Non-Linear and Unbalanced Three-Phase Load Static Compensation with Asymmetrical and Non Sinusoidal Supply

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1 Non-Lnar and nbalancd Thr-Phas Load Statc Comnsaton wth Asymmtrcal and Non Snusodal Suly Rys S. Hrrra and P. Salmrón Elctrcal Engnrng Dartmnt Escula Poltécnca Suror, nvrsty of Hulva Ctra. Palos d la Frontra, s/n. Palos d la Frontra, Hulva 89 (Esaña) -mal: Rys.Sanchz@d.uhu.s, atrco@uhu.s Abstract. P-q thory has bn wdly usd to control actv owr fltrs snc ts formulaton n 983. Th comnsaton stratgy usd by th -q thory usrs has not suffrd modfcaton; so t s usd th constant owr comnsaton stratgy. In ths way, th suly nstantanous owr aftr comnsaton s constant. Ths knd of comnsaton stratgy has obtand good rsults n th cas of balancd and snusodal voltags, howvr t has not bn arorat n th cas of unbalancd or non snusodal voltags. In ths ar t has bn rsarchd about th modfcatons ncssary, nto th -q thory fram, to gt control stratgs whch allow to attack th unt owr factor comnsaton or th balancd and snusodal currnts comnsaton. Kywords. Instantanous ractv owr, owr qualty, thr-has systms, harmoncs, actv owr fltrs.. Introducton. Th ntgraton of owr lctronc n th common lf has orgnatd a consdrabl ncras of non lnar currnts. Th roblms assocatd wth ths currnts hav mad of ts comnsaton a rortary task. Thr ar svral ways to achv th targt (non lnar currnts comnsaton), and on of th most usd currntly s th actv owr fltr (APF). Ths s a convrtr whch suls th non usful art of th currnt rqurd by th load. Imosng a comnsaton targt to th control crcut, t calculats th currnt suld by th comnsator accordng to th nstantanous ractv owr thory. In ths ar, th control stratgy has bn ald to a thr-has four-wr systm smlar to that rsntd n fgur. It s a thr-has balancd and non snusodal voltag sourc. Th load nvolvs a rgulator comosd by b-drctonal thyrstors and srs rsstors wth dffrnt valus n ach lg to obtan asymmtrcal condtons. Th comnsator njct th non usful art of th currnt rqurd by th load. Along th ar, lowr cas rrsnts nstantanous masur, ur cas avrag masur, subndx L load rqurmnt, subndx c comnsator suly and subndx s sourc suly. From ths thory dffrnt formulatons aard durng th last dcad, ths work s focusd on th frst of thm, calld -q u orgnal nstantanous ractv owr thory.. Th Orgnal Instantanous Ractv Powr Thory. It was formulatd at th bgnnng of th ghts, [], and ths s th formulaton whch has xrmntd a largst dffuson along ths yars. That s th rason to b th most usd as control stratgy for th APF. Th -q thory was dvlod to b ald to thr-has thrwr systms wth balancd and snusodal sourc voltags. Th comnsaton targt assumd was gttng a constant nstantanous sourc owr. 33 Fgur. rfrnc systm c L u L Ths thory s basd on a coordnats transformaton from th has rfrnc systm (3) to th systm, fgur. Th transformaton matrxs assocatd ar as follows: Fgur. Comnsaton dagram

2 u = u u3 () = () Lookng at ths quatons t can b dducd that: = + + = 3 (3) 3 Thy ar dfnd th dffrnt owr as follows: = = [ P ] q whr s th zro squnc (ral) nstantanous owr, s th ral nstantanous owr and q s th magnary nstantanous owr. By man of th [P ] nvrs matrx t s ossbl to calculat th currnt comonnts knowng th dffrnt owr trms. Th xrsson s shown n th nxt quaton: = whr = + q Th stratgy assumd by ths thory snc ts concton has bn th obtnton of a constant nstantanous owr n th sourc sd wth th only rstrcton of gttng a null actv owr xchangd by th comnsator, P c. In ths way, th frst row n th tabl I shows th control stratgy drvd from ths thory as t was dvlod orgnally. In fact, to calculat th comnsator currnt, and accordng to th fgur rfrncs, t s vrfd that: c L s = L PLu (4) () = (6) Whr P Lu s th total actv owr consumd by th load. Th quaton (6) mans that: L c L PL = ~ L P = ~ = (7) = (8) c L L L whch, ffctvly, fulfls that < c >=, whr ~ rrsnts th L varabl art. On th othr sd, q c q = (9) L Th scond column n th tabl nforms whthr thory lmnats th nutral currnt and th fourth on nforms about th actv owr suld by th sourc aftr comnsaton, sayng whthr t s constant n tm. Th stratgy shown n th frst row dos not lmnat th nutral currnt but th actv owr xchangd by th comnsator s null. To mos ths nw rstrcton, n th -q thory contxt, t s only ncssary a sml modfcaton: th zro squnc owr usd to calculat th comnsator currnt s th zro squnc owr rqurd by th load, []: L c = () c = L () Th thrd column shows that, ffctvly, ths nw stratgy lmnats th nutral currnt. Morovr, th nstantanous owr suld by th sourc gos on bng constant. 3. Othr Comnsaton Stratgs. A comnsaton stratgy wdly usd along th tm s th unt owr factor comnsaton. Th targt s to gt sourc currnts wth th sam dstorton and symmtry condtons as sourc voltags. It mans that th sourc currnts ar collnar to th suly voltags. In ths stuaton, th sourc suls th load actv owr, but th nstantanous ral owr s not constant aftr comnsaton, [3]. Whn sourc voltags ar snusodal and balancd, constant owr and unty owr factor comnsatons rsults ar th sam. In fact, for snusodal and balancd sourc voltags, f comnsaton currnts cancl ractv comonnts, load currnt dstorton and asymmtry, sourc currnts wll b snusodal and balancd n th sam way as voltag. And for that rason, nstantanous owr suld by th sourc wll b constant. Howvr, whn suly voltag s unbalancd and non snusodal th stuaton won t b th sam. Th tabl I thrd row shows th control stratgy, basd on th -q thory, whch gts unt owr factor n th sourc sd aftr comnsaton. now t has not bn lannd ths stratgy adoton n th -q thory fram. Now, th sourc currnt must b roortonal to th voltag suld:

3 r r = G () s To go on satsfyng that th actv owr xchangd by th comnsator s null, th roortonalty constant must b fxd to th nxt valu: PLu G = (3) whr E s th squar of voltag vctor rms valu. Th comnsator nstantanous owr s th dffrnc btwn th total ral nstantanous owr rqurd by th load and th art suld by th sourc as t has bn xland abov. Tabl I rsnts th comnsator currnts quatons n coordnats. Const. S. Powr wthout Elmnatng Nutral Const. S. Powr Elmnatng Nutral nt Powr Factor wthout Elmnatng Nutral nt Powr Factor Elmnatng Nutral Snusodal Sourc c c c = TABLA I. Control Stratgs. Control Stratgy ~ ~ L ql L () L t PL c = c ~ L c ql PLu L c = PLu () c t L c q L L c P Lu PLu c = L () c ql t = c c c q L L PLu + + ( + ) + PLu + + ( ) + L Nutral Sourc Powr s = ct = s = ct s ct = s ct = s ct Ths control stratgy dos not rmt to lmnat th nutral currnt. To satsfy ths rstrcton, th stratgy must b chang to th form shown n th tabl I fourth row. In ths cas, th owr factor s a lttl lowr than th unt. Nvrthlss, th nutral currnt s lmnatd vn n th cas that voltags ar not balancd or snusodal. For last yars, both comnsaton objctvs rsntd n ths ar ar not nough to solv th nw roblms aard. It s ncssary to dfn a nw comnsaton targt basd on an dal rfrnc cas. Du to th fact that th lctrcal comans roduc, gnrally, th lctrcal owr as snusodal and drct squnc voltags, t has bn got as rfrnc condton n th suly. A rsstv load, balancd and lnar s consdrd as th dal rfrnc load. A rfrnc voltag ald to an dal rfrnc load wll gnrat snusodal and balancd currnts, n has wth voltags, [4]-[]. Anythng whch roducs a no conformty wth ths rfrnc condtons suoss a oor lctrc owr qualty. Comnsaton dvcs must achv that th sourc currnt ar snusodal, balancd, on ostv squnc and n has wth voltags ostv squnc fundamntal comonnt. It must b fulflld for any suly voltags condtons and load knds. Thos

4 rqurmnts nvolv th dal rfrnc condtons for suly currnt and thy ar th currnts that dfn th last comnsaton objctvs assumd n ths ar. If voltags ar balancd and snusodal, dal currnts wll b (as n arlr stratgy) roortonal to th voltags. Th roortonalty constant must hav th valu stablshd n th quatons () and (3), too. If voltags ar balancd no snusodal, th dal rfrnc currnt must b roortonal to ts fundamntal comonnt. Th roortonalty constant s fxd to satsfy th rstrcton of havng null actv owr xchangd by th comnsator. Th dal currnt xrsson s as follows, [6]-[7]: E r P r s = u (4) u u r rrsnts th voltag fundamntal comonnt and th squar of th voltag fundamntal comonnt rms valu. If voltags ar unbalancd non snusodal, th sourc currnt xrsson bcoms as shown n quaton (). + u r comonnt and r P r u u + s = + () rrsnts th voltag ostv squnc fundamntal + E + th squar of th voltag ostv squnc fundamntal comonnt rms valu. Now, r s n () s roortonal to th voltag ostv squnc fundamntal comonnt. Th roortonalty constant valu s got mosng th rstrcton of null actv owr xchangd by th comnsator as shown n th arlr quaton. In th tabl I last row t s shown th comnsaton quatons to gt balancd and snusodal suly currnts. Thy ar, as arlr stratgs, n coordnats. In ths cas t must b notd that, at frst tm, t s ncssary to mos rstrctons not only to th nstantanous ral owrs (t) and (t), vn bsds to th nstantanous magnary owr q (t). Ths nw rstrcton s ncssary to gt snusodal and balancd sourc currnts vn voltags ar non snusodal. Morovr, ths stratgy achvs to lmnat th nutral currnt wth a null actv owr xchangd by th comnsator. 4. Smulaton Practcal Rsults Th fv stratgs rsntd n rvous tms and shown n tabl I hav bn ald to th smulaton latform shown n fgur. Th mthod usd to obtan voltag fundamntal comonnt ncssary to by th last stratgy s basd n th us of ass band fltrs, multlrs and addrs as t s shown n fgur 3. v (t) snωt v (t) cosωt X X Fgur 3. Procdur to gt voltag fundamntal comonnt Th roduct comosd by voltag vctor on has v (t) and th sgnal snωt has a constant trm: ½V cosϕ bng V th has voltag fundamntal comonnt amltud and ϕ th has voltag fundamntal comonnt has. On th othr hand, th roduct comosd by th sam voltag vctor v (t) and th sgnal cosωt has a constant trm: ½V snϕ. Multlyng both constant trms r and r snωt and cosωt, rsctvly, as t can b sng n fgur 3, has voltag fundamntal comonnt wavform s bult. To gt th othr two hass, th rocdur may b th sam. Th rsults got from th dffrnt stratgs smulatons ar rsntd n fgur 4. All of thm rrsnt th has sourc voltag and currnt wavforms corrsondnt to th dffrnt stratgs smulaton. Th frst grah n fgur 4 shows th voltag and currnt rqurd by th load or th sourc currnt bfor comnsaton. It s a strongly non lnar currnt load. Th scond grah rsnts th currnt sourc aftr comnsaton accordng to th frst control stratgy shown n th tabl I frst row. Ths stratgy looks for gttng constant nstantanous owr suld by th sourc aftr comnsaton. Th sourc currnt s non snusodal and t stll rsnts a hgh dstorton. Nvrthlss, th thrd grah shows that, wth th only addtonal rstrcton of lmnatng th nutral currnt n th sam stratgy (gttng constant th nstantanous owr suld by th sourc), th currnt dstorton s wdly rducd. Ths scond stratgy dos not gt snusodal sourc currnt as can b sn n th thrd grah. Both stratgs gt constant nstantanous owr suld by th sourc aftr comnsaton. Th fourth grah n fgur 4 rsnts th currnt sourc aftr comnsaton accordng to th thrd stratgy shown n th tabl I thrd row. Ths s th unt owr factor stratgy wthout takng nto account th nutral currnt lmnaton. X X + + v (t)

5 Voltag (V) Voltag (V) Voltag (V) Voltag (V) Voltag (V) Voltag (V) Bfor Comnsaton Constant Sourc Powr Wthout Elmnatng Nutral Constant Sourc Powr Elmnatng Nutral ntary Powr Factor Wthout Elmnatng Nutral ntary Powr Factor Elmnatng Nutral Snusodal Sourc Voltag Fgur 4. Sourc Voltaj and Phas Got from th Control Stratgs Smulatons (A) (A) (A) (A) (A) (A) In ths cas, th sourc currnt wavform s collnar to voltag. Imosng th addtonal rstrcton of lmnatng th nutral currnt, both wavforms (voltag and currnt) ar not collnar. Ths nw stratgy s shown n tabl I fourth row. Nvrthlss, th dstorton s low n ths cas, too. Both stratgs rsntd n ths aragrah do not gt constant nstantanous owr suld by th sourc aftr comnsaton. Th owr factor s th unt n th fst on and almost th unt n th othr. Th currnt wavform shown n fgur 4 sxth grah s th only snusodal on. It corrsonds to th stratgy rsntd n tabl I last row, whch moss gttng snusodal and balancd sourc currnts n has wth th voltags and carryng th actv owr rqurd by th load. Ths stratgy dos not gt constant nstantanous owr or unt owr factor aftr comnsaton. Howvr, ths s th only cas whch obtans a sourc currnt wavform wthout any dstorton.. Concluson It has bn rsntd n ths ar an xhaustv analyss of th -q nstantanous ractv owr thory. Bsds, t has bn carrd out a nw analyss whr t s shown that, wth fw modfcatons n th control stratgy dvlod by th authors of th orgnal thory, t may b achv any comnsaton targt mosd to th systm. Th rsults ar corroboratd by th corrsondnt smulatons n th Matlab-Smulnk nvronmnt. Rfrncs []. H. Akag, Y. Kanazawa, and A. Naba, Instantanous Ractv Powr Comnsators Comrsng Swtchng Dvcs Wthout Enrgy Storag Comonnts, IEEE Trans. Ind. A.,vol.IA-,No.3,.6-63, 984. []. H. Akag, S. Ogasawara, H. Km, Th Thory of Instantanous Powr n Thr-Phas Four-Wr Systms: A Comrhnsv Aroach, Conf.Rc.of IEEE IAC, Vol.,999, [3]. A. Cavalln, G. C. Montanar. Comnsaton Stratgs for Shunt Actv-Fltr Control. IEEE Transactons on Powr Elctroncs, Vol. 9, No. 6, Novmbr 994, [4].P. Salmrón Rvulta, R. S. Hrrra, Alcaton of th Instantanous Powr Thors n Load Comnsaton wth Actv Powr Fltrs, EPE 3. []. Fang Z. Png, Lon M. Tolbrt, Comnsaton of Non- Actv n Powr Systms Dfntons from Comnsaton Standont -, Powr Engnnrng Socty Summr Mtng,, Volumn:,, [6]. P. Salmrón, J.C. Montaño, J. R. Vázquz, J. Prto and A. Pérz, Practcal Alcaton of th Instantanous Powr Thory n th Comnsaton of Four-Wr Thr-Phas Systms, IECON, Svlla,, Volumn 4, 6-6. [7]. P. Salmron, J.C. Montaño, J.R. Vázquz, J. Prto and A. Pérz, Comnsaton n Non-Snusodal, nbalancd Thr- Phas Four-Wr Systms wth Actv Powr Ln Condtonr, IEEE Trans on Powr Dlvry, n rss. [8]. A. Horn, L. A. Pttorno, J. H: R. Ensln, Evaluaton of Actv Powr Fltr Control Algorthms ndr Non-Snusodal and nbalancd Condtons, Proc. of th 7 th Intrnatonal confrnc on Harmoncs and Qualty of Powr, ICHQP 996, 7-4. [9]. H. Km, Hrofum Akag, Th Instantanous Powr Thory on th Rotatng -q-r Rfrnc Frams, PEDS 99.

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