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Equations and Variables in the ATB

Calculated Variables Equations

Levelized Cost of Energy (LCOE): a summary metric that combines the primary technology cost and performance parameters: capital expenditures, operations expenditures, and capacity factor

$$ LCOE = \frac{FCR \times CAPEX + FOM}{CF \times 8,760 (hours/yr)} + VOM + FUEL - PTC $$

Fixed Charge Rate (FCR): amount of revenue per dollar of investment required that must be collected annually from customers to pay the carrying charges on that investment 

$$ FCR = CRF \times ProFinFactor $$

Capital Expenditures (CAPEX): expenditures required to achieve commercial operation of the generation plant 

$$ CAPEX = ConFinFactor \times (OCC + GCC) $$

Capital Recovery Factor (CRF): ratio of a constant annuity to the present value of receiving that annuity for a given length of time

$$ CRF = WACC \times \left [ \frac{1}{\left ( 1 - \frac{1}{(1 + WACC)^{t}} \right )} \right ] $$

Weighted Average Cost of Capital (WACC): average expected rate that is paid to finance assets

$$ WACC = \frac{1 + [1-DF]\times[(1+RROE)(1+i)-1] + DF\times[(1+IR)(1+i)-1]\times[1-TR]}{1+i}-1 $$

Project Finance Factor (ProFinFactor): technology-specific financial multiplier to account for any applicable differences in depreciation schedule and tax policies; the investment tax credit (ITC) appears first to reduce the depreciable basis and a second time to apply the value of the credit (Ho et al., 2021).

$$ ProFinFactor = \frac {(1 - TR \times PVD) \times (1 - \frac{ITC}{2}) - ITC}{(1 - TR)} $$

Construction Finance Factor (ConFinFactor): portion of all-in capital cost associated with construction period financing

$$ ConFinFactor = \sum_{y=0}^{y=C-1} FC_{y} \times AI_{y} $$

Accumulated Interest (AI): a portion of the construction finance factor due to interest on a construction loan.

$$ AI_{y} = 1 + (1 - TR) \times [(1 + IDC)(year + 0.5) -1] $$

Production Tax Credit (PTC): a before-tax credit that reduces LCOE; credits are available for 10 years, so it needs to be adjusted for a 10 year CRF relative to the full CRF of the project.

$$ PTC = \frac {PTC_{full} \times (1 - TR) \times CRF } {CRF_{10yrs}} $$

Total Battery System Cost: the total overnight capital cost of battery system after accounting for storage duration.

$$ TotalBatterySystemCost($) = TotalBatteryStorageCost($) + TotalBatteryPowerCost($) $$

$$ TotalBatteryStorageCost($) = BatteryStorageCapacity(kWh) \times BatteryEnergyCost($/kWh) $$

$$ TotalBatteryPowerCost($) = BatteryPowerCapcity(kW) \times BatteryPowerCost($/kW) + BatteryPowerConstant($) $$

Total Utility-Scale PV-Plus-Battery System Cost: LCOE equation for PV-plus-battery systems, which accounts for the potential effects of the PV and battery systems using different tax credits, as well as the costs of charging the battery from the grid.

$$ PVPlusBatterySystemCost($/MWh) = FixedCosts($/MWh) + VOM($/MWh) + ChargingCosts($/MWh) $$

$$ FixedCosts = \frac {[AnnualPVSystemCost + AnnualBatterySystemCost + FOM($/kW-yr)]\times 1,000(kW/MW)}{CF\times 8,760(hours/yr)} $$

$$ AnnualPVSystemCost($/kW) = RROE \times PFF(PV) \times ConFinFactor \times [ PVSystemCost($/kW) \times 1 + GCC($/kW)] $$

$$ AnnualBatterySystemCost($/kW) = RROE \times PFF(Battery) \times ConFinFactor \times BatteryStorageCost($/kW) \times 1 $$

Input Variables

Input Variable Equations

Depreciation Factor (DepFac): a function of the discount rate:

$$ DepFac = \frac {1}{[(1+WACC) \times (1 + i)]^{y}} $$

Real Interest Rate (IR): assumed interest rate on debt

$$ IR = \frac {1 +d}{1 + i} -1 $$

Present Value of Depreciation (PVD): a function of FD, f, and y:

$$ PVD = \sum (MACRS~FD) \times \frac {1}{[(1 + WACC) \times (1 + i)]^{y}} $$

Other Input Variables

Input Variables and Their Values and Definitions

Input VariableValueDefinition/Description
Construction duration (C)Varies by technologyNumber of years in construction period
Capacity factor (CF)Varies by technologyGenerally, the ratio of actual annual output to output at rated capacity for an entire year
Charging CostsFor PV-plus-battery systems  ($/MWh)Average cost of electricity to charge a PV-plus-battery system during the lowest cost hour of each day. The default assumes the battery charges 75% of its energy from the PV system.
Nominal debt rate (d)Varies by technologyThe nominal debt rate varies by technology. It is fixed over time for the R&D financials and varies with time in the Markets & Policies financials.
Debt fraction (DF)Varies by technologyFraction of capital financed with debt; 1-DF is assumed financed with equity; also referred to as the leverage ratio
Expected market return on premium (EMRF)Varies by technologyDifference between the expected return on a market portfolio and the risk-free rate
Capital fraction (FC)Varies by technologyFraction of capital spent in each year of construction, 1 to C
Depreciation fraction (FD)Varies by technologyFraction of capital depreciated in each year, 1 to M
Fixed operation and maintenance expenses (FOM)ATB input ($/MW-yr)Annual expenditures to operate and maintain equipment that are not incurred on a per-unit-energy basis
FuelHeat rate (MMBtu/MWh) *Fuel Costs($/MMBtu)Fuel costs, converted to $/MWh, using heat rates
Grid connection costs (GCC)Varies by technology ($/kW)Overnight capital cost includes a nominal-distance spur line (<1 mi) for all technologies, and for offshore wind, it includes export cable and construction period transit costs for a 30-km distance from shore. Project-specific costs lines that are based on distance to existing transmission are not included.
Inflation rate (i)2.5%Assumed inflation rate based on historical data
Interest during construction (IDC)Varies by technology Assumed nominal interest rate during project construction
Investment tax credit (ITC)Varies by technology (%)Tax credit received on the CAPEX of the system
Depreciation period (M)Varies by technology (years)Number of years in modified accelerated cost recovery system (MACRS ) depreciation schedule
Overnight capital costs (OCC)Varies by technology ($/kW)CAPEX if plant could be constructed overnight (i.e., excludes construction period financing); includes on-site electrical equipment (e.g., switchyard), a nominal-distance spur line (<1 mi), and necessary upgrades at a transmission substation
Production Tax Credit (PTC)Varies by technology ($/MWh)Production tax credit value received by the project, normalized by the capital recovery period
Capital regional multiplier (CapRegMult)1Multiplier to account for regional variation in capital costs; not used in the ATB
Rate of return on equity (RROE)Varies by technologyAssumed rate of return on the share of assets financed with equity
Economic lifetime (t) or cost recovery period30 years (default); 20 years and technical life also availableLength of time for paying off assets
Tax rate (TR)25.7%Combined assumed marginal state and federal tax rate before application of available federal tax credits for renewable generators
Variable operation and maintenance (VOM)Varies by technology ($/MWh)Operation and maintenance costs incurred on a per-unit-energy basis
Year index (y)

Assumptions common to all technologies include the following:

  • Variables are defined in the Financial Definitions worksheet in the ATB data spreadsheet, where two sets of financial assumptions are available in the:
    • R&D Only Financial Assumptions Case (R&D Only Case)
    • Market + Policies Financial Assumptions Case (Market + Policies Case).
  • Though the tax rate has been updated to include the changes in corporate taxes in the Market + Policies Case, the federal/state blended tax rate is not assumed to vary by technology in our calculations; in practice, depreciation schedules vary by technology based on the tax code.

References

The following references are specific to this page; for all references in this ATB, see References.

Ho, Jonathan, Jonathon Becker, Maxwell Brown, Patrick Brown, Ilya (ORCID:0000000284917814) Chernyakhovskiy, Stuart Cohen, Wesley (ORCID:000000029194065X) Cole, et al. “Regional Energy Deployment System (ReEDS) Model Documentation: Version 2020.” Golden, CO: National Renewable Energy Laboratory, June 9, 2021. https://doi.org/10.2172/1788425.

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