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Computing the Global Irradiation over the Plane of Photovoltaic Arrays: A Step-by-Step Methodology

By Oswaldo A. Arraez-Cancelliere, Nicolás Muñoz-Galeano and Jesús M. López-Lezama

Submitted: October 23rd 2019Reviewed: December 11th 2019Published: January 6th 2020

DOI: 10.5772/intechopen.90827

Downloaded: 24

Abstract

The quality of solar resource data is critical for the economic and technical assessment of solar photovoltaic (PV) installations. Understanding uncertainty and managing weather-related risk are essential for successful planning and operating of solar electricity assets. The input information available for PV designers is usually restricted to 12 monthly mean values of global horizontal irradiation (GHI) and average temperature, which characterize solar climate of locations. However, for calculating the energy production of a photovoltaic system, the global irradiation over the plane of the PV array is necessary. For this reason, this book chapter presents a methodology to appropriately determine the global irradiation over the plane of photovoltaic arrays. The methodology describes step by step the necessary equations for processing the data. Examples with numerical results are included to better show the data processing.

Keywords

  • global horizontal irradiation (GHI)
  • photovoltaic (PV)
  • energy production
  • solar resource
  • data processing

1. Introduction

Renewable energy resources have become a promissory alternative to overcome the problems related to high pollution and limited sources of conventional energy. So, the analysis of energy resources and their economic feasibility is a concern topic for researchers around the world [1, 2, 3, 4]. In this context, photovoltaic power plants have become one of the most important renewable sources of energy that have rapidly spread in the last decade. However, the assessment of the solar resource is not a topic usually approached by engineers and researchers due to the complexity in the process of computing the data, being extensive when the global horizontal irradiation is processed to obtain the global tilde irradiation. Therefore, this book chapter provides a step-by-step methodology for computing the global irradiation over the plane of photovoltaic arrays.

The quality of solar resource data is critical for economic and technical assessment of solar power installation. Understanding uncertainty and managing weather-related risk are essential for successful planning and operating of solar electricity assets. High-quality solar resource and meteorological data can be obtained by two approaches: high-accuracy instruments installed at a meteorological station and complex solar meteorological models, which are validated using high-quality ground instruments [5].

The input information available for PV designers is usually restricted to the 12-monthly mean values of global horizontal irradiation (GHI) and average temperature, which characterize solar climate of locations. However, the global irradiation over the plane of the PV array is necessary to calculate the energy production of a photovoltaic system.

The assessment of radiation arriving on an inclined surface, using global horizontal data as input, raises two main problems; first, to separate the GHI into their direct and diffuse components (decomposition) and, second, from them, to estimate the radiation components falling on an inclined surface (transposition) [6]. To solve these problems, it is important to describe in detail parameters such as declination angle, solar hour angle, solar zenith, solar altitude, solar azimuth, angle of incidence, solar constant, extraterrestrial irradiance over the horizontal surface, clearness index, and diffuse fraction index [6, 7, 8].

As regards decomposition models, in Ref. [9], authors reviewed and compared four decomposition models for monthly average of horizontal daily irradiation: a linear relationship proposed by Page [10], two polynomial equations defined by the authors of Refs. [11, 12], and a local correlation proposed by Macagnan et al. [13]. The authors of Ref. [9] stand out for the Page decomposition model of Ref. [10] in combination with the Perez transposition model [13] as a good performance combination used for passing from global horizontal irradiation to effective in-plane irradiance when it is started from monthly average of daily irradiation values. After applying the decomposition models, direct, diffuse, and albedo components of irradiation can be obtained.

For obtaining radiation components falling on an inclined surface, geometric considerations can be taken into account [9]. In Ref. [9], eight transposition models were reviewed, obtaining the best results using the Perez model [13]. Similar results were obtained in Ref. [14], where four transposition models were compared and validated with two-year data measured at site, and Liu and Jordan isotropic sky model is used in this book chapter due to its simplicity of implementation and good results reported in Ref. [15].

This book chapter describes in detail a methodology to determine the global irradiation over the plane of photovoltaic arrays. The chapter is organized as follows: Section 2 contains the proposed methodology, Section 3 includes experimental results that include data processing, and Section 4 presents the conclusions.

2. Proposed methodology

The proposed methodology includes the data processing and also the definitions to understand it. Therefore, some basic definitions are presented below:

2.1 Declination angle (δ)

It is the angle between the equatorial plane and a straight line drawn between the center of the Earth and the center of the sun. It may be considered as approximately constant over the course of any day. It can be calculated using Eq. (1), where dnis the day number counted from the beginning of the year [7].

δ=23.45°×sin360×dn+284365E1

2.2 Solar hour angle (ω)

It is the difference between noon and the selected moment of the day in terms of a 360° rotation in 24 h. ωis equal to 0 at midday of each day, and it is counted as negative in the morning and positive in the afternoon. The solar hour angle is given as:

ω=15×Tsolarh12E2
Tsolarh=Tlocalh+4LstLloc+EoT60E3
EoTmin=9.87sin2B7.53cosB1.5sinBE4
Bn=360365dn81E5
Lst=15×ΔTgmtE6

where Tsolarand Tlocalare the solar time and the local clock time, respectively. Lstand Llocare the standard meridian for the local time zone and the longitude of the location (east positive and west negative). ΔTgmtis the local time zone (e.g., Bogotá, −5). EoTstands for equation of time, which is the time difference between the apparent solar time for people and the real mean solar time, and takes into account the perturbation of the earth’s rotation [7].

2.3 Solar zenith (θzs) and solar altitude (γs)

Solar zenith is the angle between the vertical and the incident solar beam, and it can also be described as the angle of incidence of beam radiation on a horizontal surface [6]. The complement of the zenith angle is called the solar altitude, γs. These angles can be calculated using Eq. (7) and are function of the declination angle (δ), the latitude (ϕ) (north positive and south negative), and the true solar time (ω).

cosθzs=sinδsinϕ+cosδcosδcosω=sinγsE7
γs=90θzsE8

Eq. (7) can be used to find the sunrise angle (ωs)since at sunrise γs=0, hence.

ωs=cos1tanδtanϕE9

In accordance with the sign convention, ωsis always negative. The sunset angle is equal to ωs, and the length of the day is equal to:

Tdhour=2×absωs15°E10

2.4 Solar azimuth (ψs)

It is the angle between the meridians of the locations and the sun. It can also be described as the angular displacement from noon to the projection of beam radiation on the horizontal plane. The solar azimuth is given by:

cosψs=sinγs×sinϕsinδcosγscosϕE11

In the Northern Hemisphere, true solar is the reference of the system, and it is defined as positive toward the west, that is, in the evening, and negative toward the east, that is, in the morning [6]. In Figure 1, the solar zenith, solar altitude, and solar azimuth are described.

Figure 1.

Position of the sun relative to a fixed point on the earth defining the solar azimuth ( ψ s ), the solar zenith ( θ zs ), and the solar altitude ( γ s ).

2.5 Angle of incidence (θi)

Most practical applications require the position of the sun relative to an inclined plane to be determined. The angle of solar incidence between the sun’s rays and the normal to the surface is given by:

cosθi=sinδsincosβsignsinδcossinβcosα+cosδcoscosβcosω+signcosδsinsinβcosαcosω+cosδsinαsinωsinβE12

where βis the tilt of the inclined plane (the angle formed with the horizontal), and αis the surface azimuth angle conventionally measured clockwise from the south (See Figure 2) [8]. The signfunction is 1 when the latitude is greater than 0 and is −1 otherwise.

Figure 2.

Definition of angles used as coordinates for an element of sky radiation to an inclined plane of tilt β and oriented to α .

2.6 Solar constant (B0)

It is the amount of solar radiation received at the top of the atmosphere on a normal plane at the mean Earth-sun distance [6]. A good approximation of this value is:

B0=1367W/m2E13

Extraterrestrial irradiance over the horizontal surface (B00): Extraterrestrial radiation over a horizontal surface varies over the day, and it is given by:

B00=B0×1+0.033cos360365dncosθzsE14

If Eq. (14) is integrated over the day, the following expression is obtained:

B0d0=24πB01+0.033cos360365dncosϕcosδsinωs+π180ωssinϕsinδE15

Hence, average daily extraterrestrial irradiation in a month over a surface is obtained by:

B0dm0=1dn2dn1+1dn1dn2B0d0E16

The value calculated in Eq. (16) is used to estimate the clearness index (KTm) [2].

2.7 Clearness index (KTm)

It is the relation between the solar radiation at the Earth’s surface and the extraterrestrial radiation over the horizontal plane. The clearness index KTmfor each month is given by:

KTm=Gdm0B0dm0E17

where Gdm0is the average daily horizontal global irradiation of a month, which is usually an input value [6].

2.8 Diffuse fraction index (Kd)

It is the relation between the diffuse radiation over the horizontal plane and the global radiation over the horizontal plane. This index is widely used on decomposition models to separate the global radiation into its direct and diffuse components.

Kdm=Ddm0Gdm0E18

The modeling process for calculating the effective in-plane hourly irradiation when starting from monthly average of horizontal daily irradiation and using monthly average daily irradiance profiles is shown in Figure 3 [9].

Figure 3.

Calculation of daily irradiation on an inclined surface.

The daily irradiance profile can be defined in terms of irradiance divided by daily irradiation and on assuming that the profile of the extraterrestrial horizontal solar radiation translates directly into the profile of the diffuse component while an empirical correction is needed for global radiation [11]. The following equations describe the model to calculate the daily irradiance profile starting from daily average monthly values:

G0=rG×Gdm0E19
D0=rD×Ddm0E20
B0=G0D0E21

with

rD=B00B0d0=πT×cosωcosωsπ180×ωs×cosωssinωsE22
rG=rD×a+b×cosωE23
a=0.4090.5016×sinωs+60E24
b=0.6609+0.4767×sinωs+60E25

where ωand ωsare expressed in degrees, and T is the day length, usually expressed in hours (24 h). The unit of indexes rDand rGis T1, and they can be used to calculate irradiation during short periods centered on the considered instant ω. Subscripts “d” and “m” refer to the daily and monthly average of daily values, respectively.

The diffuse component of the average daily irradiation, Ddm0, is derived from a decomposition model consisting of an empirical relationship between the clearness index, KTm, and the diffuse fraction, Kdm. In Ref. [9], the authors review and compare four decomposition models for monthly average of horizontal daily irradiation: a linear relationship proposed by Page [10], two polynomial equations defined by the authors of Refs. [11, 12], and a local correlation proposed by Macagnan et al. [13]. The authors stand out the Page decomposition model in combination with the Perez transposition model [13] as a good performance combination used for passing from global horizontal irradiation to effective in-plane irradiance when it is started from monthly average of daily irradiation values.

The decomposition model proposed by Page consists of a linear equation that correlated the diffuse fraction index and the clearness index using data from locations situated between 40°N and 40°S, and it is given by:

Kdm=11.13KTmE26

Once the global horizontal irradiance is separated into direct and diffuse components and the daily irradiance profile is obtained, it is necessary to calculate the effective irradiance on the plane of the array. The irradiance over the plane with a tilt β, in degrees, and oriented to angle α, conventionally measured clockwise from the south, can be obtained by:

Gβα=Bβα+Dβα+ALβαE27

where G, B, D, and ALrepresent global, direct, diffuse, and albedo components, respectively. The irradiance components over the plane are given by:

Bβα=B0×rBE28
Dβα=D0×rDE29
ALβα=G0×rALE30

The beam transposition factor is calculated straightforward from simple geometric considerations [9]:

rB=max0cosθicosθzsE31

Assuming isotropic albedo radiation, the corresponding transposition factor is given by:

rAL=ρ1cosβ2E32

where ρis the ground reflection factor. The albedo radiation is scarcely relevant and rarely measured. A general reflection value of 0.2 is considered since this is extendedly used on practice [9].

The diffuse transposition factor depends on the assumption made for the sky radiance distribution. In Ref. [9], eight transposition models are reviewed, obtaining the best results using the Perez model. Similar results are obtained in Ref. [14], where four transposition models are compared and validated with two-year data measured at site, and the most accurate results were obtained by the Hay and Davies transposition model and the Perez transposition model. In this work, the Liu and Jordan isotropic sky model is used due to its simplicity of implementation and good results as reported in Ref. [15].

In the transposition model proposed by Liu and Jordan, the diffuse radiation is given by an isotropic component coming from the entire celestial hemisphere. The diffuse transposition factor is given by:

rD=1+cosβ2E33

In summary, the global irradiation over the tilted plane is calculated by:

Gβα=B0×rB+D0×rD+G0×rALE34

3. Experimental results: Data processing

Santa Cruz del Islote in Colombia was used as a location for the case study. Data that consist on the monthly average daily global horizontal irradiance (Gdm) and the monthly average ambient temperature (Tamb,m) were provided by Solargis through its pvPlanner platform, see Table 1.

LocationIslote de Santa Cruz, Colombia
Latitude9.79
Longitude−75.859167
Time zone (GTM)−5
Tilt10Plane inclination
Azimuth0Plane orientation (South = 0°; West 90°)
GRF0.2Ground reflection factor (0–1) Default = 0.2
MonthDaily sum of global irradiation [Wh/m2]Average diurnal (24-h) air temperature [°C]
January5922.627.8
February6271.427.6
March6267.727.6
April5906.727.7
May5367.727.9
June5396.728.5
July5587.128.7
August5538.728.5
September5363.328.2
October5025.827.9
November4970.027.8
December5200.028.0
Average5568.128.0

Table 1.

Meteorological input parameters [16].

A MATLAB routine was used to compute the monthly global irradiance over the horizontal and over the plane. Results were also compared with the data that can be processed from Solargis, and Table 2 shows the results obtained. Comparison shows that the results are good enough for the purpose of this work.

Global horizontal irradiation (kWh/m2) SolargisGlobal horizontal irradiation (kWh/m2) calculatedDev (%)Global tilted irradiation (kWh/m2) SolargisGlobal tilted irradiation (kWh/m2) calculatedDev (%)
January183.6182.0−0.88%201.9198.6−1.65%
February175.6174.2−0.81%186.9184.3−1.41%
March194.3193.0−0.68%198.5196.2−1.14%
April177.2176.1−0.65%175172.9−1.17%
May166.4165.2−0.70%160.1158.8−0.83%
June161.9160.8−0.71%153.6152.6−0.65%
July173.2172.0−0.69%165.3163.9−0.84%
August171.7170.6−0.65%167.8166.1−1.03%
September160.9159.8−0.70%162160.1−1.17%
October155.8154.4−0.91%162.4159.5−1.79%
November149.1147.7−0.96%160.5157.1−2.13%
December161.2159.7−0.93%177.8174.1−2.09%
Year2030.92015.3−0.77%2071.82044.1−1.34%

Table 2.

Global horizontal and tilted irradiations.

In Figure 4, it is shown the daily global profile on the horizontal and on the tilted plane for the first 4 days of the year calculated for the selected location. As expected, global tilde irradiation is higher than the global horizontal irradiation.

Figure 4.

Daily global irradiation calculated for “Islote de Santa Cruz.”

4. Conclusions

This book chapter presented a methodology that describes in detail the data processing to obtain the global irradiation over the plane of photovoltaic arrays. The methodology includes definitions and equations necessary to perform the processing of the data. The following parameters were described and along with their equations: declination angle, solar hour angle, solar zenith, solar altitude, solar azimuth, angle of incidence, solar constant, extraterrestrial irradiance over the horizontal surface, clearness index, and diffuse fraction index.

The obtention of the global irradiance over a tilde plane requires decomposition models, which provide direct, diffuse, and albedo components of irradiation. This book chapter provides global horizontal irradiation to effective in-plane irradiance when it is started from monthly average of daily irradiation values.

This book chapter is extensive in the use of geometric considerations. This is due to the fact that input data are global horizontal irradiation and average temperature, while output data are the global irradiation over a tilde PV plane. Therefore, equations and schemes were provided for facilitating the explanation.

Acknowledgments

The authors gratefully acknowledge the financial support provided by the Colombia Scientific Program within the framework of the call “Ecosistema Científico” (Contract No. FP44842-218-2018). The authors also want to thank the “Programa de Sostenibilidad de la UdeA.”

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Oswaldo A. Arraez-Cancelliere, Nicolás Muñoz-Galeano and Jesús M. López-Lezama (January 6th 2020). Computing the Global Irradiation over the Plane of Photovoltaic Arrays: A Step-by-Step Methodology [Online First], IntechOpen, DOI: 10.5772/intechopen.90827. Available from:

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