### Sunshine duration-based models

Different regression models to estimate global solar radiation on the horizontal surface from sunshine duration is proposed in the literature (Almorox and Hontoria 2004; Angstrom 1924; Prescott 1940; Akinoglu and Ecevit 1990; Ertekin and Yaldiz 2000; Ampratwum and Dorvlo 1990; Sen 2007). The models we have examined in this work are given in Table 2. We have used the general A–P model and also modified the model for seasonal variations. We did not take higher-degree models greater than third degree as those models fluctuates a lot and do not give proper fit.

In the models in Table 2, *H* is the global solar radiation on horizontal surface (kW h/m^{2}), \(H_0\) is the extraterrestrial radiation (kW h/m^{2}), *S* is the sunshine duration (h), \(S_0\) is the day length (h), and *a*, *b*, *c*, and *d* are the empirical coefficients those we have to find.

Extraterrestrial solar radiation \(H_0\) can be calculated according to Duffie and Beckman (2006):

$$\begin{aligned} H_0&= {} \frac{(24\times 3600G_{\mathrm{sc}})}{\pi } \left( 1+0.033 \cos \frac{360D}{365} \right) \nonumber \\&\quad \times \left( \cos \phi \cos \delta \sin \omega _{\mathrm{s}}\right) +\frac{\left( \pi \omega _{\mathrm{s}}\right) }{180} \sin \phi \sin \delta \end{aligned}$$

(1)

where \(G_{\mathrm{sc}}\) is the solar constant (1367 W/m^{2}), *D* is the day number of the year counting from first January, \(\phi\) is the latitude of the place (°), \(\omega _{\mathrm{s}}\) is the sunset hour angle (°), \(\delta\) is the solar declination (°).

The value of \(\delta\) and \(\omega _{\mathrm{s}}\) can be calculated from Eqs. (2) and (3), respectively (Duffie and Beckman 2006):

$$\delta= 23.45\sin \left[ \frac{360(D+284)}{365}\right]$$

(2)

$$\omega _s= \arccos \left[ -\tan (\delta )\tan (\phi )\right]$$

(3)

The day length \(S_0\) can be calculated using the value of \(\omega _{\mathrm{s}}\) as follows (Duffie and Beckman 2006):

$$S_0= \frac{2}{15} \omega _{\mathrm{s}}$$

(4)

Now, the ratio \(H{/}H_0\) can be found by substituting the values of measured global solar radiation (*H*) and extraterrestrial radiation \(H_0\). This ratio \(H{/}H_0\) is known as clearness index which gives the percentage deflection by the sky of the incoming global radiation and changes in the atmospheric conditions in a given locality (Iqbal 1983).

### Sunshine duration and other meteorological parameter-based models

Solar radiation does not only depend on sunshine duration. It also depends on temperature deviation, precipitation, cloud cover, or extraterrestrial radiation. So, to derive better models for solar radiation estimation, in this study we have proposed few models combining the effects of sunshine duration and other meteorological parameters. Table 5 lists all these models and their coefficients with statistical evaluation.

### Cloud cover-based models

Black (1956) used data of 88 stations from all over the world to develop an empirical relationship between solar radiation and cloud cover. He used the mean monthly values for 88 stations and performed a regression analysis to find the following relationship between \(H{/}H_0\) and *C*:

$$\frac{H}{H_0}=0.803-0.340C-0.458C^{2}$$

(5)

Unfortunately, this relationship can produce substantial amount of error as according to Black himself (Black et al. 1954): (1) Mean monthly values were calculated from the maximum numbers of years, and (2) different instruments have been used in different stations, and no attempts have been made to reduce these instruments to a common standard.

Therefore, we have done a regression analysis to establish some new models to estimate solar radiation (*H*) directly from cloud fraction (*C*) for Bangladesh. To do this, we have established a relationship between clearness index, \(H{/}H_0\), and cloud fraction, *C*. The value of \(H{/}H_0\) can be determined similarly as shown in the previous section.

### Temperature-based models

There is a clear relationship between solar radiation and temperature. Higher insolation increases the temperature, and low or no insolation decreases the temperature significantly. Therefore, some relationships can be established between solar radiation and temperature deviation to estimate global solar radiation. Bristow and Campbell (1984) expressed the relationship between clearness index \((H{/}H_0)\) and difference of maximum and minimum air temperatures (\(T_{\mathrm{max}}\) and \(T_{\mathrm{min}}\), respectively) \(\Delta T\) as:

$$\frac{H}{H_0} = a\left[ 1-\hbox {exp}\left( -b \Delta T^{\mathrm{c}}\right) \right]$$

(6)

where *a*, *b*, *c* are empirical coefficients to be found.

Hargreaves and Samani (1982) proposed another relationship between clearness index and temperature difference:

$$\frac{H}{H_0} = a \left( \sqrt{\Delta T} \right)$$

(7)

Later on, this relationship was modified by various researchers such as Chen et al. (2004) who gave the following relationship:

$$\frac{H}{H_0} = a \left( \sqrt{\Delta T} \right) + b$$

(8)

Table 7 shows all the empirical models based on temperature to estimate solar radiation. Few new models have also been proposed here in this study.

### Temperature- and extraterrestrial radiation-based models

Extraterrestrial radiation, \(H_{0}\), is the amount of solar radiation reaching the atmosphere from the sun. After coming through the atmosphere, it is absorbed by various particles like water vapor and air molecules or reflected from the earth surface. Extraterrestrial radiation is the theoretically possible maximum radiation that would reach the earth surface if the atmosphere was absent (Duffie and Beckman 2006). The temperature difference, \(\Delta T\), occurs because of the variation of insolation reaching the earth surface which is directly related with solar radiation, *H*. It can now be assumed that there is a clear and close relationship between solar radiation, extraterrestrial radiation, and temperature difference. Goodin et al. (1999) have proposed the following relationship:

$$\frac{H}{H_0}= a \left[ 1-\hbox {exp} \left( -b \frac{\Delta T^{\mathrm{c}}}{H_0} \right) \right]$$

(9)

Table 7 shows few empirical models based on temperature and extraterrestrial radiation and their associated statistical evaluation.

### Temperature- and precipitation-based models

Precipitation surely decreases the solar insolation level significantly. Temperature deviation also occurs because of precipitation. DeJong and Stewart (1993) have given the following relationship based on precipitation and temperature difference:

$$\frac{H}{H_0} = a\left( \Delta T^{\mathrm{b}}\right) \left( 1+cP+dP^2\right)$$

(10)

In this study, we have also proposed few more models based on precipitation and temperature as shown in Table 7.

### Data

In this work, the measured daily global solar radiation data are taken from Institute of Energy (previously known as Renewable Energy Research Center), University of Dhaka, and Bangladesh Meteorological Department (BMD). Measured time series data of global solar radiation was available for five sites, and sunshine duration, precipitation, cloud cover, and temperature data were available for 34 stations of BMD. Table 1 summarizes the detailed information of the stations and the period of observations of the relevant data. Figure 2 shows the distribution of the stations over the country.

Institute of Energy (RERC) measured global solar radiation from January 2003 to December 2005 using two Eppley PSP pyranometers at 1-min interval for 24 h. Sunshine duration data were recorded using Campbell–Stokes sunshine recorders by both BMD and Institute of Energy. We have used the monthly averaged daily solar radiation data and their corresponding other parameter data, such as sunshine duration, precipitation, and temperature to perform the regression analysis. Only five sites, where measured solar radiation data are available, are taken to find the parameters of the models. Data of all the other sites are used to test the models. DLR satellite time series data (German Aerospace Center 2015) of global solar radiation were also available for ten sites from Solar and Wind Energy Resource Assessment (SWERA) project database (Schillings et al. 2004). NASA’s Surface meteorology and Solar Energy (SSE) (NASA 2015) data which are the 22-year average satellite data are also collected. As the DLR data and NASA SSE data are actually estimated data, they were not used to find model parameters, rather they are just used to show a comparison among the results.

### Statistical evaluation tools

In this study, six different statistical quantitative indicators were used to evaluate different models. These quantitative indicators are: the coefficient of determination \((R^{2}),\) mean percentage error (MPE), mean bias error (MBE), root mean square error (RMSE), mean absolute relative error (MARE), and *t* statistic (*t* stat). These indicators can be calculated as follows (Despotovic et al. 2015):

$$R^2= 1-\frac{\sum \nolimits _{i=1}^{n} \left( H_{i,{\mathrm{m}}}-H_{i,{\mathrm{c}}}\right) ^2}{\sum \nolimits _{i=1}^{n} \left( H_{i,{\mathrm{m}}}-H_{m,{\mathrm{avg}}}\right)^2}$$

(11)

$$\hbox{MPE}= \frac{1}{n}\sum \limits _{i=1}^{n} \left( \frac{H_{i,{\mathrm{c}}}-H_{i,{\mathrm{m}}}}{H_{i,{\mathrm{m}}}}\times 100\right)$$

(12)

$$\hbox {MBE}= \frac{1}{n} \sum \limits _{i=1}^{n} \left( H_{i,{\mathrm{m}}}-H_{i,{\mathrm{c}}}\right)$$

(13)

$$\hbox {RMSE}= \sqrt{\frac{1}{n} \sum _{i=1}^{n} \left(H_{i,{\mathrm{m}}}-H_{i,{\mathrm{c}}}\right) ^2}$$

(14)

$$\hbox {MARE}= \frac{1}{n} \sum _{i=1}^{n} \left| \frac{H_{i,{\mathrm{m}}}-H_{i,{\mathrm{c}}}}{H_{i,{\mathrm{m}}}}\right|$$

(15)

$$t\,\hbox{stat}= \sqrt{\frac{(n-1)\hbox {MBE}^2}{\hbox {RMSE}^2-\hbox {MBE}^2}}$$

(16)

where \(H_{i,{\mathrm{m}}}\) and \(H_{i,{\mathrm{c}}}\) are the *i*th measured and calculated values, respectively (kW h/m^{2}), \(H_{\mathrm{m,avg}}\) is the average of the calculated and measured values (kW h/m^{2}), and *n* is the number of observations. For better data modeling, MPE, MBE, MARE, *t* stat, and RMSE should be closer to zero, but \(R^{2}\) should approach 1.0 as closely as possible.