Calculation of losses in electric power cables as the base for cable temperature analysis

Transcription

1 Calculation of losses in electric ower cables as the base for cable temerature analysis I. Sarajcev 1, M. Majstrovic & I. Medic 1 1 Faculty of Electrical Engineering, University of Slit, Croatia Energy Institute Hrvoje Pozar Zagreb, Croatia Abstract Power losses refer to the heat generated in cable conducting arts (hase conductors and sheaths) and in cable insulating arts. It is necessary to know the exact data regarding heating owers for the calculation of heath transfer and cable temeratures. Heating ower in hase conductors and sheaths mainly deend on current values. Exact calculation of those owers is very difficult. This aer develos a mathematical model of heating ower calculation in three hase single-core cable conductors and sheaths. This model is used to determine filament currents and heating owers in hase conductors and sheaths. Geohysical features of the cable route are also considered. Three-hase single core electric ower cables of 35 kv rated voltage are taken as an examle. Two laying conditions (trefoil and flat formation) are considered. Sheathes are bonded and grounded at both ends. Calculation results for cables in flat configuration show that heating owers in cable sheaths do not have equal magnitude and increase with distance. The least heating ower occurs in the sheath of the middle cable. Heating owers in sheaths of outer cables are of unequal magnitude too. Thereby, the cable sheath of the lag hase has a higher ower. Our research has shown that in some cases heating owers in sheaths could be greater than heating owers in hase conductors. The method resented in this aer is used to determine heating owers of all filaments over the crosssectional area. Calculation results show that sheath filament heating owers are not radial symmetric.

2 530 Advanced Comutational Methods in Heat Transfer VI 1 Introduction Heating ower is identical to an electrical ower loss and occurs during the cable oeration. In general, there are two tyes of owers generated in a cable: currentdeendent owers and voltage-deendent owers. Current-deendent owers refer to the heat generated in metallic cable comonents (conductors, sheaths etc.). Voltage-deendent owers refer to the owers in cable insulation. These owers belong to two grous: dielectric owers and owers caused by the charging current. The energy generated by the above- mentioned owers is converted to other energy forms, redominantly heat. This heat energy tends to increase the temeratures of the associated electrical and unelectrical comonents. The heating owers roduced in cable insulation deend on oeration voltage. In general, their calculations are simle, esecially for single-core cables. They will not be analyzed in this aer. Current-deendent owers are comosed of conductor owers and sheath owers. They are a function of the load current. Skin and Proximity Effects, Cable Comonents, Laying Conditions and Sheath Earthing have to be taken into account when making the heating ower calculation. Exact calculation of current-deendent owers is very comlicated. The mathematical model of heating ower calculation in conductors and sheaths is develoed in this aer. The transmission line is comosed of three single core cables with sheaths earthed at both ends. This method is based on the segmentation into filaments of both the conductor and the sheath cross-sectional area. The filament has a small cross-sectional area thus we can assume the uniform density of currents flowing through the cross-sectional area of each filament. The method of Geometric Mean Distance is used too. Electrical and thermal characteristics of soil are also considered. Mathematical model The transmission line comosed of three single-core cables in a quasi-stationary oeration is considered (Figure 1). This line is a art of the directly earthed network. The sheaths of these cables are earthed at both ends. Conductors and sheaths are divided into N and Ns filaments, resectively. The total number of filaments is N=3 (N+Ns). Unknown currents ( I i, 1,,,N) flow through these filaments. The cables are connected to the huge Power System (infinite connection oint) with three hase voltages V R =V R /φ R, V S =V S /φ S, V T =V T /φ T (1) Three-hase assive network is connected to the other cable terminals. This network is reresented by imedance Z l R, Z l S and Z l T. This aroach enables the g l alication of various load tyes. Z e and Z e are earthing imedances at generator and load cable terminals, resectively (Figure 1).

3 Advanced Comuational Methods in Heat Transfer VI 531 Unknown filament currents are calculated in the loo frame of reference. The indeendent loo consists of the filament with earth return. Matrix equation can be written as follows: V = Z I () where V - column vector of loo voltages Z - matrix of self and mutual imedance of filaments with earth return I - column vector of filament currents Figure 1: Single-core cable circuits with earthing arrangement of cable sheaths. Elements of matrix Z are calculated as follows: - filaments of hase conductors g l z ik Zik + Z (i,k) + Ze + Ze = α (3) where α(i,k) are: for R i,k 3 N. 1 i,k N, for S N +1 i,k N, for T N +1 - filaments of sheaths g l z ik Zik + Ze + Ze = (4)

4 53 Advanced Comutational Methods in Heat Transfer VI Mutual imedance between i and k circuits is calculated as follows: ω µ o l ω µ o l 658 ρe Z = + ik j ln (5) 8 π dik f where j = 1, ω = π f - angular frequency, f - frequency of system, Hz, 7 µ o = 4 π 10, Vs / Am - vacuum ermeability, l - length of the cable route, m, d ik - geometric mean distance between filaments i and k, m, ρ e - earth electrical resistivity, Ωm. Self imedance of the circuit filament with earth return i is: ω µ o l ω µ o l 658 ρe Z = + + ii R i j ln (6) 8 π dii f where R i - resistance of filament i, Ω, d ii - geometric mean radius of filament i, m. The resistance of filament i is, as follows: ρf l R i = (7) Ai where A i - cross-sectional area, m, ρ f - filament electrical resistivity, Ωm. Unknown filament currents are from eqn (): I = Z -1 V (8) These currents are taken into the heating ower calculation. The heating ower of filament i is: Pi i R i = I (9) Heating owers of conductors and sheaths are: P R N N 3N = I i R i, PS = Ii R i, PT = I i 1 N + 1 N + 1 R i

5 Advanced Comuational Methods in Heat Transfer VI 533 P sr 3N + Ns 3N + Ns N = I i R i, PsS = Ii R i, PsT = Ii 3N + 1 3N + N + 1 N + N + 1 s s R i Total heating ower is as follows: (10) T T = P i + Psj R j= R P (11) Once we have calculated the heating ower the quasi-state heat transfer and temerature field can be calculated either by electrical analogue rocedure (Ohm s law) or by finite element method. 3 An Examle The method resented is alied in the transmission line of three single-core cables. Electric ower cable is of IPHA 04, (FKS [6]) tye. Cross-sectional area of the coer stranding conductor is 400 mm. The sheath is made of 1.5 mm thick aluminium ie with outside diameter of 44.6 mm. Oil-imregnated aer tae insulation is 8 mm thick. mm thick PVC is used for non-metallic outer sheath. External diameter D k =48.5 mm. Rated voltage is 35 kv. The cable route is 1000 m long. Three single-core cables are connected to the infinite ower buses. Their voltages are: V R =0.07/0 kv, V S =0.07 /4π/3 kv, V T =0.07 /π/3 kv. Three-hase balanced load is connected to the other terminal. The load imedance is: Z R l = Z S l = Z T l =43 +j 0 Ω. The current rating of cables in flat formation with the searation of s o =70 mm is 450 A. For cables in trefoil formation the current rating is 500 A. Conductors and sheaths are divided into five (N =5) and sixty filaments (N s =60), resectively. The stranded conductor contains 61 ( ) wires whose diameter is.9 mm. The conductor filament resitivity at 60 o C is ρ f = µωm. The sheath filament resitivity at 55 o C is ρ s = µωm. The earth electrical resistivity is ρ e =50 Ωm. Cables in flat (Figure ) and trefoil (Figure 3) formation are analyzed. During the calculations the cable searation is changed for cables in flat formation as follows: s o =0 mm and s o =70 mm and s o =140 mm. Earthing imedance at generator and load cable terminals are neglected (Z e g = Z e l =0+j0 Ω). In the case of laying cables in flat formation the calculation results show that the electromagnetic symmetry exists for y=0 and it does not exist for x o =s o +D k flatness. Heating owers of sheath filaments er unit length (one meter) are resented in Figure 4. Cable searation is s o =0 mm. For laying cables in trefoil formation these owers are resented in Figure 5. Calculation results of total heating owers in conductors and in sheaths are resented in Table 1. These owers calculated again this time using IEC recommendation (IEC [7]) are shown in Table 1, too. A good agreement is seen between the two ways of calculation. We have got the same results for heating

6 534 Advanced Comutational Methods in Heat Transfer VI owers of three hase conductors for cables laid in trefoil formation. A similar conclusion can be drawn for sheaths. Figure : Cables in flat formation. Figure 3: Cables in trefoil formation. The calculation of laying cables in flat formation has given the similar results for three-hase conductors. Influence of the searation s o on the owers is negligible. However, heating owers in sheaths differ among different sheaths and they deend on searation s o. These owers increase with the increase of s o. It is ossible that in some cases these owers are greater than conductor owers. The least ower occurs in the sheath of the middle cable. The sheath owers of external cables are not equal and they deend on the hase order. This order is resented in Table. For each hase order the cable with the greatest heating ower is marked with a bold-italic letter (gray cells). In Figure cable ositions are marked I, II, III, resectively.

7 Advanced Comuational Methods in Heat Transfer VI 535 Table 1: Conductor and sheath total heating owers er unit length. s o (mm) P R P S P T P sr P ss P st P Calc. IEC Error (%) Calc. Flat formation IEC Error (%) Calc. IEC Error (%) Trefoil formation Error Calc. IEC (%)

8 536 Advanced Comutational Methods in Heat Transfer VI Figure 4: Sheath filament heating owers of cables in flat formation. Figure 5: Sheath filament heating owers of cables in trefoil formation.

9 Advanced Comutational Methods in Heat Transfer VI 537 Table : The greatest sheath heating owers in hase order function. I II III R S T R T S S R T S T R T R S T S R 4 Conclusion The mathematical model resented in this aer rovides an exact calculation of conductor and sheath heating owers in the transmission line comosed of three single-core cables. Thereby, there are no constraints by cable laying. Heating owers can be calculated for both cases of balanced and unbalanced quasi-steady loads since they are not affected by ower factor. The least heating ower occurs in the sheath of the middle cable in flat formation. Furthermore, heating owers in sheaths of outer cables are not of equal magnitude. Between them the cable sheath of lag hase has a higher ower than the other one. In some cases heating owers in sheaths could be greater than heating owers in hase conductors. This knowledge is imortant for cable temerature analysis because these owers are acting as heating sources. The method described in this aer is used to determine filament heating owers over the cross-sectional area of ower cables. It can be considered also as the first ste in calculations of temerature rise for more comlex, nonhomogeneous bodies. The calculation includes lack of radial symmetry of sheath filaments heating owers. References [1] Heinhold, L. Power Cables and their Alications, Siemens: Berlin, [] Sarajcev, I. Gubici snage kabelskog renosa, Ph. d. disertation, Elektrotehnički fakultet: Zagreb, [3] Carson, J.,R. Ground Return Imedance: Underground Wire with Erth Return, Bell System Tech. J., Vol. 8, 84, 199. [4] Siemens. Formel-und Tabellenbüch fur Starkstrom-Ingenieure, Girordet: Essen, [5] Haznadar Z., Matjan J. Određivanje rasodjele struja i roračun gubitaka i imedancija u sustavima ravnih vodiča, X Savjetovanje elektroenergetičara Jugoslavije: Dubrovnik, [6] FKS. Kataloški odaci tvornice kabela Moša Pijade, FKS: Svetozarevo, [7] IEC. Publication 87/ [8] Anders G., J. Rating of Electric Power Cables, IEEE Press: New York, 1997.