"Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

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1 Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE OWER CABLES * B. AHIDI ** AND A. MOHAMMADZADEH FAKHR DAOOD Dept. of Electrcal Engneerng, Amrkabr Unversty of Technology Tehran, I. R. of Iran, Emal: Abstract Determnaton of the electrc feld n nsulated cable wll lead to an optmum desgn and a better selecton of both conductor sze and nsulaton thckness. A smple numercal method usng the Charge Smulaton Method (CSM) s used to calculate electrc feld stresses n hgh voltage cables. An mage charge for each fcttous charge s consdered n such a way that the potental of sheath s always kept at zero. The effect of cable sheath s consdered and results of the calculaton are shown. Keywords Three phase cable, feld dstrbuton, charge smulaton method 1. INTRODUCTION Insulaton strength s lmted by the presence of partcles and other contamnaton and vods n synthetc materals. Electrcal dscharges are ntated n the vod or n the vcnty of nsulaton due to the presence of a localzed hgh electrc feld. Therefore, the computaton of the electrc feld n a three-core cable s mportant for the proper desgn and safe operaton of power cables [1-3]. Salama et al. [1] ntroduced the applcaton of charge smulaton method (CSM) for the computaton of the electrcal feld n hgh voltage cables. In [] M.Salama and R.Hackam consder a non shelded nsulated conductor placed at a varyng dstance from a conductng plane usng a small number of fcttous charges. In the other paper [1] they used CSM to calculate the electrc feld n three core belted cables surrounded by a grounded sheath. Salama et al. dd not consder the effects of the poston of the charges on the accuracy of computaton, and ths s a weak pont of ther paper. Ghourab et al. used optcal fber sensors for onlne fault and problem detecton n an XLE cable [3]. The nserton of these sensors, whch generally possess electrcal propertes dfferent from those of the cable nsulaton, s expected to nfluence the electrcal feld dstrbuton n the space surroundng the H conductors. Ths stuaton, n turn, affects the electrc stress on the H nsulaton. Ths stuaton s consdered n ther paper. In the present paper, the electrc feld at the surface of conductors and on the surface of grounded sheath are obtaned usng the charge smulaton method. The conductors are smulated by an euvalent system of fcttuous charges. The method of mage generaton, n combnaton wth charge smulaton, was used to calculate the electrc feld.. CHARGE SIMULATION METHOD (CSM) The calculaton of electrc felds reures the soluton of Laplace s and ossons s euatons wth the boundary condtons satsfed. Ths can be done by ether analytcal or numercal methods. In many Receved by the edtors January 11, 004; fnal revsed form Aprl 4, 006. Correspondng author

2 790 B. ahd / A. Mohammadzadeh Fakhr Davood crcumstances, the stuaton s so complex that analytcal solutons are dffcult or mpossble, and hence numercal methods are commonly used for engneerng applcatons. The charge smulaton method s one of them [4-7]. Ths method s smple and accurate. In a smple example of Fg. 1: 7 + = 1 = Q Q = 1 13 = Φ Q = Φ ( = 1, ) ( = 3 ) Φ s potental at contour. are the potental coeffcents. When boundary condton s appled, a smlar condton can be appled to contours on the surface of electrode number. In these condtons and show the contour number and charge number respectvely. (1) Fg. 1. Typcal poston of charges and contour ponts n mult delectrc system (flled crcles and stars show charges and contour ponts respectvely) When the boundary condton s appled for the uncton of two nsulaton: E n1 and E n are normal components of the electrc feld to the nsulaton surface. Euatons are solved to determne unknown charges. 10 ε = () 0εr1En1 ε0εren 13 Q Q = 0 ( = 8,9,10) (3) = 8 = ε = + r1 FQ ε r FQ FQ (4) = 1 = 1 = 11 F are feld coeffcents n a drecton whch s normal to delectrc boundary at the respectve contour ponts. In order to determne the accuracy of computaton some checkponts wll be consdered. By computng the potental n these ponts and comparng t wth actual values, the accuracy of computaton wll be obtaned. 3. USING CSM FOR THE CALCULATION OF THE ELECTRIC FIELD IN THREE CORE BELTED CABLE The electrc feld, due to each phase of a three-phase cable, can be smulated by dfferent methods. In ths paper, each conductor s represented by an nfnte lne type charge. For makng sheath potental zero, a seres of mage charges were used. In order to save computaton tme, nstantaneous values of potental and electrc feld were consdered. In ths case, the voltages at the conductors are as E. (5). Iranan Journal of Scence & Technology, olume 30, Number B6 December 006

3 Applcaton of charge smulaton method to electrc 791 The worst case was chosen, n whch one of the three-phase voltages assumes ts peak value. The modelng method for a three-phase cable s shown n Fg. [1]. a b c = Cos = Cos = Cos ( ωt) o ( ωt - 10 ) o ( ωt + 10 ) Fg.. Modelng of three-phase cable, (1,,3) are charges, ( 1,, 3 ) are mage charges One of the most mportant problems s fndng the exact postons of the mage charges. δ s dstance of charge locaton from the center of the conductor. θ s the azmuth angle of the test pont. In order to fnd the exact locaton of the mage charge, Fgure 3 has to be consdered. In ths fgure only one phase s shown. Sheath s an eupotental surface wth zero value. If conductor charge and mage charge are ρ l and ρ l respectvely (Fg. 3), the locaton of the mage charge wll be determned n a way that makes an eupotental surface ( = 0) wth a radus of R. Fg. 3. Method for fndng locaton of mage charge otental of a pont (M) of ths surface s obtaned as follows: ρl r0 ρl r0 M = Ln + Ln πε0 r πε0 r (6) where r 0 s the dstance of M from reference potental. For smplcty: ρ l = ρ l Then M ρl r = Ln (7) πε r 0 In order to delete the effect of r 0, the reference potental pont s chosen the same dstance from ρ l and ρ l. For eupotental surface on a crcle wth radus R December 006 Iranan Journal of Scence & Technology, olume 30, Number B6

4 79 B. ahd / A. Mohammadzadeh Fakhr Davood Therefore d = R S R T + d d H = S, S =, R = S + T + + g (8) S 3 For determnng the value of the charge, sx test ponts (number of charges) were chosen on conductors and the metal sheath (Fg. 4). In order to check the computaton some checkponts were consdered. Fg. 4. Three-phase cable for showng the locaton of mage charge, checkponts and test ponts To satsfy the boundary condton otental coeffcents are gven by [ ][ Q] [ ] = (9) ( X () X ( ) ) + ( Y () Y ( ) ) ) = Ln (10) (Xp, Yp), (X, Y) are coordnates of the test pont and test charge locaton respectvely. Smlarly, for the electrc feld: F Q = E (11) F F X Y = = [ ][ ] [ ] Iranan Journal of Scence & Technology, olume 30, Number B6 December 006 X ( ) X ( ) () X ( ) + Y Y () Y ( ) () X ( ) + Y ( X ) ( () Y ( ) ) ( X ) ( () Y ( ) ) In Fg. 4, ponts 1 to 3 and 4 to 6 are shown as charge locaton and mage charge locaton respectvely. 1 to 6 show test ponts and 1 to 16 are check ponts. In ths fgure charges are n the center of conductors (δ=0). 4. RESULTS OF SIMULATION By usng CSM, t s found that the most mportant parameters n potental and electrcal feld dstrbuton are T/d and δ/d. The results of smulaton show that error n potental calculaton wll decrease wth ncreasng T/d rato. For 1.0 < T/d < 3.0 the error wll be less than 0.1%. For ths range of potental, the error on sheath wll be less than 0.4%. (1)

5 Applcaton of charge smulaton method to electrc 793 These values are for the worst case, (phase angle of conductor A s zero, at tme t=0) whch n ths condton conductor A has the maxmum potental. Fgure 5 shows the relaton between smulatng charges and Τ/d. In Fg. 5, Q are smulatng charges and curves are for 0.5 < T/d <3.0. These curves show that Q wll ncrease f T/d decreases. Excessve electrc stress can ntate potental dscharge and damages the nsulaton of conductors. Fgures 6a, 6b, 6c and 6d show electrcal feld dstrbuton on the surface of conductors and grounded sheath for dfferent δ/d. Fg. 5. Effect of Τ/d on value of smulatng charge (conductor A) Fgure 6a shows that the electrc feld ncreases at a certan pont, where the dstance between conductor A and sheath has ts least value. Fgures 6-b and 6-c show that at 300 and 60 degrees, whch are the nearest ponts to the sheath for conductor B and C respectvely, the electrc feld has ts mnmum value on the conductors surface. The nterestng pont s the nfluence of δ/d on the electrc feld around the sheath. Wth ncreasng δ/d, the electrc feld around the sheath wll decrease, because the charges are far from the sheath, but around the conductors (see Fg. 6d). Fg. 6. a) Electrc feld on surface of conductor A Fg. 6. b) Electrc feld on surface of conductor B Fg. 6. c) Electrc feld on surface of conductor C Fg. 6. d) Electrc feld on surface of grounded sheath December 006 Iranan Journal of Scence & Technology, olume 30, Number B6

6 794 B. ahd / A. Mohammadzadeh Fakhr Davood Effect of δ/d on potental dstrbuton s shown n Fg DISCUSSION In the present paper, Fg. 7 shows the effect of δ/d on the computaton of potental dstrbuton around the conductor, whch has one perunt potental (the other two phases have 0.5 p.u. potental). Accordng to Fg. 8, t can be udged that at the pont (180 degree) whch s close to the sheath when the charge moved far from center of the conductor (greater δ/d), the potental at that pont wll decrease. By comparng Fg. 5 wth Fg. 5 of [1], Fg. 6a wth Fg. of [3] and Fg. 6d wth Fg. 4 of [3], the results of the present paper are verfed. Fg. 7. otental dstrbuton on conductor A versus θ for dfferent δ/d Fg. 8. Confguraton of conductors n three-phase cable 6. CONCLUSION In ths paper the electrc feld and potental dstrbuton n a three-phase cable wth a grounded sheath have been dscussed. As shown n ths paper, electrc feld on conductors and sheath surfaces are not lnearly dstrbuted and n some ponts, the nsulaton s under more stress. As much as δ/d ncreases, the potental dfference of dfferent ponts around the conductor wll ncrease (these are not mentoned n other papers). In ths paper, the potental around A phase s computed and dscussed (these are not mentoned n other papers). In the present paper as well as others, the electrc feld around phase A, B, C and the effect of δ/d on charge values are dscussed. From the results, t can be uged that the maxmum electrc feld n a three core cable occurs at specfc locatons n the cable. REFERENCES 1. Salama, M. M. et al. (1984). Methods of calculaton of feld stresses n a three core power cable. IEEE Trans. On AS, AS-103(1), Salama, M. M. & Hackam, R. (1984). oltage and electrc feld dstrbuton and dscharge ncepton voltage n nsulated conductors. IEEE Trans. On AS, AS-103(1), Ghourab, M. E. & Ans, H. I. (1998). Electrc stresses n three-core cables eupped wth optcal sensors. IEEE Trans. On Delectrcs and Electrcal Insulaton, 5(4), Wefang, J., Humng, W. & Kuffell, E. (1994). Applcaton of the modfed surface charge smulaton method for solvng axal symmetrc electrostatc problems wth floatng electrodes. roc. Of 4 th nt. Conf. On ropertes and Applcatons of delectrc materal, Brsbane, Qld, Australa1, Chakravort, S. & Mukheree,. K. (199). Effcent feld calculaton n three-core belted cable by charge smulaton usng complex charges. IEEE Trans. On Electrcal Insulaton, 7(6), Malk, N. H. (1989). A revew of the charge smulaton method and ts applcaton. IEEE Trans. On Electrcal Insulaton, 4(1), Snger, H. et al. (1974). A charge smulaton method for the calculaton of hgh voltage felds. IEEE Trans. On AS, 93, Iranan Journal of Scence & Technology, olume 30, Number B6 December 006

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