*Article* **Estimation of the Hourly Global Solar Irradiation on the Tilted and Oriented Plane of Photovoltaic Solar Panels Applied to Greenhouse Production**

**Francisco J. Diez 1,\*, Andrés Martínez-Rodríguez <sup>1</sup> , Luis M. Navas-Gracia <sup>1</sup> , Leticia Chico-Santamarta <sup>2</sup> , Adriana Correa-Guimaraes <sup>1</sup> and Renato Andara <sup>3</sup>**


**Abstract:** Agrometeorological stations have horizontal solar irradiation data available, but the design and simulation of photovoltaic (PV) systems require data about the solar panel (inclined and/or oriented). Greenhouses for agricultural production, outside the large protected production areas, are usually off-grid; thus, the solar irradiation variable on the panel plane is critical for an optimal PV design. Modeling of solar radiation components (beam, diffuse, and ground-reflected) is carried out by calculating the extraterrestrial solar radiation, solar height, angle of incidence, and diffuse solar radiation. In this study, the modeling was done using Simulink-MATLAB blocks to facilitate its application, using the day of the year, the time of day, and the hourly horizontal global solar irradiation as input variables. The rest of the parameters (i.e., inclination, orientation, solar constant, albedo, latitude, and longitude) were fixed in each block. The results obtained using anisotropic models of diffuse solar irradiation of the sky in the region of Castile and León (Spain) showed improvements over the results obtained with isotropic models. This work enables the precise estimation of solar irradiation on a solar panel flexibly, for particular places, and with the best models for each of the components of solar radiation.

**Keywords:** extraterrestrial solar irradiation; global, beam and diffuse solar components; groundreflected solar radiation; horizontal, tilted and oriented solar irradiation

### **1. Introduction**

The solar irradiation incident on the surface of a solar panel is the fundamental parameter for the design of photovoltaic systems that are best integrated into greenhouses for agricultural production and for determining the amount of electrical energy that is produced by such a panel, as well as for the simulation of its operation with the required precision. The value of the solar radiation that affects the solar panels is the main variable needed to determine the performance of a photovoltaic (PV) system, together with the ambient temperature, humidity and the speed and direction of the wind (see Pérez-Alonso et al. [1]).

In modern agriculture, greenhouses are intended to increase the productivity, quality, and precocity of crops that are characterized by the intensive use of land and of other means of production and inputs (see Yano and Cossu [2]). Greenhouses, except those located in large, protected production areas, are usually located in rural off-grid areas, and connection to the grid can be very expensive for technical, economic, or environmental reasons; therefore, an autonomous power source is required (see Chaurey and Kandpal [3] and

**Citation:** Diez, F.J.; Martínez-Rodríguez, A.; Navas-Gracia, L.M.; Chico-Santamarta, L.; Correa-Guimaraes, A.; Andara, R. Estimation of the Hourly Global Solar Irradiation on the Tilted and Oriented Plane of Photovoltaic Solar Panels Applied to Greenhouse Production. *Agronomy* **2021**, *11*, 495. https:// doi.org/10.3390/agronomy11030495

Academic Editors: Miguel-Ángel Muñoz-García, Luis Hernández-Callejo and Byoung Ryong Jeong

Received: 30 December 2020 Accepted: 3 March 2021 Published: 6 March 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Qoaider and Steinbrecht [4]). Thus, an efficient framework is needed to use solar/diesel systems in off-grid greenhouses (see Cai et al. [5]). On the other hand, the highest electrical consumptions in greenhouses correspond to ventilation, refrigeration, and pumping equipment (water and nutrients). These agricultural structures are usually located in open spaces where they receive large amounts of direct solar radiation. Hence, the largest demand for electricity occurs during periods in which solar irradiation is available in large quantities, thus matching the demand and supply, which makes the use of solar energy viable (see Al-Ibrahim et al. [6]).

To estimate the incident solar irradiation, a pyranometer can be used [7], installed in the same solar panel plane that is to be studied [8], and if such a sensor is not available, its value can be estimated with the measurements of pyranometers installed in nearby meteorological stations, from which such measurements are normally taken on the horizontal surfaces.

The solar irradiation received by a solar panel inclined at a certain angle with respect to the horizontal surface and oriented with a deviation towards the east or west with respect to the equator, with respect to the solar irradiation that reaches the horizontal surface, which is usually measured and recorded in meteorological stations, depends on various variables and parameters. Furthermore, this transformation is performed by treating the three components of solar radiation separately, namely the direct radiation received in the direction of the sun; the diffuse radiation, coming from all directions of the celestial vault when the plane of the panel is inclined horizontally; and the albedo, which is the solar radiation reflected from the surroundings of the earth's surface.

The effect of the inclination of the solar panel on its electrical production performance has been studied by different authors. Hafez et al. [9] detailed most of the design criteria for a solar collector, suggesting a low optimal angle of inclination for summer and spring and a high one for winter and autumn. In addition, photovoltaic systems show their best performance with an optimal annual angle of inclination for which the solar tracking system is not a necessary element. In addition, the solar irradiation incident on an inclined surface has been studied depending on the geographical location. In the Mediterranean Region, Darhmaoui and Lahjouji [10] found the optimal angle of inclination to achieve the maximum annual solar energy collection, starting from the latitude of the place and the values of the daily global solar irradiation on the horizontal surface, assuming a correct south orientation.

In India, Pandey and Katiyar [11] studied the variation of the hourly global solar irradiation for surfaces inclined at intervals of 15◦ , where the one received with an angle of inclination equal to the latitude of the place was the optimum throughout the year, by using, for its simplicity, the isotropic model of Liu and Jordan [12] to estimate the monthly mean hourly global solar irradiation on inclined surfaces. The same model developed by Liu and Jordan [12] was used by Klein [13] to calculate the monthly mean daily solar irradiation on inclined surfaces, regardless of the orientation of the collecting surface. In the same period, Temps and Coulson [14] estimated the values of solar irradiation on the inclined and oriented plane, using the solar flux model developed by Robinson [15].

From the beginning, the technological development of photovoltaic and thermal solar energy has included scientific and technical work to estimate the solar irradiation available on the horizontal surface from easily measurable parameters. More recently, Gómez and Casanovas [16] proposed a model that applied to Spanish conditions of solar irradiation on inclined surfaces arbitrarily oriented based on procedures of fuzzy logic. This model considers overlapping classes, thus allowing a better description of the sky situations in the transition zones between contiguous categories. Other studies have been published analyzing the performances of different models of global solar irradiation (e.g., Loutzenhiser et al. [17], Evseev and Kudish [18], El Mghouchi et al. [19], and Li et al. [20]).

The direct component of the solar irradiation incident on an inclined plane can be calculated trigonometrically, but it is also necessary to know the diffuse component of the available solar irradiation on the horizontal surface. In some places, the global and

diffuse solar irradiation on the horizontal surface is measured but, generally, only global data are measured or inferred from satellite data. In South Korea, Yoon et al. [21] evaluated 20 cases (five solar radiation models for each of the four albedo models) and proposed the photographic method with two factors (sky view and ground view) acquired from the pyranometer; the precision was improved, mainly by increasing the angle of inclination (i.e., considering the influence of obstacles against solar radiation); this improved the prediction accuracy for diffuse irradiation. However, the prediction accuracy of direct radiation was not improved.

The most widely available solar energy data are the measurements of global solar irradiation on a horizontal surface and thus these are the main models used to estimate diffuse solar irradiation on the horizontal surface, utilizing the horizontal global solar irradiation. After the first studies, numerous models emerged utilizing a method that provided a relationship for solar irradiation (diffuse vs. global) on a horizontal surface. These models are generally expressed in terms of polynomials from the first to fourth degrees, relating the diffuse fraction to the clearness index. Validity is discussed in these studies in order to apply the findings at different locations from where the data have been used for their development and for different climatic conditions or other geographical latitudes. The original correlations were developed for daily values, but in this study the hourly diffuse fraction vs. hourly clearness index was used, as it is the hourly solar irradiation incident on the surface of the solar panel that is a fundamental input required in the simulation of a more comprehensive design of a photovoltaic system.

Due to the lack of data series for solar irradiation measured on an inclined surface, several models have been used to estimate the solar irradiation incident on the surface of the solar panel from the measurement of global irradiation on a horizontal surface. This estimation requires previous knowledge of the components (direct and diffuse) of the global horizontal irradiation. Normally, they are not recorded at measurement stations, so the search for these components is generally done through estimation models. In the case of diffuse irradiation, the most widely used models or correlations are those that refer to the diffuse fraction k<sup>d</sup> and the clearness index k<sup>t</sup> on an hourly, daily, or monthly average basis. For the case of hourly fractions k<sup>d</sup> vs. k<sup>t</sup> , state-of-the-art models can be classified as first-order models (e.g., Boland et al. [22]), second-order models (e.g., Hawlader [23]), thirdorder models (e.g., Karatasou et al. [24]), and fourth-order models (e.g., Soares et al. [25], in this case utilizing an artificial neural network technique). Muneer and Munawwar [26], with a wide network of stations in Europe and Asia, show that the conventional model (k<sup>d</sup> vs. kt) for solar irradiation diffusion produces a high dispersion and therefore it is not satisfactory. For Australia, Ridley et al. [27] developed multiple predictions, using the hourly and daily clearness indexes as predictors, along with the solar height, the apparent solar time, and a measure of the persistence of the global solar irradiation level, suggesting its use as a universal model. For Spain, Posadillo and López [28,29] studied the dependence of k<sup>d</sup> and k<sup>t</sup> on solar height for their generalization to different places. Other experimental studies concerning diffuse solar irradiation on the horizontal surface can be found, such as those of Elminir [30], Ruíz-Arias et al. [31], and Torres et al. [32].

The method required for modeling the components of solar irradiation (beam, diffuse, and ground-reflected) to estimate the incident on the solar panel is extensive and its application can be complicated, so this study intended to make its use easier by providing a methodology using Simulink-MATLAB blocks. This methodology was applied with hourly horizontal global solar irradiation data from an agrometeorological station near to a greenhouse, resulting in a better approximation thanks to the use of anisotropic models of the diffuse solar irradiation.

### **2. Materials and Methods**

This section describes the databases used, the component models applied, and the methodology developed with Simulink-MATLAB.

### *2.1. Materials*

The hourly horizontal global solar irradiation data used in this study were recorded in 2011 in an agrometeorological station that belongs to the Agroclimatic Information System for Irrigation (SIAR), located in Mansilla Mayor (León, Castile and León region, Spain) with the following geographical coordinates: 42◦3004300 N and 5◦2604600 W, altitude 791 mamsl and local time GMT-21.725555. SIAR is a project of the Ministry of Environment and Rural and Maritime Areas of Spain, managed by the Agricultural Technological Institute in Castile and León (ITACyL), which, through the InfoRiego service for irrigating information, provides farmers with management recommendations for the use of water for irrigation [33]. The sensor used was a silicon photocell that measures the solar irradiation incident in the spectrum band between 350–1100 nm in the Skye SP1110 photovoltaic pyranometer (Campbell Scientific, Inc., North Logan, UT, USA).

The hourly horizontal diffuse solar irradiation data used in this study were taken in 2011 from the State Meteorological Agency database (AEMet is its name in Spanish) of the Ministry for Ecological Transition of Spain [34], registered in the meteorological station located in La Virgen del Camino (León, Castile and León region, Spain) with the geographical coordinates: 42◦3501800 N and 5◦3900400 W, altitude 912 mamsl.

The solar irradiance data measured on the 45◦ inclined plane and oriented towards the equator, which were used for comparison with the results obtained by the estimates of the methodology proposed here, were recorded at the facilities of the University of León (León, Castile and León region, Spain) with the geographical coordinates: 42◦3605000 N and 5 ◦3303900 W, altitude 848 mamsl. The thermoelectric sensor used generated a voltage of 10 mV/(kW·m<sup>2</sup> ), with a measurement range between 0–2000 W/m<sup>2</sup> and a spectral field of 305–2800 nm, and was deployed as part of a 1st class LP PYRA 02 AC pyranometer (Delta OHM Srl, Padova, Italy), manufactured under the ISO 9060 standard following the recommendations of the World Meteorological Organization (WMO).

### *2.2. The Components of the Solar Irradiation Incident on an Inclined Plane*

The evaluation of solar irradiation reaching an inclined plane is crucial because, usually, only solar irradiation data recorded on the horizontal surface is available. The methodology used for its estimation must determine the amount of received solar irradiation (direct and diffuse) and, for a good simulation of the photovoltaic system, it must be calculated with values for a minimum period of one hour. Methods mentioned in the literature to calculate each of the components of the solar irradiation that affect the solar panel are described below: directly from the sun; reflected from the ground; and diffused from the sky. These are generally deployed separately before their subsequent union into a global measurement.

### 2.2.1. The Beam Irradiation of the Sun Incident on an Inclined Plane

The direct solar irradiation incident on an inclined plane results from the relationship among the components of the solar beam irradiation (extraterrestrial, horizontal, and inclined), for which Iqbal [35] assumed that the direct irradiation on a surface (inclined vs. horizontal) is the same on the surface of the Earth as it is at the maximum height of the atmosphere, as shown in Equation (1) and also detailed in Equation (2), where r<sup>b</sup> is the ratio of solar irradiation on a plane (inclined/horizontal) at the maximum height of the Earth's atmosphere (I0β/I0) ≈ (cos θ0/cos θz).

$$\mathbf{I\_{b\beta\gamma}} = \mathbf{I\_b} \frac{\mathbf{I\_{0\beta\gamma}}}{\mathbf{I\_0}} \tag{1}$$

$$\mathbf{I\_{b\beta}}\_{\mathbf{b}\beta\gamma} = \mathbf{I\_{b}} \frac{\cos\theta}{\cos\theta\_{\mathbf{z}}} = \mathbf{I\_{b}} \ \mathbf{r\_{b}}\tag{2}$$

where:

Ib: direct hourly irradiation incident on a horizontal surface;

Ibβγ: direct hourly irradiation incident on an inclined and oriented plane; rb: ratio of irradiation on an inclined plane and the horizontal surface at the maximum of the earth's atmosphere I0<sup>β</sup> I0 ≈ cosθ<sup>0</sup> cosθz .

### 2.2.2. The Radiation Reflected by the Earth Incident on an Inclined Plane

The solar radiation that reaches the ground has direct and diffuse components. The word "earth" here refers to the surface of the earth that the solar panel inclined plane sees. Depending on the type of land cover, the albedo of solar irradiation (direct and diffuse) is not the same, so the total irradiation reflected by the ground can be described, following Iqbal [35], by Equation (3). As a result, two cases of reflection (isotropic and anisotropic) can happen and are presented below.

$$\mathbf{I\_r = (I\_b \ \rho\_b + I\_d \ \rho\_d)A\_g} \tag{3}$$

where:

Ir: diffuse hourly irradiation reflected by the earth incident on an inclined plane; Id: diffuse hourly irradiation incident on a horizontal surface; ρb: albedo of the soil due to direct irradiation; ρd: albedo of the soil due to diffuse irradiation; Ag: total area of the terrain seen by the inclined plane. − Albedo with Isotropic Reflection

The isotropic reflection albedo refers to the perfectly diffuse reflection that occurs when the global solar irradiation is mainly composed of diffuse irradiation (e.g., with a cloudy sky) and/or when the ground cover is a perfectly diffuse reflector (e.g., a floor of concrete). Then, by using the ratio of solar irradiation on an inclined plane to that on a horizontal surface, a configuration factor relating the ground to the inclined plane can be obtained, as developed by Iqbal [35] in Equation (4).

$$\mathbf{I}\_{\mathbf{r}} = \frac{1}{2}\mathbf{I}\,\boldsymbol{\varrho}(1-\cos\beta)\tag{4}$$

where:

ρ: albedo of the ground (irradiation reflected from the ground/irradiation incident on the ground).

− Albedo with Anisotropic Reflection

The anisotropic reflection albedo refers to the imperfect diffuse reflection that occurs when global solar irradiation is mainly composed of direct irradiation (e.g., with a clear sky) and/or when the ground is wet or there are shiny surfaces. Then, the isotropic model can be corrected with the following factors, as described by Iqbal [35], in Equation (5).

$$\mathbf{I}\_{\mathbf{r}} = \frac{1}{2}\mathbf{I}\,\boldsymbol{\varrho}(1-\cos\beta)\left[1+\mathrm{sen}^{2}\left(\frac{\theta\_{\mathbf{z}}}{2}\right)\right](|\cos\Delta|)\tag{5}$$

where:

∆: azimuth of the inclined surface to that of the Sun; this angle is reduced to ω for surfaces inclined towards the equator.

2.2.3. The Diffuse Irradiation of the Sky Incident on an Inclined Plane

The empirical formulations for the diffuse solar irradiation of the sky incident on an inclined surface are well-developed for each category of the sky (i.e., clear, cloudy, and partly cloudy).

− Circumsolar Model

The circumsolar model is applied with a clean and clear sky and assumes that all solar irradiation that reaches the horizontal surface comes from the direction of the Sun, and thus that it can be treated in the same way as direct irradiation, as in Equation (6) from Iqbal [35].

$$\mathbf{I\_s = I\_d \ r\_b} \tag{6}$$

where:

Is: diffuse irradiation of the hourly sky incident on an inclined plane.

− Isotropic Model

The isotropic model is applied with a cloudy sky and assumes that the diffuse solar irradiation of the sky is uniform throughout the celestial dome. The diffuse irradiation of the sky incident on an inclined plane can thus be obtained with the Liu and Jordan model [36], as in Equation (7).

$$\mathbf{I}\_{\mathbf{s}} = \frac{1}{2} \mathbf{I}\_{\mathbf{d}} (1 + \cos \beta) \tag{7}$$
