Prediction of domestic warm-water consumption

Abstract

The paper presents methodologies able to predict dynamic warm water consumption in district heating systems, using time-series analysis. A simulation model according to the day of a week has been chosen for modeling the domestic warm water consumption in a block of flats with 60 apartments. The analysis of the residuals indicates good simulation and prediction models for the cases studied. Double-cross validation was done using data collected by the SCADA system from District Heating Company of Iasi.

Full text

Prediction of Domestic Warm-Water Consumption
ELENA ŞERBAN1, DANIELA POPESCU2
Department of Computer Science, Department of Fluid Mechanics
Technical University Gheorghe Asachi Iasi
Bd. D. Mangeron nr. 53A, Iasi, 700050
1eserban@cs.tuiasi.ro, 2daniela_popescu@tuiasi.ro, http://www.tuiasi.ro
Abstract: - The paper presents methodologies able to predict dynamic warm water consumption in district
heating systems, using time-series analysis. A simulation model according to the day of a week has been
chosen for modeling the domestic warm water consumption in a block of flats with 60 apartments. The
analysis of the residuals indicates good simulation and prediction models for the cases studied. Double-cross
validation was done using data collected by the SCADA system from District Heating Company of Iasi.
Key-Words: - district heating systems, domestic warm water, time series models, autoregressive model,
simulation, prediction.
1 Introduction
Increasing energy efficiency is an important issue
for District Heating Companies [1]. A source of
saving energy is production of thermal energy
according to demand. Very good information about
future consumption are needed in order to do good
production plans. Dynamic simulations enable the
ongoing development of operational optimization
models, but unfortunately, estimation of heat
demands is a complex task. To make full advantages
of the district heating network modeling, the
systems must have measurement points. Nowadays,
SCADA systems were implemented in Romania and
historical data both regarding space heat and
domestic warm water consumption are available. It
is time to use them, not only for monitoring but also
for a better management by doing realistic scenarios
of future consumption.
For a typical district heating system, four load
components shape the total heat-load: space heating
for buildings, domestic warm-water, distribution
loss, additional work day loads [2]. Werner
investigated six Swedish district heating systems
using 5-11 years monitoring data. He found out that
60% of the heat was consumed for space heating,
30% for domestic warm-water preparation, 6-8% by
distribution losses, the rest representing loads that
are dependent on the day of the week [3].
There are computer programs such as CONDOR,
EcNetz, RNET, SYSTEM RORNET, TERMIS,
BoFiT, ANSYS, DH SIM that can make space heat
load prediction [4]. Scientific literature also suggests
interesting models. For instance, Dotzauer [5]
proposes a simple model based on the insight that
the space heat demand is mainly affected by the
outdoor temperature and the social behavior of the
consumers. Artificial neural networks represent an
alternative. The approach used by Ian Beausoleil-
Morrison and Moncef Krarti used a multilayer
feedforward neural network with the back-
propagation learning algorithm [6] and leads to very
good results. Prediction of thermal performance of a
hot water system is presented in [7]. Time-series
analysis is used to model systems and to predict
their behavior in different area (alone or combined
with other methods) [8], [9], [10].
Even if domestic warm water is used within a
house for bath/shower, wash hand basin, dish
washing, clothes washing, studies about the thermal
energy consumed for preparing it are only a few
[11]. The energy used for heating domestic warm
water depends on the human behavior most of all, a
factor that is difficult to control. Usually, prediction
means an average value for the monthly/daily fluid
flow rates per person to be taken into consideration.
It is not enough for a real-time operation of a district
heating system. At least hourly heat load profiles are
needed.
The objective of this work is to develop and
analyze methodologies able to predict dynamic
warm water consumption in district heating systems
(DHS), using time-series analysis. Validation of the
methods was performed by comparing the modeling
results with acquired data via a monitoring system
from the District Heating Company of the city of
Iasi (Romania).
The theoretical bases are presented in section 2,
the simulation algorithm in section 3, section 4
presents the statistical analysis of computational
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results and finally in Section 5 some conclusions are
given.
2 Theoretical basis
In order to predict dynamic warm water
consumption in district heating systems, a
simulation model using experimental data must be
done.
The experimental data are sequence of values,
each value corresponding to a time moment.
Graphical representation of experimental data in
time is a time history representation, and the
sequence of data is a time series.
A time-series can be model as the output of a
system that has as input a white noise signal. With
this observation, the general form of a time-series
model is [13]:
() () ()(
teitecityaty ca n
ii
n
ii+−=−+ ∑∑ == 11
)
(1)
In equation (1) t represents the discretized time
as multiple of sampling interval , ; and
are model parameters (scalars); is the value
of the observed time-series at the moment t and
is the value of the white noise considered as
system input at the moment t. and are the
number of model parameters for considered time-
series and model residuals, respectively.
tΔZt ∈i
a
i
c
()
ty
()
te
a
nc
n
Equation (1) describes an ARMAX type model
(Auto Regressive Moving Average with eXogenous
inputs model).
The model (1) highlights the fact that value of
modeled output at moment t depends on the past
time-history of time-series and of the exogenous
input.
The following notation is made:
()
(
)
tyqty 1
1−
=− (2)
where is considered as a back shift (delay)
operator.
1−
q
With notation (2) the model (1) becomes:
()
()
(
)
()
teqCtyqA 11 −− = (3)
where and are polynomials in the
delay operator and they have the following
forms:
(
1−
qA
) )(
1−
qC
1−
q
(
)
()
nc
nc
na
na
qcqcqcqC
qaqaqaqA
−−−−
−−−−
++++=
++++=
K
K
2
2
1
1
1
2
2
1
1
1
1
1 (4)
where naiai,1, = and ncici,1, = are coefficients of
model polynomials, and
(
1−
qA
)
(
)
1−
qC ,
respectively. They are referred as parameters of the
model. The degrees of the respective polynomials
are and , respectively.
a
nc
n
The variance of the white noise is considered
to be
()
te
σ
.
The order of the model is the pair .
()
ca nn ,
If
(
)
1
1=
−
qC then we have an autoregressive (AR)
model and the model (3) becomes:
(
)
(
)(
tetyqA =
−1
)
(5)
If
θ
is the parameters vector defined in equation
(6):
[
]
T
ncna ccaa KMK 11
=
θ
(6)
and
(
)
t
ϕ
is defined in equation (7):
(
)
(
)
(
)
(
)
(
)
[
]
ca ntetentytyt
−
−
−
−
=
KMK 11
ϕ
(7)
an equivalent form of equation (1) is obtained:
(
)
(
)(
ttty T
εθϕ
+=
)
(8)
Choosing the appropriate order structure for the
ARMA model is an iterative procedure which
involves data analysis, model parameters estimation,
model analysis, selection and validation.
In order to obtain a measure of the model fitness
some quantities are calculated in equations (
The percentage of the output model variation that
is explained by the model is represented by the fit
function described in equation (9)
100
~
1⋅
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
−
−
−= yy
yy
Fit (9)
where is the observed (real) time-series, y y
~
is the
simulated time-series and y is the mean value of
time series calculated applying the formula: y
∑
=
=N
ii
y
N
y
1
1 (10)
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where is the number of samples from time series
under consideration and is i-th sample of the
time-series .
N
i
y
y
In equation (9) w is the Euclidean norm of the
time-series calculated with the formula:
w
∑
=
=n
ii
ww
1
2 (11)
where is the number of samples of the time-
series and is the value of the i-th sample of
the time-series .
N
wi
w
w
Another criterion for model evaluation is the loss
function. Loss function is calculated as the
determinant of the covariance matrix of the
prediction errors (residuals of the model). The loss
function V is calculated with the following relation:
() ()
(
)
⎟
⎠
⎞
⎜
⎝
⎛
=∑
=
=
Ni
i
T
ii
N
V
1
ˆˆ
1
det
θεθε
(12)
where is the estimate of the parameters vector
θ
ˆ
θ
and i
ε
is the i-th value of the model residuals time-
series.
The Akaike Information Criterion (AIC) for an
estimated model is defined as the value of the
negative log-likehood function at the estimated
parameters plus the number of estimated parameters
[12]. If the disturbance source is Gaussian with
covariance matrix , the logarithm of likehood
function is:
Λ
() () ()
C
N
LN
ii
T
i+Λ−Λ=Λ ∑
=
−detlog
22
1
,
1
1
θεθεθ
(13)
where is a constant. Maximizing this analytically
with regard to and then maximizing the results
with regard to
CΛ
θ
, gives
() ()
V
NpN
CL log
22
,+
⋅
+=Λ
θ
(14)
where
p
is the number of outputs and V is the loss
function defined in equation (12). After removing
constants and suitable normalization, the following
expression is reached:
N
d
VAIC 2
)log( += (15)
where is the number of estimated parameters.
d
Akaike Final Prediction Error (FPE) [12] for an
estimated model is calculated with the formula:
N
d
N
d
VFPE
−
+
⋅=
1
1
(16)
It is technically possible for FPE to become
negative if the number of estimated parameters
exceed the number of data. In such a case it is better
to use AIC.
The estimation
θ
ˆ of the parameters vector is
calculated using the condition [14]:
(17)
()
θθ
θ
Vminarg
ˆ=
where the criterion function
()
θ
V is defined in
equation (18):
(18)
( ) () () ()
[]
∑∑
==
−== N
t
N
t
TttytV
11
2
2
θϕεθ
The solution of equation (17) is:
(19)
() () () ()
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=∑∑ =
−
=
N
t
N
t
Ttyttt
1
1
1
ˆ
ϕϕϕθ
In equation (19) the following matrix:
(20)
() ()
∑
=
N
t
Ttt
1
ϕϕ
has to admit an inverse and to be semi-positive
defined.
Another form of the solution of equation (17),
equivalent with solution (19) is:
[
]
Y
TT ΦΦΦ= −1
ˆ
θ
(21)
where
(
)()
[
]
() ( )
[]
T
T
N
NyyY
ϕϕ
K
K
1
1
=Φ
= (22)
The matrix (21) admits inverse if the elements of
the vector
(
)
t
ϕ
are linear independent.
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In order to predict the behavior of the system an
on-line determination of the systems parameters can
be made. This procedure is based on the recursive
algorithm for the parameters estimation [14].
If is the estimation of the parameters of the
system (8), the estimation based on the first
measurements, then:
()
t
θ
ˆ
t
(23)
() () () ()()
⎥
⎦
⎤
⎢
⎣
⎡
⎥
⎦
⎤
⎢
⎣
⎡
=∑∑ =
−
=
t
s
t
s
Tsyssst
1
1
1
ˆ
ϕϕϕθ
Estimation (23) satisfies, by definition, the
equation
() () () () ()()
sssysst T
t
s
θϕεεθ
θ
−== ∑
=
,minarg
ˆ
1
2 (24)
The weighted form of the criterion (18) is
considered:
(25)
() ()
∑
=
−
=t
s
st tV
1
2
ελθ
In equation (25)
λ
is the forgetting factor. This
way the criterion function uses, for parameters
estimation, the last
1,
1
1<
−
λ
λ
(26)
measurements. The other measurements, before
these, are almost ignored.
The recursive algorithm is described by the
following equations:
() ( ) () ()
() () () ( )
() ( ) () () ( ) ()
[]
() ( ) () () ( )
[]
λϕ
ϕϕλϕ
θϕε
εθθ
11
11
1
ˆ
1
ˆˆ
−−−=
−+−=
−−=
+−=
tPttKtPtP
ttPtttPtK
tttyt
ttKtt
T
T
T
(27)
where is of form (28), depending on the
covariance matrix of the estimator:
()
tP
() () ()
1
1
−
=
−⎥
⎦
⎤
⎢
⎣
⎡
=∑
t
s
Tst sstP
ϕϕλ
(28)
The recursive algorithm (28) must be initialized
with and . The value of the forgetting
factor
()
0
ˆ
θ
()
0P
λ
must be also established.
If there are no a-priori information about
θ
, the
following initializations are made:
(
)
00
ˆ=
θ
and
()
IP
α
=0 (29)
where
I
is the unity matrix of appropriate order and
α
is constant. A value too small for
α
leads to a
slow convergence of the estimates, and a value too
big for
α
has as consequence oscillation with high
amplitude of the parameters values (which are
elements of vector
θ
). If the analyzed system has
time-varying parameters, then the chosen value for
λ
must be less then 1 (smaller if the system
parameters have a fast variation in time). Usual,
[
]
]995.0,95.0
∈
λ
.
3 Numerical Simulation
The simulation modeling assumes measurements
of data for a given period.. Data collected during
three months by a datalogger were used in this
study. The conversion of the pulse signal to a
voltage signal appeared not to be as constant in time
as desired and to give meaningful results the heat
meter signal had to be filtered.
Figure 1 shows the consumption of domestic
warm water for a block of flats. Like in other studies
described by scientific literature, variation is large
and unsystematic. In the present case, the monthly
quantity of heat consumed for domestic warm water
preparation was 243 KWh/month per apartment.
Other studies took into consideration different
values. For instance, Bohm B. investigated
consumption of domestic warm water in
Copenhagen and found out that the thermal energy
used was 106 kWh/month per apartment. Yang used
data from an apartment having a thermal energy
consumption for preparing warm water of 200-
222kWh/month, and Lawaetz used values in the
domain 108-233kWh/month per apartment [15].
Yao R. and Steemers K. analyzed typically hourly
domestic warm water load profile for an average
size domestic house hold from U.K [11]. They took
into consideration five types of consumers according
to different occupancy scenarios: unoccupied from
9.00-13.00, unoccupied from 9.00-18.00,
unoccupied from 9.00-16.00, all day occupied,
unoccupied from 13.00-18.00. They found out a
typical consumption profile and used it in a
computer program for prediction of daily load
profile for households from U.K.
The building studied in the present paper has 60
apartments; therefore it is difficult to choose a
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model specific for all the inhabitants. A simulation
model according to the day of a week has been
chosen. Usually, studies assume that the behavior of
consumers depends most of all on the fact that the
day is a working one or not. What about the same
type of day? Is really the consumption similar in
non-working days?
Fit Function
Model Saturday Sunday
AMX(2,2) 75.86 67.81
AMX(2,4) 81.59 75.41
AMX(2,6) 81.48 76.34
AMX(4,4) 81.65 75.52
AMX(4,10) 82.06 76.64
This work studies consumption of domestic warm
water during two nonworking days: Saturday and
Sunday. The simulation model was built using data
for one single day and another day was used for
validation. Also cross validation was done checking
out the accuracy of predictions.
Table 2
FPE
Model Saturday Sunday
AMX(2,2) 2.03443 2.0763
AMX(2,4) 1.21434 1.2456
AMX(2,6) 1.26239 1.1708
AMX(4,4) 1.23486 1.2605
AMX(4,10) 1.29019 1.2339
In order to choose a simulation model for
modeling the warm water consumption many
ARMA model were taken into consideration. The
criteria for choosing one model or another were the
value of fit function (9), the Akaike Final Prediction
Error (16) and Akaike Information Criterion (15).
For fit function larger values are better, while for
FPE and AIC smaller values are better.
Table 3
AIC
Model Saturday Sunday
AMX(2,2) 0.7102 0.7306
AMX(2,4) 0.1942 0.2196
AMX(2,6) 0.2330 0.1577
AMX(4,4) 0.2109 0.2315
AMX(4,10) 0.2544 0.2101
In Table 1 are presented the fit functions values
for analyzed models, Table 2 contains values for
FPE and Table 3 contains values for AIC. In the
model name the first number represent the value for
and the second the value for .
a
nc
n
Table 1
Figure 1. Daily thermal power consumption of domestic warm water during a month.
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Figure 2. Simulation model for Saturday.
Figure 3. Simulation model for Sunday.
The chosen model for Saturday is described by
equation (3) and the polynomials of the models are:
A(q) = 1 - 1.006 q^-1 - 0.2217 q^-2 +
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+0.1405 q^-3 + 0.09357 q^-4 (30)
C(q) = 1 + 0.5169 q^-1 + 0.3459 q^-2 +
+ 0.1216 q^-3 + 0.1592 q^-4 - 0.8185 q^-5 –
-0.3647 q^-6 - 0.293 q^-7 - 0.09375 q^-8 -
- 0.2134 q^-9 - 0.1543 q^-10 (31)
Estimated using PEM from data set zi1s.
Loss function 1.0917 and FPE 1.29019.
Sampling interval: 300 s.
Figure 2 presents the experimental data and the
simulation output for Saturday.
The chosen model for Sunday is described by
equation (3) and the polynomials of the models are:
A(q) = 1 - 1.431 q^-1 + 0.0099 q^-2 +
+ 0.5751 q^-3 - 0.1506 q^-4 (32)
C(q) = 1 + 0.08072 q^-1 - 0.3723 q^-2 –
- 0.03749 q^-3 - 0.0645 q^-4 - 0.9458 q^-5 –
- 0.0696 q^-6 + 0.4276 q^-7 +
+0.08007 q^-8 + 0.05085 q^-9 - 0.04453 q^-10
(33)
Estimated using PEM from data set zis2.
Loss function 1.08164 and FPE 1.2783.
Sampling interval: 300 s
It can be noticed that the polynomials of the
models and the shape of a graph are different even if
both days are non-working days. Concluding, a
different simulation model has to be done for every
day of the week.
4 Statistical analysis
In order to validate the model, the residual
analysis is used. Residuals represent the portion of
the validation data not explained by the model.
Residual analysis consists of two tests: the whiteness
test and the independence test. According to the
whiteness test criteria, a good model has the residual
autocorrelation command inside the model
confidence interval, indicating that the residuals are
uncorrelated. According to the independence test
criteria, a good model has residuals uncorrelated
with past inputs. Evidence of correlation indicates
that the model does not describe how part of the
output relates to the corresponding input [12].
4.1 Simulation model
This work contains the autocorrelation of
residuals for Saturday and Sunday, presented in
Figure 4 and Figure 5, respectively. The horizontal
dotted lines represent the limits of the confidence
interval set to 99%.
It can be noticed that almost all values of the
residuals autocorrelation are inside the confidence
interval indicating a good simulation model for both
days.
Figure 4. Autocorrelation of residuals for simulation
model. Day Saturday..
Figure 5. Autocorrelation for residuals for
simulation model. Day Sunday.
4.2 Prognosis model
The main purpose of modeling is forecasting
consumption in order to have an efficient
production. Space heating is obviously the first
option of scientists’ works [16], [17]. Forecast of
domestic warm water consumption is a different
problem depending on other parameters, so different
methods are needed [18]. The present work tries to
do it using time-series analysis.
Previous paragraphs present two different
simulation models: one for a Saturday day and the
other for a Sunday day. The next step is to check
with experimental data for other days of Saturday
and Sunday, if these time-series simulation models
are or not appropriate for prognosis.
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An analysis of residuals when the simulation
model is used for prediction should give the answer
to this problem. Each simulation model was used for
prognosis of consumption for two different future
days. Figure 6 presents the residuals of Saturday’s
forecasting and figure 7 presents the residuals of
Sundays’ forecasting. The confidence interval was
set to 99%.
For both figures prediction for the first future day
is represented with solid line, and the second future
day is represented with dashed line. The confidence
interval limits are the dotted lines.
Figure 6. Autocorrelation of residuals for prediction
models. Prognosis for Saturday days.
Figure 7. Autocorrelation of residuals for prediction
models. Prognosis for Sunday days.
It can be noticed that the prediction is very good.
Even if the confidence interval is tight, practically
all the residuals are within it.
The prediction method described above considers
that the system's parameters are constant in time. If
these parameters vary in time, the recursive method
described by equations (27) and (28) has to be
applied. The recursive method has also the
advantage that in any moment, the next value of the
time series can be calculated.
In Figure 7 the evolution in time of the Saturday
model parameters are presented. Figure 8 presents
the experimental data (the blue line) and predicted
data (green line) obtained applying the recursive
algorithm for data for Saturday.
Figure 7. Model coefficient time-history for
Saturday.
Figure 8. Experimental and recursive predicted data
for Saturday
In Figure 9 the evolution in time of the Sunday
model parameters are presented. Figure 10 presents
the experimental data (the blue line) and predicted
data (green line) obtained applying the recursive
algorithm for data for Sunday.
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Figure 9. Model coefficient time-history for Sunday.
Figure 10. Experimental and recursive predicted
data for Sunday
Neither of these two models (for Saturday and for
Sunday) does not have model parameters, and a
recursive method for model parameters evaluation
might be more appropriate.
Figure 11 presents the time-history for model
parameters for the days of the week-end and Figure
12 presents the observed data and predicted data for
the days of the weekend. As one can notice
observing Figure 11, in this case the model
parameters vary in time, but, after a time, they tend
to have constant values.
Figure 11. Model coefficient time-history for
weekend days.
Figure 12. Experimental and recursive predicted
data for weekend days
5 Conclusions
The paper presents a methodology for prognosis of
domestic warm water consumption in district heating
systems based on time series analysis. Double cross
validation was done; the simulation model was used
for prediction in other two cases. Recursive
algorithm was applied highlighting that the model
parameters are not constant in time. The results of
the statistical analysis are very good pointing out that
daily simulation models using time series analysis
are a powerful and appropriate tool for the prognosis
of consumption in district heating systems.
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WSEAS TRANSACTIONS on COMPUTERS
Elena Serban, Daniela Popescu
ISSN: 1109-2750
2041
Issue 12, Volume 7, December 2008