Wallace MPC

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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

a r t i c l e i n f o

Article history:

Received 4 March 2011

Received in revised form

5 July 2011Accepted 8 July 2011Available online 29 July 2011

Keywords:

Vapor compression cycle

Temperature control

Building control

Energy efficient control

Model predictive control

EnergyPlus

a b s t r a c t

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 efficientbuilding 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) andaccount 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

http://-/?-http://www.elsevier.com/locate/ceshttp://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.ces.2011.07.023mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.ces.2011.07.023http://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.ces.2011.07.023mailto:[email protected]://localhost/var/www/apps/conversion/tmp/scratch_1/dx.doi.org/10.1016/j.ces.2011.07.023http://www.elsevier.com/locate/ceshttp://-/?-


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expansion valve, and an evaporator. In the VCC, the refrigerant

circulates through the four components, undergoing various

thermodynamic changes, which, in turn, influence external con-

ditions. Mathematically, the dynamics of the refrigerant states

and external conditions are modeled using a set of coupled

nonlinear ordinary differential and algebraic equations, resulting

in a complex differential-algebraic equation system. The control

objectives are typically defined in terms of degrees of superheat

in the refrigerant at the evaporator exit and the air temperature atthe evaporator exit (the supply air temperature). Ensuring that

superheated refrigerant exits the evaporator is of utmost impor-

tance in preventing physical damage in the VCC, as liquid

refrigerant can damage the mechanical components used in the

compressor. The manipulated variables include the compressor

speed, the air flow rates and the expansion valve opening. While

research activity has been strong for a long time on the design/

material and, to a smaller extent, modeling, of the various

components of the VCC (see Rasmussen, 2005 for details), there

has been a recent trend towards improved control of the VCC for

improved 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 latter

limits overall efficiency due to large power requirements and

significant thermal inertia during start-up transients while the

former’s efficiency is limited by extensive interactions and non-

linearity in the VCC system dynamics and the presence of input

constraints. Approaches to improve the performance of conven-

tional PID/PI controllers can be categorized into those which

attempt to decouple the VCC dynamics to improve SISO controll-

ability (Keir and Alleyne, 2007; Jain et al., 2010) and adaptive

control approaches which attempt to account for the process

nonlinearity (i.e. time-varying process gains) (Lin and Yeh, 2007;

Zhu et al., 2001).

Among the decoupling approaches, the most straightforward

extension has been to employ linear decouplers to remove

interactions among the individual control loops. However, the

effectiveness of this approach is entirely contingent on the model

accuracy, and a poor model can lead to closed-loop performance

degradation. Another decoupling approach has been to use linear

combinations of available VCC measurements as the controlled

variables 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 dynamics

by updating the PI/PID tuning parameters online using a model of

the process. However, these approaches are typically restricted to

linear models (for computational reasons), implying the tuning

parameter updates may be erroneously updated since the true

process is highly nonlinear, leading to poor control performance.

Despite these improvements, PI/PID control designs remain

inherently based on a single-input–single-output framework

and do not account for the presence of constraints and optimality.The control action prescribed by a controller that does not

account for input constraints can result in performance degrada-

tion or even closed-loop instability.

One control method well suited to handling constraints and

optimality 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 in

determining the control action by using a model of the process. In

model predictive control, a nonlinear or linear process model is

used to evaluate the effect of candidate manipulated input trajec-

tories via an objective function, and an optimization problem is

solved to yield the manipulated input trajectory that minimizes the

objective function while satisfying any constraints. Only the first

piece of the input trajectory is implemented and the problem is

re-solved at the next sampling time, using the new measured

values of the process variables. One of the strengths of the MPC

framework is the flexibility (and scope) in specifying optimality

objectives through various terms in the objective function, or

through constraints on the variables of interest. This, and the

results available on the stability and feasibility properties of MPC

formulations (see, e.g., Mhaskar et al., 2005, 2006; Mhaskar, 2006)

make MPC a preferred candidate to be evaluated for possible use

within building control structures. Recently, there have been manyexamples in the literature of the application of MPC for regulating a

wide range of VCC or HVAC systems. In addition to the nature of

the system being regulated, the major differentiating feature

among these MPC approaches is the complexity of the model used

for 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 empirically

identified linear (Huang et al., 2009; Ma et al., 2010) or nonlinear

model (Xi et al., 2007), or a first-principles non-linear model

(Leducq et al., 2006; Ma et al., 2010; Sarabia et al., 2007). The

majority of the MPC application examples have utilized a linear

VCC model (either a linearized state-space model or an empirically

identified input–output model). For example, in Sandipan et al.

(2010), an experimental chiller network, consisting of two chillers

and multiple external heat exchangers, is regulated to satisfy the

cooling load in addition to minimizing electricity costs. In Huang

et al. (2009), empirical first-order time-delay (FOTD) models are

used in a robust MPC formulation for improving temperature

regulation of an air-conditioning system. Specifically, several FOTD

models are identified at various operating points, and based on the

current operating conditions, the most appropriate FOTD model is

used in the MPC optimization. Linear MPC applications in the

context of building control include the work in Morosan et al.

(2010) where a distributed MPC design is used to regulate the

temperature of multiple zones in a building and minimize power

consumption. That is, each zone is served by a separate HVAC unit

under the control of a zone-specific MPC design to regulate the

internal zone temperature. Another example of building control

using (linear) MPC is available in Ma et al. (2010) where weather

data is incorporated into the design to determine the building zone

temperature set-point. This allows for pre-cooling during non-peak

periods and reduced power consumption compared to traditional

pre-programmed HVAC unit control strategies. The existing results

notwithstanding, there still exists a lack of results on the applica-

tion of a MPC design to a detailed model of a VCC unit coupled with

a realistic building model to evaluate the control performance in

the presence of disturbances.

Motivated by the above, this work evaluates the performance

of an integrated temperature control framework via simulations.

Specifically, we design a predictive controller for a stand-alone

VCC unit and utilize it in a cascade control structure for tempera-

ture control. The proposed control structure is implemented on a

realistic building model that accounts for varying weather condi-tions and internal heat loads throughout the course of a day and

the results are compared with a PI-based control structure. The

rest of this manuscript is organized as follows. In Section 2, we

give an overview of the VCC and building models used in this

work. Then, in Section 3, we develop a control strategy for

temperature control in a building zone. To this end, we first

estimate an input–output model for the VCC in Section 3.1 and

then design an offset free predictive controller for a stand-alone

VCC unit in Section 3.2. This controller is subsequently incorpo-

rated in a cascade control strategy to control a specific room

temperature and then implemented on a realistic building model

in Section 3.3. The performance of the proposed control strategy is

compared with a conventional PI-based control strategy. Finally,

we summarize our results in Section 4.

M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–5846


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2. Preliminaries

In this section, we give an overview of an existing VCC model

used in this work and point out the key limitations of the models

and modifications. Next, we describe the building model and the

software used for interfacing the VCC model with the building

model. Note that model development is not the focus of the

present work. A detailed model of the VCC and the building is

used 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 energy

absorption in an evaporator. An overlay of a VCC and the

corresponding pressure-volume diagram of the refrigerant is

shown in Fig. 1.

In a VCC unit, the refrigerant enters the compressor as a

superheated vapor and is compressed to a higher pressure,

resulting in the superheated vapor having a higher temperature

than the ambient temperature. From the compressor, the super-heated refrigerant vapor enters a condenser (typically placed

outdoors), condensing to a sub-cooled liquid at the condenser

exit as a fan blows the ambient air over the condenser. The high

pressure sub-cooled liquid then flows into an expansion valve

which decreases the pressure and temperature of the refrigerant,

causing a liquid–vapor mixture to form. Then, the two-phase

refrigerant mixture enters an evaporator that is exposed to the

environment to be cooled. The environment temperature is above

the temperature of the refrigerant, resulting in the evaporation

and subsequent heating of the refrigerant to a superheated vapor

at the evaporator exit. The air, in turn, is cooled and available as

primary air to be distributed for cooling. The superheated vapor

from the evaporator exit then flows into the compressor, com-

pleting the cycle.

The VCC model used in this work is adapted from the existing

simulation package, Thermosys, developed at the Air Conditioning

& Refrigeration Center (ACRC) at the University of Illinois at

Urbana-Champaign. In this simulation package, the refrigerant is

R-134a and is assumed to be cooling an air medium. The

simulator 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 following

subsections, a brief overview of the model components is pro-

vided followed by a general mathematical representation. For a

full description of the model components and a complete list

of the equations and parameters, the reader is referred to

Rasmussen (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 work

required 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 a

model of a variable speed compressor to illustrate the improved

efficiency achievable by model-based control designs and the key

interpretations remain applicable for other compressor types (on/

off compressors, etc.). Demonstrating improved energy efficiency

on other cooling units is the subject of future work and outside

the scope of the present manuscript.

The efficiency of the compressor depends on the pressure ratio

of the outlet to inlet stream and the compressor RPM, ok. Theyare obtained via (experimentally obtained) lookup tables of

efficiencies at various operating conditions. The mass flow rate

in the compressor is modeled using the following static equation:

_mk ¼ okV krkZV ð1Þ

where V k denotes the swept volume (a compressor parameter)

and rk is the refrigerant density at the inlet. The term, V krkZV ,characterizes the compressor capacity in terms of inlet refrigerant

conditions. For the compressor energy balance, the dynamics of

the heat transfer during the transport of the refrigerant from the

compression cylinder (where the compression takes place) to the

shell (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

k þhok ¼ K ðZk,h

ik,h

o

kÞ ð2Þ

where hok is the enthalpy of the outlet refrigerant, tk is a time

constant (a compressor parameter), and K ðÞ is the gain. The gainis constant during integration and a nonlinear function of the

isentropic efficiency, inlet refrigerant enthalpy, hik, and the ideal

isentropic outlet enthalpy, ho

k, which is determined by the

refrigerant 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 (most

existing compressors utilize an on/off strategy). Due to the

limited range of operating speeds for an on/off compressor, the

startup and shutdown of a setup equipped with this compressor

type can draw considerably more energy during these operating

conditions than a setup equipped with a variable speed compres-

sor. Demonstrating the improvement over traditional on–off

setups (using a good model of such a unit) does remain an

objective 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), as

well as to the control design to ensure the implementation of the

MPC control structure is feasible. In particular, any proposed

control design for the VCC must account for the discrete nature of

the compressor operation. For instance, in the present work, the

compressor RPM is a manipulated variable for VCC control and

treated as a continuous variable. With an on/off type compressor,

the compressor RPM is fixed when it is on and zero otherwise. In a

model predictive control framework, this can be modeled either

indirectly (by choosing the on/off durations as input variables) or

explicitly using binary variables in the optimization problem. On

the other hand, classical control approaches, such as PI control,

have limited options in handling units with discrete operation.

Volume

P r e s s u r 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.

M. Wallace et al. / Chemical Engineering Science 69 (2012) 45–58 47


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Another limitation of the current VCC model is the range of

compressor speeds over which the model is valid. In particular,

the model does not remain valid for low RPM values, and as a

result, the simulations have been carried out with the RPM

restricted to this range to preserve the validity of the results.

Enhancing the range of validity of the model also remains another

direction 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 mass

flow rate given by the following expression:

_mv ¼ C d

ffiffiffiffiffiffiffiffiffiffiffiffiffiffirvDP v

q ð3Þ

where C d is the valve discharge coefficient, DP v is defined as the

pressure difference between the inlet and outlet refrigerant, and

rv is the maximum of either the sub-cooled liquid density or thesaturated liquid density at the inlet operating conditions. The

discharge coefficient is determined by the valve opening and

pressure differential and obtainable from experimental lookup

tables. The sub-cooled and saturated liquid refrigerant densities

are also obtained from lookup tables of the refrigerant’s thermo-

dynamic properties.

Remark 2. An electronic expansion valve (EEV) is used in this

VCC model, where the valve position is adjustable and in agree-

ment with electronic expansion valves used in practice. The valve

position of an actual EEV is proportionally adjusted through

varying the frequency of a built-in step-motor, where the energy

draw of this motor is minimal relative to the other energy-

consuming components of a practical VCC (i.e., compressor and

fan motors).

2.1.3. Heat exchangers

The dynamics of the VCC are dominated by the condenser and

evaporator. Both heat exchangers are modeled as a long thin

horizontal 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 multiple

phase changes; accordingly, the refrigerant is modeled using a

lumped parameter, moving boundary approach which accounts

for different fluid regions (superheated vapor, saturated vapor–

liquid, or sub-cooled liquid) and their time varying boundaries. In

this approach, each fluid region is represented as a separate

control 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 condenser

has three fluid regions: a vapor region followed by a two-phase

and a sub-cooled region. In each two-phase region, the refrigerant

fluid properties are taken as the weighted combination of the

saturated liquid and vapor properties. The mean void fraction, g ,which is defined as the ratio of the vapor volume in a region to

the total region volume, is used to weight the properties. For

instance, the refrigerant density in a two-phase region is given by

gr f þ ð1gÞr‘ where r f and r‘ are the saturated vapor and liquid

densities, respectively. In the superheated and sub-cooled

regions, the refrigerant properties, such as density and tempera-

ture, are determined using the heat exchanger pressure (assumed

constant) and the average regional enthalpy (the average of the

inlet and outlet enthalpies).

The mass and energy balance ordinary differential equations

(ODEs) for each fluid region are derived from the governing

partial differential equations (PDEs) for fluid flow in a tube. To

yield a set of ODEs from the PDEs, the spatial dependence fromthe PDEs is removed after applying simplifying assumptions and

Leibnitz’s rule on any differential with respect to the spatial co-

ordinate, z . The full details of the modeling approach are available

in Rasmussen (2005). Eqs. (4) and (5) represent the governing

refrigerant mass and energy balance PDEs (respectively) of a

specific fluid region.

@ðr Ac Þ@t

þ @ _m

@ z ¼ 0 ð4Þ

@ðr Ac h Ac P Þ

@t þ

@ð _mhÞ

@ z ¼ piaiðT wT r Þ ð5Þ

where r, _m, and h denote the refrigerant density, mass flow rate,and specific enthalpy (respectively), Ac is the heat exchanger

cross-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, T w is

the wall temperature, and T r is the refrigerant temperature. These

PDEs are coupled with the following wall energy balance for each

region:

ðc pr AÞw _T w ¼ piaiðT r T wÞ þ poaoðT aT wÞ

where ðc pr AÞw 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 the

surrounding air with temperature T a. 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 in

the following matrix form:

Zhð xh,uÞ _ xh ¼ f hð xh,u,d Þ

where ZhðÞ and f hðÞ are a matrix and vector, respectively, and the

heat exchanger state variables, xh, include: the length of the

superheat, Lc ,1, and two-phase, Lc ,2, regions in the condenser,

the condenser wall temperatures in all three regions, T w,c ,1, T w,c ,2,

and T w,c ,3, the constant condenser pressure, P c , the condenser

outlet refrigerant enthalpy, hoc , the length of the two-phase region

in the evaporator, Le,1, the evaporator wall temperatures for both

regions, T w,e,1, T w,e,2, the constant evaporator pressure, P e, the

evaporator outlet refrigerant enthalpy, hoe, and the compressor

outlet refrigerant enthalpy, hok. The input vector, u, elements are

the compressor RPM, ok, and valve opening, vo. Note that in some

VCC systems, the fan speeds for the air being blown over theevaporator and condenser (and therefore the mass flow rate of

air) may also be available for adjustment; however, for this VCC

model, these are assumed constant. The disturbance vector, d , is

constituted of two measurable temperatures: the temperature of

Two-phase region Two-phase region Sub-cooled regionSuperheat region Superheat region

Fig. 2. Heat exchanger schematics showing the different fluid regions (adapted from Rasmussen, 2005) which are modeled using the moving boundary approach.

(a) Evaporator with two fluid regions. (b) Condenser with three fluid regions.



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the air to be blown over the evaporator, T ia,e, which is commonly

referred to as the mixed air temperature, and the air temperature

at the inlet of the condenser. The latter temperature is simply the

ambient air temperature since the condenser is assumed to be

outdoors and henceforth will be denoted by T amb. The air supplied

to 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 of

80% 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 VCC

model ODEs and algebraic expressions from the source files and

expressed the model as a differential-algebraic equation (DAE)

system. In order to integrate the DAE system, algebraic equations

are required to be satisfied at all integration steps. Integrating the

DAE system in Matlab as opposed to running the Simulink model

files yielded significant computational benefits with execution

times for the same test period being reduced by over 70%. The

VCC DAE can be expressed in the following general form:

Zð x,uÞ _ x ¼ f ð x,u,d Þ

g ð x,uÞ ¼ 0

y ¼ hð x,d Þ

where ZðÞ and f ðÞ again denote a matrix and vector, respectively,

x is the VCC state vector, u and d were previously defined in

Section 2.1.3, g ð x,uÞ represents the algebraic expressions (used for

modeling the piping and expansion valve), and y denotes the VCC

outputs. The VCC outputs are defined to be the superheat of the

refrigerant exiting the evaporator, T s,e, and the air temperature at

the evaporator exit, the so-called supply air temperature, T oa,e. The

superheat is defined as the number of degrees by which the

refrigerant temperature at the evaporator exceeds its saturation

temperature. As mentioned in Section 1, T s,e is required to be

maintained above 0 1C to protect against any liquid refrigerant

entering 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, hðÞ, denotes

the (nonlinear) output mapping function. The mapping function

for the superheat is relatively straightforward whereas the func-

tion to compute the supply air temperature consists of finding the

root of a nonlinear equation as discussed next.

The following discussion contains modifications of the supply

air temperature calculation procedure found in the original

Thermosys model. Specifically, we make corrections to the pro-

cedure in the event of any condensation of the water vapor

content in the air. To obtain the supply air temperature, T oa,e, an

energy balance for the wall side of the evaporator is solved.

Assuming no energy accumulation in the evaporator walls, the

heat absorption by the evaporator wall must equal the heat loss of the air:

ao Ac X2i ¼ 1

Le,iLe

ðT a,eT w,e,iÞ ¼ H loss ð6Þ

The term, ao Ac P2

i ¼ 1 ðLe,i=LeÞðT a,eT w,e,iÞ, represents the energy

absorption by the evaporator where T a,e is the average air

temperature around the evaporator:

T a,e ¼ 12ðT

ia,e þT

oa,eÞ

In the most general case (i.e., assuming there is condensation of

the water vapor content in the air), the heat loss of the air, H loss , is

given by

H loss ¼ _

ma,

ec p,

a,

eðT

i

a,

eT

o

a,

eÞ þ _

ma,

eðw

i

a,

eh

i

W ,

ew

o

a,

eh

o

W ,

eÞh‘ ,

e ð7Þ

where _ma,e and c p,a,e denote the mass flow rate and specific heat

capacity of the dry air being blown over the evaporator (assumed

to remain constant), wia,e and woa,e denote the humidity ratio,

which is defined as the ratio of the mass of water vapor in the air

to the total dry air mass, of the inlet and outlet air (respectively),

and hiW ,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 in

this term is T o

a,e (the variable of interest). The second term is theenergy loss of the water vapor content in the air. In this term, the

inlet humidity and water vapor enthalpy are readily computable

from the known temperature (and pressure). If no condensation

occurs, there is no change in the humidity ratio and wia,e ¼ woa,e. In

the case of condensation, the outlet air is saturated, implying the

relative humidity at the outlet, foa,e, is 1.

1 To compute the

humidity ratio, its relationship with the relative humidity can

be used to derive (Dincer and Rosen, 2007, Chapter 6):

woa,e ¼ 0:622 f

oa,eP

oW ,e,sat

P afoa,eP

oW ,e,sat

where P a is the known air pressure and P oW ,e,sat is the saturation

pressure of water at T oa,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 h f ,sat þ

Z T oa,eT ref

c p,W ðT Þ dT

where h f ,sat is the heat of saturated water vapor at the air

pressure, T ref is a reference temperature, and c p,W ðT Þ is the

(possibly) temperature-dependent specific heat capacity of water

vapor.2 The third term in Eq. (7), h‘ ,e, represents the heat content

in the condensed water if condensation occurs. Note that the

negative sign is required in front of h‘ ,e since heat losses are

written as positive energies in Eq. (7). The heat content in the

water is given by

h‘ ,e ¼ _ma,eðwia,ew

oa,eÞc p,W T

oa,e ¼ _m‘ ,ec p,W T

oa,e

where the product _ma,eðwia,ew

oa,eÞ equals the mass of condensed

water (follows from the definition of humidity ratio) or _m‘ ,e and

c p,W is the constant heat capacity of liquid water.

Having defined all the terms/variables in Eq. (6) and their

dependence on the unknown supply air temperature, T oe,a, a root

finding algorithm can be applied to Eq. (6) to compute T oe,a.

Alternatively, an iterative (i.e., direct-substitution) procedure

can be used where an initial guess for the supply air temperature

is made, H loss is computed, and then the left hand side of Eq. (6) is

solved for T oa,e. If the difference between the newly computed

supply air temperature and the initial guess exceeds a pre-defined

tolerance, the newly computed value can be used to initialize the

next iteration. Note also that when solving Eq. (6), an assumption

regarding the occurrence of condensation has to be made. In thiswork, we first solve Eq. (6), assuming no condensation (i.e., with

no h‘ ,w term), which is correct only if T oa,e4T dp where T dp is the

dew-point temperature (computable from the air pressure). If

T oa,erT dp, the necessary condensation term is added to H loss prior

to solving Eq. (6) and the equation system is re-solved.

2.1.5. VCC cooling capacity

For proper regulation of a building zone temperature, the

corresponding VCC unit for the zone must meet the cooling

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 how,e and h

iw,e

such that T ref disappears in Eq. (7).



6/14

capacity dictated by the highest possible ambient conditions and

heat load disturbances. The nominal Thermosys VCC model has a

maximum cooling capacity of 1127 W or 0.32 ton of refrigeration.

This capacity is in agreement with that of a small experimental

VCC used to validate the nominal model. Exploratory simulations

revealed that this cooling capacity is insufficient (even with

perfect control) to achieve the desired control objectives in terms

of 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 the

cooling capacity increases to 0.85 ton. Specifically, the mass flow

rate of the circulating refrigerant, _mr , along with the compressor

volume, V k, are first increased. Next, the length of the evaporator,

Le, is increased to allow longer contact of the supply air with the

evaporator wall. The inner and outer cross-sectional areas of the

evaporator, Ae,i and Ae,o respectively, and the mass of the eva-

porator, M e, are then increased by the same factor. Then, the

diameter 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 heat

absorbed by the refrigerant in the evaporator could be dissipated

into the surrounding environment at the condenser, the same

parameters 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 variable

speed compressor, enabling the VCC to achieve a varying cooling

capacity, our current model does not capture the total cooling

range associated with either an experimental or an industrial

cooling unit. Specifically, the operating conditions corresponding

to operating the VCC near its upper and lower cooling extremes

are not captured in the scaled VCC model, as inaccuracies arose

due to two factors: (1) the experimentally populated lookup

tables for certain component and thermodynamic parameters

(ZV , Zk, C d, etc.) corresponded to different operating conditionsfor the scaled and original VCC, resulting in a smaller feasible

operating range for the scaled system, and (2) the limiting natureof the EEV caused the minimal operating conditions associated

with the compressor to be higher. Further reductions in the

compressor RPM cause the refrigerant mass flow rate to decrease,

however, the refrigerant mass flow rate will only converge to a

steady-state value as long as the static valve opening is able to

achieve the same decrease in flow. Eventually, the compressor

RPM will reach a value where the static valve opening will not be

able to reduce the refrigerant mass flow rate to the exit conditions

corresponding to the specific RPM value, causing the refrigerant

to never reach a steady-state value throughout the cycle, which

will eventually result in liquid flowing into the compressor (i.e.,

evaporator superheat region going to zero). This factor is solely a

contribution of the choice of valve opening in the VCC and not

affected by the current VCC model used. Future work will explore

the 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 air

temperature (the air temperature at the condenser inlet) andmixed air temperature (the air temperature at the evaporator

inlet). The ambient air temperature is naturally dictated by the

outdoor weather conditions while the mixed air temperature is

influenced by a variety of interacting factors including the degree

of active heating/cooling in the room, the heating/cooling in

adjacent rooms (if any), and various heat load disturbances,

including the ambient air temperature. In this work, we utilize

the EnergyPlus simulation package to provide realistic mixed and

ambient air conditions based on a detailed building model and

actual weather data.

The building model in EnergyPlus accounts for building con-

struction, surface geometries, and HVAC systems with the details

based on the U.S. Department of Energy reference small office

building model (U.S. Department of Energy). An important featureof the EnergyPlus building model is that it accounts for the typical

daily variation of the internal gains in a building. Internal gains

capture heat variations caused by a variety of realistic heat loads

such as the movement of people and lighting schedules. As the air

in a building is exposed to these internal gains, varying amounts

of heat transfer occur from/to the air, causing fluctuations in the

temperature and humidity of the zone temperature. This is

reflected in the VCC unit as variations in the mixed air conditions.

Recall that the mixed air is a mixture of the zone temperature and

the ambient air temperature.

The EnergyPlus building model used in this work considers a

small (511 m2) single story office building in Chicago, Illinois, on a

typical July day. The building is assumed to be divided into five

occupied 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-ceiling

height in all zones is 3 m. In total, the building houses 28 people

at a standard occupant density of 5:38=100 m2 per zone. During

peak operation, the building is occupied between the hours of

8:00 and 18:00 with the highest levels of occupancy. In this work,

we assume that the thermal environment of perimeter zone 2 is

regulated by the detailed VCC model described in Section 2.1. All

remaining thermal zones are assumed to be controlled by

separate 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, T amb, relative humidity,

and the zone 2 internal gains over the course of the July day

considered for the building model. The ambient conditions are

obtained from historical data (in a data file) consisting of hourly

measurements of the temperature and relative humidity.

2.2.1. VCC-building model interface

To link the building model in EnergyPlus with the VCC unit

model 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 is

accomplished 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 two

are mixed (80% zone air with 20% ambient air) to form the mixed

air conditions for the VCC. Ideally, the data exchange sequence

Table 1

Nominal and re-scaled VCC model parameters to allow for a greater cooling

capacity.

Parameter Nominal Re-scaled Units

_mr 7.76 103 1.13 102 kg/s

V k 3.04 105 1.52 104 m3

Le 11.46 57.29 m

Ae,o 3.07 15.34 m2

Ae,i 0.32 1.60 m2

M e 1.55 7.74 kg

De 8.90 103 3.57 102 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

M c 4.66 23.30 kg

Dc 8.10 103 3.24 102 m



7/14

should be as follows: (1) subsequent to computing the mixed air

conditions, the EnergyPlus client is paused momentarily, (2) the

mixed and ambient air conditions are sent to Matlab, (3) in Matlab,

the VCC model is integrated (with computed input values from a

given 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 forward

using the newly computed cooling load. However, one limitation of

the interfaced environment is that concurrent to sending the mixed

and ambient air conditions to Matlab, the BCVTB software requires

a cooling load from Matlab. That is, data is sent to and read from

Matlab simultaneously because there is no effective way to pause

the EnergyPlus model until the newly computed cooling load

(corresponding to the sent data) is computed. Instead, during the

simultaneous data exchange, the cooling load from the previous

time step is read and implemented in EnergyPlus, thereby intro-

ducing an input delay (of one sampling instant). To minimize the

effects of this delay, the fastest available sampling time of 60 s is

used for the EnergyPlus model.

The cooling load of a VCC quantifies the heat absorption by the

refrigerant in the evaporator from the inlet air. The accurate

computation of the cooling load is essential for properly interfa-

cing the VCC model. The total VCC cooling load is the sum of the

sensible and latent cooling loads, which correspond to changes in

the evaporator dry air temperature and humidity, respectively.

The sensible cooling load is equivalent to the first term in H loss in

Eq. (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, the

humidity ratio of the air does not change (as mentioned in Section

2.1.4); thus, the sensible cooling load equals the total cooling

load. The energy change associated with the humidity change (or

equivalently the condensation of the water vapor content in the

air) is the latent cooling load, X‘ , and is given by

X‘ ¼ _m‘ ,eDh f ‘

where Dh f ‘ is the enthalpy of water condensation and _m‘ ,erepresents the mass of condensed water and depends on the

supply air temperature, which can be computed using the

procedure described in Section 2.1.4.

3. Temperature control

In this section, we propose a temperature control framework

for regulating the air temperature of zone 2 in the EnergyPlus

building model (interfaced with Matlab). To this end, we first

identify an auto-regressive exogenous (ARX) model for the VCC

outputs using simulation data. Next, we utilize the model in an

offset free predictive control design for the stand-alone VCC unit

and compare its performance against PI control. Finally, we

integrate the proposed predictive controller in a cascade control

structure for regulating the zone temperature and implement the

control structure on the interfaced building model.

8

24

26

28

30

Time (h)

A m b i e n t t e

m p e r a t u r e ,

T a m b

( ° C )

60

70

80

90

A m b i e n t

r e l a t i v e h u m i d i t y ( % )

200

300

400

500

I n t e r n a l G a i n s ( W )

10 12 14 16 18 8

Time (h)

10 12 14 16 18

8

Time (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

Integrate

VCC model

E x c h a n g e D o u b l e w i t h S o c k e t . m Read values

Output

variables

Integrate

building model

Matlab BCVTB EnergyPlus

Fig. 4. Schematic of the energy Plus-Matlab interface.



8/14

3.1. ARX VCC model

In the ARX type modeling approach, the process outputs at a

specific sampling instant are assumed to depend linearly on the

previous process conditions (defined by the process outputs and

inputs). Mathematically, ARX models are defined as

y ðkÞ ¼ Xn y

i ¼ 1

Ai y ðkiÞ þ Xnu

i ¼ 1

BiuðkiÞ þ Xnd

i ¼ 1

Cid ðkiÞ þv ðkÞ ð8Þ

where y ðkÞ and uðkÞ are the process output and input vectors at

sampling instant k (respectively), d ðkÞ is a vector of measurable

disturbances, Ai, Bi, and Ci are model coefficient matrices (that are

estimated using least-squares regression), v ðkÞ is the noise vector,

and n y, nu, and nd denote the (maximum) number of time lags in

the outputs, inputs, and disturbances (respectively) and define

the order of the ARX model. For specific outputs, inputs, or

disturbances which do not require the maximum number of lags,

the appropriate elements in the coefficient matrices can be set to

zero. For the VCC, the outputs, inputs, and measurable distur-

bances were previously defined in Section 2.1.4 as follows:

y ¼ ½T s,e T oa,e

T, u ¼ ½ok voT, and d ¼ ½T amb T ia,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 a

portion of the PRBS data) and subsequently implemented on the

nonlinear stand-alone VCC model. Using the System Identification

Toolbox in Matlab (which essentially solves the linear regression

problem to compute the model coefficient matrices), the ARX

model coefficient matrices for numerous lag choices are esti-

mated. Among these models, the lag choice representing a good

trade-off between the prediction accuracy and number of model

parameters is summarized in Table 2. Fig. 6 compares the output

prediction by the ARX model with the training data from the

nonlinear model, demonstrating the prediction capability of the

identified model.

Remark 4. A key objective of this work was to study the

applicability of a predictive control based scheme for temperature

control in the presence of realistic disturbances. In this work, we

opt for an empirically identified input–output VCC model as the

predictive model in the control design instead of a linearized

state-space model (coupled with a state estimator). In general, a

linearized state-space model of a nonlinear system at a specific

operating point only captures the local dynamics around the

linearization point and therefore calls for successive linearizationtechniques when used in an MPC framework to maintain reliable

predictions. Another limitation of using a first-principles (deter-

ministic) model as the foundation of the control design is that the

model’s reliability is subject to the accuracy of numerous physical

parameters (i.e., thermodynamic properties, etc.), which may not

be known accurately. Additionally, many of the simplifying

assumptions made during the model development can be violated

in practice, further decreasing its validity. These reliability issues

together with the inherent error introduced by linearizing a

nonlinear model motivated the use of an empirical model for this

work. From an industrial perspective, if a sufficiently large

number of identical packaged units are produced, it may make

economic sense to invest in the effort to develop a dedicated first

principles model, or alternatively, generation of enough data to

capture the model characteristics in a data-driven model.

0

1

1.5

·103

Time (h)

R P M , ω k

0

11

11.5

12

12.5

Time (h)

V a l v e o p e n i n g , v o

( % )

20

22

24

26

M i x e d a i r t e m p e r a t u r e , T i a

, e ( ° C )

20

25

A m b i e n t a i r t e m p e r a t u r e ,

T a m b ( ° C )

10 20 30 40 50 10 20 30 40 50

0

Time (h)

0

Time (h)

10 20 30 40 50 10 20 30 40 50

Fig. 5. Portion of the input profiles used to generate output data for ARX model identification.

Table 2

Final ARX model lag structure.

Output Lags

T s,e T oa,e ok vo T amb T ia,e

T s,e 2 2 2 2 2 2

T oa,e 1 1 1 1 1 1



9/14

Remark 5. Using an ARX model with a measurable disturbance

vector as one of the predictors in a predictive control design

effectively incorporates an element of feed-forward control into

the design. That is, the control algorithm utilizes the measured

disturbance vector to anticipate its effect and takes corrective

action and further improve upon the achieved energy efficiency.

However, for an MPC design with a prediction horizon greater

than one, future disturbance measurements are required to make

predictions over the horizon. In this case, the current measure-

ments of the disturbances can be assumed to hold for the length

of horizon. This is a common assumption used in MPC formula-

tions that utilize disturbance measurements and is meaningful in

the present context due to the different time scales at which the

VCC evolves (small time scale) and the disturbance variables

change (larger time scale).

Remark 6. The estimated ARX model predicts the VCC output

behavior relatively well; however, the nonlinear nature of the

process dynamics (i.e., varying process gains) cannot be fully

captured using a single linear model. One approach to capture

this nonlinearity is to identify multiple local linear models at

various operating points and combine them with an appropriateweighting function during prediction. Recently, in Aumi and

Mhaskar (in press), a data-driven modeling methodology was

proposed that unifies the concepts of ARX modeling, latent

variable regression techniques, fuzzy c -means clustering, and

multiple local linear models in an integrated framework capable

of capturing process nonlinearities. Specifically, plant data is first

clustered using fuzzy c -means clustering to identify the most

suitable points for linearization and come up with a continuous

weighting 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 for

dimensionality and noise reduction. The same weighting function

is then utilized to weight the individual models given an initial

condition and inputs. The proposed modeling methodology hasbeen shown to be effective for identifying accurate models for use

in MPC formulations (Aumi and Mhaskar, in press; Aumi et al.,

submitted), and represents one possibility for developing

improved data-based models for use in the control design.

3.2. Stand-alone VCC control

In this section, we design and implement a predictive controller

on the nonlinear VCC model using the model identified in the

previous section and compare the closed-loop simulation results

against PI control. The control objectives for stand-alone VCC

control are to track a given set-point trajectory of the supply air

temperature, to maintain reliable/safe operation by maintaining

the 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-loop

performance is evaluated in terms of the integral of squared error,

ISESA, between the supply air temperature, T oa,e, and its set-point

trajectory, T oa,e,SP:

ISESA ¼D

t X

K

i ¼ 1½T

o

a,

e,

SPðiÞT

o

a,

eðiÞ2

where i indexes the sampling instant, Dt is the sampling period

(60 s), and K is the total number of sampling instants in the

simulation. To quantify the energy demand associated with a

control design, the instantaneous compressor power is summed

over the simulation time, yielding a measure of the total energy

consumption, TEC (see Section 2.1.1 for the variable definitions):

TEC¼Dt XK i ¼ 1

_mkðiÞ½hokðiÞh

ikðiÞ

Z

where Z is the combined total efficiency of the compressor, whichis the product of the power and the mechanical efficiencies (known

parameters).

3.2.1. MPC control design and implementation

Consider a VCC system for which the ARX model for its outputs

has been computed. For the proposed predictive control design,

the inputs to the VCC at sampling instant i are computed by

solving the following constrained optimization problem:

minuminruðkÞrumax

XP k ¼ 1

J ^ yn

2ðkÞ y2,SPðkÞJQ þJu1Jr þJDuJR

subject to : DuminrDuðkÞrDumax

^ y ðkÞ ¼Xn yi ¼ 1

Ai ^ y ðkiÞ þXnui ¼ 1

BiuðkiÞ þXndi ¼ 1

Cid ðkiÞ for k ¼ 1, . . . , P

^ y n

ðkÞ ¼ ^ y ðkÞ þaþbðiÞ

y1,minr ^ yn

1ðkÞr y1,max

a ¼ k½ y ð0Þ ^ y ð0Þ

b1ðiÞ ¼ b1ði1Þ þ g 1 maxf0,½ y1,min y1ð0Þgþ g 2 maxf0,½ y1ð0Þ y1,maxg

b2ðiÞ ¼ b2ði1Þ þ f ½ y2ð0Þ y2,SPð0Þ

where the notation, J JQ , refers to the weighted norm, defined by

J xJQ ¼ xTQ x and Du denotes a vector in which each element is the

difference between successive input moves. The weighting

matrices are diagonal and used to trade-off the relative impor-

tance of the different control objectives. The plant measurement

at the current sampling instant i corresponds to k ¼0 or y ð0Þ.

In this MPC formulation, the control objective of supply air

temperature set-point tracking is addressed by penalizing the

0

10

15

20

25

Time (h)

S u p e r h e a t , T

s , e

( ° C )

Nonlinear model

ARX model

14

16

18

20

22

S u p p l y

a i r t e m p e r a t u r e ,

T

o a , e

( ° C )

Nonlinear model

ARX model

10 20 30 40 50 0

Time (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.



10/14

deviation between the predicted supply-air temperature from its

set-point, y2,SPðkÞ, weighted by Q . The predicted superheat is also

bounded between y1,min and y1,max. To reduce the energy consump-

tion associated with the control action, the absolute value of RPM is

also penalized using the weight r . The inputs are constrained in a

range for which the nonlinear VCC model is known to be valid. In

addition to using hard constraints for the input rates, excessive

input movements are penalized in the objective function using a

move suppression factor with the weighting matrix, R . Whentuning the different weighting matrices, the highest importance

was 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 is

added to the model predictions that is expressed by combining

two 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 instant

i from the previous control calculation and the measured outputs

at i . The bðiÞ term is the summation of tracking errors up to and

including sampling instant i. For the superheat (output 1 or y1), a

non-zero tracking error at i is used only if the current measure-

ment exceeds the minimum or maximum superheat. The b term

essentially ‘persists’ and influences the control action until the

offset is eliminated. It can be understood as operating the same

way as the integral mode in a PI controller. The tuning para-

meters, g 1, g 2, and f , are used to trade-off the input aggressiveness

and the amount of offset. The list of tuning parameters which

resulted in offset free performance while maintaining relatively

smooth input behavior is tabulated in Table 3 along with the

constraint bounds. Fig. 7 demonstrates the effect of the a and b. Inthe nominal case (no corrections), there is considerable offset in

the supply air temperature. After adding the feedback term to

account for plant-model mismatch, this offset is significantly

reduced but not eliminated. Zero offset is only achieved after

also including the error accumulation term in the formulation.

Next, closed-loop simulation results for MPC and PI control are

compared. For these simulations, constant disturbances are

assumed. That is, the ambient air conditions (temperature and

humidity) and the inlet air temperature to the evaporator (the

mixed air temperature) are maintained at constant values. Using

the results in Keir and Alleyne (2007), for the PI loop pairing, the

supply air temperature is paired with the compressor RPM while

the superheat is paired with the expansion valve opening. The

superheat set-point for the PI controller is specified to be 10 1C

(see Remark 7). The PI controllers are initially tuned using the

internal model control tuning method and fine-tuned to minimize

the integral of absolute error while maintaining relatively smooth

input trajectories.

Fig. 8 displays the closed-loop VCC input and output variable

responses for the two control strategies and Table 4 summarizes

their control performances using the metrics previously discussed

and also the settling times for the supply air temperature, t settSA , for

the different set-point step changes. As shown in Fig. 8, the

proposed MPC design is able to provide better tracking performance

of the supply air temperature for the different set-point changes

with similar settling times and lower energy consumption. The

third supply air set-point change (to approximately 23.7 1C) is an

infeasible set-point for the VCC cooling capacity, but note that the

predictive controller is able to drive the supply air temperature

closer to this set-point compared to the PI controller. Note, how-

ever, that the infeasibility is merely a result of the model not being

valid 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 PI

control, the superheat is maintained at the constant safety margin

of 10 1C. This additional ‘degree of freedom’ for the predictive

controller leads to more accurate tracking and better overall

control performance. Note that if the superheat was prescribed

to be maintained at a constant value of 10 1C for the MPC design

as well, the corresponding closed-loop results would be similar to

those obtained when using the PI controller. In regard to the

energy efficiency, the MPC design required 8% less energy

compared to the PI controller. This is a consequence of using

higher valve openings and lower RPM values resulting from the

multivariable nature of the MPC controller and the ability to allow

the superheat to ‘float’ between acceptable values.

Remark 7. For the PI closed-loop simulation, the safety margin for

the superheat is specified to be 10 1C. This represented a rough

lower bound for the superheat set-point for reliable simulations

under PI control. When the VCC model is interfaced with the

building model and the superheat set-point is prescribed to be less

than 10 1C, the PI controller drives the superheat to a negative

value, resulting in a failed simulation. Note that in practice, a VCC

unit has protections to ensure against negative superheat values;

however, such protections are not considered in the existing VCC

model. Simulation studies also revealed that by increasing the

superheat set-point to 20 1C, the supply air tracking performance

can be substantially improved. However, maintaining the superheat

at a higher safety margin requires lower valve openings, which, in

turn, results in the PI controller prescribing higher RPM values to

Table 3

MPC tuning parameters.

Parameter Value

P 4

Q 950

R diag{0.004,0.5}

r 350/17002

f y1,min, y1,maxg {3.5, 20}

fumin,umaxg {[678.8 6]T, [1700 15]T}

fDumin,Dumaxg f½200 1T,½200 1Tg

k ½0:2 0:50T

f g 1 , g 2g {6, 0.3}

f 0.01

0

23.1

23.3

23.4

23.5

Time (h)

S u p p l y a i r t e m p e r a t u r e ,

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.



11/14

track the supply air temperature. Thus, there is a trade-off between

the improved tracking performance and increased power consump-

tion. It is also worth noting that the MPC design still offered better

tracking performance (in addition to being more energy efficient by

23%) 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 the

entire range of RPM, it is possible that the PI controller could be

used to keep the superheat value at a fixed, low set-point, this

would result in the RPM being able to change over the entirerange to provide minimum cooling where required, and at other

times, providing additional cooling using minimal RPM. In such a

scenario, the energy efficiency of a PI control structure would be

comparable 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 (and

could) be incorporated within the MPC controller to provide more

efficient operation over conventional control structures.

3.3. Energy efficient temperature control framework

In this section, we integrate the proposed MPC design for

stand-alone VCC control in a cascade control structure for energy

efficient temperature control in zone 2 of the EnergyPlus building

model. Note that the main purpose of the interfacing is to

demonstrate superior control of the cooling device subject to

realistic disturbances (induced by the interfacing and use of

weather data). The primary control objective we consider is to

maintain the zone 2 temperature, T zone, within acceptable comfort

standards in the presence of disturbances brought on by varying

ambient conditions, changes in the internal gains, and zone

interactions (see Section 2.2). The secondary control objectives

are the stand-alone VCC control objectives listed in Section 3.2.

The comfort standards we consider are inspired by those used by

the American Society of Heating, Refrigeration and Air-Condition-

ing Engineers (ASHRAE). For typical summer conditions, assuming

that the room occupants are wearing light clothing, the ASHRAE

comfort standards (ASHRAE 55-2004) entail maintaining the zone

temperature between 22.3 and 24.7 1C and for this work, a zone

temperature set-point, T zone,SP, of 24 1C is selected. Another

important ASHRAE comfort standard is that the zone temperature

not drift, which is defined as the temperature violating a band

around the set-point for longer than 15 consecutive minutes. In

this work, a 70.5 1C band around the set-point is considered.

The proposed control structure for meeting the temperature

control objectives is shown in Fig. 9. The cascade control structurewas motivated by the time-scale of the VCC dynamics compared

to the zone temperature dynamics. Step tests in the VCC inputs

(compressor RPM and valve opening) revealed the supply air

and superheat temperatures evolve roughly in the same time

scale (1–10 min) whereas the zone air temperature dynamics

were significantly slower (nearly 50 min). Varying internal gains

and ambient conditions act as disturbances to the zone air

temperature; however, by using a cascade control structure, the

relatively faster dynamics of the inner loop are exploited to

eliminate these disturbances (using the VCC inputs) before they

significantly affect the zone temperature.

In this cascade control structure, the inner loop consists of a

stand-alone VCC controller (either the predictive controller or PI

controllers designed in Section 3.2.1). The outer loop is used to

0

5

10

15

20

25

Time (h)

S u p e r h e a t , T s , e

( ° C )

MPCPI

23

23.2

23.4

23.6

23.8

S u p p l y

a i r t e m p e r a t u r e ,

T

o a , e

( ° C )

SP

MPCPI

1

·103

R P M , k

MPCPI

6

8

10

12

14


( % )

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.

Table 4

Stand-alone VCC closed-loop performance metrics.

Metric Control strategy

PI control MPC

ISESA (s 1C2) 837 222

t settSA (s) 1800, 1800, 900, 1620 1020, 1800, 1140, 4440TEC (kJ) 10017 9217



12/14

regulate the zone temperature and is connected to the inner loop

via the supply air temperature set-point. Based on the error

between the zone temperature and its set-point (T zone,SP ¼

24 1C), the outer loop prescribes a supply air set-point, T oa,e,SP, to

the inner loop controller. The zone temperature is sampled every

15 min, which also corresponds to the frequency of the supply air

set-point updates. Faster sampling times led to excessive fluctua-

tions in the prescribed supply air set-point, resulting in poor

tracking performance by the inner loop controller. The outer loop

was tuned iteratively such that it yielded trackable supply air set-

points by the VCC. The outer loop tunings were kept consistent for

both control strategies in the inner loop.

The zone temperature response for each control strategy is

shown in Fig. 10, which compares the efficacy of each control

strategy in meeting the primary control objective. The MPC-based

strategy is able to satisfy the comfort standards for the entire test

period (any zone set-point violations lasted less than 15 min)

while the comfort standards are violated for approximately the

last 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 decrease

in the internal gains owing to a decrease in the zone occupancy

and also a decrease in the ambient temperature. The zone

temperature response after 4:00 P.M. indicates that the MPC-

based design is able to respond to these disturbances more

effectively 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 two

inner loop control strategies. The performance metrics of the

inner loops are shown in Table 5. Similar to the results in the

stand-alone VCC case, for the MPC-based design, the superheat is

allowed to ‘float’ between its bounds and ended up evolving

closer to its lower bound, allowing for better supply air tempera-

ture tracking using considerably less compressor power. As

shown 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 achieve

this offset free performance (in addition to improved zone air

temperature regulation), aggressive control action is prescribed.

We note, however, that the key idea in the results with the

interfaced system is not so much to demonstrate improved

control of the zone conditions (which depends on several factors,

including the ‘outer loop’) but more to evaluate the performance

of 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 predictive

controller. The main requirement for this extension is to identify a

model between the supply air temperature and the zone air

temperature. This can be identified through step-tests or more

desirably, by generating PRBS-like sequences of the supply air.

However, in any case, the resulting model will be dependent on

the closed-loop dynamics of the stand-alone VCC controller. Due

to the large variation in the time scales between the zone and VCC

dynamics, in addition to the zone air being affected through a

single VCC output variable (supply air temperature), it is advan-

tageous to use separate MPC designs for each level of the cascade

rather than using one model predictive controller to regulate the

zone conditions. As discussed in Remark 5, a weather model/estimator can also be incorporated in the design by including an

ambient temperature component in the model. In this case,

through an economic objective function that considers varying

electricity costs for the outer loop controller, operating costs

can be reduced by pre-cooling as necessary. In addition to

optimality, the benefits of using MPC in the outer loop include

explicitly incorporating comfort specifications and accounting

for the VCC cooling capacity in computing the supply air set-

points. Finally, while we use temperature as the comfort measure

in this work, other measures of comfort, such as a Predicted Mean

Vote (Federspiel and Asada, 1994; Hanna, 1997; Brager and de

Dear, 1998; Jones, 2002; Ye et al., 2003; Baus et al., 2008) can be

readily incorporated in the objective function in the MPC

formulation.

+

−

T zone,SPPI MPC/PI

T oa,e,SP

Condenser Evaporator

Compressor

Valve

EnergyPlus

Model

k

vo

T s,e ,T oa,e

T zone

DisturbancesVCC

Fig. 9. Proposed cascade control structure for energy efficient temperature control.

8

24

Time (h)

Z o n e a i r t e m p e r a t u r e ( ° 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.



13/14

4. Conclusions

In this work, control strategies were implemented on a

realistic building model interfaced with a detailed VCC model

which provided the cooling load for a specified zone in the

building. The detailed VCC model was subject to realistic dis-

turbances in the ambient and mixed air conditions as a result of

interfacing the two models. The zone air temperature in the

building model was also subject to realistic ambient air condi-

tions (obtained from actual weather data) and typical internal

load variations. A cascade control structure was proposed to

regulate the zone air conditions in the building model. In the

proposed control structure, an outer loop regulates the zone air

temperature by adjusting the set-point of the VCC supply air

temperature using PI control. The inner loop regulates the VCC

supply air and superheat by manipulating the compressor RPM

and 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-based

cascade structure. Also the predictive control strategy was more

energy efficient. The improved performance of the MPC-baseddesign stemmed from its incorporation of the coupled nature of

VCC dynamics (through the ARX model) in computing the control

action. The ability in the MPC to ‘trade-off’ optimality considera-

tions with tracking requirements enabled achieving improved

set-point tracking while operating at lower RPM values (where

possible) resulting in better energy efficiency.

Notation

Variable Description

V volume

r density

Z efficiency

8

21

22

23

24

Time (h)

S u p p l y a i r t e m p e r a t u r e , T

o a , e

( ° C )

SPMPC

21

22

23

24

25

S u p p l y a i r t e m p e r a t u r e ,

T o

a , e

( ° C )

SPPI

5

10

15

20

25

S u p e r h e a t , T

s , e

( ° C )

MPCPI

1

·103

R P M , k

MPCPI

6

8

10

12

14


( % )

MPC

PI

1.2

0.8

10 12 14 16 18 8

Time (h)

10 12 14 16 18

8

Time (h)

10 12 14 16 18

8

Time (h)

10 12 14 16 18

8

Time (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 5

Inner 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



14/14

o compressor RPMt time constanth specific enthalpy

DP pressure drop

C d discharge coefficient

g volumetric mean void fraction_m mass flow rate

A cross-sectional area

P pressure p perimeter

a heat transfer coefficientT temperature

x state variable

u input variable

d disturbance variable

vo valve opening

c p 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 saturationdp 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 Research

and Development Program (in collaboration with Johnson Con-

trols Inc.) is gratefully acknowledged.

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