Internatonal Journal of Smart Grd and Clean Energy Tme Doman smulaton of PD Propagaton n XLPE Cables Consderng Frequency Dependent Parameters We Zhang a, Jan He b, Ln Tan b, Xuejun Lv b, Hong-Je L a * and Chongxn Lu a a X an Jaotong Unversty, No.28 Xannng West Road, X an, 749, Chna b Lny Power Supply Ltd., No.3 Jnque Hll Road, Lny, 276, Chna Abstract Partal dscharge (PD) detecton and locaton s of great sgnfcance for the power cable nsulaton condton montorng, where analyss of the PD propagaton s needed. Ths paper presents the sngle-core model of cross-lnked polyethylene (XLPE) cable consderng the nfluence of skn effect and sem-conductng screens. To analyss the propagaton of PD pulse n tme doman, the model frstly s ratonal approxmated usng vector fttng method. Based on the fttng results, the π secton lumped parameters model s then proposed. Examples on two cables wth dfferent permttvty approxmatng forms of semconductng screens are presented. Keywords: Sem-conductng screen, state-space model, vector fttng (VF), XLPE cable. Introducton The sem-conductng screens can have sgnfcant mpact on hgh frequency wave propagaton n power cables [], [2]. The hgh frequency cable model that can be used over Hz s frstly descrbed n [], and more accurate model for extruded cables s developed n [3]. A few works on the characterzaton of sem-conductng layers n cables are carred out, and the complex permttvty of the dspersve layers s usually represented by Cole-Cole and Debye models [4], [5]. Although there s good performance between the hgh frequency cable model and the actual measurements, t s stll a tough problem to solve these models n tme doman. Vector fttng (VF) s a robust numercal method whch can be appled to fttng of measured or calculated frequency responses wth ratonal functon approxmatons [6]. Ths methodology has captured ncreasng nterest n transent electromagnetc analyss and ts fttng model may be easly represented by equvalent crcut wth passve elements [7]., Alumnum conductor wth radus r ; 2, Conductor screen wth thckness t 2 ; 3, XLPE nsulaton wth thckness t 3 ; 4, Insulaton screen wth thckness t 4 ; 5, Screen bed wth radus r 5 ; 6, Copper wre screen wth radus ρ; 7, Polyethylene jacket. Fg.. Geometry of XLPE cable. * anuscrpt receved June 3, 22; revsed August 4, 22. Correspondng author. Tel.: +86-29-8266 423; fax: +86-29-8266 8626; E-mal address: hjl@mal.xjtu.edu.cn.
26 Internatonal Journal of Smart Grd and Clean Energy, vol. 2, no., January 23 Ths paper ntroduces the VF fttng model nto the tme doman analyss of sngle-core power cable models consderng the nfluence of both skn effect and sem-conductng screens. Through whch, the generc π crcut secton of the power cable s proposed. Usng the data from [4] and [5], hgh frequency pulse propagaton s smulated by state-space method. 2. Power Cable odel The sngle-core XLPE cable to be studed s shown n Fg., and ts equvalent crcut model per unt length (p.u.l.) consst of seres mpedance Z( ω ) and shunt admttance Y ( ω ) are gven as follows [3],[4]. 2. Seres mpedance Z(ω) Consderng the skn effect n the conductor and metallc screen, the seres mpedance Z( ω) s expressed by: jωμ jωμ r jωμ Z( ω) ln 2πr σ 2π r 2πρn σ 5 = + + () 6 where σ and σ 6 are the conductvtes of alumnum and copper respectvely, μ s the permeablty of free space, n s the total number of copper wres n the metallc screen. 2. Shunt admttance Y(ω) The sem-conductng screens and XLPE nsulaton contrbute to the shunt admttance Y ( ω) of the extruded cable, whch s gven by: Y ( ω) = 5 = 2 Y and 2πεε Y = jω ln ( r r ) (2) (3) where ε s the permttvty of free space, r and r are the outer and nner radus of layer respectvely, and ε = ε jε s the complex permttvty for layer, and for the XLPE nsulaton: ε =2.3 and ε =.. In addton, the complex permttvty of the sem-conductng screens obtaned through measurements s descrbed n [4], whch s represented by two Cole-Cole functons and some attached terms, whereas the complex permttvty of the dspersve layers s descrbed by Debye model n [5]. No matter what form of the complex permttvty n the cable model s, t s hard to analyss n tme doman. 3. Fttng odel and Equvalent Crcuts Assume that functon F( ω ) represents the frequency response of a frequency dependent system, whch can be ratonal approxmated n the followng form: N c r Fft ( ω) = + jωh+ d = jω + h + d = jω a = jω p (4)
where r and We Zhang et al.: Tme Doman smulaton of PD Propagaton n XLPE Cables 27 p are resdues and poles whch may be ether real or conjugate complex numbers, h and d are real. All the coeffcents n equaton (4) can be calculated by the VF procedure descrbed n [6]. Once the fttng models are obtaned, they can be easly assocated wth equvalent crcuts wth passve elements. Reference [7] shows the equvalent crcut of the p.u.l. seres mpedance of a lossy transmsson lne. Smlarly, we can obtan the equvalent crcut of the p.u.l. shunt admttance of the sem-conductng layers. To facltate the analyss, the total admttance Ysc ( ω ) of the semconductor layers s modeled. Ysc ( ω) has the same form as (3) not ncludng the XLPE part. Applyng the VF method for Z( ω) and Ysc ( ω ), they wll be ftted n the form of (4), and ther correspondng equvalent models are shown n Fg.2. In whch, Fg.2(a) represents the equvalent model of Z(ω) and R = d, L = h, R = r, L = r p,( = N), whle admttance Ysc ( ω) s represented n Fg.2(b) and G = d, C = h, G = r, ( = ). oreover, t should be noted that R and G are the correspondng parameters n the DC condtons (ω=) for Z(ω) and Y ( ω ), respectvely. 4. State Space odel of Power Cables Based on the equvalent crcut n Fg.2, the generc k-th π crcut secton of power cable s represented n Fg.3. In whch, k and (k+) are currents flowng through Z( ω ) n the k-th and (k+)-th π crcuts, whereas v k and v (k+) are the voltage across Y ( ω ) ; G and C are the conductance and capactance of the XLPE nsulaton, respectvely. sc (a) (b) Fg.2. Equvalent crcuts of fttng model for power cable. (a) Seres mpedance; (b) Admttance of sem-conductng layers. Fg.3. Equvalent π crcut for power cable. Assume that k, k, k 2,, kn are currents through L, L, L 2,, L N, and v k, v k 2,, v k are voltages across C, C 2,, C, respectvely. Consderng Kπ sectons connected n cascade, the state dfferental equatons n tme doman can be wrtten as: X = AX+ B v s (5) where X and B are vectors wth ( + N + 2) K dmensons, and Α s ( + N + 2) K ( + N + 2) K dmensonal matrx, v s exctaton source, ther specfc forms are as follows: [ 2 K ] [ L ] s T X = X X X (6) B T = (7)
28 Internatonal Journal of Smart Grd and Clean Energy, vol. 2, no., January 23 Α A 2 Α2 Α22 Α23 A = (8) Α( K )( K 2) Α( K )( K ) Α( K ) K ΑK( K ) ΑKK And for the k-th π secton, the related parameters are: [ v v v v ] X T = (9) k k k k2 kn k k k2 k The generc submatrx A kk wth ( +N+ 2)dmensons s expressed by: A kk N R = R R2 RN L L L L L R R L L R2 R2 L2 L2 RN RN LN LN = G G G = G G + C C C C C C C G G G C C C C G2 G2 C2 C2 G G C C = 2 A k(k-) and A (k-)k havng the same dmensons as A kk are very hghly sparse, and there are only a few non-zero elements n each of them, where: () Αk(k-) (, ( N + 2) K) = L Α(k-)k (( N + 2) K, ) = ( ), (( N 2) K, ) C + Α C + = C (k-)k () The state equatons (5) are frst-order lnear ordnary dfferental equatons and they can be easly solved wth numercal methods. In ths paper, Runge-Kutta method was used to solve the state space model.
5. Calculaton Examples and Dscusson We Zhang et al.: Tme Doman smulaton of PD Propagaton n XLPE Cables 29 Ths secton presents example for the purpose of applyng the proposed model. The XLPE cables to be studed are from the medum voltage cable called cable n [4] (VC) and the hgh voltage cable n [5] (HVC). The complex permttvty forms of whch are Cole-Cole and Debye models, respectvely. The detals of the cables nvestgated are n Table. Table. Cable parameters used n calculaton (All unts are n mm). Parameter VC HVC Conductor radus ( r ) 7.3 5 Conductor screen thckness (t 2 ).6.3 Insulaton thckness (t 3 ) 5.5 8 Insulaton screen thckness (t 4 ).4.3 Screen bed radus (r 5 ) 5.8 35.6 5.. odellng results by vector fttng In the ratonal approxmaton of the seres mpedance Z( ω) and the total admttance Ysc ( ω ) of the semconductor layers for both VC and HVC, vector fttng procedure was mplemented n the range of.hz to Hz. Fg.4 shows the fttng results after 2 teratons usng 2 pars of complex startng poles. It s seen that very good approxmatons have been acheved. The maxmum of the root-mean-square (RS) errors was found to be.88e-8. (a) (b) Fg.4. Vector Fttng results of VC and HVC. (a) agntude of Z(ω) p.u.l.; (b) agntude of Y ( ) sc ω p.u.l.. Fg.5. Equvalent π crcut for power cable.
3 Internatonal Journal of Smart Grd and Clean Energy, vol. 2, no., January 23 Table 2. Fttng Parameters of Seres Impedance Shown n Fg.3. Resstance (Ω/m) Inductance (H/m) Parameter VC HVC Parameter VC HVC R 9.7332-4 3.9298-5 L.626-7.7527-7 R 3.47-4 2.935-4 L.2757-7 3.79-8 R2 5.486-5 4.266-5 L2 3.266-7.949-7 R3.4774-5.48-5 L3 7.5397-7 3.659-7 R4 8.9996-6 5.3953-6 L4 4.77-6.9469-6 Table 3. Fttng Parameters of Shunt Admttance Shown n Fg.3 Conductance (S/m) Capactance (F/m) Parameter VC HVC Parameter VC HVC G.54-3 7.4673-3 C 2.482 -.775 - G.599 2 5.2223 C 4.8327-9 2.388-7 G2 5.9236-9 4.945-2 C2.977.38 - G3 2.5524-8.59-6 C3 7.38-2.2574 2 G4 3.5475-8 3.664-22 C4.323 -.983-4 G5 3.862-8 3.824-23 C5.743-2 2.92-2 Also, the frequency dependent parameters n Fg.3 (=N=4) are gven n Table 2 and Table 3 n Appendx. 5.2. Transent smulaton In the followng calculatons, the length of the extruded power cables s meters. Determnng the number of π sectons s mportant n the transent analyss of system represented by lumped-parameter π crcuts connected n cascades. In ths paper, the statstcal correlaton method was adopted whch was mplemented by the followng procedure. Step ) Choose K as the ntal number of π sectons. Step 2) Calculate the voltages V and V 2 at the observaton ponts wth K and K + ΔK π sectons consdered, respectvely. Step 3) Compute the Pearson correlaton coeffcent C of V and V 2. If C s greater than a settng value, accept K as the fnal result; otherwse set K = K +Δ K, go to Step 2. Based on the method above, the cable was fnally dvded nto π crcuts connected n cascades. Smulaton was carred out between to μs wth ns as the calculaton step. Gaussan pulse wth frequency wdth of 2Hz s njected nto the XLPE cables, and the open-crcut voltages at the recevng end are shown n Fg.5. From whch we can obtan the propagaton veloctes of VC and HVC are about 62m/μs and 79 m/μs, respectvely. Furthermore, there s hgher attenuaton n VC for the same njected pulse. 6. Concluson The characterstcs of frequency dependent parameters n power cables can be effcently ratonal approxmated by VF method. Based on the equvalent crcuts proposed n ths paper, t s easy to obtan the state space model n tme doman for extruded cable models consderng skn effects and sem-conductng screens. References [] Stone G, Boggs S. Propagaton of partal dscharge pulses n shelded power cable. In: Proc. of IEEE Conference on Electrcal Insulaton and Delectrc Phenomena, 982:275-28.
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