# 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.

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

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

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.