SIMPLE MAGNETIC FIELD MODEL FOR COMPLEX THREE DIMENSIONAL POWER TRANSMISSION LINE CONFIGURATIONS

Transcription

1 SIMPLE MAGNETIC FIELD MODEL FOR COMPLEX THREE DIMENSIONAL POWER TRANSMISSION LINE CONFIGURATIONS D.R. Swatek Major Transmission Planning Department Manitoba Hydro 820 Taylor Ave., P.O. Box 815 Winnipeg, Manitoba, Canada R3C 2P4 and l.r. Ciric Department of Electrical and Computer Engineering The University of Manitoba Winnipeg, Manitoba, Canada R3T 2N2 INTRO DUCT I ON Public interest in the possible health effects of 60 Hz magnetic fields from power transmission lines has sparked development of more complex and detailed computer models for their evaluation. Such models depart from the conventional representation of the transmission line as infinitely long straight parallel conductors. to a three-dimensional representation capable of accommodating conductor sags, bends, transpositions, and non-parallel circuits. Such a program has recently been developed by the Electric Power Research Institute (EPRI) [ 11. The EPRI program models the field by summing the exact solution to finite straight current segments which roughly trace the sag of the conductor. The position of each finite current segment is computed from given tower attachment points and sag data specified by the user for each conductor. However, such an input scheme may become impractical for complex multi-circuit transmission corridors. In this paper, a new program is proposed (titled MAGPLAN) which minimizes user input through a single-line plan view representation of the transmission corridor. Models may be constructed quickly by assembling circuits from a library of standard three-phase transmission towers. Computation time is reduced through numerical Gauss-Legendre integration along the parabolic conductor sag. Calculated magnetic flux density is compared to field measurements taken within a Manitoba Hydro 115 kv I 66 kv transmission corridor. CALCULATION METHOD The magnetic flux density in the vicinity of a multi-circuit three phase power transmission corridor is computed through evaluation of the Biot-Savart integral over the paths of the individual conductors, ( 1 )

2 where C is the path of the filementary current, /di is the vector differential current element, R is the vector from the current element to the observation point, µ 0 is the permeability of free space, and ff is the magnetic field intensity. A parabolic approximation to the conductor catenary is assumed in this paper. It is common to approximate ( 1) by summing the exact solution for finite straight thin current segments arranged to trace out the sag of the conductors [ 1,2,31. An alternate method -- proposed in this paper -- is to numerically integrate ( 1) along the parabolic conductor span using an efficient quadrature scheme such as Gauss Legendre [4). In both methods, the integration path (C) is closed by appending onto the end points of the detailed interval of integration, the analytical solution for semi-infinite current rays which approximate the remaining spans of current. The recent EPRI program, locates these current rays at the last tower attachment points. MAGPLAN constructs an additional parabolic segment from the last tower down to the point of average conductor height from which the semi-infinite ray is projected. Biot-Savart calculations in MAGPLAN are performed in vector form via three-dimensional vector algebra functions. The maximum rms value of the magnetic flux density, Bm. is taken to be../2 times Jess than the semi-major axis of the instantaneous field ellipse [5 ). COMPARISON OF INTEGRATION SCHEMES Two integration schemes are presented for comparison: the exact integration of straight current segments which approximate the parabolic sag of the conductor and a numerical Gauss-Legendre (G-L) integration of the exact parabolic conductor sag. A typical structure of a single circuit 230 kv line loaded to 500 A rms is considered. The test line uses a flat phase configuration with 7.31 Sm spacing between adjacent phases. Conductor height is 23.47m at the towers with a midspan clearance of 8.56Sm. One span of 300m is modeled in detail, with semi-infinite spans on either end, giving a total of nine parabolic spans, and six semi-infinite rays. Bm is calculated at 1S1 locations along a mid span cross section. In each case the span must be divided into several segments in order to attain a desired accuracy. The lengths of adjacent segments vary with the curvature of the conductor. As segmentation is successively doubled, the spatial maximum of Bm approaches a saturation level. For the purpose of this study, the integration is said to have converged when Bm is within O.OS~ of this saturation level. In Fig. 1, maximum values of Bm are plotted against the CPU time as segmentation is increased. A second order G-L integration converges in fewer segments and Jess CPU time than the straight line approximation (see Table 1 ). In fact, the second order G-L executes faster than the straight line approximation even for an equal number of segments.

3 Further reductions in CPU time are realized by using higher order G-L quadratures which require fewer segments to converge. As an example, a fourth order G-L integration requires 41.7 % less CPU time to converge than the straight line approximation. Although no attempt is made in this paper to find an optimum order of the G-L quadrature, enhanced computation time is clearly demonstrated. Both integration schemes converge to the analytical solution when the conductors are assumed to be infinitely long and parallel to the earth. However, with the conductors sagged as described above, the G-L integration converges to a 0.2 'X. higher value than the straight line approximation (see Fig. 1 ). This difference is most likely due to the geometric representation, since the G-L integration retains the smooth parabolic conductor shape. Thus, the G-L method is not only faster but likely more accurate. Table 1 s am_q 1 e R un T' 1mes f or t h e T est L' me (SUNSPARC station 2) Number of Order of CPU time s~ments/~an quadrature (seconds) Straight line approximation S6 16.8* S.9t Gauss-Legendre S * S.9t * 28 4 l 3.7t ( -- convergence to within O.OS%, t -- saturation to six significant digits) DATA ORGANIZATION AND MODEL CONSTRUCTION Recent software for the three-dimensional calculation of 60 Hz power line magnetic fields requires the user to define individual tower attachment points (in Cartesian coordinates) for each conductor [ 1). This form of modeling is extremely tedious and provides no simple means to modify the circuit representation. In MAGPLAN, individual tower attachment points are calculated, by the program, from a simple single-line plan view representation of the transmission corridor. The input data file is organized as follows: 1) The calculation grid is defined in the xy plane at a specified height (z). 2) A library of three-phase tower configurations is included which defines individual tower geometry in terms of conductor bundle attachment points in the xz plane and sub-conductor configuration.

4 3) Node points are defined which specify tower type, location and orientation in the xy plane. An additional vertical offset to the tower height may also be specified. 4) Node points are connected to form three-phase spans, which in turn are grouped into circuits. Balanced three-phase loading of the transmission lines is assumed. The phase order -- specified at both "to" and "from" nodes -- corresponds to the order in which conductor attachment points are defined in the tower library. Clearance of the lowest hanging conductor (or bundle) and integration parameters (number of segments and order of Gauss-Legendre integration) are specified for each span. Semi-infinite spans are requested by declaring either the "to" or "from" node to be "-1 ", and specifying average clearance and direction of the remaining spans. This organization allows the engineer to examine multiple "what if" scenarios by selecting different tower constructions from the library, or altering the path of a circuit simply by moving node points. Transpositions can be handled by altering phasing and tower types between nodes. FIELD TESTING In order to verify the modeling techniques used in MAGPLAN, actual field measurements are compared with calculated magnetic flux density values for the case of a complicated Manitoba Hydro transmission corridor. Fig. 2 provides a brief description of the test site including the location of the measurement cross section. Two double circuit 66 kv transmission lines run parallel to two double circuit 115 kv lines for 23Sm. One of the double circuit 66 kv lines makes a 90 bend to cross under the 11 S kv circuits, and then turns 90 to once again run parallel to the 11 S kv. Phase differences between the 11 S kv and 66 kv circuit currents can only be estimated through "load flow" programs. An EMDEX C-146 magnetic field meter was used to simultaneously measure and store the rms magnitude of the three orthogonal field components. For the purpose of comparison, calculation of Bm in MAGPLAN was simplified accordingly. Conductor clearances were measured by an ALTRASONIC HEIGHT METER, manufactured by SUPARULE. The MAGPLAN model consists of 14 different tower types ("left" and "right" hand sides of 2 double circuit 11 S kv towers and S double circuit 66 kv towers), 36 node points, and 8 circuits totaling 43 spans. Each span is modeled by a 24 segment fourth order Gauss-Legendre integration. Calculations and field measurements are in good agreement. A correlation is obtained between calculated and measured data (see Fig.

5 3). Deviations may be attributed to a lack of detailed engineering drawings of the test site and limited knowledge of the phase differences between the 66 kv and 115 kv line currents. CONCLUSIONS A new computer program (MAGPLAN) is presented for the calculation of the magnetic flux density in the vicinity of complex three-dimensional power transmission corridors. This model represents an advance over existing software in the areas of enhanced useability and computation time. User input is minimized through a simple singleline plan view representation of the transmission corridor. Calculation time is reduced through Gauss-Legendre integration which is shown to be faster and likely more accurate than conventional straight line segmentation. Program results are in good agreement with fields measured within a complex Manitoba Hydro transmission corridor. REFERENCES [ 1 I B.A. Clairmont and L.E. Zaffane11a, "Transmission Line Magnetic Field Design Options", presented at the EPRI sponsored "US - USSR Workshop on Innovation in High Capacity EHV and UHV Transmission", Palo Alto, CA. Spring [2) R.G. Olsen, D. Deno, and R.S. Baishiki. "Magnetic Fields From Electric Power Lines, Theory and Measurements'', paper 88 WM 078-8, submitted for the IEEE/PES 1988 Winter Meeting, New York. N.Y.. pp [3) W.H. Hayt, Engjneerjng E!ectromagnetics -- Fourth Edjtjon, McGraw-Hill, New York, N.Y. pp [4) A.H. Stroud and D. Secrest, Gaussian Quadrature Formulas, Prentice - Hall. Englewood Cliffs. N.] [5) EPRI. Iransmjssjon Line Reference Book kv and Aboye Second Edjtjon Reyjsed, EPRI. Palo Alto. CA, pp , 1987.

6 c straight segments --2nd order G-L th order G-L "' ~... 0 u 5 ~ s l::q 11.60! - - i i - i.....i... _ I I I! i i i i : : : : l I f ~ CPU Time (seconds) Fig. 1. Comparison of convergence time for straight line and Gauss Legendre approximation of the Biot-Savart integral over a 230 kv test line loaded to 500 Arms. -50 meters Ill, I 66 kv line kv line tower ~L Measurement cross section 0 50 All lines.... double circuit 1 ; meters Fig. 2. Description of test site. 1.5 Fig. 3. Comparison of measured field data and MAGPLAN calculation along the cross section indicated in Fig t.o ~,! e 1 j o;.tonce Along Croeo S.ct;,,n (m) (doto meaoured 91/11/14) 100