of heorem Proof: Let ( ( ( r. If r r r ( 4d ( 4d ( ( 4d 3 4d 4 d hold the t clear that heorem hold. ; Let g( ;. Apply ducto o r. ( r. ( r. From ( ( It elf-evdet that. It ca be verfed that ( 3 ( (. ( we ow ( hece (. (3 r 3. It ca be verfed that ( ( 3 3 ( ( 3 4 ( ( ( ( ( ( Becaue ( have the ame g g( =g( ( ( 3 3. Geerally peag we have ( ( r ( r ( ( Becaue r r r r r r r. ( r ( r ( r r have the ame g ( ( 4 g( r g( r ( g( r 4. I ummary we have d. ( ( ( ( 4d 4d 4d 3 4d 4 REFERECES [] S. Kouro M. Malow K. Gopaumar J. Pou L. G. Fraquelo B. Wu J. Rodrguez M. A. Pérez ad J. I. Leo Recet advace ad dutral applcato of multlevel verter Idutral Electroc IEEE raacto o Vol. 57 o. 4 pp Jul.. [] I. Cola E. Kabalc ad R. Baydr Revew of multlevel voltage ource verter topologe ad cotrol cheme Eergy Covero ad Maagemet Vol. 5 o. pp. 4-8 Feb.. [3] H. S. Patel R. G. Hoft Geeraled techque of harmoc elmato ad voltage cotrol of thyrtor verter: Part I-harmoc elmato Idutry

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