Sunday, March 04, 2007

The terrible cost of moving electricity

Wind, solar, and hydro electrical generation are all intermittent fluctuating power sources that require long distance power lines to transfer the generated power to end users. It's a little difficult to know how feasible it is to transmit power across thousands of miles. On the one hand, it's obvious that if you make the conductors thick enough, you can reduce the losses as much as you like. On the other hand, it isn't done already on the massive scale necessary to support intermittent fluctuating power sources.

First, how much does overhead transmission wire cost?

Consider ACSS/AW: soft aluminum, supported by aluminum-clad steel. The largest size that Southwire sells (Joree) is 1.88 inches in diameter, 2.38 pounds per foot of aluminum, .309 pounds per foot steel, .0066 ohms/1000 ft DC @ 20 C, rated for 3407 amps at 200 C. As of Dec 1, 2006, it costs $322/CWT. CWT is 100 pounds, so that's $8.66/foot.

Now lets consider how much wire we need to move 10 gigawatts across 1000 miles. The more wire (cross section) we use, the less resistance we'll have and the less power will be lost. The optimal point for these kinds of problems is when the marginal cost of the wire is equal to the marginal cost of the electricity lost to resistance. After this point, when you add wire, the cost of the wire increases faster than the value of the power saved, so that you have lost money.

Let's assume the electricity costs $0.04/kw-hr and that we're transmitting an RMS average of 10 gigawatts. The RMS (root mean square) part of this last assumption lets us estimate power losses. Finally, lets assume we transmit with a +/- 500 kV high-voltage DC transmission system, which is the lowest-loss long-distance transmission system available today.

To convert ongoing electrical costs into a present value we can compare to the cost of the wire, assume a discount rate of 5%.

The optimal point for 10 GW is 4 conductors each way (8 total conductors).
  • wire cost: $366 million
  • resistance: 8.72 ohms
  • power lost: 871 megawatts
  • P.V. lost electricity: $305 million
Here the wire cost doesn't quite equal the present value of the lost electricity because the number of conductors is quantized, and I'm only considering one type of conductor. But, it's close.

One interesting thing about electrical transmission is that the optimal point for wires used doesn't change with distance. Double the distance, double the resistance, double the power lost, and double the wire cost. The total cross section of conductors used is the same. So we can talk about how much more electricity costs after it has moved a distance.

The electricity transmitted has three costs: the cost of the power lost, the rent on the money borrowed to build the transmission lines, and the maintenance and depreciation on the power lines. We just showed the first two will be equal, and the last will be smaller - electric power lines are like dams and bridges, they last for a long time. So the total cost of transmission will be a bit more than twice the cost of the power lost.

This is a really nice rule of thumb because it reduces away the actual costs of power and interest rates and so forth. We can now convert a distance into a cost multiplier. For the geeks among you, the multiplier is (1+power lost)/(1-power lost). Note that power lost is a function of the relative costs of copper and electricity, so that hasn't been reduced away, but merely hidden.

After 1000 miles, 8.71% is lost, and delivered power costs at least 19% extra.
After 2000 miles, 17.4% is lost, and delivered power costs at least 42% extra.
After 3000 miles, 26.1% is lost, and delivered power costs at least 71% extra.
After 4000 miles, 34.8% is lost, and delivered power costs at least 107% extra.

This, in a nutshell, is the argument for locating generators near their loads.

There is a hidden assumption above: that the average power distributed (this goes in the denominator for loss%) is equal to the RMS power distributed (this goes in the numerator for loss%). If the power transmitted is peaky, like from an intermittent wind farm, then the average power will be smaller than the peak power, the power lost % grows, and delivered power costs even more.

Delivered costs are actually even worse: typically, when a transmission line is built, its capacity isn't used immediately. In the years until the capacity is reached, you pay for the capacity you are not using. In fact, you always want some reserve capacity, which drives the price up even more.

So, there you have it. If you spread your wind farms over the whole continent, and interconnect them with a high-capacity power grid, then the cost of that power once delivered is substantially more than the cost of producing it. Not only does wind power have to be as cheap as coal, even after you divide by availability, but it has to overcome the extra and substantial cost of distribution.

And the same goes for solar and hydro too.

I'll leave with a note of hope. Hydro ended up being cheap enough that the cost of distribution could be overcome. Maybe solar or wind can get that cheap as well.