HVDC Against HVAC Over Distance

What is modelled

A DC line has no charging current, no skin effect and no reactive power, but each end needs a converter station with its own losses. An AC line has cheap transformers, but its capacitance draws a charging current that grows with length and voltage and uses up the conductor current limit, and unloaded it raises the voltage at the far end (Ferranti effect). The page compares the two for the same power, and finds the length where their efficiencies cross.

Equations

• DC bipole: sending power 2 V I = P + 2 R I², line loss 2 R I² with R = ρ L / A.
• AC line: load current I = P /(√3 V cosφ), loss 3 R (I² + (Ic L)²/3), charging current per km ω C V/√3.
• Open-end voltage ratio Vr/Vs = 1 / cos(β L) with β = ω √(L′ C′), reduced by shunt compensation.

Checks run in the page

• The DC current by iterating the power balance against the quadratic solution.
• The line equation integrated numerically against the cosine solution.
• The break-even length by bisection against a fine scan.
• The AC reach where charging current alone meets the thermal limit.

Honest simplifications

Lumped line constants for a generic overhead line, the same conductor area per pole or phase, constant resistivity, a 0.98 power factor, a flat shunt-compensation model, no corona, no skin effect, no cable and no cost, reliability or right-of-way. Real projects choose between AC and DC mostly on cost and system needs, so the break-even here is an efficiency comparison only.

• Press Short link, 200 km to see AC win on efficiency with its cheap transformers.
• Press Very long, 2500 km to see AC run out of conductor capacity before any power flows.
• Press AC uncompensated to see how the open-end voltage rise grows without reactors.