ASHRAE ST-16-010-2016 Control and Optimization of Vapor Compression Systems Using Recursive Estimation.pdf

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1、102 2016 ASHRAEABSTRACTBuilding operations account for approximately 40% ofUSenergyuseandcarbonemissions,andvaporcompressioncyclesaretheprimarymethodbywhichrefrigerationandair-conditioning systems operate. Representing a significantportion of commercial and residential building energyconsumption, va

2、por compression cycles are a target forimprovement in efficiency and savings. This paper presents adata-driven approach to find the optimal operating conditionsof single- and multievaporator systems to minimize energyconsumption while meeting operational requirements such asconstant cooling or const

3、ant evaporator outlet temperatures.Theproblemliesinthedevelopmentofacontrolarchitecturethatwillminimizetheenergyconsumedwithoutrequiringanymodelsof the system or expensive mass flow sensors. The application ofthepresentedapproachimprovesefficiencyandisdemonstratedin simulation and on an experimental

4、 system.INTRODUCTIONThe first working vapor compression cycle was built in1834 by Jacob Perkins (Balmer 2011). Initially, it was just aprototype, and it took another 20 years before James Harrisonbuilt a practical version to be used in a commercial ice-makingmachinein1854inAustralia(Bruce-Wallace196

5、6).Sincethen,vaporcompressionsystemshavespreadtoallpartsoflifetoday.The heart of refrigeration, vapor compression systems can befound in homes, restaurants, research labs, industrial facilities,automobiles, aircraft, and anything that has air conditioning.Probabilistically, in industrialized areas,

6、a vapor compressionsystem can be found within 100 yards of any single point. Dueto their prevalence, they represent a significant piece of energyused by vehicles and buildings. Specifically, buildings accountfor 40% of energy use in the United States (EIA 2012), with50% of that due to heating and co

7、oling (DOE 2011). As such,theyareagreattargetforimprovementsinefficiencytogenerateenergy savings.Therehasbeenmuchresearchintoefficientcontrolofvaporcompression systems over the past several decades. Braun et al.(1989) and Ahn and Mitchell (2001) formulated methodologiesfor the optimal control of chi

8、lled-water plants. They used aquadratic function of continuous control and uncontrolled vari-ablestorepresentthepowerconsumptionofacoolingplant.Theyused another quadratic function of the load and the differencebetween condenser and evaporator water temperatures to deter-mine the power consumption of

9、 a chiller. Also, they showed thatthe power of fans and pumps can be estimated with a quadraticfunctionofcontrolvariablesandflowrates.Whiletheywereabletoshowanincreaseinenergysavingsusingtheiroptimalcontrol,the quadratic models required significant amounts of data for thenumerous relationships. This

10、 data was required around the opti-malsetpoints,demandingtimeandprovingdifficulttoimplementon systems that experience large changes in operating conditionsor changing model parameters. Massie (2002) developed aneural-network-based controller to minimize cost for an ice ther-malstoragesystem.Thecontr

11、ollerhadfourneuralnetworks:oneasaglobalcontrollerandthreetomapequipmentbehavior.Whilethe controllers are self adapting and do not require tuning overtime,thereisasignificantsetuptimeassociatedwiththenetworkslearning of the relationships of the various inputs and outputs.Biquadratic polynomial models

12、 of chillers and cooling towers tooptimize condenser-water setpoints were presented by Austin(1993).Anobjectivefunctionforglobaloptimizationformulatedfrom mathematical models of the systems components andcontrolwasimplementedthroughanadaptiveneuralfuzzyinfer-Control and Optimizationof Vapor Compress

13、ion SystemsUsing Recursive EstimationChristopher Bay Avinash Rani BryanRasmussen, PhD,PEStudent Member ASHRAE Member ASHRAEChristopher Bay is a doctoral candidate and Byan P. Rasmussen is an associate professor of mechanical engineering at the Department ofMechanical Engineering at Texas Ahowever,th

14、erobustnessofsuchmethodsisanissueinprac-tice, especially in cases where systems operate at a range notcoveredbytrainingdata.Leducqetal.(2006)developedanonlin-ear predictive optimal control algorithm for vapor compressionsystems. The difficulty of this approach comes from the need fora nonlinear mode

15、l, which can be complicated to produce accu-rately.Also,thenumberofequationsrequiredcanbecomeexten-sive, increasing the difficulty of design and control.Larsen et al. (2003); Larsen et al. (2004); and Larsen andThybo (2004) investigated controlling setpoints and increasingefficiency of refrigeration

16、 systems through the minimization ofa convex cost function. While Larsens method (Larsen et al.2004;LarsenandThybo2004)showedanincreaseinefficiency,it required the use of refrigerant mass flow sensors, which canbeexpensiveanddifficulttoinstallonsystemsalreadyinplace.Yaoetal.(2004)showedenergysavings

17、withtheuseofoptimalsetpoints by defining a system coefficient of performance(SCOP) and maximizing this SCOP with optimal setpointsdetermined from empirical models. This method again requiresmodels developed around the optimal setpoints and does notperform well because system parameters change over t

18、ime.However, Yan et al. (2008) did propose an adaptive optimalcontrol model that uses recursive least squares to estimateparameters, a fuzzy forgetting factor for varying operatingconditions over time, and a genetic algorithm for optimizingusing a fitness function. This method showed moderate energy

19、savings, but requires the development of several functions andrules for proper performance.Theaimofthispaperistopresentasimplealgorithmthatcanbegeneralizedforanyvaporcompressioncycle,adaptsappropri-ately to changes in operating conditions, does not require expen-sive refrigerant mass flow sensors, a

20、nd maximizes systemperformance. The algorithm uses a data-driven approach toformulate a cost function for the power consumption of thesystem in terms of controlled variables, namely condenser andevaporator pressures, using recursive least-squares estimation.The remainder of the paper is organized as

21、 follows. First, a briefbackground on vapor compression systems will be given. Next,the application of the recursive least-squares estimation will beexplained.Thesimulationsthatwerecompletedwillbepresentedwithresults,followedbyadescriptionoftheexperimentalsystemand discussion of experimental results

22、. In closing, future workand conclusions from the work will be discussed.BACKGROUND ONVAPOR COMPRESSION SYSTEMSAs mentioned in the Introduction, vapor compressionsystems are extremely prevalent in todays developed society.Vapor compression cycles are used to provide refrigeration forhomes, commercia

23、l buildings, and industry processes, amongother applications. Refrigeration, or removal of heat, is accom-plished through the compression and expansion of a workingfluid, which in many cases is a refrigerant. A simple vaporcompression system is shown in Figure 1(a), and the pressure-enthalpy diagram

24、 of an ideal vapor compression cycle is shownin Figure 1(b). The cycle can be described as follows.Low-temperature, low-pressure refrigerant vapor passesthrough the compressor, being compressed to a high-tempera-ture,high-pressurevapor,shownasmovingfromPoint1toPoint2 on Figure 1(b). The refrigerant

25、then passes through thecondenser, where a fan moves air across the condenser. Themoving air helps transfer heat from the refrigerant to the outsideair. This is shown as the movement from Point 2 to Point 3 onFigure 1(b). This air could be replaced with any fluid, such aswater, in which case the fan

26、would be replaced by a pump. Exit-ing as a low-temperature, high-pressure liquid, the refrigerantmovesthroughtheexpansionvalvethatservestocontroltheflowof refrigerant to the evaporator. Passing through the valve, thepressureoftherefrigerantisreduced,resultinginalow-tempera-ture,low-pressuretwo-phase

27、fluid,shownbymovingfromPoint3 to Point 4 on Figure 1(b). Moving through the evaporator, therefrigerant absorbs heat from space to be cooled and evaporatesto a low-temperature, low-pressure vapor, moving from Point 4to Point 1 on Figure 1(b).Multi-evaporator systems are a variant of the basic vaporco

28、mpression system, where different configurations ofcomponents are used to effectively deliver different amountsofcoolingtodifferentcoolingzones.Theexperimentalsystemthat the algorithm was applied to is a multievaporator system,shown in Figure 7. Figure 1(b) shows multiple cycle pathsbetween Points 4

29、 and 1, representing multiple evaporatorsoperating at different cooling loads and at different pressures.APPLICATION OFRECURSIVE LEAST-SQUARES ESTIMATIONIn order to effectively and efficiently control a vaporcompression system, this paper uses recursive least-squaresestimation to identify the parame

30、ters of a cost function that isthen minimized to achieve optimal performance. Recursiveleast squares (RLS) is an identification algorithm that is mostfrequently used when parameters are to be identified fromrecurring, real-time data (Lewis et al. 2006). By using RLS,parametersareupdatedon-line,thusa

31、ccountingforchangesinoperating conditions of the system due to environmentalshifts, changes in efficiency of equipment with age, and otheruncontrolled variables. The cost function used in the optimi-zation will now be discussed.Published in ASHRAE Transactions, Volume 122, Part 2 104 ASHRAE Transact

32、ionsAssembly of Cost FunctionIn order to maximize system performance, we need todevelop a cost function that minimizes power consumed by thesystem while maximizing the cooling produced. The total powerconsumed by the system can be written as the sum of the powerconsumedbythecompressorandfans/pumpsse

33、rvicingtheevap-orator(s) and condenser. The power consumed by the fan/pumpmoving the fluid across the evaporator(s) and condenser areapproximatedasacubicfunctionoftheirspeedinrevolutionsperminute (rpm). The relationship between the power consumed Pand the fan/pump rotor speed is shown in Equation 1.

34、(1)For the experimental system, the pumps were controlledby a voltage that varied between 7.512 volts. As such, it wasnecessary to determine the relationship between the voltagesignal and the speed of the pumps. Relationships in the formshown in Equation 2 were found, where is a proportionalityconst

35、ant.(2)The compressor power was determined from empirical effi-ciencymapsprovidedfromthemanufacturer.Thesearetypicallypolynomialrelationshipsbetweencompressorpower,evaporatortemperatures, condenser temperatures, and compressor speed.The total power of the system is described by Equation 3:(3)The coo

36、ling produced by the system is determined bycalculatingtheheatabsorbedfromthesecondaryfluidmovingovertheevaporator.ThecoolingiscalculatedwithEquation4(4)where is the rate of cooling, is the mass flow rate of thesecondary fluid, Cpis the specific heat of the secondary fluid,and Teiand Teoare the temp

37、eratures at the inlet and outlet of theevaporator,respectively.Themassflowofthefluidiscalculatedusinganempiricalrelationshiprelatingthefan/pumpspeedandthe volumetric flow rate. This data is published by many manu-facturers. In the case that this information is not published, therelationship can be d

38、etermined experimentally. Using theempirical relationship, the system does not require the installa-tion of expensive refrigerant mass flow sensors. This results incost savings and ease of implementation on existing systems.Fortheexperimentalsystem,atestwasconductedtodeter-mineifthecoolingcalculated

39、fromthesecondaryfluidsidewasan accurate representation of the cooling calculated from therefrigerant side. During the test, the compressor was turned onand run at a constant speed. The cooling from both the second-ary fluid and the refrigerant side were measured and plotted in(a)(b)Figure 1 (a) An i

40、deal single vapor compression system. (b) Pressure-enthalpy diagram of an ideal vapor compression cyclewith multiple evaporators operating at different evaporation pressures.Pf3=P 3=Ptotal=PcompressorPcondenser pumpPevaporator pump+QmCpTeiTeo=QmPublished in ASHRAE Transactions, Volume 122, Part 2 AS

41、HRAE Transactions 105Figure 2. The initial discrepancy due to the transient nature ofthe system start-up is expected. Upon reaching steady state, theerror between the two calculations reduced to 0.2%.A relationship between the power consumed by thesystem and cooling produced is needed for the cost f

42、unction.This is easily realized with the use of the systems coefficientof performance (COP). Although the overall objective is tomaximize the SCOP, the solution can be formulated as a stan-dardminimizationproblembydefiningtheinversecoefficientof performance (ICOP), shown in Equation 5.(5)Finding the

43、 ICOP from measured values is simple on afully instrumented laboratory system but not on systems usedin practice. Thus, an estimation of the ICOP needs to be madefrom measurements that are frequently available on systemscurrently in use and then related to the operating conditionsused by the compone

44、nt controllers. In this case, the ICOP iscalculated from the available measurements usingEquations 3 and 4. Then this ICOP is correlated to the systempressures and the required cooling load. Equation 6 shows asimple polynomial estimation of ICOP using the evaporatorpressure Pe, condenser pressure Pc

45、, and the cooling Q.(6)To validate the estimation given by Equation 6, anothertest was conducted where the ICOP of the experimentalsystemwascalculatedfromEquations5and6forafixedcool-ing load. The measured ICOP from Equation 5 is compared tothe predicted ICOP from Equation 6 in Figure 3.The error bet

46、ween the measured values and thepredicted values is shown to be approximately within thebounds of 10%. This level of accuracy is satisfactory forpredictingtheICOPforusewiththerecursiveleast-squaresalgorithm in this paper.In order to minimize the systems ICOP, the optimalvalues of Peand Pcneed to be

47、determined for a given coolingload. This occurs where the partial derivatives of Equation 6are zero:(7)(8)Using a steepest-descent method, the optimal setpointsfor the evaporator and condenser pressures can be determinedrecursively. The steepest-decent method calculates the largestgradient in the do

48、wnward direction at the systems currentposition on the ICOP curve. A step can then be taken in thatdirection where the step size can be adjusted based on theparticular systems capability to respond to changingsetpoints. Care must taken to choose an appropriate step size.Too large, and the system can

49、 experience abrupt changesresultinginunwantedbehavior.Toosmall,andthesystemmaynot converge to the optimal setpoints in a timely manner. Tofurther reduce unwanted oscillations in the setpoints deter-mined by the algorithm, a low-pass filter is added to eachsetpoint, shown in Equations 9 and 10 where k is the samplingindex and is a weighting factor:(9)(10)Figure 2 Comparing the cooling calculated from thesecondary