Rates and efficiencies

Chain per-unit numbers, turn a heat rate into an efficiency, and follow the losses

Math toolbox K2 of Energy Systems, Step by Step: work out the average power a home buys from the grid, chain a gas plant’s heat rate into fuel cost and CO₂ per MWh, turn a heat rate into an efficiency, follow energy through a battery’s round trip, and get a wind farm’s year from its capacity factor.
  • Math toolbox
  • 10 min guided · about 10 min practice
  • After K1

Rates and efficiencies, in five moves

  1. A rate is one amount per another: 10,791 kWh bought in a year ÷ 8,760 h = 1.232 kW, an average of 1,232 W bought from the grid.
  2. Rates chain when the units cancel: 6.226 MMBtu per MWh × $3.85 per MMBtu = $23.97 per MWh, and × 52.91 kg CO₂ per MMBtu = 329 kg per MWh.
  3. Efficiency = 3.412 MMBtu per MWh ÷ heat rate: 54.8% for the combined-cycle plant, 37.3% for the peaker (heat rate 9.142).
  4. Losses multiply, and dividing undoes them: an 85% battery gives back 85 of 100 MWh; 100 MWh back needs 100 ÷ 0.85 = 117.6 MWh in, not 115.
  5. A capacity factor is energy made ÷ the most possible over the same period; for a year, 150 MW × 8,760 h × 33.8% = 444,132 MWh.
Rate Unit Worked on this page
Average power bought from the grid kW = kWh per hour 10,791 kWh ÷ 8,760 h = 1.232 kW
Heat rate MMBtu per MWh 6.226 (combined-cycle), 9.142 (peaker)
Fuel cost $ per MWh 6.226 × $3.85 = $23.97
CO₂ at the plant kg per MWh 6.226 × 52.91 = 329 kg
Efficiency % (out ÷ in) 3.412 ÷ 6.226 = 54.8%
Round trip % (back ÷ in) 117.6 MWh in, 100 MWh back at 85%
Capacity factor % (made ÷ most possible) 150 MW × 8,760 h × 33.8% = 444,132 MWh

With JavaScript on, you work these out yourself, step by step.

  • A home's 10,791 kWh a year: EIA, 2022 data, as in lesson 1.
  • Heat rates 6.226 and 9.142 MMBtu per MWh: EIA, AEO2026 assumptions, Table 3 (April 2026). Gas $3.85 per MMBtu: EIA, 2025 average. Both as in lesson 7.
  • 52.91 kg CO₂ per MMBtu of gas: EIA (checked 2 November 2026), as lessons 12 and 13 use it.
  • CO₂ boundary: gas burned at the plant only. Producing and delivering the gas, and building the plant, are outside it.
  • Lesson 0's town uses a rounded, illustrative 0.37 t (metric tonnes) of CO₂ per MWh instead.
  • 3,412 Btu per kWh, so 3.412 MMBtu per MWh: EIA FAQ on plant efficiency (checked 3 November 2026).
  • 85% round trip: lesson 13's battery, from NREL's 2025 battery cost study (June 2025); lesson 9's illustrative battery uses 80%.
  • Prairie Town's 150 MW wind farm is illustrative; 33.8% is U.S. wind in 2025: EIA, Table 6.07.B (released 23 October 2026).

What you just did, as diagrams

Rate
One amount per another, such as $ per MWh; its unit says what was divided by what.
Chaining rates
Multiply rates when a unit cancels: MMBtu per MWh × $ per MMBtu gives $ per MWh.
Heat rate and efficiency
Efficiency = 3.412 ÷ heat rate in MMBtu per MWh: a higher heat rate means a lower efficiency.
Losses in a chain
Stage efficiencies multiply: 54.8% × 85% = 46.6%. To go backwards, divide.
Capacity factor
Energy made ÷ the most possible in the same hours: a ratio with no unit.
Go deeper · AnalystEIA's own units, and percent against percentage points
  • Btu per kWh. EIA quotes heat rates in Btu per kWh. 6,226 Btu per kWh is the same rate as 6.226 MMBtu per MWh: a million Btu per thousand kWh.
  • Efficiency in those units. \(\eta = 3{,}412 \div 6{,}226 = 54.8\%\), exactly as on this page.
  • Points. From the peaker’s 37.3% to the combined-cycle plant’s 54.8% is 17.5 percentage points.
  • Percent. Per MWh, the peaker burns \(9.142 \div 6.226 = 1.47\) times the gas: 47% more. Say which one you mean.
  • Ratios hide a period. A capacity factor needs its hours: lesson 5 uses 8,760 for a year.
Go deeper · ModelerRates as fixed numbers in a model (a preview)
  • A simplified example. One gas plant and one battery, in one-hour steps, so every flow is in MWh per hour.
  • Decision. The model chooses \(g\), the MWh the gas plant makes in an hour: a decision variable.
  • Goal. The least total cost. Each MWh costs its fuel, 6.226 × $3.85 = $23.97, plus variable O&M (VOM: variable operation and maintenance, in $ per MWh).
  • Rules. The CO₂, 0.329 t per MWh made, adds up under a yearly cap; the battery gives back 85% of each MWh charged.
  • One line. \(\text{cost in the hour} = (\mathrm{HR} \times p + \mathrm{VOM}) \times g\), with HR the heat rate (MMBtu per MWh) and \(p\) the gas price ($ per MMBtu).
  • Why fixed rates. Each term is then a fixed number times a decision, so the model stays a linear program. A real plant’s heat rate changes with its output; a fixed one is a modelling choice.
  • Where next. Lesson 13 writes the full model: two gas plants, CO₂ in tonnes, each modelled hour weighted by the hours it stands for. M3 builds one in a notebook.

Check yourself

Myth or fact? “A plant with a higher heat rate is more efficient.”

  • Myth
  • Fact
Answer

Myth. Heat rate is gas burned per MWh, so higher means more gas: 9.142 gives 37.3%, 6.226 gives 54.8%.

Myth or fact? “To get 100 MWh out of an 85% battery, charge 115 MWh.”

  • Myth
  • Fact
Answer

Myth. 115 × 0.85 = 97.75 MWh. Divide instead: 100 ÷ 0.85 = 117.6 MWh.

The peaker burns 9.142 MMBtu of gas per MWh, and gas costs $3.85 per MMBtu. Its fuel cost per MWh?

  • $2.37
  • $35.20
  • $12.99
  • $0.42
Answer

$35.20: 9.142 × 3.85, and MMBtu cancels. $2.37 divides instead; $12.99 adds numbers with different units.

A 100 MW plant made 438,000 MWh in a year of 8,760 hours. Its capacity factor?

  • 1,200%
  • 5%
  • 50%
  • 0.5%
Answer

50%: 438,000 ÷ (100 × 8,760) = 438,000 ÷ 876,000. Dividing by 365 days instead of 8,760 hours gives 1,200%, which no plant can reach.

Numbers, sources and what’s next

Numbers to remember

Number What it means Source
3.412 MMBtu of energy in one MWh of electricity (3,412 Btu per kWh): efficiency = 3.412 ÷ heat rate. EIA FAQ · checked 3 November 2026
6.226 · 9.142 Heat rates in MMBtu per MWh of a new combined-cycle plant and a new peaker: 54.8% and 37.3% efficient. EIA, AEO2026 assumptions, Table 3 (PDF) · April 2026
52.91 kg CO₂ per MMBtu of natural gas burned. EIA · checked 2 November 2026
85% Round trip of lesson 13's lithium-ion battery: 85 MWh back for 100 MWh in. NREL, 2025 battery cost study (PDF) · June 2025
8,760 h Hours in a year of 365 days: the denominator of a yearly capacity factor or average. 365 × 24

Next

K3 · Exponentials and discounting

Sources and further reading

Spotted an error, or stuck? Write to hlli@ksu.edu. Corrections are made in place and dated here.

Text licensed CC BY 4.0, code MIT: teachers and students may reuse and adapt it, with credit to Honglin Li.