Energy Efficient Buildings
The Degree-Day Method and Inverse Modeling
Introduction
The principles used in creating spreadsheet simulation models that calculate heating
and cooling energy use for each hour during a year can also be used in less
computationally-intensive methods of calculating annual energy use. The best of these
methods is called the degree-day method. In addition to being less computationally
intensive, the degree-day method supports inverse modeling, in which building
performance coefficients are extracted from actual energy use data. Inverse modeling is
one of the most powerful tools available for analyzing and improving the energy
efficiency of existing buildings. This chapter describes the degree-day method and
inverse modeling.
The Degree Day Method
The degree-day method uses the concept of overall heating coefficient and cooling
coefficients of a building to calculate annual heating and cooling energy use. To
illustrate how to build a degree-day model, consider my house.
Overall UA Values
The UA values for my house when I moved in for summer and winter were:
 Btu 
 Btu 
UA 
UA 

 hr  F 
 hr  F 
Win ter Frac Tot  Summer Frac Tot 
UA walls  A walls  R walls  1802  10.62 
UA win  A win  R win  221  1.5 
UA ceil  A ceil  R ceil  953  16.4 

UA inf  N V ρ c p  1.0  14005  .018 
UA ground  UA ground  2  293  2 
UA tot

170 22% 
147 19% 
58 7% 
252 32% 
147 19% 
 Btu 
774 
 hr  F 
170 33% 
147 28% 
58 11% 
0 0% 
147 28% 
 Btu 
522 
 hr  F 
The internal heat gains from people, electricity and solar gain through the windows
were:

 Btu 
Q people,sen  4  210  840 
 hr 
1

 kWh 
1 h
 Btu 
 Btu 
Q elec  5,500
 3,413

 2,143




 kWh  8,760  yr 
 hr 
 yr 

Q sol,win INov Mar   A winE IE  A winS IS  A winW IW  A winN IN  SHGC
 45  446   83  811  27  427  68  278 
.65
 Btu 
 3,191
24
 hr 

Q sol,sum  assume same from exterior shading




 Btu 
Q internal gains  Q people  Q elec  Q sol  840  2,143  3,191  6,174 
 hr 
Balance Temperature
A steady-state energy balance on the house during winter gives:




Toa


Q ig
Q i g  Q fur  Q UA  Q i g  Q fur  UA Tia  Toa   0

Tia
Q UA

Q fur
Next, define the “balance-point temperature” as the outside air temperature when the
house needs no heating. To calculate the balance temperature, set Qfur to zero and
replace Toa with Tbal. Doing so gives:



Q ig  Q fur  UA Tia  Tbal   0  Tbal  Tia 
Qi g
UA

Tbal, w  Tia, w
For my house:
Qig
6174

 72 
 64.0F
UA w
774

Tbal,s  Tia,s
Qig
6174

 76 
 64.2F
UA s
552
This means:
Whenever Toa < 64.0°F, my house needs heating
Whenever Toa > 64.2°F, my house needs cooling
Calculating Degree Days
In ancient times, average daily temperatures were readily available, but hourly
temperatures were not. Thus, it was common to use average daily temperatures to
estimate heating and cooling energy use. To estimate annual heating energy use, sum
the difference between the balance and average daily outdoor air temperatures for all
2
days during the year when Toa < Tbal. Doing so gives the heating degree days per year for
a given location and balance temperature.
HDDTbal    Tbal  Toa,i  " Heating Degree Days"
365

i1
For house in this example with Tbal,h = 64.0 F, and using average daily temperatures from
Daytdyus.dat, the annual heating degree days are:
 F  day 
HDD Tbal  64.0F  5,356

 yr 
To estimate cooling energy, calculate the cooling degree days by summing the
difference between the balance and average daily outdoor air temperatures for all days
during the year when Toa > Tbal.
CDDTbal    Toa,i  Tbal  " Cooling Degree Days"
365

i1
For the house in this example with Tbal,c = 64.2 F, and using average daily temperatures
from Daytdyus.dat, the annual cooling degree days are:
 F  day 
CDDTbal  64.2F  959

 yr 
Calculating Annual Furnace & Air Conditioning Energy Use
The annual heat supplied by the furnace and cooling energy removed from the house by
the air conditioner can be calculated using the overall UA value and heating and cooling
degree days.

 F  day 
 hr 
 MMBtu 
 Btu 
Q fur  UA w  HDD  774 
 5356
 24 
 99.5




 hr  F 
 yr 
 day 
 yr 

 F  day 
 hr 
 MMBtu 
 Btu 
Q AC,sen  UA s  CDD  552
 959 
 24 
 12.7




 hr  F 
 yr 
 day 
 yr 
Assuming the furnace is 80% efficient and the COP of the air conditioner is 3.0, the
annual furnace fuel and air conditioner electricity use are about:

Q ng, fur 
(UA) W
99.5  MMBtu 
 MMBtu 
 HDD 
 124 



EFUR
0.8  yr 
 yr 
3
(UA) S
12.7
 MMBtu 
 kWh 
 CDD 
 1,174 



E AC
3.0  3,412Btu/kWh   yr 
 yr 
Base 65-F Degree Days
In this example, both the heating and cooling balance temperatures were about 64 F.
This result is typical of many houses. So typical, that it is common to assume the
balance temperature is 65 F and calculate degree days based on this assumption. Base
65-F heating and cooling degree days for a given time period are often posted on the
internet and reported by weather services. The loss of accuracy from using base 65 F
degree days to calculate energy use depends on the difference between the actual
building balance temperature and the assumed 65 F balance temperature.

WE, AC 
Inverse Modeling
Now, take DD models of natural gas and electrical energy use and imagine what XY graphs of
natural gas versus outdoor air temp and electricity versus outdoor air temp should look like.
Elec
NG
UA,h
η fw
UA,c
η AC
NG,fur
Eac
Elec,lights
&plug
NG,hw
NG,hw
Elec,lights&plug
Tbal,h
Toa
Tbal,c
Toa
For my house, the actual graphs look like this.
4
Lonsdale Gas Use vs. Outdoor Air Temperature
With January gas use removed.
Lonsdale Electricity Use vs. Outdoor Air Temperature
With August electricity use removed from regression.
5
These graphs were created by:
Ascii data file of 13 months of ng and elec use from bills
AscII data file of actual avg daily temps from same period as utility bills from
www.engr.udayton.edu/weather
Combining data files in “Energy Explorer” Software
Modeling NG vs Temp with “3PH” model
Modeling Elec vs Temp with “3PC” model
Next, compare calculated coefficients and energy use to actual building coefficients and energy
use as derived from statistical analysis of the measured natural gas and electricity use.
Heating
Tbalance
Tbal,h (measured) = 63 F
Tbal,h (calculated) = 64 F
Measured and calculated are very close.
UA
(UA/Mfur )(meas) = 6.96(ccf/monthsF)
UA (meas) = 6.96 (ccf/mof) Mfur\
UA (meas) = 6.96 (ccf/mof) x .80 x 100,000(Btu/ccf) x (mo/(8760/12)hr)
UA (meas) = 763 (Btu/hrF)
UA (calculated) = 774(Btu/hrF)
Measured and calculated are very close.
Qng,fur
Qng, hotwater (meas) = 15.0 ccf/month
Qng, hw(meas) 15.0(ccf/month) x (12 months/yr) x (100,000 Btu/ccf)
Qng, hw (meas) = 18.0 mmBtu/yr
Qng,(meas) = 1,345 ccf/yr x 100,000(Btu/ccf) = 135 (mmBtu/yr)
Qng, hw (meas) = 18.0 (mmBtu/yr)
Qng, fur (meas) = Qng – Qng, hw = (135 – 18.0) (mmBtu/yr) = 117 (mmBtu/yr)
Qng, fur (calc) = 124 (mmBtu/yr)
Measured and calculated are very close, especially considering December data
not included in measured results
Cooling
Tbalance
Tbal, c (measured) = 61.5 F
Tbal, c (calculated) = 64.2 F
Measured and calculated are very close.
UA
6
UA/COP (meas) = 35.5 (kwh/monthF)
UA (meas) 35.5 (kwh/Mof)(3.0)3413 (Btu/kwh)(MO/(8760/12)hr)
UA (meas) 498 (Btu/hf)
UA (calc) = 522 (Btu/hf)
Measured and calculated are very close.
Welec, ac
Welec, other (meas) = 464 (kWh/mo) x 12 (mo/yr) = 5569kwh
Welec, tot (meas) = 6,082 kWh
Welec, ac (meas) = 6,082 – 5,568 = 514 (kWh/yr)
Welec, ac (calc) = 1,174 kWh/yr
Results consistent considering Welec, ac (meas) ≈850 (kWh/yr) if here for
July/Aug
This example shows the power of the degree-day modeling approach. Simple calculations of
UA, Tbal, HDD, CDD can accurately predict natural gas and electrical energy use.
Forward and Inverse Modeling
Traditional “forward” modeling is:
Estimated physical characteristics of building + Estimated weather Data  Estimated Energy use
But one can also do “inverse” modeling to estimate physical characteristics of building:
Estimated characteristics of building  Actual weather + Actual Energy Use
7
Simulated Natural Gas Use for House with
(Tset = 72 ) and (Tset = 72 F from 8 am to 10 pm and Tset = 60 F from 11 pm to 7 am)
Simulated Natural Gas Use for House with
(Inf = 0.8 ACH, Rwall = 4, Rceil = 16) and (Inf = 0.4 ACH, Rwall = 14 and Rceil = 26)
8
Simulated Natural Gas Use for House with
(Vhw = 5 gal/hr, Thw = 140 F) and (Vhw = 3 gal/hr, Thw = 120 F)
9