Fast Functional Modelling of Diode-Bridge Rectifier Using Dynamic Phasors

Transcription

1 Fast Functonal Moellng o Doe-Brge Recter Usng Dynamc Phasors T. Yang S.. Bozho G. M. Asher The Unersty o Nottngham Nottngham NG7 RD UK Ths paper s a postprnt o a paper submtte to an accepte or publcaton n IET Power Electroncs an s subject to Insttuton o Engneerng an Technology Copyrght. The copy o recor s aalable at IET Dgtal Lbrary Corresponng Author: Dr Tao Yang Power Electroncs Machne an Control Group Faculty o Engneerng The Unersty o Nottngham Nottngham NG7 RD Unte Kngom Phone: + ( Emal Tao.Yang@nottngham.ac.u Abstract In ths paper a unctonal moel or oe-brge recters s eelope base on the ynamc phasor concept. The eelope moel s sutable or accelerate smulaton stues o the electrc power systems uner normal unbalance an lne aulty contons. The hgh accuracy an ecency o the eelope moel hae been emonstrate by comparson aganst threephase tme-oman moel an aganst the moel employng synchronous space-ector representatons. The expermental ercaton o the eelope moel s also reporte. In aton an error analyss shows that the error o the eelope moel s less than % at the most seere unbalance contons. The prme purpose o the moel s or the smulaton stues o moreelectrc arcrat power archtectures at a system leel; howeer t can be rectly apple or smulaton stuy o any other electrcal power system nteracng wth uncontrolle oe brge recters. Inex Terms AC-DC power conerson Arcrat Power Systems Conerters Dynamcs Moellng Recters ectors

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3 I. INTRODUCTION There has been seen a sgncant penetraton o power electroncs nto electrc power systems (EPS n recent years. For example terrestral EPS s partcularly at strbuton leel promse a multplcty o power electronc conerters to hanle renewable sources energy storage or EPS contonng. A smlar scenaro pertans or solate an moble EPS such as more-electrc arcrat (MEA an more-electrc shp. In the MEA the electrc power conerson s reure to manage power strbuton lanng gear lght actuaton an other unctons []. The use o large numbers o power electronc eces brngs sgncant moellng challenge at the EPS system leel ue to the system complexty an the we araton n tme constants. The challenge s to balance the smulaton spee aganst the moel accuracy an ths s epenent on the moellng tas. Four erent moellng layers are ene accorng to the moellng banwths.e. archtectural moels unctonal moels behaoural moels an component moels [] []. The archtectural layer computes steay state power low an s use or weght cost an cablng stues. In the unctonal leel the system components are moelle to hanle the man system ynamcs up to 5Hz an the error shoul be less than 5% n respect o the behaour moel accuracy. The behaoural moel uses lumpe-parameter subsystem moels an the moellng reuences can be up to hunres o Hz. The component moels coer hgh reuences electromagnetc el an electromagnetc compatblty (EMC behaour an perhaps thermal an mechancal stressng. The banwth o component moels can be up to n MHz regon reure. Targetng or acceptable smulaton tmes or system-leel EPS moellng a number o approaches hae been nestgate. Aerage state-space moels [] are a stanar technue or conserng only the unamental wae conerter behaour. Aerage moellng o ac strbuton systems noles transormng the three-phase ac sgnals to a synchronous rotatng rame henceorth terme the moel. Ths metho has been use n moellng MEA electrc power systems an s proe to be an eecte technue []. A moel wth three-phase ac arables s henceorth calle an abc moel. One o the saantages o the approach s that uner aulty an unbalance contons ouble-reuency components appear an the smulaton tme steps must be rastcally reuce to mantan accuracy. An alternate approach that can aress ths problem s ynamc phasors (DPs [5 6]. The DP metho has been apple to the moellng o terrestral EPS systems nclung mbalance regmes [7 8]. Comparate stuy o a smple EPS wth lne aults carre out n abc an DP oman s gen n [9] where the ecency o the DP approach s emonstrate. The DP metho has also been apple or moellng o electrcal machnes [ ] an lexble ac transmsson systems [] nclung acte lters [] an statc synchronous compensator (STATCOM []. In [5] an [6] a DP moel or thyrstor-base hgh-oltage rect current (HDC transmsson systems s reporte although only balance contons are consere.

4 Derent moels or uncontrolle oe-brge recters (DBs hae been eelope n recent publcatons. A comprehense analytcal moel s eelope n [7] an [8] where erent operaton moes o DBs hae been thoroughly stue. Another mathematcal moel or analysng DBs base on swtchng unctons s ntrouce n [9]. An aerage moel or DB s propose n []. Howeer all these moels are only or steay-state applcatons an the power supply s assume to be balance. Ths paper ams to eelop a DP moel or DBs. Ths moel can be use or both steay-state an transent stues uner balance an unbalance contons. At the tme o ths stuy eelopment no report on DB moellng n DP oman has been publshe. The most smlar to DBs moelle usng DP approach s a thyrstor-base HDC conerter [6]. In aton the moel reporte n [6] s sutable only or balance operatons. The DB unctonal moel presente n ths paper s base on the DB relatons between ac an c termnal arables. The negate seuence uner unbalance or ault contons s treate as a sturbance n the moel. Both the n an 6 th harmonc on the c-ln oltage are conenently nclue n the reporte DP moel. Ths enables to coer the ac mbalance oltage an c rpple oltage or both contnuous an scontnuous operaton. The man results o the paper are summarse as ollows: - The DP approach has been successully apple to eelop moel an uncontrolle DB recter applcable or accelerate smulaton stues o complex EPS both balance an unbalance an s consstent wth the unctonal moellng layer speccatons n []. The moel s nepenent on other EPS eces an may be use as a lbrary element nteracng wthn an extene three-phase EPS moels - The moel s ere expermentally uner both balance an unbalance operaton - The computatonal eecteness o the eelope moel s proe through comparson aganst tme-oman abc swtchng an unctonal non-swtchng moels an a sgncant computatonal acceleraton s emonstrate. II. DYNAMIC PHASORS Ths Secton n orer to eep the paper sel-contanment an to assst reaablty shortly reses the basc concepts that are employe or the eelopment o the oe brge moel n DP oman. For the reaers who are not amlar wth the concept t s recommene to reer to the basc DP theory [5] snce t wll be ntensely use throughout the rest o the paper. The DP concept assumes that a tme-oman nearly-peroc waeorm x(τ can be represente on the nteral τ(t- T t] by a ollowng Fourer seres:

5 j s x( X ( t e ( t T t] ( where ω s =π/t an T the unamental pero o the waeorm. X (t s the th Fourer coecent n complex orm reerre to as a ynamc phasor an etermne as ollows: t j s X t x e T ( t T ( x ( where s the DP nex an angular bracets are use to enote DP-oman arables. In contrast to the tratonal Fourer Transormaton (FT these Fourer coecents are tme-aryng as the ntegraton nteral (wnow sles through tme. As t ollows the DP represents the araton n specc reuency component oer tme. The reure accuracy o the tmeoman arable approxmaton can be achee by approprate selecton o a DP set K or a partcular moellng tas. For example or c-le arables an sgnals the nex set only nclues the component = an or purely snusoal ones =. A ey actor n eelopng ynamc moels base on DP s the relaton between the erates o the arable x(τ an the erates o th Fourer coecents [5]: x t X ( t js X ( t ( t Ths can be ere usng ( ( an may be use n ealuatng the th phasor o tme-oman moel. Another mportant property o DP s that the th phasor o a prouct o two tme-oman arables can be obtane a the conoluton o corresponng DPs: xy x y ( The propertes ( an ( play a ey role when transormng the tme-oman moels nto DP oman. Algebrac manpulatons n ths paper wll also explot the ollowng property o real unctons x(τ: X ( X * t ( t (5 where the notaton * enotes a complex conjugate. III. UNCONTROLLED DIODE-BRIDGE RECTIFIER A ey objecte o ths paper s eelopment o a computatonally ecent moel that can represent the DB behaour n both balance an unbalance supply contons nclung lne aults. The unctonal leel moellng oes not

6 reure conseraton o swtchng behaour []. Hence our stuy s base on the non-swtchng moel that s reewe n ths Secton. A three-phase oe brge recter s well-ocumente ac/c conerter an t s shown n Fg.. In general case t s supplemente by an output c lter L c C an a ront-en nuctance L s representng eeng cables ac choes etc. The c loa s shown as an eualent resstance R L. Ths conerter employs uncontrolle oes; hence the swtchng nstants are etermne by the crcut conton exclusely. The tme-oman non-swtchng DB moel uner balance contons s well eelope an ocumente n many publcatons or example [7 ]. Doe brge recter c DC lter as L s a a D D D 5 L c bs b b c C R L cs c c D D 6 D Fg. Three-phase oe brge recter Uner the symmetrcal balance supply the recter termnal oltages can be represente as a cos( t abc b m cos( t / (6 c cos( t / where m s the oltage magntue ω s the supply electrcal reuency an φ s the ntal phase angle. The unamental o oes swtchng uncton then s gen as ollows []: Sa cos( t S abc Sb cos( t / (7 Sc cos( t / The swtchng uncton (7 enes the nput-output relatons or the oe brge: T c S abcabc m (8 I S (9 abc abc c where I abc =[ a b c ] T - an nput ac current n ector-matrx orm c an c recter output c oltage an current. As t ollows rom (9 an (7 the nput current s n phase wth the nput oltage (6 an ts magntue s:

7 m c ( In a presence o ront-en nuctance L s the c-ln oltage (8 reuces ue to the commutaton eect []. Normally ths eect s taen nto account by ntroucng n (8 an atonal term corresponng to a oltage rop across the resstance o r alue: r 6 L s ( In the ollowng paper sectons we wll also ntensely use the oltage (6 expresse as a space ector: a e b j / j / ce ( It s conenent to analyse three-phase power electronc crcuts n synchronously rotatng rame[] []. Dene the rame such that t rotates wth the gr electrcal reuency ω. Any three-phase arable abc can be expresse n the rame such that: T abc ( where T s a transormaton matrx: cos t cos( t cos( t T ( sn t sn( t sn( t Combnng (8-( (- ( yels the ollowng: c (5a m c (5b where an are an components o oltage ector (. Snce the nput current unamental s n phase wth the nput oltage ts an components are as ollows: cos ; sn (6 m m Uner the balance supply φ s eual to the ntal phase o the nput oltage an can be ere rom oltage - an - components: tan (7 Euatons (5-(7 comprse the relatons between the DB nput oltage/current expresse through ther components an

8 the output c oltage/current. These euatons wll be employe or the eelopment o DB moel n the DPs oman. I. DYNAMIC PHASOR DB MODEL DEELOPMENT In ths Secton the non-swtchng DP-oman DB moel s eelope. We wll start wth establshng how to map the generally unbalance supply oltage ector ( expresse n terms o synchronous rame rom tme oman nto the reuency oman o DPs an how the corresponng DPs can be ere rom tme-oman alues o nual phase oltages a b c or both balance an unbalance contons. Then the DB nput-output relatons (5 wll be transorme nto DPs an the DP-oman DB moel wll be assemble usng the ere relatons. A. Input oltages an current n ynamc phasors oman In general cases the DB termnal oltage can be represente by: cos( t =abc (8 where a b an c are phase oltage magntues an φ a φ b φ c ther phase angles. Applyng the Euler ormulae each o these components can be rewrtten as: e j e jt e j e jt a b c (9 The DPs o phase oltages can be ere applyng the enton ( an selectng the DPs set as = snce (8 nclues only the unamental component: * j j e e a b c ( Combnng euatons ( (8 (9 an re-arrangng the terms results n the ollowng relaton: j / j / jt * * j / * j / a b e c e e a b e c e jt e ( It s clearly seen that the two rght-han se terms ene the poste an the negate seuences o nput oltage ector. The latter wll appear uner unbalance phase oltages only. In terms o synchronous rame ene by ( the oltage ector ( can be ere as: j e jt j / j / jt * * j / * j / a b e c e e a b e c e ( Re-arrangng the rght-se terms results n:

9 jt j j e j ( where arables an can be calculate as: Re Im Re Im j / j / a b e c e j / j / a b e c e * * j / * j / a b e c e * * j / * j / a b e c e (a (b (c ( The - an - axes components o the nput oltage ector an can be ere rom ( by separaton o real an magnary parts: cos t cos t sn t sn t (5 The ollowng mportant conclusons shoul be mae analysng the result gen by (5: uner balance contons the oltage -components become c-le: = an = ; the supply oltage s unbalance the rame components o the oltage ector wll also nclue the secon harmoncs; the DP set K n orer to represent the supply oltage n rame uner both balance an unbalance contons shoul nclue zero an n harmoncs.e. K (6 The mappng o an nto DPs s establshe n Table I. The current an can be ere n the same manner. TABLE I. DYNAMIC PHASORS FOR DB INPUT OLTAGE AND CURRENT IN SYNCHRONOUSLY ROTATING FRAME arable Dynamc phasors = = ( j / ( j / I ( I ji / I ( I ji /

10 B. DC-ln oltage n ynamc phasors oman In ths Secton we wll transorm the DB oltage nput-output relaton (5a nto the DP oman. The man challenge s ue to a non-lnear nature o ths euaton that maes the rect applcaton o DP enton ( non-analytc. The approach we propose n ths paper expans (5a nto a Taylor seres wth respect to an ollowe by the transormaton o the truncate seres nto the DP oman. Conserng the DB output oltage as a uncton o two arables an as ollows: ( c (7 Approxmatng (7 by the Taylor seres reures selecton o the operaton pont. The natural choce s the par { } uner balance contons. Hence the operatng pont s ene as { } accorng to (5. The Taylor expanson o (7: c (! (! (! (! 5 (! (8 where are constants epenng on the selecte operaton pont. These can be calculate usng ( (7 an are gen n Appenx I. The seres (8 can be conerte nto the reuency oman ater sutable truncaton. Snce an nclue harmoncs up to the secon orer n our moel we truncate r -orer an hgher terms n (8. From (8 n the balance conton ( = an = the c-oltage c wll be constant an eual to whch s assocate wth the nput oltage poste seuence. Uner unbalance contons the negate seuence wll appear an sturb the oes swtchng uncton (7. The mpact o the negate seuence can be represente as a sturbance to c (7 n a orm o the secon harmonc appearng accorng to (5 n both an. Applyng the DP nex set (6 an employng the conoluton property ( to the truncate Taylor seres o (8 the DPs o c-ln oltage c are ere as ollows: * * * * c 5 ( (9a c (9b The DPs an are gen n the Table I n a preous Secton. Hence the DPs or c-ln oltage are ully ene.

11 C. Accountng or the c-oltage rpple The conerter output oltage calculate usng (9 exhbts only c-component uner balance contons an nclues the n harmonc uner unbalances. For unctonal-leel smulaton stues ths may be sucent. Howeer there s a room or conenent moel mproement by ncluson nto a moel the 6 th harmonc on the c se oltage that exhbts uner balance contons n the recter uner conseraton []. Ths secton emonstrates how ths component can be ntrouce nto the DP-base moel reure. Uner the balance operaton the 6 th harmonc n the c oltage s ue to the 5 th an 7 th harmonc n the swtchng uncton [] that can be gen as: cos(5 t 5 cos(7t S a 57 S b 57 S c 57 cos(5 t 5 cos(7t cos(5 t 5 cos(7t (a (b (c Combnng ( wth (8 (9 an ( the magntue o the 6 th harmonc n c oltage c6 an ts phase angle φ c6 are ere as ollows: c6 5 c6 6tan / 7 ( ( In a way smlar to ( one can ere the corresponng DP: c jc6 6 c6e ( Ths extra DP can be ae to the preously ere set (9 to represent the DB c-ln oltage n the DP oman. The tme-oman alue o c-ln oltage can be calculate usng the enton (: c cos t m jt ( t ( t e e sn t ( c c c where ={6}. Hence the c-ln oltage ( nclues the c component as well as the n an the 6 th harmoncs. Uner balance contons the n harmonc s absent. Howeer as the unbalance or lne ault occurs t becomes the omnant harmonc component.

12 D. Recter ac currents The lnear relatonshp between the magntue o the current ector m an the c-ln current c was gen n (5b an can be transorme nto DP oman as ollows: m c (5 The ac nput currents o the recter are epenent on the c loa current whch s etermne jontly by the c-ln oltage c an by the loa tsel. In the propose moel the c current s ere rom ts tme-oman alue measure at the moel output an then conerte nto the DP. The DP nex or c or lnear loas shoul be chosen accorng to c.e. ={6}. Howeer usng the ollowng euaton c c (6 allows us to ao calculatng o c an c 6 wth the DP enton ( an thus no cumbersome calculaton o t t-t xe -jωt t s neee. Wth (6 all the c-ln current normaton wll be nclue n c. The luctuaton o c wll be relecte to the unamental DPs abc through c an ths wll be llustrate later n the paper. From (5 the same DP set = wll apply to m. The DP or the nput current - an -axs components can be ere usng (6 as ollows: m cos (7 m sn The man challenge n calculaton o (7 eals wth the establshng the DPs or non-lnear unctons snφ an cosφ. As n preous Secton the approxmaton s execute by a Taylor seres. The non-lnear terms are expresse a - an - axes oltage components: cos ( (8a sn ( (8b Selectng the operaton pont { } usng the same technue as that ealng c n (7 eres * * * * cos h h h h (a 5

13 cos h h (b * * 5 * * sn g g g g (c sn g g ( where h an g are constant coecents that epens on a selecte operaton pont an gen n Appenx I. Hence the DPs or the non-lnear unctons (8 are ere. Then the DPs or DB nput current - an -components can be easly ere usng (7. Fnally the moellng tas nstea o - an - current components reures the current n a orm o three-phase ac arables then the DPs (7 hae to be transorme nto abc rame as escrbe n the ollowng Secton. E. DQ to ABC transormaton n ynamc phasors oman Applyng the conoluton property to the abc/ transorm gen by ( an ( yels: * c b a T T ( where T - s the generalze nerse matrx T: sn( cos( sn( cos( sn cos t t t t t t T ( Calculaton o T - reures the DPs or cosωt an snωt. As ω s constant the DP nex or these wll nclue the only component = hence employng ( one can easly ere:.5.5 j j j j j T ( Usng ( the DP o nput currents n abc rame can be ere easly. Furthermore applyng enton ( the corresponng tme-oman current alues can be calculate as well. F. Moel assemblng The euatons ere n the sectons aboe can be combne together n orer to bul the DP-oman moel o the uncontrolle brge recter as t s shown n n Fg..

14 a b c a b c Transorm En.( a b c /abc Transorm ( Current Relaton (7 ( an Table I c (6 c c + C ω a b c DP calculator En. ( a b c Transorm ( an Table I oltage Relaton (9 ( an Table I c 6 ( c + Fg. DP moel o the DB recter The current low shown n Fg. llustrates the mappng o c nto the AC currents abc. Wth (6 all the normaton n c s resere n the DP c. Ths maes abc a uncton o the tme-aryng current c an allows the harmonc characterstcs n the ac currents to be represente by the unamental DPs abc. The DP moel shown n Fg. can be use n EPS smulatons wth no nee or the user to unerstan DP theory. In case stues when the DB recter s e through an nucte lne a small capactor shoul be ae at nput termnals to ao moel state reunancy an relate numercal problems. For the user who wshes to bul the entre EPS moel n the DP oman the moel oes not nee the nterace blocs (coloure n grey n Fg. snce the DP arables are alreay aalable an the DB moel can be rectly nterace to the other EPS moel blocs.. MODEL ERROR ANALYSIS In the oe-brge DP-oman moel ere n preous Sectons we employe lnearzaton o (7 an (8 aroun the operaton pont etermne by the oltage poste seuence; the negate seuence was treate as a sturbance. Wth the ncrease o the negate seuence ths conseraton may result n some moellng error that s a subject o analyss n the current paper Secton. Conser the moellng error ene as: T c_ BM % ( t T t T c_ BM c_ DPM t where c_bm s the c-ln oltage calculate by the benchmar moel (ths s a ull swtchng moel n three-phase coornates an c_dpm s the same calculate by the DP moel. The analyss has been perorme or the scontnuous

15 moe case escrbe n the preous Secton wth the set o phase oltages as: a 8cos( t sn( t 8 cos( t / (5 b b b c Ealuaton o the error ( has been perorme by seres o smulatons aryng the magntue o phase b oltage b rom to an ts phase b rom to π. Thus all balance an unbalance contons are coere. The results o the error analyss are epcte n Fg. as a uncton o b b an the unbalance actor λ. The latter s ene as the rato o between the magntues o negate-seuence oltage n an the poste-seuence one p an wrtten as: n / p (6 5 % 8% 6% % % p/ p/ p b (ra Fg.. Analyss o the DP moel: (a - D presentaton o = ( b b λ= λ( b b; (b D graph showng = ( b λ wth b= It s clearly seen that the moel error greatly epens on unbalance actor λ. Ths s reasonable snce wth the ncrease o λ the sturbance rom the negate seuence becomes more seere an thus the operaton pont shts rom { } as selecte or lnearzaton o (7 an (8. The moellng error has a maxmum when b =5π/6 an b = at these alues λ taes an extreme alue. In practce the negate seuence becomes omnant.e. λ> the operaton pont shoul be selecte usng the negate seuence { } conserng the poste seuence as a sturbance. Thus the moel can be conenently aopte or ths case. Hence n the error analyss we only hae to conser the cases when λ[] (below the horzontal lne λ= n Fg. (b. For the case o balance operaton λ= the moellng error s less than %. The lne-to-lne ault can be consere as the case when λ= an b =; the moellng error or ths eent s less than % as ollows rom Fg. (b. For the case o phase-to-groun ault ( b = the moellng error s ery small an s always less than %. The moellng accuracy analyss conrms that the eelope DP moel s well sutable or the unctonal moellng o uncontrolle oe-brge recter. The accuracy o the DP moel wll be expermentally alate an scusse n the

16 ollowng secton. I. MODEL EXPERIMENTAL ALIDATION A test rg shown n Fg. wth parameters gen n Appenx III has been employe or the moel ercaton. A programmable source Chroma II has been use to apply balance an unbalance oltages to the DB. The moel has also been ere n contnuous an scontnuous current moes as reporte below. In the smulaton moel small capactors (C=e-8F were ae between the oe brge an the lne nuctors as suggeste n Secton I.F aboe. as R s L s a a c D D D 5 L DC + Sw CHROMA II Programmable power supply bs cs b c b c C c R L R L D D 6 D Fg.. The expermental rg A. Contnuous moe In the experment the ntal alue o balance supply oltage was set to alues a = b = c =/5Hz to ao source protecton trggerng when mtatng the ault. The phase A oltage s set to zero at t=.s n orer to perorm the lneto-groun ault to the system. A contnuous current moe s ensure wth a heay loa on c-ln se by swtchng on Sw an parallelng R L an R L. The smulaton an experment alues o c-ln oltage nput currents an the nuctor current are compare n Fg. 5. Fg. 5. The c-ln oltage (a nput current (b an nuctor current (c n contnuous moe wth phase A loss at t=.s As can be seen n Fg. 5(a uner the balance conton t<.s c has the 6 th harmonc component. Wth the 6 th

17 harmonc nclue n the DP moel the results rom the experment an DP moel are well matche. Ater the lne-togroun ault the c-ln oltage luctuates at ouble reuency. In ths case the n harmonc s nclue n the DP moel an the results rom the experment an smulaton are well matche. The DP moel an experment results or abc are shown n Fg. 5(b an they are well matche beore an ater the ault occurs. The DC-ln current c s shown n Fg. 5(c an t changes rom a contnuous current moe (CCM to a scontnuous moe (DCM when the ault occurs. It can be seen that uner CCM an DCM contons the DP moel emonstrates goo perormance n both cases. B. Dscontnuous moe In ths case the swtch Sw s open hence a large loa resstance R L results n a small c-ln current whch maes the nuctor current scontnuous. The experment starts wth balance oltages set a = b = c =8/5Hz ollowe the loss o phase A supply ( a = at t=.s. The smulaton an experment alues o c-ln oltage nput currents an the nuctor current are compare n Fg. 6. Fg. 6 The c-ln oltage (a nput current (b an nuctor current (c n scontnuous moe wth phase A loss at t=.s As can be seen n Fg. 6(a c has a 6 th harmonc uner the balance conton an a n harmonc uner the lne ault conton. In both cases the results rom the DP moel are well matche wth experment. The ac se currents rom the experment an the DP moel shown n Fg. 6(b are well matche beore an ater the ault occur. The c-ln current c shown n Fg. 6(c ncates that the recter wors uner the DCM uner both normal an aulty contons. The results rom the experment an DP moel are well matche n both cases. Followng the results n ths secton the propose moel successully ere uner balance an unbalance operaton n both contnuous an scontnuous current moe.

18 II. COMPUTATION TIME STUDIES Snce the accuracy o the DP moel has been alate ths secton wll ocus on the assessment o the computatonal perormance o the eelope DP moel. For ths purpose erent moellng technues hae been apple to an example EPS gen n Fg.7 an the CPU tme taen or smulaton run has been compare. The smulate scenaro nclues balance an unbalance operaton as well as contnuous an scontnuous DB moes as etale below. The system parameters are gen n the Appenx III. a b DP ault njector [6] a b c Fault Rc Lc Cc Rc Lc Cc L C R R A B C DP RLC cable [6] DP RLC cable [6] a b c DP DB moel L C R R Cable Cable ω c A B C Fault Rc Lc Cc Rc Lc Cc Cable Cable a b DP DB c c moel - b ABC to DP nterace a DP/nt CRU moel L C R R D Q = ω RLC cable n [] ault njector [] RLC cable n [] n DB moel [] L C R R Fg. 7 EPS moels or comparson: (a abc moel; (b DP moel; (c DP/nt moel; ( moel. The ollowng EPS moels hae been eelope an compare: - A three-phase EPS wth a swtchng DB recter moel (eal swtches llustrate by Fg.7(a an reerre to as an abc moel; - An EPS wth all the elements are DP moels as shown n Fg.7(b. There are no three-phase tme-oman arables n ths moel an t s reerre to as DP moel; - An EPS n whch only the DB s moelle usng DPs wth three-phase nteraces as shown n Fg.7(c. The EPS s seen by the user as a three-phase system an the DP ormulatons are not sble. Ths s terme the DP/nt moel; - An EPS wth all the elements n rame [ 5] as shown n Fg.7(. A. Moels comparson uner balance operaton The smulaton scenaro n ths case assumes the DB operaton n scontnuous moel ollowe by mpact o c loa mang the c-ln current contnuous at t=.s. The c-ln oltage an current or all the moels are shown n Fg.8(a. As one can see the DP an DP/nt moels match ery well to abc moel n both scontnuous an contnuous

19 moe. In contrast the moel elers correct aerage c an c alues o c only n contnuous moe howeer n scontnuous moe t emonstrates a sgncant screpancy compare to moels. Ths s because n scontnuous moe when c becomes zero the capactor scharges to the loa mpactng the aerage alue o c howeer the moel oes not nclue the eect o atonal chargng recee ue to the 6 th harmonc. The CPU tme taen by erent moels s compare n Table II. The DP an DP/nt moels n balance operaton are more than tmes aster than abc moel howeer the moel s much aster. The cumulate CPU tme urng the smulaton s shown n Fg.8(b. Note that when the contnuous moe occurs the abc moel slows own ramatcally but DP an DP/nt moels mantan the smulaton spee. Ths s because n scontnuous operaton there are no commutatng peros n DB an only two oes conuct at any nstant howeer n contnuous moe there are commutaton peros reurng rom the soler reucton o ntegraton tme-steps thus slowng own the smulaton. (a (b (c Fg. 8 Smulaton result comparson between erent moels. Balance contons (a c-ln oltage an current; (b ( computaton tme; Unbalance contons (c c-ln oltage; ( computaton tme

20 TABLE II. CPU TIME TAKEN FOR BALANCED SCENARIO SIMULATION Moel: abc DP DP/nt CPU tme taen: Perormance nex: 8 Summarzng n balance contons the DP an DP/nt moels are slower than moel ue to complexty an ncluson o hgher harmoncs howeer the moel s much less accurate especally n scontnuous moe. B. Lne-to-lne ault The unbalance regme s smulate as a lne-to-lne ault (phases A an B are shortene a the ault resstor R ault =e-ω at t=.s. The swtch Sw s on to ensure the contnuous moe pror the ault. The c-ln oltage s shown n Fg.8(c. As one can conclue the DP an DP/nt moels match well to abc moel uner both balance an unbalance regmes. The moel n the balance conton (t<.s relects the oltage aerage alue. The CPU tme taen by erent moels s compare Table III. As one can conclue the ecency o the moel s lost n unbalance operaton. Ths s clearly seen rom the cumulate CPU tme graph n Fg.8(: the slope corresponng to moel s the largest when t>.s. In contrast the DP-base moels mantan ther smulaton spee an or the entre scenaro they are tmes aster o compare to the abc moel an tmes aster than the moel. Table III. CPU tme taen or unbalance scenaro smulaton Moel: abc DP DP/nt CPU tme taen: Perormance nex: Hence the results aboe hae conrme a ery goo ecency o the eelope DB moel compare to other moellng technues n partcular or smulaton stues o EPS n unbalance/lne ault contons. III. CONCLUSION Ths paper extens the DP concept nto moellng a three-phase oe recter. Conserng the act that the 6 th harmonc s the omnant harmonc on the DC-ln se uner balance contons an the n harmonc or unbalance contons the eelope DP moel achee hgher accuracy by embracng the n an 6 th harmoncs n the DC-ln arables. Compare to the tratonal aerage moel ( moel the eelope DP-base moel s more accurate n the balance conton an much more tme-ecent n the unbalance conton. The accuracy o eelope more s alate usng smulaton an experment. In aton our analyss shows that the maxmum error s less than % at the most seere

21 unbalance contons. The moel can be conenently nterace wth other stanar three-phase tme-oman moel an thus can be wely use n the accelerate electrc power system smulaton stues. APPENDIX I. TAYLOR SERIES COEFFICIENTS IN (8 ( ; ( ; ( ; ( ; ( ; ( 5. APPENDIX II. TAYLOR SERIES COEFFICIENTS IN (9 ( h ; h ( ; h ( ; ( h ; ( h ; h ( 5 ; ( g ; g ( ; g ( ; ( g ; ( g ; g ( 5. APPENDIX III. EXPERIMENTAL RIG PARAMETERS Power source: Chroma moel 65. DB recter: IRKD-. AC supply reuency 5Hz. Input mpeance (ac cable an choes: L s =mh R s =.Ω. DC-ln: C=μF L c =μh. Loas: R L =Ω R L=9Ω. APPENDIX I. EPS IN FIG.8 PARAMETERS Cable: R c =.Ω L c =μh C c =pf; DC-ln: L=μH C=5μF R L =Ω R L =9Ω; AC gr reuency Hz. ACKNOWLEDGMENT Ths research was conucte n the rame o CleanSy JTI Project a FP7 European Integrate Project -

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