Modelling of Solar Radiation for Photovoltaic Applications

Abstract

This chapter explores the different ways in which solar radiation (SR) can be quantified for use in photovoltaic applications. Some solar radiation models that incorporate different combinations of parameters are presented. The parameters mostly used include the clearness index (Kt), the sunshine fraction (SF), cloud cover (CC) and air mass (m). Some of the models are linear while others are nonlinear. These models will be developed for the estimation of the direct (Hb) and diffuse (Hd) components of global solar radiation (H) on both the horizontal and tilted surfaces. Models to determine the optimal tilt and azimuthal angles for solar photovoltaic (PV) collectors in terms of geographical parameters are equally presented. The applicable, statistical evaluation models that ascertain the validity of the SR mathematical models are also highlighted.

1. Introduction

Solar radiation is essentially a flux of photons originating from the sun and radiating in all directions of space. These photons exhibit electromagnetic wave properties and travel at the speed of light over an average distance of about 149.4 million km to reach the earth’s surface while suffering diverse attenuations from the components of space and the earth’s atmosphere.

Many devices are being employed to measure SR but the scope of such measurements over space and time is limited. As a consequence, it is mandatory to develop alternative heuristic models to qualify and quantify solar radiation.

Data on global solar radiation (GSR) is readily available in most meteorological stations around the world but data on the diffuse and beam components of SR is rare and needs to be estimated by alternative means. Measurements of SR are mostly done on horizontal surfaces while real-time solar PV receivers require tilting from the horizontal position for optimal harvesting of the SR [1]. Information on both the direct and diffuse components of SR is necessary to accurately characterise the irradiance intercepting a solar collector or receiver.

GSR is short wavelength radiation that can characteristically be either broadband or spectral. From this premise, SR is modelled using either broadband or spectral models. Besides, satellite-based models have also been developed. The broadband models are suitable for ground-based measurements. A plethora of sub-models, with varying levels of complexity, now exist and will be presented in the subsections that follow.

The general trend over the past decades pioneering with the work of John K. Page [2], is the development of models which have been severally tested and improved upon. The common approach in the models is to predict either the diffuse and/or the direct SR components from measured GSR data. Alternatively, some models use meteorological parameters like temperature, sunshine hours and relative humidity, together with the GSR data to predict the direct or diffuse components.

Except in the subsection(s) where we treat the aspect of tilt angle, every occurrence of radiation henceforth will be considered to mean radiation measured (or predicted) regarding a horizontal surface.

To ascertain the accuracy of the models, some statistical tools for the evaluation of the models have been presented. These include the mean bias error (MBE), the root mean square error (RMSE) and t-statistics [3, 4, 5]. A. S. Angstrom [3] disclosed that these statistical tools collectively combine to establish the consistency of the models.

This chapter will be organised as follows: After this introduction, we will present in the next section the statistical tools applicable for testing of the model’s performance. This is followed in Section 3 by an exploration of the different approaches used in modelling solar radiation. Given that our emphasis is on photovoltaic technology, we present in the last section the modelling of tilt and azimuth angles in connection with solar photovoltaic energy applications. This is followed by the concluding remarks on the chapter.

2. Statistical evaluation methods for photovoltaic solar radiation models

The prediction efficiency of the models being presented in this chapter needs testing to ensure their validity and reliability. This is achieved using some statistical tools. These include: the mean bias error (MBE), the mean relative error (MRE), the root mean square error (RMSE) and the t-statistic (t-stat) error [5].

2.1 Mean bias error

The MBE is expressed as [6, 7]:

$$\mathit{MBE}=\frac{1}{k}\sum _{i=1}^k\left(y_i-x_i\right)$$E1

where x_i is the i^th observed value, y_i the i^th predicted value and k the total number of observations.

The mean bias error (MBE) is a pointer of the long-term performance of a correlation. This is achieved by calculating the real deviation between the predicted and measured values term wise. Ideally, an MBE value of zero is the best indicator. A positive MBE indicates an over-estimation while a negative MBE indicates under-estimation. Under practical conditions, vanishingly small MBE values are desirable for a good model’s performance.

2.2 Root mean square error (RMSE)

The RMSE is expressed as [5, 6, 7]:

$$\mathit{\text{RMSE}}=\sqrt{\frac{1}{k}{\sum }_{i=1}^k{\left(y_i-x_i\right)}^2}$$E2

The root mean square error (RMSE) is determinant for the short-term performance of a regression model. The RMSE estimates the differences between observed and predicted results of some quantity being modelled, which in this case is the solar radiation. RMSE is a good measure of precision and its value is always positive, representing zero in the ideal case [6].

2.3 Mean relative error

The mean relative error (MRE) tests the linearity between the measured and the estimated values. It is expressed in the form [8];

$$\mathit{MRE}=\frac{1}{k}\sum _{i=1}^k\left|\frac{y_i-x_i}{x_i}\right|$$E3

The MRE is always positive, approaching zero in the ideal case.

Each statistical assessment tool considered alone might not be a sufficient pointer of a model’s validity. It is likely to have a large RMSE value and at the same time a small MBE (a large scatter about the line of estimation). It is also possible to have a relatively small RMSE and a relatively large MBE (consistent over-estimation or underestimation).

Although these statistical indicators generally provide a reasonable tool for model performance, they do not objectively indicate whether the model’s estimates are statistically significant. An additional statistical indicator, the t-statistic can be used.

2.4 The t-statistical method

Stone [9] demonstrated that the MBE and the RMSE separately do not represent a reliable assessment of the model’s performance and can lead to the false selection of the best model from a set of candidates. To determine whether or not the equation estimates are statistically significant, Stone [9] proposed the t-stat expressed as:

$$t-\mathit{\text{stat}}=\sqrt{\frac{\left(n-1\right){\mathit{MBE}}^2}{{\mathit{\text{RMSE}}}^2-{\mathit{MBE}}^2}}$$E4

T-stat values are always positive and vanishingly small values indicate a better model’s performance. The parameter, n, represents the numbers of observations and corresponds to the twelve months (n = 12) of the year if average monthly measurements are used. This statistical indicator compares models and at the same time indicates whether the model’s estimates are statistically significant at a particular confidence level [9, 10]. Consequently, the t-statistic is used in combination with the RMSE and MBE to give a more reliable prediction [11]. After the estimation of a coefficient, the t-statistic for that coefficient expresses the ratio of the coefficient to its standard error.

3. Approaches in solar radiation modelling

3.1 Introduction

Before reaching the earth’s surface, SR suffers some of the attenuations from air particles, aerosols, water vapour and clouds. This causes the GSR to be split into three components: the reflected, the direct (or beam) and the diffuse SR components.

Several forms of SR data exist, which could be used for a variety of purposes in the design and development of solar PV systems. Daily data is often available and hourly radiation data can be estimated from available daily data.

The monthly average daily extraterrestrial radiation on a horizontal surface is expressed as [12, 13]:

$$H_0=\frac{\left(24\right)\left(3600\right)}{\pi }{G}_0\left(\mathit{\text{cosφ cosδ}}\mathit{sin}{\omega }_s+\frac{\pi }{180}{\omega }_s\mathit{\text{sinφ sinδ}}\right)$$E5

Here, $G_0$, is the extraterrestrial radiation (SR incident on the outside of the earth’s atmosphere) given by:

$$G_0=I_{\mathit{sc}}\left(1+0.034\mathit{cos}\left(\frac{360n_{\mathit{day}}}{365.25}\right)\right)$$E6

Where $I_{\mathit{sc}}$ is the solar constant and has a value of 1.367kWm⁻² [14], φ is the latitude of the site, δ is the solar declination angle, ${\omega }_s$ is the sunshine hour angle for the month and $n_{\mathit{day}}$ is the number of days of the year starting from January 1st. Figure 1 presents the variation of H₀ for Bamenda (latitude 5.96°N and longitude 10.15°E).

Figure 1.

Correlation between the estimated and observed values of the monthly mean daily diffuse solar radiation using a twenty-year (1985–2005) monthly mean daily clearance index for the area of Yokadouma, Cameroon [15].

The solar declination (δ), the mean sunshine hour angle for the month (${\omega }_s$) and the maximum possible sunshine duration ($S_0$) may be calculated from the Cooper [16] formula [13, 16, 17]:

$$\delta =23.45\mathit{sin}\left(\frac{360\left(n_{\mathit{day}}+284\right)}{365}\right)$$E7

$${\omega }_s={\mathit{cos}}^{-1}\left(-\mathit{\text{tanδ tanφ}}\right)$$E8

$$S_0=\frac{2}{15}{\omega }_s$$E9

A calculation of these parameters for Bamenda (latitude 5.96°N and longitude 10.15°E) is presented in Table 1 below.

Parameter	Value
Average sunshine hours per day	6.7 hours
Solar constant, Isc	1367Wm^−2
Latitude of site, φ	5.96^0N
Longitude of site	10.15°E
Linear regression Constants a and b	0.19 and 0.52
Declination, δ from (Eq. (7))	23.18^0
Sunshine hour angle, ωs from (Eq. (8))	92.56^0
Day length (mean sunshine hour), from (Eq. (9))	12.34 hours
$H_0$ from (Eq. (5))	35.6068 MJ/m^2/day
The estimated value of global solar radiation, H, for Bamenda for the month of June 2005	16.7756 MJ/m^2/day
The measured value of global solar radiation, H, for Bamenda for the month of June 2005	16.452 MJ/m^2/day

Table 1.

Solar radiation parameters for Bamenda, Cameroon [10].

3.2 Modelling of the direct and diffuse components of solar radiation from GSR measurements, a.k.a. decomposition models

The input parameters for these models are diffuse ratio (K), the clearness index (K_t), the diffuse transmittance index (K_d), and the beam transmittance index (K_b). These parameters are expressed as follows: [18]

$$K=\frac{H_d}{H}$$E10

$$K_t=\frac{H}{H_0}$$E11

$$K_d=\frac{H_d}{H_0}$$E12

$$K_b=\frac{H_b}{H_0}$$E13

Where: $\mathrm{H}$ is the monthly average daily GSR, ${\mathrm{H}}_{\mathrm{d}}$ is the Monthly average daily diffuse component of GSR, ${\mathrm{H}}_{\mathrm{b}}$ is the Monthly average daily direct component of GSR, ${\mathrm{H}}_0$ is the Monthly average daily extraterrestrial radiation; $H_d$ is the monthly average daily diffuse radiation received on a horizontal surface, H is the monthly average daily total (direct plus diffuse) radiation received on a horizontal surface, and $H_o$ is the extraterrestrial daily insolation received on a horizontal surface.

3.2.1 Models based on the diffuse ratio- clearness index regressions

The diffuse component of SR can be predicted using GSR data as initially done by Liu and Jordan [19]. The time scales used in this class of models range from monthly average to daily and hourly averages. For monthly average SR, John K. Page [2], estimated the monthly mean values of daily total short wave radiation on vertical and inclined surfaces from sunshine records for latitudes 40° N - 40° S. It consisted of a linear model relating K and ${\mathrm{K}}_{\mathrm{t}}$. Other similar models have been developed relating K and ${\mathrm{K}}_{\mathrm{t}}$ ranging from quadratic to higher-order polynomial models [20].

For the choice of time scale, some models relate the daily clearness index and daily diffuse SR ratio, for different geographical locations [18, 21, 22]. The approach here consists of developing a piece-wise fit between K and ${\mathrm{K}}_{\mathrm{t}}$. This is done for overcast, partly-cloudy and clear skies.

For overcast skies, the regression equation is linear and expressed as.

$$\mathrm{K}={\mathrm{a}}_0+{\mathrm{a}}_1{\mathrm{K}}_{\mathrm{t}}\text{for}{\mathrm{K}}_{\mathrm{t}}<{\mathrm{K}}_{\mathrm{ta}}$$E14

Where ${\mathrm{K}}_{\mathrm{ta}}$ is some critical value beyond which partly-cloudy conditions dominate.

Other models assume a constant value of K in the event of overcast skies, i.e.:

$$\mathrm{K}={\mathrm{a}}_0\text{for}{\mathrm{K}}_{\mathrm{t}}<{\mathrm{K}}_{\mathrm{ta}}$$E15

In the situation of partly cloudy skies, a polynomial fit in ${\mathrm{K}}_{\mathrm{t}}$ of order three or four is used, expressed as:

$$\mathrm{K}={\mathrm{b}}_0+{\mathrm{b}}_1{\mathrm{K}}_{\mathrm{t}}+{\mathrm{b}}_2{{\mathrm{K}}_{\mathrm{t}}}^2+{\mathrm{b}}_3{{\mathrm{K}}_{\mathrm{t}}}^3\text{for}{\mathrm{K}}_{\mathrm{ta}}<{\mathrm{K}}_{\mathrm{t}}<{\mathrm{K}}_{\mathrm{tb}}$$E16

Lastly, for a clear sky situation, K takes a constant value, expressed as:

$$\mathrm{K}={\mathrm{c}}_0\text{for}{\mathrm{K}}_{\mathrm{t}}>{\mathrm{K}}_{\mathrm{tb}}$$E17

Instead of the piecewise regression as outlined above, a single polynomial regression equation can be chosen that can adequately fit the available data. A nonlinear empirical expression has also been used [23], and given as:

$$\mathrm{K}=\mathrm{a}+\left(1-\mathrm{a}\right)exp\left[-\mathrm{b}{{\mathrm{K}}_{\mathrm{t}}}^c/\left(1-{\mathrm{K}}_{\mathrm{t}}\right)\right]$$E18

where, for the location of Macerata the constants take values: a = 0.154, b = 1.062 and c = 0.861.

It should be mentioned that seasonal models in which seasonal variations for daily regressions are treated exist [18, 20, 22].

The third variant consists of the models based on hourly SR measurements. Here the procedure of Liu and Jordan [19], is used with the exception that the correlation between K and ${\mathrm{K}}_{\mathrm{t}}$ is done on an hourly basis [20].

For the performance of these models, Figure 1 presents the correlation between the estimated and observed values of the monthly mean daily diffuse solar radiation using a twenty-year (1985–2005) monthly mean daily clearance index for the area of Yokadouma, Cameroon, (Latitude 3.15°, longitude 15.050° and at an altitude of 488 m) [15]. For this caption, the correlation equations are expressed in the linear and quadratic forms as: (${\mathrm{H}}_d=\mathrm{H}\left(1.265-1.463{\mathrm{K}}_{\mathrm{t}}\right)$ and ${\mathrm{H}}_d=\mathrm{H}\left(1.282-1.53{\mathrm{K}}_{\mathrm{t}}+0.063K_t^2\right)$) [15].

Figure 1 demonstrates a coefficient of determination between the estimated and the observed values close to one (0.92–0.99), which indicates an excellent agreement between the estimated and the observed diffuse fraction. Figure 2 further shows the correlation between the estimated and observed values of the diffuse fraction for the same location; Yokadouma. Even though the results of the different models follow the same trend, we notice that the predictions of Lealea T. et al. [15] are closest to the observed data. This suggests that these models are location-dependent, performing well in some locations and not in others.

Figure 2.

Comparison of the observed and estimated values of monthly mean diffuse solar radiation predicted by some existing models for Yokadouma, (Cameroon). Modified from [15].

An evaluation of these models based on the statistical indicators for Yokadouma, (Cameroon) is presented in Table 2.

Reference	MBE (Wh/m^2/day)	RMSE (Wh/m^2/day)	t-statistics
Lealea T. et al. [15] (Linear)	37.8	2615.4	0.228
Lealea T. et al. [15] (Quadratic)	−29.73	2613	0.18
J.K. Page [2]	−13,781	14,261	59.37
Liu and Jordan [24]	−21,751	22,612	55.63
Iqbal [12]	−8092	9824	22.91
Erbs et al. [20]	−19,039	19,737	57.84

Table 2.

Performance of the diffuse ratio- clearness index models using the statistical indicators for Yokadouma, Cameroon [15].

The RMSE values here reveal that the model of Lealea T. et al. [15] is best for short-term performance. Meanwhile, the MBE and the RMSE cannot adequately account for the validity of a model, the t-statistics evaluation here indicates that the results of Lealea T. et al. [15] are the most statistically significant for the study location.

3.2.2 Correlation between diffuse transmittance index and clearness index regression

The hourly diffuse SR was predicted from measured hourly GSR on a horizontal surface by Iqbal [25]. It consisted of a correlation between the hourly diffuse transmittance index, ${\mathrm{k}}_{\mathrm{d}}$, (ratio of diffuse to extraterrestrial radiation), and hourly clearness index, $\mathrm{k}$, (ratio of global to extraterrestrial radiation). The results indicated that the models depend on particular geographical sites.

3.2.3 Correlation between direct transmittance index and clearness index regressions

This approach was spearheaded by Maxwell [26] in an attempt to improve the findings of several investigations which have shown that the use of a single regression function does not sufficiently portray the connection between direct beam transmittance (${\mathrm{K}}_{\mathrm{b}}$) and the actual global horizontal transmittance (${\mathrm{K}}_{\mathrm{t}}$). The Direct Insolation Simulation Code (DISC) that uses an exponential relationship between ${\mathrm{K}}_{\mathrm{b}}$ and air mass with parameter ${\mathrm{K}}_{\mathrm{t}}$, was developed [26]. This procedure acceptably relates ${\mathrm{K}}_{\mathrm{b}}$ with ${\mathrm{K}}_{\mathrm{t}}$ for a variety of stations around the globe and seasons. The validation of the DISC Model exhibited considerable improvements in the correctness of hourly values, substantial decreases in monthly RMS errors, and the corresponding monthly MBE. Further modification of the DISC model integrated the effects of cloud-cover, water vapour, and albedo. Perez et al. [27] adopted and improved on Maxwell’s model. The method comprised primarily in using a zenith-angle independent clearness index and by employing a time-varying GSR. Consequently, the diffuse irradiance can be gotten from the difference between the GSR and the beam component once the beam component is known. These components are related by the equation:

$$\mathrm{H}={\mathrm{H}}_d+{\mathrm{H}}_{b}sin{\theta }_h$$E19

Where ${\theta }_h$ is the solar height or solar altitude.

Hence Perez et al. [27] improved on the two main shortcomings related to the reliance of the clearness index on solar height and also its slow response to sudden changes in hourly sky conditions.

3.3 Prediction of diffuse solar radiation from the beam or direct component of solar radiation

3.3.1 ASHRAE model

The ASHRAE model considers only clear cloudless days [28]. It proceeds in two steps: the first consists of calculating the intensity of the direct normal solar radiation component and next it computes the hourly direct and diffuse solar radiation on both the horizontal and slanted surfaces. The model equations are:

$${\mathrm{H}}_{\mathrm{bn}}=\text{A}\text{exp}\left(-\mathrm{B}/cos{\mathrm{\theta }}_{\mathrm{z}}\right)$$E20

$${\mathrm{H}}_{\mathrm{d}}={\mathrm{CH}}_{\mathrm{bn}}$$E21

where ${\mathrm{H}}_{\mathrm{bn}}$ is the normal beam component of SR, ${\mathrm{H}}_{\mathrm{d}}$ is the diffuse component of SR,${\mathrm{\theta }}_{\mathrm{z}}$ is the zenith angle and A, B, C are monthly mean values of empirically chosen constants.

Extensions of the ASHRAE model where the model coefficients were re-determined using cloudless data at different locations exist [29].

3.3.2 Regression models using the direct transmittance index and the diffuse transmittance index

These models are formulated using an empirical monthly regression equation between the ratio of the daily diffuse SR to the daily extraterrestrial radiation (${\mathrm{K}}_{\mathrm{d}}$) and the ratio of the daily beam SR to the daily extraterrestrial radiation (${\mathrm{K}}_{\mathrm{b}}$). An implementation in the localities of Beer Sheva and Sde Boker (Israel), is expressed as [30]:

$${\mathrm{K}}_{\mathrm{d}}=\mathrm{a}\left[exp\left(\mathrm{b}{\mathrm{K}}_{\mathrm{b}}+\mathrm{c}{\mathrm{K}}_{\mathrm{b}}^2\right)\right]$$E22

where, the constants a, band c, are monthly values of empirically determined coefficients. For the month of January at Beer Sheva: a = 0.2155, b = 3.1713 and c = −8.1261.

3.4 Prediction of solar radiation from meteorological input parameters

These models are formulated using the clearness index, $k_t$ with the input parameter being the GSR. They present a shortcoming in that the clearness index alone cannot account for changes in the diffuse component of the SR. Extensions of the model exist that can address the associated drawback that will be explored in the following subsections.

3.4.1 Prediction of solar radiation from the sunshine fraction

The first attempt that expresses SR in terms of the sunshine fraction is the linear equation [3]:

$$\frac{\mathrm{H}}{{\mathrm{H}}_0}=\mathrm{a}+\mathrm{b}\left(\frac{\mathrm{S}}{{\mathrm{S}}_0}\right)$$E23

where a and b are the two constants, H is the monthly average daily SR, ${\mathrm{H}}_{\mathrm{o}}$ is the monthly average daily extra-terrestrial radiation, S is the monthly average daily measured sunshine duration.

As an extension of this equation and to improve the accuracy, the nonlinear polynomial models, were derived. This form is given as follows [7]:

$$\frac{\mathrm{H}}{{\mathrm{H}}_0}=\mathrm{a}+\mathrm{b}\left(\frac{\mathrm{S}}{{\mathrm{S}}_0}\right)+\mathrm{c}{\left(\frac{\mathrm{S}}{{\mathrm{S}}_0}\right)}^2+\mathrm{d}{\left(\frac{\mathrm{S}}{{\mathrm{S}}_0}\right)}^3+\dots$$E24

The values of a, b, c and d, vary depending on location and month of observation. Their values may be affected by atmospheric air pollution. As the daily total amount of SR and sunshine duration vary widely, daily totals averaged over a month are used to derive the values of a, b, c and d. This can be done by the least square regression analysis.

These models have been very popular all the time because of the abundance of data on sunshine duration in most locations on earth. This eases the prediction of GSR even where measurements are absent. The mostly used equation is that proposed by John K. Page [2], expressed as:

$$\mathrm{H}={\mathrm{H}}_0\left(\mathrm{a}+\mathrm{b}\frac{\mathrm{n}}{\mathrm{N}}\right)$$E25

where H and ${\mathrm{H}}_0$ are the monthly-average daily terrestrial and extraterrestrial radiation, n is the average daily hours of bright sunshine and N is the day length. Variants of these models are linear, quadratic, third-degree polynomial, exponential and logarithmic (Figures 3 and 4).

Figure 3.

Linear relationships between the monthly average values (H/H₀ versus S/S₀) for the city of Bamenda, Cameroon [10].

Figure 4.

Comparison of the estimated and observed monthly average daily horizontal GSR data for Bamenda using the linear, quadratic and cubic models.

To test these models, we present results for both the linear, the quadratic, and the third-degree polynomial models for the city of Bamenda in Cameroon whose parameters have been presented in Table 1.

3.4.2 Cloud cover radiation models (CRM)

For cloud cover radiation models, the choice parameters used are the monthly mean values of the fraction of the sky covered by clouds, $\overline{N}e$, and duration of bright sunshine hours, N. The sunshine duration is calculated from the cloud cover data and the cloud derived sunshine data, monthly mean values of global and diffuse SR are calculated. The model equations are expressed as [31]:

$$G_{t,0}=Wsin{\theta }_h-X$$E26

$$\frac{G_t}{G_{t,0}}=1-Y{\left(\frac{N_e}{8}\right)}^Z$$E27

Where, ${\mathrm{G}}_{\mathrm{t},0}$ is the global irradiance for a cloudless sky in W$/m^2$, ${\mathrm{G}}_{\mathrm{t}}$ is the hourly global irradiance for any given cloud amount ($N_e$, in eighths) in $W/m^2$, ${\theta }_h$ is the solar height in (°), and W, X, Y and Z are empirical regression constants.

A linear model equation that correlates monthly average diffuse transmittance index, ${\overline{K}}_d$, to monthly average daily cloud cover (${\overline{N}}_e$, in eighths), is given by [32]:

$${\overline{K}}_d=a_7+b_7\frac{{\overline{N}}_e}{8}$$E28

$${\overline{K}}_d=a_8+b_8\left(1-\frac{{\overline{N}}_e}{8}\right)$$E29

Where $a_7$, $b_7$, $a_8$ and $b_8$ are regression constants.

As an extension, several empirical models for the prediction of GSR from the daily mean of cloud cover, temperature extremes (minimum and maximum) and extraterrestrial SR have been proposed [33].

3.4.3 Models based on atmospheric transmittance (ATM)

Constituents that affect the transmittance of the atmosphere include scatterers, which consist of air molecules responsible for Rayleigh scattering, aerosols causing Mie scattering and absorbers like water vapour, atmospheric gases, dust and clouds. The atmospheric transmittance models attempt to establish some parametric relationships between these parameters. These models can either be classified as broadband or spectral based/ radiative transfer models [34].

3.4.3.1 Meteorological radiation model (MRM)

The most popular broadband ATM is the Meteorological Radiation Model (MRM). The input data for this model consist of the dry- and wet- bulb temperature and a sunshine fraction (used to generate hourly SR components for all-sky conditions like overcast and clear skies) [31].

The model equations for MRM in the case of non-overcast skies are given by

$$\mathit{DBR}=0.285k_b^{-1.00648}$$E30

$$G_b=\left(\mathit{SF}\right)G_0{\tau }_r{\tau }_{\alpha }{\tau }_g{\tau }_o{\tau }_w$$E31

Where: $\mathrm{DBR}$ is the hourly diffuse to beam ratio, ${\mathrm{k}}_{\mathrm{b}}$ is the direct transmittance index, ${\mathrm{G}}_{\mathrm{b}}$ is the beam/direct irradiance, $\mathrm{SF}$ is the hourly sunshine fraction, ${\mathrm{\tau }}_{\mathrm{r}}$, ${\mathrm{\tau }}_{\mathrm{\alpha }}$, ${\mathrm{\tau }}_{\mathrm{g}}$, ${\mathrm{\tau }}_{\mathrm{o}}$, ${\mathrm{\tau }}_{\mathrm{w}}$ are the transmittances respectively due to Rayleigh and Mie scattering, mixed gases, ozone and water vapour. Empirical equations are used to determine the transmittance indices and the coefficients are obtained through data fitting.

The proposed model exists for overcast skies where the diffuse irradiance is assumed to be equal to GSR [35]. Gueymard [36] proposed another similar model referred to as the Reference Evaluation of Solar Transmittance (REST). Though similar to the other models, the particularity of REST is that it introduces an additional transmittance term ${\tau }_n$ to account for the total absorption of ${\mathrm{NO}}_2$.

3.4.3.2 Spectral models

The measurement of the solar spectrum is quite challenging necessitating models that can accurately provide the solar radiation incident at different parts of the earth’s surface.

Spectral models are particularly suitable for such applications that are prone to small changes in wavelength. These models are spurred on the one hand by the challenges encountered in measurements of the electromagnetic spectrum. On the other hand, there is a need for models capable of accurately reproducing the incident radiation at the earth’s surface. This aim is achieved by solving the radiative transfer equations as a function of the wavelength intervals as well as unit atmospheric layer intervals [37, 38]. The first of these models are the SPECTRAL and the Simple Model of the Atmospheric Radiative Transfer of Sunshine (SMARTS) developed by Bird [37]. The second is a modified version of SPECTRAL to SPECTRAL2 developed eventually by Bird and by Riordan [38]. These models apply simple mathematical expressions on tabulated look-up tables to generate the direct-normal and diffuse horizontal irradiance.

The SPECTRAL2 determines the beam component of solar radiation perpendicular to the earth surface for some wavelength λ through the equation:

$${\mathrm{H}}_{\mathrm{b\lambda }}={\mathrm{H}}_{\mathrm{o\lambda }}\mathrm{D}{\mathrm{T}}_{\mathrm{r\lambda }}{\mathrm{T}}_{\mathrm{a\lambda }}{\mathrm{T}}_{\mathrm{w\lambda }}{\mathrm{T}}_{\mathrm{o\lambda }}{\mathrm{T}}_{\mathrm{u\lambda }}$$E32

Where for some given wavelength λ and for some mean earth-sun distance: ${\mathrm{H}}_{\mathrm{o\lambda }}$ is the extraterrestrial irradiance; $\mathrm{D}$ is a correction factor; and ${\mathrm{T}}_{\mathrm{r\lambda }}$, ${\mathrm{T}}_{\mathrm{a\lambda }}$, ${\mathrm{T}}_{\mathrm{w\lambda }}$, ${\mathrm{T}}_{\mathrm{o\lambda }}$, and ${\mathrm{T}}_{\mathrm{u\lambda }}$ are functions expressing the transmittance of the atmosphere for molecular Rayleigh scattering, attenuation by aerosols, absorption by water vapour, absorption by ozone, and absorption by uniformly mixed gases, respectively. The beam component of solar irradiation on a horizontal surface is given by the product of (Eq. (32)) and the cosine of the solar zenith angle, ${\theta }_z$.

The parameter, D in (Eq. (32)) is expressed as

$$D=1.00011+0.034221\mathit{cos}{\omega }_d+0.00128\mathit{sin}{\omega }_d+0.000719\mathit{cos}2{\omega }_d+0.000077\mathit{sin}2{\omega }_d$$E33

Where ${\omega }_d$ the day angle in radians given by

$${\omega }_d=2\mathrm{n}\left(n_{\mathit{day}}-\mathrm{l}\right)/365$$E34

Three components make up diffuse solar radiation on a horizontal surface. The first $H_{\mathit{r\lambda }}$ results from the Rayleigh scattering, the second $H_{\mathit{a\lambda }}$ is caused by the aerosol scattering and the third $H_{\mathit{g\lambda }}$ originates from multiple reflections of solar radiation between the earth surface and the atmosphere. The resultant solar radiation caused by scattering is expressed as:

$${\mathrm{H}}_{\mathrm{s\lambda }}={\mathrm{H}}_{\mathrm{r\lambda }}+{\mathrm{H}}_{\mathrm{a\lambda }}+{\mathrm{H}}_{\mathrm{g\lambda }}$$E35

Obtaining the spectral solar radiation on inclined surfaces is a straight forward process achieved by combining the spectral beam and diffuse radiation components calculations as presented above. The spectral global solar radiation on a slanted surface is then given by

$$\begin{array}{l}{\mathrm{H}}_{\mathrm{T\lambda }}\left(\mathrm{t}\right)={\mathrm{H}}_{\mathrm{b\lambda }}cos\mathrm{\theta }+{\mathrm{H}}_{\mathrm{s\lambda }}\left[\left({\mathrm{H}}_{\mathrm{b\lambda }}cos\mathrm{\theta }/{\mathrm{H}}_{\mathrm{o\lambda }}\mathrm{D}cos{\mathrm{\theta }}_{\mathrm{z}}\right)+0.5\left(1+cos\mathrm{\beta }\right)\left(1-{\mathrm{H}}_{\mathrm{b\lambda }}/\left({\mathrm{H}}_{\mathrm{o\lambda }}\mathrm{D}\right)\right)\right]\\ +0.5{\mathrm{H}}_{\mathrm{T\lambda }}{\mathrm{r}}_{\mathrm{g\lambda }}\left(1+cos\mathrm{\beta }\right)\end{array}$$E36

where $\theta$ is the angle of incidence of the beam component on the tilted surface, $\beta$ is the angle of tilt of the slanted surface relative to the horizontal surface and ${\theta }_z$ is the solar zenith angle. The following expression holds for the spectral global solar radiation on a horizontal surface:

$${\mathrm{H}}_{\mathrm{T\lambda }}={\mathrm{H}}_{\mathrm{b\lambda }}cos{\mathrm{\theta }}_{\mathrm{z}}+{\mathrm{H}}_{\mathrm{s\lambda }}$$E37

The first term in (Eq. (36)), is the direct component on the inclined surface. The second term has two components: the first is circumsolar and the second is a diffuse component. The last term represents the radiation reflected from the earth surface which is distributed isotropically. A component that is missing from this model is the horizon-brightening radiation.

Gueymard [39] improved the SMARTS model to the SMARTS2. The spectral transmittance is expressed as a function of the processes responsible for radiation extinction in the atmosphere such as water vapour, Rayleigh scattering, uniformly mixed gases, absorption by ozone, aerosol extinction and Nitrogen dioxide. These functions are then used to calculate the beam component of the radiation in the shortwave range. Data obtained from spectroscopic measurements have been used to calculate coefficients for the extinction processes due to absorption by gases that depend on both temperature and pressure. The coefficient of absorption resulting from the dependence in temperature is captured in the modelling of the extinction caused by nitrogen dioxide, both in the visible and UV regions of the electromagnetic spectrum. The two-tier Angstrom methodology is used to compute the extinction resulting from absorption by aerosols. Data of visibility measured at the airport and further refined based on a prototype of the Shettle and Fenn [40] function is used to evaluate the turbidity effect of aerosols. A further improvement is introduced by expressing the wavelength exponent and some coefficients that characterise the individual aerosol components as a multivariable parametric function of the relative humidity and the wavelength. SMARTS2 is also equipped with an optional function that corrects the circumsolar radiation which together with two functions that smoothen and filter the spectral solar radiation equip it with the possibility to mimic real-time spectro-radiometers. As a result, confronting the results of modelling with observed data becomes easy. An initial evaluation of the validity of SMARTS2 revealed considerable agreement for the direct component of solar radiation obtained both from thorough and standard solar radiation schemes and from spectro-radiometric measurements. The possibility of incorporating into SMARTS2 the ability to estimate solar spectra under the canopy of clouds is further suggested in a later work by Gueymard et al. [41].

3.5 Satellite-based models

Geographical and climate parameters vary widely across the globe and consequently impute differences in the amount of SR intercepting the earth’s surface. To capture all these differences would require an infinite number of ground measuring stations. This difficulty is alleviated by the use of meteorological satellites which provide SR data over a wide geographical coverage with high spatial resolution. Models based on such data have been developed to take advantage of such ubiquitous data. The models range from: subjective, empirical (statistical and physical based), objective and theoretical (broadband and spectral) [42].

3.5.1 Subjective methods

Methods that involve some subjective interpretation of the satellite data fall under this category. For the method to provide some quantitative measure for solar radiation, it has to be associated with other methods. This method has been applied manually to estimate cloud cover from hard-copy images using a gridded overlay [43].

3.5.2 Empirical methods

Here functional relations are developed using simultaneous and co-located satellite and SR data. The methods permit some level of transferability in which the derived equations can be applied to other geographical locations, but as pointed out in [42], such a process may be uncertain due to the empirical parameters involved in the equations. In the subsequent development of these methods, two approaches are followed: the first is a statistical approach and the second is a physical approach.

The statistical approach relies upon choosing the independent variable merely based on their facility to capture the trend in the SR based on the geographical location of interest. In what follows, the physical-based approach will be prioritized and developed.

3.5.2.1 Physical based methods

These methods originate from an attempt to achieve a radiation balance between the earth and its surrounding atmosphere. A formulation presented in [42], expresses the balance as follows:

$$E_0\downarrow -E_0\uparrow -E_a-E_g\downarrow \left(1-\rho \right)=0$$E38

Where ${\mathrm{E}}_0\downarrow$ is the extraterrestrial solar irradiance, ${\mathrm{E}}_0\uparrow$ is the SR reflected back to space, ${\mathrm{E}}_{\mathrm{a}}$ is the SR absorbed by the earth’s atmosphere, ${\mathrm{E}}_{\mathrm{g}}\downarrow$ is the solar irradiance at the earth’s surface, and $\mathrm{\rho }$ is the surface albedo.

Dividing by $E_0\downarrow$ and rearranging terms results in:

$${\rho }_p=1-q_a-q_t\left(1-\rho \right)$$(39)E39

where ${\rho }_p$ is the planetary albedo (the fraction of the incident SR reflected to space); $q_a$ is the portion of the incident SR absorbed by the atmospheric constituents; $q_t$ is the transmitted portion of the incident SR through the atmosphere. Using an argument whereby $q_t$ and the spatially averaged values of ${\rho }_p$ are highly correlated, enables the use here of a statistical equation of the form

$${\rho }_p=a+b{q}_a$$E40

where a and b are some empirical coefficients equal to 0.63 and −0.64, respectively [44].

It can be deduced from (Eq. (39)) that,

$$a=1-q_a$$E41

and

$$b=-\left(1-\rho \right)$$E42

This results in average values of 0.37 and 0.36 for $q_a$and $\rho$, respectively. However, it was shown in [44] that the satellite data could have undervalued ${\rho }_p$ resulting in an overestimation of the atmospheric absorption values inferred.

Eq. (39) can be alternatively expressed in the form

$$q_t=\left(1-{\rho }_p-q_a\right)/\left(1-\rho \right)$$(43)E43

If all the quantities in this equation are obtained from appropriate measurements, then $q_t$ can be calculated.

An alternative approach was followed in [45] to develop a model in which there is a very high correlation between the planetary albedo and the SR absorbed at the earth’s surface, thereby implying that the column integrated atmospheric absorption is highly conservative. On this basis, the model is expressed as:

$${\mathrm{q}}_{\mathrm{t}}\left(1-\mathrm{\rho }\right)=\mathrm{a}+\mathrm{b}{\rho }_p$$E44

Where it can be deduced from equation (Eq. (39)), that:

$$\mathrm{a}=1-{\mathrm{q}}_{\mathrm{a}}$$E45

and

$$\mathrm{b}=-1$$E46

The conservative aspect of the regression parameters was revealed by using data from different geographical locations. This was further substantiated theoretically leading to the conclusion that even clouds cannot severely change the atmospheric absorption.

Other empirical models have been developed based on the radiation balance between the earth and its surrounding atmosphere [46, 47]. One approach followed in [48] and [49] consists of rearranging Eq. (39) in the form:

$${\mathrm{q}}_{\mathrm{t}}=\left(1-{\rho }_p-q_a\right)/\left(1-\mathrm{\rho }\right)$$E47

This equation can be rewritten in the form:

$${\mathrm{q}}_{\mathrm{t}}=\mathrm{a}+\mathrm{b}{\mathrm{\rho }}_{\mathrm{p}}$$E48

Where,

$$\mathrm{a}=\left(1-{\mathrm{q}}_{\mathrm{a}}\right)/\left(1-\mathrm{\rho }\right)$$E49

and

$$\mathrm{b}=-1/\left(1-\mathrm{\rho }\right)$$E50

A comprehensive analysis in [50] and [51] led to expressing the parameters, a and b as multivariable functions given by:

$$\mathrm{a}=\mathrm{f}\left({\mathrm{q}}_{\mathrm{a}},{\mathrm{\rho }}_{\mathrm{p}},{\mathrm{\rho }}_{\mathrm{p}}^{\prime },{\mathrm{\rho }}_{\mathrm{c}}^{\prime },{\mathrm{\rho }}^{\prime }\right)$$E51

$$\mathrm{b}=\mathrm{f}\left({\mathrm{q}}_{\mathrm{a}},{\mathrm{q}}_{\mathrm{a}}^{\prime }{\mathrm{\rho }}_{\mathrm{p}},{\mathrm{\rho }}_{\mathrm{p}}^{\prime },{\mathrm{\rho }}_{\mathrm{c}},{\mathrm{\rho }}_{\mathrm{c}}^{\prime },\mathrm{\rho },{\mathrm{\rho }}^{\prime }\right)$$E52

where: ${\rho }_c$ is cloud reflectivity and the primes indicate that the variable is calculated when the satellite sensor is in some spectral interval (typically 0.55–0.75 μm). They revealed that b is less conservative than a, as a consequence of the relatively strong reliance on aerosol absorptivity which is one component of $q_a$ and $q_a^{\prime }$. The other two most important parameters (cloud reflectivity and water vapour absorptivity) neutralize each other. Hence, the combined effect of these latter variables may be very inconsequential.

Cano et al. [52] developed a model that deviates from the previous ones and can serve as a transition between the empirical and theoretical models. In their approach, the cloudless sky albedo is computed iteratively by a procedure that minimizes the variance in the difference between the satellite inferred value of $E_g\uparrow$ and a calculated value obtained from:

$${\mathrm{E}}_{\mathrm{g}}\uparrow ={\mathrm{\rho }}_{\mathrm{p}0}{\mathrm{I}}_{\mathrm{T}}{\left(cos{\mathrm{\theta }}_{\mathrm{z}}\right)}^{1.15}$$E53

In this approach, ${\mathrm{\rho }}_{\mathrm{p}0}$, is the planetary albedo for a cloudless target. ${\mathrm{I}}_{\mathrm{T}}$ is the solar constant corrected for actual sun-earth distance. A cloud cover index (n_s) is computed for high surface albedo (e.g., with snow cover) and for infrared radiances for wavelengths in the interval between 10.5 and 12.5 μm as

$${\mathrm{n}}_{\mathrm{s}}=\frac{\mathrm{I}-{\mathrm{I}}_0}{{\mathrm{I}}_{\mathrm{c}}-{\mathrm{I}}_0}$$E54

Where $\mathrm{I}$ is the observed infrared radiance, $I_0$ is the observed infrared radiance for cloudless sky conditions and.

${\mathrm{I}}_{\mathrm{c}}$ is the observed infrared radiance for overcast sky conditions.

The atmospheric transmission was assumed to be a linear combination of the respective values for cloudless and overcast skies resulting in

$${\mathrm{q}}_{\mathrm{t}}=\left(1-{\mathrm{n}}_{\mathrm{s}}\right){\mathrm{q}}_{\mathrm{t}0}+{\mathrm{n}}_{\mathrm{s}}{\mathrm{q}}_{\mathrm{tc}}={\mathrm{q}}_{\mathrm{t}0}+{\mathrm{n}}_{\mathrm{s}}\left({\mathrm{q}}_{\mathrm{tc}}-{\mathrm{q}}_{\mathrm{t}0}\right)$$E55

In a similar regression model

$${\mathrm{q}}_{\mathrm{t}}=\mathrm{a}+\mathrm{b}{\mathrm{n}}_{\mathrm{s}}$$E56

with

$$\mathrm{a}={\mathrm{q}}_{\mathrm{t}0}$$E57

and

$$\mathrm{b}={\mathrm{q}}_{\mathrm{tc}}-{\mathrm{q}}_{\mathrm{t}0}$$E58

According to Cano et al. [52], if data were stratified hourly, absolute values of the correlation coefficients are typically greater than 0.80, thereby supporting their use of the preceding model. Although the parameters, a and b could be calculated analytically, values were determined empirically. This is in line with the fact that the regression parameters also account for many other effects, including those resulting from the characteristic response of the satellite sensors.

3.5.3 Theoretical methods

These models endeavour to simulate explicitly solar radiant energy exchanges occurring between the earth and the atmosphere. Unlike the statistical counterpart, they do no incorporate an empirical calibration of the model parameters resulting in location free models. The models however need to be provided with additional environmental data which are time and location dependent. As a short cut to this limitation, climatological and standard atmosphere data are sometimes used, often without seriously impacting negatively on model performance.

Based on the degree of simplification and realism, two general classes of models can be distinguished: broadband models formulated based on the earth’s radiation balance and spectral models which rely on results generated by the solution of the radiative transfer equation.

3.5.3.1 Broadband models

One of the pioneers in this approach is Gautier et al. [53], who developed a model that has been widely used and makes it a reference for broadband models. In their model, the solar flux that exits the earth’s atmosphere and is measured by the satellite is given by:

$$E_0\uparrow =E_0\downarrow -E_a-E_g\downarrow \left(1-\rho \right)$$E59

By starting with a cloudless sky and minimizing the effects of multiple reflections down to first-order and assuming that scattering occurs before absorption, Gautier et al. rewrote Eq. (59) in terms of broadband absorption and scattering coefficients to give

$${\mathrm{E}}_0\uparrow ={\mathrm{E}}_0\downarrow \left({\mathrm{\rho }}^{\prime }+\left(1-{\mathrm{\rho }}^{\prime }\right)\left(1-{\mathrm{a}}_1\right)\left(1-{\mathrm{a}}_2\right)\left(1-{\mathrm{\rho }}^{\ast }\mathrm{\rho }\right)\right)$$E60

The only unknown is the surface albedo, $\rho$. $E_0\uparrow$ can be inferred from the satellite measurements, ${\rho }^{\prime }$ and ${\rho }^{\ast }$ are obtainable from Coulson [42] and $a_1$ and $a_2$ can be calculated given an estimate of atmospheric water vapour content (commonly climatological data or relationships involving surface humidity are used). Making $\rho$ the subject in the last equation gives:

$$\mathrm{\rho }=\frac{{\mathrm{E}}_0\uparrow -{\mathrm{E}}_0\downarrow }{{\mathrm{E}}_0\downarrow \left(1-{\mathrm{\rho }}^{\prime }\right)\left(1-{\mathrm{a}}_1\right)\left(1-{\mathrm{a}}_2\right)\left(1-{\mathrm{\rho }}^{\ast }\right)}$$E61

A rearranged and expanded version of (Eq. (60)) was then used to express the solar irradiance at the earth’s surface (assuming cloudless skies) in terms of known variables

$${\mathrm{E}}_{\mathrm{g}}{\downarrow }_0={\mathrm{E}}_0\downarrow \left(1-{\mathrm{\rho }}^{\prime }\right)\left(1-{\mathrm{a}}_1\right)\left(1-{\mathrm{a}}_2\right)\left(1+{\mathrm{\rho }}^{\ast }\mathrm{\rho }\right)$$E62

This model was revised by Diak and Gautier [54], where they included the effects of ozone absorption while Gautier and Frouin [55] also incorporated the effects of both aerosol and all orders of multiple reflections. Additional revisions investigated the consequences of ignoring spectral dependencies in both atmospheric attenuation and satellite radiometers.

Diak and Gautier [54] recognized that: (1) the Rayleigh scattering optical depth is wavelength dependent and therefore values of ${\rho }^{\prime }$and ${\rho }^{\ast }$ must be evaluated for both the entire solar spectrum and the wavelengths covered by the visible sensors. (2) Ozone absorption must be considered, especially given its significance in the visible part of the spectrum. These considerations are captured in the calculations of both the albedos and surface irradiances. Gautier and Frouin [55] provided the following equation for the surface irradiance during cloudless skies:

$${\mathrm{E}}_{\mathrm{g}}{\downarrow }_0={\mathrm{E}}_0\downarrow exp\left(-\frac{{\mathrm{c}}_2}{cos{\mathrm{\theta }}_{\mathrm{z}}}\right)\left(1-{\mathrm{a}}_{01}\right)\left(1-{\mathrm{a}}_{03}\right)\left(1-{\mathrm{a}}_1\right)/\left(1-{\mathrm{c}}_3\mathrm{\rho }\right)$$E63

Gautier et al. [53] revised and extended their model to include the effects of clouds by assuming a plane-parallel atmosphere composed of three layers. Thirty per cent of the water vapour and all the Rayleigh scattering were assumed confined to the top cloud layer. Similar procedures to those considered in the clear sky model provided estimates of the cloud top albedo (${\rho }_0$) with which the cloud absorption ($a_0$) was parameterized using a linear function ranging from zero for no cloud to 20% absorption for maximum target brightness.

Multiple reflections were not considered in the derivation of the following equation for the irradiance at the surface under overcast conditions [53]

$$E_g{\downarrow }_c=E_0\downarrow \left(1-{\rho }^{\prime }\right)\left(1-a_{1a}\right)\left(1-{\rho }_c\right)\left(1-a_c\right)\left(1-a_{1b}\right)$$E64

We notice a striking similarity with (Eq. (62)) (for clear skies) except that the first order of multiple reflections was included in that formulation. Note also that to be consistent with the definition of ${\rho }_c$ and $a_c$, the term $\left(1-{\rho }_c\right)\left(1-a_c\right)$ should be replaced by $\left(1-{\rho }_c-a_c\right)$.

A revised equation for $E_g{\downarrow }_c$ [54] captured the effects of ozone – zone absorption and revised the formulation of cloud attenuation to render it consistent with the definition of the absorption and reflection coefficients

$$E_g{\downarrow }_c=E_0\downarrow \left(1-a_{01}\right)\left(1-a_{03}\right)\left(1-{\rho }^{\prime }\right)\left(1-a_{1a}\right)\left(1-{\rho }_c-a_c\right)\left(1-a_{1b}\right)$$E65

Further attempts were made to approach reality in the parameterization of absorption by cloud, primarily to incorporate the occurrence of both finite and sub-field-of-view clouds. A decision to limit ${\rho }_c$ to values greater than 7% was arrived at based on comparisons between measured and calculated surface irradiances. For analogous reasons, the maximum value of $a_c$ was set to 7%. The basis for these decisions to an extent weakens the claim of zero empiricism in physically-based models.

Gautier and Frouin [55] upgraded their analysis to capture the effects of both aerosols and multiple reflections resulting in

$$E_0{\downarrow }_c=c_1{E}_0\downarrow \mathit{exp}\left(-\frac{c_2}{cos{\theta }_z}\right)\left(1-{\rho }_c-a_c\right)\left(1-a_{01}\right)\left(1-a_{03}\right)\left(1-a_1\right)/\left(1-c_3\rho \right)\left(1-c_4\rho \right)$$E66

Gautier et al. [53] have described the procedures for deciding whether to implement the clear or overcast sky routines when calculating $E_g\downarrow$, for a given location (pixel) and the technique for combining these estimates for partly cloudy conditions.

From the foundational investigations of Gautier et al. [53], other works have followed and revised their models to address some of the shortcomings associated with their approach such as the investigations in [56].

3.5.3.2 Spectral models

The conceptual basis for modelling SR by a spectral model of radiative transfer is best captured in the technique developed by Halpern [57]. The solution of the radiative transfer equation for an atmosphere tending to be absorbing and scattering requires some simplifying assumptions. Dave and Braslau [58] used a direct numerical solution of the spherical harmonics approximation for the axially symmetric but highly anisotropic phase functions which describe the scattering properties of liquid water drops (cloud) and aerosol. Halpern [57] used Dave and Braslau [58] results to construct tables of the downward ground-level flux and the upward flux at the top of the atmosphere. These initial attempts have been refined in two main aspects. The limitations imposed by the discrete nature of the Halpern approach are avoided, through the use of parameterizations based on data provided by explicit solutions of the radiative transfer equation for a wide range of atmospheric conditions. The algorithms are typically independent of conventional data sources, with all site and time-specific environmental data being abstracted from the digital imagery.

Moser and Raschke [59] also using a radiative transfer model developed the following parameterizations for several model atmospheres and a wide range of boundary conditions

$${\mathrm{E}}_{\mathrm{g}}{\downarrow }_0=\mathrm{f}\left({\mathrm{\theta }}_{\mathrm{z}}\right)$$E67

$${\mathrm{E}}_0{\uparrow }_{\mathrm{c}}=\mathrm{f}\left({\mathrm{\theta }}_{\mathrm{z}},{\mathrm{h}}_{\mathrm{c}}\right)$$E68

$$\frac{{\mathrm{E}}_{\mathrm{g}}\downarrow }{{\mathrm{E}}_{\mathrm{g}}{\downarrow }_0}=1-\mathrm{f}\left({\mathrm{\theta }}_{\mathrm{z}},{\mathrm{L}}_{\mathrm{n}}\right)$$E69

The cloud height ($h_c$) was determined using simultaneous satellite measurements in the infrared (10.5–12.5) μm while $L_n$ (the normalized reflected radiance) was determined from

$${\mathrm{E}}_{\mathrm{g}}\downarrow ={\mathrm{E}}_0\uparrow \left[1-\mathrm{f}\left({\mathrm{\theta }}_{\mathrm{z}},{\mathrm{L}}_{\mathrm{n}}\right)\right]$$E70

3.6 Classification and comparative study of the models

We summarise here the models presented in the previous sections aiming to show the interrelationship amongst the models and the input and output parameters of each (Table 3).

Model	References	Input parameters	Output parameters	Linearity
Models based on the diffuse ratio- clearness index regressions	[2, 15, 18, 19, 20]	Clearness index (Kt)	Diffuse ratio (K)	Linear and Nonlinear
Models based on diffuse transmittance index and clearness index regression	[25]	Hourly clearness index, $\mathbf{k}$	Hourly diffuse transmittance index, ${\mathbf{k}}_{\mathbf{d}}$	Linear
Models based on direct transmittance index and clearness index	[26, 27].	Clearness Index (${\mathbf{K}}_{\mathbf{t}}$)	Direct beam transmittance (${\mathbf{K}}_{\mathbf{b}}$), air mass, cloud-cover, water vapour, and albedo	Nonlinear
ASHRAE model	[28, 29]	Zenith angle, ${\mathbf{\theta }}_{\mathbf{z}}$	Diffuse radiation, Hd	Nonlinear
Models using the direct transmittance index and the diffuse transmittance index	[30]	Diffuse transmittance index (Kd)	Beam transmittance index (${\mathbf{K}}_{\mathbf{b}}$)	Nonlinear
Models based on the sunshine fraction	[2, 3, 10]	Solar fraction, S/S0	Clearness Index (${\mathbf{K}}_{\mathbf{t}}$)	Linear and Nonlinear
Cloud cover radiation models (CRM)	[32, 33]	Monthly average daily cloud cover (${\overline{\mathit{N}}}_{\mathit{e}}$, in eighths)	Monthly average diffuse transmittance index, ${\overline{\mathit{K}}}_{\mathit{d}}\mathit{=}{\overline{\mathit{H}}}_{\mathit{d}}/\overline{\mathit{H}}$,	Linear
Meteorological Radiation Model (MRM)	[31, 35, 36]	SF, ${\mathbf{\tau }}_{\mathbf{r}}$, ${\mathbf{\tau }}_{\mathbf{\alpha }}$, ${\mathbf{\tau }}_{\mathbf{g}}$, ${\mathbf{\tau }}_{\mathbf{o}}$, ${\mathbf{\tau }}_{\mathbf{w}}$, ${\mathbf{k}}_{\mathbf{b}}$	$\mathbf{DBR}$, ${\mathbf{G}}_{\mathbf{b}}$	Nonlinear

Table 3.

Comparative study of the Solar radiation models.

3.6.1 Classification of the models

See Figure 5.

Figure 5.

Classification of solar radiation models. Modified from [60].

3.6.2 Comparative study of the models

4. Tilt and azimuth angles in solar photovoltaics energy applications

4.1 Introduction

The aims in this section is to present the optimum tilt angles calculation methods required for the optimal and best design of solar PV systems. Some techniques applicable for solar tilt calculations have been elaborated in [61, 62]. Some valuable excerpts from these references are considered in this section.

4.2 Optimal tilt angles for global solar radiation components

Like on horizontal surfaces, the total daily radiation falling on tilted surfaces ($G_T$) is the sum of three components: the direct ($G_{B_t}$), diffuse ($G_{D_t}$) and ground reflected ($G_{R_t}$). This is expressed as [62] (Figure 6)

Figure 6.

Zenith angle (${\theta }_z$), slope (β), surface azimuth angle (γ) and solar azimuth angle (γs) for a tilted surface. Modified from [63].

$$G_T=G_{B_t}+G_{D_t}+G_{R_t}$$E71

These three components are respectively related to direct, diffuse and total radiation on horizontal surfaces through the three expressions

$$G_{B_t}=R_b{G}_B$$E72

$$G_{D_t}=R_d{G}_D$$E73

$$G_{R_t}=R_{r}G$$E74

$R_b$, $R_d$ and $R_r$ are the quotients of the daily solar radiation incident on a slanted surface to that incident on a horizontal surface for the beam, the diffuse and the reflected components respectively. $G_B$, $G_D$ and $G$ are the beam, diffuse and total daily SR on a horizontal surface. (Eq. (71)) then asumes the expression:

$$G_T=R_b{G}_B+R_d{G}_D+R_{r}G$$E75

The calculation of the direct and diffuse components of GSR needed for the estimation of GSR on slanted surfaces was well elaborated in subSection 3.1.

In terms of the albedo, $\rho$, and the tilt angle of the horizontal surface $\beta$, $R_r$ is expressed as:

$$R_r=\rho \left(\frac{1-cos\beta }{2}\right)$$E76

Here $R_b$ depends on the transmittance of the atmosphere which is in turn affected by the atmospheric cloud cover, water vapour and concentration of atmospheric particles.

R_b for fixed slope surfaces oriented towards the equator in the northern hemisphere is expressed in [61, 64] as

$$R_b=\frac{cos\left(\phi -\beta \right)cos\delta sin{\omega }_{\mathit{st}}+{\omega }_{\mathit{st}}sin\left(\phi -\beta \right)sin\delta }{cos\phi cos\delta sin{\omega }_{\mathit{st}}+{\omega }_{\mathit{st}}sin\phi sin\delta }$$E77

Where ${\omega }_{\mathit{st}}$ is the sunset hour angle for the tilted surface, for the mean day of the month, which is given by

$${\omega }_{\mathit{st}}=\mathit{min}\left[{\mathit{cos}}^{-1}\left(-tan\phi tan\delta \right),{\mathit{cos}}^{-1}\left(-tan\left(\phi -\beta \right)tan\delta \right)\right]$$E78

For surfaces in the southern hemisphere sloped towards the equator, the equations are [61, 65]:

$$R_b=\frac{cos\left(\phi +\beta \right)cos\delta sin{\omega }_{\mathit{st}}+{\omega }_{\mathit{st}}sin\left(\phi +\beta \right)sin\delta }{cos\phi cos\delta sin{\omega }_{\mathit{st}}+{\omega }_{\mathit{st}}sin\phi sin\delta }$$E79

$${\omega }_{\mathit{st}}=\mathit{min}\left[{\mathit{cos}}^{-1}\left(-tan\phi tan\delta \right),{\mathit{cos}}^{-1}\left(-tan\left(\phi +\beta \right)tan\delta \right)\right]$$E80

It is possible to alternatively estimate $R_b$ as the quotient of the daily extraterrestrial radiation on the slanted surface to that on a horizontal surface, $G_0$ [24]. As a consequence the following relation has been proposed for $R_b$ [62, 66]:

$$\begin{array}{l}{R}_b={\int }_{{\omega }_{\mathit{rt}}}^{{\omega }_{\mathit{st}}}cos\theta \left(\omega \right)\mathit{d\omega }/{\int }_{{\omega }_r}^{{\omega }_s}cos\alpha \left(\omega \right)\mathit{d\omega }\\ =\left(cos\beta sin\delta sin\varphi \right)\left(\frac{\pi }{180}\right)\left({\omega }_{\mathit{st}}-{\omega }_{\mathit{rt}}\right)-\left(sin\delta cos\varphi sin\beta cos{\alpha }_t\right)\left(\pi /180\right)\left({\omega }_{\mathit{st}}-{\omega }_{\mathit{rt}}\right)\\ +\left(cos\varphi cos\delta cos\beta \right)\left(sin{\omega }_{\mathit{st}}-sin{\omega }_{\mathit{rt}}\right)+\left(cos\delta cos{\alpha }_{t}sin\varphi sin\beta \right)\left(sin{\omega }_{\mathit{st}}-sin{\omega }_{\mathit{rt}}\right)\\ +\left(cos\delta sin\beta sin{\alpha }_t\right)\left(cos{\omega }_{\mathit{st}}-cos{\omega }_{\mathit{rt}}\right))/\left(2\left(cos\varphi cos\delta sin{\omega }_s+\left(\frac{\pi }{180}\right){\omega }_{s}sin\varphi sin\delta \right)\right)\end{array}$$E81

Where ${\omega }_r$ and ${\omega }_s$ are the sunrise and sunset hour angles over the horizon respectively in degrees. Also, ${\omega }_{\mathit{rt}}$ and ${\omega }_{\mathit{st}}$ are sunrise and sunset hour angles over tilted plane surface calculated as follows:

if ${\alpha }_t<0$,

$${\omega }_{\mathit{rt}}=-\mathit{min}\left({\omega }_s,{\mathit{cos}}^{-1}\left(\frac{\mathit{PQ}+\sqrt{P^2-Q^2+1}}{P^2+1}\right)\right)$$E82

$${\omega }_{\mathit{st}}=min\left({\omega }_s,{\mathit{cos}}^{-1}\left(\frac{\mathit{PQ}-\sqrt{P^2-Q^2+1}}{P^2+1}\right)\right)$$E83

else

$${\omega }_{\mathit{rt}}=-min\left({\omega }_s,{\mathit{cos}}^{-1}\left(\frac{\mathit{PQ}-\sqrt{P^2-Q^2+1}}{P^2+1}\right)\right)$$E84

$${\omega }_{\mathit{st}}=min\left({\omega }_s,{\mathit{cos}}^{-1}\left(\frac{\mathit{PQ}+\sqrt{P^2-Q^2+1}}{P^2+1}\right)\right)$$E85

Where,

$$P=cos\varphi /\left(sin{\alpha }_{t}tan\beta \right)+sin\varphi /tan{\alpha }_t$$E86

$$Q=tan\delta \left(cos\varphi /tan{\alpha }_t-sin\varphi /\left(sin{\alpha }_{t}tan\beta \right)\right)$$E87

4.3 Diffuse radiation models on tilted surfaces

Both isotropic and anisotropic models exist for estimating the ratio of diffuse SR on a tilted surface to that on a horizontal surface. The isotropic models assume the intensity of diffuse sky radiation to be uniform over the skydome. As a consequence, the diffuse radiation incident on a tilted surface is a function of the fraction of the skydome it sees. The anisotropic models on the other hand assume the anisotropy of the diffuse sky radiation in the circumsolar region (portion of sky near the solar disk) plus and isotropically distributed diffuse component from the rest of the skydome. Some of these models are summarised in Table 4.

4.4 Optimization of Tilt angle techniques

For PV modules to furnish maximum output power, there is a need to optimize the tilt angle. We present here (Table 5) a non-exhaustive summary of some optimal tilt angle equations while the details are obtainable from the indicated references. Taking into consideration the functional relationship of the solar declination, $\delta$, with the day of the year through (Eq. (7)), we also include the optimum tilt angle data (monthly, seasonally and yearly) as applicable alongside the models (Tables 4 and 5).

Reference	Optimum Tilt Angle expressed in terms of altitude
Isotropic models
[65]	$R_d=\frac{3-cos2\beta }{4}$
[67]	$R_d=1-\frac{\beta }{180}$
[68]	$R_d=\frac{2+cos\beta }{3}$
[24]	$R_d=\frac{1+cos\beta }{2}$
Anisotropic models
[69]	$R_d=\frac{G_B}{G_{B_n}}{R}_B+\left(1-\frac{G_B}{G_{B_n}}\right)\left(1+\frac{cos\beta }{2}\right)\left(1+\sqrt{\frac{G_B}{G_{B_t}}}{\mathit{sin}}^3\left(\frac{\beta }{2}\right)\right)$
[70]	$R_d=\frac{G_B}{G_{B_n}}{R}_B+\text{Ωcos}\beta +\left(1-\frac{G_B}{G_{B_n}}\right)\left(1+\frac{cos\beta }{2}\right)\left(1+\sqrt{\frac{G_B}{G_{{\mathit{B}}_t}}}{\mathit{sin}}^3\left(\frac{\beta }{2}\right)\right),$; Where $\mathrm{\Omega }=max\left[0,\left(0.3-2\frac{G_B}{G_{B_n}}\right)\right]$
[71]	$R_d=0.51R_b+\frac{1+cos\beta }{2}-\frac{1.74}{1.26\pi }\left[sin\beta -\left(\beta \frac{\pi }{180}\right)\mathit{\text{cosβ}}-\pi {\mathit{sin}}^2\left(\frac{\beta }{2}\right)\right]$
[72]	$R_d=\frac{G_B}{G_{B_n}}{R}_B+\left(1-\frac{G_B}{G_{B_n}}\right)\left(1+\frac{cos\beta }{2}\right)$
[73]	$R_d=\left(\frac{1+cos\beta }{2}\right)\left(1+F{\mathit{sin}}^3\left(\frac{\beta }{2}\right)\right)\left(1+F{\mathit{cos}}^2\theta {\mathit{sin}}^2{\theta }_z\right)$; Where, $F=1-{\left(\frac{G_{D_t}}{G_B}\right)}^2$,; $cos\theta =sin\delta sin\left(\phi -\beta \right)+cos\delta cos\left(\phi -\beta \right)\mathit{\text{cosωsin}}{\theta }_z$; $=\sqrt{1-{\left(sin\delta sin\phi +cos\delta cos\mathit{\text{φcosω}}\right)}^2}$; $\omega ={\mathit{cos}}^{-1}\left(-\mathit{\text{tanφ tanδ}}\right)$

Table 4.

Solar radiation diffuse factor, $R_d$, expressed in terms of the tilt angle, $\beta$. Modified from [61].

Reference	Location	Optimum tilt angle equation	Optimum tilt angle
			Interval	Data
[74]	Romania	${\beta }_{\mathit{opt}}=\phi -\delta$	/	/
[75]	Ghana	${\beta }_{\mathit{opt}}=\phi -{17}^0$	Monthly	${40}^0$(Jan.); ${28.7}^0$(Feb.);${10.6}^0$(march)
			Yearly	${26.8}^0$
[76]	Brisbane, Australia	NA	Yearly	${26}^0$ N
[77]	Abu Dhabi, UAE	NA	Monthly	June $-9^0$, December ${52}^0$
[78]	8 provinces of Turkey	${\beta }_{\mathit{opt}}=34.783-1.4317\delta -0.0081{\delta }^2+0.0002{\delta }^3$	Monthly	0° (June) and 65° (December)
			Seasonally	21.17° (spring), 5.67° (summer), 46.48° (autumn), 57.29° (winter)
[79]	Ontario, Canada	${\beta }_{\mathit{opt}}=\phi$	Yearly	${\beta }_{\mathit{opt}}=\phi$
	Ottawa, Canada	${\beta }_{\mathit{opt}}=\phi -\left(7^0\to {12}^0\right)$	Yearly	${\beta }_{\mathit{opt}}={36}^0\to {38}^0$;$\phi ={45}^0$
	Toronto, Canada	${\beta }_{\mathit{opt}}=\phi -\left(7^0\to {12}^0\right)$	Yearly	${\beta }_{\mathit{opt}}={32}^0\to {35}^0$;$\phi ={44}^0$
[80]	United States	${\beta }_{\mathit{opt}}=\phi -\left(1^0\to {10}^0\right)$	Yearly	Optimum orientation (Tilt/Azimuth); Orlando, FL, ${29.1}^0/{7.6}^0$E; Dallas, TX, ${30.5}^0/{5.9}^0$E; Phoenix, AZ, ${33.4}^0/{0.3}^0$W; Los Angeles, CA, ${32.4}^0/{3.8}^0$W; St. Louis, MO, ${34.8}^0/{1.0}^0$W
[81]	Zahedan, Iran	${\beta }_{\mathit{opt}}=09.17\phi \pm {0.321}^0$	Bi-Annual	$5^0$, ${50}^0$
[82]	Madinah, Saudi Arabia	${\beta }_{\mathit{opt}}=\phi$	Monthly	${40}^0$ (January); ${10}^0$ (June); ${40}^0$ (December)
			Seasonally	${37}^0$ (Winter); ${12}^0$ (Summer); ${17}^0$ (Spring); ${28}^0$ (Autumn)
			Yearly	${23.5}^0$, Equal to latitude, ${\beta }_{\mathit{opt}}=\phi$
[83]	New Delhi, India	Summer; ${\beta }_{\mathit{opt}}=\phi -{60}^0$; Winter; ${\beta }_{\mathit{opt}}=\phi +{90}^0$	Seasonally	Summer;$13^0$$\left(\phi -{60}^0\right)$; Winter;${47.5}^0$ ($\phi +{90}^0$)

Table 5.

Solar radiation diffuse factor $R_d$, with respect to tilt angle $\beta$. Modified from [61].

5. Conclusion

In this chapter, we have presented the different models of SR geared towards photovoltaic applications. Solar radiation models can be distinguished based on the type of measurement of input data used. Based on this we have models that use ground measured data and models that use satellite measured data. These models can be further sub-classified as either broadband or spectral models according as they are based on the earth’s radiation balance or on results generated by the solution of the radiative transfer equation. The baseline objective of the SR models presented is to predict the three components of GSR (the beam, the diffuse and the reflected components) incident on some PV collector surface at the ground level. Consequently, in the models presented, careful attention was given to show how we could quantify these three components. Results for some geographical locations have been given to show the correlation of equations for the models based on the diffuse ratio- clearness index regressions as well as for those based on sunshine fraction.

The broadband models were prioritized in the presentation given the ubiquitous occurrence of the input data for such models. Space restrictions conditioned the presentation of the models to be summarised to the strict minimum so our readers are encouraged to consult the cited literature in addition. A comparison of the models has been presented to highlight the input and the output parameters. In addition, a flow chart to show the interrelatedness of the models has been presented.

The approaches for getting the optimal tilt and azimuthal angles of the PV panel are summarized. Based on literature sources, the optimal tilt angles for some global geographical locations have been presented.

The statistical procedures to ascertain the validity of the regression analyses are summarily treated and applied in some cases where results have been presented.

We expect this chapter to be a valuable tool for scientist and engineers specialized in solar PV research and applications.

Acknowledgments

The authors wish to acknowledge the Cameroon Ministry of Higher Education for financing this research through the research allowance paid to all its staff of Higher Education in Cameroon.

Conflict of interest

No conflict of interest.