Formulas
This document contains the formulas implemented in the
npa-howtopay model, organized by functional area.
Variable Definitions¶
Customer and Conversion Variables¶
| Variable | Description | Units |
|---|---|---|
| \(N_{gas}(t)\) | Number of gas customers in year \(t\) | customers |
| \(N_{gas}(0)\) | Initial number of gas customers | customers |
| \(N_{electric}(t)\) | Number of electric customers in year \(t\) | customers |
| \(N_{electric}(0)\) | Initial number of electric customers | customers |
| \(N_{converts}(t)\) | Cumulative number of converted households including non-npa households through year \(t\) | customers |
| \(N_{NPAconverts}(t)\) | Sum of NPA households converters in year \(t\) (excludes scattershot electrification) | customers |
| \(N_{customers}(t)\) | Total number of customers in year \(t\) | customers |
Energy Usage and Efficiency Variables¶
| Variable | Description | Units |
|---|---|---|
| \(U_{gas}(t)\) | Total gas usage in therms in year \(t\) | therms |
| \(U_{electric}(t)\) | Total electric usage in kWh in year \(t\) | kWh |
| \(Q_{heating,therms}\) | Average per-customer heating need in therms | therms/customer |
| \(Q_{electric,kWh}\) | Average per-customer electric need in kWh | kWh/customer |
| \(K_{therm\to kWh}\) | Conversion factor from therms to kWh | kWh/therm |
| \(\eta_{HP}\) | Heat pump efficiency | dimensionless |
Gas System Variables¶
| Variable | Description | Units |
|---|---|---|
| \(C_{gas,var}(t)\) | Gas variable costs in year \(t\) | $ |
| \(C_{gas,fixed}(t)\) | Gas fixed costs in year \(t\) | $ |
| \(C_{gas,overhead}(t)\) | Gas fixed overhead costs in year \(t\) | $ |
| \(C_{gas,maintenance}(t)\) | Gas maintenance costs in year \(t\) | $ |
| \(C_{gas,NPA,opex}(t)\) | Gas NPA OPEX in year \(t\) | $ |
| \(C_{gas,opex}(t)\) | Gas OPEX costs in year \(t\) | $ |
| \(R_{gas}(t)\) | Gas revenue requirement in year \(t\) | $ |
| \(RB_{gas}(t)\) | Gas ratebase in year \(t\) | $ |
| \(ror_{gas}\) | Gas rate of return | % |
| \(D_{gas}(t)\) | Gas depreciation expense in year \(t\) | $ |
| \(P_{gas}(t)\) | Gas generation cost per therm in year \(t\) | $/therm |
Electric System Variables¶
| Variable | Description | Units |
|---|---|---|
| \(C_{electric,var}(t)\) | Electric variable costs in year \(t\) | $ |
| \(C_{electric,fixed}(t)\) | Electric fixed costs in year \(t\) | $ |
| \(C_{electric,overhead}(t)\) | Electric fixed overhead costs in year \(t\) | $ |
| \(C_{electric,maintenance}(t)\) | Electric maintenance costs in year \(t\) | $ |
| \(C_{electric,NPA,opex}(t)\) | Electric NPA OPEX in year \(t\) | $ |
| \(C_{electric,opex}(t)\) | Electric OPEX costs in year \(t\) | $ |
| \(R_{electric}(t)\) | Electric revenue requirement in year \(t\) | $ |
| \(RB_{electric}(t)\) | Electric ratebase in year \(t\) | $ |
| \(ror_{electric}\) | Electric rate of return | % |
| \(D_{electric}(t)\) | Electric depreciation expense in year \(t\) | $ |
| \(P_{electric}(t)\) | Electricity generation cost per kWh in year \(t\) | $/kWh |
| \(P_{peak,kw}\) | Distribution cost per peak kW increase | $/kWh |
Revenue and Inflation Variables¶
| Variable | Description | Units |
|---|---|---|
| \(R(t)\) | Revenue requirement in year \(t\) | $ |
| \(R_{adj}(t)\) | Inflation-adjusted revenue requirement in year \(t\) | $ |
| \(R_{electric,adj}(t)\) | Inflation-adjusted electric revenue requirement in year \(t\) | $ |
| \(R_{gas,adj}(t)\) | Inflation-adjusted gas revenue requirement in year \(t\) | $ |
| \(d\) | Discount rate | % |
| \(t_0\) | Start year | year |
Tariff and Billing Variables¶
| Variable | Description | Units |
|---|---|---|
| \(C_{electric,fixed,per\_customer}\) | Electric fixed charge per customer (user input) | $/customer |
| \(C_{electric,var,kWh}(t)\) | Electric volumetric tariff per kWh in year \(t\) | $/kWh |
| \(C_{gas,fixed,per\_customer}\) | Gas fixed charge per customer (user input) | $/customer |
| \(C_{gas,var,therm}(t)\) | Gas volumetric tariff per therm in year \(t\) | $/therm |
| \(B_{per\_customer}(t)\) | Bill per customer in year \(t\) | $/customer |
| \(B_{electric,converts}(t)\) | Electric bill per customer for converts in year \(t\) | $/customer |
| \(B_{electric,nonconverts}(t)\) | Electric bill per customer for non-converts in year \(t\) | $/customer |
| \(B_{total,converts}(t)\) | Total bill per customer for converts in year \(t\) | $/customer |
| \(B_{total,nonconverts}(t)\) | Total bill per customer for non-converts in year \(t\) | $/customer |
| \(B_{gas,nonconverts}(t)\) | Gas bill per customer for non-converts in year \(t\) | $/customer |
Depreciation and Project Variables¶
| Variable | Description | Units |
|---|---|---|
| \(f_{dep}(t, t_p, L)\) | Depreciation fraction for project \(p\) in year \(t\) | dimensionless |
| \(RB(t)\) | Total ratebase in year \(t\) | $ |
| \(D(t)\) | Total depreciation expense in year \(t\) | $ |
| \(C_{maintenance}(t)\) | Total maintenance costs in year \(t\) | $ |
| \(t_p\) | Project year for project \(p\) | year |
| \(L_p\) | Depreciation lifetime for project \(p\) | years |
| \(C_p\) | Original cost of project \(p\) | $ |
| \(C_{p_{non-NPA}}\) | Original cost of non-NPA project \(p\) | $ |
| \(m_{pct}\) | Maintenance cost percentage | % |
Synthetic Initial Project Variables¶
| Variable | Description | Units |
|---|---|---|
| \(W_{total}\) | Total weight for synthetic initial projects | dimensionless |
| \(L\) | Depreciation lifetime | years |
| \(RB_{init}\) | Initial ratebase | $ |
| \(C_{est}\) | Estimated original cost per year | $/year |
Gas System Calculations¶
Gas Number of customers¶
The number of gas customers decreases over time as customers convert to heat pumps. There is no growth in the number of gas customers:
Variables:
- \(N_{gas}(t)\): Number of gas customers in year \(t\)
- \(N_{gas}(0)\): Initial number of gas customers
- \(N_{converts}(t)\): Sum of NPA households and scattershot converters in year \(t\).
Total Gas Usage¶
Total gas usage is calculated as the product of remaining gas customers and their heating needs:
Variables:
- \(U_{gas}(t)\): Total gas usage in therms in year \(t\)
- \(Q_{heating,therms}\): Average per-customer heating need in therms
Gas Variable Costs¶
Variable costs of the utility are based on total usage and generation costs:
Variables:
- \(C_{gas,var}(t)\): Gas variable costs in year \(t\)
- \(P_{gas}(t)\): Gas generation cost per therm in year \(t\)
Gas Fixed Costs¶
Fixed costs include overhead, maintenance, and NPA OPEX:
Variables:
- \(C_{gas,fixed}(t)\): Gas fixed costs in year \(t\)
- \(C_{gas,overhead}(t)\): Gas fixed overhead costs in year \(t\)
- \(C_{gas,maintenance}(t)\): Gas maintenance costs in year \(t\)
- \(C_{gas,NPA,opex}(t)\): Gas NPA OPEX in year \(t\)
Gas OPEX Costs¶
Total gas operating expenses combine fixed and variable costs:
Variables:
- \(C_{gas,opex}(t)\): Gas OPEX costs in year \(t\)
- \(C_{gas,fixed}(t)\): Gas fixed costs in year \(t\)
- \(C_{gas,var}(t)\): Gas variable costs in year \(t\)
Gas Avoided LPP Spending¶
The avoided LPP spending in a given year is the sum of the pipe value per customer times the number of converts for each NPA project in that year:
Variables:
- \(C_{gas,LPP,avoided}(t)\): Gas LPP spending avoided in year \(t\)
- \(P(t)\): Set of NPA projects in year \(t\)
- \(N_{converts}(p)\): Number of converts in project \(p\)
- \(V_{pipe,user}(p)\): Pipe value per user for project \(p\)
Gas Revenue Requirement¶
The gas revenue requirement includes ratebase return, OPEX, and depreciation. In the performance incentive scenario, it also includes performance incentives:
Variables:
- \(R_{gas}(t)\): Gas revenue requirement in year \(t\)
- \(RB_{gas}(t)\): Gas ratebase in year \(t\)
- \(ror_{gas}\): Gas rate of return
- \(C_{gas,opex}(t)\): Gas OPEX costs in year \(t\)
- \(D_{gas}(t)\): Gas depreciation expense in year \(t\)
- \(PI_{gas}(t)\): Gas performance incentive in year \(t\) (performance incentive scenario only)
Electric System Calculations¶
Electric Number of customers¶
The number of electric customers is the initial number of electric customers. There is no growth in the number of electric customers: \(\(N_{electric}(t) = N_{electric}(0)\)\)
Variables:
- \(N_{electric}(t)\): Number of electric customers in year \(t\)
- \(N_{electric}(0)\): Initial number of electric customers
Total Electric Usage¶
Electric usage includes both base electric needs and heating loads from converts:
Variables:
- \(U_{electric}(t)\): Total electric usage in kWh in year \(t\)
- \(N_{electric}(0)\): Initial number of electric customers
- \(N_{converts}(t)\): Cumulative electrification converts through year \(t\)
- \(Q_{electric,kWh}\): Average per-customer electric need in kWh
- \(Q_{heating,therms}\): Average per-customer heating need in therms
- \(K_{therm\to kWh}\): Conversion factor from therms to kWh
- \(\eta_{HP}\): Heat pump efficiency
Electric Variable Costs¶
Electric utility variable costs are based on total usage and generation costs:
Variables:
- \(C_{electric,var}(t)\): Electric variable costs in year \(t\)
- \(U_{electric}(t)\): Total electric usage in kWh in year \(t\)
- \(P_{electric}(t)\): Electricity generation cost per kWh in year \(t\)
Electric Fixed Costs¶
Electric fixed costs include overhead, maintenance, and NPA OPEX:
Variables:
- \(C_{electric,fixed}(t)\): Electric fixed costs in year \(t\)
- \(C_{electric,overhead}(t)\): Electric fixed overhead costs in year \(t\)
- \(C_{electric,maintenance}(t)\): Electric maintenance costs in year \(t\)
- \(C_{electric,NPA,opex}(t)\): Electric NPA OPEX in year \(t\)
Electric Operating Expenses Costs¶
Total electric operating expenses combine fixed and variable costs:
Variables:
- \(C_{electric,opex}(t)\): Electric OPEX costs in year \(t\)
- \(C_{electric,fixed}(t)\): Electric fixed costs in year \(t\)
- \(C_{electric,var}(t)\): Electric variable costs in year \(t\)
Electric Revenue Requirement¶
The electric revenue requirement includes ratebase return, OPEX, and depreciation:
Variables:
- \(R_{electric}(t)\): Electric revenue requirement in year \(t\)
- \(RB_{electric}(t)\): Electric ratebase in year \(t\)
- \(ror_{electric}\): Electric rate of return
- \(C_{electric,opex}(t)\): Electric OPEX costs in year \(t\)
- \(D_{electric}(t)\): Electric depreciation expense in year \(t\)
Electric Grid Upgrades¶
The cost of electric grid upgrades is determined by the peak kW increase from heat pumps and air conditioning, multiplied by a per-peak kW distribution cost:
where the peak kW increase is:
Variables:
- \(C_{grid,upgrade}(t)\): Grid upgrade capital cost in year \(t\)
- \(\Delta kW_{peak}(t)\): Total peak kW increase in year \(t\)
- \(P_{peak,kw}\): Distribution cost per peak kW increase
- \(P(t)\): Set of NPA projects in year \(t\)
- \(N_{NPAconverts}(p)\): Number of converts in project \(p\)
- \(kW_{HP,peak}\): Peak kW per heat pump
- \(kW_{AC,peak}\): Peak kW per air conditioner
- \(a_{pre}\): Pre-NPA air conditioner adoption rate
Bill Cost Calculations¶
Inflation-Adjusted Revenue Requirement¶
Revenue requirements are adjusted for inflation using the discount rate:
Variables:
- \(R_{adj}(t)\): Inflation-adjusted revenue requirement in year \(t\)
- \(R(t)\): Revenue requirement in year \(t\)
- \(d\): Discount rate
- \(t_0\): Start year
Bill Per customer¶
The basic bill per customer calculation averages the inflation-adjusted revenue requirement over the number of customers:
Variables:
- \(B_{per\_customer}(t)\): Bill per customer in year \(t\)
- \(R_{adj}(t)\): Inflation-adjusted revenue requirement in year \(t\)
- \(N_{customers}(t)\): Number of customers in year \(t\)
Electric Fixed Charge Per customer¶
The fixed charge of customers bills is a model input. We hold this constant over time:
- \(C_{electric,fixed,per\_customer}\): Electric fixed charge per customer in year \(t\) (user input)
Electric Customer Volumetric Tariff Per kWh¶
Electric volumetric tariffs are allocated per kWh on the total inflation-adjusted revenue requirement minus the fixed charge revenue:
Variables:
- \(C_{electric,var,kWh}(t)\): Electric volumetric tariff per kWh in year \(t\)
- \(U_{electric}(t)\): Total electric usage in kWh in year \(t\)
- \(R_{electric,adj}(t)\): Inflation-adjusted electric revenue requirement in year \(t\)
- \(C_{electric,fixed,per\_customer}\): Electric fixed charge per customer in year \(t\) (user input)
- \(N_{customers}(t)\): Number of customers in year \(t\)
Gas Fixed Charge Per customer¶
The fixed charge of customers bills is a model input. We hold this constant over time:
- \(C_{gas,fixed,per\_customer}\): Gas fixed charge per customer in year \(t\) (user input)
Gas Customer Volumetric Tariff Per Therm¶
Gas volumetric tariffs are allocated per therm on the total inflation-adjusted revenue requirement minus the fixed charge revenue:
Variables:
- \(C_{gas,var,therm}(t)\): Gas volumetric tariff per therm in year \(t\)
- \(R_{gas,adj}(t)\): Inflation-adjusted gas revenue requirement in year \(t\)
- \(C_{gas,fixed,per\_customer}(t)\): Gas fixed charge per customer in year \(t\) (user input)
- \(N_{customers}(t)\): Number of customers in year \(t\)
- \(U_{gas}(t)\): Total gas usage in therms in year \(t\)
Converts Electric Bill Per customer¶
Electric bills for converts include both base electric needs and heating loads:
Variables:
- \(B_{electric,converts}(t)\): Electric bill per customer for converts in year \(t\)
- \(C_{electric,fixed,per\_customer}\): Electric fixed charge per customer (user input)
- \(C_{electric,var,kWh}(t)\): Electric volumetric tariff per kWh in year \(t\)
- \(Q_{electric,kWh}\): Average per-customer electric need in kWh
- \(Q_{heating,therms}\): Average per-customer heating need in therms
- \(K_{therm\to kWh}\): Conversion factor from therms to kWh
- \(\eta_{HP}\): Heat pump efficiency
Non-Converts Electric Bill Per customer¶
Electric bills for non-converts include only base electric needs:
Variables:
- \(B_{electric,nonconverts}(t)\): Electric bill per customer for non-converts in year \(t\)
- \(C_{electric,fixed,per\_customer}\): Electric fixed charge per customer (user input)
- \(C_{electric,var,kWh}(t)\): Electric volumetric tariff per kWh in year \(t\)
- \(Q_{electric,kWh}\): Average per-customer electric need in kWh
Converts Total Bill Per customer¶
Total bills for converts include both gas and electric components:
Variables:
- \(B_{total,converts}(t)\): Total bill per customer for converts in year \(t\)
- \(B_{electric,converts}(t)\): Electric bill per customer for converts in year \(t\)
Non-Converts Total Bill Per customer¶
Total bills for non-converts include both gas and electric components:
Variables:
- \(B_{total,nonconverts}(t)\): Total bill per customer for non-converts in year \(t\)
- \(B_{gas,nonconverts}(t)\): Gas bill per customer for non-converts in year \(t\)
- \(B_{electric,nonconverts}(t)\): Electric bill per customer for non-converts in year \(t\)
Depreciation and Inflation Calculations¶
Net Present Value (NPV) Calculation¶
The NPV calculation evaluates utility investments by considering both the return on ratebase and depreciation recovery over the investment lifetime:
where:
- \(R_t = C_0 \times f_{dep}(t) \times ror\) (return on remaining ratebase value)
- \(D_t = \frac{C_0}{L}\) (annual depreciation recovery)
- \(f_{dep}(t) = \max(0, 1 - \frac{t}{L})\) (remaining value fraction)
Variables:
- \(NPV\): Net present value of the investment
- \(C_0\): Initial investment cost
- \(L\): Investment lifetime in years
- \(R_t\): Annual return on ratebase in year \(t\)
- \(D_t\): Annual depreciation recovery in year \(t\)
- \(ror\): Rate of return on ratebase
- \(r\): NPV discount rate
- \(f_{dep}(t)\): Depreciation fraction in year \(t\)
Ratebase Calculation¶
The depreciation fraction determines how much of a project's original cost remains in the ratebase:
The total ratebase is the sum of all projects' remaining values:
Variables:
- \(f_{dep}(t, t_p, L)\): Depreciation fraction for project \(p\) in year \(t\)
- \(RB(t)\): Total ratebase in year \(t\)
- \(t\): Current year
- \(t_p\): Project year for project \(p\)
- \(L\): Depreciation lifetime
- \(L_p\): Depreciation lifetime for project \(p\)
- \(C_p\): Original cost of project \(p\)
Depreciation Expense¶
Annual depreciation expense uses straight-line depreciation:
Variables:
- \(D(t)\): Total depreciation expense in year \(t\)
- \(t\): Current year
- \(C_p\): Original cost of project \(p\)
- \(L_p\): Depreciation lifetime for project \(p\)
- \(t_p\): Project year for project \(p\)
Maintenance Costs¶
Annual maintenance costs are calculated as a percentage of original project costs for non-NPA capex projects still in service:
Variables:
- \(C_{maintenance}(t)\): Total maintenance costs in year \(t\)
- \(t\): Current year
- \(m_{pct}\): Maintenance cost percentage
- \(C_{p_{non-NPA}}\): Original cost of non-NPA project \(p\)
- \(t_p\): Project year for project \(p\)
- \(L_p\): Depreciation lifetime for project \(p\)
Synthetic Initial Projects¶
For synthetic initial projects representing existing infrastructure, the total weight calculation is:
The estimated original cost per year is:
This creates a table of synthetic project with uniform original costs of that would results in the inital ratebase value in year 0. Variables:
- \(W_{total}\): Total weight for synthetic initial projects
- \(L\): Depreciation lifetime
- \(RB_{init}\): Initial ratebase
- \(C_{est}\): Estimated original cost per year
Performance Incentive Calculations¶
Performance Incentive Mechanism¶
The performance incentive mechanism models shared savings between utilities and ratepayers for NPA (Non-Pipe Alternative) projects. This mechanism treats NPA costs and avoided BAU (Business-As-Usual) costs differently:
NPA Costs Treatment: - NPA costs are treated as gas OpEx (operating expenses) - Collected in the year costs are incurred - No rate of return earned on NPA investments
BAU Costs Treatment: - Avoided BAU costs (what would have been spent replacing LPP) are treated as normal CapEx - User sets ROR and depreciation period equal to pipeline lifetime - Earns return on ratebase over depreciation lifetime
Cost Savings Calculation: Cost savings are calculated as the NPV difference between avoided BAU costs and NPA costs:
where: - \(NPV_{BAU}(t)\) = NPV of avoided LPP spending in year \(t\) - \(C_{gas,NPA,opex}(t)\) = NPA investment costs in year \(t\)
Performance Incentive Distribution: 30% of cost savings are collected from ratepayers, evenly split over a 10-year period:
Variables:
- \(Savings_{NPV}(t)\): NPV savings from NPA projects in year \(t\)
- \(NPV_{BAU}(t)\): NPV of avoided BAU costs in year \(t\)
- \(C_{gas,NPA,opex}(t)\): NPA OPEX in year \(t\)
- \(PI_{annual}(t)\): Annual performance incentive payment in year \(t\)
Performance Incentive Revenue Requirement Impact¶
In the performance incentive scenario, the gas revenue requirement includes the annual performance incentive payments:
where the summation includes performance incentive payments from all active savings projects (those initiated in years \(t-9\) through \(t\)).
Variables:
- \(R_{gas,PI}(t)\): Gas revenue requirement including performance incentives in year \(t\)
- \(R_{gas}(t)\): Standard gas revenue requirement in year \(t\)
- \(PI_{annual}(t-i)\): Annual performance incentive payment from savings project initiated in year \(t-i\)