Forecasting Electrical Energy Consumptions
K. STOILOVA1, T. STOILOV1, G. ANGELOVA1, R. PETROV2, I. KLISUROV2
1Institute of Information and Communication Technologies – Bulgarian Academy of Sciences
1113 Sofia Acad. G. Bonchev str. bl.2,
BULGARIA
2Wildlife Rehabilitation and Breeding Centre-Green Balkans,
P.O. Box 27, 6006 Stara Zagora,
BULGARIA
Abstract: - Electrical energy management of an organization is related to the appropriate planning of required
payments and allocation of resources accordingly. It is important for good management to account for past
costs and to forecast future costs. An intelligent solution for the mathematical formalization of this process is
presented in the study. The research object is the Wildlife Rehabilitation and Breeding Centre “Green Balkans”
in Bulgaria, in which the correct allocation of resources is significant due to the minimal financial support.
Mathematical models have been developed for forecasting electricity consumption based on data for the past
three-year period. The applied methodology is based on linear regression analysis. Two mathematical models
were synthesized, which were compared and analyzed depending on the length of the historical data interval.
The models represent an intelligent solution for the Centre’s electrical energy management.
Key-Words: - Electrical energy management, Planning resources, Mathematical models, Statistical methods,
Auto regression, Forecasting.
Received: April 9, 2024. Revised: September 3, 2024. Accepted: October 6, 2024. Published: November 7, 2024.
1 Introduction
In Bulgaria, a Wildlife Rehabilitation and Breeding
Centre has been actively operating since 1992,
which is a specialized unit of the activity of Green
Balkans, related to the treatment, rehabilitation,
reproduction, and return to nature of rare and
endangered wild animals, as well as environmental
education. During the cold months of the year, the
center’s patients are about 100 per month, while
during the warm months, they exceed 400. Patients,
in addition to a lot of care and professional skills,
require several resources such as food, medicine,
and adequate heating, both in summer and winter,
according to the different species of wild animals
that are accepted at the Centre. The Centre’s
management faces several problems in allocating its
limited resources and funding. This requires proper
use of available resources and the application of
scientific methods such as planning, optimization,
and forecasting for their redistribution. The present
study aims to apply scientific approaches to the
functioning of the Centre for its better management.
Here, a management approach integrating
forecasting and optimization is proposed to
reallocate available resources.
Forecasting models are statistical tools designed
to make predictions about future events based on
historical data and trends. One of the most used
forecasting methods is the Time Series Model. Time
series represent a set of measurements of a certain
variable made at regular intervals. Time is the
independent evaluation variable. Time Series Model
analyses historical data to predict future trends.
An overview of Time series analysis and
forecasting is presented in [1]. It introduces the
different Time series forecasting methods, starting
with Time series decomposition, data-driven
moving averages, and exponential smoothing,
and discusses model-driven forecasts including
regression, Autoregressive Integrated Moving
Average methods, and machine learning-based
methods using windowing techniques. Time Series
forecasting can be classified into four broad
categories of techniques [1]: Forecasting based on
Time series decomposition, smoothing-based
techniques, regression-based techniques, and
machine learning-based techniques. In this study,
according to this classification, models are based on
regression analysis. The Time series model uses
historical data and forecasting models like Auto
regressive model (AR), the Moving Average (MA)
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K. Stoilova, T. Stoilov, G. Angelova, R. Petrov, I. Klisurov
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model, and their combination Auto-Regressive
Integrated Moving Average ARIMA model [1].
Prediction models are the subject of research in
many publications. In [2], the characteristics of five
regression techniques for forecasting the sales of a
retail chain are considered. The linear weighted
method and an artificial neural network have been
applied to predict the security of supply and demand
in the forestry industry, [3]. An overview of new
product forecasting techniques is presented in [4].
New research directions are proposed to improve
the performance of new products, supporting
managers in decision-making.
In [5], three models for forecasting future cash
flows are compiled. The data used are from
Tunisian trading companies. Learning-based short-
term forecasting models for smart grids are
reviewed in [6]. Different 41 models were used to
predict wind speed based on a data set from the site
of Jodhpur, India. In [7], solar forecasting
approaches by applying machine-learning
techniques are given. A predictive convolutional
neural network model for source-load forecasting in
smart grids is presented in [8]. Prediction of the
Long-Term Electrical Energy Consumption in
Greece Using Adaptive Algorithms is described in
[9]. Three short term load forecasting models, which
aim to predict system load over an interval of one
day or one week based on Grey System theory are
presented in [10]. Forecasting Electricity Price
Using Seasonal Arima Model is given in [11]. A
review and evaluation of current wind power
prediction technologies is described in [12].
The purpose of this research is to integrate
intelligent solutions into the management policy of
the Centre. Achieving this goal is related to solving
the following tasks: choosing an appropriate method
to formalize a better management policy;
determining an appropriate mathematical
formalization; numerical simulations; and model
validation. The work consists of the following
sections: Mathematical models; Integrating linear
regression models with a shifting predictive
approach; Numerical simulations and results; and
Conclusions.
2 Mathematical Models
The quantitative formalization of the forecasting
process is based on statistical analysis and more
specifically on linear autoregression (AR) analysis.
Our goal is to construct first- and second-order
linear autoregression models (so-called AR(1) and
AR(2)) and to analyze their behavior.
2.1 AR(1) Linear Regression Model
The AR(1) model has the following formalizations
󰇛󰇜󰇛󰇜 (1)
In (1), the known value is y(t), the future
unknown value is y(t+1) and parameters a and b are
to be determined.
When there is a set of m data values,
dependence (1) can be represented as a system of m
linear equations:
󰇛󰇜󰇛󰇜 (2)
󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜.
In the system of equations (2), the coefficients a
and b are unknown. They should be determined in
such a way as to provide the best approximation for
the linear system. In this case, we will apply the
method of least squares for the approximation.
With known coefficients a and b, we denote the
predicted value by
󰇛󰇜󰇛󰇜. (3)
The difference between the actual and predicted
value is denoted by δ
. (4)
We aim to minimize the approximation error δ
by applying the least squares method, whose
formalization is by (5):

 󰇛󰇜
 (5)
The AR(1) model of type (3) can be presented in
matrix form:
, (6)
where the matrices are in the form:
; 󰈏󰈏
. (7)
Problem (5) for minimizing the error when
applying the matrix form (6) of the AR(1) model
leads to:

 . (8)
To solve (8), we differentiate both sides of (8):
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
=0 . (9)
The unknown regression coefficients or (a, b)
are determined from (9):
󰇛󰇜 . (10)
In this way, the model AR(1) in matrix form (6)
has certain linear coefficients a and b according to
(10), so that the future values can be determined
based on the available data from the dependence (6)
or (2).
Further, the predicted values are determined
using the formalization of the AR(1) model (3) or
(6) and the unknown coefficients (a and b) for the
approximation is determined with optimal accuracy
based on the method of least squares (5).
2.2 AR(2) Linear Regression Model
In the second-order linear regression model AR(2),
the predicted value at time t+1 depends on the
available data both at the previous time t and on the
data at the earlier time t-1. The AR(2) model has the
following mathematical notation:
󰇛󰇜󰇛󰇜󰇛󰇜 (11)
where y(t+1) is the forecasting value. The
unknown parameters are and that must be
determined. Since the unknown parameters are
three, at least four historical data are needed to make
a prediction. For the AR(1) model, the unknown
parameters are two, so at least three historical data
must be available.
An AR(2) model can be represented as a system
of m linear equations when m data values are
available:
󰇛󰇜󰇛󰇜󰇛󰇜(12)
󰇛󰇜󰇛󰇜󰇛󰇜
󰇛󰇜󰇛󰇜󰇛󰇜.
The unknown parameters a, b1, b2 can also be
determined here by the method of least squares in
order to better approximate the forecast. We denote
the predicted value from (12) by 󰇛󰇜:
󰇛󰇜󰇛󰇜󰇛󰇜. (13)
The error of approximation δ represents the
difference between the actual value y and the
predicted value
.
We want to minimize the error of approximation
δ. For this purpose, we apply the method of least
squares with the following formalization:

󰇛󰇜
 . (14)
We can present the model AR(2) of (13) in
matrix form:
,
where
;
.
(15)
The least squares problem (14) in matrix form is
given by:


 . (16)
The solution is obtained by differentiating both
sides of (16):

=0 .
After transformations, the unknown parameters
are determined by the dependence:
󰇛󰇜 . (17)
Relation (17) is further applied to determine the
unknown linear regression coefficients a, (or
) of the AR(2) model.
3 Integrating Linear Regression
Models with a Shifting Predictive
Approach
The Wildlife Rehabilitation and Breeding Centre
Green Balkans provided us with data on the
electricity consumption of the center for 3 years or
36 months. We use this data as consumption history
to predict the next costs of the Centre. It is
important for the manager and the operative at the
Centre to know what resources to budget for the
next month when planning expenses. To formalize
the prediction process, we will apply the two linear
regression models AR(1) and AR(2) proposed above.
The available electricity consumption data (in
kW) are for three years (36 months), given in Table
1. The prediction approach consists of the
following. When we use the AR(2) model, the
unknown regression coefficients are three: a, b1, b2
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and to determine them, at least 4 historical data are
needed. For the AR(1) model, the unknowns are 2:
a, b, so the history data could be at least two to
determine the unknown coefficients a and b. Since
the data of the first two months are included in the
determination of the unknown coefficients of (2),
the prediction can start as early as the third month.
Let the historical data interval include four records
for better approximation. We use the first 4 months'
data from Table 1 as history to predict the energy
consumption for the fifth month . The predicted
value is then compared with the actual value for the
fifth month, which is given in Table 1, and the error
δ of the prediction is determined. According to the
applied least squares method, we minimize this error
δ (see (8) and (16)) for both models and as a result
determine new values of the linear regression
coefficients for AR(1) and AR(2).
Table 1. Electricity consumption in the Centre
Month
2021
2022
1
January
2855
4029
2
February
2498
1351
3
March
2860
2873
4
April
2713
2424
5
May
2103
2498
6
June
2412
2453
7
July
2391
2545
8
August
2938
3367
9
September
2287
1691
10
October
2570
2600
11
November
2511
3075
12
December
3072
1815
These values are entered into the new regression
models AR(1) and AR(2), an offset of one month is
made and for months 2-5 (4-month history), the data
from Table 1 is used to predict the expenditure for
the 6th month. In other words, at each step of the
sliding procedure, new linear regression coefficients
are determined for the two models AR(1) and AR(2)
in an optimal way (according to the optimization
problem) and, accordingly, the predicted values are
different from the previous ones. The approach to
sequentially shift the origin of the data history and
apply the prediction is schematically presented in
Figure 1.
The historical data period varies from 4 to 8
months to assess the impact of this interval on
forecast accuracy. A comparison is made for the
accuracy of the approximation depending on the
different lengths of the historical data.
Fig. 1: Forecasting with the sliding procedure at
history 4 months
The next study, again changing the period of
history, is related to the accuracy of the two AR(1)
and AR(2) models. A comparison is made between
the two models and the results are presented
graphically.
4 Numerical Simulations and Results
The proposed prediction policy is based on the
integration of several methods:
- Linear regression analysis, through which the
AR(1) and AR(2) models are compiled,
- Least squares method, which minimizes the
prediction error and objectively determines the
values of the regression coefficients as a solution to
an optimization problem, and
- A sliding prediction procedure to sequentially
move the data used as history.
This prediction approach is applied to both
AR(1) and AR(2) models. A comparison was made
of the obtained results in terms of prediction
accuracy. The influence of the size of the data
interval n used as history for the prediction was also
investigated.
4.1 Simulations and Graphical Results for
the AR(1) Model with a History of 4
Months (n=4)
To simplify the simulations without reducing the
accuracy, relative data were used. The data were
normalized, as for each year from Table 1 the data
were summed and divided by 12.
The actual values of electricity consumption are
presented in the black solid line in Figure 2. The
black dashed line is the average of the actual costs.
The predicted costs according to the AR(1) model
are in the blue solid line, and their average value is
in the blue dashed line.
2
Historical
period – 4
months
Predicted value
for month 6
. . .
1
3
4
5
6
Time
Historical
period – 4
months
Predicted value
for month 5
5
. . .
Time
1
2
3
4
0
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Fig. 2: Actual and predicted values for model AR(1),
n=4
From Figure 2 it follows that the predicted
values are almost like the real ones. Figure 3 shows
the difference between real and predicted values for
model AR(1), n=4. A dashed line indicates the
average value of the difference.
Fig. 3: Dynamics of prediction error with a model
AR(1), n=4
The magnitude of the vertical axis is multiplied
by 10 to the minus 3rd power, indicating a very small
prediction error, with a mean value of zero.
With the same interval of 4 months for
forecasting history, model AR(2) was applied.
4.2 Simulations and Graphical Results for
the AR(2) Model with a History of 4
Months (n=4)
The actual values of electricity consumption are
presented in the black solid line in Figure 4. The
black dashed line is the average of the actual costs.
The predicted costs according to the AR(2) model
are in the red solid line, and their average value is in
the red dashed line.
The predicted values are close to the actual
values according to Figure 4. The prediction with
this model is quite precise because the mean values
of the actual and predicted values are almost the
same (dashed lines).
Figure 5 shows the difference between real and
predicted values (the prediction error) for model
AR(2), n=4. A dashed line indicates the average
value of the difference.
Fig. 4: Actual and predicted values for model
AR(2), n=4
Fig. 5: Dynamics of prediction error for model
AR(2), n=4
Analogous simulations were done for both
models on historical data for 6 months, n=6.
4.3 Simulations and Graphical Results for
the AR(1) Model with a History of 6
Months (n=6)
The actual values of electricity consumption are
presented in the black solid line in Figure 6. The
black dashed line is the average of the actual costs.
The predicted costs according to the AR(1) model
are in a blue solid line, and their average value is in
a blue dashed line .
Fig. 6: Actual and predicted values for model AR(1),
n=6
Figure 7 shows the difference between real and
predicted values for model AR(1), n=6. A dashed
line indicates the average value of the difference.
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Fig. 7: Dynamics of prediction error for a model
AR(2), n=6
This graph shows that the average error is no
longer zero as in the prediction when n=4, but
slightly larger. If at n=4 the average value of the
error is zero, then at n=6 it is 0.5.10-3. This means
that a larger interval as a history leads to a more
inaccurate prediction with the same model.
4.4 Simulations and Graphical Results for
the AR(2) Model with a History of 6
Months (n=6)
Figure 8 presents the actual values of electricity
costs denoted by the black solid line and the
predicted values red solid line for the AR(2) model.
The mean values are given in black dashed and red
dashed lines, respectively.
Fig. 8: Actual and predicted values for model AR(2),
n=6
Figure 9 shows the difference between real and
predicted values (the prediction error) for model
AR(2), n=6. A dashed line indicates the average
value of the difference.
Fig. 9: Dynamics of prediction error for model
AR(2), n=6
Comparing the average error values when n=4
(Figure 5) and n=6 (Figure 9) follows the same
conclusion for the model AR(2) as for AR(1). As the
size of the historical interval increases, the accuracy
of the prediction decreases.
Simulations were made at n=7 and n=8 months.
Here we will present the results when n=8.
4.5 Simulations and Graphical Results for
the AR(1) Model with a History of 8
Months (n=8)
The actual values of electricity consumption are
presented in the black solid line in Figure 10. The
black dashed line is the average of the actual costs.
The predicted costs according to the AR(1) model
are in the blue solid line, and their average value is
in the blue dashed line.
Fig. 10: Actual and predicted values for model
AR(1), n=8
Figure 11 shows the difference between real and
predicted values for model AR(1), n=8. A dashed
line indicates the average value of the error.
Fig. 11: Dynamics of prediction error for model
AR(1), n=8
This graph shows that the average error is no
longer zero as in the prediction when n=4, but
slightly larger. If at n=4 the average value of the
error is zero, then at n=6 it is 0.5.10-3, and at n=8 it
is about 1.10-3. Therefore, the larger history interval
leads to a more inaccurate forecast.
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4.6 Simulations and Graphical Results for
the AR(2) Model with a History of 8
Months (n=8)
The same notations are used as in the previous
simulations. Figure 12 compares the actual (in
black) and predicted (in red) values and their
respective average values (with dashed line).
Fig. 12:Actual and predicted values for model
AR(2), n=8
The variation of the error for the AR(2) model
at n=8 is given in Figure 13.
Fig. 13: Dynamics of prediction error for model
AR(2), n=8
Comparing the average error values when n=4
(Figure 5), n=6 (Figure 9), and n=8 (Figure 13)
follows the same conclusion: prediction accuracy
decreases as the historical interval increases. As a
reason for the decrease in forecast accuracy when
the historical interval increases can be said the
following. For a smaller time interval, the total error
between actual and forecast value has a smaller
value because a shorter period is considered. Slow
processes are taken into account and faster
processes are averaged.
As a confirmation of this conclusion, Figure 14
presents an illustration of the variation of the
prediction error as a function of the size of the story.
For the AR(1) model, the error variation is in the
blue solid line, with the mean value in the blue
dashed line. For the AR(2) model, these changes are
in red.
Fig. 14: Comparison of prediction error for AR(1)
and AR(2) depending on the size of the historical
interval
The following conclusions can be drawn from
the comparison of the prediction error of the two
models. The smallest prediction error for both
models is at the smallest historical interval: n=4. As
the size of the historical interval increases, the error
increases for both models. The explanation of this
dependence is that with a shorter history, the next
value is predicted more accurately because the
dynamics of the process for the next time value is
preserved.
From the comparison between the model AR(1)
and AR(2) it follows that the prediction error is
greater in the model AR(1) compared to the model
AR(2). Here the explanation is that with AR(2) two
previous values are taken from the data while with
AR(1) only one value at the previous time is used,
therefore the model AR(2) is more accurate
compared to AR(1).
5 Conclusion
The study develops an intelligent solution for the
Organization Manager to better plan future
resources. Based on the 36-month electricity
consumption data, two types of models have been
developed to forecast future costs. Statistical linear
regression was applied to formalize the process. A
sliding-sequential forecasting procedure was applied
at different historical data intervals from 4 to 8
months. The predicted values are compared with the
existing values. The minimized prediction error is
determined. As a result of the optimized error, new
values of the linear regression coefficients are
determined for each step of the sliding procedure.
This results in higher model accuracy/less prediction
error. This error results from the application of the
least square method and represents the solution of
the optimization problem (5). According to this
solution, the approximation coefficients used for
forecasting for the next month are determined. For
each simulation, the prediction error is plotted. This
error represents a maximum of ±0.006 which we
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rate as a very good result. The accuracy of the
forecast was analyzed depending on the size of the
historical interval. A comparison is made between
the two models AR(1) and AR(2). The second
model gives better predictive accuracy). Further
research is related to the inclusion of more
predictive indicators.
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WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.34
K. Stoilova, T. Stoilov, G. Angelova, R. Petrov, I. Klisurov
E-ISSN: 2224-350X
407
Volume 19, 2024
Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
- Krasimira Stoilova, compiled the first model, the
optimization and simulation, and writing the
original draft.
- Todor Stoilov compiled the research
methodology, formal analysis, and creation of the
second model.
Galia Angelova was responsible for supervision.
- Rusko Petrov collected the data included in the
simulations
- Ivailo Klisurov was responsible for the resources.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
This research has received funding from the
Bulgarian Ministry of Education and Science under
the National Science Program Intelligent Animal
Husbandry, grant agreement № D01-62/18.03.2021,
and from the EU’s Horizon Europe Widening
program via COALition project “Promoting
Innovation Excellence in Transformation of Coal
Regions to Climate-Neutral, Thriving Economies”,
grant agreement № 101087022.)
Conflict of Interest
The authors have no conflicts of interest to declare.
Creative Commons Attribution License 4.0
(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
https://creativecommons.org/licenses/by/4.0/deed.en
_US
WSEAS TRANSACTIONS on POWER SYSTEMS
DOI: 10.37394/232016.2024.19.34
K. Stoilova, T. Stoilov, G. Angelova, R. Petrov, I. Klisurov
E-ISSN: 2224-350X
408
Volume 19, 2024