Air Conditioning Using Matlab

www.elsevier.com/locate/simpat

Simulation Modelling Practice and Theory 12 (2004) 61–76

Simulink and bond graph modelingof an air-conditioned room

B. Yu *, A.H.C. van Paassen

Energy in Built Environment, Energy Technology, Delft University of Technology, Delft, The Netherlands

Received 6 May 2002; received in revised form 24 December 2002; accepted 23 December 2003

Abstract

Dynamic models of the heating, ventilation and air-conditioning (HVAC) systems in the

building are very useful for controller design, commissioning, and fault detection and diagno-

sis. Different applications have different requirements on the models and different modeling

approaches can be applied. Mathematical modeling with two different approaches, block-wise

Simulink and bond graph, is discussed. Advantage and disadvantage of both approaches are

expressed. It is shown that combination with two approaches to realize complicated models of

building HVAC system for the application of model-based fault detection and diagnosis is a

good solution.

� 2004 Elsevier B.V. All rights reserved.

Keywords: Bond graph; Simulink; Air-conditioned room; Modeling; Fault detection; Diagnosis

1. Introduction

A significant amount, as much as 30%, of all energy consumed by commercial

buildings in the US is related to inefficient and improper operation of building equip-

ment [1]. It is from the inability to optimally control, maintain, detect and diagnose

problems with the buildings and their systems, e.g. HVAC and others. Many of these

problems are often ignored or left unresolved because building staff and operatorslack the enough information to address them.

Model-based fault detection and diagnosis (FDD) is a good solution to handle this

problem [5] and general modeling concept has been raised to hierarchically realize

* Corresponding author. Tel.: +31-15-2786662; fax: +31-15-2787204.

E-mail address: [email protected] (B. Yu).

1569-190X/$ - see front matter � 2004 Elsevier B.V. All rights reserved.

doi:10.1016/j.simpat.2003.12.001

mail to: [email protected]

62 B. Yu, A.H.C. van Paassen / Simulation Modelling Practice and Theory 12 (2004) 61–76

FDD [6]. However, real buildings are diverse from each other. They have different

properties like sizes, locations and materials, etc. This puts higher requirement for

the models. The contents of a model depend on the application context for which

it is intended. As a consequence, models cannot easily be reused and exchanged.

At present, efficiently constructing high-quality models requires special skills and

experience. Computer support is only available at the computational and mathemat-

ical level. Every model has to be built from scratch, which means that modeling is

very labor and cost intensive and error prone.Nevertheless, reusing and exchanging models between applications is possible

since we accept a certain degree of similarity, to guarantee an adequate answer

(an exact answer is hardly ever required). Moreover, by complying with basic do-

main principles a certain level of confidence can be guaranteed without full valida-

tion, with respect to measured data. Provided that we acknowledge the underlying

assumptions for a particular modeling problem, models may be partially reapplied

in different situations. This means that for reuse of models the following require-

ments have to be fulfilled:

• Models have to be constructed in a modular way. The model fragments should be

pluggable: they can be substituted solely by respecting their interfaces (rather than

their contents).

• The model construction process is to be controlled by the application: model

selection must be controlled by assumption, i.e. explicit modeling experience.

• Hiding complexity. The simulation code is to be generated rather than hand

crafted, internal details are normally not shown. Modeling should occur at theconceptual rather than computational level.

Engineering and design phases are becoming dominant cost factors in the indus-

trial production cycle. This is particularly true for the automotive industry. There-

fore, in the early 1990’s a number of European industries and research institutes

initiated the OLMECO project. The aim of the project was to conceive and construct

an Open Library for models of Mechantronic COmponents. A rather unconven-

tional view of modeling was developed in the project [4]. Its core was the bond graphmodeling language for physical systems.

Two simulation methods, block-wise Simulink and bond graph, are two interest-

ing tools for modeling. Simulink is a software package of Matlab� for modeling and

simulating dynamical systems in academia and industry. It uses graphic user inter-

face (GUI) for building models as block diagrams and adopts click-and-drag mouse

operations. Nowadays, lots of toolboxes are available in Simulink. Bond graph is a

systematic way to represent power interactions between the models’ components.

IMMS [7], whose core concept is bond graph approach, is used to realize the mod-eling process. �Bonds’, the connection lines between different components, carry bothpower variables and causalities between power variables. Paynter [2] started the

bond graph technique and used for modeling dynamic multiport systems. This ap-

proach adopted circuit diagram concept to develop a general theory for engineering

system. It suggested that energy and power are the fundamental dynamic variables

B. Yu, A.H.C. van Paassen / Simulation Modelling Practice and Theory 12 (2004) 61–76 63

which represent all physical interactions and transactions. Rosenberg and Karnopp

[3] expressed the theoretical basis and definitions of the method. From then on, bond

graphs become a good modeling tool for different kind of dynamic systems.

Pseudo-thermal bond graph method is used. The pseudo-thermal bond graph

below represents a heat storage process

C

021

The constitutive relation of the C element is

T ¼ QC

where T is the pseudo-thermal effort temperature [K], Q is the pseudo-thermal stateheat [J] and

C ¼ m � c

where C is the heat capacity [JK�1], c is the material parameter specific heat capacity[JK�1 kg�1], which depends on the material selected by the modeler and

m ¼ q � V
where m is mass [kg], q is material parameter density [kgm�3], which, like specific
heat capacity c, depends on the material chosen by the modeler and V is geometricparameter volume [m3], which must be specified by the modeler or depends on thestandard geometry chosen by the modeler.

For the heat conduction process the following bond graph is appropriate

R

1 21

The 1 junction represents effort difference, in this pseudo-thermal case, tempera-

ture difference.

The constitutive relation of the R element is

Q0 ¼ TR

where Q0 is the pseudo-thermal flow heat flow [J s�1], T is the pseudo-thermal efforttemperature [K] and

R ¼ lk � A

where R is the resistance [K s3 m�2 kg�1], l is geometric parameter length [m], k ismaterial parameter heat conduction coefficient [m kg s�3 K�1] and A is geometricparameter area [m2], which, like length l, must be specified by the modeler or de-pends on the standard geometry chosen by the modeler.


Different applications ask different requirements on modeling. Some applications

require simple model and less quantitative results and some other applications re-

quire more accurate and complicated model. Some models are only used for specific

applications and some models need higher flexibility and reusability. The real build-

ings are diverse but with some similarity. In order to build up model-based fault

detection and diagnosis, the models with higher reusability and good communication

with Building Management System (BMS) are necessary. In this paper, two model-

ing approaches are studied to analyze the differences.

2. Modeling analysis on an air-conditioned room

A typical room, no. 104, in an office building is analyzed. The scheme of this floor

is shown in Fig. 1. The northeast and northwest walls insulate the room and outside

environment. The southeast and southwest walls separate the room with the neigh-

bors. Scheme of the fundamental air-conditioning components in the room, e.g. insu-lation (walls), zone (air), heating (radiator), cooling ceiling and window system

(shutters and windows) is shown in Fig. 2.

Following are the modeling analyses for these components with Simulink and

bond graph approaches.

2.1. Walls

All walls, floor and ceiling form the insulation system of the room. Lumpedparameter assumption is usually adopted for insulation models. This is realized by

splitting the wall into some layers. In each layer, the parameters, like temperature,

properties, are the same. Since the parameters are actually different, the more layers

the wall is split, and the closer the model is to the reality. However, too many layers

make the models more complicated and lower simulation speed. It is a trade-off.

Fig. 1. Scheme of the floor of room 104.

Fig. 2. Scheme of fundamental air-conditioning components of room 104.


A typical wall and its division are shown in Fig. 3.

Electrical R–C analysis chart for this wall is as follow.The equations of inner and outer surface temperatures has the form as

ð0:5qV Þ dTiodt

¼ Aqsolar þX

j

ar � Fshutter;j � AjðTj � TioÞ þ ac � AðTair � TioÞ

þX

i

as � Fradiator;j � AradiatorðTradiator � TioÞ � A � kdðTio � T1Þ ð1Þ

The equations of each layer has similar form as

dT1dt

¼aðTin � T1Þ þ k

D

� �1ðT2 � T1Þ

0:5ðqcdÞ1i ¼ 1 ð2Þ

Tn……T1

TiTo

T2 Tn-1

Fig. 3. Typical wall and its division.


dTidt

¼kD

� �iðTi�1 � TiÞ þ k

D

� �iþ1ðTiþ1 � TiÞ

0:5ðqcdÞi þ 0:5ðqcdÞiþ1i ¼ 2; . . . ; n� 1 ð3Þ

dTndt

¼aðTout � TnÞ þ k

D

� �nðTn�1 � TnÞ

0:5ðqcdÞni ¼ n ð4Þ

Simulink block representation for Fig. 4 with four layers is shown in Fig. 5 and for

each layer is shown in Fig. 6.

From the view of bond graph, the wall has radiation heat exchange with the other

walls and radiator. Meantime, all layers are constructed with storage, conduction

and power summation. Bond graph representation for the same wall is shown inFig. 7.

3

Qrad

2To

1

Ti

T1

T3T2

layer3

Ti

T2T1

layer2

Trad.

T1Ti

layer1 T2

Tout

alf a&q_s olar

To

gasbet.sierbet.1

Mux

Mux

f(u)

Fcn

3

W&qz

2

Trad

1

Tout

3

2

1

T1

T3T2

Ti

T2T1

Trad.

T1Ti

T2

Tout

alfa&q_solar

To

Mux f(u)

3

2

1

Fig. 5. Simulink block representation for a wall.

Ti ToT1 T2 Tn-1 Tn1/ δ/λ δ δ/λ λ/α 1/α

C1 C2 Cn-1 Cn

Fig. 4. Electrical R–C analysis chart for the wall.

1

T2

-K-

k3

-K-

k1k1

kSum3

Sum2

Sum11/s

Int

2T3

1T1

Fig. 6. Simulink block for certain layer.

2 Layer Wall

Heat Storage Heat Storage Heat StorageHeat Conduction Heat Conduction

Fig. 7. Bond graph structure for a two layer wall.

Fig. 8. StorageConduction module.


In Fig. 7, the bond relations are represented with some standard modules from

library. This is useful for reusable modeling.

All these modules are fundamental bond graph composition. For example, Sto-

rageConduction module is shown in Fig. 8. In the figures, the symbols of � and P

are connection plugs that use to split a bond into two parts.

2.2. Room air

Room air has considered as a lumped parameter. Its state is determined by many

other thermodynamic relations, e.g. heat exchanges with walls, radiator, shutters,

windows, cooling ceiling and so on. For the mathematical equation of room air

model, all heat exchanges should be included to make the equation achieve energy

conservation

ðqair � cp;air � V ÞdTadt

¼ Qinf :in þ Qinf :out þ acon:s:r: � As:ne:ðTs:ne: � TaÞþ acon:s:r: � As:se:ðTs:se: � TaÞ þ acon: � AneðTne � TaÞþ acon: � AseðTse � TaÞ þ acon: � AswðTsw � TaÞþ acon: � AnwðTnw � TaÞ þ acon: � AflðTfl � TaÞþ acon: � AceiðTcei � TaÞ þ acon: � AdoorðTdoor � TaÞþ acon: � Agl:dr:ðTgl:dr: � TaÞ þ acon: � AcolðTcol � TaÞ þ Qcoolþ Qintern þ Qwarm þ Qh þ Qhr ð5Þ


Simulink realizes the mathematical equations directly. All heat exchanges are treated

as inputs of integration. Fig. 9 shows the Simulink component model for room air.

For the bond graph modeling, power information (temperature, heat flow rate)

are inside the bonds. After the bonds are connected from room air node to the other

components, the thermodynamic relations are modeled in the meantime. Therefore,

the bond graph node model of room air is very simple as Fig. 10.

Fig. 9. Simulink model of room air.

Fig. 10. Bond graph node model of room air.


2.3. Heating system

Heating system is realized by radiator. By means of adjusting the opening rate of

hot water valve, heating capacity added into room is controlled.

Supposing the lumped temperature of radiator is the average of inlet and outlet

water temperature. Analyzing the control domain of radiator in Fig. 11.

Q is the total energy exchanges include radiation heat exchanges to all solid sur-faces (walls, ceiling, floor, windows, etc.) and convection heat exchange to the roomair. Mathematical equations of radiator are

T

Tw

CwMw

dTRdt

¼ _mwCwðTw;in � Tw;outÞ � Q ð6Þ

Q ¼ ARaðTR � TaÞ þ QR;wall ð7Þ
The mass flow rate _mw depends on the valve-opening rate that is controlled withproportion strategy.
Simulink model of radiator is shown in Fig. 12.

Like room air node, the bond graph model of radiator is relatively simple as fol-low, Fig. 13.

2.4. Cooling system

In the summer days, cooling system is necessary. It is the cooling ceiling. Cooling

ceiling has two functions. One function is cooling-down the recycle air. Another

Tw,in Tw,out

Q

Fig. 11. Control domain of radiator analysis.

2Qr.conv

1T rad.f(u)

conv. coef

Saturation1

f(u)

Qr.convection

Mux

Mux1

Mux

Mux

Low-Pass Filterwith Initiate

LPFIs

1

Integrator

7

Gain2

1/7

Gain1

f(u)

Fcn

4

air

3Qrad total

2mass flow

1

ater in

Fig. 12. Simulink model of radiator.

Fig. 13. Bond graph model of radiator.


function is supplying fresh air from outside. There is almost no capacity to accumu-

late energy at cooling ceiling node. Main thermodynamic process is heat exchange.

The amount of heat exchange is

QC ¼ Qrecycle þ Qfresh ð8Þ

Qrecycle ¼ ðTa � TwÞcðfactor1þ factor2ÞL ð9Þ

Qfresh ¼ GfreshqCpðTa � TfreshÞL ð10Þ
This cooling capacity will add into room air. In Eq. (9), c is controlled by the dif-ference between setpoint and real room air temperature. When the temperature of
the room is lower than setpoint, no flow in the tube and c will be zero.

2.5. Window system

Window system includes windows and shutters. Solar energy enters room throughwindows and shutters. Due to the distribution effect, the solar energy will be ab-

sorbed by all the surfaces.

Since the capacity of glasses and shutters are small, the heat transfers are instant

in the nodes of windows and shutters. This means they are algebraic equations.

For window, shutter and the air in-between window and shutter, the node equa-

tions are as follow

aoutðTout � TglassÞ þ aaðTa;between � TglassÞ þ qsolar ¼ 0 ð11Þ

aaðTglass � Ta;betweenÞ þ aaðTshutter � Ta;betweenÞ þ ainfiltrationðTa � Ta;betweenÞ ¼ 0 ð12Þ

1W&


aaðTa;between � TshutterÞ þ aaðTa � TshutterÞ þX

reðT 4wall � T 4shutterÞ

þ reðT 4radiator � T 4shutterÞ þ qsolar ¼ 0 ð13Þ

Simulink models of these components are shown in Fig. 14. Bond graph structures

for these components and sub-structures of shutter, window and the air inside the

channel are shown in Fig. 15 and Figs. 16–18. In these figures, the real bond graphsare inside each pluggable block to make the modeling reusable.

2Ta a

1Tb a

Mux

Mux2

Mux

Mux1

Mux

Mux

f(u)

Fcn2

f(u)

Fcn1

f(u)

Fcn

Demux

Demux

f (z) zSolvef(z) = 0

Algebraic Constraint2


Algebraic Constraint1


Algebraic Constraint

W Alf a o

Alfa out

5To

4Tw

3Tr

2Ta

qz

Fig. 14. Simulink models of window and shutter system.

Fig. 15. Bond graph structure of window and shutter system.

Fig. 16. Bond graph structure of shutter.

Fig. 17. Bond graph structure of the air between shutter and window.

Fig. 18. Bond graph structure of window.



2.6. Example results

After building the models, the simulation can be fulfilled. Fig. 19 shows the room

temperature results for 11 days and Fig. 20 shows the temperature difference of Fig.

19. It shows the model can simulate the reality properly.

0 50 100 150 200 250 30015

16

17

18

19

20

21

22

23

Time (h)

Tem

pera

ture

(o C)

measured datamodel output

Fig. 19. Example result of the simulation of room temperature.

0 50 100 150 200 250 300-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (h)

Tem

pera

ture

(o C)

Fig. 20. Temperature difference between measurement and model output.


3. Comparison between Simulink and BG modeling

Simulink and bond graph modeling use different ways to realize the simulation.

The comparison between these two methods has been made as follow:

Bond graph models can be realized as a block (like Matlab function or S-function)of Simulink models. So that it can take advantage of the Simulink toolbox to real-

ize more complicated functions. Because two modeling methods have their own

advantages and disadvantages, the combination with them would be a good solu-

tion. By means of bond graph method, models that directly represent the physical

relation are made. Then the BG models will be translated into S-function of Simu-link. Combined with the toolbox in Simulink, more functions like control, fuzzy

Simulink Bond graph

Use mathematical modeling concept Use physical modeling concept

Realize the mathematical equa-tions and relations with block

diagram. It’s a direct expression

for the mathematical models

Realize the physical meanings andrelations with bond

The first step of Simulink modeling

is trying to list the corresponding

mathematical equations for all

sub-domains after physical

assumption. For each sub-domain, analyzing the energy

conservation of the control

domain to obtain the mathemati-

cal equations

The first step of bond graph modeling

is analyzing the physical relationship

among the sub-domains. For each

sub-domain, determine the 0-, 1-junc-

tion, corresponding C-, R-elements andsuitable connection plugs

The temperature and heat flow

calculations are modeled sepa-

rately. Sometimes, this will cause

modeling bugs of non-balance ofenergy in the models

With pseudo-thermal BG modeling,

temperature and heat flow calculations

are modeled in the meantime. Inside a

bond, the temperature and heat floware exchanged simultaneously. The

bonds will always keep the energy

balance. Bugs of non-balance of energy

will be avoided

Simulink is easier to realize some

mathematical tricks and assump-

tions since it directly realize the

mathematical equations

Bond graph models are easier to

understand the physical relation

between components

Simulink integrates a lot of tool

box for mathematical analysis,

e.g. signal process, fuzzy logic,

neural network and so on

No toolbox available

Physical relationanalysis

Bond Graph modelling

s-function y=f(x,u)generation

final simulationmodel

Simulink block

Fig. 21. Suggested way of modeling combined with Simulink and bond graph.


logic, neural network algorithm can be easily realized. The flowchart is shown in

Fig. 21.

4. Conclusions

Model-based fault detection and diagnosis technology is a possible solution to de-

crease the energy consumption in building HVAC system. Different applications

have different requirements on the models and different modeling approaches can

be applied. Reusability is an important characteristic for the model of building that

applied for model-based FDD. By developing the modeling procedure, models of an

air-conditioned room in office building are built in two approaches, block diagram-

wise Simulink and bond graph. The comparison between two methods is made.Combination with two methods is a suggested way to build the models for fault

detection and diagnosis in building HVAC system.

Acknowledgements

This paper is written within the framework of Ecoview project (BTS97252) spon-

sored by Dutch Senter organization and with close cooperation with ing. H. Rijgers-

berg and Prof. J.L. Top of A&F BV.

References

[1] S. Katipamula, M. Brambley, Automated diagnostics: improving building system and equipment

performance energy user news 23 (4) April 1998.

[2] H.M. Paynter, Analysis and Design of Engineering Systems, MIT Press, Cambridge, Mass, 1961.

[3] R.C. Rosenberg, D.C. Karnopp, Introduction to Physical System Dynamics, McGraw-Hill, 1983.

[4] A.P.J. Breunese, J.L. Top, J.F. Broenink, J.M. Akkermans, Libraries of reusable models: theory and

application, Simulation 71 (1) (1998) 7–22.


[5] B. Yu, A.H.C. van Paassen, State-of-the-art of energy fault diagnosis for building HVAC system, in:

International Symposium of Air Conditioning in High Rise Buildings 2000, Shanghai, China, October

2000.

[6] B. Yu, A.H.C. van Paassen, S. Riahy, General modeling for model-based FDD on building HVAC

system, Simulation Modelling Practice and Theory 9 (6–8) (2002) 387–397.

[7] H. Rijgersberg, J.L. Top, HVAC library modeling in IMMS, EcoView Progress Report 1.1, ATO-

DLO, June 1999.

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