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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.
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