New controllability criteria for 3-phase 4-wire inverters applied to shunt active power filters

New Con ollabili y C i e ia o 3-phase 4-wi e in e e s applied o Shun Ac i e Powe Fil e s Manuel A. Pe ales, Luis Te ón, Juan A. Sánchez, An onio de la To e, Juan M. Ca asco, Leopoldo G. F anquelo Dep . de Ingenie ía Elec ónica. Escuela Supe io de Ingenie os. Uni e sidad de Se illa Camino de los Descub imien os s/n Isla de la Ca uja. 41092-Se illa . SPAIN [email p o ec ed] Abs ac – In Shun Ac i e Fil e applica ions, he 3-phase 4- wi e opology is equen ly used when dealing wi h unbalanced loads con aining ze o sequence componen s. A new design c i e ia o his opology is p esen ed, based on he well known exis ing me hod o he 3-phase 3-wi e sys em. Simula ion and expe imen al esul s con i ms he alidi y o his new c i e ia, p o iding an easy me hod o he design o he eac i e elemen s in ol ed in a Shun Ac i e Fil e . I. INTRODUCTION Ac i e Fil e s a e a as g owing ield in he powe elec onics a ea, due o he new egula ions and s anda ds on powe quali y. In pa icula , Shun Ac i e Fil e s a e he mos common s uc u e, ha ing nowadays a lo o p ac ical implemen a ions ins alled. The mos common s uc u e in his ield is based on he Cu en Con olled Vol age Sou ce In e e (CC-VSI). When designing his in e e , a special ca e ha e o be aken in he elec ion o he eac i e elemen s, ha is, he capaci ance and nominal ol age o he DC-Link and he alue o he smoo hing induc ance. When he load o be compensa ed is a 3-phase load, wi hou neu al connec ion, a 3-phase 3-wi e CC-VSI is commonly used. Fo his kind o in e e s, a design c i e ia was de eloped long ago [1] ha is e y use ul o selec he co ec alues o his pa ame e s, using he well known α-β ans o ma ion. In his pape , an ex ension o his me hod is p esen ed, co e ing he 3-phase 4-wi e opology, also e y employed in he Shun Ac i e Fil e s ield [2] [3] [4]. In his case, he α-β ans o ma ion is no so use ul, as i will be s a ed, because o he necessi y o adding ano he componen , he 0 axis, making he g aphical analysis oo much complica ed. In Sec ion II, he 3-phase 4-wi e opology is b ie ly p esen ed and analysed, showing he RST-αβ0 ans o ma ion o he in e e simpli ied equa ions. Then in Sec ion III he con ollabili y condi ion is de ined, showing he di icul y on he assessmen o his condi ion in αβ0 axis. In Sec ion IV a new change o axis is s a ed, called IMC ans o ma ion, ha g ea ly simpli ies he g aphical analysis o he con ollabili y p oblem. A e ha , in Sec ion V simula ion and expe imen al esul s a e p esen ed, ha con i ms he alidi y o he p oposed me hod, and inally in Sec ion VI some conclusions a e ex ac ed. II. 3-PHASE 4-WIRE TOPOLOGY The 3-phase 4-wi e opology is cha ac e ised by he connec ion o he neu al o he middle poin o he DC-Link. In his si ua ion, he h ee phases a e independen o each o he , and he cu en lowing o each leg depends only on he posi ion o he associa ed swi ches and i s phase ol age. In ig. 1 a basic scheme o his opology is p esen ed. I should be no ed ha he cu en is conside ed posi i e lowing ou o he in e e . L L L DC DC R S T i FR S 1 S 2 S 3 S 4 S 5 S 6 N Load i FS i FT Fig 1. Basic scheme o a 3-phase 4-wi e based Shun Ac i e Fil e So, he equa ions desc ibing he beha iou o he phase cu en s could be exp essed as (1) DCF FT FS FR F T S R ·(k) d d ·L i i i ; SiV;iV+−=           =           = (1) Whe e S(k) is a ec o ial disc e e unc ion depending on he s a e (k) o he in e e , wi h he ollowing alues: TABLE I VALUES OF S(K) S a e (k) Swi ch. ON S(k)R S(k)S S(k)T 0 S2,S4,S6 - 1 -1 -1 1 S1,S4,S6 1 -1 -1 2 S1,S3,S6 1 1 -1 3 S2,S3,S6 -1 1 -1 4 S2,S3,S5 -1 1 1 5 S2,S4,S5 -1 -1 1 6 S1,S4,S5 1 -1 1 7 S1,S3,S5 1 1 1 As he h ee phase a e decoupled, i is no possible o educe i o a wo componen sys em, i.e. using he αβ ans o ma ion. Ins ead, a h ee componen e e ence sys em mus be applied. I we use he αβ0 ans o ma ion, de ined by he ma ix Mαβ0 (2), he equa ions can be ansla ed o (3):                 − −− = 2 1 2 1 2 12 3 2 3 0 21211 3 2 0 // Mαβ (2) )k(VU d d L+−= F i (3) Whe e he ec o s iF, U and V(k) a e de i ed om (4): DC T S R T S R FT FS FR F F F V· )k(S )k(S )k(S M )k(V )k(V )k(V )k(V ; M U U U U; i i i M i i i           =           =           =           =           =           = 0 0 0 0 0 0 αββ α αββ α αββ α F i (4) I is impo an o no e ha all ec o s V(k) ha e a componen on he 0 axis, as i can be seen in (5). Tha leads o wo conclusions: • Vec o s 0 and 7 a e no in e changeable, as i happened in he 3-wi e opology. • The hexagon o med o he ec o s in he 3-wi e opology is now a cube, o a ed in he αβ0 coo dina es.                           − −−− − −− − − =                         2 3 2 1 2 1 2 1 2 1 2 1 2 12 3 3 2 00 31 31 02 31 31 02 00 7 6 5 4 3 2 1 0 DC V· )(V )(V )(V )(V )(V )(V )(V )(V (5) Fig 2. S a e ec o s in αβ0 coo dina es, gene a ing he con ollabili y cube This cube is he h ee dimensional equi alen o he well known wo-dimensional hexagon p oduced by he ec o s in he αβ coo dina es and will play an impo an ole on he de ini ion o he con ol capabili y o he in e e in ollowing a ce ain cu en e e ence, so we called i con ollabili y cube. III. CONTROLLABILITY CRITERIA IN αβ0 COORDINATES Using he same echnique as in [1] bu o he h ee dimensional case, le us de ine i*F as he cu en e e ence o be injec ed, and hen he acking e o (∆iF) could be exp essed as: FFF i*ii −=∆ (6) The condi ion o assu e ha he e e ence can be acked is ha he acking e o dec eases wi h he ime. So he con ollabili y condi ion will be ha he ime de i a i e o his e o is nega i e. Making his de i a i e, and using (3): )k(VU d d L d d L−      += ∆FF *ii (7) I we name e0 he e m in pa en hesis, (7) can be ew i en in he ollowing o m: )k(V d d L U d d L −= ∆ += 0 F F 0 e i *i e (8) Analysing (8) i is clea ha he con ollabili y depends on which ec o is g ea e , e0 o V(k). To ensu e he con ollabili y in he h ee phases simul aneously, he h ee componen s o he ec o e0 should be smalle han hose o he V(k) ec o . I a Space Vec o modula ion is implemen ed, V(k) will ep esen he ol age ec o o he in e e , and i will be included on he p e iously de ined con ollabili y cube. Then, he con ollabili y c i e ia could be exp essed as: Fo gi en alues o VDC and L, he so de ined in e e will be capable o acking a ce ain cu en e e ence only i he ec o e0 de ined in (8) is con ained on he inside o he con ollabili y cube desc ibed by he ec o s V(k). I is impo an o no ice ha , e en in he case ha he load has no ze o sequence componen , is no applicable he wo dimensional c i e ia, because he in e sec ion o he con ollabili y cube wi h he αβ plane is no equal o he hexagon desc ibed by he p ojec ion o he s a e ec o s o e he αβ plane. This p ojec ion is equal o he hexagon used in [1], bu he in e sec ion o he cube wi h he αβ plane, whe e should be loca ed e0 i i do no has ze o sequence o ensu e con ollabili y, is an hexagon smalle han ha , wi h i s e ices in he middle o he sides o he i s , as i is shown in ig. 3. α β 1 2 3 4 5 6 0,7 Fig 3. Con ollabili y hexagon in he wo dimensional case (solid) and in e sec ion o con ollabili y cube wi h he αβ plane(dashed) IV. NEW CONTROLLABILITY CRITERIA IN IMC COORDINATES A. IMC ans o ma ion The complexi y o he applica ion o his new con ollabili y c i e ia lays on he di icul y o de e mine whe he e0 is con ained o no in he con ollabili y cube, as i is o a ed om he e e ence sys em. To o e come his p oblem, a new ans o ma ion is p oposed, called IMC ans o ma ion (whose name comes om In e se o Ma ix o Con ollabili y), s a ed as: • The new coo dina es will be de ined by he uni a y ec o s I, M, C, in he di ec ion o V(0)-V(5), V(4)-V(5) and V(6)-V(5) espec i ely. • The cen e o he e e ence sys em will be placed on V(5). • The axis will be scaled down so he ec o s V(0)-V(5), V(4)-V(5) and V(6)-V(5) will ha e uni a y modulus. To make his coo dina es change, i is use ul o de ine he Ma ix o Con ollabili y (MC), cons uc ed by columns wi h he ec o s V(0)-V(5), V(4)-V(5) and V(6)-V(5) exp essed in αβ0 coo dina es:           − = ⇒        =− =− −=− 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 022 2 0256 254 250 DC DC DC DC VMC ),,(V)(V)(V ),,(V)(V)(V ),,(V)(V)(V (9) No e ha he columns o his ma ix ep esen s h ee o hogonal ec o s placed in h ee sides o he con ollabili y cube, and he h ee o hem ha e he same modulo: 2VDC. Now, o exp ess a gene ic ec o Y αβ0 in he IMC e e ence sys em, i is only necessa y o sub ac om i he V(5) ec o , in αβ0 coo dina es, and hen p e-mul iply his by he in e se o he ma ix o con ollabili y, esul ing: ))(VY(MCYIMC 00 15αβαβ −= − (10) Ope a ing wi h (10), (9) and (5), he change o coo dina es om αβ0 o IMC can be s a ed inally as:                       − − − −                       − − =           2 1 3 2 0 2 3 2 1 2 3 2 1 3 1 102 1 1 3 1DC DC C M I V Y Y Y · V Y Y Y β α (11) B. Con ollabili y cube on IMC axis The i ue o his new coo dina e sys em is ha , on IMC axis, he con ollabili y cube will be a uni a y cube (wi h all i s sides measu ing 1) and wi h i s aces pa allel o he coo dina e planes IM, IC and MC, as ep esen ed in ig.4 Fig 4. Con ollabili y cube ep esen ed in IMC axis. In his si ua ion, he accomplishmen o he con ollabili y c i e ia can be e y easily checked, only by es ing ha he IMC componen s o he e0 ec o emains be ween 0 and 1 always. I hey do ha , he ec o will be con ained in he con ollabili y cube. C. Con ollabili y C i e ia in IMC coo dina es Using he new coo dina es ans o ma ion, he con ollabili y c i e ia de ined in III may be ew i en in his way: Fo gi en alues o VDC and L, he so de ined in e e will be capable o acking a ce ain cu en e e ence only i he IMC coo dina es o he ec o e0 de ined in (8) emains be ween 0 and 1. No e ha e0 depends no only on he cu en e e ence de i a i es, bu also on he alue o he smoo hing induc ance, and e en on he g id ol age. On he o he hand, V(k) depends on he DC-Link ol age. Consequen ly, o a gi en cu en e e ence and g id ol age, his wo pa ame e s (VDC and L) could be adjus ed o comply wi h his con ollabili y c i e ia. V. SIMULATION AND EXPERIMENTAL RESULTS To alida e he c i e ia s a ed be o e, many simula ion and expe imen al esul s we e ob ained. Fi s a h ee phase unbalanced load, wi h ze o sequence componen , we e used as load o compensa e, doing ex ensi e simula ions o de e mine he op imal alues o VDC and L. Then , a h ee phase balanced load we e simula ed and compensa ed expe imen ally, o e ing he posibili y o compa ing he simula ion and expe imen al esul s. A. Simula ion wi h an unbalanced load A 3-phase 4-wi e load wi h unbalance, ze o sequence, cu en ha monics and eac i e powe was modelled om expe imen al measu es o a eal one. The wa e shapes o he h ee phase cu en s and neu al cu en a e d awn in ig. 5. I is no iceable he magni ude o he neu al cu en , ela ed o he ze o sequence componen . Fig 5. Unbalanced load simula ed. (a) Load cu en s in R-S-T phases. (b) Neu al Cu en Simula ions we e made calcula ing he ec o e0 needed o compensa e adequa ely his load, o many alues o VDC and L. In ig. 6, he h ee componen s o e0 in IMC coo dina es a e plo ed o wo pa icula condi ions (VDC=250, L=1.5mH and V DC=300, L=1.2mH). I can be seen ha in he i s si ua ion he con ollabili y c i e ia will no be ul illed, and i will in he second case. Nex , in ig. 7, he op imal alues o VDC depending on he alue o L is d awn. I can be seen ha a quad a ic ela ionship is almos ollowed (plo ed in solid) by he calcula ed pai s (plo ed as ci cles). Fig 6. plo o e0 in IMC coo dina es. (a) VDC=300V, L=2mH. (b) VDC=300V, L=1mH Fig 7. Op imal alues o VDC, o a gi en alue o L. The cu e ob ained in ig. 7 gi es us a design c i e ia o a Shun Ac i e Fil e ha migh ha e o compensa e his pa icula load: ixing he DC-Link ol age, due o swi ching o capaci o limi a ions, he induc ance equi ed can be calcula ed by means o his quad a ic app oxima ion. B. Simula ions wi h a load wi hou ze o sequence Nex , simula ions o balanced load wi hou ze o sequence we e pe o med, in o de o p o e ha , in his case i a 3- phase ou -wi e in e e is employed, he con ollabili y hexagon applicable is he one de ined in ig.3, and no he bigge one s a ed in [1]. The load simula ed was modelled o m labo a o y measu es, and is equal o he eal one ha will be compensa ed using an expe imen al Shun Ac i e Fil e . The wa e shape o he phase cu en is plo ed in ig. 8. In ig. 9, he h ee componen s o e0 a e plo ed, o h ee alues o VDC and he same alue o L, showing ha only he hi d case (VDC=300V) complies wi h he con ollabili y c i e ia. Then, in ig.10 he pola ep esen a ion o he α, β componen s o e0 a e plo ed, o he same condi ions as abo e. Fig 8. Load cu en used in simula ions No ice ha , as iL has no ze o sequence componen , nei he i*F will, so supposing ha g id ol age has no ze o sequence componen s, e0 will be loca ed on he αβ plane. Then, he con ollabili y c i e ia could be simpli ied o ha eo(αβ) emains in he hexagon de ined in ig. 3. I is impo an o poin ou ha , i he c i e ia de ined o he 3-phase 3-wi e case should be used, he si ua ion ep esen ed in (b) should be conside ed as con ollable e oneously, aking he ou e hexagon (dashed) ins ead o he inne hexagon (solid) as he con ollabili y hexagon. Fig 9. Componen s o e0 in IMC coo dina es. (a) VDC=225V, L=2.4mH. (b) VDC=250V, L=2.4mH. (c) VDC=300V, L=2.4mH. Fig 10. Pola ep esen a ion o e0αα,e0ββ wi h he new con ollabili y hexagon (solid) and he 3-phase 3-wi e case hexagon (dashed), o he same alues as abo e. C. Expe imen al esul s To alida e he con ollabili y c i e ia s a ed in his pape , expe imen s we e made, using he same load han in he p e ious simula ions. Fo his expe imen s, an Ac i e Powe Fil e de eloped by he au ho s was employed [5]. The con ol algo i hm implemen ed was based on Sel -Tuned Vec o Fil e s [6], able o compensa e cu en ha monics e en in p esence o a highly dis o ed ol age g id. The mos impo an pa ame e s o he powe sys em a e shown in Table I . TABLE I. POWER SYSTEM PARAMETERS Topology 3-phase 4-wi e Vol age (phase-neu al) 150V DC-Link capaci y 2 x 13.2 mF DC-Link ol age (max) 2 x 350V Smoo hing induc ances 2.4mH Powe swi ches IGBT’s (SKM300 GB 123 D) Commu a ion equency 20kHz Nominal powe (load) 12kW Finally, in ig. 11 expe imen al cu es o he compensa ed cu en a e shown. I can be obse ed easily ha in he i s and second cases a peak appea s on he compensa ed cu en , due o he loose o con ollabili y. This peak is g ea e in he i s case (VDC=225V), and do no appea s in he hi d case (VDC=300V). These esul s ma ch p ecisely he p edic ed con ollabili y o he sys em in e ms o complying he con ollabili y c i e ia s a ed, alida ing his way he p oposed me hod. VI. CONCLUSIONS The 3-phase 4-wi e opology is widely used in he implemen a ion o Shun Ac i e Fil e solu ions, ha ing he abili y o compensa e unbalanced loads wi h ze o sequence componen s. I is impo an he e o e o ha e a design c i e ia, based on a con ollabili y c i e ia in he same way han o he 3-phase 3-wi e opology. In his pape , a new con ollabili y c i e ia is p esen ed, as an ex ension o he wo dimensional case o a h ee dimensional analysis. By means o a ans o ma ion o coo dina es, he c i e ia can be easily checked, allowing he use o i o es ima e he co ec alues o he ol age o he DC-Link (VDC) and he smoo hing induc ance (L). The ma ch be ween heo e ical and expe imen al esul s con i ms he alidi y o his echnique, ha should be used when a 3-phase 4-wi e in e e is being designed o an Ac i e Fil e applica ion, e en i he load o compensa e has no ze o sequence componen s. REFERENCES [1]. H. Akagi, Y. Tsukamo o, A. Nabae. “Analysis and design o an ac i e powe il e using quad-se ies ol age sou ce PWM con e e s,” IEEE ans. on Indus y applica ions Vol. 26 nº 1, pp 93 –98. Jan-Feb. 1990 [2]. Am M. A. Amin “Fou -Wi e S a ic Reac i e Powe Compensa o ” p oceedings o he 26 h Annual Con e ence o he IEEE Indus ial Elec onics Socie y IECON-2000, pp 960-964. [3]. Na a-Segu a, G. Mino-Aguila “Fou -B anches In e e Based Ac i e-Fil e o Unbalanced 3-Phase 4-Wi es Elec ical Dis ibu ion Sys ems” p oceedings o he Indus y Applica ions Socie y con e ence IAS’2000. [4]. C.A. Quinn, N. Mohan “Ac i e Fil e ing o Ha monic Cu en s in 3-phase , 4-wi e sys ems wi h 3-phase, single-phase nonlinea loads” P oceedings o he con e ence APEC’92, pp 829-836. [5]. J.M. Ca asco, M. Pe ales, E. Gal án, L. G. F anquelo, S. Gu ié ez, E.G. de Mendí il, ”Dsp Con ol o an Ac i e Powe Line Condi ioning Sys em”, p oceedings o he con e ence EPE97 Vol. 3, pp. 1021-1026. 1997 [6]. M.A. Pe ales, J.L.Mo a, L. Te ón, J.M. Ca asco, L.G. F anquelo “T ansien Response and Dynamic Cha ac e iza ion o a New Ac i e Fil e Algo i hm based on Sel - uned Vec o il e ”, o be held in he Powe Elec onics Specialis Con e ence PESC02 on june,2002. Fig 11. Expe imen al esul s. Compensa ed cu en o di e en alues o VDC, and L=2.4mH. (a) VDC=225V. (b) VDC=250V. (c) VDC=300V