1、 I STD-ASHRAE SRCH IJHVAC b-1-ENGL 2000 D 0759650 0546771 431 International Journal of Heating, Ventilating, Air-conditioning and Refrigerating Research Editor John W. Mitchell, Ph.D., P.E. Professor of Mechanical Engineering University of Wisconsin-Madison, USA Associate Editors James E. Braun, Ph.
2、D., P.E., Associate Professor, Ray W. Herrick Laboratories, Alberto Cavallini, Ph.D., Professor, Dipartmento di Fisicia Tecnica, University of Padova, Italy Arthur L. Dexter, D.Phil., C.Eng., Reader in Engineering Science, Department of Leon R. Glicksman, Ph.D., Professor, Departments of Architectur
3、e and Ralph Goldman, Ph.D., Chief Scientist, Comfort Technology, Inc., Framingham, Massachusetts, USA Hugo Hens, Dr.Ir., Professor, Department of Civil Engineering, Laboratory of Building Physics, Katholieke Universiteit, Belgium Anthony M. Jacobi, Ph.D. Associate Professor and Associate Director AC
4、RC, Department of Mechanical and Industrial Engineering, University of Illinois, Urbana-Champaign, USA Jean J. Lebrun, Ph.D., Professor, Laboratoire de Thermodynamique, Universit de Lige, Belgium Reinhard Radennacher, Ph.D., Professor and Director, Center for Environmental Energy Jean Christophe Vis
5、ier, Ph.D., Professor, Centre Scientifique et Technique du Btiment, School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, USA Engineering Science, University of Oxford, United Kingdom Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, USA Engineering, D
6、epartment of Mechanical Engineering, University of Maryland, College Park, USA and Head of the Energy Management Automatic Controller Division, Mame La Valle, France Policy Committee Richard H. Rooley, chair Jack B. Chaddock Mano Costantino John W. Mitchell Frank M. Coda W. Stephen Comstock Editoria
7、l Assistant Publisher ASHRAE Staff Jennifer A. Haukohl W. Stephen Comstock Robert A. Parsons, Handbook Editor Scott A. Zeh, Publishing Services Manager Nancy F. Thysell, Typographer 82000 by the American Society of Heating. Refrigerating and Air-Con- ditiontng Engineers. Inc., 1791 Tullie Circle. At
8、lanta, Georgia 30329. All rights reserved. Periodicals postage paid at Atlanta. Georgia, and additional mailing offices. HVAC nor may any part of ihis book be reproduced, stored in a rebievai system, or msmitted in any form or by any means-electronic, photocopying, recording, or other-without permis
9、sion in writing from ASHRAE. AbsractoAbstracted and indexed by ASHRAE Abstract Center; Ei (Engineering Information, Inc.) Ei Compendex and Engineering Index; IS1 (Institute for Scientific Information) Web Science and Research Alen: and BSRIA (Building Services Research and 2. The “Actual Savings” ov
10、er a certain time period in which the baseline model is driven with actual monitored outdoor temperature under post-ECM conditions. The sum of the differ- ences between these values and the observed post-ECM values over that time period yields the savings (Kissock et al. 1998, IPMVP 1997). This stud
11、y is concerned primarily with (2) above. In this case, the uncertainty in the baseline consumption as determined from the statistical model becomes the major determinant of the uncertainty in the resulting measured savings. The uncertainty is of major interest to building owners and financial instit
12、utions associated with energy service contracts that are intended to reduce energy cost of operating specific buildings. The uncertainty in the baseline consumption is in turn directly related to the goodness of fit of the baseline models. Little work has been done to establish sound statistical lim
13、its for gauging the goodness-of-fit of the baseline model. The most widely used criterion is that proposed by Reynolds and Fels (1988). They proposed (1) that models with coefficient of determination, i.e., R2 values 0.7 and coefficient of variation (CV values) Pre-retrofit period Post-retrofit peri
14、od 9 I, *-.,./, 1 Savings 3 L Time I Epsi Figure 4. Conceptual plot showing how energy savings are determined operational data that does not cover a whole annual cycle was not considered (Reddy et al. 1998). Hence the baseline model is assumed to be identified from yearlong data with no model extrap
15、olation errors. Conceptually, as shown in Figure 4, actual savings (as against “normalized” savings) E, over m periods (hours, days or months, depending on the type of energy use data available) into the retrofit period are calculated as follows: where in = number of periods (hour, day, or month) in
16、 the post-retrofit period EPrP = pre-retrofit energy use predicted by the baseline model per period, and E, = measured post-retrofit energy use per period. In order to make the equations more readable, we shall rewrite Equation (6) as: where, E, now denotes the energy use over m periods. For example
17、, Epre,rri denotes the sum of m individual model predicted values of baseline energy use. With the assumption that model prediction and measurement errors are independent, the total variance is the sum of both: The total prediction uncertainty increases with ni, i.e., as the post-retrofit period get
18、s longer. However, as the amount of energy savings also increases with m, a better indicator of the uncer- tainty is the fractional uncertainty defined as the energy savings uncertainty over ni periods divided by the energy savings over m periods: STDmASHRAE SRCH IJHVAC b-L-ENGL 2000 H 0759b50 05467
19、811 380 H 10 HVAC&R RESEARCH where F is the ratio of energy savings to pre-retrofit energy use, i.e., Equation (9) provides a means of calculating the fractional uncertainty in the “actual” sav- ings, which consists of a term representative of the regression model prediction uncertainty and another
20、term representative of the measurement error in the post-retrofit energy use. The mea- surement error in the pre-retrofit energy use is inherently contained in the model goodness-of-fit parameter (namely, the mean square error (MSE) statistic), and should not be introduced a sec- ond time. In case t
21、he measurement uncertainty is small (for example, when electricity is the energy channel its associated error is of the order of 1 to 2%), the fractional uncertainty in our savings measurement can be simplified into - where Eyre is the mean pre-retrofit energy use during the selected period. This ex
22、pression can be cast into a more useful form by making certain simplifying assump- tions. The effect of measurement errors in the post-retrofit data was neglected and it was assumed that model prediction errors are the only source of prediction uncertainty (the various sources of errors as applied t
23、o building energy analysis are described by Reddy et al. 1998). Two separate cases were considered: (a) models with uncorrelated residuais which one would encounter when analyzing utility bills, and (b) models with correlated residuals often encoun- tered with models based on hourly or daily data (R
24、uch et al. 1999). Models with Uncorrelated Residuals and Additive Errors The prediction uncertainty of a simple linear model identified from random data is given in statistical textbooks (Draper and Smith 198 i). The regression model prediction uncertainty for an individual observation during the po
25、st-retrofit period, for a model with constant variance and uncorrelated model residual behavior, is: AE, = J - MSE 1 I+-+ n I (Tj-T)2 i (Ti - T)2 i=l 10.5 n a2 vhere MSE = variance of the model error = (Ei -Ei) - p) (13) = individual value of outdoor dry-bulb temperature during the prediction or pos
26、t-retrofit period T = mean value of Ti ( i.e., mean value of the outdoor temperature) during model identification or i= 1 pre-retrofit period STD.ASHRAE SRCH IJHVAC b-1-ENGL 2000 0759b50 0546782 217 W VOLUME 6, NUMBER 1, JANUARY 2000 Il The retrofit saving? methodology is not based on individual pre
27、dictions of Ej , but the sum over rn days of the Ej values. The prediction error of a sum of m future observations (as is needed for determining energy savings) is given by Theil (1971): where X is the matrix of regressor variables during the pre-retrofit period, X denotes the trans- pose of X, Z is
28、 an identity matrix, and the subscript “post“ indicates post-retrofit period. Note that pre and post multiplication of the matrix within the square brackets by unit matrices 1 is akin to summing all the elements of the matrix. Equation (14) involves matrix algebra and was simplified using numerical
29、trials to form the following equation that holds to within about 10% accuracy: II j=i MSEI m+-+ 0.5 0.5 = ?MSE(rn 1.26 + E + (15a) nn mEpreF ri, m T.,T. = average outdoor dry-bulb temperature values during post-ECM and during model identification = number of observations in the baseline and the post
30、-ECM periods, respectively Ir (Le., pre-ECM) periods, respectively. Finally, for baseline models with additive errors, Equation (15a) can be expressed in terms of the standard CV statistic (either the CVRMSE or the CVSTD as appropriate) as: / qo.s r/ - 170.5 I- 126cv 1/2F when n is large (say, n 60)
31、 m Note that the above expression yields the fractional energy savings uncertainty at one stan- dard error (i.e., at 68% confidence level). For other confidence levels, say 95%, the bounds have to be multiplied by the student t-statistic evaluated at 0.025 significance level and (n-p) degrees of fre
32、edom (Draper and Smith 1981). Models with Uncorrelated Errors and Multiplicative Errors As illustrated in Figure 3, model residuals will provide the necessary indication as to whether the errors are additive or multiplicative. Reddy et al. (1997a) had proposed the following empir- ical modification
33、to Equation (15) for the prediction uncertainty Usa, in the case of multipli- cative errors: STD-ASHRAE SRCH IJHVAC L-Ii-ENGL 2000 D 0759b50 054b783 L53 12 HVAC&R RESEARCH (MSE,m, + MSE2m2) + MSE- usa ve,m n where mI and m2 are the number of data points on either side of the change point during the
34、post-retrofit period, MSEI and MSE2 are the mean square errors of the data points on either side of the change point as shown in Figure 2, and MSE is given by Equation (13). MSE, (and MSE2 by analogy) can be calculated from pre-retrofit data: MSE,m, +MSE m 0.5 S,“, - 126cv( MSEm +)I nm Esave,m F Fin
35、ally, the fractional uncertainty in the annual savings is: which, to be consistent with Equation(lSb), can be re-written as: where WCV is termed the weighted CV value for a change point model with multiplicative errors. and is defined as: wcv = cv Defining the WCV is this manner allows the uncertain
36、ty AESa, of change point baseline models with either additive or multiplicative errors to be treated in the same manner. In the remainder of the paper, the term CV to denote either the CV or the WCV is used as appropriate. Models with Correlated Residuals Equations (15) or (19) are appropriate for r
37、egression models without serial correlation in the residuals. This would apply to models identified from utility (i.e., monthly) data. When models are identified from hourly or daily data, previous studies (Ruch et al. 1999) have shown that serious autocorrelation often exists. These autocorrelation
38、s may be due to ( 1) “pseudo“ pat- terned random behavior due to the strong autocorrelation in the regressor variables (for example, outdoor temperature from one day to the next is correlated), or (2) to seasonal operational changes in the building and HVAC system not captured by an annual model. Co
39、nsequently, the uncertainty bands have to be widened appropriately. Accurate expressions for doing so have been proposed by Ruch et al. (1999). In the framework of this study, a number of numerical tri- STDaASHRAE SRCH IJHVAC b-L-ENGL 2000 0757650 0546784 09T VOLUME 6, NUMBER I. JANUARY 2000 13 als
40、were performed and found that the following simplified treatment yields results accurate to within 20% of those proposed by Ruch et al. (1999). From statistical sampling theory, the number of independent observations nof n observations with constant variance but having a lag 1 autocorrelation p equa
41、ls (Neter et al. 1989): For example, if n = 365 (i.e., one year of daily data), and p = 0.85, then n = 365 x 0.081 = 29.6, which implies that there were effectively only about 30 independent observations. Equa- tion (15) in the presence of serial autocorrelation can be expressed as: The CV value cal
42、culated using n degrees of freedom has been renormalized by the new degrees of freedom n. Discussion Equations (15), (19) and (22) provide a more rational means of evaluating the accuracy of our baseline model to determine savings. Figure 5 that has been generated based on Equation (1%) allows easy
43、determination of fractional savings uncertainty. Consider a change point baseline model based on daily energy use measurements with a CV (or WCV) of, for example, lo%, we wish to assess the uncertainty in savings for 6 months into the post-retrofit period for a retrofit measure that is supposed to s
44、ave 10% of the pre-retrofit energy use (i.e., F = 0.1). Then for m = 6 months = 182.5 days, and CV = IO%, the y-ordinate value from Figure 5 is 0.01, i.e., AE,yu,.e/E,y(,.e = (0.01/0.1) x 100 = 10%. On the other hand, if a baseline model is relatively poor, with, for example, a CV of 30%, and the re
45、trofit is supposed to save 40% of the preretrofit energy use, then from Figure 5, AE,FuL?JE.Tuve = (0.028/0.4) x 100 = 7.0%. Hence, the first model, despite having a CV value three times lower than that of the second model, leads to a larger frac- tional uncertainty in the savings than the second ca
46、se. The above example serves to illustrate how viewing the retrofit savings problem in the perspective outlined in this paper is more rele- vant than merely looking at the baseline model goodness-of-fit. This example is based on 68% confidence level (Le., t-statistic = i). Suitable corrections need
47、to be made for different confi- dence levels. The variation of the fractional uncertainty M.rave/EFuve with CV, n, m, and Fis of interest to energy managers and energy service companies while negotiating energy conservation service contracts. If utility bills are the means of savings verification an
48、d if yearlong pre- and post-ret- rofit data are available, then Table 1 provides an indication of how AEJu,e/EFu,e varies with CV (or WCV) and F. For example, if both negotiating parties are comfortable with a fractional uncertainty in energy savings of 10% at the 68% confidence level, and if the re
49、trofits are expected to save 20% (i.e., F = 0.2), then a baseline model with a CV of 5% or less will satisfy the expectations of savings verification when one year of pre-retrofit and one year of post-ret- rofit data are available. Table 2 provides the same information as Table 1 but assuming that daily monitored data is available for savings verification (i.e., we now have 365 data points instead of i2 data points only during either period) and that no residual autocorrelation is present (i.e., p = O). For the abo
