1、884 ASHRAE TransactionsABSTRACT This paper examines several methods for calculatingheating and cooling degree-days to any base by relying ontemperature statistics such as the monthly mean temperatureand the standard deviation of daily average temperature. It isfound that the method developed by Scho
2、enau and Kehrigworks best, and is usually able to estimate monthly heating andcooling degree-days to within 3C/day or 5.4F/day (in a root-mean-square error sense) of their true value. The methodclearly outperforms other widely used methods, such as the onederived by Erbs et al. (1983). The paper als
3、o presents an exten-sion of the Schoenau and Kehrig method to the case when onlymonthly mean temperatures are known.INTRODUCTIONFor a number of decades, heating and cooling degree-days, calculated to various bases, have been used extensivelyin a variety of fields of engineering and science. For exam
4、ple,heating degree-day base 18.3C (65F) is often used to esti-mate the heating energy required to heat buildings via aproportionality factor specific to each building (ASHRAE2005a). Similarly, a cooling degree-day base of 10C (50F) isoften used to correlate cooling energy requirements to thelocal cl
5、imate. In agriculture, degree-days are often referred toas “growing degree-days” and are used to correlate the devel-opment of crops, pests, or parasites to the local climate (Wiki-pedia 2010). The use of degree-days has somewhat diminished inrecent years, as the development of information technolog
6、ieshas made hourly, or even subhourly, temperature data widelyavailable to a larger audience. The use of sophisticated hourlymodels for energy monitoring or building simulation has nowbecome routine where degree-day-based methods were oftencalled upon. Nevertheless, degree-days retain their usefulne
7、ss,particularly because they constitute an elegant and concisedescriptor of the climate. For instance, ANSI/ASHRAE Stan-dard 90.1-2007 (ASHRAE 2007) relies mostly on heatingdegree-days base 18.3C (65F) and cooling degree-days base10C (50F) to classify locations into climate zones.Bases other than 10
8、C (50F) or 18.3C (65F) are oftendesired. The base temperature used in energy modeling meth-ods can be interpreted as the “balance temperature,” i.e., theoutside temperature at which the buildings internal heatgeneration offsets its losses. As modern buildings are betterinsulated, tend to have higher
9、 internal heat generation rates,and are often kept at lower temperatures than in the past, theuse of base temperatures other than 18.3C (65F) or 10C(50F) becomes necessary. Similarly, in agricultural applica-tions, the base temperature often depends on the type of cropor parasite considered (Wikiped
10、ia 2010). EnvironmentCanada (2010) provides heating degree-day bases of 0C, 5C,10C, 15C, 18C, and 24C (32F, 41F, 50F, 59F, 64.4F,and 75.2F), and cooling degree-day bases of 0C, 5C, 10C,15C, and 18C (32F, 41F, 50F, 59F, and 64.4F).The “Climatic Information” chapter of ASHRAE Hand-bookFundamentals (AS
11、HRAE 2005b) has long containedtables of climatic data suitable for the design of heating andair-conditioning systems. Recent editions have included datafor both North American and international locations. Becauseof their usefulness, it was decided to include heating and cool-ing degree-days in the t
12、ables of the 2009 edition. However,because of space constraints, only two base temperaturescould be included; the most commonly used temperatures,18.3C (65F) and 10C (50F), where chosen. Since degree-Methods for EstimatingHeating and Cooling Degree-Days to Any Base TemperatureDidier Thevenard, PhD,
13、PEngMember ASHRAEDidier Thevenard is principal of Numerical Logics, Inc., Waterloo, Ontario, Canada.LV-11-0212011. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions, Volume 117, Part 1. For personal use only. Additional
14、 reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAES prior written permission.2011 ASHRAE 885days to other bases may be required by user, it was decided toalso provide in the Handbook a method suitable for the eval-uation of degree-days to oth
15、er bases. The selection and vali-dation of such a method is the object of this paper.TRUE AND ESTIMATED DEGREE-DAYSHeating and cooling degree-days are defined as the sumof the differences between daily average temperatures and thebase temperature. For example, the number of heating degree-days in a
16、month, HDDb, is calculated as(1)where N is the number of days in the month, Tbis the basetemperature to which the degree-days are calculated, and is the mean daily temperature. The “+” superscript indicatesthat only positive values of the bracketed quantity are takeninto account in the sum. Similarl
17、y, monthly cooling degree-days CDDbare calculated as.(2)For historical reasons, the mean daily temperature isoften calculated by adding the maximum and minimumtemperatures for the day, then dividing by 2. This is the methodused in this study. Other calculation methods are sometimesused, such as thos
18、e listed by CIBSE (2006). Yearly degree-days are simply the sum of monthlydegree-days over the 12 months of the year.When hourly temperature records or daily min/max arenot available, it becomes necessary to estimate degree-daysfrom other meteorological variables. The first and still widelyused meth
19、od is the one proposed by Thom (1954a, 1954b,1966). Thom defined two variables h and l as follows:(3)(4)where is the monthly mean temperature and is the stan-dard deviation of monthly average temperature. By plotting lvs. h for 12 meteorological stations distributed across the US,Thom observed that
20、all (h, l) pairs of points aligned on auniversal curve that he tabulated experimentally (Thom1954a), and for which he later proposed a mathematical fit(Thom 1966).Another widely used method is that of Erbs et al. (1983).They experimentally fitted a curve to the temperature distri-bution functions of
21、 nine US stations, then calculated from thefit the degree-days to any base as.(5)They further simplified the inputs required by the methodby regressing against the standard deviation of themonthly average ambient temperature from the annual aver-age ambient temperature as follows:(6)with(7)More rece
22、ntly, Schoenau and Kehrig (1990) derived asimple method for calculating degree-days to any base. Theironly assumption is that the daily mean temperatures arenormally distributed around the monthly mean. Heatingdegree-days HDDbto base Tbare expressed as,(8)where Zbis the difference between base tempe
23、rature Tbandmonthly average temperature, normalized by the standarddeviation of the daily average temperature sd:(9)Function f is the normal (Gaussian) probability densityfunction, with mean 0 and standard deviation 1, and functionF is the equivalent cumulative normal probability function:(10)(11)No
24、te that f and F are readily available as built-in functionsin many scientific calculators or spreadsheet programs, sotheir manual calculation is rarely warranted.Cooling degree-days CDDbto base Tbare calculated bythe same equation,(12)except that Zbis now expressed as.(13)HDDbTbTi()+i 1=N=TiCDDbTiTb
25、()+i 1=N=TihTbTm()Nm-=lHDDbTm Tb()()Nm-=TmmHDDbmN32/h2-1.698h()cosh()ln3.396- 0.2041+=myrTyrm1.45 0.0290 Tm0.0664 yr+=yrTmTyr()2m 1=1212-=HDDbNsdZbFZb()fZb()+=ZbTbTmsd-=fZ()12-Z22-exp=FZ() fz() dzZ=CDDbNsdZbFZb()fZb()+=ZbTmTbsd-=886 ASHRAE TransactionsThe method derived by Schoenau and Kehrig (1990)
26、 iselegant, compact, and requires knowledge of only two variables,and sd, both of which can easily be calculated from hourlyor daily temperature records. The only potential criticism is thatin their paper, Schoenau and Kehrig (1990) tested their methodwith only a very small number (four) of US stati
27、ons. Further-more, they used typical meteorological year (TMY) data for thetests; since TMYs are single years made of months statisticallyselected from long-term records, there is a potential that thetemperature statistics derived from the TMYs would not berepresentative of the long-term statistics
28、derived from the fullperiod of record. In the rest of this paper, the Schoenau andKehrig (1990) method is validated using a very large data set anda long period of record; this will also enable the derivation ofsome statistics about the accuracy of the method.VALIDATION OF THE SCHOENAU AND KEHRIG (1
29、990) METHODEquations 8 through 13 were tested with the calculation ofheating and cooling degree-day bases of 0C, 5C, 10C,15C, 20C, and 25C (32F, 41F, 50F, 59F, 68F, and77F) for all 12 months, for over 1000 stations randomlyselected from the stations listed in the “Climatic DesignConditions” tables i
30、n the 2009 ASHRAE HandbookFunda-mentals (ASHRAE 2009). The stations included 189 sites inthe US, 55 sites in Canada, and 766 sites throughout the restof the world, with latitudes ranging from 69S to 81.6N. Inall, this test included nearly 150,000 monthly degree-dayvalues. Although not all these valu
31、es were useful in derivinginformation about model performance, as will be seen later,the number was large enough to be able to statistically assessthe validity of the method for a very wide range of conditions.The original data consisted of hourly or three-hourly inte-grated surface data set (ISD) d
32、ata for stations from around theworld provided by NCDC for the period 1982 to 2006 (Lott etal. 2001; NCDC 2003), and hourly weather records for theperiod 1981 to 2005 for Canadian locations from the Canadianweather energy and engineering data sets (CWEEDS),produced by Environment Canada (2007). The
33、period ofrecord used in the calculations spanned, in most cases, 25years, but the actual number of years used varied by station,depending on the number of missing data, and for some mayhave been as little as 8 years. For each station and each month,the degree-days were calculated with Equations 1 an
34、d 2, alongwith the monthly average and standard deviation of the dailyaverage temperatures. In the rest of this paper, degree-dayscalculated from the original time series will be called “true”degree-days, in opposition to those calculated by the Schoe-nau and Kehrig (1990) method or other methods, w
35、hich will becalled “estimated” degree-days.Equations 8 and 12 suggest that there exists a quasi-universal relationship between where DD is thenumber of either heating or cooling degree-days and Zbisexpressed either through Equation 9 or Equation 13; that is,according to Equation 8 or Equation 12,.(1
36、4)Figure 1 provides a scatter plot of DD / (N sd), calculatedwith true DD values, versus Zb. All pairs of points align them-selves neatly along the theoretical curve, Zb F(Zb) + f(Zb),shown separately for clarity in the inset of Figure 1. Thisconfirms the validity of the relationship (Equation 14) f
37、or thetest data set.One should notice that in Thoms original method (Equa-tions 3 through 4), the term is used as a substitute forthe standard deviation of the daily average temperature sd. Theuniversal relationship (Equation 14) is therefore nothing otherthan an analytical derivation of Thoms empir
38、ical (l, h) rela-tionship, and Figure 1 is simply the equivalent of the originalfigure in Thoms paper (Thom 1954a) from which he hadderived his empirical relationship.It should be noted that for Zb 2, the equation becomesasymptotically , whereas for Zb2. Instead, the data wereTmDD N sd()Figure 1 Plo
39、t of DD/N sd vs. Zb. In inset, the theoreticalcurve Zb F(Zb) + f(Zb).DDNsd- ZbFZb()fZb()+=NmDD N sd()()ZbDD N sd()()0HDDbNTbTm()=2011 ASHRAE 887binned according to the value of Zb, and model performancewas estimated separately inside each bin according to its meanbias error, MBE, and root mean squar
40、e error, RMSE:(15)(16)where DDestrepresents degree-days estimated with Equations8 or 12, DDtruerepresents true degree-days, Nbinis the numberof data in the bin, and the summation is taken over all pointsin the bin. Model evaluation is shown in Figure 2. Mean biaserror is small, at less than 1C/day (
41、1.8F/day) for all bins; androot mean square error is also quite acceptable, at less than3C/day (5.4F/day) for all bins. In view of this, the Schoenau and Kehrig (1990) methodwas added to the 2009 ASHRAE HandbookFundamentals;the parameters it requires (monthly average temperature andstandard deviatio
42、n of daily average temperature for all 12months of the year) were added to the tables of climatic designinformation for the more than 5400 stations worldwideincluded in the Handbook.ESTIMATION OF DEGREE-DAYS WHEN ONLY MONTHLY AVERAGE TEMPERATURES ARE KNOWNFor comparison, a similar graph (Figure 3) w
43、as generatedwith the Erbs et al. (1983) formula (Equations 56). Errors arelarger, with MBEs as high as 10C/day (18F/day), andRMSEs as high as 15C/day (27F/day). An advantage of theErbs et al. (1983) method, however, is that it requires only themonthly average temperatures for all 12 months of the ye
44、ar,whereas the Schoenau and Kehrig (1990) method requires theadditional knowledge of the standard deviation of daily aver-age temperature. When the latter is unknown, it is still possibleto estimate its value by regressing sdagainst the monthly meantemperature and the standard deviation of themonthl
45、y average ambient temperature from the annual aver-age ambient temperature , following an approach similarto the one of Erbs et al. (1983). The proposed regression,derived from the sample of 1000 stations described above, is.(17)Values of sdcalculated this way compare only moderatelywell to true val
46、ues of sd, as can be seen in Figure 4. The result-ing “hybrid” model still uses Schoenau and Kehrigs (1990)Equations 813, but with sdestimated from Equation 17. TheMBE1Nbin- DDestDDtrue()=RMSE1Nbin- DDestDDtrue()2=TmyrTyrsd3.228 0.0796 Tm0.1733 yr+=Figure 2 Error on the estimation of monthly degree-
47、days bythe Schoenau and Kehrig method, according tothe value of Zb.Figure 3 Error on the estimation of monthly degree-days bythe Erbs et al. method, according to the valueof Zb.Figure 4 Comparison of true values of sdto valuescalculated from monthly average temperatures.888 ASHRAE Transactionshybrid
48、 models MBE and RMSE are evaluated in Figure 5. Itcan be seen that the performance of the hybrid model falls half-way between that of the full Schoenau and Kehrigs (1990)model and that of the Erbs et al. (1983) model; it has a verysmall MBE (10C/day or 18F/day for some bins).EXAMPLETable 1 compares true degree-days base 18.3C (65F/day) and 10C (50F/day) for Vancouver, BC (Canada), withdegree-days estimated with the Schoenau and Kehrig (1990),Erbs et al. (1983), and hybrid formulae. The table e
