Overhead lines
Medium line nominal-π: sending-end voltage and current
- where
- total series impedance per phase (Ω)
- resistance per phase (Ω/km)
- reactance per phase (Ω/km)
- line length (km)
- total shunt admittance per phase (S)
- shunt susceptance per phase (µS/km)
- ABCD constant A (= D)
- sending- and receiving-end phase voltage phasors (V)
- sending- and receiving-end current phasors (A)
Explain this equation
Gives sending-end voltage, current, power and regulation for a medium-length line (roughly 80–250 km) using the nominal-π equivalent. Use it for load-flow sense checks or for reactive planning on long 132 kV and 275/400 kV circuits.
Z is the total series impedance and Y the total shunt admittance; half of Y is placed at each end of the π. B is susceptance per km, ωC, in µS/km, so ×10⁻⁶ converts to siemens. A (equal to D) is the voltage ratio constant, and Y(1 + YZ/4) is the C constant of the ABCD set.
The sending-end quantities come straight from the two-port equations V_s = AV_r + BI_r and I_s = CV_r + DI_r with B = Z. Regulation uses the no-load receiving voltage |V_s|/|A|, because the shunt capacitance alone lifts the open-end voltage.
Sense check: A is slightly below 1 (about 0.97–0.99). On light load I_s can exceed I_r because of charging current. Glover’s 345 kV, 200 km example gives about 8.7 % regulation.
Watch out: above about 250 km use the exact hyperbolic model; the lumped π then loses accuracy.
Sending-end voltage, current and power of a medium-length line using the nominal-π ABCD constants, given the receiving-end load.
Enter your values to see the result.
Good to know
The nominal π lumps half the line shunt admittance at each end. A = D = 1 + YZ/2, B = Z, C = Y(1 + YZ/4). Regulation uses the no-load receiving voltage |Vs|/|A|. Susceptance B = 2πfC: about 2.8–3.5 µS/km for overhead lines at 50 Hz.
Common mistake
Entering total line capacitance instead of susceptance per km, or forgetting that the no-load receiving voltage is |Vs|/|A|, not |Vs|.
Limits
Nominal-π model, accurate to about 250 km at 50 Hz for overhead lines. Balanced, fully transposed line; positive sequence; conductance neglected.
Sources
- Glover, J.D., Overbye, T.J. & Sarma, M.S. Power System Analysis and Design (Cengage), chapters 4–5: transmission line parameters and steady-state operation
- Grainger, J.J. & Stevenson, W.D. Power System Analysis (McGraw-Hill), chapters 4–6: inductance, capacitance and current-voltage relations of a transmission line
For competent high-voltage engineers. Results are calculations from the published methods shown and depend entirely on the data you enter. They are not a substitute for a system study, manufacturer data, the applicable standards and Engineering Recommendations, or your organisation’s safety rules. Never use a result to justify working on or near live HV equipment.