An interpolation scheme for Phase-I DHP calibration
Josh, Christian, Ben, Adam
May 2014
This note describes a simple method for obtaining calibration adjustments for ductless heat pumps
based on metering data collected in the NEEA/Ecotope 2013 DHP study. The main focus is on “Phase I”
adjustments, which refer to total heating energy in homes whose billing data shows reasonably strong
heating energy signatures. Phase II adjustments related to supplemental heat and weak signatures will
be discussed separately.
In December of 2013, the RTF approved updated Phase I adjustments for single-family SEEM calibration
based on RBSA data. These adjustments depend on heating zone, envelope insulation and tightness,
and primary heating equipment (electric resistance versus gas or heat pump). Since DHPs did not exist
in high numbers in the RBSA dataset, we derive DHP calibration adjustments from the DHP metering
study. Since this analysis is based on only 95 sub-metered homes in the DHP metering study, we need
to ensure that sample idiosyncrasies do not wield undue influence on our calibration adjustments. Prior
to installing DHPs, the sampled homes all had zonal electric resistance heat, so the previously-approved
SF SEEM calibration tells us how to obtain calibrated heating energy estimates for our sample in the preDHP period. Because of this, we can define the present objective as follows: We seek DHP calibration
factors that, on average, yield
(DHP factor)×(SEEM.DHP.69 kWh)
(Elec-resistance factor)×(SEEM.FUR.69 kWh)
≈
Post-DHP VBDD kWh
Pre-DHP VBDD kWh
Here, SEEM.DHP.69 refers to SEEM’s output when equipment type is set to DHP1, DHP2, or DHP3 as
appropriate, and the thermostat input is set at 69⁰F (day) and 64⁰F (night). SEEM.FUR.69 is the output
with the same thermostat settings but equipment type set to represent zonal electric. Other SEEM
inputs are site-specific in both cases. The VBDD estimates are derived from actual billing data using a
variable-base degree-day algorithm.
Since the 95 study homes all had electric-resistance heat prior to installing DHPs, the numerator on the
right-hand-side of the expression is the actual calibrated-SEEM kWh estimate under the pre-DHP
conditions. The expression is therefore saying that we want our DHP calibration factors to yield
estimates that relate to billing data in about the same way as the previously-approved calibration.
The current calibration depends on climate zone, equipment type, and envelope quality. Ideally, our
DHP calibration should have a similar form—it should depend on non-equipment site parameters in the
same way as the existing calibration. To accomplish this, we seek to formulate DHP calibration factors
that interpolate between electric-resistance calibration factors and gas/HP calibration factors. For
example, a home in heating zone 1 that is moderately well-insulated (say Uo = 0.15) has a Phase-I
calibration of 0.77 if it has electric-resistance heat and 0.99 if it has gas heat or a heat pump. If this
home has a DHP, its calibration factor should fall somewhere between 0.77 and 0.99. (At this point in
the analysis, we don’t really know whether it will fall between electric-resistance and heat pump, but
the previous DHP calibration effort resulted in a thermostat setting that fell between the two, so we’re
betting this DHP adjustment factor will fall between, too.)
Mathematically, we seek an interpolation constant 𝜃̂ which we can use to calculate DHP calibration
factors according to this expression:
DHP factor = (𝜃̂) × (Gas/HP factor) + (1 − 𝜃̂) × (Elec-res. factor)
Using more compact notation, we write this as
𝑓DHP = (𝜃̂) × 𝑓G/HP + (1 − 𝜃̂) × 𝑓E.Res
Our intention is that 𝜃̂ should be a single fixed value that applies to all DHP homes; the DHP calibration
factors will depend on individual site characteristics only through 𝑓G/HP and 𝑓E.Res (which vary with
heating zone and envelope quality). To understand the interpolation expression, note that (𝜃̂) ×
𝑓G/HP + (1 − 𝜃̂) × 𝑓E.Res is a linear combination of the factors 𝑓G/HP and 𝑓E.Res. The expression would be
equal to 𝑓G/HP if 𝜃̂ was one, and it would equal 𝑓E.Res if 𝜃̂ was zero; it would be half-way between the
two 𝜃̂ was one-half.
We now restate our basic objective in terms of the interpolation expression: We seek 𝜃̂ that, on
average, yields
((𝜃̂) × 𝑓G/HP + (1 − 𝜃̂) × 𝑓E.Res ) ×
SEEM.DHP.69
SEEM.FUR.69
≈ 𝑓E.Res ×
Post-VBDD
Pre-VBDD
One can imagine several reasonable versions of 𝜃̂—with a little algebra, we could set up the estimation
in terms of a regression, a ratio estimator, a mean, or something else.
A. Average of interpolations
The tab in the DHP_SEEM_calibration workbook called CalibrationCalcs derives 𝜃̂ as the mean of 95 sitespecific interpolation values. The steps are as follows:
1. For 𝑖 = 1, 2, … , 95, let 𝜃𝑖 be the value that yields equality in the interpolation expression with
the data from site 𝑖. In other words,
((𝜃𝑖 ) × 𝑓G/HP, 𝑖 + (1 − 𝜃𝑖 ) × 𝑓E.Res, 𝑖 ) ×
With a little algebra this becomes
SEEM.DHP.69𝑖
Post-VBDD𝑖
= 𝑓E.Res, 𝑖 ×
SEEM.FUR.69𝑖
Pre-VBDD𝑖
(𝜃𝑖 ) × (𝑓G/HP, 𝑖 − 𝑓E.Res, 𝑖 ) + 𝑓E.Res, 𝑖 = 𝑓E.Res, 𝑖 ×
SEEM.FUR.69𝑖 Post-VBDD𝑖
×
SEEM.DHP.69𝑖
Pre-VBDD𝑖
Or, equivalently,
(
𝜃𝑖 =
𝑓E.Res, 𝑖 × SEEM.FUR.69𝑖
) × Post-VBDD𝑖 − 𝑓E.Res, 𝑖 ×SEEM.DHP.69𝑖
Pre-VBDD𝑖
𝑓G/HP, 𝑖 × SEEM.DHP.69𝑖 − 𝑓E.Res, 𝑖 × SEEM.DHP.69𝑖
2. Set 𝜃̂ equal to the mean of the 𝜃𝑖 .
The result is 𝜃̂ ≈ 0.60, which says that DHP calibration factors should be about half-way between the
factors for heat pumps and those for electric-resistance (but a little closer to the heat pump values). This
value is the mean of the 𝜃𝑖 for 78 study sites that passed filters similar to those used in the original
Phase I calibration.
B. Interpolation of averages
The tab called CalibrationCalcs also attempts to derive 𝜃̂ by interpolating averages (the version above
did the reverse – it averaged interpolations). In other words, we take averages first, and then identify
the 𝜃 value that interpolates the averages.
Remember, we want to set up a DHP calibration that yields this:
DHP - Calibrated SEEM.DHP.69 kWh
ER - Calibrated SEEM.FUR.69 kWh
≈
Post-DHP VBDD kWh
Pre-DHP VBDD kWh
As before, we are shooting for DHP calibration factors that interpolate between our existing Gas/HP and
electric-resistance factors. To set this up in terms of averages, we need a slightly different notation
than used in the previous scheme. To talk about calibrated-SEEM averages, our notation must
distinguish between the equipment type used in the SEEM simulations (FUR in the pre-cases versus DHP
in the post-cases) and the equipment type used to derive calibration adjustment factors (electric
resistance versus HP). For this, we use:
95
ER-Cal.FUR.69
=
∑
𝑖=1
95
ER-Cal.DHP.69
=
∑
𝑖=1
95
HP-Cal.DHP.69 =
∑
𝑖=1
𝑓E.Res, 𝑖 × SEEM.FUR.69𝑖
95
𝑓E.Res, 𝑖 × SEEM.DHP.69𝑖
95
𝑓E.HP, 𝑖 × SEEM.DHP.69𝑖
95
In addition, we will use VBDD.pre and VBDD.post to refer to the mean of the billing-data estimates in
the pre and post cases, respectively. In this notation, our objective is to derive theta with
(𝜃̂) × HP-Cal.DHP.69 + (1 − 𝜃̂) × ER-Cal.DHP.69
VBDD.post
=
ER-Cal.FUR.69
VBDD.pre
With a little algebra this becomes
(𝜃̂) × (HP-Cal.DHP.69 − ER-Cal.DHP.69) + ER-Cal.DHP.69
=
ER-Cal.FUR.69 ×
VBDD.post
VBDD.pre
Or, equivalently,
𝜃̂ =
ER-Cal.FUR.69
( VBDD.pre ) × VBDD.post − ER-Cal.DHP.69
HP-Cal.DHP.69 − ER-Cal.DHP.69
The result is 𝜃̂ ≈ 0.55, which says that DHP calibration factors should be about half-way between the
factors for heat pumps and those for electric-resistance (a little closer to the heat pump values). As
before, the sample is limited to the 78 sites that passes filters similar to those used in the original Phase
I calibration.
Average of Interpolations or Interpolation of Averages?
Confucius says, “Man with one watch always know exactly what time it is…man with two watches never
quite sure.”
We pursued two distinct interpolation approaches to support our QC checks and verify robustness (if
they yielded wildly different answers we would know we have a problem). From a modelling
standpoint, we see no clear reason to prefer one method over the other—both are reasonably welljustified.
To choose a method, and to verify the calibration aligns with the previous T-stat based calibration, we
re-ran the existing RTF-approved DHP savings with both versions of 𝜃. The results are shown in the
following table:
Category Name
Measure
Name
noscr_noscr_HZ1CZ1
heat
noscr_noscr_HZ2CZ1
heat
noscr_noscr_HZ3CZ1
heat
noscr_noscr_HZ1CZ2
heat
noscr_noscr_HZ2CZ2
heat
noscr_noscr_HZ3CZ2
heat
noscr_noscr_HZ1CZ3
heat
noscr_noscr_HZ2CZ3
heat
noscr_noscr_HZ3CZ3
heat
noscr_noscr_HZ1CZ1
cool
noscr_noscr_HZ2CZ1
cool
noscr_noscr_HZ3CZ1
cool
noscr_noscr_HZ1CZ2
cool
noscr_noscr_HZ2CZ2
cool
noscr_noscr_HZ3CZ2
cool
noscr_noscr_HZ1CZ3
cool
noscr_noscr_HZ2CZ3
cool
noscr_noscr_HZ3CZ3
cool
MLBscr_Pass_HZ3CZ1
heat
MLBscr_Pass_HZ3CZ2
heat
MLBscr_Pass_HZ3CZ3
heat
MLBscr_Fail_HZ3CZ1
heat
MLBscr_Fail_HZ3CZ2
heat
MLBscr_Fail_HZ3CZ3
heat
MLBscr_Pass_HZ3CZ1
cool
MLBscr_Pass_HZ3CZ2
cool
MLBscr_Pass_HZ3CZ3
cool
Original
2,719
2,604
311
2,719
2,604
311
2,719
2,604
311
-19
-19
-19
142
142
142
412
412
412
1,339
1,339
1,339
157
157
157
-19
142
412
Savings
THETA
=0.55
2,593
2,565
337
2,593
2,565
337
2,593
2,565
337
-19
-19
-19
142
142
142
412
412
412
1,449
1,449
1,449
169
169
169
-19
142
412
% Difference from Original
THETA
=0.6
2,528
2,479
320
2,528
2,479
320
2,528
2,479
320
-19
-19
-19
142
142
142
412
412
412
1,376
1,376
1,376
161
161
161
-19
142
412
THETA =0.55
-5%
-1%
8%
-5%
-1%
8%
-5%
-1%
8%
0%
0%
0%
0%
0%
0%
0%
0%
0%
8%
8%
8%
8%
8%
8%
0%
0%
0%
THETA =0.6
-7%
-5%
3%
-7%
-5%
3%
-7%
-5%
3%
0%
0%
0%
0%
0%
0%
0%
0%
0%
3%
3%
3%
3%
3%
3%
0%
0%
0%
MLBscr_Fail_HZ3CZ1
cool
MLBscr_Fail_HZ3CZ2
cool
MLBscr_Fail_HZ3CZ3
cool
-19
142
412
-19
142
412
-19
142
412
0%
0%
0%
Since both methods are equally valid, and since 𝜃 = 0.55 (Interpolation of Averages) gives us savings
values closer to the original DHP savings values (except for in Heating Zone 3), we will choose that
method.
0%
0%
0%