Generally, diffuse radiation models for inclined surfaces can be classified into two groups: isotropic and anisotropic models. They differ in the division of the sky into regions with normal and elevated diffuse radiation intensities. Isotropic models assume there is uniformity in the distribution of diffuse radiation intensity over the sky. Anisotropic models include appropriate modules for representing areas of elevated diffuse radiation.
4.2. Anisotropic Models
● Bugler’s Model
Bugler (1977) added modules for the diffuse radiation emanating from the sun’s disc and alternative components of the sky counting on the sun’s angular height over the horizon [52
]. Bugler’s equation is:
● Temps and Coulson’s Model
Temps and Coulson (1977) modified the isotropic model of Liu and Jordan and introduced two terms that represent diffuse radiation by assuming a clear sky condition [53
is the vicinity of the sun’s disc and
is the sky radiation from the region near the horizon.
● Hay’s Model
Another anisotropic model is the one proposed by Hay and Davies, which is commonly referred to as the Hay model. Two primary sources are assumed to be the origins of sky diffuse radiation, namely the disc of the sun disc and the rest of the sky with isotropic diffuse radiation [54
]. The two components are described by the anisotropy index
Based on the Hay model, the equation for the intensity of diffuse radiation on an inclined plane has the form
● Reindl’s Model
] proposed a model for the diffuse radiation emitted from the areas near the horizon line described by the Hay model. Reindle found that with increasing overcast sky, there is a decrease in the diffuse radiation intensity originating from the given region. Therefore, the modulating function
was included in the module:
The Reindl equation is:
● Klucher’s Model
Anisotropic model of Klucher’s is based on the models by Temps and Coulson and and Liu and Jordan [56
]. Klucher found that Liu and Jordan’s isotropic model provides fruitful results for overcast skies but overlooks radiation for some sky conditions, such as partly overcast and clear skies. Such conditions are distinguishable by a rising intensity in the proximity of the circumsolar sky and horizon region. To overcome such a limitation, the Temps and Coulson model was refined by introducing a function
that determines the degree of cloud cover.
● Klucher’s Model Is Described by the Following Equation:
● The HDKR Model (the Klucher and Reindl, and Hay and Davies’ Model)
The HDKR model was developed with the aim of analyzing the beam reflection and all diffuse radiation terms, such as isotropic, circumsolar, and horizon brightening, by adding them to the solar radiation equation. Although originating from the Hay and Davies model, HDKR introduces the term “horizon brightening” similar to Klucher. As a result, this model was named HDKR (Hay, Davies, Klucher, Reindl) by Duffie and Beckman [10
● Skartveit and Olseth’s Model
Solar radiation measurements carried out by Skartveit and Olseth in Bergen (Norway) partly indicated the fact that sky diffuse radiation originates from the part of the sky surrounding the zenith under overcast sky conditions. This effect disappears with the disappearance of cloud cover. To overcome this effect, Skartveit and Olseth refined Hay’s model [57
is the correcting factor.
The effect of barriers blocking the horizon and obscuring part of the diffuse radiation incident on a sloped plane, is represented by the term
. This term is usually neglected as data are typically derived from radiometric stations. Situated in open terrains, radiometric stations face insignificant natural or artificial obstacles [2
● Steven and Unsworth’s Model
The anisotropic model of Steven and Unsworth is defined as diffuse radiation on a plane inclined at a b
angle. The source is considered to be the heliocentric radiation of the gleaming horizon and the sun’s disk.
● Wilmott’s Model
Another anisotropic model is Willmott’s model, which adapted the model proposed by Hay and defined a new anisotropy index [58
is in radians, and
is the solar constant.
● Perez’ Model
The basis of the Perez model is an in-depth applied mathematic analysis of the sky’s diffuse components. This model divides diffuse radiation into three components: isotropic background, circumsolar, and horizon zones [59
]. The governing equation is:
In this equation
represent solid angles occupied by the circumsolar region, weighted by its average incidence radiation on an angled and horizontal surface, respectively;
, are the dimensionless horizon brightness and the the circumsolar coefficients respectively. The two factors are defined as follows:
These increasing factors set the radiation magnitude values within the two anisotropic regions relevant to those in the major a part of dome. Within the model, the degree of anisotropy could be a performance of solely these two regions. Thus, the model will perform both as an isotropic configuration (
), and collectively incorporating circumsolar and/or horizon brightening equivalent time.
is the air mass (dimensionless), Id
is the hourly diffuse radiation on a horizontal surface and Io
is the extraterrestrial radiation at normal incidence (W/m2
). The ε
is a function of hourly diffuse radiation
which is given in Table 4
and direct beam radiation
]. The required coefficients
are obtained from Perez et al., as seen in Table 5
The details and mathematical relationships of diffuse models on inclined surfaces including isotropic and anisotropic models are provided in Table 6
4.3. Recognizing Accurate Models for Estimation of Diffuse Radiation on Angled Surfaces
A number of researchers have attempted to find the most accurate models among anisotropic and isotropic models for estimating diffuse solar radiation on oblique surfaces. To find the most suitable model for a specific location, the amount of estimated diffuse radiation on an inclined surface at various angles is compared with the value of the diffuse radiation on an angled surface measured by a pyranometer with a shadow band at the same angle.
New models for estimation of diffuse solar radiation on an inclined surface were suggested by the Laboratory of Solar Radiometry of Botucatu-UNESP (latitude 22°9′ S, longitude 48°45′ W) in Brazil. The results of the proposed models were compared with some isotropic and anisotropic model results. The results indicated that the isotropic and anisotropic models were more accurate than the proposed models [61
A study was done in Singapore, at 1°37′ N, 103°75′ E. The values of radiation sensors facing 60° NE, tilted at 10°, 20°, 30°, and 40° and vertically tilted radiation sensors facing south, north, west, and east in Singapore were measured. A pyranometer with a shadow band measured the diffuse horizontal radiation while another pyranometer measured the global radiation on a horizontal surface. The direct radiation value on a horizontal surface was calculated using the difference between global radiation on a horizontal surface and diffuse horizontal radiation. The diffuse radiation was calculated for an inclined surface using an isotropic model (Liu and Jordan’s model) and two anisotropic models (Klucher and Perez’ models) and was compared with the measured values. The model of Perez et al. was proposed as the best model for Singapore [62
In a study conducted by the Royal Meteorological Institute of Belgium in UCCLE (latitude 50.79° N, longitude 4.35° E) the diffuse solar radiation on an inclined surface was measured by three isotropic models (Liu and Jordan, Korokanis, Badescu) and 11 anisotropic models (Bugler, Hay, Skartveit and Olseth, Willmott, Reindl, Temps and Coulson, Klucher, Perez, Iqbal, Muneer and Gueymard). The data collected over a period of eight months (April 2011 to November 2011) were utilized to define the relative capacity of 14 different models to estimate the global solar radiation on an inclined surface facing south as a distinctive element of sky conditions. It was identified that Bugler’s model performed the most effectively under all sky conditions, such as partly clear and clear, while Willmott’s model was observed to provide the most accurate results under overcast and partly cloudy conditions. Finally, Perez’ model most closely fit the estimation under overcast conditions [4
In a study conducted by Gulin et al. at the University of Zagreb (45°80′ N, 15°87′ E), global radiation on a horizontal surface was measured using a pyranometer, while direct radiation was measured by a sun tracker pyrheliometer with an extra sensor for closed-loop tracking of the sun. Three isotropic models (Liu and Jordan, Korokanis and Badescu) and six anisotropic models (Skartveit and Olseth, Willmott, Temps and Coulson, Bugler, Hay and Klucher) were used to estimate solar diffuse radiation on an inclined surface at 5°, 30°, 55°, and 80°. Gulin et al. developed three different neural network models for predicting the solar radiation incident on an oblique surface. These models’ performance was then evaluated in light of the three isotropic and six anisotropic rival models for tilted surfaces [63
In Botucatu region of the state of São Paulo, Brazil (22°53′ S, 48°26′ W), twenty models were used to estimate the hourly diffuse radiation incident on angled surfaces facing north at and 32.85° 22.85°, and 12.85°, under various cloudy conditions. The most promising results were obtained using the anisotropic models of Ma and Iqbal, Hay, Reindl et al., and Willmott, while the circumsolar models and the isotropic models of Badescu and Koronakis proved to be the best [64
In a study carried out in Poland, Polish researchers selected numeric models of both isotropic and anisotropic nature for estimation of the diffuse solar radiation on photovoltaic module planes. They clarified which model was most appropriate for central Poland. Isotropic models (Liu-Jordan, Badescu, Koronakis, Tian) and anisotropic models (Hay, Steven and Unsworth) were used to estimate the distribution of radiation power on photovoltaic planes slanted at 30°, 45°, and 60° facing south. The outcomes demonstrated that the anisotropic models facilitated obtaining higher radiation throughout the year in comparison to isotropic models for the Polish latitude [65
Włodarczyk and Nowak carried out a study and statistically analyzed 14 major models for the solar radiation intensity on an inclined plane. Models with various degrees of complexity were analyzed, from the simplest classical isotropic model to the most complex anisotropic model (the Perez model). They compared the model results with data collected at 35° and 50° angles from the actinometrical station laboratory in Wrocław, Poland. The analyzed models of diffuse solar radiation [66
A research done in Egypt suggested that the Perez model is suitable for that area. This model was selected among the Tamps and Coulson, Bugler and Perez models [67
]. Another comparative study addressed the performance of one isotropic and nine anisotropic models, where actual data were employed to estimate the solar radiation diffusion of inclined surfaces. The data were obtained from the province of Valladolid, Spain, on a south-facing surface inclined at 42°. It became clear that the best model was Hay’s, followed by Muneer and Willmott’s models [68
]. Based on a daily analysis, Perez’ model and the isotropic model showed an average performance and the Temps-Coulson model had the least satisfying results.
Mehleri et al. [69
] developed a new neural network model (RBF) to estimate diffuse radiation on inclined surfaces for the Athens region. The RBF model was suggested as the most accurate model for the location.
A summary of studies on identifying the most accurate diffuse models on inclined surfaces for different locations is given in Table 7
4.6. Combination of Diffuse Estimation Models for Horizontal and Inclined Surfaces
As previously explained, to measure diffuse radiation received by a horizontal surface, a diffuse shadow band on the horizontal surface or a pyrheliometer is compulsory. Furthermore, to measure diffuse radiation on an inclined surface, a diffuse shadow band is necessary. The values of direct or diffuse radiation are measured infrequently at meteorological stations, while the global radiation values on horizontal surfaces are usually available. Considering the above-mentioned limitations, some researchers have used combinations of diffuse models for horizontal surfaces, whether decomposition or parametric models and diffuse models for an inclined surface, or isotropic or anisotropic models. Finally, the estimated global radiation values on inclined surfaces are compared with the measured global radiation values on inclined surfaces.
A group of researchers in Bhopal, India, used a decomposition model in order to determine the diffuse radiation on a horizontal surface. They also used three isotropic and three anisotropic models to calculate diffuse radiation on a sloped surface. They found that Badescu’s model is the best isotropic model for India [74
More recently, a study was done in the south of Sindh region, Pakistan, to find the best combination of models for diffuse radiation received by horizontal and angled surfaces. To achieve this objective, nine new models were developed based on function of clearness index models [75
] Erbs’ model [21
], Liu and Jordan’s model [30
], a cubic polynomial model and a quadratic polynomial model [76
], a sunshine fraction model (Barbara’s model) [77
], and Haydar et al.’s model [78
] in order to estimate diffuse radiation on a horizontal surface and an isotropic model to estimate diffuse radiation on an inclined surface [79
]. The calculated and measured values showed that the best is a combination of the sunshine fraction model with Liu and Jordan’s model [80
In South Korea, a study was conducted with the objective of predicting solar irradiation on inclined surfaces in reference to horizontal measurements. This study included measurements of the accuracy of the two established models. Using the first and second types of models, respectively, diffuse horizontal radiation from global components and global radiation on angled planes from diffuse and global components on horizontal surfaces were quantified. The solar radiation was gradually reduced as the inclined angle was increased from horizontal to vertical surfaces, apart from south-facing orientations. The maximum value was observed at inclination angles between 20° and 40° [81
In a case study carried out in Karaj, Iran, combinations of a decomposition model (Miguel et al.) and 12 models including four isotropic models (Badescu, Koronakis, Tian, and Liu and Jordan), and some anisotropic models (Reindl, Skartveit and Olseth, Hay, Steven and Unsworth, Temps and Coulson, Klucher and Perez) were investigated. The results indicated that the Skartveit and Olseth, Hay, Reindl, and Perez models made the most accurate predictions for south-facing surfaces [82
A similar research was done to quantify the level of accuracy of different models. The first type consisted of seven diffuse radiation models for a horizontal surface. The second type with fifteen models differentiated between measurements of global radiation on inclined planes and global components on horizontal surfaces. The study combined two model classes and calculated the level of adequacy of each association for the data, which was collected hourly at Ajaccio, a French Mediterranean site. The result of each combination was compared with the value of data collected on 45° and 60° tilted surfaces. The best combination was Maxwell + Klucher for the 45° inclination angle. For 60° inclination, the most efficient model was Skartveit and Olseth’s model combined with Klucher’s model [83
A study was done in Padova, Italy, to find the best model for estimating diffuse radiation on horizontal and tilted surfaces. To reach this goal, four decomposition models were used to estimate diffuse radiation on horizontal surfaces, and an isotropic model and three anisotropic models were used to estimate diffuse radiation on tilted surfaces. Erbs, Perez, Erbs and HDKR were suggested as the best combination [84
]. A summary of studies on identifying the most accurate combination models to estimate diffuse radiation on horizontal and inclined surfaces for different locations is given in Table 8