Influence of introducing various meteorological parameters to the Angström–Prescott model for estimation of global solar radiation

Abstract

This study aims to recognize that whether introducing various meteorological parameters to the Angström–Prescott (A–P) model eventuates in enhancing the precision of monthly mean global solar radiation estimation in cities of Bandar Abbas and Jask, situated in the south coast of Iran. To identify the significance of the average, maximum and minimum ambient temperatures, average and maximum relative humidity as well as water vapor and sea level pressures, seven models have been chosen from the literature. Using the long-term measured data and via statistical regression technique, the new regression coefficients have been developed for the original A–P model and the other seven nominated models. The models’ performances have been appraised via commonly utilized statistical indicators. The results indicated that the new models provided only minor improvements over the traditional A–P model; therefore, as more complexity is associated with introducing different meteorological parameters, their applications are not appealing practically. Making comparisons with the existing models developed using PSO (particle swarm optimization) technique demonstrated the superiority of the new established A–P models of this study; consequently, even without any improvement, the simple A–P models are indeed qualified for accurate estimation of global solar radiation in cities of Bandar Abbas and Jask and their neighboring.

Introduction

Solar energy possesses a high rank among all renewable energy sources owing to the fact that it enjoys the greatest accessibility. Solar energy is regarded as a clean source for supplying the energy demand in the current and future of the world as it does not bring perilous pollutants and gasses to the environment (Khorasaninejad and Hajabdollahi 2014; Casares et al. 2014). Although 99.9 % of the extraterrestrial solar energy is conveyed between the wavelengths of zero and 8 µm, due to atmospheric attenuation of solar radiation, the wavelength of terrestrial solar radiation that is important to solar energy applications is between 0.29 and 2.5 μm (Duffie and Beckman 2006). During the recent decades there have been numerous global efforts toward solar energy harnessing via various technologies. In fact, the growth of solar energy industry has been significant recently. Also, the number of global policies and targets to support the development of solar energy technologies is increasing rapidly. Nevertheless, prior to performing any attempt for installation of a solar energy system, precise and reliable knowledge of solar radiation data is of indispensable importance for solar experts (Coskun et al. 2011; Boland et al. 2013; Diagne et al. 2013; Yacef et al. 2014).

Accessibility to the measured solar radiation data is a fundamental prerequisite to simulate, optimize and monitor the performance of different solar energy systems including solar water collectors, solar air collectors, photovoltaic panels and etc. Unfortunately, owing to scarcity of instruments as well as fiscal issues the solar radiation data are scarce in many locations across the globe, particularly in developing countries (Benmouiza and Cheknane 2013). Thus, the development of proper models for accurate prediction of solar radiation is a must.

In the absence of measured data, the estimated solar radiation predicted by such developed models can be utilized for short or long-term simulation and performance evaluation of different solar systems. On this account, a large number of models have been developed so far to estimate global solar radiation using different input elements mainly meteorological parameters. If proper input parameters are considered, the models’ capability would be generally favorable. Nevertheless, solar radiation in a particular region is generally functions of the geographical position and the climatic conditions. In fact, the regression constants of the models obtained by measured solar radiation data of a particular location are dependent upon many factors such as the solar radiation characteristic, climatic conditions and latitude of that location. Thus, the solar radiation model developed or established for a specific location can only be utilized to estimate solar radiation at the considered location and its neighboring region only if the details of its weather conditions are similar. Otherwise, application of such models is not viable, owing to the large errors attained compared to the real solar radiation data. In such cases, utilization of the unsuitable models in terms of either inappropriate inputs or unjust application will lead to incorrect design and inaccurate performance evaluation of solar energy systems, which subsequently eventuates in technical and economical fails of such systems.

In the realm of solar radiation models, the first model suggested for calculating the monthly mean daily global solar radiation on a horizontal surface is that of Angström (1924). The functional form of this model has been extensively utilized in so many studies. The preliminary suggested model correlated the ratio of monthly mean daily horizontal global solar radiation to its corresponding value in a clear day with the ratio of monthly mean daily relative sunshine duration. However, the fundamental difficulty associated was attaining the correct value of global solar radiation in a clear day. To overcome this defectiveness, Prescott (1940) recommended utilizing the extraterrestrial horizontal global solar radiation instead of the clear day solar radiation. The achieved model globally known as Angström–Prescott model is:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) $$

(1)

where the regression coefficients of a and b can be obtained by finding the best fit between ${{\bar{H}} \mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}} \right. \kern-0pt} {\bar{H}_{o} }}$ and ${{\bar{n}} \mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}} \right. \kern-0pt} {\bar{N}}}$ via the regression analysis.

It is worthwhile to state that under overcast sky condition the (${{\bar{n}} \mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}} \right. \kern-0pt} {\bar{N}}}$) is equal to zero. Furthermore, the regression coefficients of “a” and ‘‘b”, respectively, illustrate the overall atmospheric transmissions compared to that of total cloudy conditions and the rate of increasing (${{\bar{H}} \mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}} \right. \kern-0pt} {\bar{H}_{o} }}$) with (${{\bar{n}} \mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}} \right. \kern-0pt} {\bar{N}}}$). Technically, the sum of regression coefficients, (a + b), can be inferred as the clear transmissivity (${{\bar{n}} \mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}} \right. \kern-0pt} {\bar{N}}} = 1$). In general, the variability of these coefficients is contingent upon many factors including the geographical location and the atmospheric conditions.

The Angström–Prescott model has been the subject of regular alterations to enhance its preciseness and validity. These modifications involve introduction of extra variables such as meteorological and geographical parameters (Glover and McCullogh 1958; Garg and Garg 1982; Ojosu and Komolafe 1987; Halouani et al. 1993; Ododo et al. 1995; Elagib and Mansell 2000; Jin et al. 2005; Demirhan 2014), increasing the order of relative sunshine duration (${{\bar{n}} \mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}} \right. \kern-0pt} {\bar{N}}}$) (Ogelman et al. 1984; Bahel et al. 1987; Akinoglu and Ecevit 1990; Tasdemiroglu and Sever 1991; Ulgen and Hepbasli 2002) or utilizing different functional forms (Newland 1988; Almorox and Hontoria 2004; Bakirci 2009). Developing new regression coefficients for the Angström–Prescott model for utilization in a particular region is of little significance compared with trying to modify it. Nonetheless, it is unquestionably debatable whether or not it is rewarding to introduce more input parameters to the traditional Angström–Prescott for the sake of enhancing the prediction precision. In fact, there are two contradictory issues: achieving more accuracy versus losing the model’s simplicity and convenience. On this ground, several investigations have been carried out to realize whether the newly modified models enjoy more capability than the traditional Angström–Prescott model. However, reviewing the literature reveals that due to location dependency of global solar radiation estimation there is no chance of making a firm conclusion.

This study aims at evaluating the influence of introducing several metrological parameters to the original Angström–Prescott model for the sake of accurately predicting the monthly mean daily horizontal global solar radiation. The south coastal region of Iran is nominated as the case study due to its great solar energy potential, its particular weather conditions as well as specific importance in the country. For this purpose, among all major cities in this region, the two port cities of Bandar Abbas and Jask are nominated, since the long-term measured daily global solar radiation on a horizontal surface is available for them. Seven models previously presented in the literature are nominated to recognize the notability of some meteorological variables such as average, maximum and minimum ambient temperatures, average and maximum relative humidity as well as water vapor and the sea level pressures. These variables have already been introduced by different researches to enhance the exactness of Angström–Prescott model. Several widely applied statistical indicators consisting of the absolute percentage error (APE), the mean absolute percentage error (MAPE), the mean absolute bias error (MABE), the root mean square error (RMSE) and the correlation coefficient (R) are utilized for appraising the models’ performances.

Models review

In this section, the nominated models from the literature are introduced and reviewed briefly. The first one is the well-known Angström–Prescott model and the other seven models have been chosen among numerous existing models based upon some specific criteria. These criteria include simplicity of the functional form, accessibility ease of the meteorological parameters utilized as inputs and successful history in estimating the global solar radiation in other locations across the globe.

Model 1: Angström–Prescott model

The well-known Angström–Prescott model was thoroughly defined in the introduction section. Its functional form proposed initially by Angström (1924) and modified later by Prescott (1940) is:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) $$

(1)

Model 2: El-Sebaii et al. model

El-sebaii et al. (2009) using long-term measured data for city of Jeddah, situated in Saudi Arabia, proposed a new correlation by adding the average ambient temperature to the original Angström–Prescott model as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{T}_{\text{avg}} $$

(2)

Model 3: El-Sebaii et al. model

By introducing the average relative humidity to the Angström–Prescott model, the following correlation was proposed by El-Sebaii et al. (2009) for calculating the monthly mean horizontal global solar radiation in the city of Jeddah, Saudi Arabia, as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{R}_{h} $$

(3)

Model 4: Abdalla model

To estimate the monthly mean global solar radiation in Bahrain, Abdalla (1994) related the global solar radiation to not only the relative sunshine duration but also the average ambient temperature and the relative humidity as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{T}_{\text{avg}} + d\,\bar{R}_{h} $$

(4)

Model 5: Ojosu and Komolafe model

Ojosu and Komolafe (1987), by utilizing different meteorological parameters including relative sunshine duration, average minimum and maximum ambient temperatures as well as average and maximum relative humidity, suggested the following correlation for estimation of global solar radiation in south western area of Nigeria as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\left( {{\raise0.7ex\hbox{${\bar{T}_{min}}$} \!\mathord{\left/ {\vphantom {{\bar{T}} {\bar{T}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{T}_{max}}$}}} \right) + d\left( {{\raise0.7ex\hbox{${\bar{R}_{h}}$} \!\mathord{\left/ {\vphantom {{\bar{R}} {\bar{R}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{R}_{h,max}}$}}} \right) $$

(5)

Model 6: Ododo et al. model

Ododo et al. (1995), using available data for nine stations located in different geographical and climatic zones in Nigeria, recommended a new multiple variables model. For prediction of global solar radiation in the nominated zones, this model emphasizes on the importance of maximum ambient temperature and maximum relative humidity as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{T}_{\hbox{max} } + d\,\bar{R}_{h} + e\bar{T}_{\hbox{max} } \left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) $$

(6)

Model 7: Abdalla model

Abdalla (1994) added the sea level pressure divided by the water vapor pressure to the model 4 and recommended the following model for Bahrain as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{T}_{\text{avg}} + d\,\bar{R}_{h} + e\left( {{\raise0.7ex\hbox{${\bar{P}}$} \!\mathord{\left/ {\vphantom {{\bar{P}} {\bar{P}_{V} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{P}_{V} }$}}} \right) $$

(7)

Model 8: Trabe and Shaltout model

To enhance the global solar radiation prediction in five locations in Egypt, Trabe and Shaltout (2000) modified the Abdalla model, expressed by Eq. (7), and introduced the water vapor pressure as a variable as:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = a + b\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + c\,\bar{T}_{\text{avg}} + d{\kern 1pt} {\kern 1pt} \bar{P}_{V} + e\,\bar{R}_{h} + f\left( {{\raise0.7ex\hbox{${\bar{P}}$} \!\mathord{\left/ {\vphantom {{\bar{P}} {\bar{P}_{V} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{P}_{V} }$}}} \right) $$

(8)

In the proceeding nominated models a, b, c, d, e and f are the regression coefficients. The parameters $\bar{H}$, $\bar{H}_{o}$, $\bar{n}$, $\bar{N}$, $\bar{T}_{\text{avg}}$, $\bar{T}_{\hbox{min} }$, $\bar{T}_{\hbox{max} }$, $\bar{R}_{h}$, $\bar{R}_{h,\hbox{max} }$, $\bar{P}$ and $\bar{P}_{V}$ presented in Eqs. (1)–(8) have been defined in the nomenclature.

The monthly mean daily extraterrestrial solar radiation on a horizontal surface ($\bar{H}_{o}$) expressed by Duffie and Beckman (2006) is:

$$ \bar{H}_{o} = \frac{24 \times 3600}{\pi }G_{sc} \left( {1 + 0.033\cos {\mkern 1mu} \frac{{360n_{day} }}{365}} \right) \times \left( {\cos \varphi {\mkern 1mu} \cos \delta {\mkern 1mu} \sin \omega_{s} + \frac{{\pi \omega_{s} }}{180}{\mkern 1mu} \sin \varphi \sin \delta } \right) $$

(9)

where G _sc is the solar constant assumed equal to 1367 W/m² and n _day is the average day of each month (Duffie and Beckman 2006).

δ and ω _s are the monthly mean daily solar declination and sunset hour angle, respectively, defined as (Duffie and Beckman 2006):

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

(10)

$$ \omega_{s} = \cos^{ - 1} \left( { - \tan \varphi \,\tan \delta } \right) $$

(11)

The monthly mean daily maximum possible sunshine duration is (Duffie and Beckman 2006):

$$ \bar{N} = \frac{2}{15}\cos^{ - 1} \left( { - \tan \varphi \,\tan \delta } \right) $$

(12)

Locations description, data collection and refinement

The nominated south coastal region of Iran enjoys great potential of solar energy, particular climate conditions as well as significant strategic and mercantile position in the country. Among all major cities in the region, the two port cities of Bandar Abbas and Jask were selected, as for them the long-term measured daily global solar radiation on a horizontal surface was available. Figure 1 illustrates the map of Iran including the locations of Bandar Abbas and Jask in the Hormozgan province. Port of Bandar Abbas, the capital of the Hormozgan province, is situated in the southern part of the country on the Persian Gulf coast at geographical location of 27°13/N and 56°22/E, and its elevation is 9.8 m above the sea level. Port of Jask, also one of the major cities of the Hormozgan province, is situated in south eastern part of the province on the coast of Gulf of Oman at 25°38/N and 57°46/E, and its elevation is 5.2 m above the sea level. Long warm season and short cool season are the climatic characteristics of the region. Basically the region is a desert zone with extremely low level of atmospheric precipitation. More information regarding the climatic conditions of Bandar Abbas and Jask are offered in “Bandar Abbas” and “Jask”, respectively.

Fig. 1

Map of Iran illustrating the south coastal cities of Bandar Abbas and Jask (Hormozgan province has been highlighted)

The long-term daily global solar radiation on a horizontal surface and sunshine duration as well as some other meteorological parameters including minimum, average and maximum ambient temperatures, average and maximum relative humidity, sea level pressure and water vapor pressure provided by the Iranian Meteorological Organization (IMO) for Bandar Abbas and Jask was utilized.

The precision of the models which are established is mainly influenced by the quality of raw data utilized. The data cleaning procedure generally aim at enhancing the data quality by checking and filtering them from any uncertainty or erroneous. In horizontal global solar radiation data used in this study, there were differences in the number and distributions of the years for which data existed for Bandar Abbas and Jask. Moreover, the solar data for these cities contained some missing and incorrect values. The missing values related to some days, months and sometimes years that data have not been recorded and incorrect values related to possible malfunction of instruments. In the present study to refine the data the following procedure was applied:

• 1. To recognize which global solar radiation values were not correct, the concept of daily clearness index was used as a benchmark. For this purpose all daily values which produced a daily clearness index (K _T) out of range of 0.015 < K _T < 1 were eradicated (Gani et al. 2015).

• 2. A global solar radiation data collection for a month involving more than 5 days missing or incorrect values was fully withdrawn. Furthermore, for a month involving less than 5 days missing or incorrect values, the proper values, obtained using interpolation, were substituted (Gani et al. 2015).

• 3. To attain more precise and consistent data sets for the omitted months, all of the sunshine duration as well as other meteorological parameters were also excluded from the data series.

The available data sets for this study were divided into two parts, one part for establishing the models and the other for evaluating them. The establishing dataset was used to build and calibrate the models, whereas the evaluating dataset was utilized to measure the capability of the models and to provide a basis for making comparisons among them. Table 1 presents the considered time periods for each city including the number of years and months from which data sets provided by IMO were used to establish and evaluate the models.

Table 1 The considered time periods of data sets utilized for establishing and evaluating the models

After completing the data refinement step, an average was obtained for each day of the year using the measured daily values related to different years in each data series. Then the achieved daily values of each month were averaged over the corresponding months to obtain the monthly mean daily values.

Statistical indicators

The absolute percentage error (APE), the mean absolute percentage error (MAPE), the mean absolute bias error (MABE), the root mean square error (RMSE) and the correlation coefficient (R) have been nominated as widely applied statistical indicators to evaluate and compare the models’ performance. These indicators are described briefly in the following.

The APE shows the absolute relative deviation between the calculated and the measured values. It is utilized to express the relative agreement between the measured and estimated values in percent. The APE is expressed as:

$$ {\text{APE}} = \left| {\frac{{\bar{H}_{i,c} - \bar{H}_{i,m} }}{{\bar{H}_{i,m} }}} \right| \times 100 $$

(13)

where $\bar{H}_{i,c}$ is the ith calculated value and $\bar{H}_{i,m}$ is the ith measured value.

The MAPE represents the mean absolute percentage deviation between the calculated and measured values and the MABE gives the mean absolute value of bias error. The MAPE and MABE values are indications of the goodness of the models given, respectively, as:

$$ {\text{MAPE}} = \frac{1}{x}\sum\limits_{i = 1}^{x} {\left| {\frac{{\bar{H}_{i,c} - \bar{H}_{i,m} }}{{\bar{H}_{i,m} }}} \right| \times 100} $$

(14)

and

$$ {\text{MABE}} = \frac{1}{x}\sum\limits_{i = 1}^{x} {\left| {\bar{H}_{i,c} - \bar{H}_{i,m} } \right|} $$

(15)

where x is the total number of observations.

The RMSE gives information on the short-term performance of the models, by allowing a term by term comparison of the actual deviation between the calculated and measured values, and its value is always positive. The RMSE is defined as:

$$ {\text{RMSE}} = \sqrt {\frac{1}{x}\sum\limits_{i = 1}^{x} {\left( {\bar{H}_{i,c} - \bar{H}_{i,m} } \right)}^{2} } $$

(16)

The correlation coefficient, R, used to test the linear relationship between the calculated and the measured values is defined as:

$$ R = \frac{{\sum\nolimits_{i = 1}^{x} {\left( {\bar{H}_{i,c} - \bar{H}_{{c,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}} } \right) \times } \left( {\bar{H}_{i,m} - \bar{H}_{{m,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}} } \right)}}{{\sqrt {\left[ {\sum\nolimits_{i = 1}^{x} {\left( {\bar{H}_{i,c} - \bar{H}_{{c,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}} } \right)^{2} } } \right]\left[ {\sum\nolimits_{i = 1}^{x} {\left( {\bar{H}_{i,m} - \bar{H}_{{m,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}} } \right)^{2} } } \right]} }} $$

(17)

where $\bar{H}_{{c,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}}$ and $\bar{H}_{{m,{\text{a}}{\kern 1pt} {\text{v}}{\kern 1pt} {\text{g}}}}$ are the averaged calculated and measured values, respectively.

It should be mentioned that the smaller values of APE, MAPE, MABE and RMSE show more accuracy of the predictions and in an ideal case they are zero. The R takes values between −1 and +1. While its values around −1 or +1 show perfect linear relationship between the measured and the calculated values, a value around zero indicates the absence of the linear relationship.

Results and discussion

By applying the statistical regression technique (SRT) and based on the long-term data from the establishing data series, the new regression coefficients of the selected models were developed for Bandar Abbas and Jask. The obtained regression coefficients are presented in Table 2. Although Bandar Abbas and Jask have a general similar climate, the regression coefficients of the models for Bandar Abbas differ significantly from those achieved for Jask, particularly for the models with higher number of meteorological parameters. This is due to possible different timing distribution of meteorological variables and either overcast or cloudy hours in Bandar Abbas and Jask. Accordingly, the models do not behave the same for these two nominated cities. Likewise, for the original Angström–Prescott model (model 1), the regression coefficients of “a” and “b” in Bandar Abass and Jask are different. However, for Bandar Abbas and Jask the sum (a + b) takes the close values of 0.6327 and 0.6262, respectively. It should be noted that the sum (a + b) represents the clear-sky atmospheric transmissivity; thus can be used to estimate the clear-sky global solar radiation. Bandar Abbas and Jask are located in the same climate zone and enjoy the same atmospheric conditions, hence have very close global solar radiation.

Table 2 Regression coefficients of the models for Bandar Abbas and Jask

A comparison between the predicted values of global solar radiation by the models (1–8) and the averaged measured global solar radiation values from the evaluation date series was conducted and the corresponding statistical indicators of MAPE, MABE, RMSE and R are presented in Table 3. For the nominated cities of Bandar Abbas and Jask, the best models among all are the models 8 and 6, respectively, which have been introduced in Table 3 in bold.

Table 3 Statistical indicators of the models for Bandar Abbas and Jask

To evaluate the influence of introducing different meteorological parameters properly and compare the models’ capability for Bandar Abbas and Jask, in the following subsections after a brief description about climatic conditions of each city the achieved results are analyzed and discussed separately.

Bandar Abbas

For Bandar Abbas, the long-term monthly mean daily terrestrial and extraterrestrial horizontal global solar radiations, the sunshine duration and the other meteorological parameters utilized as well as the calculated monthly mean daily maximum possible sunshine duration are presented in Fig. 2. For Bandar Abbas, the long-term monthly average ambient temperature lies between 17.4 and 34.3 °C. The difference between the monthly mean maximum and minimum temperatures is in the range of 6.6 and 12 °C. The monthly mean relative humidity values are between 60.4 and 72.4 % and the difference between the maximum and average relative humidity in different months varies between 13.6 and 22.2 %. Also, the monthly mean sunshine duration varies between 6.9 and 10.7 h. The sea level and water vapor pressures are very high in all months and they vary between 996.6 and 1018.0 mb and between 13.3 and 36.9 mb, respectively. Based upon Köppen classification, the climate condition of Bandar Abbas is categorized as BWh which relates to arid desert hot (Kottek et al. 2006). The daily horizontal global solar radiation for Bandar Abbas is in the range of 10.01–23.81 MJ/m². Also, the city enjoys 3152 sunny hours throughout the year.

Fig. 2

The monthly mean daily global solar radiation, sunshine hours and other meteorological parameters as well as the maximum possible sunshine hours and extraterrestrial solar radiation for Bandar Abbas

According to the results presented in Table 3, for predicting the monthly mean global solar radiation on a horizontal surface in Bandar Abbas, all models except model (2) perform better than the original Angström–Prescott. Therefore, introducing the average ambient temperature solely does not result in improving the preciseness of the Angström-Prescott model in Bandar Abbas. Although the model (8), in which the average ambient temperature, water vapor pressure, relative humidity and the ratio of sea level pressure to water vapor pressure have been added as variables, performs best, nonetheless its comparative goodness needs to be evaluated. For this purpose, in Fig. 3 the predictions of the Angström-Prescott model (model 1) have been compared with the predictions of the model (8) as well as the long-term measured data for Bandar Abbas. As seen in Fig. 3, the differences between the predictions of these two models compared with the measured data are not significant such that the APE of the model (1) varies between 1.57 % in August and 4.61 % in December while the APE of the model (8) varies between 0.31 % in February and 4.95 % in December. Moreover, the highest APE deference between the predictions of the model (8) and those of the model (1) is 2.26 % for January and the lowest is 0.23 % for September. Also, the average of all APE differences between those of the models (1) and (8) is 0.84 % and there are only small differences as for the other statistical indicators. Accordingly, it can be concluded that the model (8), as a complex multi-linear model, which utilizes seven different meteorological parameters, is less applicable compared to the simple and linear Angström–Prescott model. In fact, if greater precision with less complexity is required the model (3) may be employed, as it provides a little extra accuracy with only one more input parameter of relative humidity.

Fig. 3

Global solar radiation predicted by the model (1) compared with those of the model (8) and the averaged long-term measured values for Bandar Abbas

To verify the accuracy and applicability of the models established for Bandar Abbas, the proposed best model of Behrang et al. (2011), developed using PSO (particle swarm optimization) technique, was tested. Behrang et al. (2011) model is:

$$ {\raise0.7ex\hbox{${\bar{H}}$} \!\mathord{\left/ {\vphantom {{\bar{H}} {\bar{H}_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{H}_{o} }$}} = 0.7426 - 0.7684\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right) + 0.7559\left( {{\raise0.7ex\hbox{${\bar{n}}$} \!\mathord{\left/ {\vphantom {{\bar{n}} {\bar{N}}}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\bar{N}}$}}} \right)^{2} $$

(18)

Based on the evaluating data series of this study, the statistical indicators for this model are: MAPE = 5.7656 %, MABE = 0.9474 MJ/m², RMSE = 0.9896 MJ/m² and R = 0.9967. Comparing these statistical indicators with those of the established models, presented in Table 3, reveals that all of the models established in this study, including the Angström–Prescott, perform better than the best model of Behrang et al. (2011). Consequently, even without any improvement, the Angström–Prescott model established in this study appears qualified for prediction of horizontal global solar radiation in Bandar Abbas. The superiority in predication of global solar radiation achieved by the established models of this study for Bandar Abbas compared to the Behrang et al. (2011) model may be due to the data cleaning procedure conducted to filter the data from any uncertainty or erroneous.

Jask

Figure 4 illustrates the monthly mean daily horizontal global solar radiation and extraterrestrial horizontal global solar radiations, the sunshine duration and the other meteorological parameters utilized as well as the calculated monthly mean daily maximum possible sunshine hours in Jask. The long-term monthly average ambient temperature is in the range of 20.4 and 32.2 °C. The difference between the monthly mean maximum and minimum temperature varies between 2.6 and 6.9 °C. The monthly mean relative humidity varies between 59 and 82.3 %, and the difference between the maximum and average relative humidity in different months varies between 5.0 and 10.8 %. Also, the monthly mean sunshine duration takes values from 7.1 to 10.0 h. The sea level pressure and water vapor pressure, which are really high in different months of the year, are in the range of 997.1–1017.4 mb and 15.0–38.3 mb, respectively. For Jask, the weather condition in terms of Köppen classification is also categorized as BWh which relates to arid desert hot (Kottek et al. 2006). The monthly mean daily horizontal global solar radiation varies between 11.7 and 23.5 MJ/m². Jask enjoys from 3126 sunny hours throughout the year. Making a comparison between the weather conditions of Bandar Abbas and Jask demonstrates that, although these cities are located in the same province and enjoy from a general similar climate conditions based upon Köppen classification, there are some differences in the detail of their weather conditions, particularly in terms of their ambient temperature and relative humidity.

Fig. 4

The monthly mean daily values of global solar radiation, sunshine hours and other meteorological parameters as well as the maximum possible sunshine hours and extraterrestrial solar radiation in Jask

The results presented in Table 3 for Jask reveal that all models except model (3) have the ability of predicting the monthly mean horizontal global solar radiation more accurate than the original Angström–Prescott model. In fact, the model (3), in which along with the sunshine hours the relative humidity plays role as a variable, shows poorer performance compared to the traditional Angström–Prescott model. As a consequence, the relative humidity solely does not result in improving the precision of the Angström–Prescott model applied for Jask. Statistical indicators presented in Table 3 demonstrate that for Jask the model (6) shows the best performance among all of the models studied. Therefore, it is observed that introducing the combination of maximum ambient temperature, relative humidity as well as the product of maximum ambient temperature and sunshine hours to the original Angström–Prescott model eventuates in further exactness. However, to show the level of its goodness, in Fig. 5 the predictions of Angström–Prescott model (model 1) have been compared with the predictions of the model (6) as well as the long-term measured data in Jask. As seen from Fig. 5, the differences between the estimated values by the models and the measured data are not considerable such that the APE for the model (1) varies between 0.49 % for July and 5.16 % for December, whereas APE of the model (6) varies between 0.03 % for August and 4.94 % for June. In addition, the APE differences for the model (6) and those of the model (1) vary between 0.23 % for September and 2.26 % for January and the average of all differences is 0.84 %. Similar to the conclusion extracted for Bandar Abbas it is visible that, despite some improvements achieved, the model (6) is not really appealing in practice, because of its more complexity and its insignificant precision enhancement compared to the Angström–Prescott model.

Fig. 5

Global solar radiation predicted by the model (1) compared with those of the model (6) and the averaged long-term measured values for Jask

To check the performance and ability of the established models in this study for Jask, the suggested model by Behrang et al. (2011), developed using PSO (particle swarm optimization) technique especially for Jask, was tested. The Behrang et al. (2011) model is:

$$ {\raise0.7ex\hbox{$H$} \!\mathord{\left/ {\vphantom {H {\mathop H\nolimits_{o} }}}\right.\kern-0pt} \!\lower0.7ex\hbox{${\mathop H\nolimits_{o} }$}} = 0.4511 - 1.0081\cos \left( {0.5134\left( {{\raise0.7ex\hbox{$n$} \!\mathord{\left/ {\vphantom {n N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right) + 0.5967\sin \left( {0.5134\left( {{\raise0.7ex\hbox{$n$} \!\mathord{\left/ {\vphantom {n N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right) + 0.5770\cos \left( {1.0268\left( {{\raise0.7ex\hbox{$n$} \!\mathord{\left/ {\vphantom {n N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right) + 0.5272\sin \left( {1.0268\left( {{\raise0.7ex\hbox{$n$} \!\mathord{\left/ {\vphantom {n N}}\right.\kern-0pt} \!\lower0.7ex\hbox{$N$}}} \right)} \right) $$

(19)

On the basis of the evaluating data series utilized for this study, the statistical indicators for this model are MAPE = 8.0281 %, MABE = 1.5049 MJ/m², RMSE = 1.6786 MJ/m² and R = 0.9878. By making a comparison between these values and the attained values for Jask reported in Table 3, it is found that all of the models, including the Angström–Prescott model, show a higher capability compared to the model recommended by Behrang et al. (2011). Thus, the simple Angström–Prescott model calibrated for this study is capable to predict the monthly mean global solar radiation on horizontal surface in Jask with a satisfactory level of precision. It should be mentioned that the dominance of the models established in this study for Jask over the Behrang et al. (2011) model may be owing to the quality of the refined data utilized in this study.

Conclusion

The Angström–Prescott model, as a widely used global solar radiation model, has been the subject of different modifications so far. In this research, to recognize the notability of various meteorological parameters such as the average, maximum and minimum ambient temperatures, the average and maximum relative humidity as well as the water vapor and sea level pressures to boost the performance of the Angström–Prescott models, seven proposed models were chosen from the literature. As a case study, the long-term measured global solar radiation on a horizontal surface as well as other meteorological parameters provided by the Iranian Meteorological Organization (IMO) for two south coastal cities of Iran, named Bandar Abbas and Jask was utilized. Using the long-term data and via statistical regression technique (SRT), the new regression coefficient was established for all of the models. The performance and enhancement level of the models were examined via commonly used statistical indicators of the absolute percentage error (APE), the mean absolute percentage error (MAPE), the mean absolute bias error (MABE), the root mean square error (RMSE) and the correlation coefficient (R). The results disclosed that, although these cities both have BWh climate type, due to some differences in detail of the weather conditions the models performed differently and the attained regression constants for these cities differed vastly. Also, the best model recognized for each city, in terms of the variables utilized, was different. For Bandar Abbas the model 8 and for Jask the model 6, utilizing several meteorological parameters, showed the best performance. Nonetheless, owing to only small differences in predictions of these models compared to the original Angström–Prescott model and more complexity associated with them their applications in practice did not seem truly attractive. Furthermore, the results illustrated that the average ambient temperature and the relative humidity solely are not important parameters for prediction of global solar radiation in Bandar Abbas and Jask, respectively. By comparing the performance of the established models of this study for Bandar Abbas and Jask with two previously proposed models the superiority of the models of this study, particularly the Angström–Prescott model, was confirmed; as a consequence, even without any modification the Angström–Prescott model is totally eligible to estimate the monthly mean daily global solar radiation in the both nominated cities and their neighboring.