Energy efficient model predictive building temperature control $ Matt Wallace a , Ryan McBride a , Siam Aumi a , Prashant Mhaskar a, , John House b , Tim Salsbury b a Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L8 b Johnson Controls Inc., 507 E. Michigan Street, Milwaukee, WI 53202, United States article info Article history: Received 4 March 2011 Received in revised form 5 July 2011 Accepted 8 July 2011 Available online 29 July 2011 Keywords: Vapor compression cycle Temperature control Building control Energy efficient control Model predictive control EnergyPlus abstract Many systems used in buildings for heating, ventilating, and air-conditioning waste energy because of the way they are operated or controlled. This paper explores the application of model predictive control (MPC) to air-conditioning units and demonstrates that the closed-loop performance and energy efficiency can be improved over conventional approaches. This work focuses on the problem of controlling the vapor compression cycle (VCC) in an air-conditioning system, containing refrigerant which is used to provide cooling. The VCC considered in this work has two manipulated variables that affect operation: compressor speed and the position of an electronic expansion valve. The system is subject to constraints, such as the range of permissible superheat, and also needs to regulate temperature variables to set points. An MPC strategy is developed for this type of system based on linear models identified from data obtained from a first-principles model of the VCC. The MPC strategy incorporates economic measures in the objective function as well as control objectives. Tests are carried out on a simulated VCC system that is linked to a simulation of a realistic building that is developed in the U.S. Department of Energy Computer Simulation Program, EnergyPlus. The MPC demonstrated significantly better tracking control relative to conventional approaches (a reduction of 70% in terms of the integral of squared error for step changes in the temperature set-point), while reducing the VCC energy requirements by 16%. The paper describes the control approach in detail and presents results from the tests. & 2011 Elsevier Ltd. All rights reserved. 1. Introduction Environmental concerns as well as increased fuel prices have brought energy efficiency to the forefront of research priorities. Canada currently ranks as the world’s sixth largest user of primary energy (such as fossil fuels, nuclear fuels, hydro power, etc.). In Canada, approximately 30% of the energy obtained from primary sources of energy is consumed in the commercial and residential sectors of the economy (Behidj et al., 2009). In these sectors, a significant portion of the energy is used towards meeting the thermal and electrical energy demands in buildings. Recent government reports estimate that through more efficient building operation, the total energy consumption by the com- mercial and residential sectors can be reduced by 15–20% (Behidj et al., 2009). The operating efficiency of a building is influenced by many factors and can be improved at various points over its lifespan. Prior to construction, using design standards that incorporate energy and environmental concerns is often the first step to achieving an energy efficient building design. However, these design standards alone are not sufficient to ensure that a building remains energy efficient in response to changing energy and environmental standards. Once constructed, with the advent of more energy efficient technology (i.e., EnergyStar certified tech- nology), the building can be appropriately retrofitted to meet more stringent energy and environmental standards. Finally, the energy efficiency of existing buildings can be improved through better control of their heating, ventilation, and air-conditioning (HVAC) systems (see American Society of Heating, for a detailed description of the common components of an HVAC system), which regulate building comfort (temperature and humidity) and account for 30–50% of the total energy consumption in buildings (Albieri et al., 2009). The focus of the present work is to demonstrate (via simulations) this improved efficiency achievable through use of advanced (model based) control techniques. To this end, we utilize an existing model of a VCC and couple it with a building model to approximate a (reasonably) realistic scenario of a roof top unit providing cooling to a room in a building using air as the only cooling medium. A vapor compression cycle (VCC) refers to a type of thermo- dynamic machinery that transfers heat using a compressible fluid referred to as the refrigerant. The most common realization of a VCC consists of four components: a compressor, condenser, Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ces Chemical Engineering Science 0009-2509/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ces.2011.07.023 $ This work is a collaborative effort between Johnson Controls Inc. and the McMaster Advanced Control Consortium. Corresponding author. E-mail address: [email protected] (P. Mhaskar). Chemical Engineering Science 69 (2012) 45–58
Transcript
Chemical Engineering Science 69 (2012) 45–58
Contents lists available at ScienceDirect
Chemical Engineering Science
0009-25
doi:10.1
$This
McMast� Corr
E-m
journal homepage: www.elsevier.com/locate/ces
Energy efficient model predictive building temperature control$
Matt Wallace a, Ryan McBride a, Siam Aumi a, Prashant Mhaskar a,�, John House b, Tim Salsbury b
a Department of Chemical Engineering, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4L8b Johnson Controls Inc., 507 E. Michigan Street, Milwaukee, WI 53202, United States
Many systems used in buildings for heating, ventilating, and air-conditioning waste energy because of
the way they are operated or controlled. This paper explores the application of model predictive control
(MPC) to air-conditioning units and demonstrates that the closed-loop performance and energy
efficiency can be improved over conventional approaches. This work focuses on the problem of
controlling the vapor compression cycle (VCC) in an air-conditioning system, containing refrigerant
which is used to provide cooling. The VCC considered in this work has two manipulated variables that
affect operation: compressor speed and the position of an electronic expansion valve. The system is
subject to constraints, such as the range of permissible superheat, and also needs to regulate
temperature variables to set points. An MPC strategy is developed for this type of system based on
linear models identified from data obtained from a first-principles model of the VCC. The MPC strategy
incorporates economic measures in the objective function as well as control objectives. Tests are carried
out on a simulated VCC system that is linked to a simulation of a realistic building that is developed in
the U.S. Department of Energy Computer Simulation Program, EnergyPlus. The MPC demonstrated
significantly better tracking control relative to conventional approaches (a reduction of 70% in terms of
the integral of squared error for step changes in the temperature set-point), while reducing the VCC
energy requirements by 16%. The paper describes the control approach in detail and presents results
from the tests.
& 2011 Elsevier Ltd. All rights reserved.
1. Introduction
Environmental concerns as well as increased fuel prices havebrought energy efficiency to the forefront of research priorities.Canada currently ranks as the world’s sixth largest user ofprimary energy (such as fossil fuels, nuclear fuels, hydro power,etc.). In Canada, approximately 30% of the energy obtained fromprimary sources of energy is consumed in the commercial andresidential sectors of the economy (Behidj et al., 2009). In thesesectors, a significant portion of the energy is used towardsmeeting the thermal and electrical energy demands in buildings.Recent government reports estimate that through more efficientbuilding operation, the total energy consumption by the com-mercial and residential sectors can be reduced by 15–20% (Behidjet al., 2009).
The operating efficiency of a building is influenced by manyfactors and can be improved at various points over its lifespan.Prior to construction, using design standards that incorporateenergy and environmental concerns is often the first step to
ll rights reserved.
nson Controls Inc. and the
ar).
achieving an energy efficient building design. However, thesedesign standards alone are not sufficient to ensure that a buildingremains energy efficient in response to changing energy andenvironmental standards. Once constructed, with the advent ofmore energy efficient technology (i.e., EnergyStar certified tech-nology), the building can be appropriately retrofitted to meetmore stringent energy and environmental standards. Finally, theenergy efficiency of existing buildings can be improved throughbetter control of their heating, ventilation, and air-conditioning(HVAC) systems (see American Society of Heating, for a detaileddescription of the common components of an HVAC system),which regulate building comfort (temperature and humidity) andaccount for 30–50% of the total energy consumption in buildings(Albieri et al., 2009). The focus of the present work is todemonstrate (via simulations) this improved efficiency achievablethrough use of advanced (model based) control techniques. Tothis end, we utilize an existing model of a VCC and couple it witha building model to approximate a (reasonably) realistic scenarioof a roof top unit providing cooling to a room in a building usingair as the only cooling medium.
A vapor compression cycle (VCC) refers to a type of thermo-dynamic machinery that transfers heat using a compressible fluidreferred to as the refrigerant. The most common realization of aVCC consists of four components: a compressor, condenser,
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5846
expansion valve, and an evaporator. In the VCC, the refrigerantcirculates through the four components, undergoing variousthermodynamic changes, which, in turn, influence external con-ditions. Mathematically, the dynamics of the refrigerant statesand external conditions are modeled using a set of couplednonlinear ordinary differential and algebraic equations, resultingin a complex differential-algebraic equation system. The controlobjectives are typically defined in terms of degrees of superheatin the refrigerant at the evaporator exit and the air temperature atthe evaporator exit (the supply air temperature). Ensuring thatsuperheated refrigerant exits the evaporator is of utmost impor-tance in preventing physical damage in the VCC, as liquidrefrigerant can damage the mechanical components used in thecompressor. The manipulated variables include the compressorspeed, the air flow rates and the expansion valve opening. Whileresearch activity has been strong for a long time on the design/material and, to a smaller extent, modeling, of the variouscomponents of the VCC (see Rasmussen, 2005 for details), therehas been a recent trend towards improved control of the VCC forimproved energy efficiency.
Traditional VCC control strategies have included PID/PI decen-tralized control (i.e., multiple independent single-input–single-output (SISO) controllers) and simple on/off control. The latterlimits overall efficiency due to large power requirements andsignificant thermal inertia during start-up transients while theformer’s efficiency is limited by extensive interactions and non-linearity in the VCC system dynamics and the presence of inputconstraints. Approaches to improve the performance of conven-tional PID/PI controllers can be categorized into those whichattempt to decouple the VCC dynamics to improve SISO controll-ability (Keir and Alleyne, 2007; Jain et al., 2010) and adaptivecontrol approaches which attempt to account for the processnonlinearity (i.e. time-varying process gains) (Lin and Yeh, 2007;Zhu et al., 2001).
Among the decoupling approaches, the most straightforwardextension has been to employ linear decouplers to removeinteractions among the individual control loops. However, theeffectiveness of this approach is entirely contingent on the modelaccuracy, and a poor model can lead to closed-loop performancedegradation. Another decoupling approach has been to use linearcombinations of available VCC measurements as the controlledvariables instead of traditional controlled variables (Jain et al.,2010). The adaptive control approaches, on the other hand,attempt to account for the nonlinear nature of the VCC dynamicsby updating the PI/PID tuning parameters online using a model ofthe process. However, these approaches are typically restricted tolinear models (for computational reasons), implying the tuningparameter updates may be erroneously updated since the trueprocess is highly nonlinear, leading to poor control performance.Despite these improvements, PI/PID control designs remaininherently based on a single-input–single-output frameworkand do not account for the presence of constraints and optimality.The control action prescribed by a controller that does notaccount for input constraints can result in performance degrada-tion or even closed-loop instability.
One control method well suited to handling constraints andoptimality is model predictive control (MPC). MPC is an optimiza-tion-based control approach in which the coupled, multiple-input–multiple-output nature of complex systems can be accounted for indetermining the control action by using a model of the process. Inmodel predictive control, a nonlinear or linear process model isused to evaluate the effect of candidate manipulated input trajec-tories via an objective function, and an optimization problem issolved to yield the manipulated input trajectory that minimizes theobjective function while satisfying any constraints. Only the firstpiece of the input trajectory is implemented and the problem is
re-solved at the next sampling time, using the new measuredvalues of the process variables. One of the strengths of the MPCframework is the flexibility (and scope) in specifying optimalityobjectives through various terms in the objective function, orthrough constraints on the variables of interest. This, and theresults available on the stability and feasibility properties of MPCformulations (see, e.g., Mhaskar et al., 2005, 2006; Mhaskar, 2006)make MPC a preferred candidate to be evaluated for possible usewithin building control structures. Recently, there have been manyexamples in the literature of the application of MPC for regulating awide range of VCC or HVAC systems. In addition to the nature ofthe system being regulated, the major differentiating featureamong these MPC approaches is the complexity of the model usedfor predictions. Specifically, the predictive model may be a linear-ized version of a non-linear state-space model (Schurt et al., 2009;Sandipan et al., 2010; Morosan et al., 2010), an empiricallyidentified linear (Huang et al., 2009; Ma et al., 2010) or nonlinearmodel (Xi et al., 2007), or a first-principles non-linear model(Leducq et al., 2006; Ma et al., 2010; Sarabia et al., 2007). Themajority of the MPC application examples have utilized a linearVCC model (either a linearized state-space model or an empiricallyidentified input–output model). For example, in Sandipan et al.(2010), an experimental chiller network, consisting of two chillersand multiple external heat exchangers, is regulated to satisfy thecooling load in addition to minimizing electricity costs. In Huanget al. (2009), empirical first-order time-delay (FOTD) models areused in a robust MPC formulation for improving temperatureregulation of an air-conditioning system. Specifically, several FOTDmodels are identified at various operating points, and based on thecurrent operating conditions, the most appropriate FOTD model isused in the MPC optimization. Linear MPC applications in thecontext of building control include the work in Morosan et al.(2010) where a distributed MPC design is used to regulate thetemperature of multiple zones in a building and minimize powerconsumption. That is, each zone is served by a separate HVAC unitunder the control of a zone-specific MPC design to regulate theinternal zone temperature. Another example of building controlusing (linear) MPC is available in Ma et al. (2010) where weatherdata is incorporated into the design to determine the building zonetemperature set-point. This allows for pre-cooling during non-peakperiods and reduced power consumption compared to traditionalpre-programmed HVAC unit control strategies. The existing resultsnotwithstanding, there still exists a lack of results on the applica-tion of a MPC design to a detailed model of a VCC unit coupled witha realistic building model to evaluate the control performance inthe presence of disturbances.
Motivated by the above, this work evaluates the performanceof an integrated temperature control framework via simulations.Specifically, we design a predictive controller for a stand-aloneVCC unit and utilize it in a cascade control structure for tempera-ture control. The proposed control structure is implemented on arealistic building model that accounts for varying weather condi-tions and internal heat loads throughout the course of a day andthe results are compared with a PI-based control structure. Therest of this manuscript is organized as follows. In Section 2, wegive an overview of the VCC and building models used in thiswork. Then, in Section 3, we develop a control strategy fortemperature control in a building zone. To this end, we firstestimate an input–output model for the VCC in Section 3.1 andthen design an offset free predictive controller for a stand-aloneVCC unit in Section 3.2. This controller is subsequently incorpo-rated in a cascade control strategy to control a specific roomtemperature and then implemented on a realistic building modelin Section 3.3. The performance of the proposed control strategy iscompared with a conventional PI-based control strategy. Finally,we summarize our results in Section 4.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 47
2. Preliminaries
In this section, we give an overview of an existing VCC modelused in this work and point out the key limitations of the modelsand modifications. Next, we describe the building model and thesoftware used for interfacing the VCC model with the buildingmodel. Note that model development is not the focus of thepresent work. A detailed model of the VCC and the building isused only to illustrate the control design and is described in thissection for completeness.
2.1. VCC model overview
An ideal VCC consists of four processes: isentropic compres-sion in a compressor, isobaric energy dissipation in a condenser,isenthalpic expansion in an expansion valve and isobaric energyabsorption in an evaporator. An overlay of a VCC and thecorresponding pressure-volume diagram of the refrigerant isshown in Fig. 1.
In a VCC unit, the refrigerant enters the compressor as asuperheated vapor and is compressed to a higher pressure,resulting in the superheated vapor having a higher temperaturethan the ambient temperature. From the compressor, the super-heated refrigerant vapor enters a condenser (typically placedoutdoors), condensing to a sub-cooled liquid at the condenserexit as a fan blows the ambient air over the condenser. The highpressure sub-cooled liquid then flows into an expansion valvewhich decreases the pressure and temperature of the refrigerant,causing a liquid–vapor mixture to form. Then, the two-phaserefrigerant mixture enters an evaporator that is exposed to theenvironment to be cooled. The environment temperature is abovethe temperature of the refrigerant, resulting in the evaporationand subsequent heating of the refrigerant to a superheated vaporat the evaporator exit. The air, in turn, is cooled and available asprimary air to be distributed for cooling. The superheated vaporfrom the evaporator exit then flows into the compressor, com-pleting the cycle.
The VCC model used in this work is adapted from the existingsimulation package, Thermosys, developed at the Air Conditioning& Refrigeration Center (ACRC) at the University of Illinois atUrbana-Champaign. In this simulation package, the refrigerant isR-134a and is assumed to be cooling an air medium. Thesimulator consists of dynamic models for the condenser, eva-porator, and compressor and static models (i.e., algebraic equa-tions) for the expansion valve and piping. In the followingsubsections, a brief overview of the model components is pro-vided followed by a general mathematical representation. For a
Volume
Pre
ssur
e
Gas
Liquid
Wet vapor(saturated conditions)
Evaporator Compressor
Condenser
Heat from refrigerant
Heat from process
Valve
Fig. 1. VCC overlay on a pressure–volume diagram of a typical refrigerant.
full description of the model components and a complete listof the equations and parameters, the reader is referred toRasmussen (2005).
2.1.1. Compressor
The compressor in the VCC model is a reciprocating compres-sor defined by its isentropic efficiency, Zk, which is the ratio of thework required for ideal adiabatic compression to the workrequired for actual compression, and its volumetric efficiency,ZV , which is the ratio of induced gas volume to the discharged gasvolume (swept volume). Note that in the present work we use amodel of a variable speed compressor to illustrate the improvedefficiency achievable by model-based control designs and the keyinterpretations remain applicable for other compressor types (on/off compressors, etc.). Demonstrating improved energy efficiencyon other cooling units is the subject of future work and outsidethe scope of the present manuscript.
The efficiency of the compressor depends on the pressure ratioof the outlet to inlet stream and the compressor RPM, ok. Theyare obtained via (experimentally obtained) lookup tables ofefficiencies at various operating conditions. The mass flow ratein the compressor is modeled using the following static equation:
_mk ¼okVkrkZV ð1Þ
where Vk denotes the swept volume (a compressor parameter)and rk is the refrigerant density at the inlet. The term, VkrkZV ,characterizes the compressor capacity in terms of inlet refrigerantconditions. For the compressor energy balance, the dynamics ofthe heat transfer during the transport of the refrigerant from thecompression cylinder (where the compression takes place) to theshell (where the refrigerant exits) are taken into consideration.Specifically, the energy dynamics are modeled as a linear (in K),first-order differential equation:
tk_h
o
kþhok ¼ KðZk,hi
k,ho
kÞ ð2Þ
where hok is the enthalpy of the outlet refrigerant, tk is a time
constant (a compressor parameter), and Kð�Þ is the gain. The gainis constant during integration and a nonlinear function of theisentropic efficiency, inlet refrigerant enthalpy, hi
k, and the idealisentropic outlet enthalpy, h
o
k, which is determined by therefrigerant thermodynamic properties and inlet enthalpy.
Remark 1. In general, the compressor type (variable speed or on/off) is dependent on the specific application of the VCC (mostexisting compressors utilize an on/off strategy). Due to thelimited range of operating speeds for an on/off compressor, thestartup and shutdown of a setup equipped with this compressortype can draw considerably more energy during these operatingconditions than a setup equipped with a variable speed compres-sor. Demonstrating the improvement over traditional on–offsetups (using a good model of such a unit) does remain anobjective of future work, but is outside the scope of this manu-script. If an on/off type compressor is used in the VCC, modifica-tions can be made to the model (as presented in this section), aswell as to the control design to ensure the implementation of theMPC control structure is feasible. In particular, any proposedcontrol design for the VCC must account for the discrete nature ofthe compressor operation. For instance, in the present work, thecompressor RPM is a manipulated variable for VCC control andtreated as a continuous variable. With an on/off type compressor,the compressor RPM is fixed when it is on and zero otherwise. In amodel predictive control framework, this can be modeled eitherindirectly (by choosing the on/off durations as input variables) orexplicitly using binary variables in the optimization problem. Onthe other hand, classical control approaches, such as PI control,have limited options in handling units with discrete operation.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5848
Another limitation of the current VCC model is the range ofcompressor speeds over which the model is valid. In particular,the model does not remain valid for low RPM values, and as aresult, the simulations have been carried out with the RPMrestricted to this range to preserve the validity of the results.Enhancing the range of validity of the model also remains anotherdirection of future work.
2.1.2. Expansion valve
The expansion valve is modeled as an isenthalpic process,meaning the inlet and outlet enthalpies are identical, with a massflow rate given by the following expression:
_mv ¼ Cd
ffiffiffiffiffiffiffiffiffiffiffiffiffiffirvDPv
qð3Þ
where Cd is the valve discharge coefficient, DPv is defined as thepressure difference between the inlet and outlet refrigerant, andrv is the maximum of either the sub-cooled liquid density or thesaturated liquid density at the inlet operating conditions. Thedischarge coefficient is determined by the valve opening andpressure differential and obtainable from experimental lookuptables. The sub-cooled and saturated liquid refrigerant densitiesare also obtained from lookup tables of the refrigerant’s thermo-dynamic properties.
Remark 2. An electronic expansion valve (EEV) is used in thisVCC model, where the valve position is adjustable and in agree-ment with electronic expansion valves used in practice. The valveposition of an actual EEV is proportionally adjusted throughvarying the frequency of a built-in step-motor, where the energydraw of this motor is minimal relative to the other energy-consuming components of a practical VCC (i.e., compressor andfan motors).
2.1.3. Heat exchangers
The dynamics of the VCC are dominated by the condenser andevaporator. Both heat exchangers are modeled as a long thinhorizontal tube with one-dimensional fluid flow, negligible pres-sure drop (due to momentum change and viscous friction), andnegligible axial conduction.
In both heat exchangers, the refrigerant may undergo multiplephase changes; accordingly, the refrigerant is modeled using alumped parameter, moving boundary approach which accountsfor different fluid regions (superheated vapor, saturated vapor–liquid, or sub-cooled liquid) and their time varying boundaries. Inthis approach, each fluid region is represented as a separatecontrol volume (see Fig. 2) with corresponding states and para-meters. For the evaporator, there are two fluid regions: a two-phase region followed by a superheat region while the condenserhas three fluid regions: a vapor region followed by a two-phaseand a sub-cooled region. In each two-phase region, the refrigerantfluid properties are taken as the weighted combination of thesaturated liquid and vapor properties. The mean void fraction, g,which is defined as the ratio of the vapor volume in a region tothe total region volume, is used to weight the properties. Forinstance, the refrigerant density in a two-phase region is given bygrf þð1�gÞr‘ where rf and r‘ are the saturated vapor and liquid
Two-phase region Superheat region S
Fig. 2. Heat exchanger schematics showing the different fluid regions (adapted from
(a) Evaporator with two fluid regions. (b) Condenser with three fluid regions.
densities, respectively. In the superheated and sub-cooledregions, the refrigerant properties, such as density and tempera-ture, are determined using the heat exchanger pressure (assumedconstant) and the average regional enthalpy (the average of theinlet and outlet enthalpies).
The mass and energy balance ordinary differential equations(ODEs) for each fluid region are derived from the governingpartial differential equations (PDEs) for fluid flow in a tube. Toyield a set of ODEs from the PDEs, the spatial dependence fromthe PDEs is removed after applying simplifying assumptions andLeibnitz’s rule on any differential with respect to the spatial co-ordinate, z. The full details of the modeling approach are availablein Rasmussen (2005). Eqs. (4) and (5) represent the governingrefrigerant mass and energy balance PDEs (respectively) of aspecific fluid region.
@ðrAcÞ
@tþ@ _m
@z¼ 0 ð4Þ
@ðrAch�AcPÞ
@tþ@ð _mhÞ
@z¼ piaiðTw�TrÞ ð5Þ
where r, _m, and h denote the refrigerant density, mass flow rate,and specific enthalpy (respectively), Ac is the heat exchangercross-sectional area, P is the fluid region pressure, pi is the innerperimeter of the heat exchanger, ai is the heat transfer coefficientbetween the refrigerant and the heat exchanger inner wall, Tw isthe wall temperature, and Tr is the refrigerant temperature. ThesePDEs are coupled with the following wall energy balance for eachregion:
ðcprAÞw_T w ¼ piaiðTr�TwÞþpoaoðTa�TwÞ
where ðcprAÞw is the thermal capacitance of the tube wall per unitlength, po is the outer perimeter of the heat exchanger, and ao isthe heat transfer coefficient between the tube wall and thesurrounding air with temperature Ta. After integrating @ _m=@z
and @ð _mhÞ=@z along the length of the tube using Leibnitz’s rule,the final set of ODEs for the heat exchangers can be arranged inthe following matrix form:
Zhðxh,uÞ _xh ¼ f hðxh,u,dÞ
where Zhð�Þ and f hð�Þ are a matrix and vector, respectively, and theheat exchanger state variables, xh, include: the length of thesuperheat, Lc,1, and two-phase, Lc,2, regions in the condenser,the condenser wall temperatures in all three regions, Tw,c,1, Tw,c,2,and Tw,c,3, the constant condenser pressure, Pc, the condenseroutlet refrigerant enthalpy, ho
c, the length of the two-phase regionin the evaporator, Le,1, the evaporator wall temperatures for bothregions, Tw,e,1, Tw,e,2, the constant evaporator pressure, Pe, theevaporator outlet refrigerant enthalpy, ho
e, and the compressoroutlet refrigerant enthalpy, ho
k. The input vector, u, elements arethe compressor RPM, ok, and valve opening, vo. Note that in someVCC systems, the fan speeds for the air being blown over theevaporator and condenser (and therefore the mass flow rate ofair) may also be available for adjustment; however, for this VCCmodel, these are assumed constant. The disturbance vector, d, isconstituted of two measurable temperatures: the temperature of
Two-phase region Sub-cooled regionuperheat region
Rasmussen, 2005) which are modeled using the moving boundary approach.
1 The relative humidity is defined as the ratio of the partial pressure of the
water vapor to the saturation pressure of water at the system temperature.2 The same reference temperature is used for the computation of ho
w,e and hiw,e
such that Tref disappears in Eq. (7).
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 49
the air to be blown over the evaporator, Tia,e, which is commonly
referred to as the mixed air temperature, and the air temperatureat the inlet of the condenser. The latter temperature is simply theambient air temperature since the condenser is assumed to beoutdoors and henceforth will be denoted by Tamb. The air suppliedto the VCC evaporator is typically a mix of the zone (i.e., room)and ambient air. For example, the mixed air may be a mixture of80% zone air and 20% ambient air.
2.1.4. Mathematical representation
The Thermosys VCC model comes in the form of a Simulink-based toolbox in Matlab. For this work, we extracted the VCCmodel ODEs and algebraic expressions from the source files andexpressed the model as a differential-algebraic equation (DAE)system. In order to integrate the DAE system, algebraic equationsare required to be satisfied at all integration steps. Integrating theDAE system in Matlab as opposed to running the Simulink modelfiles yielded significant computational benefits with executiontimes for the same test period being reduced by over 70%. TheVCC DAE can be expressed in the following general form:
Zðx,uÞ _x ¼ f ðx,u,dÞ
gðx,uÞ ¼ 0
y¼ hðx,dÞ
where Zð�Þ and f ð�Þ again denote a matrix and vector, respectively,x is the VCC state vector, u and d were previously defined inSection 2.1.3, gðx,uÞ represents the algebraic expressions (used formodeling the piping and expansion valve), and y denotes the VCCoutputs. The VCC outputs are defined to be the superheat of therefrigerant exiting the evaporator, Ts,e, and the air temperature atthe evaporator exit, the so-called supply air temperature, To
a,e. Thesuperheat is defined as the number of degrees by which therefrigerant temperature at the evaporator exceeds its saturationtemperature. As mentioned in Section 1, Ts,e is required to bemaintained above 0 1C to protect against any liquid refrigerantentering the compressor and therefore required for safe andreliable compressor operation. In practice, the superheat is main-tained above 0 1C with a safety margin. The vector, hð�Þ, denotesthe (nonlinear) output mapping function. The mapping functionfor the superheat is relatively straightforward whereas the func-tion to compute the supply air temperature consists of finding theroot of a nonlinear equation as discussed next.
The following discussion contains modifications of the supplyair temperature calculation procedure found in the originalThermosys model. Specifically, we make corrections to the pro-cedure in the event of any condensation of the water vaporcontent in the air. To obtain the supply air temperature, To
a,e, anenergy balance for the wall side of the evaporator is solved.Assuming no energy accumulation in the evaporator walls, theheat absorption by the evaporator wall must equal the heat loss ofthe air:
aoAc
X2
i ¼ 1
Le,i
LeðT a,e�Tw,e,iÞ ¼Hloss ð6Þ
The term, aoAcP2
i ¼ 1ðLe,i=LeÞðT a,e�Tw,e,iÞ, represents the energyabsorption by the evaporator where T a,e is the average airtemperature around the evaporator:
T a,e ¼12ðT
ia,eþTo
a,eÞ
In the most general case (i.e., assuming there is condensation ofthe water vapor content in the air), the heat loss of the air, Hloss, isgiven by
Hloss ¼ _ma,ecp,a,eðTia,e�To
a,eÞþ _ma,eðwia,ehi
W ,e�woa,eho
W ,eÞ�h‘,e ð7Þ
where _ma,e and cp,a,e denote the mass flow rate and specific heatcapacity of the dry air being blown over the evaporator (assumedto remain constant), wi
a,e and woa,e denote the humidity ratio,
which is defined as the ratio of the mass of water vapor in the airto the total dry air mass, of the inlet and outlet air (respectively),and hi
W ,e and hoW ,e denote the specific water vapor enthalpy at the
inlet and outlet air conditions (respectively). The first term in Eq.(7) is the heat loss of the dry air and the only unknown variable inthis term is To
a,e (the variable of interest). The second term is theenergy loss of the water vapor content in the air. In this term, theinlet humidity and water vapor enthalpy are readily computablefrom the known temperature (and pressure). If no condensationoccurs, there is no change in the humidity ratio and wi
a,e ¼woa,e. In
the case of condensation, the outlet air is saturated, implying therelative humidity at the outlet, fo
a,e, is 1.1 To compute thehumidity ratio, its relationship with the relative humidity canbe used to derive (Dincer and Rosen, 2007, Chapter 6):
woa,e ¼ 0:622
foa,ePo
W ,e,sat
Pa�foa,ePo
W ,e,sat
where Pa is the known air pressure and PoW ,e,sat is the saturation
pressure of water at Toa,e, which can be computed using Antoine’s
equation. Meanwhile, the outlet water vapor enthalpy is com-puted using the standard formula:
hoW ,e ¼ hf ,satþ
Z Toa,e
Tref
cp,W ðTÞ dT
where hf ,sat is the heat of saturated water vapor at the airpressure, Tref is a reference temperature, and cp,W ðTÞ is the(possibly) temperature-dependent specific heat capacity of watervapor.2 The third term in Eq. (7), h‘,e, represents the heat contentin the condensed water if condensation occurs. Note that thenegative sign is required in front of h‘,e since heat losses arewritten as positive energies in Eq. (7). The heat content in thewater is given by
h‘,e ¼ _ma,eðwia,e�wo
a,eÞcp,W Toa,e ¼ _m‘,ecp,W To
a,e
where the product _ma,eðwia,e�wo
a,eÞ equals the mass of condensedwater (follows from the definition of humidity ratio) or _m‘,e andcp,W is the constant heat capacity of liquid water.
Having defined all the terms/variables in Eq. (6) and theirdependence on the unknown supply air temperature, To
e,a, a rootfinding algorithm can be applied to Eq. (6) to compute To
e,a.Alternatively, an iterative (i.e., direct-substitution) procedurecan be used where an initial guess for the supply air temperatureis made, Hloss is computed, and then the left hand side of Eq. (6) issolved for To
a,e. If the difference between the newly computedsupply air temperature and the initial guess exceeds a pre-definedtolerance, the newly computed value can be used to initialize thenext iteration. Note also that when solving Eq. (6), an assumptionregarding the occurrence of condensation has to be made. In thiswork, we first solve Eq. (6), assuming no condensation (i.e., withno h‘,w term), which is correct only if To
a,e4Tdp where Tdp is thedew-point temperature (computable from the air pressure). IfTo
a,erTdp, the necessary condensation term is added to Hloss priorto solving Eq. (6) and the equation system is re-solved.
2.1.5. VCC cooling capacity
For proper regulation of a building zone temperature, thecorresponding VCC unit for the zone must meet the cooling
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5850
capacity dictated by the highest possible ambient conditions andheat load disturbances. The nominal Thermosys VCC model has amaximum cooling capacity of 1127 W or 0.32 ton of refrigeration.This capacity is in agreement with that of a small experimentalVCC used to validate the nominal model. Exploratory simulationsrevealed that this cooling capacity is insufficient (even withperfect control) to achieve the desired control objectives in termsof temperature control (see Section 3.3) for the ambient condi-tions and heat load disturbances considered in the simulations. Asa result, the VCC model parameters are re-scaled such that thecooling capacity increases to 0.85 ton. Specifically, the mass flowrate of the circulating refrigerant, _mr , along with the compressorvolume, Vk, are first increased. Next, the length of the evaporator,Le, is increased to allow longer contact of the supply air with theevaporator wall. The inner and outer cross-sectional areas of theevaporator, Ae,i and Ae,o respectively, and the mass of the eva-porator, Me, are then increased by the same factor. Then, thediameter of the evaporator pipe, De, and (dry) air mass flow rate,_ma,e, are increased to allow for more heat transfer from the air
passing over the evaporator. To ensure that the additional heatabsorbed by the refrigerant in the evaporator could be dissipatedinto the surrounding environment at the condenser, the sameparameters for the condenser are increased by the same factors.Table 1 lists the nominal model parameters and the new ‘re-scaled’ parameters.
Remark 3. While the VCC model includes an EEV and variablespeed compressor, enabling the VCC to achieve a varying coolingcapacity, our current model does not capture the total coolingrange associated with either an experimental or an industrialcooling unit. Specifically, the operating conditions correspondingto operating the VCC near its upper and lower cooling extremesare not captured in the scaled VCC model, as inaccuracies arosedue to two factors: (1) the experimentally populated lookuptables for certain component and thermodynamic parameters(ZV , Zk, Cd, etc.) corresponded to different operating conditionsfor the scaled and original VCC, resulting in a smaller feasibleoperating range for the scaled system, and (2) the limiting natureof the EEV caused the minimal operating conditions associatedwith the compressor to be higher. Further reductions in thecompressor RPM cause the refrigerant mass flow rate to decrease,however, the refrigerant mass flow rate will only converge to asteady-state value as long as the static valve opening is able toachieve the same decrease in flow. Eventually, the compressorRPM will reach a value where the static valve opening will not beable to reduce the refrigerant mass flow rate to the exit conditionscorresponding to the specific RPM value, causing the refrigerantto never reach a steady-state value throughout the cycle, whichwill eventually result in liquid flowing into the compressor (i.e.,
Table 1Nominal and re-scaled VCC model parameters to allow for a greater cooling
capacity.
Parameter Nominal Re-scaled Units
_mr 7.76�10�3 1.13�10�2 kg/s
Vk 3.04�10�5 1.52�10�4 m3
Le 11.46 57.29 m
Ae,o 3.07 15.34 m2
Ae,i 0.32 1.60 m2
Me 1.55 7.74 kg
De 8.90�10�3 3.57 �10�2 m_me,a 0.243 2.43 kg/s
Lc 10.7 53.5 m
Ac,o 2.79 13.97 m2
Ac,i 0.28 1.38 m2
Mc 4.66 23.30 kg
Dc 8.10�10�3 3.24�10�2 m
evaporator superheat region going to zero). This factor is solely acontribution of the choice of valve opening in the VCC and notaffected by the current VCC model used. Future work will explorethe potential benefit of using a non-adjustable valve in the VCC.
2.2. Building model
The key disturbances in the VCC model are the ambient airtemperature (the air temperature at the condenser inlet) andmixed air temperature (the air temperature at the evaporatorinlet). The ambient air temperature is naturally dictated by theoutdoor weather conditions while the mixed air temperature isinfluenced by a variety of interacting factors including the degreeof active heating/cooling in the room, the heating/cooling inadjacent rooms (if any), and various heat load disturbances,including the ambient air temperature. In this work, we utilizethe EnergyPlus simulation package to provide realistic mixed andambient air conditions based on a detailed building model andactual weather data.
The building model in EnergyPlus accounts for building con-struction, surface geometries, and HVAC systems with the detailsbased on the U.S. Department of Energy reference small officebuilding model (U.S. Department of Energy). An important featureof the EnergyPlus building model is that it accounts for the typicaldaily variation of the internal gains in a building. Internal gainscapture heat variations caused by a variety of realistic heat loadssuch as the movement of people and lighting schedules. As the airin a building is exposed to these internal gains, varying amountsof heat transfer occur from/to the air, causing fluctuations in thetemperature and humidity of the zone temperature. This isreflected in the VCC unit as variations in the mixed air conditions.Recall that the mixed air is a mixture of the zone temperature andthe ambient air temperature.
The EnergyPlus building model used in this work considers asmall (511 m2) single story office building in Chicago, Illinois, on atypical July day. The building is assumed to be divided into fiveoccupied thermal zones, in which there is a conditioned floor areaof 150 m2 in the core zone, 113 m2 in perimeter zones 1 and 3,and 67 m2 in perimeter zones 2 and 4. The ground-to-ceilingheight in all zones is 3 m. In total, the building houses 28 peopleat a standard occupant density of 5:38=100 m2 per zone. Duringpeak operation, the building is occupied between the hours of8:00 and 18:00 with the highest levels of occupancy. In this work,we assume that the thermal environment of perimeter zone 2 isregulated by the detailed VCC model described in Section 2.1. Allremaining thermal zones are assumed to be controlled byseparate air-conditioning units (pre-)modeled in EnergyPlus,and their zone temperatures are maintained at a constant set-point temperature of 24 1C (to minimize inter-zone heat transfer).Fig. 3 shows the ambient temperature, Tamb, relative humidity,and the zone 2 internal gains over the course of the July dayconsidered for the building model. The ambient conditions areobtained from historical data (in a data file) consisting of hourlymeasurements of the temperature and relative humidity.
2.2.1. VCC-building model interface
To link the building model in EnergyPlus with the VCC unitmodel in Matlab, data is exchanged between the two environ-ments over sockets using the Building Controls Virtual Test Bed(BCVTB) middle-ware (Wetter and Haves, 2008). This exchange isaccomplished using a Matlab script file (exchangeDoublewith-Socket.m), which is included in the BCVTB library (see Fig. 4).
In the EnergyPlus client, the ambient air and the air of zone twoare mixed (80% zone air with 20% ambient air) to form the mixedair conditions for the VCC. Ideally, the data exchange sequence
8
24
26
28
30
Time (h)
Am
bien
t tem
pera
ture
, Tam
b (°
C)
60
70
80
90
Am
bien
t rel
ativ
e hu
mid
ity (
%)
200
300
400
500In
tern
al G
ains
(W
)
10 12 14 16 18 8Time (h)
10 12 14 16 18
8Time (h)
10 12 14 16 18
Fig. 3. Variations in the ambient temperature, relative humidity, and internal gains, which act as disturbances in the zone 2 EnergyPlus building model.
Cooling load
Read values
IntegrateVCC model
Exc
hang
eDou
blew
ithS
ocke
t.m
Read values
Outputvariables
Integratebuilding model
Matlab BCVTB EnergyPlus
Fig. 4. Schematic of the energy Plus-Matlab interface.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 51
should be as follows: (1) subsequent to computing the mixed airconditions, the EnergyPlus client is paused momentarily, (2) themixed and ambient air conditions are sent to Matlab, (3) in Matlab,the VCC model is integrated (with computed input values from agiven control algorithm) and the corresponding cooling load iscomputed, (4) the cooling load is sent to the EnergyPlus model, and(5) the EnergyPlus model is un-paused and integrated forwardusing the newly computed cooling load. However, one limitation ofthe interfaced environment is that concurrent to sending the mixedand ambient air conditions to Matlab, the BCVTB software requiresa cooling load from Matlab. That is, data is sent to and read fromMatlab simultaneously because there is no effective way to pausethe EnergyPlus model until the newly computed cooling load(corresponding to the sent data) is computed. Instead, during thesimultaneous data exchange, the cooling load from the previoustime step is read and implemented in EnergyPlus, thereby intro-ducing an input delay (of one sampling instant). To minimize theeffects of this delay, the fastest available sampling time of 60 s isused for the EnergyPlus model.
The cooling load of a VCC quantifies the heat absorption by therefrigerant in the evaporator from the inlet air. The accuratecomputation of the cooling load is essential for properly interfa-cing the VCC model. The total VCC cooling load is the sum of thesensible and latent cooling loads, which correspond to changes inthe evaporator dry air temperature and humidity, respectively.The sensible cooling load is equivalent to the first term in Hloss inEq. (7). If the inlet air has a sufficiently high water vapor content,condensation may result, causing a humidity change and there-fore a non-zero latent cooling load. If no condensation occurs, thehumidity ratio of the air does not change (as mentioned in Section2.1.4); thus, the sensible cooling load equals the total coolingload. The energy change associated with the humidity change (orequivalently the condensation of the water vapor content in theair) is the latent cooling load, X‘ , and is given by
X‘ ¼ _m‘,eDhf ‘
where Dhf ‘ is the enthalpy of water condensation and _m‘,e
represents the mass of condensed water and depends on thesupply air temperature, which can be computed using theprocedure described in Section 2.1.4.
3. Temperature control
In this section, we propose a temperature control frameworkfor regulating the air temperature of zone 2 in the EnergyPlusbuilding model (interfaced with Matlab). To this end, we firstidentify an auto-regressive exogenous (ARX) model for the VCCoutputs using simulation data. Next, we utilize the model in anoffset free predictive control design for the stand-alone VCC unitand compare its performance against PI control. Finally, weintegrate the proposed predictive controller in a cascade controlstructure for regulating the zone temperature and implement thecontrol structure on the interfaced building model.
Table 2Final ARX model lag structure.
Output Lags
Ts,e Toa,e ok vo Tamb Ti
a,e
Ts,e 2 2 2 2 2 2
Toa,e 1 1 1 1 1 1
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5852
3.1. ARX VCC model
In the ARX type modeling approach, the process outputs at aspecific sampling instant are assumed to depend linearly on theprevious process conditions (defined by the process outputs andinputs). Mathematically, ARX models are defined as
yðkÞ ¼Xny
i ¼ 1
Aiyðk�iÞþXnu
i ¼ 1
Biuðk�iÞþXnd
i ¼ 1
Cidðk�iÞþvðkÞ ð8Þ
where yðkÞ and uðkÞ are the process output and input vectors atsampling instant k (respectively), dðkÞ is a vector of measurabledisturbances, Ai, Bi, and Ci are model coefficient matrices (that areestimated using least-squares regression), vðkÞ is the noise vector,and ny, nu, and nd denote the (maximum) number of time lags inthe outputs, inputs, and disturbances (respectively) and definethe order of the ARX model. For specific outputs, inputs, ordisturbances which do not require the maximum number of lags,the appropriate elements in the coefficient matrices can be set tozero. For the VCC, the outputs, inputs, and measurable distur-bances were previously defined in Section 2.1.4 as follows:y¼ ½Ts,e To
a,e�T, u¼ ½ok vo�
T, and d¼ ½Tamb Tia,e�
T.To identify the ARX model coefficient matrices, pseudo ran-
dom binary sequences (PRBS) are generated for the inputs anddisturbances for the typical operating range (see Fig. 5 for aportion of the PRBS data) and subsequently implemented on thenonlinear stand-alone VCC model. Using the System IdentificationToolbox in Matlab (which essentially solves the linear regressionproblem to compute the model coefficient matrices), the ARXmodel coefficient matrices for numerous lag choices are esti-mated. Among these models, the lag choice representing a goodtrade-off between the prediction accuracy and number of modelparameters is summarized in Table 2. Fig. 6 compares the outputprediction by the ARX model with the training data from thenonlinear model, demonstrating the prediction capability of theidentified model.
0
1
1.5
·103
Time (h)
RPM
, ωk
20
22
24
26
Mix
ed a
ir te
mpe
ratu
re, T
i a,e
(°C
)
10 20 30 40 50
0Time (h)
10 20 30 40 50
Fig. 5. Portion of the input profiles used to gener
Remark 4. A key objective of this work was to study theapplicability of a predictive control based scheme for temperaturecontrol in the presence of realistic disturbances. In this work, weopt for an empirically identified input–output VCC model as thepredictive model in the control design instead of a linearizedstate-space model (coupled with a state estimator). In general, alinearized state-space model of a nonlinear system at a specificoperating point only captures the local dynamics around thelinearization point and therefore calls for successive linearizationtechniques when used in an MPC framework to maintain reliablepredictions. Another limitation of using a first-principles (deter-ministic) model as the foundation of the control design is that themodel’s reliability is subject to the accuracy of numerous physicalparameters (i.e., thermodynamic properties, etc.), which may notbe known accurately. Additionally, many of the simplifyingassumptions made during the model development can be violatedin practice, further decreasing its validity. These reliability issuestogether with the inherent error introduced by linearizing anonlinear model motivated the use of an empirical model for thiswork. From an industrial perspective, if a sufficiently largenumber of identical packaged units are produced, it may makeeconomic sense to invest in the effort to develop a dedicated firstprinciples model, or alternatively, generation of enough data tocapture the model characteristics in a data-driven model.
0
11
11.5
12
12.5
Time (h)
Val
ve o
peni
ng, v
o (%
)
20
25
Am
bien
t air
tem
pera
ture
, Tam
b (°
C)
10 20 30 40 50
0Time (h)
10 20 30 40 50
ate output data for ARX model identification.
0
10
15
20
25
Time (h)
Supe
rhea
t, T
s,e
(°C
)
Nonlinear modelARX model
14
16
18
20
22
Supp
ly a
ir te
mpe
ratu
re, T
o a,e
(°C
)
Nonlinear modelARX model
10 20 30 40 50 0Time (h)
10 20 30 40 50
Fig. 6. Comparison of the output prediction by the ARX model with the nonlinear model for the input and disturbance profiles in Fig. 5.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 53
Remark 5. Using an ARX model with a measurable disturbancevector as one of the predictors in a predictive control designeffectively incorporates an element of feed-forward control intothe design. That is, the control algorithm utilizes the measureddisturbance vector to anticipate its effect and takes correctiveaction and further improve upon the achieved energy efficiency.However, for an MPC design with a prediction horizon greaterthan one, future disturbance measurements are required to makepredictions over the horizon. In this case, the current measure-ments of the disturbances can be assumed to hold for the lengthof horizon. This is a common assumption used in MPC formula-tions that utilize disturbance measurements and is meaningful inthe present context due to the different time scales at which theVCC evolves (small time scale) and the disturbance variableschange (larger time scale).
Remark 6. The estimated ARX model predicts the VCC outputbehavior relatively well; however, the nonlinear nature of theprocess dynamics (i.e., varying process gains) cannot be fullycaptured using a single linear model. One approach to capturethis nonlinearity is to identify multiple local linear models atvarious operating points and combine them with an appropriateweighting function during prediction. Recently, in Aumi andMhaskar (in press), a data-driven modeling methodology wasproposed that unifies the concepts of ARX modeling, latentvariable regression techniques, fuzzy c-means clustering, andmultiple local linear models in an integrated framework capableof capturing process nonlinearities. Specifically, plant data is firstclustered using fuzzy c-means clustering to identify the mostsuitable points for linearization and come up with a continuousweighting function for the individual models. Using this weight-ing function, the local linear model coefficients are simultaneously
estimated using latent variable regression tools, which allow fordimensionality and noise reduction. The same weighting functionis then utilized to weight the individual models given an initialcondition and inputs. The proposed modeling methodology hasbeen shown to be effective for identifying accurate models for usein MPC formulations (Aumi and Mhaskar, in press; Aumi et al.,submitted), and represents one possibility for developingimproved data-based models for use in the control design.
3.2. Stand-alone VCC control
In this section, we design and implement a predictive controlleron the nonlinear VCC model using the model identified in theprevious section and compare the closed-loop simulation resultsagainst PI control. The control objectives for stand-alone VCCcontrol are to track a given set-point trajectory of the supply airtemperature, to maintain reliable/safe operation by maintainingthe superheat above 0 1C (see Section 2.1.4), and to maximize the
energy efficiency by minimizing the compressor energy consump-tion (the largest energy consumer in the VCC). The closed-loopperformance is evaluated in terms of the integral of squared error,ISESA, between the supply air temperature, To
a,e, and its set-pointtrajectory, To
a,e,SP:
ISESA ¼DtXK
i ¼ 1
½Toa,e,SPðiÞ�To
a,eðiÞ�2
where i indexes the sampling instant, Dt is the sampling period(60 s), and K is the total number of sampling instants in thesimulation. To quantify the energy demand associated with acontrol design, the instantaneous compressor power is summedover the simulation time, yielding a measure of the total energyconsumption, TEC (see Section 2.1.1 for the variable definitions):
TEC¼DtXK
i ¼ 1
_mkðiÞ½hokðiÞ�hi
kðiÞ�
Z
where Z is the combined total efficiency of the compressor, whichis the product of the power and the mechanical efficiencies (knownparameters).
3.2.1. MPC control design and implementation
Consider a VCC system for which the ARX model for its outputshas been computed. For the proposed predictive control design,the inputs to the VCC at sampling instant i are computed bysolving the following constrained optimization problem:
where the notation, J � JQ , refers to the weighted norm, defined byJxJQ ¼ xTQ x and Du denotes a vector in which each element is thedifference between successive input moves. The weightingmatrices are diagonal and used to trade-off the relative impor-tance of the different control objectives. The plant measurementat the current sampling instant i corresponds to k¼0 or yð0Þ.
In this MPC formulation, the control objective of supply airtemperature set-point tracking is addressed by penalizing the
0
23.1
23.3
23.4
23.5
Time (h)
Supp
ly a
ir te
mpe
ratu
re, T
o a,e
(°C
)
SPNominal�� + �
1 2 3 4 5 6 7 8
23.2
23
Fig. 7. Supply air temperature responses using various combinations of the bias
terms in the proposed MPC design.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5854
deviation between the predicted supply-air temperature from itsset-point, y2,SPðkÞ, weighted by Q . The predicted superheat is alsobounded between y1,min and y1,max. To reduce the energy consump-tion associated with the control action, the absolute value of RPM isalso penalized using the weight r. The inputs are constrained in arange for which the nonlinear VCC model is known to be valid. Inaddition to using hard constraints for the input rates, excessiveinput movements are penalized in the objective function using amove suppression factor with the weighting matrix, R. Whentuning the different weighting matrices, the highest importancewas initially given to tracking the supply air set-point. Subse-quently, the remaining weighting matrices were adjusted appro-priately to achieve relatively smooth input behavior.
To achieve offset free performance, a disturbance/bias term isadded to the model predictions that is expressed by combiningtwo constant terms, a and bðiÞ. The first term, a, is the disturbancedue to plant-model mismatch at the current sampling instant,multiplied by a tuning parameter, k. Specifically, a is defined asthe difference between the predicted outputs at sampling instanti from the previous control calculation and the measured outputsat i. The bðiÞ term is the summation of tracking errors up to andincluding sampling instant i. For the superheat (output 1 or y1), anon-zero tracking error at i is used only if the current measure-ment exceeds the minimum or maximum superheat. The b termessentially ‘persists’ and influences the control action until theoffset is eliminated. It can be understood as operating the sameway as the integral mode in a PI controller. The tuning para-meters, g1, g2, and f, are used to trade-off the input aggressivenessand the amount of offset. The list of tuning parameters whichresulted in offset free performance while maintaining relativelysmooth input behavior is tabulated in Table 3 along with theconstraint bounds. Fig. 7 demonstrates the effect of the a and b. Inthe nominal case (no corrections), there is considerable offset inthe supply air temperature. After adding the feedback term toaccount for plant-model mismatch, this offset is significantlyreduced but not eliminated. Zero offset is only achieved afteralso including the error accumulation term in the formulation.
Next, closed-loop simulation results for MPC and PI control arecompared. For these simulations, constant disturbances areassumed. That is, the ambient air conditions (temperature andhumidity) and the inlet air temperature to the evaporator (themixed air temperature) are maintained at constant values. Usingthe results in Keir and Alleyne (2007), for the PI loop pairing, thesupply air temperature is paired with the compressor RPM whilethe superheat is paired with the expansion valve opening. Thesuperheat set-point for the PI controller is specified to be 10 1C(see Remark 7). The PI controllers are initially tuned using theinternal model control tuning method and fine-tuned to minimizethe integral of absolute error while maintaining relatively smoothinput trajectories.
Table 3MPC tuning parameters.
Parameter Value
P 4
Q 950
R diag{0.004,0.5}
r 350/17002
fy1,min ,y1,maxg {3.5, 20}
fumin ,umaxg {[678.8 6]T, [1700 15]T}
fDumin ,Dumaxg f½�200 �1�T ,½200 1�Tg
k ½0:2 0:50�T
fg1 ,g2g {6, 0.3}
f 0.01
Fig. 8 displays the closed-loop VCC input and output variableresponses for the two control strategies and Table 4 summarizestheir control performances using the metrics previously discussedand also the settling times for the supply air temperature, tsettSA , forthe different set-point step changes. As shown in Fig. 8, theproposed MPC design is able to provide better tracking performanceof the supply air temperature for the different set-point changeswith similar settling times and lower energy consumption. Thethird supply air set-point change (to approximately 23.7 1C) is aninfeasible set-point for the VCC cooling capacity, but note that thepredictive controller is able to drive the supply air temperaturecloser to this set-point compared to the PI controller. Note, how-ever, that the infeasibility is merely a result of the model not beingvalid at low RPM (or as low as required) to provide less cooling.
For the MPC design, the superheat is permitted to ‘float’between its minimum and maximum value whereas for PIcontrol, the superheat is maintained at the constant safety marginof 10 1C. This additional ‘degree of freedom’ for the predictivecontroller leads to more accurate tracking and better overallcontrol performance. Note that if the superheat was prescribedto be maintained at a constant value of 10 1C for the MPC designas well, the corresponding closed-loop results would be similar tothose obtained when using the PI controller. In regard to theenergy efficiency, the MPC design required 8% less energycompared to the PI controller. This is a consequence of usinghigher valve openings and lower RPM values resulting from themultivariable nature of the MPC controller and the ability to allowthe superheat to ‘float’ between acceptable values.
Remark 7. For the PI closed-loop simulation, the safety margin forthe superheat is specified to be 10 1C. This represented a roughlower bound for the superheat set-point for reliable simulationsunder PI control. When the VCC model is interfaced with thebuilding model and the superheat set-point is prescribed to be lessthan 10 1C, the PI controller drives the superheat to a negativevalue, resulting in a failed simulation. Note that in practice, a VCCunit has protections to ensure against negative superheat values;however, such protections are not considered in the existing VCCmodel. Simulation studies also revealed that by increasing thesuperheat set-point to 20 1C, the supply air tracking performancecan be substantially improved. However, maintaining the superheatat a higher safety margin requires lower valve openings, which, inturn, results in the PI controller prescribing higher RPM values to
0
5
10
15
20
25
Time (h)
Supe
rhea
t, T
s,e
(°C
)
MPCPI
23
23.2
23.4
23.6
23.8
Supp
ly a
ir te
mpe
ratu
re, T
o a,e
(°C
)
SPMPCPI
1
·103
RPM
, �k
MPCPI
6
8
10
12
14
Val
ve o
peni
ng, v
o (%
)MPCPI
1.2
1.1
0.9
0.8
0.7
5 10 15 0
Time (h)
5 10 15
0
Time (h)
5 10 15 0
Time (h)
5 10 15
Fig. 8. Closed-loop output and input profiles for the VCC under MPC and PI control.
tsettSA (s) 1800, 1800, 900, 1620 1020, 1800, 1140, 4440
TEC (kJ) 10017 9217
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 55
track the supply air temperature. Thus, there is a trade-off betweenthe improved tracking performance and increased power consump-tion. It is also worth noting that the MPC design still offered bettertracking performance (in addition to being more energy efficient by23%) than the PI controller by 10% even with the increased super-heat set-point of 20 1C.
Remark 8. Note that if the model is allowed to operate over theentire range of RPM, it is possible that the PI controller could beused to keep the superheat value at a fixed, low set-point, thiswould result in the RPM being able to change over the entirerange to provide minimum cooling where required, and at othertimes, providing additional cooling using minimal RPM. In such ascenario, the energy efficiency of a PI control structure would becomparable with the MPC in the current form. In such a scenario,however, possible nonlinear (and more importantly, non-mono-tonic) dependence of energy efficiency on the RPM would (andcould) be incorporated within the MPC controller to provide moreefficient operation over conventional control structures.
3.3. Energy efficient temperature control framework
In this section, we integrate the proposed MPC design forstand-alone VCC control in a cascade control structure for energyefficient temperature control in zone 2 of the EnergyPlus building
model. Note that the main purpose of the interfacing is todemonstrate superior control of the cooling device subject torealistic disturbances (induced by the interfacing and use ofweather data). The primary control objective we consider is tomaintain the zone 2 temperature, Tzone, within acceptable comfortstandards in the presence of disturbances brought on by varyingambient conditions, changes in the internal gains, and zoneinteractions (see Section 2.2). The secondary control objectivesare the stand-alone VCC control objectives listed in Section 3.2.The comfort standards we consider are inspired by those used bythe American Society of Heating, Refrigeration and Air-Condition-ing Engineers (ASHRAE). For typical summer conditions, assumingthat the room occupants are wearing light clothing, the ASHRAEcomfort standards (ASHRAE 55-2004) entail maintaining the zonetemperature between 22.3 and 24.7 1C and for this work, a zonetemperature set-point, Tzone,SP, of 24 1C is selected. Anotherimportant ASHRAE comfort standard is that the zone temperaturenot drift, which is defined as the temperature violating a bandaround the set-point for longer than 15 consecutive minutes. Inthis work, a 70.5 1C band around the set-point is considered.
The proposed control structure for meeting the temperaturecontrol objectives is shown in Fig. 9. The cascade control structurewas motivated by the time-scale of the VCC dynamics comparedto the zone temperature dynamics. Step tests in the VCC inputs(compressor RPM and valve opening) revealed the supply airand superheat temperatures evolve roughly in the same timescale (1–10 min) whereas the zone air temperature dynamicswere significantly slower (nearly 50 min). Varying internal gainsand ambient conditions act as disturbances to the zone airtemperature; however, by using a cascade control structure, therelatively faster dynamics of the inner loop are exploited toeliminate these disturbances (using the VCC inputs) before theysignificantly affect the zone temperature.
In this cascade control structure, the inner loop consists of astand-alone VCC controller (either the predictive controller or PIcontrollers designed in Section 3.2.1). The outer loop is used to
+
−
Tzone,SPPI MPC/PI
T oa,e,SPCondenser Evaporator
Compressor
Valve
EnergyPlus
Model
�k
vo
Ts,e ,T oa,e
Tzone
DisturbancesVCC
Fig. 9. Proposed cascade control structure for energy efficient temperature control.
8
24
Time (h)
Zon
e ai
r te
mpe
ratu
re (
°C)
MPCPI
24.6
24.4
24.2
23.8
23.6
23.4
23.2
9 10 11 12 13 14 15 16 17 18
Fig. 10. Air temperature in zone 2 when using a cascade control structure for
temperature control with a PI or predictive controller in the inner loop.
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5856
regulate the zone temperature and is connected to the inner loopvia the supply air temperature set-point. Based on the errorbetween the zone temperature and its set-point (Tzone,SP ¼
24 1C), the outer loop prescribes a supply air set-point, Toa,e,SP, to
the inner loop controller. The zone temperature is sampled every15 min, which also corresponds to the frequency of the supply airset-point updates. Faster sampling times led to excessive fluctua-tions in the prescribed supply air set-point, resulting in poortracking performance by the inner loop controller. The outer loopwas tuned iteratively such that it yielded trackable supply air set-points by the VCC. The outer loop tunings were kept consistent forboth control strategies in the inner loop.
The zone temperature response for each control strategy isshown in Fig. 10, which compares the efficacy of each controlstrategy in meeting the primary control objective. The MPC-basedstrategy is able to satisfy the comfort standards for the entire testperiod (any zone set-point violations lasted less than 15 min)while the comfort standards are violated for approximately thelast 40 min of the test period when using the PI-based controller.From Fig. 3, after 16:00 or 4:00 P.M., there is a significant decreasein the internal gains owing to a decrease in the zone occupancyand also a decrease in the ambient temperature. The zonetemperature response after 4:00 P.M. indicates that the MPC-based design is able to respond to these disturbances moreeffectively than the PI-based control strategy.
With regard to the secondary control objectives, Fig. 11 dis-plays the closed-loop VCC input and output profiles for the twoinner loop control strategies. The performance metrics of theinner loops are shown in Table 5. Similar to the results in thestand-alone VCC case, for the MPC-based design, the superheat isallowed to ‘float’ between its bounds and ended up evolvingcloser to its lower bound, allowing for better supply air tempera-ture tracking using considerably less compressor power. Asshown in the supply air profiles in Fig. 11, in contrast to the PI-based design, the MPC-based controller provides an offset freesupply air temperature profile for the majority of the feasible set-point values prescribed by the outer loop controller. To achievethis offset free performance (in addition to improved zone airtemperature regulation), aggressive control action is prescribed.We note, however, that the key idea in the results with theinterfaced system is not so much to demonstrate improvedcontrol of the zone conditions (which depends on several factors,including the ‘outer loop’) but more to evaluate the performanceof the VCC control structure subject to realistic disturbances.
Remark 9. One natural extension of the proposed control struc-ture is to replace the outer PI loop with a model predictivecontroller. The main requirement for this extension is to identify amodel between the supply air temperature and the zone air
temperature. This can be identified through step-tests or moredesirably, by generating PRBS-like sequences of the supply air.However, in any case, the resulting model will be dependent onthe closed-loop dynamics of the stand-alone VCC controller. Dueto the large variation in the time scales between the zone and VCCdynamics, in addition to the zone air being affected through asingle VCC output variable (supply air temperature), it is advan-tageous to use separate MPC designs for each level of the cascaderather than using one model predictive controller to regulate thezone conditions. As discussed in Remark 5, a weather model/estimator can also be incorporated in the design by including anambient temperature component in the model. In this case,through an economic objective function that considers varyingelectricity costs for the outer loop controller, operating costscan be reduced by pre-cooling as necessary. In addition tooptimality, the benefits of using MPC in the outer loop includeexplicitly incorporating comfort specifications and accountingfor the VCC cooling capacity in computing the supply air set-points. Finally, while we use temperature as the comfort measurein this work, other measures of comfort, such as a Predicted MeanVote (Federspiel and Asada, 1994; Hanna, 1997; Brager and deDear, 1998; Jones, 2002; Ye et al., 2003; Baus et al., 2008) can bereadily incorporated in the objective function in the MPCformulation.
8
21
22
23
24
Time (h)
Supp
ly a
ir te
mpe
ratu
re, T
o a,e
(°C
)
SPMPC
21
22
23
24
25
Supp
ly a
ir te
mpe
ratu
re, T
o a,e
(°C
)
SPPI
5
10
15
20
25
Supe
rhea
t, T
s,e
(°C
)
MPCPI
1
·103
RPM
, �k
MPCPI
6
8
10
12
14
Val
ve o
peni
ng, v
o (%
)
MPCPI
1.2
0.8
10 12 14 16 18 8Time (h)
10 12 14 16 18
8Time (h)
10 12 14 16 18
8Time (h)
10 12 14 16 18
8Time (h)
10 12 14 16 18
Fig. 11. Closed-loop output and input profiles for the VCC when interfaced with the EnergyPlus building model under MPC and PI control.
Table 5Inner loop performance metrics in the cascade control structure.
Metric Control strategy
PI control MPC
ISESA (s 1C2) 70 978 20 987
TEC (kJ) 5080 4284
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 57
4. Conclusions
In this work, control strategies were implemented on arealistic building model interfaced with a detailed VCC modelwhich provided the cooling load for a specified zone in thebuilding. The detailed VCC model was subject to realistic dis-turbances in the ambient and mixed air conditions as a result ofinterfacing the two models. The zone air temperature in thebuilding model was also subject to realistic ambient air condi-tions (obtained from actual weather data) and typical internalload variations. A cascade control structure was proposed toregulate the zone air conditions in the building model. In the
proposed control structure, an outer loop regulates the zone airtemperature by adjusting the set-point of the VCC supply airtemperature using PI control. The inner loop regulates the VCCsupply air and superheat by manipulating the compressor RPMand valve opening using an ARX-model-based predictive control-ler. The proposed control strategy demonstrated better distur-bance rejection ability in the zone air temperature than a PI-basedcascade structure. Also the predictive control strategy was moreenergy efficient. The improved performance of the MPC-baseddesign stemmed from its incorporation of the coupled nature ofVCC dynamics (through the ARX model) in computing the controlaction. The ability in the MPC to ‘trade-off’ optimality considera-tions with tracking requirements enabled achieving improvedset-point tracking while operating at lower RPM values (wherepossible) resulting in better energy efficiency.
Notation
Variable
Description
V
volume r density Z efficiency
M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5858
o
compressor RPM t time constant h specific enthalpy
DP
pressure drop
Cd
discharge coefficient g volumetric mean void fraction _m mass flow rate
A
cross-sectional area P pressure p perimeter a heat transfer coefficient T temperature x state variable u input variable d disturbance variable vo valve opening cp specific heat capacity L length w humidity ratio D diameter M mass t time
Subscript
Description
k
compressor v valve h heat exchanger e evaporator c condenser f vapor ‘ liquid r refrigerant w wall a air amb ambient W water sat saturation dp dew point V volumetric s superheat SA supply air SP set-point
Superscript
Description
i
inlet o outlet
Acknowledgments
Financial support from the Natural Sciences and EngineeringResearch Council of Canada through the Collaborative Researchand Development Program (in collaboration with Johnson Con-trols Inc.) is gratefully acknowledged.
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