i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9

www. i ifi i r .org

ava i lab le at www.sc iencedi rec t . com

journa l homepage : www.e lsev i er . com/ loca te / i j r e f r ig

Multivariable control-oriented modeling of a directexpansion (DX) air conditioning (A/C) system

Qi Qi, Shiming Deng*

Department of Building Services Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong SAR

a r t i c l e i n f o

Article history:

Received 28 June 2007

Received in revised form

17 October 2007

Accepted 18 October 2007

Published online 9 January 2008

Keywords:

Cooling system

Air conditioning

Direct expansion

Modelling

Simulation

Comparison

Experiment

* Corresponding author. Tel.: þ852 2766 5859E-mail address: besmd@polyu.edu.hk (S.

0140-7007/$ – see front matter ª 2007 Elsevidoi:10.1016/j.ijrefrig.2007.10.009

a b s t r a c t

A dynamic mathematical model for a DX A/C system has been developed. The dynamic

model, written in state-space representation which was suitable for designing multivari-

able control, was linearized at steady state operating points. The linearized model has

been validated by comparing the model simulation results with the experimental data ob-

tained from an experimental DX A/C system. The simulated results agreed well with the

experimental data, suggesting that the model developed was able to capture the transient

characteristics of the DX A/C system modeled. It is expected that the model developed can

be useful in designing a multi-input multi-output (MIMO) controller to simultaneously

control indoor air temperature and humidity in a space served by a DX A/C system.

ª 2007 Elsevier Ltd and IIR. All rights reserved.

Modelisation aux variables multiples axee sur la regulationd’un systeme de conditionnement d’air a detente directe

Mots cles : Systeme frigorifique ; Conditionnement d’air ; Detente directe ; Modelisation ; Simulation ; Comparaison ; Experimentation

1. Introduction

Direct expansion (DX) air conditioning (A/C) systems are

widely used in small- to medium-scaled buildings in recent

decades. Compared to central chilled water-based A/C sys-

tems, the use of DX A/C systems is advantageous since they

are simpler in configuration, more energy efficient and gener-

ally cost less to own and maintain. In the US, according to

; fax: þ852 2765 7198.Deng).er Ltd and IIR. All rights

Department of Energy, packaged rooftop DX A/C systems

accounted for approximately 60% of the total installed cooling

capacity (Bordick and Gilbridge, 2002).

Residential buildings are most likely served by DX A/C sys-

tems, but controlling indoor humidity at an appropriate level

using a DX A/C system is both challenging and important

since this directly affects occupants’ thermal comfort and in-

door air quality (IAQ) (Fanger, 2001). Most DX A/C units are

reserved.

Nomenclature

A, B, C coefficient matrices (in Eq. (18))

A1 heat transfer area of the DX evaporator in dry-

cooling region, m2

A2 heat transfer area of the DX evaporator in wet-

cooling region, m2

Cp specific heat of air, kJ kg�1 K�1

f air volumetric flow rate, m3/s

M moisture load in the conditioned space, kg/s

Mref mass flow rate of refrigerant, kg/s

Pr Prandtl number

Qload sensible heat load in the conditioned space, kW

Qspl heat gain of supply fan, kW

SH super heat of refrigerant, �C

T1 temperature of air leaving the DX evaporator, �C

T2 air temperature in the conditioned space, �C

T3 air temperature leaving the dry-cooling region of

the DX evaporator, �C

Tw temperature of the DX evaporator wall, �C

V volume of the conditioned space, m3

Vh1 air side volume of the DX evaporator in dry-cooling

region on air side, m3

Vh2 air side volume of the DX evaporator in wet-

cooling region on air side, m3

Vcom swept volume of the rotor compressor, m3

W1 moisture content of air leaving the DX evaporator,

kg/kg dry air

W2 moisture content of air-conditioned space, kg/kg

dry air

a1 heat transfer coefficient between air and the DX

evaporator wall in dry-cooling region, kW m�2 �C�1

a2 heat transfer coefficient between air and the DX

evaporator wall in wet-cooling region, kW m�2 �C�1

r density of moist air, kg/m3

hfg latent heat of vaporization of water, kJ/kg

hr1 enthalpy of refrigerant at evaporator inlet, kJ/kg

hr2 enthalpy of refrigerant at evaporator outlet, kJ/kg

je1, je2 Colburn factors

kspl coefficient of supply fan heat gain, kJ/m3

vs specific volume of superheated refrigerant,

m3 kg�1

e rotor eccentricity, m

l stroke of cylinder, m

r radius of rotor, m

s speed of compressor, rpm

l compressor’s displacement coefficient

lsys eigenvalue (in Eq. (23))

3 rotor relative eccentricity

Superscript

w evaporator wall

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9842

currently equipped with single-speed compressors and supply

fans, relying on on–off cycling compressors as a low-cost

approach to maintain only indoor air dry-bulb temperature,

resulting in either space overcooling or an uncontrolled equi-

librium indoor relative humidity (RH) level.

Recent developments in variable speed drive (VSD) tech-

nology offer tremendous opportunities for improving indoor

thermal comfort and energy efficiency for DX-based space

air conditioning. Compressor speed can be continuously var-

ied to modulate the output cooling capacity to match the ac-

tual thermal load. The supply fan speed can be also altered

to affect both sensible heat and latent heat transfer rate across

heat exchangers. Therefore it is possible to improve indoor

thermal comfort control using DX A/C systems equipped

with variable speed compressor and supply air fan.

In the open literature available, a considerable number of

previous investigations have focused on the dynamic model-

ing of vapor compression refrigeration cycles. He et al. (1997)

developed an overall dynamic model for a vapor compression

refrigeration cycle, and the simulation results indicated that

there were strong cross-couplings among system inputs and

outputs. Linear Quadratic Gaussian (LQG) technique was

then used to design a multi-input multi-output (MIMO) con-

troller with guaranteed stability and robustness (He et al.,

1998). The possibility of using a model-based nonlinear con-

troller was also investigated numerically for a vapor compres-

sion refrigeration system (Tao et al., 2004, 2005). Rasmussen

and Alleyne (2004) presented a reduced order dynamic model

of a transcritical vapor compression cycle. It was demon-

strated that the reduced order model was adequate for predict-

ing the dominant system dynamics. Therefore the reduced

order model with minimal loss in accuracy was very useful

for designing an MIMO controller. Shah et al. (2004) developed

a model for the vapor compression refrigeration cycle in an

automotive air conditioning system with a variable speed

compressor, and applied the multivariable adaptive control

strategy to the air conditioning system to improve its capacity

control and system efficiency. Lin and Yeh (2007) developed

a low-order linear model for an air conditioning system

through system identification. Experimental results indicated

that an MIMO-based controller can both achieve satisfactory

transient responses in indoor air temperature and improve

energy efficiency at steady states. However, the model estab-

lished through system identification was only valid for certain

particular systems.

In mechanical cooling based on A/C systems, dehumidifi-

cation is less straightforward because of the dual function of

cooling and dehumidification taking place in cooling coils.

This has led to the controlled variables of air temperature

and humidity becoming coupled, which was confirmed by ex-

perimental investigations (Li and Deng, 2007a). Krakow et al.

(1995) suggested that space air temperature and relative hu-

midity could be controlled by varying compressor speed and

varying evaporator fan speed, separately, using a propor-

tional–integral–derivative (PID) control method. However,

the study focused on the feasibility of such a PID control

method, without looking at the coupling effect of air temper-

ature and humidity by treating the two controlled variables

separately. A DDC-based control algorithm developed by Li

and Deng (2007b,c) considered the coupling effect of air tem-

perature and humidity and used space sensible heat ratio

(SHR) as a controlled variable to simultaneously control space

Thermal space

EEVVariable speedcompressor

Evaporator

Condenser

Variable speedsupply fan

T1,W1

T2,W2 Supply air

hr1hr2

Fig. 1 – The schematic diagram of the experimental DX A/C

system.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9 843

air temperature and relative humidity. However, the sensitiv-

ity of the DDC-based controller was poor as it had to wait for

a long time for the controller to be fed with the required infor-

mation to take the next control action.

It can therefore be seen that the previous studies mainly

focused on either the modeling and control of vapor compres-

sion refrigeration system or the control of air temperature and

humidity by using conventional control such as PID control

method without much consideration on the coupling effect

of air temperature and humidity. The PID control method

was in fact to have two separate control loops, i.e., controlling

indoor air temperature by varying compressor speed and in-

door air humidity by varying supply fan speed. The two con-

trol loops have been traditionally treated as two separate

single-input single-output (SISO) systems, while the coupling

effect between the two parameters has been often ignored.

The performance of the conventional SISO control has been

inherently poor. Therefore developing an MIMO control strat-

egy for coupled air temperature and humidity is urgently re-

quired. Consequently, dynamic modeling of a DX A/C

system suitable for developing MIMO control algorithms

which consider the coupling effect of air temperature and hu-

midity becomes highly necessary.

This paper presents the development of a dynamic mathe-

matical model for a DX A/C system, written in state-space repre-

sentation which was suitable for designing multivariable

control. The organization of the paper is as follows. Section 2 de-

scribes briefly an experimental DX A/C system. The develop-

ment of the dynamic model of the experimental DX A/C

system and its linearization process are presented in Section

3. Simulation results using the linearized model and the results

obtained for the experimental DX A/C system are compared in

Section 4 for model validation. Section 5 presents conclusions.

2. Description of the experimentalDX A/C system

The experimental DX A/C system was mainly composed of

two parts, i.e., a DX refrigeration plant (refrigerant side) and

an air-distribution sub-system (air side). Its simplified sche-

matic diagram is shown in Fig. 1. The major components in

the DX refrigeration plant included a variable speed rotor com-

pressor, an electronic expansion valve (EEV), a high-efficiency

tube-louver-finned DX evaporator and an air-cooled tube-

plate-finned condenser. The evaporator was placed inside

the supply air duct to work as a DX air cooling coil. The design

air face velocity for the DX cooling coil was 2.5 m/s. The nom-

inal output cooling capacity from the DX refrigeration plant

was 9.9 kW (2.8 RT). The working fluid of the plant was refrig-

erant R22, with a total charge of 5.3 kg.

The air-distribution sub-system included an air-distribution

ductwork with return air dampers, a variable speed centrifugal

supply fan, and a conditioned thermal space. Inside the space,

there are sensible heat and moisture load generating units

(LGUs). The units are intended to simulate the cooling load in

the conditioned space.

The experimental DX A/C system has been fully instru-

mented. High-precision sensors/transducers were used for

measuring all operating parameters including temperatures

and flow rates of both air and refrigerant, pressures in the

DX A/C unit, etc. All measurements were computerized, so

that all the measured data can be recorded for subsequent

analysis.

3. Dynamic modeling of the experimentalDX A/C system

The dynamic mathematical model for the DX A/C system was

mainly derived from the energy and mass conservation prin-

ciples. The following assumptions were made in developing

the mathematical model: (1) perfect air mixing inside all

heat exchangers and the thermal space, and no fresh air in-

take to the system; (2) two regions on the air side of the DX

evaporator, i.e., dry-cooling region and wet-cooling region;

and (3) negligible thermal losses in air ducts.

In the DX A/C system to be modeled, the temperature

and moisture content of the air leaving the DX cooling

coil, as shown in Fig. 1, were T1 and W1, respectively. With

the air perfect mixing assumption, air temperature, T2, and

air moisture content, W2, leaving the conditioned space

can be regarded as being equal to those in the conditioned

thermal space. Based on the energy conservation principle,

the sensible energy balance equation for the conditioned

space was:

CprVdT2

dt¼ CprfðT1 � T2Þ þ Qload þ Qspl (1)

where V was the volume of the conditioned space, Qload the

space sensible load, Qspl heat gain of the supply fan, f the air

volumetric flow rate. The heat gain of supply fan increased

with the air flow rate.

Qspl ¼ ksplf (2)

On the other hand, the moisture mass balance inside the

conditioned space was

rVdW2

dt¼ rfðW1 �W2Þ þM (3)

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9844

where M was the moisture load generation in the conditioned

space.Corresponding to the assumed two regions on the air side,

the refrigerant side in the DX A/C systems was assumed to

have a two-phase region and a superheated region, as shown

in Fig. 2.

At the air side of the evaporator, with the assumption of no

fresh air, the temperature and moisture content of the air en-

tering evaporator were T2 and W2, respectively. The air tem-

perature decreased along the evaporator wall and was equal

to T3 at the end of dry-cooling region. Since the dry-cooling

region was usually small, the temperature of the entire evap-

orator wall was assumed at the same Tw. Applying the energy

balance principle in the dry-cooling region on the air side

yielded

CprVh1dT3

dt¼ CprfðT2 � T3Þ þ a1A1

�Tw �

T2 þ T3

2

�(4)

On the other hand, in the wet-cooling region, there was not

only sensible heat transfer between the air and evaporator

wall but also latent heat transfer. The coupled air cooling

and dehumidification mainly took place in the wet-cooling re-

gion. Therefore the energy balance in the wet-cooling region

can be written in enthalpy form:

rVh2dh1

dt¼ rfðh3 � h1Þ þ a2A2

�Tw �

T3 þ T1

2

�(5)

The relationship among air enthalpy, temperature and

moisture content was

h ¼ CpTþ hfgW (6)

where hfg was the latent heat of vaporization of water.

Substitute Eq. (6) into Eq. (5) to obtain

CprVh2dT1

dtþ rVh2hfg

dW1

dt¼ CprfðT3 � T1Þ þ rfhfgðW2 �W1Þ

þ a2A2

�Tw �

T3 þ T1

2

�(7)

The degree of refrigerant sub-cooling in a condenser with

a receiver is normally rather small, and the refrigerant in the

receiver can be assumed to be the saturated liquid refriger-

ant at condensing pressure. Therefore, after knowing the

real-time measured condensing pressure, the enthalpy of

refrigerant leaving the receiver, hre2, can be obtained using

the R22 State Equations (Cleland, 1986). Neglecting the en-

ergy loss in the refrigerant line and approximating the

wT3

dry-region

T2,W2

Air side

Tw

h2 superheated region two-p

Fig. 2 – The schematic di

refrigerant throttling process in an EEV as being isenthalpic,

the enthalpy of refrigerant entering the DX evaporator is

given by

hr1 ¼ hre2 (8)

The enthalpy of superheated refrigerant at compressor suc-

tion, hrc1, can be evaluated based on the real-time measured

pressure and the temperature of superheated refrigerant us-

ing the R22 State Equations. Neglecting the energy loss in

the refrigerant line between DX evaporator and compressor

suction owing to good thermal insulation, the enthalpy of

the refrigerant leaving the DX evaporator is given by

hr2 ¼ hrc1 (9)

The swept volume of the rotor compressor, Vcom, was calcu-

lated using the related compressor’s geometric parameters as

follows:

Vcom ¼ pr2l3�2� 3

�(10)

where l is the stroke of cylinder; r the radius of rotor and 3 the

rotor relative eccentricity.

The compressor displacement coefficient, l, was given by

l ¼ 1� 0:015

"�Pc

Pe

�1b

�1

#(11)

where b is the compression index which was assumed to be

constant at 1.18. Pc and Pe were the condensing pressure and

evaporating pressure, respectively.

Therefore, the refrigerant mass flow rate can be deter-

mined by

Mref ¼sVcom

vs

�1� 0:015

hðPc=PeÞ

1b�1

i�(12)

where s is the compressor speed, vs specific volume of super-

heated refrigerant, which can be obtained from measured

pressure and temperature of refrigerant at compressor suc-

tion using the R22 State Equations.

Due to the significant difference in thermal inertia for both

refrigerant and air, dynamic responses to changes on the air

side were much slower than that on the refrigerant side. When

the airside waited for a long time to fully respond, the refrigerant

side was already in its steady state for a quite while. Thus the

same refrigerant mass flow rate at both the inlet and the outlet

of the DX evaporator was assumed. Therefore the energy

balance equation for the evaporator wall can be written as

Refrigerant side

h1

Evaporator wall

T1,W1

et-region

hase region

agram of evaporator.

0 200 400 600 800 1000 120023.00

23.25

23.50

23.75

24.00

24.25

24.50

Simulation

Experiment

Tem

pera

ture

(ºC

)

Time (s)

Fig. 3 – Simulated and measured air temperature in the

conditioned space in response to a step change in

compressor speed.

11.2

11.4

11.6

cont

ent

(g/k

g)

Simulation

Experiment

Table 1 – Numerical values of the system parameters

Cp 1.005 kJ/kg A1 4.14 m2

r 1.2 kg/m3 A2 17.65 m2

hfg 2450 kJ/kg Vh1 0.04 m3

V 77 m3 Vh2 0.16 m3

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9 845

�CprV

�w

dTw

dt¼ a1A1

�T2 þ T3

2� Tw

�þ a2A2

�T3 þ T1

2� Tw

��Mrefðhr2 � hr1Þ (13)

The airside convective heat transfer coefficients for the

louver-finned evaporator in both dry-cooling and wet-cooling

regions were evaluated as follows (Chen, 2005):

a1 ¼ je1ryCp

Pr23

(14a)

a2 ¼ je2ryCp

Pr23

(14b)

where je1, je2 are the Colburn factors, y the air velocity.

The supply air leaving the DX evaporator was assumed to

be at 95% saturated. The relationship between air moisture

content and temperature can be derived by plotting and curv-

ing fitting:

W1 ¼�0:0198T2

1 þ 0:085T1 þ 4:4984��

1000 (15a)

Therefore

dW1

dt� ð2� 0:0198T1 þ 0:085ÞdT1

dt=1000 ¼ 0 (15b)

Eqs. (1), (3), (4), (7), (13) and (15b), all of which were first order

differential equations, formed the dynamic model of DX A/C

system. Since the objective of the model development was

to assist the design of a multivariable controller, it was sug-

gested (Tewari, 2002; Skogestad and Postletheaite, 1996) that

these differential equations should be written in state-space

representation, such that it did not formally distinguish be-

tween a multivariable system and a single variable system,

allowing an efficient design and analysis for a multivariable

system in the same manner as for a single variable system.

Hence the model in state-space representation may be

expressed in the following compact format:

_X ¼ D�1$g1

�X;U

�þ D�1$g2

�Z�

(16)

where the state variables X ¼ ½T1;T2;T3;Tw;W1;W2�T and_X ¼ ðdX=dtÞ, the input variables U ¼ ½f ; s�T, and the disturbance

variables Z ¼ ½Qload;M�T, g1, g2 are the functions, defined as

follows:

Table 2 – Operating condition of the DX A/C system

T1 13.25 �C Pc 1.812� 106 Pa

W1 9.03/1000 kg/kg dry air Pe 0.486� 106 Pa

T2 24 �C Qload 4.49 kW

W2 11.35/1000 kg/kg dry air M 0.96/1000 kg/s

T3 17 �C Mref 0.042 kg/s

Tw 13 �C s 3960 rpm

SH 6 �C f 0.347 m3/s

g1ðX;UÞ¼

26666664

CprfðT1�T2ÞþksplfrfðW1�W2ÞCprfðT2�T3Þþa1A1

�Tw�T2þT3

2

�CprfðT3�T1ÞþrfhfgðW2�W1Þþa2A2

�Tw�T3þT1

2

�a1A1

�T2þT3

2 �Tw

�þa2A2

�T3þT1

2 �Tw

��s vs

Vcomlðhr2�hr1Þ

0

37777775

(17a)

g2ðZÞ¼

26666664

Qload

M0000

37777775

(17b)

D¼

26666664

0 CprV 0 0 0 00 0 0 0 0 rV0 0 CprVh1 0 0 0

CprVh2 0 0 0 rVh2hfg 00 0 0

�CprV

�w

0 01 0 0 0 ð2�0:0198T1þ0:085Þ=1000 0

37777775

(18)

0 200 400 600 800 1000 120010.6

10.8

11.0

Moi

stur

e

Time (s)

Fig. 4 – Simulated and measured air moisture content in

the conditioned space in response to a step change in

compressor speed.

0 200 400 600 800 1000 120012.00

12.25

12.50

12.75

13.00

13.25

13.50

Tem

pera

ture

(ºC

)

Time (s)

Simulation

Experiment

Fig. 5 – Simulated and measured temperature of the air

leaving evaporator in response to a step change in

compressor speed.

0 200 400 600 800 1000 120058.0

58.5

59.0

59.5

60.0

60.5

61.0

RH

(

)

Time (s)

Simulation

Experiment

Fig. 7 – Simulated and measured relative humidity in the

conditioned space in response to a step change in

compressor speed.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9846

The developed dynamic model expressed in state-space

representation, i.e., Eq. (16), was nonlinear since the relation-

ship between state variables and input variables was nonlin-

ear. In most cases, a DX A/C system was designed to operate

in the vicinity of a predetermined set point given that thermal

space cooling load did not significantly change. As long as the

control system can properly regulate the dynamic deviation of

the controlled objectives from the set points, the controlled

system can be well represented by a linearized model around

the set points. Hence, the state variables, X, and control in-

puts, U, can be expressed as follows, respectively:

X¼xþx0 (19a)

U¼uþu0 (19b)

where x0 and u0 are the state vector and input vector, both evalu-

ated at a steady state operating point, and x and u represent the

small dynamic deviation from x0 and u0, respectively. For the DX

A/C system to be modeled, the sensible and moisture content load

disturbance can be regarded as being constant at a steady state.

Therefore the linearized model describing the system’s dynamic de-

viation at an operating point can be written as

0 200 400 600 800 1000 12008.0

8.2

8.4

8.6

8.8

9.0

9.2

Moi

stur

e co

nten

t (g

/kg)

Time (s)

Simulation

Experiment

Fig. 6 – Simulated and measured moisture content of the

air leaving evaporator in response to a step change in

compressor speed.

_x¼vg1

vX

x0 ;u0

xþvg1

vU

x0 ;u0

u¼A�x0;u0

�xþB

�x0;u0

�u (20)

Therefore the linearized dynamic model of the DX A/C sys-

tem in state-space representation, which is highly suitable for

designing multivariable control, can be written as8<:

_x¼AxþBu

y¼Cx(21)

where the output variables y¼½dT2;dW2�T, the dynamic devia-

tions of air temperature and moisture content from their set

points, respectively, and A, B, C were the coefficient matrices.

At a particular operating point, where T2¼ 24 �C,

W2¼ 0.0135 kg/kg dry air, T1¼ 13.25 �C, W1¼ 0.00903 kg/kg

dry air and the air flow rate f¼ 0.347 m3/s, the system matrices

A, B and C were calculated as follows:

A ¼

26666664

�5:731 0 0:0756 4:1883 �5287 52870:0045 �0:0045 0 0 0 0

0 4:6577 �12:692 8:0346 0 00:0139 0:0067 0:0206 �0:0412 0 00:0006 0 0 0 0 0

0 0 0 0 0:0045 �0:0045

37777775

(22a)

0 200 400 600 800 1000 12000

1

2

3

4

5

6

7

8

Latent cooling capacity

Sensible cooling capacity

Coo

ling

capa

city

(kW

)

Time (s)

Total cooling capacity Simulation

Experiment

Fig. 8 – Simulated and measured output cooling capacities

in response to a step change in compressor speed.

0 200 400 600 800 100011.0

11.5

12.0

12.5

13.0

13.5T

empe

ratu

re (

ºC)

Time (s)

Simulation

Experiment

Fig. 9 – Simulated and measured temperature of the air

leaving evaporator in response to a step change in supply

fan speed.

0 200 400 600 800 100023.00

23.25

23.50

23.75

24.00

24.25

24.50

Tem

pera

ture

(ºC

)

Time (s)

Simulation

Experiment

Fig. 11 – Simulated and measured air temperature in the

conditioned space in response to a step change in supply

fan speed.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9 847

B ¼

26666664

55:035 0�0:098 0172:5 0

0 �5:9310 0

�0:00003 0

37777775; C ¼

0 1 0 0 0 00 0 0 0 0 1

�(22b)

The eigenvaluesofthe linearizedmodel for theDXA/Csystem

are shown in Eq. (23) for the operating point. All the eigenvalues

have the negative real parts, suggesting that the DX A/C system

represented by the linearized model was asymptotically stable.

lsys ¼

26666664

�2:46e� 017�4:51e� 003�2:85e� 002�6:15e� 001�5:12eþ 000�1:27eþ 001

37777775

(23)

4. Model validation

Simulation results using the linearized dynamic model, i.e.,

Eq. (18), have been compared with the experimental data

0 200 400 600 800 10008.0

8.2

8.4

8.6

8.8

9.0

9.2

Moi

stur

e co

nten

t (g

/kg)

Time (s)

Simulation

Experiment

Fig. 10 – Simulated and measured moisture content of the

air leaving evaporator in response to a step change in

supply fan speed.

obtained from the experimental DX A/C system for the pur-

pose of model validation. The simulation results and experi-

ment results were the open-loop responses to step changes

in compressor and supply fan speeds, respectively.

When the system was operating around a steady state con-

dition, step changes were introduced to the controllable in-

puts such as compressor speed and supply fan speed. The

same operating conditions and step changes were also input

to the model to obtain simulation results to facilitate the

comparison.

The following comparisons were based on the steady state

operational condition of around 24 �C indoor air temperature

and 11.3 g/kg moisture content, or 60% RH in conditioned

space. The numerical values of both the system parameters

used in the simulation and the operating condition of the

DX A/C system are given in Tables 1 and 2, respectively. The

linearization of the model was also based on this operating

condition. Figs. 3–8 present the comparisons between the sim-

ulation results and experimental data in response to a step

change in compressor speed from 3960 rpm to 4488 rpm

(from 66 Hz to 75 Hz), introduced at 420 s. When the compres-

sor speed increased, the temperature and moisture content of

0 200 400 600 800 100011.00

11.05

11.10

11.15

11.20

11.25

11.30

11.35

11.40

Moi

stur

e co

nten

t (g

/kg)

Time (s)

Simulation

Experiment

Fig. 12 – Simulated and measured air moisture content in

the conditioned space in response to a step change in

supply fan speed.

0 200 400 600 800 100058.0

58.5

59.0

59.5

60.0

60.5

61.0

RH

(

)

Time (s)

Simulation

Experiment

Fig. 13 – Simulated and measured relative humidity in the

conditioned space in response to a step change in supply

fan speed.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9848

the air in conditioned space decreased due to the increased

output cooling capacity of the DX A/C system, as shown in

Figs. 3 and 4. As seen in both figures, there existed a good

agreement between the simulated responses and the experi-

mental results. Similar observations can be obtained for other

operating parameters such as the temperature and moisture

content of air leaving the DX evaporator (Fig. 5 and 6), indoor

air relative humidity (Fig. 7) and output cooling capacities

(Fig. 8). In Fig. 8, it is seen that the increase in output latent

cooling capacity was more than that in output sensible cooling

capacity, when the total output cooling capacity was in-

creased due to the increase in compressor speed, which

resulted in a lower evaporating temperature or a lower evap-

orator wall temperature. Hence, as seen in Fig. 7, indoor air

RH was reduced due to the increased output latent cooling

capacity.

On the other hand, the comparisons between the simula-

tion results and experimental data, in response to a step

change in supply fan speed from 2448 rpm to 2160 rpm (from

41 Hz to 36 Hz), introduced at 290 s, are illustrated in Figs. 9–14.

When the supply fan speed was reduced, the temperature and

moisture content of the air leaving the DX evaporator de-

creased, as shown in Figs. 9 and 10, respectively. Again, in

both figures, a good agreement between the simulated results

0 200 400 600 800 10000

1

2

3

4

5

6

7

8

Latent cooling capacity

Sensible cooling capacity

Cap

acit

y (k

W)

Time (s)

Total cooling capacity

Simulation

Experiment

Fig. 14 – Simulated and measured output cooling capacities

in response to a step change in supply fan speed.

and measured experimental data can be observed. Similar ob-

servations of agreement can also be found for other systems

parameters as shown in Figs. 11–14.

The comparisons shown in Figs. 3–14 confirmed that the

developed model after linearization was experimentally vali-

dated to be able to both capture the transient change of sys-

tem parameters in a timely manner, and to represent the

steady state operation with an acceptable accuracy. Although

there were a number of points where there existed noticeable

differences between the measured and the simulated re-

sponses possibly due to the fact that the model developed

was a simplified one, the general trends for both were consis-

tent. Hence, this model was a good representation of the DX A/

C system during both the steady state and transient operating

conditions. More importantly, the model was written in state-

space representation, thus suitable for multivariable control-

ler design.

5. Conclusions

A dynamic mathematical model of a DX A/C system has been

developed based on the principle of energy and mass conser-

vation, and is reported in this paper. The dynamic model writ-

ten in state-space representation was linearized at its

operating point, which makes it highly suitable for designing

a multivariable control algorithm, such as MIMO control.

The linearized model was experimentally validated. It is

expected that the validated model could pave the way for fu-

ture work of designing an MIMO controller for simultaneously

controlling indoor air temperature and humidity in a space

served by a DX A/C system.

Acknowledgments

The authors acknowledge the financial supports from both

the Research Grant Council of Hong Kong (B-Q796) and The

Hong Kong Polytechnic University for the work reported in

this paper.

r e f e r e n c e s

Bordick, J., Gilbridge, T.L., 2002. Focusing on buyer’s needs: DOE’sengineering technology program. Energy Engineering 99 (6),18–38.

Chen, W., 2005. Modeling and Control of a Direct Expansion (DX)Variable-air-volume (VAV) Air Conditioning System. PhDthesis, Department of Building Services Engineering, HongKong Polytechnic University.

Cleland, A.C., 1986. Computer sub-routines for rapid evaluation ofrefrigerant thermodynamic properties. International Journalof Refrigeration 9, 346–351.

Fanger, P.O., 2001. Human requirements in future air-conditionedenvironments. International Journal of Refrigeration 24, 148–153.

He, X.D., Liu, S., Asada, H.H., 1997. Modeling of vaporcompression cycles for multivariable feedback control ofHVAC systems. Transaction of the ASME: Journal of DynamicSystems, Measurement, and Control 119, 183–191.

i n t e r n a t i o n a l j o u r n a l o f r e f r i g e r a t i o n 3 1 ( 2 0 0 8 ) 8 4 1 – 8 4 9 849

He, X.D., Liu, S., Asada, H.H., Itoh, H., 1998. Multivariable controlof vapor compression systems. HVAC & R Research 4 (3),205–230.

Krakow, K.I., Lin, S., Zeng, Z.S., 1995. Temperature and humiditycontrol during cooling and dehumidifying by compressor andevaporator fan speed variation. ASHRAE Transactions 101 (1),292–304.

Li, Z., Deng, S.M., 2007a. An experimental study on the inherentoperational characteristics of a direct expansion (DX) airconditioning (A/C) unit. Building and Environment 42, 1–10.

Li, Z., Deng, S.M., 2007b. A DDC-based capacity controller ofa direct expansion (DX) air conditioning (A/C) unit forsimultaneous indoor air temperature and humidity control.Part I: Control algorithms and preliminary controllability tests.International Journal of Refrigeration 30, 113–123.

Li, Z., Deng, S.M., 2007c. A DDC-based capacity controller ofa direct expansion (DX) air conditioning (A/C) unit forsimultaneous indoor air temperature and humidity control.Part II: Further development of the controller to improvecontrol sensitivity. International Journal of Refrigeration 30,124–133.

Lin, J.L., Yeh, T.J., 2007. Modeling, identification and control of airconditioning systems. International Journal of Refrigeration30, 209–220.

Rasmussen, B.P., Alleyne, A.G., 2004. Control-oriented modelingof transcritical vapor compression systems. Transaction of theASME: Journal of Dynamic Systems, Measurement, andControl 126, 54–64.

Skogestad, S., Postletheaite, I., 1996. Multivariable FeedbackControl Analysis and Design. John Wiley & Sons Ltd.

Shah, R., Rasmussen, B.P., Alleyne, A.G., 2004. Application ofa multivariable adaptive control strategy to automotive airconditioning systems. International Journal of AdaptiveControl and Signal Processing 18, 199–221.

Tao, C., He, X.D., Asada, H.H., 2004. Nonlinear observer design fortwo-phase flow heat exchangers of air conditioning systems.Proceedings of the American Control Conference, USA, 1534–1539.

Tao, C., He, X.D., Asada, H.H., Kasahara, S., 2005. Heat exchangerdynamic observer design. ASHRAE Transactions 111, 328–335.

Tewari, A., 2002. Modern Control Design with Matlab andSimulink. John Wiley & Sons Ltd.