J.M. Cano, G.A. Orcajo, C.H. Rojas, M.G. Melero, M.F. Cabanas

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1 Anlyss of the AC-DC hrmon nterton etween the power networ nd unontrolled 6 pulse retfers worng n the dsontnuous onduton mode usng Newton s method J.M. Cno, G.A. Orjo, C.H. Rojs, M.G. Melero, M.F. Cns Eletrl Engneerng Deprtment Unversty of Ovedo Cmpus de esques s/n 3324 Gjón, Asturs (Spn) phone: , fx: , e-ml: jmno@orreo.unov.es Astrt. Ths pper summrses study ompletely developed n [] for the nlyss of AC-DC hrmon nterton etween the power networ nd non-ontrolled three-phse retfers opertng n dsontnuous mode. The presented method s n dptton for the se of unontrolled three-phse retfers of the proedure desgned y Smth nd Arrllg [2] for the nlyss of three-phse ontrolled retfers. The new lgorthm hs een developed n the frequeny domn, nd therefore ntrodues some mportnt dvntges for overll power system nlyss. The nfluene of ftors suh s unlne or hrmon dstorton n the power supply voltge, hrmon dstorton n the DC sde voltge nd unlne n the AC sde flterng retnes n esly e nlysed. Moreover, very flexle modellng of the ruts n oth the DC nd AC sde n e heved wth mnmum effort. Key words Hrmons, retfers, dsontnuous onduton mode, Newton s method.. ntroduton n the desgn nd nlyss of eletr power systems t s extremely mportnt to fnd out the effets of the nterton etween the lner networ nd the non-lner deves. To heve ths, t s neessry to otn very detled models of suh non-lner deves n order to ntegrte them nto the networ for n tertve mthemtl resoluton proess. For the se of sx pulse retfers, severl tehnques exst to lulte the soluton for the stedy stte n the frequeny domn. The frst of these tehnques s the fxed pont terton method. Ths lgorthm uses the supply voltge of the retfer s strtng pont to predt the urrent t njets nto the power system. These new urrents, when mposed to the AC rut generte new voltge system tht s then used s the strtng pont for new terton. The mn drw of ths fxed-pont tehnque s ts lmted onvergene pty. n ft, when the power system s we n the ouplng pont nd/or the flterng retnes present low vlue, the onvergene of the lgorthm nnot e ssured [3]. Severl solutons hve een proposed to overome ths prolem [2,pp.242]. However, ll of them onsderly nrese the omplexty of the lulton proess. For ths reson, present reserh tendenes fous on the susttuton of the fxed-pont lgorthm for lterntve lulton methods rther thn fndng solutons to ts nherent onvergene lmttons. The Equvlent Norton Method [2] models the rut s non-lner deves durng the tertve proess s urrent soures prllel onneted to n dmttne. Ths dmttne must e lulted to represent the deve lnerzton round ts opertng pont. n ths wy, the response of the lgorthm durng the tertve proedure s muh fster nd ts onvergene prolty s hgher. n ths method, n dmttne mtrx must e lulted to represent the dependeny tht exsts t the opertng pont, etween the supply urrent nd every sngle hrmon voltge of the non-lner deve. Although ths method n e suessfully ppled for the modelng nd nlyss of ertn eletrl deves suh s the study of sturton n power trnsformers, t s not sutle for the study of AC/DC onverters sne the nfluene of the DC sde s negleted. n 989 E..Lrsen [4] presented method tht sought to solve the nterton prolem etween the AC nd DC sdes of three-phse ontrolled retfers. Ths method, ntlly lled HMAT (Hrmon Mtrx Clulton) s now nown s the ABCD prmeters method. n ths method, the lnerzton of the onverter round ts operton pont s lso ppled. To do ths, set of new mtrl equtons s defned. These equtons nlude the mthemtl reltonshps etween the onverter supply voltge nd urrents, s well s the reltonshp of these vrles wth the DC voltge nd urrent. The presene of ontrol ngle modulton, negleted n the Norton Equvlent Method, s lso nluded n ths model. Although the mtrxes tht represent the nrements n the system vrles re otned n numerl fshon, ths method neglets the effets of delyed ommutton. For

2 ths reson, f ths effet s to e nluded, the system of equtons must e deoupled to e solved. n ths se, the lgorthm dverges f the retfer supply voltges nlude even hrmons [5]. The Newton method, whh wll e desred n the next seton, solves the drws relted to the ove lulton tehnques. However, omputton tme for ths lgorthm s hgher thn for the ones referred to ove. 2. The Newton Method n 996 Brue C. Smth [6] proposed soluton exlusvely sed on the Newton method for the modellng of the three-phse ontrolled retfer opertng n ontnuous onduton mode. The mn ontruton of hs study ws the overll smultneous tretment of oth the retfer s nternl vrles nd ther nterton wth the AC nd DC sdes of the system. The theoretl prnples of the Newton method re very smple, sne t onsst of the resoluton of non-lner equton system nludng ll the eletrl reltonshps etween ll the eletrl vrles of the retfer. The overll pplton of ths lgorthm s presented n [2]. The new formulton needed for the pplton of the Newton method to non-ontrolled retfers opertng n dsontnuous mode hs een developed n [] y usng the model presented n [7]. The AC sde rut s modelled s the three phse Thevenn s equvlent shown n Fgure. Ths equvlent rut n e used for the smplfed representton of ny lner power supply. [Z] g g g Fg.. Thevenn s equvlent of the AC sde of the system The DC rut wll lso e represented y mens of ts Thevenn s equvlent. Ths rut wll e otned for every supply frequeny s the seres omnton of n mpedne nd n del voltge soure (Fg. 2). Although suh rut s lner, the pplton of ddtonl tertve methods provdes soluton even for non-lner equvlents. + - Zth dth Fg. 2. Thevenn s equvlent of the DC sde of the system A. Error Funtons n ths seton, the omplete set of equtons tht must e smultneously solved n order to otn the vlues of ll the system vrles wll e desred. These funtons re ll defned n the form f(x)= nd re lled error funtons. The error funtons n e lssfed nto the followng groups: Error funtons for the strtng of onduton (F θ ) Error funtons for the end of onduton (F φ ) Error funtons for the AC rut ( F α ) Error funtons for the rpple n the DC rut ( ) F d Error funtons for the DC rut ( F d ) Error funtons for the men vlue of DC voltge ( F d ) Tle shows the numer of equtons tht must e solved for every type of error funton. The ndex N represents the numer of hrmons nluded n the soluton. TABLE. Numer of equtons to e solved Group of Error Funton Numer F θ 6 F φ 6 F α 6N F d 2N F d TOTAL 8N+3 B. Mthemtl desrpton of the method The junton of ll the error funtons wll produe system of equtons whose vrles wll e the AC supply voltges, the DC voltge nd the strtng nd end of onduton tmes. The AC voltges re omplex vrles, for ths reson ts rel nd mgnry prts must e ten nto ount n the soluton of the system. n ths wy, the totl numer of equtons tht must e solved s α 8N+3, whh orresponds to:, d, θ, φ, d. The pplton of the Newton method to the ove equton system produes mtrl equton. [ S] n [ S] n + [ J ] ( ) n [ F] n + = () n ths equton [S] s olumn vetor formed y ll the system vrles, [F] s olumn vetor formed y ll error funtons, nd [J] s the Jon Mtrx of the equton system. The ndex n represents the numer of tertons.

3 { d } R R R R d φ θ { d } n+ { d } R R R = R d φ θ { d } n + n ([ J ] ) R R R R F F d φ θ d d n Equton 2 represents the expnson of equton. n ths new expresson the orderng of the dfferent vrles nsde the mtrxes n e oserved. n the next fgure flux dgrm summrzng the dfferent steps tht t s neessry to follow n order to mplement the desred lgorthm s presented. The lo nmed ntl estmton of ll the vrles nttes the lulton usng the followng ssumptons: the retfer supply voltge s dentl to the Thevenn s equvlent voltge nd the voltge rpple n the DC sde s nl. The vlue of the DC voltge s estmted s slghtly lower thn the mxmum possle vlue. Ths estmton mples good pproxmton to ts rel vlue espelly n DC ruts wth ptve flterng stge. The tmes of strtng nd end of onduton wll e extly lulted from the retfer supply voltge nd the DC sde voltge fter solvng y the Newton-Rphson method the error funtons y. Strtng F θ NTAL ESTMATON OF ALL THE ARABLES Numer of ter= CALCULATON OF THE ERROR FUNCTONS Convergene? NO YES Numer of ter=numer of ter+ JACOBAN CALCULATON ARABLES CALCULATON F φ (2) RESULTS OUTPUT STOP Fg. 3. Flux dgrm for Newton s method pplton The onvergene of the tertve method wll e rehed when ll the error funtons produe output vlues lose to. n ths wy, t s possle to estlsh reltve lmts for every error funton ordng to the followng expresson: F X < ε (3) X Where ε defnes the degree of preson heved n the soluton. By mens of ths reltve error prmeter the sme preson n e seleted for ll the hrmon omponents ndependently of ther order nd/or mpltude. Therefore, the ext estmton of ll the hrmon omponents, even those tht exht lower mpltudes, wll e esly heved. However, only the onvergene of hrmon omponents whose mpltudes surpss ertn mnmum vlue n e fored. f ths restrton s not ten nto ount the tertve soluton of the system would produe flse hrmon omponents tht re not relly present n the system nd ths stuton would gve rse to numerl prolems durng the terton proess. When t s not neessry to otn the sme exttude n the lulton of ll the hrmon omponents, other dfferent pprohes for the lulton of the soluton re more sutle. For nstne, estlshng lmt for the sum of the solute vlues of the devton n every error funton s ondton tht n e very effently evluted ordng to the followng equton: 8N + 3 = F < ε (4) C. Clulton of the Jon mtrx of the system For the tertve soluton of the equton system usng the Newton method t s neessry to lulte the Jon mtrx. Ths mtrx, whh s formed y the prtl dervtves of every funton wth respet to every vrle, n e lulted n two dfferent wys. The frst possle proedure onssts of the numerl lulton of every prtl dervtve. n ths se, the dervtve of n error funton F wth respet to the vrle x wll e lulted y evlutng the error funton n x nd x+ x ordng the equton (5): F F = lm x x ( x + x) F( x) F( x + x) F( x) x x The pplton of ths lulton proedure to every prtl dervtve wll led to the omplete omputton of the Jon mtrx. However, ths pproh to the prolem s not prtl sne the omputton tme requred for the lulton of ll the prtl dervtves s extremely hgh. For ths reson, dfferent non-numerl method must e ppled. The lterntve to the numerl lulton of the prtl dervtves s to otn nlytl expressons y mens of the dervton of the error funtons. Although ths method ntlly nvolves huge mthemtl effort, t drmtlly redues omputton tme of the Jon mtrx. n the present study the mthemtl expressons of ll the prtl dervtves of the error funtons hve (5)

4 een lulted nd n e heed n []. Moreover, ll of them hve een verfed y usng numerl lulton method. One the nlytl expresson of the prtl dervtve of every error funton hs een otned ts numerl vlue ws ompred wth the numerl method defned y equton (5), nd perfet djustment etween oth results ws otned. The exttude of ll the mthemtl lultons ws thus verfed. D. nterpretton of the Jon mtrx The Jon mtrx otned from the tertve soluton of the equton system represents lnerston of the represented eletrl system round ts opertng pont. Fgure 4 shows the three-dmensonl vew of the Jon mtrx otned, nludng 3 hrmon omponents. n ths fgure the omponents of the mtrx wth vlues hgher thn.5 hve een trunted n order to pprete the nfluene of the less sgnfnt omponents. JACOBAN wth the mxmum reltve mportne whle the l ones re ssgned to omponents wth mnmum reltve mportne. Severl onlusons n e rehed from these grphs: All the elements elongng to the mn dgonl of the mtrx re relevnt, sne every vrle hs n mportnt nfluene on ther own error funtons. The vrton n ny -order hrmon n the AC sde exerts prtulrly menngful nfluene over the DC sde hrmon omponents wth orders - nd + nd vevers. Ths stuton leds to the pperne of dgonl strutures n ertn zones of the mtrx. Fgure 6 shows zoom of the grph presented n fgure 5 n the zone orrespondng to AC-DC nterton. The dgonl lnes nsde the mtrx n e oserved n ths fgure. R{ } {Fd} Asolute vlues Funtons, rles Fg. 4. Three-dmensonl vew of the Jon mtrx F 2 3 F F 7 8 F 9 d { JACOBAN Fφ { Fθ F d rles Fg. 5. Top vew of the Jon mtrx 8 6 Funtons A lerer evluton of the reltve mportne of every omponent n the Jon mtrx n e otned y usng the representton dopted n fgure 5, where the top vew of the Jon mtrx s shown. n ths grph the olour sle represents the reltve mportne of eh omponent. The whte ponts elong to omponents 3 Fg. 6. Zoom of the top vew of the Jon mtrx for the zone orrespondng to the AC-DC nterton The tmes of end of onduton φ, exert we nfluene over the rest of the vrles. n ft, ther effet s only relevnt to ther own error funtons. The tmes of strtng of onduton hve strong nfluene on the rest of the system vrles, espelly on ther own error funtons nd on the error funtons of end of onduton. The two lst onlusons n esly e understood f t s rememered tht the vrtons n the tmes of strtng of onduton ffet oth the preursor urrents nd the onvoluton pulses. However, the hnges n the end of onduton tmes only ffet the onvoluton pulses. Ths stuton s shown n grph form n fgure 7. φ t Fg. 7. The nfluene of vrton n the strtng nd endng onduton tmes θ t

5 E. Methods for the elerton of the lgorthm Severl tehnques hve een proposed for the ontrolled retfer opertng n the ontnuous mode to elerte the exeuton of the lgorthm [2]. The extrpolton of these tehnques to the unontrolled retfer worng n the dsontnuous onduton mode s lso possle. n ft, two dfferent proedures re sutle for the elerton of the lgorthm for operton n the dsontnuous onduton mode. Sngle lulton of the Jon mtrx Sne the Jon mtrx s lnerston of the system round n operton pont, t s possle to ssume tht the vlues of the prtl dervtves wll not present drst hnges from one terton to nother. For ths reson t s possle to me only the ntl lulton of the Jon mtrx nd eep ts vlues onstnt untl the fnl onvergene of the method. f the sngle lulton of the Jon mtrx s ppled, lrger numer of tertons must e mde efore rehng onvergene. However, the svng of tme nd omputton resoures s onsderle sne every terton of ll the elements n squre mtrx of order 8N+3 need not e lulted. Although these methods llows the Newton lgorthm to onverge fster, t should e noted tht under ertn rumstnes the l of updtng n the Jon mtrx durng the tertve proess n use dvergene of the method, prtulrly n those ses where resonne n our. Prtl updtng of the Jon mtrx A less drst proedure for the elerton of the lgorthm onssts of the prtl updtng of the Jon mtrx. n ths pproh, only the elements of the Jon mtrx tht surpss ertn solute vlue re updted fter the frst terton. Usully the elements to e updted elong to the AC/DC dgonl of the system nd re relted to the strtng nd end of onduton tmes. A dret pplton of the method usng sequene omponents, [8], [9] s lso possle. Ths proedure permts the dret lulton of the soluton wthout ddtonl onversons n the se when the desred formt of the result re the sequene omponents, whh s very frequent stuton n the nlyss of power systems. 3. Prtl Applton of the Method n ths seton, prtl se for the pplton of the method wll e presented. The seleted se wll nlude resonne frequeny n the nlysed spetrum n order to demonstrte the roustness of the method. The next fgure shows the system tht wll e nlysed. The supply system wll hve.3% unlne nd the dret sequene omponent wll e delyed 4º wth respet to the nverse sequene omponent. The omplete defnton of the supply voltge s presented n tle. g g Rl Rl Ll Ll g Rl Ll L T4 T6 T2 Cl Cl Cl L L T Fg. 8. Test system TABLE. - Sequene omponents of supply voltge Postve sequene Negtve sequene d rms () 22 rms () 22,3 Phse ngle (º) Phse ngle (º) 4 The eletrl system onssts of three-phse unontrolled retfer suppled through the equvlent mpedne of the power networ. A ptor ttery for power system ompenston hs lso een nluded n the onneton of the retfer to the power system. The vlues of ll these omponents re shown n tle. TABLE. - lues of the AC prmeters Flterng retnes Power system mpedne L (mh) L (mh) L (mh) Rl (mω) Ll (µh) Cl(µF) The vlue gven for Rl n the ove tle tes nto ount the vrton used n the resstne y the sn nd proxmty effets. Although more detled pproh to ths phenomenon n e found n [2,pp.4-46], n the present smulton the orreton ftor 5 hs een nluded n order to show the smplty of nludng the dependene of ertn system prmeters on frequeny when the smulton s rred out n the frequeny domn. However, n the se of tme domn smulton ths nd of effet n hrdly e modelled. On the DC sde of the system n ndutve seres mpedne onneted to voltge soure ws nluded. The soure wll hve DC level n order to produe dsontnuous onduton mode operton n the retfer. A ertn rpple level wll lso e nluded n the soure s presented n tle. T3 TABLE. - DC prmeters Thevenn oltge ( Th ) Thev. mped. (Z Th ) Hrmons Th =522,72 Rl=,Ω Ll=,2mH Order 2 5 Th Ampltude () Phse ngle (º) 45 6 All the frequenes lower thn 3 KHz hve een nluded n the resoluton of the system equtons. Ths resoluton s equvlent to ountng for the frst 6 hrmon omponents. For the present system t n e esly demonstrted tht resonne frequeny exsts t 265 Hz. Therefore, ths exmple s n exellent test for the onvergene propertes of the method. T5 R L d

6 The exmple ws solved usng two dfferent methods. On one hnd, the sngle lulton of the Jon mtrx ws ppled n order to elerte the lgorthm onvergene. On the other hnd, the omplete method of totl updtng of the Jon mtrx ws lso used n order to otn the optml onvergene hrtersts nd ensure the orret lulton of the soluton even under resonne ondtons. The seleted onvergene rteron onssts of lmtng the totl sum of the solute error vlue to ertn vlue n every error funton. The next equton represents the mthemtl formulton of ths rteron N F < (6) = n the resoluton of ths exmple, y pplyng the totl updtng of the Jon mtrx n every terton, 8 omplete tertons were neessry. However, n the se of sngle lulton of the Jon mtrx 24 terton were needed. n spte of the hgher numer of tertons needed n the seond se, ts omputton tme ws pproxmtely 3 tmes shorter, whh shows the huge svng of mthemtl lulton heved wth the use of elerton methods. Fgure 9 shows omprson of the evoluton of the error vs. numer of tertons nd omputton tme for the two dfferent tehnques. n fgure the fnl pperne n the tme domn of the retfer supplyng urrents nd the power system urrents up from the ptor n s presented. Fgure shows the retfer supplyng urrents s they re otned from the dret pplton of the method n the frequeny domn. The postve sequene hrmons hve een represented on the postve sse xs nd the negtve sequene ones on the negtve xs Error (d) Totl u p dtng LMT SELECTED FOR CONERGENCE Sngle lulton 336 tme unts tme unts Numer of tertons Fg. 9. Error evoluton vs. numer of tertons nd tme The nlysed se s useful for the study of the roustness of the lgorthm. However, t should e ponted out tht the onneton of pure ptor n, s ndted n fgure 8, would not e sutle soluton for n ndustrl nstllton euse t ould gve rse to resonne prolems. n ft, nthrmon ndutnes re ommonly used n ndustrl plnts. These ndutnes re seres onneted wth the ptor n n order to djust the resonne frequeny to vlues lower thn the expeted hrmon frequenes (typlly under the 5 th hrmon). The use of nthrmon ndutnes lso elmntes the nfluene on the resonne frequeny of the lne ndutnes, whh re n ths se muh lower thn the nthrmon ones. (A) (A) t(m s) Fg.. Retfer supply urrents (top). Power system urrents efore the ptor n (ottom) Ampltude (d) ngle(º) Order Fg.. Retfer supply urrents n the frequeny domn n order to present results tht re more relst, the smulton ws repeted fter seres onnetng three nthrmon ndutnes to the thee-phse ptor n. To lulte the vlue of the ndutnes, equton 7 n e ppled. f r = (7) 2π LC A vlue of 6,7 mh ws seleted n order to tune the resonne frequeny elow the 5 th hrmon. The results lulted for the new system re presented n fgure 2. n ths fgure the effet used y the nthrmon

7 ndutnes n e oserved (elmnton of the rpple present n the power system urrents). Ths rpple ws used y the proxmty etween the resonne frequeny nd some relevnt hrmon omponents n the lne urrent (A) (d) t(m s) Order Fg. 2. Power system urrents fter the nstllton of nthrmon ndutnes 4. Conlusons A new model for the lulton of the AC/DC nterton n 6 pulse unontrolled retfers hs een presented. Ths new model, y mens of n tertve lgorthm, llows the nlyss nd quntfton of the effets used to the system y ths non-lner lods, tng nto ount even the nfluene of frequeny n systems prmeters. The Guss method hs proved tself to e the smplest; however, ts onvergene s muh slower nd the mthemtl resoluton of the prolem s not possle f the strtng pont for terton s not lose to the rel soluton. Ths stuton strongly redues the sope of pplton of the method, espelly n the lulton of DC vrles where the presene of moderte level of voltge rpple n led to the dvergene of the lgorthm f n del DC voltge s ten for the frst terton. The Newton method provdes seres of mportnt mprovements on the Guss method. Convergene s fster nd the prolty of dvergene of the lgorthm s muh more lmted. The mn drw of ths method s the need for very prese nowledge of the system n order to otn n urte lulton of the Jon mtrx. The numerl lulton of ll prtl dervtves needed to otn the Jon mtrx hs shown tself to e proess tht nvolves huge onsumpton of tme nd omputtonl resoures. A possle soluton to ths prolem s the nlytl lulton of the mthemtl reltons etween vrles. Ths nd of lulton nvolves g ntl effort, ut one the nlytl expressons hve een otned the reduton n omputton tme s relly remrle. Although the usul elerton tehnques for the Newton s lgorthm hve proved ther usefulness n the reduton of omputton tme nd resoures, ts effet on onvergene must e lso ten nto ount, espelly n those ses where the posslty of resonne phenomen exst. Anowledgement The uthors would le to grtefully nowledge the fnnl support gven y the Eduton nd Culture Deprtment of the Regonl Government of Asturs through funds from the Reserh, Tehnologl Development nd nnovton Pln (+D+) of Asturs Referenes [] J.M. Cno. (2) "Funones de Trnsferen pr el nálss en dstnts ondones de funonmento de l dstorsón rmón sod retfdores trfásos ", Tess dotorl, Unversdd de Ovedo. [2] J. Arrllg, B.C. Smth, N.R. Wtson, A.R.. Wood. (997) "Power System Hrmon Anlyss", John Wley & Sons. [3] C. Cllghn, J. Arrllg. (994) " A doule tertve lgorthm for the nlyss of power nd hrmon flows t -d termnls ", Pro. EE, 36(6), [4] E.. Lrsen, D.H. Ber, J.C. Mver. (989) " Low-order Hrmon ntertons on AC/DC Systems ", Trnstons on Power Delvery, ol. 4, Nº, pp [5] B.C. Smth. (996) "A hrmon domn model for the nterton of the Hd onvertor wth nd d", Tess dotorl, Unversdd de Cnterury, Nuev Zelnd. [6] B.C. Smth, N.R. Wtson, A.R. Wood, J. Arrllg. (996) "Newton soluton for the stedy-stte nterton of AC/DC systems", EE Pro.-Gener. Trnsm. Dstr., ol. 43, Nº 2, pp [7] J.M.Cno, G.A.Orjo, J.G.Myordomo, R.Asens, M.F.Cns, M.G.Melero. (2) "Dstorton Anlyss n AC/DC retfers opertng n dsontnuous mode under unlned ondtons", Pro. Power Qulty Conferene PQC, pp [8] B.C. Smth, N.R. Wtson, A.R. Wood, J. Arrllg. (997) "A Sequene omponents model of the /d onverter n the hrmon domn", EEE Trnstons on Power Delvery, ol. 2, Nº 4, pp [9] B.C. Smth, J. Arrllg. (999) "Power Flow Constrned Hrmon Anlyss n AC-DC Power Systems", EEE Trnstons on Power Systems, ol. 4, Nº 4, pp

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