6. Transmission Line Models Wire over Earth capacitance b Point B a = h Electric Field Point A Volts per meter bi ≈ 2h ai = h Distance b = radius r Distance bi ≈ 2h for h >> r. 2π • 8.854 pF per meter length = 55.6 pF / 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 pF/m Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015

Reasonable estimate is 1 μH/m 6. Transmission Line Models, cont. Wire over Earth inductance Amperes per meter . Magnetic field intensity H 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

6. Transmission Line Models, cont. Symmetric bundles have an equivalent radius Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015 Figure 8. Equivalent Radius for Three Common Types of Bundled Phase Conductors

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

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

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. 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. 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

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

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

6. Transmission Line Models, cont. Ready for Use! R jω L jωC/2 1 jω C/2 + 200kVrms - P + jQ I Q C1 produced 2 L absorbed C2 V / δ One circuit of the 345kV line geometry, 100km long Prof. Mack Grady, TAMU Relay Conference Tutorial, Topic 6, March 31, 2015

6. Transmission Line Models, cont. 345kV 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

4.67” corresponds to 30 m 2.7” 0.8” 1” 138 kV, 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 Ground wire

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

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 HW3_Create a loadflow “base case” for the network, using the above numbers as load MW. Assume power factor = 0.85,