Theoretical prediction and validation of global horizontal solar irradiance for a tropical climate in India

Abstract

This paper aims to propose monthly models responsible for the theoretical evaluation of the global horizontal irradiance of a tropical region in India which is Sivagangai situated in Tamilnadu. The actual measured global horizontal irradiance hails from a 5 MW solar power plant station located at Sivagangai in Tamilnadu. The data were monitored from May 2011 to April 2013. The theoretical assessment was conducted differently by employing a programming platform called Microsoft Visual Basic 2010 Express. A graphical user interface was created using Visual Basic 2010 Express, which provided the evaluation of empirical parameters for model formulation such as daily sunshine duration (S), maximum possible sunshine hour duration (S₀), extra terrestrial horizontal global irradiance (H₀) and extra terrestrial direct normal irradiance (G₀). The proposed regression models were validated by the significance of statistical indicators such as mean bias error, root mean square error and mean percentage error from the predicted and the actual values for the region considered. Comparison was made between the proposed monthly models and the existing normalized models for global horizontal irradiance evaluation.

Keywords

global horizontal irradiance (GHI) / mean bias error / root mean square error / mean percentage error / coefficient of regression / Visual Basic 2010 Express

Introduction

The knowledge on projections and forecast of solar radiation distribution over the globe forms an essential tool for choosing the exact location of the solar energy distribution system by which it runs efficiently in gaining the maximum extraction of solar energy output from the sun. In addition, the evolution of solar atlas with radiation measurements across the world is also made easily available. Furthermore, the forecast of solar radiation is a vital indicator for economic policy formulation. Nevertheless, the solar irradiation measurements are not always available for all the developing countries because of the lack of investment in precise measurement devices which help the measurement of global, direct and diffuse irradiations reaching the surface of the earth. Hence it becomes necessary to develop models for theoretical assessment of radiation measurements from the most available metrological data [ 1]. India is bestowed with an overall annual average of direct normal irradiance of up to 4.5 kWh/(m²·d) from 2002 to 2013, out of which Sivagangai measures individually in the range of 5.0 to 5.5 kWh/(m²·d). Similarly the annual average of global horizontal irradiance (GHI) for Sivagangai amounts approximately to 5.1 kWh/(m²·d). According to the Koppen classification, the Sivagangai district is of tropical semi arid climate where the rate of evaporation of water is higher than that of moisture received through precipitation.

Researchers have developed models based on astronomical, geometrical, geographical, physical and metrological factors which are cleverly classified into sunshine-based models, cloud-based models, temperature-based models and other metrological models [ 2] for the prediction of GHI for a particular location under study. Sunshine-based models employ bright sun shine hours for calculating the daily GHI. The basic sun shine model was given by Angstrom in 1924 [ 3] which was further modified by Prescott in 1940 [ 4]. The modified quadratic model and the cubic models were introduced by Ögelman et al. in 1984 [ 5] and Samuel in 1991 [ 6]. Cloud cover prevents the solar radiation from reaching the surface of the earth and hence the GHI predicted is a function of the mean cloud cover in cloud-based models. Examples of the same model include the ones proposed by Paltridge and Proctor in 1976 [ 7], Daneshyar in 1978 [ 8], Samimi in 1994 [ 9], Badescu in 1999 [ 10], and Sabziparvar in 2008 [ 11]. Temperature-based models were evolved due to the scarce availability of cloud cover where the calculated GHI was a function of the minimum and maximum air temperature. Some of the models include the ones proposed by Bristow and Campbell in 1984 [ 12], Allen in 1997 [ 13], Thornton and Running in 1999 [ 14] and Almorox et al. in 2011 [ 15].

Other successful metrological models for GHI include the Swartman model [ 16] involving relative humidity, the Glover and McCulloch model [ 17] involving latitude and average sun shine hour duration (S/S₀), and the angular model suggested by Gopinathan [ 18] dealing with latitude, altitude and average sun shine hour duration. Models incorporating latitude, longitude, altitude and average sun shine hour duration were developed by Chen et al. [ 19]. Artificial neural network were also used for predicting GHI by AI-Alawi and AI-Hinai [ 20], Mohandess et al. [ 21] and Mubiru and Banda [ 22]. Though many recent models based on neural network [ 20- 22] and numerical weather prediction [ 23] give more prediction forecast of less than a day than regression models, the difference between the regression coefficient and root mean square error (RMSE) for the sunshine regression model and sunshine-based artificial neural network (ANN) model is as less as 0.16% and 1.7e^-3 respectively [ 24], proving better dependency of regression models. Additionally, the ANN and numerical weather prediction (NWP) methodology requires prior input knowledge base and training, spatial resolution and post filtering techniques respectively and implementation complexity. Thus there occurs a trade-off between the prediction accuracy and complexity. Furthermore, basic Angstrom-Prescott assessment models are not concentrated for southern regions of India, especially Tamilnadu, where a great amount of solar potential exists.

This paper deals with the development of modified Angstrom-based models for Sivagangai to estimate the monthly average daily GHI considering two years of data. The inputs to the model are programmed in Microsoft Visual Basic 2010 Express. Better comparison and validation is made by comparing the proposed monthly models with the existing models such as the Glover and McCulloch model [ 17], the Logarithmic model [ 25] and the Pandey model [ 26].

System measurements and methodology

The actual GHI was measured at the site of a 5 MW solar power plant in Sivagangai located at an altitude of 102 m along the latitudinal and longitudinal belt of 9.47°N and 78.87°E, respectively. A monitoring weather station is available at the site which further includes a pyranometer and a temperature sensor whose output port is connected to a SCADA or a data acquisition unit where the GHI is saved and further converted to usable excel (.xlsv) format. The GHI is monitored from May 2011 to April 2013.

Angstrom [ 3] first derived the theoretical model for the calculation of GHI which was further modified by Prescott [ 4]. Angstrom [ 3] predicted that the direct clearness index K_t which is the measure of measured GHI to the calculated extraterrestrial irradiance is a function of the average daily sunshine duration which is expressed as

(1) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=f\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)

The modified Angstrom model [ 4] is given by

(2) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=a+b\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)

The basic form of the quadratic and cubic model as given by Refs. [ 5] and [ 6]

(3) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=a+b\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)+c{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}

(4) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=a+b\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)+c{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+d{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}

where H refers to the monthly mean of the daily measured GHI, H₀ refers to the monthly mean of the daily calculated extra terrestrial horizontal radiation under the absence of atmosphere, S corresponds to the monthly mean of the daily calculated sunshine hour and S₀ is the monthly mean of the daily calculated maximum possible sunshine hour, and a and b are correlation constants.

The extra terrestrial horizontal global irradiance is given by

(5) H_{\mathrm{0}}={\displaystyle \frac{24}{\pi }}{G}_{\mathrm{0}}\left(\cos L\cos \delta \sin {\omega }_s+{\displaystyle \frac{\pi }{\mathrm{180}}}{\omega }_s\sin L\sin \delta \right),

(6) G_{\mathrm{0}}=I_{\mathrm{sc}}\left(\mathrm{1}+\mathrm{0.34}\cos {\displaystyle \frac{\mathrm{360}{D}_n}{\mathrm{365.25}}}\right),

where L represents the latitude of the location, δ represents the declination angle, ω_s represents the sunset hour angle in degrees, and D_n denotes the day of the year.

(7) \delta =\mathrm{23.45}\sin {\displaystyle \frac{\mathrm{360}(D_n+\mathrm{284})}{\mathrm{365}}},

(8) {\omega }_s={\cos }^{-\mathrm{1}}(-\tan \delta \cdot \tan L),

(9) S_{\mathrm{0}}={\displaystyle \frac{\mathrm{2}}{\mathrm{15}}}{\omega }_s.

The daily sunshine can be calculated by [ 27]

(10) S={\displaystyle \frac{h}{\mathrm{360}}}\arccos \left(\tan L\tan \mathrm{23.5}\cos \left({\displaystyle \frac{\mathrm{360}{D}_n}{\mathrm{365.25}}}\right)\right),

where h represents a day in hours (24 h); L corresponds to the latitude of the monitored site.

The site lies in the latitude and longitudinal range of 9.47°N and 78.87°E at an altitude of 102 m respectively. The empirical constants a, b and c vary with respect to the location. They are much affected by the air pollution produced by urban activity. The values of empirical constants are derived from the direct clearness index and the sunshine hour, which will be described in sections below.

Determination of empirical inputs to estimate GHI

The determination of inputs such as sunshine hour (S), declination angle (δ), hour angle (ω_s), extra terrestrial direct normal (G₀), global irradiance (H₀), and maximum sunshine hour (S₀) were computed by developing a graphical user interface which makes it possible to store the data with ease in Visual Basic 2010 Express. Equations (5)–(10) were used in programming the necessary output, which is finally the extra terrestrial GHI (H₀). The programming steps include declaration of set of class as Public Class, declaration of the private variables employed in the equation as Dim S as Double, employing Eqs. (5)–(10) and setting the output value to a Text Box as S = TextBox1·Text, and end class.

Figure 1 shows a debugged GUI after building the code in the code editor. Intermediate mathematical values can also be made available by assigning text box corresponding to the value to be stored.

Determination of empirical constants a, b and c for monitored period

Employ the linear Prescott Angstrom model as given in Eq. (2) which is of the linear form where

y={\displaystyle \frac{H}{H_{\mathrm{0}}}},x={\displaystyle \frac{S}{S_{\mathrm{0}}}}.

The variation of polynomial coefficients x and y in accordance to the model type employed for model formulation is given in Table 1. Apply the linear fit equation by the least square method in statistics, Eqs. (11) and (12) can be obtained.

(11) {\displaystyle \sum y=na+b{\displaystyle \sum x},}

(12) {\displaystyle \sum xy=a{\displaystyle \sum x+b{\displaystyle \sum x^{\mathrm{2}}}}}.

where n represents the number of data pairs which varies in accordance with the type of model proposed, such as monthly average and daily models. Yearly variation of two data sets forms a data pair. As monthly models are proposed for a period of 2 years (2 data sets) from 2011 to 2013, n is considered to be 1.

Thus, the monthly average GHI from 2011 to 2013 as monitored by the Kipp and Zonn pyranometer are averaged to obtain the monthly average daily GHI for the monitored period of two years as illustrated in Fig. 2 [ 28]. Then, on applying the above linear first order Eqs. (11) and (12) employing the measured GHI, we obtain the regression coefficients a and b for each month.

Proposed monthly correlations for the annual monitored period on employing Eqs. (11) and (12) are given as

(13) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{3.7128}-\mathrm{6.5433}\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)(\mathrm{for}\mathrm{January}),

(14) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.4212}+\mathrm{0.1220}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{February}),

(15) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.4024}+\mathrm{0.2722}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{March}),

(16) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.4141}-\mathrm{1.6406}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{April}),

(17) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.5469}+\mathrm{0.3204}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{May}),

(18) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=-\mathrm{73.2777}+\mathrm{148.22}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{June}),

(19) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{2.3878}-\mathrm{3.3469}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(for\; July),

(20) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{2.7144}-\mathrm{4.1258}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{August}),

(21) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=-\mathrm{0.2124}+\mathrm{1.6042}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{September}),

(22) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{2.0898}-\mathrm{3.2109}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{October})

(23) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.0661}-\mathrm{1.3140}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{November}),

(24) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{30.2653}-\mathrm{59.6734}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(\mathrm{for}\mathrm{December}).

Proposed second order quadratic equation model

Applying the statistical analysis to the quadratic model, Equation (25) can be obtained.

(25) y=a+bx+c{x}^{\mathrm{2}},

which replicates the actual quadratic model {\displaystyle \frac{H}{H_{\mathrm{0}}}}=a+b{\displaystyle \frac{S}{S_{\mathrm{0}}}}+c{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}.

On comparing the equations, Equation (26) is obtained,

(26) y={\displaystyle \frac{H}{H_{\mathrm{0}}}}\mathrm{a}\mathrm{n}\mathrm{d}x={\displaystyle \frac{S}{S_{\mathrm{0}}}}.

The regression constants are obtained from the least square method by employing Eqs. (27)–(29).

(27) {\sum }^{}y=na+b{\sum }^{}x+c{\sum }^{}{x}^{\mathrm{2}},

(28) {\sum }^{}xy=a{\sum }^{}x+b{\sum }^{}{x}^{\mathrm{2}}+c{\sum }^{}{x}^{\mathrm{3}},

(29) {\sum }^{}{x}^{\mathrm{2}}y=a{\sum }^{}{x}^{\mathrm{2}}+b{\sum }^{}{x}^{\mathrm{3}}+c{\sum }^{}{x}^{\mathrm{4}}.

The proposed quadratic monthly regression coefficient model for the annual monitored period is

(30) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.9934}-\mathrm{0.7888}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.5963}{\displaystyle \frac{S}{S_{\mathrm{0}}}}(for\; January),

(31) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.2700}+\mathrm{0.6500}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.4500}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(for\; February),

(32) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=-\mathrm{0.192}+\mathrm{0.9488}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.0512}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(for\; March),

(33) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.1208}+\mathrm{0.3433}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.2452}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(for\; April),

(34) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.5024}+\mathrm{0.3240}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.176}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{May}),

(35) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=-\mathrm{0.0433}+\mathrm{1.0521}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.0521}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{June}),

(36) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.6024}+\mathrm{0.1424}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.1424}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{July}),

(37) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.4816}-\mathrm{2.6316}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.3616}{\frac{S}{S_{\mathrm{0}}}}^{\mathrm{2}}(\mathrm{for}\mathrm{August}),

(38) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.0941}+\mathrm{0.8527}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.2842}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{September}),

(39) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.1493}-\mathrm{0.2239}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{2.2389}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{October}),

(40) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.0455}+\mathrm{0.8123}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.1877}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{November}),

(41) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.0944}+\mathrm{1.1696}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{1.1696}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}(\mathrm{for}\mathrm{December}).

Proposed third order or cubic model

The equation for the cubic model is represented as

(42) y=a+bx+c{x}^{\mathrm{2}}+d{x}^{\mathrm{3}},

which represents the actual cubic model given by

(43) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=a+b{\displaystyle \frac{S}{S_{\mathrm{0}}}}+c{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+d{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}.

The regression or empirical constants a, b, c and d are obtained by the least square method by taking Eqs. (44)–(47) into account.

(44) {\sum }^{}y=na+b{\sum }^{}x+c{\sum }^{}{x}^{\mathrm{2}}+d{\sum }^{}{x}^{\mathrm{3}},

(45) {\sum }^{}xy=a{\sum }^{}x+b{\sum }^{}{x}^{\mathrm{2}}+c{\sum }^{}{x}^{\mathrm{3}}+d{\sum }^{}{x}^{\mathrm{4}},

(46) {\sum }^{}{x}^{\mathrm{2}}y=a{\sum }^{}{x}^{\mathrm{2}}+b{\sum }^{}{x}^{\mathrm{3}}+c{\sum }^{}{x}^{\mathrm{4}}+d{\sum }^{}{x}^{\mathrm{5}},

(47) {\sum }^{}{x}^{\mathrm{3}}y=a{\sum }^{}{x}^{\mathrm{3}}+b{\sum }^{}{x}^{\mathrm{4}}+c{\sum }^{}{x}^{\mathrm{5}}+d{\sum }^{}{x}^{\mathrm{6}}.

Hence substituting the x and y in the above equations pertaining to each month, the monthly models proposed are derived as

(48) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.1505}-\mathrm{1.0067}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.7508}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}-\mathrm{0.0819}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{January}),

(49) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.3420}-\mathrm{0.3198}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.6015}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{1.2293}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{February}),

(50) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=-\mathrm{0.0376}-\mathrm{0.1130}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.3491}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{2.4449}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{March}),

(51) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.5392}-\mathrm{1.8911}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.4995}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{1.0000}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{April}),

(52) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.1896}+\mathrm{0.7658}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.2920}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{0.5578}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{May}),

(53) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.8154}-\mathrm{0.0952}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.4881}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}-\mathrm{1.1665}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{June}),

(54) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.3857}+\mathrm{0.1828}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.5526}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{0.7396}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{July}),

(55) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{1.2049}-\mathrm{1.7723}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{1.1121}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{0.3398}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{August}),

(56) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.7019}+\mathrm{0.0416}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.1908}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}-\mathrm{0.6141}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{September}),

(57) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{2.0914}-\mathrm{3.2158}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.0012}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{0.0046}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{October}),

(58) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.0848}+\mathrm{0.3007}{\displaystyle \frac{S}{S_{\mathrm{0}}}}+\mathrm{0.2159}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}+\mathrm{0.9151}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{November}),

(59) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.4123}+\mathrm{0.4503}{\displaystyle \frac{S}{S_{\mathrm{0}}}}-\mathrm{0.9222}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{2}}-\mathrm{0.1573}{\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)}^{\mathrm{3}}(\mathrm{for}\mathrm{December}).

As seen from Fig. 2, a slight deviation between the predicted and the actual GHI occurs for the proposed linear and quadratic model where as for the cubic model it lies exactly on the same line. The accuracy of the proposed models is yet proved by the coefficient of the best fit which is 0.999 for linear and quadratic models and exactly 1 for the cubic model. As seen from the proposed cubic model in Fig. 2, the error between the predicted and actual GHI vary from a minimum of –0.00063 kWh/(m²·d) to 0.00552 kWh/(m²·d) which reflects and justifies the value of regression coefficient for the same as stated above.

Statistical comparative analysis of validation

The accuracy or the closeness between the predicted and the actual GHI is evaluated by enumerating the statistical indicators such as mean bias error (MBE), RMSE, mean percentage error (MPE) and t-static value commonly available in Refs. [ 29- 32].

(60) \mathrm{MBE}={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum }_{i=\mathrm{1}}^N(H(\mathrm{pred})-H(\mathrm{meas}))

(61) \mathrm{RMSE}=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum }_{i=1}^N{\left(H\left(\mathrm{pred}\right)-H\left(\mathrm{meas}\right)\right)}^2}

(62) \mathrm{MPE}={\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum }_{i=\mathrm{1}}^N\left({\displaystyle \frac{H(\mathrm{pred})-H(\mathrm{meas})}{H(\mathrm{meas})}}\right)\times \mathrm{100}\%

(63) t-static=\left({\displaystyle \frac{(n-\mathrm{1}){(\mathrm{MBE})}^{\mathrm{2}}}{{(\mathrm{RMSE})}^{\mathrm{2}}-{(\mathrm{MBE})}^{\mathrm{2}}}}\right)

MBE gives the accurate information of the long-term performance of the model. A low MBE is always desired for better accuracy of the proposed model. A positive MBE shows an over-estimate while a negative MBE an under-estimate by the model [ 33]. The RMSE test gives the information of the short-term performance of the proposed model by allowing a term-by-term comparison of the actual deviation between the predicted and the measured GHI. Although the MBE and RMSE provide a reasonable methodology to compare models, the statistical significance of the model is objectively indicated from the measured counterpart [ 34]. The smaller the static t-static value is, the better the performatric accuracy of the prediction model is. To determine whether the estimates of the model are statistically significant, a critical t-static value has to be estimated from the standard statistical table, that is t_α/2 at α level of significance and (n–1) degrees of freedom which should be always greater than the calculated t-static value.

Results and discussion

Monthly monitored average results

Figure 2 shows a monthly average variation of the measured GHI for Sivagangai in the monitored period. The GHI varies from a minimum of 4.309 kWh/(m²·d) in December to a maximum of 5.942 kWh/(m²·d) in March. Figure 4 shows the average monthly average daily GHI in the total monitored period. Table 2 is a comparison between the measured GHI for different locations reviewed in the past and the present study. A lesser GHI results ultimately in the reduction of efficiency of solar photovoltaic and thermal systems.

The bright sunshine hour of a day is conventionally recorded by employing a sunshine recorder. But in this paper, it is calculated theoretically by employing Eq. (10). The seasonal change throughout the month affects the sunshine hours. The overall peak sunshine hour of 6.28 h occurs on July 2, 2011. The monthly ambient temperature and module temperature measured for the location of 9.47 N and 78.26 E are demonstrated in Fig. 5. The monthly average daily module temperature for the monitored period varies from a minimum of 39.11°C to a maximum of 46.63°C whereas the ambient temperature varies from 26.1°C to 30.3°C.

Module temperature [ 37] is calculated by

(64) T_m=T_a+(\mathrm{NOCT}-\mathrm{20}){\displaystyle \frac{G}{\mathrm{800}}}

T_m and T_a represent the module and ambient temperature, NOCT refers to the normal operating cell temperature, and G marks the instantaneous irradiation. The module temperature is always observed to be higher than the ambient temperature. The average maximum temperature of 46.49°C occurs in April.

As the day of the year progresses, the extra terrestrial GHI is found to increase at the start of the year and decreases further and increases again through the whole year as shown in Fig. 6.

The extra terrestrial GHI and sunshine hours are inversely proportional with respect to the day of the year. The ratio of measured GHI to the extra terrestrial GHI gives the value of clearness index represented as K_t . The measure of available solar energy is usually characterized by the clearness index. Yousif et al. [ 38] in 2013 proposed four different intervals of K_t which determines the clearness of the sky.

For cloudy condition,

The overall average of clearness index for the tropical climate of Sivagangai depicts the sky to be partly cloudy according to Yousif’s classification enumerating to 0.57. The K_t value showed a greater variation from the transition of summer to winter that is attributed to the increase in water vapor content which is the indication of the winter rain pattern in Tamilnadu. The variation of K_t with respect to the relative sunshine hour is displayed in Fig. 7.

Validation of proposed model for assessment of GHI

The statistical indicators such as MBE, RMSE and MPE are evaluated and compared with the prevalent monthly average models proposed for different locations under study in Table 3. A negative average MBE is observed for the proposed linear, quadratic and cubic model, suggesting an estimate of GHI to be less than the measured GHI. It is inferred from the above comparison that the proposed linear, quadratic and cubic models have a lower MBE, RMSE and MPE than the existing monthly average models.

In addition, the proposed models are compared with the existing models for assessment of GHI such as linear model (suggested for India), logarithmic or Newland model and multiple parameter linear model. First, the angular model which was derived from modified Angstrom models in which the horizontal irradiance is represented as a function of the latitudinal angle of the site under investigation proposed by Glower and McCulluch in 1958 is considered as

{\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.29}\cos \phi +\mathrm{0.52}{\displaystyle \frac{S}{S_{\mathrm{0}}}},

valid conditionally for where \phi represent the latitudinal angle of the particular location.

A theoretical logarithmic model was proposed by Newland in 1983 as

(65) {\displaystyle \frac{H}{H_{\mathrm{0}}}}=\mathrm{0.34}+\mathrm{0.40}\left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right)+\mathrm{0.17}\log \left({\displaystyle \frac{S}{S_{\mathrm{0}}}}\right).

A linear regression model for predicting the solar irradiance in India was proposed in 2013 [ 25] represented in Eq. (66),

(66) H=H_{\mathrm{0}}\left(\mathrm{0.1382}+\mathrm{0.5564}{\displaystyle \frac{S}{S_{\mathrm{0}}}}\right).

The GHI obtained from the angular, logarithmic linear regression model and proposed monthly models are compared with the measured GHI for validating the accuracy of the proposed model. The statistical indicators are also compared for the same which is aimed at concluding the agreement between the measured and the predicted GHI for the location under study.

It is observed from Table 4 that the proposed linear, quadratic and cubic models have remarkably better agreement with the actual or the measured GHI than the other compared models. The average RMSE ranges from 0.000923 kWh/(m²·d) to 0.0055 kWh/(m²·d). The negative MBE of the proposed models indicate an under estimation of GHI by the models. The average MPE of the proposed model ranges from –0.00467% to –0.0877%. Thus, the proposed model is much better for the location of Sivagangai than the Glover&McCullah, Newland and Pandey et al. models. The comparison of the proposed GHI obtained through empirical models and the actual GHI are also graphically compared in Fig. 8.

(67) \mathrm{Error}\mathrm{in}\mathrm{deviation}(e)=\left({\displaystyle \frac{(H(\mathrm{pred})-H(\mathrm{actual})}{H(\mathrm{actual})}}\right)\times \mathrm{100}

In this case the error in deviation [ 42- 44] of predicted estimates with the measured data was found to be the same as the MPE as the number of data pairs is 1. Thus the error in deviation is found to be within the acceptable range of –10% to+ 10% for the proposed models.

Conclusions

Thus the above work projects the theoretical evaluation of GHI. The empirical input parameters are programmed differently by employing Visual Basic 2010 Express. The modified Angstrom, quadratic and cubic models are proposed for the theoretical assessment of GHI with the derivation of regression constants by using the least square method. Moreover, the bright sunshine hours are also theoretically calculated to reduce the dependence of measured values for the model formulation. The proposed models are also compared with the selected existing GHI models such as Glover&McCullah, Newland and Pandey model with the justification of statistical analysis. The actual GHI is obtained from a 5 MW solar power plant in Sivagangai. The following strategic conclusions are obtained for the location under study with a latitude and longitudinal angle of 9.47°N and 78.87°E, respectively.

For the evaluation of GHI in Sivagangai the proposed models hold good as justified by the statistical indicators indicated in Tables 3 and 4. Excellent accuracy is observed in the cubic model which is most suited for the assessment of the monthly average daily GHI. The average MBE, RMSE and MPE for the proposed cubic model is –0.00067 kWh/(m²·d), 0.00092 kWh/(m²·d) and –0.01203 kWh/(m²·d), respectively. The regression coefficient or the best fit R² of the same also turns out to be 1 which is higher than the proposed linear and quadratic models. The prediction of the most accurate model for GHI varies with location. As seen in Ref. [ 28] the accurate model for prediction of Jodhpur, Calcutta, Bombay was linear.

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