Issue 052 - Energy - Capital-to-capacity scaling
How much electricity does a $1.9 billion nuclear-plant restart buy?
The U.S. Department of Energy has approved a loan of up to $1.9 billion to help NextEra Energy restart Iowa's Duane Arnold nuclear plant, which shut down in 2020. The reactor has a generating capacity of about 615 megawatts and is expected to supply electricity partly under a long-term agreement with Google.
The problem
Estimate how much electricity a 615 MW nuclear plant could generate over its remaining operating life if successfully restarted.
Then estimate the approximate market value of that electricity and compare it with the $1.9 billion federal loan.
You will need to estimate how much of the year a nuclear plant actually operates, how many additional years the restarted reactor might remain in service, and the average wholesale value of the electricity it generates.
As a second step, estimate roughly how many years of electricity production would have a gross market value equal to $1.9 billion.
Because Fermi problems target an order of magnitude, I normally use no more than two significant digits and write most calculations in scientific notation; the Fermi reference explains both conventions.
Before checking sources
Matt's first pass
I ignored the given data at the bottom indicating that the 615 MW plant could power about 500,000 homes.
First I took 615 MW and multiplied that by 8,800 hours, about how many hours there are in a year, to get an annual production of about 5.5 x 10^9 kWh as the maximum possible production of that power plant when running year-round.
maximum annual generation
~= 615 MW x 8.8 x 10^3 h/year
~= 5.5 x 10^6 MWh/year
~= 5.5 x 10^9 kWh/yearUsing about 10,000 kWh as the average annual energy consumption of a home, that gives about 550,000 homes that could be powered by this plant.
If I assume an average monthly retail energy cost of about $150 per month, or $1,800 annually, then the annual retail cost of the energy that could theoretically be produced by the plant is about $990 million.
home equivalents
~= 5.5 x 10^9 kWh/year / 1 x 10^4 kWh/home-year
~= 5.5 x 10^5 homes
retail value
~= 5.5 x 10^5 homes x $1.8 x 10^3/home-year
~= $9.9 x 10^8/yearNext, I assumed the wholesale cost of that energy is about 70% of retail, or about $690 million. Then I assumed the plant might only operate six months out of the year, producing a wholesale energy amount valued at about $350 million.
annual wholesale value after uptime assumption
~= $6.9 x 10^8/year x 0.5
~= $3.5 x 10^8/year
years to equal loan value
~= $1.9 x 10^9 / $3.5 x 10^8/year
~= 5.4 yearsAt this annual rate, it would take about 5.4 years for the plant to produce an energy amount whose wholesale value is equal to the $1.9 billion investment or loan.
I expect the lifespan of the plant will be at least another 10 to 20 years, so that more than pays back the investment.
Calibration Score
Matt's Calibration Score: 70 / 100
Higher is better: earn points for accurate pegs, sound models, correct math, and a result close to the sourced answer. The image shows percent full of it: 100 minus the Calibration Score.
Pegs: 10/30. The 8,800 hours/year, household electricity use, and retail bill pegs were strong. The big misses were assuming a nuclear plant might operate only half the year and using a wholesale price much closer to retail than to typical wholesale electricity prices.
Model: 30/30. Capacity times hours times capacity factor times price is the right model for gross electricity value, and the household-equivalent cross-check was useful.
Math: 10/10. The arithmetic followed cleanly from the assumptions.
Result: 20/30. The 5.4-year answer is close to the corrected 7-to-10-year range, comfortably within one order of magnitude.
Grounding facts
DOE announced a loan of up to $1.9 billion to support the restart of Duane Arnold and says the plant would add 615 MW of baseload power, enough to power nearly 500,000 homes.
NextEra says the 615 MW facility stopped operating in 2020, is being restarted through regulatory and readiness work, and is planned to return no later than the first quarter of 2029 if approvals are secured. The company also cites a 25-year economic-benefit frame for the restart.
EIA's power-plant example gives the key conversion: a 100 MW plant running continuously for a full year produces 876,000 MWh. That is the same as saying 1 MW running all year produces about 8,760 MWh.
DOE says U.S. nuclear plants have historically operated at maximum power more than 92% of the year, with refueling outages every 18 to 24 months. A Fermi capacity factor around 90% is a better peg than 50% for a working nuclear unit.
EIA reports average U.S. residential electricity use around 1.1 x 10^4 kWh per household-year, which makes DOE's nearly-500,000-homes statement a useful independent check.
After checking sources
Check and recalibrate
Start with the nameplate output, convert power to annual energy, and then apply a high nuclear capacity factor.
annual generation
~= 615 MW x 8.8 x 10^3 h/year x 0.9
~= 4.9 x 10^6 MWh/year
~= 4.9 x 10^9 kWh/yearThat amount of electricity lines up with the home-equivalent claim:
home equivalents
~= 4.9 x 10^9 kWh/year / 1.1 x 10^4 kWh/home-year
~= 4.5 x 10^5 homesFor gross market value, a useful wholesale peg is not the retail bill paid by households. A round wholesale value near $50/MWh, with large regional swings, gives this scale:
gross wholesale value/year
~= 4.9 x 10^6 MWh/year x $50/MWh
~= $2.5 x 10^8/yearThen compare that annual gross value with the loan:
years of gross electricity value equal to loan
~= $1.9 x 10^9 / $2.5 x 10^8/year
~= 7.6 yearsThe best Fermi answer is therefore roughly 5 billion kWh per year, enough for about 450,000 to 500,000 homes, with gross wholesale electricity value around $250 million per year if priced near $50/MWh.
At that rate, the $1.9 billion loan is comparable to about 8 years of gross electricity value, not just a few months and not most of a 20-to-25-year operating horizon. Over 20 to 25 years, the restarted plant could generate about 100 to 120 TWh and perhaps $5 billion to $6 billion of gross wholesale electricity before operating costs, fuel, capital recovery, financing, licensing, outage risk, and contract terms.
Post-check reflection
Matt's reflection
Looks like I landed on just about the right answer despite two errors with assumed pegs. I nailed it on the theoretical power output and the retail cost of that energy, but my assumption about wholesale rates was far too high. Then I assumed the plant might only operate half the year and that was far too low. Honestly, if the question did not include a suggestion to consider how much of the year the plant was operational, I would have assumed all year long, so the question led me astray.
Ultimately, as is so often the case, the errors cancelled out and I landed at a reasonably good answer. Also, it does not surprise me that the plant should be able to produce more than enough energy to offset the value of the loan. Loans are not often given unless the return value is expected to be greater than the loan amount, and this was about the scale I would have expected for a 615 MW nuclear reactor.
Recommended memory peg
For power-plant scale problems, remember: 1 MW running all year produces about 8.8 x 10^3 MWh. A nuclear plant is often near 90% capacity factor, and U.S. wholesale electricity prices are often on the order of tens of dollars per MWh, with $50/MWh a useful rough peg.
Reader results
Bars show how submitted estimates sort into the answer choices from the gut-check prompt.