Developing a new solar radiation estimation model based on Buckingham theorem

Results in Physics 9 (2018) 263–269

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Developing a new solar radiation estimation model based on Buckingham theorem Can Ekici a,b,⇑, Ismail Teke a a b

Yildiz Technical University, Department of Mechanical Engineering, Istanbul, Turkiye Turkish Standard Institution, Gebze Calibration Laboratory, Kocaeli, Turkiye

a r t i c l e

i n f o

Article history: Received 2 November 2017 Accepted 25 February 2018 Available online 1 March 2018 Keywords: Daily total global solar radiation Netherlands North Dakota Prediction, modeling

a b s t r a c t While the value of solar radiation can be expressed physically in the days without clouds, this expression becomes difficult in cloudy and complicated weather conditions. In addition, solar radiation measurements are often not taken in developing countries. In such cases, solar radiation estimation models are used. Solar radiation prediction models estimate solar radiation using other measured meteorological parameters those are available in the stations. In this study, a solar radiation estimation model was obtained using Buckingham theorem. This theory has been shown to be useful in predicting solar radiation. In this study, Buckingham theorem is used to express the solar radiation by derivation of dimensionless pi parameters. This derived model is compared with temperature based models in the literature. MPE, RMSE, MBE and NSE error analysis methods are used in this comparison. Allen, Hargreaves, Chen and Bristow-Campbell models in the literature are used for comparison. North Dakota’s meteorological data were used to compare the models. Error analysis were applied through the comparisons between the models in the literature and the model that is derived in the study. These comparisons were made using data obtained from North Dakota’s agricultural climate network. In these applications, the model obtained within the scope of the study gives better results. Especially, in terms of short-term performance, it has been found that the obtained model gives satisfactory results. It has been seen that this model gives better accuracy in comparison with other models. It is possible in RMSE analysis results. Buckingham theorem was found useful in estimating solar radiation. In terms of long term performances and percentage errors, the model has given good results. Ó 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:// creativecommons.org/licenses/by-nc-nd/4.0/).

Introduction

Solar declination angle is given in Eq. (1). J is the calendar day of the year [2,3].

Solar radiation is an important variable for daily life and applications. Therefore, it is important that this value is known correctly. Estimation models are in effect when solar measurements cannot be performed [1]. In this study, Buckingham theorem based solar radiation estimation model will be established. Applications for this model will be done and error analysis will be performed. The model will be calibrated in the different regions of the world.

sind ¼ 0:39785  sin½278:97 þ 0;9856J þ 1:9165  sinð356:6 þ 0; 9856JÞ

Main mathematical expressions about solar radiation Main mathematical expressions about solar radiation will be given in this section. ⇑ Corresponding author at: Yildiz Technical University, Department of Mechan-

ð1Þ Sunrise hour angle is given in Eq. (2). xs is sunrise angle; ø is the latitude [2,3].

xs ¼ cos1 ½tanø  tand

ð2Þ

Eccentricity correction factor is called eccentricity factor; Eo. The simple expression of the eccentricity factor is given in Eq. (3) [2,3].

E0 ¼ 1 þ 0:033  cos


 2p  J 365

ð3Þ

Mathematical expression of extraterrestrial radiation is given in Eq. (4) [2,3]. Isc is the solar constant; 4.921 MJ/day.m2 [2].

ical Engineering, Istanbul, Turkiye. E-mail address: [email protected] (C. Ekici). https://doi.org/10.1016/j.rinp.2018.02.064 2211-3797/Ó 2018 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Abbreviations Daily total global solar radiation, units of MJm2day1 Extraterrestrial solar radiation, units of MJm2day1 Measured daily total global solar radiation, units of MJm2day1 Calculated daily total global solar radiation, units of MJm2day1

H H0 Hm Hc

H0 ¼

24

p

 Isc  E0  sinø  sind 

h

p  180

 xs  tanxs

i

ð4Þ

Error analysis methods Error analysis will be done with the help of meteorological data for the model derived using Buckingham theorem in this study. When performing these error analyzes, the percentage errors, short-term performance, long-term performance, and compatibility between the model and the meteorological data will be examined. Error analysis methods will be able to evaluate these performances. RMSE analyzes provide information on the short-term performance of the models. MBE analyzes provide information on systematic errors and long term performance. The MPE error analysis gives the percentage error between the measured values and the values obtained from the models. As the results of this analysis approach to zero, the models become perfect [3,4]. These error analysis methods are given in Eqs. (5)–(7).

MPE ¼

 n  1X Hi;c  Hi;m  100 N i¼1 Hi;m

ð5Þ

Pn

 Hi;m N v ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u Pn u ðHi;c  Hi;m Þ2 RMSE ¼ t i¼1 N MBE ¼

i¼1 H i;c

ð6Þ

NSE ¼ 1  Pi¼1

Using Buckingham’s Pi theorem, it is seen that the solar radiation expression can be mathematically modeled [6]. First of all, the solar radiation will be mentioned in terms of the parameters available in the literature, in the models and in theory for the correct expression. Many models in the literature appear to have the expression of solar radiation together with extraterrestrial radiation [7–12]. It is also seen in many examples in the literature; global solar radiation can be expressed using the maximum and minimum air temperature differences, which is usually the square root of the temperature difference in mathematical formulas [9,12,13,14]. The global solar radiation value varies with meteorological factors when reaching the Earth’s surface from the atmosphere. Radiation is subjected to multiple attenuation processes; the permeability of the atmosphere is related to the attenuation processes [2]. Transmittance, which is a proportional expression, will be another parameter to be considered in this theorem. Using Buckingham theorem, models were developed using different temperature and humidity parameters (daily maximum, minimum and mean values) and error analysis were made with the station data. Mathematical expressions that are not satisfied with the results will not be shared here. The expressions to be used in the formation of P’s are given above. It is possible to express the global solar radiation functionally as follows, in which some parameters are written for use in nondimensioning the above-mentioned parameters.

H ¼ f ðH0 ; s; DT; T min ; RHÞ ð7Þ

The Nash-Sutcliffe equation is also an error evaluation method. A model is more efficient when NSE converges to 1. NSE equation is given in Eq. (8) [10].

Pn

It should be noted here again; the P parameters are without units. Both global solar radiation and extraterrestrial solar radiation expresses in energy per square meter (MJm2day1). Hence, it is possible to write the first P parameter as in Eq. (5).

Y 1

ðHi;m  Hi;c Þ
m Þ2 H

ð8Þ

Modeling the solar radiation

Y Y Y Y  ¼U ; ; ; . . . ; 2 3 4 kr

As the first step in the Buckingham Pi theorem, it is desirable to list the variables that affect the problem [5].

H H0

ð5Þ

The second pi parameter is obtained by nondimensioning the difference between the maximum and minimum daily temperatures (Eq. (6)). Here, the minimum temperature will be used for the nondimensioning process.

Y According to the Buckingham theorem, all terms collected in an equation must have the same dimensions. This situation is expressed as dimensional homogeneity law. This law guarantees that all terms in an equation are in the same dimension. Unit analysis has to be performed for nondimensionalization process [5]. Dimensionless parameters are often referred to as P. The relationship between dimensionless pi numbers that are related to each other is expressed as follows [6]. 1

¼

2

n i¼1 ðH i;m

Y

Daily minimum temperature, units of °C Daily maximum temperature, units of °C Relative humidity, units of %rh

Tmin Tmax RH

2

¼


 T min DT

ð6Þ

Finally, the atmospheric transmissivity expression will be used in the derivation of the third P as in Eq. (7). Since atmospheric permeability is a proportional expression, the relative humidity, which is another proportional expression, will be used. The maximum, minimum, or mean value of the relative humidity will be used to modeling the solar radiation for these nondimensioning. This choice will be made in future.

Y 3

¼ 100 

s RH

ð7Þ

As the result of the processes, the situation in Eq. (8) finds out according to the Buckingham Pi theorem.

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C. Ekici, I. Teke / Results in Physics 9 (2018) 263–269 Table 1 Coefficients derived in the use of Eq. (12) for Netherlands. Stations

A

b

c

d

e

g

h

w

q

Using Celcius for temperature

Eindhoven Groningen Maastricht Rotterdam Twente

0.008907 0.03427 0.05183 0.1323 0.207

0.9212 0.527 0.6733 0.6832 0.1891

1.402 0.5635 1.178 1.211 0.6474

0.2366 0.2238 0.1777 0.2003 0.2186

0.05516 0.01442 0.2248 0.2899 0.06611

0.2483 0.1227 0.1827 0.09311 0.106

0.3584 0.123 0.2037 0.1252 0.19

0.1836 0.1781 0.1551 0.1648 0.1667

0.5335 1.664 0.934 1.053 0.7744

Using Kelvin for temperature

Eindhoven Groningen Maastricht Rotterdam Twente

0.2069 1.268 1.436 0.8955 0.3331

0.8023 0.01491 0.117 0.1902 1.117

1.586 0.1535 2.819 2.436 2.828

0.1844 1.656 0.1716 0.18 0.0506

0.5632 2.465 0.1077 0.6039 2.199

0.002098 0.01337 0.007172 0.005708 0.00775

0.002971 0.006669 0.000752 0.00829 0.004167

0.9323 0.3275 0.4723 0.3646 0.7313

0.6876 2.035 2.521 1.872 1.042

Table 2 Geographical coordinates of the meteorological stations in Holland. Station name

Latitude

Longitude

Altitude

Eindhoven Groningen Rotterdam Maastricht Twente

51.451° 53.125° 51.962° 50.906° 52.274°

5.377° 6.585° 4.447° 5.762° 6.891°

22.6 m 5.2 m 4.3 m 114.3 m 34.8 m

Using Celcius for temperature

Using Kelvin for temperature

Eindhoven

MBE RMSE MPE NSE

0.48 2.84 12.48 0.88

0.50 2.14 10.39 0.88

Groningen

MBE RMSE MPE NSE

0.40 2.03 13.27 0.87

0.42 1.81 10.58 0.88

Maastricht

MBE RMSE MPE NSE

0.67 2.51 14.84 0.88

0.77 2.48 13.09 0.88

Rotterdam

MBE RMSE MPE NSE

0.50 2.12 13.51 0.86

0.63 2.01 12.90 0.87

MBE RMSE MPE NSE

0.43 2.19 11.53 0.87

0.49 2.13 10.52 0.87

Twente





H ¼U H0

  T min 100  s ; DT RH

2 RH RH þ 0; 6976 þ 0; 1467  100 100

DT < 10  C ! s=ð11  DTÞ

RH 2 RH DT P 10 C ! 0; 7451  100 þ 0; 1467  100 þ 0; 6976 

ð10Þ

s ¼ 0; 019  RH þ 1; 576

ð8Þ




ð9Þ

In addition, Spokas proposed an equation that provides atmospheric transmissivity correction when the maximum and

ð11Þ

Up to this point, one conditional equation (Eq. (10)) and one derivation equation (Eq. (11)), which can be used to determine atmospheric transmissivity, are given. Using these equations, P ’s were calculated and tested in solar radiation estimation. Many of these applications have not yielded satisfactory results and are thought to have failed to estimate solar radiation. It has been seen that the use of the conditional equation gives better results in the models. In addition, while the temperature units are nondimensioning, this process is done for both Kelvin and Celsius. A satisfactory equality will be shared here, followed by error analysis results in terms of both Kelvin and Celsius temperature units. Eq. (12) gives the most effective result in the applications. In the equation, q, a, b, c, d, e, g, h, and w are empirical coefficients.



   H s b s d ¼ q a 1  exp 100  þ c  100  H0 RH RH

   s  T min  w þ g  cos w þ e  cos 100  RH DT

  T min w þ h  sin DT

In the literature, there are studies that show the relationship between atmospheric transmissivity and the relative humidity. In a study in the literature, two methods were presented using other studies in the literature [15]. The first method was compiled from Spokas’s study. In the study, atmospheric transmissivity was expressed in terms of relative humidity ranges [15,16]. This table was tried to be formulated within the scope of this study and the formulated form of the table is given in Eq. (9).

s ¼ 0; 7451 

(

s0 ¼

Another mathematical equation that can be used to determine transmissivity depending on relative humidity was compiled from a study in Athens [15,17]. This expression is shown in Eq. (11).

Table 3 Error analyzes of the model to select temperature unit. Station

minimum air temperature difference is less than 10 °C [16]. In this case, a conditional equation will be used, the conditional equation is given in Eq. (10).

ð12Þ

It is useful to talk about units in this section. Metric unit systems based on meters and kilograms first appeared in the French Revolution. In 1799 two platinum standards for meters and kilograms were produced. In 1946, the MKSA metering system (meters, kilograms, seconds, amps) was accepted by the countries connected to the Meter Convention. In 1954, Kelvin and Candela were added to these units. As a result of this expansion, this system is called the International System of Units (SI) [18]. As a result of investigations in the field of measurement science, changes can be made in the definitions and realization of the units as the probability of reaching a higher level of certainty. It is possible to give the basic SI units as moles for the amount of substance, kilogram for mass, meter for length, seconds for time, Kelvin for thermodynamic temperature, Ampere for electric current, and Candela for light intensity. Each basic SI unit has a physical description. The meter is the distance that the light travels in the vacuum at 1/299,792,458 fractional time interval. A kilogram

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Fig. 1. Application of model to Eindhoven city using Celsius temperatures.

Fig. 2. Application of model to Eindhoven city using Kelvin temperatures.

Table 4 Geographical coordinates of the meteorological stations in North Dakota. Station

Latitude

Longitude

Altitude

Baker Berthold Bottineau

48.167° 48.380° 48.821°

99.648° 101.822° 100.760°

512 m 649.8 m 450.8 m

equals the mass of the international kilogram prototype. Second, corresponds to the 9.192.631.770 period of the transition beam between the two super-thin levels of the Cesium-133 atom in the lowest energy level. Amperage is a constant current if two straight parallel conductors of infinite length are placed at a distance of 1 m in a vacuum environment and can produce power between these conductors equal to 2  107N/m. Mole is the

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C. Ekici, I. Teke / Results in Physics 9 (2018) 263–269 Table 5 Empirical coefficients for temperature-based models in the literature for North Dakota. Station

Model

a coefficient

b coefficient

c coefficient

Baker

Hargreaves Allen Bristow-Camp. Chen

0.1382 0.158 0.6748 0.2028

0.06147 – 0.110 0.04335

– – 1.1 –

Berthold

Hargreaves Allen Bristow-Camp. Chen

0.1305 0.1583 0.668 0.1967

0.0875 – 0.179 0.05875

– – 0.9242 –

Bottineau

Hargreaves Allen Bristow-Camp. Chen

0.1271 0.1519 0.6952 0.1996

0.0823 – 0.117 0.03891

– – 1.055 –

Table 6 Error analysis for temperature-based models in the literature for North Dakota. Station

Model Name Hargreaves

Allen

Bristow Camp.

Chen

Baker

MBE RMSE MPE NSE

0.27 3.92 17.18 0.77

0.39 3.88 16.62 0.78

0.04 3.78 12.05 0.79

0.27 3.91 16.04 0.77

Berthold

MBE RMSE MPE NSE

0.34 3.98 19.23 0.77

0.52 3.94 18.36 0.77

0.35 3.96 18.13 0.77

0.35 3.97 18.17 0.77

Bottineau

MBE RMSE MPE NSE

0.37 3.97 16.96 0.77

0.57 3.96 16.50 0.78

0.64 3.97 17.31 0.78

0.37 3.97 15.89 0.78

amount of substance in a system that contains atoms in 0.012 kg of Carbon-12. Kelvin is 1/273,16 fraction of the thermodynamic temperature of the water’s triple point [18]. Eq. (12) was derived in accordance with Buckingham’s theorem. It seems more appropriate to use the basic SI units for the Buckingham theorem. From this point of view, error analysis was performed by applying Eq. (12) in terms of both Celsius and Kelvin temperature units. Applications for Kelvin and Celsius will be based on Dutch meteorological data. These data are shared free of charge by the Royal Netherlands Institute of Meteorology. The data are believed to be reliable. First, the model was calibrated with the help of this data. As a result of this calibration process, empirical coefficients were derived. The derivation of the coefficients was done with the help of MATLAB. The empirical coefficients for this application are given in Table 1. Also, the geographical coordinates of the Dutch meteorological stations are seen in Table 2. Pi values were obtained with both Celsius temperatures and Kelvin temperatures. Example applications were made with data from Dutch meteorological stations. Error analysis was applied for pi values obtained from both temperature units. The use of Kelvin for temperatures in the total MPE analyzes yielded better results. When Celsius is used as the unit of temperature, MPE analyzes showed higher amount of percentage errors. At the same time, the RMSE values are lower in the application for Kelvin. The error analysis is given in Table 3. As a result of these applications, Kelvin is thought to be more suitable to be taken as the unit of temperature. Screenshots of the analyzes for the applications are shown in Figs. 1 and 2. Looking at the point accumulations in shapes on the graphs, Kelvin-type temperature usage is thought to be more suitable and more inclusive. In the graph of Celsius use, it is seen that points are concentrated in certain regions.

Application of the model In this section, the obtained model will be applied using meteorological data. Thus, the quality of the results of the model will be seen. The temperature-based models available in the literature will be compared with the model that obtained in the study. Analyzes will be applied through meteorological data obtained from the North Dakota state of the United States. Firstly, the coefficients of the model will be derived for the stations, then the errors for the models will be examined. The meteorological data for the stations were taken between the first half of 2010 and the first half of 2016. The model that was derived in this study will be applied to North Dakota and commonly used temperature based solar radiation estimation models in the literature will be applied. Allen, Hargreaves, Bristow-Campbell and Chen models were used in these applications and models will be compared. Hargreaves model is given in Eq. (13), Allen model is seen in Eq. (14) [12,13]. Empirical coefficients are a and b.

H ¼ a  ðT max  T min Þ0:5 þ b H0

ð13Þ

H ¼ a  ðT max  T min Þ0:5 H0

ð14Þ

Bristow-Campbell and Chen models are given in Eqs. (15) and (16) [1,14]. a”, ‘‘b” and ‘‘c” are empirical coefficients.

H ¼ a  ½1  expðbDT c Þ H0

ð15Þ

H ¼ a  lnðT max  T min Þ þ b H0

ð16Þ

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Table 7 Empirical coefficients of Eq. (12) for North Dakota. Station

a

b

c

d

e

g

h

w

q

Baker Berthold Bottineau

0.3376 1.996 0.9731

0.787 0.1615 0.07425

4.154 4.301 1.789

0.05491 0.195 0.1333

3.064 0.405 0.0691

0.00443 0.00125 0.002172

0.02091 0.01614 0.000656

0.449 0.4533 0.7074

0.7688 1.216 3.871

Conclusions

Table 8 Error analysis of Eq. (12) for North Dakota.

MBE RMSE MPE NSE

Baker

Berthold

Bottineau

0.09 1.52 14.37 0.80

0.03 1.44 15.79 0.80

0.03 1.85 14.10 0.80

The North Dakota State of US has an agricultural climate network that named North Dakota Agricultural Weather Network. The data recorded on the stations are shared publicly on hourly, daily, monthly, weekly basis. In the applications, the data of the three stations which have uninterrupted measurement records were used. The geographical information of meteorological stations in North Dakota is shown in Table 4. Four models in the literature were calibrated for the state of North Dakota. As a result of these calibration procedures, empirical coefficients were obtained. These coefficients are shown in Table 5. These experimental coefficients were derived separately for each station. Error analysis was performed between the values calculated by using these coefficients and the measured values. These error analyzes are given in Table 6. The model given in this study was also applied for the same stations. The empirical coefficients derived by this application are given in Table 7. The error analysis applied to Eq. (12) is shown in Table 8. As a result of the applications, the error analysis of the models in the literature is higher than Eq. (12). It can be said that Eq. (12) gives better results in terms of percentage errors, MBE and NSE analysis. In addition, Eq. (12) gives much better results in terms of RMSE error analysis. In terms of RMSE values, there are big differences between the error results of the models in the literature and the error results of Eq. (12). Eq. (12) is considered to be satisfactory in terms of short-term performance.

Discussions Buckingham theorem is used to obtain mathematical expressions by non-dimensionizing the physical quantities. A mathematical model was given in this study for estimating the daily total global solar radiation where solar radiation cannot be measured or where measurement records are not available. This model was derived by MATLAB with the help of Buckingham theorem. While this model is derived, the temperature units were taken separately as Celsius and Kelvin. These applications were made with the help of the data sets taken from Netherlands Royal Meteorological Institute. It is thought; it is appropriate to construct the pi parameters by taking Kelvin temperature unit. Applications were done with the aim of comparing the models in the literature with the model that is derived in the study. The applications were implemented using data obtained from North Dakota’s agricultural climate network. In these applications, the models obtained within the scope of the study are thought to give better results. Adding the relative humidity parameter to the model can also be effective in improving the results.

In most developing countries, there are cases where solar radiation measurements are not available. These measurements may not be taken due to reasons such as calibration costs, installation costs, repair and maintenance requirements. Estimation models are used when solar radiation data are not available. In this study, applications were made on solar radiation estimation and one solar radiation prediction model was presented. This solar radiation prediction model was derived with the help of Buckingham theorem. As a result of these applications, it was seen that this model gave satisfactory results. Applications were made for the state of North Dakota, and this model was compared with other commonly used temperature-based solar radiation estimation models in the literature. Allen, Hargreaves, Chen and Bristow-Campbell models were chosen for the applications. In terms of short-term performances, it has been seen that this model gives a high degree of accuracy in comparison with the other models. It is possible to see this situation from the RMSE analysis results. As a result of the study, Buckingham theorem was found to be useful in estimating solar radiation. In terms of long term performances and percentage errors, the model has given good results.

Declaration Authors declare that there is no conflict of interest regarding the publication of this article.

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