# 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 Variable | Value | Definition/Description |
---|---|---|

Construction duration (C) | Varies by technology | Number of years in construction period |

Capacity factor (CF) | Varies by technology | Generally, the ratio of actual annual output to output at rated capacity for an entire year |

Charging Costs | For 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 technology | The 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 technology | Fraction 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 technology | Difference between the expected return on a market portfolio and the risk-free rate |

Capital fraction (FC) | Varies by technology | Fraction of capital spent in each year of construction, 1 to C |

Depreciation fraction (FD) | Varies by technology | Fraction 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 |

Fuel | Heat 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) | 1 | Multiplier to account for regional variation in capital costs; not used in the ATB |

Rate of return on equity (RROE) | Varies by technology | Assumed rate of return on the share of assets financed with equity |

Economic lifetime (t) or cost recovery period | 30 years (default); 20 years and technical life also available | Length 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.