Currents and Voltages Induced by Electric Field in Two Converging Single-wire Overhead Transmission Lines

Three methods for specifying the function of partial mutual capacitance between converging single-wire overhead transmission lines (OTL) $i$ and $k$ depending on the distance along the line are considered: in graphic, analytical and interpolation modes. The practical coincidence of the mutual capacitance values along the converging lines calculation results for the considered methods is shown. Schemes and algorithms for current and voltage distribution along $i$ and $k$ converging single-wire OTL calculation are developed using Carson's equation for line self-inductance, taking into account the ground finite conductivity, and with the replacement of partial mutual conductance and phase’ electromotive force (EMF) to elementary current source function. Calculations of the induced current and voltage distribution along $i$ disconnected line under its grounding at near and far from $k$ line end were carried out. It is shown that induced voltage at the ungrounded end of disconnected line is higher than at the grounded one. It is noted that when $i$ line is grounded at far end from $k$ line, the induced voltage at the ungrounded end is higher than when $i$ line is grounded at near from $k$ line end. Current voltage values induced by electric field in $i$ line grounded an no grounded ends depending on the line convergence “p” section length, minimum “a0” distance between the converging lines and “Θ” angle of the line convergence are carried out. Boundary p, a0 and Θ values, one of which violation allows us to neglect the converging OTL electric effect.


INTRODUCTION
Personnel working on overhead transmission lines (OTL) under induced voltage health maintenance requires these voltages limit values compliance. These requirements compliance ensure is possible by use the developed algorithms for calculation the current and voltage distribution along the disconnected grounded single-wire (single-phase) i line, induced by electric field (EF) of operating single-wire (single-phase) k line, converging with the i line under Θ angle.

PARTIAL CAPACITANCE BETWEEN CONVERGENT SINGLE-WIRE I AND K LINES
Consider first two parallel single-wire i and k OTLs located from each other at a distance and at h i = h and h k = H height above the ground (Figure 1, 1(a)). Figure 1(b) shows the scheme of i and k single-wire OTLs connection between lines and with the ground by partial capacities.
C ii0 , C kk0 and C ki0 specific partial capacities are determined by Maxwell's formulas first group potential factors use (1).
where: H and h are the heights above the ground; R and r are k and i line wires radiuses, respectively. Potential factors α matrix has the form (2): (2) Matrix α inversion leads to obtain the Maxwell's formulas second group potential factors β matrix (3): Potential factors have dimension is meter per farad [m/F], and capacity factors dimension is farad per meter [F/m].
Consider lines k and i, converging at Θ angle ( Figure 2). We know: i line length l i = p, the distance between k and i lines at the beginning a 0 and at the end a(p) of i line, and so at the and a(p) of line we know k and iΘ converging angle, because: Let's count, that a 0 = 5 m and a(p) = 100 m. Lines k and i parameters are: H = 19 m, h = 17.5 m, R = r = 0.014 m. Partial capacity values between k and i lines are determined by the expression [1, 2] C ki0 = −β ki = −β 12 = −β 21 , then on expressions (1) ÷ (3) we obtain C ki0 (a) values with a change from 5 m to 100 m in 5 m increments, as it is shown in Table 1.
It should be noted that to obtain the values of partial capacities between phases and lighting wires of converging working and disconnected three-or more-phase power lines, C ki0 (a) value calculation must be carried out taking into account all of these OTL phases and lighting wires. C ki0 (a) values can be calculated for case from one to six three-phase OTL by computer program "OTL EMF" [3].
To represent C ki0 (a) capacity value changes from a distance in the graphical form, fill in the MathCAD the matrix named Z, containing one row and c = 20 columns, C ki0 values from the Table 1. Next, we transform it into column matrix and, choosing the boundaries and the step of ψ j argument changing we obtain Z j curve with C ki0 values linear interpolation. To smooth Z j curve, create ZZ(y) function and set the limits and the step of argument a changing (4): ZZ(a) function allows carrying out numerical mathematical operations with it, but it is not analytically determined. Define an analytical expression for C ki0 (a) capacity value.
For this purpose, in MathCAD, using the expressions (1) and (2) at ε 0 ≡ ε, we carry out the operation of matrix α analytical inversion analytical equal sign (→) using: Then analytically obtained C An ki0 (a) capacity value can be written (5): C An ki0 (a) function disadvantage is complexity, and complexity degree increases dramatically with increasing number of converging lines.
Will hold a rough interpolation (approximation) ZZ(a) curve function (6) using: The coincidence of ZZ(a), C An ki0 (a) and C Int ki0 (a) curves is good. Thus, when calculating transverse voltage induced on the disconnected and grounded single-wire i line by converging single-wire k line electric field, partial capacity between these lines, depending on the distance between them, can be described by C ki0 (a) = ZZ(a), C An ki0 (a) and C Int ki0 (a) functions.

CURRENTS AND VOLTAGES DISTRIBUTION ALONG GROUNDED AT ONE END I LINE CONVERGING TO K LINE
OTL k l k length is in idle mode under voltageU k =Ė k = 127 kV, and i line l i < l k length is disconnected and grounded at one end remoted from k line substation (in this case, the left end) (Figure 4). Figure 4:
The distance a(l) between k and i OTLs change (Figures 2 and 5) is described by the expression (7): Substituting in C ki0 (a) expression as well as expressions (4) ÷ (6) instead of variable a a(l) function, get the equations for C ki0 (l) = ZZ(a(l)), C An ki0 (l) and C Int ki0 (l). Let us replace electromotive forceĖ k and specific capacitance jωC ki0 (l)dl with elementary current source dİ(l) ( Figure 5): dİ(l) = jωC ki0 (l)Ė k dl.  Because dİ(l) = jωC ki0 (l)Ė k dl, the current in iline find by expressions in graphic, analytical and interpolation forms: VoltageU A / (l) value to point A / , which potential is assumed to be 0, and voltageU a b (l) value along i line find by equations:      I(l) current in graphic, analytical and interpolation forms are presented in (8): Since there is no current in Z e0 resistance, the voltage isU a b (l) =U A / (l):  (Figure 7(a)) when it is grounded at near to at k line end ( Figure 5). This is mathematically explained by larger area under the curveİ(l) in Figure 9(a), than in Figure 7(a), calculated by the integral´İ(l)dl, included in the expression of voltageU A / (l) andU a b (l). The physical explanation is that in case of i line grounded at far from k end induced currentİ(l) module from the beginning to the end of i line is higher than in case of i line grounded at near k line end. In expression (7) a function variables will be a 0 , l and Θ: a(a 0 , l, Θ). Partial capacity between lines we present in analytical form: C ki0 (a) = C An ki0 (a). Substituting a(a 0 , l, Θ) function in expression (5) instead of variable a get C An ki0 (a 0 , l, Θ). Similarly, by introducing additional variables into formulas (7) and (8), we obtain equations for current and voltage: İ(a 0 , l, Θ) andU a b (a 0 , l, Θ).
Take the angle Θ = 0.0573 • . Figure 11 shows the distribution of current modulesİ(a 0 , l = p, Θ) (a) and voltage modulesU A / (a 0 , l = p, Θ) =U a b (a 0 , l = p, Θ) (b), for case of grounded (l = p) i line, as well as Figure 12 shows voltageU A / (a 0 , l = 0, Θ) =U a b (a 0 , l = 0, Θ) module values for case of ungrounded (l = 0) i line end l i = p = 100, 50 and 10 km length with minimum distance a 0 between k and i lines change from 5 to 100 m.

CONCLUSION
The results of calculation show that transverse voltage induced in unconnected i line by electric field of converging with itk line under voltage can be neglected in case of violation from the values of any tested parameters presented in Table 2 as well as in [7]: the length of i line converging part l i = p, minimal distance a 0 between converging lines, convergence angle Θ.