COMPUTATION OF ELECTRIC CHARGE ON POWER TRANSMISSION LINES Aleš Berkopec Fakulteta za elektrotehniko, Ljubljana, Slovenia Key words: quasi-static cliarge computations, power transmission lines Abstract: A system of parallel lines above a conducting or insulating plane serves as a model of a transmission line system. We present a few computational steps and results that address the question of the synchronicity of electric potential and charge on a given wire of the power line system. The differences in phase angles of the oscillating charge and the associated potential depend on the geometry of the system. For a benchmark and three additional cases the charges on the wires were computed using the described procedure. They are presented in the results section. Izračun električnega naboja na močnostnih prenosnih linijah Kjučne besede: kvazistatičnl izračuni električnega naboja, daljnovodnl sistemi Izvleček: Dvodimenzionalni sistem vzporednih vrvi končnih polmerov, ki se razpenjajo nad prevodno ali včasih neprevodno ravnino, služi kot osnovni model pri obravnavi sistemov daljnovodnih napetostnih vodov. V članku pokažemo, da naboj danega vodnika in njegov potencial v splošnem ne nihata sofazno. Razlika med faznima kotoma potenciala in njemu pripadajočega naboja je odvisna od geometrije sistema. Predlagamo ustrezno pot do iskanih nabojev. Le-ti so osnova za Izračun ostalih električnih količin, predvsem električne poljske jakosti v okolici sistema, in prikažemo nekaj rezultatov za izbrane postavitve daljnovodnih vrvi. 1 Introduction The most basic among the models of power transmission line systems is two-dimensional. It consists of conducting parallel straight lines with known phase angles and r.m.s. values of electric potentials. The diameters of the lines are small compared to the distances between wires. The task is to determine linear charge densities on the lines from the given electric potentials of the lines. There are more realistic models of powertransmission lines, for instance, the diameter of the conductors may not be small compared to the distances between the lines, or the gravity and string forces may be included, which distort the straight lines into the chain curves. Further more, some computer programs consider the electrical properties and geometry of the pylons and even terrain. For all the cases mentioned the computational algorithm is basically the same, although the matrix coefficients may be a way more difficult to compute and the size of the matrix tends to grow considerably /1 / . 2 Methods The notation used in this article for a-priori known quantities is: r,-: position of the /-th wire, (x;, y,), V/: electrical potential of the /-th wire, l^/: phase angle of the electrical potential V,, and for a-priori unknowns: Qk'. linear charge density of the k-th wire, cpk'. phase angle of the linear charge density qk . The potential of the i-th wire has a form V/ ■ cosicot+ß,), where co =2nv and v is the frequency. The frequency v = 50 Hz justifies a quasi-static approach for power transmission lines, so at any given time the potential of /-th wire may be written as a superposition of the charge on all wires /5/: V\ ■ cos{ivt+ = qi ■ COS{ujt + ) 1 27reo In — + r-io y-v gk ■ COS {LJt+ (,0k) _1 2vreo |ri- (i; i-rkl The introduction of parameter Pik defined as Pik = Zttso ^ io 1 In 1 . 2tt£O " ' |ri - rk| gives a shorter form of equation {1): i = k i ^ k V, ■ cos{ujt+^i )= ^ /ik ■ % ■ cos{üjt+ipi^). k (2) The identity cos (a + ß) = cos a cos ß - sin a sin ß and equation (2) lead to two separate parts of the system of equations, the first oscillating as cos(CLtf),the second as sin(arf): v; ■ cosi9i = ^i^k-gk-cos¥Jk Vi ■ sinili = Y^Fik qk-siiKpk (3) (4) Any attempt to solve equations (3) and (4) directly for unknown Qk and pk is bound to fail for almost any set of input parameters r,-, V,, and -ö;. However, with the introduction of new variables, as shown in the following paragraph, the system of equations can be linearized, its matrix becomes diagonally dominant, and therefore suitable for further numerical manipulation. The system of equations (3) and (4) can be linearized by introducing variables Ok = gk ■ oos(^ bk = Qk- sinc^k Vc;i = "K ■ COSl?i = M-sini?:, and takes the form of two separate sets of linear equations: Vc = P-a Vs = P-b, where a = (ai, 32,...), b = (bi, b2,...) (5) (6) P= P21 P22 \ ■ and \ ■7 Since noCki-rkl for each / and k it follows that P is diagonally dominant. After solving (5) and (6) one can obtain the unknown linear charge densities q^ and phase angles (pk as: 9k 'Pk = ^flf^k , ök = arctan —. fflk Results This section presents computational outcomes - the linear charge densities and their phase angles - for three different systems of power transmission lines. All cases deal with 400 kV systems, with wires of 1 centimeter in diameter, but differ in some other aspects. The zeroth example serves as a benchmark. It is followed by the first case, which is a realistic example of the 400 kV system. The second example and the third example are a bit exotic: the second only because of the geometry chosen, while the third deals also with the number of the wires and their potentials that can hardly be found in practice. In a view of the conductivity of the ground both extreme possibilities were taken into account. When the ground is considered to be a perfect insulator the computations are performed as explained in the previous section, and their results in the examples section may be found under qmm/ {2neo) and cpmin- With the ground as a perfect conductor the solution is obtained by applying the method of images, and the results for each wire in these cases may be found in qmax/{27ceo) and cpmax columns. No additional unknowns are introduced in the case of a perfectly conducting ground, since the image charge of q/rth linear charge density at time t has a value Qk ■ cos(cot + cpk + n). Each of the following examples has three parts: input data, two-dimensional (x, y) sketch of the wires with the ground. and the resulting qk and (pk for all the wires in cases of insulating and conducting grounds. 3.1 Example 0 Table 1 input data i x[m] y [m] l^ma« [V] en 0 0.0 10.0 100000.0 0.0 ...............i.............®............. -10 Table 2 -5 0 X [m] 10 output data i II Žg-M 1 fmin[°] 11 ^[V] 1 max[°] 0 i| 18873.91 0.0 II 12056.81 0.0 Consider a wire with potential V= Vo- cos{cot), where Vo = 100 kV. We can obtain an analytical result for the charge on the wire if it is suspended above a conducting plane by the method of images In"", 27r£o 7-0 (7) where ro is the radius of the wire, and h is the ele-vation of the wire above the ground. The solution of (7) gives 90 = 0, where for given ro = 5 mm and h = 10 m we get: 7o/ln?^=12.057kV. The results of the computational algorithm below give the same result for qmax/(27ieo). The charge of the wire above a conducting ground matches the analytical result, but the result for an insulating ground should be ignored, since a single infinite conducting line does not have a uniquely defined electric potential. The numerical value Qmin in the output data equals ^ in{1/ro) and cannot be connected to the electric potential of the wire. 3.2 Example 1 Six parallel wires serve as a first model for a double 400 kV power transmission line system. Table 3 input data i X [m] y [m] VnnaK [V] en 0 -1.0 6.0 230940.0 0.0 1 0.0 6.0 230940.0 -120.0 2 1.0 6.0 230940.0 120.0 3 -1.0 10.0 230940.0 120.0 4 0.0 10.0 230940.0 -120.0 5 1.0 10.0 230940.0 0.0 10 8 — 6 ^ 4 2 0 "oSa- Table 6 -10 Table 4 0 X [m] 10 output data i qmin . 27rE" M

Ca ......................... 0 f h ---■(3....... ®......... J; -10 Table 8 -5 0 X |m| output data i qmin , . m