---
title: "Why HVDC over HVAC: mathematical comparison"
subject: "Power Electronics"
chapter: 7
key: "semester_7/power_electronics:hvdc-vs-hvac-comparison"
source: https://www.bctengineer.com/q/semester_7/power_electronics/hvdc-vs-hvac-comparison
site: BCT Engineering
---

# Why HVDC over HVAC: mathematical comparison

Power Electronics · Chapter 7: HVDC power transmission

- Asked in: 2076 Asoj, 8 marks; 2075 Chaitra, 8 marks; 2071 Chaitra, 8 marks; 2069 Chaitra, 8 marks (asked 4 times)
- Video: [Economics of Power Transmission | Comparison of EHV AC and DC Transmission HVDC & AC #aktu #aktutor](https://www.youtube.com/watch?v=bwKqCdVKv5M) — AKTutor
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_7/power_electronics/hvdc-vs-hvac-comparison
- Whole chapter: https://www.bctengineer.com/md/semester_7/power_electronics/chapter-7

## Question

Why HVDC transmission is preferred over HVAC transmission for transport of large power over long distance? What are the differences between HVAC and HVDC transmission lines? Perform a mathematical comparison between HVAC and HVDC lines.

## Short answer

HVDC is preferred for long-distance bulk power because beyond the break-even distance its lower line cost and losses outweigh converter cost. It has no stability limit, no reactive power, and lower losses. Mathematically, with same insulation and conductor current, a 2-conductor DC line carries the same power as a 3-conductor AC line at power factor 0.943, and total losses are only 2/3.

## Answer

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.

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

#### Differences between HVAC and HVDC lines

| Point | HVAC | HVDC |
|---|---|---|
| Conductors | 3 (3-phase) | 2 (bipolar), 1 (monopolar) |
| Stability limit | Yes, $P \propto \sin\delta / X$ | None |
| Reactive power | Charging current, needs compensation | None in line |
| Skin effect / corona | Present / more | Absent / less |
| Terminal cost | Low (transformers) | High (converters, filters) |
| Line cost per km | High | Low |
| Interconnection | Synchronous only | Asynchronous possible |
| Fault current | High | Limited by converter control |
| Voltage change | Easy (transformer) | Needs converter |
| Harmonics | Little | Converter harmonics, filters needed |

#### 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.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. Why does HVDC become preferable beyond the break-even distance?
   A. Converter stations become cheaper with length
   B. The saving in line cost and losses exceeds the extra converter terminal cost
   C. AC lines stop working
   D. DC needs no insulation
   Answer: B. HVDC terminals cost more, but per-km line cost is lower, so beyond about 500-800 km (overhead) total DC cost is less. (Key point: Break-even distance)

2. Why does AC transmission have a stability limit that DC does not?
   A. AC power P = V1V2 sin(delta)/X depends on line reactance and load angle; DC has neither
   B. DC lines have more resistance
   C. AC has no insulation limit
   D. DC power depends on frequency
   Answer: A. AC power transfer falls as reactance X grows with length; DC has no reactance or load angle, so only the thermal rating limits it. (Key point: Stability limit)

3. With equal conductor size, current, and peak voltage to ground, what is P_dc/P_ac for a 2-conductor DC and 3-conductor AC line?
   A. 1.414/cos(phi)
   B. 0.943/cos(phi)
   C. 0.707/cos(phi)
   D. 2/3
   Answer: B. P_dc = 2VmI and P_ac = 2.121 Vm I cos(phi), so the ratio is 0.943/cos(phi). (Key point: Power ratio)

4. For the same loss per conductor, what is the DC to AC total loss ratio?
   A. 3/2
   B. 1
   C. 2/3
   D. 1/2
   Answer: C. DC has 2 conductors and AC has 3, each with loss p, so the ratio is 2p/3p = 2/3. (Key point: Loss ratio)

## Related in Power Electronics

- [HVDC vs HVAC line conductor saving](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ahvdc-conductor-saving) — Power Electronics, chapter 7, asked 2×
- [HVDC advantages and power transfer capacity](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ahvdc-power-transfer-capacity) — Power Electronics, chapter 7, asked 2×
- [Compare HVAC and HVDC; 2/3 power loss](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Acompare-ac-dc-two-thirds-loss) — Power Electronics, chapter 7
- [HVDC power loss is 2/3 of HVAC](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ahvdc-two-thirds-loss) — Power Electronics, chapter 7, asked 2×
- [When HVDC is economical; equal transmission capacity](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ahvdc-economical-bulk-power) — Power Electronics, chapter 7
- [HVDC two conductors vs HVAC three conductors](https://www.bctengineer.com/question.md?key=semester_7%2Fpower_electronics%3Ahvdc-two-vs-three-conductors) — Power Electronics, chapter 7, asked 2×

## Related in other subjects

- [HVDC for long distance and B2B significance](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Ahvdc-long-distance-b2b) — High Voltage Engineering, chapter 1
- [Infinite bus and HVDC vs HVAC comparison](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Ainfinite-bus-hvdc-vs-hvac) — Power System Analysis I, chapter 1
- [HVDC vs HVAC and HVDC scope in Nepal](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Ahvdc-hvac-compare-nepal) — High Voltage Engineering, chapter 1
- [Criteria for choosing HVDC or HVAC](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Ahvdc-hvac-choice-criteria) — High Voltage Engineering, chapter 1