Five-Level Single-Leg Flying Capacitor Converter Voltage Balance Dynamics Analysis

Transcription

1 Fe-eel Sngle-eg Flyng Capacor Conerer Volage Balance Dynamcs Analyss () A. uderman ( ) and B. eznko () () Elmo Moon Conrol d., General Saelle Corporaon aruderman@elmomc.com, reznkob@spb.gs.ru Absrac he arace feaure of a flyng capacor conerer s he naural olage balance propery. he repored olage balance dynamcs analycal research mehods are based on heay frequency doman ransformaons (Fourer ransform, Bessel funcons ec) and are raher algorhmc and dffcul o use n an eeryday engneerng pracce. Suggesed me doman approach uses schng of pece-wse analycal soluons for consecue swchng nerals. he small parameer analyss of a fe-leel sngle-leg conerer yelds physcally meanngful, smple, and accurae expressons for aerage olage balance dynamcs gng an n-deph nsgh no parameers, carrer frequency, and modulaon sraegy mpac for boh and AC. I. INODUCION Mulleel conerers are beng progressely used for medum and hgh olage / power applcaons [, ]. Flyng Capacor (FC) mulleel conerer s an arace choce due o he naural olage balance propery. For Mulple on Clamped (MC) conerer used n fron end applcaons, s possble o achee bus capacors olage balance usng opmal (neares rual hree) olage modulaon sraegy only for a lmed modulaon ndex load power facor operaon enelope. Maxmal possble modulaon ndex M may be acheed for pure reace (nduce) load only. As he load power facor s approachng uny, maxmal modulaon ndex s compromsed for boh sngle- and hreephase conerers due o capacor olage balance mposed lmaons []. Full bus olage dynamc range ulzaon for hgh power facors s only possble a he expense of sgnfcanly compromsng olage qualy []. I s recognzed ha FC conerer olage balance process s dren by load curren hgh order harmoncs [-9]. More specfcally, he olage balance essence consss n capacor excesse "olage unbalance" energy dsspaon n conerer load by swchng harmoncs. hough s a common pracce o consder a prme seres -load (copper loss), he olage balance dynamcs s expeced o be nfluenced by addonal hgh frequency loss mechansms lke eddy curren ron core loss, skn [9] and proxmy effecs ec. FC conerer analycal olage balance research mosly deals wh an AC modulaon [-9]. As an FC conerer may be well used for / power conerson, s reasonable o ge sared wh ganng an n-deph undersandng of he aerage olage balance dynamcs for more smple case. he repored FC conerer olage balance analyss mehods are essenally based on frequency doman ransformaons. Howeer, seems somewha unnaural o use "nermedae" heay frequency doman mehods for deraon of lnear me naran models for he purpose of me doman analyss. As a resul, he repored olage balance analyss mehods are raher algorhmc han rue analycal and, herefore, don' ge a horough nsgh no olage balance process physcal naure. he alernae approach based on me doman schng of analycal ransen soluons for consecue perod swchng nerals s more adequae for undersandng olage balance mechansm []. A basc sngle-leg hree-leel FC conerer sudy was carred ou resulng n a smple accurae expresson for a olage balance dynamcs me consan ha s ald under praccal assumpons of nducance lmed load rpple curren and reasonably low capacor rpple olage []. In [, ] hs approach was appled o hree-leel H-brdge and four-leel sngle-leg conerers. In hs paper, he me doman olage balance mehodology s appled o a fe-leel sngle-leg FC conerer ha yelds ery smple, comprehense, physcally meanngful soluon for he aerage olage balance dynamcs. he resuls show he real power of he rue analycal soluon ha reeals n a clear ransparen way he olage balance process fundamenal dependences on he load parameers, carrer frequency, and olage modulaon sraegy for boh and AC modulaon. II. SINGE-EG FIVE-EVE FC CONVEE OOOGY AND MODUAION SAEGY Sngle-leg -leel FC conerer opology s shown n Fg.. Volage modulaon sraegy for normalzed olage command.< s demonsraed n Fg.. Insananeous olage command V s scanned by he four rangular wae carrer COM sgnals o generae he swchng nsans. he carrer sgnal C (C C C) s responsble for generang he swchng nsans for he complemenary swch par S S ( S S S S S S ). Fg.,b shows conerer swchng saes and oupu olage waeform assumng deal swches and he balanced capacors olages V /, V /, V /. A swchng perod s comprsed of egh nerals. Inerals,,, opologes generae he load olage Vdc/ (Fg.) four occasons of neral he load olage of Vdc/. For he opology, he conerer load s drecly conneced o he pose half of he power supply all he "pose" swches S, S, S, and S are n conducance sae and all he capacors are dsconneced.

2 S C C C S / / / S C C C C C C S S C C C V / / S Fg.. FC conerer opologes for.<: a, b, c, d Fg.. Sngle-leg fe-leel FC conerer wh -load Swchng nerals duraon () ( /, where - perod D V COM / V - normalzed olage command (Fg.) () ( D.) /. For he olage commands range of <., he four occasons of he neral (Fg.) are replaced by he four zero oupu olage opologes,,, (Fg.). Swchng nerals duraon for hs case () D /, (. /. () III. FC CONVEE VOAGE BAANCE DYNAMICS MODEING he sngle-leg fe-leel FC conerer s a h order lnear swched sysem. On each swchng neral, s descrbed by V COM S S V he sae-space equaon X ( ) A ( ) X () + B ( ) V X ( ) j j / [ ( ) ( ) ( ) ( ) ] j,...,8. he sae-space marces and ecors n () are obaned by analycally solng on he separae swchng nerals he frs and second order lnear ordnary dfferenal equaons []. In our calculaons, we assume an oscllang sep response of he nddual C-crcus (Fg., ) ha, n fac, s equalen o relaely small ressance (relaely small capacance). For he nerals wh wo capacors conneced n seres (nerals,,, ), he soluon s frs obaned for he second order crcu wh an equalen capacance and nal olage condon. Inddual capacor olages are hen found usng he law of charge conseraon. A dscree FC conerer model for calculang he sae-space arables a he swchng nsans: X ( ) A ( ) X () + B ( ) V / X (... X ( 8 X ( 9 ) A ( ) A ( ) X ( ) A ( ) X ( ) X ( ) + B ( ) + B ( ) V ) V ) + B ( ) V /... / 8 7 / C C / / C () (8) V / / / C V / C C Fg.. Volage modulaon process (a) and oupu olage waeform wh swchng saes (b) for normalzed olage command D.7 Fg.. FC conerer opologes for <.: a, b, c, d

3 where Example. Consder sngle-leg fe-leel FC conerer: V Ohm. mh C 6uF C uf C uf us 6 (.8kHz ). Conerer dynamcs swched smulaon resuls obaned by programmng formulas (8), (9) are presened n Fg.. he aerage load curren and capacor olages conerge o ( ) VD /() () ( ) V / ( ) V / ( ) V / for any se of nal condons for D>. For D, only he olage conerges o s balanced alue whle and do no (Fg.,a) hough her dfference conerges o Vdc /. hs s because he wo capacors are conneced n seres (Fg.,b,d) and, herefore, her olages are lnearly dependen. In he examples hroughou hs paper, we assume deal Curren and Volages Curren and Volages oad Curren FC Conerer oad Curren and Capacor Volages (D) oad Curren V Capacor Volage V Capacor Volage me, s a V Capacor Volage FC Conerer oad Curren and Capacor Volages (D.) V Capacor Volage V Capacor Volage me, s b V Capacor Volage Fg.. FC conerer curren and capacor olages smulaon for zero nal condons and dfferen olage commands: a - D b - D. (9) swches wh reerse olage blockng capably. hs s no correc for he real lfe swches wh an-parallel dodes. As he naural olage balance condon < < ends o be olaed due o unbalanced nal condons and oscllang response, he an-parallel dodes come no play and he lnear model s no more ald due o non-lnear clampng effecs. Howeer, f he capacor olages are balanced (some knd of precharge s requred n pracce) and he bus dsurbance s moderae, hen he aboe condon s neer olaed and he lnear model s ald for he real lfe swches [9]. IV. FC CONVEE VOAGE BAANCE FO MODUAION Suppose we ake he nal condons of he neral and subsue hem, along wh he neral duraon, no he neral dynamc soluon. hs wll ge us nal condons for he nex neral. By subsung hem wh he neral duraon no he neral dynamc soluon, we oban nal condons for he neral. Carryng ou he same procedure for he res of subnerals -----, we complee a perod and oban a lnear dfference ecor equaon for.< n he form X + ) A( X ( ) + B( V /, () ( where A( A A A A A A A A B( A ( A ( A ( A ( A ( A + B ) + B ) + B ) + B. ( A B + B ) + B ) + B ) + () Whle calculang marx A and ecor B n (), we sared wh he swchng neral. Oher seen possbles o ge sared wll produce dfferen sysem marces bu he same ransen behaor (he same egenalues). A smlar procedure may be carred ou for <.. Assumng sysem () sably, a seady sae soluon may be found as X ( ) ( I A( ) B( /, () I - uny marx. Generally speakng, he soluon () s a funcon of he normalzed olage command D and also s dependen on he seleced nerals order , , ,.... Howeer, for a good praccal conerer, he curren and olage rpples should be low and he soluon () s supposed o be close o () for any of he egh possble neral orders. Usng "on aerage" deraes approxmaon d ( + ) ( ) d ( + ) ( ), () d d from () he FC conerer aeraged dynamcs connuous me model s n he form of dfferenal equaon

4 dx d ( A( I) X + B( V /. () An accurae soluon and swched smulaon wll show four mes swchng frequency rpples n load curren and capacor olages ha are flered ou n he aeraged models (), (). Our mehodology o analyze he FC conerer aeraged olage balance dynamcs comprses he followng seps. Frs, we oban he characersc polynomal of marx A( () ( λ ) λ + bλ + b λ + bλ+ b. (6) As he marx A( s abou mulplyng 8 x marces, hs could hardly be done manually and requres an asssance of some symbolc calculaons ool (we use Maple). Second, each coeffcen of (6) s expanded no power seres of a small parameer (mulpled by some funcon of β, (7) ω << where ω ω ω /( ), α /( ). α C he small parameer assumpon s ald for mos praccal conerers ha hae low rpple curren (nducance domnaed load) and low rpple olage (reasonably large capacance). hough we assume ha all C-crcus n Fg., hae an oscllang sep response (relaely small capacance), our resuls are general and hold for relaely large capacances C> / as well hough, mos lkely, hs s no a praccal FC conerer case []. he selecon of he seres expanson maxmum power for he coeffcens (6) s dcaed by he accuracy consderaons for he polynomal roos wh a modulus close o uny. hrd, he roos of (6) wh he seres expanded coeffcens are found analycally usng Ferrar formulas []. For he small parameer approxmaon (7), may be shown ha here are always wo real and wo complex conjugae roos. he aboe calculaons mus be performed on separae for <. and.<. he auhors can prode he deals of he small parameer calculaons upon reques. he real roos are pose < λ < λ < and produce he aeraged connuous sysem me consans, / ln( λ, ). (8) he frs small me consan s acually he load one ( / (9) ha does no depend on he normalzed olage command D because an aerage load olage praccally almos does no depend on he nddual capacor ones []. ( ) exp ( ) exp ( ) exp { () cosω CC ( C + C ) C C[ () ()] { cosω+ C+ C C[ () ()] { cosω C+ C he second large me consan s approxmaed as 8 ( C+ C) A(,. D ( () 8 ( C+ C) A(,., ( (D ) does no depend on C and s symmercal n D around D. endng o nfny wh D approachng zero and uny. he complex roos par of characersc equaon (6) λ mexp( ± jθ ),< m (), delers equalen connuous sysem olage balance angular frequency and dampng me consan ωθ / / ln( m). he olage balance dynamcs parameers amoun o: angular frequency ( D ω( 6 C C ( ω( 8 C C CC C ( C + C ) CC C ( C + C ) ),.,., () () where he equalen capacance CC C () C+ C dampng me consan for D <. ( 8( C+ C) () [6D ( F+ 6) D + ( F+ )] for. D < 8( C+ C) (, (') ( [(8D+ ) F+ 6] where F C + C C C + F( C, C, C) +. (6) C C C he hree pecewse analycal funcons (), (), and () are connuous wh her frs deraes a D.. Once he olage balance dynamcs parameers are found, an aerage olage balance free process approxmae analycal soluon can be wren down. he olage balance dynamcs approxmaon (7) assumes ha here s no sgnfcan fas ransen due o an unbalanced curren nal condon ha may cause essenal nal capacor olages redsrbuon. [ () ()]snω} C () + C() () snω} + exp C+ C C () + C() () snω} + exp C+ C A A. (7)

5 Curren and Volages Curren and Volages Accordng o our calculaons, for conerer parameers ha prode reasonably low load curren and capacor olage rpples, he equaons (7) hold wh a good praccal accuracy. he physcal meanng of (7) s ha he mddle capacor C exchanges he excesse unbalance energy hrough he load nducance wh he equalen seres capacor comprsed of C and C. Oscllang olage componens of C and C are n he oppose phase. Oscllang olage of C leads ha of C and lags ha of C by he quarer of he perod. In general, he oscllaory par of he olage balance process s smlar o he behaor of a seres low ressance C-crcu ha s characerzed by he elecromagnec energy exchange beween he nducance and capacance. Once he oscllaory par of he ransen s oer, C olage reaches s balanced alue wh C and C capacor olages equalzed. Afer ha, C and C equal olages exponenally reach her balanced alues wh a common mode aperodc slow me consan () ha does no depend on C. In he small parameer approxmaon, he soluon (7) may be ewed as frs obaned for zero load ressance whle nonzero ressance s laer on accouned for by he dampng erms. o ncorporae a non-zero bus olage (forced soluon) no (7), one has o add he balanced capacor olage alues () and o replace he nal condons accordng o () () V / () () V / (8) () () V / FC Conerer oad Curren and Capacor Volages (D.) oad Curren V Capacor Volage V Capacor Volage me, s a V Capacor Volage FC Conerer oad Curren and Capacor Volages (D.) oad Curren V Capacor Volage V Capacor Volage me, s b V Capacor Volage Fg. 6. FC conerer dynamcs for zero supply olage (D.): a zero C and C common mode olage b pure C and C common mode olage Example. For he FC conerer wh he parameers from Example and zero supply olage, he accurae capacor olages obaned from swched smulaon and her analycal approxmaons (7) (n pnk as n he res of hs paper) for D. and zero nal curren are shown n Fg.6. he angular frequency and me consans accordng o (), (), and (): ω (.) 8rad / s, (.) 67ms, A(.) 6ms. he small frequency dscrepancy may be elmnaed by he frequency correcon ha s proporonal o he fourh power of perod and does no depend on load ressance. V. FC CONVEE VOAGE BAANCE FO AC MODUAION For AC, he FC dynamcs models become lnear mearable. Howeer, as he load fundamenal frequency s relaely hgh, we can oban a olage balance soluon for AC by he me aeragng of he connuous sysem egenalues along he olage command rajecores []. In oher words, he angular frequency and me consans (), (), () n (7) mus be replaced by her respece alues "aeraged" on a fundamenal perod accounng for her symmerc, een behaor for < and -<. For AC wh a modulaon ndex M, <M<, an nsananeous olage command D( ) M sn( ) (9) ω f and angular frequency and nerse me consans mus be aeraged on a fundamenal perod π / ω. hs way, for <M<. he angular frequency s ( M ( M ) 6 C C ) ω, () me consans, for example, for C C C C 6 C ( M ) () A M M 9π 6 C ( M ). () 6 M + M 9π For.<M<, he expressons for olage balance dynamcs parameers become a lle b bulky o be presened here. Example. Consder an FC conerer wh he parameers from Example. Fg.7 presens olage balance dynamcs for M. and fundamenal frequency ω rad s. f / AC olage balance parameers: ω (.) 6rad / s (.) 8ms A(.) ms. Smulaons show ha here s praccally no fundamenal frequency nfluence on aerage olage balance dynamcs for a fundamenal perod smaller han olage balance oscllaons perod and me consans ha enables effece aeragng. f

6 Curren and Volages FC Conerer oad Curren and Capacor Volages (M.) oad Curren V Capacor Volage V Capacor Volage me, s Fg. 7. Volage balance dynamcs for AC modulaon VI. AENAIVE VOAGE MODUAION SAEGY So far, we assumed he carrer based olage modulaon sraegy (Fg.) ha can be referred o as he lead one because he carrer wae C leads C, C leads C, and C leads C. he lag modulaon sraegy (he carrer wae C leads C, C leads C, and C leads C) generaes nerse swchng saes sequences compared wh he lead one. he effec of he lag sraegy may be formally accouned for by changng he sgns of he hree snusodal erms n (7). For he lag sraegy, oscllang olage of C leads ha of C and lags ha of C by he quarer of he perod. Volage balance dynamcs comparson for he wo sraeges s gen n Fg.8. Curren and Volages Curren and Volages V Capacor Volage FC Conerer oad Curren and Capacor Volages (D.) oad Curren V Capacor Volage V Capacor Volage me, s a V Capacor Volage FC Conerer oad Curren and Capacor Volages (D.) oad Curren V Capacor Volage V Capacor Volage me, s b V Capacor Volage Fg. 8. Comparson of olage balance dynamcs for lead and lag modulaon sraeges for modulaon: a lead sraegy b lag sraegy VII. CONCUSION For a modulaed fe-leel sngle-leg FC conerer, smple and accurae expressons for olage balance naural frequency and me consans and capacor olage balance dynamcs were obaned by applyng he me doman analyss echnque and ulzng he small parameer naurally arses for praccal conerers wh low curren and olage rpples. he frequency and me consans formulas along wh he aerage olage balance dynamcs expressons clearly reeal he dependences on nduce load parameers, carrer frequency, olage command, and modulaon sraegy. An nsgh no he olage balance mechansm ganed from consderaon s defnely useful for AC as well. Accurae AC frequency and me consans dependences on modulaon ndex are obaned by aeragng on AC fundamenal perod. raccally, AC fundamenal frequency has no mpac on olage balance dynamcs. ACKNOWEDGMEN he auhors graefully acknowledge Elmo Moon Conrol and General Saelle Corporaon managemen for on-gong suppor gen o adanced appled power elecroncs research. EFEENCES [] J. S. a and F. Z. eng, Mulleel Conerers - New Breed of ower Conerers, roc. IEEE Ind. Appl. Socey Annual Meeng, Oc 99, pp [] D.G. Holmes and.a. po, ulse Wdh Modulaon for ower Conerers: rncples and racce. Hoboken, NJ: John Wley,. [] J. ou,. ndado, and D. Boroyech, "Volage-balance lms n fourleel dode-clamped conerers wh passe fron ends," IEEE rans. Ind. Elecron., ol., Feb., pp [] S. Busques-Monge, S. Alepuz, J. ocaber, J. Bordonau, ulsewdh modulaons for he comprehense capacor olage balance of n-leel dode-clamped conerers, roc. IEEE ower Elecroncs Specalss Conference (ESC), hodes, Greece, June 8, pp []. Meynard, M. Fadel, and N. Aouda, Modelng of Mulleel Conerers, IEEE rans. Ind. Elec., ol., no., June 997, pp.6-6. [6] X. Yuang, H. Semmler, and I. Barb, Self-Balancng of he Clampng- Capacor-Volages n he Mulleel Capacor-Clampng-Inerer under Sub-Harmonc Modulaon, IEEE rans. ower Elecron., ol. 6, no., March, pp [7]. Wlknson, H. de Mouon, and. Meynard, Naural Balance of Mulcell Conerers: he wo-cell Case, IEEE rans. ower Elecron., ol., no. 6, No. 6 pp [8]. Wlknson, H. de Mouon, and. Meynard, Naural Balance of Mulcell Conerers: he General Case, IEEE rans. ower Elecron., ol., no. 6, No. 6, pp [9] B.. McGrah and D.G. Holmes, Analycal Modelng of Volage Balance Dynamcs for a Flyng Capacor Mulleel Conerer, roc. IEEE ower Elecroncs Specalss Conference (ESC), Orlando, F, June 7, pp [] A. uderman, B. eznko, M. Margalo, Smple Analyss of Flyng Capacor Conerer Volage Balance Dynamcs for Modulaon, roc. In. ower Elecroncs and Moon Conrol Conf. (EE-EMC), oznan, oland, Sep. 8, pp [] A. uderman and B. eznko, hree-eel H-Brdge Flyng Capacor Conerer Volage Balance Dynamcs Analyss, roc. h European Conference on ower Elecroncs (EE), Barcelona, Span, Sep. 9. [] B. eznko and A. uderman, Four-eel Sngle-eg Flyng Capacor Conerer Volage Balance Dynamcs Analyss, roc. h European Conference on ower Elecroncs (EE), Barcelona, Span, Sep. 9. [] G.A. Korn and.m. Korn, Mahemacal Handbook for Scenss and Engneers, nd Edon, New York: Doer,.