The Maury Equation

I’m claiming my 15 microseconds of fame right… now! The Maury Equation of the Economics of Solar Power demonstrates that the price of space based power will never be competitive. I intend to prove this, below.

Some background

The price of electricity from any power system is a combination of a few basic inputs. These include:

1) capital costs
2) fixed operational costs (building maintenance, etc)
3) variable operational costs (turbine maintenance depends on delivered power)
4) fuel input costs

If you add these up for the lifetime of the system, you end up with the total cost of the power produced. Now divide that by the amount of power produced during that same lifetime, and you get the cost of that power on a per-whatever basis. I like cents per kilowatt-hour (c/kWh), because that’s what appears on people’s bills.

The result of this calculation is the “Levelised Cost of Electricity” (or Energy, depending on the source). It’s widely used when comparing power sources because it offers a simple apples-to-apples comparison.

…and how we’ll use it

In the case of solar power, the LCOE inputs for (2), (3) and (4) are vanishingly small. The cost of power is dominated entirely by the capital costs of the equipment.

As the sun’s output doesn’t change much, one can easily predict the amount of power produced on a yearly basis, multiply by the expected lifetime, and then estimate the total cost of power. For large solar farms in the US southwest, that’s somewhere in the range of 15 c/kWh assuming a 40 year life span, or 20 c/kWh if you factor in short lifetimes of the inverters and plan for two replacements over the lifetime of the panels.

In the case of space based power, there are the launch costs to consider. This is likely much more than the cost of the panels, but in effect it factors into the overall capital costs. Yet those same panels will produce more power. This makes it difficult to directly compare the two cases… well difficult until you actually try it. So let’s try it!

Go ahead and try this at home…

Lets’s start with the basic capital/construction cost calculation. Let’s define P as the cost of buying enough panels to produce 1 kW of peak power at “standard” conditions, STC. This number is the same for ground and space. But for this discussion, we also need to consider the cost of shipping those panels:

S_ground is the price of shipping and installing those panels on the ground
S_space is the price of shipping and installing those panels in space

… then the total capital cost of installing the panels, greatly simplified, we’ll define as C:

C = P + S

Now let’s look at the income side of things. Panels generate power from the “insolation”, which we’ll call I. This is normally expressed in terms of power generated over the period of a year given a set of panels that would generate 1 kW under STC. So if that’s the power it generates in a year, then we just need to multiply by the number of years the system is in operation, L. Presto, that’s the lifetime electrical generation, E. So:

E = I x L

There’s one more thing to consider in this particular case, and that’s the cost of transmission. In the case of space based power, and often for ground based as well, the systems are located at long distances from the consumption, and there will be losses on the way between the two. We’ll call this T. So then for our calculations:

E = I x L x T

Ok, so then we’re ready to go, the levelised cost of electricity from the systems we’ll be considering is simply:

LCOE = E / C

Now let’s talk numbers

So great, we have an equation, but what are we supposed to plug into it? That’s the part I’ll do for you, with a little Google-Fu

P is currently about $3000 per kWp (for comparison, hydro is about $2000, and nuclear is about $11000).

S_ground is about $1 a pound, S_space is about $12,000 a pound.

I_ground is about 1600 for fixed-plate collectors in the US southwest, I_space is about 8600 in GEO

L_ground is about 40, L_space  is roughly 12 (space is a nasty place!)

T_ground is small, maybe 10% in the worst case, while around T_space is around 50%, and then has to add T_ground. Let’s leave it at 50% net for now.

So then:

E_ground is 1650 * 40 * .90 ~= 60,00
E_space is 8600 * 12 * .5 ~= 52,000

So contrary to the space power boosters’ basic claim, space based power generally produces less power over the lifetime of the system.

So…

Let’s put it all together…

LCOE_ground = C_ground / E_ground = ($3000 + nothing) / 60,000 = 5 cents per kWh on capital alone

LCOE_space = C_space / E_space = ($3000 + (100 lbs/kWp * $12,000) / 52,000 ~= 125,000 / 52,000 = $2.40 per kWh

What’s interesting about this set of equations is that it is utterly dominated by the transit costs. Doubling the efficiency of transmission, for instance, does nothing to address “the problem”. Moreover, note that the price of the panels isn’t even a factor, which means that improving the technology on the panel side does nothing — both sides improve by the same amount.

So I simply state it flat out. The numbers suggest that space based power cannot ever become competitive with ground based solar unless the cost of launch falls by three orders of magnitude. I consider this to be “effectively impossible”: although it is not specifically impossible, the chance of it happening is much lower than the chance of an entirely different invention coming along that renders the entirely argument moot (say fusion power).

Should you trust these numbers? Well, the US the DOE is in the process of driving LCOE to 6 cents by 2020, including all factors, like the cost of land, labor, everything that we ignored above. Those too are much more expensive in space, including “land” – orbital slots cost a lot more than Mohave desert.

Fight!

I ask all proponents of space solar power to attack this as hard as they can. All I ask is that you use the formulas above (or suitably modified version), present your numbers for each one, and reference why you believe that number can be supported.

The numbers used above are all fully referenced in previous posts (suddenly I wish there was a wiki-easy way of doing refs here!) but I’d be happy to present them again if anyone needs them.