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Optimal coordination of heat pumpcompressor and fan speeds andsubcooling over a wide range of loadsand conditionsTea Zakula a , Peter Armstrong b & Leslie Norford a ca Building Technology Program, Massachusetts Institute ofTechnology (MIT), Cambridge, MAb Mechanical Engineering Program, Masdar Institute of Science andTechnology, Abu Dhabi, United Arab Emiratesc Department of Architecture, Massachusetts Institute of Technology(MIT), Cambridge, MAAccepted author version posted online: 10 Aug 2012.Version ofrecord first published: 14 Nov 2012.

To cite this article: Tea Zakula, Peter Armstrong & Leslie Norford (2012): Optimal coordination ofheat pump compressor and fan speeds and subcooling over a wide range of loads and conditions,HVAC&R Research, 18:6, 1153-1167

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Optimal coordination of heat pump compressor and fanspeeds and subcooling over a wide range of loads

and conditionsTea Zakula,1,∗ Peter Armstrong,2 and Leslie Norford1,3

1Building Technology Program, Massachusetts Institute of Technology (MIT), Cambridge, MA2Mechanical Engineering Program, Masdar Institute of Science and Technology, Abu Dhabi, United Arab Emirates

3Department of Architecture, Massachusetts Institute of Technology (MIT), Cambridge, MA∗Corresponding author e-mail: tzakula@mit.edu

Advanced cooling systems with high-temperature cooling (sensible only), night precooling, and highlyefficient variable-speed compressors, fans, and pumps can benefit from on-line model predictive controlto find and continually update optimal daily (or weekly) precooling sequences. As a part of the modelpredictive control, it is desirable to use optimized plant-specific control laws to match compressor, fan, andpump speeds to required capacity. A previous work presented a modular heat pump model that simulatedsteady-state performance over wide ranges of lift (external pressure ratios from <1 to 6) and capacity(10:1 turndown) that could be explored in the search for optimal solutions. This article describes theadaptive grid search technique used to map optimal heat pump performance as a function of the capacityand indoor and outdoor temperatures. The grid search finds optimal condenser and evaporator airflowsand optimal subcooling at each operating point. The method is illustrated for a number of cases, includingtwo-compressor systems and refrigerants R410A, R600 (propane), and R717 (ammonia). The non-linearityof optimal fan-speed control laws is demonstrated. The impact of zero subcooling with respect to optimalsubcooling is assessed for the single compressor machines. The specific power at optimal fan speeds, asa function of capacity and indoor–outdoor temperature, is compared for R410A, propane, and ammonia-charged machines. Finally, the question of optimal sizing of optimally controlled variable-speed heatpumps is explored, and it is shown that modest oversizing is desirable. These findings suggest that therelative sizing of heat pump components—compressor, compressor motor, condenser, and evaporator—aswell as the sizing of the heat pump itself relative to design load, may benefit from a thorough reassessmentof current practice.

Introduction

One attractive way to achieve efficient cool-ing in buildings is to combine efficient, optimallycontrolled thermal energy storage (TES); high-temperature distribution, such as chilled beams; and

Received February 22, 2012; accepted June 25, 2012Tea Zakula, Student Member ASHRAE, is PhD Candidate. Peter Armstrong, PhD, Member ASHRAE, is Associate Professor.Leslie Norford, PhD, Member ASHRAE, is Professor.

a cooling plant that operates efficiently over a widerange of lift and part-load fractions. This combi-nation of components and controls may be called alow-lift cooling system (LLCS). In simulations withidealized TES, annual cooling system energy sav-ings of up to 75% were found compared to a baseline

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HVAC&R Research, 18(6):1153–1167, 2012. Copyright C© 2012 ASHRAE.ISSN: 1078-9669 print / 1938-5587 onlineDOI: 10.1080/10789669.2012.713832

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ASHRAE 90.1-2004 VAV system (Jiang et al. 2007;Armstrong et al. 2009a, 2009b; Katipamula et al.2010). Initial verification of these results was pro-vided by Gayeski (2010) for a typical summer weekfor Atlanta and Phoenix in a climate chamber exper-iment using the identical outdoor unit (compressor,condenser, and fan) for both the low-lift and baselineconfigurations. Since optimization and control arecritical for maximizing the benefits of the LLCS,an important part of the low-lift cooling technol-ogy is model predictive control, an algorithm thatoptimizes the system’s operation given a thermal re-sponse model of the building and weather forecasts.Another key optimization question is the coordina-tion of condenser and evaporator fan and/or pumpspeeds with compressor speed to satisfy any givenload under any given conditions. This process isknown in the HVAC literature as static optimiza-tion (ASHRAE 2011, Chap. 42). Although the useof empirical data (Gayeski 2010) is unavoidable formodeling building transient thermal response, heatpump performance can, in principle, be more easilyand reliably characterized by the use of engineeringmodels (Gayeski et al. 2011; Verhelst et al. 2012).However, heat pump manufacturers’ data are oftenonly available for a limited range of operating con-ditions and capacity. This makes it nearly impos-sible to analyze systems that operate outside thoseconditions or systems that are still commercially un-available. Therefore, to perform static optimization,a heat pump model that is accurate, yet computa-tionally inexpensive, is required.

The most detailed physics-based heat pumpmodel found in the static optimization literature isthat developed by Armstrong et al. (2009a) (Jianget al. 2007). Two optimization variables are theevaporating and condensing temperatures, whichcan then be related to the optimal evaporator fan,condenser fan, and compressor speeds. The modelassumes constant evaporating and condensingtemperatures without evaporator superheating, con-denser subcooling, or heat pump pressure drops. Italso assumes constant conductance (U-value) forthe evaporator and condenser, independent of refrig-erant and air/water flow rates. Zakula et al. (2011)showed that even neglecting pressure drops can leadto serious errors in power consumption predictions,and therefore, this model would need to be extendedfor more accurate performance predictions. Thereare numerous physical models found in the liter-ature that do not perform optimization but that docalculate steady-state heat pump performance and,

hence, could potentially be adopted for optimiza-tion purposes. However, they vary from complexmodels that are computationally expensive andrequire a large number of input parameters tomodels that are similar to Armstrong et al.’s (2009a)model, in that they are relatively simple and fastbut do not take into account certain importantphenomena. A more detailed literature review onheat pump modeling is given in Zakula (2010).

Though they do not describe the optimization ofa heat pump’s performance, related works on the op-timization of large chiller plants can be found in theliterature. Lau et al. (1985) developed a TRNSYS(transient system simulation program; Klein et al.2010) model to analyze different control strategiesfor an existing chiller plant with four centrifugalchillers, a cooling tower, and chilled water tanks.For a given cooling load and wet-bulb temperature,the cooling tower fan speed, condenser pump flow,and number of chillers were optimized for mini-mal power consumption. The power consumptionof each chiller is characterized as a function of thecooling load, leaving chilled water temperature, andleaving condenser water temperature using curvefits to manufacturer’s data. Braun et al. (1987a) in-vestigated the performance and optimal control of alarge chiller plant equipped with a cooling tower. Asimplified model was used to find near-optimal con-trol, with the cooling tower airflow and condenserwater flow rates as the control variables. For an indi-vidual chiller, measured data from the existing plantwere fit to curves that define the chiller power as afunction of the cooling load and temperature dif-ference between the leaving condenser and chilledwater flows. In subsequent work (Braun et al. 1989),the system was extended to include the chilled wa-ter loop and the air handlers with the five indepen-dent control variables of supply air set temperature,chilled water set temperature, cooling tower airflow,condenser water flow, and the number of operatingchillers. Braun’s more recent work on chiller plantoptimization (Braun 2007) analyzed near-optimalcontrol strategies for a hybrid cooling plant pow-ered by electricity and natural gas. The optimizationobjective function was the operating cost, which in-cludes the electrical and gas energy cost, electri-cal demand cost, and maintenance cost. For a givencooling load, return air (zone) temperature, wet-bulbtemperature, and state of charge, the model of Kingand Potter (1998) optimized chilled water and sup-ply air temperature set-points for the lowest systempower consumption, including the chiller, pumps,

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Figure 1. Grid search optimization algorithm.

cooling-tower fan, and supply- and return-air fans.Similar to the previous chiller plant optimizationmodels, performance of an individual chiller is cap-tured using curve fits to manufacturer’s data. Jiangand Reddy (2007) developed a general methodologyfor optimization of HVAC&R plants and showed itsapplication on a cooling plant that consists of threechillers (one of which is absorption chiller) and three

variable-speed cooling towers. The semi-empiricalGordon–Ng model and effectiveness-NTU (Num-ber of Transfer Units) model were used to modelchillers and cooling towers. Given total cooling loadand required supply temperature and flow rate, theload allocated to each chiller and cooling tower out-let water temperature of each chiller were used asoptimization variables.

The objective of this study is to use optimizationto better understand the extent to which heat pumpdesign and control improvements can impact theannual energy use of advanced cooling systems.

Model description

Optimization is performed using the steady-stateheat pump model developed by Zakula et al. (2011)that can simulate the performance of different heatpump types, such as air-to-air heat pumps and air-and water-cooled chillers. Two evaporator sub-models that describe finned-tube air-to-refrigerantand brazed-plate water-to-refrigerant heat exchang-ers are modeled using the heat balance equations andε-NTU method for evaporating and superheatingregions. The condenser is modeled in a similarmanner, except it consists of desuperheating,condensing, and subcooling regions. The heattransfer coefficients are calculated separately forthe air/water stream and two-phase and single-phaserefrigerant flows. The compressor sub-model calcu-lates the compressor speed, compressor power, anddischarge temperature for a given mass flow rate,compressor inlet state, and outlet pressure. The shaft

Figure 2. Search loop for the optimal airflow (B-grid) (color figure available online).

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Figure 3. Grid search step adaptation (color figure available online).

speed is calculated using a volumetric efficiencymodel, and the compressor power is calculated asthe power required for isentropic work, correctedby the combined efficiency that takes into accountlosses in the compressor and motor. The compressoroutlet temperature is calculated from the compres-sor heat balance, through which the lubricating oil isassumed to pass in a constant mass fraction. A liquidreceiver is assumed to maintain the necessary chargebalance, which is not modeled. The heat pumpmodel takes into account pressure drops in refrig-erant piping and heat exchangers, the dependenceof heat transfer coefficients on flow rates, super-heating in the evaporator and desuperheating, and

subcooling in the condenser. A modular approachoffers the possibility of choosing between differentsimulation options (level of complexity) and makesthe model easy to expand and customize. The modelcan be used for a single-, two-, or variable-speedcompressor, a single compressor, multiple parallelcompressors, evaporators or condensers, as well asfor different refrigerants. The inverse heat pumpmodel with compressor frequency as an inputhas also been developed and is used to optimizea heat pump with a two-speed compressor. Thetwo-speed heat pump serves as a base case forannual energy consumption assessments presentedlater.

Figure 4. Optimal (a) evaporator and (b) condenser airflow as a function of part-load ratio for To = 30◦C (86◦F) (color figureavailable online).

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Figure 5. Optimal compressor frequency as a function of part-load ratio for To = 30◦C (86◦F) (color figure available online).

The optimization input parameters are the cool-ing load (Q), zone temperature (Tz), and outsidetemperature (To), and the optimization variables arethe evaporator airflow rate (Vz), condenser airflowrate (Vo), and condenser area fraction devoted tosubcooling (ysc). If one wants to optimize only oneor two variables, the other variables need to be givenas an input; e.g., one may want to know the perfor-mance impact of the optimal subcooling as opposedto zero subcooling, in which case a zero subcoolingarea is specified and the condenser and evaporatorair or water flow rates are optimization variables.All other heat pump operating variables arefunctions of the optimization variables; in partic-ular, for optimal control, one is mainly interestedin the optimal evaporator fan speed, condenserfan speed, and compressor speed. One may alsobe interested in the related refrigerant mass flowrate, evaporating temperatures and pressures, con-densing temperatures and pressures, suction anddischarge state, subcooling temperature difference,and total power consumption.

The optimization algorithm, using the gridsearch method shown in Figure 1, has the advan-tage of avoiding gradient calculations. Gradientcalculations can be computationally expensiveand challenging for this type of problem due tonon-linearities and possible convergence issues.Furthermore, if a grid step is appropriately chosen,the grid search is more reliable in finding the globalminima than the gradient search method.

For each set of conditions (Q, Tz, To), there aretwo loops, the outer loop for the optimal subcooling

Figure 6. Optimal subcooling as a function of part-load ratio forTo = 30◦C (86◦F) (color figure available online).

area ratio search and the inner loop for the opti-mal flow rates search. The initial 3-by-1 grid (A-grid) and 3-by-3 grid (B-grid) are created for theouter and inner loops, respectively. First, the totalpower consumption is calculated for each of the nineB-grid points and ysc = ysc{1}. If the lowest poweris anywhere on the B-grid boundaries, the grid isextended according to Figure 2, and total powersare evaluated for new points. The process contin-ues until the lowest power is in the middle B-gridpoint (B{2,2}), in which case the algorithm movesto the second A-grid point (ysc{2}). Similar to theB-grid, if the sub-optimization finishes for all threeA-grid points, and the lowest power is on the A-grid

Figure 7. Specific power as a function of part-load ratio forTo = 30◦C (86◦F) (color figure available online).

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Figure 8. Optimal: (a) zone and evaporating temperature difference and (b) condensing and ambient temperature difference as afunction of part-load ratio for T = 30◦C (86◦F) (color figure available online).

boundaries, the A-grid extends until the optimum isat the middle A-grid point (A{2}).

The optimization process is further acceleratedwith the step adaptation, in which after finding theoptimal variables with a larger step, a new 5-by-5B-grid is created using a smaller step (half the largestep) and Vz opt and Vo opt as the central grid points(Figure 3). Since the power consumption in 9 pointsis already known from the previous (large step) gridsearch, only 16 additional function evaluations areperformed. The point with the lowest power is as-signed as the final optimal point (new Vz opt and

Vo opt). The same is applied for the optimal subcool-ing search.

Performance map results

For a given cooling rate, outside, and zonetemperature, the result of the static optimizationprovides the optimal set of the evaporator andcondenser airflow rates, compressor speed, and sub-cooling for which the power consumption will bethe lowest. To show the broad utility and potential

Figure 9. Optimal evaporator and condenser airflow-to-compressor-frequency ratio over a wide range of conditions and loads (colorfigure available online).

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benefits of optimization using a component-basedmodel the optimization is performed for severaldifferent scenarios:

a) air-to-air heat pump with a single compressor(variable-speed rolling-piston compressor) andR410A as a refrigerant;

b) air-to-air heat pump with two parallel com-pressors, evaporators, and condensers (twovariable-speed rolling-piston compressors) andR410A as a refrigerant;

c) air-to-air heat pump with a single compressor(variable-speed rolling-piston compressor) andammonia (R717) as a refrigerant; and

d) air-to-air heat pump with a single compressor(variable-speed rolling-piston compressor) andpropane (R600) as a refrigerant.

The optimization is performed with a 0.025 m3/s(53 cfm) step for the airflows (0–0.3 and 0–0.7 m3/s[0–635.7 and 0–1483.2 cfm] airflow range for theevaporator and condenser fan, respectively) and a0.001 step for the condenser area ratio devoted tosubcooling.

For the heat pump with a variable-speed com-pressor and R410A as a working fluid, the optimalset of evaporator and condenser airflow rates (Fig-ure 4), compressor speed (Figure 5), and subcool-ing (Figure 6), for which the power consumption forcooling (Figure 7) will be the lowest, is shown forthe outside temperature To = 30◦C (86◦F). Figure 4shows that the evaporator and condenser airflows area strong function of part-load ratios. Furthermore,it was noticed that when optimizing the airflows,the parameter indirectly being optimized is the tem-perature difference (Figure 8). For a given coolingrate, the optimizer tries to maintain the optimal tem-perature difference on the evaporator (between theevaporating and air temperature) and the condenser(between the condensing and air temperature), re-gardless of the zone or ambient temperature. FromFigure 7, which shows the specific power as a func-tion of part-load ratios, it can be seen that the heatpump efficiency increases with lower part-load ra-tios. This raises the question of the appropriate heatpump “external sizing,” since with modest oversiz-ing, the heat pump will run at higher efficiencies.However, because there is a cost penalty associatedwith a size increase, both size and initial cost needto be carefully balanced. The optimal “external siz-ing” for the lowest energy consumption is discussed

Figure 10. Optimal subcooling over a wide range of conditionsand loads (color figure available online).

later in this article. In addition to “external sizing,”which refers to selecting a heat pump capacity ap-propriate for the load, another interesting topic tobe addressed is “internal sizing,” which is the sizingof each heat pump component, primarily the evapo-rator, condenser, and compressor. Although not in-cluded as a part of this analysis, the presented heatpump optimization algorithm could be extended tothe component-sizing problem given a joint distri-bution of cooling load and operating conditions.

The optimal results are presented here over awide range of loads (0.1–1 part-load ratio) andtemperature differences (0◦C–30◦C [0◦F–54◦F]difference between the zone and ambient). Al-though Figure 4 indicates nearly linear trends for

Figure 11. Specific power over a wide range of conditions andloads (color figure available online).

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Figure 12. COP relative difference: (a) optimal versus zero subcooling case and (b) optimal versus fixed air flow and zero subcoolingcase, all for Tz = 22◦C (71.6◦F) (color figure available online).

the evaporator and condenser airflows, it can beseen from Figure 9 that the same is not true forthe optimal airflow-to-compressor-frequency ratio.Similar to optimal airflows, Figure 10 shows astrong dependence of the optimal subcooling on thepart-load ratio. Finally, it can be seen from Figure 11that the optimal specific power is almost solelya function of part-load ratio and (for a moderaterange of evaporating temperature) of temperaturedifferences between the zone and outside. Thisis in agreement with the study done by Braunet al. (1987b) that concluded that the chiller powerconsumption is primarily a function of cooling loadand the temperature difference between the leavingcondenser and chilled water streams. It is importantto point out that the optimal evaporator airflows athigh part-load ratios are significantly higher thanthe maximum feasible for the specific heat pump(the airflow rate at the maximum evaporator fanspeed was around 0.15 m3/s [317.8 cfm] for thereal heat pump). Because manufacturers primarilyoptimize this type of heat pump for simultaneoussensible cooling and dehumidification, in which theheat pump would run with much lower evaporatorfan speeds, their use for only sensible coolingresults in fan speeds that are far from optimal.The consequence of this for the total energyconsumption is described later in an example.

Subcooling has been used as one of the opti-mization variables considering that some heat pumpmanufacturers already do control subcooling byplacing an additional valve between the condenser

and liquid receiver. The impact of optimal subcool-ing with respect to zero subcooling is assessed. Fig-ure 12a shows that the coefficient of performance(COP) differences, although relatively small, in-crease for higher part-load ratios and temperaturedifferences. As a result, the average annual COPof a system that operates at lower part-load ratioswould be much less influenced by non-optimizedsubcooling compared to prevailing systems that de-liver most of their annual cooling effect at high

Figure 13. Specific power as a function of part-load ratio forsingle-compressor machine (solid line) and two-compressor ma-chine with each compressor sized for half Qmax (dashed line),all for To = 30◦C (86◦F). Specific power is the same for bothmachines above ∼50% part-load (color figure available online).

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Figure 14. Economizer mode (dashed line) and compressor mode (solid line) for: (a) To = 15◦C (59◦F) and (b) To = 20◦C (68◦F)(color figure available online).

part-load ratios. The COP relative differences inFigure 12a are calculated as

COP relative difference

= COPoptimized − COPzero subcooling

COPoptimized. (1)

An additional analysis was done to assess theimpact of the optimized airflows with respect tofixed airflows. Fixed airflows were set to the maxi-mum feasible for the specific heat pump (0.15 and0.77 m3/s [317.8 and 1631.5 cfm] for the evaporatorand condenser fan, respectively), and the subcoolingwas set to zero. The results (Figure 12b) show sig-nificant differences in the specific power betweenthe optimal and non-optimal cases. This leads tothe conclusion that optimizing the airflows playsa significant role in the heat pump performanceand that heat pumps optimized for simultaneoussensible cooling and dehumidification can signifi-cantly underperform when used for sensible coolingonly. The COP relative differences in Figure 12b arecalculated as

COP relative difference

= COPoptimized − COPnon-optimized

COPoptimized. (2)

It can be seen in Figure 7 that due to inverterlosses, the specific power increases for part-load ra-

tios less than 0.2. The inverter losses account fora small portion of the total power at higher part-load ratios but become more important as the cool-ing load decreases. Recently, manufacturers havestarted to design heat pumps with two, rather thanone, variable-speed compressors, which can help toavoid high inverter losses at lower part-load ratiosand result in better overall heat pump performance.This case has been analyzed by optimizing the heatpump with 2× scaling of evaporators, condensers,and piping and two variable-speed rolling-pistoncompressors. The optimization algorithm tests bothcases and decides if it is more efficient to runboth or just one of the compressors. The resultsof the analysis (Figure 13) show that the heat pumpdesign with two parallel compressors can signif-icantly improve the performance at low part-loadratios by running only one of the two inverter-compressor subsystems.

When the outside temperature is lower than thezone temperature, a heat pump can run in an econo-mizer mode, in which only the evaporator and con-denser fans are running and the compressor is turnedoff. In some cases, liquid may flow from the con-denser to evaporator by gravity, while in others, theflow may be assisted by a small efficient hermeticpump (New Technology Demonstration Program[NTDP] 1994). Having a lower outside temperature,however, does not necessarily imply that running inthe economizer mode is more energy efficient. Forsmall temperature differences relative to a part-loadratio, the total sum of the fan and compressor power

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Figure 15. Combined COP (compressor and economizer mode)for Tz = 22◦C (71.6◦F) (color figure available online).

can be lower than the sum of fan powers when oper-ating in the economizer mode due to high fan speeds.Therefore, without having detailed heat pump mapsfor both modes (Figure 14), it is hard to develop anappropriate control algorithm that regulates switch-ing between the two. Figure 14a illustrates the slopediscontinuity, where the optimal compressor-modeand economizer-mode performance surfaces inter-sect, and Figure 14b shows how the intersectionmoves for different outside temperatures. The com-bined heat pump map is created (Figure 15) usingthe optimization results and selection of the leastpower mode when To < Tz.

The heat pump model and the optimization al-gorithm can also be used to analyze how changing

refrigerants might influence the heat pump perfor-mance. When switching to refrigerants other thanR410A, the heat pump geometry (primarily the com-pressor displacement volume and refrigerant piping)was adapted to account for differences in density andheat of vaporization. The compressor displacementvolume is scaled using

Drefrigerant = DR410AρR410A

ρrefrigerant

hlg,410A

hlg,refrigerant. (3)

The largest pressure drop in the evaporator thatcorresponds to the 4◦C (7.2◦F) drop in the satura-tion temperature (at Tz = 18◦C [64.4◦F], To = 45◦C[113◦F], and Q = Qmax) for R410A was used to findthe appropriate scaling factors for the refrigerantpiping. The scaling factors found to give the pres-sure drops that correspond to the same 4◦C (7.2◦F)drop in the saturation temperature are 0.90 for am-monia and 1.37 for propane. The relative differencesin the COP for different refrigerants are shown inFigure 16. The COP relative differences in Figure 16are calculated as

COP relative difference

= COPrefrigerant − COPR410A

COPR410A. (4)

The results show higher COP values for propaneand ammonia compared to R410A, which is inagreement with the results of theoretical perfor-mance analysis for different refrigerants (ASHRAE

Figure 16. COP relative difference for: (a) ammonia versus R410A case and (b) propane versus R410A case, all for Tz = 22◦C(71.6◦F) (color figure available online).

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2009; Chap. 29, Table 9). Differences in the COP ofa theoretical Rankine cycle are caused by throttling-induced irreversibility and additional work requiredfor the superheated-vapor horn (Domanski 1995).For refrigerants operating closer to the critical point,as in the case of R410A, these irreversibilities arehigher. Differences due to suction density and en-thalpy of vaporization have been largely eliminatedby scaling the heat exchanger (HX) channels, com-pressor displacement, and interconnect piping.

Annual performance results

This section demonstrates how switching to aheat pump with a single two-speed compressor,two parallel compressors, different refrigerants, ora heat pump with non-optimal airflows or non-optimal subcooling can influence annual energyconsumption. Cooling loads used for this analysisare the results of the study done by Armstrong et al.(2009b) for a typical office building in Chicago. Theload scenarios are presented in terms of full-load-equivalent operating hours (FLEOH), as a functionof part-load ratio and outside temperature. In Fig-ure 17 Load distribution A represents instantaneousloads for a building without TES, while load dis-tribution B represents loads that are shifted towardlower part-load ratios and lower outdoor temper-atures using TES and daily optimized precooling.Load distribution B assumes hydronic radiant cool-ing, ideal thermal storage, a variable-speed chiller,and a dedicated outdoor air system for ventilation.

Although the shifted load distribution will be af-fected by the exact performance characteristic of theheat pump or chiller, it is assumed that the distri-bution presented here is representative of the classof static-optimized machines with high-turndownvariable-speed compressors, pumps, and fans.

The two cooling load scenarios may be convolvedwith several different heat pump configurations toestimate the influence of heat pump design on theannual energy consumption. Performance maps arecreated for the following heat pumps:

1. variable-speed compressor heat pump with opti-mized airflows and subcooling;

2. variable-speed compressor heat pump with opti-mized airflows, assuming zero subcooling;

3. variable-speed compressor heat pump with non-optimized airflows and subcooling (assumingzero subcooling and maximal airflows Vz =0.15 m3/s, Vo = 0.77 m3/s);

4. two-speed compressor heat pump (assumingtwo-speed fans); and

5. dual variable-speed compressor heat pump withoptimized airflows and subcooling.

For the heat pump with a two-speed compressor,subcooling is optimized for compressor speeds off = 0.5f max and f = f max. For this case, the fanswere also assumed to be two-speed, with airflowsset at the optimal values found for the variable-speed heat pump (Case 1) at 0.5Qmax (used at thelow-frequency compressor speed) and Qmax (usedat the high-frequency speed). For part-load ratiosother than Q = 0.5Qmax and Q = Qmax, it has been

Figure 17. Sensible cooling load distribution for: (a) Case A (without TES) and (b) Case B (with TES) (color figure available online).

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Table 1. Annual energy savings for different cases with R410A.

Optimized versus zero subcooling (B1 versus B2) (Ezero sc – Eoptimized)/Ezero sc 0.4%Optimized versus non-optimized (B1 versus B3) (Enon-optimized – Eoptimized)/Enon-optimized 49.6%Dual versus single compressor (B1 versus B5) (Esingle – Edual)/Esingle 11.9%

assumed that the compressor cycles between f = 0,f = 0.5f max, and f = f max. The specific power(1/COP) for those cases is calculated using equa-tions given in Armstrong et al. (2009a):

tH = 2Q

Qmax−1, (5)

where tH is the high-speed duty fraction: if tH < 0,

if tH < 0,

1

COP= 1

COP (at To and f = 0.5fmax); (6)

else

1

COP= (1 − tH )

1

COP (at To and f = 0.5 fmax)

+ tH1

COP (at To and f = fmax). (7)

For cases 1 through 5, additional optimizationwas done for the refrigerant-side economizer mode.Combined heat pump maps, similar to the onein Figure 15 (compressor mode and economizermode), were created using an algorithm that decideswhich mode uses less power for a given cooling load,zone, and outside temperature.

The annual energy consumption is calculatedby convolving FLEOH distributions (distribution Aor B) with different heat pump performance maps(Case 1–Case 5) as follows:

E =∑

i

∑

j

FLEOHi, j1

COPi, jQmax, (8)

where i,j is the cell index in which i refers to theQ/Qmax grid and j refers to the To grid. AlthoughFLEOH and the COP are generally functions of Tz,To, and part-load ratio (Q/Qmax), in this analysis,fixed Tz = 22◦C (71.6◦F) is assumed.

Ten possible cases, A1-5 and B1-5, have beendefined and introduce different refrigerants resultsin additional permutations. However, only severalof the more interesting case results will now be de-scribed and compared.

The results presented in Table 1 show that withprecooled TES, airflow optimization is considerablymore important than subcooling optimization. Theannual energy savings for the optimal case com-pared to the non-optimal case with fixed airflowsand zero subcooling were around 50%. The savingswere significantly less (around 0.4%) for the optimalcase compared to the case with optimized airflowsand zero subcooling. Table 1 also shows that a heatpump with two parallel compressors saves a signif-icant amount of energy, especially for systems thatoperate at lower part-load ratios most of the time.

The results in Table 2 show that a variable-speedcompressor heat pump (Case 1) achieves large en-ergy savings compared to a two-speed compressorheat pump (Case 4), whether one is mostly operatingat higher (distribution A) or at lower (distributionB) part-load ratios. Armstrong et al.’s study (2009a,2009b) assumed a two-speed compressor heat pumpfor distribution A and a variable-speed compressorheat pump for distribution B. The results of thisanalysis (Table 2, column 4) show savings (around20%) achieved by operating the heat pump at lowerpart-load ratios in addition to the variable-speedcompressor savings (around 30%). However, asshown in Table 2, not all heat pumps benefit

Table 2. Annual energy savings for distribution A and distribution B using two-speed and variable-speed compressor heat pumps.

A1 versus A4 B1 versus B4 A4 versus B1(EA4 – EA1)/EA4 (EB4 – EB1)/EB4 (EA4 – EB1)/EA4

R410 29.4% 34.5% 48.6%Ammonia 20.1% 21.6% 38.3%Propane 7.4% 12.6% 28.6%

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HVAC&R RESEARCH 1165

Table 3. Annual energy savings for ammonia and propane with respect to R410A.

A1 scenario B1 scenario(ER410A – Erefrigerant)/ER410A (ER410A – Erefrigerant)/ER410A

Ammonia versus R410A 14.0 8.7%Propane versus R410A 7.8 2.4%

equally from variable-speed compressor usage oroperation at lower part-load ratios. For a betterunderstanding of the heat pump power consumptionwhen switching to different refrigerants (Table 3)the optimization algorithm presented here hasshown to be extremely advantageous.

Using the same annual FLEOH distributions(Figure 17), an optimization is performed to findthe heat pump sizing factor that minimizes energyconsumption for scenarios A1 (distribution A withvariable-speed heat pump), A4 (distribution A withtwo-speed heat pump), and B1 (distribution Bwith variable-speed heat pump). Figure 18 showshow different sizing factors influence the total an-nual energy savings compared to the nominal sizesystem. The gradient-based optimization algorithmin MATLAB determined for this particular case thatthe optimal sizing factor for B1 would be 1.17,meaning that in this scenario, the heat pump hasnear-optimal size. Furthermore, the additional en-ergy savings of the optimally sized system werevery small compared to the nominally sized system.The heat pumps for A1 and A4 scenarios, on the

Figure 18. Heat pump optimal sizing for FLEOH distribution Awith variable-speed heat pump (blue diamond), FLEOH distri-bution A with two-speed heat pump (green square), and FLEOHdistribution B with variable-speed heat pump (red circle) (colorfigure available online).

other hand, are far from the optimum. The optimumthat would be achieved with a 2.5 sizing factor forA1 and a 1.9 sizing factor for A4 would result inenergy savings of approximately 20% for A1 and17% for A4 compared to the nominal-sized system.Note that heat pump capital cost is not consideredhere, but also that the savings for an optimally sizedheat pump are achieved almost entirely by increasedevaporator and condenser size. Hence, its capitalcost can be reduced without much impact on annualenergy consumption by using a much smaller com-pressor (Arthur D. Little, Inc. [ADL] 2000; Levinset al. 1997).

Summary

An optimization algorithm that uses an adaptivegrid search method is developed for heat pump staticoptimization over a wide range of conditions andloads. For a given cooling rate, zone, and outsidetemperature, the algorithm finds the optimal evapo-rator airflow, condenser airflow, and subcooling forwhich the total power consumption in minimal. Allother heat pump parameters, such as the compressorfrequency, refrigerant mass flow rate, temperatures,and pressures, are functions of the given inputs andoptimization variables. The algorithm can be usedto optimize an air-to-air, water-to-air, and water-to-water heat pump with any refrigerant supported bysoftware REFPROP (Lemmon et al. 2007). It canalso be used for different heat pump types, such asheat pumps with a single or multiple evaporators,compressors, and condensers, heat pumps withsingle-, two-, or variable-speed compressors, aswell as heat pumps with different compressortypes. Although only sensible cooling optimizationresults were presented here, the heat pump modeldeveloped by Zakula et al. (2011) can also beused to optimize simultaneous sensible and latentcooling.

The optimization results for an air-to-air heatpump with a single rolling-piston compressor andR410A as a working fluid have been presented overa wide range of conditions and loads. It is shown

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that the COP and optimal subcooling are almostsolely functions of a part-load ratio and temperaturedifference between a zone and outside. It has alsobeen shown that the COP increases for lower part-load ratios, which suggests that modest oversizingof optimally controlled variable-speed heat pumpsis desirable.

The case with zero subcooling and optimizedairflows has been compared to the fully optimizedheat pump performance. Although relatively small(0–5%), the COP benefits of subcooling increase forhigher part-load ratios and larger temperature dif-ferences. The importance of airflow optimizationwas demonstrated, analyzing the case with fixedairflows, in which case the COP values were sig-nificantly lower compared to the optimal case.Furthermore, the results show that a heat pumpprimarily optimized for simultaneous sensible cool-ing and dehumidification can significantly under-perform when used for sensible cooling only in, forexample, an application where dedicated outdoor airsystem (DOAS) handles latent loads.

The specific power of a heat pump with a singlecompressor increases for very low part-load ratios(less than 0.2 in this example), mainly because ofinverter losses. One of the possible improvementsanalyzed here is a heat pump with two parallel com-pressors. For part-load ratios lower than 0.5, thecontrol algorithm decides if it is more efficient torun one or two compressors, while for the higherpart-load ratios, both compressors operate in par-allel. The results of this analysis show significantimprovement at lower part-load ratios when two par-allel compressors are used.

Although an economizer mode can be used whenthe outside temperature is lower than the inside tem-perature, it has been shown that the economizermode is not always more economical. The optimiza-tion algorithm can be used here to decide whetherthe compressor mode or economizer mode uses theleast amount of energy.

Using the optimization algorithm, it is possibleto predict the performance of a heat pump whenswitching to different refrigerants. This was demon-strated by comparing the power consumption of am-monia and propane heat pumps against an R410Aheat pump.

The annual energy consumption has been com-pared among different cases using the sensible cool-ing load distribution for a typical office in Chicago.For this particular example, the system with opti-mized variables annually saves approximately 50%of energy compared to the non-optimized system,

which indicates the importance of heat pump staticoptimization and the model predictive control ofTES. The system with a variable-speed compressorthat operates at night by precooling and at lowerpart-load ratios most of the time saves a significantamount of energy compared to the system with two-speed compressor that operates at higher part-loadratios. It is also shown that the system can ben-efit from a heat pump with two compressors thatwork in parallel, since this reduces inverter lossesat low part-load ratios. Furthermore, optimizationhas been used to find the optimal heat pump siz-ing. The shape of oversizing curves and the optimalsize are shown to be very dependent on a particularapplication. For the system that operates at higherpart-load ratios most of the time, the optimal sizingresulted in approximately 17–20% annual energysaving compared to the nominal system size.

In conclusion, the algorithm presented here canbe used to optimize heat pump performance over awide range of operating conditions and loads. Thegradient method was then used to find an optimalheat pump sizing for a particular application. Theseprocedures are particularly beneficial for novel sys-tems, for which curve fits to the optimization resultscould easily be implemented as a part of a buildingpredictive control. The algorithm can also be usedto analyze the influence of different optimizationvariables or to compare heat pumps with a differentcombination of components or different refriger-ants.

Acknowledgments

The authors wish to acknowledge the MITEnergy Initiative, the Masdar Institute of Sci-ence and Technology, the National Science Foun-dation (EFRI-SEED award 1038230), and theMartin Family Society of Fellows for Sustainabilityfor their support of this research. They also thankPacific Northwest National Laboratory for sharingthe results of its study of low-lift cooling technol-ogy, Eric Lemmon from the National Institute ofStandards and Technology for his support with thesoftware REFPROP, and Dr. Piotr Domanski for hishelp regarding the appropriate sizing when switch-ing to different refrigerants.

Nomenclature

COP = coefficient of performance (—)D = compressor displacement, m3 (ft3)E = energy, J (Btu)

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f = compressor shaft speed, Hzh = enthalpy, J/kg (Btu/lb)J = power, W (Btu/hr)Q = cooling rate, W (Btu/hr)T = temperature, ◦C (◦F)tH = high-speed duty fraction (—)y = condenser area percentage (—)ρ = refrigerant density at suction conditions,

kg/m3 (lb/ft3)

Subscripts

c,avg = average condensinge,avg = average evaporatinglg = refers to change from saturated liquid to

saturated vapormax = maximalo = outsideopt = optimalsc = subcooling region of condenserz = zone

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