An Evidential Solar Irradiance Forecasting Method Using Multiple Sources of Information

Abstract

In the context of global warming, renewable energy sources, particularly wind and solar power, have garnered increasing attention in recent decades. Accurate forecasting of the energy output in microgrids (MGs) is essential for optimizing energy management, reducing maintenance costs, and prolonging the lifespan of energy storage systems. This study proposes an innovative approach to solar irradiance forecasting based on the theory of belief functions, introducing a novel and flexible evidential method for short-to-medium-term predictions. The proposed machine learning model is designed to effectively handle missing data and make optimal use of available information. By integrating multiple predictive models, each focusing on different meteorological factors, the approach enhances forecasting accuracy. The Yager combination method and pignistic transformation are utilized to aggregate the individual models. Applied to a publicly available dataset, the method achieved promising results, with an average root mean square error (RMS) of 27.83 W/m² calculated from eight distinct forecast days. This performance surpasses the best reported results of 30.21 W/m² from recent comparable studies for one-day-ahead solar irradiance forecasting. Comparisons with deep learning-based methods, such as long short-term memory (LSTM) networks and recurrent neural networks (RNNs), demonstrate that the proposed approach is competitive with state-of-the-art techniques, delivering reliable predictions with significantly less training data. The full potential and limitations of the proposed approach are also discussed.

Full text

Citation: Mroueh, M.; Doumiati, M.;
Francis, C.; Machmoum, M. An
Evidential Solar Irradiance
Forecasting Method Using Multiple
Sources of Information. Energies 2024,
17, 6361. https://doi.org/10.3390/
en17246361
Academic Editor: Daniel Pérez-López
Received: 5 November 2024
Revised: 11 December 2024
Accepted: 13 December 2024
Published: 18 December 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
Article
An Evidential Solar Irradiance Forecasting Method Using
Multiple Sources of Information
Mohamed Mroueh 1, Moustapha Doumiati 2,* , Clovis Francis 3and Mohamed Machmoum 4
1Triskell Consulting, 32 Rue Arago, 92800 Puteaux, France; mohamed.mroueh@non.se.com
2
IREENA Lab UR 4642, Electrical and Electronics Department, ESEO, 10 Bd Jeanneteau, 49100 Angers, France
3Arts et Métiers Paris Tech, Châlons en Champagne, Department of Design, Industrialization, Risk, and
Decision (CIRD), 51000 Châlons en Champagne, France; clovis.francis@ensam.eu
4IREENA Lab UR 4642, Nantes University, 37 Bd de l’université, 44602 Saint Nazaire, France;
mohamed.machmoum@univ-nantes.fr
*Correspondence: moustapha.doumiati@eseo.fr
Abstract: In the context of global warming, renewable energy sources, particularly wind and solar
power, have garnered increasing attention in recent decades. Accurate forecasting of the energy
output in microgrids (MGs) is essential for optimizing energy management, reducing maintenance
costs, and prolonging the lifespan of energy storage systems. This study proposes an innovative
approach to solar irradiance forecasting based on the theory of belief functions, introducing a novel
and flexible evidential method for short-to-medium-term predictions. The proposed machine learning
model is designed to effectively handle missing data and make optimal use of available information.
By integrating multiple predictive models, each focusing on different meteorological factors, the
approach enhances forecasting accuracy. The Yager combination method and pignistic transformation
are utilized to aggregate the individual models. Applied to a publicly available dataset, the method
achieved promising results, with an average root mean square error (RMS) of 27.83 W/m
2
calculated
from eight distinct forecast days. This performance surpasses the best reported results of 30.21 W/m
2
from recent comparable studies for one-day-ahead solar irradiance forecasting. Comparisons with
deep learning-based methods, such as long short-term memory (LSTM) networks and recurrent
neural networks (RNNs), demonstrate that the proposed approach is competitive with state-of-the-art
techniques, delivering reliable predictions with significantly less training data. The full potential and
limitations of the proposed approach are also discussed.
Keywords: machine learning; solar energy; belief functions theory; information fusion
1. Introduction
1.1. Context and Motivations
In recent years, interest in renewable energy sources has increased to overcome the
depletion of the world’s traditional energy supplies [
1
] (fuel, natural gas, coal, and even
uranium). In 2019, the European Commission presented the Green Deal [
2
] for Europe, a
road map to reach climate neutrality by 2050. To ensure reaching this goal, the European
Union (EU) executives set up the “Fit for 55” package [
3
] in July 2021. This package
includes, among other topics, a greater calling: to increase the share of renewable energies
on the continent to 40% by 2030, up from the initial 32%. These are some of the reasons
why electric utilities are focusing more on efficient, ecologically friendly, and cost-effective
solutions [4–9]
. One of the most significant advances associated with the energy transition is
implementation of the microgrid (MG) [
10
–
12
], which is a network structure that integrates
localized energy management systems with renewable energy sources in a decentralized
manner. An MG is a system which can be controlled and supplies a nearby region with
electric power by combining loads and distributing sources and power converters. Over the
course of the last several years, the prevalence of the MG idea has seen a substantial uptick.
Energies 2024,17, 6361. https://doi.org/10.3390/en17246361 https://www.mdpi.com/journal/energies

Energies 2024,17, 6361 2 of 31
The intermittent nature of renewable energy sources makes it difficult for MGs to keep up
with customer demand [
11
,
12
]. Consequently, it is necessary to have an energy management
system (EMS) which also includes an energy storage system (ESS). In most cases, ESSs
manage the power balance between production and consumption by storing electricity
during low-cost or off-peak hours and discharging it during high-cost or peak hours. This
allows the ESS to retain the power balance between generation and consumption. The
majority of the time, when it comes to energy storage technologies, having an accurate
prediction of the amount of power which will be produced enables a more effective use of
ESS units, which in turn extends their lifetime. Therefore, forecasting the production of
energy is essential for an EMS, as it may result in large cost savings, simpler maintenance
of grid components, and the avoidance of eventual faults in the MG.
Photovoltaic generators are one of the most well-known techniques for producing
renewable energy. Using the characteristics of semiconductors such as silicon, the solar
radiation is converted into electrical energy [
13
,
14
]. The generation of solar energy is highly
reliant on both the sun’s position and the meteorological conditions. Hence, utilizing mete-
orological characteristics and astronomical data about the sun is often required to forecast
solar energy production. However, since the model of energy production is dynamic,
nonlinear, and parameter-dependent, it is exceedingly difficult to forecast the amount of
electric power which will be generated. In addition, the proliferation of decentralized
energy resources raises the level of uncertainty in power networks, thus increasing the
complexity to provide an exact estimate of the amount of generated electricity. Further-
more, in real-world applications, the penalties for underpredictions and overpredictions are
drastically different, depending on the corresponding financial applications, which makes
prediction assessment more subjective. To tackle the challenges posed by the inherent
uncertainties in solar power generation, robust forecasting models can be constructed
utilizing time series weather data.
The primary contribution of this study is the proposal of a novel predictive method
based on evidence theory. This method demonstrates the capability to operate effectively
even when confronted with an incomplete learning database. It also exhibits low compu-
tational complexity, making it efficient even with limited training data. Additionally, the
suggested method is flexible and modular, and it proves to be competitive when compared
to other state-of-the-art machine learning approaches applied for energy solar production.
1.2. Brief Overview of Forecasting Techniques and Time Horizons
Accurate generation forecasting is essential for renewable energy systems, particularly
in MGs, where energy production is heavily influenced by natural variability and cannot be
easily controlled. According to [
14
–
16
], forecasting models are categorized based on their
prediction horizons, each tailored to specific operational and planning needs within MGs:
•
Very short-term forecasting covers a from few minutes to 1 h, being critical for real-time
MG operations, electricity market clearing, and immediate regulatory actions.
•
Short-term forecasting spans from 1 h to 1–2 days, supporting MG energy dispatch,
operational security, and load-balancing decisions.
•
Medium-term forecasting encompasses 5–7 days, aiding decisions on resource alloca-
tion, unit commitment, and storage optimization in MGs.
•
Long-term forecasting extends beyond 1 week, facilitating maintenance scheduling,
strategic planning, and overall operational management in MG systems. Forecasting
uncertainty increases with longer time horizons due to the unpredictable nature of
influencing factors.
Forecasting techniques can be broadly classified into three categories based on their ap-
proach and computational requirements [14]:
•
Physical models [
17
] rely on meteorological data, such as numerical weather predic-
tion (NWP), to estimate solar irradiance. These models incorporate local physical
influences, adapting data through solar PV models and power curves. While they pro-

Energies 2024,17, 6361 3 of 31
vide high accuracy, especially for long-term forecasting, they often require significant
computational resources, making them less practical for real-time MG applications.
•
Statistical models [
18
] use historical data to forecast solar irradiance and are well
suited for short-term predictions. These methods analyze time series data to identify
patterns, which are then used to forecast future values. Common statistical approaches
include regression models, persistence models, moving averages, and ARIMA. Al-
though computationally efficient, statistical models may lack precision compared with
physical models, particularly when dealing with complex and nonlinear data.
•
Intelligent techniques [
14
,
19
] are ideal for handling non-stationary and erratic time
series data, such as solar irradiance. These techniques, including neural networks, ge-
netic algorithms, and belief function theory (BFT), offer greater flexibility and reduced
computational complexity. Machine learning models, including hybrid approaches
which combine multiple techniques, are particularly effective for predicting solar be-
havior based on historical data, accommodating uncertainty, and improving forecast
accuracy in a dynamic MG environment.
Intelligent techniques and hybrid models have become increasingly favored due to their
ability to manage uncertainty, adapt to real-time changes, and optimize energy distribution
with minimal computational overhead. In this context, this study proposes an innovative,
intelligent evidential approach [20,21] to solar irradiance forecasting which is particularly
suitable for short-to-medium-term scenarios within MGs, enabling more reliable energy
management and decision making. The strategy consists of constructing many predictors,
each based on a different meteorological component, and then proceeding to the fusion
of all information sources using the BFT framework. Predictors utilize past information
(historical data) to predict the solar irradiance values.
1.3. Related Works
In the early 2000s, Cao et al. [
22
] presented a combination of an artificial neural
network and wavelet analysis to predict solar irradiance. This approach is typical of
sample data preparation utilizing wavelet transformation for forecasting. The anticipated
solar irradiance is simply the summation of all the forecasted components acquired by
the various recurrent neural networks (RNNs), whose time–frequency domains match
appropriately. When adjusting the weights and biases of the networks during network
training, discount coefficients are employed to account for the influence of various time
steps on the accuracy of the final prediction. On the basis of a mix of recurrent BP networks
and wavelet analysis, an enhanced model for forecasting solar irradiance was constructed.
However, the optimal updating of weights and biases was not studied.
More recently, in 2018, Qing et al. [
23
] provided a solar prediction method for hourly
day-ahead solar irradiance prediction using weather forecasting data, where the prediction
problem was formulated as a structured output forecasting model which independently
predicted several outputs at once. The prediction model was trained using long short-term
memory (LSTM) networks, which accounted for the interdependence between consecutive
hours within the same day. The proposed method was evaluated on a dataset collected on
the island of Santiago in Cape Verde. The results demonstrated that the proposed algorithm
was 18.34% more accurate than backpropagation algorithm in terms of root mean square
error (RMSE) by using about 2 years of training data to predict the half-year testing data.
In the following year, Husein et al. [
24
] opted to rely exclusively on meteorological
variables, such as the dry-bulb temperature, dew point temperature, and relative humidity,
for generating their solar irradiance forecasts, addressing the challenge posed by the ab-
sence of historical solar irradiation data in certain scenarios. To achieve this, they employed
a deep recurrent neural network architecture incorporating both long-term and short-term
memory (LSTM-RNN). This model was designed to predict the hourly solar irradiance
one day ahead, effectively capturing temporal dependencies in the meteorological data. To
comprehensively evaluate the proposed approach, six experiments were conducted using
weather station data from Germany, the USA, Switzerland, and South Korea, representing

Energies 2024,17, 6361 4 of 31
diverse climate types. The results demonstrated that the proposed method outperformed
feedforward neural networks (FFNNs) and achieved an RMSE as low as 60.31 W/m
2
.
Additionally, compared with the persistence model, it achieved an average forecast skill of
50.90%, with improvements of up to 68.89% on specific datasets.
Within the same year, Yu et al. [
25
] published an article presenting a technique based
on LSTM for short-term forecasts considering a timeline which spans global horizontal
irradiance (GHI) one hour in advance and one day in advance. A clearness index was
introduced as input data for the LSTM model to improve prediction accuracy on cloudy
days and to classify the type of weather by k-means during data processing, where cloudy
days were classified as cloudy and mixed (partially cloudy). This information was derived
from an analysis of the results of an ANN and SVR, as reported in the literature. The
authors stated that inaccurate forecasts typically took place on cloudy days.
After a short period, Wojtkiewicz et al. [
26
] evaluated the use of gated recurrent units
(GRUs) to predict solar irradiance and reported the results of utilizing multivariate GRUs
to estimate the hourly solar irradiance in Phoenix, Arizona. Using purely historical solar
irradiance data as well as the inclusion of external meteorological factors and cloudiness
data, the authors compared and assessed the performance of GRUs and LSTM. Their
discussions prove that the proposed deep learning methods could be further improved by
incorporating more detailed cloud cover-related features.
At the end of the same year, Aslam et al. [
27
] published a comparative study of
various deep learning approaches for the purpose of forecasting one-year-ahead hourly
and daily solar radiation using historical solar radiation data and clear sky global hori-
zontal irradiance. These approaches include gated recurrent units (GRUs), LSTM, RNNs,
feedforward neural networks (FFNNs), and support vector regression (SVR). According
to this study, the two most effective and state-of-the-art deep learning models are LSTM
and GRUs. The GRU has two gates unlike LSTM, which has three gates. This results in
reducing the complexity of the structure. Therefore, less operation is needed for GRUs
compared with LSTM.
In 2020, Hui et al. [
28
] proposed a probabilistic hybrid approach for solar irradiance
forecasting which integrates a deep recurrent neural network with residual modeling.
Specifically, an LSTM-based point forecast is generated using historical data along with
relevant features. These point predictions are subsequently utilized as inputs for estimating
residual distributions. The parameters of these residual distributions are then determined
via maximum likelihood estimation. The final probabilistic forecast is obtained by simulta-
neously considering both the point prediction and the corresponding residual distribution.
In the same year, Byung-ki et al. [
29
] suggested a model based on an algorithm for
long-term and short-term memory which made use of restricted input data as well as data
from other locations. The authors stated that it is feasible to develop a model with one-time
learning utilizing national and international data, and the suggested model has the ability
to predict solar irradiance by using weather predictions for the next day provided by the
Korea Meteorological Administration as well as daily solar irradiance.
More recently, the authors of [
30
] proposed similarity-based forecasting models
(SBFMs) for predicting photovoltaic (PV) power at a high temporal resolution, utiliz-
ing weather variables which were measured at a lower temporal resolution. As a case study,
the model forecasted the PV power generated by the solar panels installed on the rooftop of
a commercial building for the following day in five-minute intervals, considering various
scenarios of available weather data. The results indicated that the proposed SBFMs could
achieve a greater forecasting accuracy than several benchmark models by relying on only
two or three weather variables.
In [
14
], the authors presented an extensive review of forecasting models and perfor-
mance metrics documented in the literature, with a particular emphasis on short-term
forecasting for wind and solar power generation. It included a detailed analysis of the
data duration used by each model and provided a comparative evaluation of their perfor-
mance metrics through a comprehensive overview. In [
13
], various deep neural network

Energies 2024,17, 6361 5 of 31
(DNN)-based models were investigated for the estimation of prediction intervals (PIs) in the
context of regional solar and wind energy forecasting. Another research work [
31
] applied
deep learning techniques to predict photovoltaic energy generation in residential systems.
The study leveraged real-world data to evaluate the effectiveness of LSTM networks, convo-
lutional neural networks (CNNs), and hybrid convolutional-LSTM models across different
forecasting horizons for photovoltaic power production. The models were trained using
aggregated historical data from hundreds of residential PV systems within a region, and
their performance was assessed in terms of predicting energy generation both for the entire
region and for individual systems. The authors of [
19
] presented a novel CNN-CatBoost
hybrid approach for predicting solar radiation using 1 h weather observation data from
the Korean Weather Data Open Portal. The model predicts solar radiation based on extra-
atmospheric solar radiation and three weather variables: temperature, relative humidity,
and total cloud volume. The hybrid model surpassed the CNN-single model, achieving
improved average absolute mean error accuracy in solar radiation prediction. The research
presented in [
32
] introduced a systematic, robust approach to improving resilience against
missing features in energy forecasting applications using robust optimization. Specifically,
the authors developed a robust regression model designed to effectively manage missing
features during test time. The authors of [
33
] presented a comprehensive evaluation of
four prominent machine learning models, namely the bidirectional gated recurrent unit
(BiGRU), bidirectional long short-term memory (BiLSTM), the simple bidirectional recur-
rent neural network (BiRNN), and unidirectional LSTM, in the context of solar power yield
time series forecasting.
1.4. Contributions
As discussed previously, solar energy forecasting research has traditionally focused
on neural network architectures, with an increasing emphasis on deep learning techniques.
Although these methods achieve high accuracy, their performance is heavily dependent
on the availability of large, high-quality datasets. However, in practical applications, data
quality and availability are frequently hindered by challenges such as network latency, sen-
sor malfunctions, or data integrity issues. These challenges can lead to incomplete datasets
and reduced forecasting reliability, underscoring the need for alternative approaches. This
study introduces an innovative method grounded in the BFT framework which incorpo-
rates multiple predictive models to address critical issues in solar energy forecasting. The
proposed approach excels in managing uncertainty, effectively handling missing data while
maximizing the utility of available information. It also supports rapid real-time execution,
delivering reliable short-term forecasts in few seconds, a crucial advantage for dynamic
decision-making systems. The key contributions of this research work are as follows:
•
Competitive performance: The method demonstrates accuracy on par with traditional
techniques, achieving reliable forecasts even in challenging data environments. The
proposed method demonstrates promising results, achieving a root mean square error
(RMS) of 27.83 W/m
2
compared with the best result of 30.21 W/m
2
, reported in [
29
],
for one-day-ahead solar irradiance forecasting
•
Robust handling of incomplete data: The approach efficiently manages missing or
partial datasets, maintaining forecasting accuracy despite real-world data limitations.
•
Reduced data requirements: Compared with deep learning models, the method
delivers reliable results with significantly smaller datasets, addressing the practical
constraints of data availability.
By addressing these challenges, this study presents a resilient and efficient alternative to
conventional methods, enhancing the practicality and reliability of solar energy forecast-
ing systems.

Energies 2024,17, 6361 6 of 31
2. Material and Methods
2.1. Database
In this study, we used data collected from the Saaleaue weather station managed by
the Max Planck Institute of Meteorology [
34
]. This dataset was chosen in this study for
its completeness and public availability. Moreover, as the dataset includes a wide range
of features, it enabled meaningful comparisons of the proposed method’s performance
with other techniques in the literature which utilize different subsets of features. It is worth
noting that the proposed method would remain operational if applied to a different dataset.
This subsection provides geographical features and a structural presentation of the
employed database.
2.1.1. Geographical Location
The geographical characteristics are offered so that the outcomes of this research may
be properly interpreted. The weather station installation (see Figure 1) is located at the
coordinates 50°57
′
04.8
′′
N, 11°37
′
29.0
′′
E in the Jena Experiment field [
35
], near the Saale
River in Germany.
Germany
France
Austria
Czechia
Saaleaue WS
6°E 9°E 12°E 15°E
48°N
50°N
52°N
54°N
0 km 150 km
N
(a) (b)
Figure 1. Geographical location (a) and photograph (b) of the Saaleaue weather station in
Germany [34]
.
Table 1gives some relevant details about this region’s solar potential, and Figure 2
shows the solar trajectory from the perspective of the weather station during 2022.
Table 1. Solar features of the Saaleaue weather station location [36].
Feature Value Unit
Specific photovoltaic power output 1071.6 kWh/kWp
Direct normal irradiation (DNI) 995.0 kWh/m2
Global horizontal irradiation (GHI) 1082.3 kWh/m2
Global tilted irradiation (GTIopta) 1265.8 kWh/m2
Optimum tilt of PV modules (OPTA) 38/180 °
Air temperature 9.9 °C
Terrain elevation 138.0 m

Energies 2024,17, 6361 7 of 31
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
0
20
40
60
80
Elevation (in °)
140
150
160
170
Azimuth(in °)
Solar angles across the year
Elevation Azimuth
(a)
Observer
Sun Traj.
150°
120°
210°
E
240°
W
S
(b)
Figure 2. Solar angles (a) and geographical solar trajectory (b) from the Saaleaue weather station
location at 12:00 p.m. through the year 2022.
2.1.2. Data Structure
The database used in this study is presented as a compilation of freely accessible CSV
files. Each column represents a particular meteorological or solar component, whereas each
row corresponds to measurements made at the same time. The measurement’s date and
time are listed in the first column. The shortwave downward radiation (SWDR) column,
which represents the solar irradiance this work aims to predict, was considered the most
pertinent feature in our analysis. It holds significant relevance as it directly influences the
solar energy received at the Earth’s surface and plays a crucial role in various applications,
including solar energy forecasting. Therefore, this feature held paramount importance in
our analysis and served as the primary focus for prediction in this study. Based on their
availability ratio in the database, their predictability according to current climatographical
models, and their correlation with the SWDR values, we identified a particular set of
weather features useful for this study among those available (see Table 2).

Energies 2024,17, 6361 8 of 31
Table 2. Selected weather features.
Feature Unit
Atmospheric pressure mbar
Temperature °C
Relative humidity %
Specific humidity g/Kg
Dew point °C
Saturation pressure mbar
Vapor pressure mbar
Deficit pressure mbar
Vapor concentration mmol/mol
Air tight g/m3
Wind speed m/s
Precipitation mm
2.1.3. Preprocessing
There are different predispositions which must be taken into consideration in order
to use this database. Extreme outliers were eliminated first since they are indicative of
technical data acquisition issues. Then, a two-hour moving average filter was applied to
the SWDR signal to remove high frequencies caused by measurement noise and sensor
limitations. Next, the SWDR signal was resampled with a period of one hour to accom-
modate the needs of this study. Note that some gaps are present in the database. For
convenience, artificial measurement points were inserted in the series with no meaningful
values (NaN). For the rest of this paper, this processed signal is referred to as the “solar
irradiance”, denoted as yiwith i∈N.
2.2. Feature Extraction
In the context of this study, every irradiance value
yi
recorded at time
ti
(
i∈ {
0, 1, 2
. . . I}
)
was linked to a feature vector x
i
. The dimension of this vector represents the number of
calculated features, and the integer
i
refers to the index of the measure in the database. Since
it expresses measures with regularly sampled discrete signals, one can express
ti
as a function
of
i
and
T
, the time separating two consecutive samples (i.e., the sampling period), where
ti=t0+iT. In our case, Twas set to 1 h.
In this work, a set of 15 features are calculated, of which the first 12 represent normal-
ized values of every selected weather feature (see Table 2). Let W
i
be a weather feature
vector of dimension 12, corresponding to the instant
ti
and
Wi,j
, the
jth
component of W
i
.
The normalization method used here is based on unit scaling with scaling bounds defined
by lower and upper quantile at a 1/1000 level (see Equation (1)). This level is established
while considering the remaining outliers which need to be eliminated.
xi,j=Wi,j−aj
bj−aj
(1)
where
aj
and
bj
are quantiles of
Wi,j
for all
i
valuesat levels of 0.001 and 0.999, respectively,
and j∈ {1, 2 . . . , 12}.
The remaining three features represent normalized versions of time-related variables.
The “day cursor” is a normalized representation of the time of day (see Equation (2)), scaled
between 0 and 1 (e.g., 3:00 p.m. becomes 0.625). The “year cursor ” (see Equation (3)) is
a normalized representation of the day of the year, also scaled between 0 and 1 (e.g., 4
February becomes 0.096). Lastly, the “previous day irradiance” feature (see Equation (4))
directly uses the solar irradiance measured 24 h prior. The choice of a 24 h lag was
motivated by the significant daily cycle observed in the solar irradiance data, as indicated
by a prominent peak in the frequency spectrum (see Figure 3).

Energies 2024,17, 6361 9 of 31
xi,13 =ti(mod 1 day)
1 day (2)
xi,14 =ti(mod 1 year)
1 year (3)
xi,15 =yi−24 (4)
5 10 15 20 25 30
Period (in hours)
0
5
10
Amplitude
(in W/m2)
FFT on the solar irradiance signal
from 01/07/2015 to 01/07/2018
Figure 3. Fast Fourier transform of a three years of records on solar irradiance.
The objective of the forecasting task is to determine the value of solar irradiance
yi
based on the feature vector x
i
available at time
ti
. This explains the selection criteria
for the weather features (see Table 2), which rely on straightforward predictability. For
the proposed model to achieve accuracy, x
i
must consist of predictable variables derived
from validated climatological models. It is also important to highlight that the choice
of the predictive model affects the performance of the proposed forecasting approach,
emphasizing the need for further research to identify the most suitable model. This issue,
however, falls outside the scope of the present work. In the following, let
tp
represent
the present time and
tk
(
k>p
) denote the prediction time. The objective is to estimate
yk
at
tk
given x
k
, which is derived from predicted meteorological data and climatological
models, as well as x
i
and
yi
for all
(p−ℓ)<i≤p
. Here,
ℓ
corresponds to the length of the
training period divided by the sampling interval
T
, representing the number of samples
in the training dataset. This study primarily focuses on forecasting solar irradiance 1 day
ahead and 1 h ahead. However, the full potential of the proposed method is explored in
Section 3.4.
2.3. Building Basic Probability Assignment
This section presents a proposed technique for creating basic probability assignments
(BPAs) from the training dataset. For a comprehensive understanding of the belief function
theory, readers are referred to [
20
,
21
,
37
]. Since the objective was to estimate the value
of
yi
based on the feature vector x
i
, the issue could be split into separate estimations
of
yi
based on each component
xi,j
, making a substantially unjustified independence
assumption between features. However, this study reveals that this assumption has a
significant detrimental impact on the prediction performance. Therefore, the proposed
method suggests a different approach. Due to the high regularity in the signal to be
predicted, which is caused by some connections to physical and astronomical aspects, the
feature day cursor (indexed 13) is significantly crucial and needs to work with every other
feature. Therefore, there are 14 distinct subproblems, each of which is based on a feature
combined with the day cursor one. For each component except for the day cursor, a single
BPA was used as the basis for the estimation. Therefore, the subproblem indexed by
j

Energies 2024,17, 6361 10 of 31
(1
≤j≤
15,
j=
13) was defined to estimate
yk
with the only inputs being
xk,13
and
xk,j
as
the testing instance and xi,13,xi,j, and yifor all (p−ℓ)<i≤pas the training data.
In order to solve this subproblem, the first step was to keep only pertinent data
from the available training dataset for analysis. To accomplish this, a selection criterion
was derived using the distance between features in the training dataset (i.e.,
xi,j
,
xi,13
∀(p−ℓ)<i≤p
) and the ones provided at the time of prediction (i.e.,
xk,j
and
xk,13
). This
criterion includes the contribution of the day cursor feature with the actual one. Based
on the Euclidean distance metric (see Equation (5)), a selection of
yi
was retained for the
next step by comparing
di,j,k
with a tuning parameter
α
(see Equation (6)). The objective of
this distance function is to take into account only irradiance values measured around the
same time of day. Using this distance metric, only relevant information were selected from
the learning database, specifically for the purpose of feeding the mass synthesis approach
proposed in this research. The impact of the parameter
α
is investigated later in a dedicated
section (see Section 3.1.1):
di,j,k=q(xi,13 −xk,13))2+ (xi,j−xk,j))2(5)
Sj,k=nyi|di,j,k<αowith j=13 (6)
Second, using the kernel density estimation (KDE) method with a Gaussian kernel for
this subset of solar irradiance values (i.e.,
Sj,k
), a probability distribution function was fitted
(see Equation (7)). KDE is a non-parametric method for calculating a random variable’s
probability density function (PDF) which is widely used in the field of renewable energy
time series prediction [
38
,
39
]. Thanks to its flexibility, this method was suitable for the
application under inquiry:
fj,k(y) = 1
nσj,k√2π∑
yi∈Sj,k
e−1
2y−yi
σj,k2
(7)
with
n
being the number of elements in
Sj,k
and
σj,k
being a smoothing parameter also
known as the “bandwidth”, which is defined in this work by a rule-of-thumb formula
(see Equation (8)) [40]:
σj,k=(4/3)1/5
n1/5(n−1)1/2
v
u
u
u
t∑
yi∈Sj,k
y2
i−1
n
∑
yi∈Sj,k
yi

2
(8)
Next, the PDF curve was horizontally sliced to produce a series of areas, with the
surface of each representing a probability value. A tuning parameter designated as
N
predetermined the total number of slices. The impact of this parameter is further covered
in Section 3.1.4. Each slice was assigned to a set of values and a mass value (structure of a
BPA) as described by Equations (9) and (10). The mechanism of the BPA building step is
shown in Figure 4:
I(p)
j,k=ny|fj,k(y)>p∆j,ko∀p∈ {1, ..N−1}(9)
with
∆j,k=1
Nmaxnfj,k(y)∀y∈I(0)
j,ko
and
I(0)
j,k
representing the BPA universe set, which
was defined in our case as the interval [0,1000] (W/m2):
mj,kI(p)
j,k=ZI(p)
j,k
ming(p)
j,k(y),∆j,kdy(10)
with g(p)
j,k(y) = maxfj,k(y)−p∆j,k, 0.

Energies 2024,17, 6361 11 of 31
Selection
Training
irradiance
values
Testing
feature
value
Training
feature
values
Selected
irradiance
values
     

Fitting
PDF 
󰇛󰇜
Slicing
BPA

Figure 4. Mechanism of the BPA building step.
Example 1. In the example shown in Figure 5, the PDF was fitted using solar irradiance values
(i.e.,
yi
) selected based on their associated relative humidity values (i.e.,
xi,3
) when similar to the
actual one at the time 4:00 p.m. on 8 August 2017. The probability density curve was then divided
into three sections, with the bases of each section serving as the BPA’s focal elements and their
surface representing the BPA’s masses values. The BPA was then defined as follows:
m([220, 375])=0.14
m([81.4, 399])=0.345
m([0, 1000])=0.515

Energies 2024,17, 6361 12 of 31
0 100 200 300 400 500
Solar irradiance (in W/m2)
0
1
2
3
4
5
Probability density
10-3
Fitted pdf for relative humidity
Time : 08/08/2017 16:00
51.5 %
34.5 %
14.0 %
Figure 5. Slicing method to build BPA.
2.4. Complexity Simplification
In the previous section, when generating a BPA for each feature (aside from the
day cursor one), the number of focal elements was predefined by the tuning parameter
N
. We could anticipate that the computation entailed iteration over
N14
mass products
for the combination of evidence. The proposed approach became impractical due to the
exponential relationship’s potential to significantly increase calculation complexity. For
this reason, before moving on to the combination step introduced later in this section (see
Section 2.6), it is convenient to simplify each BPA whenever possible. Hence, any pair of
focal elements with enough similarities to be treated as one focal element will be found
using the simplification method. The two sets were then merged by taking their union,
after which the masses were added together. A set similarity metric (see Equation (11))
was used to measure the similarity across sets, and the measurement was compared to the
tuning parameter
β
. The effect of the parameter
β
on the overall method performance is
examined later in Section 3.1.2. The process is described in greater detail in Algorithm 1,
where the objective is to combine intervals which are similar, as keeping them separate only
provides a minimal amount of information that does not justify the increase in complexity:
SI(p1)
j,k,I(p2)
j,k=



I(p1)
j,k∆I(p2)
j,k


∀p1,p2∈ {0, 1, . . . N−1}(11)
where
| · |
is the normalized length of the set
·
and
∆
is the symmetrical set difference
operator. The normalization was performed over the BPA universe set [0,1000] (W/m2).
2.5. Discounting
In the BFT framework, this operation is heavily used. In general, and especially
when dealing with a large number of BPAs, it aims to reduce the confidence level of
BPAs and gives relatively good improvement to the combination. Typically, to operate
discounting, all BPA masses are reduced by the same ratio (i.e., discounting factor), and
lost masses are transferred to the mass of the ignorance set (i.e., universe set). In this study,
a novel discounting strategy is brought forth in which the discounting factor varies and
is dependent on the size of each set (see Equation (12)). The idea behind this discounting
method is that small sets need to be discounted more than bigger sets. This assumption is
compounded by the fact that small sets contribute more to the combination process and
can become unsettling if too much faith is put in them. Furthermore, the discounting factor
must be higher when the BPA mass calculation employs more data (i.e., when the selection

Energies 2024,17, 6361 13 of 31
phase retains more data). This was addressed by using a parametric relation, defined in
Equation (13):
m∗
j,kI(p)
j,k= (1−ζj,k)e−31−

I(p)
j,k

mj,kI(p)
j,k(12)
where
| · |
is the normalized length of the set
·
and
ζj,k
is the discounting factor [
41
]. In
addition, we have
ζj,k=1−
1−

Sj,k


ℓ

γ
(13)
where
| · |
is the number of elements in
·
,
ℓ
is the training dataset length, and
γ
is the
discounting power coefficient representing a tuning parameter for the proposed method.
This parameter’s effect on the efficiency of the suggested method is investigated later in
Section 3.1.3.
Algorithm 1 Complexity simplification
1: procedure SIMPL IFY(BPA, β)
2: n←|BPA.Sets|
3: for i←1to N−1do
4: if Massesi=−1then
5: j←i+1
6: while j≤ndo
7: if Massesj=−1 and i=jthen
8: s←S(BPA.Setsi, BPA.Sets j)▷Equation (11)
9: if s>βthen
10: BPA.Setsi←BPA.Seti∪BPA.Setsj
11: BPA.Massesi←BPA.Massesi+BPA.Massesj
12: BPA.Setsj←∅
13: BPA.Massesj← −1
14: j←1
15: else
16: j←j+1
17: end if
18: else
19: j←j+1
20: end if
21: end while
22: end if
23: end for
24: for i←1to Ndo
25: if BPA.Massesi=−1then
26: Delete BPA.Setsi
27: Delete BPA.Massesi
28: end if
29: end for
30: return BPA
31: end procedure
2.6. Combination
After determining all of the mass functions resulting from the 14 distinct subproblems,
we intended to combine them so as to make use of the knowledge which was brought on
by each of them individually. There are different combination methods which could have
been used, depending on the reliability of each source of information. In our instance, all
of the information sources were trustworthy, but there were many sources of information.
The Yager combination method [
42
–
46
] was utilized to prevent fading of the ignorance

Energies 2024,17, 6361 14 of 31
set mass across successive mass function combinations. Algorithm 2presents the pseudo-
code of the used combination method. It is important to keep in mind that the procedure
for simplifying the complexity was executed between each combination in order to stop
exponential development in the number of components in the final mass variables.
Algorithm 2 Combination
1: procedure COMBINE(BPA1, BPA2)
2: n1←|BPA1.Sets|
3: n2←|BPA2.Sets|
4: BPA.Set ←Empty Set
5: BPA.Masses ←Empty Set
6: k←1
7: for i←1to n1do
8: for j←1to n2do
9: BPA.Setsk←BPA1.Setsi∩BPA2.Setsj
10: BPA.Massesk←BPA1.Massesi×BPA2.Massesj
11: if BPA2.Massesk=∅then
12: BPA2.Massesk←[0, 1000]
13: end if
14: k←k+1
15: end for
16: end for
17: return BPA
18: end procedure
19: procedure COMBI NE ALL(BPAs)
20: n←|BPAs|
21: BPA.Set ← {[0, 1000]}
22: BPA.Masse ← {1}
23: for i←1to ndo
24: BPA ←COMBINE(BPA, BPAsi)
25: BPA ←SIMPLIFY(BPA, β)
26: end for
27: return BPA
28: end procedure
2.7. Pignistic Transformation
The integration of several mass functions resulted in a source of information which
was deeper in detail. Nevertheless, it was also portrayed by a mass function. It was required
to proceed to the transformation of the mass function in order to transfer the information
brought by it to a probabilistic representation. In most cases, the pignistic transformation
distributes any given mass value to a collection of events across the individual components
which make up that set in an equal manner. In the context of our research, this was carried
out in a continuous form, as demonstrated by Equation (14):
fk(y) = ∑
y∈I(p)
k
mkI(p)
k


I(p)
k


(14)
where m
k
is the final mass function obtained from the combination of all
m∗
j,k
(1
≤j≤
15,
j=
13), I
(p)
k
is the focal elements in m
k
,
|·|
is the normalized set length of
·
, and f
k
is the
PDF estimated for the predicted solar irradiance value at time tk[47].
2.8. Decision Making
Given the PDF derived in the previous step, the decision making phase entailed
picking a value for the solar irradiance to use as the expected one (i.e.,
ˆ
yk
). In most

Energies 2024,17, 6361 15 of 31
cases, this was accomplished by determining the solar irradiation value which resulted in
the highest possible probability density expressed in f
k(·)
. In our situation, the method
for making decisions was based on a customized calculation (see Equation (16)). This
calculation determined the expected value based only on the values which provided a
probability density that was greater than a predetermined threshold, fixed in our case to
90% of the maximal probability density in fk(·)) (see Equation (15)):
Ek=ny|fk(y)≥0.9 f(max)
ko(15)
ˆ
yk=REkyfk(y)dy
REkfk(y)dy(16)
2.9. Summary
A series of procedures was followed in order to create a forecast using the suggested
technique for the impending values of the solar irradiance. In the first step, the problem was
subdivided into 14 smaller issues, each of which was assigned a single feature in addition to
the one pertaining to the day cursor. Each individual sub-problem was supposed to result
in a distinct BPA. When developing a BPA, it is necessary to compare the dedicated feature
values in the training dataset to the one supplied at the time of prediction in order to choose
relevant solar irradiance values. These data were gathered, and then by using the KDE
approach, a PDF was produced. After this, the PDF was chopped horizontally, and each
of the produced pieces represented a different component of the BPA. The underside of
each chunk was considered to be a BPA set, whereas the surface was given the mass value
corresponding to that BPA set. After this, the BPA was made simpler to prevent complex
computations. The resulting mass function was then discounted by a designated factor,
and the Yager combination technique was applied to combine the results with the BPAs
which were previously produced. The final outcome was a new mass function which was
more informative than the starting ones. The pignistic transformation technique was used
to convert this mass function into a PDF, and then a prediction value was calculated using
a selected weighted average to obtain the final result. Table 3provides a comprehensive
display of all the symbols employed in the presentation of the proposed method. On the
other hand, Figures 6and 7illustrate the complete workflow of the proposed method,
offering a visual representation of the entire process. Tuning parameters are available in
Table 4.
Table 3. List of symbols.
Symbol Description
TSampling period
iIndex of measurement in the dataset
INumber of measurements in the dataset
tiith measurement time in the dataset
yiith measurement irradiance value in the dataset
xiFeature vector calculated at time ti
jIndex of feature or weather component
WiWeather features vector at time ti
Wi,jWeather feature jat time ti
xi,jFeature jat time ti
tpPresent time (0 ≤i≤I)
tkPrediction time (p<k≤I)
ℓNumber of elements in the training dataset
di,j,kDistance metric based on features jand 13
Sj,kSubset of data used for subproblem j
fj,k(·)Fitted probability density function from Sj,k
σj,kBandwidth for the fit process of Sj,k

Energies 2024,17, 6361 16 of 31
Table 3. Cont.
Symbol Description
I(p)
j,kBPA sets from subproblem jat tk(0 ≤p≤N−1)
mj,k(·)Mass function obtained from subproblem jat tk
S(·,·)Similarity function between two sets
m∗
j,k(·)Discounted mass obtained from subproblem jat tk
ζj,kDiscounting factor of subproblem jat tk
mk(·)Combination of all m∗
j,k(·)mass functions
fk(·)Calculated probability density function from mk(·)
I(p)
kFocal elements of mk(·)
Table 4. Proposed method’s tuning parameters.
Symbol Description Domain
αThreshold distance between features [0, 1]
βThreshold hamming distance between sets [0, 1]
γDiscounting power coefficient [0, +∞[
NNumber of focal elements in single BPA N
Combination
BPA
1
BPA
2
BPA
12
BPA
14
BPA
15
…
Mass function
Prediction
Transformation
PDF
Decision making
𝒎𝑘(⋅)
𝒇𝑘(⋅)
ො𝑦𝑘
Figure 6. Mechanism of the proposed method.

Energies 2024,17, 6361 17 of 31
Solar DB
Measured
Features
PDFs BPAs
Mass
function
Final PDF
Predicted
value
Selection
Slicing
Combining
TransformationDecision
Figure 7. Entire mechanism of the proposed method.
3. Results and Discussion
Several analyses addressing various aspects of the proposed prediction approach are
presented in this section. First, the parameters of the method were examined to determine
their impact on the overall performance. Subsequently, the key concept of aggregating mul-
tiple predictors was emphasized by comparing the accuracies of the individual predictors,
each based on distinct sources of information, with the predictor obtained by combining
them. The overall performance of the method was then compared to several recent state-of-
the-art techniques, validating its effectiveness. Finally, the method was tested to its limits to
showcase its full predictive potential. Simulations were performed on an Intel(R) Core(TM)
i7-8550U CPU 1.80 GHz 1.99 GHz processor using MATLAB R2021a software.
The evaluation of the prediction performance relied on various metrics, such as the
absolute mean error (AME), mean percentage error (MPE), mean bias error (MBE), and root
mean square error (RMSE). Among these, the RMSE is the most commonly used metric
in recent performance analyses, making it ideal for comparative studies. Consequently,
the RMSE (see Equation (17)) was selected as the primary performance measure in this
study [48]:
RMSE =v
u
u
t
1
h
p+h−1
∑
i=p
(yi−ˆ
yi)2(17)
with
tp
referring to the present time,
h
being the prediction horizon,
yi
being the real
irradiance at time
ti
, and
ˆ
yi
being the predicted irradiance. The other performance metrics
can be formulated as follows:
AME =1
h
p+h−1
∑
i=p|yi−ˆ
yi|(18)
MPE =1
h
p+h−1
∑
i=p
|yi−ˆ
yi|
yi
(19)
MBE =1
h
p+h−1
∑
i=p
(yi−ˆ
yi)(20)

Energies 2024,17, 6361 18 of 31
3.1. Impact of Parameters
The parameters of the proposed method are listed in Table 4. To assess the impact
of each parameter individually, each one was varied within a suitably designed range,
while the remaining parameters were kept constant based on a benchmark set of values.
The objective of these simulations was to provide deeper insight into the parameter cal-
ibration process of the proposed method and to justify the selection of the benchmark
values. The results from these simulations would serve as a basis for parameter calibra-
tion in the subsequent tests presented in this section. For each tuning parameter, solar
irradiance forecasting simulations using the proposed methodology were performed over
seven consecutive random days in August 2017. The forecast horizon was set to one day
ahead, and the training set consisted of data from the preceding 5 years. For each of the
seven simulations, the RMSE was calculated, and the average RMSE was evaluated across
different parameter values.
3.1.1. Threshold Distance Between Features (α)
The threshold distance between features, denoted by
α
, regulates the number of data
points selected from the training set to construct the BPA. Increasing
α
resulted in the
selection of more data points, while decreasing
α
led to fewer selected points. If
α
is too
small, then the model may choose an insufficient number of data points, causing overfitting
and reducing the generalization capability. Conversely, if
α
is set too high, excessive data,
including irrelevant points, may be included, diminishing the precision of the regression.
Therefore, an appropriate balance for αwas essential.
Using the simulation set-up detailed in Section 3.1, we evaluated the RMSE for differ-
ent values of α, as depicted in Figure 8.
0 0.05 0.1 0.15 0.2
values
0
200
400
RMSE (in W/m2)
Avg. RMSE according to
Figure 8. Impact of αon the proposed method’s performance.
The plot illustrates that for
α=
0, corresponding to no selected data points and leading
to a BPA of full ignorance, the prediction error was considerably higher than for positive
values of
α
. As
α
increased beyond 0.1, the RMSE began to rise, suggesting that too much
irrelevant data were being considered. From a methodological perspective, the optimal
value of αminimizes the RMSE, and in this case, the best performance was achieved with
α∗=0.025.
3.1.2. Threshold Hamming Distance Between Sets (β)
In the complexity reduction process (see Section 2.4), a set distance metric was intro-
duced to quantify the similarity between two sets within a BPA. This metric was compared
against a threshold
β
, which determined whether these sets should be aggregated into
a single set within the BPA. The selection of the
β
threshold is crucial because it directly
affects the computational complexity of the proposed method. If
β
is set too high, then no
sets are considered similar, making the simplification ineffective and leaving the problem
at its original complexity. Conversely, if
β
is too low, then an excessive number of sets are

Energies 2024,17, 6361 19 of 31
aggregated, leading to a loss of information within the BPA, which increases ignorance
and diminishes the performance of the method. As such, optimizing βrequires balancing
two competing objectives: minimizing computational complexity while maintaining the
performance of the proposed technique.
In Figure 9, the RMSE and execution time are evaluated across various values of
β
. The purpose of this measurement is to assess the impact of different
β
values on the
performance and computational efficiency of the proposed method.
0.95 0.96 0.97 0.98 0.99 1
values
54
56
58
60
RMSE (in W/m2)
Avg. RMSE according to
(a)
0.95 0.96 0.97 0.98 0.99 1
values
0
10
20
30
Execution time
(in seconds)
Avg. execution time according to
(b)
Figure 9. Impact of βon the proposed method performance (a) and the execution time (b).
The results show that the RMSE decreased as
β
increased up to 0.975, where it reached
its minimum, but rose for
β>
0.975, indicating that too much aggregation began to affect
the performance. Additionally, the execution time grew exponentially with an increasing
β
.
Since the trade-off between prediction accuracy and execution time is dependent on the
application, determining an objective optimal
β
may not be feasible. To address this, we
plotted the Pareto front in Figure 10, which highlights the bi-objective optimization problem.
In this analysis, the chosen threshold
β∗=
0.975 was found to be optimal, as 4.083 s of
execution time was negligible compared with the forecast’s usage frequency.

Energies 2024,17, 6361 20 of 31
1.5 2 2.5 3 3.5 4 4.5
Execution time (in seconds)
54
56
58
60
RMSE (in W/m2)
Pareto front
Figure 10. Pareto front with the average RMSE and the execution time according to variable β.
3.1.3. Discounting Power Coefficient (γ)
In the proposed method outlined in this paper, the BPA undergoes a discounting phase
(see Section 2.5) where additional ignorance is introduced to moderate its influence in the
combination process. This adjustment is controlled by a discounting coefficient, denoted as
γ
. If
γ
is set too low, then there is a risk of overreliance on potentially inaccurate outputs.
Conversely, excessively high
γ
values lead to an undesirable loss of critical information.
Thus, tuning γto find an optimal balance is essential.
Figure 11 illustrates the variation in the average RMSE as a function of different
γ
values. The high RMSE at
γ=
0 emphasizes the necessity of the discounting step. As
observed, the RMSE generally decreased as
γ
increased, though with greater instability
compared with other parameters. The optimal
γ
value was identified to be
γ∗=
90, as this
corresponded to the minimum RMSE value, highlighting its importance in achieving better
prediction accuracy.
0 20 40 60 80 100
values
48
50
52
54
RMSE (in W/m2)
Avg. RMSE according to
Figure 11. Impact of γon the proposed method’s performance.
3.1.4. Number of Focal Elements in Single BPA (N)
In the proposed method, the number of focal elements, denoted by
N
, is predefined
during the BPA construction phase (cf. Section 2.3). This parameter plays a key role in bal-
ancing the BPA’s expression of ignorance and its conformity to the underlying probability
density function. Additionally,
N
impacts computational complexity, requiring bi-objective
optimization analysis to weigh the trade-offs between performance and computational
efficiency. When
N
was set to larger values, the BPA became more dependent on the knowl-
edge encapsulated in the fitted probability density function, thereby reducing its level of
ignorance. However, excessive adherence to the probability density function is undesirable,
as it may not perfectly capture the underlying distribution. Furthermore, increasing
N

Energies 2024,17, 6361 21 of 31
led to a rise in computational complexity. Conversely, smaller
N
values injected more
ignorance into the BPA, which diminished the efficient utilization of available information.
The average RMSE across different values of
N
is presented in Figure 12a. It can be
observed that for
N≥
4, the RMSE increased, which indicates overreliance on the data
source during BPA computation. This undermines the effectiveness of using belief function
theory (BFT) to represent ignorance. The execution time for a one-day-ahead prediction
with various
N
values is shown in Figure 12b, demonstrating an exponential increase in the
computation time as
N
increased. This resulted from the higher complexity of combining
BPAs with more focal elements.
345678
N values
48
50
52
RMSE (in W/m2)
Avg. RMSE according to N
(a)
345678
N values
0
20
40
Execution time
(in seconds)
Avg. execution time according to N
(b)
Figure 12. Impact of N on the proposed method’s performance (a) and the execution time (b).
Given the objectives of minimizing the error and computational time, the values
N=
3
and
N=
4 were optimal for the Pareto front. The difference in execution time between
these two values was negligible, and the prediction process in the underlying application
required frequent updates. Thus, N∗=4 was chosen as the optimal value.
3.1.5. Conclusions
In this study on parameter impacts, a pragmatic individual optimization approach
was applied to each parameter. Through this method, simulations were conducted over
seven consecutive days for a one-day-ahead prediction, providing a reliable indicator of
the proposed method’s accuracy. Table 5presents the local variation in the mean RMSE for
parametric changes around the identified optimal values, thus offering insights into the
sensitivity of the predictor’s accuracy with respect to each parameter. This analysis helped
highlight which parameters exerted the most influence on the model’s performance and
required more precise tuning.

Energies 2024,17, 6361 22 of 31
Table 5. Study results for impact of parameters.
Symbol Optimum Sensibility Features
α0.025 3.1606 -High sensibility
-Bounded
β0.975 0.4294
-Affects execution time
-User-defined (Pareto)
-Bounded
γ90 0.1561 -High instability
-Low sensibility
N4 2.741
-Affects execution time
-High sensibility
-Discrete
3.2. Performance Analysis: Individual Versus Combined Predictors
In this section, the individual contributions of certain predictors are thoroughly ex-
amined to showcase the significance of combining the efforts of multiple forecasters. It is
essential to recognize that each predictor is associated with a specific feature. The primary
aim of this study was to illustrate the collaborative functionality of various predictors
when they were combined, thereby emphasizing the synergistic effects resulting from
their integration. For instance, the relative humidity-based predictor was anticipated to
demonstrate reduced accuracy during the winter months, as precipitation exacerbates
the decoupling between solar irradiance and humidity during this period. Conversely,
the atmospheric pressure predictor was expected to perform better in winter due to the
enhanced correlation it exhibits with solar irradiance under cloudy conditions.
To exemplify the seasonal behavior of these predictors, we present the performance
of two distinct predictors: one based on the relative humidity and the other based on
atmospheric pressure. This was accomplished by executing a one-day-ahead prediction
simulation for a randomly selected day in summer, as illustrated in Figure 13a, and for
another randomly selected day in winter, shown in Figure 13b.
The figures clearly demonstrate that the atmospheric pressure-based predictor exhib-
ited greater accuracy during the wet and cold winter months compared with the dry and
hot summer months, while the relative humidity predictor yielded the opposite results
during these same periods. This performance discrepancy underscores the necessity for em-
ploying a diverse set of predictors. Moreover, Table 6shows the performance of individual
forecasting predictors based on specific meteorological criteria compared to their combi-
nations. Notably, as shown in the last column of Table 6, the combined approach, which
incorporates all features, consistently outperformed most individual predictors in terms of
overall performance across the year. The simulations presented in Figure 13 are comple-
mented by the prediction curve generated from the integration of all predictors, as depicted
in Figure 14. It is worth noting that since this study focused on short-to-medium-term
forecasting rather than ultra short-term forecasting, we did not assess the computational
cost or benefit of using the full set of weather features versus a subset. However, the
execution time of the proposed method, which integrates all features, was discussed in the
previous section, demonstrating its suitability for short-to-medium-term forecasting, with
execution times consistently remaining below one minute.

Energies 2024,17, 6361 23 of 31
00:00 06:00 12:00 18:00 00:00
Time Aug 08, 2017
0
200
400
600
Irradiance (in W/m2)
One day ahead prediction using
Atm. Pressure (RMSE : 200.8 W/m2)
Rel. humidity (RMSE : 42.6 W/m2)
Real
Atm. pr.
Rel. hum.
(a) Summer
00:00 06:00 12:00 18:00 00:00
Time Dec 29, 2017
0
100
200
300
400
Irradiance (in W/m2)
One day ahead prediction using
Atm. Pressure (RMSE : 16.5 W/m2)
Rel. humidity (RMSE : 111.9 W/m2)
Real
Atm. pr.
Rel. hum.
(b) Winter
Figure 13. One-day-ahead predictions utilizing two different predictors: (a) summer and (b) winter.

Energies 2024,17, 6361 24 of 31
00:00 06:00 12:00 18:00 00:00
Time Aug 08, 2017
0
200
400
600
Irradiance
(in W/m2)
One day ahead prediction
RMSE : 29.11 W/m2
Real
Predicted
(a) Summer
00:00 06:00 12:00 18:00 00:00
Time Dec 29, 2017
0
200
400
Irradiance
(in W/m2)
One day ahead prediction
RMSE : 9.53 W/m2
Real
Predicted
(b) Winter
Figure 14. Prediction one day ahead with the proposed method in summer (a) and in winter (b).
Table 6. Performance (RMSE in W/m
2
) of several individual predictors in comparison with their
combination in 2017.
Date Precip. Atm.
Pressure
Rel.
Humidity Wind Speed Overall
29 Jan 23.36 20.06 119.80 19.41 28.09
18 Mar 27.69 20.52 31.64 25.98 15.96
4 May 21.30 54.31 20.43 42.71 25.35
21 Jun 270.40 261.20 113.80 260.40 74.28
8 Aug 209.10 200.80 42.60 199.60 29.11
24 Sep 40.09 33.84 40.55 37.98 21.30
16 Nov 11.75 10.78 10.19 10.23 10.12
29 Dec 8.475 16.49 111.90 16.45 9.535
3.3. Comparison with Other Recent State-of-the-Art Methods
This section aims to evaluate the performance of the proposed approach by conducting
a comparative analysis with recent studies addressing similar forecasting challenges. Given
that the existing methods in the literature employ varying input parameters, training
periods, and forecast horizons, we performed multiple simulations tailored to the specific
parameters of each state-of-the-art method. This ensured a fair and equitable comparison

Energies 2024,17, 6361 25 of 31
across the different forecasting approaches. To obtain a robust evaluation metric, the
performance of the proposed method was quantified using the mean of a set of RMSE
values derived from eight distinct forecast days evenly distributed throughout the year
at intervals of 45 days. This methodology was designed to yield performance metrics
representative of each method’s effectiveness across different seasons, thereby mitigating
the potential biases which could arise from season-dependent results. For methods which
implement one-day-ahead predictions, as referenced in previous works [
22
–
24
,
27
–
29
], the
comparative results are summarized in Table 7. The methods were organized according
to their year of publication, and the input parameters and training periods were also
outlined. It is essential to underscore that the proposed method was structured to learn
exclusively from a designated set of meteorological parameters, contingent upon the
specific input features utilized by each method. Furthermore, it is noteworthy that the
proposed approach may omit certain weather-related features if the comparative method
incorporates parameters not present in the utilized database [
34
], such as cloud cover. This
flexibility allows the proposed method to adapt to varying data availability, potentially
enhancing its applicability across different forecasting scenarios. Table 2lists all available
weather features in this study. It can be noticed that the proposed method performed better
than all of the other listed state-of-the-art methods.
Table 7. Comparison of the proposed method accuracy on a one-day-ahead forecast horizon with
other state-of-the-art methods.
References Model
Proposed Method’s Input Parameters
Adapted to Match State-of-the-Art
Methods for a Fair Comparison) Training Period
RMSE (in W/m2)
State of the Art Proposed
Cao et al. [22]RNN -Solar irradiance 6 years 44.326 30.60
Qing et al. [23]LSTM
-Dew point
-Humidity
-Temperature
-Visibility *
-Wind speed
2.5 years 76.245 34.17
Husein
et al. [24]LSTM
-Cloud cover *
-Humidity
-Precipitation
-Temperature
-Wind direction *
-Wind speed
15 years 60.310 32.68
Aslam et al. [27]
LSTM
GRU
RNN
-Solar irradiance 10 years
55.277
55.821
63.125
29.53
Hui et al. [28]LSTM
-Atmospheric pressure
-Cloud cover *
-Relative humidity
-Temperature
-Wind speed
10 years 62.540 30.63
Byung-ki
et al. [29]LSTM
-Atmospheric pressure
-Cloud cover *
-Humidity
-Precipitation
-Solar irradiance
-Temperature
-Wind speed
5 years 30.210 27.83
* Unavailable feature for the proposed method. Bold text refers to productive results. State of the art results
source: [49].
Several methodologies in the literature focus on forecasting solar irradiance for a
one-hour-ahead horizon [
25
–
27
]. The proposed method was similarly configured to predict

Energies 2024,17, 6361 26 of 31
solar irradiance in one-hour intervals, ensuring that the input features and training duration
were aligned with those of the established state-of-the-art methods.
To achieve a comprehensive assessment, the proposed method was rigorously evalu-
ated across various hours of the day. This approach allowed for a nuanced understanding
of its performance throughout different times, capturing the diurnal variability in solar
irradiance. The results of this evaluation are presented in Table 8, which illustrates that
the proposed method consistently outperformed all of the other methods analyzed, with
the exception of the latest method in the comparison. Notably, this latter method showed
only a marginal improvement of 5 W/m
2
in prediction accuracy, and this enhancement is
restricted solely to predictions made during midday. Additionally, the proposed method
was subjected to comparison against other studies which employed varied input datasets,
as outlined in Table 9. This comparative framework not only highlights the robustness
of the proposed methodology but also contextualizes its performance within the broader
body of research. To further facilitate external performance evaluations, Table 10 presents
additional assessment metrics which provide insight into various dimensions of the predic-
tion accuracy and reliability. This comprehensive approach underscores the significance of
the proposed method in advancing the field of solar irradiance forecasting.
Table 8. Comparison of the proposed method’s accuracy on a one-hour-ahead forecast horizon with
other state-of-the-art methods.
Authors
and
References
Model
Proposed Method’s
Input Parameters
(Adapted to Match
State-of-Art Methods for
a Fair Comparison)
Training
Period
RMSE (in W/m2)
State of
the Art Proposed
Average 07:00 a.m. 12:00 p.m. 5:00 p.m. Average
Yu et al.
[25]LSTM
-Air temperature
-Cloud type *
-Dew point
-GHI *
-Precipitation
-Relative humidity
-Solar zenith angle *
-Wind direction *
-Wind speed
5 years 41.370 27.14 46.31 32.25 35.23
Wojtkiewicz
et al. [26]
GRU
LSTM
-Air temperature
-GHI *
-Relative humidity
-Solar zenith angle *
11 years 67.290
66.570 33.48 51.43 19.95 34.95
Aslam
et al. [27]
LSTM
GRU
RNN
-Solar irradiance 10 years 108.89
99.722
105.28
2.930 39.76 21.71 21.46
* Unavailable feature for the proposed method. Bold text refers to productive results. State of the art results
source: [49].
Table 9. Method performances with respect to other state-of-the-art methods (one day ahead).
Authors and
References Model Database RMSE (in W/m2)
Sujan et al. [50] CSVR Daystar Energy Solar Farm 25.14
Minli et al. [51] EELM University of Macau PV System 46.97
Proposed Evidential Saaleaue WS 17.81
3.4. Full Potential of the Proposed Method
In this section, the proposed methodology is subjected to rigorous testing to assess
its limits by extending the forecast horizon beyond practical constraints and reducing
the availability of training data. The primary objective of this analysis was to gain a
comprehensive understanding of the inherent capabilities of the proposed method. It

Energies 2024,17, 6361 27 of 31
was generally anticipated that performance would deteriorate as the forecast horizon was
lengthened and the training duration was curtailed, but this section aims to provide a
quantitative characterization of this relationship.
Table 10. Performance of the proposed method using different evaluation metrics (RMSE was also
evaluated in Table 6).
Date RMSE
(in W/m2)
AME
(in W/m2)
MPE
(in %)
MBE
(in W/m2)
29 Jan 28.09 15.58 71.80 % −8.73
18 Mar 15.96 8.12 27.75 % −4.46
4 May 25.35 32.37 80.22 % −24.89
21 Jun 74.28 33.02 18.14 % −6.84
8 Aug 29.11 21.18 10.42 % −17.56
24 Sep 21.30 12.55 22.55 % −5.34
16 Nov 10.12 12.70 53.45 % −3.31
29 Dec 9.535 3.43 26.45 % 0.39
Figure 15a illustrates the RMSE as a function of the forecast horizon. Notably, the
RMSE increased significantly with extended forecast horizons; specifically, it rose from
15.14 W/m
2
in the 1 week forecast to 76.5 W/m
2
in the 23 week forecast when trained on
data spanning 1 year. Conversely, when the training period was reduced to just 4 days, the
RMSE escalated from 84.21 W/m
2
to 146.9 W/m
2
over the same forecast horizon. Further-
more, Figure 15b depicts the RMSE plotted against the training duration using a reverse
logarithmic scale. This figure reveals that transitioning from a training period of
1 year
to just 2 days led to a substantial increase in the RMSE from 27.22 W/m
2
for the 3 week
forecast to 138.7 W/m2and from 67.36 W/m2to 151.1 W/m2for the 4 month forecast.
0 5 10 15 20 25
Forecast horizon (in weeks)
0
100
200
RMSE (in W/m2)
Avg. RMSE according to forecast horizon
Training on 1 year
Training on 4 days
(a)
100
101
102
103
Training period (in days)
0
100
200
RMSE (in W/m2)
Avg. RMSE according to training period
3 weeks ahead forecast
4 months ahead forecast
(b)
Figure 15. Proposed method performance according to forecast horizon (a) and training period (b).

Energies 2024,17, 6361 28 of 31
To further elucidate the interactions between the forecast horizon and the training
period, a color map representation is provided in Figure 16. This visual aid facilitates a
deeper understanding of the simultaneous effects of varying both parameters on the overall
performance of the proposed forecasting method.
Avg. RMSE according to forecast
horizon and training period
0 10 20 30 40 50
Training period (in weeks)
0
10
20
Forecast horizon
(in weeks)
50
100
150
RMSE (in W/m2)
Figure 16. Proposed method performance according to both forecast horizon and training period.
4. Conclusions and Perspectives
This study presented an innovative evidential approach for predicting solar irradi-
ance. The method was formulated in a BFT framework, and its major contributions are
listed below:
•
Enhanced flexibility: The integration of belief functions with machine learning allows
for better management of missing data, making the model more adaptable to real-
world scenarios.
•
Improved data utilization: The method effectively utilizes available data even when
some data points are missing, improving overall prediction accuracy.
•
Integration of multiple predictive models: By incorporating various predictive mod-
els, each addressing different meteorological factors, the approach provides a more
comprehensive and accurate forecast.
•
Competitive accuracy: Performance evaluations show that the method achieved
an average root mean square (RMS) error of 27.83 W/m
2
, outperforming the latest
comparable method with an RMS error of 30.21 W/m2[29].
•
Short-to-medium-term forecasting capability: The approach is specifically designed
for short-to-medium-term solar irradiance forecasting, making it suitable for practical
applications in energy management and forecasting.
Future Directions and Applications
•
Feature selection: Future directions should focus on a more detailed examination of
the relationship between meteorological variables and solar radiation, potentially em-
ploying techniques such as principal component analysis to address multicollinearity
and enhance feature selection for more accurate predictions.
•
Real-time testing and sensor faults: Future efforts will focus on real-time testing of the
proposed model, with considerations for potential sensor faults.
•
Extension to other applications within the MG environment: The method’s versatility
may be extended to other domains, such as wind energy production and electrical
load forecasting.
•
Privacy in load forecasting: Although the datasets used are publicly available, ap-
plications like load forecasting may require decentralized learning approaches such
as federated learning to safeguard data privacy. The parametric nature of the BFT
framework supports its adaptability to decentralized and distributed applications,
making it particularly suitable for privacy-sensitive environments.

Energies 2024,17, 6361 29 of 31
Author Contributions: M.M. (Mohamed Mroueh): Conceptualization, Methodology, Software, Vali-
dation, Writing—original, and Validation. M.D.: Project administration, Conceptualization, Funding
acquisition, Supervision, Writing—review and editing, Validation. C.F.: Project administration, Su-
pervision, Conceptualization and Validation. M.M. (Mohamed Machmoum): Project administration,
Supervision, Funding acquisition, and Validation. All authors have reviewed and approved the final
version of the manuscript for publication.
Funding: This research work was funded by Fonds Européen de Développement Régional FEDER
and Région Pays de la Loire, France in the context of DETECT Project AAP RFI WISE. APC was
funded by Triskell Consulting.
Data Availability Statement: The original contributions presented in the study are included in the
article. Further inquiries can be directed to the corresponding author.
Conflicts of Interest: The authors declare no conflicts of interest.
Abbreviations
AME Absolute mean error
ANN Artificial neural network
BFT Belief function theory
BPA Basic probability assignments
CNN Convolutional neural network
DNN Deep neural network
EMS Energy management system
ESS Energy storage system
GRU Gated recurrent unit
KDE Kernal density estimation
LSTM Long short-term memory
MBE Mean bias error
MG Microgrid
MPE Mean percentage error
PDF Probability density function
PV Photovoltaic
RMSE Root mean square error
RNN Recurrent neural network
SBFM Similarity-based forecasting model
SWDR Shortwave downward radiation
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