A toll is an option on the spark spread in every hour; priced on the flat forward the plant is worthless, priced on the hours it earns a fee.
A tolling fee is priced from both ends. The toller, who supplies the gas and takes the power, can pay at most the margin the plant earns in the hours it runs, less a haircut for risk; the owner needs at least its fixed costs plus a return on the plant. On an illustrative 400 MW gas plant the hourly dispatch margin is 30,349,019 a year, so the toller bids 5.37 per kW-month after a 15 per cent haircut. The owner needs 9.84 to earn 9 per cent on 750 per kW. Priced on the flat annual forward, the plant would be worth nothing to the toller at all.
Worked in full in Energy Trading by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Under a tolling agreement the owner is paid a fixed capacity fee for making the plant available, and the toller takes the commodity risk: it delivers gas and carbon allowances, decides when the plant runs and sells the output. The fee is therefore the price of an option on the spark spread, struck at the plant's marginal cost, exercisable in every hour. The way to price an option on hours is hour by hour.
| Input | Value |
|---|---|
| Capacity | 400 MW |
| Heat rate | 7.2 MMBtu/MWh |
| Gas price | 8.60 per MMBtu |
| Carbon: 0.0531 t per MMBtu at 30.00 per tonne | 11.47 per MWh |
| Variable O&M | 3.00 per MWh |
| Availability | 92% |
| Fixed O&M (owner) | 25 per kW-year |
| Plant value, owner's target return, life | 750 per kW, 9%, 15 years |
| Toller's risk haircut on expected margin | 15% |
Fuel is 7.2 × 8.60 = 61.92 per MWh. Carbon is 7.2 × 0.0531 = 0.382 t per MWh at 30.00, or 11.47. Add 3.00 of variable O&M and the plant's marginal cost is 76.39 per MWh. The year's average power price on the curve below is 72.66, so at the flat annual forward the spark spread is minus 3.73 and the plant would never run.
| Hours | Price | Spread over 76.39 | Margin earned |
|---|---|---|---|
| 100 | 260 | 183.61 | 6,756,863 |
| 400 | 140 | 63.61 | 9,363,451 |
| 900 | 105 | 28.61 | 9,475,764 |
| 1,500 | 85 | 8.61 | 4,752,941 |
| 2,000 | 70 | -6.39 | 0 |
| 2,000 | 60 | -16.39 | 0 |
| 1,260 | 48 | -28.39 | 0 |
| 600 | 20 | -56.39 | 0 |
| 8,760 | 72.66 | 30,349,019 |
Dispatch margin = Σ max(price − marginal cost, 0) × MW × hours × availability
The plant runs 2,900 hours and generates 1,067,200 MWh, earning 30,349,019: 75.87 per kW-year, or 6.32 per kW-month. In Excel, with hours in A2:A9 and prices in B2:B9: =SUMPRODUCT((B2:B9>76.39)*(B2:B9-76.39)*A2:A9)*400*0.92
More than half the value, 53.1 per cent, comes from the 500 dearest hours. That concentration is why the toller does not pay the whole expected margin: one mild winter removes a large share of it. A 15 per cent haircut, 4,552,353, leaves 25,796,666 a year, the toller's maximum fee of 5.37 per kW-month.
Owner's minimum fee = fixed O&M + plant value × capital recovery factor
Capital recovery at 9 per cent over 15 years is 0.1241, so 750 per kW needs 93.04 a year. With 25 of fixed O&M the owner needs 118.04 per kW-year, 9.84 per kW-month, 47,217,665 a year on a 300,000,000 plant.
The bid is 4.46 per kW-month below the ask. Read the other way, the toller's fee, 64.49 per kW-year, leaves 39.49 after fixed O&M, which supports a plant value of 318 per kW, 42.4 per cent of the 750 the owner is carrying it at. A gas plant whose margin lives in a few hundred scarce hours is worth what those hours pay, not what it cost to build.
| Gas, per MMBtu | Marginal cost | Dispatch margin | Before haircut | After haircut |
|---|---|---|---|---|
| 6.60 | 61.99 | 51,612,353 | 10.75 | 9.14 |
| 7.60 | 69.19 | 38,629,313 | 8.05 | 6.84 |
| 8.60 | 76.39 | 30,349,019 | 6.32 | 5.37 |
| 9.60 | 83.59 | 22,665,179 | 4.72 | 4.01 |
| 10.60 | 90.79 | 18,177,198 | 3.79 | 3.22 |
With the power curve held fixed, cheaper gas widens every spread and the toll rises; the owner's 9.84 is reached only below 6.30 per MMBtu. In reality the power curve moves with gas, which is why tolls are better compared through the implied heat rate than the gas price, as in how to calculate the implied heat rate of a power price. Scarcity matters as much: cap the 100 dearest hours at 140 instead of 260 and the margin falls 14.6 per cent, to 25,933,019, or 5.40 per kW-month before the haircut.
Pricing the toll on the annual forward. At 72.66 against a 76.39 marginal cost the flat spread is negative, and a desk that values the plant as a baseload block puts nothing on a contract worth 30,349,019 a year before haircut. The opposite error is paying for the whole hourly margin: the expected value of 100 scarcity hours is not a cash flow that can fund a fixed fee. Price from the duration curve, haircut the concentrated top, and set the result beside the owner's capital charge before negotiating anything else.
Real tolls add start costs, ramp limits, minimum run times, outage penalties and often capacity market revenues, which can shift the fee materially. The curve and plant are illustrative; nothing here is investment advice.
The same plant dispatched hour by hour off the curve, and valued four ways, is in the free workbook for this book. The spread itself is worked in how to calculate a clean spark spread.
A contract in which a toller pays a plant owner a fixed capacity fee, supplies the fuel and carbon allowances, dispatches the plant and takes the output. The owner keeps operating risk, the toller takes market risk. On the illustrative 400 MW plant the toller can pay up to 5.37 per kW-month, about 25,796,666 a year.
Because the owner is paid for capacity made available, not energy generated: the fee is fixed whether the plant runs 2,900 hours or none. Converting the toller's annual value into per kW-month, 30,349,019 divided by 400,000 kW and 12 months is 6.32 before the risk haircut.
Holding the power curve fixed, the toller's fee after a 15 per cent haircut reaches the owner's 9.84 per kW-month only below a gas price of about 6.30 per MMBtu. At 8.60 it bids 5.37. In practice power prices move with gas, so the implied heat rate is the better guide.
This article is one calculation from Energy Trading. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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