1 Introduction

The development and utilisation of renewable energy resources, which will serve as the fundamental future energy source for the world, is essential for generating sustainable energy technologies (Dincer, 2000; Elliott, 2000; Falayi & Rabiu, 2011; Ulgen & Hepbasli, 2009). Renewable energy resources, such as solar energy, offer critical advantages over fossil fuels, as they reduce the emission of greenhouse gases and a wide range of air pollutants and are a more sustainable source of energy. To fully utilise solar energy, it is important to have a precise knowledge of the amount of solar radiation and its components available at each location. This information is crucial for the design and optimisation of various solar energy systems, including those used in human-clothing-environment systems, and for biological studies of vitamin D. Additionally, this knowledge is essential for the development of solar energy conservation devices, such as water desalinators, concentrating collectors, solar furnaces, and solar dryers (Bakirci, 2009, 2015; Chandrashekara & Yadav, 2017; Falayi & Rabiu, 2011; Krzyścin et al., 2011; Salhi et al., 2020; Shimazaki et al., 2017; Taşdemiroğlu & Sever, 1991; Tiris et al., 1996; Zarezade & Mostafaeipour, 2016).

Despite the widespread availability of pyranometers, solarimeters, and pyrheliometers to measure incoming solar radiation, measuring diffuse solar radiation is not as common, even though it has significant applications in solar energy systems, home energy analysis, and building heat transfer (Berrizbeitia et al., 2020; Gopinathan & Soler, 1995; Jacovides et al., 1996; Soneye et al., 2019; Taşdemiroğlu & Sever, 1991; Ulgen & Hepbasli, 2009). Several factors contribute to the limited measurements of diffuse solar radiation, including the impact of geographic, climatic, and atmospheric conditions and the high cost of equipment maintenance (Ayoola et al., 2014; Jamil & Akhar, 2017; Soneye et al., 2019). As a result, many theoretical and empirical models have been developed to estimate this parameter (Bakirci, 2015; Duffie & Beckham, 2006; Khorasanizadeh et al., 2014).

In 1960, Liu and Jordan established the first empirical relationship for estimating diffuse solar radiation, which has served as a baseline. Since then, many scientists (including Arslanoglu, 2016; Boland et al., 2008; Falayi & Rabiu, 2011; Gopinathan, 1988; Haydar et al., 2006; Iqbal, 1979; Jamil & Akhar, 2017; Jin et al., 2004; Karatasou et al., 2003; Khahro et al., 2015; Khalil & Shaffie, 2013; Oliveira et al., 2002; Salhi et al., 2020; Tarhan & Sari, 2005; Ulgen & Hepbasli, 2009; Wattan & Janjai, 2016; Yousuf & Umair, 2018, etc.) have proposed various empirical models for estimating diffuse solar radiation at different locations using input variables like pressure, temperature, humidity, clearness index, sunshine duration, and incoming solar radiation. Nevertheless, the most commonly used parameters include extra-terrestrial solar radiation, clearness index, sunshine duration, and incoming solar radiation (Duffie & Beckham, 2006; Ulgen & Hepbasli, 2009; Yousuf & Umair, 2018). Various types of regression analysis, including linear, quadratic, cubic, exponential, logarithmic, and inverse, which all take into consideration the least squares method, are the most widely used techniques for developing diffuse solar radiation models (Yousuf & Umair, 2018).

In Nigeria, there are limited meteorological stations that measure diffuse solar radiation. Considering the significance of diffuse solar radiation in solar energy applications, this paper aims to develop empirical models for estimating monthly diffuse solar radiation for different classifications. These models are formulated based on the relationship between the cloudiness index and diffusion coefficient, which are functions of the clearness index and/or relative sunshine duration. In order to find the best model for estimating monthly diffuse solar radiation in the study area, we determine the accuracy of the developed empirical models through a comprehensive statistical error analysis between the modelled and observed data. Furthermore, the performance of five models chosen from existing literature based on their broad applicability and/or similarity to the climatic conditions is evaluated and compared against measured data.

2 Methodology

2.1 Location of the Study Area

This study utilised a two-year dataset (January 2016–December 2017) of incoming solar radiation obtained at a meteorological site in the Teaching and Research Farm of Obafemi Awolowo University in Ile-Ife (7.53° N; 4.54° E) in the southwestern region of Nigeria. According to Köppen's climate classification system, this region falls under the wet and dry climatic zones of West Africa and receives a high amount of solar radiation throughout the year, with sunrise and sunset occurring at approximately 07:00 and 19:00 local time (LT), respectively (Ayoola et al., 2014; Griffiths, 1974; Soneye et al., 2019; Soneye-Arogundade, 2021). The dry season runs from November to February, while the rainy season typically starts in March/April and ends in October, with average annual rainfall ranging between 1000 and 1500 mm. The mean relative humidity and temperature in the area are about 80% and 27 °C, respectively (Griffiths, 1974; Hayward and Oguntoyinbo, 1987; Ayoola et al., 2014; Soneye et al., 2019; Soneye-Arogundade, 2021). In general, the diffuse components of solar radiation increase during the wet months due to the increasing prevalence of clouds such as altostratus, thin cirrus, and altocumulus clouds (Iziomon & Aro, 1998; Okogbue et al., 2009; Soneye, 2021).

2.2 A description of the Meteorological Sensors and Data Used for the Study

The high-quality SR01 pyranometer (ISO-class) with a spectral range between 0.3 and 2.8 µm facing upward, incorporated into a four-component net radiometer (model NR01, Hukseflux, USA) sensor, was used to measure the incoming solar radiation. These measurements were recorded as one-minute values in Wm−2 and stored in a datalogger (model CR1000, Campbell Scientific, USA), as shown in Fig. 1. Additional information regarding sensor sensitivities, calibration, and maintenance can be found in Soneye et al. (2019) and Soneye-Arogundade (2021). The one-minute data was used to estimate the monthly average daily values of incoming solar radiation in MJ m−2 day−1. The data was then divided into two annual sets.

Fig. 1
figure 1

Acquisition and processing of radiation data at Ile-Ife, Nigeria

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2.3 Atmospheric and Astronomical Parameters

The clearness index (\({C}_{t}\)) is the ratio of the measured global solar radiation at the surface of the Earth to the extra-terrestrial solar radiation at the top of the atmosphere. This index is used to determine the transparency of the Earth's atmosphere to solar energy and how clouds and aerosols in the atmosphere affect the amount of solar radiation reaching the Earth's surface (Akhlaque et al., 2009; Augustine & Nnabuchi, 2009; Kuye & Jagtap, 1992; Liu & Jordan, 1960; Okogbue et al., 2009; Poudyal et al., 2012; Soneye, 2021). The clearness index also takes into account changes in atmospheric conditions and the amount of solar radiation at a given location. Theoretically, the index ranges between 0 and 1, but in practise, it varies from almost 0 on cloudy days to about 0.8 in clear-sky conditions (Soneye, 2021). The daily values of the clearness index were computed using:

$${C}_{t}=\frac{{S}_{i}}{{S}_{0}}$$
(1)

where \({S}_{i}\) is the measured incoming solar radiation (\({\mathrm{W m}}^{-2}\)) and \({S}_{0}\) is the daily extra-terrestrial solar radiation at the top of the atmosphere (\({\mathrm{W m}}^{-2}\)).

The daily extra-terrestrial solar radiation is the intensity of the solar radiation at the top of Earth's atmosphere, and it varies throughout the year due to the elliptical orbit of the Earth (Iqbal, 1983; Duffie & Beckham, 2006; Soneye, 2021). The intensity of the extra-terrestrial solar radiation was computed using:

$${S}_{0}=\frac{24\times 3600}{\pi }\times {G}_{0}\times {E}_{0}\times \left(\mathrm{sin}{\omega }_{s}\mathrm{cos}\phi \mathrm{cos}\delta +\frac{\pi {\omega }_{s}}{180}\mathrm{sin}\phi \mathrm{sin}\delta \right)$$
(2)

where \({G}_{0}\) is the solar constant (\(1367 {\mathrm{W m}}^{-2}\)), \({E}_{0}\) is the eccentricity correction factor of the Earth’s orbit, \(\phi\) is the latitude (positive north), \({\omega }_{s}\) is the sunrise hour angle, \(\delta\) is the solar declination angle and \(\pi = 3.14286\). The units of \(\phi\), \({\lambda }_{e}\) and \(\delta\) are degrees \(\left(^\circ \right)\).

The solar declination angle is the angle between the equatorial plane and the line connecting the centres of the sun and Earth. This angle constantly changes, with a value of zero during the autumnal and vernal equinoxes, and approximately equal values of \(-23.5^\circ\) and \(+23.5^\circ\) at the winter and summer solstices, respectively (Iqbal, 1983). The solar declination angle was determined using the equation developed by Cooper (1969) and (Soneye, 2021):

$$\delta ={\phi }_{L}\mathrm{sin}\left[\frac{2\pi \left({d}_{n}-{d}_{r}\right)}{{d}_{y}}\right]$$
(3)

where \({\phi }_{L}\) is the latitude of the Tropic of Cancer \(= 23.45^\circ\), \(\pi =180^\circ\), \({d}_{n}\) is the Julian day number of the year, which ranges from 1 on 1 January to 365 on 31 December, \({d}_{r}\) is a constant = 284, \({d}_{y}\) is the average number of days per year with a value of \(365\).

The eccentricity correction factor was computed using (Spencer, 1971; Iqbal, 1983; Soneye, 2021):

$${E}_{0}=1.000110+0.034221 \mathrm{cos} \Gamma+0.001280\mathrm{sin}\Gamma+0.000719\mathrm{cos}2\Gamma+0.000077\mathrm{sin}2\Gamma$$
(4)

where \(\Gamma\) (unit is degrees) is the day angle and can be expressed as:

$$\Gamma=\frac{360\left({d}_{n}-1\right)}{365}$$
(5)

The sunrise hour angle was estimated using:

$${\omega }_{s}={\mathrm{cos}}^{-1}\left(-\mathrm{tan}\phi \mathrm{tan}\delta \right)$$
(6)

The daily values of the diffuse solar radiation (\({S}_{d}\), unit \(W{m}^{-2}\)), for the study area were calculated using Eq. (7) when \({\omega }_{s}>81.4^\circ\) and \(0.3\le {C}_{tAVG}\le 0.8\) (Duffie & Beckham, 2006):

$${C}_{d}=\frac{{S}_{d}}{{S}_{i}}=1.311-3.022{C}_{tAVG}+3.427{{C}_{tAVG}}^{2}-1.821{{C}_{tAVG}}^{3}$$
(7)

where \({C}_{d}\) is the cloudiness index and \({C}_{tAVG}\) is the average value of the clearness index.

The direct solar radiation was calculated from the difference between incoming solar radiation and diffuse solar radiation values.

To determine the daily sunshine duration (\(N)\) in the absence of measured data, eighteen solar radiation models, listed in Table 1, were utilised using the techniques of Aras et al. (2006), Ulgen and Hepbasli (2009), and Sabzpooshani and Mohammadi (2014). These methods involve selecting various diffuse solar radiation models from the existing literature to estimate this parameter. The resulting data was then averaged to estimate diffuse solar radiation, as selecting the most appropriate model without experimental data can be challenging. The purpose of this approach was to eliminate variations between the estimates generated by the models (Aras et al., 2006; Ulgen & Hepbasli, 2009).

Table 1 Models used for obtaining sunshine duration (\(\mathrm{N}\)) data
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2.4 Development of Diffuse Solar Radiation Models

In the past, various empirical models have been developed by different authors, such as Aras et al. (2006), Gopinathan (1988), Ulgen and Hepbasli (2009), Katiyar and Pandey (2010), Jamil and Akhar (2017), and others, to estimate monthly mean values of diffuse solar radiation. These models typically use extra-terrestrial radiation, measured incoming solar radiation, and measured sunshine duration or relative sunshine duration as input variables. In this study, a correlation approach was used to develop 21 models that fall under five categories, as presented in Table 2. Multivariate regression analysis was used to generate these models. Models with several input variables and an order of up to two in each input variable were more accurate. However, higher-order models were not considered due to the increased complexity of the correlation equations. Five previously published models (listed in Table 3) were selected from the literature and compared with the measured diffuse solar radiation. These models were selected due to their broad applicability and/or similarity to climatic conditions.

Table 2 Classes of the models developed in this paper for estimating monthly mean values of diffuse solar radiation
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Table 3 Solar radiation models selected from the literature
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2.5 Method of Statistical Evaluation

The aim of this study is to determine the best model for estimating solar radiation at the study location. According to Sen (2008) and Gueymard (2012), previously developed solar radiation estimation models do not have a perfect correlation coefficient of R = 1 since the measured data are prone to calibration uncertainty. Hence, this study evaluates the accuracy and performance of all newly developed models using the statistical tools listed in Table 4.

Table 4 Statistical errors
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3 Results and Discussion

The solar radiation data measured at Ile-Ife from January 2016 to December 2017 is presented in Fig. 2. The height of the bar chart corresponds to the average monthly solar radiation, while the red and green portions represent the monthly means of the diffuse and direct solar radiation components, respectively. The incoming solar radiation reached its maximum value of 19.36 MJ m−2 day−1 in April of the first year, while its minimum values of 13.75 MJ m−2 day−1 and 11.63 MJ m−2 day−1 were recorded in the wet months of July and August, respectively. In the second year, 2017, the values for July and September were practically the same, with approximately 13.40 MJ m−2 day−1, with the lowest value of 11.53 MJ m−2 day−1 recorded in August and the maximum value of 18.267 MJ m−2 day−1 in March, followed by April and May. The increase in incoming solar radiation during these months (April 2016 and March 2017) can be attributed to the reduction in atmospheric turbidity resulting from the rain scavenging aerosol particles suspended in the atmosphere. This leads to a reduction in the attenuation of incoming solar radiation, less atmospheric cloud cover, and more direct solar energy reaching the earth's surface (Ayoola et al., 2014; Chukwuemeka & Asiegbu, 2017; Soneye, 2021; Soneye et al., 2019; Udo, 2000). The minimal values of incoming solar radiation in July and August for both years can be attributed to the predominance of high moisture content and high cloudiness throughout the rainy season (Afiesimama et al., 2006; Ayoola et al., 2014; Fagbenle, 1992; Maduekwe & Chendo, 1997; Ohunakin et al., 2015; Soneye, 2021; Soneye et al., 2019; Udo, 1978).

Fig. 2
figure 2

Variation of the monthly mean solar radiation for January-December 2016 and January–December 2017 at Ile-Ife, Nigeria

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In 2016 and 2017, the variations in direct solar radiation followed the same pattern as the incoming solar radiation, with the minimum values occurring in August and the highest values recorded in April and March, respectively. The highest values of diffuse solar radiation, approximately 8.20 MJ m−2 day−1 were observed from June to September for both years, while the lowest values for both years were measured in December. The increase in cloud cover during the rainy months caused the high values of diffuse solar radiation observed between June and September of both years. This diffuse radiation is mostly caused by the intense forward scattering of solar radiation due to clouds such as altostratus, thin cirrus, and altocumulus (Ayoola et al., 2014; Iziomon & Aro, 1998; Okogbue et al., 2009; Soneye, 2021; Soneye et al., 2019). The lowest value obtained indicates that molecule scattering and, to a lesser extent, surface albedo are the primary causes of diffuse radiation that reaches the Earth's surface (Iziomon & Aro, 1998). These results are similar to those published for Abia, Nigeria, by Chukwuemeka and Asiegbu (2017), and Aligarh City, India, by Jamil and Akhtar (2017).

The monthly averaged values of extra-terrestrial radiation, incoming solar radiation, and direct and diffuse solar radiation for the period from January 2016 to December 2017 are shown in Fig. 3. A minimum value of 32.33 MJ m−2 day−1 was recorded in December, while a maximum value of 39.61 MJ m−2 day−1 was recorded in April for extra-terrestrial solar radiation. The low value in December is due to the high turbidity of the atmosphere, resulting from wind-borne harmattan dust of Saharan origin and smoke from bush burning activities (Butt et al., 2010; Falaiye et al., 2014; Maduekwe & Chendo, 1997; Ohunakin et al., 2015; Okogbue et al., 2009; Soneye et al., 2019). The average value of 11.58 MJ m−2 day−1 and the maximum value of 18.39 MJ m−2 day−1 of incoming solar radiation were recorded in August and April, respectively. The variation in direct solar radiation followed a similar pattern to that of incoming solar radiation, with a minimum average value of 3.30 MJ m−2 day−1 and a maximum average value of 10.89 MJ m−2 day−1 recorded in August and April, respectively. The maximum values of all three parameters were observed in April due to less attenuation of solar radiation caused by a reduction in aerosol particle concentration due to rainfall. The average minimum (6.85 MJ m−2 day−1) and maximum (8.35 MJ m−2 day−1) values of measured diffuse solar radiation were recorded in December and July, respectively. The decrease in direct and incoming solar radiation in August and an increase in diffuse solar radiation in July were due to increased cloudiness caused by an increase in the number, height, thickness, and type of clouds during those months (Ayoola et al., 2014; Butt et al., 2010; Soneye et al., 2019). This is also due to the effect of relatively moist air from the ocean due to the movement of the moist south-westerly wind from the ocean (Pal et al., 2000; Lawal, 2010; Ohunakin et al., 2015; Soneye, 2018). The yearly average values of extra-terrestrial, incoming, direct, and diffuse solar radiation at the research location were 37.05 MJ m−2 day−1, 15.89 MJ m−2 day−1, 8.21 MJ m−2 day−1 and 7.68 MJ m−2 day−1, respectively.

Fig. 3
figure 3

Variation of the monthly mean extra-terrestrial radiation, incoming solar radiation, and direct and diffuse solar radiation for January-December 2016 and January–December 2017 at Ile-Ife, Nigeria

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The monthly and yearly average changes in the clearness index, cloudiness index, and diffusion coefficient during the observation period are shown in Fig. 4. The cloudiness index fluctuates in a similar manner to the diffusion coefficient, with the minimum and maximum values observed during the dry/transition and wet months, respectively. As shown in the figure, the cloudiness index had its lowest yearly mean value of 0.44 in December and its highest yearly mean value of 0.72 in August. Similarly, the diffusion coefficient exhibited its lowest yearly mean value of 0.19 in April and its highest yearly mean value of 0.21 in July. The high values for both the diffusion coefficient and cloudiness index seen in July and August can be attributed to the overcast conditions, an increase in relative humidity, and frequent thunderstorm activity, which are all related to the rainy season. These factors lead to significant attenuation of incoming solar radiation reaching the Earth's surface (Ayoola et al., 2014; Díaz-Torres et al., 2017; Soneye et al., 2019). Additionally, the concentration of diffuse coefficient values between 0.19 and 0.21 indicates that diffuse radiation accounts for only a small portion of extraterrestrial radiation (Che et al., 2006). The yearly averages for the clearness index, cloudiness index, and diffusion coefficient were 0.43, 0.52, and 0.21, respectively. These results are consistent with those reported by Udo (2000), Che et al. (2006), Okogbue et al. (2009), and Ohunakin et al. (2015). On the other hand, the clearness index, which depicts the availability of solar radiation and meteorological conditions at a particular area on the earth’s surface, showed a minimum monthly mean value of 0.27 in August and a maximum monthly value of 0.51 in November, as shown in the figure. This shows that the study site is not fully overcast in November, since \({C}_{t}>0.5\), but mostly cloudy in August (\({C}_{t}<0.5\).) The trend observed mirrors that described by Adaramola (2012) at Akure, in the southern part of Nigeria, and Ohunakin et al. (2013). The highest clearness index value recorded in November suggested that the incoming solar radiation received at the Earth's surface consisted mainly of the direct radiation component due to minimal cloudiness and turbidity caused by the removal of suspended aerosol particles by rain showers (Babatunde & Aro, 1995; Iziomon & Aro, 1998; Soneye, 2021; Udo, 2000). This indicates that the sky conditions at the study area are favourable with high solar radiation, suggesting that there is possible utilisation and high performance of renewable energy systems at the location almost year-round, apart from only December and January. The minimum values recorded for the two parameters are attributed to the presence of strongly developed altostratus, thin cirrus, and altocumulus clouds during the peak of the rainy season (Iziomon & Aro, 1998; Okogbue et al., 2009; Soneye, 2021).

Fig. 4
figure 4

Variation of the monthly and yearly mean of cloudiness index, clearness index, and diffusion coefficient for January-December 2016 and January–December 2017 at Ile-Ife, Nigeria

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The new diffuse solar radiation models developed in this paper (see Table S1 in the supplementary material) use the curve estimate technique and the mathematical expressions for each class (Table 2). The details of the diffuse solar radiation models developed in this paper are discussed in the supplementary material (Figure S1 and Table S2). The estimated values from all the models follow a similar pattern to the measured values, with peak values ranging from 8.34 to 8.99 MJ m−2 day−1 in June and July and lower values between 5.19 and 7.27 MJ m−2 day−1 in December. The maximum measured value is 8.35 MJ m−2 day−1 in July, while the lowest is 6.81 MJ m−2 day−1 in December. Model 6 (Class 1), Model 9 (Class 2), Model 11 (Class 3), Model 16 (Class 4), and Model 21 (Class 5) perform the best, with high R- and d-values, the lowest errors, and the best statistics. Model 11 in Class 3 is the best-performing model among all the classes. The statistical results presented in Table S2 confirm that diffuse solar radiation is overestimated in Class 1, there is poor agreement between estimated and measured values in Classes 2, 4, and 5, and close variation and good agreement in Class 3.

The statistical results of all the models developed in this paper under the five classes are presented in Table S2. The results show that most models have good correlation coefficients (R > 0.7), with about five models slightly below (0.5 ≤ R ≤ 0.7). Model 17, which has a significant value of MBE = 0.0247 MJ m−2 day−1, performs best according to the MBE statistical evaluator. The values of MAE (0.0680–0.7207 MJ m−2 day−1), RMSE (0.0825–0.8740 MJ m−2 day−1), and RSE (0.2720–0.3067) indicate good agreement between estimated and measured values. Most models underestimated the diffuse solar radiation, as indicated by the negative values of MPE recorded, except for Model 16, with a value of 0.2884, which is close to zero. The Akaike Information Criteria (AIC) and Global Performance Index (GPI) methods were used to rank the models according to their suitability for estimating diffuse solar radiation at the study site, with the model having the lowest AIC and GPI values performing the best.

The results of the AIC and GPI values for all the models developed in this paper are presented in the supplementary material (Table S3), while the overall ranking of all the developed models based on their respective AIC and GPI values is presented in Table 5. As shown in Table S3, Model 11 (quadratic function) from Class 3 with the clearness index as the input variable performed better than all the other models developed in this study, with a GPI of − 2.17957 and AIC, AICC, and ∆AICC values of 1.8099, 4.8099, and 0.0000, respectively. In addition, as shown in Table 5, models with only one input variable performed better than those with two input variables.

Table 5 Ranking of the models developed in this paper based on the Global Performance Index and Akaike Information Criteria
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The monthly variations of the measured and estimated diffuse solar radiation using the best-performing model (Model 11) and the worst performing model (Model 8) are presented in Fig. 5. As shown in the figure, the estimated values of the diffuse solar radiation obtained from Model 11 essentially duplicate the pattern of the measured radiation. The values of the measured data range between 6.81 and 8.36 MJ m−2 day−1, while those estimated from Model 11 range between 6.84 and 8.37 MJ m−2 day−1. The minimum and maximum values for both the measured and estimated data were observed in the months of December and July, respectively. However, Model 11 slightly underestimated the diffuse solar radiation between January and March, May, September, and October, while coinciding with the measured data in the other months. In contrast, the estimated values from Model 8 ranged from 5.52 MJ m−2 day−1 (December) to 8.99 MJ m−2 day−1 (July), with overestimation in January, March, May–August, and underestimation in February, April, September, November, and December. The only month where the estimated values from Model 8 were similar to the measured data was October. The high correlation coefficient (0.9906) and index of agreement (0.9930) obtained for Model 11 indicate good conformity and agreement between the values estimated from this model and the measured data. Conversely, the low correlation coefficient (0.5335) and index of agreement scores (0.6188) for Model 8 suggest significant discrepancies and very poor agreement between the estimated and measured data.

Fig. 5
figure 5

Variations of diffuse solar radiation, both measured and estimated using the best performing-model (Model 11) and the worst performing model (Model 8)

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The monthly variations of the diffuse solar radiation measured and estimated using the five published models selected from the literature (listed in Table 3) are presented in Fig. 6. As shown in this figure, Model 22 (Class 2) produces values that closely follow the pattern of the measured radiation but overestimate diffuse solar radiation in all months, including December, which had the lowest values for both the measured (6.81 MJ m−2 day−1) and estimated (7.11 MJ m−2 day−1) radiation. The highest values for both the measured and estimated (using Model 22) diffuse solar radiation, 8.36 MJ m−2 day−1 and 8.74 MJ m−2 day−1, respectively, were recorded in July. The values estimated by the other models deviate significantly from the measured values. The lowest values, 7.22, 7.12, 5.80, and 2.81 MJ m−2 day−1, were recorded for Models 23, 24, 25, and 26, respectively, in December. The highest values, 11.08, 8.41, and 9.61 MJ m−2 day−1 for Models 23, 25, and 26, respectively, were recorded in July, while the highest value of 10.54 MJ m−2 day−1 for Model 24 was recorded in August. Models 23 and 24 overestimated diffuse solar radiation in all the months, while Model 25 underestimated it. Additionally, Model 26 underestimated diffuse solar radiation between September and June but overestimated it in July and August. The under- and overestimation of diffuse solar radiation by these models is supported by the high values of MBE (~ 1.50 MJ m−2 day−1), MAE (~ 1.93 MJ m−2 day−1), RMSE (~ 2.24 MJ m−2 day−1), RRMSE (~ 29.20%), MPE (~ 21.47%), SSRE (~ 1.51), RSE (~ 0.35), U95 (~ 6.71 MJ m−2 day−1), and t-stat (~ 11.03), listed in Table 6. The table shows that most of the models have a correlation coefficient and index of agreement less than 0.5, indicating a high deviation of the estimated values from the measured values. Model 22 performed the best, with the highest correlation coefficient of 0.9922 and an index of agreement of 0.9276, as well as the lowest values of MAE (0.2783 MJ m−2 day−1), RMSE (0.2906 MJ m−2 day−1), RRMSE (3.7831%), and U95 (0.6762 MJ m−2 day−1). The differences between the measured and estimated values may be due to various factors such as weather conditions, geographical location, precipitation, aerosol mass concentration, atmospheric turbulence, cloud formation, pollution levels, and differences in the intensity of global and diffuse solar radiation between Ile-Ife and the other sites, South Africa, Turkey, Italy, and India, where these models were developed (Alaa et al., 2017; Falayi et al., 2011; Soneye, 2021; Soneye-Arogundade, 2021).

Fig. 6
figure 6

Variations of diffuse solar radiation, both measured and estimated using models obtained from the literature

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Table 6 Statistical results of the solar radiation models selected from the literature
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The monthly variations of the measured and estimated diffuse solar radiation using the best-performing model (Model 11) developed in this paper and the best-performing model (Model 22) obtained from the literature are presented in Fig. 7. Also, the statistical results of Model 11 and Model 22 are presented in Table 7. As shown in the figure, the estimated values of the diffuse solar radiation obtained from the two models essentially duplicate the pattern of the measured radiation. The results show that Model 11 provided estimated values of diffuse solar radiation that closely matched the measured data (6.81 and 8.36 MJ m−2 day−1), with values ranging from 6.84 to 8.37 MJ m−2 day−1. On the other hand, Model 22 overestimated the diffuse solar radiation, with values ranging from 7.11 to 8.74 MJ m−2 day−1. The minimum and maximum values of measured and estimated data were observed in December and July, respectively. However, Model 11 slightly underestimated the diffuse solar radiation between January and March, May, September, and October, but deviated only by 5.84–9.28% from the measured values. The statistical analysis presented in Table 7 indicates that Model 11 outperformed Model 22, with lower errors and better statistics, suggesting good conformity and agreement between the estimated and measured data. This implies that none of the models selected from the literature was suitable for estimating diffuse solar radiation at the study site; thus, the Model 11 developed in this study is recommended for estimating diffuse solar radiation not only at the study site but also at other locations with similar climatic conditions.

Fig. 7
figure 7

Variations of diffuse solar radiation, both measured and estimated using the best-performing-model (Model 11) developed in this paper and the best-performing model (Model 22) obtained from the literature

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Table 7 Statistical results of Model 11, the best-performing model developed in this paper, and Model 22, the best-performing model selected from the literature
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4 Conclusions

In this study, diffuse solar radiation models were developed and estimated using sunshine duration and clearness index data for Ile-Ife, a tropical site in Nigeria. Additionally, the best model suitable for estimating monthly diffuse solar radiation in the study area was identified using detailed statistical analysis.

The values of incoming solar radiation for 2016 and 2017 varied from 11.53 to 19.36 MJ m−2 day−1, with minimum values in August and maximum values in April and March. Direct incoming solar radiation showed the same pattern. Diffuse solar radiation had its highest values from June to September and its lowest values in December, while extra-terrestrial solar radiation had low values during the dry months, with the lowest recorded in December. The study suggests that cloudiness, moisture content, and atmospheric turbidity have a significant impact on the variation of solar radiation.

The cloudiness index and diffusion coefficient have a similar pattern, with the lowest values occurring during the dry or transition months and the highest during the wet months. The diffusion coefficient had concentration values ranging from 0.19 in April to 0.21 in July, indicating that only a small part of the extra-terrestrial radiation is diffuse. In contrast, the clearness index had a minimum monthly value of 0.27 in August and a maximum monthly value of 0.51 in November, suggesting that the site is not overly cloudy. The yearly average values were 0.43, 0.52, and 0.21 for the clearness index, cloudiness index, and diffusion coefficient, respectively.

The best results were obtained from Model 6 (Class 1), Model 9 (Class 2), Model 11 (Class 3), Model 16 (Class 4), and Model 21 (Class 5). These models yielded MBE, MAE, and RMSE < 0.6 MJm−2 day−1, RRMSE < 7.0%, MPE < 2.5%, and d > 0.8. In terms of the GPI (− 2.1796), AIC (1.8099), AICC (4.8099), and ∆AICC (0.0000) values, Model 11 (a quadratic function) from Class 3 with clearness index as the input variable performed better than all other models developed in this paper, while Model 8 (GPI = 3.0089, AIC = 15.9159, AICC = 24.7673, and ∆AICC = 19.9575) from Class 2 with clearness index and sunshine duration as input variables shows pretty poor performance. This implies that one input variable model performed better than the two input variable models. The use of a one-input variable model is recommended due to the increased complexity of models with two or more input variables and the high amount of requisite data. The clearness index is more significant than the relative sunshine duration in estimating diffuse solar radiation at the study site.

The monthly variations of the diffuse solar radiation estimated from the five published models selected from the literature deviate significantly from the measured values. However, the estimated values obtained from Model 22 (Class 2) duplicate the pattern of the measured radiation and show fairly good results, although they continuously overestimate the diffuse solar radiation.

The evaluated literature models were found to be unsuitable for estimating diffuse solar radiation at the study site. Therefore, the models developed in this paper performed better than the selected models. Among these models, Model 11 (Class 3) was found to be the best-performing model. It is a quadratic function with the clearness index as an input variable and provided estimated values of diffuse solar radiation that were in close agreement with the measured data. The estimated values ranged from 6.84 to 8.37 MJ m−2 day−1, with measured data at 6.81 and 8.36 MJ m−2 day−1. Conversely, Model 22 was found to overestimate the diffuse solar radiation, with values ranging from 7.11 to 8.74 MJ m−2 day−1. The statistical analysis of Model 11, including GPI (− 2.1796), AIC (1.8099), AICC (4.8099), and ∆AICC (0.0000), demonstrated good conformity and agreement between the estimated and measured data. This suggests that Model 11 outperformed Model 22. The proposed Model 11 with the expression \({C}_{DD}=0.0896+0.4790\left({C}_{t}\right)-0.4674{\left({C}_{t}\right)}^{2}\), is recommended for estimating diffuse solar radiation not only at the study site but also at other locations with similar climatic conditions. Overall, it is suggested to use this Model 11 for sites in other climatic regions as well as to test its broader applicability and versatility for potential other applications where no radiation measurements are available. Our approach emphasises the usefulness of evaluating a suite of different diffuse solar radiation models to arrive at a statistically well-informed assessment for the recommendation of the best available model.