New approach to estimate 5-min global solar irradiation data on tilted planes from horizontal measurement

Journal Pre-proof New approach to estimate 5-min global solar irradiation data on tilted planes from horizontal measurement

Abdelatif Takilalte, Samia Harrouni, Mohamed Rédha Yaiche, Llanos Mora-López PII:

S0960-1481(19)31185-1

DOI:

https://doi.org/10.1016/j.renene.2019.07.165

Reference:

RENE 12077

To appear in:

Renewable Energy

Received Date:

11 March 2019

Accepted Date:

31 July 2019

Please cite this article as: Abdelatif Takilalte, Samia Harrouni, Mohamed Rédha Yaiche, Llanos Mora-López, New approach to estimate 5-min global solar irradiation data on tilted planes from horizontal measurement, Renewable Energy (2019), https://doi.org/10.1016/j.renene.2019.07.165

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.

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New approach to estimate 5-min global solar irradiation data on tilted

2

planes from horizontal measurement

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Abdelatif Takilalte1.2*, Samia Harrouni1, Mohamed Rédha Yaiche2, Llanos Mora-López3

4 5 6 7 8 9 10 11 12 13 14

[email protected],[email protected],[email protected],[email protected] Laboratoire d’Instrumentation – LINS, Faculté d’Electronique et d’Informatique, Université des Sciences et de la Technologie Houari Boumediene-USTHB, P.0. Box 32, El Alia, Bab Ezzouar, 16111, Algiers, Algeria.

1

Centre de Développement des Energies Renouvelables, CDER BP 62 Route de l'Observatoire, Bouzaréah, 16340, Algiers, Algeria 2

Departamento de Lenguajes y Ciencias de la Computación, E.T.S.I. Informática, Universidad de Málaga, Campus de Teatinos, 29071 Malaga, Spain

3

15 16

Abstract

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This paper presents a new methodology to estimate global tilted irradiation in 5-min steps using only

18

global irradiation on the horizontal plane. The methodology is based on a combination of the two well-

19

known conventional Perrin Brichambaut and Liu and Jordan models. Tilted irradiation is of great

20

importance for the design and short-term performance assessment of fixed-tilt flat-plate collectors and

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photovoltaic (PV) systems. Intermediate key parameters of the state of the sky, referred to here as

22

cloudiness factors, are determined and introduced to transform isotropic models into an anisotropic

23

model. The results of the proposed model for all sky conditions with regard to normalized root mean

24

square error (nRMSE), relative percentage error (RPE), normalized mean absolute error (nMAE) and

25

coefficient of correlation (R2) range from 4.7 to 6.41%, 5.5 to 5.9%, 3.07 to 4.73% and 0.97 to 0.99,

26

respectively, which are very accurate results, especially for such a short time step. A comparison with

27

the best conventional models and even artificial neural network (ANN) models described in the literature

28

has confirmed that the developed model has smaller errors.

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Keywords: 5-min time step, global tilted irradiation, estimation, cloudiness factor.

30 31

1. Introduction

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The depletion of fossil resources and their direct influence on global environmental issues such as

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pollution and climate change have given rise to serious problems related to energy consumption. As a

34

result, the future of the world’s energy system will increasingly depend upon renewable energy. Among

35

renewable energy sources, solar energy is considered the most attractive due to its wide availability and

36

exploitation feasibility in many regions across the globe, particularly in sunny regions of the world such

37

as Algeria [1][2]. Solar energy can be converted into electricity by photovoltaic panels (PV) or into

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useful thermal energy by thermal solar collectors [2]. The output of these solar systems is calculated 1

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using global solar radiation on the tilted plane (GI), which can usually be estimated from the global

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horizontal irradiance (GHI) [3].

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Tilted global solar data are of great importance in various fields of research and for scientific designers

42

in order to size, simulate and optimize the performance of solar systems, as well as for other engineering

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applications [4]. However, due to the high cost of installing many pyrometers for different inclinations,

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there is a scarcity of this type of data and most radiometric and meteorological stations only record

45

radiation on horizontal surfaces [5]. Hence, researchers are resorting to increasingly precise

46

mathematical models to determine the tilted global solar radiation based on the measured horizontal

47

radiation [4][6].

48

It has been shown in the literature [7][8][9] that very short time-step estimations of solar global

49

irradiation on inclined collectors are required given that the finer the time scale, the higher the

50

coincidence between the results obtained in the solar system simulation and its reality. Therefore, the

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monthly average, or even better daily data, can provide an overall view of the performance of solar

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absorbers, but it is not sufficient as such data may lead to a significant under-sizing due to the impact of

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averaging [10]. In addition, hourly or more accurate 5-minute calculations are necessary and very useful

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for a more correct sizing and precise prediction of the behaviour, which enhances the management

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strategies and economic attractiveness of solar systems [11][12].

56

However, it is a difficult task to perform short-step calculations of the global radiation on the tilted

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plane from the horizontal plane due to the non-linear relationship, which is mainly related to the

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anisotropic phenomenon [13][14]. This phenomenon describes the inequality in situations regarding the

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inclination or measurement direction, where solar radiation properties or intensity systematically vary.

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In our context, we note that, contrary to horizontal surfaces, solar collectors mounted at a tilt for a

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specific orientation could not all face the sky [14] and hence receive less direct radiation. Moreover, it

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is important to take into account the incoming diffuse radiation reflected by the ground and scattered by

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clouds or aerosols. This anisotropic phenomenon is more pronounced when the time scale of conversion

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is smaller, especially when it is 5-min [8]. It is therefore difficult or even impossible to develop a

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straightforward model to ensure great accuracy when performing this conversion [8][11].

66

Alternatively, the mean values of solar irradiation for large time steps, such as monthly, daily or even

67

hourly, can be converted by many empirical models or linear regression models due to the effect of time

68

averaging and compensating. The distribution thus becomes rather isotropic [8].

69

Large-scale evaluation surveys of many conversion models have been reported in the literature, which

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mostly deal with monthly, daily and hourly [14][15][6][16] mean values of solar irradiation and, to a

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much lesser extent, with sub-hourly data such as 10-min or 5-min [11][8][17].

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Tilted global irradiation (GI) can be determined by “conventional” methods which generally combine

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two models [18]. Some of these methods require that the global horizontal irradiation (GHI), direct

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normal irradiation (DNI) and diffuse horizontal irradiation (DHI) reflected on a horizontal surface are

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known [18]. Separation models may predict DHI, and thus DNI or direct horizontal irradiation (𝑆ℎ) if 2

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they are not available, using measured GHI and other observable parameters (such as cloud fraction and

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clear sky irradiance) [10]. Other models transpose the horizontal irradiation components to the tilted

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global irradiation. The differences between the conventional methods used to estimate GI based on the

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available data are summarized in Fig. 1.

80

Horizontal plane

81

𝐺𝐻𝐼

82

GHI :

Global horizontal irradiation

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DHI :

Diffuse horizontal irradiation

84

DNI :

Direct normal irradiation

85

Sh :

Direct horizontal irradiation

86

h:

Solar altitude

87

GI :

Tilted global irradiation

88 89

𝐺𝐻𝐼 = 𝐷𝐻𝐼 + 𝑆ℎ

(4) (1)

𝐷𝐻𝐼

𝐷𝐻𝐼 = 𝐺𝐻𝐼 ― 𝑆ℎ

Separation

(5)

𝐷𝑁𝐼 𝑜𝑟 𝑆ℎ

(2)

(3)

𝑆ℎ = 𝐷𝑁𝐼 sin (ℎ)

Transposition

𝐺𝐼 Tilted plane

Measured data (input) Measured or estimated data Predicted data (objective)

90 91 92 93 94 95 96

Performed methods (5): Separation of DNI from GHI for further transposition to reach GI [19] . (4): Separation of DHI from GHI for further transposition to reach GI [20]. (2) (3): Transposition models, DH and DNI available [10]. (1), (2): Transposition model, GHI and DHI available [21]. (1), (4), (2): Separation model + Transposition model, only GHI available [14] [18]. (1): Direct transposition of GHI to reach GI [6].

Fig. 1. Differences between conventional methods to predict global tilted irradiation.

In the review of [19], 140 separation models

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to predict DNI from GHI were validated at 54 research-class stations from 7 continents. In [10], a total

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of 26 transposition models were reviewed and arranged using 18 case studies of tilt and azimuth angles

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from 4 sites. Most of the models focus on the diffuse irradiance received by a sloped plane because the

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diffuse solar component is more complicated to calculate [10]. The tests show that no universal model

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can be concluded.

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Another method used for this purpose consists of successively applying two types of models, the first

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of which estimates DHI from the measured GHI. Seven models of this type were tested in [20]. The

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second model calculated the tilted global irradiation from both the global horizontal and diffuse solar

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irradiations. In this context, 15 models were validated by the same authors [21]. Based on the results of

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the two previous studies, the authors evaluated 94 combinations for Ajaccio, France, in [18]. The

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optimum model showed a normalized root mean square error (nRMSE) of around 6% and a normalized

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mean absolute error (nMAE) of around 3.5%. The performance of 12 combinations was evaluated by

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[14] for hourly solar data collected from Padova, Italy and found to be between 6% and 7.2% for the

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nMAE and 6.4% to 8.7% for the nRMSE.

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Olmo et al. developed a direct and simple method in [6], which allows the hourly tilted global

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irradiation to be calculated using only the horizontal global irradiation, the sun’s azimuth and elevation

113

as input parameters. Their method achieved an accuracy of 10% for nMAE and 27% for nRMSE. In

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[16], the monthly average of daily global irradiation on a horizontal surface was first estimated with

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bright sunshine duration using empirical models in Terengganu, Malaysia, and the Olmo model is was

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then applied, obtaining an nRMSE of around 20%.

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The performance of empirical and traditional methods has proven to be limited [11][22]. Alternatively,

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artificial intelligence methods, particularly artificial neural networks (ANN), attracted increasing

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attention in the literature for estimating global solar radiation on tilted surfaces from the horizontal one

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and/or other parameters, given their efficiency in handling non-linear and complex relationships [23][5].

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Researchers have used ANNs extensively as an alternative tool, which can improve the accuracy of such

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a conversion, particularly on a short time scale [11][9][5]. Mehleriet al. [23] proposed an ANN model

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for predicting hourly slope radiation using global solar radiation and extraterrestrial radiation on a

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horizontal surface, the solar zenith angle and the solar incidence angle on a tilted plane. The authors

125

compared a selection of the most precise conventional models for the city of Athens, Greece, and showed

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that the ANN technique is more realistic as it performs better with a nRMSE equal to 15.35% and a R2

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of around 0.96. Similarly, Notton et al. developed an ANN based on total solar irradiance on a horizontal

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surface, extraterrestrial radiation, the zenith, the declination and inclination angles and time as the main

129

input variables to estimate the global radiation on the inclined surface [11][13][9]. The model was

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optimized using 5 years of data collected from Ajaccio, France, on horizontal, 45° and 60° inclined

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planes. Its performance was evaluated and compared to that of conventional empirical correlations for

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one-hour and 10-minute time steps in [11] and [9], respectively. Furthermore, the same configuration

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has been applied for other data in other sites [8] and [5]. The ability to estimate GI values on 36° tilted

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absorbers was tested in [8] using two years of 5-min solar data from Algiers, Algeria, which is the same

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data used in the present study. Moreover, in the case study performed on Mashhad, Iran, in [5], an ANN

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was developed and optimized using hourly data for one year of horizontal, 45° and 60° inclined global

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irradiance. The nRMSE of the optimal configuration for these studies was around 6% to 10 %, which is

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considered a good accuracy for such a short time step.

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Moreover, a comparative study was performed between a support vector machine (SVM) and ANN in

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[17] to make this conversion to 16° and 37.5° tilted angles at a 5-min time step for two locations in Saudi

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Arabia. The models were used to predict solar radiation on inclined surfaces. The authors concluded

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that the SVM is more robust than ANN in terms of calculation accuracy and rapidity. In addition, they

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showed that the SVM is easy to use as it is more stable during calculation and does not require a large

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amount of data for building the model.

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In sum, a wide range of studies have been performed to determine global solar irradiation on tilted

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planes from horizontal planes, but the majority of these methods have been applied to hourly or daily

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data. Our approach contributes to closing the gap in this research area where the 5-minute resolution is

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rarely addressed in the literature and in which algorithms based on hourly data do not perform

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satisfactorily at this high time resolution [9].

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Furthermore, the proposed method is very easy to implement as it uses simplified equations that do not

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require previous data (training data) to build the model, such as when ANNs or SVM are used. Instead,

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the proposed model only requires the current value of global horizontal radiation data to estimate its

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corresponding value on the tilted surface.

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The estimation is performed by modifying the clear sky models of both Perrin Brichambaut and Liu

155

and Jordan and taking into account new direct and diffuse cloudiness factors in order to improve the

156

accuracy. This hybridization option is commonly used in the literature to achieve the best models in all

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cases. In this study, we only use the first part of Perrin Brichambaut’s model, which describes the solar

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irradiation on horizontal plane. The direct and diffuse data obtained by the model serve as input to the

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transposition of the Liu and Jordan model to calculate its inclined values.

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Although it is possible to use only one of these models to calculate the clear sky horizontal and tilted

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irradiation, several studies have shown that the coupling of Perrin Brichambaut’s and Liu and Jordan’s

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models is efficient and leads to higher fitting of the clear sky irradiation [2], [24], [25], [26].

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This choice is also justified by the fact that Perrin Brichambaut uses many complex formulas to calculate

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inclined diffuse data unlike its counterpart transposition of Liu and Jordan, which is simpler and more

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accurate. This new methodology includes new direct and diffuse cloudiness factors that enable the clear

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sky models to model all type of skies with high accuracy.

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The rest of this paper is organized as follows: section 2 describes the materials and methods, the data

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used, the models on which our proposal is based, the proposed methodology and the metrics used for

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validation. The results and performance evaluation of the proposed model are discussed in section 3 and

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a comparison is made with other models in the literature. Finally, section 4 concludes and sets out some

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future perspectives.

172 173 174

2. Materials and methods 2.1. The sites and solar radiation databases

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The data available for this study are the 5-min time-step measurements of global irradiation on the

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horizontal plane and tilted surface. The data were collected at two stations in Algiers and Ghardaia,

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Algeria, and one station in Malaga, Spain. The data from Algiers were collected at the Renewable

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Energies Development Centre (CDER), which has a meteorological and radiometric station in

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Bouzareah-Algiers, while the data from Ghardaia were recorded at the Applied Research Unit for

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Renewable Energies (URAER). The data from Malaga were recorded at the Photovoltaic Laboratory of 5

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the University of Malaga. The measurements of global irradiation on the horizontal and on the tilted

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surface were taken every minute using Kipp & Zonen CM11 pyranometers and the 5-min energy was

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the recorded data.

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Table 1. Geographic characteristics of the sites, data record period and inclination angle of the surface. Stations Algiers Ghardaia Malaga

Geographic characteristics Latitude (°) 36.8 32.37 36.46

Longitude (°) 3.17 3.77 4.29

Altitude (m) 347 450 60

Data record period (month/day/year)

Tilted Angle (°)

04/08/2011 to 04/08/2013 01/01/2005 to 01/31/2006 10/01/2012 to 09/30/2014

36 32 32

185 186 187

Table 2. Monthly and annual values of daily global horizontal irradiation, ambient temperature and relative humidity in the locations under study.

Algiers Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. Nov. Dec. Ann.

Malaga

Ghardaia

Irradiation (kWh/m2)

Temperature (°C)

Humidity (%)

Irradiation (kWh/m2)

Temperature (°C)

Humidity (%)

Irradiation (kWh/m2)

Temperature (°C)

Humidity (%)

2.19 2.97 4.13 4.91 6.01 6.17 7.05 6.36 5.12 3.53 2.72 2.06 4.43

11.56 12.5 15.5 18.2 18.5 21.5 24.3 25.2 23.3 19.4 15.1 12.3 18.1

76.4 77.9 76.3 74.6 74.3 70.0 68.5 68.8 70.0 72.5 74.7 77.1 73.42

3.7 5.2 5.6 5.3 6.3 7.1 7.4 7.0 6.9 5.8 4.2 4.2 5.7

15.7 15.7 17.2 19.3 22.6 25.6 29.0 29.5 27.1 23.1 17.1 15.7 21.5

67.6 57.9 57.8 66.9 58.3 58.7 47.5 55.0 53.4 60.5 60.1 57.3 58.4

3.9 5.0 6.0 7.3 7.6 8.1 7.7 7.1 7.0 5.9 4.2 3.8 6.1

16.6 17.0 18.1 22.1 28.2 32.4 35.7 34.3 28.89 24.6 17.0 16.5 24.2

52.8 42.4 30.3 30.4 22.6 18.8 17.4 21.0 32.6 36.5 39.7 52.6 33.1

188 189

Table 1 summarizes the main geographical characteristics of the stations as well as the available time

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period for ground measurements and the inclination angles of the tilted planes in each station.

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These three locations have different climatic conditions. Algiers is located in the northern part of Algeria

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near the Mediterranean Sea. Consequently, it has a Mediterranean climate with dry and hot summers

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and damp and cool winters. Ghardaia is a province of Algeria situated in the southern and sunny part of

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the country. It is characterized by an arid climate, which exhibits mild and dry climatic conditions.

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Malaga is a city in southern Spain located on the Mediterranean coast. It is characterized by a

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Mediterranean climate with hot summers and warm winters. Table 2 shows the monthly and as well as

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the annual mean values of the main daily radiometric and meteorological parameters of the three sites.

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The parameters for the three sites indicate their different climatic conditions. Specifically, the annual

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global irradiation values range from 4.4 to 6.1, temperatures range from 18.1 to 24.2 and humidity values

200

range from 33.1 to 73.42.

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The quality of the raw data used is a crucial factor in the precision of the developed model [27], [28].

202

Generally, the data cleaning procedure aims to enhance the data quality by checking and filtering any

203

uncertainty or errors possibly due to instrument malfunctioning [28]. To overcome this issue and extract

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missing or unreliable values, data are checked every 5 minutes [8]. Therefore, we excluded from the

205

data series the outlier identified as the values for the clearness index (Kt) outside the range

206

0.015 < Kt < 1. It is worth mentioning that the clearness index (Kt) is calculated as the ratio between

207

GHI to the incoming solar radiation on a horizontal surface at the top of the earth’s atmosphere (Go):

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Kt = GHI/Go [29]. Moreover, to discard data collected close to sunrise or sunset and to avoid the mask

209

effect of the environment and the non-reliable response of pyranometers at a low elevation angle (cosine

210

effect) that introduce some errors [30], we only use records with a solar elevation angle h > 15° [30]

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following the proposal of [31][30]. This choice is also justified because there is no significant solar

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radiation to utilize the received irradiance during these periods [31].

213

2.2. Clear sky approach

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Solar global radiation under clear sky conditions can be estimated from direct solar radiation and

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diffuse solar radiation using Perrin Brichambaut’s model. The global irradiation, Gh, received on a

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horizontal plane is given by: 𝐺ℎ = 𝑆ℎ + 𝐷ℎ

(1)

217

Where Sh is the direct solar radiation and Dh the diffuse solar radiation.

218

Direct solar radiation under clear sky conditions obtained on a horizontal plane is given by [32]:

[

9.4

𝑆ℎ = 𝐼0𝐶𝑡 ― 𝑠 𝑒𝑥𝑝 ― 𝑇𝐿∗ (0.9 + 0.89𝑧sin (ℎ))

―1

]cos (𝑖),

(2)

219

where i is the incidence angle. Knowing that for a horizontal plane, we have: cos (𝑖) = sin (ℎ).

220

𝐼0: the solar constant, which is defined as the energy flux received by a unit surface. In our case, the

221

value that was selected is 1367 W/m2.

222

𝐶𝑡 ― 𝑠: earth-sun distance correction.

223

The atmospheric turbidity factor allows us to calculate the direct and diffuse irradiation received on a

224

horizontal plane in clear sky. Absorption and diffusion caused by the constituents of the atmosphere can

225

be expressed by this factor [32][1][33]. In this model, the Linke turbidity factor 𝑇𝐿∗ is given by the

226

following relation: 𝑇𝐿∗ = 𝑇0 + 𝑇1 + 𝑇2,

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(3)

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𝑇0: is the atmospheric turbidity factor caused by gas absorption and by fixed components of the

228

atmosphere to ozone and especially by steam. A model of this factor based on only geo-astronomical

229

parameters allowed us to put forward the following expression: 𝑇0 = 2.4 ― 0.9sin (𝜑) + 0.1(2 + sin (𝜑))𝐴ℎ𝑒 ― 0.2𝑧 ― (1.22 + 0.14𝐴ℎ𝑒)(1 ― sin (ℎ))

230

(4)

Where: 𝐴ℎ𝑒 = sin ((360/365) (𝑁 ― 121)),

(5)

231

𝑧: the altitude (km),

232

𝑁 : the number of days in the year,

233

𝜑: the latitude of the site (°),

234

ℎ: the height of the sun (°),

235

𝑇1: the atmospheric turbidity corresponding to the absorption by atmospheric gases (O2, CO2, and O3)

236

and molecular Rayleigh scattering given by the approach: 𝑇1 = 0.89𝑧

(6)

237

𝑇2: the atmospheric turbidity relative to the aerosol scattering coupled with a slight absorption

238

(depending on both the nature and the amount of aerosols), a factor which is a function of the Ångström

239

atmospheric turbidity coefficient. In the absence of atmospheric turbidity measurements the following

240

formulation is adopted: 𝑇2 = (0.9 + 0.4𝐴ℎ𝑒)(0.63)𝑧

241

(7)

The diffuse solar radiation on a horizontal plane is given by the following expression[2]: 𝐷ℎ = 𝐼0exp ( ―1 + 1.06log (sin (ℎ))) + 𝑎 ― 𝑎2 + 𝑏2

(8)

𝑎 = 1.1

(9)

𝑏 = log (𝑇𝐿∗ ― 𝑇0) ― 2.8 + 1.02(1 ― sin (ℎ))2

(10)

242

This theoretical approach, particularly the solar irradiance incident on a horizontal surface, has been

243

successfully validated in several papers for a completely clear sky [26][25].

244 245

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2.3. Estimation of solar radiation on tilted surface

247

The values obtained from the Perrin Brichambaut model (the direct and diffuse horizontal irradiation

248

𝑆ℎ and 𝐷ℎrespectively) are used to achieve the best possible estimation of the solar radiation incident on

249

an inclined plane by an angle 𝛽 using Liu and Jordan’s model.

250

The general expression of Liu and Jordan is as follows [2]: 𝐺𝑖= 𝑆𝑖 + 𝐷𝑖 + 𝐷𝑟𝑒,

251

The direct solar radiation incident on an inclined plane is expressed by the following relationship: 𝑆𝑖 = 𝑆ℎ𝑅𝑏,

252

(12)

Where the factor of inclination 𝑅𝑏of the direct radiation is [2]: 𝑅𝑏 =

cos (𝜑 ― 𝛽)cos (𝛿)cos (𝜔) + sin (𝜑 ― 𝛽)sin (𝛿) , cos (𝜑)cos (𝛿)cos (𝜔) + sin (𝜑)sin (𝛿)

253

With:

254

𝛽: the inclination angle of the plane (°).

255

𝛿: declination of the sun (°),

256

𝜔: the hour angle (°),

257

On the other hand, the diffuse solar radiation on an inclined surface is [2]:

(1 + cos2 (𝛽))

𝐷𝑖 = 𝐷ℎ 258

(11)

(13)

(14)

The following expression represents the reflected solar radiation on a tilted plane [2]: 𝐷𝑟𝑒 = 𝜌(𝑆ℎ + 𝐷ℎ)

(1 ― cos2 (𝛽))

(15)

259

Where 𝜌 is the albedo of the site.

260

Regarding the estimation of solar irradiance for different incident angles and directions for a completely

261

clear sky, the model of Liu and Jordan has also been selected and validated [26][25][34].

262

2.4. Model to estimate the cloudiness of the sky

263

R. Yaiche et al. in [2] developed a method that allows solar radiation incident on a horizontal or tilted

264

plane to be calculated for different orientations from hours of sunshine. The method is based on the

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theoretical models of both Perrin Brichambaut and Liu and Jordan [32], which are only valid for a

266

completely clear sky.

267

Their proposal is a new approach to determine the cloudiness of the sky. They propose the use of

268

different types of sky and use 𝑁𝑖 to denote the direct cloudiness factor and 𝑁𝑑 to denote the diffuse

269

cloudiness factor. The values they propose corresponding to different types of sky are summarized in

270

Table 3 and the proposed equations have been proven and used in [2] and [24]. In [2], the authors attempt

271

to draw the annual global solar irradiation maps for Algeria at any inclination and orientation based on

272

sunshine duration using this model. In [24], they carry out an annual estimation of global irradiation on

273

the horizontal plane with a comparative study using satellite data for Algeria.

274

Table 3. The direct cloudiness factor and diffuse cloudiness factor values [2].

Number of okta 0 0/1 0/1 0/1 0/1 0/1 1 1/2 1/2 1/2 1/2 1/2 2 2/3 2/3 2/3 2/3 2/3 3 3/4 3/4 3/4 3/4 3/4

Type of sky

𝑁𝑖

𝑁𝑑

Completely clear sky Clear sky

1 0.9792 0.9583 0.9375 0.9167 0.8958 0.8750 0.8542 0.8333 0.8125 0.7917 0.7708 0.7500 0.7292 0.7083 0.6875 0.6667 0.6458 0.6250 0.6042 0.5833 0.5625 0.5417 0.5208

1 1.0208 1.0417 1.0625 1.0833 1.1042 1.1429 1.1637 1.1845 1.2054 1.2262 1.2470 1.3333 1.3542 1.3750 1.3958 1.4167 1.4375 1.6000 1.6208 1.6417 1.6625 1.6833 1.7042

Partly cloudy sky Partly cloudy sky Partly cloudy sky

Partly cloudy sky

Partly cloudy sky Partly cloudy sky

Number of okta 4 4/5 4/5 4/5 4/5 4/5 5 5/6 5/6 5/6 5/6 5/6 6 6/7 6/7 6/7 6/7 6/7 7 7/8 7/8 7/8 7/8 7/8 8

Type of sky

𝑁𝑖

𝑁𝑑

Moderately cloudy sky Cloudy sky

0.5000 0.4792 0.4583 0.4375 0.4167 0.3958 0.3750 0.3542 0.3333 0.3125 0.2917 0.2708 0.2500 0.2292 0.2083 0.1875 0.1667 0.1458 0.1250 0.1042 0.0833 0.0625 0.0417 0.0208 0

2.0000 2.0208 2.0417 2.0625 2.0833 2.1042 2.1250 2.1458 2.1667 2.1875 2.2083 2.2292 2.2500 2.2708 2.2917 2.3125 2.3333 2.3542 2.3750 2.3958 2.4167 2.4375 2.4583 2.4792 2.5

Cloudy sky Cloudy sky

Cloudy sky Cloudy sky Very cloudy sky Very cloudy sky

Covered sky

275 276

2.5. Proposed methodology

277

There are many different types of sky and many different mathematical methods used to describe them

278

in terms of cloud cover as this is a major factor in the Earth’s climate. Cloudiness refers to the fraction

279

of the sky obscured by clouds when observed from a particular location and is expressed in octas [2]. In

280

our contribution, the first step is to estimate the cloudiness factors (𝑁𝑖∗ , 𝑁𝑑∗ ) every 5 minutes (see Step

10

Journal Pre-proof 281

1 in Fig. 2) by comparing the measured horizontal global irradiance data, 𝐺ℎ𝑚and to the calculated values

282

given by the following model of R. Yaich, which is a modified version of Perrin Brichambaut’s clear

283

sky model [2][24]: 𝐺ℎ𝑐 = 𝑁𝑖𝑆ℎ + 𝑁𝑑𝐷ℎ

(16)

284

The comparison is performed by inserting in Eq. 16 the different possible couples of (𝑁𝑖, 𝑁𝑑) from Table

285

3 and adopting the factor couple (𝑁𝑖∗ , 𝑁𝑑∗ ) that is the nearest value obtained of 𝐺ℎ𝑐 to the measured one

286

𝐺ℎ𝑚. That is, (𝑁𝑖∗ , 𝑁𝑑∗ ) = 𝑀𝐼𝑁(𝑁𝑖, 𝑁𝑑){𝑒𝑟𝑟 = |𝐺ℎ𝑚 ― 𝐺ℎ𝑐(𝑁𝑖, 𝑁𝑑)|}

(17)

287

Our proposal is to introduce corrections to the clear sky models of Liu and Jordan to obtain the 5-min

288

tilted global solar radiation data in all possible sky conditions using the formula (see Step 2 in Fig. 2): 𝐺𝑖𝑐∗ = 𝑁𝑖∗ 𝑆𝑖 + 𝑁𝑑∗ 𝐷𝑖 + 𝐷𝑟𝑒𝑓

(18)

289

As an example of the process, the different parameters proposed were estimated for a 5-minute,

290

horizontal global radiation value, Ghm. Let Ghm = 74.55 Wh/m2. Using equations (2) and (8), the values

291

of Sh and Dh, respectively, have been estimated (Sh= 69.74 Wh/m2 and Dh = 8.25 Wh/m2, horizontal

292

surface). To estimate the key parameters 𝑁𝑖∗ (direct) and 𝑁𝑑∗ (diffuse), the calculated values Sh and Dh

293

were then multiplied by all the Ni and Nd pairs obtained from Table 3 (from clear sky, whose values are

294

1 and 1 to overcast sky whose values are 0 and 2.5) using Equation (16) to obtain all possible values of

295

Ghc. Among all these estimated values of Ghc, the one that is most similar to the real value of Ghm was

296

selected (Eq. 17). In the example, this corresponds to the pair 𝑁𝑖∗ = 0.9375 and 𝑁𝑑∗ = 1.0625. Then, the

297

transposition formulas of Liu and Jordan were used to calculate the clear sky direct Si (Eq. 12), diffuse

298

Di (Eq. 14) and reflected Dref (Eq. 15) solar irradiations on the tilted plane from the previously estimated

299

clear sky direct and diffuse horizontal irradiation, Sh and Dh. These values are Si = 79.99 Wh/m2,

300

Di = 7.43 Wh/m2 and Dref = 3.48 Wh/m2. The sum of these components is the clear sky global irradiation

301

on the tilted plane (Eq. 11). Finally, the estimated value of the global irradiation on the current sky

302

situation can be obtained by multiplying the clear sky direct irradiation, Si, and diffuse irradiation, Di,

303

by the estimated direct and diffuse cloudiness factors 𝑁𝑖∗ and 𝑁𝑑∗ , respectively (Eq. 18):

304

𝐺𝑖𝑐∗ = 0.937 ∗ 79.99 + 1.06 ∗ 7.43 + 3.48 = 86.36 𝑊ℎ/𝑚2

305

The range of clearness indices in which the proposed model is valid have also been checked in order

306

to establish the applicability of the model, especially for very cloudy skies. In this sense, 𝐺𝑖𝑐∗ could be

307

considered a provisional value. In a very cloudy sky, the global irradiation on the inclined plane has

308

approximately the same intensity level compared to its values on the horizontal plane, at least on similar

309

inclinations, as in our case. 11

Journal Pre-proof 310

To clearly highlight the initial hypothesis, we plotted the variation in the measured data for the horizontal

311

irradiation (𝐺ℎ𝑚 ), the tilted global irradiation (𝐺𝑖𝑚) (black and blue curves, respectively) and the

312

estimated values of the tilted global irradiation 𝐺𝑖𝑐∗ (red curve above) on the same graph (Fig. 3 upper

313

subplot); while the corresponding clearness index (𝐾𝑡) is represented by the black curve in the lower

314

subplot of Fig. 3. The model in Eq.18 shows a very good fit with the measured data for the periods

315

where the clearness index value is over a certain threshold (represented here by the dotted line in the

316

lower subplot of Fig. 3); for instance, the first three days shown in Fig. 3. Moreover, the model relatively

317

fails to reflect reliable values of 𝐺𝑖𝑚in other time intervals characterized by lower clearness index values

318

(e.g. the last day shown in Fig. 3), but matches approximately to 𝐺ℎ𝑚.

319 𝑴𝒆𝒂𝒔𝒖𝒓𝒆𝒅 𝒅𝒂𝒕𝒂 𝑯𝒐𝒓𝒊𝒛𝒐𝒏𝒕𝒂𝒍 𝒈𝒍𝒐𝒃𝒂𝒍 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏

𝑪𝒍𝒆𝒂𝒓𝒏𝒆𝒔𝒔 𝒊𝒏𝒅𝒆𝒙

𝐺ℎ𝑚

𝐾𝑡

𝑻𝒊𝒍𝒕𝒆𝒅 𝒈𝒍𝒐𝒃𝒂𝒍 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏

𝐺𝑖𝑚

(𝟏)𝑬𝒔𝒕𝒊𝒎𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝐜𝒍𝒐𝒖𝒅𝒊𝒏𝒆𝒔𝒔 𝒇𝒂𝒄𝒕𝒐𝒓𝒔 𝐶𝑙𝑜𝑢𝑑𝑖𝑛𝑒𝑠𝑠 𝑓𝑎𝑐𝑡𝑜𝑟𝑠

(𝑁𝑖∗ , 𝑁𝑑∗ )

𝑷𝒆𝒓𝒇𝒐𝒓𝒎𝒂𝒏𝒄𝒆 𝒂𝒔𝒔𝒆𝒔𝒔𝒎𝒆𝒏𝒕

{(𝑁𝑖, 𝑁𝑑)} 𝑯𝒐𝒓𝒊𝒛𝒐𝒏𝒂𝒍 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏 𝒃𝒚 𝑷𝒆𝒓𝒓𝒊𝒏 𝑩𝒓𝒊𝒄𝒉𝒂𝒎𝒃𝒂𝒖𝒕

𝐿𝑎𝑡𝑖𝑡𝑢𝑑𝑒 (𝜑), 𝑙𝑜𝑛𝑔𝑖𝑡𝑢𝑑𝑒 (L)

𝐺𝑖𝑐∗

𝐺ℎ

𝐾𝑡𝑡ℎ𝑟𝑒

𝐺𝑖𝑐

𝑁𝑢𝑚. 𝑜𝑓 𝑑𝑎𝑦𝑠 (𝑁)

𝑇𝑖𝑙𝑡𝑒𝑑 𝑎𝑛𝑔𝑙𝑒 (𝛽)𝐴

𝑻𝒉𝒓𝒆𝒔𝒉𝒐𝒍𝒅𝒊𝒏𝒈

𝐺𝑖

𝑙𝑏𝑒𝑑𝑜 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑖𝑡𝑒 (𝜌)

𝒕𝒊𝒍𝒕𝒆𝒅 𝒈𝒍𝒐𝒃𝒂𝒍 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏 (𝟐) 𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒊𝒐𝒏 𝒐𝒇 𝒑𝒓𝒐𝒗𝒊𝒔𝒊𝒐𝒏𝒂𝒍

𝑻𝒊𝒍𝒕𝒆𝒅 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏

𝒕𝒊𝒍𝒕𝒆𝒅 𝒈𝒍𝒐𝒃𝒂𝒍 𝒓𝒂𝒅𝒊𝒂𝒕𝒊𝒐𝒏

𝒃𝒚 𝑳𝒊𝒖 & 𝐽𝑜𝑟𝑑𝑎𝑛

𝑪𝒂𝒍𝒄𝒖𝒍𝒂𝒕𝒆𝒅 𝒄𝒍𝒆𝒂𝒓 𝒔𝒌𝒚 𝒅𝒂𝒕𝒂

320 321

(𝟑) 𝑭𝒊𝒏𝒂𝒍 𝒆𝒔𝒕𝒊𝒎𝒂𝒕𝒊𝒐𝒏 𝒐𝒇

Fig. 2. Flowchart of the proposed method.

12

Solar global irradiation(Wh/m2)

Journal Pre-proof

Measured of horizontal global irradiation Measured of inclined global irradiation(36.8°) Calculated of inclined global irradiation(36.8°) without thresholding

105 90 75 60 45 30 15

8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15

Time(hours)

Clearness index

1 Clearnes index data Threshold

0.8 0.6 0.4 0.2 0

8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15 8 11 15

Time(hours)

322 323

Fig. 3. Evolution of provisional estimated values of tilted global irradiation against the measured data (top) with respect to the corresponding clearness index and its threshold (bottom).

324

Therefore, our proposal is to introduce the notion of the clearness index threshold, 𝐾𝑡𝑡ℎ𝑟𝑒, to distinguish

325

between the two weather situations and the final estimated values of the inclined global solar irradiation

326

(𝐺𝑖𝑐) with thresholding (see step 3 in Fig. 2), which is expressed by:

{

𝐺 ∗ 𝑖𝑓 𝐾𝑡 ≥ 𝐾𝑡 𝐺𝑖𝑐 = 𝐺𝑖𝑐 𝑖𝑓 𝐾𝑡 < 𝐾𝑡 𝑡ℎ𝑟𝑒 ℎ𝑚 𝑡ℎ𝑟𝑒

327

(19)

328 329

2.6. Performance metrics

330

Several criteria were used to evaluate the accuracy of the proposed models. The expected data are

331

compared to the observed data and the assessment metrics are calculated as given in Table 4, where N

332

is the total number of the observations for the period in question, 𝑦𝑖 is the ith measured value, 𝑦𝑖 is the ith

333

estimated value and 𝑦 is the mean of the measured values. These criteria parameters are reviewed briefly

334

in Table 4 [27].

335

The RMSE (Eq. 20) is a frequently used measure of the differences between values predicted by a model

336

and the observed values. RMSE is a good measure of accuracy but only to compare predicting errors of

337

different models for a particular dataset or variable. The MBE gives the mean value of bias error.

338

Negative and positive values of MBE show underestimation and overestimation, respectively. MAE is

339

a positive quantity used to measure how close estimated data series are to experimental data series. The

340

normalized version of these metrics in percentages (i.e. nRMSE (Eq. 21), nMBE (Eq. 22) and nMAE

341

(Eq. 23)) is obtained by dividing the mean measured values. These metrics are preferred to comparing 13

Journal Pre-proof 342 343

Table 4. Performance metrics. Metrics Root Mean Squared Error (RMSE)

Mathematical expressions 1 𝑁 ∑ 𝑁 𝑖 = 1(𝑦𝑖

𝑅𝑀𝑆𝐸 =

2

(20)

― 𝑦𝑖)

Normalized Root Mean Squared Error (nRMSE)

𝑛𝑅𝑀𝑆𝐸 (%) = (𝑅𝑀𝑆𝐸 𝑦) ∗ 100

Normalized Mean Bias Error (nMBE)

𝑛𝑀𝐵𝐸 =

Normalized Mean Absolute Error (nMAE)

𝑛𝑀𝐴𝐸 = (𝑁 ∑𝑖 = 1|𝑦𝑖 ― 𝑦𝑖| 𝑦) ∗ 100

Relative Percentage Error (RPE)

𝑅𝑃𝐸 = (𝑁 ∑𝑖 = 1|𝑦𝑖 ― 𝑦𝑖| 𝑦𝑖 ) ∗ 100

Correlation coefficient (𝑅2)

𝑅2 = 1 ― ∑𝑖 = 1(𝑦𝑖 ― 𝑦𝑖) ∑𝑖 = 1(𝑦𝑖 ― 𝑦)2

(

1 𝑁

1

1

)

𝑁

∑𝑖 = 1(𝑦𝑖 ― 𝑦𝑖) 𝑦 ∗ 100

𝑁

𝑁

𝑁

2

(21)

(22)

(23)

(24)

𝑁

(25)

344 345

the predictive performance of the models over different datasets in other studies [35].

346

The RPE (Eq. 24) measures the ratios as a percentage of the absolute errors of the estimation relative to

347

the magnitude of the exact values.

348

The R2 (Eq. 25) is a useful parameter that takes possible values between 0 and 1. The higher the R2, the

349

better it represents the linear relationship between the estimated and the measured values.

350

3. Results and discussions

351

3.1. Estimation of the threshold clearness index

352

In order to use the proposed model to estimate solar global radiation on tilted surfaces, it is necessary

353

to previously estimate the optimum threshold. Our proposal is to analyse the data for one location in

354

detail, since the relationship between this parameter and the global solar radiation on tilted surfaces does

355

not depend on the location. Other authors have proposed the use of a clearness index threshold to

356

determine which model should be used [36], such as Liu and Jordan’s model or Iqbal’s model for

357

estimating monthly mean values of diffuse daily solar radiation from global radiation. Liu and Jordan

358

proposed a model valid for clearness index values ranging from 0.3 to 0.7, whereas Iqbal proposed an

359

expression valid for clearness index between 0.3 and 0.6. Likewise, Orgill and Hollands [37] proposed

360

a different expression for estimating hourly values of diffuse radiation depending on the clearness index

361

value.

14

Journal Pre-proof 362

In our study, the optimum threshold 𝐾𝑡𝑡ℎ𝑟𝑒 has been adjusted using a sensitivity analysis for the data

363

of Algiers and for a specific inclination angle of the plane, which is equal here to 36.8°. This step is

364

performed once and for the rest of the time. Therefore, the possible values of 𝐾𝑡𝑡ℎ𝑟𝑒 are taken in order

365

to choose the optimal value and exhibit its influence on the performance of the proposed model in terms

366

of the previously described metrics. The results are summarized in Table 5 and the best ones are

367

underlined and marked in bold.

368

Furthermore, the variations of only two main criteria to avoid congestion, such as the nRMSEs and the

369

RPE with respect to the thresholding is presented in Fig. 4 to better highlight the results, but the same

370

shape is obtained for the other metrics.

371

As 𝐾𝑡 range from 0 to 1, the extreme values of 𝐾𝑡𝑡ℎ𝑟𝑒 represent particular cases of the proposed model

372

where:

373

𝐾𝑡𝑡ℎ𝑟𝑒 = 0 ; which indicates that all values of the inclined global irradiance are calculated from the

374

model in Eq. 18; 𝐺𝑖𝑐 = 𝐺𝑖𝑐∗ .

375

𝐾𝑡𝑡ℎ𝑟𝑒 = 1; which indicates that the inclined global irradiance is taken directly from the values of the

376

measured horizontal irradiance;𝐺𝑖𝑐 = 𝐺ℎ𝑚 .

377

According to Table 5 and Fig. 4, it appears that even if the model described in Eq. 18 ( 𝐺𝑖𝑐 = 𝐺𝑖𝑐∗ ) is used

378

to estimate all the data, the results are not completely disappointing. Moreover, they can compete

379 380

Table 5. Model performance versus the threshold to highlight its influence (Algiers)

Threshold 𝑲𝒕𝒕𝒉𝒓𝒆

nMAE (%)

nMBE (%)

RMSE (Wh/m2)

nRMSE (%)

RPE (%)

R2

0 (𝐺𝑖𝑐 = 𝐺𝑖𝑐∗ ) 0.1 0.2 0.3 0.4 0.5

5.81 4.52 3.41 3.07 3.13 3.54

-3.34 -2.04 -0.89 -0.45 -0.18 0.14

4.29 3.18 2.25 2.05 2.13 2.58

9.82 7.28 5.14 4.7 4.88 5.91

39.77 13.43 6.84 5.55 5.63 6.39

0.972 0.984 0.992 0.993 0.993 0.990

0.6 0.7 0.8 0.9 1 (𝐺𝑖𝑐 = 𝐺ℎ𝑚 )

4.89 9.78 16.74 16.94 16.96

0.60 3.27 8.76 8.92 8.94

3.88 7.59 11.15 11.26 11.28

8.87 17.34 25.49 25.75 25.78

8.84 13.70 18.07 18.19 18.21

0.977 0.913 0.813 0.809 0.800

381 382

and outperform certain results reviewed (see Introduction and later in this section), with

383

nRMSE = 9.82%, nRMAE = 5.81% and R2 = 0.972. However, the use of RPE, an uncommon indicator

384

in similar studies (conversion from 𝐺ℎ𝑚to 𝐺𝑖𝑐, see Table 8), is poor in terms of the accuracy of the exact

15

Journal Pre-proof 385

values, with a RPE = 39.77%. This is due to the limitation of the model in Eq.18 when representing low

386

values of few data that are usually overestimated (see Fig. 3), thus confirming our hypothesis.

387

40

388 389 391 392 393 394 395 396 397 398 399

30

Performances(%)

390

NRMSE(%) RPE(%)

35

25 20 15 10 5 0

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Clearness index threshold

Fig. 4. Choice of the optimum clearness index threshold (𝐾𝑡𝑡ℎ𝑟𝑒) based on nRMSE and RPE criteria.

400 401

By filtering the modelling via thresholding (Eq. 19), the amount of errors decreases quickly and

402

considerably. For example, the RPE obtained for 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.1 is equal to 13.43% and the performance

403

continues to improve until the best performance is obtained for 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.3 (circled points in Fig. 4),

404

with RPE = 5.55%, nMAE = 3.07%, nRMSE = 4.7% and R2 = 0.993. Subsequently, whenever we move

405

the 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.3 away, which means more horizontal solar irradiation data being adopted as tilted data,

406

the results gradually worsen up to 𝐾𝑡𝑡ℎ𝑟𝑒 = 1, as expected. From the analysis, we conclude that the

407

interval of the threshold clearness index ranges between [0.2, 0.6], thus indicating good accuracy. 𝐾𝑡𝑡ℎ𝑟𝑒

408

= 0.3 is the optimum threshold, indicating that the proposed model is excellent in our case.

409 410

3.2. Estimation of solar global radiation on tilted surface

411

We now go a step further in the analysis of our proposed model for 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.3. Fig. 5 plots the

412

evolution of the experimental global tilted irradiation compared to the modelled irradiation for part of

413

the validation period which is arbitrarily chosen. As can be observed in the figure, we test the different

414

climatic conditions, which range from a completely clear to a cloudy sky. Moreover, the scatter plot in

415

Fig. 6 illustrates the degree of the linear relationship between the measured data and the estimated data.

416

The estimated data calculated from Eq. 18 (𝐺𝑖𝑐 = 𝐺𝑖𝑐∗ ), seen that 𝐾𝑡 ≥ 0.3, are represented in red and the

417

green starred points represent the alternative case where 𝐺𝑖𝑐 = 𝐺ℎ𝑚 are taken under 𝐾𝑡 < 0.3.

418

It is worth mentioning that the percentage of the values of the tilted irradiations found here are 𝐾𝑡 < 0.3

419

, is: 17.07 %. Table 6 shows the results obtained for both thresholding states: 𝐾𝑡 ≥ 0.3 and 𝐾𝑡 < 0.3.

16

Journal Pre-proof 420

As can be seen, the data modelled by 𝐺𝑖𝑐∗ , perform nearly 1% better than the figures obtained for all the

421

data. On the other hand, those supposed to be equal to 𝐺ℎ𝑚 give yield quite satisfactory results but

422 423 425 426 427 428 429 430 431 432 433 434

5-min solar irradiation on 36.8° plane(Wh/m2)

424 100 Measured data Estimated data for Kt>0.3 Estimated data for Kt<0.3

90 80 70 60 50 40 30 20 10 0

6200

6300

6400

6500

435 436 437

6600

6700

6800

6900

Time(5-min) Fig. 5. Comparison of the measured 5-min tilted global solar irradiation for Algiers and their values estimated by the proposed model for 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.3, during part of the validation.

438 100 Kt>0.3 Kt<0.3

Estimated values(Wh/m2)

90 80 70 60 50 40 30 20 10 0

0

10

20

30

40

50

60

70

80

90

100

2

Measured values(Wh/m ) Fig. 6. Scatter plot of the measured tilted global dataset of the site under study (Algiers) against their estimated values using the proposed model for 𝐾𝑡𝑡ℎ𝑟𝑒 = 0.3.

17

Journal Pre-proof 439 440

worse than in the case of 𝐾𝑡 ≥ 0.3, as the errors increase from 22% to 28% and the coefficient of

441

correlation decreases by around 1%. In order to further improve the quality of this model, future research

442

should focus on the modelling of the data for the clearness index under the threshold 𝐾𝑡 < 𝐾𝑡𝑡ℎ𝑟𝑒).

443 444 445

Table 6. Main statistical parameters applied on the two data sets of Algiers corresponding to Kt ≥ Ktthre and Kt < Ktthre obtained by the proposed model.

𝑲𝒕 ≥ 𝟎.𝟑 𝒘𝒉𝒆𝒓𝒆𝑮𝒊𝒄 = 𝑮𝒊𝒄∗ nRMSE RPE nMAE (%) (%) (%) 4.17

4.08

2.69

𝑲𝒕 < 0.3 𝑤ℎ𝑒𝑟𝑒𝑮𝒊𝒄 = 𝑮𝒉𝒎 nRMSE RPE nMAE (%) (%) (%)

R2

0.993

15.14

11.76

11.05

R2

0.939

446 449 450

Table 7. Metrics obtained for each site when the Perrin Brichambaut, Liu and Jordan and hybrid models are used.

Sites

447

Models nMAE(%) nRMSE(%) R2 RPE(%) Perrin 7.83 10.86 0.96 11.14 Brichambaut Algiers Liu and Jordan 4.89 7.52 0.98 8.55 Hybrid 3.07 4.70 0.99 5.55 Perrin 6.52 8.89 0.96 7.84 Brichambaut Malaga Liu and Jordan 6.68 9.06 0.96 8.03 Hybrid 4.37 6.13 0.98 5.6 Perrin 5.52 7.45 0.96 6.74 Brichambaut Ghardaia Liu and Jordan 5.46 6.88 0.97 6.56 Hybrid 4.73 6.41 0.97 5.94 Using the value of 0.3 for 𝐾𝑡𝑡ℎ𝑟𝑒, the proposed model has been used to estimate the global solar radiation

448

for the three sites used in this study.

451

Moreover, we have checked the two models individually (Perrin Brichambaut and Liu and Jordan [32])

452

in order to determine if they produce similar results to the proposed hybrid model. Table 7 shows the

453

results for each site.

454 455

As can be seen in Table 7, the proposed hybrid model clearly improves the results obtained with the

456

other two models when they are used individually. The proposed methodology using the hybrid model

457

obtains satisfactory results for all locations with a nMAE and a nRMSE below 5% and 6.5%,

458

respectively. The R2 values also indicate the good performance of the proposed model. For instance,

459

according to Figs. 5 and 6, the results for Algiers also confirm the excellent efficacy of the developed 18

Journal Pre-proof 460

model. The estimated data clearly show a very good fit with the measured data even in very cloudy

461

skies, with total errors equal to 3.07% and 4.7% for nMAE and nRMSE, respectively. The estimated

462

and measured values show a close match with a total R2 = 0.993 (Fig. 6).

463 464 465

3.3. The new model versus its counterparts in the literature

466

To show the effectiveness of the proposed model with respect to the other principal models in the

467

literature, Table 8 summarizes the results of recently published papers that have studied the conversion

468

of horizontal global solar irradiation into tilted solar irradiation for different time scales.

469

It is important to note that most studies in the scientific literature use empirical methods for time steps

470

equal or up to one hour [18][14][6][16][38] with a nRMSE generally ranging from 8% to 28% and a R2

471

from 0.923 to 0.993.

472

ANN techniques have recently been introduced [11][8][23][9] and applied for one hour and even in

473

sub-hour time steps (10 minutes and rarely in 5 minutes), thus improving performance twice-fold. This

474

is the case, for example, of two studies performed for the same site with the same slope angles in [18]

475

and [11] or the comparative study in [23].

476

However, the performance of these models usually decreases when the time resolution is increased.

477

For instance, in [9], where 10-minute data were used, the ANN model has twice the amount of errors

478

compared to [11], which used hourly data; and it is more degraded when 5-minute data are used [8].

479

This can be explained by the limitation of the time-averaging effect and data compensation for smaller

480

time steps, such as 5-min. The distribution thus becomes rather anisotropic.

481

Table 8 shows that the proposed method clearly outperforms all its counterparts on the different time

482

scale. The same conversion can be performed with up to two times fewer errors than with the ANN

483

models applied to the same data (site: Algiers, period of time, time-step and tilt) in the survey carried

484

out by Dahmani et al.

485

which exhibit more isotropic nature than 5-min data. This difference in performance is more pronounced

486

compared to conventional methods applied to hourly and monthly data. That difference is up to six-fold,

487

which makes the proposed model the best.

488

The superiority of the proposed method is due to the fact that it features a combination of the

489

advantages of the ANN model and the conventional models. Given the inclusion of adapted cloudiness

490

factors, our model is similar to the ANN model in terms of the non-linearity description of the solar

491

radiation data, which is non-existent in the conventional models. This is particularly true in the very

492

short time step. Moreover, the traditional models consist of equations that physically describe the

493

intrinsic characteristics of the solar irradiation and the energy conversion. Thus, in this model, cloud

19

Journal Pre-proof 494

cover data is estimated dynamically from measured global horizontal irradiation, which is then added to

495

modify the conventional clear sky models to estimate tilted solar irradiance in overcast conditions.

496

The proposed model also outperforms its principal opponent (ANN) in many aspects as it is considered

497

to be straightforward and its construction seems to be part of a more principled framework compared to

498

the ANN model due to [5][17]:

499

- The requirement of the availability of numerous reliable data in the training procedure to determine

500

the ANN parameters (weights, threshold), which is a constraint for their development and decreases

501 502

Table 8. Comparison of the performance of the proposed model with various published models for some inclinations and at different time resolutions.

Author

Our study

[8]

Location Algeria (Algiers) Malaga (Spain) Ghardaia (Algiers) Algeria (Algiers)

Time step

Tilt

Model type

36.8° 32° 5-min

Proposed model

32°

36.8°

ANN

Saudi Arabia (Jaddah)

16°

[9]

France (Ajaccio)

45° 60°

France (Ajaccio)

45° 60°

ANN

[11]

Iran (Mashhad)

45° 60°

ANN

[17]

[5]

nMAE (%) 3.07

Parameters nRMSE R2 (%) 4.70 0.99

4.37

6.13

0.98

5.60

4.73

6.41

0.97

5.94

6.67

8.88

0.994

4.94 5.89

8.18 10.22

0.0360.096 0.9190.976 0.996 0.994

2.79 3.42

5.28 6.28

0.998 0.998

ANN SVM

10-min

ANN

20

RPE (%) 5.55

0.930 0.924

28.0192.72 33.8951.64

Journal Pre-proof

[18]

45°

-94 comb. of conv. models -Olmo model

8.1117.17 12.14

0.9740.993 0.988

60°

10.7127.62 17.01 15.35 20.928.73

0.9680.989 0.982 0.960 0.9230.938

France (Ajaccio)

[23]

Greece (Athens)

[14]

Italy (Padova)

20°-30°

-94 comb. of conv. models -Olmo model -ANN -10 conv. models 12 comb. of conv. models

[6]

Spain (Granada)

44°

Olmo model

17.8

90°

Olmo model

27

5.10°

Empirical model + Olmo model

20

[38]

[16]

1 hour

Spain (Madrid) Malaysia (Terengganu )

1 month

32°

6-7

6.4 -8.7 0.996

0.973

503 504

[8]. There is a huge difference between our study compared to [17] when tested in the same time step.

505

Our model shows a significant improvement over the ANN models in hourly data such as [9][5][11][23],

506

their interest [8]. Conversely, we can estimate the tilted data at every time step using the proposed model

507

without prior knowledge of the data and only requiring the present global horizontal irradiation.

508

-The variability of the initial weights and bias, which yield different results from one test to another for

509

the same data.

510

- It must prove their generalization ability and prevent overfitting. Therefore, an optimization study

511

should be performed to choose the structure of the network, the optimum number of hidden layers, the

512

optimum number of neurons in each layer, the adequate learning algorithm, the adequate transfer

513

functions, the adequate percentage for the training and the rest will serve for the testing and the best

514

inputs with good correlation to the output.

515

4. Conclusions

516

A new methodology to estimate both the cloudiness factor and the global solar irradiation received on

517

the tilted plane for any sky situation using two well-known semi-empirical models is proposed. The

518

developed method is easy to implement as it consists of simplified equations. It is also economic given

519

that it takes into account only one type of measured data, the global solar irradiation on the horizontal

520

plane, as well as a few available parameters such as geographical parameters and the albedo of the site

521

to calculate the solar irradiations in clear sky.

522

The proposed model has been tested under difficult and complex contexts as we have checked all types

523

of skies (not only clear skies) and faced anisotropic issues. A non-linear relationship between the 21

Journal Pre-proof 524

measured data used as input and the target data was found. Moreover, we have used a very short time

525

resolution (5-min) where there is no compensation or averaging effects as occurs when monthly data are

526

used. Despite all these issues, our proposed methodology has proven to be robust in resolving these

527

kinds of problems with great accuracy.

528

An excellent match was found between the calculated and measured values for data from different sites

529

with different climatic conditions. The satisfactory results are reflected in the main assessment metrics

530

nRMSE, RPE and nMAE, which range from 4.7 to 6.41%, 5.5 to 5.9% and 3.07% to 4.73% respectively.

531

This shows the superiority of the model studied here over traditional models or even techniques recently

532

developed in the literature such as ANN for the same order or for higher time scales. In addition, our

533

model construction seems to be part of a more principled framework than ANN due to the variability of

534

the initial weights for the ANN and the fact that it requires a large amount of reliable data in the training

535

phase to set the model parameters (weights, threshold).

536

Theoretically, conventional models are universal and efficient regardless of the location as they contain

537

intrinsic characteristics of the regions under study. By exploiting the two conventional models in

538

conjunction with the cloudiness factor, our model shows potential to be extendable everywhere,

539

especially to sites with similar Mediterranean and arid climatic conditions.

540 541

Acknowledgments

542

This work has been supported by “Programme National Exceptionnelle 2018-2019”; scholarship fully

543

funded by the Algerian Ministry of Higher Education and Scientific Research. It has been also supported

544

by the project RTI2018-095097-B-I00 at the 2018 call for I+D+i Project of the Ministerio de Ciencia,

545

Innovación y Universidades, Spain.

546 547

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24

Journal Pre-proof Research Highlights

Title of the article: “New approach to estimate 5-min global solar irradiation data on tilted

planes from horizontal measurement”



A model to estimate solar global radiation on tilted plane for 5 minutes is proposed.



The proposed model only uses global solar radiation as input and simplified equations.



The nRMSE range from 4.7 to 6.41% for three sites with different climatic conditions.



The prediction errors are lower than those obtained with ANN and classical methods.



The proposed model can be used for all sky conditions.