Mode switching in modal domain models of overhead lines and underground cables

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1 Mo swithing in mol omin mols o ovrh lins n unrgroun ls I. Zltunić, S. Voopij, R. Goić Astrt- This ppr prsnts ri rviw o irnt mthos or lultion o ignvlus n ignvtors or pplition in th mol trnsmission lin thory. In this il o prtiulr intrst is th numril phnomnon known s mo swithing. Mo swithing is rlt to vry spii rqunis n must proprly trt in mol omin mols o ovrh lins n unrgroun ls. This ppr summrizs th mthos n intiis th irns mong thm. Impn n mittn mtris o ovrh lins n unrgroun ls r lult in wi rng o rqunis in orr to ompr irnt mthos. Kywors: Eignvlu, Eignvtor, Mol trnsmission lin thory, Mo swithing, Mol omin, Ovrh lin, Unrgroun l. U I. INTRODUCTION nrgroun ls n ovrh lins r in EMTPs simultion progrms mol s rqunypnnt lmnts known s FD (rquny-pnnt) or ULM (Univrsl lin mol). A numr o pprs r stritly ling with moling ltromgnti trnsint hvior o ovrh lins n ls []-[7]. Most o thm us mol omposition thory to oupl phs systm in quivlnt mol systm s i it is onsist o singl phs lins. In this pross it is vry importnt to vlop suitl lgorithm tht will lult ignvlus longing to th sm st o ignvtors. On o th most importnt onition tht hs to ulill is tht ignvtors lult rom Z r ontinuous n smooth throughout wi rng o rqunis. I th stnr routins r us or lultion o ignvlus n ignvtors it is oun tht ignvlus n ignvtors r not sort proprly n invitl swithing twn mos n our. For xmpl, this n our i w us stnr Mtl untion ig() or implmnting si QR or powr mtho lgorithm or lultion o ignvlus n ignvtors. To voi this prolm thr mthos whih r ommonly us in rport litrtur []-[3] r prsnt. Th irst on uss moii Joi lgorithm [], [2]. Joi I. Zltunić n S. Voopij r working t Frtl Lt Split, Croti, n oth r post grut stunts t th Fulty o Eltril Enginring, Mhnil nginring n Nvl rhittur, Dprtmnt o Eltril Enginring, Split, Croti (-mil:ivo@rtl.hr n stip@rtl.hr). R. Goić is n Assoit Prossor t th Fulty o Eltril Enginring, Mhnil nginring n Nvl rhittur, Dprtmnt o Eltril Enginring, Split, Croti, (-mil:rgoi@s.hr). Ppr sumitt to th Intrntionl Conrn on Powr Systms Trnsints (IPST25) in Cvtt, Croti Jun 5-8, 25 lgorithm is ommonly us or solving ignvlus n ignvtors prolms o rl symmtri mtris n with rtin hngs it n us or solving symmtril omplx mtris [8]. Th son on is th Nwton-Rphson mtho [3]. This mtho is not otn us in rport litrtur ut n riv smooth n ontinuous ignvtors n ignvlus throughout wi rng o rqunis. Sin this routin utilizs th rsults rom prvious rquny s strting vlu or nxt rquny it is rommn to us nothr lgorithm in orr to gt strt with Nwton-Rphson mtho. This will work iintly only i th ignvlus n ignvtors o not vry wily rom on rquny to th othr. Th thir on lso utilizs stnr routins (powr mtho, Joi mtho, mtl untion ig n othrs) with orrltion thniqu [3]. In this s mo swithing is invitl ut n rogniz using orrltion thniqu. Th orrltion thniqu hks i th ignvtors longing to th sm st o ignvlus r orthogonl rom on rquny to th othr. Bsi sription n lgorithm o ormntion mthos r provi in this ppr. For this purpos, rquny pnnt impn n mittn mtris r lult or unrgroun ls n ovrh lins using qutions rport in [4]-[6]. Th min ojtiv o this ppr is to rviw rport mthos or solving ignvlu n ignvtor prolm in mol omposition thory n to provi omprhnsiv sription n implmnttion o thm. This ppr is strutur s ollows: Stion II outlins th prolm o mo swithing. Stion III sris si inormtion o mthos rport in litrtur or solving prolm o mo swithing. Stion IV unrlins th ommonly us normliztion routins or ignvtors mtris. Stion V prsnts simultion xmpls n rsults or simpl iruits o ovrh lins n ls. Stion VI onlus with isussion o th otin rsults. II. MODE SWITCHING IN THE MODAL DOMAIN MODELS Figur. shows n xmpl o multipl ignvlu swithovrs whn lulting mol vloity or unrgroun l givn in stion V. Mol vloity is otin or six mos o propgtion -. In mol thory o unrgroun ls, mo in ig. prsnts zro squn mo o onutor whih is nrgiz y injting unit urrnt into h l onutor n xtrting it rom orrsponing shth. Mos n rprsnt intronutor mos o propgtion. Mo is zro squn shth mo n

2 Mol vloity (m/s) mos n r intrshth mos o propgtion. Mo hs lowst mol vloity u to high inutiv impn o soil pth. Mos - hv mount o mol vloity in mount o 2/3 sp o light in ss o rquny lrgr thn Hz. Eignvlus n ignvtor or vry spii rquny r lult using stnr Joi routin. Almost th sm rsults n otin whn using othr stnr routins s mtl untion ig(), QR lgorithm, powr mthos n othrs. In this s whn ignvlus swithing hs ourr it lso mns tht ignvtors swith pls t rtin rqunis. Figur. shows n xmpl o swithovrs with mos, n n t rqunis nr Hz, 3 Hz, 4 Hz, 5 Hz, n 5 khz x Frquny (Hz) Fig.. Mol vloity hrtristis o nturl mos o propgtion or th kv unrgroun l with mo swithing Aoring to [], [2] two onitions hv to ulill in orr to mol FD mol o unrgroun ls n powr lins in mol omin: Th ignvtors lult rom prouts Z hv to ontinuous n smooth throughout wi rng o rqunis. In orr to otin minimum-phs shit untions, ignvtors o Z mtris n to normliz so tht on o thir lmnts oms rl n onstnt throughout wi rng o rqunis. It is ovious tht irst onition nnot ulill with stnr ignprolm routins. For this rson spil mthos r vlop in orr to l with th irst onition. Th son onition n sily mng using pproprit normliztion routin with ignvtors. III. DEALING WITH MODE SWITCHING This stion prsnts si inormtion out thr ormntion mthos. In orr to voi unsir ovrlow rrors proprly sling is utiliz iviing h lmnt o mtrix with lmnt (-ω 2 ε μ ) or lling ignvlu n ignvtor routin. At th n lult ignvlus must multipli with sm lmnt in orr to otin goo rsults. A. Moii Joi mtho Stnr Joi mtho uss Joi rottions to igonliz rl n symmtri mtris n to in pproprit ignvlus n ignvtors [], [2], [7]. Th irst ssumption tht hs to ulill is tht mtrix prout Z is igonlizl. Our xprin n t prsnt in litrtur show tht in prtil ss Z or powr ls n lins is lwys igonlizl throughout wi rng o rqunis (.- MHz). I this is not s it is still possil to otin ignvtor mtris using prturtion thniqus in orr to insur tht ignvlus r istint [2]. In orr to utiliz Joi mtho qut trnsormtion is n to insur tht mtrix is symmtri. Sin mtrix prout Z is symmtri, ignprolm () hs to trnsorm in (2). ZQ Q HR R H S S ZS R Q S S whr: - mittn mtrix, Z - impn mtrix, Z - prout o n Z mtris, Q - ignvtors o mtrix Z, λ ignvlus o mtrix Z, H symmtri mtrix, R ignvtors o mtrix H, S ignvtors o mtrix, λ - ignvlus o mtrix. Eignprolm (2) mns tht: Th mtrix H is symmtri throughout wi rng o rqunis n th ignvlus λ o H r th sm s ignvlus o Z. Sin H is symmtri, th ignvtor mtrix stisis R - =R T. Eignvtors Q o mtrix prout Z n otin using qution (4). Eignvtors S o n sily otin using stnr routins. Amittn mtrix is lmost igonl mtrix n ignprolm n sily solv. Using this pproh it is possil to lult ignvlus λ n ignvtors R pplying stnr Joi routin. Atr lulting ignvtor mtrix R it is possil to lult ignvtors o mtrix Z using qution (4). In orr to insur ontinuous n smooth ignvtors n ignvlus throughout wi rng o rqunis it is nssry to utiliz ignvtors otin in prvious rquny stp. This is on y qution (6) in whih mtrix H or prsnt rquny is prn post multipli with ignvtors lult or prvious rquny. R HR P P Hn 6

3 Whr: R P is ignvtor lult or prvious rquny. Invrsion o R P is not n sin it is orthogonl n stisis R - =R T. Hn is nw mtrix whih is lmost igonl n us or lulting ignvlus n ignvtors or prsnt rquny. Sin mtrix Hn is lmost igonl Joi rottions will lmost unit mtrix n this will tivly spup lultions. This mtho is tst or unrgroun ls in [], [2]. B. Nwton Rphson mtho In Nwton-Rphson mtho thr is no n or symmtri mtrix to lult ignvlus n ignvtors o Z. It utilizs qution (7) in orr to in ignvlus n ignvtors 7 ( Z U) Q kk k Whr: Q k is ignvtor longing to th ignvlu λ kk. U is unit mtrix. Eqution (7) is rpt or ll ignvtors n ignvlus in orr to lult xt vlus. As n xmpl or thr phs powr lin or l ths qutions n writtn s: ( Z ) Q Z Q Z Q Z Q ( Z ) Q Z Q Z Q Z Q ( Z ) Q In qutions (8) thr r our unknowns n thr qutions. In orr to otin solution it is possil to rpl lrgst vlu t th initil rquny n st it qul to on or ll rqunis. Aoring to [3] this normliztion is not goo us it n somtims l to unsirl rrors. Inst, thy rommn spiying sums o th squrs o th lmnts o th ignvtor to unity (8.4) Q Q Q Using ormntion our qutions (8. to 8.4) n stnr Nwton Rphson mtho nw untion G(x) n in s: G( x) x J( x) F( x) 9 Whr J(x) is th Join mtrix n F(x) is mtrix orm y qutions (8). Sin Nwton Rphson is itrtion mtho, initil vlus o ignvlus n longing ignvtors is n. Oviously th ignvlus n ignvtors otin rom prvious rquny r us s strting vlus or prsnt rquny. Also this mtho is not sl-strting mtho n nothr routin must us to lult ignvlus n ignvtors or irst rquny. This mtho is tst or ovrh lins in [3]. C. Corrltion thniqu Corrltion thniqu n us to voi multipl swithovrs twn ignvtors. In orr to in out swithing rqunis it is nssry to trk ignvtors throughout th wi rquny rng. Th orrltion thniqu propos in [3] is s on th t tht th ignvtors longing to th sm st o ignvlu r orthogonl rom on rquny to th othr. Th lgorithm or orrltion thniqu propos in [3] n summriz in iv stps: Clult th ignvtor mtrix Q or prsnt rquny. Otin omplx onjugt trnspos mtrix Q T *. Clult mtrix prout (Q T *)Q p. Q p is ignvtor mtrix rom prvious rquny. In h row o th ormntion prout in th lrgst lmnts. Th row numr o this lmnt ins th olumn numr o th ignvtor rom th prvious rquny. Convrsly, olumn numr ins olumn numr o th ignvtor t th prsnt rquny. This mns i mo swithing hs not ourr, ll lrgst numrs will sort in igonl o ormntion mtrix prout. I ignvtor swithing hs ourr it is nssry to swith ignvtors n ignvlus in orr to mth prvious rquny. Our xprin shows tht this mtho works iintly with stnr ignprolms routins (powr mtho, mtl untion ig() n joi mtho) ut lso shows tht it nnot ppli with QR mtho. IV. EIGENVECTOR NORMALIZATION Aoring to [], [2] uthor rommns to normliz ignvtors o Z mtris in orr to otin minimum phs shit untions. For unrgroun ls uthor rommns to multiply h ignvtor y tor suh tht on o its lmnts oms onstnt n rl throughout whol rquny rng. This sling pross utomtilly ors ll lmnts o ignvtor to minimum-phs-shit-untions. In [3] uthors rommn using sums o squrs o th lmnts o th ignvtors to unity. In this ppr uthor prsnts tht oring on lmnt to rl n onstnt n somtims tk wy nturl vrition hvior n prou unsirl rrors. In this ppr or th purpos o omprison in stion V th irst normliztion routin will utiliz or ll mthos. In utur rsrh it is rommn to in optiml normliztion routins or ovrh lins n unrgroun ls. V. SIMULATION EXAMPLES AND RESULTS For simultion purposs two xmpls r prsnt. First simultion xmpl with rsults is prsnt or kv unrgroun ls n son on or kv ovrh lins. Mtris n Z or unrgroun ls with shths r lult throughout wi rng o rqunis (. Hz

4 MHz) with 5 points or vry in log sl. Mtl untion is implmnt to lult lmnts o impn n mittn mtris using wll known qutions [4]. Frquny pnnt intrnl impn o or n shths r lult using ull lssil ormuls. Erth-rturn impn n mutul rth-rturn impn r lult solving Pollzk intgrl with onvrgnt sris n y implmnting numril intgrtion mtho using utious, ptiv Romrg xtrpoltion [4]. Also lultions r implmnt n ompr with pproximt ormuls givn in [4]. All rsults r otin s on homognous soil rsistivity. Physil n gomtry t or kv thr phs unrgroun ls with shths n isoltion r shown in igur 2. Th ls us s n xmpl in igur 2. r singl or, with luminium phs onutors, with ross link polythyln insultion (XLPE) n smi-onutiv lyr nth n ovr insultion. Outr insultion is high-nsity polythyln (PEHD). Angl ( ) Amplitu Angl ( ) Amplitu Mol vloity (m/s) x Frquny (Hz) Fig. 3. Mol vloity hrtristis o nturl mos o propgtion or th kv unrgroun l without mo swithing r r 2 r 3 r h=m s=5 m ρ= Ωm r = 9 mm r 2= 35 mm r 3= mm r 4= 43 mm ρ C= Ωm ρ S=.72-8 Ωm tnδ XLPE =. tnδ PE-HD =. ε rxlpe=2.3 ε rpe-hd=2.5 μ r= L=5 km Fig. 2. Dt n gomtry or kv thr phs unrgroun ls For omprison purposs smooth n ontinuous untions o mplitu n ngl o ignvtors or ls r prsnt in Figur 4. Figur 3 shows mol vloity or six nturl mos o propgtion (-). Corrltion thniqu is us with stnr mtl untion ig() n sling pross o ignvtors is m oring to stion IV. Simultion with irnt kins o ls gomtry n physil t shows tht ll mthos giv goo n lmost intil rsults. Nwton-Rphson mtho n moii Joi mthos giv rsonly goo rsults tr w itrtion. Mximum rltiv rror twn ths mthos in this xmpl o not x,3%, whih is ngligil. Th irns mong orrltion thniqu n Nwton-Rphson mtho (4 itrtion us in simultions) us in this simultion r shown in igur 5. Figur 6 shows irns mong orrltion thniqu n moii Joi mtho. In this s or moii Joi lgorithm is implmnt with vrg 9 itrtion or olumn 5 ignvtor Frquny (Hz) Frquny (Hz)

5 Angl ( ) Amplitu Mgnitu irn o ignvtor olumn 5 (%) Angl ( ) Amplitu Mgnitu irn o ignvtor olumn 5 (%) Angl ( ) Angl ( ) Amplitu Amplitu Frquny (Hz) Frquny (Hz) Fig. 4. Amplitu n ngl o ignvtors to 6 or unrgroun l Frquny (Hz) Fig. 5. Mgnitu irn o Nwton-Rphson (itr. 4) ompr to Corrltion thniqu Frquny (Hz) Frquny (Hz) 8 x Frquny (Hz) Fig. 6. Mgnitu irn o Joi mtho (vrg itrtion 9) ompr to Corrltion thniqu Similrly, or ovrh lins, lultion o mtris n Z r lso implmnt in mtl throughout sm rng o rqunis. For ovrh lins xplntions out mos o propgtion r similr with ls. Figur 8 shows ontinuous n smooth untion o mol vloity or our nturl mos o propgtion (zro squn mos n intronutor mos) or ovrh lins with ssumption o uniorm soil rsistivity. At high rqunis

6 Angl ( ) Angl ( ) Amplitu Amplitu Mol vloity (m/s) Angl ( ) Amplitu Angl ( ) Amplitu mol vloity or ll mos hs mount nr sp o light. Physil n gomtry t or thr phs ovrh lin with on sky wir is shown in igur 7. Figur 9 shows lultion untions o ngl n mplitu o our ignvtors sl in orn with stion IV. 3, Sky wir 2,5 3,5 3, m 2,2 m 2,2 m 3,8 m Conutors: n= 3 r C = 9.48 mm R C =.88 Ω/km Sky wir: n= r w = 6.97 mm R w =.342 Ω/km Lin lngth: 5 km Frquny (Hz).5 ρ rth = Ωm Fig. 7. Dt n gomtry or kv thr phs ovrh lin 3 x Frquny (Hz) Fig. 8. Mol vloity hrtristis o nturl mos o propgtion or th kv ovrh lins without mo swithing Frquny (Hz) Frquny (Hz) Frquny (Hz) Fig. 9. Amplitu n ngl o ignvtors to 4 or ovrh lin VI. CONCLUSIONS This ppr provis rviw, implmnttion n simultion xmpls o mthos or ling with mo swithing in

7 mol trnsmission lin thory. Th mthos rport in litrtur r: moii Joi lgorithm, Nwton Rphson mtho n orrltion thniqu. All mthos n sily implmnt in EMTP progrms n prou smooth n ontinuous untions o ignvlus n ignvtors throughout wi rng o rqunis. Th omprison twn mthos using ormntion unrgroun ls n ovrh lins shows tht irns r ngligil ut utur nlysis with mor omplit gomtry mols o ls (pip typ l n othrs) n ovrh lins n to invstigt. Aoring to rport litrtur mo swithing is importnt to solv only in mol omins FD mols us ignvtors untions n to ontinuous n smooth throughout wi rng o rqunis. I phs mols r us thn mo swithing is not importnt. Nwton-Rphson mtho n moii Joi mtho utiliz ignvtors lultion rom prvious rquny stp n tht is rson wy untions r smooth, ontinuous n th lultion pross is vry st. Corrltion mtho uss trking lgorithm in orr to rogniz mo swithing. VII. REFERENCES [] L. Mrti, "Simultion o trnsints in unrgroun ls with rquny pnnt mol trnsormtion mtris," IEEE Trns. on Powr Dlivry, vol. 3, pp. 99-, July 988. [2] L. Mrti, "Simultion o ltromgnti trnsints in unrgroun ls with rquny pnnt mol trnsormtion mtris," Ph.D issrttion, Th Univrsity o British Columi, Nov [3] L. M. Wpohl, H. V. Nguyn, G. D. Irwin, "Frquny pnnt trnsormtion mtris or untrnspos trnsmission lins using Nwton-Rphson mtho," IEEE Trns. on Powr Systms, vol., No.3, pp , Aug [4] S. Nvrtnm, "Sris impn n shunt mittn mtris o n unrgroun l systm," Ph.D issrttion, Th Univrsity o British Columi, Nov [5] O. R. Lnos, J. L. Nro, J. A. G. Rols, "An vn trnsmission lin n l mol in Mtl or th simultion o powr-systm trnsints, Mtl A Funmntl Tool or sintii omputing n nginring pplitions,",vol, Inth, Spt. 22. [6] A. Morh, B. Gustvsn, M. Trtii, "A univrsl mol or urt lultion o ltromgnti trnsints, on ovrh lins n unrgroun ls", IEEE trns., on Powr Dlivry, July 999. [7] T. C. u, "Full rquny pnnt molling o unrgroun ls or ltromgnti trnsint nlysis," Ph.D issrttion, Th Univrsity o British Columi, Sp. 2. [8] R. W. Hornk, Numril mthos, Quntum Pulishrs, pp , 98.