Performance of Transmission Lines

A line's model depends on its length. From the model we get the sending-end voltage, regulation, efficiency and the power the line can carry.

Line models

ABCD constants: V_S = A V_R + B I_R, I_S = C V_R + D I_R, with AD − BC = 1 (and A = D for symmetric lines).

• Short line: A = D = 1, B = Z, C = 0.
• Nominal π: A = D = 1 + YZ/2, B = Z, C = Y(1 + YZ/4).
• Long line: A = cosh γl, B = Z_c sinh γl, C = sinh(γl)/Z_c.

Regulation and efficiency

• Voltage regulation = (|V_R,no load| − |V_R,full load|)/|V_R,full load|, with V_R,NL = V_S/|A|.
• Short line approximation: regulation ≈ I(R cos φ + X sin φ)/V_R (+ for lagging pf).
• Efficiency = P_R/(P_R + losses).

Ferranti effect

On a lightly loaded or open long line, the charging current through the series inductance makes the receiving-end voltage higher than the sending-end voltage. Corrected with shunt reactors.

Surge impedance and SIL

• Surge (characteristic) impedance Z_c = √(L/C) ≈ 400 Ω for overhead lines (about 40 Ω for cables).
• Surge impedance loading SIL = V_L²/Z_c. At SIL, the line's reactive power generation and absorption balance, and the voltage profile is flat.
• Velocity of propagation 1/√(LC) ≈ speed of light for overhead lines.

Power transfer

For a lossless line of reactance X:

P = (V_S V_R/X) sin δ

Maximum at δ = 90°. Reactive power flows from the higher voltage end to the lower; real power flows from the leading-angle end.

Compensation: series capacitors reduce X (raise power transfer); shunt capacitors support voltage under heavy load; shunt reactors limit voltage at light load. FACTS devices (SVC, STATCOM, TCSC) do this with electronics.

Worked example

A 400 kV line has Z_c = 400 Ω.

• SIL = (400 × 10³)²/400 = 400 MW.

Practice