ECAI 2015 – International Conf erence 7th Edition [600109]

ECAI 2015 – International Conf erence – 7th Edition
Electronics, Computers and Artificial Intelligence
25 June -27 June, 2015, Bucharest, ROMÂNIA
Buildings modeling in order to implement
optimal temperature control

Giorgian Neculoiu, Valentin Dache, Grigore Stamatescu, Valentin Sgarciu
Faculty of Automatic Control and Computers
Politehnica University of Bucharest
Bucharest, Romania
[anonimizat] , [anonimizat] , [anonimizat] , [anonimizat]

Abstract – When there is the desire to achieve optimal
temperature control for a building or its energetic
performance description, one of the steps that should be followed is to identify experimentally the dynamic model for heat transfer in that building. In these models there
are used heat equations in time domain, but as these
systems have multiple inputs and outputs, these equations can be represented in matrix form. In this manner, we can easily obtain the state-space
representation and subsequently the transfer function
representation. Furthermore, the model parameters can be determined easily. It will be used as an example for a detailed study, a normal size house for which a number
of inside and outside parameters are monitored.
Keywords – modeling; parameters identification;
optimal control; reduced order model; transfer function.
I. INTRODUCTION
Due to the growing demands of life comfort, lack
of natural resources and global climate change, an international pressure is generated on reducing energy consumption in buildings. The construction works for residential buildings should be made in accordance with thermal settlements. In case of an old building the rehabilitation or restoration should be done to improve thermal characteristics.
Currently, in the EU member states a sustainable
development policy is being established. On this line, the European authorities have adopted a strategic plan, "3×20 objectives, which includes several objectives: by 2020 the balance of renewable energy will have to represent 20% of the energy total, EU countries should target a 20% reduction in CO2 emissions and the increase of energy efficiency should reach the 20% threshold. Achieving these goals is very important because, in Europe, 40% of the consumed energy and 35% of greenhouse gases emission is due to buildings. Thus, there is a growing need for society to evaluate and quantify the thermal properties of buildings in order to reduce energy consumption and develop standard methods for their description and control.
The main purpose of this paper is to develop a
dynamic model of an experimental building that is used to implement a control strategy based on Model Predictive Control (MPC). Achieving thermal comfort using less energy is
one of the main performances to be attained in
building thermal comfort. In order to achieve this,
several advanced technologies can be applied, the MPC being one of the newest and most advanced of all. Within MPC, the weather forecast and occupation schedule of the building are taken into account for optimal control. To apply such a control strategy, initially it is necessary to achieve a dynamic model of the building that should describe closely its behavior. Such a model is usually obtained in two steps: determining the low-order model structure based on physical knowledge of the building and identifying physical parameters of the model using the least squares identification method [1].
In fact, there are many inputs that influence the
system a lot. Building a model that can better characterize the system it is necessary to choose the correct inputs and outputs which take into account the physical causality of the system. This model can be used later in the control process. The inputs and outputs are dependent on one another, and once the previous inputs vary, they will influence the evolution of the future system output [2].
II. E
XPERIMENT CONFIGURATION AND
EXPERIMENTAL DATA
Measured data used in this paper come from
measurements performed on a test house with an area of 100 m
2, located in southern Germany – Holzkirchen
– Fig. 1. These data correspond from 09.04.2014 to 28.04.2014, with a sampling period of 60 minutes – Fig. 2.

Figure 1. The plan of the reference building

Giorgian Neculoiu, Valentin Dache, Grigore Stamatescu, Valentin Sgarciu
2

Figure 2. Measured data (outdoor temperature, supply air
temperature, solar irradiance, heat flux densities)
The house was tested under the action of external
weather conditions. These data were also measured (Table I). The inside temperature is controlled and maintained at a constant value throughout the test period [3]. The block diagram of inputs and outputs for the described system is shown in Fig. 3.
III. M
ODELING OF A BUILDING
To obtain a reduced order model, the minimal
assumptions should be considered: building properties are uniformly distributed, model parameters are constant in time, etc. These assumptions lead to model simplification. The models based on physical knowledge are represented in state-space using differential equations. These equations are actually a representation of systems using differential equations for vector functions by which a model can be represented [4].
To obtain a particular model that can solve the
modeling problem, heat transfer can be represented in graphical form. The most common method is represented by heating networks. For buildings, especially when we want to control internal parameters, low-order models are usually obtained through linear network representations with focus parameters. Thus, an analogy between two different physical domains is created through the same mathematical representations. In case of a building, the model construction is represented by a linear electric circuit, and the state-space equation results from the resolution of this circuit [5-10].
In this case, the voltage source refers to
temperatures, the current source refers to the heat flux, the electrical resistance refers to heat transmission resistance and the electrical capacity replaces the thermal capacity of the model. The correspondent to a building model can be obtained by merging some circuits for walls, windows, ventilation, internal mass, etc.
The interior walls of buildings with a single area
are considered to be part of the internal mass and the exterior walls form the envelope of it.

Figure 3. The block diagram of system input/output
The most used networks for a building envelope
are 2R-C [11] or 3R-2C [12] representations. Ventilation, infiltration and building windows do not represent thermal energy accumulators and can be represented in the model as simple resistances.
The home presented in the previous chapter was
built based on an approximate model for a single-zone building whose equivalent circuit is shown in Fig. 4. The building envelope is represented through a 2R-C network. Envelope, windows, ventilation, infiltration and internal thermal mass are considered the passive components of the building. The stored thermal capacity is represented by C
w and the wall insulation is
represented by conductive resistances R p1 and R p2. The
thermal resistance that is found at the boundary between the envelope and inside air (or outside air) is represented by convective resistances R
si or R se. The
heating capacity of internal mass is represented by C z,
and the building ventilation and infiltration by the R v
resistance. Outside temperature, ventilated air temperature, solar radiation and internal heat flux are considered active components of the building.
The indoor temperature,
zθ, is the output of the
system. In the case presented above, zθ is influenced
by four inputs: outdoor temperature, solar radiation,
internal heat flux and ventilated air temperature.
The internal heat flux is a controllable quantity and
it is usually the system command. Outdoor temperature, solar radiation and ventilated air temperature are uncontrollable sources, but they are measurable.
TABLE I. MEASURED DATA IN THE EXPERIMENTAL BUILDING
Measured
variable Symbol Units Range of
measurements
Inputs
Outdoor temperature
epT [°C] [-4,93 to 20,41]
Supply air
temperature evT [°C] [11,91 to 21,23]
Solar radiation sQ [W] [0,00 to 892,71]
Internal heat
flux elpQ [W/m2] [3,43 to 1952,26]
Output
Indoor
temperature zθ [°C] [21,00 to 32,61]

Buildings modeling in order to implement optimal temperature control
3

Figure 4. Equivalent electrical network representation of a low-order thermal model of a building
To obtain state-space equations for the circuit
shown in Fig. 4, the superposition theorem must be applied through which four single-input single-output (SISO) models are extracted. Each of them corresponds to the input / output peer. Finally, these four models can be represented as a single multi-input single-output (MISO) model, as shown in Fig. 3.
A. State-space representation of the thermal model
The environment consists of an infinite set of
nodes, each of them having its own temperature. The nodes are connected by branches, each branch having its own thermal resistance. Starting from the heat balance method, the building model can be written as an infinite set of differential equations [2]:

f GbA GA CT+ + = θ θT-A . (1)
For the thermal circuit presented above, A is the
incidence matrix of the network that represents the heating circuit (the connection between the network nodes), is the transpose of the matrix A and the model parameters are represented by G – the diagonal matrix of thermal conductance and C – the diagonal matrix of thermal capacities:
⎥⎥⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢⎢⎢
⎣⎡
−−−=
011 01 01 01001000 1010 0
A
⎥⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢⎢
⎣⎡
=pz
CCC
0000 000 0000 000
⎥⎥⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢⎢⎢
⎣⎡
=
−−−−−
11
21
111
0 0 0 00 0 0 00 0 0 00 0 0 00 0 0 0
sippsev
RRRRR
G
In equation (1), the unknown variables are those
from the node temperatures vector,
[]T
p z si se θ θ θ θθ= , and the model inputs are those
from the input vector []Tfbu= , where
[ ]000ep evT Tb= represents the source
temperature vector on branches and
[ ]0 0elp s Q Qf= the heat sources vector of the
nodes. The equation (1) can be written as:
fbK K Cb+ + =θ θ . (2)
, where the following notations were used:
GAA KT−= and GA KT
b= . For the thermal circuit
presented in the paper, the values of K and bK are:
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎡
+−+−+−+−
=2 11 2
2 12 221 11
01 10101 1010 0
p pp p
p psivv si
sip si si pp sip p sese p
RRR R
R RRRR R
RR R RRR RR RRR R
K

Giorgian Neculoiu, Valentin Dache, Grigore Stamatescu, Valentin Sgarciu
4
⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢
⎣⎡
−−
=
01 10 010 0 011 10 0 00 01 10
2 121
p psi vsi pp se
b
R RR RR RR R
K
From the matrix form of K and bK the values of
11K, 12K, 21K, 22K, respectively 1bK, 2bK can be
easily deduced. As it can be seen, in the diagonal
matrix of thermal capacity, C, some elements are zero. This is way, (2) can be represented as a system of differential algebraic equations, which means the algebraic equations are those corresponding to zero values of C and the differential equations correspond to non-zero values of C. By a simple rearrangement, keeping the methodology presented in [2], with the
aim of separating the algebraic differential equations
(2) can be written as:
⎥⎦⎤
⎢⎣⎡
⎥⎦⎤
⎢⎣⎡+⎥⎦⎤
⎢⎣⎡+⎥⎦⎤
⎢⎣⎡
⎥⎦⎤
⎢⎣⎡=⎥
⎦⎤
⎢
⎣⎡
⎥⎦⎤
⎢⎣⎡
C bb
C C C ff
IIbKK
K KK K
C0
2211
21 0
22 2112 11 0
00
000
θθ
θθ
 (3)
, where CC is the diagonal matrix of non-negligible
thermal capacity.
B. Differential equations:
If the algebraic equations are eliminated from the
(3) representation, the state-space representation can be obtained in the following form:
uI KK K KKK CK KKK C
b b CC C C
] [ ) (
221
11 21 2 11
11 21122 121
11 211
− − −− −
− + − ++ + − = θ θ. (4)
, which can be write like this:
uB AS CS C + =θ θ . (5)
By comparing (4) and (5) it can be inferred that the
relationship between states matrix, SA, and (4) is:
) (22 121
11 211K KKK C AC S + − =− −. (6)
, which can be written for the studied house in the
following form:

⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢
⎣⎡
+ ++ + +−++ ++ +−
=
) )( ( ) (1) (1
) (
2 11 2
22 22
si p se p psi se p p
p si pp si z p si vzp v si
S
R R R RCR R R R
R RCR RC R RRCR R R
A.(7)
, and the relationship between input matrix,SB, and
(4) is: ] [221
11 21 2 11
11 211I KK K KKK C Bb b C s− − −− + − = . (8)
, which for the presented house, it can be written as:

⎥⎥⎥⎥
⎦⎤
⎢⎢⎢⎢
⎣⎡
+ +=
0) ( ) (1010 01
1 1 se p pse
se p pz vz
s
R RCR
R RCC RCB.(9)
C. Algebraic equations:
Like the previous method of (3), a set of algebraic
equations can be obtained to complete the state-space model:

[]
⎟⎟⎟
⎠⎞
⎜⎜⎜
⎝⎛
⎥⎥⎥
⎦⎤
⎢⎢⎢
⎣⎡
−=−
c0 11 b1 121
11 0
ffb
0 I K CK K θ θ. (10)
, which can be written as:
uD CS CS+ =θ θ0. (11)
The relationship between matrix outputs SC
presented in the equation (11) and the equations (10)
has been obtained by comparing those two as following:

121
11KK CS−−= . (12)
, and the relationship between SD matrix and (10) is:
[] 0 I K 11 b11
11−−=K DS. (13)
Considering that the system output is zθ, and
knowing the heating circuit presented in this paper has
a non-negligible thermal capacity, in order to achieve
the model of the presented house, the zθ output value
will be obtained from the matrix sC and vector cθ,
and the of the sD matrix will be zero for the studied
case in this paper:

]0000[]01[
==
SS
DC. (14)
IV. TRANSFER FUNCTION REPRESENTATION
For a system as the one presented in this paper, the
relationship between inputs and outputs can be represented as a set of transfer functions. The corresponding state-space model of the thermal circuit presented in this paper is:

Buildings modeling in order to implement optimal temperature control
5

uD CuB A
S CS zS CS C
+ =+ =
θ θθ θ. (15)
To obtain the transfer function, the Laplace
transform must be applied in zero initial conditions, and (15) can be written as:

⎩⎨⎧+ =+ =
uD CuB A s
S CS zS CS C
θ θθ θ. (16)
From the first equation of the equation system (16)
we can obtain the following relationship:
uB A sIS S C1) (−− =θ . (17)
, and using (17) and the second equation of the (16)
equation system, the output of our system can be deduced as:

uD B AsICS S S S z ) ) ((1+ − =−θ . (18)
, where the system transfer matrix is:
S S S S S D B AsIC H + − =−1) ( . (19)
In this transfer matrix each element is a transfer
function representing the system output for a given input vector entries [13]. For the heating circuit presented in this paper, the circuit includes four inputs, so four transfer functions are obtained. Finally, the building is represented by the superposition of the four transfer functions and thus it is shown how each input can affect output system (internal temperature).
V. I
DENTIFYING THE PARAMETERS OF THE MODEL
Disturbances are a very important feature when
you want to achieve control for a real building. Thus, identifying the parameters of the building is a very important issue. In order to realize this, several methods can be chosen, including iterative min-search or Least Squares Methods [1]. Identification method of least squares is used successfully to identify the parameters of the discrete transfer function of the
system, so this method was used in this paper. For the
inferior building model presented in this paper,
1m,
2m, 11n…42n parameters will be identified. The
representation below is obtained by discretizing the
continuous time transfer function of the building model and offers a linear formulation of the identification problem which ensure finding the optimal solution.
The parameters n
11 and n 12 correspond to the
system output at the interaction with air temperature variation through the ventilation system, the parameter n
22 correspond to the system output at the interaction
with the outside air temperature variation, the parameter n
32 correspond to the system output at the
interaction with varying solar flux incident on the building envelope and the parameters n 41 and n 42
corresponding output system interacting with internal heat flux variation.

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝⎛
+ +++ ++ ++ ++
=
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝⎛
=− −− −− −−− −−− −− −
−−−−−−−−
−
2
21
12
421
412
21
12
322
21
12
222
21
12
121
11
11111111
1
1111
)()()()()()()()(
)(
zm zmzn znzm zmznzm zmznzm zmzn zn
z QzzQzzTzzTz
zH
elpzszepzevz
θθθθ
. (20)

After obtaining them, it is good to know that you
have to accept that between model and measured data there is always a difference because low-order model is only an approximation of the building thermal behavior .Thus, there is some error between the measured data and the model response. The obtained error of the model response and the measured data is 8.07% and can be considered as white noise introduced by the variation of the system inputs. The system response can be also seen graphically in Fig. 5.
The regulation of the measured data from
28.04.2014 to 09.04.2014 and the response of the low- order model with the identified parameters can be calculated using the formula:

%100*
))( () ˆ(
1
2 22 2
⎟⎟⎟
⎠⎞
⎜⎜⎜
⎝⎛
−−
−=
∑∑
y mean yy y
fit
ii i . (20)

, factor that can be used to measure the quality model.
For the model presented, fitting factor is 79.38%. As higher this factor is, the best the model.
For the approximation of this work, the fitting factor
does not have such a good value, but it can be considered satisfactory for a low – order model.

Figure 5. Comparison between the measured data and the
response of the model

Giorgian Neculoiu, Valentin Dache, Grigore Stamatescu, Valentin Sgarciu
6
CONCLUSINOS
The purpose of this paper is to present a way to get
a low-order model for the studied building. Such a model allows the building thermal behavior and can be useful for temperature regulators based on models such as MPC.
The building was seen as a model with a single
thermal zone for which the state-space representation
has been calculated. In this paper, were have considered four inputs. The model output was defined as a weighted average room temperatures.
However, the identified parameters do not describe
so well the model building, and even if the result is satisfactory, the authors believe that the model can be improved.
In order to improve the results, as prospects for the
future, we want to realize a building model that takes into account two areas of it. It was found that the rooms on the south side of the building can be taken into account and other inputs that bring quite large perturbations in the calculated model.
A
CKNOWLEDGMENT
The work has been funded by the Sectoral
Operational Programme Human Resources Development 2007-2013 of the Ministry of European Funds through the Financial Agreement POSDRU/159/1.5/S/132397.
The experimental data used in this paper were
obtained by INSA Lyon, France, from Fraunhover-Institut fur Bauphysik IBP, Germany. The authors would like to express their gratitude for these data.
We also would like to express our gratitude and
appreciation to Ingo Heusler (Fraunhover-Institut fur Bauphysik IBP, Germany), Ghristian Ghiaus and Iban Naveros (INSA Lyon, France) for their support and advice.
R
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