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

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].

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:

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).

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

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:

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:

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.

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:

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 sign∅function is 1 when the latitude is greater than 0 and is −1 otherwise.

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:

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

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

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

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:

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.

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].

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:

with

where ωand ωsare expressed in degrees, and T is the day length, usually expressed in hours (24 h). The unit of indexes rDand rGis T−1, 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:

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:

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

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

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

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:

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