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Calculation of losses - EIHP

Calculation of losses in electric power cables as the base for cable temperature analysis I. Sarajcev1, M. Majstrovic2 & I. Medic1. 1. Faculty of Electrical Engineering, University of Split, Croatia 2. Energy Institute Hrvoje Pozar Zagreb, Croatia Abstract Power losses refer to the heat generated in cable conducting parts (phase conductors and sheaths) and in cable insulating parts. It is necessary to know the exact data regarding heating powers for the Calculation of heath transfer and cable temperatures. Heating power in phase conductors and sheaths mainly depend on current values. Exact Calculation of those powers is very difficult. This paper develops a mathematical model of heating power Calculation in three phase single-core cable conductors and sheaths.

Calculation of losses in electric power cables as the base for cable temperature analysis I. Sarajcev1, M. Majstrovic2 & I. Medic1 1Faculty of Electrical Engineering, University of Split, Croatia 2Energy Institute Hrvoje Pozar Zagreb, Croatia Abstract Power losses refer to the heat generated in cable conducting …

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1 Calculation of losses in electric power cables as the base for cable temperature analysis I. Sarajcev1, M. Majstrovic2 & I. Medic1. 1. Faculty of Electrical Engineering, University of Split, Croatia 2. Energy Institute Hrvoje Pozar Zagreb, Croatia Abstract Power losses refer to the heat generated in cable conducting parts (phase conductors and sheaths) and in cable insulating parts. It is necessary to know the exact data regarding heating powers for the Calculation of heath transfer and cable temperatures. Heating power in phase conductors and sheaths mainly depend on current values. Exact Calculation of those powers is very difficult. This paper develops a mathematical model of heating power Calculation in three phase single-core cable conductors and sheaths.

2 This model is used to determine filament currents and heating powers in phase conductors and sheaths. Geophysical features of the cable route are also considered. Three-phase single core electric power cables of 35 kV rated voltage are taken as an example. 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 powers in cable sheaths do not have equal magnitude and increase with distance. The least heating power occurs in the sheath of the middle cable. Heating powers in sheaths of outer cables are of unequal magnitude too. Thereby, the cable sheath of the lag phase has a higher power. Our research has shown that in some cases heating powers in sheaths could be greater than heating powers in phase conductors.

3 The method presented in this paper is used to determine heating powers of all filaments over the cross- sectional area. Calculation results show that sheath filament heating powers are not radial symmetric. 530 Advanced Computational Methods in Heat Transfer VI. 1 Introduction Heating power is identical to an electrical power loss and occurs during the cable operation. In general, there are two types of powers generated in a cable: current- dependent powers and voltage-dependent powers. Current-dependent powers refer to the heat generated in metallic cable components (conductors, sheaths etc.). Voltage-dependent powers refer to the powers in cable insulation. These powers belong to two groups: dielectric powers and powers caused by the charging current. The energy generated by the above- mentioned powers is converted to other energy forms, predominantly heat.

4 This heat energy tends to increase the temperatures of the associated electrical and unelectrical components. The heating powers produced in cable insulation depend on operation voltage. In general, their calculations are simple, especially for single-core cables. They will not be analyzed in this paper. Current-dependent powers are composed of conductor powers and sheath powers. They are a function of the load current. Skin and Proximity Effects, Cable Components, Laying Conditions and Sheath Earthing have to be taken into account when making the heating power Calculation . Exact Calculation of current-dependent powers is very complicated. The mathematical model of heating power Calculation in conductors and sheaths is developed in this paper. The transmission line is composed of three single core cables with sheaths earthed at both ends.

5 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. 2 Mathematical model The transmission line composed of three single-core cables in a quasi-stationary operation is considered (Figure 1). This line is a part of the directly earthed network. The sheaths of these cables are earthed at both ends. Conductors and sheaths are divided into Np and Ns filaments, respectively. The total number of filaments is N=3 (Np+Ns). Unknown currents ( Ii, i=1,2, ,N) flow through these filaments.

6 The cables are connected to the huge Power System (infinite connection point) with three phase voltages VR=VR / R , VS=VS / S , VT=VT / T (1). Three-phase passive network is connected to the other cable terminals. This network is represented by impedance ZRl, ZSl and ZTl. This approach enables the application of various load types. Zeg and Zel are earthing impedances at generator and load cable terminals, respectively (Figure 1). Advanced Comupational Methods in Heat Transfer VI 531. Unknown filament currents are calculated in the loop frame of reference. The independent loop consists of the filament with earth return. Matrix equation can be written as follows: V =Z I (2). where V - column vector of loop voltages Z - matrix of self and mutual impedance of filaments with earth return I - column vector of filament currents Figure 1: Single-core cable circuits with earthing arrangement of cable sheaths.

7 Elements of matrix Z are calculated as follows: - filaments of phase conductors z ik = Z ik + Z (i,k ) + Z ge + Z le (3). where (i,k) are: for R 1 i,k Np, for S Np+1 i,k 2 Np, for T 2 Np+1 . i,k 3 Np. - filaments of sheaths z ik = Z ik + Z ge + Z le (4). 532 Advanced Computational Methods in Heat Transfer VI. Mutual impedance between i and k circuits is calculated as follows: o l o l 658 e . Z ik = + j ln (5). 8 2 d ik f .. where j = 1 , = 2 f - angular frequency, f - frequency of system, Hz, o = 4 10 7 , Vs / Am - vacuum permeability, l - length of the cable route, m, dik - geometric mean distance between filaments i and k, m, e - earth electrical resistivity, m. Self impedance of the circuit filament with earth return i is: o l o l 658 e . Z ii = R i + + j ln (6).

8 8 2 d ii f .. where Ri - resistance of filament i, , dii - geometric mean radius of filament i, m. The resistance of filament i is, as follows: l Ri = f (7). Ai where Ai - cross-sectional area, m2 , f - filament electrical resistivity, m. Unknown filament currents are from eqn (2): I = Z-1 V (8). These currents are taken into the heating power Calculation . The heating power of filament i is: 2. Pi = I i Ri (9). Heating powers of conductors and sheaths are: Np 2Np 3N p PR = I i . 2 2 2. R i , PS = Ii R i , PT = Ii Ri i =1 i = N p +1 i = 2 N p +1. Advanced Comupational Methods in Heat Transfer VI 533. 3N p + N s 3N p + 2 N s N.. 2 2 2. PsR = Ii R i , PsS = Ii R i , PsT = Ii Ri i =3 N p +1 i =3 N p + N s +1 i = 2 N p + 2 N s +1. (10). Total heating power is as follows: T T.

9 P= Pi + Psj (11). i =R j= R. Once we have calculated the heating power the quasi-state heat transfer and temperature field can be calculated either by electrical analogue procedure (Ohm's law) or by finite element method. 3 An Example The method presented is applied in the transmission line of three single-core cables. Electric power cable is of IPHA 04, (FKS [6]) type. Cross-sectional area of the copper stranding conductor is 400 mm2. The sheath is made of mm thick aluminium pipe with outside diameter of mm. Oil-impregnated paper tape insulation is 8 mm thick. 2 mm thick PVC is used for non-metallic outer sheath. External diameter Dk= mm. Rated voltage is 35 kV. The cable route is 1000 m long. Three single-core cables are connected to the infinite power . buses.

10 Their voltages are: VR= kV, VS= /4 /3 kV, VT= /2 /3 kV. Three-phase balanced load is connected to the other terminal. The load impedance is: ZRl = ZSl = ZTl =43 +j 0 . The current rating of cables in flat formation with the separation of so=70 mm is 450 A. For cables in trefoil formation the current rating is 500 A. Conductors and sheaths are divided into five (Np=5) and sixty filaments (Ns=60), respectively. The stranded conductor contains 61 (1+6+12+18+24) wires whose diameter is mm. The conductor filament resitivity at 60oC is f = m. The sheath filament resitivity at 55oC is s= m. The earth electrical resistivity is e =50 m. Cables in flat (Figure 2) and trefoil (Figure 3) formation are analyzed. During the calculations the cable separation is changed for cables in flat formation as follows: so=0 mm and so=70 mm and so=140 mm.


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