ASHRAE IJHVAC 9-4-2003 HVAC&R Research《《HVAC&R研究》第9卷 4号 2003年10月》.pdf

1、VOL. 9, NO. 4 HVACT. Agami Reddy is a professor in the Department of Civil, Architectural and Environmental Engineering and DagmarNiebur is an associate professor in the Department of Electrical and Computer Engineering at Drexel University, Phila-delphia, Pa. Klaus K. Andersen is a research associa

2、te in the Department of Mathematical Modeling at the TechnicalUniversity of Denmark, Lyngby, Denmark. 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC andc. integrated automation and control of building systems and services tha

3、t are meant to assist inproper facility management, which include energy management, comfort monitoring, facilityoperation, and services billing and communication with the energy supplier.The scope of this paper is limited to the first two areas only. An increasing number of energyperformance contra

4、cts require verification by actual field monitoring of the energy and cost sav-ings resulting from implementing energy efficiency projects. The National Association ofEnergy Service Contractors developed protocols for the measurement of retrofit savings in1992, which were followed by federal protoco

5、ls, such as FEMP (1996), IPMVP (1997), andARI (1998), and, finally, ASHRAE Guideline 14 (ASHRAE 2002). There are also numerousrefereed publications in this area, for example, the ASME Special Issue (Claridge 1998) or refer-ences listed in Reddy and Claridge (2000).Investigators and service companies

6、 are being required to develop custom measurement plansand analytical procedures for each project, which increases total project costs. An importantissue during the M Katipamula et al. 1998; Reddy et al. 1998; Reddy etal. 2002) have investigated the latter option in an empirical manner and made reco

7、mmendationsas to the season (or time of the year) that is likely to yield performance models of HVAC (b) compressor power P in kWe;(c) supply chilled water temperature Tchi in K, and (d) condenser water supply temperature Tcdiin K. The data set used in the subsequent analysis contains 810 observatio

8、ns and is fullydescribed in Reddy et al. (2001) and Reddy and Andersen (2002). From the time series plots ofthe four variables shown in Figure 1, we note that there is relatively little variation in the twotemperature variables, while the load and power experience important variations. Since thechil

9、led water flow rate is constant, we have chosen to perform the analysis with the followingregressor set Tcdi, Tchi, Tcho where Tchois the chilled water temperature leaving the chiller.Chiller #2 DataThis is a 450 T centrifugal chiller located on the Drexel University campus. A comprehensivedescripti

10、on of the steady-state data (consisting of 1126 sets of observations of 15-minute dataover 14 days) is described in Reddy et al. (2001). Figure 2 depicts the time variation of the perti-nent variables, namely, the condenser and evaporator fluid temperatures and the electricalpower. The evaporator an

11、d condenser water flow rates can be assumed essentially constant sincevar b()OLS2XTX()1,=Figure 1. Time Series Data of the Four Measured Variables of Centrifugal Chiller #1(Tchi= Inlet Water Temperature to Evaporator (K); Tcdi= Inlet Water Temperature to Condenser (K); P = Electric Power Consumed by

12、 Chiller (kW); Qch= Chiller Thermal Load (kW) 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC in other words, to underline the fact that more field data does not necessarilymean more information. We shall investigate this issu

13、e in two ways: 1. Traditional statistics: One intuitive and simple way is to look at the histograms of the regres-sor variables since this would provide an indication of the variability in operating conditionsto which the chiller is exposed. A uniform distribution would indicate good coverage ofchil

14、ler operating conditions and vice versa. Figure 3 depicts such histograms for the threevariables Tcdi, Tchi, and Qchfor Chiller #1. We note that Qchvalues are fairly well distributed,while those for the two temperatures are not. For example, there are only a couple of datapoints for Tcdi 27.5C. Thes

15、e points are likely to be influence points (Cook and Weisburg1982), and whether these reflect actual operating conditions or are a result of either erroneousdata or uncharacteristic chiller operation has to be determined by the analyst from physical(as against statistical) considerations. Additional

16、 insight into the extent to which the data col-lected are repetitive in nature can be gleaned by studying the joint occurrences. Table 1shows these values for four bins of Tchiand Qcheach and for two bins of Tcdi(which exhibitFigure 3. Histogram of Number of Occurrences for the Three Regression Vari

17、ables Using the Hourly Chiller #1 Data (Total of 810 Data Points) 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVACb. the GN and VT parameter estimates are deduced for this data set;c. steps (a) and (b) are repeated a large numb

18、er of timesspecifically 400 times in this analysiswith the starting point m0being moved from 1 to 400;d. mean values of the parameter estimates and their 2.5% and 97.5% percentiles are calculatedfrom the 400 sets pertaining to window length m;e. steps (a) through (d) are repeated by incrementally ch

19、anging the window length m from m =20 to m = 400.Chiller #1 data were used with the above scheme. Figures 4 and 5 depict the results of thisanalysis. Convergence along with acceptable 2.5 and 97.5 percentiles seems to require a mini-mum of 60 to 70 data points for the VT model parameters and close t

20、o 300 for the GN modelparameters. The parameters of the VT model converge four to five times faster than the GNmodel, i.e., the physical model requires at least four to five times more data in order to obtainreasonably accurate parameter estimates. The parameter uncertainty of the two models, showni

21、n Figures 4 and 5, is partly due to the ill conditioning of the GN model structure (see Reddy andAndersen 2002), as well as due to the serial correlation in the data. The former effect is the rea-son why the convergence properties of the GN and VT models differ even when applied to thesame basic chi

22、ller data set. This important consequence of serial correlation is illustrated in Figure 6 for the GN model.The same procedure as previously is applied, but the m data points are no longer taken sequen-tially, but randomly, from the entire data set of 810 points with replacement (this is the bootstr

23、apmethod). Consequently, most of the serial-correlation in the data is removed. Inspection of Fig-ure 6 leads us to a completely different conclusion than previously. The number of observations(from 20 to 400 observations) seems to have no effect on the mean values of the model parame-ters nor on th

24、eir variance (indicated by the 2.5 and 97.5 percentiles). In other words, using about20 independent samples is just as good in terms of variance of parameter estimates as using 400data points monitored continuously on-line! Comparing Figures 4 and 6 leads us to concludethat about 20 independent samp

25、les contain as much information as about 300 serial correlatedobservations in this case. The above conclusion can be confirmed in a simple manner. From statistical sampling theory,the number of independent observations n of n observations with constant variance but having a1-lag autocorrelation is e

26、qual to (Reddy and Claridge 2000)n n1 1 +- .= 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC for Each Plot, the Middle Line is the Calculated Mean and the Upper and Lower Lines Correspond to the 97.5 and 2.5 Percentiles; the

27、X-Axis Shows the Win-dow Size, and the Y-Axis Shows the Parameter Estimates b1, b2, and b3Figure 5. Same as Figure 4 but for the Black-Box VT Model 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC I1of Equation 8 can beinterpre

28、ted as the sum of the parameter variances up to a scaling factor. 2. Another measure of information is the log of the mean of the determinant (Beck and Arnold1977).(9)where m is the number of observations used to calculate (XTX). Beck and Arnold (1977) showedthat minimizing det(XTX) or, alternativel

29、y, minimizing lndet(XTX) also minimizes the uncer-tainty of the parameter estimator vector b. Measure I2 may be better than measure I1for two reasons. First, it has a clear physical inter-pretation in that it represents the hypervolume of the hyperellipsoid of the confidence region ofthe parameter e

30、stimates of the model (Beck and Arnold 1977). Second, the logarithmic transfor-mation makes the numerical values more robust.The above notions are relevant in the optimal design of experiments as well as during fieldmonitoring. The analyst can tailor the experimental sequence so that a minimum numbe

31、r ofexperiments can be performed that will provide the necessary accuracy in the model parameterestimates. How these indices can be used for optimal experimental design for assurance testingof chillers during the factory witnessed test is described in a paper by Corcoran and Reddy(2003). Not so obvi

32、ous, however, is the value of such measures of information in the context ofnon-intrusive field-monitored data where little can be done to select data in an optimal way. Oneadvantage is that instead of tracking the uncertainty of the full set of model parameter estimatesindividually as more data are

33、 forthcoming (which is tedious), it is far simpler, both computation-ally and for decision making, to track only one overall measure. This aspect is further discussedlater in this paper.Application to Chiller Data The set of 810 observations of Chiller #1 data has been used to calculate the measures

34、 ofinformation I1and I2using Equations 8 and 9, respectively. The computation has been per-formed for three cases:1. Incremental window, i.e., an ever-increasing window length to mimic the manner in whichon-line data will be collected in actuality. Specifically, we started the computation with anini

35、tial set of ten data points, which was increased one observation at a time till the end of thedata stream was reached (represented by the point mo= 810).2. Sliding window with 100 data points, where the sliding window concept is similar to the gen-eralized moving average concept used in time series

36、analysis.3. Sliding window of 200 data points.I1trace XTX()1=I2det XTX()mln= 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC the Basic Variable Set Tcdi, Tchi, Tcho for Chiller #1 Has Been Used 2003. American Society of Heatin

37、g, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC the Basic Variable Set (Tcdi, Tchi, Qch) for Chiller #2 has Been Used 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC (b) artificial neu

38、ral network (ANN) mod-els, specifically radial basis function (RBF) and multilayer perceptron (MLP); (c) the genericphysical component (PC) model approach; and (d) the lumped physical Gordon-Ng (GN)model. All models except for (b) are linear in the parameters. A review of the engineering litera-ture

39、 identified the three following on-line training schemes as suitable for evaluation: ordinaryrecursive least squares (ORLS) under incremental window scheme, sliding window scheme, andweighted recursive least squares (WRLS) scheme, where more weight is given to newer data.The evaluation was done base

40、d on five months of data from a 220 ton field-operated chiller fromToronto (a data set of 810 data points) and fourteen days of data from a 450 ton field-operatedchiller (a set of about 1120 data points) located on Drexel University campus. The evaluationincluded a preliminary off-line or batch anal

41、ysis to gain a first understanding of the suitabilityof the various models and their particular drawbacks and then to investigate whether the differ-ent chiller models exhibit any time variant or seasonal behavior. The subsequent on-line evalua-tion consisted of assessing the various models in terms

42、 of their suitability for model parametertracking as well as model prediction accuracy (which would provide the necessary thresholdsfor flagging occurrence of faults). The former assessment suggested that parameter trackingusing the GN model parameters could be a viable option for fault detection (F

43、D) implementa-tion, while the black box models were not at all suitable given their high standard errors. Theassessment of models in terms of their internal prediction accuracy revealed that the MLPmodel was best, followed by the MP and GN models. However, the more important test of exter-nal predic

44、tive accuracy suggests that all models are equally accurate (CV about 2% to 4%) and,hence, comparable within the experimental uncertainty of the data. ORLS with incremental win-dow scheme was found to be the most robust compared to the other computational schemes. Thechiller models do not exhibit an

45、y time variant behavior since WRLS was found to be poorest.Finally, in terms of the initial length of training data, it was determinedat least with the datasets used that exhibited high autocorrelationthat about 320 and 400 data points would berespectively necessary for the MP and GN model parameter

46、 estimates to stabilize at theirlong-term values. This paper also provides a detailed discussion of the potential advantages thaton-line model training can offer and identifies areas of follow-up research.T. Agami Reddy is a professor in the Department of Civil, Architectural and Environmental Engin

47、eering, Dagmar Nie-bur is an associate professor and Paolo P. Pericolo is a former graduate student in the Department of Electrical and Com-puter Engineering, and Gaspar Cabrera is a doctoral student in the Department of Mechanical Engineering andMechanics at Drexel University, Philadelphia, Pa. Kla

48、us K. Andersen is a research associate in the Department of Math-ematical Modeling at the University of Denmark, Lyngby, Denmark. 2003. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in HVAC Pau 1975; Patton et al.1989; Pouliezos and Stavr

49、akakis 1994; Gertler 1998; Chen and Patton 1999). Many of thesetechniques could be applicable to FDD of HVAC Jiang and Reddy 2003):a. Model structure or functional form should be suitable, which would includei. generality of the model,ii. physical relevance of the model parameters, andiii. proper behavior of the model residuals.b. Model parameters should bei. stable, i.e., not change much when different data periods are used for model identifica-tion, andii. “efficient,” where “efficiency” is a statistical concept suggesting that model parameteresti