People tend to have strong reactions to Elon Musk (both positive and negative), and he provides lots of ammunition. The tweets above are an interesting example of how a salesman can stretch the truth like a pair of size 4 pantyhose over the cranium of a 6’5” tall mugger. It’s really unpleasant and neither is recognizable afterwards.
Today we’re looking at the bit in the tweet above about being able to generate more energy per square kilometer with solar panels than with a nuclear power plant. This is, to put it mildly, hella wrong. But where did he get this, and how can he say it and claim it’s true?
By treating energy and power as the same thing, using an ideal model for the solar panels, and a worst case model for the nuclear plant.
So where did Musk get that number for the solar panels?
It’s not difficult to work out where the figure of 200 MW/km² for the solar panels comes from. For those who don’t speak engineering, 200 MW means 200 megawatts, or 200 million watts. A typical hair dryer uses a lot of power – the one at Walmart that I didn’t buy because I don’t have any hair is rated for 1875 watts. So 200 MW will run over 100,000 hair dryers at the same time, generating approximately 50% as much hot air as Elon Musk.
[Before we start, here’s a quick conversion to remember: 1 MW/km² = 1 Watt/m²]
To arrive at the number of 200 MW/km², we first need to find out how much solar energy from the sun reaches the earth1. This is around 1,366 Watts/m² above the atmosphere but on the surface of the earth the peak value, at the point where the sun is directly overhead, is about 1,000 Watts/m² (or 1,000 MW/km²). Looking good so far – now let’s look at how much of that we can capture.
Of course not all the energy that hits the solar panels is converted to electricity. In fact, typical efficiency for good commercial solar panels is about 20%. So taking the 1,000 MW/km² and multiplying by 20%, we get to the 200 MW/km² from Musk’s tweet. Yay for math! Or maths, if you’re British and annoying.
So we have 200 MW/km² of power – which is not the same as energy. Power is the peak output for a single instant, and it’s not constant during each day and over the course of the year.
So how much energy actually reaches us down here?
Here’s one of those .gov websites that technology nerds at DARPA or the Department of Energy or whatever like to put together: https://pvwatts.nrel.gov/index.php
You put in your location, type of solar panels, etc. and it calculates the amount of energy you can expect to generate per year if you install solar panels. One nice thing about this calculator is that it gives us the average amount of solar energy we’ll see in kWhr/m² each day (thousands of watt hours per square meter). Since this is per day, we can divide this by 24 hours to get the average number of watts per square meter of solar panel to compare with the nuclear plants.
For this example we’re using the address of a beautiful historical site in the town of Marathon, Texas. It’s called The Chicken House2 and it has a five star rating on Google. Unfortunately the five star rating is based on a single review conducted by Mr. Martinez, a local tour guide. I don’t doubt Mr. Martinez’s sincerity, but I do doubt his judgment.
Marathon, Texas is in the desert and as far south as possible so it gets a lot of sun. I spent an hour playing with the calculator and didn’t find anything significantly better. It’s a great site for a solar facility, except for not being near anything that needs nuclear plant levels of power. Unless the chickens are working on their own Manhattan Project, in which case we have bigger problems.
The Chicken House Solar Generating Plant
Per the calculator each square meter of The Chicken House Generating Plant will see an average of 6.57 kWh/day. This equals 273.75 watts per square meter of power from the sun. (273.75 watts times 24 hours is 6,570 watt hours per day, or 6.57 kWh/day).
So we have (rounding) 274 watts/m², which is also 274 MW/km² – much less than the theoretical maximum of 1,000 MW/km². Not having sunshine for half of each day actually matters, and having cloudy days and seasons and stuff are important too.
Multiple 274 MW/km² by the 20% efficiency number for solar panels and we have about 55 MW/km² average power production as a more realistic starting number, which is about ¼ of what Musk is claiming.
As a quick aside, the energy calculator doesn’t give us this much total energy as the actual output. There are other variables, like inverter efficiency, or losses from DC to AC conversion of the power. The actual number, when all factors are accounted for, is much lower (In reality it’s probably more like 30 - 40 MW/km², but we’ll just use the 55 MW/km² for now). So we’re giving the solar setup a significant starting advantage.
What about the nuclear plant output? Let’s look at a few examples.
Clinton Nuclear Power Plant - basically a huge park
The Clinton plant3 in Illinois is located on 14,000 acres but only generates 1,065 MW, which gives us 17 MW/km² (after adjusting for efficiency). So it seems like the solar panels have this one beat.
Well, maybe not. The 14,000 acres is actually a recreation area and it includes a 5,000 acre reservoir – which is all open to the public. So the site is basically a big lake with a park around it, and the actual plant only occupies 150 acres.
The real power number is 1,619 MW/km², or about 30 times more power per area than The Chicken House Plant.
To produce as much power as the Clinton Nuclear Plant, The Chicken House Plant would need to be about 19 km² of solar panels, or over 4,700 acres. That’s about as big as the Clinton reservoir but it’s not usable as anything else - it’s just a giant swath of solar panels the size of a lake.
San Onofre Nuclear Power Plant – angry California commuters drive past it every day
The San Onofre plant4 on the California coast is located between the 5 freeway and the ocean and uses ocean water for cooling. Imagine, every year millions of people zip past within a few hundred yards.
The plant is rated for 2,150 MW and sits on a whopping 84 acres, not even enough land for my future alpaca ranch. 84 acres is about 0.34 km², and with the magic of simple division, we get an output of 5,205 MW/km² (again, after adjusting for efficiency). Hey Elon, hombre, this is much more than 200 MW. In fact, it’s 26 times as much as Musk’s Magic Solar Plant, and 95 times as much as The Chicken House.
[Notice we didn’t even need to do the earlier calculations of a theoretical best-case solar facility. Simple division (of plant output divided by area) of an existing (although now shut down) nuclear plant was enough to show Elon’s way off the mark.]
Palo Verde Nuclear Generating Station – basically the worst case example you can find, and it’s still better
The Palo Verde plant5 is located in the Arizona desert and is the only nuclear plant not located near a large body of water. Palo Verde uses treated sewage from nearby towns for cooling instead, so the site includes wastewater treatment and storage facilities.
The plant is rated for about 3,937 MW and is located on a 4,050 acre site so it has an output of 217 MW/km² (again, adjusted for efficiency). This is close to the number Elon Musk stated, but it’s still four times as much power as the chickens are producing. To match Palo Verde, the Chicken House Plant would need 17,600 gleaming acres.
Palo Verde is also treating waste water so this plant is doing double duty. But hey, at least we found an example that has output that’s lower than Musk’s number so for a brief moment each day (in the summer) an ideal solar array will output slightly more power (but not more energy).
If Elon Musk were quoting this post that’s probably the only part he would mention. (“But… we found an example… solar array will output… more… energy.”)
Okay, so Elon is full of you-know-what. What would be a more accurate statement?
Elon has left out a lot of context when he stated the 200 MW/km² number for the solar panels. This is power, not energy – it’s only true for a brief time (in the middle of the day), and in the ideal location. To make his comparison to a nuclear plant work he needs to pretend that power and energy are the same thing, then he still needs to include the cooling water source as part of the nuclear plant area. In some cases this source is an entire ocean.
A more accurate statement would be: an ideal solar plant, on the few days when it’s output is highest, and for just a short while in the middle of each day, will be generating as much power per area as some nuclear power plants - assuming we include the giant lake next to the nuclear plant as part of the plant area.
The cooling water is not toxic and comes from a lake or reservoir that has other uses. Sometimes it comes from an ocean, which is, well, an ocean.
An even more accurate statement would be: nuclear power plants produce much more energy in a given area than solar panels, even if the solar panels are in the ideal location. And it’s not just a little bit more - it can easily be 100 times as much or more.
Or maybe this: Elon Musk spews a lot of crap.
Time to GoFundMe an alpaca ranch
Let’s use Elon’s technique to get rich too. Everybody invest in my new ranch, it totally works if we use Musk Math. We’ll get a San Onofre sized plot – and I promise we can pack in 200 alpacas per acre. The 84 acres will mean 16,800 alpacas on a ranch that will only cost $840,000 because it’s in the desert next to the Palo Verde nuclear plant. (Free waste water! Plus the alpacas might glow in the dark!)
Every alpaca makes $1,000 worth of HyperWool per year, so that’s $16,800,000 (every year!) and all we need to do is shave them. The alpacas only cost $1,000 each so we get back the investment after a little more than one year! Plus, we can market it as “Nuclear Wool” and make press releases about how one square kilometer of compressed alpacas will provide enough wool to clothe all the lobbyists in D.C.
Notice how I said “we” repeatedly there? Once Alpaca Enterprises goes public my shares will be worth about $100 million, and that by itself is proof the business idea is good. To the moon, baby!
What’s the point?
Don’t trust Elon Musk. He took a power number that’s only correct for a brief time in an ideal case, called it energy, then proceeded to compare it to the worst case example for nuclear power in order to make a point that’s completely wrong.
The examples of nuclear plants are all old - we’ve been piling funds into solar panel development but not nuclear power. If Elon Musk wants us to produce more clean power, he should be using his considerable (but undeserved) influence to convince people to look more into nuclear power and less into giant batteries.
Addendum (Added 30 Aug 2022)
This type of rough analysis is something engineers do all the time. We take an ideal case of what Musk is describing (best location, no energy lost in converting to grid power, not counting the cost of storage, etc.) and compare it to existing power plants.
If it’s not close to true even in the most ideal case, then it’s just not true. Why bring this up now? In an article in Forbes, Michael Shellenberger discussed a study confirming something he has claimed in the past - that solar and wind power raise the average cost of electricity by a significant amount6. From his article:
Previous studies were misleading, the economists note, because they didn't "incorporate three key costs," which are the unreliability of renewables, the large amounts of land they require, and the displacement of cheaper "baseload" energy sources like nuclear plants.
Shellenberger brought this up recently (here’s his comment on Twitter):
Given the analysis we just did (even in a best case scenario for solar, Elon was off by a factor of 100) we have good reason to think this part of Shellenberger’s analysis is correct.
One could also point out that if the renewable energy sources were cheaper they wouldn’t require laws mandating their use, or taxpayer money subsidizing their construction. So it’s reasonable to think these sources might actually be more expensive.
But using renewable sources means Elon gets to sell us expensive batteries.
Original:
http://ecgllp.com/files/3514/0200/1304/2-Solar-Radiation.pdf
Archived Copy:
https://web.archive.org/web/20201121071833/http://ecgllp.com/files/3514/0200/1304/2-Solar-Radiation.pdf
https://www.google.com/maps/place/The+Chicken+House/@30.2050825,-103.2478548,17z/data=!3m1!4b1!4m5!3m4!1s0x86f1cb4706ec6c77:0xc6455c414352e63e!8m2!3d30.2050759!4d-103.245656
https://www.eia.gov/nuclear/state/archive/2010/illinois/
https://www.eia.gov/nuclear/state/california/index.php
https://www.eia.gov/nuclear/state/archive/2010/arizona/
In case the study disappears, here’s the current:
https://epic.uchicago.edu/wp-content/uploads/2019/07/Do-Renewable-Portfolio-Standards-Deliver.pdf
And here’s the archived copy:
https://epic.uchicago.edu/wp-content/uploads/2019/07/Do-Renewable-Portfolio-Standards-Deliver.pdf