Energy Conversion and Management 92 (2015) 353–365

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Exergy-based optimal control of a vapor compression system

http://dx.doi.org/10.1016/j.enconman.2014.12.0140196-8904/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail addresses: neerajain@purdue.edu (N. Jain), alleyne@illinois.edu

(A. Alleyne).

Neera Jain a,⇑, Andrew Alleyne b

a Purdue University, West Lafayette, IN, USAb University of Illinois at Urbana-Champaign, Urbana, IL, USA

a r t i c l e i n f o

Article history:Received 8 September 2014Accepted 3 December 2014Available online 13 January 2015

Keywords:Exergy destruction minimizationControl systemsOptimal controlDynamic modelingVapor compression systems

a b s t r a c t

Exergy-based analysis and optimization has been successfully used to design a variety of thermal systemsto achieve greater efficiency. However, the advantages afforded by exergy destruction minimization(EDM) at the design stage have not been translated to closed-loop operation of thermal systems suchas vapor compression systems (VCSs). Through online optimization and control, VCSs can effectivelyrespond to disturbances, such as weather or varying loads that cannot be accounted for at the designstage, while simultaneously maximizing system efficiency. Furthermore, in applications where VCSsencounter high frequency disturbances, such as in refrigerated transport applications or passenger vehi-cles, optimizing efficiency at steady-state conditions alone may not lead to significant reductions inenergy consumption. In this paper we design the first exergetic, or second law, optimal controller for acanonical four-component vapor compression system (VCS). A lumped parameter moving boundarymodeling framework is used to model the two heat exchangers in the VCS. A model predictive controlleris then designed and implemented in simulation using a dynamic exergy-based objective function todetermine the optimal control actions for the VCS to maximize exergetic efficiency while achieving adesired cooling capacity. Simulation results show that an exergy-based model predictive controller min-imizing exergy destruction achieves over 40% greater exergetic efficiency during operation than a com-parable first law MPC. Moreover, the distribution of exergy destruction across individual componentsoffers new insight into the effect of variable-speed actuators on system efficiency in VCSs.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Exergy analysis has been used extensively in the thermodynam-ics community to understand the source of irreversibilities in avariety of thermal systems, thereby influencing design changes atthe system or component level [1]. In the context of vapor com-pression systems (VCSs), Ahamed et al. [2] provide an extensivereview of exergy analyses that have been conducted, particularlyhighlighting the effect of different refrigerants, as well as keyparameters such as evaporating temperature, on the exergetic effi-ciency of the system. Similarly, Padilla et al. [3] use an exergy anal-ysis to evaluate the use of R413A as an alternative refrigerant in anunmodified R12 vapor compression system. More recently, Maha-badipour and Ghaebi [4] used an exergy analysis to evaluate thebetter of two designs of expander cycles for refrigeration systems.In addition to exergy analyses, researchers have used exergydestruction minimization (EDM), also known as entropy genera-tion minimization or thermodynamic optimization [5], to optimize

design and operational parameters in many thermal systems froma static point of view. For example, design parameters such as heatexchanger geometry have been optimized using EDM by Nag andDe [6] and Vargas and Bejan [7]. However, the advantages affordedby EDM at the design stage have not been translated to closed-loopoperation of thermal systems, including VCSs. In this paper, wepresent the first use of transient exergy destruction as a metricfor a closed-loop decision-making algorithm.

In order to meet the increasing demand for more efficient VCSs,effective control of these systems is required. Through online opti-mization and control, VCSs can effectively respond to disturbancessuch as changes in ambient conditions or time-varying thermalloads that cannot be accounted for at the design stage alone.Techniques from optimal control theory [8] can be used to designsystem controllers that optimally balance competing objectivessuch as performance and efficiency. However, to do so requires amathematical characterization of these quantities. Therefore, thegoal of this paper is to enable the use of EDM for feedback controlof VCSs by first modeling the transient effects of changes in controlvariables on the rate of exergy destruction in a canonical VCS andthen using this model for the design and implementation of anexergy-based model predictive controller.

Nomenclature

A area, m2

a aperture, % of maximumC cooling capacity, kWE energy, kJJ objective function, kJh specific enthalpy, kJ kg�1

L length, mm mass, kg_m mass flow rate, kg s�1

P pressure, kPa_Q heat transfer rate, kW

S entropy, kJ K�1

s specific entropy, kJ (kg K)�1

T temperature, Kt time, su control input, –UA overall heat transfer coefficient, kJ(s K)�1

V volume, m3

v velocity, m s�1

_W work transfer rate (power), kWX exergy, kJ_X exergy transfer rate, kW�x mean quality, dimensionless�y mean void fraction, dimensionlessf normalized zone length, dimensionlessg efficiency, dimensionlessq density, kg m�3

w specific flow exergy, kJ kg�1

Subscripta airc condenserc1 superheated fluid region in condenserc2 two-phase fluid region in condenserc3 subcooled fluid region in condenserCR cross-sectional areacv control volumedest destroyede evaporatore1 two-phase fluid region in evaporatore2 superheated fluid region in evaporatorg gaseous stategen generationH high temperature environmenti in (inlet)k compressorL low temperature environmentl liquid stateo outletr,R refrigerantv EEVw wall0 reference environment12 between the first and second fluid regions23 between the second and third fluid regionsI first law of thermodynamics

354 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

2. Background

2.1. Exergy

Exergy (also referred to as ‘‘availability’’) is defined as the max-imum reversible work that can be extracted from a substance at agiven state during its interaction with a given environment [9].Whereas energy is always conserved, exergy is not. Similarly toenergy, exergy can be transferred in three ways: by heat transfer,work, or through mass exchange with the environment. However,contrary to energy, exergy is destroyed during irreversible phe-nomena such as chemical reactions, mixing, and viscous dissipa-tion. The rate of change of exergy in a control volume is definedmathematically as

dXcv

dt¼ dEcv

dt� T0

dScv

dt

¼X

j

1� T0

Tj

� �_Q j � _Wcv � P0

dV cv

dt

� �þX

i

_miwi

�X

o

_mowo � _Xdest; ð1Þ

where _Qj is the heat transfer rate at the location on the control vol-ume boundary where the instantaneous temperature is Tj [10]. Thespecific flow exergy, w, is defined as

w ¼ ðh� h0Þ � T0ðs� s0Þ þv2

2þ gzþ wch; ð2Þ

where the quantities T0, P0, h0, and s0 are the temperature, pressure,specific enthalpy, and specific entropy, respectively, of the referenceenvironment. The reference environment is typically chosen as aninfinite reservoir with which the system is interacting, such as theambient environment. The amount of exergy destroyed in a systemor through a process is a measure of the loss of potential to do work

and is proportional to the amount of entropy generated in the sys-tem as stated by the Gouy–Stodola theorem shown in Eq. (3) [11].

_Xdest ¼ T0_Sgen ð3Þ

The most common efficiency metric used for VCSs is the coeffi-cient of performance (COP), defined as the ratio of the heat energyremoved to the energy consumed (by the VCS). This quantity isgenerally greater than one. However, the maximum achievableCOP is limited by the temperatures of the hot and cold environ-ments that interact with the VCS [12]. Therefore, it is more infor-mative to consider the second law, or exergetic, efficiency, gII,which characterizes efficiency relative to the maximum achievableefficiency as postulated by the second law of thermodynamics:

0 6 gII ¼ 1� exergy destroyedexergy supplied

� �< 1: ð4Þ

Eq. (4) states that minimizing exergy destroyed in a system willmaximize its exergetic efficiency.

2.2. Reversible work

In the context of optimization of thermal energy systems, it iscommon for the amount of power consumed (or generated) in asystem to be minimized (or maximized). The relationship betweenthe minimization of exergy destruction rate versus the minimiza-tion of power consumption is characterized by

_Xdest ¼ _W � _W rev ð5Þ

and is equivalent to Eq. (3) [5]. The reversible rate of change ofwork, _W rev, also called reversible power, refers to the amount ofpower that the system would consume (or produce) if there wereno irreversibilities in the system. Let us define two objective

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 355

functions, J1 and J2, where J1 is equal to the rate of exergy destruc-tion and J2 is equal to the actual power consumed (or produced).

J1 ¼ _Xdest ¼ _W � _W rev ð6ÞJ2 ¼ _W ð7Þ

Eq. (5) implies that if the quantity _W rev is constant with respectto the decision variables of an optimization problem, then the min-imizations of J1 and J2 will be equivalent as shown in Eq. (8) [5].That is to say, the minimization of exergy destruction rate andminimization of power will be equivalent.

min_Wrev¼C

ðJ1Þ ¼minðJ2Þ ð8Þ

In Section 5 we will specifically present a case study when thisequivalence does not exist and discuss its impact on closed-loopcontrol of a VCS.

2.3. VCS modeling and control

For dynamic modeling of heat exchangers, two differentapproaches have been primarily used: a finite-volume (FV)approach and a lumped parameter moving boundary (LPMB)approach [13]. In the LPMB modeling approach, the heat exchangeris modeled with a fixed number of fluid regions (defined by fluidphase), and the location of the boundary between each fluid regionis a dynamic variable, allowing the length of the fluid regions tovary. Fluid properties such as temperature and density, are lumpedin each region, and an average is used for model computations.Although this approach results in some loss in accuracy as com-pared to a FV approach, the resulting models are of low dynamicorder, making them well suited for control design. A review ofthe literature shows that the LPMB approach has been applied toa variety of VCSs, often with variations in the details of the model-ing approach [14]. The LPMB condenser and evaporator modelsthat are used as the basis for the derivation of transient exergydestruction rate in this paper are described in detail in [15,16].

The control of vapor compression systems (VCSs) variesdepending upon the available hardware. For VCSs with only afixed-speed compressor and condenser fan, on–off control prevails;typically a mechanically-controlled thermostatic expansion valveregulates evaporator superheat and users are given manual controlover the evaporator fan speed. Commercial variable-speed VCSsare typically controlled using multiple proportional–integral (PI)controllers coupled with discrete logic to handle critical con-straints and transitions between operation modes [17]. However,a range of advanced control methodologies than can better miti-gate the multivariable nature of VCSs have been proposed in the

Sub

Condenser

Evaporator

CompressorExpansion Valve

EEV

(a)

Fig. 1. (a) VCS schematic; (b) VCS schematic depicting

literature, including [18–22]. A major focus of these methodologieshas been achieving desired performance objectives (e.g. room tem-perature regulation) while minimizing energy consumption ormaximizing COP [23–25]. However, as discussed in Section 2.1,COP is limiting in its ability to characterize the true efficiency ofa VCS. Therefore, we are interested in studying exergy destructionminimization as an alternative metric for controlling vapor com-pression systems for maximum exergetic efficiency. For the currentwork, model predictive control [26,27] was chosen as an approachthat gives a fair comparison between multiple objective functionsbeing used to optimize the same system.

3. Theory

3.1. Derivation of transient exergy destruction rate

Here we consider a canonical four-component VCS (Fig. 1a).Additional components such as receiver and oil separator are notincluded to simplify this first investigation of EDM for real-timecontrol. To develop a transient (dynamic) expression for the totalrate of exergy destruction in the VCS, it is necessary to considereach component individually as a control volume as shown inFig. 1b which demonstrates the control volumes encompassingsolely the refrigerant.

The total rate of exergy destruction in a canonical VCS is a sumof the rates of exergy destruction in each individual component:

_Xdest;VCS ¼ _Xdest;k þ _Xdest;v þ _Xdest;c þ _Xdest;e: ð9Þ

In the following sections, the exergy destruction rate for eachcomponent will be derived [28]. Note that evaporator and con-denser fans are not considered in this analysis for the purpose ofillustrative clarity. They could be added if needed, but the refriger-ant-focused construct here is sufficient for characterizing the tran-sient exergy destruction rate in the VCS. The referencetemperature, T0, for the exergy calculation is assumed to be thetemperature of the high-temperature reservoir (i.e. ambient envi-ronment), TH, with which the VCS interacts.

3.1.1. Compressor and EEVIn VCS modeling, both the compressor and expansion device,

assumed here to be an electronic expansion valve (EEV), are typi-cally modeled using quasi-steady assumptions. Therefore, thecompressor and EEV control volumes can be analyzed assumingsteady state operation. The compressor is assumed to be adiabaticbut not isentropic; therefore, there is no exergy transfer by heattransfer. A control volume is defined around the refrigerant insidethe compressor as shown in Fig. 1b. The inlet and outlet mass flow

Two-phase Superheated

Evaporator

Super-heated-cooled Two-phase

Condenser

Compressor

(b)

control volumes drawn inside each component.

356 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

rates are equal to the refrigerant mass flow rate through the com-pressor. Assuming steady state operation, Eq. (1) reduces to

� � _Wk

� �þ _mr;kðwk;i � wk;oÞ � _Xdest;k ¼ 0: ð10Þ

The effects of kinetic and potential energy are assumed to benegligible. Substituting Eq. (2) into Eq. (10) and simplifying yields

_Xdest;k ¼ � � _Wk

� �þ _mr;k ðhk;ri � hk;roÞ � THðsk;ri � sk;roÞ

� : ð11Þ

Note that the work transfer rate term in Eq. (11) must be a posi-tive quantity because if the compressor was isentropic, then therate of exergy destruction would equal zero (and hk,ri � hk,ro is anegative quantity). Therefore, we write –ð� _WkÞ to emphasize thefact that the sign convention for work done on the system is neg-ative, where

_Wk ¼ _mr;kðhk;ro � hk;riÞ: ð12Þ

The rate of exergy destruction in the compressor is determinedby substituting Eq. (12) into Eq. (11) and simplifying:

_Xdest;k ¼ �TH _mr;kðsk;ri � sk;roÞ: ð13Þ

To derive the rate of exergy destruction in the EEV, a controlvolume is defined around the refrigerant in the EEV. The expansionof the refrigerant is assumed to be isenthalpic (i.e. hv,ri = hv,ro).There is only exergy transfer by mass transfer, and the inlet andoutlet mass flow rates are equal to the refrigerant mass flow ratethrough the EEV. Assuming steady state operation and regardingthe effects of kinetic and potential energy as negligible gives

_Xdest;v ¼ �TH _mr;vðsv;ri � sv;roÞ: ð14Þ

3.1.2. Heat exchangersThe remaining components in the current VCS model are the

two heat exchangers: the evaporator and the condenser. Thedynamics of these components dominate the overall dynamics ofthe cycle; consequently, transient rates of exergy destructionthrough each of these components will be derived. In the LPMBmodeling approach, the evaporator is typically modeled with twofluid regions: a two-phase refrigerant fluid region and a super-heated refrigerant fluid region. Separate lumped parameters areused to estimate the fluid properties in each of the fluid regions,thereby improving the accuracy of the estimates. Similarly, weuse two separate control volumes to derive the total exergydestruction rate through the evaporator as shown in Fig. 2.

For the two-phase refrigerant fluid region, denoted by the sub-script e1, Eq. (1) reduces to

dXe1

dt¼ 1� TH

Tw;e1

� �_Qe1 þ P0

dVe1

dtþ _mr;vðhe;ri � THse;riÞ

� _me12ðhe;g � THse;gÞ � _Xdest;e1; ð15Þ

where Tj is replaced with Tw,e1, the lumped tube wall temperature inthe two-phase fluid region, and _me12 is the refrigerant mass flowrate between the two control volumes pictured in Fig. 2. Similarly,

Two-phase

cocontrol volume ‘e1’

,r vm&em&

1eQ&

Fig. 2. Individual control volumes drawn aro

for the superheated refrigerant fluid region, denoted by the sub-script e2, Eq. (1) reduces to

dXe2

dt¼ 1� TH

Tw;e2

� �_Q e2 þ P0

dVe2

dtþ _me12ðhe;g � THsgÞ

� _mr;kðhe;ro � THse;roÞ � _Xdest;e2; ð16Þ

where Tj is replaced with Tw,e2, the lumped tube wall temperature inthe superheated fluid region. Applying superposition allows us toexpress _Xdest;e as

_Xdest;e ¼ _Xdest;e1 þ _Xdest;e2: ð17Þ

Therefore, the total exergy destruction rate through the evapo-rator is

_Xdest;e ¼ 1� TH

Tw;e1

� �_Q e1 þ 1� TH

Tw;e2

� �_Q e2 þ P0

dVe1

dtþ dVe2

dt

� �

þ _mr;vðhe;ri � THse;riÞ � _mr;kðhe;ro � THse;roÞ �dXe1

dtþ dXe2

dt

� �:

ð18Þ

where it is assumed that

_Qe ¼ _Q e1 þ _Qe2 ¼ ðUAÞe1ðTw;e1 � Tr;e1Þ þ ðUAÞe2ðTw;e2 � Tr;e2Þ: ð19Þ

In other words, it is assumed that there is no heat transferbetween the refrigerant in control volume e1 and the refrigerantin control volume e2. In Eq. (19), Tr,e1 and Tr,e2 refer to the lumpedrefrigerant temperatures in each fluid region, and (UA)e1 and (UA)e2

are the overall heat transfer coefficients between the refrigerantand tube wall in each fluid region.

An alternative method for deriving the exergy destruction rateis to perform an entropy balance on the control volume using Eq.(20), solve for the rate of entropy generation _Sgen, and then scale_Sgen by the reference environment temperature, TH.

dScv

dt¼X

j

_Qj

TjþX

i

_misi �X

o

_moso þ _Sgen ð20Þ

Because it is difficult to evaluate dXcv/dt, the entropy ratebalance given in Eq. (20) can be used to derive an expressionfor the exergy destruction rate in terms of dScv/dt instead ofdXcv/dt. Applying Eq. (20) to each control volume of the evapo-rator yields

dSe1

dt¼

_Q e1

Tw;e1þ _mr;vse;ri � _me12se;g �

þ _Sgen;e1; ð21Þ

dSe2

dt¼

_Q e2

Tw;e2þ _me12se;g � _mr;kse;ro �

þ _Sgen;e2; ð22Þ

where

_Sgen;e ¼ _Sgen;e1 þ _Sgen;e2: ð23Þ

Substituting Eqs. (21) and (22) into Eq. (23), rearranging terms,and scaling by TH yields the following alternative expression for theexergy destruction rate in the evaporator:

Superheated

ntrol volume ‘e2’

,r km&12

2eQ&

und each fluid region in an evaporator.

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 357

_Xdest;e ¼ TH_Sgen;e ¼ �TH

_Q e1

Tw;e1þ

_Q e2

Tw;e2

!� _mr;vTHse;ri � _mr;kTHse;ro �

þ THdSe1

dtþ dSe2

dt

� �: ð24Þ

Expressions for dSe1/dt and dSe2/dt will be derived in Section3.1.3.

In the LPMB modeling framework, the condenser is typicallymodeled with three refrigerant fluid regions: a superheated fluidregion, a two-phase fluid region, and a subcooled fluid region. Toremain consistent with this modeling approach, three separatecontrol volumes are used to derive the total exergy destruction ratethrough the condenser as shown in Fig. 3. Although at steady stateit can be assumed that the heat transfer out of the condenser isoccurring at the reference temperature, TH, the control volumesdefined in Fig. 3 for the condenser only contain the refrigerantflowing through the condenser tube. Therefore, the transfer of heataway from the refrigerant is occurring at the tube wall tempera-tures of each fluid region. The procedure for deriving the total exer-gy destruction rate through the condenser is analogous to theprocedure described for the evaporator, and so we present onlythe final result in Eq. (25),

_Xdest;c ¼ 1� TH

Tw;c1

� �� _Q c1

� �þ 1� TH

Tw;c2

� �� _Q c2

� �þ 1� TH

Tw;c3

� �� _Q c3

� �þ P0

dVc1

dtþ dVc2

dtþ dVc3

dt

� �þ _mr;kðhc;ri � THsc;riÞ � _mr;vðhc;ro � THsc;roÞ

� dXc1

dtþ dXc2

dtþ dXc3

dt

� �ð25Þ

which is equivalent to

_Xdest;c ¼ TH_Sgen;c

¼ TH

_Q c1

Tw;c1þ

_Q c2

Tw;c2þ

_Q c3

Tw;c3

!� _mr;kTHsc;ri � _mr;vTHsc;ro �

þ THdSc1

dtþ dSc2

dtþ dSc3

dt

� �: ð26Þ

@se1

@�ce

����P

¼�ceqe;gþ 1��ceð Þqe;l

� �qe;gse;g�qe;lse;l

� �� �ceqe;gse;gþ 1��ceð Þqe;lse;l

� �ðqe;g�qe;lÞ

�ceqe;gþ 1��ceð Þqe;l

� �2 ð32Þ

We assume that there is no heat transfer between the refriger-ant in control volumes c1, c2, and c3 so that _Qc ¼ _Qc1 þ _Qc2 þ _Qc3.Expressions for dSci/dt, i = {1, 2, 3}, will be derived in the nextsection.

3.1.3. The entropy differentialThe rate of change of entropy can be described as [29]

dScv

dt¼ scv

dmcv

dtþmcv

ð@scvÞ@h

����P

dhdtþ ð@scvÞ

@P

����h

dPdt

� �: ð27Þ

In Eq. (27) the dependent variables are chosen as specificenthalpy and pressure, but they can be chosen as any two indepen-dent thermodynamic state variables. As was mentioned in Section2.3, the LPMB modeling approach is desirable from a controls

perspective because of the low dynamic order of the resultingmodel. However, the consequence of a low order model is that onlycertain time derivatives are defined. Therefore, the ability to re-express Eq. (27) using different thermodynamic state variables isnecessary. Eq. (27) also highlights why it is helpful to define multi-ple control volumes for the heat exchangers in which a separatecontrol volume is drawn around each fluid region (recall Fig. 2and 3). This approach allows for lumped parameters to be usedto approximate scv and mcv for each control volume as is done inthe LPMB framework. The expression for dmcv/dt can be derivedfor each control volume as described in [30].

The dynamic state variables for the evaporator arexe ¼ fe1 Pe he2 Tw;e1 Tw;e2 �ce½ �T . For the two-phase fluidregion of the evaporator, denoted by the subscript e1, the timederivative of enthalpy is not available. Instead, refrigerant meanvoid fraction, �ce, and pressure, Pe, can be used to describe specificentropy for the two-phase fluid region, as shown in Eq. (28), whereme1 = qe1fe1LR,eACR,e.

dSe1

dt¼ se1

dme1

dtþme1

ð@se1Þ@�ce

����P

d�ce

dtþ ð@se1Þ

@Pe

�����c

dPe

dt

!ð28Þ

The rate of change of mass in the control volume is given by

dme1

dt¼ _mr;v � _me12ð Þ þ qe1ACR;eLR;e

dfe1

dt; ð29Þ

and the specific entropy in the control volume is given by

se1 ¼ �xese;g þ 1� �xeð Þse;l ¼�ceqe;gse;g þ 1� �ceð Þqe;lse;l

�ceqe;g þ 1� �ceð Þqe;l: ð30Þ

Mean void fraction, �c, is related to mean quality, �x, by the fol-lowing relationship:

�x ¼ �cqg

q: ð31Þ

The variables qe,l, qe,g, se,l, and se,g are all solely functions of pres-

sure. The partial derivatives @se1@�ce

���P

and @se1@Pe

����c, shown in Eqs. (32) and

(33) respectively, are derived using Eq. (30).

@se1

@Pe

�����c¼ b1�b2

b3

b1¼ �ceqe;gþ 1��ceð Þqe;l

� �

� �ceqe;gdse;g

dPeþ se;g�ce

dqe;g

dPeþ 1��ceð Þqe;l

dse;l

dPeþ se;l 1��ceð Þ

dqe;l

dPe

� �

b2¼ �ceqe;gse;gþ 1��ceð Þqe;lse;l

� ��ce

dqe;g

dPeþ 1��ceð Þ

dqe;l

dPe

� �

b3¼ �ceqe;gþ 1��ceð Þqe;l

� �2

ð33Þ

Superheated Sub-cooledTwo-phase

control volume ‘c1’

control volume ‘c2’

control volume ‘c3’

,r vm&c23m&c12m&,r km& 1cQ& 2cQ& 3cQ&

Fig. 3. Individual control volumes drawn around each fluid region in a condenser.

358 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

For the superheated (single-phase) fluid region in the evapora-tor, the time derivative of the lumped enthalpy, he2, is defined as adynamic state. Therefore, Eq. (27) is used to describe the rate ofchange of entropy in the superheated fluid region of the evapora-tor, yielding

dSe2

dt¼ se2

dme2

dtþ qe2fe2LR;eACR;e

ð@se2Þ@he2

����P

dhe2

dtþ ð@se2Þ

@Pe

����h

dPe

dt

� �ð34Þ

where

dme2

dt¼ _me12 � _mr;k �

þ qg;eACR;eLR;edfe1

dt: ð35Þ

A procedure analogous to the one described above can be usedto determine the rate of change of entropy in each of the condensercontrol volumes: dSci/dt, i = {1, 2, 3}. It is assumed that the outletrefrigerant condition of the condenser is subcooled liquid; there-fore, the condenser is characterized using three fluid regions. Asin the case of the evaporator, specific enthalpy and pressure areused to describe the rate of change of specific entropy in the sin-gle-phase (superheated and subcooled) fluid regions, and meanvoid fraction and pressure are used in the two-phase fluid region.The expressions for dSci/dt, i = {1, 2, 3} are given in Eqs. (36)–(41),respectively.

dSc1

dt¼ sc1

dmc1

dtþqc1fc1LR;cACR;c

ð@sc1Þ@hc1

����P

dhc1

dtþð@sc1Þ

@Pc

����h

dPc

dt

� �ð36Þ

dmc1

dt¼ _mr;k� _mc12 �

þqg;cACR;cLR;cdfc1

dtð37Þ

dSc2

dt¼ sc2

dmc2

dtþqc2fc2LR;cACR;c

ð@sc;2Þ@�cc

����P

d�cc

dtþð@sc;2Þ

@Pc

�����c

dPc

dt

!ð38Þ

dmc2

dt¼ _mc12� _mc23ð Þþql;cACR;cLR;c

dfc1

dtþdfc2

dt

� ��qg;cACR;cLR;c

dfc1

dt

ð39ÞdSc3

dt¼ sc3

dmc3

dtþqc3fc3LR;cACR;c

ð@sc3Þ@hc3

����P

dhc3

dtþð@sc3Þ

@Pc

����h

dPc

dt

� �ð40Þ

dmc3

dt¼ _mc23� _mr;vð Þ�ql;cACR;cLR;c

dfc1

dtþdfc2

dt

� �ð41Þ

Finally, substituting Eqs. (13), (14), (24), and (26) into Eq. (9) andsimplifying results in the following expression for the total instan-taneous exergy destruction rate in the VCS:

_Xdest;VCS ¼ TH

_Q c1

Tw;c1þ

_Q c2

Tw;c2þ

_Q c3

Tw;c3�

_Q e1

Tw;e1�

_Q e2

Tw;e2

!

þ THdSe1

dtþ dSe2

dtþ dSc1

dtþ dSc2

dtþ dSc3

dt

� �: ð42Þ

3.2. Optimal control design

In Section 3 we derived an expression for the transient rateof exergy destruction in a VCS. This enables us to characterizethe efficiency of the system dynamically from a second law

perspective. Moreover, by utilizing techniques from the field ofoptimal control theory [8], we will use Eq. (42) as a minimizationmetric for operating the VCS at its maximum exergetic efficiencywhile meeting performance requirements.

In this section, an optimal controller is designed using total exer-gy destruction as the minimization metric. While there are manydifferent optimal control algorithms, they all rely on the minimiza-tion of an objective or cost function to determine the optimalsequence of control actions for a particular system. However, manyoptimal control algorithms are applicable only to systems withoutinput or output constraints. In the case of a second law, or exergeticoptimization such as this one, it will be necessary to enforce con-straints to ensure that thermodynamic laws are not violated. More-over, for VCSs it is typical to have limits on actuators and componentoperation. Therefore, we will use model predictive control (MPC), areceding-horizon optimal control framework that allows for con-straints to be placed on input, output, and state variables [26].

3.2.1. Prediction modelMPC uses a dynamic model to predict how the system will

behave in response to a particular sequence of control decisionsover a specified prediction horizon so as to influence control deci-sions at the current time step – this model is called the predictionmodel. The prediction horizon, Np, is the number of discrete timesteps over which the system behavior is predicted. It is defined as

Np ¼thorizon

Dtð43Þ

where Dt is the length of the discrete time step and thorizon is thelength of time over which the algorithm predicts the system behav-ior. The control horizon, Nu, is the number of discrete time steps forwhich control decisions are optimized, where Nu 6 Np. Fig. 4 pro-vides a visual interpretation of the MPC algorithm.

To reduce computational complexity, a linear prediction modelis preferred. Therefore, the nonlinear first-principles VCS modelused to derive Eq. (42) is linearized about an equilibrium point ofthe system; details of the model linearization are provided in[31]. Moreover, since MPC is implemented in discrete time, the lin-earized model is discretized at a sample time of Dt and representedin a state space representation [32] as

dx½kþ 1� ¼ Adx½k� þ B1du½k� þ B2dd½k�; ð44Þ

where dx, du, and dd represent deviations from the equilibriumpoint, A is the state matrix, B1 is the control input matrix, and B2

is the disturbance input matrix. The state, input, and disturbancevectors for the VCS are described in Eqs. (45)–(47), respectively.Note that instead of treating the evaporator and condenser fanspeeds as decision variables, the air mass flow rates produced byeach fan, _ma;e and _ma;c , are used. The EEV aperture is described byav and the compressor speed is described by xk. Furthermore, tosimplify the notation, the use of d will be dropped since it isunderstood that we are discussing deviations about some nominaloperating condition when referring to state, input, and disturbancevariables.

Past Future

Prediction HorizonControl Horizon Δt

k k + 1 k + Nu k + Np. . .. . .

ReferenceMeasured OutputPredicted OutputPrevious InputPredicted Input

Fig. 4. Schematic describing model predictive control.

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 359

x¼ fe1 Pe he2 Tw;e1 Tw;e2 �ce fc1 fc2 Pc hc1 hc3 Tw;c1 Tw;c2 Tw;c3 �cc½ �T 2R15

ð45Þ

u ¼ av xk _ma;e _ma;c½ �T 2 R4 ð46Þ

d ¼ TL TH½ �T 2 R2 ð47Þ

Next we discuss a few modifications that are made to the pre-diction model. One can constrain the rate of change in control deci-sions over the control horizon by augmenting the system withadditional states defined as xu[k] = u[k � 1] where u[k] = u[k � 1]+ Du[k]. Therefore, xu [k + 1] = xu[k] + Du[k] and the augmentedstate-space representation of the system is given by�x½kþ 1� ¼ A�x½k� þ B1Du½k� þ B2d½k�, where

�x ¼xxu

� �; A ¼

A B1

0 I

� �; B1 ¼

B1

I

� �; B2 ¼

B2

0

� �: ð48Þ

For a numerical optimization, it is convenient to define theinput vector in its lifted form, DU ¼ Du½k� Du½kþ 1� � � �½Du½kþ nu � 1��T . Using the lifted input vector, DU, and the initialvalue of the state vector, �x½0�, the evolution of all of the statescan be quickly evaluated in the lifted vector X using the liftedmatrix equation X ¼ T�x½k� þ S1DUþ S2D where �X ¼ �x½k�½�x½kþ 1� � � � �x½kþ np � 1��T . The expressions for T, S1, and S2 are givenby

T¼

AA2

..

.

Anp

266664

377775;S1¼

B1 0 . . . 0

AB1. .

. . .. ..

.

..

. . ..

B1 0

Anu�1B1

..

.

Anp�2B1

Anp�1B1

� � �...

. ..

� � �

�A�B1

..

.

. ..

Anp�nu�1B1

B1

..

.

Anp�nu�1B1

Anp�nu B1

26666666666666664

37777777777777775

;

S2¼

B2 0 . . . 0

AB2. .

. . .. ..

.

..

. . ..

B2 0

Anu�1B2

..

.

Anp�2B2

Anp�1B2

� � �...

. ..

� � �

AB2

..

.

. ..

Anp�nu�1B2

B2

..

.

Anp�nu�1B2

Anp�nu B2

2666666666666666664

3777777777777777775

:

ð49Þ

The objective function, JVCS,II, that will be defined in the nextsection is a function of X.

3.2.2. Objective functionFor VCS operation we would like for the objective function to

characterize both the efficiency and performance of the system.Since the MPC considers a receding finite time-horizon, the totalexergy destroyed over the prediction horizon will be minimized.The performance objective is defined as the 2-norm of the differ-ence between the desired cooling capacity (as specified by theuser) and the cooling capacity achieved by the VCS over the predic-tion horizon. A weighting parameter, k, is used to emphasize theimportance of one objective over the other. Finally, as mentionedearlier, the MPC will be implemented in discrete time; therefore,numerical integration will be used to approximate the total exergydestroyed over the prediction horizon.

The complete second law objective function, JVCS,II, is expressedas

JVCS;II ¼ kCdes � Cachk2ð Þ|fflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflffl}Performance

Objective

þ k �XNp

k¼1_Xdest;VCS½k�

� �Dt|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

EfficiencyObjective

:ð50Þ

Substituting Eq. (3) into Eq. (50) and simplifying yields

JVCS;II ¼ Cdes � Cachk k2 þ k � TH � DtXNp

k¼1_Sgen;VCS½k�

� �¼ Cdes � Cachk k2 þ �k

XNp

k¼1_Sgen;VCS½k�

� � ð51Þ

where the reference (dead state) temperature, TH, and the discretesample time, Dt, are absorbed into the weighting parameter �k.The theoretical minimum of the objective function shown in Eq.(51) is zero.

Remark 1. The choice of the correct reference temperaturebecomes less critical in this formulation where TH is absorbed intothe weighting parameter, �k. In fact, TH plays little role in thetradeoff between the performance and efficiency objectives fromthe perspective of the optimization algorithm assuming the userheuristically tunes �k for the desired tracking performance of thecontroller. This is a feature for many thermal systems as it can bedifficult to define the correct reference state for certain applica-tions and problems [33,34]. h

Remark 2. In Eq. (50), _Xdest represents the instantaneous rate ofexergy destruction and will take on a constant value at steadystate, unlike a time differential which will become zero at steadystate. Therefore, the expression _Xdest;VCS½k� refers to the rate of exer-gy destruction in the VCS at some time instant k. Similarly, in Eq.(51), _Sgen is the instantaneous rate of entropy generation, and

360 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

_Sgen;VCS½k� refers to the rate of entropy generation in the VCS at sometime instant k. h

The desired cooling capacity, Cdes 2 RNp , is specified by the userin the optimization problem, and Cach 2 RNp is calculated using theexpression

Cach ¼ _Q e ¼ ðUAÞe1ðTw;e1 � Tr;e1Þ þ ðUAÞe2ðTw;e2 � Tr;e2Þ: ð52Þ

The efficiency objective can be expanded as

XNp

k¼1

_Sgen;VCS½k� ¼XNp

k¼1

_Qc1½k�Tw;c1½k�

þ_Q c2½k�

Tw;c2½k�þ

_Q c3½k�Tw;c3½k�

�_Q e1½k�

Tw;e1½k��

_Q e2½k�Tw;e2½k�

!

þXNp

k¼1

ge1½k� þ ge2½k� þ gc1½k� þ gc2½k� þ gc3½k�ð Þ

ð53Þ

where_Q ei½k� ¼ UAð Þei½k� � Tw;ei½k� � Tr;ei½k�

�; i 2 f1;2g; ð54Þ

_Q ci½k� ¼ UAð Þci½k� � Tr;ci½k� � Tw;ci½k� �

; i 2 f1;2;3g; ð55Þ

ge1½k� ¼ se1½k� _mr;v ½k�� _me12½k�þLR;eACR;eqe1½k�fe1½k�� fe1½k�1�

Dt

� �� �

þLR;eACR;eqe1½k�fe1½k�ð@se1Þ@�ce

½k��ce½k���ce½k�1�

Dt

� ��

þð@se1Þ@Pe

½k� Pe½k��Pe½k�1�Dt

� ��; ð56Þ

ge2½k� ¼ se2½k� _me12½k�� _mr;k½k�þLR;eACR;eqg;e½k�fe1½k�� fe1½k�1�

Dt

� �� �

þLR;eACR;eqe2½k�fe2½k�ð@se2Þ@he2

½k� he2½k��he2½k�1�Dt

� ��

þð@se2Þ@Pe

½k� Pe½k��Pe½k�1�Dt

� ��; ð57Þ

gc1½k� ¼ sc1½k� _mr;k½k�� _mc12½k�þLR;cACR;cqg;c½k�fc1½k�� fc1½k�1�

Dt

� �� �

þLR;cACR;cqc1½k�fc1½k�ð@sc1Þ@hc1

½k� hc1½k��hc1½k�1�Dt

� ��

þð@sc1Þ@Pc

½k� Pc½k��Pc½k�1�Dt

� ��; ð58Þ

gc2½k� ¼ sc2½k� _mc12½k� � _mc23½k� þ LR;cACR;cql;c½k�fc1½k� � fc1½k� 1�

Dt

��

þ fc2½k� � fc2½k� 1�Dt

��

� sc2½k� LR;cACR;cqg;c½k�fc1½k� � fc1½k� 1�

Dt

� �� �

þ LR;cACR;cqc2½k�fc2½k�ð@sc2Þ@�cc

½k��cc½k� � �cc½k� 1�

Dt

� ��

þð@sc2Þ@Pc

½k� Pc½k� � Pc½k� 1�Dt

� ��; ð59Þ

and

gc3½k� ¼ sc3½k� _mc23½k�� _mr;v ½k��LR;cACR;cql;c½k�fc1½k��fc1½k�1�

Dt

��

þfc2½k�� fc2½k�1�Dt

��

þLR;cACR;cqc3½k�fc3½k�ð@sc3Þ@hc3

½k� hc3½k��hc3½k�1�Dt

� ��

þð@sc3Þ@Pc

½k� Pc½k��Pc½k�1�Dt

� ��: ð60Þ

For clarity of notation, the rate of change of entropy in eachevaporator control volume is denoted by the expression gei wherei = {1, 2}. Similarly, in each condenser control volume, the rate ofchange of entropy is denoted by the expression gci where i = {1,2, 3}. It should also be noted that JVCS,II is not only a function ofthe states of the dynamical system representation of the VCS butalso a function of variables such as Tr,c1 and @se2/@h which are non-linear functions of the states. These variables are typically evaluatedusing data-based refrigerant look-up tables [30]. Therefore, whilethe dynamic prediction model itself is linear, the objective functionis nonlinear.

3.2.3. ConstraintsIn the MPC framework, upper and lower bound constraints can

easily be placed on the values of the control decisions at each timeinstant. The constraint values are detailed in Section 4.1 for thespecific VCS considered in the case study. Additionally, upperand/or lower bound constraints can be enforced on specific statevariables in the dynamical system. For VCS operation, we typicallyseek nonzero superheat in the evaporator and nonzero subcoolingin the condenser [12]. Therefore, the constraints defined in Eqs.(61) and (62) ensure that the normalized lengths of the super-heated fluid region in the evaporator and subcooled region in thecondenser, respectively, are maintained at some minimum fractionof the total tube length in each heat exchanger.

fe1½k� 6 0:95 8 k ð61Þfc1½k� þ fc2½k� 6 0:95 8 k ð62Þ

Finally, the following nonlinear constraints are introduced tosatisfy the second law of thermodynamics:

_WVCS½k� � _XVCS½k�P 0 8 k; ð63Þ

where _WVCS ¼ _Wk is equal to the power (energy consumption rate)of the compressor and

_Xdest;k½k�P 0 8 k;_Xdest;v ½k�P 0 8 k;_Xdest;e½k�P 0 8 k;_Xdest;c½k�P 0 8 k:

ð64Þ

Eq. (63) ensures that the reversible power is always nonnega-tive, and the inequalities shown in Eq. (64) ensure that the exergydestruction rate for each individual component of the VCS isalways nonnegative.

4. Results and discussion

In this section, a case study is presented in which the exergy-based (second law) model predictive controller (MPC) is imple-mented on a simulated VCS. The details of the case study and theclosed-loop results will be presented in Section 4.1. In Section4.2, an energy-based (first law) MPC will be designed and imple-mented on the same VCS, and the simulation results will be com-pared against the results presented in Section 4.1. The tradeoffsbetween the first law and second law optimal controllers will bediscussed, specifically in the context of transient VCS operation.

4.1. Simulation of second law (exergy-based) MPC

The exergy-based (second law) MPC was implemented insimulation on a linearized VCS model. The nonlinear VCS modelhas been validated against an experimental system at the Univer-sity of Illinois at Urbana-Champaign described in [35], and the lin-earization procedure is described in [31]. The model describes a1 kW capacity vapor compression system operating with R134a

0 50 100 150 2000.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

Time (s)

Des

ired

Coo

ling

Cap

acity

(kW

)

Fig. 5. Cooling capacity reference trajectory.

0 50 100 150 2008

10

12

Val

ve O

peni

ng(%

ope

n)

0 50 100 150 200950

1000

1050

1100

1150

Com

pres

sor

Spe

ed (

rpm

)

0 50 100 150 2000

0.2

0.4

Eva

p A

irF

low

Rat

e(k

g/s)

0 50 100 150 2000.2

0.4

0.6

0.8

Time (s)C

ond

Air

Flo

w R

ate

(kg/

s)Fig. 7. Second law (exergy-based) MPC – control input signals.

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 361

refrigerant and consists of a semi-hermetic reciprocating compres-sor, electronic expansion valve, condenser, and evaporator, alongwith variable-speed heat exchanger fans. The function fmincon inthe MATLAB Optimization Toolbox was used with a sequentialquadratic programming algorithm to implement the model predic-tive controller. The desired cooling capacity, Cdes, is shown in Fig. 5.This reference trajectory was chosen to elicit the transient behaviorthat results from high frequency loading in cooling applicationssuch as refrigerated food transport.

The length of the prediction horizon and the control horizonwere chosen as Np = Nu = 15, with a sample time, Dt, of 1 s. For thiscase study, it was assumed that the reference trajectory wasknown a priori. The weighting factor k was chosen heuristicallyas 8 � 10�3 to sufficiently weight the performance objective andachieve reasonable reference tracking performance. The constantdisturbances (Eq. (47)) were specified as TL = 18 �C and TH = 26 �C

Table 1Upper and lower bound constraints on decision variables.

Decision variable Units Lower bound Upper bound

av % open 8 11xk rpm 900 1100_ma;e kg/s 0.1 0.3_ma;c kg/s 0.3 0.65

0 50 100 150 2000.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

Time (s)

Coo

ling

Cap

acity

(kW

)

Second Law (Exergy-Based) MPCC

desired

Fig. 6. Second law (exergy-based) MPC – reference tracking performance.

where TL is the temperature of the low-temperature reservoirinteracting with the evaporator and TH is the temperature of theambient environment. Finally, the upper and lower bound con-straints on the decision variables are given in Table 1 where av isthe EEV aperture, xk is the compressor speed, and _ma;e and _ma;c

are the evaporator and condenser air mass flow rates, respectively.The tracking performance of the second law MPC is shown in

Fig. 6. The control input signals are shown in Fig. 7, and the exergydestruction rate and the exergetic efficiency are plotted in Fig. 8.

4.2. Comparison between first and second law MPC

4.2.1. Reversible work analysisRecall Eq. (5) written here in non-rate form:

Xdest ¼W �W rev: ð65Þ

0 50 100 150 200

0.2

0.25

0.3

0.35

Tot

al E

xerg

yD

estr

uctio

n R

ate

(kW

)

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

Time (s)

Exe

rget

icE

ffici

ency

Fig. 8. Second law (exergy-based) MPC – exergy destruction rate and exergeticefficiency.

150 155 160 165 170 175 1800.87

0.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

Time (s)

Coo

ling

Cap

acity

(kW

)

First Law MPCSecond Law MPCC

desired

Fig. 10. Closer view of Fig. 9.

0 50 100 150 2008

10

12V

alve

Ope

ning

(% o

pen)

First Law MPCSecond Law MPC

0 50 100 150 200950

1000

1050

1100

1150

Com

pres

sor

Spe

ed (

rpm

)

0 50 100 150 2000

0.2

0.4

Eva

p A

irF

low

Rat

e(k

g/s)

0.6

0.8

d A

ir R

ate

/s)

362 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

Using Eq. (65), the reversible work during a finite time-horizonassuming transient operation of the VCS is

W rev¼W�Xdest

¼XNp

k¼1

_mr;k½k� hk;ro½k��hk;ri½k� � �

�TH

Xnp

k¼1

_Q c1½k�Tw;c1½k�

þ_Q c2½k�

Tw;c2½k�þ

_Qc3½k�Tw;c3½k�

�_Q e1½k�

Tw;e1½k��

_Q e2½k�Tw;e2½k�

!

�TH

XNp

k¼1

ge1½k�þge2½k�þgc1½k�þgc2½k�þgc3½k�ð Þ

ð66Þ

which will not be constant with respect to the decision variables(i.e. the control input sequence DU) each time the MPC problemis solved. Therefore, we expect that an MPC designed to minimizea first law objective will produce different results than were pre-sented in the previous section. We will now design another modelpredictive controller where the objective function, JVCS,I, is formu-lated to minimize the total energy consumed over the predictionhorizon:

JVCS;I ¼ Cdes � Cachk k2ð Þ þ k � DtXNp

k¼1

_Wk½k� !

¼ Cdes � Cachk k2ð Þ þ k � DtXNp

k¼1

_mk½k� hk;ro½k� � hk;ri½k� � !

; ð67Þ

where _wk is the instantaneous power consumption (i.e. energy con-sumption rate) in the VCS. The first law MPC is designed with thesame constraints, weighting factor k, and constant disturbances thatwere specified in the second law MPC. The closed-loop results of thetwo controllers will be compared in the following section.

4.2.2. Comparison of closed-loop simulation resultsFirst the tracking of the desired cooling capacity by each opti-

mal controller is compared in Fig. 9. As expected, both controllersproduce very similar results (see Fig. 10 for a closer comparison).

The control input signals associated with each controller arecompared in Fig. 11. The primary difference is seen in the input sig-nals for the compressor and the condenser air mass flow rate. Inparticular, the first law MPC resulted in the condenser air massflow rate at its maximum allowable value (0.65 kg/s) for most of

0 50 100 150 2000.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

Time (s)

Coo

ling

Cap

acity

(kW

)

First Law (Energy-Based) MPCSecond Law (Exergy-Based) MPCC

desired

Fig. 9. Comparison of cooling capacity tracking performance using first and secondlaw MPC.

0 50 100 150 2000.2

0.4

Time (s)

Con

Flo

w (kg

Fig. 11. Comparison of control input signals using each of the two controllers.

the 200-s time horizon whereas the second law MPC dropped thecondenser air mass flow rate to its lower bound (0.3 kg/s) for muchof the latter part of the simulation. During this time the first lawMPC kept the compressor speed constant where the second lawMPC increased the value to 1100 rpm.

Remark 3. To better characterize and understand transient exergydestruction in a VCS, only the refrigerant side dynamics of the VCSwere considered in the derivation. Therefore the power consump-tion of the heat exchanger fans is not considered in either objectivefunction, implying that there is no penalty, from a first lawperspective, of choosing high evaporator and condenser air massflow rates. This can explain why the first law MPC resulted inhigher evaporator and condenser air mass flow rates than thesecond law MPC. However, operating the VCS with high air mass

0 50 100 150 2000.1

0.15

0.2

0.25

0.3

0.35

Exe

rgy

Des

truc

tion

Rat

e (k

W)

First Law MPCSecond Law MPC

0 50 100 150 2000.15

0.2

0.25

0.3

0.35

0.4

Pow

erC

onsu

mpt

ion

(kW

)

0 50 100 150 2000

0.05

0.1

0.15

0.2

0.25

Time (s)

Wdo

t,rev

Fig. 12. Comparison of exergy destruction rate, energy consumption rate, andreversible power using each of the two controllers. The same range is used for eachplot.

0 50 100 150 2000

0.05

0.1

Exe

rgy

Des

t. R

ate

in V

alve

(kW

)

First Law MPCSecond Law MPC

0 50 100 150 2000.1

0.15

0.2

Exe

rgy

Des

t. R

ate

in C

ompr

esso

r (k

W)

0 50 100 150 2000

0.05

0.1

Exe

rgy

Des

t. R

ate

in E

vapo

rato

r (k

W)

0 50 100 150 2000

0.05

0.1

Time (s)

Exe

rgy

Des

t. R

ate

in C

onde

nser

(kW

)

Fig. 13. Exergy destruction rate comparison by VCS component for each of the twocontrollers. The same range is used for each plot.

Table 2Total exergy destruction and energy consumption using each controller.

Second lawMPC

First lawMPC

Percent difference(%)

Total exergydestroyed (kJ)

45.6 52.4 �14.9

Total energyconsumed (kJ)

65.6 63.1 3.76

Table 3Total exergy destruction evaluated by component using each controller.

Total exergy destructionby component

Second lawMPC (kJ)

First lawMPC (kJ)

Percentdifference (%)

EEV 3.12 2.83 9.45Compressor 33.7 32.1 4.86Evaporator 2.06 3.81 �85.4Condenser 6.73 13.7 �103

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 363

flow rates has consequences with regards to the exergy destructionin the evaporator and condenser which are penalized by the secondlaw MPC. This will be highlighted later in Fig. 13. h

The exergy destruction rate, energy consumption rate, andreversible power resulting from each of the controllers are shownin Fig. 12. First, it is important to highlight that the reversiblepower is not equivalent between the two simulations, verifyingthe statement made earlier regarding the expected difference inthe two controllers. This is particularly important becausetransient exergy analyses are not typically applied to VCSs nor isexergy destruction minimization typically conducted using a tran-sient exergy destruction rate. These results show that when consid-ering transient operation of a VCS, an exergy-based optimalcontroller has the potential to make different decisions abouthow to operate the system than a conventional energy-based con-troller will, to meet the same performance demand. Also asexpected, the second law MPC destroyed less exergy over the200-s simulation whereas the first law MPC consumed less energyduring the same simulation. The total exergy destroyed and totalenergy consumed using each controller is compared in Table 2.Note that the percent differences were calculated relative to theperformance of the second law MPC.

Although the second law MPC consumes 3.76% more energythan the first law MPC, it destroys almost 15% less exergy. There-fore, the tradeoff between energy consumption and exergydestruction is not necessarily 1:1. To analyze this more closely,the exergy destruction rate for each individual VCS component iscompared in Fig. 13. The total exergy destroyed and energy con-sumed during the complete simulation in each component usingboth controllers is shown in Table 3.

Surprisingly, the first law MPC destroyed less exergy in the EEVand compressor. However, in the evaporator and condenser, thefirst law MPC destroyed 103% and 85.4% more exergy, respectively,than was destroyed using the second law MPC.

Remark 4. It has long been cited that the greatest exergydestruction site in a VCS is the compressor [1]. This is still thecase as shown in Table 3. However, these results show that it ispossible for exergy to be destroyed on the same order of magnitudein other components, in this case the condenser, when all fourcontrol inputs are being modulated. As variable-speed fans becomemore common in commercial VCSs, the effect of air mass flow rateon the overall efficiency of the system can be quite significant,particularly during transient operation. What is more, the irrever-sibilities being characterized here are not of the fans themselves(which are omitted from this analysis) but from heat transfer andmass transfer occurring inside the evaporator and condenser.These irreversibilities are inherently not taken into account in an

364 N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365

energy-based controller which would only be able to account forlosses (e.g. due to pressure drop) in the fans which drive air acrossthe heat exchangers. h

Finally, we evaluate two efficiency metrics for VCSs – the COPand the exergetic efficiency – for transient operation as shown inEqs. (68) and (69) respectively.

COP½k� ¼_Q e½k�_Wk½k�

ð68Þ

gII½k� ¼ 1�_Xdest;VCC½k�

_Wk½k�ð69Þ

COP is a measure of the rate at which cooling is achieved by theVCS, relative to the rate at which work is done on the system. Byvirtue of how VCSs work, the COP is generally greater than one,with a higher COP indicating greater efficiency. A major downsideof COP as a metric, however, is that it is inherently not normalized,so it cannot be used to characterize how well a system is perform-ing relative to a baseline measure of performance. Alternatively,the exergetic efficiency measures the rate at which exergy isdestroyed relative to the rate at which exergy is supplied to theVCS. This metric is defined between 0 and 1 and tells us how effec-tively the exergy supplied to this system, in this case work done onthe compressor, is used in the VCS in an absolute sense.

Fig. 14 shows that the first law MPC operates the system at ahigher COP but with a lower exergetic efficiency. On average, theCOP achieved by the second law MPC is 3.95% lower than thatachieved by the first law MPC. On the other hand, on average,the exergetic efficiency achieved by the second law MPC is 41.4%greater than that achieved by the first law MPC. These results indi-cate that although the second law MPC would operate the systemin such a way as to consume slightly more energy, that energy isbeing used by the system more effectively. To be more precise, thismeans operating the system with fewer irreversibilities, such asfriction in refrigerant flow and losses in heat transfer across finitetemperature differences. In the case of VCSs with thermal storagesystems, this may allow one to achieve cooling more efficientlyby producing and storing excess cooling at particular times, asopposed to simply minimizing energy consumption at all times.Operating the system using an exergy-based optimal controllercan also have implications on the wear of the physical componentsthemselves, a longer term objective for the operation of thermal

0 50 100 150 2002.4

2.6

2.8

Coe

ffici

ent o

f P

erfo

rman

ce

First Law MPCSecond Law MPC

0 50 100 150 2000

0.1

0.2

0.3

0.4

0.5

Time (s)

Exe

rget

ic E

ffici

ency

Fig. 14. Instantaneous COP and exergetic efficiency resulting from each of the twocontrollers.

systems which was not explicitly accounted for by either controllerconsidered in this case study.

5. Conclusion

In this paper we derived an expression for the transient rate ofexergy destruction for the refrigerant-side dynamics of a VCS thatin turn was used to design and implement an exergy-based modelpredictive controller for closed-loop operation of the VCS. Simula-tion results showed that during transient operation of the VCS, thesecond law (exergy-based) MPC specifically accounted for irrever-sibilities in each component of the system whereas a comparablefirst law MPC did not. The distribution of irreversibilities acrossthe heat exchangers, in particular, varied significantly betweenthe second law and first law model predictive controllers.Moreover, the distribution of exergy destruction across the compo-nents of the VCS changed as a function of the control inputs, dem-onstrating not only the importance of considering the dynamicexergy destruction rate but also of optimal control of VCSs with fullactuation. The results presented here consider a canonical VCS andtherefore, future work will focus on applying the tools developedin this paper to an experimental VCS to validate the use ofexergy-based optimal control.

Acknowledgement

This work was supported in part by the Department of Energy(DOE) Office of Science Graduate Fellowship Program.

References

[1] Kotas TJ. The exergy method of thermal plant analysis. London: Butterworths;1985.

[2] Ahamed JU, Saidur R, Masjuki HH. A review on exergy analysis of vaporcompression refrigeration system. Renew Sustain Energy Rev2011;15(3):1593–600.

[3] Padilla M, Revellin R, Bonjour J. Exergy analysis of R413A as replacement ofR12 in a domestic refrigeration system. Energy Convers Manage2010;51:2195–201.

[4] Mahabadipour H, Ghaebi H. Development and comparison of two expandercycles used in refrigeration system of olefin plant based on exergy analysis.Appl Therm Eng 2012;50(1):771–80.

[5] Bejan A. Fundamentals of exergy analysis, entropy generation minimization,and the generation of flow architecture. Int J Energy Res 2002;26:545–65.

[6] Nag PK, De S. Design and operation of a heat recovery steam generator withminimum irreversibility. Appl Therm Eng 1997;17(4):385–91.

[7] Vargas J, Bejan A. Thermodynamic optimization of finned crossflow heatexchangers for aircraft environmental control systems. Int J Heat Fluid Flow2001;22:657–65.

[8] Liberzon D. Calculus of variations and optimal control theory: a conciseintroduction. Princeton, New Jersey: Princeton University Press; 2012.

[9] Cengel YA, Boles MA. Thermodynamics: an engineering approach. 6thed. Boston: The McGraw-Hill Companies, Inc.; 2008.

[10] Moran MJ, Shapiro HN. Fundamentals of engineeringthermodynamics. Hoboken, New Jersey: John Wiley & Sons Inc.; 2004.

[11] Bejan A. Advanced engineering thermodynamics. Hoboken, New Jersey: JohnWiley & Sons, Inc.; 2006.

[12] Stoecker W, Jones J. Refrigeration and air-conditioning. New York: McGraw-Hill Book Company; 1983.

[13] Rasmussen BP. Dynamic modeling for vapor compression systems – Part I:Literature review. HVAC&R Res 2012;18(5):934–55.

[14] Bendapudi S, Braun JE. A review of literature on dynamic models of vaporcompression equipment. ASHRAE Report #4036-5; May 2002.

[15] McKinley TL, Alleyne AG. An advanced nonlinear switched heat exchangermodel for vapor compression cycles using the moving-boundary method. Int JRefrig 2008;31(7):1253–64.

[16] Li B, Alleyne AG. A dynamic model of a vapor compression cycle with shut-down and start-up operations. Int J Refrig 2010;33(3):538–52.

[17] Amano K. Heat pump apparatus and control method thereof, Europe PatentEP2469201 A3; 2012.

[18] He XD, Liu S, Asada HH, Itoh H. Multivariable control of vapor compressionsystems. HVAC&R Res 1998;4(3):205–30.

[19] Shah R, Rasmussen BP, Alleyne AG. Application of a multivariable adaptivecontrol strategy to automotive air conditioning systems. Int J Adapt ControlSignal Process 2004;18(2):199–221.

N. Jain, A. Alleyne / Energy Conversion and Management 92 (2015) 353–365 365

[20] Lin J-L, Yeh T-J. Control of multi-evaporator air-conditioning systems for flowdistribution. Energy Convers Manage 2009;50(6):1529–41.

[21] Rasmussen BP, Alleyne AG. Gain scheduled control of an air conditioningsystem using the Youla parameterization. IEEE Trans Control Syst Technol2010;18(5):1216–25.

[22] Hovgaard TG, Larsen LF, Edlund K, Jørgensen JB. Model predictive controltechnologies for efficient and flexible power consumption in refrigerationsystems. Energy 2012;44(1):105–16.

[23] Elliott MS, Rasmussen BP. A model-based predictive supervisory controller formulti-evaporator HVAC systems. In: Proceedings of the American ControlConference; 2009.

[24] Larsen LFS, Thybo C, Stoustrup J, Rasmussen H. A method for onlinesteady state energy minimization with application to refrigeration systems.In: Proceedings of the 43rd IEEE conference on decision and control;2004.

[25] Koeln JP, Alleyne AG. Decentralized controller analysis and design for multi-evaporator vapor compression systems. In: Proceedings of the 2013 Americancontrol conference. Washington, D.C.; 2013.

[26] Maciejowski J. Predictive control with constraints. Prentice Hall; 2000.[27] Mayne DQ, Rawlings JB, Rao CV, Scokaert POM. Constrained model predictive

control: stability and optimality. Automatica 2000;36:789–814.

[28] Jain N, Alleyne AG. Transient exergy destruction analysis for a vaporcompression system. In: International refrigeration and air conditioningconference. West Lafayette; 2014.

[29] Doty JH, Camberos JA, Yerkes KL. Approximate approach for direct calculationof unsteady entropy generation rate for engineering applications. In:Proceedings of the 50th AIAA Aerospace Sciences Meeting. Nashville; 2012.

[30] Rasmussen BP. Control-oriented modeling of transcritical vapor compressionsystems. Urbana (IL): University of Illinois at Urbana-Champaign, M.S. Thesis;2000.

[31] Jain N. Thermodynamics-based optimization and control of integrated energysystems. Urbana (IL): University of Illinois at Urbana-Champaign; 2013 (Ph.D.Thesis).

[32] Franklin GF, Powell JD, Workman M. Digital control of dynamic systems. 3rded. Half Moon Bay (CA): Ellis-Kagle Press; 1998.

[33] Sciubba E, Wall G. A brief commented history of exergy from the beginnings to2004. Int J Thermodyn 2007;10(1):1–26.

[34] Gallo WLR, Milanez LF. Choice of a reference state for exergetic analysis.Energy 1990;15(2):113–21.

[35] Rasmussen BP. Dynamic Modeling and advanced control of air conditioningand refrigeration systems. Urbana (IL): University of Illinois at Urbana-Champaign, Ph.D. Thesis; 2005.