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Calculation of earthing and screening effects of compensation conductor laid alongside underground multi-cable power lines I. Medic1, M. Majstrovic2 & P. Sarajcev1 1 Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Croatia. 2 Energy Institute "Hrvoje Požar", Zagreb, Croatia. Abstract Power substations in an urban area are interconnected by high voltage underground power cables, resulting also in interconnections of their earthing grids through cable metallic screens/sheaths. Most modern power cables with XLPE insulated core have metallic screens which are practically insulated from ground. These screens can only be (and usually are) grounded at cable ends and therefore do not act like extra earth electrodes extending from a station. In this case, almost the same positive earthing effects of an extended electrode can be obtained only by extending an auxiliary wire in the same trench and routing it closely parallel to the power line. This bare conductor also provides better screening for nearby telecommunication circuits against inductive interference. This paper presents an analytical method to determine the line-to-ground fault current distribution and related voltages in the earthing system which consists of two substation earthings interconnected by insulated metal sheaths of modern XLPE cables and one bare earthing wire laid in parallel in the same cable trench. The analytical expressions, derived by solving a set of differential equations, are based on the assumption of uniformly distributed conductance between the earthing wire and earth. The proposed mathematical model takes into account mutual inductive couplings between all conductors (phases, cable sheaths and the bare conductor). In addition the paper presents a procedure for computing the resulting nominal screening factor of multi-conductor connection between two substations. The earthing and screening effects of one bare conductor laid parallel to double cable power line are numerically illustrated and outlined. 206 Electrical Engineering and Electromagnetics VI 1 Introduction Earthing grids of power substations in an urban area are interconnected through underground cable metallic screens/sheaths. The older power cables usually have paper-insulated core, lead sheath and external semiconducting jacket of compound and jute. Lead sheath acts like an additional extended horizontal grounding electrode, which have finite and distributed conductance to ground. During line-to-ground fault in power system, the zero-sequence fault current returns to the source stations through the neutral conductors (metallic cable sheaths, overhead ground wires, auxiliary earthing wires, etc.), or through the earthing structures and earth. Parts of the fault current, flowing into the earth through earthing structures raise the potential of the surrounding earth with respect to remote earth. Therefore, with respect to both personnel safety and equipment operational compatibility, the lead sheath cables have a positive influence on a substation in reducing substation earthing impedance and earth voltage rise. Most modern power cables have XLPE insulated core, copper wire screen and insulated outer protecting covering (PVC or PE) and thereby do not have distributed conductance to ground. In this case, almost the same positive earthing effects of lead sheaths can be obtained only by extending an auxiliary bare conductor in the same trench. At the same time this conductor will also provide better screening for nearby telecommunication circuits and auxiliary cables against inductive interference. The disturbances that result from the line- to-ground fault current are of concern to both the power utility and to utilities with nearby telecomunications facilities. Therefore, it is important to make an accurate analysis of zero-sequence currents distribution among the earthing system components. In the same time it is also difficult to achieve that goal, mainly due to complexity and number of influencing factors. A number of different analytical methods have been developed in the past three decades (e.g. [1]-[4]) to accurately calculate fault current distribution. These methods yield adequate results for overhead lines but cannot be applied directly to underground cables. The method described by Guven and Sebo [5] deals with current distribution along the single neutral conductor representing underground cable sheath (or overhead shield-wire tower-footing chain). Therefore, it does not take into account mutual coupling between neutral conductors. This paper introduces an improved method of calculation of both fault current and voltage distributions along a single earthing/compensation bare conductor and arbitrary number of insulated neutral conductors buried in the same trench. The method includes mutual inductive coupling among all conductors in the trench (phases, sheaths and earthing wire). 2 Mathematical model Consider two high-voltage power substations (”A” and ”B”) which are interconnected by power cable lines with a set of M phase conductors. These power cables have insulated metallic screens or sheaths that are connected at Electrical Engineering and Electromagnetics VI 207 both terminals on the appertained earthing grid. Let the number of insulated screens/sheaths be N. Furthermore, we presume the existence of an earthing wire which is laid in the same cable trench and parallel to cable sheaths. The total number of connected neutral conductors is then N+1. To investigate this case, the scheme of Figure 1 is constructed representing the zero-sequence equivalent circuit for earthing system under consideration. Referring to Figure 1, the following set of differential equations applies to neutral conductor circuits: d − {i } = [g]⋅ {v} (1) dx d − {v } = [ z ]⋅ {i }− [z f ]⋅ {I f } (2) dx L1 L2 L3 L1 L2 L3 IfM I f M-1 I f M-2 If3 If2 If1 IN+1 IN Ik I2 “ A” I1A I1B “ B” i1(x) i1(x+ ∆x) J rA JrB v1(x) v1(x+ ∆x) IA IB VA ∆i1(x) VB ZA ZB ∆x x L-x x=0 x=L Figure 1: Equivalent scheme of multi-cable power lines earthing system. 208 Electrical Engineering and Electromagnetics VI The axis of the power line is denoted as the x-axis, and the total line length is denoted by L with the left end at x=0 and the right end at x=L. Denotations in eqns (1) and (2) have the following meaning: [z] - square (N+1, N+1) matrix of self and mutual unit-length serial impedances of neutral conductors, with earth return. [zf] - rectangular (N+1, M) matrix of unit-length mutual impedances between neutral and phase conductors, with earth return. [g] - (N+1, N+1) matrix of self and mutual unit-length shunt admittances of neutral conductors. {i} - the vector of unknown currents ik(x), k = 1, 2, ...,(N+1) of neutral conductors at point x. {v} - the vector of unknown voltages vk(x), k = 1, 2, ...,(N+1) of neutral conductors at point x. {If} - the vector of known phase conductors fault currents If k, k = 1, 2, ..., M. Matrices [z], [zf], {v} and {i} can be partitioned into blocks so that eqn (2) can be rewritten as follows: d v1 z 11 z 1s i 1 z f 1 − = ⋅ − ⋅ {I } (3) dx vs z s1 z ss I z f s f where: [zss] is (N, N) submatrix related to impedances of insulated sheaths only. The calculation method presented in this paper is based on the following assumptions: Only the fundamental frequency (e.g. 50 Hz) is considered. Mutual admittances between all conductors are neglected, so the matrix [g] is a diagonal one. Uniformly distributed parameters (serial impedance z11 and shunt admittance g11 are applied to the earthing wire circuit presentation. Self admittances of insulated neutral conductors are neglected (gkk = 0, k= 2, 3, ..,.N+1). Serial unit-length impedances with earth return [z] and [zf], are calculated by means of the Carson-Pollaczek [6], [7] equations. The medium surrounding the earthing wire and cable lines is homogeneous and characterized by soil resistivity ρ (Ωm). Conductive coupling between the earthing wire and substation earthing grids is neglected. The magnitudes of phase conductor fault currents are known from system studies. Electrical Engineering and Electromagnetics VI 209 Since the cable metal screens are insulated from ground, we may assume that gkk=0 (for k = 2, 3,…,N+1) and eqn (1) yield solutions: ik (x)= const. = Ik k = 2, 3, …, N+1. (4) Using eqns (1), (2) and (4), it follows: d 2 v1 ( x ) − z11 ⋅g11 ⋅ v1 ( x ) = 0 (5) dx 2 The equation (5) may be solved by standard methods to obtain the general solution which can be written in terms of two arbitrary constants. Using the general solution for v1(x) and eqns (3) and (4), one can obtain the general solutions for i1(x) in terms of the same two arbitrary constants plus the additional one. Furthermore, integrating eqn (3) and using eqn (4) and the general solution for i1(x), yields the set of general solutions for insulated neutral conductor voltages vk(x), k=2, 3, …,N+1, each of them in terms of a different arbitrary constant. So the total number of arbitrary constants are N+3. Next, if we assume that the boundary conditions at x=0 (i.e. at the fault point "A") are known: i1 (x=0) = I1A vk(x=0) = VA (k = 1, 2…,N+1) the arbitrary constants can be determined and the required solutions can be written, as follows: Z csh ( γx ) N +1 M v1 ( x ) = VA ch ( γx ) − I1A Z csh ( γx ) − ⋅ ∑ z1 k I k − ∑ z f 1 k I f k (6) z11 k=2 k =1 VA ch ( γx ) − 1 N +1 M i1 ( x ) = − sh( γx ) + I1A ch ( γx ) + ⋅ ∑ z1 k I k − ∑ z f 1 k I f k (7) Zc z11 k =2 k =1 z s1 N +1 z s1 ⋅ z1k vs ( x ) = v1 ( x ) − x ⋅ ∑ zsk − I k , z11 k =2 z11 (8) M z s1 ⋅ z f 1k + x ⋅ ∑ zf sk − I f + 1 − z s1 VA . k k =1 z11 z11 s = 2, 3, …, N+1 where 210 Electrical Engineering and Electromagnetics VI γ = z11g11 z11 Zc = g11 Hence, at x=L (i.e. at point “B”): Z csh( γL) N +1 M VB = VA ch ( γL) − I1A Z csh( γL) − ⋅ ∑ z1 k I k − ∑ z f 1 k I f k (9) z11 k=2 k =1 VA ch ( γL) − 1 N +1 M I1B = − sh( γL) + I1A ch ( γL) + ⋅ ∑ z1 k I k − ∑ z f 1 k I f k (10) Zc z11 k =2 k =1 Also, for the boundary condition at point "B" (x=L): vk(x=L)=VB. (k = 1, 2…,N+1) from eqn (8) we obtain: N +1 z s1 ⋅ z1k z L⋅ ∑ zsk − z11 I k −( VA − VB ) ⋅ 1 − s1 = z k =2 11 (11) M z s1 ⋅ z f 1k L ⋅ ∑ zf sk − If . z11 k k =1 s = 2, 3, …, N+1 Usually, the known values are phase currents coming from all substation transformers and power lines connected to buses L1, L2 and L3 of substations "A" and "B" during the phase-to-ground fault. In order to achieve more realistic results (as recommended in Seljeseth et al. [8]), the "reduced fault currents" JrA and JrB should be calculated. These currents which actually enter the ground through earthing structures connected at points "A" and "B" respectively, are calculated separately using power line cross-section data and power line phase or zero-sequence currents. Therefore, the boundary conditions should be modified and the terminal impedances ZA and ZB incorporated as well. According to Figure 1 these conditions can be written as: VB − Z B I B = 0 (12) VA − Z A I A = 0 (13) N +1 I B − I1B − ∑ I k = J rB (14) k =2 N +1 I A + I1A + ∑ I k = J rA (15) k =2 Electrical Engineering and Electromagnetics VI 211 where ZA and ZB are the earthing impedances of substations "A" and "B", respectively, with neutral interconection conductors excluded. Finally, to obtain the solution for voltages and currents {X} at both terminals, one has to solve the set of simultaneous linear eqns (9)-(15) which can be written in the following matrix form: [A]⋅ {X} = {B} (16) {X} = {VA {I }T } T where I1A VB I1B IA IB {I} – the column vector of sheath currents, VA, VB, I1A, I1B – earthing wire terminal voltages and currents, IA , IB – substation earthing structure currents, 3 Calculation of the resulting screening factor The current in each neutral conductor has two components. One is caused by inductive coupling between phase and neutral conductors and the other is caused by voltage difference between the two earthing grids. The currents exclusively caused by inductive coupling are of particular interest because they also reduce negative electromagnetic influence of phase conductor fault currents on neighbouring communication cables and other metallic installations that are isolated in relation to the earth. They can be derived from eqns (16) as special case when ZA = ZB = 0 or directly from eqns (2) substituting dv = 0. Thus for this case the next matrix equation is valid: [ z ]⋅ { I } = [ z f ]⋅ {I f } (17) For a multi-conductor connection, the resulting nominal screening factor (also known as the current reduction factor), can be defined as: N +1 ∑ Ii r =1− i = 1 (18) M ∑ If k k =1 and using the solution for currents from (17) it can be finally expressed as: N + 1 N + 1 M ∑ ∑ Yi j ⋅ ∑ z f jk ⋅ If i =1 j=1 k k =1 r =1− (19) N +1 ∑ If k k =1 212 Electrical Engineering and Electromagnetics VI where Yij are elements of the inverse of the matrix [z]: [Y] = [z]-1 (20) The same procedure applied to the single line power cable conductors (phase and neutral conductors) yields the value of the so called line screening factor that is frequently used for substation earthing current calculations (e.q. Seljeseth et. al. [8]). 4 Illustrative example As an example of the mathematical model application we have developed, consider the cable trench of Figure 2 in which the following conductors exist: Two groups of three single-core 110 kV cables forming two power lines. One earthing bare conductor (Cu 50 mm2) extended at depth 0,7 m. Cable type: single-core 110 kV, (ABB AXKJ 1000Al mm2), sheath 95 mm2 Cu, cable length L = 4000 m. D = 0,086 m; S1 = 0,35 m; S = 0,7 m; H=0,5 m. Other relevant data are: Phase currents: If 1 = If 4 = 7050∠-79o A; If 2 = If 3 = If 5 = If 6 = 413∠119 o A. Soil resistivity: 100 Ωm. Earthing conductor shunt admittance is: g11=0,003 S/m. Terminal reduced currents: JrA= 13503 ∠-78o A, JrB= 6308 ∠111o. Substation earthing impedances: ZA=0,15 Ω, ZB=0,1Ω. S1 S Cu 50 mm 3 6 H 1 2 4 5 D D D D Figure 2: Example of cable trench with six 110 kV single-core cables and an earthing conductor. Electrical Engineering and Electromagnetics VI 213 1400 [V], [A] 1200 1000 | V1 |, | V2 |, | V20 |, | I1 |, | I1a | 800 V1 V2 600 V20 I1 I1a 400 200 0 0 500 1000 1500 2000 2500 3000 3500 4000 x [m] Figure 3: Voltage and current distributions along the earthing wire and power cable sheaths. The main numerical results obtained using the derived mathematical model are presented graphically in Figure 3, where. V1=|V1(x)| and I1=|I1(x)| are distributions of voltage and current magnitudes along the earthing conductor, respectively. V2=|V2(x)| and V20=|V20(x)| are cable sheath voltage magnitude distributions obtained with and without earthing conductor presence, respectively. I1a=| I1 |= 863 [A] is the earthing conductor constant current obtained in case ZA = ZB = 0 Ω. Numerical results obtained show a quantitative reduction of the voltage V1(x=0) at fault point "A" as well as the reduction of cable sheath currents, caused by earthing wire connection. In this example the magnitude of the remote terminal voltage VB is lowered also. Both terminal voltages depend to a great extent on magnitudes of corresponding reduced current Jr and substation earthing impedance. The resultant nominal screening factor and the earthing wire current (both obtained in case ZA = ZB = 0 Ω) are respectively: r = 0,058∠-83,8o and I1a = 863,7∠-120o A. In case that earthing wire is absent the nominal screening factor is r = 0,076∠-78,3o. At the same time the screening factor calculated for single power line is r1 = 0,152∠-73,4o. 214 Electrical Engineering and Electromagnetics VI As differences in numerical results show, the calculations with individual line reduction factors should not be used if more than one power line exist in the same trench. 5 Conclusion This paper has derived a set of equations providing a more accurate calculation of the fault current distribution and related voltages in underground cable neutral conductors and the accompanied parallel earthing wire interconnecting two neighboring substations. These equations are based on the assumption of a uniformly distributed conductance between earthing wire and earth. Furthermore, the procedure for determining resulting nominal screening factor of multi- conductor connection between two power substations has been presented. The paper outlines the effects of inductive coupling between cable neutral conductors and earthing wire on both current and voltage distributions in the earthing system during phase-to-ground fault. The earthing wire connection is always in favor of both the substation safety and cable thermal withstanding. It also provides better screening for nearby auxiliary cables against inductive interference. The results obtained show that method using individual line screening factors cannot be considered as generally recommendable for practical applications when more power lines are in the same trench. 6 References [1] Endrenyi, J. Analysis of Transmission Tower Potentials During Ground Faults, IEEE Trans. on Power Apparatus and Systems, 86(10), pp. 1274- 1283, 1967. [2] Funk, G. Berechnung der Nullstromverteilung bei Erdkurzschluss einer Freileitung, ETZ-A, 92(2), pp. 74-80, 1971. [3] Dawalibi, F. Ground Fault Current Distribution between Soil and Neutral Conductors, IEEE Trans. on Power Apparatus and Systems, 99(2), pp. 452- 461, 1980. [4] Gooi, H. B. & Sebo, S. A. Distribution of ground fault currents along transmission lines – an improved algorithm, IEEE Trans. on Power Apparatus and Systems, 104(3), pp. 663-670, 1985. [5] Guven, A. N. & Sebo, S. A. Analysis of Ground Fault Current Distribution Along Underground Cables, IEEE Trans. on Power Delivery, 1(4), pp. 9- 18, 1986. [6] Carson, J. R. Ground Return Impedance: Underground Wire with Earth Return, Bell System Technical Journal, 8, pp. 94-98, 1929. [7] Pollaczek, F. Über das feld einer unendlich langen wechsel- stromdurchflossenen Einfachleitung, Elektr. Nachr. Technik, 3(9), pp. 339- 359, 1926. [8] Seljeseth, H., Campling, A., Feist, K. H. & Kuussaari M. Station Earthing. Safety and interference aspects, Electra, (71), pp. 47-69, 1980.

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