Paper number: #040 Analysis of Very Fast Transients in Transformer. Shigeto Fujita

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1 Paper number: #4 naly of Very Fat Tranent n Tranformer Yohazu Shbuya Department of Electrcal Engneerng Shbaura Inttute of Technology Shbaura, Mnato-u, Toyo , JPN Shgeto Fujta dvanced Technology R&D Center Mtubh Electrc Corporaton 8-- Tuaguch-Honmach magaa, , JPN Ech Tama Tranmon & Dtrbuton Tranportaton Sytem Center Mtubh Electrc Corporaton 65 Tenwa, o, , JPN btract - practcal calculaton method developed to calculate the poble voltage ocllaton n tranformer ubjected to very fat tranent overvoltage (VFTO) generated by operatng a dconnector n ga nulated wtchgear (GIS). It baed on the multconductor tranmon-lne (MTL) theory for a hell-form tranformer. In the frequency doman analy, the hgh frequency charactertc of a tranformer are calculated ung the dtrbuted contant obtaned from the wndng geometry. The calculated frequency charactertc of nterturn voltage agree wth the expermental up to everal megahertz. The nterturn voltage waveform for a gven VFTO pule are calculated ung FFT. The tme-doman reult are found to be partcularly n good agreement wth the expermental. The feature of th method that the hgh frequency repone of tranformer can be calculated drectly from the wndng geometry, o that the effect of VFTO on tranformer hould be aeed durng degnng. Keyword: Very fat tranent overvoltage, Tranformer, Multconductor tranmon lne, Tranent naly, Modellng. I. INTRODUCTION Very fat tranent overvoltage (VFTO) generated by operatng a dconnector n ga nulated wtchgear (GIS) could caue a voltage ocllaton nde the tranformer connected. The tuaton everer n the ytem that the man tranformer drectly connected to GIS nce the hgh frequency urge travel more ealy along the coaxal ga nulated bu. Partcularly, t mportant to ae the nterturn nduced voltage level nce t the mot vulnerable pont n th tuaton. Lumped crcut model have been ued to analye the tranformer tranent aocated wth the lghtnng or wtchng urge [-4]. For the analy nvolvng the VFTO of much hgher frequence, the multconductor tranmon-lne (MTL) theory appled [5]. The method the turn-to turn modellng whch requre a large computaton capacty. To avod th, the author have propoed a hybrd method combnng the ngle tranmon-lne analy of whole wndng and the MTL analy of the hghet-voltage ecton [6]. The man purpoe of preent paper to provde a practcal MTL method to calculate the hgh frequency charactertc of hell-form tranformer developng a method to reduce the number of unnown by combnng multple turn under common parameter. The nterturn voltage calculated and compared wth the expermental reult obtaned n the actual tranformer both n frequency and tme doman. The effect of VFTO on the tranformer n actual ytem are brefly dcued.. umpton II. MTL MODEL Fg. how the hell-form tranformer n whch HV wndng compoed of col and two tatc plate (SP) ndcated a SP o, SP e. It aumed that the electromagnetc wave propagate along conductor nde the core wndow jut a n a wavegude. Regardng the feld n the vcnty of N+ conductor (total turn N plu two SP) to be tranvere electromagnetc (TEM), the followng relaton obtaned for the dtrbuted capactance and nductance matrce [C], [L] of N+ dmenon [7]. [ L ]=[C ] /v Where, v the electromagnetc velocty: = / r v ε ε µ Here, ε r the delectrc contant of nulaton.. pplcaton of MTL theory The model hown n Fg. derved conderng the tranformer ubjected to a VFTO. Here, the nuodal voltage E o of frequency ω repreent the VFTO. The thc parallel lne are the nter-lned tranmon lne of average turn length a, and the dahed lne are fcttou zero -length lead. The current and voltage of lne are denoted by the vector (I (m ) (x )) and (V (m ) (x )), wth m= or ndcatng regon or. lthough the core reject hgh frequency magnetc fluxe but tll ome flux may form a common flux there. The common flux n- VFTO SPo turn E E E turn N HV col core SPe Fg.. Shell-form tranformer wndng.

2 duce the voltage E at the md pont of each turn where SP have a brea. The followng MTL equaton hold [7]. d dx d ( V dx = jω [ C] ( V = jω [ L] Ung n (3) lead to the wave equaton: d dx (3) = Γ (4) Here, Γ (=jω/ v ) the propagaton contant. The general oluton of (4) or (3) wrtten a ( ( ( V m) m) = ( ) exp( Γx) + ( ) exp( Γx) (5) = v [ L]{( exp( Γx) ( ) exp( Γx)} The vector ( ) and ( (m ) ) are unnown contant repreentng the urge current. The dpaton can be ncorporated f the followng Γ ued [6]: j ω ωtanδ ω Γ = + + (6) v v v d σ µ Th derved aumng a contant nulaton lo tangent of tanδ and a proxmty effect n the conductor of conductvty σ eparated by turn nulaton d. To the 4(N+) unnown, ( (m ) ) and ( (m ) ), the ame number of lnear equaton can be contructed by conderng the connectng condton n Fg.. In the equaton, E can be evaluated by N a E jω Lc I ( ) (7) = where, L c the nductance of the common flux. C. Reducton of MTL equaton It poble to elmnate 8 unnown n (5): o (m), o (m), e (m), e (m) (m=, ) the urge component n all ecton of SP. The reult can be wrtten a: ( V SPo turn E o N SPe ) v [ M] { γ = ( ) exp( Γx) + ( ) exp( Γx) (8) = v [ L]{( ) exp ( Γx) ( )exp( Γx)} + E ( o regon I o (x) I (x) I (x) IN (x) Ie (x) o E ( ) γ Io (x) I (x) I (x) IN (x) Ie (x) x= x=a/ x=a turn regon Fg.. MTL model of hell-form tranformer. + ( )} Where, γ = exp ( Γa ) (9) In (8), the dmenon of matrce and vector reduced from orgnal N + to N. The lat two term n the econd relaton of (8) expre the electrotatc and electromagnetc nducton from SP. The vector ( o ) and matrx [M] can be defned from the orgnal nductance matrx of N+ dmenon [8]. The vector ( o ) o-called ntal voltage dtrbuton or capactve voltage d trbuton. The connectng condton at x=a/ n (8) can be expreed n the form: ( V ( a / )) = ( a / )) = ( V ( a / )) ( a / )) + E ( ) () Where, mean the vector of all the element beng unty. The frt equaton can be atfed f the followng equaton ntroduced wth an arbtrary vector ( ): ( ) = ( ) ( ) () ( ) = ( ) + γ ( ) Th mean that the forward and bacward travellng wave are modfed or cattered by ( ) at the md-turn. The econd relaton of () demand the followng condton: v Γa E / γ () { L] [ M ]}( ) = v nh [ M ]{( ) + γ ( )} + ( ) [ Here, the approxmaton coh( Γa / ) ued: Tang, and a new unnown, the number of unnown reduced to 3N. et of 3N lnear equaton can be contructed conderng both () and the connecton equaton: V I () = E C ( a) = I VN ( a) = Ee (3) ( a) = V () N + (), V + D. Combnng turn for further reducton Prelmnary numercal calculaton ha hown that, change wth, but carcely change except at the col-end [8]. It uggeted to et a dvon compred of mu ltple turn n the nner col regon a hown n Fg. 3. Here, the combned turn are drawn a f they are of a ngle tranmon lne, n whch the ame catterng parameter aumed. The followng relaton hould hold among the comp onent of and -th turn: = = turn catterng pont ( ) turn - -th dvon turn Fg. 3. Travellng wave n a multple-turn dvon.

3 = γ =,, = γ = γ + γ (4) The frt two are from the contnuty of urge comp o- nent, and the lat two from (). From (4), the followng teratve relaton are derved. = γ γ = γ, + (5) Let o the frt turn of dvon, then thee gve: + γ γ = γ γ + + γ γ = γ + γ Here, are renamed a,., (6) If HV wndng dvded nto N d dvon, the number of unnown 3N d. Choong adequately mall N d, the number of unnown can be reduced ubtantally. III. PRELIMINRY CLCULTION. Tranformer analyed The two tranformer wndng n Table are analyed. One a -col model contructed for expermental purpoe ung the platc -flm nulated col extracted from a ga nulated tranformer. The other a complete et of HV wndng of a 5 V autotranformer wth ol/paper nulaton. For the -col model of N =84, turn-to-turn calculaton poble.e. wthout multple-turn dvon. However, for 5 V tranformer of N =363, t dffcult to obtan the oluton wthout reducng the number of unnown by ettng multple-turn dvon n the author computer envronment.. Contant of Tranformer The nductance [L] neceary for the preent analy calculated from the wndng geometry n the followng tep. Frtly, [C] calculated by the -dmenonal charge mulaton. In the calculaton, ground urface are aumed at ome dtance from the wndng a hown n Fg. 4 (the cae of -col tranformer). Then, [L] obtaned ung. Snce core wa not ued n the experment to ecure the neceary meaurng pace, ome dtance are correpondngly taen between the wndng and the grounded plane n calculatng [L]. Fg. 5 how elected lne element n [L] of the -col model. The common nductance L c et zero wndng dmenon nulaton Table. Tranformer analyed. contant -col model 5 V autotranformer number of col total no. of turn, N av. turn length, a [m] nterturn thcne, d [mm] delectrc contant, εr dpaton factor, tanδ conductor conductvty, σ [S/m] 5 ~ 7 5 ~ 7 nductance Lj (µh/m) a=4.77 (a) de vew (n m) conderng the expermental condton of wthout core. C. Valdaton of calculaton SP 56.5 aumed ground (n mm) (b) cro-ecton Fg. 4. Confguraton of -col model. = Fg. 5. Inductance L j calculated for -col model. The urge component and can be numercally determned olvng the 3N d -dmenon lnear equaton f the frequency of nput voltage E o gven (the ampltude et a E o = n calculaton). Then, current or voltage at any pont to be calculated for the contnuou nuodal nput voltage. Effect of number of dvon are examned n the cae of -col model tranformer. Fg. 6 how the current and voltage dtrbuton at.5 MHz calculated n two nd of dvon condton: 84 dvon (turn-to-turn cae) and the cae of 4 dvon ( or 3 turn are combned except at col end). Ther good correpondence ndcate the applcablty of the preent method. The patal d trbuton of current of Fg. 6a how a fluctuaton a typcally een n the wave propagatng n a tranmon lne. Th may be een a a tandng wave phenomenon. mlar fluctuaton een n the voltage dtrbuton of Fg. 6b. It hould be noted that only the abolute value are hown n thoe reult. The two broen lne n Fg. 6b are the ntal voltage dtrbuton defned by ( o ) whch expected at an extremely hgh frequency, and the lnear dtrbuton expected at a very low frequency. It jutfable that the calculated dtrbuton for.5 MHz tuated between 6 col turn no. j

4 I /E (S) V /E dvon 4 dvon turn no. 84 dvon ntal volt. dtr. lnear dtr. (a) current dtrbuton 4 dvon turn no. (b) voltage dtrbuton Fg. 6. Current and voltage dtrbuton n -col model at.5 MHz. δv /E Frequency (MHz) Fg. 7. Frequency charactertc of frt nterturn voltage for -col model. a: 84 dv (dv/turn) by MTL model a: 4 dv (dv/col) by MTL model b: by turn-to-turn lumped crcut model c: expermental c a a b thoe two curve of extreme frequence, wth ome fluctuaton due to the tandng wave phenomenon. IV. COMPRISON WITH EXPERIMENT δv /E.. b a. Frequency charactertc The nterturn voltage nduced by VFTO of the man concern n the preent analy. Partcularly, t frequency charactertc mportant nce VFTO may have a varety of hgh frequency component [9]. The frequency charactertc of nterturn voltage can be obtaned n the preent analy by calculatng δv = V V (7) + Fg. 7 and 8 are the frequency charactertc of the frt nterturn voltage δv for the two tranformer. In the cae of -col model (Fg. 7), curve a and a are calculated by the preent MTL method wth 84 and 4 dvon, repectvely. Curve b calculated ung the turn-to-turn lumped crcut model baed on the nductance and capactance matrce correpondng to thoe ued n the MTL model. The expermental curve c obtaned obervng the nterturn voltage through an optoelectronc olaton technque applyng a contnuou nuodal voltage of varou frequency [, ]. ll the charactertc n Fg. 7 have a ere of pea due to nternal reonance. mong the calculated reult, the reonance pea by MTL are hgher than thoe by the lumped crcut reult demontratng the feature of tranmon lne modellng. lthough the calculated reonance frequence are a bt hfted from the expermental, ther mlarty confrm the applcablty of the preent method. In the cae of 5 V autotranformer (Fg. 8), curve a calculated by the MTL method wth 9 dvon, and curve b obtaned expermentally. Ther correpondence Frequency (MHz) Fg. 8 Frequency charactertc of frt nterturn voltage of 5V autotranformer.. a: 9 dv (9dv/col) MTL b: expermental here le remarable, probably due to the nterference from many reonance. t leat, ther reemblance ndcate that the preent method can be ued for a rough etmaton It hould be beard n mnd that the aement ung the frequency charactertc tend to overetmate the nterturn voltage level. Th becaue the actual VFTO not a contnuou nuodal but of a damped ocllaton.. Interturn voltage waveform The VFTO compre not only ocllatng comp o- nent but alo a pule of hgher magntude [9]. It mportant to now how the tranent nterturn voltage develop by VFTO. The nterturn voltage waveform can be calculated from the frequency doman data ung FFT. For convenence, repreentng the VFTO pule by one-cycle nuodal pule of MHz, repone of the two tranformer mo del are calculated and compared wth expermental. Fg. 9 how the nput pule ued n experment. The nduced nterturn voltage recorded at elected poton ung the optoelectronc technque a already mentoned [, ].

5 Fg. 9 Snuodal pule ued a nput voltage n the experment. In the cae of -col model, Fg. how the nterturn voltage, comparng the calculated and expermental reult at 3 pont (frt, mddle and end poton n the frt col). The calculated (Fg. a) and expermental (Fg. b) waveform correpond very well. There a dtnct delay tme n the wavehape at the mddle part of col n both reult, ndcatng that travellng wave tart from the both of col end and propagate toward nner regon [, ]. ponted out prevouly, ther velocty nterpreted a the electromagnetc wave along the conductor dependng on the delectrc contant of nulaton pace ε r a decrbed by. Fg. how the frt nterturn voltage of 5 V autotranformer ubjected to the ame nuodal pule of MHz. The correpondence of the calculated and expermental atfactory. The nduced voltage level maller than that of -col model. Th becaue the ncreaed number of col decreae the voltage enterng nto a col, therefore, the nterturn voltage level. V. CONCLUDING REMRKS practcal method to analye the hgh frequency tranent n the power tranformer developed by reducng the number of unnown n applyng the MTL theory. Voltage or current at any pont n the wndng calculable both n the frequency and tme doman. The contant neceary n the calculaton can be etmated from the tranformer geometry or degn parameter. The experment ncludng an actual 5 V tranformer have confrmed the applcablty of the preent method to the analy of hgh frequency tranent of everal megahertz. Comparng the calculated and expermental waveform n Fg. and, they eem to dffer at the later tme of the fgure. Th how the dpaton repreented by (6) not accurate enough. However, ther correpondence better than the frequency doman (Fg. 7, 8). It mght be that the ome dpaton mechanm not well a V frt nterturn ( δv ) b.4 V a.5 µ mddle part ( δv) b.5 µ a3 col end ( δv4) b3 (a) calculated (b) expermental Fg.. Interturn voltage waveform of -col model at applcaton of nuodal pule..4 V.5 µ (a) calculated (b) expermental Fg.. Voltage nduced at frt nterturn δv of 5 V autotranformer ubjected to nuodal pule.

6 repreented n the preent model, but t doe not play an mportant role n the tme doman. From Fg.7, one may thn tradtonal lumped crcut model can be ued for hgh frequency analy equally well. There have been propoed other way of applyng MTL theory. Cornc et al. ued MTL n the hghet voltage ecton only and replaced the ret of wndng by an mpedance [5]. nd there a hybrd method combnng the ngle tranmon-lne analy of whole wndng and the MTL analy of the hghet-voltage ecton [6]. Thoe technque have followng problem comparng wth the preent method. Lumped crcut model: The turn-to-turn modellng requre a large computaton capacty. Cornc MTL method: The ret of wndng dffcult to be repreented by a mple mpedance. Hybrd MTL method: Reonance tend to be n exce due to the ue of ngle tranmon lne. In th ene, the preent MTL method more advantageou n analyng VFTO effect on large tranformer. nother alent feature of the preent MTL method that the hgh frequency charactertc of a tranformer can be calculated from the confguraton of HV wndng, or degn data. Fg. depct th proce. The frt tep the calculaton of capactance by a charge mulaton, then nductance. Next, the urge component (unnown) are calculated at a ere of Fourer frequence, whch gve frequency charactertc. nd fnally, the FFT calculaton turn out tme doman waveform. The VFTO generated n GIS compre varou hgh frequency ocllaton extendng to everal megahertz [9]. y the ue of preent MTL method, t poble to etmate the nterturn nduced voltage n frequency or tme doman baed on the contant determned from the wndng geometry. Th partcularly ueful n degnng phae nce the mot vulnerable nulaton nterturn n VFTO pont of vew. The tme doman calculaton ung FFT n the preent method eem to gve more accurate than the frequency INTPUT: groupng pattern INPUT: wndng confguraton CLCULTION OF CONSTNTS (capactance, nductance) FREQUENCY DOMIN CLCULTION (olver of reduced MTL equaton) OUTPUT: frequency charactertc of current & voltage INTPUT: ncomng VFTO waveform TIME DOMIN CLCULTION (FFT) OUTPUT: nduced current & voltage Fg.. Flow to calculate hgh frequency charactertc of tranformer ung MTL model. doman. Therefore, t recommended to run the tme doman calculaton for the predcted VFTO waveform n the cae hgh level of nterturn voltage are antcpated n the frequency charactertc. The author hope th technque contrbute to the better degn of tranformer and to the mproved power ytem relablty. VI. REFERENCES [] J. H. McWhrter, et al., Determnaton of mpule tree wthn tranformer wndng by computer, Tranacton of IEE, vol.76 Pt.III, 957, pp [] W. J. McNutt, et al., Repone of tranformer wndng to ytem tranent voltage, IEEE Tranacton on Power pparatu and Sytem, vol.ps-93, 974, pp [3] R. C. Degeneff, general method for determnng reonance n tranformer wndng, IEEE Tranacton on Power pparatu and Sytem, vol.ps-96, 977, pp [4] D. J. Wlcox, et al., pplcaton of modfed modal theory n the modellng of practcal tranformer, Proceedng of IEE, vol.39 Pt.C (6), 99, pp [5] K. Cornc, et al., Dtrbuton of very fat tranent overvoltage n tranformer wndng, CIGRE Report, -4, 99. [6] Y. Shbuya, et al., naly of very fat tranent overvoltage n tranformer wndng, IEE Proceedng-Generaton Tranmon and Dtrbuton, vol.44, No.5, 997, pp [7] R. P. Clayton, naly of multconductor tranmon lne, Wlly, New Yor, 994. [8] Y. Shbuya, et al., naly of Very Fat Tranent n Tranformer, to be publhed n IEE Proceedng-Generaton Tranmon and Dtrbuton. [9] Y. Shbuya, et al., Effect of very fat tranent overvoltage on tranformer, IEE Proceedng-Generaton Tranmon and Dtrbuton, vol.46, No.5, 999, pp [] S. Fujta, et al., Expermental tudy of very fat tranent phenomena n tranformer wndng, IEEE Tranacton on Power Delvery, Vol.3, No.4, 998, pp.-7. [] S. Fujta, et al., Voltage ocllaton n tranformer wndng affected by very fat tranent urge, Tranacton of IEE Japan, Vol.-, No.5,, pp