Energy Savings and Payback: Method

Change-Point Models, Savings Uncertainty and Measure Economics

Why bills are normalized

A building's monthly energy use rises in cold or hot months and settles to a base load in between, so a bill before a retrofit cannot be compared with a bill after it unless the weather is taken out. Whole-building measurement and verification (IPMVP Option C, FEMP M&V Guidelines 4.0 §4.3) fits a regression model to the baseline bills against a weather variable, applies that model to the weather of the post-retrofit months to get the baseline adjusted to the new conditions, and reports the avoided energy as the adjusted baseline less the metered use. The same model, evaluated at a typical year's degree days, gives the normalized annual consumption that lets one year or one building be compared with another.

Degree days and the balance point

Heating degree days of a period count how far, and for how long, the outdoor temperature fell below a base: HDD = Σ (T_base − T)⁺ over the days, with (·)⁺ zero when negative; cooling degree days count the excess above the base. The conventional base is 65 °F (18.3 °C), the outdoor temperature at which an average house needs neither heating nor cooling, but a building's true balance point depends on its internal gains, insulation and thermostat setting. Utilities and weather services report degree days summed from daily means; when only a period's mean temperature is known the tool uses days × (T_base − T̄)⁺, which understates the true HDD in mild months whose cold days are averaged with warm ones. The BPA guide notes this and prefers true degree days where a utility supplies them.

The Inverse Modeling Toolkit change-point models

ASHRAE research project RP-1050 produced the Inverse Modeling Toolkit (Kissock, Haberl and Claridge 2003), whose piecewise-linear models of use against outdoor temperature T have become the standard baseline forms of Guideline 14. The number of regression parameters names the model:

• 2P: y = β₁ + β₂ T, a straight line, for use that varies with temperature over the whole range (or hardly at all).
• 3PH: y = β₁ + β₂ (β₃ − T)⁺, the heating model: flat at the base load β₁ above the change point β₃ and rising with slope β₂ below it. Gas or oil heating in a building without cooling.
• 3PC: y = β₁ + β₂ (T − β₃)⁺, the cooling model: flat below the change point, rising above it. Electricity in a building with air conditioning and no electric heat.
• 4P: y = β₁ + β₂ (β₄ − T)⁺ + β₃ (T − β₄)⁺, two slopes meeting at one change point; variable-air-volume systems, buildings with large latent loads, economizer and reset schedules.
• 5P: y = β₁ + β₂ (β₄ − T)⁺ + β₃ (T − β₅)⁺, heating below one change point, cooling above another, a flat base load between them; whole-building electricity with heat pumps or electric heat and chillers.

The tool writes the heating slope as a positive number of use per degree below the change point, as the IMT does, so a 4P model's two coefficients are both positive in the usual case. The change points are not linear parameters, so the IMT finds them by a grid search: the temperature range is divided into ten steps, the linear coefficients are regressed at each candidate change point, the one with the least sum of squared residuals is kept, and the search is repeated on a finer grid about it. The tool runs the coarse grid and two refinements, so the change point is known to within a hundredth of the range; the IMT reports twice the finest step as that uncertainty, and the tool does the same. A candidate is admitted only when at least two data points lie on each sloped side of it, which keeps every slope estimable. When the model is fitted to monthly bills of unequal length, the use and the degree days are first divided by the days of the period, so that every point is a daily average; this is the practice of the IMT and of the BPA guide, and the normalized annual consumption then multiplies the base load by 365 days.

Degree-day regression

When the bills come with degree days at a known base, the model y = β₁ + β_H HDD + β_C CDD (per day) is the classical form of Fels' PRISM and of most utility programs; a building with only heating or only cooling uses the one term. It is the 3P model with the base fixed in advance rather than fitted, and the tool offers it with HDD alone, CDD alone and both.

Goodness of fit and the Guideline 14 criteria

With n points and p parameters, the root mean square error is RMSE = √(Σ(y − ŷ)²/(n − p)) and the coefficient of variation CV-RMSE = RMSE/ȳ; the net mean bias error NMBE = Σ(y − ŷ)/((n − p) ȳ) is zero for any least-squares fit that carries a constant. R² = 1 − SSE/SST measures the share of the variation explained and the adjusted R² = 1 − (1 − R²)(n − 1)/(n − p) charges for the extra parameters, as the RMSE itself does through n − p. ASHRAE Guideline 14 accepts a monthly baseline model when CV-RMSE ≤ 15 % and |NMBE| ≤ 5 % (30 % and 10 % for hourly data); FEMP M&V Guidelines 4.0 Table 4-2 adopts the same values. The tool ranks the fitted models by CV-RMSE, marks any slope whose t statistic is below 2 as weak (the dependence it describes is not established by the data), and recommends the lowest-CV model whose slopes are all significant; the analyst may choose another model from the list, and should prefer the simplest model that fits, since a 5P model fitted to twelve points has seven degrees of freedom left and a change point placed by a handful of months.

Avoided energy and its uncertainty

The adjusted baseline of a post-retrofit period is the model at that period's weather (times its days), and the saving is the adjusted baseline less the metered use; the fractional savings F is the total saved divided by the total adjusted baseline. The saving inherits the baseline model's scatter. Guideline 14 Annex B gives, following Reddy and Claridge (2000), the fractional savings uncertainty of m post-retrofit periods predicted by a baseline of n points and p parameters,

U = 1.26 · t · CV(RMSE) · √((n + 2)/(n m)) / F,

where t is the two-sided Student statistic at the chosen confidence level with n − p degrees of freedom. The factor √((n + 2)/(n m)) is the root-mean-square prediction error of a regression averaged over m new points, and the 1.26 is an empirical correction for monthly data that Reddy and Claridge found reproduces the exact matrix expression within 10 %; the 2014 and 2023 editions replace it by a polynomial in the effective number of independent points when daily or hourly data are autocorrelated. The tool reports U at 68 % (Guideline 14's compliance level: the uncertainty must be below 50 % of the annual savings) and at 90 %, and beside it the IMT's own prediction-band form, in which each period's prediction carries t · RMSE · √(1 + 2/n) and the m periods add in quadrature (Kissock, Haberl and Claridge 2003, equations 25 to 28). The IMT form has no 1.26 and treats the periods as independent, so it is the smaller figure; the two bracket what the model can claim. Neither includes meter error, sampling or the uncertainty of stipulated quantities, which FEMP adds in quadrature when a project needs a total.

Normalized annual consumption

NAC is the model evaluated over a typical year: 365 days of the base load plus the heating slope times the typical-year HDD and the cooling slope times the CDD. The degree days must be counted at the model's own base: the change point of a 3P or 4P model, the entered base of a degree-day model. Published TMY degree days are at 65 °F; when the fitted change point is far from it, degree days at the change point should be recomputed from the typical-year daily temperatures, and the tool says which base it used. For a 2P model, which has no base, the tool uses the annual mean temperature implied by the degree days, T̄ = T_base − (HDD − CDD)/365.

Measure economics

A conservation measure costs C now and saves energy every year of its life n. Its first-year saving is the energy saved times the prices plus any other saving (maintenance, water). Later years escalate: NIST Handbook 135, the life-cycle costing manual of the Federal Energy Management Program, discounts an amount A₀ growing at a constant rate e with the modified uniform present value factor UPV* = Σ ((1 + e)/(1 + d))ᵗ = (1 + e)/(d − e) · [1 − ((1 + e)/(1 + d))ⁿ], so that the present value of the savings is PV = A_elec UPV*(e_elec) + A_gas UPV*(e_gas) + A_other UPV*(e_other), end-of-year convention. The measures of worth are then:

• Simple payback SPB = C / A₁, years of first-year savings to repay the cost; it ignores escalation, discounting and life, and Handbook 135 allows it only as a screen.
• Savings-to-investment ratio SIR = PV / C (Handbook 135 equation 6-4); a measure is cost-effective when SIR exceeds 1, and independent measures competing for a budget are ranked by SIR (§7.5).
• Net present value NPV = PV − C, the net savings of Handbook 135; positive when SIR exceeds 1.
• Internal rate of return, the discount rate at which NPV is zero, found by bisection on the year-by-year cash flows; it is undefined when the savings never repay the cost.
• Discounted payback, the year in which the cumulative discounted savings first exceed the cost (Handbook 135 §6.4, Table 6-3), interpolated within that year; "beyond life" when the cost is never repaid.

Prices, escalation and the discount rate must be in the same terms: FEMP analyses use a real discount rate (3 % in 2018 to 2020) with real escalation rates from the Handbook 135 annual supplement; a nominal analysis uses a nominal discount rate and nominal escalation. CO2 avoided is the electricity saved times an emission rate in lb/MWh (the eGRID total output rate of the site's subregion; 823.1 lb/MWh is the 2022 national value) plus the gas saved at 11.7 lb per therm (EPA's 0.0053 metric tons per therm), given in short and metric tons.

Quick calculators

Lighting: a retrofit that changes the fixture input power from W_old to W_new on N fixtures saves N (W_old − W_new) × hours / 1000 kWh a year, FEMP's Option A for lighting (fixture power measured, hours metered or stipulated), and reduces the billing demand by the connected-load reduction times its coincidence with the building's peak, charged every month of the year. Interactive effects on heating and cooling are not included. Motors: a replacement motor of higher efficiency saves hp × load × 0.746 kW/hp × hours × (100/η_old − 100/η_new) kWh a year, the efficiencies in percent at the operating load (DOE Motor Systems Tip Sheet #1; the DOE MotorMaster+ method); the saving depends on the efficiency at the actual load, not the nameplate value, and the DOE fact sheet on motor load describes how to measure it.

Limits

The models take one driver, temperature or degree days; the IMT's multivariable forms (occupancy, production) are not offered, nor are daily or hourly data with their autocorrelation corrections. A 5P model on twelve months is rarely justified. The uncertainty formulas cover the baseline model's scatter alone. The degree days computed from a period's mean temperature are an approximation. Money has no unit; the economics are in whatever currency the costs and prices share, and no taxes, rebates, replacement costs or salvage values are taken.