Why HVDC over HVAC: mathematical comparison

HVDC transmission sends bulk power over long distances at high DC voltage, with a rectifier station at the sending end and an inverter station at the receiving end. It is preferred over HVAC for large power over long distances because the saving in line cost and losses exceeds the extra cost of the converter stations.

Why HVDC for large power over long distance

• Lower line cost: a bipolar DC line needs only 2 conductors instead of 3, smaller towers and a narrower right of way. Terminal (converter) cost is high, but beyond the break-even distance (about 500–800 km for overhead lines, 40–80 km for cables) the total cost of DC is lower.
• No stability limit: AC power $P = \dfrac{V_1V_2}{X}\sin\delta$ falls as line reactance grows with length; DC has no reactance or load angle, so power is limited only by thermal rating.
• No reactive power or charging current: no need for line compensation; voltage stays nearly constant along the line.
• Lower losses: no skin effect, less corona, fewer conductors.
• Asynchronous link and fast power control by firing angle.

 Cost
  |                      / AC
  |                    /
  |                  /    _____ DC
  |            _____X-----
  |     _____-    / :
  | DC -        /   :
  |           /     :
  |  AC     /       :
  |       /         :
  |     /           :
  +-----------------:------> km
             break-even distance

Differences between HVAC and HVDC lines

Mathematical comparison

(a) Power transfer. Assumptions: same conductor size and current rating (same rms current $I$ per conductor), same insulation, i.e. the same peak voltage to ground $V_m$; three-phase AC line with 3 conductors; bipolar DC line with 2 conductors at $\pm V_d$, $V_d = V_m$, $I_d = I$.

$$\begin{aligned} P_{ac} &= 3\cdot\frac{V_m}{\sqrt{2}}\cdot I\cos\phi = 2.121\,V_m I\cos\phi \\ P_{dc} &= 2V_d I_d = 2V_m I \end{aligned}$$ $$\frac{P_{dc}}{P_{ac}} = \frac{2V_m I}{2.121\,V_m I\cos\phi} = \frac{0.943}{\cos\phi}$$

Per conductor:

$$\frac{P_{dc}/2}{P_{ac}/3} = \frac{V_m I}{0.707\,V_m I\cos\phi} = \frac{1.414}{\cos\phi}$$

With $\cos\phi \approx 0.943$, a 2-conductor DC line carries the same power as a 3-conductor AC line, and at unity pf each DC conductor carries about 1.41 times as much power as an AC conductor.

(b) Losses. For the same power and the same loss per conductor, the total loss ratio is

$$\frac{\text{Loss}_{dc}}{\text{Loss}_{ac}} = \frac{2p}{3p} = \frac{2}{3}$$

(c) Conductor material. For the same power, voltage and loss, the DC line needs only $0.5\cos^2\phi$ times the conductor volume of the AC line (50% at unity pf).

(d) Insulation. AC insulation must withstand the peak voltage $\sqrt{2}V_{rms}$; DC insulation is set by $V_d$, so for the same working level a DC line needs less insulation.