Practice Problems: Transmission
Question 1
A transmission engineer is analyzing a 345 kV, 60 Hz, three-phase transmission line that is 200 miles long. The line has the following per-phase parameters: resistance = 0.05 Ω/mile, inductive reactance = 0.85 Ω/mile, and shunt capacitive reactance = 0.18 MΩ·mile. The engineer needs to determine if the line should be modeled as a short, medium, or long transmission line and calculate the total series impedance per phase. What is the total series impedance per phase?
(a) 10 + j170 Ω
(b) 15 + j160 Ω
(c) 8 + j150 Ω
(d) 12 + j180 Ω
Question 2
A power utility engineer is evaluating a 230 kV transmission line operating at 60 Hz with a line-to-line voltage of 230 kV. The line delivers 150 MW at 0.85 power factor lagging to a load. The line has a series impedance of 50 + j180 Ω per phase. What is the sending-end voltage (line-to-line) required to maintain the receiving-end voltage at 230 kV?
(a) 245.8 kV
(b) 252.4 kV
(c) 238.6 kV
(d) 260.2 kV
Question 3
A transmission line design engineer is working on a 500 kV, three-phase transmission line with bundled conductors. Each phase consists of 4 sub-conductors arranged in a square bundle with 18 inches spacing between sub-conductors. Each sub-conductor has a GMR of 0.0435 ft and a diameter of 1.165 inches. What is the equivalent GMR of the bundled conductor configuration?
(a) 0.285 ft
(b) 0.312 ft
(c) 0.268 ft
(d) 0.341 ft
Question 4
A protection engineer is analyzing fault current on a 138 kV transmission system. A three-phase fault occurs at a point where the positive-sequence impedance is 5 + j25 Ω, negative-sequence impedance is 5 + j25 Ω, and zero-sequence impedance is 8 + j40 Ω. The pre-fault voltage is 138 kV line-to-line. What is the magnitude of the three-phase fault current?
(a) 3.12 kA
(b) 2.89 kA
(c) 3.54 kA
(d) 2.65 kA
Question 5
A transmission planning engineer needs to calculate the inductance per phase per mile of a single-circuit, three-phase transmission line. The conductors are arranged in a horizontal configuration with D₁₂ = 20 ft, D₂₃ = 20 ft, and D₁₃ = 40 ft. Each conductor has a GMR of 0.0373 ft. What is the inductance per phase per mile?
(a) 1.28 mH/mile
(b) 1.42 mH/mile
(c) 1.36 mH/mile
(d) 1.51 mH/mile
Question 6
A consulting engineer is evaluating corona inception voltage for a 345 kV transmission line. The conductor has a radius of 0.75 inches, and conductors are spaced 30 feet apart in an equilateral triangle configuration. The air density factor is 0.95, and the conductor surface irregularity factor is 0.85. Using Peek's formula, what is the critical disruptive voltage (line-to-neutral) in kV?
(a) 185.6 kV
(b) 198.4 kV
(c) 172.3 kV
(d) 205.7 kV
Question 7
A system operator is analyzing power flow on a 500 kV, 300-mile transmission line connecting two substations. The line has a surge impedance of 280 Ω and is operating at its surge impedance loading (SIL). What is the power being transmitted through this line?
(a) 893 MW
(b) 756 MW
(c) 982 MW
(d) 1024 MW
Question 8
A protection relay engineer is setting up a distance relay for a 230 kV transmission line. The line is 80 miles long with a positive-sequence impedance of 0.12 + j0.85 Ω/mile. The relay is to be set for Zone 1 protection covering 85% of the line length. What impedance setting (magnitude) should be used for Zone 1?
(a) 58.2 Ω
(b) 62.8 Ω
(c) 54.6 Ω
(d) 68.4 Ω
Question 9
A transmission engineer is calculating the capacitance per phase per mile of a 500 kV three-phase line with bundled conductors. Each phase has 3 sub-conductors in a triangular bundle with 18-inch spacing. Each sub-conductor has a radius of 0.0466 ft. The phases are arranged in a horizontal plane with spacing of 35 ft, 35 ft, and 70 ft. What is the capacitance per phase per mile?
(a) 0.0145 μF/mile
(b) 0.0128 μF/mile
(c) 0.0162 μF/mile
(d) 0.0138 μF/mile
Question 10
A grid operations engineer is analyzing the performance of a 345 kV, 150-mile transmission line modeled using ABCD parameters. The line has series impedance Z = 15 + j120 Ω and shunt admittance Y = j0.001 S. Using the nominal π model approximation, what is the value of parameter A?
(a) 0.940 ∠1.8°
(b) 0.965 ∠2.2°
(c) 0.952 ∠1.5°
(d) 0.978 ∠0.9°
Question 11
A transmission line engineer is calculating sag for a 795 kcmil ACSR conductor suspended between two towers 1200 feet apart at the same elevation. The conductor weighs 1.094 lb/ft and is under a tension of 6000 lbf. Assuming a catenary curve can be approximated by a parabola, what is the sag at the midpoint?
(a) 32.9 ft
(b) 29.5 ft
(c) 35.8 ft
(d) 26.4 ft
Question 12
A utility engineer is analyzing a single line-to-ground fault on phase A of a 138 kV transmission system. The sequence impedances are Z₁ = 8 + j40 Ω, Z₂ = 8 + j40 Ω, and Z₀ = 15 + j70 Ω. The pre-fault line-to-neutral voltage is 79.67 kV. What is the magnitude of the fault current?
(a) 1.68 kA
(b) 1.52 kA
(c) 1.84 kA
(d) 1.39 kA
Question 13
A transmission engineer is designing a 765 kV line and needs to calculate the capacitive reactance to neutral per mile. The line has conductors arranged in a flat horizontal configuration with phase spacing of 45 ft between adjacent phases. Each phase uses 4 bundled conductors in a square with 18-inch spacing. Each sub-conductor has a diameter of 1.424 inches. What is the capacitive reactance per mile at 60 Hz?
(a) 0.1065 MΩ·mile
(b) 0.0945 MΩ·mile
(c) 0.1182 MΩ·mile
(d) 0.0865 MΩ·mile
Question 14
A power systems engineer is evaluating the thermal rating of a transmission line conductor. The conductor is ACSR Drake with a diameter of 1.108 inches, operating at 75°C conductor temperature with 25°C ambient temperature, 2 ft/s wind speed, and full sun conditions. The total heat loss rate is calculated to be 12.8 W/ft. If the conductor resistance at 75°C is 0.132 Ω/mile, what is the maximum allowable current?
(a) 925 A
(b) 1048 A
(c) 876 A
(d) 1152 A
Question 15
A transmission planning engineer is calculating the characteristic impedance (surge impedance) of a 230 kV transmission line. The line has an inductance of 1.32 mH/mile and a capacitance of 0.0088 μF/mile. What is the characteristic impedance of the line?
(a) 387 Ω
(b) 342 Ω
(c) 415 Ω
(d) 368 Ω
Question 16
A relay engineer is coordinating overcurrent protection on a radial 69 kV transmission system. A recloser at the source has a time-current characteristic of t = 0.14 + 2.0/(I/Ipickup - 1) seconds. A downstream fuse must clear a 2000 A fault in 0.30 seconds. If the recloser pickup current is 400 A, what coordination time interval exists at 2000 A fault current?
(a) 0.18 seconds
(b) 0.25 seconds
(c) 0.12 seconds
(d) 0.31 seconds
Question 17
A transmission engineer is evaluating transient stability of a system following a three-phase fault. A 500 MVA, 345 kV generator with H = 5.0 MJ/MVA is connected to an infinite bus through a transmission line. The generator is delivering 400 MW at 0.95 power factor leading when a fault occurs. What is the angular acceleration of the rotor immediately after the fault if all power transfer is interrupted?
(a) 144 electrical degrees/s²
(b) 158 electrical degrees/s²
(c) 132 electrical degrees/s²
(d) 171 electrical degrees/s²
Question 18
A design engineer is calculating the voltage regulation of a 230 kV, 100-mile medium-length transmission line. The line has R = 0.10 Ω/mile and X = 0.80 Ω/mile. The line delivers 200 MW at 230 kV with 0.90 power factor lagging. Neglecting shunt capacitance, what is the percent voltage regulation?
(a) 8.6%
(b) 9.8%
(c) 7.2%
(d) 10.5%
Question 19
A transmission engineer is analyzing a line-to-line fault between phases B and C on a 230 kV system. The sequence impedances are Z₁ = 5 + j30 Ω and Z₂ = 5 + j28 Ω. The pre-fault line-to-line voltage is 230 kV. What is the magnitude of the fault current?
(a) 2.28 kA
(b) 1.97 kA
(c) 2.56 kA
(d) 2.14 kA
Question 20
A transmission operations engineer needs to determine the maximum power transfer capability of a 345 kV, 180-mile lossless transmission line connecting two systems. The line has an inductive reactance of 0.75 Ω/mile. Both sending and receiving end voltages are maintained at 345 kV. What is the theoretical maximum power that can be transferred (steady-state stability limit)?
(a) 876 MW
(b) 784 MW
(c) 952 MW
(d) 1024 MW
