# Development of New Models for the Estimation of Hourly Components of Solar Radiation: Tests, Comparisons, and Application for the Generation of a Solar Database in Morocco

Authors: Mohammed Benchrifa; Rachid Tadili; Ahmed Idrissi; Hajar Essalhi; Abdellah Mechaqrane
Journal: International Journal of Photoenergy
 (2021)

---

## Abstract

All studies of solar systems need the hourly values of solar fluxes related to the different components of solar radiation. However, for most sites, measurements are not available. The goal of this paper is to establish an hourly solar radiation data bank for the entirety of Morocco. This data bank will contain the energetic components (global, diffuse, direct, and inclined solar radiation) and the spectral components (ultravioletIUV, infrared IIR, and photosynthetically active radiation IPAR). To create this database, in the beginning, we build up models for predicting the various components of the hourly solar radiation, from measurements provided by the different stations owned by the Laboratory of Solar Energy and Environment (LESE). Then, these equations will be verified by using statistical tests. Moreover, we compared our results with those obtained by similar studies. Finally, based on the daily database of global radiation provided by the Laboratory of Solar Energy and Environment, we created an hourly database that unites the various components of solar radiation over 10 years for the entirety of Morocco. Regarding the validity study, the statistical indicators showed that the used models have minimal errors that do not exceed, for the global radiation, -2.93% in Rabat, -3.9% in Tangier,-3.28% in Marrakech, and -0.85% in Fez, and for diffuse radiation, -1.09% in Rabat, -0.68% in Fez,-3.08% in Tangier, and -2.49% in Marrakech. These results show the good quality of the used estimation models. Thus, the data bank that we have realized will fill the gap of the hourly solar data and will meet the needs of engineers, installers of solar systems, and researchers who often need an extensive database.


---

## Body

## 1. Introduction

For most practical applications of solar energy (habitat, thermal conversion systems, photovoltaic conversion systems), the values of daily irradiation are insufficient, because the variations of solar irradiation during the day affect the functioning of the solar systems. Consequently, in the performance analysis, we need to know hourly data in the way to test the impact of the solar radiation variation on such systems. However, in most stations, there are no hourly values; for this reason, we need methods to estimate the various components of hourly solar radiation.To fill the lack of hourly solar radiation data, several models have been made. We can mention among them, theASHRAE model [1] for predicting the hourly global radiation, hourly diffuse radiation, and hourly beam radiation, in terms of zenith angle, latitude, hour angle, and solar declination. This model was revised by Nijegorodov [2] for Botswana, Machler and Iqbal [3] for Canada, and Parishwad et al. [4] for India.Other models were generated to predict direct normal terrestrial solar radiation through the normal extraterrestrial solar radiation and Linke turbidity factor such as thePerez et al. model [5], or in terms of the air mass, integrated Rayleigh scattering optical thickness of the atmosphere, and Linke turbidity factor such as the Kasten and Young model [6], or in terms of zenith angle, altitude, and atmospheric transmittance such as the Hottel model [7].Furthermore, many models were mad to predict diffuse radiation such as theLiu and Jordan model [8] based on data from 98 localities in the USA and Canada with latitudes between 19 and 55°N; the Orgill and Hollands model [9] based on data from Toronto, Canada; the Erbs et al. model [10] developed with measurements from five stations in the USA with latitudes between 31 and 42°N, the Spencer model [11] based on data from five stations in Australia with latitudes between 20 and 45°S; the Reindl et al. model [12] based on data from five locations in the USA and Europe with latitudes between 28 and 60°N in terms of the clearness index and the solar elevation; the Lam and Li model [13] based on data from Hong Kong with latitude equal to 22.31°N; the Hawlader model [14] using data from a tropical site in Singapore; the De Miguel et al. model [15] based on data set from several countries in the North Mediterranean Belt area; the Chandrasekaran and Kumar model [16] based on data from a tropical environment in Madras, India; and the Oliveira et al. model [17] based on data from a tropical Sao Paolo site, Brazil.Moreover, various models were developed to estimate the hourly global radiation like theWhillier/Liu and Jordan model [18], the Collares-Pereira and Rabl model [19], the Jain model [20] based on the normal distribution equation, and the Garg and Garg model [21] based on data from various Indian sites.In addition, we can find different models to approximate the spectral components of solar radiation. We can cite among them, theCanãda et al. model [22] used to predict the Iuv through the total clearness index Kt; the Escobedo model [23] used to predict IUV, IPAR, and IIR through the global solar radiation under various sky conditions at Botucatu, Brazil; and the Jacovides et al. model [24] which gives the relationship between global photosynthetically active radiation (PAR) and global solar radiation at Athalassa, Cyprus.At last, there are new methods to predict the hourly solar radiation based on artificial intelligence, for example, the neural-network methods [25], the neuro-fuzzy methods, and the methods based on the satellite image data [26].All these models and many others can be grouped into four main categories according to the type and the way of processing the used data. Therefore, we find physical models [27, 28], empirical models [29–31], models based on artificial intelligence [32–35], and hybrid models [36, 37]. All these methods represent more than 294 models around the world [38]. Recently, to find the valid estimation model for a site, several comparison works have been carried out, such as Betti et al.’s work in Croatia [39], Wang et al.’s work [40] in China, and Chukwujindu’s work in Africa [32].The paper’s goal is to establish an hourly solar radiation data bank for the entirety of Morocco. This data bank will contain the energetic components (global, diffuse, direct, and inclined solar radiation) and the spectral components (ultravioletIUV, infrared IIR, and photosynthetically active radiation IPAR).To develop this work, we will go through the following steps:(i)
Determination of the correlation equation of hourly global solar irradiation(ii)
Development of the correlation equation of hourly diffuse irradiation(iii)
Obtaining the estimation equations for the other solar components(iv)
Validating the obtained correlations(v)
Compare our results with the results of similar studiesFinally, based on the database of daily global radiation owned by LESE, we generated at first the hourly global radiation and after the other components of solar radiation for the entirety of Morocco over 10 years. According to the above goals, the remainder of the paper is structured as follows. In Section2, the description and analysis of data are introduced. In Section 3, the methods of developing the database are presented. In Section 4, some existing models and correlations are cited. Section 5 provides the method of reconstituting the hourly values. Section 6 shows the used statistical indicator. Results and discussions are presented in Section 7. Finally, the conclusion is given in Section 8.


## 2. Description and Analysis of Data

The Laboratory of Solar Energy and Environment has two types of data:(i)
The hourly data obtained directly from ten years of measurements of solar radiation components (2009–2018), taken by four stations spread over the Moroccan territory. These data have been subdivided into three intervals:(i)
An eight-year interval (2009–2016) for the development of the different models and correlations.(ii)
A second interval consisting of data for the year 2017 for the comparison and selection of the models.(iii)
A third interval relating to the data for the year 2018 reserved for the validation of the models.(ii)
The daily data obtained by extrapolation over the entire Moroccan territory over a period of ten years [41].


### 2.1. Hourly Data from Measurement Stations

The Laboratory of Solar Energy and Environment has four stations located in representative sites of the Moroccan climate (Table1). These stations measure the various components of solar radiation, over ten years (2009–2018).Table 1
Measuring station locations [42].
CitiesLatitudeLongitudeAltitudeRabat34-6.8375Fez33.56-5579Marrakech31.62-8.03463Tangier35.72-5.7515


#### 2.1.1. Description of Representative Sites of Morocco’s Climate

In Morocco, the climate is Mediterranean on the coasts, while it is desert in inland areas and continental in the highest mountains. Thus, Morocco is divided into many climatic regions, which can be represented by the following sites:(i)
Fez climateThe geographical location of Fez inland gives it a continental climate. It has a cold winter and a hot summer. The average summer temperature is 35°C. The average rainfall is 536 mm per year(ii)
Marrakech climateIn Marrakech, the climate is semi-desert, with mild winters and very hot summers. The average summer temperature is 36°C. The wind can lift sand and dust from the desert, reducing visibility. Rainfall is scarce since it amounts to 240 millimeters per year(iii)
Tangier climateTangier has a Mediterranean climate with mild winters and not hot summers. The average maximum temperature in the summer does not exceed 26°C. The average rainfall is 800 mm per year.(iv)
Rabat climateRabat lies on 75 m above sea level. The climate in Rabat is temperate. The rain in Rabat falls mostly in the winter with relatively little rain in the summer. The average annual temperature in Rabat is 17.9°C. The annual rainfall is 523 mm per year.


#### 2.1.2. Description of the Measuring Instruments

Each station measures the components of solar radiation and climatic variables. Table2 brings together these different components and the corresponding measuring instruments [42–44].Table 2
Characteristics of measuring instruments.
Meteorological variablesInstrumentsAccuracyGlobal solar radiationThe Kipp & Zonen brand SP-Lite pyranometer2%Diffuse solar radiationThe Kipp & Zonen brand SP-Lite pyranometer with a sun visor strip2%Direct solar radiationPyrheliometer type Eppley model NIP 31820 E60.5%Total solar ultraviolet radiationTotal Ultraviolet type Eppley2%Infrared solar radiationInfrared radiometer type Eppley2%Photosynthetically active radiation (PAR)Quantum sensor SKP215, type Campbell Scientific0.5%Wind speed and directionAnemometer with a wind vane model wind monitor 05103, type Campbell Scientific0.25%PrecipitationRain Gauge ARG100, manufactured by Environmental Measurements for Campbell Scientific4%Using an acquisition system, the measurements are collected every 5 seconds, then accumulated, integrated, and stored at an hourly step. Using these hourly measurements obtained by the four stations, we were able to generate models for estimating hourly solar irradiations for all Moroccan regions.


#### 2.1.3. Data Processing

The hourly database was processed using data processing software. Then, we deleted the erroneous values and replaced some missing values by extrapolation. The erroneous values generally come from a short circuit on the terminals of the instruments, mainly observed on rainy days, while the missing values occur by long-term power failure. It should be noted that all the cases of erroneous or missing values represent only 0.006% of the values in the database.


### 2.2. Data Generated by Extrapolation

In addition to the collected data, the Laboratory of Solar Energy and Environment has a database of daily global solar irradiation obtained by extrapolation [41]. This database includes data for the sites listed in Table 3. Moreover, Figure 1 gives a geographic representation of these sites.Table 3
The locations of the Moroccan sites [42].
Measuring stationLatitudeΦ (°)Longitudeλ (°)Measuring stationLatitudeΦ (°)Longitudeλ (°)Agadir30.38-9.57Midelt32.68-4.73Al Hoceima35.18-3.83Nador35.15-2.92Beni-Mellal32.37-6.4Nousar33.37-7.57Bouarfa32.52-1.95Ouarzazate30.93-6.9Casa33.57-7.67Oujda34.78-1.93Dakhla23.77-15.93Errachidia31.93-4.4El Jadida33.23-8.52Safi32.28-9.23Essaouira31.52-9.78Sidi Ifni29.37-10.18Irfan33.5-5.17Sidi Sliman34.23-6.05Kenitra34.3-6.6Tangier35.72-5.75Laayoun27.17-13.22Tantan28.6-11.08Larache35.18-6.13Taza34.22-4Marrakech31.62-8.03Tetouan35.57-5.33Meknes33.88-5.53Rabat34-6.83Fez33.9-5Figure 1
Map of the Moroccan sites.


## 2.1. Hourly Data from Measurement Stations

The Laboratory of Solar Energy and Environment has four stations located in representative sites of the Moroccan climate (Table1). These stations measure the various components of solar radiation, over ten years (2009–2018).Table 1
Measuring station locations [42].
CitiesLatitudeLongitudeAltitudeRabat34-6.8375Fez33.56-5579Marrakech31.62-8.03463Tangier35.72-5.7515


### 2.1.1. Description of Representative Sites of Morocco’s Climate

In Morocco, the climate is Mediterranean on the coasts, while it is desert in inland areas and continental in the highest mountains. Thus, Morocco is divided into many climatic regions, which can be represented by the following sites:(i)
Fez climateThe geographical location of Fez inland gives it a continental climate. It has a cold winter and a hot summer. The average summer temperature is 35°C. The average rainfall is 536 mm per year(ii)
Marrakech climateIn Marrakech, the climate is semi-desert, with mild winters and very hot summers. The average summer temperature is 36°C. The wind can lift sand and dust from the desert, reducing visibility. Rainfall is scarce since it amounts to 240 millimeters per year(iii)
Tangier climateTangier has a Mediterranean climate with mild winters and not hot summers. The average maximum temperature in the summer does not exceed 26°C. The average rainfall is 800 mm per year.(iv)
Rabat climateRabat lies on 75 m above sea level. The climate in Rabat is temperate. The rain in Rabat falls mostly in the winter with relatively little rain in the summer. The average annual temperature in Rabat is 17.9°C. The annual rainfall is 523 mm per year.


### 2.1.2. Description of the Measuring Instruments

Each station measures the components of solar radiation and climatic variables. Table2 brings together these different components and the corresponding measuring instruments [42–44].Table 2
Characteristics of measuring instruments.
Meteorological variablesInstrumentsAccuracyGlobal solar radiationThe Kipp & Zonen brand SP-Lite pyranometer2%Diffuse solar radiationThe Kipp & Zonen brand SP-Lite pyranometer with a sun visor strip2%Direct solar radiationPyrheliometer type Eppley model NIP 31820 E60.5%Total solar ultraviolet radiationTotal Ultraviolet type Eppley2%Infrared solar radiationInfrared radiometer type Eppley2%Photosynthetically active radiation (PAR)Quantum sensor SKP215, type Campbell Scientific0.5%Wind speed and directionAnemometer with a wind vane model wind monitor 05103, type Campbell Scientific0.25%PrecipitationRain Gauge ARG100, manufactured by Environmental Measurements for Campbell Scientific4%Using an acquisition system, the measurements are collected every 5 seconds, then accumulated, integrated, and stored at an hourly step. Using these hourly measurements obtained by the four stations, we were able to generate models for estimating hourly solar irradiations for all Moroccan regions.


### 2.1.3. Data Processing

The hourly database was processed using data processing software. Then, we deleted the erroneous values and replaced some missing values by extrapolation. The erroneous values generally come from a short circuit on the terminals of the instruments, mainly observed on rainy days, while the missing values occur by long-term power failure. It should be noted that all the cases of erroneous or missing values represent only 0.006% of the values in the database.


## 2.1.1. Description of Representative Sites of Morocco’s Climate

In Morocco, the climate is Mediterranean on the coasts, while it is desert in inland areas and continental in the highest mountains. Thus, Morocco is divided into many climatic regions, which can be represented by the following sites:(i)
Fez climateThe geographical location of Fez inland gives it a continental climate. It has a cold winter and a hot summer. The average summer temperature is 35°C. The average rainfall is 536 mm per year(ii)
Marrakech climateIn Marrakech, the climate is semi-desert, with mild winters and very hot summers. The average summer temperature is 36°C. The wind can lift sand and dust from the desert, reducing visibility. Rainfall is scarce since it amounts to 240 millimeters per year(iii)
Tangier climateTangier has a Mediterranean climate with mild winters and not hot summers. The average maximum temperature in the summer does not exceed 26°C. The average rainfall is 800 mm per year.(iv)
Rabat climateRabat lies on 75 m above sea level. The climate in Rabat is temperate. The rain in Rabat falls mostly in the winter with relatively little rain in the summer. The average annual temperature in Rabat is 17.9°C. The annual rainfall is 523 mm per year.


## 2.1.2. Description of the Measuring Instruments

Each station measures the components of solar radiation and climatic variables. Table2 brings together these different components and the corresponding measuring instruments [42–44].Table 2
Characteristics of measuring instruments.
Meteorological variablesInstrumentsAccuracyGlobal solar radiationThe Kipp & Zonen brand SP-Lite pyranometer2%Diffuse solar radiationThe Kipp & Zonen brand SP-Lite pyranometer with a sun visor strip2%Direct solar radiationPyrheliometer type Eppley model NIP 31820 E60.5%Total solar ultraviolet radiationTotal Ultraviolet type Eppley2%Infrared solar radiationInfrared radiometer type Eppley2%Photosynthetically active radiation (PAR)Quantum sensor SKP215, type Campbell Scientific0.5%Wind speed and directionAnemometer with a wind vane model wind monitor 05103, type Campbell Scientific0.25%PrecipitationRain Gauge ARG100, manufactured by Environmental Measurements for Campbell Scientific4%Using an acquisition system, the measurements are collected every 5 seconds, then accumulated, integrated, and stored at an hourly step. Using these hourly measurements obtained by the four stations, we were able to generate models for estimating hourly solar irradiations for all Moroccan regions.


## 2.1.3. Data Processing

The hourly database was processed using data processing software. Then, we deleted the erroneous values and replaced some missing values by extrapolation. The erroneous values generally come from a short circuit on the terminals of the instruments, mainly observed on rainy days, while the missing values occur by long-term power failure. It should be noted that all the cases of erroneous or missing values represent only 0.006% of the values in the database.


## 2.2. Data Generated by Extrapolation

In addition to the collected data, the Laboratory of Solar Energy and Environment has a database of daily global solar irradiation obtained by extrapolation [41]. This database includes data for the sites listed in Table 3. Moreover, Figure 1 gives a geographic representation of these sites.Table 3
The locations of the Moroccan sites [42].
Measuring stationLatitudeΦ (°)Longitudeλ (°)Measuring stationLatitudeΦ (°)Longitudeλ (°)Agadir30.38-9.57Midelt32.68-4.73Al Hoceima35.18-3.83Nador35.15-2.92Beni-Mellal32.37-6.4Nousar33.37-7.57Bouarfa32.52-1.95Ouarzazate30.93-6.9Casa33.57-7.67Oujda34.78-1.93Dakhla23.77-15.93Errachidia31.93-4.4El Jadida33.23-8.52Safi32.28-9.23Essaouira31.52-9.78Sidi Ifni29.37-10.18Irfan33.5-5.17Sidi Sliman34.23-6.05Kenitra34.3-6.6Tangier35.72-5.75Laayoun27.17-13.22Tantan28.6-11.08Larache35.18-6.13Taza34.22-4Marrakech31.62-8.03Tetouan35.57-5.33Meknes33.88-5.53Rabat34-6.83Fez33.9-5Figure 1
Map of the Moroccan sites.


## 3. The Methods of Developing the Database

To carry out this work, we will follow the following steps.At the beginning, using the first interval of the measurement database (2009–2016), we developed new models to estimate hourly global solar irradiation, diffuse solar irradiation, and spectral irradiations of solar radiation.Secondly, using the second interval constituted by the data for the year 2017, we compared the obtained equations and the existing models with the measurements. The statistical indicators of comparison led us to choose the appropriate models to predict solar irradiations.Then, we validated the selected models using the third interval consisting of the data for the year 2018.On the other hand, using the extrapolated database of daily global solar radiation, we generated at first the global hourly radiation and after the other components of solar radiation for the whole of Morocco over 10 years.


## 4. Existing Models and Correlations

Daily values of global irradiation are generally available at classical meteorological stations. However, in many solar applications, we need hourly values of solar radiation. Thus, it is necessary to look for methods to estimate the hourly values of the different components of solar radiation. Among these methods, we can cite the following ones.


### 4.1. Liu and Jordan’s Method for Estimating Direct Radiation

The first work for estimating direct radiation was carried out by Whillier [18] and after taken over by Liu and Jordan [8], who established a relationship that gave the ratio of the monthly average of the hourly values and the monthly average of the daily values of the direct radiation:
(1)Ib¯Hb¯=∫1hourτb¯ωIoncosθdω∫1dayτb¯ωIoncosθdω.τb¯ω is the monthly mean of the atmospheric transmittance for direct radiation.Assuming that the coefficientτb¯ω is constant during the day, they identified the ratios Ib¯/Hb¯ and Ioh¯/Hoh¯:
(2)Ib¯Hb¯=Ioh¯Hoh¯=π24cosωi−cosωcsinωi−π/180ωccosωc.However, several studies have shown that the transmission coefficient is not constant during the day, especially for partially covered days [45, 46].Therefore, this estimation model is no longer valid.


### 4.2. The Method of Collares-Pereira and Rabl for the Estimation of Global Radiation

Since daily values of solar radiation are often available at the meteorological stations, Collares-Pereira and Rabl [19] have taken a purely empirical approach to highlight a relationship between Ih¯/Hh¯/Ioh¯/Hoh¯ ratio and the next variables:
(i)
The hour angleωi, calculated in the middle of the considered hour, characterizing the hour(ii)
The hour angle of sunsetωc¯ characterizing the month:(3)Ih¯/Hh¯Ioh¯/Hoh¯=a+bcosωi,with
(4)a=0.49+0.5016sinωc¯−60°,b=0.6609+0.4767sinωc¯−60°.The coefficientsa and b have been calculated by the least-squares method.


### 4.3. The Hay’s Method

In this method, Hay [47] considered the values of solar irradiation arriving on the ground, before and after the multiple reflections between the ground, clouds, and other constituents of the atmosphere. The equations thus established are
(5)I¯−I¯′=I¯ρρan¯Nj¯+ρc1−n¯Nj¯,Id¯−Id¯′=I¯ρρan¯Nj¯+ρc1−n¯Nj¯,whereρis soil albedo, ρa is albedo of a sky without clouds, ρc is cloud albedo, Nj¯ is the maximum duration of corrected insolation, and I¯′andId¯′ are the global and diffuse hourly irradiations before the multiple reflections between the ground and the sky.Then, the obtained correlation between the componentsI¯′ and Id¯′is
(6)Id¯′I′¯=0.9702+1.668I′¯Ioh¯−21.303I′¯Ioh¯2+51.288I′¯Ioh¯3−50.081I′¯Ioh¯4+17.551I′¯Ioh¯5.The Hay method requires knowledge of the monthly average of the hourly global irradiation that is not available in general, as well as the albedo of the clouds that varies with the type, the thickness, and the number of cloud layers and finally the albedo of a cloudless sky that depends considerably on the atmospheric aerosols.


### 4.4. Orgill and Hollands Correlation for the Estimation of the Diffuse Radiation

This method represents the first correlation of this type. It was established over four years of measurement in Canada (43-48 N) and by following the state of the sky characterized by its clarity index. This method includes three equations depending on theKt intervals, and it is given by the following equations (9):
(7)IdIh=1−0.249Kt,Kt≤0.35,1.557−1.84Kt,0.35<Kt≤0.75.0.177,Kt>0.75.


### 4.5. Erbs et al. Correlation for the Estimation of Diffuse Radiation

Based on the Orgil and Hollands work [9], Erbs et al. [10] realized a new diffuse radiation correlation using the data of 5 measuring stations which are located in the United States with latitude that varies from 31° to 42° N. The duration of the database used to carry out this correlation varies from 1 to 4 years, and the obtained equations are written in the following form:
(8)IdIh=1−0.09Kt,Kt≤0.22,0.9511−0.164Kt+4.388Kt2−16.638Kt3+12.336Kt4,0.22<Kt≤0.8,0.16,Kt>0.8.


### 4.6. Correlation of De Miguel et al. for Estimating the Diffuse Radiation

The purpose of this correlation is to find a relationship between diffuse radiation and the clarity index. Thus, De Miguel et al. [15] used the measurements of eight stations located in Portugal, Spain, France, and Greece, with latitudes ranging from 37° to 44°, to obtain the following equations:
(9)IdIh=0.995−0.08Kt,Kt≤0.21,0.724+2.738Kt−8.32Kt2+4.96Kt3,0.21<Kt≤0.76,0.18,Kt>0.76.


## 4.1. Liu and Jordan’s Method for Estimating Direct Radiation

The first work for estimating direct radiation was carried out by Whillier [18] and after taken over by Liu and Jordan [8], who established a relationship that gave the ratio of the monthly average of the hourly values and the monthly average of the daily values of the direct radiation:
(1)Ib¯Hb¯=∫1hourτb¯ωIoncosθdω∫1dayτb¯ωIoncosθdω.τb¯ω is the monthly mean of the atmospheric transmittance for direct radiation.Assuming that the coefficientτb¯ω is constant during the day, they identified the ratios Ib¯/Hb¯ and Ioh¯/Hoh¯:
(2)Ib¯Hb¯=Ioh¯Hoh¯=π24cosωi−cosωcsinωi−π/180ωccosωc.However, several studies have shown that the transmission coefficient is not constant during the day, especially for partially covered days [45, 46].Therefore, this estimation model is no longer valid.


## 4.2. The Method of Collares-Pereira and Rabl for the Estimation of Global Radiation

Since daily values of solar radiation are often available at the meteorological stations, Collares-Pereira and Rabl [19] have taken a purely empirical approach to highlight a relationship between Ih¯/Hh¯/Ioh¯/Hoh¯ ratio and the next variables:
(i)
The hour angleωi, calculated in the middle of the considered hour, characterizing the hour(ii)
The hour angle of sunsetωc¯ characterizing the month:(3)Ih¯/Hh¯Ioh¯/Hoh¯=a+bcosωi,with
(4)a=0.49+0.5016sinωc¯−60°,b=0.6609+0.4767sinωc¯−60°.The coefficientsa and b have been calculated by the least-squares method.


## 4.3. The Hay’s Method

In this method, Hay [47] considered the values of solar irradiation arriving on the ground, before and after the multiple reflections between the ground, clouds, and other constituents of the atmosphere. The equations thus established are
(5)I¯−I¯′=I¯ρρan¯Nj¯+ρc1−n¯Nj¯,Id¯−Id¯′=I¯ρρan¯Nj¯+ρc1−n¯Nj¯,whereρis soil albedo, ρa is albedo of a sky without clouds, ρc is cloud albedo, Nj¯ is the maximum duration of corrected insolation, and I¯′andId¯′ are the global and diffuse hourly irradiations before the multiple reflections between the ground and the sky.Then, the obtained correlation between the componentsI¯′ and Id¯′is
(6)Id¯′I′¯=0.9702+1.668I′¯Ioh¯−21.303I′¯Ioh¯2+51.288I′¯Ioh¯3−50.081I′¯Ioh¯4+17.551I′¯Ioh¯5.The Hay method requires knowledge of the monthly average of the hourly global irradiation that is not available in general, as well as the albedo of the clouds that varies with the type, the thickness, and the number of cloud layers and finally the albedo of a cloudless sky that depends considerably on the atmospheric aerosols.


## 4.4. Orgill and Hollands Correlation for the Estimation of the Diffuse Radiation

This method represents the first correlation of this type. It was established over four years of measurement in Canada (43-48 N) and by following the state of the sky characterized by its clarity index. This method includes three equations depending on theKt intervals, and it is given by the following equations (9):
(7)IdIh=1−0.249Kt,Kt≤0.35,1.557−1.84Kt,0.35<Kt≤0.75.0.177,Kt>0.75.


## 4.5. Erbs et al. Correlation for the Estimation of Diffuse Radiation

Based on the Orgil and Hollands work [9], Erbs et al. [10] realized a new diffuse radiation correlation using the data of 5 measuring stations which are located in the United States with latitude that varies from 31° to 42° N. The duration of the database used to carry out this correlation varies from 1 to 4 years, and the obtained equations are written in the following form:
(8)IdIh=1−0.09Kt,Kt≤0.22,0.9511−0.164Kt+4.388Kt2−16.638Kt3+12.336Kt4,0.22<Kt≤0.8,0.16,Kt>0.8.


## 4.6. Correlation of De Miguel et al. for Estimating the Diffuse Radiation

The purpose of this correlation is to find a relationship between diffuse radiation and the clarity index. Thus, De Miguel et al. [15] used the measurements of eight stations located in Portugal, Spain, France, and Greece, with latitudes ranging from 37° to 44°, to obtain the following equations:
(9)IdIh=0.995−0.08Kt,Kt≤0.21,0.724+2.738Kt−8.32Kt2+4.96Kt3,0.21<Kt≤0.76,0.18,Kt>0.76.


## 5. Reconstitution of Hourly Values

### 5.1. Estimation of Hourly Values of Global IrradiationIh

To study the solar potential of a site, we need to know the hourly values of global irradiation over a long period. However, this variable is not always available. Consequently, several studies have attempted to estimate this component. Among these works, we can cite for example the Collares-Pereira and Rabl model. Hence, based on this work, we developed a new model valid for Morocco.In order to improve the quality of the estimated values generated by the Collares-Pereira model, we looked for an empirical relationship adapted for Morocco. So we calculated the coefficientsa and b of Equation (3) from measurement that led us to find a new relation between the ratioI¯h/H¯hoverI¯oh/H¯oh, ωi and ωc¯. Therefore, we proceeded to adapt the coefficients for Morocco, as follows.In the first step, we determined the monthly averages of daily global solar irradiation and the monthly averages of hourly solar irradiation using the measured data. Then, for each month, we calculated the values ofa and b by the least-squares method. That gave us, for each station, and each coefficient, 12 values projected in Table 4.Table 4
Thea and b values calculated for the representative station’s measurements.
FezMarrakechRabatTangierababababJanuary0.430.650.480.6080.320.7570.3670.735February0.5160.5780.40.70.4640.640.5460.548March0.60.3810.5180.6020.6530.4260.7840.274April0.5520.5690.6060.51470.6480.4820.6810.404May0.7550.3330.70630.3990.6640.4510.7790.3Jun0.6990.4040.6810.4350.6630.4710.7560.335July0.6520.4650.6830.4310.6040.5540.7650.318August0.620.4990.7020.3980.6010.5260.7120.379September0.5030.5990.5410.5820.5460.5550.6840.399October0.4130.6890.5290.6010.5730.5410.7010.358November0.320.830.4120.6990.5750.50.4340.670December0.4490.6280.3490.7550.3860.7210.5520.517Then, for every station, we plotted sparely the monthly values ofa and b as a function of sinωc¯−70°, which gave us presentations as Figure 2.Figure 2
Example of the presentation ofa and b as a function of sinωc¯−70°.
(a)(b)As a result, we obtained equations in the following form:(10)a=a′+b′sinωc¯−70°,b=a′′+b′′sinωc¯−70°.70° represents the minimum value of the hour angle of the sunsetThe coefficientsa′, a″, b′, and b″ are calculated from values in Table 4 by using the least-squares method. The obtained values are presented in Table 5.Table 5
The coefficient values calculated in function ofsinωc¯−70°.
Coefficientaa′b′RCoefficientba″b″RRabat0.4060.4500.9Rabat0.672-0.3540.89Fez0.3500.5890.88Fez0.735-0.5480.85Marrakech0.3240.6780.94Marrakech0.788-0.6730.93Tangier0.44060.53740.91Tangier0.677-0.60.9In order to have one equation valid for all of Morocco, we averaged the coefficients mentioned in Table5, which gave us the following equations:
(11)a=0.38+0.564sinωc¯−70°,b=0.72−0.54sinωc¯−70°.Finally, by using the equation below, we will be able to estimate the monthly average of hourly global solar radiation for all Moroccan regions:(12)I¯h/H¯hI¯oh/H¯oh=0.38+0.564sinωc¯−70°+0.72−0.54sinωc¯−70°cosωi.For advanced studies of solar systems, data of monthly average of solar irradiation is insufficient; therefore, we need to find new methods of estimating hourly values. For this, we adapted Equation (12) to be valid for estimating hourly values; thus, we changed the means of the variables I¯oh,H¯oh,H¯h,andωc¯ by Ioh, Hh, ωc, and Hoh. That gave us the following equation:
(13)Ih/HhIoh/Hoh=0.38+0.564sinωc−70°+0.72−0.54sinωc−70°cosωi.It remains to validate this adaptation by a comparison with the measurements. This will be the subject of Table6.Table 6
OurIh model validation test.
Ih estimation modelRMBE (%)OurIh model-1.9%


### 5.2. Estimation ofId and Ib

The hourly evolution of the diffuse component of solar radiation is an important data for researchers and engineers when they study the behavior of solar thermal conversion systems or photovoltaic systems. Therefore, several models have been made for estimating this component such as the Orgill and Hollands model [9], the Erbs et al. model [10], and the model of De Miguel et al. [15].In the same way, we tried to find our own model to estimateId. For this reason, we plotted KdId/Ih as a function of KtIh/Ioh as shown in Figure 3.Figure 3
Hourly correlation betweenKdId/Ih and the clarity index KtIh/IohFrom Figure3, we notice that the cloud of dots has the shape of a waterfall, which is characterized by two almost horizontal parts at the beginning and at the end and another inclined part in the middle. In order to find a correlation that represents all the points with a good fit, we divided the Kt values into three intervals, and by the least-squares method, we established separately the correlation equations that represent them.In addition, we find that the cloud of dots show a remarkable dispersion especially in theKt range between 0.3 and 0.55. This is mainly due to two causes, firstly to the large volume of used data, which is well above 12,000 values from different sites, and secondly to the fluctuation of the diffuse fraction which depends on the type and distribution of particles and clouds that are in the sky.For each interval, the calculated correlations equations are in the following form:(14)IdIh=0.9382−0.2017Kt,Kt≤0.2,0.731+1.9202Kt−6.5525Kt2+4.014Kt3,0.2<Kt≤0,0.14,Kt>0.8.Moreover, with the predicted values of hourly diffuse and global solar radiation, we can calculate easily the hourly values of direct solar radiation by using the equation below:(15)Ib=Ih−Id.


### 5.3. Estimation ofIs

The hourly inclined solar radiationIs is calculated by using the diffuse isotropic approximation cited by [48]:
(16)Is=Ibcosicosθ+Ih−Ib1+cosS2+ρIh1−cosS2,where S is the tilt angle; i and θ are, respectively, the angles of incidence of the sun’s rays on the inclined plane and on the horizontal plane; and cosi/cosθ is the transposition factor of the direct solar component calculated by the division of the incidence angles of the solar rays on the inclined plane and incidence angles of the solar rays on the horizontal plane, in the middle of the considered hour.As it is interesting to know the evolution of global hourly irradiation for different inclinations, we chose to calculate it forS=ϕ, ϕ+20, and ϕ−20,90.


### 5.4. Predicting the Values of Solar Spectral Radiation

Hourly data of global, direct, and diffuse solar radiation are insufficient to study all solar systems because other applications require the knowledge of the hourly values of spectral components.For example, the ultraviolet component is used in atmospheric studies, the photosynthetically active radiation is necessary for the plant development process, and the infrared solar radiation is used in meteorological studies.Knowing that theIPAR, IUV, and IIR measurements are not often available in the meteorological stations, for this reason, we established simple formulas for estimating these radiations from the measurement of global solar radiation.Thus, in order to establish the estimation equations of spectral component, we used measurements collected by the LESE station network, and through the least-squares method, we established relations between the global irradiation and the spectral solar components. For more precision, the equations are established according three ranges ofKt. The obtained correlations are given in Table 7.Table 7
The equations for predicting the values ofIPAR, IUV, and IIR.
Kt intervalsCorrelation equationsRKt≤0.35IUV=0.048Ih0.98IPAR=0.4889Ih0.99IIR=0.461Ih0.990.35<Kt≤0.65IUV=0.044Ih0.98IPAR=0.485Ih0.99IIR=0.469Ih0.99Kt>0.65IUV=0.041Ih0.98IPAR=0.479Ih0.99IIR=0.477Ih0.99From Table7, we notice that the hourly fractions of IUV and IPAR, namely, IUV/Ih and IPAR/Ih, increase when sky conditions change from clear to overcast; on the other hand, the fraction IIR/Ih decreases when the sky conditions change from clear to overcast. This variation is due to the absorption of infrared by cloud water vapor, contrary to IUV and IPAR, which are not affected by this attenuation. In addition, the values of the correlation coefficients (R) reflect the good quality of the carried out correlations.Finally, it should be noted that all these results are in agreement with other work carried out either in Morocco [49] or across the world [23].


## 5.1. Estimation of Hourly Values of Global IrradiationIh

To study the solar potential of a site, we need to know the hourly values of global irradiation over a long period. However, this variable is not always available. Consequently, several studies have attempted to estimate this component. Among these works, we can cite for example the Collares-Pereira and Rabl model. Hence, based on this work, we developed a new model valid for Morocco.In order to improve the quality of the estimated values generated by the Collares-Pereira model, we looked for an empirical relationship adapted for Morocco. So we calculated the coefficientsa and b of Equation (3) from measurement that led us to find a new relation between the ratioI¯h/H¯hoverI¯oh/H¯oh, ωi and ωc¯. Therefore, we proceeded to adapt the coefficients for Morocco, as follows.In the first step, we determined the monthly averages of daily global solar irradiation and the monthly averages of hourly solar irradiation using the measured data. Then, for each month, we calculated the values ofa and b by the least-squares method. That gave us, for each station, and each coefficient, 12 values projected in Table 4.Table 4
Thea and b values calculated for the representative station’s measurements.
FezMarrakechRabatTangierababababJanuary0.430.650.480.6080.320.7570.3670.735February0.5160.5780.40.70.4640.640.5460.548March0.60.3810.5180.6020.6530.4260.7840.274April0.5520.5690.6060.51470.6480.4820.6810.404May0.7550.3330.70630.3990.6640.4510.7790.3Jun0.6990.4040.6810.4350.6630.4710.7560.335July0.6520.4650.6830.4310.6040.5540.7650.318August0.620.4990.7020.3980.6010.5260.7120.379September0.5030.5990.5410.5820.5460.5550.6840.399October0.4130.6890.5290.6010.5730.5410.7010.358November0.320.830.4120.6990.5750.50.4340.670December0.4490.6280.3490.7550.3860.7210.5520.517Then, for every station, we plotted sparely the monthly values ofa and b as a function of sinωc¯−70°, which gave us presentations as Figure 2.Figure 2
Example of the presentation ofa and b as a function of sinωc¯−70°.
(a)(b)As a result, we obtained equations in the following form:(10)a=a′+b′sinωc¯−70°,b=a′′+b′′sinωc¯−70°.70° represents the minimum value of the hour angle of the sunsetThe coefficientsa′, a″, b′, and b″ are calculated from values in Table 4 by using the least-squares method. The obtained values are presented in Table 5.Table 5
The coefficient values calculated in function ofsinωc¯−70°.
Coefficientaa′b′RCoefficientba″b″RRabat0.4060.4500.9Rabat0.672-0.3540.89Fez0.3500.5890.88Fez0.735-0.5480.85Marrakech0.3240.6780.94Marrakech0.788-0.6730.93Tangier0.44060.53740.91Tangier0.677-0.60.9In order to have one equation valid for all of Morocco, we averaged the coefficients mentioned in Table5, which gave us the following equations:
(11)a=0.38+0.564sinωc¯−70°,b=0.72−0.54sinωc¯−70°.Finally, by using the equation below, we will be able to estimate the monthly average of hourly global solar radiation for all Moroccan regions:(12)I¯h/H¯hI¯oh/H¯oh=0.38+0.564sinωc¯−70°+0.72−0.54sinωc¯−70°cosωi.For advanced studies of solar systems, data of monthly average of solar irradiation is insufficient; therefore, we need to find new methods of estimating hourly values. For this, we adapted Equation (12) to be valid for estimating hourly values; thus, we changed the means of the variables I¯oh,H¯oh,H¯h,andωc¯ by Ioh, Hh, ωc, and Hoh. That gave us the following equation:
(13)Ih/HhIoh/Hoh=0.38+0.564sinωc−70°+0.72−0.54sinωc−70°cosωi.It remains to validate this adaptation by a comparison with the measurements. This will be the subject of Table6.Table 6
OurIh model validation test.
Ih estimation modelRMBE (%)OurIh model-1.9%


## 5.2. Estimation ofId and Ib

The hourly evolution of the diffuse component of solar radiation is an important data for researchers and engineers when they study the behavior of solar thermal conversion systems or photovoltaic systems. Therefore, several models have been made for estimating this component such as the Orgill and Hollands model [9], the Erbs et al. model [10], and the model of De Miguel et al. [15].In the same way, we tried to find our own model to estimateId. For this reason, we plotted KdId/Ih as a function of KtIh/Ioh as shown in Figure 3.Figure 3
Hourly correlation betweenKdId/Ih and the clarity index KtIh/IohFrom Figure3, we notice that the cloud of dots has the shape of a waterfall, which is characterized by two almost horizontal parts at the beginning and at the end and another inclined part in the middle. In order to find a correlation that represents all the points with a good fit, we divided the Kt values into three intervals, and by the least-squares method, we established separately the correlation equations that represent them.In addition, we find that the cloud of dots show a remarkable dispersion especially in theKt range between 0.3 and 0.55. This is mainly due to two causes, firstly to the large volume of used data, which is well above 12,000 values from different sites, and secondly to the fluctuation of the diffuse fraction which depends on the type and distribution of particles and clouds that are in the sky.For each interval, the calculated correlations equations are in the following form:(14)IdIh=0.9382−0.2017Kt,Kt≤0.2,0.731+1.9202Kt−6.5525Kt2+4.014Kt3,0.2<Kt≤0,0.14,Kt>0.8.Moreover, with the predicted values of hourly diffuse and global solar radiation, we can calculate easily the hourly values of direct solar radiation by using the equation below:(15)Ib=Ih−Id.


## 5.3. Estimation ofIs

The hourly inclined solar radiationIs is calculated by using the diffuse isotropic approximation cited by [48]:
(16)Is=Ibcosicosθ+Ih−Ib1+cosS2+ρIh1−cosS2,where S is the tilt angle; i and θ are, respectively, the angles of incidence of the sun’s rays on the inclined plane and on the horizontal plane; and cosi/cosθ is the transposition factor of the direct solar component calculated by the division of the incidence angles of the solar rays on the inclined plane and incidence angles of the solar rays on the horizontal plane, in the middle of the considered hour.As it is interesting to know the evolution of global hourly irradiation for different inclinations, we chose to calculate it forS=ϕ, ϕ+20, and ϕ−20,90.


## 5.4. Predicting the Values of Solar Spectral Radiation

Hourly data of global, direct, and diffuse solar radiation are insufficient to study all solar systems because other applications require the knowledge of the hourly values of spectral components.For example, the ultraviolet component is used in atmospheric studies, the photosynthetically active radiation is necessary for the plant development process, and the infrared solar radiation is used in meteorological studies.Knowing that theIPAR, IUV, and IIR measurements are not often available in the meteorological stations, for this reason, we established simple formulas for estimating these radiations from the measurement of global solar radiation.Thus, in order to establish the estimation equations of spectral component, we used measurements collected by the LESE station network, and through the least-squares method, we established relations between the global irradiation and the spectral solar components. For more precision, the equations are established according three ranges ofKt. The obtained correlations are given in Table 7.Table 7
The equations for predicting the values ofIPAR, IUV, and IIR.
Kt intervalsCorrelation equationsRKt≤0.35IUV=0.048Ih0.98IPAR=0.4889Ih0.99IIR=0.461Ih0.990.35<Kt≤0.65IUV=0.044Ih0.98IPAR=0.485Ih0.99IIR=0.469Ih0.99Kt>0.65IUV=0.041Ih0.98IPAR=0.479Ih0.99IIR=0.477Ih0.99From Table7, we notice that the hourly fractions of IUV and IPAR, namely, IUV/Ih and IPAR/Ih, increase when sky conditions change from clear to overcast; on the other hand, the fraction IIR/Ih decreases when the sky conditions change from clear to overcast. This variation is due to the absorption of infrared by cloud water vapor, contrary to IUV and IPAR, which are not affected by this attenuation. In addition, the values of the correlation coefficients (R) reflect the good quality of the carried out correlations.Finally, it should be noted that all these results are in agreement with other work carried out either in Morocco [49] or across the world [23].


## 6. Statistical Indicators

To validate the various models mentioned before, we compared the estimated values provided by models with measurements. This study was done by using the following indicators [45]:The mean difference(17)Themeandifference=∑i=0npi−miN.The mean difference is an indication of the average difference between the calculated and measured values.The relative error(18)RMBE=100∑i=0npi−mi∑i=0nmi.The RMBE tells us the percentage of deviation; when it is positive, it tells us the overestimation percentage of the model, while when it is negative, it tells us the underestimation percentage of the model.The mean absolute error(19)MAE=∑i=0n∣pi−mi∣N.The percentage of mean absolute error(20)MAPE=1N∑i=0npi−mimi∗100.The percentage of mean absolute error measures the mean size of the errors; it is always positive and less than or equal to the root mean square error.The mean squared error(21)RMSE=100m¯∑i=0npi−mi2N.The RMSE gives us information on the amplitude of the deviations and the dispersion of the points around the first bisector, wherepi and mi are the ith calculated and measured values, N is the number of observations, and m¯is the mean of measured values.


## 7. Results and Discussions

### 7.1. Selection of the Appropriate Estimation Models for Morocco

To select the appropriate estimation models for Moroccan sites, we compared the values generated by the various estimations equations with measurements by using the relative error. The result of this study is shown in Tables6, 8 and 9.Table 8
Comparison between models to estimateI¯h.
I¯h estimation modelRMBE (%)OurI¯h model-1.58%Collares-Pereira and Rabl model7.7%Table 9
Comparison between models to estimateId.
Id estimation modelRMBE (%)Orgill and Hollands model-4.05%De Miguel et al. model-2.82%Erbs et al. model-3.2%OurId model-9%


#### 7.1.1. Global Solar Radiation

From Table8, we note that the relative error calculated from data generated by our I¯h model does not exceed -1.58% which is much lower than the error obtained using the Collares-Pereira and Rabl model.This result is approved by the graphs of Figure4 where we notice in Figure 4(a) a concentration of points in a straight line called trend curve; this curve is totally confused with the first bisector, contrary to Figure 4(b) where there is a dispersion of the points along the curve with a deviation of the trend curve compared to the first bisector.Figure 4
Graphical representation of the estimated values according to the measured values for various models to estimateI¯h. Comparison between our I¯h model (a) and Collares-Pereira and Rabl model (b).
(a)(b)From this, we noticed that ourI¯h model is the most adapted model to predict monthly averages of global hourly irradiation.Using Equation (13), we generated hourly values of global irradiation. Then, we verified the quality of these estimated data through the relative error. The result is presented in Table 6.The comparison between measured values and values estimated by OurIh model showed that the relative error does not exceed -1.9% (Table 6), which is relatively low; this result is approved by the weak dispersion of the points in Figure 5 and the confusion of the trend curve with the first bisector.Figure 5
Graphical representation of the estimated values according to the measured values.These results show the good quality of estimating global radiation by ourIh model. From that, we conclude that the adaptation used to have Equation (13) works to estimate hourly global radiation.


#### 7.1.2. Diffuse Solar Radiation

From Table9, we notice that the relative error calculated from data generated by De Miguel et al. model does not exceed -2.83%, which is the lowest error compared to the other models. In addition, the graphical representation of data generated by the different models as function of the measured data shows that the cloud of dots in Figure 6(a) (De Miguel et al. model) is concentrated in the middle of the graph and its trend curve is almost confused with the first bisector with a small spacing for large values of Id. On the other hand, for the other models, the spacing is highly remarkable along the curves with a considerable dispersion of the points.Figure 6
Graphical representation of the estimated values according to the measured values for various models to estimateId: (a) Orgill and Hollands model; (b) De Miguel et al. model; (c) Erbs et al. model; (d) our Id model.
(a)(b)(c)(d)Also, from Table9 and Figure 6, we found that the model of De Miguel et al. is the most adapted for the estimation of diffuse radiation because it presents the lowest relative error.Consequently, to generate the hourly database, we selected ourIh model to estimate the hourly global solar radiation and the model of De Miguel et al. to estimate the hourly diffuse radiation.


### 7.2. Validation of the Selected Models

To examine the accuracy of the selected correlations, we compared the estimated values with measurements by using the statistical indicators.To verify the coordination of the estimations against reality, we traced the estimated values as a function of the measured values for all the considered sites.


#### 7.2.1. Validation ofIh Estimation Model

To deepen the validation study of ourIh model, we used five statistical indicators presented in Table 10 and we showed graphical representation of the estimated values according to the measured values in Figure 7.Table 10
The performance of used model for predicting the global radiation.
IhRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-2.93-0.7843.097.89.5Tangier-3.9-23.342.27.666.96Fez-0.85-1.1723.317.76.32Marrakech-3.28-11.8318.95.745.42Figure 7
Graphical representation of the estimated values according to the measured values for global radiation of Rabat (a), Tangier (b), Fez (c), and Marrakech (d).
(a)(b)(c)(d)Based on the obtained results, we can deduce that the used correlations have a very good quality, in view of the following:(i)
The average of relative error, calculated from the measured and the estimated values, does not exceed – 2.93% in Rabat, -3.9% in Tangier,-3.28% in Marrakech, and -0.85%in Fez(ii)
The mean difference between the measured and the estimated values does not exceed −0.78 Wh/m2 in Rabat, -23.3 Wh/m2 in Tangier, -11.83 Wh/m2 in Marrakech, and -1.17 Wh/m2 in Fez.In addition, from Figure7, we noticed that the first bisector and the trend curve are almost the same for all stations.


#### 7.2.2. Validation of Diffuse Estimation Model

As seen in Table6, the De Miguel et al. model is most adapted to predict diffuse solar radiation in Rabat. To extend the validity of this model for all Moroccan regions, we calculated the statistical indicators for the representative climatic regions. The results of this study are shown in Table 11.Table 11
The performance of used model for predictingId.
IdRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-1.09-1.7917.7810.2513.63Tangier-3.08-15.2617.948.8612.84Fez-0.68-3.5417.737.458.85Marrakech-2.49-5.849.766.539.49From Table10 we noticed that the annual average of relative error, calculated from the measured and the estimated values, does not exceed -1.09% in Rabat, -0.68% in Fez, -3.08% in Tangier, and-2.49% in Marrakech. In addition, from Figure 8, we noticed that the point clouds generated by the representation of estimated Id as a function of the measured Id show a remarkable reconciliation between the first bisector and the trend curve, which approves the high quality of the estimations for all stations.Figure 8
Graphical representation of the estimated values according to the measured values for diffuse radiation of Rabat (a), Tangier (b), Fez (c) and Marrakech (d).
(a)(b)(c)(d)Based on the mentioned results, we deduced that the model of De Miguel et al. to estimate the diffuse radiation give a remarkable coordinance with the measurements collected at different sites, whereas this model was made for four countries located in the north of the Mediterranean Sea with different latitudes than Morocco. All of these observations lead us to conclude that this model has a regional aspect and can be used even for other countries with a similar climate such as Algeria, Tunisia, and Italy. It remains to prove its validity for all the Mediterranean area.


#### 7.2.3. Validation of Spectral Components

To examine the quality of the used correlations for predicting the spectral components, we compared the estimated values provided by equations in Table7 with measurements, which gave us the relative errors in Table 12.Table 12
RMBE calculated from estimated and measured values ofIPAR, IUV, and IIR.
Kt intervalsIUV relative error (%)IPAR relative error (%)IIR relative error (%)Kt≤0.355.43.33.20.35<Kt≤0.651.12.22.6Kt>0.651.21.21.10≤Kt≤11.81.81.8The obtained results (Table12) reveal that the proposed model estimates the spectral components with remarkable precision since the maximum of relative errors does not exceed 5.4% for IUV, 3.3% for the IPAR, and 3.2% for the IIR.


### 7.3. The Main Steps Followed for the Creation of the Hourly Solar Database

Utilizing the database of daily values of global solar radiation and through to the selected correlation equations, we were able to make a database of all solar radiation components on the hourly scale. Figure9 is a flowchart summarizing the various steps to generate this database.Figure 9
The main steps followed for creating the hourly solar database for Morocco.


### 7.4. Samples of the Obtained Results

After having carried out, tested, and compared the solar irradiation estimation models, we chose to use the models that represent the minimum error. Thus, we used ourIh model to predict global irradiation, De Miguel et al.’s model to predict diffuse irradiation, the isotropic model to predict tilted irradiation, and our models in Table 7 to predict spectral components.The use of these models has allowed us to obtain a data bank that gathers the different components of solar radiation, either energetic (global irradiation, diffuse irradiation, direct irradiation, tilted irradiation) or spectral (ultravioletIUV, infrared IIR, and photosynthetically active radiation IPAR) on an hourly scale. This bank includes data for 10 years (2009–2018) for 29 sites, which represents the entire Moroccan territory.Given the large number of results, we have chosen to exhibit the obtained data in 3D graphics representing the hourly evolution of solar irradiation throughout the months of the year.Figure10 is an example of results generated by different models to estimate solar components for one site (Rabat): (a) global solar radiation, (b) diffuse solar radiation, (c) direct solar radiation, (d) inclined solar radiation, (e) infrared IIR solar radiation, (f) photosynthetically active radiation IPAR, and (g) ultraviolet IUV solar radiation.Figure 10
Annual variation of hourly radiation: (a) global solar radiation, (b) diffuse solar radiation, (c) direct solar radiation, (d) inclined solar radiation, (e) infraredIIR solar radiation, (f) photosynthetically active radiation IPAR, and (g) ultraviolet IUV solar radiation.
(a)(b)(c)(d)(e)(f)(g)From Figure10, we note that all components of solar irradiation are maximum in July around midday. Their values exceed 862 Wh/m2 for global radiation, 641 Wh/m2 for direct radiation, 830 Wh/m2 for inclined radiation, 280 Wh/m2 for diffuse radiation, 412 Wh/m2 for infrared radiation, 413 Wh/m2 for photosynthetically active radiation, and 36 Wh/m2 for ultraviolet radiation. We also note that the solar radiations are symmetrical relatively to the peaks of irradiations.In addition, we noticed that all solar irradiations are always present during the year with significant quantities; it results that the annual duration of insolation in Morocco varies from 2700 hours in the north of the country to more than 3500 hours in the south. The annual solar energy received by a horizontal plane varies from 1.7 to more than 2.2 MWh/m2/year.All of these observations are caused by the clarity of the sky, the height of the sun, and the geographical position of Morocco.


## 7.1. Selection of the Appropriate Estimation Models for Morocco

To select the appropriate estimation models for Moroccan sites, we compared the values generated by the various estimations equations with measurements by using the relative error. The result of this study is shown in Tables6, 8 and 9.Table 8
Comparison between models to estimateI¯h.
I¯h estimation modelRMBE (%)OurI¯h model-1.58%Collares-Pereira and Rabl model7.7%Table 9
Comparison between models to estimateId.
Id estimation modelRMBE (%)Orgill and Hollands model-4.05%De Miguel et al. model-2.82%Erbs et al. model-3.2%OurId model-9%


### 7.1.1. Global Solar Radiation

From Table8, we note that the relative error calculated from data generated by our I¯h model does not exceed -1.58% which is much lower than the error obtained using the Collares-Pereira and Rabl model.This result is approved by the graphs of Figure4 where we notice in Figure 4(a) a concentration of points in a straight line called trend curve; this curve is totally confused with the first bisector, contrary to Figure 4(b) where there is a dispersion of the points along the curve with a deviation of the trend curve compared to the first bisector.Figure 4
Graphical representation of the estimated values according to the measured values for various models to estimateI¯h. Comparison between our I¯h model (a) and Collares-Pereira and Rabl model (b).
(a)(b)From this, we noticed that ourI¯h model is the most adapted model to predict monthly averages of global hourly irradiation.Using Equation (13), we generated hourly values of global irradiation. Then, we verified the quality of these estimated data through the relative error. The result is presented in Table 6.The comparison between measured values and values estimated by OurIh model showed that the relative error does not exceed -1.9% (Table 6), which is relatively low; this result is approved by the weak dispersion of the points in Figure 5 and the confusion of the trend curve with the first bisector.Figure 5
Graphical representation of the estimated values according to the measured values.These results show the good quality of estimating global radiation by ourIh model. From that, we conclude that the adaptation used to have Equation (13) works to estimate hourly global radiation.


### 7.1.2. Diffuse Solar Radiation

From Table9, we notice that the relative error calculated from data generated by De Miguel et al. model does not exceed -2.83%, which is the lowest error compared to the other models. In addition, the graphical representation of data generated by the different models as function of the measured data shows that the cloud of dots in Figure 6(a) (De Miguel et al. model) is concentrated in the middle of the graph and its trend curve is almost confused with the first bisector with a small spacing for large values of Id. On the other hand, for the other models, the spacing is highly remarkable along the curves with a considerable dispersion of the points.Figure 6
Graphical representation of the estimated values according to the measured values for various models to estimateId: (a) Orgill and Hollands model; (b) De Miguel et al. model; (c) Erbs et al. model; (d) our Id model.
(a)(b)(c)(d)Also, from Table9 and Figure 6, we found that the model of De Miguel et al. is the most adapted for the estimation of diffuse radiation because it presents the lowest relative error.Consequently, to generate the hourly database, we selected ourIh model to estimate the hourly global solar radiation and the model of De Miguel et al. to estimate the hourly diffuse radiation.


## 7.1.1. Global Solar Radiation

From Table8, we note that the relative error calculated from data generated by our I¯h model does not exceed -1.58% which is much lower than the error obtained using the Collares-Pereira and Rabl model.This result is approved by the graphs of Figure4 where we notice in Figure 4(a) a concentration of points in a straight line called trend curve; this curve is totally confused with the first bisector, contrary to Figure 4(b) where there is a dispersion of the points along the curve with a deviation of the trend curve compared to the first bisector.Figure 4
Graphical representation of the estimated values according to the measured values for various models to estimateI¯h. Comparison between our I¯h model (a) and Collares-Pereira and Rabl model (b).
(a)(b)From this, we noticed that ourI¯h model is the most adapted model to predict monthly averages of global hourly irradiation.Using Equation (13), we generated hourly values of global irradiation. Then, we verified the quality of these estimated data through the relative error. The result is presented in Table 6.The comparison between measured values and values estimated by OurIh model showed that the relative error does not exceed -1.9% (Table 6), which is relatively low; this result is approved by the weak dispersion of the points in Figure 5 and the confusion of the trend curve with the first bisector.Figure 5
Graphical representation of the estimated values according to the measured values.These results show the good quality of estimating global radiation by ourIh model. From that, we conclude that the adaptation used to have Equation (13) works to estimate hourly global radiation.


## 7.1.2. Diffuse Solar Radiation

From Table9, we notice that the relative error calculated from data generated by De Miguel et al. model does not exceed -2.83%, which is the lowest error compared to the other models. In addition, the graphical representation of data generated by the different models as function of the measured data shows that the cloud of dots in Figure 6(a) (De Miguel et al. model) is concentrated in the middle of the graph and its trend curve is almost confused with the first bisector with a small spacing for large values of Id. On the other hand, for the other models, the spacing is highly remarkable along the curves with a considerable dispersion of the points.Figure 6
Graphical representation of the estimated values according to the measured values for various models to estimateId: (a) Orgill and Hollands model; (b) De Miguel et al. model; (c) Erbs et al. model; (d) our Id model.
(a)(b)(c)(d)Also, from Table9 and Figure 6, we found that the model of De Miguel et al. is the most adapted for the estimation of diffuse radiation because it presents the lowest relative error.Consequently, to generate the hourly database, we selected ourIh model to estimate the hourly global solar radiation and the model of De Miguel et al. to estimate the hourly diffuse radiation.


## 7.2. Validation of the Selected Models

To examine the accuracy of the selected correlations, we compared the estimated values with measurements by using the statistical indicators.To verify the coordination of the estimations against reality, we traced the estimated values as a function of the measured values for all the considered sites.


### 7.2.1. Validation ofIh Estimation Model

To deepen the validation study of ourIh model, we used five statistical indicators presented in Table 10 and we showed graphical representation of the estimated values according to the measured values in Figure 7.Table 10
The performance of used model for predicting the global radiation.
IhRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-2.93-0.7843.097.89.5Tangier-3.9-23.342.27.666.96Fez-0.85-1.1723.317.76.32Marrakech-3.28-11.8318.95.745.42Figure 7
Graphical representation of the estimated values according to the measured values for global radiation of Rabat (a), Tangier (b), Fez (c), and Marrakech (d).
(a)(b)(c)(d)Based on the obtained results, we can deduce that the used correlations have a very good quality, in view of the following:(i)
The average of relative error, calculated from the measured and the estimated values, does not exceed – 2.93% in Rabat, -3.9% in Tangier,-3.28% in Marrakech, and -0.85%in Fez(ii)
The mean difference between the measured and the estimated values does not exceed −0.78 Wh/m2 in Rabat, -23.3 Wh/m2 in Tangier, -11.83 Wh/m2 in Marrakech, and -1.17 Wh/m2 in Fez.In addition, from Figure7, we noticed that the first bisector and the trend curve are almost the same for all stations.


### 7.2.2. Validation of Diffuse Estimation Model

As seen in Table6, the De Miguel et al. model is most adapted to predict diffuse solar radiation in Rabat. To extend the validity of this model for all Moroccan regions, we calculated the statistical indicators for the representative climatic regions. The results of this study are shown in Table 11.Table 11
The performance of used model for predictingId.
IdRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-1.09-1.7917.7810.2513.63Tangier-3.08-15.2617.948.8612.84Fez-0.68-3.5417.737.458.85Marrakech-2.49-5.849.766.539.49From Table10 we noticed that the annual average of relative error, calculated from the measured and the estimated values, does not exceed -1.09% in Rabat, -0.68% in Fez, -3.08% in Tangier, and-2.49% in Marrakech. In addition, from Figure 8, we noticed that the point clouds generated by the representation of estimated Id as a function of the measured Id show a remarkable reconciliation between the first bisector and the trend curve, which approves the high quality of the estimations for all stations.Figure 8
Graphical representation of the estimated values according to the measured values for diffuse radiation of Rabat (a), Tangier (b), Fez (c) and Marrakech (d).
(a)(b)(c)(d)Based on the mentioned results, we deduced that the model of De Miguel et al. to estimate the diffuse radiation give a remarkable coordinance with the measurements collected at different sites, whereas this model was made for four countries located in the north of the Mediterranean Sea with different latitudes than Morocco. All of these observations lead us to conclude that this model has a regional aspect and can be used even for other countries with a similar climate such as Algeria, Tunisia, and Italy. It remains to prove its validity for all the Mediterranean area.


### 7.2.3. Validation of Spectral Components

To examine the quality of the used correlations for predicting the spectral components, we compared the estimated values provided by equations in Table7 with measurements, which gave us the relative errors in Table 12.Table 12
RMBE calculated from estimated and measured values ofIPAR, IUV, and IIR.
Kt intervalsIUV relative error (%)IPAR relative error (%)IIR relative error (%)Kt≤0.355.43.33.20.35<Kt≤0.651.12.22.6Kt>0.651.21.21.10≤Kt≤11.81.81.8The obtained results (Table12) reveal that the proposed model estimates the spectral components with remarkable precision since the maximum of relative errors does not exceed 5.4% for IUV, 3.3% for the IPAR, and 3.2% for the IIR.


## 7.2.1. Validation ofIh Estimation Model

To deepen the validation study of ourIh model, we used five statistical indicators presented in Table 10 and we showed graphical representation of the estimated values according to the measured values in Figure 7.Table 10
The performance of used model for predicting the global radiation.
IhRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-2.93-0.7843.097.89.5Tangier-3.9-23.342.27.666.96Fez-0.85-1.1723.317.76.32Marrakech-3.28-11.8318.95.745.42Figure 7
Graphical representation of the estimated values according to the measured values for global radiation of Rabat (a), Tangier (b), Fez (c), and Marrakech (d).
(a)(b)(c)(d)Based on the obtained results, we can deduce that the used correlations have a very good quality, in view of the following:(i)
The average of relative error, calculated from the measured and the estimated values, does not exceed – 2.93% in Rabat, -3.9% in Tangier,-3.28% in Marrakech, and -0.85%in Fez(ii)
The mean difference between the measured and the estimated values does not exceed −0.78 Wh/m2 in Rabat, -23.3 Wh/m2 in Tangier, -11.83 Wh/m2 in Marrakech, and -1.17 Wh/m2 in Fez.In addition, from Figure7, we noticed that the first bisector and the trend curve are almost the same for all stations.


## 7.2.2. Validation of Diffuse Estimation Model

As seen in Table6, the De Miguel et al. model is most adapted to predict diffuse solar radiation in Rabat. To extend the validity of this model for all Moroccan regions, we calculated the statistical indicators for the representative climatic regions. The results of this study are shown in Table 11.Table 11
The performance of used model for predictingId.
IdRMBE (%)The mean difference (Wh/m2)MAE (Wh/m2)MAPE (%)RMSE (%)Rabat-1.09-1.7917.7810.2513.63Tangier-3.08-15.2617.948.8612.84Fez-0.68-3.5417.737.458.85Marrakech-2.49-5.849.766.539.49From Table10 we noticed that the annual average of relative error, calculated from the measured and the estimated values, does not exceed -1.09% in Rabat, -0.68% in Fez, -3.08% in Tangier, and-2.49% in Marrakech. In addition, from Figure 8, we noticed that the point clouds generated by the representation of estimated Id as a function of the measured Id show a remarkable reconciliation between the first bisector and the trend curve, which approves the high quality of the estimations for all stations.Figure 8
Graphical representation of the estimated values according to the measured values for diffuse radiation of Rabat (a), Tangier (b), Fez (c) and Marrakech (d).
(a)(b)(c)(d)Based on the mentioned results, we deduced that the model of De Miguel et al. to estimate the diffuse radiation give a remarkable coordinance with the measurements collected at different sites, whereas this model was made for four countries located in the north of the Mediterranean Sea with different latitudes than Morocco. All of these observations lead us to conclude that this model has a regional aspect and can be used even for other countries with a similar climate such as Algeria, Tunisia, and Italy. It remains to prove its validity for all the Mediterranean area.


## 7.2.3. Validation of Spectral Components

To examine the quality of the used correlations for predicting the spectral components, we compared the estimated values provided by equations in Table7 with measurements, which gave us the relative errors in Table 12.Table 12
RMBE calculated from estimated and measured values ofIPAR, IUV, and IIR.
Kt intervalsIUV relative error (%)IPAR relative error (%)IIR relative error (%)Kt≤0.355.43.33.20.35<Kt≤0.651.12.22.6Kt>0.651.21.21.10≤Kt≤11.81.81.8The obtained results (Table12) reveal that the proposed model estimates the spectral components with remarkable precision since the maximum of relative errors does not exceed 5.4% for IUV, 3.3% for the IPAR, and 3.2% for the IIR.


## 7.3. The Main Steps Followed for the Creation of the Hourly Solar Database

Utilizing the database of daily values of global solar radiation and through to the selected correlation equations, we were able to make a database of all solar radiation components on the hourly scale. Figure9 is a flowchart summarizing the various steps to generate this database.Figure 9
The main steps followed for creating the hourly solar database for Morocco.


## 7.4. Samples of the Obtained Results

After having carried out, tested, and compared the solar irradiation estimation models, we chose to use the models that represent the minimum error. Thus, we used ourIh model to predict global irradiation, De Miguel et al.’s model to predict diffuse irradiation, the isotropic model to predict tilted irradiation, and our models in Table 7 to predict spectral components.The use of these models has allowed us to obtain a data bank that gathers the different components of solar radiation, either energetic (global irradiation, diffuse irradiation, direct irradiation, tilted irradiation) or spectral (ultravioletIUV, infrared IIR, and photosynthetically active radiation IPAR) on an hourly scale. This bank includes data for 10 years (2009–2018) for 29 sites, which represents the entire Moroccan territory.Given the large number of results, we have chosen to exhibit the obtained data in 3D graphics representing the hourly evolution of solar irradiation throughout the months of the year.Figure10 is an example of results generated by different models to estimate solar components for one site (Rabat): (a) global solar radiation, (b) diffuse solar radiation, (c) direct solar radiation, (d) inclined solar radiation, (e) infrared IIR solar radiation, (f) photosynthetically active radiation IPAR, and (g) ultraviolet IUV solar radiation.Figure 10
Annual variation of hourly radiation: (a) global solar radiation, (b) diffuse solar radiation, (c) direct solar radiation, (d) inclined solar radiation, (e) infraredIIR solar radiation, (f) photosynthetically active radiation IPAR, and (g) ultraviolet IUV solar radiation.
(a)(b)(c)(d)(e)(f)(g)From Figure10, we note that all components of solar irradiation are maximum in July around midday. Their values exceed 862 Wh/m2 for global radiation, 641 Wh/m2 for direct radiation, 830 Wh/m2 for inclined radiation, 280 Wh/m2 for diffuse radiation, 412 Wh/m2 for infrared radiation, 413 Wh/m2 for photosynthetically active radiation, and 36 Wh/m2 for ultraviolet radiation. We also note that the solar radiations are symmetrical relatively to the peaks of irradiations.In addition, we noticed that all solar irradiations are always present during the year with significant quantities; it results that the annual duration of insolation in Morocco varies from 2700 hours in the north of the country to more than 3500 hours in the south. The annual solar energy received by a horizontal plane varies from 1.7 to more than 2.2 MWh/m2/year.All of these observations are caused by the clarity of the sky, the height of the sun, and the geographical position of Morocco.


## 8. Conclusion

Through this work, we created a ten-year database for the hourly values that regroup all solar radiation components. This database consists of more than eleven million hourly values, spread over 29 sites, covering all regions of Morocco for the period between 2009 and 2018.The validity study has shown that the estimation models used have minimal errors:(i)
For the global radiation, relative errors are -2.93% for Rabat, -3.9% for Tangier, -3.28% for Marrakech, and -0.85% for Fez. The corresponding average differences are -0.78 Wh/m2 for Rabat, -23.3 Wh/m2 for Tangier, -11.83 Wh/m2 for Marrakech, and -1.17 Wh/m2 for Fez(ii)
For diffuse radiation, relative errors are -1.09% for Rabat, -0.68% for Fez, -3.08% for Tangier, and -2.49% for Marrakech. The corresponding average differences are -1.79 Wh/m2 for Rabat, -3.54 Wh/m2 for Fez, -15.26 Wh/m2 for Tangier, and -5.84 Wh/m2 for MarrakechOn the other hand, the point clouds generated by the representation of estimated values as a function of the measured values show a remarkable reconciliation between the first bisector and the trend curve, which proves the high quality of the used estimations.Regarding the spectral components, the calculated relative errors are equal to 5.4% forIUV, 3.3% for the IPAR, and 3.2% for the IIR, which proves that our models estimate the spectral components with remarkable precision.Considering the high quality of the results obtained from the estimation of the diffuse radiation by the De Miguel et al. model, we deduced that this model has a regional aspect and can be used even for other neighboring countries with a similar climate such as Algeria, Tunisia, and Italy. It remains to prove its validity for all the Mediterranean area.In addition, we noticed that the direct radiation is maximum during the most part of the year and it exceeds 641 Wh/m2. This result will encourage engineers and installers to move towards the technologies that use this type of radiation as solar concentrators often used for the production of thermodynamic electricity.This work gives us the possibility of evaluating the Moroccan potential of solar radiation over a large period and studying qualitatively the different energetic and spectral characteristics of solar radiation. In addition, we were able to know the various physical variables that have an effect on the solar irradiation such as the lightness factor that enters in the diffuse radiation estimation, or the angle of sunset, which is directly related to the global radiation.Thus, this work allowed us to offer a wide presentation of the hourly solar data over a period of 10 years, to fill the gap of the solar data at the hourly scale, to complete the various previous works carried out within our laboratory, and to build a complete data bank for all Moroccan territory. This database meets the needs of engineers, researchers, and installers of solar systems.



---

*Source: 8897818-2021-01-27.xml*