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‎docs/formulas.md‎

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@@ -33,25 +33,122 @@ title: Formulas - NPA How to Pay Model
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toc-title: Table of contents
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---
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# Mathematical Formulas
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This document contains the mathematical formulas implemented in the
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This document contains the formulas implemented in the
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`npa-howtopay` model, organized by functional area.
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## Variable Definitions
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### Customer and Conversion Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $N_{gas}(t)$ | Number of gas customers in year $t$ | customers |
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| $N_{gas,init}$ | Initial number of gas customers | customers |
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| $N_{electric}(t)$ | Number of electric customers in year $t$ | customers |
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| $N_{electric,init}$ | Initial number of electric customers | customers |
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| $N_{converts}(t)$ | Cumulative number of converted households including non-npa households through year $t$ | customers |
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| $N_{NPAconverts}(t)$ | Sum of NPA households converters in year $t$ (excludes scattershot electrification) | customers |
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| $N_{customers}(t)$ | Total number of customers in year $t$ | customers |
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### Energy Usage and Efficiency Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $U_{gas}(t)$ | Total gas usage in therms in year $t$ | therms |
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| $U_{electric}(t)$ | Total electric usage in kWh in year $t$ | kWh |
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| $Q_{heating,therms}$ | Average per-customer heating need in therms | therms/customer |
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| $Q_{electric,kWh}$ | Average per-customer electric need in kWh | kWh/customer |
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| $K_{therm\to kWh}$ | Conversion factor from therms to kWh | kWh/therm |
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| $\eta_{HP}$ | Heat pump efficiency | dimensionless |
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### Gas System Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $C_{gas,var}(t)$ | Gas variable costs in year $t$ | $ |
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| $C_{gas,fixed}(t)$ | Gas fixed costs in year $t$ | $ |
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| $C_{gas,overhead}(t)$ | Gas fixed overhead costs in year $t$ | $ |
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| $C_{gas,maintenance}(t)$ | Gas maintenance costs in year $t$ | $ |
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| $C_{gas,NPA,opex}(t)$ | Gas NPA OPEX in year $t$ | $ |
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| $C_{gas,opex}(t)$ | Gas OPEX costs in year $t$ | $ |
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| $R_{gas}(t)$ | Gas revenue requirement in year $t$ | $ |
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| $RB_{gas}(t)$ | Gas ratebase in year $t$ | $ |
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| $ror_{gas}$ | Gas rate of return | % |
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| $D_{gas}(t)$ | Gas depreciation expense in year $t$ | $ |
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| $P_{gas}(t)$ | Gas generation cost per therm in year $t$ | $/therm |
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### Electric System Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $C_{electric,var}(t)$ | Electric variable costs in year $t$ | $ |
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| $C_{electric,fixed}(t)$ | Electric fixed costs in year $t$ | $ |
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| $C_{electric,overhead}(t)$ | Electric fixed overhead costs in year $t$ | $ |
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| $C_{electric,maintenance}(t)$ | Electric maintenance costs in year $t$ | $ |
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| $C_{electric,NPA,opex}(t)$ | Electric NPA OPEX in year $t$ | $ |
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| $C_{electric,opex}(t)$ | Electric OPEX costs in year $t$ | $ |
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| $R_{electric}(t)$ | Electric revenue requirement in year $t$ | $ |
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| $RB_{electric}(t)$ | Electric ratebase in year $t$ | $ |
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| $ror_{electric}$ | Electric rate of return | % |
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| $D_{electric}(t)$ | Electric depreciation expense in year $t$ | $ |
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| $P_{electric}(t)$ | Electricity generation cost per kWh in year $t$ | $/kWh |
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| $P_{peak,kw}$| Distribution cost per peak kW increase | $/kWh |
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### Revenue and Inflation Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $R(t)$ | Revenue requirement in year $t$ | $ |
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| $R_{adj}(t)$ | Inflation-adjusted revenue requirement in year $t$ | $ |
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| $R_{electric,adj}(t)$ | Inflation-adjusted electric revenue requirement in year $t$ | $ |
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| $R_{gas,adj}(t)$ | Inflation-adjusted gas revenue requirement in year $t$ | $ |
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| $d$ | Discount rate | % |
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| $t_0$ | Start year | year |
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### Tariff and Billing Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $C_{electric,fixed,per\_customer}$ | Electric fixed charge per customer (user input) | $/customer |
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| $C_{electric,var,kWh}(t)$ | Electric volumetric tariff per kWh in year $t$ | $/kWh |
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| $C_{gas,fixed,per\_customer}$ | Gas fixed charge per customer (user input) | $/customer |
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| $C_{gas,var,therm}(t)$ | Gas volumetric tariff per therm in year $t$ | $/therm |
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| $B_{per\_customer}(t)$ | Bill per customer in year $t$ | $/customer |
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| $B_{electric,converts}(t)$ | Electric bill per customer for converts in year $t$ | $/customer |
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| $B_{electric,nonconverts}(t)$ | Electric bill per customer for non-converts in year $t$ | $/customer |
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| $B_{total,converts}(t)$ | Total bill per customer for converts in year $t$ | $/customer |
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| $B_{total,nonconverts}(t)$ | Total bill per customer for non-converts in year $t$ | $/customer |
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| $B_{gas,nonconverts}(t)$ | Gas bill per customer for non-converts in year $t$ | $/customer |
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### Depreciation and Project Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $f_{dep}(t, t_p, L)$ | Depreciation fraction for project $p$ in year $t$ | dimensionless |
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| $RB(t)$ | Total ratebase in year $t$ | $ |
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| $D(t)$ | Total depreciation expense in year $t$ | $ |
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| $C_{maintenance}(t)$ | Total maintenance costs in year $t$ | $ |
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| $t_p$ | Project year for project $p$ | year |
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| $L_p$ | Depreciation lifetime for project $p$ | years |
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| $C_p$ | Original cost of project $p$ | $ |
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| $C_{p_{non-NPA}}$ | Original cost of non-NPA project $p$ | $ |
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| $m_{pct}$ | Maintenance cost percentage | % |
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### Synthetic Initial Project Variables
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| Variable | Description | Units |
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|----------|-------------|-------|
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| $W_{total}$ | Total weight for synthetic initial projects | dimensionless |
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| $L$ | Depreciation lifetime | years |
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| $RB_{init}$ | Initial ratebase | $ |
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| $C_{est}$ | Estimated original cost per year | $/year |
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## Gas System Calculations
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### Gas Number of customers
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The number of gas customers decreases over time as customers convert to
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heat pumps. There is no growth in the number of gas customers:
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$$N_{gas}(t) = N_{gas,init} - \sum_{i=1}^{t} HP_{converts}(i)$$
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$$N_{gas}(t) = N_{gas,init} - \sum_{i=1}^{t} N_{converts}(i)$$
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**Variables:**
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- $N_{gas}(t)$: Number of gas customers in year $t$
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- $N_{gas,init}$: Initial number of gas customers
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- $HP_{converts}(t)$: Sum of NPA households and scattershot converters
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- $N_{converts}(t)$: Sum of NPA households and scattershot converters
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in year $t$.
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### Total Gas Usage
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**Variables:**
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- $C_{gas,opex}(t)$: Gas OPEX costs in year $t$
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- $C_{gas,fixed}(t)$: Gas fixed costs in year $t$
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- $C_{gas,var}(t)$: Gas variable costs in year $t$
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### Gas Avoided LPP Spending
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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:
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$$C_{gas,LPP,avoided}(t) = \sum_{p \in P(t)} N_{converts}(p) \times V_{pipe,user}(p)$$
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**Variables:**
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- $C_{gas,LPP,avoided}(t)$: Gas LPP spending avoided in year $t$
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- $P(t)$: Set of NPA projects in year $t$
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- $N_{converts}(p)$: Number of converts in project $p$
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- $V_{pipe,user}(p)$: Pipe value per user for project $p$
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### Gas Revenue Requirement
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- $R_{gas}(t)$: Gas revenue requirement in year $t$
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- $RB_{gas}(t)$: Gas ratebase in year $t$
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- $ror_{gas}$: Gas rate of return
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- $C_{gas,opex}(t)$: Gas OPEX costs in year $t$
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- $D_{gas}(t)$: Gas depreciation expense in year $t$
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## Electric System Calculations
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- $N_{electric}(t)$: Number of electric customers in year $t$
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- $N_{electric,init}$: Initial number of electric customers
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### Total Cumulative Converts
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The cumulative number of heat pump converts through year $t$. This does
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not affect the number of electric customers:
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$$N_{converts}(t) = \sum_{i=1}^{t} HP_{converts}(i)$$
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**Variables:**
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- $N_{converts}(t)$: Sum of NPA households and scattershot converters
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in year $t$.
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### Total Electric Usage
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- $U_{electric}(t)$: Total electric usage in kWh in year $t$
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- $N_{electric,init}$: Initial number of electric customers
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- $N_{converts}(t)$: Cumulative electrification converts through year $t$
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- $Q_{electric,kWh}$: Average per-customer electric need in kWh
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- $Q_{heating,therms}$: Average per-customer heating need in therms
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- $K_{therm\to kWh}$: Conversion factor from therms to kWh
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- $\eta_{HP}$: Heat pump efficiency
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**Variables:**
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- $C_{electric,var}(t)$: Electric variable costs in year $t$
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- $U_{electric}(t)$: Total electric usage in kWh in year $t$
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- $P_{electric}(t)$: Electricity generation cost per kWh in year $t$
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### Electric Fixed Costs
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**Variables:**
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- $C_{electric,opex}(t)$: Electric OPEX costs in year $t$
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- $C_{electric,fixed}(t)$: Electric fixed costs in year $t$
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- $C_{electric,var}(t)$: Electric variable costs in year $t$
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### Electric Revenue Requirement
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- $R_{electric}(t)$: Electric revenue requirement in year $t$
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- $RB_{electric}(t)$: Electric ratebase in year $t$
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- $ror_{electric}$: Electric rate of return
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- $C_{electric,opex}(t)$: Electric OPEX costs in year $t$
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- $D_{electric}(t)$: Electric depreciation expense in year $t$
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### Electric Grid Upgrades
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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:
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$$C_{grid,upgrade}(t) = \Delta kW_{peak}(t) \times P_{peak,kw}$$
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where the peak kW increase is:
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$$\Delta kW_{peak}(t) = \sum_{p \in P(t)} \left[ N_{NPAconverts}(p) \times \left( kW_{HP,peak} + (kW_{AC,peak} \times (1 - a_{pre})) \right) \right]$$
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**Variables:**
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- $C_{grid,upgrade}(t)$: Grid upgrade capital cost in year $t$
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- $\Delta kW_{peak}(t)$: Total peak kW increase in year $t$
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- $P_{peak,kw}$: Distribution cost per peak kW increase
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- $P(t)$: Set of NPA projects in year $t$
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- $N_{NPAconverts}(p)$: Number of converts in project $p$
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- $kW_{HP,peak}$: Peak kW per heat pump
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- $kW_{AC,peak}$: Peak kW per air conditioner
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- $a_{pre}$: Pre-NPA air conditioner adoption rate
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## Bill Cost Calculations
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### Inflation-Adjusted Revenue Requirement
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**Variables:**
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- $B_{per\_customer}(t)$: Bill per customer in year $t$
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- $R_{adj}(t)$: Inflation-adjusted revenue requirement in year $t$
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- $N_{customers}(t)$: Number of customers in year $t$
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### Electric Fixed Charge Per customer
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- $B_{electric,converts}(t)$: Electric bill per customer for converts
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in year $t$
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- $C_{electric,fixed,per\_customer}$: Electric fixed charge per customer (user input)
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- $C_{electric,var,kWh}(t)$: Electric volumetric tariff per kWh in year $t$
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- $Q_{electric,kWh}$: Average per-customer electric need in kWh
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- $Q_{heating,therms}$: Average per-customer heating need in therms
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- $K_{therm\to kWh}$: Conversion factor from therms to kWh
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- $\eta_{HP}$: Heat pump efficiency
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### Non-Converts Electric Bill Per customer
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- $B_{electric,nonconverts}(t)$: Electric bill per customer for
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non-converts in year $t$
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- $C_{electric,fixed,per\_customer}$: Electric fixed charge per customer (user input)
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- $C_{electric,var,kWh}(t)$: Electric volumetric tariff per kWh in year $t$
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- $Q_{electric,kWh}$: Average per-customer electric need in kWh
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### Converts Total Bill Per customer
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- $B_{total,converts}(t)$: Total bill per customer for converts in
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year $t$
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- $B_{electric,converts}(t)$: Electric bill per customer for converts in year $t$
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### Non-Converts Total Bill Per customer
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in year $t$
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- $B_{gas,nonconverts}(t)$: Gas bill per customer for non-converts in
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year $t$
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- $B_{electric,nonconverts}(t)$: Electric bill per customer for non-converts in year $t$
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## Depreciation and Inflation Calculations
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- $f_{dep}(t, t_p, L)$: Depreciation fraction for project $p$ in year
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$t$
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- $RB(t)$: Total ratebase in year $t$
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- $t$: Current year
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- $t_p$: Project year for project $p$
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- $L$: Depreciation lifetime
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- $L_p$: Depreciation lifetime for project $p$
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- $C_p$: Original cost of project $p$
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**Variables:**
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- $D(t)$: Total depreciation expense in year $t$
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- $t$: Current year
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- $C_p$: Original cost of project $p$
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- $L_p$: Depreciation lifetime for project $p$
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- $t_p$: Project year for project $p$
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### Maintenance Costs
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**Variables:**
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- $C_{maintenance}(t)$: Total maintenance costs in year $t$
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- $t$: Current year
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- $m_{pct}$: Maintenance cost percentage
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- $C_{p_{non-NPA}}$: Original cost of non-NPA project $p$
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- $t_p$: Project year for project $p$
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$$C_{est} = \frac{RB_{init}}{W_{total}}$$
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This creates a table of synthetic project with uniform original costs of that would results in the inital ratebase value in year 0.
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**Variables:**
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- $W_{total}$: Total weight for synthetic initial projects

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