Implicit Multilinear Modeling of Air Conditioning Systems

Torben Warnecke

1 a

and Gerwald Lichtenberg

2 b

Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, 22607 Hamburg, Germany

Faculty of Life Sciences, Hamburg University of Applied Sciences, Ulmenliet 20, 21033 Hamburg, Germany

Keywords:

Multilinear Algebra, Differential-Algebraic Models, State-Space Models, Tensor Decomposition, HVAC,

Air-Conditioning, Building Automation, Building Simulation.

Abstract:

The publication explores the applicability of implicit multilinear model approaches in air conditioning systems.

Implicit multilinear time-invariant models offer a structure that allows for the representation of most of the

fundamental physical equations of HVAC systems. Since the implicit multilinear time-invariant model class

is closed, it enables a component-based modeling approach to represent various types of HVAC systems with

different combinations of components. Multilinear time-invariant models are usually represented by tensors.

With HVAC-Systems having a large number of inputs and states, the models can be efﬁciently represented

in a decomposed manner, resulting in a matrix representation. As an example, the model of a precision

climatization hutch with a PI controller is derived and simulated.

1 INTRODUCTION

In this paper, an example model is provided to demon-

strate the applicability of implicit multilinear time-

invariant (iMTI) models for HVAC-Systems (Heat-

ing, Ventilation and Air Conditioning) working with

humid air. The energy equations of HVAC-Systems

mostly consist of linear combination of mass ﬂows

and thermodynamic states, such as temperatures or

enthalpies (speciﬁc energy content of mass), e.g. in

the ﬁrst law of thermodynamics applied to ﬂuids.

Such equations typically exhibit a multilinear time-

invariant structure, as shown in (Lichtenberg et al.,

2022), (Pangalos et al., 2013) and (Pangalos et al.,

2014). A multilinear time-invariant structure can be

derived from the fundamental physical equations of

thermodynamic states and the energy conservation

principles of humid air systems. For the example of a

precision climatization hutch, typical thermodynamic

approaches, e.g. shown in (Yao and Yu, 2018), were

used. Usually, these systems are approximated using

linearized state-space models. However, this paper

proposes a modeling method with fewer approxima-

tions, allowing for direct utilization of most of the

fundamental physical equations.

https://orcid.org/0009-0004-3037-8634

https://orcid.org/0000-0001-6032-0733

2 MULTILINEAR MODELS

Recently, the implicit multilinear model framework

has been introduced. Firstly in an explicit form in

(Pangalos et al., 2013) and an implicit form in (Licht-

enberg et al., 2022). The multilinear model method

serves as a compromise between slow but accurate

nonlinear and fast but linear models. Furthermore,

linear and binary models can be fully represented us-

ing the multilinear structure. The multilinear func-

tions enable the utilization of tensor algorithms and

multilinear algebra, since the model structure and

parameter-space can be represented by tensors. This

facilitates the usage of tensor decomposition methods

and enhances the efﬁcient use of multilinear models

in simulations or other applications, like control and

automation.

2.1 Implicit Multilinear State-Space

Modeling

Expanding the explicit multilinear time-invariant

(eMTI) model format into the implicit multilinear

time-invariant (iMTI) format allows the multiplica-

tion of state derivatives

x ∈ R

with states x ∈ R

inputs u ∈ R

and/or outputs y ∈ R

. This enables

the representation of broken rational functions within

implicit multilinear models. More importantly, the

iMTI model class is a closed model class, mean-

440

Warnecke, T. and Lichtenberg, G.

Implicit Multilinear Modeling of Air Conditioning Systems.

DOI: 10.5220/0012138200003546

In Proceedings of the 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH 2023), pages 440-447

ISBN: 978-989-758-668-2; ISSN: 2184-2841

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

ing that any combination with different iMTI mod-

els results in the overall model still being an iMTI

model. This includes representation of series, par-

allel and feedback connections, as demonstrated in

(Lichtenberg et al., 2022). An iMTI model can be ex-

pressed as the inner tensor product of the model tensor

H ∈ R

2n+m+r

z }| {

2 × ... ×2×e

, with e being the application spe-

ciﬁc number of equations, and the monomial tensor

x,x,u,y) ∈ R

2n+m+r

0 = h H | M(

x,x,u,y)i . (1)

The model tensor can be decomposed using the

canonical polyadic (CP)-representation, shown in

(Kruppa, 2017), and normalization methods proposed

in (J

ores et al., 2022) into the norm-1 structure matrix

F ∈ R

(2n+m+r)×R

and the parameter matrix Φ ∈ R

e×R

with R as the model rank given by the number of over-

all terms

H =



F,Φ



. (2)

In the decomposed form, the implicit system of equa-

tions can be constructed from the structure matrix F

and parameter matrix Φ without reconstructing the

full tensor by inserting the matrices into the normal-

ized factored polynomial

0 =

∑

k=1

j,k

2n+m+r

∏

i=1

(1− | F

i,k

| +F

i,k

) (3)

with the variables v =













for all j ∈ {1,2...e}.

3 PHYSICS OF AIR

CONDITIONING SYSTEMS

To simplify the following equations and provide

a rough overview of the given dynamics, an iso-

baric, respectively ambient pressure, system is con-

sidered, allowing the use of isobaric material proper-

ties. Because kinetic and potential energy differences

in air streams of AC (air conditioning)-systems are

small compared to enthalpy differences, those are ne-

glected.

3.1 Enthalpy of Wet Air

The state of humid air can be adequately described by

the independent states of temperature and humidity

along with the isobaric material properties of dry air

and water vapor. The speciﬁc enthalpy

h = c

ϑ + c

µϑ + h

µ (4)

of wet air can be calculated with the temperature ϑ

◦

C and absolute humidity µ in kg

water

/kg

air

, where

represents the heat capacity of dry air, c

the heat

capacity of water vapor in the air and h

the evapora-

tion enthalpy. Equation (4) is a multilinear function of

temperature ϑ and humidity µ due to the second term

of the equation, whereas the ﬁrst and third terms are

linear.

3.2 Energy and Humidity Transfer of a

Wet Air Stream

To describe the energy absorbed by a mass ﬂow q

of air, the ﬁrst law of thermodynamics for stationary

ﬂowing ﬂuids can be applied. To calculate the change

in enthalpy h

and humidity µ

in a given reference

volume V , the following differential equations can be

obtained

ρV

= q

− h

out

), (5)

ρV ˙µ

= q

(µ

− µ

out

), (6)

with ρ as density of the air, h

as enthalpy of the en-

tering and h

out

as enthalpy of the exiting air.

The assumption is made that the air is well-mixed in-

side the reference volume and the air states are the

same throughout the volume (principle of concen-

trated masses and lumped parameters). This assump-

tion holds for small distances/volumes and high ve-

locities. This means that the exiting air states are

equal to the air states of the reference volume and the

equations (5) and (6) can be rewritten as

ρV

= q

− h

), (7)

ρV ˙µ

= q

(µ

− µ

). (8)

For higher resolution or larger systems, multiple in-

terconnected volumes (cells) can be used, as in CFD

or FEM calculations.

To calculate the temperature ϑ

of the reference vol-

ume, the additional algebraic equation

= c

+ c

+ h

(9)

is needed. Eq. (4) can be used for calculating the

enthalpy h

of the entering air stream from its tem-

perature ϑ

and humidity µ

. Substituting (4) and

(9) into (7) results in

ρV

+ h

+ c

nµ

− q

(10)

The differential equations (8) and (10) can

be transformed into an implicit state-space model

with the states x =





, inputs u =





and output y = ϑ

as follows

0 =ρV ˙x

+ u

− c

− h

− c

(11)

0 =ρV ˙x

+ u

− u

(12)

Implicit Multilinear Modeling of Air Conditioning Systems

441

with the additional algebraic output equation

0 = x

− c

y − c

− h

. (13)

The structure matrix F of the iMTI model is

F =







0 0 0 0 0 0 0 0 0 0 1

0 0 0 0 0 0 0 0 0 1 0

0 0 0 0 0 0 0 1 1 0 0

0 0 0 0 1 1 1 0 0 0 0

0 1 1 1 0 0 1 0 1 0 0

0 0 1 1 0 0 0 0 0 0 0

0 1 0 1 0 0 0 0 0 0 0

1 0 0 0 0 1 0 0 0 0 0







(14)

The parameter matrix Φ is







0 0 −c

−1 −h

0 −c

0 0 −c

0 0 −h

1 0 0

0 0 1

0 1 0

ρV 0 0

0 ρV 0







. (15)

In the upcoming chapter, an iMTI model of an ac-

tual air conditioning system is developed to further

investigate the practical applicability of this modeling

approach.

4 EXAMPLE: CLIMATIZATION

HUTCH

A precision climatization hutch for magnetic ﬁeld

measurements of magnets needed in particle ac-

celerators will act as example for this study. The

temperature in the climatization hutch needs to main-

tain at 21±0.1

◦

C, to prevent warping of the magnets

due to thermal expansion during the measurements.

Therefore, the hutch is cooled by a recirculating air

conditioning system, which is operated with a con-

stant air ﬂow and a high air exchange rate of approx.

/h (which means that the air volume is completely

replaced 20 times per hour). The system is shown

in the Figure 1. The hutch itself is located inside an

industrial hall. The air conditioning system consist

of a duct system, a water-air heat exchanger with

a mixing-valve operated with cooling water and a

fan. The mixing valve is controlled by a PI-controller.

Figure 1: Climatization hutch with a recirculating air con-

ditioning system (in grey).

4.1 Component-Based Modeling

The model consists of interconnected sub-models, in-

cluding the room, air duct, dry water-air heat ex-

changer, mixing valve, walls, ﬂoor and PI controller.

In the following sections each component will be de-

scribed by thermodynamic equations.

4.1.1 Room

The thermodynamic behavior of the room can be de-

scribed by the change

room

of energy content of the

room air. Often climatization zones are described by

multiple cells, e.g. a 3-cell model with an air-supply,

working and air-return cell, as shown in (Yao and Yu,

2018). To reduce the number of equations in this pa-

per, a simpler one-cell model is represented. As be-

fore such a cell is considered well-mixed.

The example includes the energy exchange Q

vent

due

to ventilation, energy losses Q

wall

towards the walls,

energy losses Q

f loor

towards the ﬂoor and internal

heat sources Q

int

(like the magnet and the magnet

power supplies). The heat losses from the fan mo-

tor are part of the internal heat sources, since it is a

closed system.

The change of energy content of the room can be fur-

ther described by the mass m

times the change

enthalpy of the room air. The dynamic energy conser-

vation law results in

= Q

vent

+ Q

wall

+ Q

f loor

+ Q

int

. (16)

The water content of the air is modeled by the dy-

namic mass conservation law, also called the conti-

nuity or transport equation. The change ˙µ

in abso-

lute humidity (mass of water per mass of air) can be

calculated with the entering and exiting water vapor

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

442

mass ﬂows. For simpliﬁcation no internal gain rates

of water vapor are modeled, only the exchanged vapor

mass ﬂow q

m,µ,vent

due to ventilation is considered.

The transport equation for the water vapor results in

˙µ

= q

m,µ,vent

. (17)

Eq. (4) can be used to calculate the temperature ϑ

implicitly from the enthalpy h

and humidity µ

of the

room air

= c

+ c

+ h

. (18)

4.1.2 Air Duct

The air duct system supplies and removes air from

the room. The energy rate Q

vent

between the supplied

air by the AC-unit and the room air is modeled using

the ﬁrst law of thermodynamics for stationary ﬂowing

ﬂuids

vent

m,air

(ϑ

sup

− ϑ

) + c

(µ

sup

− µ

)

+ h

(µ

sup

− µ

)),

(19)

where q

m,air

is the mass ﬂow of the exchanged air,

sup

the temperature of the supplied air and µ

sup

the

absolute humidity of the supplied air.

The exchange rate q

m,µ,vent

of water vapor due to ven-

tilation is calculated by

m,µ,vent

= q

m,air

(µ

sup

− µ

). (20)

4.1.3 Dry Water-Air Heat Exchanger

A heat exchanger can be modeled by the energy con-

tent of the 2 liquids inside the exchanger and the en-

ergy storage of the exchange material itself. Over-

all the model has 3 energy storages. The concept of

concentrated masses and lumped parameters is used

and each storage is represented by one cell with one

according mass. It is also possible to use multi-cell

approaches for the different materials.

In the climatization hutch, the heat exchanger is

used for cooling the air, but it is only operated with

temperatures above 14°C, so the heat exchanger al-

ways operates in dry operation mode and no conden-

sation appears.

The energy content of the air is modeled based on

its average enthalpy h

a,av

and air mass m

inside the

heat exchanger. The airﬂow q

m,air

is similar in the

whole system. The incoming air has the states of the

room air (temperature ϑ

and humidity µ

) since it is

directly extracted from the room. Duct heat losses are

neglected. Since the heat exchanger is operated in a

dry mode, the exiting humidity µ

sup

equals the incom-

ing humidity µ

. The average enthalpy h

a,av

is calcu-

lated by the mean value of the entering and exiting

enthalpies h

and h

sup

a,av

+ h

sup

. (21)

The enthalpy h

can be calculated with (18). The

supply temperature ϑ

sup

can be calculated implicitly

from the enthalpy h

sup

and humidity µ

sup

= c

sup

+ c

sup

+ h

. (22)

The change m

a,av

in energy content of the air in-

cludes heat dissipated from the airﬂow and heat trans-

fer to the heat exchange material. The power balance

for the air results in

a,av

m,air

− h

sup

)

+ k

(ϑ

ex,w

− ϑ

a,av

(23)

where k

is the heat transfer coefﬁcient of the air-side,

the heat transfer surface of the air-side, η

the heat

exchange efﬁciency and ϑ

ex,w

the surface temperature

of the heat exchanger on the water-side.

The change in energy content of the heat exchange

material is modeled by the change

ex,w

in tempera-

ture of the material on the water-side, the mass m

the material and its heat capacity c

. The change in

energy content includes the heat transfer between the

water and heat exchange material, as well as the heat

transfer between the air and heat exchange material

ex,w

(ϑ

w,av

− ϑ

ex,w

)

+ k

(ϑ

a,av

− ϑ

ex,w

(24)

with the heat transfer coefﬁcient k

of the water-side,

the heat transfer surface A

of the water-side and the

average temperature ϑ

w,av

of the water inside the heat

exchanger.

The energy content of the water is modeled by its

mass m

, its heat capacity c

and its average tempera-

ture ϑ

w,av

, which is calculated by the mean of the inlet

and outlet temperatures ϑ

w,in

and ϑ

w,out

w,av

w,in

+ ϑ

w,out

. (25)

As for the air, the change in energy content of the

water is similar to the heat absorbed by the water ﬂow

m,w

and the heat transfer from the heat exchange ma-

terial

w,av

m,w

(ϑ

w,in

− ϑ

w,out

)

+ k

(ϑ

ex,w

− ϑ

w,av

(26)

4.1.4 Mixing Valve

A valve inside the water circuit is used to control the

output of the heat exchanger. For precision cooling

usually a mixing circuit with a mixing valve is used.

Implicit Multilinear Modeling of Air Conditioning Systems

443

The 3-way mixing valve is positioned at the inlet wa-

ter stream, as it can be seen in Figure 1. With the

mixing valve, the hotter outlet water can be mixed in

the cold feed water, raising the inlet water tempera-

ture of the heat exchanger. With an additional pump

the circuit is operated with a constant water ﬂow q

m,w

so unnecessary oscillation can be avoided. To avoid

damaging pressure spikes the actuator response time

valve

is high. The change ˙x

valve

of the valve posi-

tion is modeled as an integrator over the difference

between the valve position x

valve

and control signal

controller

divided by the response time T

valve

˙x

valve

controller

− x

valve

. (27)

The water inlet temperature ϑ

w,in

of the heat ex-

changer is calculated using the mixing equation

w,in

= x

valve

w, f eed

+ (1 − x

valve

)ϑ

w,out

, (28)

with ϑ

w, f eed

as the feed temperature of the primary

cooling water supply and ϑ

w,out

as the water outlet

temperature of the heat exchanger.

4.1.5 Wall and Floor

The walls and ﬂoor act as additional thermal energy

storages and transfer heat with the room and environ-

ment. The change in energy content of these struc-

tures is modeled by the change

wall,i

of their tem-

peratures times their masses m

wall,i

and heat capacity

wall,i

. The transferred heat is linearly depending on

the temperature differences between the wall and ad-

jacent zones. The power balance results in

wall,i

in,i

wall,i

(ϑ

− ϑ

wall,i

)

+ k

out,i

wall,i

(ϑ

amb,i

− ϑ

wall,i

(29)

with the inner heat transfer coefﬁcient k

in,i

, the heat

transfer surface A

wall,i

, the outer heat transfer coefﬁ-

cient k

out,i

and the ambient temperature ϑ

amb,i

, which

is the hall temperature in case of the walls and the

ground temperature in case of the ﬂoor.

The heat Q

wall,i

transferred between the room and the

wall/ﬂoor can be calculated by

wall,i

= k

in,i

wall,i

(ϑ

wall,i

− ϑ

). (30)

In the example, the walls have been modeled as one

wall, with their surfaces added to each other. The

ﬂoor is modeled separately.

4.1.6 PI Controller

A PI controller is implemented in the model to main-

tain the desired room temperature. The controller sig-

nal u

controller

consists of a proportional and an inte-

grating part. The proportional part is calculated via a

gain of K

on the control error between the set tem-

perature ϑ

set

and the room temperature ϑ

. The inte-

grating part is calculated via a gain of K

on the inte-

grated control error. The equations of the PI controller

result in

˙x

=ϑ

set

− ϑ

, (31)

controller

(ϑ

set

− ϑ

) + K

. (32)

These equations describe the behavior of the com-

ponents of the precision climatization hutch and can

be used for simulation and analysis.

4.2 Overall Model

The overall model is built from the implicit state-

space models of the single components. In the fol-

lowing those state-space models and the connection

equations will be shown.

Since no condensation or humidity sources are

modeled, the humidity is constant and could be

treated as a parameter for the model. For demon-

stration purposes, it remains as a variable, it is ex-

pected that in further research those effects can be

represented with iMTI models (see chapter 7).

The state-space model of the room has the states









, inputs



... u





vent

wall

f loor

int

m,µ,vent



and out-

puts









. The equations of the

state-space model are

0 = m

˙x

− u

0 = m

˙x

− u5,

0 = c

− x

+ h

+ c

0 = x

− y

(33)

The air duct state-space model has the inputs



... u





m,air

sup



and outputs







vent

m,µ,vent



. The

equations of the state-space model are

0 =c

− c

− y

− h

+ h

− c

+ c

0 =u

− u

− y

(34)

The duct model is connected to the room model

via the connection equations

0 = u

− y

0 = u

− y

0 = u

− y

0 = u

− y

0 = u

− y

(35)

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

444

The state-space model of the heat ex-

changer has the states







a,av

ex,w

w,av



, inputs



... u





m,air

m,w

w,in



, outputs







sup

w,out



and auxiliary z

= ϑ

a,av

The equations of the state-space model are

0 =2c

− 2u

− m

˙x

+ 2h

+ 2c

+ A

− A

0 =A

− x

+ A

)

− c

˙x

+ A

0 =A

− A

− c

˙x

+ 2c

− 2c

0 =h

− x

+ c

0 =2x

− c

− 2h

− c

0 =2x

− u

− y

(36)

The heat exchanger model is connected to the rest of

the model with

0 = u

− y

0 = u

− y

0 = u

− y

(37)

The state-space model of the walls has the state x

wall

, the inputs







amb



and out-

put y

= Q

wall

. The equations of the state-space

model are

0 =A

wall

wall,in

− A

wall

wall,in

+ k

wall,out

+ A

wall

wall,out

− c

wall

˙x

0 =A

wall

wall,in

− A

wall

wall,in

− y

(38)

The state-space model of the ﬂoor has the state x

f loor

, inputs







ground



and out-

put y

= Q

f loor

. The equations of the state-space

model are

0 =A

f loor

f loor,in

− A

f loor

f loor,in

+ k

f loor,out

+ A

f loor

f loor,out

− c

f loor

˙x

0 =A

f loor

f loor,in

− A

f loor

f loor,in

− y

(39)

The wall and ﬂoor model is connected to the rest with

0 = u

− y

0 = u

− y

0 = u

− y

0 = u

− y

(40)

The state-space model of the mixing valve has

the state x

= x

valve

, inputs







controller

w, f eed

w,out



and output y

= ϑ

w,in

The equations of the state-space model are

0 =u

− x

− T

valve

˙x

0 =u

− y

+ u

− u

(41)

The valve model is connected to the system model

with

0 = u

− y

0 = u

− y

(42)

The controller state-space model has the state x

= x

inputs







set



and output y

controller

. The equations of the state-space model are

0 =u

− u

− ˙x

0 =K

− y

− K

+ K

(43)

The controller model is connected to the system

model with

0 = u

− y

0 = u

− y

(44)

The resulting input and disturbance signals of the

overall model are

open

= [

]

= [

int

m,air

m,w

amb

ground

w, f eed

set

]

(45)

The redundant inputs u

and u

have been replaced

by one input. The linear connection equations have

been reduced by exchanging the variables y

and u

with one similar auxiliary variable z

. In case of the

variables ϑ

and µ

, they have been replaced by simi-

lar output variables.

The overall model has 9 states, 7 inputs, 2 out-

puts and 9 auxiliaries. The structure matrix F can be

stored as sparse boolean matrix and has the dimen-

sions R

36×47

with 63 elements of one. The parameter

matrix Φ can be stored as a sparse matrix with dimen-

sions R

20×47

with 76 nonzero elements. The values

for the model parameters are shown in the appendix.

This model construction process demonstrates

that it is possible to directly represent a thermody-

namic modeling approach of an HVAC system in an

iMTI model. In the next chapter, simulation results

are provided to showcase a practical application of the

model.

5 SIMULATION

For simulation, the system of equations (3) is solved

using the ode15i-solver from Matlab version 2022b.

Implicit Multilinear Modeling of Air Conditioning Systems

445

Therefore initial values of the states and input trends

are needed. The simulation represents the operation

of 4 hours of an imaginary summer day in Germany.

5.1 Initial Conditions

For this example, the initial states are set as follows

43488 0.008 43488 23 23

23 17 0 0

]

(46)

This refers to initial temperatures of 23

◦

C for the

room and walls, as well as the air and water inside

the heat exchanger and the heat exchange material it-

self. The initial ﬂoor temperature is set to 17

◦

C, since

it is directly exposed to the ground. The initial room

humidity is set to 8 × 10

−3

water

/kg

air

. The mixing

valve is completely closed and the initial controller

signal is set to zero.

With the states and ﬁrst input signals initial

guesses for the state derivatives and outputs can be

calculated with Eq. (3). The numerical solver

vpasolve for symbolic equations is used.

Figure 2: Input temperatures, mass ﬂows and heatload.

5.2 Input Signals

The trends of the input signals are shown in Figure

2. The ﬁgure displays the temperature input signals

amb

, ϑ

ground

, ϑ

w, f eed

and ϑ

set

, which are nearly held

constant for better traceability of the model behavior,

as well as the mass ﬂows q

m,air

and q

m,w

of the air and

water. Also the inner heatload Q

int

of the climatiza-

tion hutch is shown and represented as an rectangular

function (in kW ). This could represent when a magnet

is powered and measurements of the magnetic ﬁeld

are conducted.

5.3 Results

Figure 3 presents the computed temperature trends,

including the states: ϑ

wall

, ϑ

w,av

, ϑ

ex,w

, and the output

and auxiliary variables: ϑ

, ϑ

sup

, ϑ

w,in

, ϑ

w,out

. Figure

4 shows the controller signal u

controller

and the valve

position x

valve

. Based on experience with precision

climatization, the results appear reasonable.

Notably, the responses to the up and down step

of the inner heatload differ qualitatively from each

other, particularly in the trend of the valve opening

valve

. This difference could be attributed to the non-

linearities of the mixing valve’s sub-model, as other

variables in multilinear terms are held constant.

Further research into other air conditioning sys-

tems, especially those using variable airﬂow and sys-

tems with humidiﬁcation and dehumidiﬁcation re-

quirements, may reveal more of such behavior.

Figure 3: Simulated temperature signals.

Figure 4: Simulated control signal and valve opening.

SIMULTECH 2023 - 13th International Conference on Simulation and Modeling Methodologies, Technologies and Applications

446

6 CONCLUSION

Implicit MTI models prove to be a suitable exten-

sion of the eMTI class for modeling complex HVAC-

systems. These models can directly represent most

physical laws and equations of HVAC systems, and

offer the advantage of being a closed model class,

simplifying component-based modeling and the use

of standardized methods. They can be utilized in

model-based applications, like nonlinear model pre-

dictive control to optimize competing goals, like en-

ergy consumption, comfort and technical compli-

ances.

However, there are some limitations of the appli-

cability of the iMTI class in HVAC systems. Non-

integer exponents can’t be represented by the iMTI

formula, which may arise from empirical analysis for

calculating convection heat transfer coefﬁcients or dy-

namic ﬂow resistances.

7 OUTLOOK

To account for the cooling of air below the saturation

temperature ϑ

sat

, which would involve the air cooler

operating in a wet mode where condensation occurs

and the air dehumidiﬁes, additional binary signals are

required. The resulting system would be a hybrid sys-

tem with both binary and continuous signals, as well

as additional inequality constraints. Future research

will focus on investigating how binary variables can

be implemented and simulated within the iMTI class.

Furthermore, the overall system can be potentially

reduced further by redundant equations, variables and

unnecessary states. This reduces storage capacity and

computation time. Future research will explore auto-

mated methods for reduction of iMTI models.

ACKNOWLEDGMENTS

T.W. (DESY) acknowledges funding of the technical

design phase of Petra IV granted by the Beh

orde f

Wissenschaft, Forschung, Gleichstellung und Bezirke

(BWFGB) of the Freie und Hansestadt Hamburg

under the contract BWFG/F|97236, as well as by

the Bundesministerium f

ur Bildung und Forschung

(BMBF) under the contract DES21TDR.

REFERENCES

ores, N., Kaufmann, C., Schnelle, L., Y

nez, C., Pangalos,

G., and Lichtenberg, G. (2022). Reduced cp represen-

tation of multilinear models. Proceedings of the 12th

International Conference on Simulation and Modeling

Methodologies, Technologies and Applications.

Kruppa, K. (2017). Comparison of tensor decompo-

sition methods for simulation of multilinear time-

invariant systems with the mti toolbox * *this work

was partly supported by the project observe of the

federal ministry for economic affairs and energy ger-

many (grant-no.: 03et1225b). IFAC-PapersOnLine,

50(1):5610–5615.

Lichtenberg, G., Pangalos, G., Cateriano Y

nez, C., Luxa,

A., J

ores, N., Schnelle, L., and Kaufmann, C. (2022).

Implicit multilinear modeling. at - Automatisierung-

stechnik, 70(1):13–30.

Pangalos, G., Eichler, A., and Lichtenberg, G. (2013). Ten-

sor systems - multilinear modeling and applications.

Proceedings of the 3rd International Conference on

Simulation and Modeling Methodologies, Technolo-

gies and Applications.

Pangalos, G., Eichler, A., and Lichtenberg, G. (2014). Hy-

brid multilinear modeling and applications. Advances

in Intelligent Systems and Computing, page 71–85.

Yao, Y. and Yu, Y. (2018). Modeling and control in air-

conditioning systems. Springer Berlin.

APPENDIX

Parameters

f loor

20.8m

102.3m

5.22m

wall

240.8m

1006

/kgK 477

/kgK

f loor

900

/kgK 1860

/kgK 4200

/kgK

wall

650

/kgK 2.501 × 10

/kg 600

f loor,in

f loor,out

K 2

K −2 × 10

−4

/Ks

wall,in

−0.1

/K 1000

K 3

wall,out

K 0.4341kg 42.5kg

f loor

70000kg 418.8kg 27.75kg

wall

valve

770.4kg 120s 0.85

Implicit Multilinear Modeling of Air Conditioning Systems

447