Space Based Solar Power

The Wikipedia's take on the topic is here https://en.wikipedia.org/wiki/Space-based_solar_power

Power Satellites Economics

In the absence of other forces such as legal requirements, power satellites compete in the energy market. Energy, particularly electrical energy, is the ultimate standard commodity. When a customer plugs in a toaster, energy is just there. At the end of the month, they pay the bill at the rate set by the state utility commission. One level up, the power companies are hemmed in by regulations that they buy (or make) the lowest cost electricity, with exceptions that they have to purchase certain amounts of renewable power.

Space based solar power is renewable. That should make it easier to sell power from space at a premium. However, governmental energy polity changes unpredictably over time. An alternative would be a "design to cost" where the target cost of power is low enough to get a large market share without government intervention. Competing on cost is the way discount suppliers of many commodities obtained a substantial market share. (Examples, Southwest Airlines, GIECO, Charles Schawb.)

Levelized cost of power

The formula for the levelized cost of electricity is here; https://en.wikipedia.org/wiki/Cost_of_electricity_by_source

The below spreadsheet assumes $1,600,000 per MW as the initial cost and 10% per year of the cost for maintenance. Power satellites run supplying base load, here assumed ~91% of the time, it may be higher.

The discount rate used in the model is 6.8%, same as the government uses for other sources. The accounting period is 20 years and no salvage value is assumed.

https://docs.google.com/spreadsheets/d/1wDvn369EudkYGsPK3jNt4FmBFpNFtt0ZwDZl_lt_SNM/edit#gid=1481425448

The ratio between the $1600/kW cost and the cost that comes out of the formula (~2 cents per kWh) is close enough to 80,000 to one. Electric power cost is proportional to the cost of a power satellite (or any power source that has no fuel cost) in this ratio for this discount rate and years of service.

The UK government has determined that 3.5% discount is proper for projects of this kind. Using 3.5%, the electric cost comes out at just over 1.5 cents per kWh and ~100,000 to one. Extending the accounting period to 30 years at 3.5% brings the cost of power down to 1.24 cents per kWh and a cost of power to cost of investment ratio to ~130,000 to one. It's a live spreadsheet, try your own numbers. A ratio of 80,000 to one is conservative.

To take market share from coal will require the cost to be less than 4 cents per kWh. Three cents per kWh allows a capital cost of $2400/kW.

Top level place holder cost allocations

The current model has this split out as

$200/kW for the rectanna ($1 B for 5 GW),

$900/kW for the cost of parts and minor labor in space and

$1300/kW (6.5 kg/kW and $200/kg) for the cost of transport to GEO.

Power Satellite Types

Photovoltaic (PV)

Most designs for power satellites since the 1970s have been PV. PV has advantages, long experience powering communication satellites being important. However, PV suffers from relatively low efficiency (20%) and degradation from radiation. There are proposals by Forward and Holt on draining van Allen belt, http://en.wikipedia.org/wiki/HiVolt, however, just the presence of a substantial number of power satellites in GEO is expected to greatly mitigate the radiation from particles trapped in the Earth's magnetic field. (There are only about 3 kg of protons trapped in the belts.) There are PV cells that range up to 40% efficient, but they require concentrated light and cooling.

Thermal

Thermal (heat engines) power satellites are expected to eventually range up to 60% efficient, similar to combined cycle plants on earth. This means they need about 1/3 of the light interception area, which reduces station keeping from light pressure. However, they also need radiator area that is about twice the sunlight interception area, (Bejan, 1997, pg 495, ref Bejan, A. Advanced Engineering Thermodynamics, 2nd ed. New York: Wiley, 1997.) Counting both sides of the radiator makes the exposed area for thermal and photovoltaic power satellites about the same.

Common considerations

Mass

The original studies done in the late 1970s came up with a mass of ~10 kg/kW. More recent realistic studies have averaged around 7 kg/kW. A few studies have proposed designs under one tenth of a kg/kW. Very light designs require a lot of station keeping against light pressure where designs in excess of 5 kg/kW can average the light pressure over a year. Because a substantial fraction of the construction cost is for transport to GEO, the mass of a power satellite is an important number as is the lift cost to GEO. This analysis will use 6.5 kg/kW. The number can be adjusted in the spreadsheets.

Energy transmission loss

How efficient is the transmission of the energy with the microwave beam?

For economic analysis, 50%. The loss chain might be a little better with technical improvements, but not much. It means you generate two kW in space for one kW on the rectenna bus. That's been assumed in all the analysis here.

Energy payback time

Most of the energy embedded in a power satellite comes from the fuel.

One of the metrics used to evaluate energy projects is EROEI or energy return on energy invested. This is expressed as a ratio, and for shallow oil wells decades ago was typically 100 to one or higher. I.e., number of barrels of oil needed to drill an oil well divided into the number it produced. An alternate way more applicable to renewable sources is the energy payback time. It's typically about two years for PV.

In his book on power satellites, John Mankins gives a number of 8 weeks. I have determined similar numbers, but they depend on highly efficient transport. John uses an energy from physics, about 12 kWh/kg (IIRC, it's actually 14.75 kW/kg)and doubles it twice for his estimate. Is that reasonable?

Falcon heavy

Consider the projected Falcon Heavy, 53,000 kg to LEO, 121,400kg of RP-1. That's about 2.3 kg of fuel for every kg lifted to LEO (here I assume that the entire mass in LEO is either reaction mass for the trip up or can be converted to power satellite parts--this may not be reasonable). RP-1 is 42 MJ/kg, so the energy expended to LEO is about 96.2 MJ/kg or 26.7 kWh/kg. You also have to lift the reaction mass for the LEO to GEO leg so the fuel cost needs to be increased by around 20% giving 33.4 kWh/kg. If the reaction mass is hydrogen, it's energy content is not zero, hydrogen is close to 50 kWh/kg to make it and another 20 kWh/kg to make it into a liquid. So to move a kg from LEO to GEO would take 1/4 kg of LH2 at 70 kWh/kg or 17.5 kW/kg, for a total of ~51 kWh/kg. At 10 kg/kWe (on the ground) 510 kWh will be required to move a kW of power sat to GEO. At 100% on time, that's about 23 days to repay the energy. Unless I made an error, John's number is close enough. If only half of a Falcon Heavy payload becomes power satellite parts, John's estimate is very close.

Skylon

Much of the analysis been done using 50 kWh/kg as a ballpark number for hydrogen. It's good enough for rough calculations and it what it takes to make hydrogen if you are making it with electric power.

Combustion of hydrogen is 286 kJ/mol. Mole is 2.016 gm, so 496 moles in a kg, 143 MJ/kg or 39.4 kWh/kg. That's quite a bit less. The rough analysis have also been using 20 kWh/kg to liquify the hydrogen. That's way off. If you Google for "A Future Energy Chain Based on Liquefied Hydrogen" Berstad and go down to page 20, they cite a number of ~6.5 kWh/kg. We might do a little better because we can use warming the LNG to cool the hydrogen. That makes the energy content of the LH2 about 46 kWh/kg. It's hard to say exactly where you should measure the input energy, but the process to make hydrogen from natural gas is efficient, and it only uses about 1.7% of the energy in a kg of NG to liquify it. I.e., measuring at the well head will not make a great deal of difference. Still, this Wiki article needs to go into the thermodynamics of steam reforming.

The current LEO to GEO hydrogen consumption (using electric propulsion) is 4000 tons to get 15,000 tons to high orbit. On average, that makes a 15 ton Skylon payload 11.84 tons cargo and 3.16 tons of hydrogen reaction mass. At launch the average Skylon has 59 plus 3 or 62 tons of LH2. That makes the hydrogen per ton of payload 62/11.84 or 5.25 kg/kg of payload.

(Why hydrogen and not an inert gas? For a given amount of energy you can get considerably more exhaust velocity out of hydrogen.)

The solar energy used to power the LEO to GEO transport is not included since it is outside the system. (The capital cost of the propulsion power satellite is included in the financial models.)

Using my supported but perhaps optimistic number of 6.5 kg/kW, an installed kW would use a little over 34 kg of LH2 to get it in place. Using 46 kWh/kg for the LH2, that's around 1566 kWh per installed kW.

Without a detailed list of the parts materials and mass in a power satellite, it's not easy to say how much energy goes into the parts. Aluminum has the most embedded energy at 15 kWh/kg. But not very much will be aluminum. If we estimate 10 kWh/kg, the energy in the parts would be ~650 kWh, for a total of 2216 kWh per kW of capacity.

When the power satellite is turned on, it takes 92 days (at 100% on time) to repay the energy used to build it and move it to GEO.

That's a three month energy repayment time. I don't know of any other proposal that is even close. And the energy from a power satellite not intermittent.

If the power satellites last 30 years, you get an ERoEI of 120 to one.

This is subject to adjustment as we settle on more accurate numbers in the design phase, but it should be reasonably close.

ERoEI is an interesting metric, but the more important figure is the cost. Target cost is 3 cents a kWh falling to 2 cents within a few years. This just barely meets Gail's requirements. For a first approximation, the embedded energy in the parts and the energy required to make LOX are so small in comparison to RP1 or LH2 that they can be ignored.

It's surprising that even with a lower kg/kW number the energy payback time is longer for the Skylon.

EROEI and Maximum Growth Rate

For a repayment of about 3.6 times a year and a lifetime of 30 years, the EROEI is a little over 100 to one. This is as good as the best conventional oil.

By the criteria given here https://en.wikipedia.org/wiki/Energy_cannibalism and here https://web.archive.org/web/20090817071517/http://www.climate2008.net/?a1=pap&cat=1&e=61 a power satellite growth rate of lower than 360% per year would result in lowering GHG emissions.

Reliability

Electric power needs to be reliable. First, the power satellite size is only 5 GW. The target number is 3000 for 15 TW. There would be spares sending power to low priority loads that could be switched in less than a second to replace a higher priority failed one. Plus we would still have the grid to distribute electricity from the remaining powered rectennas, and for a long time, there would be other generation in the mix.

None the less, there are ways we could lose the whole fleet of them in an instant if they were not designed to deal with it. In the year 774 or 775 the Earth seems to have been hit with either a unprecedented solar flare or a fairly close gamma ray burst. The latter are typically a few seconds, the former might take a few hours. In any case, it put a serious kink in the carbon 14 for the next growing season. Such an event would take out the controls for any power satellite that did not have enough shielding around the control computers. The shielding needed against these 1000 year events is considerably more than the worst solar flares observed to date.

Unlikely as they are, GRB or intense high energy solar flares are a concern that requires mitigation and recovery strategies such as watch dog timers and hardened reboot memory. Radiation resistance is an argument in favor of rotating machines rather than PV.

A solar flare can be seen coming and outside workers would probably have time to reach shelter. A GRB would be over before the workers had a chance to move. Neither are serious problems for people behind shielding good enough to stop cosmic rays.

Transport Earth to LEO

SpaceX

SpaceX will not get the transport cost down low enough. It has to be to be SSTO or possibly TSTO runway operations.

SpaceX _will_ eventually get the cost to GEO down by a full order of magnitude, a remarkable achievement. Unfortunately this won't do it for power from space, it takes _two_ orders of magnitude reduction. Elon Musk knows this. It might be why he is so down on power satellites.

Skylon

Until Reaction Engines demonstrated their high performance precooler, there were no realistic SSTO proposals out there.

www.reactionengines.co.uk/space_skylon.html

www.reactionengines.co.uk/space_skylon.html

LEO to GEO

Space Junk

Back in the late 70s, Boeing proposed building power satellite in LEO and then self powering to GEO (using ion engines or arcjets). They even did some nice artwork. But even in those days there was enough space to make this a very risky move and it was abandoned. HKH redid the math in recent years using the density of space junk given in the Wikipedia article. The result was that a power satellite would be hit almost 40 times while self powering to GEO. Almost all the hits are below 2000 km altitude.

Building power satellites in LEO would work, it's just they could not be moved to where they are useful unless the plan includes cleaning up the space junk first.