Direct Current Transmission Lines

Ryan Hamerly
October 22, 2010

Submitted as coursework for Physics 240, Stanford University, Fall 2010

Fig. 1: Demand for electric lighting was a major factor in the expansion of electrical coverage in the early 20th century. (Source: Wikimedia Commons)

In 2008, the world's inhabitants consumed energy at an average rate of 15 terawatts (i.e. 1.5 × 1013 W) total. In a given day, we consume the energy equivalent of 40 million tons of coal, or equivalently, 30,000 ten-kiloton atom bombs. And because most consumers do not want to live right next to a power plant or an oil rig, transmitting this energy to the end user is an especially important challenge.

Throughout most of history, energy was harvested locally, primarily in the form of wood. The transition from wood to coal power was one of the major causes of the Industrial Revolution, and indeed some scholars have asserted that the abundance of wood and coal in the United States and Britain played a dominant role in the growth of these nations' economies during this period. Yet coal, oil, gas, and wood are not very convenient sources of energy. In order to convert them into mechanical work, one needs a combustion engine, which in addition to making a lot of noise and putting out unpleasant odors, poses a very real safety hazard in either the home or the workplace. In order to convert them into light, one needs an oil lamp, which a very inefficient lightsource and a notorious fire hazard. Indeed, for almost any energy application besides heating, fuels are a very cumbersome way to consume energy.

Electricity fills this niche well. The invention of the AC motor and the incandescent lamp allowed electricity to be harnessed to provide both lighting and mechanical work to thousands of end users without the hazards and inefficiencies associated with burning fuels. [1,2] Electricity generation could be centralized to power plants, where economies of scale allow it to be generated far more efficiently than in the individual household or workplace, and transmitted to the end users via electrical cables.

AC Transmission

There are two different approaches to electrical transmission: Direct Current (DC, proposed by Edison) and Alternating Current (AC, proposed by Tesla). Direct Current works by applying a constant electric voltage, from which most devices will draw a constant electric current. Batteries are a common source of direct current, and most modern electronics require direct current in order to operate. In an alternating current scheme, the voltage oscillates as a function of time - usually at a rate of 50 or 60 Hz.

Fig. 2: Schematic of an electrical transmission line, broken up into discrete segments. In the continuum limit, L and C are replaced by the conductance and capacitance per unit length.

Both types of electricity are equally capable of powering light bulbs, electric motors, and most types of appliances. What makes alternating current superior from a power transmission point of view is the fact that due to the principle of magnetic induction, it is very easy and cheap to raise or lower the voltage by means of an iron-core transformer. Since the power dissipated in a transmission line scales as the square of the current, and since increasing the voltage using a transformer decreases the current, one can dramatically reduce losses by using high-voltage power transmission. The principle of magnetic induction only works for alternating currents, and this is the reason that, for the past century, almost all commercial electricity has been produced and transmitted via alternating current.

An electrical cable can be modeled as a transmission line, or equivalently, as an infinite chain of capacitors and inductors, as shown in the figure, where C and L refer to the capacitance and inductance per unit length. Kirchoff's Equations give:

Taking the continuum limit, we find alternating-current wave solutions of the form


are the wave propagation speed and characteristic impedance. (The voltage and current are both real quantities. Since the system is linear, the real part of a complex solution is a solution itself). The power transmitted across the line at any given point is

Fig. 3: Major components of a DC electrical transmission system. Realistic systems may also include a ground wire connecting the rectifier and inverter, which allows the device to function at half-power if one of the cables is damaged.

The voltage is limited by the breakdown strength of the dielectric medium (air about 3 MV/m). This fundamentally limits the power that can be carried in any given wire. The result is independent of the oscillation frequency, from which we can infer that the same limit must constrain DC cables as well as AC cables.

There is another slightly more subtle limitation to AC transmission power and efficiency - the skin effect. This effect, unique to AC systems, prevents current from flowing in the interior of the conducting cables. The effect is more pronounced the higher the frequency, so DC cables do not suffer from this limitation. In general, the skin effect limits the practical diameter of cables to 3 centimeters. Thinner cables transmit electricity less efficiently than thick cables, and as a result the skin effect has a negative impact on power line efficiency.

Overview of DC Transmission

Direct Current offers an alternative to the conventional AC transmission that cures many of AC's defects. It should not be thought of as a replacement for AC transmission, which in most contexts works fine, but rather as an alternative for particular applications in which AC lines are impractical or costly. Such applications include:

  1. Connections between unsynchronized power grids,
  2. Underground or underwater connections exceeding 50 km in length,
  3. Above-ground connections exceeding 800 km in length,

Connections over regions where land costs are a dominant factor. [3,4]

Fig. 4: Diagram of a thyristor. (Source: Wikimedia Commons)

A direct current transmission system typically consists of three parts. Electric power enters the system in the form of alternating current - generated, for instance, at a local power station, is up-converted to high-voltage AC using standard AC transformers, and converted into DC power by way of a circuit referred to as a rectifier. The electrical power is then transferred down the DC power cables and converted back into alternating current by way of an inverter. It is worthwhile to note that while the current flowing through the wires is DC, both the input and output of the system is AC, and so DC cables can be quite seamlessly integrated into preexisting alternating current power grids. [3]

Direct current systems are advantageous for several reasons. First, they can transmit slightly more power per cable as compared to AC systems of equivalent voltage. Second, control of the rectifier and inverter circuits makes it easy to synchronize the transmission input and output to the respective power grids. In addition, DC circuits can often function at partial capacity even when one of the lines is down. Yet in many circumstances, these advantages must be weighed against the increased costs of the AC-to-DC conversion equipment.

The first modern DC power transmission line was a submarine cable connecting Gotland Island to Sweden in 1954. The DC current was generated using mercury arc valves, a technology that has since been largely replaced with solid-state thyristors. [5]
Fig. 5: Top: Three AC voltage sources. By triggering the thyristors at times t1, t2, and t3, the rectifier produces an output that matches the maximum voltage of all three sources. Bottom: Simplified schematic of a 6-pulse converter bridge. The device converts a 3-pole AC source to a DC output. Colors correspond to the voltages plotted on the top panel.

AC-DC Conversion

The major technical hurdle of DC transmission is that of converting alternating current to DC and vice versa so that the transmission line can interface with existing power grids. Currently, this is handled with circuits called rectifiers and inverters, which use a high-voltage triggered diode called a thyristor.

A thyristor is composed of four alternating layers of N- and P-type semiconductor. It functions as a diode with a trigger; before the device is triggered, it will not conduct, but thereafter it will conduct electricity for as long as the thyristor remains forward-biased. Once the forward bias is removed, the device will stop conducting and can only resume by being subsequently triggered.

The simplest AC-to-DC converter would consist of a single thyristor, an inductor, and an AC current supply. The thyristor is triggered midway through the cycle and conducts current for a fraction of a period. Being a diode, however, it stops conducting when the voltage points the other way, and waits for the remainder of the period before being triggered again. In this way, the thyristor acts in the same way as a typical diode. The resulting voltage, far from an ideal DC source, is a periodic series of positive pulses. However, unlike in the AC case, all of the pulses have the same polarity. By properly combining together pulses from different AC sources, we can smooth out the bumpy signal and create a much more suitable DC output.

This can be done with three synchronized AC sources, oscillating 120 degrees out of phase with each other. Instead of a single thyristor, one employs six units, each set to trigger once a cycle. Take the positive pole in the converter shown in Fig. 5. At time t1, the first thyristor triggers and current flows through the first AC line. A third of a period later, at time t2, the second thyristor triggers, and since at this point the potential at the second line exceeds that at the first line, the first thyristor is negative-biased and shuts off. A third of a period later, the third thyristor triggers, the third line delivers the final third of the DC power for the cycle, and the cycle repeats. A similar repetition pattern can be traced out for the negative pole. The voltages at these poles are still nonuniform, but nevertheless, this circuit, called a 6-pulse converter bridge, delivers a far more consistent DC power source than the one-thyristor model described above. [3]

Commercial DC transmission systems do even better than this by using a 12-pulse converter bridge that smooths out the signal even more; and to eliminate any remaining wiggles in the line, rectifiers put band-pass filters on both the AC and DC ends of the circuit. Applying a similar scheme, one can use thyristors to convert DC power back to AC - i.e. to perform the inverter function. [3]

Fig. 6: Development of Thyristor and IGBT Technology. [8]

Early DC transmission lines relied on mercury arc valves, but by the 1970's, thyristors had taken over the market. [3] Over the last 20 years, power companies have been pushing the state of the art in thyristor technology, more than doubling the device's power throughput and increasing its voltage by 50%. Further advances, such as the Voltage Sourced Converters (VSC) and Insulated Gate Bipolar Transistors (IGBT) may soon make smaller-scale DC transmission schemes economical. With such advances in the underlying technology, the future of DC power transmission is promising.


The cost of a high-voltage transmission scheme depends on four principal factors: the cost of the transformers, the cost of the cables and towers, the cost of the land over which the lines lie, and the cost of losses due to ohmic heating in the power lines. [6] As to the first count, AC wins hands-down. Unlike the cheap iron-core power transformers, the AC-to-DC rectifiers are extremely expensive. [7] However, the price of the transformers does not depend on the length of the wire, so if DC lines are to prove cheaper or more efficient than AC lines, then there will exist some break-even point beyond which DC becomes the better option.

DC's increased efficiency for particularly large projects stems from the fact that the voltage on a DC wire is constant. This constant, Vmax, is related to various engineering concerns as well as the geometry of the power line configuration and the electric breakdown strength of air. Similar constraints set the size of AC wires, and as a result a 3-cable 500 kV AC tower is about the 1.5 times as large as a 2-cable 500 kV DC tower. [4] This roughly gives rise to a 30% savings in line costs. But for very long lines, system stability limitations put AC at an additional disadvantage; moreover, intermediate switching stations are usually required for long-distance AC lines, further increasing costs. From these facts, we a arrive at the general conclusion that a DC line can carry at least twice as much power as an AC line of the same voltage. [8]

As the Fig. 7 shows, this means that a DC power line will be significantly smaller than its AC equivalent, roughly by a factor of two. Since AC and DC use roughly the same types of cables, this translates into reduced line costs and reduced tower costs. [6] In addition, the ground clearance required is reduced considerably. In regions where land is expensive and regulations are strict, obtaining a contiguous strip of ground can be as much of a challenge as putting up the lines themselves -- and DC is 40-50\% cheaper in this respect. DC was chosen in the Rihand-Delhi project (India) and the Queensland project (Australia) in part the towers were more compact. [5]

Fig. 7: Three high-voltage configurations with 2 GW capacity. The clearance required for the DC line is somewhat less than that required for the AC equivalents. [4]

In addition, DC benefits from decreased line losses due to ohmic heating. No doubt this is at least in part due to the fact that DC lines experience no skin effect, allowing cables to be made thicker to decrease the resistance of the line. Indeed, high-voltage DC lines generally have a loss rate of around 3%, as compared to the 6% rate for AC lines. However, except for the longest lines, where the loss rates grow significantly larger than this figure, costs due to inefficiencies are only a minor factor in the economics of power lines. [8]

Reliability will play an additional role in the debate. It is possible (although not desirable) for a DC system to run at half power if one of the cables or transformers fails. This is achieved by running a ground wire (which need not be insulated) along the power-carrying wires and using the ground to transmit power from the surviving power-carrying line to form a closed circuit at half the original voltage. Alternatively, one can forego this extra wire altogether and simply conduct the circuit through the ground in the event of a failure. However, these advantages need to be balanced against the fact that the thyristor transformers are generally less reliable than traditional AC transformers, which may make DC systems less reliable.

Fig. 8: Break-even distance as a function of voltage. [4,5]

In sum, while the fixed cost of DC transmission, due to transformer stations, is much greater than the fixed cost for AC transmission, the per-kilometer cost for DC lines is significantly less. Therefore, a break-even line length exists beyond which DC becomes the cheaper choice. For above-ground cables, the break-even length is on the order of 800 kilometers. For undersea cables, it is much shorter -- typically around 50 km, because alternating current dissipates quickly underwater. [9]


Because HVDC breaks even for particularly long lines, it has found a niche connecting larger power grids to isolated power stations. Many energy sources, particularly renewables, are concentrated in regions where there is little or no population. Transporting this energy to population centers requires long, high-voltage lines often exceeding 1000 km. Offshore wind farms provide one such example, and indeed the first such facility constructed in the UK was linked to the island by way of a 26-km high-voltage DC cable. [9] On a much larger scale, over a third of the 22 GW produced by the Three Gorges Dam in China is transported to East and South China by way of 500 kV DC cables. [10]

For the same reasons, high-voltage DC may provide inexpensive power to isolated power consumers throughout the world. For instance, the islands of southeast Alaska, remote areas in northern Canada, and mining communities in western Australia currently depend on local oil-fired generators for their electricity. High-voltage DC lines, particularly those developed with the new VSC technology, may make it economical to connect these isolated load centers to the main grid. [3]

Fig. 9: Eskimos in the Alaskan islands may one day rely on DC electricity to heat their igloos. (Source: Wikimedia Commons)

Both Europe and North America are spanned by a number of different power grids. These grids do not all operate in phase, and in Europe in particular, they often do not operate at the same frequency either. So-called "back-to-back" DC connectors - connectors that include a modest-voltage rectifier and inverter, but whose transmission line is so short as to be negligible -- can be built to connect these grids, allowing power companies to supply needed electricity to regions with especially high demand, to arbitrage electricity price fluctuations, and to prevent blackouts. Such business strategies would not be possible with traditional AC connections, since connecting two out-of-phase power grids can easily result in overloading and blackouts. A back-to-back DC connector, on the other hand, can be trivially synchronized to the grids it connects.


While alternating current has dominated the power lines for a century, solid-state thyristors have made direct current a viable alternative in certain circumstances. While it is unlikely to replace alternating current as the dominant form of electricic power, it has become more economical than AC for overhead lines exceeding 800 km and undersea or underground lines exceeding 50 km - making DC the ideal choice for especially long connections and connections to isolated power plants and consumers. Still further, back-to-back DC connections allow asynchronous power grids to be connected, providing increased price stability and protection against blackouts.

© Ryan Hamerly. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.


[1] N. Tesla, "Electric Motor," Patent No. 416194 (1889).

[2] T. Edison, Improvement in Electric Lights, Patent No. 214636 (1879).

[3] J. Cochrane and R. Hauth, "HVDC Power Transmission," in Wiley Encyclopedia of Electrical and Electronics Engineering (Wiley, 2001).

[4] P. Hartley, "HVDC Transmission: Part of the Energy Solution?," James A. Baker III Institute for Public Policy, Rice University, May 2003.

[5] R. Rudervall, J. P. Charpentier and R. Sharma, "High Voltage Direct Current (HVDC) Transmission Systems," Energy Week 2000, Washington, D.C., Mar 7-8 2000.

[6] M. P. Bahrman and B. K. Johnson, "The ABCs of HVDC Transmission Technologies," IEEE Power and Energy Magazine 5, No, 2 (2007).

[7] M. Rabinowitz, "Power Systems of the Future. 4", Power Eng. Rev. 20, No. 8, 4 (2000).

[8] M. P. Bahrman, "Overview of HVDC Transmission," in Proc. IEEE Power Systems Conference and Exposition 2006 (IEEE, 2006), p. 18.

[9] P. Bresesti et al., "HVDC Connection of Offshore WindFarms to the Transmission System," IEEE Trans. on Energy Conversion 22, 37 (2007).

[10] H. Guo, "Survey of Three Gorges Power Grid (TGPG)," Proc. 2000 IEEE Power Eng. Soc. Winter Meeting 1, 3 (2000).