Resistance is the property of a material to oppose (resist) current flow through the material, causing some of the electricity to be lost (e.g., as heat). For example, electric wires have resistance which causes some of the electricity that is transmitted through the wire to be lost.

All materials have some resistance. Conductors have less resistance than insulators, which is what makes them conductors. But even conductors have resistance, just less resistance than insulators.

For a given resistance of a material, denoted R, the amount of power lost is given by the following formula:

PR = I 2 · R

where PR denotes power that is lost to resistance, I 2 denotes the current squared (current times itself), and R denotes resistance of the material (measured in Ohms). This is known as the “I2R Loss”.

Notice that the I2R loss increases as a square of the current (an exponential relationship): doubling the current quadruples the loss, tripling the current increases the loss nine-fold, etc. Thus, to reduce loss, we need to reduce current by increasing voltage, which is possible with transformers.

Reducing current by increasing voltage is a linear relationship, not an exponential relationship, given by this formula from the previous page of this report:

P = V · I

where P denotes power, V denotes voltage, and I denotes current.

For a given initial electrical power P0 we are trying to transmit, the power that is transmitted PT becomes the initial power minus the I2R loss PR

PT = P0 – ( I 2 · R )

with P = V · I holding throughout: for the power that is transmitted; for the power we are trying to transmit; and for the I2R loss.

For a given initial electrical power P0 we are trying to transmit, say we want to reduce the I2R loss of transmitting that electrical power by doubling the voltage. In that case, the current is halved (according to the formula P = V · I), keeping the power we are trying to transmit constant. Then, plugging the new current level, which is half the current (at double the voltage), into the I2R formula shows that the power lost will be one-fourth of the power lost.

Likewise, reducing the current to one-third of the current (by tripling the voltage) reduces I2R loss to one-ninth of the loss. Quadrupling the voltage drops the I2R loss to 1/16 of the loss, etc.

Huge amounts of electricity are prevented from being lost by using transformers to step up the voltage for transmission and step it back down when needed for end uses. That is why AC transmission became popular, because transformers only work on AC (see previous page of this report).

There will always be a need for AC transmission, to be able to use transformers. The role of DC will be to transmit bulk electricity long distances. Electrical transmission systems need both AC and DC.

Figure 1: AC transformer for HVDC line that connects Denmark and Sweden. [ABB]

One of the disadvantages of trying to use AC for bulk electricity transmission is that larger wires are needed because AC electricity migrates toward the edges of the conductor, not utilizing the interior of a wire to transmit electricity. This is called the skin effect.

“when a conductor is transmitting alternating current, the current-density across the conductor cross section is nonuniform and is a function of the AC frequency. This phenomenon, known as the skin effect, causes the ac resistance to be greater than the dc resistance.”

—

Syed A. Nasar, Electric Power Systems, p. 26.

The skin effect is caused by induced emfs opposing current flow.

Electricity flowing in a wire magnetizes the wire which creates an electric field that opposes AC reversing directions. The effect is greater in the portion of the wire that has more wire surrounding it (the center of the wire).

This does not happen with DC. Direct current transmits electricity uniformly across the cross section of a wire, using the entire wire to transmit electricity.

For AC wires, the transverse distance in from the edge of the wire, where most of the current is flowing, is called the skin depth.

The skin depth, denoted δ (small delta), is the wire edge depth where 1/e (38.6 %) of the intensity of the wire current flows. For transmission line frequencies, this can be calculated with the following formula (eq. 9 in Riba, see References below):

δ = 1 ∕ k = 1 ∕ sqrt( π f μσ )

where f is the AC frequency (50 or 60 cycles per second), and μ and σ are material constants (and of course π is 3.14… which is also constant). The skin depth is constant for a given frequency (and conductor conditions). Note: The denominator is a constant, denoted k, which can be used as a scalar in subsequent calculations, not to be confused with the k of (i,j,k) vectors.

Frequency of AC is measured in cycles per second which is also called hertz (abbreviated Hz). 50 Hz has a deeper skin depth than 60 Hz, resulting in less resistive line losses. For that reason, Europe has switched to 50 Hz, and Asian countries have been adopting 50 Hz. Note: The US still uses 60 Hz.

The following map shows examples of HVDC lines and back-to-back stations around the world that connect two AC grids together, specifying which AC grids are 50 Hz and which are 60 Hz.

Figure 2: HVDC interconnects that link two AC grids together. Orange denotes that both ends of the HVDC interconnect are 60 Hz AC grids. Yellow denotes that both ends are 50 Hz AC grids. Combination orange and yellow indicates that one end is a 60 Hz AC grid, and the other end is a 50 Hz AC grid.

Harting (see References below) gives an example of calculating skin depth and resistive line loss for an ideal aluminun wire at the same conditions but different frequencies (50 Hz and 60 Hz). The line loss at 50 Hz was 3.34 %, while at 60 Hz it was 3.66 %.

Another problem with long distance AC lines is reactance compensation. Reactance is electrical power that is not normally used but helps motors to start without degrading electrical grid supply for other users.

Reactance is the phase shift of AC voltage and current (i.e., the voltage and current levels in each cycle no longer coincide).

Medium length AC lines lose more electricity, not just from increased I2R losses, but also because of reactance, requiring more electricity to operate. And installation and maintenance costs per km go up, because reactance compensation devices become necessary.

Long distance AC lines get even worse, requiring power plants along the way to provide reactive power. Very long distance AC lines are not practical, because so many power plants would be needed along the way that it is better to just use those power plants directly (instead of for reactance compensation).

High Voltage DC (HVDC) lines do not need reactance compensation as a function of distance, require less electrical polarity conductors (only two poles instead of three), and can even operate single pole if the second conductor fails (until repairs can be made).

“single pole operation of dc transmission systems is possible for extended period, while in ac transmission, single phase operation (or any unbalanced operation) is not feasible for more than a second.”

—

Vijay K. Sood, HVDC and FACTS Controllers: Applications of Static Converters in Power Systems, p. 9

The AC distribution grid is a forest of directed tree graphs o with electricity generation at each root node, and electricity usage at the leaf nodes. Following shows a simple example of an AC distribution grid as a single rooted tree graph.

Figure 3: Simple AC distribution grid.

The root node of Figure 3 is a Power Plant (labeled 1) that generates electricity. From there, a short distance link takes the electricity to a Step Up Transformer (2) which increases the voltage many fold (substantially reducing the current) for efficient long distance transmission.

From Step Up Tranformer 2, high voltage power lines on Towers 3 carry the electric power to Transmission Substation 4, which steps down the voltage.

The electric power is then transmitted to multiple neighborhood Distribution Substations 5, which further lower the voltage and distribute the electricity radially via Poles and small Transformers (6) to Homes (7).

A more accurate example is shown in the following schematic (see also Matias, p. 6):

Figure 4: AC distribution grid with two root nodes.

The electrical symbol for AC power generation is the sine wave in a circle, in this example a hydro power dam and a wind farm, and the symbol for a transformer is circles that partially overlap (symbolizing transformer wire winding field).

The hydro power dam in this example generates more electricity than the wind farm, and is further away. The dam generates electricity at 20 kV (kiloVolts). A large transformer substation near the dam steps up the voltage to 400 kV for long distance transmission.

Many kilometers later, another large transformer substation steps down the voltage to 130 kV, for shorter distance transmission in areas that do not have enough room for the higher voltage AC lines which require much more Right of Way.

After the electric power is transmitted on power lines at 130 kV, it reaches another transformer substation that is not as large as the first two substations. That station steps the voltage down to 50 kV.

Meanwhile, wind turbines have been generating AC electricity which is aggregated into a 10 kV line that goes to an even smaller substation that steps the voltage up to 50 kV so that it can be combined with the now 50 kV line from the hydro dam.

The two lines are joined together in yet another substation, to step the voltage down to 10 kV. This substation is comparable in size to the just mentioned wind power substation. Thus, the electric power from both the dam and wind farm are now on the same line, with reactance compensation and voltage regulation performed at all of the substations just mentioned.

This 10 kV line is ready to supply electricity to loads. Industrial users (e.g., factories) that have their own tranformers can tap this line for electricity.

For homes, another tranformer steps the voltage down to 400 V so that the homes do not need their own transformer. This is referred to as low voltage.

For safety and stability of the grid, utilities need to put as little equipment in homes as possible, usually only a meter to record electricity usage. The low voltage grid is not even monitored. Sensors are located at all the medium and high voltage substations, not on the low voltage part of the grid, which would be expensive.

“It is now common practice in developed countries to monitor the primary distribution system down to 10–15 kV and to display alarm, voltage and power flow conditions in a control room; and in the event of an incident, repair crews are dispatched quickly. Repairs to the low voltage system are still dependent, however, on consumers notifying a loss of supply.”

—

B.J. Cory, “The power system”, chapter 13 in D.F. Warne, ed., Newnes Electrical Power Engineer’s Handbook Second Edition, p. 387

For increased stability, grid operators do not alter the low voltage grid infrastructure other than for special cases.

“Another grid operator stated that individual tap changing of the local distribution transformers, in conjunction with wide-area control, is technically sufficient and economically more efficient than the use of voltage-regulated local distribution transformers.”

—

Bayer, et al., p 136 (see References below)

Tap changing refers to changing the effective number of windings of a transformer (in order to change the voltage). This can be done on the fly, with tranformers that have different taps built into the windings which can be switched to automatically. These types of transformers operate at medium voltage, not at the low voltage level. A special geared switch may be implemented that can contact two taps at a time while changing taps, gradually gearing through resistive taps on the way to a final tap change.

Note: In the United States, the low voltage transformer steps down the voltage to 200 V instead of 400 V, increasing line losses (by doubling the current). All discussion of 400 V in this report refers to 200 V in the US.

1. Jordi-Roger Riba, “Calculation of the ac to dc resistance ratio of conductive nonmagnetic straight conductors by applying FEM simulations”, 2017. pdf

2. Curt Harting, “AC Transmission Line Losses”, 2010. html

3. Jose Matias, “Reactive Power Compensation”, 2013. pdf

4. Benjamin Bayer, Patrick Matschoss, Heiko Thomas, Adela Marian, “The German experience with integrating photovoltaic systems into the low-voltage grids”, Renewable Energy, 119 (2018): 129-141. doi/pdf

Introduction

Electricity

Transmission (this page)

Conversion

Stations

Station Layouts

Higher Voltage

Overhead Lines

Pacific DC Intertie

Electricity

Transmission (this page)

Conversion

Stations

Station Layouts

Higher Voltage

Overhead Lines

Pacific DC Intertie

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2022–Jan–18 02:05 UTC

Arc Math Software, P.O. Box 221190, Sacramento CA 95822 USA Contact

2022–Jan–18 02:05 UTC