Calculation of Building Thermal Response Factors (BTLRF) as Wiener Filter Coefficients T, Kusuda National Bureau of Standards * Washington, D, c. Recent advances in the application of computers for environmental engineering problems have brought forth a number of sophisticated computer programs for simulating the hour by hour thermal performance of buildings, These programs not only calculate hourly thermal load of the building spaces, but also simulate the operation of energy distribution systems and mechanical equipment. When applied to a large building, however, the amount of computations to be performed become formidable even for the modern high speed and large memory computers, One way to reduce the computational requirement and to save the computer time (and cost) is to use building thermal response factors, (BTLRF), which are secondary sets of numbers derived from the limited amount of detailed calculations which are obtained from the exact thermal analysis, Presented in this paper is a preliminary attempt to apply time series analysis for obtaining BTLRF of a single room building. It is pointed out that BTLRF could also be obtained from measured thermal performance data or energy consumption data. Key Words: Building thermal load response factors, energy requirements, heating and cooling load calculation, Wiener filters 1. Background Calculations to determine the heating and cooling load for use in predicting building energy requirements can now be done by many digital computer programs, Although differing in minor technical details, most of the current computer programs for energy calculation obtain the hourly thermal load in conjunction with hourly weather tape data as provided, for example, by the National Weather Record Center, This hour by hour calculation of energy requirements, based upon detailed simulation of building thermal response, has been considered more accurate for a wider type of buildings than other simplified techniques, corrnnonly known as the ''degree day method", the "equivalent load factor method 11 and the 11 bin method"**/. These simplified methods are based upon the assumption that the building thermal performance can be calculated by a simple linear function of outdoor air temperature, particularly the temperature difference between the out- and indoor air. The temperature difference concept of the simplified techniques ignores the fact that the building thermal load is also dependent upon the other factors, such as solar radiation, moisture content of air, internal heat generation, and heat storage of the building structure, The simplified methods based upon the linear temperature difference concept have been, however, considered relatively accurate for use in residential applications, mainly due to the fact that the effect of solar radiation is relatively small and the internal heat generation is small and relatively constant as compared with commercial or industrial buildings. * Senior Mechanical Engineer, Environmental Engineering Section, Building Research Division ''* Although numerous references are available for these traditional methods, the most convenient one will be the ASHRAE Guide and Data Book, Systems 1970, Chapter 40, pp. 619-634. 117
The factors that have justified the use of the simple temperature difference concept become increasingly inappropriate as the building becomes larger and operational and occupancy characteristics grow complex. For example, the solar radiation effect becomes extremely important when an exterior wall of a modern office building is largely glass. For another case, internal heat generation due to the heavy lighting power per square foot of floor area tends to eclipse the indoor-outdoor temperature difference effect on the thermal load, Added to the complexity of these characteristics of the modern large scale building is the sophisticated nature of the heating and cooling system and its controls, for distributing excess internal heat of the building core to the periphery to limit the heating requirements during the winter. The hourly load simulation method with the use of computers performs an algorithmic operation which is designed to follow the actual thermal performance of buildings under realistic or randomly fluctuating outdoor weather conditions. One of the difficulties involved in using the sophisticated and exact hourly calculation of building thermal performance is that a large amount of computer and memory time is needed. For examplel/, the computer program developed by the U.s. Post Office Department requires a 100 K core storage computer (if applied to large postal facilities) and approximately 2 minutes of UNIVAC 1108 time to obtain an energy requirement estimate for heating and cooling of one room for a period of 365 days, If the calculation is to be performed for a large building consisting of, say 100 different rooms, the total computation time becomes prohibitive. This is particularly so when the building characteristics under consideration are complex, and when the accuracy requirement is such that the simplification of the computational efforts may be risky. The reduction of the computational effort is usually accomplished in two different ways. The first is to simplify the algorithms such as to delete refined calculation routines (thermal storage effect and infiltration effect). The second method is to simplify the building structure such as to treat a multi-room building as a single room building by ignoring the heat exchange among rooms and by ignoring details of the building structure. However, it has not been wellestablished under what conditions these computational shortcuts are justified. But in addition to these two, also presented in this paper is a preliminary attempt to study a third alternative for the reducing computational requirement, with a reasonable accuracy. Its objective is to obtain a secondary set of numbers called the building thermal load response factors (BTLRF) from the results of a limited number of detailed calculations, 2. Building Thermal Response Factors (BTLRF) These building response factors are basically regression coefficients as it becomes clear in the later discussion. The primary assumption imposed upon this technique is that the building thermal loads are a linear function of various excitation parameters such as~qutdoor temperature, solar radiation, internal heat generation and as well as the indoor temperatur~/, It is also assumed that the stochastic characteristics of the thermal load as well as the excitation time parameters are stationary, meaning that their basic means and standard deviation do not change with respect time, As a matter of fact, it is important to point out that the basic technique used to derive these building response factors can be applicable to any time series relationship, whether it be the heating/ cooling load, energy requirement, space thermal load or building thermal load, The time series, therefore, could be the observed energy usage values rather than the calculated values as mentioned previously. The idea is to derive regression coefficients from any input and the output time series by a suitable regression technique, For example, the hourly room thermal load may be calculated by a detailed computer program for a predetermined period (say N hours), The calculated hourly thermal load is then the desired output time series whereas the dry-bulb temperature, solar radiation and the internal heat generation may be considered inpu-t time series or the excitation time series. Denoting the hourly values of the thermal load, outdoor dry-bulb temperature, room temperature, solar radiation and the internal heat generation by q, DB, T, SOL and LT, respectively, it is assumed that the following linear relationship exists among them. M =l L s=o ( t-s t-s DB - T J SOL t-, LT t-s ' t 1, 2 N (1) The constant factor relating the degree days data to the energy requirement is a simplified BTLRF w hen the temperature difference is the major contributor to the energy requirement, 118
In this expression f 1 (s), f (s) and f (s) for s ~ 0, 1, 2 Mare the regression coefficients for the 2 excitation parameters, temperature, solar 3 radiation and the internal heat generation respectively. Subscript t in eq. (1) refers to the hour at which q is calculated and t-s refers to DB, T, SOL and LT evaluated at t-s hour. These regression coefficients are called the Wiener filters [1] if they are determined in such a way that N 0 ) (q u t t=l (2) in mlnlmum, whereby the q' is the value obtained by the exact calculations taking into account the building details and, or the desired output time series, whereas q is the value approximated by eq, (1) solely on the basis of time series analysis of participating v~riables, In eq. (2) N is the total number of data points to be analyzed to arrive at the least squares regression coefficients or Wiener filters. For example, if the two weeks data were used for the hourly thermal load calculations, N should be 336, A computer program to obtain the Wiener filters coefficients has been developed and published by E. A. Robinson [2]. The program utilizes a recursion type solution of multi-channel normal equations of the data to be processed. Given in the following section are examples of the application of the Robinson's computer program to the heating and cooling load calculation by the thermal analysis program [2] of the U. S. Post Office Department (USPOD). 3. Sample Calculations In order to examine the feasibility of the use of the Wiener filter routine to obtain BTLRF as the least square regression coefficients, hourly heating and cooling loads of a one-room building was first computed for 336 hours by the USPOD program. The weather data used for the calculation were for January 1949 of Washington, D. c. Figure 1 shows the trend of the excitation functions, namely _the dry-bulb temperature, solar radiation and internal heat generation during the computation periods, In order to simplify the calculation, the room temperature, T, in eq. (1) was assumed constant at 75 F. When the calculated thermal load was plotted against the ofitdoor temperature and against the solar radiation, they showed very much scatter as shown in figures 2 and 3 respectively. Figure 2, for example, suggests a danger of estimating hourly thermal load by a linear relationship with outdoor air temperature alone, The Wiener filtering technique was applied to the calculated thermal load regressed with (DB-75)t-s' SOLt-s and LTt-s for eq. (1) for s = 0, 1, 2, M. The value M in equation (1) is called the filter length and is related to the delayed reaction of the thermal load qt With respect to the excitation parameters. A satisfactory value for M may be determined by letting M = a, 1, 2 in eq. (1) until further increase does not significantly decrease the value of 0. In this particular example, values of M up to 20 have been tried and it was found that the optimum value is 3 for all the practical purposes. In Order to illustrate building response factors form~ 3, the filter coefficients for a one-room building are listed as follows: 1 (0) 31.913 2 (0) 3.807 3 (O) 4.308 1 (1) -.426 2 (1) -.056 3 (1) 1.809 1 (2) -.267 2 (2) 1. 777 3 (2) 1. 762 1(3) -.245 2 (3) 1.110 3 (3) 2.639 119
Normalized valuesl/ of 0 form= 0, 1, 2. 10 respectively for a similar analysis are 0.219,.137,.093,.067,,062,,057,,054,,053,,050, and.047, which show the diminishing return form beyond 3, It should be pointed out that it is difficult to draw physically meaningful conclusions from these coefficients, since they were derived solely by numerical data manipulation. Nevertheless, they simulate thermal load very accurately for the period where the original data were analyzed, Also to be pointed out is the reduction of mathematical operation manifested in a simple algebraic formula of equation (1) against a detailed thermal analysis program consisting of approximately 2000 Fortran statements. It is, however, to be expected from the theor~/ of heat conduction equation that the absolute values of BTLRF should start to decrease steadily1 as the value of s increases beyond a certain value, say Smax' such that f1 (8+3) I < I f1 (8+2) I < I f1 (8+1) I < I f1 (8) I whens~s max This decreasing trend was not observed for this sample calculation even when M was carried up to 20, although it is possible that filter coefficients of more physically consistent nature might have been obtained, had a suitable smoothing technique been applied to the input data. Although these response factors did reproduce the original data very well, a true test of the response factors would be when they are applied in a predictive manner, Figure 4 shows the same response factors applied to eq, (1) for the climatic data beyond the period when the original thermal load was calculated. Figure 5 is in turn the thermal load calculated by the USPOD program for the same weather record period, If the response factors are ideal, figures 4 and 5 should match each other well for the entire period. By overlaying figure 4 on figure 5 it can be shown that the two curves match almost perfectly for the first 336 hours during which period the response factors were generated. The same two curves, hewn ever, begin to differ considerably as the time goes beyond the first 336 hours and particularly during the summer period, although general trend of the increase of the mean thermal load is obtained by the response factor calculation. The increase of the diurnal amplitude of the thermal load during the summer, however, \V"as not well represented by the calculation using BTLRF. The similar calculation repeated for 336 hours (two weeks period) data of thermal load and accomn parrying weather data during the last week of June yielded another set of building response factors such as: 1 (0) 39.497 f2 (0) 11.558 f3 (0) 7.206 f1(1) 15.488 f2 (1) -4.362 f3 (1) 1.668 f1 (2) -so. 894 f2 (2) -4.177 f3 (2) 1. 789 f1 (3) 38,594 f2 (3) 10.768 f3 (3) 1.650 These values were in turn used again to calculate the hourly building thermal load from January to June by eq. (1), results of which are shown in figure 6. The agreement between the thermal loads obtained by the detailed calculation with use of USPOD program and those approximated by eq. (l) is poor during the winter this time. The decrease of the average values and amplitudes of the building thermal load during the winter is not well reproduced. of time, These dwo sets of calculations and figures 4 and 6 suggest that the BTLRF can be made a function It is assumed that they will change from set (3) to (4) by a linear fashion such that: (5) :~~~eti!~~ and [fs] represent the winter and summer building response factors and [f] is those adjusted 120
The value of St in eq, (5) was assumed to be a step time function representing: Integer part of (t/336) 12 0) for the 12 bi-weekly periods spanning the beginning of January through the near end of June. The detailed 4 and 6, function result of this calculation is shown in figure 7, and indicates a better agreement with the calculations (figure 3) obtained by the USPOD program throughout the period than figures The agreement should be further improved if the values of BTLRF were made a more complex of time than a simple linear function. 4. Summary A possible new approach to enhance the use of computers for calculating building thermal load is the application of Wiener-type filter coefficients which are called in this paper the BTLRF or the building thermal load response factors, It is pointed out in this paper that BTLRF can be obtained either from the heating/cooling load calculated by the very comprehensive computer program (simulating entire building heat transfer processes) or from the experimentally observed values for a limited period of time, say two weeks, Once determined, these BTLRF can permit the calculation of the thermal load by one simple linear algebraic equation, This results in drastic reduction of the computational effort as well as the core requirement on the computers, from a computer program needing a few thousand Fortran statements and 100 K core storage computer to a program of a few Fortran statements that can be executed on a mini-computer, A rough estimate of computer time reduction is from 2 minutes per room of a building to a few seconds per room for a computation covering 365 days, This paper presents one result of an exploratory investigation to derive BTLRF by the use of Wiener Filter Technique to the heating and cooling load calculated by the U, S, Post Office Energy Analysis Computer Program, The BTLRF were found to be dependent on time if they were to be applicable for the calculation of hourly building thermal load over as long as a half year's period. This consideration is necessary because building thermal load characteristics cannot be considered stationary if the time span is as long as a hal! year. The time span of the hourly data used to determine the BTLRF was 336 hours for the calculation illustrated in this report, although.it could most possibly have been shortened to 168 hours or even less. A satisfactory length of the filter appeared to be 4 terms (j = 0, 1, 2, 3). Although BTLRF provide a relatively good estimate in load calculation by a very simple algebraic operation, the coefficients obtained by the Wiener filtering technique did not follow the expected trend that the absolute value would eventually start decreasing steadily. Further work is being performed at the Environmental Engineering Section of the National Bureau of Standards to obtain building thermal load response factors which do follow this expected trend and which are therefore more amenable to physical interpretation. 5. References r1] U. S, Post Office Department Report "Computer Program for Analysis of Energy Utilization in Postal Facilities 11, Copies obtainable from J. M. Anders of the U, S, Post Office Department, Washington, D. C. 20260, 1970. [2] E. A. Robinson, Multi-channel Time Series Analysis with Digital Computer Programs, Holden-Day, San Francisco, 1967, p. 249. [3] T. Kusuda, "Thermal Response Factors for Multi-layer Structures of Various Heat Conduction Systems", ASHRAE Transactions, pp. 246-271, 1969, Chicago, Illinois. 121
r Dry-bulb Temperature,F Solar En~rgy, Btu per sq.ft,hr Internal Heat Generation, Btu per hr 48 96 144 Mon Tue Wed Thu Fri Sat 192 Sun Mon 240 Tue Wed Thu 288 Hours Figure 1. Excitation functions used for the thermal load calculation by USPOD computer program for the first two weeks of January. 122
-3000 WASHINGTON. 0. c.. JANUARY 1-7 1949. ~ -2500 m m 0::: 9 :r: " w ~ :::::J-2000 I w " " m m 9 f- m CD I e m I m I m m I 9 D-1500 m il a: m. ~ 9 0 9 j w 0-1000 e 9 z ~ f- e a: w -500 g :r: m >- j 0::: :::::J 0 0 m :r: 500 I e 10oq 0 15 20 25 30 35 40 DRY BULB TEMPERATURE F Figure 2. Relationship between the calculated hourly thermal load and outdoor air dry-bulb temperature. 123
-3000 WASHINGTBN. 0. c.. JANUARY 1-7. 1949-2500 ex:: I ' =:J-2000 f- e (D m " D-1500 a: 0 J c:.-1000 - :z "' f- a: -500 w I.. >- 1!1 0 I!J 1!11!1 J 0 d' ex:: =:J 0 ol' I 500 '\, 10000.. 1!1..... "" 20 40 60 80 100 SOLAR HEAT. BTU/HR. so. FT. Figure 3 Relationship between the calculated hourly thermal load and solar radiation over the south facing wall. 124
_,J:~~~~... X 1 SOLA!! RADIATION X 1 LIGHTING POWER =,~~-~-~~AM~" ~.~--~~~..---~~"~~ DJ,YS Figure 4. Thermal load calculated by the winter BTLRF for January through June. "" Figure 5. Thermal load calculated by the detailed computer program of USPOD for January through June.
CAl. 0L.ATED BY BUILDING RESPONSE x,oo-7~ Xo SOLAR RA!)IATION X." LIGHTING POWER JANUARY FEBRUARY DAYS Figure 6. Thermal load calculated by the summer BTLRF for January through June. CALCULATED BY BUILDING RESPONSE ~RS LF4 Xt DB-7& XSOLAR IIAOIATION X, LlaHTING POWER FEBRUARY MARCH ""'' APRIL Figure 7. Thermal load calculated by the BTLRF which is a linear function of time.