Overhead lines
Short line: sending-end voltage, regulation and losses
- where
- sending- and receiving-end phase voltage phasors (V); V_r from the line-to-line kV input ÷ √3
- load current phasor (A), from P, V_r and power factor
- resistance per phase (Ω/km)
- reactance per phase (Ω/km)
- line length (km)
- voltage regulation (%)
- three-phase line losses (MW)
Explain this equation
Gives sending-end voltage, voltage regulation and I²R losses for a short overhead line supplying a balanced load. Short means shunt capacitance can be ignored, generally below about 80 km at 50 Hz.
V̄_r is the receiving-end phase voltage (line-to-line ÷ √3), taken as reference. Ī is the load current found from P, voltage and power factor, lagging for positive pf and leading for negative. (R + jX)l is the total series impedance per phase. Losses are three phases of I²R; ÷10⁶ gives MW.
With only series impedance the same current flows at both ends, so the sending voltage is just the receiving voltage plus the phasor drop. With no load, receiving voltage equals sending voltage, so regulation is (|V_s| − |V_r|)/|V_r|.
Sense check: the drop is close to I(R cos φ + X sin φ). A 10 MW, 0.9 pf lagging load on 20 km of 33 kV line (0.18 + j0.38 Ω/km) gives roughly 6–7 % regulation. A leading load can make regulation negative.
Watch out: use resistance at the operating temperature, not 20 °C, if losses matter.
Sending-end voltage, voltage regulation and I²R losses of a short overhead line (series impedance only) supplying a balanced load.
Enter your values to see the result.
Good to know
For a short line the no-load receiving voltage equals the sending voltage, so regulation = (|Vs| − |Vr|)/|Vr|. The familiar approximation ΔV ≈ I(R cosφ + X sinφ) is close to this exact phasor result for small drops. Leading loads can give negative regulation.
Common mistake
Using line-to-line voltage with per-phase impedance without the √3, or entering the power factor sign the wrong way round for a leading load.
Limits
Short-line model (shunt admittance neglected), valid up to about 80 km overhead at 50 Hz. Balanced load, constant-power at the stated receiving voltage. Use conductor resistance at operating temperature.
Sources
- Grainger, J.J. & Stevenson, W.D. Power System Analysis (McGraw-Hill), chapters 4–6: inductance, capacitance and current-voltage relations of a transmission line
- 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
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.