6 Transmission Line Models Wire over Earth capacitance

6. Transmission Line Models Wire over Earth capacitance b Point B Electric Field a=h Point A Volts per meter bi ≈ 2 h ai = h Distance b = radius r Distance bi ≈ 2 h for h >> r. 2π • 8. 854 p. F per meter length = 55. 6 p. F / meter ln(10000) = 9. 2, C = 6. 0 pf/m ln(1000) = 6. 9, C = 8. 1 pf/m ln(100) = 4. 6, C = 12. 1 pf/m For our purposes, a reasonable estimate is 10 p. F/m Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 1

6. Transmission Line Models, cont. Wire over Earth inductance Magnetic field intensity H Amperes per meter. for h >> r. (4π • 10 -7) / 2π Henries per meter length = 0. 2 μH/m ln(10000) = 9. 2, L = 1. 8 μH/m ln(1000) = 6. 9, L = 1. 4 μH/m ln(100) = 4. 6, L = 0. 92 μH/m Reasonable estimate is 1 μH/m Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 2

6. Transmission Line Models, cont. Symmetric bundles have an equivalent radius Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 3

6. Transmission Line Models, cont. Three phases If the transmission line is symmetric, then the “P matrix” has the equal diagonal, equal off-diagonal property that permits 0 -1 -2 analysis rather than a-b-c analysis Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 4

6. Transmission Line Models, cont. If the transmission line is symmetric, then the “L matrix” has the equal diagonal, equal off-diagonal property that permits 0 -1 -2 analysis rather than a-b-c analysis Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 5

6. Transmission Line Models, cont. Summary of Positive/Negative Sequence Capacitance and Inductance Calculations Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 6

6. Transmission Line Models, cont. Summary of Positive/Negative Sequence Capacitance and Inductance Calculations Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 7

6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations meters Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 8

6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 9

6. Transmission Line Models, cont. Summary of Zero Sequence Capacitance and Inductance Calculations Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 10

6. Transmission Line Models, cont. Ready for Use! QL absorbed P 1 + j. Q 1 I 1 + 200 k. Vrms - R 1 jωC/2 QC 1 produced jω L 1 jω C/2 QC 2 produced P 2 + j. Q 2 I 2 + VR / δR - One circuit of the 345 k. V line geometry, 100 km long Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 11

6. Transmission Line Models, cont. 345 k. V Double-Circuit Transmission Line 22. 9 m at tower, sags down 10 m at mid-span to 12. 9 m. Double conductor phase bundles, bundle radius = 22. 9 cm, conductor radius = 1. 41 cm, conductor resistance = 0. 0728 Ω/km Single-conductor ground wires, conductor radius = 0. 56 cm, conductor resistance = 2. 87 Ω/km Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 12

Ground wire 1” 4. 67” corresponds to 30 m 0. 8” 1” 2. 7” • • • 138 k. V, single circuit. Bundled conductors are 0. 5 m apart. Phase conductors have 1. 5 cm radius, 0. 1 Ω/km Sag depth at mid-span is 5 m. Ground wire is steel, with radius 0. 5 cm, 3 Ω/km

Connect County Seats. Each segment is 50 km, 138 k. V, single circuit. Fort Worth Dallas Cleburne Waxahachie Hillsboro Waco Belton Georgetown Austin

Relative Load Levels (sum = 30) • • • Dallas, 10 Fort Worth, 6 Waxahachie, 1 Cleburne, 1 Hillsboro, 1 Waco, 3 Belton, 1 Georgetown, 1 Austin, 6 Relative Generation Levels (sum = 40) • • • Dallas, 10 Fort Worth, 3 Waxahachie, 2 Cleburne, 2 Hillsboro, 3 Waco, 10 Belton, 2 Georgetown, 2 Austin, 6 HW 3_Create a loadflow “base case” for the network, using the above numbers as load MW. Assume power factor = 0. 85,