Analytical model of high-voltage transmission line subjected to the downburst wind with rainfall

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1 Rsarch Articl Analytical mdl f high-vltag transmissin lin subjctd t th dwnburst wind with rainfall Advancs in Mchanical Enginring 1 13 Ó Th Authr(s) 2015 DOI: / aim.sagpub.cm Cha Zhu and Yibing Liu Abstract Thr hav bn many rprts rcntly n unanticipatd gallping and cllaps accidnts f twr-lin systms du t dwnburst wind with rainfall. Althugh wt dwnburst is charactrizd by high-vlcity wind with rainfall, vry littl rsarch wrk is invlvd with gallping f high-vltag transmissin lin inducd by th dwnburst wind with rainfall. Thus, this articl prpss a prliminary thrtical study aiming t prvid an analytical mdl f th high-vltag transmissin lin subjctd t th dwnburst wind with rainfall t xplain sm phnmna bsrvd frm fild masurmnts. Thrugh wind r rain wind tunnl xprimnts, w btaind ardynamic charactristics f th high-vltag cnductr with diffrnt yaw angls and rainfall rats. Cnsidring th variatins f svral factrs such as wind vlcity, rainfall rat, yaw angl, and attack angl, th prpsd analytical mdl was cratd by finit lmnt mthd and cntral diffrncs with th btaind ardynamic cfficints f th high-vltag cnductr. Th thrtical rsults accrd wll with th xprimntal data. Th analytical mdl nabls bttr cmprhnsin f th gallping f th high-vltag transmissin lin subjctd t th dwnburst wind with rainfall. Kywrds Dwnburst wind, high-vltag transmissin lin, finit lmnt mthd, rainfall Dat rcivd: 15 Nvmbr 2014; accptd: 2 January 2015 Acadmic Editr: Jia-Jang Wu Intrductin Th plannd high-vltag transmissin twr-lins ar prsnt in th suthwst rgin f China whr thundrstrms ccur frquntly. Th currnt trnd is t dsign and cnstruct much tallr twrs with lngr spans than vr, which maks nsuring scur and stabl pratin f th high-vltag transmissin twrlins a challng fr lctric pwr nginring. 1,2 Failur rcrds f high-vltag transmissin twrlins in Amrica, Australia, Suth Africa, and svral thr rgins shw that mr than 0% f wathrrlatd failurs ar causd by th dwnbursts with th ccurrnc f rainfall. 3 Althugh thr ar a larg numbr f th dwnbursts which typically cnsists f prcipitatin (calld wt dwnburst), vry littl attntin was paid t th ffct f th dwnburst wind cupld with rainfall n gallping f high-vltag transmissin lin. Mrvr, th cnvntinal layr vrtical prfil f wind vlcity is n lngr valid fr th dwnbursts, and nrmal dsign mthds f th transmissin twrlins vidntly ar nt adquat fr this ff-dsign cnditin. Schl f Enrgy and Mchanical Enginring, Nrth China Elctric Pwr Univrsity, Bijing, China Crrspnding authr: Cha Zhu, Schl f Enrgy and Mchanical Enginring, Nrth China Elctric Pwr Univrsity, Bijing , China. zhucha@ncpu.du.cn Crativ Cmmns CC-BY: This articl is distributd undr th trms f th Crativ Cmmns Attributin 3.0 Licns ( which prmits any us, rprductin and distributin f th wrk withut furthr prmissin prvidd th riginal wrk is attributd as spcifid n th SAGE and Opn Accss pags ( pnaccss.htm).

2 2 Advancs in Mchanical Enginring In rcnt yars, many rsarchrs hav studid th diffrnt aspcts f this subjct. Olivr t al. 4 dscribd a prbabilistic mdl fr dsign transmissin lin systms against th dwnburst typ winds and dvlpd a practical mdl t stimat th risk f a strik f damaging dwnburst n a high-vltag transmissin lin with spcifid lngth and rintatin in Australia. Savry t al. 5 carrid ut a prliminary study n th structural rspns f a typical lattic transmissin twr subjctd t a micrburst. Th rsults shwd that th micrburst did nt caus failurs du t its lwr intnsity and lngr duratins. Lin t al. 6 intrducd an arlastic mdl f a singl span f supprt lattic twr-lins with scal rati f 1:100, which mts th nd f th similarity thry fr wind tunnl tsts. Dirct cmparisn f twr and lin rspns t synptic wind prfil vrsus dwndraft utflw wind prfil indicatd that transmissin lin failurs frm dwndraft winds wr mst likly causd by pak twr lads that culd xcd that f bundary layr winds by an rdr f magnitud, and pak upstram cnductr lads that culd rach svral tims mr than th pak dwnstram cnductr lads. Shhata t al. 7, mplyd th finit lmnt mthd t idntify th critical micrburst paramtrs that lad t maximum frcs in varius mmbrs f a transmissin twr structur. Th rsult shwd that th pak valus bviusly xcd th transmissin twr structur just ncuntrd with nrmal wind lads, and th pak valus wr snsitiv t th dwnburst lcatin with rspct t th twr. Zhang t al. 9 stablishd finit lmnt mdls fr singl twr and transmissin twr-lin systm t simulat wind-inducd prgrssiv cllaps. Th simulatin rsults dmnstratd that th transmissin twr-lin systm cllaps mchanism dpnds n th numbr, psitin, and last dfrmatin f damag lmnts, and th ffcts f th cnductr and th grund culd nt b ignrd. Mara and Hng 10 invstigatd th inlastic rspns f a slf-supprtd lattic transmissin twr undr bundary layr wind and dwnburst wind and wind lading at diffrnt dirctins rlativ t th twr. And nnlinar static pushvr analysis was usd t btain th capacity curv f th twr, dfind by th frc dfrmatin rlatinship at ach cnsidrd wind dirctin. Th rsult shwd that th yild and maximum capacitis vary with wind dirctin and wuld b usful fr th valuatin f th adquacy f xisting twrs undr dwnburst vnts. Kikuchi and cllagus 11,12 singld ut a blw-dwn-typ wind tunnl t study th influnc f havy rainfall cnditins n th ardynamic drag cfficint f vrhad pwr lin. Th drag cfficints wr masurd with diffrnt valus f wind vlcity, incidnt angl, surfac rughnss, and turbulnc intnsity. Th xprimntal rsults btaind in th wind tunnls indicatd that th nwly lctric pwr lin, LP110, was vry ffctiv t rduc th drag frc in a practical us. Yang t al. 13 stablishd a tim-dpndnt failur prbability mdl t valuat shrt-trm rliability f vrhad lins undr th impact f strng wind and rain lads, and tk IEEE-79 systm as an xampl t simulat th rliability f th vrhad lins. Th rsults shwd that th impact f th strng wind and rain lads wuld sriusly affct rliability indics f th transmissin systm, and rain lads hav bvius ffct n th rliability f transmissin lin. Chi 14 studid wind-drivn rain and driving-rain cfficint during thundrstrm and nnthundrstrm vnts at a wind-drivn rain-masuring statin in Singapr. Th bsrvatin f th driving-rain intnsity cfficint indicatd that th idal cfficint calculatd basd n drp siz distributin was nt apprpriat fr lw altituds cls t th grund. Li and Bai 15,16 prpsd a calculatin apprach f rain lad with cmbinatin principl f vrhad transmissin lins and stablishd a rain-wind-inducd dynamic mdl f transmissin twr systm with finit lmnt mthd. By mans f numrical simulatin, thy fund that th rainfall influncs n th rspns f vrhad transmissin lin wr vidnt, which psssss th xcitatin fatur f simultanus actin with wind turbulnc. Zhu t al. 17,1 stablishd an analytical mdl f rain-wind-inducd vibratin f th high-vltag transmissin lin and invstigatd th ffct f wind vlcity, rivult mtin, raindrp vlcity, and tim-varying mass n th vibratin amplitud. Th rsults shwd that th largst amplitud f th high-vltag cnductr nly ccurrd within a crtain rang f wind vlcity and prsntd a vlcityrstrictd vibratin rspns. In th abv litraturs, studis abut th dynamic charactristics f transmissin twr-lins mainly fcusd n th dynamic charactristics f th twr subjctd t dry dwnburst, but th dynamic charactristics f th transmissin lins subjctd t wt dwnburst (dwnburst wind with rainfall) wr rarly invstigatd. Fr this rasn, this articl aims t validat an analytical mdl f th high-vltag transmissin lin subjctd t th dwnburst wind with rainfall t clarify th gallping mchanism. By wind r rain wind tunnl xprimnts, w btaind ardynamic cfficints f th high-vltag cnductr at diffrnt yaw angls and rainfall rats. Cnsidring th variatins f svral factrs, such as wind vlcity, rainfall rat, yaw angl, and attack angl, th prpsd analytical mdl is cratd by mans f finit lmnt mthd and cntral diffrncs with th btaind ardynamic charactristics f th high-vltag cnductr.

3 Zhu and Liu 3 Gnral faturs f th dwnburst wind and rainfall during a thundrstrm Wind vlcity f th dwnburst A dwnburst is a svr lcalizd dwndraft frm a thundrstrm, such that its structur, scal, intnsity f wind, and rainfall cannt b masurd in fild with cnvntinal rcrding statins. 19,20 Fr purpss f analysis, th wind vlcity ccurring at any tim t and any hight y within a dwndraft can b assumd as a summatin f a man wind vlcity and a fluctuating wind vlcity as fllws Uy, ð t Þ= U ðy, tþ+ uy, ð tþ ð1þ whr U(y, t) is th wind vlcity at any tim t and any hight y, U(y, t) is th man wind vlcity, and u(y, t) is th turbulnt wind vlcity. W assum that th man wind vlcity f any tim at any hight can b factrizd as th prduct f a vrtical prfil and a tim functin as fllws U ðy, tþ= VðyÞ3 fðþ t ð2þ whr f (t)= jv c (t) j= maxjv c (t) j, v c is th wind vlcity acting n th bsrving pint f th cnductr, V max is th maximum wind spd at hight y, and y max is th hight at which th maximum vlcity ccurs, 21 and V(y)=1:22 3 ½ 0:15y=y max 3:2175y=y max Š 3 V max. As shwn in Figur 1, th dwnburst is assumd t mv frward alng its straight track with translatinal vlcity v t, th distanc f th dwnburst frm th bsrving pint N is r with yaw angl u. Th crdinat systm in Figur 1 is fixd t th dwnburst cntr f O and mvs with th dwnburst. S, at any tim t, th wind vlcity acting n th bsrving pint f th cnductr is v c = v N + v t ð3þ Th fluctuating wind vlcity u(y, t) can b btaind by amplitud-mdulating sm statinary prcss with pwr spctral dnsity functin as fllws Twr z O v r Dwnburst Twr Cnductr Figur 1. Hrizntal diagram f th dwnburst with transmissin twr-lin. r θ N v t v c x uy, ð t Þ= aðy, tþkðy, tþ ð4þ whr a(y, t) is th mdulatin functin, and w assum a(y, t) =0:25 U(y, t) basd n full-scal and wind tunnl data by rsarchrs at th Wind Scinc and Enginring Rsarch Cntr f Txas Tchnlgy Univrsity. 20 k(y, t)= Ð + ivt dy(y, v) 22 is a statinary Gaussian stchastic prcss, and Y(y, v) is an rthgnal-incrmntal prcss fr a givn y. Raindrp spctra and rain lads Wt dwnbursts ar cratd by thundrstrms with mdrat r havy prcipitatin. 23 Th raindrps hav impact n th transmissin twr-lins with grat nrgy, which aggravats th vibratin f th cnductr. Th nrgy f a raindrp ffct n th cnductr is rlatd t its diamtr, vlcity, and rainfall intnsity. 24 A larg numbr f bsrvatins shw that th raindrp siz bys a ngativ xpnntial distributin. Th Marshall Palmr xpnntial siz distributin (rfrrd t as M-P spctrum) is widly usd 25 as fllws nd ð Þ= N 0 xpð DDÞ ð5þ whr N 0 = (m 3 =mm), gradint factr D = 4:1I 0:21, D is th raindrp diamtr, and I is th rainfall intnsity pr hur as shwn in Tabl 1. As th rang f raindrp diamtr in natural rainfall is mm, 26 t simplify th analysis, w tak six typs f intrmittnt raindrp diamtr t simulat cntinuus distributin f raindrp diamtr f rainfall. Diamtrs f ach typ f raindrp and its rprsntd rang f diamtr ar shwn in Tabl 2. Crrspndingly, th vlum ccupancy rati f th diffrnt sizs f raindrps in wind fild can b xprssd as g = ð6:0 0:1 1 6 pd3 nd ð ÞdD ð6þ And th trminal vlcity f th raindrp, V m, with th diffrnt sizs f raindrps, 27 can b xprssd as :77 D ffiffiffi 2 p D\1 mm D V m = pffiffiffiffiffiffiffiffiffiffi ð17:2 0:44DÞ 0:1D 1 mm\d\3 mm >: D ð0: :045DÞ 3 mm\d\6 mm ð7þ Onc a raindrp hits th cnductr, th vlcity f th raindrp bcms zr within an xtrmly shrt tim t. Basd n Nwtn s scnd law, th calculatin frmula can b drivd as

4 4 Advancs in Mchanical Enginring Tabl 1. Intnsity f rainfall (mm/h). Lvl S M Larg H Hu Rainfall rat S: slight; M: mdrat; L: larg; H: havy; Hu: hug rain. Tabl 2. Rprsntd rang f diamtr with ach typ f raindrp (mm). D R D: diamtr; R: rprsntd rang. FðÞ= t mv m = 1 t 6t r 1pD 3 V m ðþ whr r 1 is th dnsity f th raindrp, and assuming t = D=2V r, th prssur frc acting n th cnductr pr unit lngth by th raindrps can b xprssd as A y z x v w l B F r = FðÞ t A bg = 2 9 r 1pnD 3 Vm 2 ð9þ whr th actin ara f a raindrp is A = pd 2 =4, sctinal width f lading structur is b, and n = Ð 0 n(d)dd is th dnsity f ttal numbrs f raindrps pr unit vlum. Crss-sctin f cnductr z y Usinθ w& U r β v& Dwnburst wind with rainfall Analytical mdl f high-vltag cnductr with finit lmnt discrtizatin Analytical mdl f high-vltag cnductr subjctd t dwnburst wind with rainfall Lt us us a rigid and unifrm cabl t rprsnt a suspndd high-vltag cnductr with small sag, and th tw twrs (A and B) at bth nds f cabl ar assumd t b fixd (s Figur 2). Th crdinats f any pints alng th cnductr ar rprsntd by using Cartsian systm, which dfins that th mtin in Oxy is in-plan vibratin, whil th mtin in Oyz is ut-f-plan vibratin. Th dynamic displacmnts at y and z dirctins ar v and w, rspctivly. Cnsidring th in-plan and ut-f-plan mtin vlcitis f th cabl, _v and _w, th rlativ >: m 2 v t 2 + c v s m 2 w t 2 Figur 2. Schmatic rprsntatin f th cnductr and its crss sctin subjctd t th dwnburst with rainfall. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U r = ðu sin u _w Þ 2 + _v 2 ð10þ Th attack angl btwn th rlativ vlcity U r and th dwnburst wind vlcity U is dfind as b, which can b xprssd as _v b = arctan ð11þ U sin u _w Taking small displacmnts and larg dfrmatins as th basic assumptin fr th cnductr, th analytical mdl f th cnductr subjctd t dwnburst wind with rainfall is built by partial diffrntial quatin as fllws t ðh + hþ dy x + dv = 1 2 dru r 2 ðc L ðbþcs b + C D ðbþsin bþ+ 2 9 r 1pnD 3 ðv m _v Þ 2 + c w s t ðh + hþ dw = 1 x 2 dru r 2 ðc D ðbþcs b C L ðbþsin bþ+ 2 9 r 1pnD 3 ðu sin u _w Þ 2 ð12þ vlcity f th dwnburst wind t th cabl can b xprssd as whr y = mg(xl x 2 )=2H is th prfil f th cnductr givn by th parabla; 2 H and h ar th hrizntal

5 Zhu and Liu 5 static tnsin and dynamic tnsin f th cnductr, rspctivly; c s is th damping rati f th cnductr in th y and z dirctins, rspctivly; r is th air dnsity; d is th diamtr f th cnductr; and C L and C D ar th lift and drag cfficints, rspctivly. Th frcs in th right-hand sid f quatin (12) ar th functin f th cnductr vlcity, _v and _w, but nt f its displacmnt r acclratin, sinc it is basd n th quasi-stady apprach. Thrfr, th frcs ar xpandd int Taylr s sris at b = 0, and th itms highr than first rdr ar nglctd. Whn b 0 and U sin u _w, b =arctan(_v=(u sin u _w)) _v=(u sin u). Thus >: m 2 v t 2 + c v y m 2 w t 2 t ðh + hþ dy x + dv + c w z t ðh + hþ dw = F z x = F y ð13þ whr F y = (1=2)drU 2 sin 2 uc L +(2=9)r 1 pnd 3 Vm 2, F z =(1=2)drU 2 sin 2 u(c D +((dc D =db) C L )(_v=u))+f 0, and F 0 =(2=9)r 1 pnd 3 U 2 sin 2 u. c y = c s + c a y and c z = c s + c a z dnt th ttal damping cfficints at crdinats f Oxy and Oyz, rspctivly. Ardynamic damping cfficints at crdinats f Oxy and Oyz ar dntd by c a y =(1=2)drU sin u((dc L=db)+C D )+ (4=9)r 1 pnd 3 V m and c a z = (4=9)r 1pnD 3 U sin u, rspctivly. Th shap functin, N(x), is dscribd as NðÞ= x ½N 1 ðþe x 3 N 2 ðþe x 3 N 3 ðþe x 3 Š ð16þ N 1 = ð1 2x=l Þð1 x=l Þ 4x N 2 = l ð1 x=l Þ ð17þ x >: N 3 = l ð1 2x=l Þ whr E 3 is th idntity matrix, N 1 =(1 2x=l )(1 x=l ), N 2 = 4x=l (1 x=l ), N 3 = x=l (1 2x=l ), and l is th lngth f th lmnt. Thus, in-plan and ut-f-plan mvmnts f th high-vltag cnductr f quatin (13), subjctd t a distributd wind rain lading, can b rprsntd by th fllwing quatins m 2 vx, ð tþ vx, ð tþ t 2 + c y ðh + ht ðþþ dv x, t + = F y ðx, y, tþ t x m 2 wx, ð tþ wx, ð tþ t 2 + c z dw x, t ðh + ht ðþþ >: = F z ðx, y, tþ t x ð1þ Finit lmnt discrtizatin A cnductr suspndd btwn tw twrs is discrtizatd int a numbr f thr-nd isparamtric lmnts. Th cabl lmnt is rfrrd t th crdinats f Oxyz as shwn in Figur 3. Th displacmnt vctr Bundary cnditin: v(x, t)=w(x, t)=0j x = 0 and f lmnt nds, q, is dfind as v(x, t)=w(x, t)=0j x = l and initial cnditin: v(x, t)=v(x)j t = 0 and w(x, t)=w(x)j t = 0. q = fq 1, q 2, q 3 g T ð14þ Cnsquntly, quatin (1) is slvd by Galrkin mthd lading t a dcras in th intgratin rdr. whr q i = fq u i, qv i, qw i gt i = 1, 2, 3. Th typical lmnt pndrd rsidual quatin can b Isparamtric rprsntatins f th crdinats f writtn as fllws Oxyz and th displacmnts vr an lmnt ar givn, ð rspctivly, by Rx, ð t; aþn i ðþ x = 0 fx, y, zg = P3 N i ðx i, y i, z i Þ T ð19þ i = 1 ð15þ whr >: fu, v, wg = P3 N i ðu i, v i, w i Þ T i = 1 m 2 vx, ð tþ vx, ð tþ t Rx, ð t; aþ= + c y ðh + ht ðþþ dv x, t + F y ðx, y, tþ t x m 2 wx, ð tþ wx, ð tþ t 2 + c z dw x, t ðh + ht ðþþ >: F z ðx, y, tþ t x H y A z Wind N ( x, y, z ) u, v, w i 1 i 1 i 1 i 1 i 1 i 1 i 1 l N ( x, y, z ) i i i i u, v, w i+ 1 i+ 1 i+ 1 N ( x, y, z ) i+ 1 i+ 1 i+ 1 i+ 1 Figur 3. Discrtizatd mdl f th high-vltag cnductr with finit lmnt mthd. u, v, w i i i B H x

6 6 Advancs in Mchanical Enginring Th intgral part can b writtn as fllws Ð N i ðþm x 2 v ðx, tþ t + Ð v N 2 i ðþc x ðx, tþ y t Ð h i N i ðþ x x ðh + ht ðþþ dyðþ x dv x, t + ð Þ = Ð N i ðþf x y ðx, y, tþ Ð N i ðþm x 2 w ðx, tþ t + Ð w N 2 i ðþc x ðx, tþ z t Ð h i N i ðþ x dw x, t >: x ðh + ht ðþþ ð Þ = Ð N i ðþf x z ðx, y, tþ ð20þ Taking an apprximat slutin f th prblm such that vx, ð t; aþ= Pn q v j ðþn t j ðþ x j = 1 >: wx, ð t; aþ= Pn q w j ðþn t j ðþ x j = 1 ð21þ Substituting quatin (21) int quatin (20), th fllwing can b writtn ½MŠ d 2 q v ðþ t dt 2 >: ½MŠ d 2 q w ðþ t dt 2 whr F w ðþ t = K ij = ð + ½C v + ½C w dn i Š dq v ðþ t dt Š dq w ðþ t dt + ½KŠ q v ðþ= t F y ðþ t + ½KŠ q w ðþ= t F z ðþ t ðþ x ðh + ht ðþþ dn j ðþ x ð M ij = C v ij = ð N i N i ðþmn x ðþ x ðþc x y N ðþ x j j ð22þ ð dw x, t N i ðþf x z ðx, y, tþ + ðh + ht ðþþ ð Þ xi + 1 N i ðþ x ð F y ðþ t dv x, t = N i ðþf x y ðx, y, tþ + ðh + ht ðþþ ð Þ xi + 1 N i ðþ x x i ð + N i ðþ x ðh + ht ðþþ dyðþ x x C w ij = ð N i Aftr assmbling all lmnts ½MŠ f q g+ ½CŠ _q ðþc x z N ðþ x f g+ ½K j Šfqg= ffg x i ð23þ whr q i (t) rprsnts th displacmnts f ach nd, and quatin (23) can b numrically slvd using th cntral diffrnc mthd. At th cntral tim t n 1 ½MŠ f qðþ tg n 1 + ½Ct ðþ n 1 Š f_qðþ tg n 1 ð24þ + Kt fqt ðþgn 1 = fft ðþg n 1 ðþ n 1 Th tw drivativs can b apprximatd by cntral diffrncs f_qðþ tg n 1 = fqt ðþ f qðþ tg n 1 = fqt ðþ g n g n f qt ðþ 2Dt g n 2 2 f qt ðþg n 1 + fqt ðþ Dt 2 g n 2 ð25þ ð26þ Substituting quatins (25) and (26) int quatin (24) yilds 1 Dt 2 ½MŠ+ 1 2Dt ½Ct ðþš n 1 fqt ðþg n = ½Kt ðþš n 1 2 Dt 2 ½MŠ fqt ðþg n 1 1 Dt 2 ½MŠ 1 2Dt ½Ct ðþš n 1 fqt ðþg n 2 + fft ðþg n 1 ð27þ By slving quatin (27), it is pssibl t find th displacmnts f ach nd, and thn ach q i (t) can b summd accrding t quatin (21) t prduc v(x, t) and w(x, t), rspctivly. Exprimntal tst and numrical studis In rdr t clarify th gallping mchanism f th highvltag transmissin lin subjctd t th dwnburst wind with rainfall, xprimntal tsts shuld b carrid ut with tw cnditins, that is, n is th cnductr ncuntrd nly with th dwnburst wind, and th thr is th cnductr ncuntrd nly with th dwnburst wind and rainfall. Crrspndingly, th prpsd analytical mdl is applid t simulat th mtin f an actual cnductr with th btaind ardynamic cfficints frm th xprimntal tsts and is vrifid by th cmparisn with fild masurmnts.

7 Zhu and Liu 7 Th high-vltag cnductr ncuntrd nly with th dwnburst wind T start with th simplst cas, in this sctin, w invstigat ardynamic bhavirs f th high-vltag cnductr subjctd t dry dwnburst wind. Whn rainfall is nt cnsidrd, th raindrp lad is st t zr, and th mtin f th high-vltag cnductr is n lngr rlativ t th rainfall. Th mtin quatin f th high-vltag cnductr ncuntrd nly with th dwnburst bcms ring givs th instantanus sctinal fluid frc within th tap ring sctin. Th lift and drag cfficints f th tst mdl vrsus attack angl b with diffrnt yaw angls u f 10, 20, 30, and 40 ar shwn in Figur 5. In gnral, it can b sn that as attack angl b incrass du t th incras in th wind vlcity, th magnituds f th drag cfficint stay mr r lss in th sam rang, whras th magnituds f lift cfficint incras gradually. As th attack angl b rachs a crtain lvl (b 29 ), th drag m 2 v t 2 + c s dru sin u dc L v db + C D t ðh + hþ dy x + dv = 1 2 dru 2 sin 2 uc L m 2 w t 2 + c w s t ðh + hþ dw = 1 x 2 dru 2 sin 2 u C D + dc D db C _v >: L U sin u ð2þ Whn rainfall is nt cnsidrd, th ttal damping cfficints and th xtrnal frc may chang with th yaw angl u, and th attack angl b that, in turn, dpnds n th cnductr vlcity, _v, as sn frm quatin (2). As shwn in Figur 4, fr th sak f rlatins btwn th yaw angl u, th attack angl b, and th ardynamic cfficints f th high-vltag cnductr, a wind tunnl tst is st up in an pn-circuit tunnl with an riginal wrking sctin f 1.3 m (width) m (hight) and a maximum wind vlcity f 50 m/s. In this wind tunnl tst, an aluminum stl cnductr is usd as th tst mdl, which is placd in a hrizntal plan prpndicular t ncming wind dirctin and yawd tward it by an angl u. Fr th purps f cmparisn with arlir studis, 17,29 th chsn paramtrs f th tst mdl is 1. m in lngth and 0.03 m in diamtr, th mass f th cnductr mdl is 1.65 kg, th wind vlcity is frm 5 t 50 m/s, and k=d 0:02. Th tst mdl is hangd up by tw pairs f springs at bth nds, and th springs ar rstd n a vrtical squar-shapd fram. By sliding th squar-shapd fram, th yaw angl u can b adjustd. Th tw pairs f springs ar in th plan prpndicular t th tst mdl axis and prpndicular t ach thr. Th springs ar dsignd t simulat th in-plan and ut-f-plan tst mdl mtins, f which th frquncis ar slightly diffrnt and cntrlld by th stiffnss f th springs. At ach nd f th tst mdl, tw nn-cntact-typ lasr vlcity snsrs and tw cntact-typ displacmnt snsrs ar installd t masur th vlcitis and displacmnts alng th in-plan and ut-f-plan dirctins, rspctivly. Thr prssur tap rings, cntaining 16 taps circumfrntially, ar arrangd at diffrnt lngitudinal lcatins. Th plan f ach tap ring is prpndicular t th tst mdl axis, and th timvarying surfac prssur f all th taps n th sam tap cfficint C D drps sharply, and ngativ slps f th lift cfficint curvs ccur. Whn b 29, th wind vlcity is narly 26 m/s and R 5: (k=d 0:02), which is cls t th critical Rynlds numbr rang. 29 Whn th wind angl f attack is b.29, th magnituds f drag cfficint stay mr r lss cnstant again, and th magnituds f lift cfficint incras gradually. As ach curv assciatd with a spcific yaw angl u, th magnituds f ardynamic cfficints at diffrnt yaw angls can b idntifid. By th cmparisn f ths curvs, it can asily b fund that th magnitud f th lift cfficint at a yaw angl f 30 is apparntly largr than that at yaw angls f 10, 20, and 40, and th critical yaw angl pssibly is in th vicinity f yaw angl f 30. T invstigat th capability f th analytical mdl fr rvaling th gallping mchanism, th abv finit lmnt mthd is applid n a high-vltag transmissin lin. As an xampl, th ky paramtrs f th cnductr ar as fllws: diamtr f 30 mm, mass pr unit lngth f 1.65 kg/m, span f 255 m, lasticity cfficint f 73 kn/mm 2, tnsin frc f :31 kn, structural damping rati f 2%, and dnsity f air f 1.25 kg/m 3. Basd n th typical curvs btaind frm Figur 5, w xpand th drag cfficint and lift cfficint in quatin (2) as functins by th first thr trms f Taylr s sris C L ðbþ= a i + b i b + c i b 2 + d i b 3 C D ðbþ= i + f i b + g i b 2 + h i b 3 ð29þ whr i = 10, 20, 30, 40. Applying th functins f th drag cfficint and lift cfficint (quatin (29)) int quatin (2) at diffrnt yaw angls, w can btain th rspns amplitud f th cnductr. Frm Figur 6(a), it can b fund that th in-plan larg rspns amplitud ccurs nc th wind vlcity rachs a crtain lvl f abut 26 m/s, but ut f this rang, th cnductr has vry small

8 Advancs in Mchanical Enginring Figur 4. Exprimntal stup f wind tunnl tst: (a) phtgraph f wind tunnl tst dvic with tst mdl and (b) schmatic rprsntatin f tst mdl in wind tunnl tst.

9 Zhu and Liu 9,CL CD Ardynamic cfficint θ = 10 θ = 20 θ = 30 θ = 20 θ = 30 θ = Attack angl β (in dgrs) vibratin amplitud. This rsult is in gd agrmnt with fild masurmnts f 500-kV flashvr causd by windag yaw at th wind vlcity f m/s. 30 This phnmnn can b xplaind by th fact that at th critical vlcity f 26 m/s, th Rynlds numbr is cls t th critical rang, and th ttal damping cfficint gts a ngativ valu. Th in-plan amplitud f vibratin f th cnductr, at th critical vlcity f 26 m/s, varis at diffrnt yaw angls. This is bcaus th valus f lift cfficint ar smwhat diffrnt with th yaw angls f 10, 20, and 40, and th critical yaw angl pssibly ccurs in th vicinity f yaw angl f 30. Figur 6(b) shws th ut-f-plan vibratin amplitud f th cnductr vrsus wind vlcity. Th vibratin amplitud f th cnductr incrass with th wind vlcity, and th larg rspns amplitud ccurs nc th wind vlcity rachs a crtain lvl f abut 15 m/s. Th larg amplitud vibratin rmains cnstant with furthr incras in th wind vlcity, whil a sharp drp f th vibratin amplitud ccurs at th critical wind vlcity f 26 m/s. This rsult is in gd agrmnt with fild masurmnts f accidnts f 500-kV flashvr causd by windag yaw at th wind vlcity f m/s. 30 Bynd th critical wind vlcity, th amplitud vibratin rmains cnstant again with furthr incras in th wind vlcity. This rsult can b xplaind by th fact that th valus f th drag cfficint incras with th wind vlcity (whn U \ 15 m/ s), and almst rmain cnstant within a crtain rang f wind vlcity (15 \ U \ 26 m/s), and suddnly drp at th critical wind vlcity f 26 m/s. Thy rmain cnstant again as th wind vlcity is bynd th critical valus. As shws in Figur 6(a) and (b), th attack angl b almst incrass linarly with th wind vlcity but has a suddn chang at a wind vlcity f 26 m/s, and btains a valu f 29. Th rasn why th attack angl CD CL θ = 40 θ = 40 Figur 5. Ardynamic cfficints vrsus attack angl b. In-plan vibratin amplitud(m) Out-f-plan vibratin amplitud(m) θ = 30 θ = 20 θ = 10 θ = Wind vlcity (m/s) 0.30 θ = β (a) Wind vlcity (m/s) (b) θ = 40 θ = 30 b has a suddn incras is that whn th wind vlcity is cls t th critical wind vlcity f 26 m/s, th ttal damping cfficint btains a ngativ valu, and cnductr vlcity _v has a suddn incras. Th high-vltag cnductr ncuntrd nly with th dwnburst wind and rainfall T invstigat th capability f th analytical mdl fr prdicting th mtin f th cnductr subjctd t th dwnburst wind with rainfall, a rainfall simulatr is urgntly ndd t study th ardynamic cfficints f th tst mdl at varid wind vlcity, with diffrnt rainfall intnsitis. As shwn in Figur 7, th facility f rainfall similitud xprimnt is muntd n th abv wind tunnl tst and lcatd windward in th tst mdl. Th uppr watr pip is dsignd t cnsist f tw vrtical FullJt nzzls, and th nzzls ar assmbld by thr diffrnt sizs f 1/, 2/, and 3/. Figur shws a phtgraph f simulatd raindrps in wind tunnl, and th raindrps ar rlasd frm th nzzls. By cmbinatin f fur nzzls with diffrnt sizs, w achiv a rainfall intnsity adjustabl dvic frm slight rain t havy rain, which maks th rainfall simulatr similar t natural rainfall. A sssin f an xprimnt bgins with submrsibl pumps t pump watr up frm th sink t watr pips. Cntrl valvs installd at th lwr watr pips ar pnd in a scnd β θ = 20 Figur 6. (a) In-plan vibratin amplitud vrsus wind vlcity and (b) ut-f-plan vibratin amplitud vrsus wind vlcity Attack angl β (dg) Attack angl β (dg)

10 10 Advancs in Mchanical Enginring Figur 7. Schmatic rprsntatin f rain wind tunnl tst. Ardynamic cfficint CD,CL mm/h 2.5 mm/h mm/h mm/h 2.5 mm/h 0 mm/h 16 mm/h CD CL 0 mm/h Attack angl β (in dgrs) 60 Figur 9. Ardynamic cfficints vrsus attack angl b (whn u = 30 ). Figur. Phtgraph f rain wind xprimnt with nzzls. and thn simulatd raindrps ar drivn by th prssur frc and th gravitatinal frc, flwing frm th nzzls t wind tst. Th vlcity f th simulatd raindrps can b changd discrtly by th cntrl valvs. As simulatd raindrps fall int wind fild and th tst cnductr mdl is placd in th middl f wind tunnl tst, a rain wind tunnl tst is stablishd. Th lift and drag cfficints f th tst mdl vrsus wind angl f attack b, at a yaw angl f 30, varying with rainfall rats f 0, 2.5,, and 16 mm/h, ar shwn in Figur 9. Fr rainfall rats f 2.5,, and 16 mm/h, it can b fund that as attack angl b incrass with th incras in th wind vlcity, th magnituds f th drag cfficint gradually dcras, whras th magnituds f th lift cfficint gradually incras. Manwhil, th magnitud f th drag cfficint stays mr r lss in th sam rang at a rainfall rat f 0 mm/h. Pssibl rasn

11 Zhu and Liu 11 fr this diffrnc may b th dcras in th surfac rughnss f th tst mdl du t th rainfall. Whn th angl b is narly 22, th drivativs f lift cfficints with rainfall rats f and 16 mm/h hav a suddn chang frm psitiv valus t ngativ valus, whras th drivativs f drag cfficints chang frm ngativ valus t psitiv valus. This phnmnn is smhw alik th rain-wind-inducd vibratin, 17 fr th rasn is whn th wind vlcity is abut 10 m/s and th attack angl is narly 22, and rivults asily ccur n th surfac f th tst mdl at th rainfall rats f and 16 mm/h. At th rainfall rats f 2.5,, and 16 mm/h, whn th attack angl b rachs a crtain lvl (b 45 ), th drag cfficint C D drps sharply, and ngativ slps f th lift cfficint curvs ccur again. Whn th attack angl b 45, th wind vlcity is abut 40 m/s and R (k=d 0:02), which is cls t th critical Rynlds numbr rang. 29 Whn th wind angl f attack b.45, th lift and drag cfficints stay mr r lss cnstant. By cmparing ths curvs, it can asily b fund that critical Rynlds numbr f th tst mdl with rainfall (2.5,, and 16 mm/h) is vidntly highr than that f th tst mdl with rainfall (0 mm/h). This diffrnc can b xplaind that th rainfall allviat th surfac rughnss f th tst mdl. T study th capability f th analytical mdl fr th high-vltag cnductr ncuntrd nly with th dwnburst wind and rainfall, th abv drivd finit lmnt mthd is applid. Fr cmparisn purps, th ky paramtrs f th cnductr ar similar t th xampl shwn in th abv-mntind sctin. Basd n th typical curvs btaind frm Figur 9, xpand th drag and lift cfficints in quatin (13) as functins with rspct t b using th first thr trms f Taylr s sris C L ðbþ= A i + B i b + C i b 2 + D i b 3 C D ðbþ= E i + F i b + G i b 2 + H i b 3 ð30þ whr i = 0, 2:5,, 16. Applying th functins f th drag cfficint and lift cfficint (quatin (30)) int quatin (13) with diffrnt rainfall rats, w can btain th rspns amplitud f th cnductr. Frm Figur 10(a), it can b fund that th in-plan largst amplitud ccurs nc th wind vlcity rachs a crtain lvl f abut 10 m/s, but ut f this rang, th cnductrs btain smallr vibratin amplitud. This rsult is in gd agrmnt with th xprimntal rsults. 17 Th rsult can b xplaind by th fact that at th critical vlcity f 10 m/s, th rivults ar frmd n th surfac f th cnductrs at th rainfall rats f and 16 mm/h, whras rainfall is nt nugh t frm rivult whn rainfall rats ar 0 and 2.5 mm/h. Furthrmr, inplan largr rspns amplitud ccurs again nc th In-plan vibratin amplitud(m) In-plan vibratin amplitud(m) mm/h 2.5 mm/h 0 mm/h 0 Yaw angl θ = mm/h Wind vlcity (m/s) (a) mm/h 2.5 mm/h Wind vlcity (m/s) (b) 0 mm/h wind vlcity rachs a crtain lvl f abut 40 m/s, but ut f this rang, th cnductr has vry smallr vibratin amplitud. This rsult is in gd agrmnt with fild masurmnts f accidnts f twr cllapss f 500-kV Rnshang 5237 transmissin lin causd by dwnburst at wind vlcity f m/s. 31 Th rasn can b xplaind by th fact that at th critical vlcity f 40 m/s, th Rynlds numbr is cls t th critical rang, and th ttal damping cfficint btains a ngativ valu. Furthrmr, th in-plan amplitud f vibratin f th cnductr, at th critical vlcity f 40 m/s, varis with diffrnt rainfall rats. This is bcaus th valus f lift cfficints ar smwhat diffrnt with th rainfall rats f 2.5,, and 16 mm/h, and th critical rainfall rat pssibly is in th vicinity f rainfall rat f mm/h. Figur 10(b) shws th ut-f-plan vibratin amplitud f th cnductr vrsus wind vlcity. Th vibratin amplitud f th cnductr incrass with th wind vlcity, and th larg rspns amplitud f th rainwind-inducd vibratin ccurs nc th wind vlcity rachs a crtain lvl f abut 10 m/s, but ut f this rang, th cnductr has lwr vibratin amplitud. Aftr th critical wind vlcity f 10 m/s, th vibratin amplitud f th cnductr incrass with th incras in th wind vlcity f up t 15 m/s (rainfall rat f 0 mm/h) r 19 m/s (rainfall rats f 2.5,, and 16 mm/h). Bynd this thrshld, th amplitud vibratin narly rmains cnstant with furthr incras in th wind vlcity, whras a sharp drp f th vibratin β 0 Yaw angl θ = 30 β 16 mm/h Figur 10. (a) In-plan vibratin amplitud vrsus wind vlcity and (b) ut-f-plan vibratin amplitud vrsus wind vlcity Attack angl β (dg) Attack angl β (dg)

12 12 Advancs in Mchanical Enginring amplitud ccurs at th critical wind vlcity f 26 m/s (rainfall rat f 0 mm/h) r 40 m/s (rainfall rats f 2.5,, and 16 mm/h). This rsult can b xplaind by th fact that th valus f th drag cfficint incras with th wind vlcity (whn U \ 15 m/s, at th rainfall rat f 0 mm/h), and almst rmain cnstant within a crtain rang f wind vlcity (15 \ U \ 26 m/s), and suddnly drp at th critical wind vlcity f 26 m/s. Thy rmain cnstant again as th wind vlcity is bynd th critical valus. Manwhil, th valus f th drag cfficint incras with th wind vlcity (whn U \ 19 m/s, fr rainfall rats f 2.5,, and 16 mm/h), and almst rmain cnstant within a crtain rang f wind vlcity (19 \ U \ 40 m/s), and suddnly drp at th critical wind vlcity f 40 m/s. Thy rmain cnstant again as th wind vlcity is bynd th critical valus. As shws in Figur 10(a) and (b), th attack angl b (with diffrnt rainfall rats) almst incrass linarly with th wind vlcity but has a suddn chang at th wind vlcitis f 10 and 40 m/s, rspctivly. Whn th wind vlcity is 10 m/s, th attack angl b btains a pak valu f 22. This phnmnn is smhw alik th rain-wind-inducd vibratin, and th rivults asily ccur n th surfac f th tst mdl. Whn th wind vlcity is 40 m/s, th attack angl b btains a pak valu f 45. This is bcaus th wind vlcity is cls t th critical Rynlds numbr rang, th drag cfficint C D drps sharply, and ngativ slps f th lift cfficint curvs ccur again. Cnclusin An analytical mdl fr dscribing gallping f th high-vltag transmissin lin subjctd t dwnburst wind with rainfall is prpsd in this articl. Th analytical mdl is applid t simulat th mtin f an actual cnductr with th btaind ardynamic cfficints frm th xprimntal tsts and is vrifid by th cmparisn with fild masurmnts. 1. Th analytical mdl is validatd by cmparing numrical rsults with th xprimntal data, which can captur main charactristics f gallping vibratin f th high-vltag cnductr ncuntrd with th dwnburst wind with rainfall. 2. Whn high-vltag cnductr ncuntrd nly with th dwnburst wind, in-plan r ut-fplan larg rspns amplitud ccurs nc th wind vlcity rachs a crtain lvl f abut 26 m/s, but ut f this rang, th cnductr has vry small vibratin amplitud. Furthrmr, th critical yaw angl pssibly ccurs in th vicinity f yaw angl f Whil high-vltag cnductr ncuntrd nly with th dwnburst wind with rainfall, in-plan r ut-f-plan largst amplitud ccurs nc th wind vlcity rachs a crtain lvl f abut 10 m/s, but ut f this rang, th cnductr btains smallr vibratin amplitud. Furthrmr, in-plan r ut-f-plan largr rspns amplitud ccurs again nc th wind vlcity rachs a crtain lvl f abut 40 m/s, but ut f this rang, th cnductr has vry smallr vibratin amplitud. It shuld b ntd that th prpsd analytical mdl is still a prliminary mdl. Th fluctuating wind, twrs, and span ffcts ar nglctd in this articl. Sm assumptins in th mdl nd t b rlasd in th furthr study. Mr ralistic wind rain tunnls r fild masurmnts guidd by th prpsd analytical mdl nd t b dvlpd. Dclaratin f cnflicting intrsts Th authrs dclar that thr is n cnflict f intrsts rgarding th publicatin f this articl. Funding Th rsarch prsntd in this articl was supprtd by th Natural Scinc Fundatin f Yuth Fund f China (n ), th Bijing Natural Scinc Fundatin (n ), and th Fundamntal Rsarch Funds fr th Cntral Univrsitis (n. 2014ZD07). Rfrncs 1. Su YB and Zhang WL. Rsarch f ky tchnlgis fr UHV transmissin lin. Chin J Elctr Eng 2007; 27(31): Li HN and Bai HF. High-vltag transmissin twr-lin systm subjctd t disastr lads. Prcss Nat Sci 2006; 16(9): Dmpsy D and Whit H. Winds wrak havc n lins. Transm Distrib Wrld 1996; 4(6): Olivr SE, Mriarty WW and Hlms JD. A risk mdl fr dsign f transmissin lin systms against thundrstrm winds. Eng Struct 2000; 22(9): Savry E, Park G, Zinddini M, t al. Mdling f trnad and micrburst-inducd wind lading and failur f a lattic transmissin twr. Eng Struct 2001; 23(4): Lin WE, Savry E, McIntyr RP, t al. Th rspns f an vrhad lctrical pwr transmissin lin t tw typs f wind frcing. J Wind Eng Ind Ard 2012; 100(1): Shhata AY, El Damatty AA and Savry E. Finit lmnt mdling f transmissin lin undr dwnburst wind lading. Finit Elm Anal Ds 2005; 42(1): Shhata AY, Nassf AO and El Damatty AA. A cupld finit lmnt-ptimizatin tchniqu t dtrmin

13 Zhu and Liu 13 critical micrburst paramtrs fr transmissin twrs. Finit Elm Anal Ds 200; 45(1): Zhang ZQ, Li HN, Li G, t al. Th numrical analysis f transmissin twr-lins systm wind-inducd cllapsd prfrmanc. Math Prbl Eng 2013; 2013: Mara TG and Hng HP. Effct f wind dirctin n th rspns and capacity surfac f a transmissin twr. Eng Struct 2013; 57: Eguchi Y, Kikuchi N, Kawabata K, t al. Drag rductin mchanism and ardynamic charactristics f a nwly dvlpd vrhad lctric wir. J Wind Eng Ind Ard 2002; 90(4): Kikuchi N, Matsuzaki Y, Yukin T, t al. Ardynamic drag f nw-dsign lctric pwr wir in a havy rainfall and wind. J Wind Eng Ind Ard 2003; 91(1): Yang Q, Xing XF, Wi YN, t al. Shrt-trm rliability valuatin f transmissin systm undr strng wind and rain. J Pwr Enrg Eng 2014; 2(4): Chi ECC. Wind-drivn rain and driving rain cfficint during thundrstrms and nn-thundrstrms. J Wind Eng Ind Ard 2001; 9(3): Li HN and Bai HF. Dynamic bhavir and stability f transmissin twr-lin systm undr wind (rain) frcs. China Civ Eng J 200; 41(11): Bai HF and Li HN. Dynamic rspns f vrhad transmissin lins t scillatin causd by wind r rainfall lads. Pwr Syst Tchnl 2009; 33(2): Zhu C and Liu YB. Analytical mdl f rain-wind inducd vibratin f high-vltag transmissin lin. Adv Mch Eng 2014; 6: Zhu C, Liu YP and Sng YW. Mchanism and mdling f rain-wind inducd in-plan vibratin n highvltag transmissin lin. J Mch Sci Tchnl 2014; 2(4): Hlms JD and Olivr SE. An mpirical mdl f a dwnburst. Eng Struct 2000; 22(9): Chn L and Ltchfrd CW. A dtrministic-stchastic hybrid mdl f dwnbursts and its impact n a cantilvrd structur. Eng Struct 2004; 26(5): Vicry DD. Assssmnt f micrburst mdls fr dwndraft stimatin. J Aircraft 1992; 29(6): Pristly MB. Evlutinary spctra and nn-statinary prcsss. J Ry Stat Sc B Stat Mth 1965; 27(2): Fujita TT. Th dwnburst: micrburst and micrburst. Chicag, IL: Univrsity f Chicag, 195, vl Abuku M, Janssn H, Psn J, t al. Impact absrptin and vapratin f raindrps n building facads. Build Envirn 2009; 44(1): Gunn R and Kinzr GD. Th trminal vlcity f fall fr watr drplts in stagnant air. J Mtrl 1949; 6(4): Yang JT and Lu WJ. Rsarch n wind-drivn rain CFD simulatin and mthd calculating man rain lad. Acta Ardyn Sin 2011; 29(5): Gng L. Ersin calculatin f raindrps s kintic nrgy f lss platau s rainfall. J Lanzhu Jiatng Univ (Nat Sci) 2005; 24(4): Shn SC, Xu CB and Zha C. Dsign f cabl structurs. Bijing, China: China architctur & Building Prss, 1997, pp Simiu E and Scanlan RH. Wind ffcts n structurs: fundamntals and applicatins t dsign. Nw Yrk: Jhn Wily & Sns, Xia DB. Analysis and prvntin f 500 kv flashvr causd by windag yaw. Pwr Syst Tchnl 2009; 33(5): Xi Q, Zhang Y and Li J. Invstigatin n twr cllaps f 500 kv Rnshang 5237 transmissin lin causd by dwnburst. Pwr Syst Tchnl 2006; 30(10):