Control of D-STATCOM during unbalanced grid faults based on DC voltage oscillations and peak current limitations

UPCommons Po al del coneixemen obe de la UPC h p://upcommons.upc.edu/e-p in s Aques a és una còpia de la e sió au ho ’s inal d a d'un a icle publica a la e is a IEEE T ansac ions on indus y applica ions . URL d'aques documen a UPCommons E- p in s: h p://hdl.handle.ne /2117/113793 A icle publica / Published pape : A ash Khoshooei; Ja ad S. Moghani; Ignacio Candela; Ped o Rod iguez (2017) Con ol o D-STATCOM du ing unbalanced g id aul s based on DC ol age oscilla ions and peak cu en limi a ions. IEEE T ansac ions on indus y applica ions., Vol. PP, Iss. 99, p. 1-10. Doi: 10.1109/TIA.2017.2785289 Con ol o D-STATCOM Du ing Unbalanced G id Faul s Based on DC Vol age Oscilla ions and Peak Cu en Limi a ions A ash Khoshooei, Ja ad S. Moghani, Ami kabi Uni e si y o Technology Teh an, I an Khoshooei@au .ac.i , Moghani@au .ac.i , Ignacio Candela1, Ped o Rod iguez1,2 1. Technical Uni e si y o Ca alonia (UPC) 0822, Ba celona, Spain [email protected], p od i[email p o ec ed]pc.edu 2. Depa men o Enginee ing, Loyola Uni e si y Andalusia 41014, Se ille, Spain p od [email protected] Abs ac — The sa e ope a ion o g id connec ed powe con e e s du ing abno mal condi ion is a key issue in o de o gua an ee i s ope a ion and o a oid undesi ed ips. In his pape di e en con ol s a egies o he ope a ion o a D-STATCOM a e in oduced, whe e he e e ence cu en s a e de e mined in such a way ha no only none o he phase cu en s goes o e he limi s, bu also he DC ol age luc ua ions emain in sa e ope a ion limi . Fluc ua ing ac i e powe in e change, du ing unbalanced condi ion leads o DC ol age oscilla ion. Se e e unbalanced condi ion and small DC capaci o selec ion ( o mee he size and cos cons ain s) in ensi y he DC ol age oscilla ion. The e o e, he con ibu ion o his pape lays on he combina ion o he DC ol age oscilla ions and he cu en limi con ol. The e ec i eness o h ee p oposed con ol s a egies a e e i ied by simula ing a D-STATCOM ied o an indus ial dis ibu ion ne wo k. Mo eo e a scaled scena io has been ep oduced expe imen ally which shows ha he esul s cope well wi h he analy ical equa ions and he simula ion esul s. Index Te ms— Cu en con ol; DC ol age oscilla ions; D-STATCOM; nega i e sequence; eac i e powe ; sa e ope a ion. I. INTRODUCTION G id codes wo ldwide a e becoming mo e es ic i e day by day [1]. The inc easing ins alla ion o Dis ibu ed Gene a ion Powe Supplies (DGPS), based on powe con e e s b ings he oppo uni y o u ilizing hei unique ea u es. G id suppo ing unc ionali ies, e en unde se e e ansien condi ions, such as g id aul s is an ou s anding capabili y o DPGS. Nowadays, when a g id aul occu s, g id connec ed powe con e e s a e equi ed no only o emain connec ed o he g id bu also hey mus educe hei ac i e powe deli e y and inc ease he eac i e powe injec ion o suppo ing he g id [2]. Nume ous esea ch wo ks ha e epo ed di e en powe con ol s a egies o DGPS o shun connec ed powe elec onics con e e s, like D-STATCOMs, o ope a ing unde abno mal g id condi ions[3]-[5]. Since mos o he g id aul s a e unbalanced aul s, se e al esea ch wo ks ha e done o ol age p o ile egula ion by injec ing unbalanced eac i e cu en s o boos he posi i e sequence ol age as well as minimizing he nega i e sequence componen . Conside ing he impedance o he Poin o Common Coupling (PCC), a con ol algo i hm is p oposed o PCC ol age egula ion in [6]. The e ec i eness o STATCOMs o enhance he s abili y ma gin o a ixed speed wind powe plan s unde unbalanced aul s is p esen ed in [7]. Th ee eac i e cu en injec ion s a egies o in luence on posi i e and nega i e ol age sequences a e minal o wind powe plan s ha e de eloped in [8]. Di e en s a egies o injec ing a coo dina ed combina ion o posi i e and nega i e sequence cu en s in D-STATCOMs a e in oduced in [9]-[11]. In a aul condi ion, he PCC ol age and injec ed cu en s a e unbalanced. The e o e, he in e ac ion be ween posi i e and nega i e sequences in he ol age and hei coun e pa s in he injec ed cu en esul s in ac i e powe luc ua ions and consequen ly DC link ol age oscilla ions. Rega dless o he con ol s a egy objec i e, a sa e ope a ion o he con e e om he pe spec i e o maximum ins an aneous phase cu en s, as well as he maximum ins an aneous o e ol age o DC bus because o luc ua ions is c i ically impo an . Su passing ei he o he a o emen ioned limi s would gi e ise o an undesi ed con e e ipping. Con olling he maximum phase cu en o a STATCOM encoun e ing an unbalanced g id aul s was in oduced in [12]. Respec ing he maximum phase cu en c i e ion, [13] has s udied he maximum ac i e and eac i e powe deli e y o a DGPS. Maximum phase cu en cons ain in low ol age ide h ough o a DGPS and eac i e cu en injec ion a e espec i ely p esen ed in [14] and [15]. DC ol age oscilla ion issue is no add essed in he abo e men ioned esea ch wo ks. In associa ion wi h DC ol age oscilla ions, he e ec s o unbalanced supplying ol age on a con en ional con olled D-STATCOM and i s e ec s on DC ol age oscilla ions is discussed in [16]. Fo a 48 pulse STATCOM esponsible o he egula ion o posi i e and nega i e sequence ol ages, [17] p oposes o use a single phase in e e , in se ies wi h he DC link capaci o o elimina ing he DC ol age oscilla ions du ing he aul pe iod. Elimina ion o DC ol age oscilla ions in a ansmission le el STATCOM [18] is ackled by in oducing a second o de e m o he angle con olle o he con e e . DC ol age oscilla ion educ ion in a HVDC sys em is discussed in [19]. On he o he hand, DC link capaci o s play an impo an ole in size, cos and ailu e a e o he con e e . Wi h he indus y end o use high eliable as well as cos e ec i e DC link capaci o s, high eliable ilm capaci o s a e used ex ensi ely [20]. Howe e , o an a o dable p ice, hei ene gy densi y is low. Op imal DC side capaci o design which copes wi h s ingen eliabili y and cos cons ain s mo es owa d minimiza ion o he capaci o size [21]. In a con e e wi h a educed size DC link capaci o , he amoun o DC ol age oscilla ions in aul condi ion is qui e high. Mo eo e , in a ol age sou ce con e e wi h a ixed modula ion algo i hm, high amoun o oscilla ions supe imposed on he DC ol age, in oduce non-cha ac e is ics ha monics in he ou pu ol age spec um [22]. The e o e, i is necessa y o in ol e he DC ol age oscilla ion cons ain in accompany wi h peak cu en limi a ion in calcula ion o e e ence cu en . A con ol algo i hm which conside s bo h c i e ia, DC bus ol age oscilla ions limi as well as phase cu en limi a ion, has no been s udied in deep. Mo eo e , up o now li le wo k has been done on he limi a ion o DC ol age o D-STATCOMs acing se e e unbalanced si ua ions. In his esea ch, h ee s a egies o eac i e powe injec ion a e in oduced which ul ill no only he phase cu en limi a ion bu also DC ol age oscilla ion cons ain , o ensu e a secu e ope a ion o D-STATCOM while iding h ough he aul . This wo k is an ex ended e sion o [23] wi h u he simula ions and mo e discussion. This h ee eac i e powe injec ion s a egies a e named: A e age Ac i e Reac i e Con ol(AARC), Balanced Posi i e Sequence Con ol (BPSC) and Posi i e Nega i e Sequence Con ol (PNSC). Fo each s a egy, a couple o eac i e powe e e ence alues a e calcula ed which sa is y he peak cu en limi a ion and maximum DC ol age oscilla ions c i e ia espec i ely. By compa ing his e e ence alues, he inal eac i e powe e e ence is chosen, which will espec bo h he DC ol age oscilla ion and peak cu en limi a ion. The o ganiza ion o he pape is as ollows. Sec ion II discusses he basics o he h ee di e en eac i e powe con ol s a egies. The de i a ion o ac i e powe luc ua ions and consequence DC ol age oscilla ions a e p esen ed in sec ion III. Sec ion IV is de o ed o calcula ion o maximum phase cu en s. The o e all con ol sys em is discussed in sec ion V and he pe o mance o a D-STATCOM, connec ed o a weak indus ial ne wo k expe iencing aul condi ion is analyzed in sec ion VI. Finally he expe imen al e alua ion o a labo a o y scaled D-STATCOM conside ing bo h limi ing c i e ia is shown in sec ion VII, jus be o e he conclusions. II. DIFFERENT REACTIVE POWER CONTROL STRATEGIES In an a bi a y h ee phase ne wo k wi h unbalanced a iables { , } i ∈ and supposing a h ee wi e sys em as well as he a ailabili y o a ∆ connec ion in one o he windings o in e acing ans o me , as shown in Fig. 1, he ze o sequence ol ages and cu en s a he poin o i Fig. 1. S uc u e o a D-STATCOM connec ed o he g id connec ion o he con e e o he g id will be elimina ed. The e o e, by using a cons an ampli ude Cla k T ans o ma ion, we can w i e: ( ) 1 1 2 ( ) 3 3 3 ( ) 1 1 ( ) 0 ( ) 3 3 a b c α β − − = −                         (1) whe e { } ( ), , , i i a b c ∈a e phase a iables ( ol ages and cu en s), u he mo e each a iable in s a iona y e e ence ame can be decomposed in o a couple o balanced se s o posi i e(+) and nega i e( − ) a iables as shown below: ( ) ( ) ( ) α α α + − = + (2) ( ) ( ) ( ) β β β + − = + (3) In ac , i is e y common o use a couple o in-quad a u e 90o shi ed ec o s o de elop he eac i e powe de ini ion: ( ) ( ) ( ) α α α + − ⊥ ⊥ ⊥ = − (4) ( ) ( ) ( ) β β β + − ⊥ ⊥ ⊥ = − (5) Fig. 2 ep esen s sys em a iables in he s a iona y e e ence ame. F is he o a ing space ec o and ⊥ F is i s in-quad a u e coun e pa . + F and − F a e he posi i e and nega i e sequence componen s espec i ely. Fig. 2. Vec o ep esen a ion in s a iona y e e ence ame Acco ding o Fig. 2, he ime exp essions o he posi i e and he nega i e sequences o bo h he eal and in-quad a u e ec o s can be w i en as: ( ) ( ) ( ) ( ) ( ) ( ) . ( ) ( ) ( ) ( ) ( ) ( ) .cos( ) .cos( ) .cos( ) 2 .cos( j F F F F β α αβ αβ αβ ω θ ω θ π ω θ ω                                 =                                 + + + − − − +++ ⊥⊥⊥ −−− ⊥⊥⊥ + + − − + + − = + + − + + − − + F F F F .cos( ) 2 .cos( ) . 2 .cos( ) ) .cos( ) 2 F F j F F π ω θ π ω θ ω θ π π θ ω θ π                                                 + + − − + + −− − + − − + − + + − −− + − (6) In case o using cons an ampli ude Cla k T ans o ma ion, ac i e and eac i e powe s can be w i en as: 3 ( ) . 2 p= i (7) 3 ( ) . 2 q⊥ = i (8) whe e , ⊥ and i a e ol age , in-quad a u e ol age and cu en ec o s espec i ely. In A e age Ac i e Reac i e Con ol (AARC) s a egy, ac i e and eac i e cu en componen s a e o ien ed ac oss he ol age space ec o and i s in-quad a u e ec o espec i ely. The modulus o and ⊥ emain cons an h oughou g id pe iod. O ien a ion o e e ence cu en ac oss he posi i e sequence ol age leads o a balanced cu en injec ion in Balanced Posi i e Sequence Con ol (BPSC). A se o unbalanced cu en s a e injec ed o he g id in Posi i e Nega i e Sequence Con ol (PNSC). The e e ence cu en ec o is di ec ed in a way ha cancel ou he oscilla ions in he ins an aneous powe s injec ed in o he g id. De ails o AARC, BPSC and PNCS schemes and hei cha ac e is ics a e gi en in [24] and he e e ence cu en s a e shown in Table I. * P and * Q a e ac i e and eac i e powe se poin s and V + and V − a e he ol age posi i e and nega i e sequence ampli udes espec i ely. Table I. Re e ence cu en ec o s o di e en powe injec ion schemes Scheme Re e ence Cu en Vec o AARC * * * 2 2 2 2 ( ) ( ) ( ) ( ) ( ) ( ) 2 3 2 3P Q V V V V ⊥ + − + − = + + + i (9) BPSC * * * 2 2 ( ) ( ) ( ) ( ) 2 3 2 3P Q V V + + ⊥ + + = + i (10) PNSC * 2 2 * * (2 3) ( ) ( )] ( ) ( ) [P Q V V + − + − ⊥ ⊥ + − = − + + − i (11) III. EFFECT OF DIFFERENT REACTIVE POWER CONTROL STRATEGIES ON DC BUS VOLTAGE OSCILLATIONS This sec ion is de o ed o he calcula ion o ac i e powe luc ua ions in he h ee a o emen ioned eac i e powe con ol s a egies conside ing unbalanced ol age condi ion. Fu he mo e, a s ep by s ep de i a ion o he DC ol age oscilla ions, based on he p inciple o ene gy conse a ion is p esen ed. Finally, some hin s o p ope DC capaci o selec ion a e p esen ed. A. Ac i e Powe Fluc ua ions Acco ding o he ins an aneous powe heo y [25], he ac i e powe luc ua ions a he e minal o a powe con e e could be w i en as: ( ) (3 2 )( ) p i i i i α α α α β β β β + − − + + − − + = + + +  (12) Fo ex ac ing he ol age sequence componen s used in (12), he main p inciples o se e al esea ch wo ks, such as [26] is conside ed. I could be in e ed om (12) ha he ac i e powe luc ua ion is a consequence o he di e en sequence ol ages and cu en s in e ac ion. In o he wo ds, o a balanced ol age and pu e balanced posi i e sequence cu en injec ion, he e is no powe luc ua ion. A he o he ex eme, when he ol age is balanced and he con e e only injec s a nega i e sequence cu en o he g id, he ampli ude o he powe luc ua ions eaches i s maximum alue. The majo pa o con e e cu en is alloca ed o nega i e sequence cu en . Hence, he 2nd and 4 h e ms in (12) a e negligible. In con as , 1s and 3 d e ms a e signi ican and he powe luc ua ion eaches i s maximum. This condi ion is e y p obable when he D-STATCOM wo ks in a load cu en balancing mode. Unde unbalanced g id aul condi ions, when he D-STATCOM wo ks in g id ol age suppo ing mode, posi i e sequence ol age is always highe han he nega i e sequence ol age, he e o e, he s a egies which injec mo e nega i e sequence cu en , p oduces highe ac i e powe luc ua ions. In (12) he cu en componen s a e gene a ed by he con ol block wi h espec o he eac i e powe injec ion scheme. Re e ence cu en s o each a o emen ioned s a egy could be achie ed by inse ing he a bi a y ol ages o (6) in o (9)-(11). The D-STATCOM ohmic losses compa ed wi h i s a ed V.A is insigni ican so he e e ence ac i e powe is almos ze o ( * 0 P ≈ ). Inse ing he calcula ed e e ence cu en s as well as he ol age componen s in (12), he ac i e powe luc ua ions o di e en schemes a e in oduced in Table II, whe e λ is he Vol age Unbalance Fac o (VUF) as a measu e o se e i y o ol age imbalance which is de ined as: V V λ − + =(13) Rega dless o he AARC ha p esen s no luc ua ions in ac i e powe , wo la e schemes expe ience a 2nd o de componen luc ua ions wi h he ampli udes in luenced om eac i e powe se poin s and he ol age unbalance ac o . Table II. Ac i e powe luc ua ions o di e en schemes Scheme Ac i e Powe Fluc ua ion s AARC ( ) 0 p =  (14) BPSC ( ) . *sin(2 ) p Q λ ω θ θ + − = + − (15) PNSC 2 2 *. ( ) sin(2 ) 1 Q p λ ω θ θ λ + − = + − − (16) B. DC Capaci o Vol age Oscilla ions Neglec ing he con e e losses and acco ding o he ene gy conse a ion heo y, he DC link powe abso p ion ( ( ) c p ) is he same as he inpu powe , he e o e: ( ) ( ) c p p =  (17) The DC link capaci o ol age is: ( ) ( ) c c c V = +  (18) whe e his ol age is a composi ion o a cons an componen ( c V ) and a luc ua ing componen ( ( ) c  ), as a esul : ( ) ( ) ( ). ( ) ( ). c c c c c d p i C d = = (19) by subs i u ing (18) in (19) : ( ) ( ) ( ) ( ) ( . ( ). ) . . c c c c c c c d d d p C V C V d d d = + ≈    (20) in he abo e equa ion, he second e m in compa ison o he i s one is negligible he e o e, by in eg a ing (20) an equa ion o he DC ol age oscilla ions is a ained: 1 1 ( ) ( ) ( ) . . c c c c p d p d C V C V = = ∫ ∫   (21) DC ol age oscilla ions, p opo ionally ela e o he ac i e powe luc ua ions. In con as , highe he DC ol age alue o capaci ance, lowe is he DC ol age oscilla ions. Using (14)-(16) in (21), a supe imposed second o de oscilla ions on he a e age DC alue o all he a o emen ioned con ol schemes a e lis ed in Table III. I is clea ha he highe ol age unbalance ac o , he highe is he DC ol age de ia ion. The de ia ion abo e he a e age alue is mo e impo an han he unde going ol age. O e ol age has de imen al e ec s on he semiconduc o swi ches and he DC link capaci o , migh Table III. DC ol age oscilla ions o di e en schemes Scheme DC Vol age Oscilla ions AARC ( ) 0 c =  (22) BPSC *. ( ) cos(2 ) 2 . c c Q CV λ ω θ θ ω + − − = + − (23) PNSC 2 *. ( ) cos(2 ) . (1 ) c c Q CV λ ω θ θ ω λ + − − = + − − (24) ac ua e he DC o e ol age p o ec ion uni . Fo a speci ied pe missible DC o e ol age, he maximum eac i e powe can be de e mined. DC ol age oscilla ions ampli ude o a ypical 4MVA D-STATCOM, deli e ing a ed and 50% o a ed V.A, wi h espec o he ol age unbalanced ac o is p esen ed in Fig. 3. I is i idly shown ha i he eac i e powe e e ence is no educed he DC ol age de ia ion would no be ole a ed. Beside, he ac i e powe luc ua ions is no occu ed in AARC s a egy and i is he ines s a egy o p e en ing he DC ol age oscilla ions. On he o he hand, PNSC s a egy su e s om high DC ol age de ia ion in la ge VUFs and i he eac i e powe se -poin is no educed p ope ly i migh esul in con e e ipping. C. DC Capaci o Selec ion o Mee he C i e ia The main c i e ia o DC capaci o sizing is o be su e abou he D-STATCOM capabili y in he egula ion o ol age du ing ansien s. Di e en esea ch wo ks ha e p esen ed di e en me hods o sizing he capaci o wi h ega ds o ansien pe o mance equi emen s [27]-[28]. Howe e , aul ide h ough pe o mance o he D-STATCOM and he e ec o capaci o size on he DC ol age oscilla ions is no conside ed in p e ious wo ks. The main p inciple o all he me hods used o capaci o sizing lays on he ac ha he change in he capaci o ’s s o ed ene gy should be equal o a mul iplica ion o he D-STATCOM a ed powe ( a ed S ) by a speci ied pe iod o ime, e.g. 0.5-1 cycle. A ypical ela ion is : 2 2 ,max ,min 1 . 2( ) . c c s a ed an C k S T V V− = (25) whe e ,max c V and ,min c Va e he maximum and he minimum pe missible alues o DC ol age. s k is a coe icien ha de e mines he sha e o D-STATCOM con ibu ion o a speci ic ansien ime, an T . Fo limi ing he ampli ude o he DC ol age oscilla ions, a le el o immuni y could be de ined like: ( ) . c c k V ≤ (26) Fig. 3. DC ol age oscilla ions o a ypical 4MVA D-STATCOM 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 VUF DC Vol age Oscilla ion (PU) DC Vol age Oscilla ion(%) wi h espec o VUF AARC BPSC (Q a ed ) BPSC (0.5*Q a ed ) PNSC (Q a ed ) PNSC (0.5*Q a ed ) S=4MVA C=20mF Vdc=1150 V whe e ( ) c  is he ampli ude o DC ol age oscilla ions and k is he allowed pe cen age o nominal DC ol age. In he AARC s a egy, DC ol age oscilla ions a e ze o and he alue o capaci ance is de i ed om (25). By inse ing he oscilla ions ampli ude om (23) in (26), he minimum capaci ance o mee DC ol age oscilla ions o BPSC is: 2 *. 2 . . c Q C k V λ ω ≥(27) In he same way by combining (24) and (26) o PNSC, he minimum capaci ance alue is calcula ed as: 22 *. . . (1 ) c Q C k V λ ω λ − ≥(28) I is e iden ha he capaci ance alue in e sely ela e o squa e alue o he DC ol age. Fu he mo e, he capaci ance is a unc ion o ol age unbalanced ac o ( λ ). The maximum alue o he calcula ed capaci ance among (25) and (27)-(28), mee s bo h he ansien esponse equi emen as well as limi a ion o DC ol age oscilla ions. Conside ing size and cos cons ain s, selec ing a capaci o ha main ains he ampli ude o 2nd o de oscilla ions below he le el o immuni y o all alues o λ is no sensible. The e o e, in he design s age, he capaci o is sized o an assumed maximum alue o λ . I in p ac ice an unbalanced condi ion wi h la ge λ appea s, he con olle calcula es he e e ence eac i e powe in a way ha DC ol age oscilla ion does no su pass he immune alue. IV. MAXIMUM PHASE CURRENT IN DIFFERENT REACTIVE POWER CONTROL STRATEGIES Conside ing an unbalanced ol age condi ion, i he eac i e powe se poin is no educed, i is likely ha cu en s in one o mo e phases pass o e hei nominal alues and he o e cu en p o ec ion o he con e e would be ac i a ed. This sec ion concen a es on he de i a ion o new eac i e powe se poin o each s a egy in which he maximum o phases cu en s kep in a sa e egion acco ding o he nominal cu en . Assuming a se o a bi a y equa ion o phase cu en s in na u al (abc) ame as: ( ) cos( ) ( ) cos( ) ( ) cos( ) a a a b b b c c c i I i I i I ω ϕ ω ϕ ω ϕ + + +         =             (29) o each s a egy he magni ude o maximum phase cu en acco ding o he posi i e and nega i e sequence ol age componen s a e ex ac ed and hen he pe missible amoun o e e ence eac i e powe is calcula ed. A. Maximum Phase Cu en o AARC S a egy Conside ing (9), he e e ence cu en o AARC is: 1 1 1 * * * i . . .b b b i i α α β β β α β β α α + − ⊥ + − ⊥       +   = = = =         −− −               (30) whe e 1 b is an ins an aneous suscep ance and de ined as: * 2 2 1 ( ) ( ) ( ) 2 3 Q b V V + − =+(31) pu ing he ime domain posi i e and nega i e ol age componen s om (6) in (30), magni ude o maximum phase cu en a e calcula ed as: 2 2 1 ( ( 2 . cos( ) ) ) a bI V V V V δ π + − + − = + + + (32) 2 2 1 ( ( 2 . cos( 3 ) ) ) b bI V V V V δ π + − + − = + + − (33) 2 2 1 ( ( 2 . cos( 3 ) ) ) c bI V V V V δ π + − + − = + + + (34) whe e δ θ θ + − = + which is a ailable a he ou pu o sequence ex ac ion block. The maximum sa e ampli ude o he phase cu en s is he nominal one. Fo a speci ic unbalanced condi ion he maximum pe missible eac i e powe in which none o he phase cu en s su pass he limi a ion could be de e mined. By inse ing (13) and (31) in (32) o (34), he maximum allowed eac i e powe as a unc ion o posi i e sequence ol age and VUF could be ob ained. This ela ion is p esen ed in Fig. 4 o 0 δ = . I is clea ha in case o aul y condi ion he eac i e powe se poin mus be dec eased o main ain he phase cu en less han he a ed alues. I is wo h men ioning ha some o he poin in his g aph a e no achie able in p ac ice. B. Maximum Phase Cu en o BPSC S a egy The e e ence cu en o BPSC s a egy is inspi ed om (10) and is exp essed as: 2 2 * * * i . .b b i i α β α α β β + + ⊥ + + ⊥       = = =       −             (35) Fig. 4. Maximum pe missible eac i e cu en se poin in AARC s a egy 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.5 1 1.5 |V+| VUF Q e whe e 2 b is de ined as: * 2 2 ( ) ( ) 2 3 Q V b+ =(36) In his s a egy all he phases ha e same ampli ude which is calcula ed as: * ( ) 2 3 a b c Q V I I I + = = = (37) F om (37) i could be inspi ed ha o keeping he phase cu en s sa ely o a ed alue, he maximum e e ence eac i e powe mus be educed in p opo ion o V + . C. Maximum Phase Cu en o PNSC S a egy Acco ding o (11), in PNSC s a egy he cu en con olle mus ack he ollowing e e ence cu en : 3 3 * * * i . .b b i i α α β β α α α β β β + − + − ⊥ ⊥ + − + − ⊥ ⊥   +     + = = =       +− −             (38) whe e 3 b is de ined as: * 2 2 3 ( ) ( ) ( ) 2 3 Q V V b + − =−(39) By applying componen s o (6) in (38) and applying e e se Cla k ans o ma ion, he phase cu en ampli udes a e ob ained as: 2 2 3 ((2 . cos( ) ) ) a I b V V V V δ + − + − = + + (40) 2 2 3 ( ( 2 . cos( 2 3 ) ) ) b I b V V V V δ π + − + − = + + + (41) 2 2 3 ( ( 2 . cos( 2 3 ) ) ) c I b V V V V δ π + − + − = + + − (42) Assuming 0 δ = and combining (13) wi h (40)-(41) esul s in Fig. 5 which p esen s he d op o e e ence eac i e powe as a unc ion o ol age unbalanced condi ion o PNSC s a egy. Fig. 5. Maximum pe missible eac i e cu en se poin in PNSC s a egy 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Reac i e Powe (PU) VUF PNSC AARC BPSC 0.8 V PU += 0 δ = Fig. 6. Pe missible sa e ope a ing eac i e powe e e ence compa ison Fo a simila amoun o ol age dip ( 0.8 V PU + =), Fig. 6 isually has compa ed he maximum pe missible eac i e powe o a o emen ioned h ee s a egies. I is clea ha in case o PNSC s a egy, as he VUF inc eases, he a e age eac i e powe descends in o de o keep he phase cu en in a sa e band. In con as , as BPSC s a egy does no ca e abou VUF, i dec eases he eac i e powe p opo ional o he posi i e sequence ol age. In case o AARC he d op o e e ence powe is mo e han BPSC in low VUFs bu o se e e VUFs he a e age e e ence eac i e powe is highe o AARC. I should be men ioned ha o di e en alues o δ , he pa e n o he eac i e powe emains app oxima ely he same o di e en s a egies, simila o Fig. 6. V. OVERALL CONTROL SCHEME The o e all con ol sys em is buil up wi h he agg ega ion o ol age limi a ion and sa e cu en injec ion limi a ion as a uni ied con olle ha no only ca es abou peak cu en limi a ion bu also DC ol age oscilla ions as well. A simpli ied block diag am o he p oposed con ol s a egy is shown in Fig. 7. A ol age sequence ex ac ion block based on Double Second O de Gene alized In eg a o (DSOGI) accompanied by a F equency Locked Loop (FLL) p esen ed in [29] is esponsible o he posi i e and nega i e sequence ol age ex ac ion in s a iona y e e ence ame. Fig. 7. Block diag am o he D-STATCOM con ol 0 0.2 0.4 0.6 0.8 1 0 0.5 1 0 0.2 0.4 0.6 0.8 1 |V + | VUF Q e , αβ αβ ⊥ +− +− abc max _ max ,DC I V dc * P limi ed * Q * i αβ abc i m αβ abc αβ αβ αβ 1 6 ... p p abc αβ * dc ne Z 1 T 2 T 3 T Fig. 8. Connec ion o a D-STATCOM o a dis ibu ion g id The DC ol age o he capaci o is kep on i s nominal a e age alue ia a DC ol age con ol loop. Fo a as and accu a e acking o he gene a ed e e ence cu en s a couple o P opo ional-Resonan (PR) con olle s as well as a eed- o wa d ol age om he poin o common coupling (PCC) is embedded in he con olle . Space Vec o Modula ion (SVM) is u ilized o gene a e he ga ing pulses o he swi ches in a wo le el in e e . VI. PERFORMANCE SIMULATION OF D-STATCOM IN A WEAK DISTRIBUTION GRID To alida e he beha io o he p oposed con ol s a egy, he ope a ion o a 4MVA D-STATCOM in a weak dis ibu ion g id which is shown in Fig. 8, is analyzed. The DC link nominal ol age and capaci ance a e 1150V and 20mF espec i ely. In his s udy case, when he con e e is supplying a 0.17 PU eac i e powe , a Single Line o G ound (SLG) aul happens in he middle o one o he pa allel lines. The beha io o DC ol age, ac i e and eac i e powe s and hei maximum de ia ions o all he h ee a o emen ioned con ol s a egies a e p esen ed in Fig. 9. As i can be seen, he e is a good ma ching be ween he analy ical calcula ions shown in Table IV and he oscilla ions cap u ed in Fig. 9. Main aining he peak cu en and he DC ol age in hei secu e ope a ion egions is in oduced in Fig. 10. I can be seen ha in his aul scena io he cu en limi c i e ion each as e han he o e ol age limi in he DC bus. The ype o aul as well as i s loca ion leads o di e en unbalance cha ac e is ics. Based on unbalance cha ac e is ics, ei he o maximum phase cu en limi a ion o DC ol age limi a ion c i e ia could a ise i s . Table IV. Analy ical expec a ion o ampli ude o ac i e powe luc ua ions and DC ol age oscilla ions (a) (b) (c) Fig. 9. DC ol age oscilla ions and ac i e / eac i e powe s o a) AARC, b) BPSC, c)PNSC s a egies Fig. 10. DC ol age and phase cu en s a e kep in a secu e ange Fig. 11 p esen s he esul s o happening a SLG aul a he sending end o pa allel lines. In his unbalanced scena io, he eac i e powe se poin is domina ed by DC ol age limi ing sub-algo i hm. The maximum pe missible DC 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 1000 1150 1300 Vdc(V) AARC Algo i hm 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 -2 0 2 x 106 P(W) 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0 2 4 6x 106 Q(Va ) 0.69 Q MVAR = 0 c   0 p   2.67 Q MVAR = 1.07 q MVAR =  0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 1,000 1,150 1300 Vdc(V) BPSC Algo i hm 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 -2 0 2x 10 6 P(W) 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0 2 4 x 10 6 Q(Va ) (s) 0.69 Q MVAR = 2.91 Q M VAR = 0.64 q MVAR  0.64 p MW =  44.5 c V =  0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 1,000 1150 1300 PNSC Algo i hm 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 -2 0 2 x 10 6 P(W) 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0 2 4 6x 10 6 Q(Va ) (s) 2.37 Q MVAR = 0.69 Q MVAR = 81 c V =  1.165 p MW =  0 q   0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 950 1050 1150 1250 1350 (V) DC Bus Vol age (PNSC Algo i hm) 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 -4730 0 4730 (A) ISTATCOM (s) ˆ4.73I nom kA = 81 c V =  AARC BPSC PNSC | ( ) | p  0  0.64 MW 1.142 MW | ( ) | c  0  44.4 V 79.4 V Fig. 11. DC ol age limi a ion each as e han phase cu en s limi a ion. SLG aul happens a he sending end o he lines ol age oscilla ion (k=10% and  115.5 c =) is eached bu , he maximum o phase cu en (4.5kA) is less han he limi ing alue (4.73kA). VII. Expe imen al Resul s The p oposed con ol s a egies a e implemen ed in dSPACE DS1103 pla o m and applied o a 5KVA ,400V in e e wi h a 700V DC bus and DC capaci ance o 4.7mF. The swi ching equency is chosen o be 10kHz. The expe imen al pla o m is demons a ed in Fig. 12. The pe o mance o he con ol s a egies, conside ing he DC ol age and phase cu en limi a ions, a e e alua ed acing a D ype ol age sag. U ilizing a powe ampli ie commanded om OPAL-RT eal ime simula o a D- ype ol age sag wi h a cha ac e is ics o 0.3 35 ∠ − ° is applied o he e minal o he con e e . The ol age sag occu ed when he con e e was deli e ing 3KVAR (7A peak cu en ) o he g id. Fig. 13 shows he unbalanced ol age and he injec ed cu en s when using he AARC s a egy and Fig. 14 is p esen ing he ac i e powe , eac i e powe as well as DC ol age oscilla ions in his s a egy. Du ing he aul , he phase which expe iences mo e dip has he maximum cu en and cu en peaks do no su pass he maximum se poin (7A he e). Fig. 12. Expe imen al pla o m The e is no luc ua ion in ac i e powe and no oscilla ion in DC ol age ei he . The e e ence eac i e powe dec eased om 3KVA o 1.7KVA which is supe imposed by a 100Hz oscilla ions. Fig. 15 and Fig. 16 a e belonging o BPSC s a egy. Fig. 13. PCC ol age and injec ed cu en s in AARC s a egy Fig. 14. Ac i e / Reac i e powe and DC ol age oscilla ions in AARC Fig. 15. PCC ol age and injec ed cu en s in BPSC s a egy 0.05 0.1 0.15 0.2 0.25 950 1050 1150 1250 1350 (V) DC Bus Vol age (PNSC Algo i hm) 0.05 0.1 0.15 0.2 0.25 -3000 0 3000 (A) ISTATCOM (s) max 4.50 ( 4.73 ) Limi I kA I kA = = 115.5 c =  a i b i c i a b c ( ) abc i ( ) abc 3 Q kVAR = 1.70 Q kVAR = a i b i c i a b c ( ) abc i ( ) abc