What is the relationship between a power line's voltage and the amount of power it can transmit?

Question

I am considering a classic power distribution network using 3-phase AC power. There is usually power lines of various voltages ranging from 10kV to 400kV. These values change country-by-country. For example in France, the distribution network is usually of 20kV, and the transmission network is made of 63kV, 90kV, 225kV and 400kV lines.

I wonder the relationship between the power line's voltage and the amount of power it can carry. For example, I wonder how many 200kV lines would make the same job as a single 400kV line.

Arguments towards a linear relationship : P = U*I so assuming the same current, by doubling the powerline voltage, we double the amount of power transmitted.

Arguments towards a quadratic relationship : A powerline designed with twice the voltage has everything larger, higher pylons, thicker conductors. Therefore a line designed with twice the voltage can have its current that can go through the cables double as well. Thus a line with 2x the voltage will carry 4 times more power.

Arguments towards a cubic relationship : If the wires of a power line for twice the voltage have their diameter twice larger, it means their surface is 4 times as much, thus supporting 4 times the current, hence 8 times the power.

Answer 1

There are a lot of factors at play in the increase of the capacity of overhead transmission lines as the nominal system voltage increases. By "capacity", I mean the thermal limit based on heating of the conductors and subsequent increasing in the sag.

• Power transferred is proportional to voltage
• Due to the way impedance reflects through transformers, higher voltage lines have lower "per-unit" impedance even for the same physical construction. This effect goes with the square of voltage. This does not change the current-carrying capacity of the line, but it does mean that in a networked system, the higher voltage lines will tend to carry more load, and therefore will be designed with greater capacity.
• Due to corona effects, high voltage lines (approximately 230 kV and above) are designed with minimum conductor sizes and utilize bundled conductors to reduce the electric field gradient around the conductors and reduce the inductive reactance of the line
• At the highest voltages, cost of terminal equipment at substations is relatively higher, and lines tend to be very long, so the load that the line is able to carry tends to be limited by system stability concerns and the inductive reactance of the line rather than the thermal limits of the line. This means there is less incentive to keep increasing thermal limits.

I do not currently have access to any utility's line rating database, but I did pull together some publicly available transmission system models that are based on real utility data. Below is a plot of the empirical relationship between system nominal voltage and transmission line capacity.

I fit a power curve and a quadratic curve (with y-intercept fixed at 0) to the mean points at each voltage level. For a power relationship, the best overall fit is a power of 1.24. In relation to your original question, it shows the relationship is between linear and quadratic.

The quadratic curve is a better fit to the data, but its hard to relate it to your original question.

Unfortunately, I do not have handy access to similar data for lower voltage systems, so I'm not able to include distribution-level line designs.

Here's my source code to do the analysis and generate the plot.

Answer 2

Arguments towards a quadratic relationship : A powerline designed with twice the voltage has everything larger, higher pylons, thicker conductors.

Making the conductors thicker is not necessary to design for higher voltage. (Making the insulators thicker, is required)

If you make the conductors thicker then you are changing the design to allow for higher current.

If you design for both higher voltage and higher current, then yes, the power carrying capability will increase more than if you just design for higher voltage.

As the designer you have the freedom to do this in any proportion, you don't have to double the current if you double the voltage. You could even design for lower current at higher voltage, keeping the power the same (for example, this might save you on the cost of the wires).

If you're talking about cross-country transmission lines you might be constrained by things like the weight-bearing capacity of the pylons, wind loading, etc.

Arguments towards a cubic relationship : If the wires of a power line for twice the voltage have their diameter twice larger, it means their surface is 4 times as much, thus supporting 4 times the current, hence 8 times the power.

It's not the total surface area, but the surface area per unit length, (aka, the circumference) that matters. And the circumference only increases in proportion to the diameter, not as the square of the diameter.