

Abstract—The objective of this study is to analyze and

compare classical time series and deep learning models for energy

load prediction. Energy predictions are important for

management and sustainable systems. After analyzing the

climacteric factors impact on energy load (a case study in

Albania) we considered classical and deep learning models to

perform forecasts. We have used hourly and daily time series for

a period of three years. In total respectively 26,280 hours and

1095 days. Average temperature is considered as external

variable in both statistical and deep learning models. The

dynamic evolution of hourly (daily) load is correlated with hourly

(daily) average temperature. The performance of the proposed

models is analyzed and evaluated based on accuracy

measurements (MSE, RMSE, MAPE, AIC, BIC etc.) and

graphics results of statistical tests. In-sample and out-of-sample

accuracy is evaluated. The models show competitive performance

to some recent works in the field of short-and medium-term

energy load forecasts. This work may be used by stakeholders to

optimize their activities and obtain accurate forecasts of energy

system behavior.

Keywords— time series, forecasting, electric energy

consumption, deep learning.

I. INTRODUCTION

The Mediterranean basin is one of the key points of energy

efficiency production and use. Every country's energy

consumption is specially affected by its economic and industry

development, climatic conditions and its energy production

sources. Numerous and diverse sources of energy have

undergone significant evolution in the last 30 years. There has

been a decline in coal use and a significant increase in natural

gas use. Although climate change has affected this region over

the last decades again the Mediterranean basin is an area

which benefits from a mild climate with mild and warm

winters and sunny summers. This climate offers the region a

great potential for energy production from renewable energy

[1]. Albania is a country in which the energy produced by

hydropower plants occupies almost 90% of the energy

produced in the country. Given that this energy produced relies

on the availability of water in large reservoirs of cascades

located mainly in the northern part of the country, or the

intensity of the flow of rivers that supply these cascades,

precipitation and snowmelt. High summer temperatures and

droughts are limiting production by hydropower plants.

What is noticed in recent years in the Mediterranean region

and in Albania is also the fact that based on the above factors

the utilization rate of hydropower plants has decreased. This

decline has been followed by an increase in interest in solar

energy which is mainly influenced by surface solar radiation

whose variations depend mainly on the atmospheric

composition (aerosols, water vapor) and clouds [2], [3].

An increase in solar radiation has been observed in Europe [4]

and especially for the Mediterranean basin these solar sources

are seen with special interest as one of the areas with medium

to high solar radiation on the continent [5]. Exactly at the

beginning of winter in 2021, the region was involved in an

energy crisis and not only. Experts emphasize the importance

of a safe and sufficient energy, especially when the energy

sources are not numerous and diverse. In this context, they

suggest the addition of new and clean energy sources, in the

same time they highlight to focus on the importance of optimal

management of existing resources. Climate change associated

with drought can reduce power generation and result in less

electricity produced by the hydro power plants. Significant

changes in production and consumption have been observed

which have influenced government policies to provide optimal

and long-term solutions. Many European countries are part of

the energy crisis and have already had wide-ranging impacts

on their economy and environment.

There is a lot of work done regarding prediction in different

areas. In their work [6] have presented most of the challenges

the prediction field has faced with during 25 years of

forecasting. More than one decade ago they pointed out the

necessity of computational ability for the high complexity

amount of data to become the power of prediction in many

areas. The relationship between energy consumption and

economic growth was analyzed in a considerable number of

countries in Europe by [7]. They indicate that attention is

required to the relation between the efficiency use of resources

and climate change in consequence the global warming.

Researchers are provided with a systematic literature review of

a considerable number of articles on energy demand modeling.

Reference [8] reviewed and offered a classification of different

techniques used in energy demand. There is also a lot of work

done especially in machine learning (ML) techniques which

A comparative study of statistical and deep

learning models for energy load prediction

1E. Gjika, 2L. Basha

Department of Applied Mathematics, Faculty of Natural Science, University of Tirana,

Tirana, Albania

1eralda.dhamo@fshn.edu.al, 2lule.hallaci@fshn.edu.al

Volume 4, 2022

Received: July 14, 2021. Revised: December 22, 2021. Accepted: January 17, 2022. Published: January 26, 2022.

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

rely on historic data and are extensively used to short-term

forecasting [9]. Classical statistical techniques on energy

prediction are often used as a benchmark for many techniques

but in many cases depending on the nature of the data used and

exogenous variables these techniques perform comparable

with engineering-based models or ML models [2], [3].

Given the high ability of deep learning techniques to deal

with the change of power generation and load there are many

neural network structures which have been utilized to obtain

short term load predictions [10]. The study undertaken by [11]

presents a forecasting model for hourly load consumption

considering external variables unitizing convolutional neural

networks (CNNs) to extract the features of variables used.

They also show in their study their ability to deal with short-

term and long-term memories. ANNs (Artificial Neural

Networks) are used by [12] for Greek interconnected power

system. They point out that the accuracy of the ANNs’

prediction depends on the quality and availability of the

training data. An analysis of the accuracy of ML and statistical

techniques for Albanian energy sector is done by [2]. In their

work they consider the energy production by hydropower’s

which is the main source of production in the country. They

came in the conclusion that neural networks have handle with

seasonal patterns of monthly energy produced by HPP and

provide accurate forecasts in short-term. Reference [13]

proposed a short-term load forecasting method for hourly data

using long short-term memory (LSTM) algorithm as an

algorithm which have shown to deal with regularity of

historical data. They use encoded external factors to predict

the load in the next half an hour and showed accurate results.

In their work [14] present forecasting methodology for daily

electricity demand using weather ensemble predictions. They

show that weather ensemble predictions can improve the

accuracy of electricity demand forecasts. When forecasting

energy demand, it's often a good practice to use temperature as

an exogenous variable. Reference [3] present a methodology

of ensemble models to predict energy production by

hydropower relying in exogenous variables such as

temperature, precipitation, water inflow. The show accurate

results of combining statistical and machine learning models

for monthly data.

Depending on many factors when dealing with energy data

studies have shown that in special circumstances such as:

geographical position, climate conditions , variables taken into

considerations, seasonality patterns and frequency of data

there are not consensus on the “best” model used in the energy

situations. Going through the results of [15], they show the

efficiency of STL decomposition (Seasonal-Trend

decomposition using LOESS) when used as a pre-processing

step in statistical models. Another study which shows the

efficiency of statistical time series models is the one proposed

by [16] which is a simple procedure combining time series

models dealing with multi seasonality. In reference [17] the

authors offer a new approach for forecasting time series with

complex seasonal.

Although there is plenty of material, research and

competitions about recommended models for forecasting in

different fields [18] there is still discussion of the conditions

under which different methods perform best.

A. Data

In our study we use hourly time series of energy load for a

period of three years (2016-2018) in total there are 26,280

hours and 1095 days. Together with energy load we have

considered also the average temperature (hourly and daily).

In this material we have used the terms described below:

Hour: The time of day for which the variables are

expressed. The time is expressed as an integer with values

ranging from 0 to 23

Load: The aggregated energy load (consumption) observed

each hour (measured in Mwh).

Temperature: The hourly average value (in Celsius) of the

temperature of the day.

When dealing with hourly time series data there are a few

models that one can try. Since hourly time series contain

multiple seasonal patterns (daily, weekly, and yearly); in your

case it contains all these seasonality’s because it contains 3

years of hourly data. Many time series exhibit complex

seasonal patterns.

Fig. 1 Hourly energy load (unit MWh)

In Fig. 1 is shown the hourly energy load for a period of

three years. What is clearly observed is the fact that the time

series has multi seasonality patterns. After an accurate

investigation of each year we observe a high load at the

beginning of the year which corresponds to the winter season

and accompanied by a noticeable decrease during the spring

season. Further an upward trend for the period of summer

which in the Mediterranean climate is accompanied by high

temperatures, and with a marked decline during the autumn.

Patterns are distinct from year to year due to the extreme

temperatures and weather situations during the winter months

mainly in the entire Mediterranean region and especially in

Albania for that year. This behavior can be observed in Fig.2,

Fig.3 and Fig.4.

Volume 4, 2022

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

Fig. 2 Hourly energy load (2016)

Fig. 3 Hourly energy load (2017)

Fig. 4 Hourly energy load (2018)

The hourly average temperature (measured in Celsius

degree) for the period under consideration is shown in Fig.5.

The Mediterranean climate would certainly not be absent in

the seasonal behavior of the average temperature. High

temperatures are observed during the summer months (up to

42 degrees Celsius) and low temperatures during the winter

months (up to -7 degrees Celsius).

Fig. 5 Hourly average temperature (Celsius degree)

These extreme values are especially noticeable during 2016,

and there is a decrease in values for both high and low

temperatures for 2017 and further for 2018 which is also

confirmed by world indicators related to climate change and

global warming. Fig.6 shows the different levels of daytime

peak energy demand by month. The situation is displayed for

three years in separate and we may notice a flattened pattern

from May to September and a clear three peaks for other

months. The lower peak in energy demand is observed from

midnight to 5am and then a rapid increase of the demand from

5am to 8am, then a steady situation which culminates with the

evening hour 8pm and then a decrease again to the lowest

levels of the day. Especially for January and December the

morning “jump” load is more distinctive and very fast in levels

from 20000 Mwh to 35000 Mwh. It is also observed a slightly

increase of energy load levels from 2016 to 2018.

Sep

Oct

Nov

Dec

May

Jun

Jul

Aug

Jan

Feb

Mar

Apr

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

20000

30000

40000

20000

30000

40000

20000

30000

40000

Hour

Energy Load (MwH)

2016 2017 2018

Fig. 6 Twenty-four-hour load by month (MWh)

The twenty-four-hour load helps us to investigate the levels

of daytime and peak loads which depends also on the solar

penetration conditions and variations.

Sep

Oct

Nov

Dec

May

Jun

Jul

Aug

Jan

Feb

Mar

Apr

0 5 10 15 20 0 5 10 15 20 0 5 10 15 20 0 5 10 15 20

Hour

Average Temp (Celcius)

2016 2017 2018

Fig. 7 Twenty-four-hour load by month (MWh)

The Mediterranean climate of Albania show a correlation of

energy demand and average temperature. This can be easily

Volume 4, 2022

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

observed also by 24 hour evolution of these variables faced by

month as shown in Fig.6 and Fig.7. In both variables we

observe presence of seasonality and one high peak for the

average temperature which obviously is reached in midday and

high average temperature levels for the summer and low levels

for winter. No noticeable differences of levels from 2016 to

2018.

A correlation analysis of energy load and climacteric factors

such as temperature is important. Pearson correlation

coefficient is a measure of linear correlation between two sets

of data. It is defined as:

cov( , )





and takes values

from -1 to 1. A value close to -1 or 1 indicates a significant

(negative or positive) relationship between X and Y.

This correlation analysis between load and temperature is

illustrated graphically in Fig.8. The interesting part which is

observed is the fact that aggregated energy load displays a

significant correlation with the average temperature (by hour)

when observed by month. We notice a significant positive

correlation between these two variables especially for the

summer season (it varies from 0.64 up to 0.72).Another pattern

we clearly observe is the density plot which corresponds to

daily and night hours of the day. These findings may be used

to focus on the research of seasonal power load prediction in

order to satisfy and optimize power supply and demand. In this

study we have not taken into consideration the seasonal

modeling by hour or month which can be further studied.

Corr: -0.074***

Jan: 0.150***

Feb: 0.074***

Mar: 0.278***

Apr: 0.305***

May: 0.474***

Jun: 0.648***

Jul: 0.728***

Aug: 0.693***

Sep: 0.527***

Oct: 0.409***

Nov: 0.224***

Dec: 0.304***

Load

full_temp

Load

full_temp

400 800 1200 1600-10 0 10 20 30 40

0.000

0.001

0.002

0.003

-10

Fig. 8 Density plot and correlation of energy load (consumption) and

average temperature by month

Fig.9 shows the correlation of energy load (consumption)

and average temperature faced by year and also colored by

month. It is clear that the same behavior is observed for each

year taken into observation. What makes different the spread

are the registered values of the average hourly temperature

which clearly display a compression in the amplitude for the

last year 2018. The scatterplot of hourly energy load and

average temperature shows two clusters which correspond to

daily and night hours. The behavior is almost the same apart

the shift of the daily cluster above the night cluster. This

shift corresponds also to the higher differences in

temperature which suggest the need of electricity due to

heating or cooling in respect also to the month or season.

2016

2017

2018

-10 0 10 20 30 40 -10 0 10 20 30 40 -10 0 10 20 30 40

400

800

1200

1600

Average Temperature

Energy Consumption (MWh)

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Fig. 9 Correlation of energy consumption and average temperature faced by year (hourly observations)

Volume 4, 2022

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Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

II. METHODOLOGY

A. Data preprocessing

The data was organized in training and testing dataset

respectively (80% and 20%). The testing dataset was used for

validation which corresponds to the forecast horizon. Some

machine learning methods face difficulties when dealing with

missing observations, but this was not our case. The daily

energy load was the aggregated load of 24 hours and the

average temperature was the average calculated for 24 hour of

that corresponding day. Given the complexity of the data and

the luck of other external variables which can handle and

better explain multi seasonal patterns of energy load we

switched to daily aggregated time series of energy load and

daily average temperature. Based on the above analysis of the

data and the literature review we have proposed some of the

models which can deal with the multi seasonality pattern of the

energy load and which can handle exogenous variables. In our

case we have tried the average daily temperature as external

variable in some of our models.

Fig. 10 Daily energy load ((aggregated energy load MW/day))

B. Statistical Techniques

First we focused our attention on statistical forecasting

models such as: Naïve, moving average, ARIMA, Exponential

Smoothing and Seasonal Naïve. In many situations is shown

that STL is a successful try on decomposing the time series

into their seasonal, trend, and remainder components. After

that STL can be used for modeling purposes. In this study we

have used the Hyndman-Khaldakar algorithm for performing

STL and ARIMA models [19], [20], [21]. In reference of [23]

the STL decomposition shows good performance when used in

statistical methods and time series with monthly seasonalities

but was not performing well in machine learning methods. The

choice of the best algorithm depends on the nature of the data

and the frequency as well. In precence of seasonality patterns

and trend ARIMA and exponential smoothing methods are the

ones which can perform good in prediction. In their work [20]

present a complete modeling framewrok for time series

exponential smoothing models.

The autoregressive integrated moving average

(ARIMA(p,d,q)) processes are a combination of

autoregressive (AR(p), where p-the degree of autoregressive

model) and moving average (MA(q), where q- degree of

moving average model) processes and d is the degree of

differences [22]. For implementation of ARIMA models in R

we have used forecast package in R which combines unit root

tests, minimization of the AICc and MLE to obtain the

ARIMA parameters and coefficient estimates [21]. There are

many models which use STL (Seasonal and Trend

decomposition using Loess- a method for estimating nonlinear

relationships.) to understand seasonal data and fit appropriate

models [24].

C. Deep Learning Techniques

Machine learning techniques and deep learning are

attracting more and more attention from researchers of many

fields. Especially in the forecasting field these methods have

passed through many competitions such as the M Competitions

[25], [26], [27]. Artificial neural networks are forecasting

methods that are based on simple mathematical models of the

brain. They allow complex nonlinear relationships between the

response variable and its predictors. There are many studies of

using deep learning methods in energy prediction and

reviewed by [28]. In their study [29] present a neural network

approach for short-term energy load prediction paying

attention to seasons and using temperature as an external

variable. They achieved reliable results for hour ahead load

prediction. In reference to the work presented by [30] they

agreed on the weakness of NNs when dealing with seasonality.

Many researches suggest removing seasonality before

modeling, to achieve better predictions. Testing were made by

[31] on this topic and they showed that for clear seasonality

patterns RNNs are adequate but when this is not the case then

a deseasonalisation technique should be used.

In this study we have considered Recurrent Neural Nets

(RNNs). The scheme of how this network performs is shown

in Fig.11.

Fig.11 RNN Architecture

Here, x’s in yellow are predictor variables, h’s in green are

hidden layers, and y’s in blue are predicted values.

Recurrent Neural Nets are essentially a bunch of neural nets

stacked on top of each other. The output of the model at

h1 feeds into the next model at h2 as shown. The goal of the

learning process is to find the best weight

matrices U, V and W that give the best prediction of y^(t),

starting from the input x(t), of the real value y(t).

Volume 4, 2022

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

Neural Network AutoRegression (NNAR) models are

developed on the principle of using lagged observations as

inputs to a neural network. They are feed-forward networks

with one hidden layer. These models perform good with

seasonal data, where it adds as input the last observed values

from the same season. In general the model NNAR(p, k) uses p

lagged inputs and k nodes in the single hidden layer. Seasonal

NNAR(p, P, k): with k-neurons in the hidden layer. and input

1, 2 2

( ,..., , , , )

t t t p t m t m t Pm

y y y y y y

     

. NNAR(p, P, 0)m model is

equivalent to an ARIMA(p, 0, 0)(P,0,0)m model but without

stationarity restrictions. More generally, an

NNAR(p,P,k)m model has inputs :

1, 2 2

( ,..., , , , )

t t t p t m t m t Pm

y y y y y y

     

and k neurons in the hidden

layer. If the values of p and P are not specified, they are

selected automatically [21]. For seasonal time series, the

default values of P is 1 and p is chosen from the optimal linear

model fitted to the data after seasonally adjusted. If k is not

specified then it is set to k=(p+P+1)/2 (which is rounded to

the nearest integer).

D. Evaluation metrics

The performance of the models presented in this study were

evaluated in terms of a number of metrics. The selection of the

most accurate model is made by analyzing and comparing

error measurements and information criteria for in-sample and

out-of-sample data. As well as extending personal judgment to

the advantages offered by each model based in the nature of

the data. The metrics used to assess and compare the various

methods are :

1ˆ

MAE X X

n





Maximum Absolute Error

1ˆ

()

ME X X

n





Mean Error

1100%

MAPE nX















Mean Absolute Percentage

Error

1ˆ

()

RMSE X X

n





Root Mean Square Error

,in sample naive

MAE

MASE MAE 



Mean Absolute Scaled Error

In many research studies there are many arguments of using

different accuracy measurements of the model. This depends

of course on the nature of the data and their complexity. In

reference of [32] MASE offers a straightforward indication on

the relative model performance compared with the naïve

benchmark. It is a scale-independent measure where a value

less than one indicates that the performance of the model is

better than the naïve benchmark on average. And a value

greater than one indicates the opposite. What is important is

the fact that this critical value should not conclude the

performance of the model but further analysis are suggested.

III. ANALYSIS OF RESULT

This section provides a comprehensive analysis of the

results obtained from the modeling process. Results in terms of

all error metrics used to evaluate the performance of the

models are shown in Table 1. The abbreviations used to denote

the “top model” selected from the work done in this study are

respectively: ‘Snaive’- seasonal naïve method, ‘STL+ARIMA’-

STL decomposition with ARIMA errors, ’Hybrid’- ensemble

model with combination of statistical and deep learning

models , ‘NNAR’- Neural network with autoregression,

‘NNAR-Xreg’- neural network with autoregression and average

temperature as external variable.

In Table 1, the best model based on the error is indicated in

boldface respectively for training and testing dataset.

Analyzing the values of error metrics for each model we

observe that for the training dataset STL+ARIMA seem to

perform better than seasonal naïve. On other hand, NNAR

with daily average temperature as regressor seem to perform

better than NNAR without regressors. The difference between

NNAR and NNAR-Xreg is not significant. In this situation we

may suggest adding other exogenous variables (such as

humidity) to explain daily energy load. Overall for training

dataset the neural network with average temperature as

external variable is significantly better compared to other

statistical models.

The MASE value for all proposed models is lower than 1

which suggest that all the models perform better than the naïve

benchmark on average. The situation changes apparently for

the testing data where we have approximately seven months of

observations (20% of the three years taken into consideration).

Investigating the lowest value of error measurements in this

part STL+ARIMA shows significantly better performance

compared to the other models. MASE is higher than 1 but

close to this value. Comparing the MASE value of

STL+ARIMA for the training data and testing data we may

gain confidence that this model outperforms the other models.

Again between NNAR and NNAR with exogenous variable the

first has a slightly difference in error values.

For a better understanding and comparison of the error

metrics for training and testing data we plotted the

performance displayed in Fig. 12.

Volume 4, 2022

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

TABLE I.

MODEL PERFORMANCE ERROR METRICS FOR TRAINING AND TESTING DATASET

Train

Test

Model

RMSE

MAE

MPE

MAPE

MASE

RMSE

MAE

MPE

MAPE

MASE

Snaive

1134

2105

1671

5.67

8.28

1.00

-6255

6255

-35.36

3.74

STL+ARIMA

6.98

401

298

0.00

1.57

0.18

-260

3185

2405

-2.58

1.44

Hybrid

277

646

501

1.36

2.49

0.30

-17959

18008

17959

-2561

2561

10.74

NNAR

1.16

201

138

-0.02

0.70

0.08

-18048

18132

18048

-2577

2577

10.80

NNAR-Xreg

0.93

160

115

-0.01

0.60

0.07

-18529

18620

18529

-2645

2645

11.09

500

1000

1500

2000

Hybrid NNAR NNAR-Xreg Snaive STL+ARIMA

Model

Value

Error

MAE

MAPE

MASE

MPE

RMSE

Training Error

-20000

-10000

10000

20000

Hybrid NNAR NNAR-Xreg Snaive STL+ARIMA

Model

Value

Error

MAE

MAPE

MASE

MPE

RMSE

Testing Error

Fig. 12 Model performance error metrics for training and testing dataset

In the left side of Fig.12 are displayed the metrics for the

training data and right the metrics for testing data. It is clear

that the NNAR with external variable outperformed the

traditional univariate methods in training dataset and it is

comparative to the hybrid model. The hybrid model was

obtained as a combination with equal weights of four models:

nnetar, stlm, tbats and snaive [33].

16000

20000

24000

2016 2017 2018 2019

Time

Energy load (MWh)

series

Test

Forecasts from NNAR(22,1,12)[365]

Fig. 13 Energy load forecast from NNAR with daily average

temperature as regressor (aggregated energy load MW/day)

In Fig. 13 is displayed the daily energy load prediction

obtained from a neural network model where daily average

temperature is used as an external variable.

As we mentioned above these models perform good with

seasonal data, where the last observed values from the same

season are added as input. For energy load the model has k=12

neurons in the hidden layer and use

1, 2 2

( ,..., , , , )

t t t p t m t m t Pm

y y y y y y

     

observations as input

where p=22,P=1 and m=365 daily seasonality.

Fig. 14 shows the energy load point forecast and confidence

intervals (80% and 95%) from STL+ARIMA model.

10000

15000

20000

25000

30000

35000

2016 2017 2018 2019

Date

Energy load (MWh)

series

Test

Forecast from STL+ARIMA(5,1,4)

Fig. 14 Energy load forecast from STL+ARIMA model with no-

regressor (aggregated energy load MW/day)

IV. CONCLUSIONS

Energy supply and demand plays an important role in the

economy of a country and the region. Predictions are

important for energy management and sustainable systems.

The motivation for this study was to address some of the key

issues with related to the ability of predicting energy load

using statistical and deep learning models. The work presented

here can be used as a reference from researchers and

practitioners working in the energy field and especially in the

Volume 4, 2022

ISSN: 2769-2507

Ιnternational Journal of Electrical Engineering and Computer Science (EEACS)

Balkan region.

We showed that due to the high complexity of the hourly

data and multiple seasonalities it was easy as a start of our

analysis to work with daily data. We performed many

statistical and machine learning models which are capable of

handling seasonality in time series. In our work we took into

consideration a decomposition method which would give

better performance of the modeling process. The models were

evaluated on error metrics and comparative view of in sample

and out of sample dataset. NNAR architecture was able to

outperform the statistical techniques for in sample data in

terms of all error metrics used at the performance evaluation

phase but STL decomposition with ARIMA error was the best

model when evaluated to the testing data. The proposed

models can be used as a short term or medium term prediction

models for energy load. Other exogenous variables can give a

better effect to the models.

ACKNOWLEDGMENT

The authors want to thank Faculty of Natural Science,

University of Tirana, Albania which has financially supported

the presentation of this work at the conference.

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