Tools and Techniques to Calibrate Electric Chiller Component Models

Mark Hydeman, P.E. Kenneth L. Gillespie, Jr.Member ASHRAE Associate Member ASHRAE

AC-02-9-1

ABSTRACT

Chilled water plants are complex and dynamic systemsthat are difficult to accurately model. Core to an accurate anal-ysis of a chilled water plant is the component models used forthe chilled water plant equipment. This paper documents tech-niques to create calibrated simulation models of electric chill-ers from manufacturers’ and short-term monitored data. Theauthors present two techniques: (1) standard linear regressionfor well-formed data sets and (2) a reference-curve methodthat can be used with small or limited data sets through theapplication of a library of well-formed regression models.These techniques, incorporated into a publicly available auto-mated tool, have been applied to hundreds of chillers. Thispaper describes the authors’ experience with these techniquesand provides insight into the lessons learned in their applica-tion.

OBJECTIVE

Chilled water plants are difficult to accurately model dueto the complex interaction of the plant components andcontrols. Nevertheless, mechanical designers, energy servicecontractors, and plant operators are faced with the challenge ofdetermining the appropriate mix of equipment, design criteria(such as design temperatures or flows), control setpoints, andcontrol algorithms to optimize their performance. For aspecific plant, issues such as the optimal condenser watercontrol setpoint, the cost-effectiveness of variable speeddriven chillers, and the life-cycle cost-effective condenserwater flow rate are highly dependant on the performance ofindividual pieces of equipment, the configuration of thepiping, and the control system design. These design and oper-

ational issues can only be answered accurately through simu-lation.

The accuracy of the simulations depends in part on thecalibration of the component models. Although the literatureis full of case studies, there are few general references thatdocument procedures or techniques (Haberl and Bou-Saada1998). This paper describes techniques to develop accuratecalibrated electric chiller models from short-term monitoredand/or manufacturers’ data. This research was part of a largereffort to provide design tools for the optimization of chilledwater plant performance (PEC 1998; Hydeman et. al. 1997).Project research on the propagation of uncertainty ineconomic analysis provided a target model accuracy of ±6%.1

According to the research, the uncertainty in cost benefit anal-ysis of chilled water plants is most strongly affected by thecomponent model accuracy and the accuracy in predicting theeconomic rate of return (Kammerud et al. 1999).2 Asdiscussed in this paper, the authors successfully developedcalibration techniques that meet or exceed this target uncer-tainty from readily accessible data.

METHOD

Prior to developing a reliable calibration technique, theauthors had to select a fundamental electric chiller model.Desired model attributes included availability for use in publictools, accuracy, availability of the data used to calibrate the

1. See Equations 13 and 16 for definitions of component modelaccuracy.

2. In this study, a 6% component model error propagated to a 40%error in the cost-benefit analysis. These numbers were based onthe evaluation of a new plant. The benefit to cost analysis uncer-tainty is expected to reduce for a simple chiller replacement.

THIS PREPRINT IS FOR DISCUSSION PURPOSES ONLY, FOR INCLUSION IN ASHRAE TRANSACTIONS 2002, V. 108, Pt. 1. Not to be reprinted in whole or inpart without written permission of the American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc., 1791 Tullie Circle, NE, Atlanta, GA 30329.Opinions, findings, conclusions, or recommendations expressed in this paper are those of the author(s) and do not necessarily reflect the views of ASHRAE. Writtenquestions and comments regarding this paper should be received at ASHRAE no later than January 25, 2002.

Mark Hydeman is a principal at Taylor Engineering, LLC, Alameda, Calif. Kenneth L. Gillespie, Jr. is a technologist at Pacific Gas and Elec-tric Company, San Ramon, Calif.

model, and ease of implementing the model in a central plantsimulation engine. A literature search uncovered four candi-date models.

• The model utilized in the simulation program DOE2(DOE 1980)

• The model presented in the ASHRAE primary systemstoolkit (Lebrun et al. 1996)

• The Gordon and Ng model utilized in ASHRAEResearch Project 827 (Brandemuehl et al. 1996)

• A generic regression model developed at a national lab-oratory as part of a commissioning project (Sezgan et al.1996)

As shown in Table 1, 45 data sets of manufacturers’ elec-tric chiller data were tested on three of the subject models. The45 data sets represented water-cooled centrifugal and screwcompressors from all of the major manufacturers. It alsoincluded centrifugal chillers with and without variable speeddrives. With these data sets and models the predicted powerroot mean squared (rms) error was smallest in the DOE2format, as shown in Table 1.

The authors selected the DOE2 electric chiller model asit was well documented, publicly available, and provided thedesired accuracy across a wide range of chiller configurations.The model from ASHRAE’s primary systems toolkit wasdeemed unsuitable because it required data about chillers thatwere not directly available to our target market of designprofessionals, ESCOs, or facility managers.3

Having selected a model, the team developed calibrationtechniques. To do this, they had to account for lack of readilyavailable performance data for electric chillers. Althoughmanufacturers once published extensive performance data ontheir electric chillers, with the advent of the ARI Standards550 and 590 (ARI 1992a, 1992b), they largely withdrew thatdata from publication. At present, performance data for mostelectric chillers are available only through the manufacturer’s

sales representatives with their proprietary software.4

Although it takes a broad range of data to accurately create achiller model through linear regression, a typical report froma manufacturer's representative will provide only 4 to 10points of data. Through experience we learned that moreperformance data could be readily obtained from the manu-facturers during the bid process if a data request form was writ-ten into the bid specifications (Taylor et al. 1999).

Further problems were encountered when evaluating theperformance of existing chillers in the field. The DOE2 modelrequires that the performance data be definitively separatedinto full-load and part-load conditions. Although in theory onecould monitor the signals to chiller controls (e.g., actuators forinlet vanes or slide valves), these signals are not readily avail-able in the field. A technique had to be developed to calibratemodels from mixed data sets.

In the end, the authors developed and tested two tech-niques: one for rich data sets where the data could be separatedinto full- and part-load conditions and another suitable foreither small data sets or any amount of mixed full- and part-load data. The first technique employs a standard least-squareslinear regression method to develop the model coefficientsdirectly from the data. The second technique employs a libraryof fully developed regression curves to extrapolate thechiller’s performance beyond the range of the calibration dataset. Both of these techniques have been tested over a range ofair-cooled and water-cooled electric chillers, includingscrews, scrolls, and reciprocating and centrifugal compres-sors. They have also been tested over a range of manufacturesand refrigerants and for centrifugal machines with variablespeed drives. Each of these techniques is described below.

DOE2 Electric Chiller Model

Prior to describing the calibration techniques, some back-ground is presented here on the basic format of the DOE2 elec-tric chiller simulation model.

The DOE2 model consists of the following three curves.

• CAPFT—a curve that represents the available capacityas a function of evaporator and condenser temperatures

• EIRFT—a curve that represents the full-load efficiencyas a function of evaporator and condenser temperatures

• EIRFPLR—a curve that represents the efficiency as afunction of the percentage unloading

In the case of the CAPFT and EIRFT, the model employsheat exchange fluid temperatures as a proxy for the refrigerantoperating pressure in the evaporator and condenser. Thechilled water supply temperature is used for the evaporatorconditions of all electric chillers. The condenser water supplytemperature is used for the condenser conditions of all water-

3. Details such as the surface area of the heat exchangers, compres-sor characteristics, and engineering data for the expansion device.

TABLE 1 RMS Power Prediction Error* of

Alternative Chiller Models†

* See Equation 13 for the definition of RMS power prediction error.† Note: Developed from 45 sets of manufacturer’s data representing a full rangeof electric chillers (Hydeman et al. 1997).

DOE2 Gordon and Ng Sezgen et. al.

Minimum 0.7% 1.6% 1.1%

Maximum 4.2% 37.0% 17.3%

Median 1.6% 4.1% 6.0%

Average 1.8% 5.1% 6.9%

4. This is the case for water-cooled centrifugal and screw chillers;data on packaged air-cooled chillers and water-cooled reciprocat-ing and scroll chillers are still available in either electronic or printform.

2 AC-02-9-1

cooled electric chillers. The outdoor dry-bulb temperature isused for the condenser conditions of all air-cooled electricchillers.

The format of the curves is as follows:

CAPFT = a1 + b1 × tchws + c1 × t2chws + d1 × tcws/oat

+ e1 × t2cws/oat + f1 × tchws × tcws/oat (1)

EIRFT = a2 + b2 × tchws + c2 × t2chws + d2 × tcws/oat

+ e2 × t2cws/oat + f2 × tchws × tcws/oat (2)

EIRFPLR = a3 + b3 × PLR + c3 × PLR2 (3)

(4)

where

tchws = the chilled water supply temperature (°F),

tcws/oat = the condenser water supply temperature (°F) for water-cooled equipment or the outdoor air dry-bulb temperature (°F) for air-cooled equipment,

Q = the capacity (ton),

Qref = the capacity (ton) at the reference evaporator and condenser temperatures where the curves come to unity,

PLR = a function representing the part-load operating ratio of the chiller.

ai, bi, ci, di, ei, and fi are the regression coefficients.

Using Equations 1 to 4, the power under any conditions ofload and temperatures can be found from the following equa-tion.

P = Pref × CAPFT(tchws, tcws/oat) × EIRFT(tchws, tcws/oat) × EIRFPLR(Q, tchws, tcws/oat) (5)

where

P = the power (kW) and

Pref = the power (kW) at the reference evaporator and condenser temperatures where the curves come to unity.

A given chiller performance model is defined by theregression coefficients (ai, bi, ci, di, ei, and fi), the referencecapacity (Qref), and the reference power (Pref).

TECHNIQUE 1: LEAST-SQUARES LINEAR REGRESSION MODEL

The first technique employs standard least-squares linearregression techniques. This technique is applicable to rela-tively large data sets (20 to 30 records) where the data fullycovers the range of conditions that will subsequently be simu-lated. Failure to fully represent the simulation conditions willlead to model extrapolation with highly unpredictable andinaccurate results. Further data requirements are imposed by

the mechanics of developing the 13 regression coefficients. Asummary of all the required data characteristics follows.

• The data can be separated into full-load and part-loadconditions.5

• The full-load data represent the range of condenserwater supply temperatures and chilled water supplytemperatures to be simulated by the model.

• There must be at minimum five distinct full-load datapoints representing at a minimum two distinct condi-tions of both chilled-water and condenser-water supplytemperatures.

• The part-load data represent the full range of unloadingto be simulated by the model.

• There must be a minimum of three distinct part-loaddata points.

The steps to developing the regression curve coefficientsin this technique are identical for air- and water-cooledmachines and are as follows.

1. Select a reference full-load condition from the data set (Qref,Pref). This can be any full-load point but is often the designcondition.

2. Calculate the values for the CAPFT and EIRFT functionsfor each of the full-load data records.

3. Use standard least-squares linear regression to calculate thecurve coefficients for the CAPFT and EIRFT functions.

4. Calculate the values for the PLR and EIRFPLR functionsfor the combined full- and part-load data.

5. Use standard least-squares linear regression to calculate thecurve coefficients for the EIRFPLR function.

These steps are described in detail below.

Selecting Reference Full-Load Conditions

The reference full-load conditions can be arbitrarilyselected from the full-load data as the choice will not have anyimpact on the accuracy of the model. Nonetheless, the resultswill be more intuitive for the user if the machine’s “designconditions” are used as the reference—the reference capacity(Qref) will equal the full-load capacity of the chiller at designconditions, and the design efficiency (kW/ton) will equal theratio of the reference power to capacity (Pref / Qref). Theseresults can be checked by hand, which helps reduce the risk ofmisapplication of the model resulting from user-introducederror.

Calculating Intermediate Curve Values for CAPFT and EIRFT

Development of the intermediate curve values is a prelim-inary step for calculating the actual CAPFT and EIRFT curvecoefficients. The CAPFT function (Equation 1) represents the

PLRQ

Qref CAPFT tchws tcws oat⁄,( )×----------------------------------------------------------------------------≡

5. As previously discussed, this is seldom the case with field-measured data.

AC-02-9-1 3

ratio of the full-load capacity at one set of temperatures to thereference capacity at the reference temperatures. This curve istypically domed with a peak at the evaporator and condenserconditions (saturated temperatures) for which the compressoris optimized. The curve will typically decrease at high lift.6

Figure 1 depicts a sample CAPFT curve as a function of evap-orator and condenser temperatures.

The CAPFT is calculated from the full-load performancedata as follows:

(6)

The EIRFT function is the ratio of the efficiency (kW/ton)of the fully loaded chiller at one set of temperatures over theefficiency of the fully loaded chiller at the reference condi-tions. This curve typically slopes from high kW/ton at high liftconditions to low kW/ton (high efficiency) at low lift. Figure2 depicts a sample EIRFT curve as a function of evaporatorand condenser temperatures.

The EIRFT is calculated as follows:

(7)

Equations 6 and 7 are applied to each full-load datarecord. These equations scale the curves to unity at the refer-ence conditions. This normalization of the curves is particu-larly useful in the reference curve method where curves aresubsequently scaled for use in modeling other chillers.

Calculating Coefficients for CAPFT and EIRFT Curves

With intermediate values calculated in the previous step,an input matrix of full-load data is created for use in a standardleast-squares linear regression routine.7 Figure 3 shows theCAPFT input matrix and the required fields.

The input matrix structure for EIRFT is identical to thatshown in Figure 3 for calculation of the CAPFT coefficients.

Developing Intermediate Curve Values for EIRFPLR

Having solved the full-load equations, we are now readyfor the EIRFPLR. Similar to the process applied to the CAPFTand EIRFT equations, the first step is to calculate intermediatecurve values for each record. This step can be performed oneither the part-load data alone or on the combination of thefull- and part-load data. The authors have tried both and foundthe combined data to produce the best results. For each dataentry, calculate the CAPFT and EIRFT as described in Equa-tions 1 and 2. Note that you are now using the actual regressioncoefficients and the values will differ slightly from those

6. Lift is a measure of the difference between the condenser andevaporator operating pressures (or saturated temperatures).

Figure 1 CAPFT curve.

CAPFTi

Qi

Qref----------=

EIRFTi

Pi

Qi-----

Pref

Qref----------

----------Pi

Pref CAPFTi×------------------------------------= =

7. A mathematical description of least-squares linear regression canbe found in most general texts on statistics. Many spreadsheetsand databases provide automated functions that will perform thistype of regression.

Figure 2 EIRFT curve.

Figure 3 CAPFT input matrix.

4 AC-02-9-1

produced in Equations 6 and 7. The part-load ratio (PLR) func-tion is the ratio of the present capacity over the available full-load capacity at the same temperature conditions. It is impor-tant to note that the divisor is not simply the reference full-loadcapacity, it is the available capacity at the same evaporator andcondenser temperatures as the part-load point. The PLR iscalculated as follows:

(8)

The EIRFPLR function is the ratio of power for the part-load condition to power for the full load at the same temper-ature conditions. It is parabolic, sloping from 100% power at100% part-load ratio (PLR) down to a higher percentagepower than the percentage load at low load conditions. Figure4 depicts a sample EIRFPLR curve.

The EIRPLR is calculated from Equation 5 as follows:

(9)

After calculating the CAPFT, EIRFT, PLR, and EIRF-PLR for each data record, we can calculate the final curvecoefficients.

Calculating Coefficients for the EIRFPLR Curve

As we did for the CAPFT and EIRFT functions, we nowcreate an input matrix of data that can be used in a standardleast-squares linear regression routine. Figure 5 shows theEIRFPLR input matrix and the required fields.

This completes the calculation of the curve coefficientsby the linear regression technique.

TECHNIQUE 2: REFERENCE CURVE METHOD

As previously stated, the reference curve method relies ona library of well-developed regression curves. Prior to devel-opment of this technique and bundling of the calculationengine with a library of curves in a publicly available tool(PEC 1998), users with insufficient data for the developmentof curve coefficients by the least-squares linear regressionmethod were forced to develop calibrated models usingpublicly available curve coefficients and fitting them to asingle point of data.8 The resulting models were often improp-erly fit due to the misunderstanding of the reference datapoint.9

The steps for the reference curve method are as follows.

1. Select the subset of curves in the library that are applicableto the target chiller.

2. Calculate the reference capacity for each curve from thesubset.

3. Calculate the reference power for each curve from thesubset.

4. Calculate the power prediction error for each curve fromthe subset.

5. Select the curve with the smallest error.

Each of these steps is discussed below.

Filtering Curves for Applicability

The purpose of this step is to select a subset of the avail-able chiller curves that reasonably represent the type ofmachine that the user intends to model. Possible parametersfor selection include the following:8. The DOE2 program (DOE 1980) only provides four electric

chiller models—hermetic centrifugal, hermetic reciprocating,open centrifugal, and open reciprocating. There were no curvesprovided for air-cooled chillers, screw or scroll compressors, orchillers controlled by variable speed drives. Coefficients for sevenair- and water-cooled chillers with screw, scroll, reciprocating,and centrifugal compressors were published in the first publicreview of ASHRAE/IES Standard 90.1-1989 (ASHRAE 1996)and subsequently in the California Energy Commission's ACMManual (CEC 1998).

PLRi

Qi

Qref CAPFTi×------------------------------------=

EIRFPLRi

Pi

Pref CAPFTi× EIRFTi×------------------------------------------------------------=

9. The published DOE2 electric chiller curves all go to unity at 44°Fchilled-water supply temperature and 85°F condenser watersupply temperature (DOE 1980). As this often differs from thedesign conditions, the user needs to provide a reference capacityand efficiency that are different from the design conditions. Theseadjustments are described in Equations 5 and 12. Review of utilityrebate applications and energy code compliance submissionsrevealed that these corrections were seldom made.

Figure 4 EIRFPLR curve.

Figure 5 EIRFPLR input matrix.

AC-02-9-1 5

• Type of compressor (centrifugal, screw, scroll, or recip-rocating)

• Type of condenser (air, water, or evaporatively cooled)• Type of unloading mechanism (staging, unloaders, slide

valve, inlet vanes, or inlet vanes with variable-speeddrive)

• Refrigerant• Range of data on which the curve was developed (evap-

orator temperatures, condenser temperatures, and loadratio)

• Manufacturer or model line• Chiller size

The authors have tested filtering by compressor type,condenser type, the unloading mechanism, the refrigerant, andthe range of data for which the curves were developed. Ofthese, all but the refrigerant appear critical to the accuracy ofthe results. The data range on which the curve was developedis critical as it protects the user from operating the model in aregion where its performance is extrapolated and highlyunpredictable. The challenge is that the user is often employ-ing this technique because they have a limited set of data. Thedata they bring to the calibration may not fully define the rangeof conditions that they will use to exercise the underlyingmodel in a simulation.

Calculating Reference Capacity

The reference capacity can be selected in one of severalways:

• Assume that at least one record represents the machineas fully loaded.

• Select the maximum percentage unloading for the dataset.

• Specify a reference fully loaded data record.

If you assume that the data set contains at least one recordwith full-load operation, the reference capacity can be calcu-lated as follows:10

for all data records. (10)

The assumption of full-load operation is probably accu-rate where the data source is the manufacturer. Where fieldmeasured data is employed, it may be better to allow the userto either define a maximum percentage full-load ratio(PLRmax) that they feel the chiller achieved during the measur-ing period or to define a rated full load capacity at specificconditions.

To calculate a reference capacity with a user-definedmaximum percentage full-load ratio, one would use thefollowing equation:

(11)

for all data records.To calculate a reference capacity with a user-defined rated

full-load capacity you would use Equation 12.

(12)

Calculating Reference Power

Reference power is calculated to minimize the predictedpower error. The format of root mean square error utilized bythe authors follows.

(13)

In Equation 13, the only unknown is the reference power(Pref). We seek the minimum of this error function, whichshould occur when the first derivative goes to zero.

(14)

where

This minimum can be found where the equation in thenumerator goes to zero. This happens where

. (15)10. This equation is derived from Equation 4 with the PLR set equal

to one. The maximum function guarantees that the selected refer-ence capacity will create PLRs across the data set of less than orequal to one with at least one record fully loaded.

Qref Maximum Qi

CAPFTi-------------------- =

Qref Maximum Qi

CAPFTi PLRmax×---------------------------------------------- =

Qref

Quser

CAPFT tchws_user tcws oat_user⁄,( )----------------------------------------------------------------------------------=

Error

Pref CAPFTi EIRFTi× EIRFPLRi×( ) Pi–×Pi

------------------------------------------------------------------------------------------------------------

2

i 1=

n

∑

n-------------------------------------------------------------------------------------------------------------------------------=

f Pref( )=

∂f∂Pref-------------

2 A× Pref B+×( )

2 A Pref2 B Pref C+×+××

---------------------------------------------------------------------- 0= =

A

CAPFTi EIRFTi× EIRFPLRi×( )Pi

---------------------------------------------------------------------------------

2

i 1=

n

∑

n---------------------------------------------------------------------------------------------------≡

B

2–CAPFTi EIRFTi× EIRFPLRi×( )

Pi---------------------------------------------------------------------------------

i 1=

n

∑×

n-------------------------------------------------------------------------------------------------------------≡

C 1≡

PrefB

2 A×------------ –=

CAPFTi EIRFTi× EIRFPLRi×( )Pi

---------------------------------------------------------------------------------

i 1=

n

∑

CAPFTi EIRFTi× EIRFPLRi×( )Pi

---------------------------------------------------------------------------------

2

i 1=

n

∑

---------------------------------------------------------------------------------------------------=

6 AC-02-9-1

This mathematical solution has been validated throughtests using binomial search and other numerical nonlinearequation solution methods as a basis for comparison. In alltests, Equation 15 produced superior results (lower error).

Calculating the Power Prediction Error

For each curve in the subset and with reference capacityand power calculated as described above, we calculate a totalerror on the predicted power across the data set. The equationused for the error calculation is shown in Equation 13. Thecurve with the lowest prediction error is selected for the finalmodel.

DISCUSSION

In the five years since these techniques have been devel-oped, they have been tested against hundreds of chillers. Ingeneral, the authors have found the least-squares regressiontechnique to produce results between ±1% and ±3% RMSerror. The reference curve method has been tested against bothsmall sets of manufacturer’s data and mixed sets of field datawith general results between ±3% and ±6% RMS error. In theretrofit of a 17,000-ton chilled water plant, these techniqueswere validated by field-measured data (Hydeman et al. 1999).Validated on four months of post-retrofit data, a regressionmodel of a newly installed chiller predicted its measuredperformance within –2.8%. This error was within instrumen-tation uncertainty tolerances.

An automated tool incorporating these techniques wastested by nearly 200 design professionals (PG&E 1999).These beta testers were subsequently surveyed. Of the groupthat used the tool on real projects, the majority expressed satis-faction with the results and rated the software superior to otherexisting software or techniques.11

The authors have uncovered several limitations with theunderlying DOE2 model. Chillers with variable speed drivesappear to have a relatively high error (~10% power prediction)at low loads (>20%) and condenser temperatures (below~60°F). This error reduces if the EIRFPLR curve (Equation 3)is expanded to a cubic equation. However, this does not fullysolve the problem. It is likely that the curve needs to be refor-mulated to include both temperature and load terms. As thiserror only occurs at low loads and temperatures, it generallyhas a minor impact on the chiller energy usage in an annualsimulation.

The DOE2 model does not directly address variation ofevaporator and condenser flows. The authors have tested themodel on widely varying evaporator flows (at the full range of

flows certified by the manufacturer) and have seen no signif-icant increase on the model RMS error (still within the ±1% to±3% range). Changes in the heat transfer coefficient in theevaporator appear to directly offset the changes in lift withvariations in flow.

Condenser flow variations are an entirely different story.Although the authors have not extensively tested the modelunder varying condenser flows, we expect the present formatwill be inadequate. The problem is not likely to impact mostplants, as there are very few plant configurations with variablecondenser flow systems. Modeling of different condenserwater flow regimes (e.g., 2 gpm/ton versus 3 gpm/ton) ispossible, but a different regression model for the chiller needsto be developed for each regime. The authors plan to test asimple reformulation of the CAPFT and EIRFT equationsusing the leaving condenser water temperature in place of theentering condenser water temperature for water-cooled equip-ment. With flooded condensers, the leaving temperature is abetter indication of heat exchanger approach.

Another source of error comes from the ARI Standard550/590 tolerances allowed in the manufacturer’s certifiedprograms. The authors have verified through zero-tolerancechiller bids with factory witness tests that all of the manufac-turers use these tolerances to overstate the capacity and under-state the efficiency of their chillers. By comparing bidsubmissions from the manufacturers with both zero tolerancedata and the data from the ARI certified programs, the authorshave noted routine derating of the capacity by 3% to 5% andderating of the performance (kW/ton) by the full toleranceprovided by the ARI Standard. This error does not apply tofield measured data.

Equation 13 is just one formulation of power predictionerror. An alternate form of error calculation for calibratedsimulations is the coefficient of variation of the root meansquared error (CVRMSE) (Haberl et al. 1998). Subsequent tothe development of the authors’ techniques, CVRMSE hasgained in popularity and is making its way into standards. It isgiven by Equation 16.

(16)

Haberl et al. (1998) also present mean bias error (MBE),which is a measure of the systematic error of a simulationmodel. This is given by Equation 17.

11. 54% rated the software “superior” and 32% rated it about the sameas existing software and techniques. The remainder did not havean opinion.

CVRMSE

Pref CAPFTi EIRFTi× EIRFPLRi×( ) Pi–×( )2

i 1=

n

∑

n------------------------------------------------------------------------------------------------------------------------------

Pii 1=

n

∑

n---------------

----------------------------------------------------------------------------------------------------------------------------------=

AC-02-9-1 7

(17)

Using the CVRMSE metric, the reference curve methodwould calculate a different reference power than the onepresented in Equation 15. The appropriate reference power forCVRMSE is given as follows:

. (18)

Both techniques are subject to bias from the selection ofcalibration data by the user. As presently employed in the auto-mated tool, both techniques apply equal weight to each recordin the data set. If the entire data set is not representative of theoperation of the chiller in the simulation model, the resultingmodel could be overly accurate in areas of little interest andnot very accurate in areas of extended operation. For the least-squares regression technique, the optimal data set would beequally distributed in a grid across the entire range of opera-tion (i.e., a data record starting at the extremes of temperaturewith equal distribution of samples from one extreme to theother). For the reference curve method, a binning of like datapoints with one or more representatives from each bin mayhelp to reduce any bias. Alternately, in either method the usercould be allowed to assign weights to each data record.

CONCLUSIONS

The techniques presented have proven robust throughfield verification and have achieved the author’s goals foracceptable electric chiller equipment model accuracy. Withadequate data and care to select a representative data sample,the least-squares regression method will always be the moreaccurate of the two techniques. The reference curve methoddepends on having and selecting a representative set of curvesfor the target machine. The automated tool referred to by theauthors now has over 200 curves in the library representing awide range of manufacturers, models, and design conditions.As the library of curves grows through the user base, in theoryit should improve accuracy through time.

The greatest challenges in the use of either technique arethe proper selection of data and the proper application of theresults. An automated tool must run a careful balance betweenexcessive data requirements on the user and protecting theuser from running the curves beyond their well-formedregions.

Overall, the authors have been very pleased with thesetechniques and are presently working to extend these to otherequipment models including cooling towers, gas-fired chill-ers, and thermally fired chillers.

ACKNOWLEDGMENTS

The CoolTools project was made possible throughpublic-goods funds administered by the Pacific Gas and Elec-tric Company on behalf of the ratepayers of California. Theauthors would like to thank Peter Turnbull the project managerfor his guidance and support, the CoolTools Project Advi-sory Board for their direction and support, Ernie Limperis,who programmed the automated tool, and Dr. RonKammerud, Steven Gates, Dr. Agami Reddy, Dr. Jeff Haberl,and Steve Taylor for their technical assistance and criticalreview.

REFERENCES

ARI. 1992a. ARI Standard 550-92, Standard for centrifugaland rotary screw water-chilling packages. Arlington,Va.: American Refrigeration Institute.

ARI. 1992b. ARI Standard 590-92, Standard for positive dis-placement compressor water-chilling packages. Arling-ton, Va.: American Refrigeration Institute.

ASHRAE. 1996. ASHRAE/IESNA 90.1-1989R, Energy stan-dard for buildings except low-rise residential buildings,First Public Review Draft, March 1996. Atlanta: Ameri-can Society of Heating, Refrigerating and Air-Condi-tioning Engineers.

Brandemuehl, M.J., M. Krarti, and J. Phelan. 1996. Method-ology development to measure in-situ chiller, fan andpump performance, Final report. Boulder, Colo.: JointCenter for Energy Management, publication JCEM TR/96/3.

CEC (California Energy Commission). 1998. The nonresi-dential alternative calculation method (ACM) approvalmanual for compliance with California's 1998 energyefficiency standards. Sacramento, Calif.: CaliforniaEnergy Commission, Publication Number P400-98-011,April.

DOE (Department of Energy). 1980. DOE 2 Reference man-ual, part 1, version 2.1. Berkeley, Calif.: Lawrence Ber-keley National Laboratories.

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8 AC-02-9-1

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