A simple modulation approach for interfacing three-level Neutral-Point-Clamped converters to the grid

A Simple Modula ion App oach o In e acing Th ee-Le el Neu al-Poin -Clamped Con e e s o he G id An onio Ven osa-Cu illasa,∗, Pablo Mon e o-Robinaa, Fede ico Cues aa, F ancisco Go dilloa aDepa amen o de Ingenie ´ıa de Sis emas y Au om´a ica. Escuela T´ecnica Supe io de Ingenie ´ıa. Uni e sidad de Se illa. Camino de los Descub imien os, s/n. 41092 Se illa Abs ac Mul ile el con e e s a e nowadays an enabling key in he in eg a ion o elec ic powe in o he g id as hey in oduce less dis o ion and, hus, hey a e mo e compli- an wi h he g id s anda ds, among o he bene i s. A well-known opology is he h ee-le el Neu al-Poin -Clamped (NPC) whose con ol equi es o deal wi h he capac- i o s ol age unbalance. This pape p esen s a modula ion app oach whe e he injec ion o a common componen in he modula ed ol age is s udied in o de o achie e such ol age balance. An op imiza ion p oblem ha , apa om he ol age balance, aims he lowes numbe o commu a ions and can be sol ed e y e icien ly wi h up o i e compu a ions o he cos unc ion is o mula ed. The main ad an ages o he p oposed modula ion s a egy a e i s simplici y and i s lexibili y, since i is also alid o unbalanced g id condi ions and, wi h li le added complexi y, o low (and e en ze o) powe ac o condi ions. Simula ion esul s unde unbalanced g id condi ions a e p o ided in o de o show i s alidi y unde his scena io. The s a egy is e alua ed and compa ed wi h a space- ec o -based app oach in an expe imen al se up, yielding simila o al ha monic cu en dis o ion (4.4%), a 30% educ ion in he numbe o commu a ions, and be - e ol age balance pe o mance o lowe powe ac o condi ions. Keywo ds: Synch onous ec i ie applica ion, neu al-poin -clamped (NPC) con e e , mul ile el con e e , g id-connec ed ec i ie , ol age balancing, op imum ze o-sequence ol age injec ion. 1. In oduc ion Elec onic powe con e e s a e nowadays he mos used de ices o in e acing sys- ems ha abso b o gene a e ene gy wi h he g id. Thanks o hei capabili y o ans- o ming he cha ac e is ic o he elec ical ene gy, whe he i i is ac, dc o in e mi en , ∗Co esponding au ho Email add esses: [email p o ec ed] (An onio Ven osa-Cu illas ), [email p o ec ed] (Pablo Mon e o-Robina), [email p o ec ed] (Fede ico Cues a), [email p o ec ed] (F ancisco Go dillo) P ep in submi ed o Ene gy Feb ua y 28, 2020 hey handle he powe low o ul ill he equi emen s o hei applica ions. One o he5 ypical uses comp ehends dealing wi h he ac na u e o he g id o ei he injec /abso b ac i e powe o/ om a dc load/sou ce o injec /abso b eac i e powe in o he g id. No ma e he aim, he quali y o he handled powe o g id-connec ed applica ions is c u- cial [1, 2]. In such applica ions, mul ile el con e e s o e se e al ad an ages compa ed wi h he basic wo-le el con igu a ion: lowe cu en dis o ion and smalle ol age s ess10 o he semiconduc o de ices among o he s [3]. These ea u es make hem e y appeal- ing o medium and high powe applica ions whe e he ol age bounda ies o he powe de ices a e e y limi ing [4]. Among he b oad choices o di e en mul ile el con e e opologies, h ee-le el diode-clamped con e e —also known as Neu al-Poin -Clamped con e e (NPC)—has been widely accep ed in he indus y and many applica ions ha e15 been de eloped o i [5–7]. Howe e , he use o h ee-le el NPC con e e s gi es ise o a new con ol p oblem wi h espec o he wo-le el case: he dc-link capaci o s ol age balance. This is an inhe en ea u e in mul ile el con e e s as mo e capaci o s appea when he numbe o le els inc eases and hei ol age migh no emain he same unde no mal ope a ing condi ions. The e o e, i is c ucial o conside how his unbalance can20 be compensa ed and o design a s a egy o achie e he capaci o s ol age equalisa ion. Conside ing he h ee-le el NPC opology, he unbalance issue appea s due o une en cu en s ha go h ough each capaci o , i.e. cu en en e ing/exi ing he neu al poin (NP) [5]. A oiding he cu en going h ough he NP will, howe e , neglec he bene i s o he h ee-le el con e e . Tha is why plen y o esea ch ha e been ca ied ou as he25 solu ion o his p oblem is no unique. A common app oach is o sepa a e he con ol sys em in o wo s ages: he powe low con ol and he modula ion. The i s one is simila in mul ile el and wo-le el opologies and de ines some e e ence signals o he second s age such ha he powe low is handled as desi ed. The second s age is in cha ge o commanding he swi ching de ices in o de o implemen he p e ious con ol30 signals in o he sys em, hus de e mining which capaci o s a e used. In his s age, he e a e emaining deg ees o eedom ha can be used o he capaci o s ol age balance objec i e. In a g id-connec ed powe con e e , he modula ion can be ca ie -based pulse wid h modula ion (CBPWM) [8] o space- ec o pulse wid h modula ion (SVPWM) [9], among35 o he s [10, 11]. Ca ie -based app oaches use one o se e al ca ie signals o modula e he desi ed ou pu signal o smalle equency. The ou pu signals o modula e a e sampled a an speci ic a e— e e ed as swi ching equency ( sw)—, and hese sam- ples a e compa ed wi h he ca ie signals—usually saw oo h o iangula shaped— whose alue goes along i s ull ange e e y pe iod o sw. Thus, he swi ching signals40 change in he in e sec ions o hese samples and he ca ie . In his kind o modula ion o h ee-le el, h ee-wi e g id-connec ed con e e s, he e exis s a common componen called ze o-sequence ha can be added/sub ac ed o he h ee ou pu signal samples simul aneously wi hou a ec ing he modula ed phase- o-phase ol age bu allowing o modi y he dc-link poin s ha a e connec ed. This ac is specially in e es ing o con ol-45 ling he capaci o s ha a e cha ged/discha ged [12]. Al e na i ely, SVPWM gene ally uses he nea es h ee- ec o (NTV) s a egy [7, 13] o he nea es h ee- i ual- ec o (NTV2) one [14] o achie e modula ion. The main idea is o conside simul aneously he ou pu signals o he h ee phases by using se e al conca ena ed swi ching ec o s, al hough, he e exis edundan swi ching ec o s ha p o ide he same ou pu ol age50 2 alue bu hey use di e en le els. The e o e, he exploi a ion o his deg ee o eedom also makes possible o modi y he connec ed poin s [6]. In summa y, in o de o deal wi h he modula ion and he capaci o s ol age balancing, CBPWM uses ca ie s and he ze o-sequence componen , while SVPWM uses conca ena ed swi ching ec o s, in- cluding edundan ones [15]. In spi e o being di e en echniques, se e al wo ks in he55 li e a u e ha e ound equi alences be ween he ze o-sequence injec ion and he edundan swi ching ec o s use [16]. In ea ly CBPWM app oaches, he ze o-sequence componen was compu ed as a hi d ha monic signal ha , in summa y, en ails he injec ion o a cu en in o he NP ha co ec he unbalance [17]. This app oach, howe e , equi es o know he phase o he60 g id in addi ion o cos ly igonome ic compu a ions. Nowadays, se e al app oaches exis ha conside ably educe he compu a ional bu den, sol e he balancing issue and mi iga e he low- equency ol age oscilla ions o he NP [12, 16, 18–22]. Pape [20] sum- ma izes he di e en kind o ze o-sequence ol age injec ion con ol in o: 1) Cons an injec ion; 2) Cons an cha ging injec ion; and 3) Maximum cha ging injec ion. Re e ence65 [19] p esen s wo app oaches based on he ze o-sequence ol age injec ion, which exhibi hei simplici y in e ms o compu a ional bu den compa ed wi h o he exis ing me hods a he expense o inc easing he numbe o commu a ions. F om he NP cu en pe - spec i e, in [22], an analysis o he ze o-sequence compu a ion is made in o de o ob ain he ela ionship be ween i and he neu al cu en . In his way, a p ecise algo i hm is70 p oposed ha compu es he exac alue o he ze o-sequence o achie e he desi ed neu- al cu en . Howe e , such accu acy may no be necessa y o a eal applica ion as he ipple a s eady s a e ha would esul om no doing so could be negligible. La e , in [12], an analysis o he p e ious pape is de eloped esul ing in a simple lowcha wi h less compu a ion bu den. In [23], he equi alences among se e al neu al poin cu en 75 con ol app oaches ha achie e balancing wi h CBPWM and SVPWM a e p esen ed. Howe e , none o he app oaches p esen ed in [12, 19, 22, 23] aims o educe as much as possible he numbe o commu a ions a he same ime ha he ol age balance is achie ed. The e exis s a con ollabili y issue in he capaci o s ol age balancing wi h ze o-80 sequence injec ion when he ope a ing condi ions ha e a powe ac o di e en om uni y. Depending on his alue and he modula ion index, he ze o-sequence deg ee o eedom may no be enough o achie e ol age balancing a all ime [18]. In his ega d, [18] analyses he ze o-sequence injec ion wi h an addi ional ca ie in CBPWM such ha his issue is o e come. The addi ional ca ie in oduces he use o ano he le el wi hin85 he swi ching pe iod, which inc eases he o al numbe o commu a ions. Rega ding SVPWM, i s ly, i is necessa y o loca e he ol age ec o wi hin he h ee-le el Space Vec o hexagon, i.e. he sec o whe e i is loca ed, in o de o know which swi ching ec o s, among he 27 a ailable o NPC con e e s, should be used o modula e i . Then, hese swi ching ec o s a e sequenced and hei s du y a ios a e90 compu ed. The swi ching ec o selec ion and sequencing, howe e , is no unique and plen y o c i e ia can be used [9, 14, 15, 24, 25]. Indeed, his is one o he ad an ages o SV-based app oaches as many deg ees o eedom a e exhibi ed, al hough i is mo e complex o implemen compa ed wi h CBPWM app oaches [26] as se e al s eps ha e o be aken in o accoun : sec o loca ion, ec o s selec ion, du y a ios compu a ion,95 3 ec o s sequencing and ha dwa e implemen a ion. Pape [9] p esen s a s aigh o wa d applica ion o SVPWM o h ee-le el NPC ol age and cu en in e e s whe e hese asks a e achie ed. This applica ion does no conside capaci o s ol age balancing, hough. In NPC con e e s, he swi ching ec o s a e classi ied as la ge, medium o sho . The i s ones do no c ea e unbalance; he second ones injec one phase cu en in o he NP; while100 he hi d ones come in pai s ha p oduce he same e ec on he sys em powe low con ol bu injec he opposi e cu en in o he NP. The e o e, o achie e capaci o s ol age balancing and wi hou changing he swi ching ec o posi ions, he edundan swi ching ec o s can be used. In his way, [15] p oposes a SVPWM wi h balancing capabili ies by combining he medium ec o s— ha c ea es unbalances—wi h he edundan ec o s.105 Al e na i ely, [25] p esen s a simila app oach bu aiming a educ ion in he numbe o commu a ion. The e exis s a modi ica ion o SVPWM known as i ual space ec o (VSV) [14]. I basically c ea es inhe en swi ching ec o s in he hexagon by combining ixed du y a ios o su ounding swi ching ec o s. Thus, pape [24] modi ies he VSV in o de o achie e balancing capabili ies by conside ing he edundan swi ching ec o s in110 he c ea ion o swi ching ec o s. This app oach will be e e ed as mVSV in his pape . As i was men ioned o CBPWM app oaches, he e exis s an issue in he balancing when low powe ac o ope a ing condi ions a e gi en. Using only he edundan swi ch- ing ec o deg ee o eedom o achie e balancing migh be no enough depending on he modula ion index and he powe ac o alue [25]. Thus, none o he ci ed SVPWM115 app oaches [9, 14, 15, 24, 25] p esen s an s aigh solu ion o his issue. Compa ing SVPWM wi h CBPWM, ca ie -based app oaches a e usually much easie o implemen han hose based on Space-Vec o -Modula ion (SVM). Tha is why in [16] an analysis o he SVPWM app oach is p esen ed in o de o ob ain a me hod based on CBPWM ha p o ides he same esul s. Thus, an algo i hm based on se e al checks and120 he knowledge o he sec o whe e he ol age ec o is loca ed wi hin he space ec o hexagon is p oposed such ha he ze o-sequence componen is de e mined. In his pape , a CBPWM app oach is conside ed whe e he ze o-sequence de e mina- ion is exp essed as an op imiza ion p oblem based on he minimiza ion o a piecewise- linea unc ion o one a iable. In his way, he de i a ion o he ze o-sequence compu a-125 ion algo i hm is simple and easie o ollow han he p e ious app oaches. In con as o [12, 14, 15, 19, 22–24], he minimum amoun o commu a ions is achie ed by keeping one phase a one le el while he o he wo commu e be ween wo adjacen le els each. Fo his, i su ices o e alua e he p oposed cos unc ion jus in he cases when one phase s ays a one le el and choose he minimum o hem. The e o e, a simple op i-130 mal CBPWM app oach, which has led o an in e na ional pa en [27], ha educes he commu a ions and ac i ely sol es he unbalance is p esen ed along wi h i s jus i ica ion. Rega ding unbalanced g id condi ions, he p oposed s a egy does no ely on any balanced-g id- ol age assump ion. The e o e, he s a egy may be sui able o his con- di ions keeping i s simplici y compa ed wi h o he CBPWM app oaches [28].135 This pape also conside s an enhancemen o he base algo i hm o imp o e he bal- ancing capabili ies. This is conside ed o hose ope a ion poin s whe e using he ze o- sequence injec ion wi h one phase ixed a one le el du ing he sampling pe iod and he o he wo phases a wo adjacen le els each is no enough o achie e balancing a all ime. Thus, one phase is allowed o use he h ee le els while he o he wo ollow140 4 Figu e 1. Schema ic diag am o he h ee-phase h ee-le el NPC ec i ie . he same p inciple han he base algo i hm. This modi ica ion equi es o compu e a new exp ession o he cos unc ion ha s ill keeps he p ope y o being piece-wise linea . In con as o [18], no addi ional ca ie is equi ed, and he enhancemen is only applied un il he capaci o s ol age di e ence is small enough, educing he numbe o commu a ions a s eady-s a e.145 On he whole, he p oposed me hod is easie o ollow and simple o implemen han he SVM-based app oaches [9, 14, 15, 24, 25] as no swi ching ec o sequencing and la ge du y compu a ions a e equi ed. The ze o-sequence alue and he du y a ios, o bo h he p oposed algo i hm and i s enhancemen , can be compu ed s aigh o wa dly wi h no added di icul y in compa ison wi h o he CBPWM app oaches [16, 18, 28], a 150 he same ime ha bo h look o he lowes amoun o commu a ions while achie ing capaci o s ol age balancing. Besides, simula ion esul s show i s capaci o s ol age balancing alidi y unde unbalanced g id ol ages scena ios, due o i s inhe en lack o assump ion o balanced g id ol ages. Fu he mo e, he enhancemen seeks co e ing hose ope a ion poin s wi h powe ac o lowe han 1, whe e, a some ins an s, he base155 algo i hm ails o achie e capaci o s ol age balancing. The esul s a e u he alida ed by expe imen s unde di e en ope a ing condi ions. The ou line o he pape is as ollows: Sec ion 2 p esen s he con e e dynami- cal model along wi h i s sys em a iables and componen s, and he equi ed con olle s co esponding o he s age p e ious o he modula ion; Sec ion 3 in oduces he modu-160 la ion s age whe e his pape ocuses on and whe e he main con ibu ion is p esen ed, in addi ion o he ad an ages wi h espec o o he me hods; besides, an enhancemen is in oduced in o de o imp o e he ol age balancing capabili ies unde lowe powe ac o ope a ing condi ions; Sec ion 4 depic s he expe imen al esul s and compa isons wi h mVSV [24]; and Sec ion 5 d aws some inal conclusions.165 5 2. Model and Con ol o he Sys em As i has been said be o e, his pape ocuses on g id-connec ed h ee-phase, h ee- le el NPC con e e s wo king as ec i ie wi h a esis i e load in he dc side (Fig. 1). The sys em should handle he ac i e powe o ma ch he one abso bed by he load such ha he dc-side ol age is equal o i s e e ence, which is speci ied by170 he use . The sys em should also ack he speci ied eac i e powe acco ding o he powe ac o condi ions equi ed by he applica ion. Wi h e e ence o Fig. 1, poin s {a, b, c}a e he ou pu s o he con e e whose ol age le els a e se by swi ching signals {Si 1, Si 2} o i={a, b, c}. The g id ol ages a e deno ed by si, o i={a, b, c}, while phase cu en s a e ii. Induc i e il e s a e conside ed {L1, L2, L3}175 whose induc ances a e assumed o ha e he same alue L. Two capaci o s cons i u e he dc-link side {C1, C2}wi h he same capaci ance alue Cand whose ol ages a e exp essed by a iables { c1, c2}. To exp ess he unbalance, a iable dis de ined as c1− c2. The h ee connec ion poin s o he dc-link a e e e ed o as {p, o, n}and he esis i e load R is connec ed in pa allel o i . In his way, dc-link ol age dc is equal o c1+ c2.180 Based on [29], a model o his con e e in αβγ coo dina es—ob ained by ans- o ming he abc a iables using he powe -in a ian Cla ke ans o ma ion—is ob ained. An a e aged model o e a swi ching pe iod is conside ed by using du y a ios dkj o k={α, β, γ}and le el j={p, o, n}—i.e. his du y a ios exp ess he amoun o ime in a swi ching pe iod ha ou pu o componen kis connec ed o le el j. In his way, Ldiα d = sα −(dαp −dαn) dc 2−(dαp +dαn) d 2(1) Ldiβ d = sβ −(dβp −dβn) dc 2−(dβp +dβn) d 2(2) Cd dc d = (dαp −dαn)iα+ (dβp −dβn)iβ−2 dc R(3) Cd d d = (dαp +dαn)iα+ (dβp +dβn)iβ. (4) The objec i e in he ollowing sec ions will be o p esen a con ol algo i hm which p o- ides he alues o dkj o k={α, β, γ}as a unc ion o he measu ed a iables si, ii, c1 and c2. No ice ha dij in abc can be compu ed di ec ly om dkj by applying he in e se powe -in a ian Cla ke ans o ma ion. No ice also ha du y a ios o componen γdo no appea in (1)–(4), and he e o e hei alues can be conside ed as deg ees o eedom ha can be used o o he con ol objec i es. Simila ly, du y a ios dko do no appea in he model bu hey can be compu ed om he es o du y a ios in abc ame acco ding o he cons ain s X j={p,o,n} dij = 1, dij ∈[0,1]; i={a, b, c}.(5) To in oduce he con olle design, and inspi ed by (1)–(2), h ee con ol a iables 6 a e de ined uα=dαp −dαn (6) uβ=dβp −dβn (7) uγ=dγp −dγn, (8) whe e uγis di ec ly ela ed o he homopola componen o he dij a iables o i= {a, b, c}. By in oducing hese a iables in o he cu en dynamics (1)–(2) and assuming ha a iable dis small enough o be neglec ed, a simpli ied model is ob ained Ldiα d = sα −uα dc 2(9) Ldiβ d = sβ −uβ dc 2.(10) No ice ha hese equa ions ecall he cu en dynamics o a simple wo-le el con e e wi h uαand uβas con ol inpu s. Mo eo e , a iable uγdoes no appea in he cu en dynamics and he e o e, i is le as a deg ee o eedom. A his poin , he con e e equi es a con ol algo i hm o handle he powe aking in o accoun he model dynamics. In his ega d, wo cascaded185 con olle s a e usually used [30]: an ou e loop o compu e he powe equi ed p o keep dc owa ds i s e e ence dc; and an inne as e loop in cha ge o d i e he ac i e and eac i e powe s p, q close o he p e ious e e ence and he desi ed eac i e powe q , espec i ely. In spi e o he ac ha hese s ages a e no he aim o his pape , he algo i hms used in he simula ion190 and expe imen s a e p esen ed he e o he sake o comple eness. 2.1. Dc-link Vol age Regula ion Loop The o al capaci o s ol age o he dc-link ( dc) should each i s e e ence ( dc) a s eady s a e. To achie e so, a PI con olle is used o compu e he equi ed amoun o powe (p )[30], p =kdc p( dc 2− 2 dc) + kdc iZ 0 ( dc 2− 2 dc)dτ, (11) whe e {kdc p, kdc i}a e con ol pa ame e s o be uned, which can be done using any app oach exis ing on li e a u e [30, 31]. A e his, he e e ences o he cu en con ol {i α, i β}a e ob ained by applying he ins an aneous powe heo y o {p , q },195 whe e q is he use -de ined eac i e powe e e ence. 2.2. Cu en Con olle Once {i α, i β}a e compu ed, a cu en con olle s age is implemen ed such as o educe he cu en acking e o {e iα,e iβ}={i α−iα, i β−iβ}as much as possible. In 7 his way, conside ing (9)-(10),a non-ideal p opo ional- esonan con olle [32] uned a he g id equency is implemen ed. GP Rω(s) = kp+2k ωcs s2+ 2ωcs+ω2(12) uα uβ=2 dc −GP Rωge iα e iβ+ sα sβ ,(13) whe e {kp, k }a e, espec i ely, he p opo ional and esonan con ol gains; ωcis he cu -o equency o he low-pass il e implemen ed in o he esonan pa ; ωis he esonan equency—in his case, uned a he g id one (ωg= 2π g id)—; and { sα, sβ}200 a e he g id ol ages in he αβγ e e ence ame. Bea in mind ha his pape does no ocus on his s age and ha any o he exis ing con olle in he li e a u e (e.g. [21, 33]) could be used ins ead as long as i p o ides he modula o wi h he alues o uαand uβ. 3. Modula ion S age wi h Vol age Balance Capabili ies205 This sec ion p esen s he main con ibu ions o his pape by p o iding an algo i hm ha compu es he du y a ios dij such ha he esul s o he cu en con olle , namely {uα, uβ}, a e implemen ed op imizing he numbe o commu a ions a he same ime ha he capaci o s ol age a e balanced. In he ollowing subsec- ions he base algo i hm, i s e alua ion unde unbalanced g id condi ions and210 an enhancemen o imp o e i s balancing capabili es a e shown. 3.1. Ze o-sequence Componen Compu a ion As i has been said in he p e ious sec ion, he ou pu o he cu en con olle is he alue o uαand uβ. In o de o con e hem o abc coo dina es he e is a emaining deg ee o eedom, namely he homopola componen uγ. Fo simplici y, in he ollowing,215 a iable xis in oduced as x˙= uγ/√3. In his way,   ua ub uc = 2 3   1 0 1 √2 −1 2 √3 2 1 √2 −1 2−√3 2 1 √2     uα uβ uγ =    q2 3uα+x −q1 6uα+1 √2uβ+x −q1 6uα−1 √2uβ+x     ˙=   ηa+x ηb+x ηc+x , (14) whe e a iables ηa, ηb, ηcha e been de ined. These a iables a e known once uαand uβ a e compu ed by he cu en con olle . No ice ha he maximum and minimum alues o xa e imposed by he cons ain ui∈[−1,1] o i={a, b, c}—ob ained om (5)220 and ui=dip −din—in conjunc ion wi h (14) esul ing in x∈[−1−min(ηa, ηb, ηc),1− max(ηa, ηb, ηc)] ˙=[xmin, xmax]. Mo eo e , i can be seen ha he easibili y condi ion xmin ≤xmax is equi alen o he equi emen ha ec o (uα, uβ) is inside he SVM hexagon. Equa ion (14) can be in e p e ed wi h he help o Fig. 2 [34]. To pu hings in225 con ex , in he le g aph o his igu e a e e ence ec o is shown in he usual space 8 ec o hexagon. This e e ence ec o co esponds o pa icula alues o uα, uβ ha , wi h he app oach used in his pape , a e gi en by (13). This e e ence ec o mo es as ime mo es on. In he same way, a iables ηa, ηb, ηcimplici ly de ined in (14) depend on ime. In s eady s a e, hey desc ibe a sinusoidal wa e as shown in he cen al g aph o 230 Fig. 2 whe e = 1co esponds o he same ins an han he e e ence ec o shown in he le g aph. The igh g aph co esponds o his ime ins an . In i , he ac ual disc e e le els o he h ee-le el con e e a e ep esen ed by he alues {−1,0,1}and he leaning lines depic he alues o ua(x), ub(x), uc(x) as a unc ion o he alue o x o be chosen. The dashed e ical lines ep esen he bounda ies xmin ≤x≤xmax. No ice also ha 235 o x= 0, ui=ηi. As ime ad ances, hese h ee lines mo e acco ding o he mo emen o ηiin he cen al g aph. One ad an age o he app oach can be spo ed on he igh g aph whe e he deg ee o eedom associa ed wi h he homopola componen uγ(o , analogously x) is explici , while i is no so e iden in he space ec o ep esen a ion o he le g aph. The black do s in he igh g aph will ha e an impo an ele ance below.240 An anima ion o he ime e olu ion o he h ee lines o he igh g aph o Fig. 2 and i s ela ionship wi h ηa, ηband ηcand he space ec o hexagon can be obse ed in he ollowing link: h p://g upo.us.es/ ep102/lines_anima ion.mp4. Figu e 2. Le g aph: a e e ence ec o in he Space Vec o ep esen a ion. Cen al g aph: e olu ion o ηi( ), i =a, b, c; he e ical line co esponds o he same ime ins an = 1 han he le g aph. Righ g aph: Rep esen a ion o equa ion (14) a = 1. Back o he capaci o s ol age unbalance dynamic o mula ion (4), i can also be ans o med o abc, esul ing in245 Cd d d =ia(dap +dan) + ib(dbp +dbn) + ic(dcp +dcn).(15) 9 The p oposed enhanced algo i hm coincides wi h he base one when a leas one alue o cos o he cases o Table 1 is nega i e. In he case when no poin s ge s nega i e alues, he enhancemen is implemen ed o dec ease i . The ol age balance365 capabili ies a e imp o ed bu a he cos o inc easing he numbe o le els used and, he e o e, he numbe o commu a ions and he swi ching losses. To a oid his, in con as o [18], and conside ing ha i is no necessa y ha dgoes o ze o bu i su ices ha i is small enough, a band o alue ζis de ined such ha only when he alue o | d|is ou side o i , he enhancemen is applied. Consequen ly, he370 inc eased numbe o commu a ions occu s only du ing he ansien pe iod when dis la ge and, hus, he base algo i hm is used a s eady s a e. A b ie low cha wi h he enhancemen implemen a ion is plo ed in Fig. 8. Figu e 8. Flow cha o he implemen a ion o he algo i hm enhancemen 4. Expe imen al Ve i ica ion This sec ion aims o show he beha iou o he p oposed algo i hms in an expe imen al375 p o o ype o 12 kVA (Fig. 9). The ci cui and con ol pa ame e s o he con e e a e shown in Table 4, whe eas he pa ame e s ha change along he expe imen a e gi en in Table 3 oge he wi h he ime in e al whe e hey ake place. The sys em consis s o a g id-connec ed h ee-le el con e e con olled by a eal- ime a ge machine. The sys em powe a e and all componen s ha e been designed acco ding380 o he limi s o he a ailable in as uc u e. The sys em ac s as a ec i ie capable o injec ing o sub ac ing eac i e powe om he g id. Besides, he esis o connec ed o he dc-link uses a choppe ci cui o demand di e en powe a es which allows o emula e di e en esis ance alues. Conside ing his, he sys em is sui able o e alua ing he algo i hm p oposed in his385 pape . Se e al ope a ing poin s a e emula ed by modi ying use -inpu a iables: dc,q ; and ci cui pa ame e : R. Unless o he wise s a ed, q is assumed o be ze o o achie e uni y powe ac o . Fu he mo e, o p o ide a compa ison wi h o he published app oaches, he esul s o he expe imen s using a modi ied e sion o SVPWM app oach wi h balancing390 capabili ies [24] a e also included in he igu es— e e ed as modi ied i ual-space- ec o (mVSV) om now on. In he expe imen s, bo h dc-link and cu en con olle s a e kep he same o he mVSV app oach and he p oposed ones. 16 Figu e 9. Expe imen al p o o ype o he h ee- le el NPC ec i ie . Table 3. Expe imen Pa ame e s Va ia ion Time In e al R dc 0→0.8s120 Ω 700 V 0.8→1.5s60 Ω 700 V 1.5→2.2s60 Ω 700 →800 V 2.2→3.8s60 Ω 800 V 3.8→4.5s120 Ω 800 V The phase cu en s, ins an aneous ac i e powe and swi ching s a es o phase aa e plo ed in Fig. 10 o bo h he base algo i hm and he mVSV app oach. I can be seen395 ha he base algo i hm shows a e y simila beha iou compa ed wi h mVSV excep o he numbe o commu a ions. Conside ing he esul s p esen ed in Fig. 10, he base algo i hm yields 265 commu a ions—numbe o 1-le el ansi ions— pe g id pe iod, while he mVSV algo i hm yields 375. Rega ding he cu en dis o ion, Fig. 11 depic s he ha monic spec um and he o al400 ha monic dis o ion (THD) alue o he cu en s o Fig. 10 o he base algo i hm and he mVSV one. I can be seen ha he p oposed algo i hms and he mVSV ha e e y simila cu en dis o ion, he e o e his p oposal does no wo sen he cu en quali y Table 4. Expe imen Pa ame e s Pa ame e Value Pa ame e Value G id equency g id 50 Hz Sampling equency ( s) 10 kHz G id Vol age sa, sb, sc 230 VRMS Swi ching equency ( sw) 10 kHz Fil e Induc ance L2 mH Cu en con ol P gain kp5 Capaci ance C3300 µF Cu en con ol R gain k 100 dc ol age con ol P gain kdc p0.05 dc ol age con ol I gain kdc i1 Enhancemen pa ame e 0.1 Cu en con ol esonan cu -o equency ωc1 ad/s Enhancemen band ζ10 V 17 Figu e 10. Expe imen al esul s a s eady s a e wi h dc = 800 Vand R= 60 Ω: ( op) base algo i hm, (bo om) mVSV algo i hm: (le ) h ee phase cu en s a s eady s a e wi h dc = 800 Vand R= 60 Ω; (cen e ) e olu ion o sys em powe (p) and powe e e ence (p ) along di e en ope a ing poin s; ( igh ) swi ching s a e o phase aou pu when compa ed wi h o he space- ec o -based algo i hm. In summa y, a s eady s a e he base algo i hm p esen s less numbe o commu a ions, and hus i gene a es less405 losses, wi h simila cu en dis o ion when compa ed wi h he mVSV app oach [24]. In e ms o balancing capabili ies, wo expe imen al es s s a ing om an unbalanced si ua ion wi h di e en ope a ion poin s a e conside ed as depic ed in Fig. 12: a) uni y powe ac o , and b) ze o powe ac o . Figu e 12a) shows a compa ison o he balancing pe o mance be ween he p oposed algo i hm and he mVSV, once he cu en con olle 410 is a s eady s a e, wi h dc = 700 V,R= 120 Ω and uni y powe ac o . I can be seen ha he balanced si ua ion is achie ed almos a he same ime o bo h app oaches, making he p oposed algo i hm sui able o balancing pu poses when compa ed wi h o he exis ing solu ion. On he o he hand, in o de o exhibi he ele ance o he enhanced algo i hm, Fig. 12b) is depic ed o powe ac o equal o ze o. I shows415 how, unde his condi ion, he balancing capabili y o he base algo i hm is slowe ed, inc easing he ime i akes o each he balanced condi ion. Simila ly, he mVSV algo i hm p esen s much slowe beha iou in e ms o balancing capabili ies. On he con a y, by implemen ing he enhanced algo i hm, he balancing capabili y is conside ably imp o ed whene e | d|> ζ. In his way, a sui able app oach o balancing420 pu poses unde di e en ope a ing poin s is p oposed. No ice ha a s eady s a e, he base and he enhanced algo i hm show simila beha iou as he di e ences be ween hem 18 Figu e 11. Expe imen al esul s: Ha monic spec um o cu en s o phase ashown in Fig. 10 and co esponding THD alue o he p oposed algo i hms and he mVSV one only appea when | d|> ζ. Ne e heless, his modi ica ion en ails an inc emen in he amoun o commu a ions as depic ed in Fig. 13. This inc ease is due o he occasional use o a hi d le el du ing a ansien pe iod whe e | d|> ζ and he ope a ing condi ions yield425 cos >0 o he base algo i hm. In his ega d, du ing his ansien s age, he enhanced algo i hm yields 627 commu a ions pe g id pe iod acco ding o Fig. 13. Ne e heless, his inc eased numbe only occu s du ing a sho ansien (less han 0.1 seconds in Fig. 12b) un il a iable dis wi hin he band |ζ|. 5. Concluding Rema ks430 In his pape , he capaci o s ol age balance o g id-connec ed h ee-le el NPC con e e s has been add essed as an op imisa ion p oblem based on a cos unc ion di ec ly ela ed o he balance dynamic equa ion. The p oposed s a egy minimizes he numbe o commu a ions a he same ime ha he ol age balance is ackled. Fu he mo e, gi en ha he p oposal only depends on435 he no malised ou pu ol age, i can be implemen ed wi h any kind o cu en /powe con ol. Besides, an enhancemen in he algo i hm assis s in he capaci o s ol age balancing when he p e ious app oach could yield la ge balancing imes. In his way, he capaci o s ol age balance is imp o ed o di e en ope a ion poin s.440 The alidi y o he p oposal has been es ed in an expe imen al se up and com- pa ed wi h a modi ied e sion o i ual space ec o modula ion wi h ol age balance capabili ies (mVSV). This compa ison shows ha he p oposed app oach and 19 Figu e 12. Expe imen al esul s: E olu ion o he capaci o s ol age s a ing om an unbalanced si ua ion: a) Base algo i hm compa ed wi h mVSV wi h dc = 700 Vand R= 120 Ω; b) Base algo i hm compa ed wi h he enhanced one and mVSV wi h dc = 700 V,R= in Ω and ζ= 10 V. Figu e 13. Expe imen al esul s: Swi ching s a e o phase aa s eady s a e o he base algo i hm wi h dc = 800 Vand R= 120 Ω (le ) and he enhanced one du ing a ansien s a e whe e d> ζ wi h dc = 800 Vand R= in Ω ( igh ). i s enhancemen achie e simila balancing capabili ies o a wide ange o powe ac o alues, wi h no de e io a ion o he cu en dis o ion (4.4%445 THD o mVSV and he p oposed algo i hm) bu wi hou he added di i- 20 cul y o implemen ing space ec o modula ion. Indeed, he implemen a ion complexi y is as simple as hose o ca ie -based. Besides, he numbe o commu a ions is educed by 30% a s eady-s a e in compa ison wi h mVSV. The e o e, a alid app oach o in e acing NPC ec i ie s wi h he g id is p esen ed. In450 addi ion, he ex ension o his app oach o h ee-le el NPC in e e s is s aigh o wa d conside ing jus he co esponding change o he cu en con ol and he di ec ion o he cu en s. Acknowledgemen s This wo k has been unded unde g an s MINECO-FEDER DPI2016-75294-C2-1-R455 and FEDER Andaluc´ıa US-1264655. 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