Tigray, Ethiopia
Estimation of Daily Solar Radiation Using SDF and ANN Regression Models
Yacob Gebreyohannes12*, Mulu Bayray1, Johan Lauwaert2
1Thermal and Energy Systems Chair, School of Mechanical and Industrial Engineering, Mekelle University, Endayesus Campus, Mekelle, Ethiopia
2Department of Electronics and Information Systems, Faculty of Engineering and Architecture, Ghent University, iGent-tower, Technologiepark-Zwijnaarde 15, 9052 Ghent-Zwijnaarde
Abstract— Like many developing countries, properly recorded real-time measured solar radiation data are seldom available in Ethiopia. As such, using different solar radiation estimation models that use geographical and climatological parameters of locations is necessary. In this study, the solar duration fraction (SDF) empirical model of Angstrom-Prescott linear correlation and the artificial neural networks (ANNs) model of Generalized Regression Neural Network (GRNN) were considered for the estimation of daily solar radiation.
Correlation coefficients for these models were determined using measured (irradiance and sunshine hours) and calculated Sun-Earth parameters in Mekelle, Ethiopia. The performance of these models was evaluated and verified using the methods of statistical analysis. The study resulted in the respective H/Ho = -0.02 + 0.82 (n/N) and HGRNN = 5 + 0.74 (H) correlations for the optimal estimation of daily solar radiation in Mekelle areas and somewhere else with comparable climatic conditions. The results indicated the ability of the proposed models to reasonably predict the nature of available solar radiation with their own advantages and drawbacks. Thus, further tests should be conducted with possible adjustments in the input features in order to tune the study results for effective estimations of both models.
Keywords: Solar radiation, Sunshine hours, Estimation models, SDF, ANN, Mekelle
1. INTRODUCTION
The solar radiation received by a surface on Earth varies on seasonal (daily or monthly) and hourly basis [1]. Thus, accurate information on the extent of solar radiation at a given location is important for the expansion of solar energy applications. For some particular researches, the real-time measurement which records the solar radiation data on smaller time intervals (i.e. one minute and several minutes) is preferred. However, the calibration and maintenance works are costly in addition to the greatly affected values by the time and location. As a matter of fact, in most developing countries, properly recorded real-time measured solar radiation data are not available [1]-[2].
Without properly recorded real-time measured solar radiation data, estimation models are necessary to convert the available data into solar radiation. Through investigating the literature, most of the studies are to estimate the daily and hourly global solar radiation [2]- [4]. The effective estimations of daily and hourly solar energy capacity through the estimation models play a significant role in the design and sizing of solar energy applications [1][5]. Of course, the hourly average daily solar radiation is more useful, but several studies have replaced it with a daily average in the process of data reduction. Diurnal average daily solar radiation is more accurate than the monthly average because it has recorded detailed changes of solar radiation day by day, but the difficulty also increases, because it changes rapidly along with the weather condition [2].
Estimation models are formulated using geographical parameters, climatological parameters, and the concept of sun-earth angle. Among various models and parameters, the solar duration fraction (SDF) model of Angstrom–Prescott correlation has been widely and commonly used to determine daily radiation. The model proposes a correlation between the ratios of actually available and maximum possible daily global solar radiation and daily sunshine duration [6]-[13]. Many studies have also applied Artificial Neural Networks (ANNs) to estimate daily global solar radiation [2]. A much more efficient algorism for small datasets, Generalized Regression Neural Network (GRNN), is one of the most important methods for searching relationships among variables and is chosen for this task [14]-[15].
The most typical statistical gauges Mean Absolute Error (MAE) and Root Mean Square Error (RMSE), as well as the correlation coefficient (R), have been used to assess and test predictions of the estimation models. The statistically tested correlations of the models could be then used for the estimation of daily average global solar radiation of the location under study and other locations with comparable climatic conditions.
The objective of this study is, therefore, to analyze and validate SDF and ANN estimation models using measured data of irradiance and sunshine hours. The study also aims to show the effectiveness of SDF and ANN models for solar radiation estimation. Data was collected in Mekelle University's main campus, Ethiopia. A description of selected models was made from literature.
This was followed by the determination of the model’s unknown coefficients and statistical tests.
Computational spread sheet Microsoft Excel and programming language Python was used for data conditioning, treatment, and analysis. The output of the study envisages achieving the most appropriate models and will have significant importance in terms of promoting solar energy applications.
2. MATERIALS and METHODS
2.1.Extraterrestrial and Ground Measured Solar Radiation 2.1.1. Earth-Sun Parameters
Location is represented by geographical coordinates as latitude (φ) and longitude (λ) in degrees.
Latitude is the angle between the location and the equator. Longitude is the angle between the meridian of the location and the standard meridian. The Sun motion is represented by solar angle parameters represented as the declination (δ) and sunset hour (ωs) angles in degrees. These are defined using equations in Table 1 [16]-[21].
Table 1: Equations of Sun motion and Earth trajectory parameters
Parameter Equation Remark
Declination angle ( (
δ is the angle between the sun's direction and the equatorial plane of the earth.
nd is day number.
Sunset hour angle ( ωs is the sunshine duration.
Equation of time EoT = 9.87sin (2𝐵) − 7.53cos (𝐵) − 1.5sin (𝐵) 𝐵 = (3600/365) (nd − 81)
The EoT is the difference between apparent and mean solar times, both taken at a given longitude for the same real instant of time.
𝐵 is a factor.
Sunshine duration
( ( ( Maximum possible bright sunshine hours.
The correction of the
Earth-Sun distance (
The mean Earth-Sun distance varies from 144 (21 December) to 154 million km (21 June).
2.2.Daily extraterrestrial solar radiation
The daily extraterrestrial solar radiation, Ho (J/m2day), on a horizontal plane outside the atmosphere is given by (1):
[ (
)] [
] ( 1)
Where Gsc, solar constant (i.e 1367W/m2) is the amount of solar irradiance at the mean distance of Earth-Sun perpendicular to the direction of propagation outside the atmosphere [16]-[21].
3. STUDY AREA AND SOLAR RADIATION DATA
The measurement site considered is located in Mekelle University's main campus at an altitude of 2208m. Here, there are two major seasons, namely dry (October to May) and wet (June to September) to represent the clear sky and overcast days. Sunshine hours and solar irradiance data were collected every ten minutes from March 2018 to October 2019. The sunshine hours and solar irradiance are recorded by SPN1 type sunshine Pyranometer recorder with an accuracy of +/-5%.
The threshold irradiance for sunshine duration measurement of SPN1 is 120 Wm-2, which is recommended as the reference for finding a precise physical definition by the World Metrological Organization (WMO).
4. SOLAR RADIATION DATA PROCESSING AND ANALYSIS
Computational spread sheet Microsoft Excel was employed to process the data as required for analysis. The measured data are checked for inconsistencies and errors. Errors resulted from missing single data and data logger time shifts are corrected and all other null and unnecessary data out of the objective were canceled for easy and fast handling of missing values. Then, a computer program using a programming language Python was employed for data analysis. From the raw data set and sun-earth equations, daily average statistics were made for the solar irradiance and sunshine hour data.
5. SOLAR RADIATION ESTIMATION MODELS 5.1.Sunshine Duration Fraction (Sdf) Model
The simplest classical form of Angstrom–Prescott has been proposed to estimate daily solar radiation. The correlation given in (2) is first suggested by Angstrom in 1924 and modified to its present form by Prescott in 1940 [6],[16]-[21]:
̅
̅ ( ̅
̅) (2)
Where ̅ is average daily ground solar radiation on a horizontal surface, ̅ is average daily extraterrestrial solar radiation on a horizontal surface, ̅ is average daily bright sunshine hours, ̅ is average maximum possible daily bright sunshine hours, a and b are correlation coefficients.
5.2.Artificial Neural Networks (ANNs) Model
The basic principle of ANN is to correlate the input features with the target (output features) in applying different approaches. The calculation units of ANN are interconnected neurons in the layers. In terms of the data manipulation process, it mainly involves two stages: training stage and testing stage. During the training stage, the ANN completes the learning and storing of the pattern information for the existing database. In the testing stage, the ANN recalls the stored information to produce output data based on a particular input database [2]. GRNN uses lazy learning that does not require iterative training but just stores parameters and uses them to make predictions [14]-[15].
The methods for the suggested GRNN model are listed in Table 2.
Table 2: GRNN model methods
SN Method Remark
1 Normalization The network is sensitive when one input feature has higher values than the other one. Input data normalization is required before training.
2 Standard deviation (Std) The network is sensitive to the learning rate (Std) value. Std should be on the range of input features for good prediction.
3 train(x_trai, y_train) The network stores all the information about the data (x, y) for the prediction.
4 y_test = predict(x_test) The network returns prediction per each sample in the input (x_test).
5.3.Statistical Evaluation Parameters
The performance of different models was compared by Mean Absolute Error (MAE) and Root Mean Square Error (RMSE). The depth of the correlation between the estimated and ground measured radiation was compared using the coefficient of correlation (R) [22].
The MAE yields the deviation of the measured and estimated radiation values defined using (3).
( ) ∑| | (3)
Where n is the number of data considered, is estimated value of mean daily global radiation, and is the measured value of mean daily global radiation.
The RMSE yields the same idea of deviation between measured and estimated radiation values.
This is defined using (4):
√∑( (4)
R is defined using the statistical formula in (5):
∑( ̅ ( ̅
√∑( ̅ ∑( ̅ ( 5)
The values of MAE can be negative and positive for under and overestimations by models, respectively. To obtain the mean, errors are added up neglecting the signs. As such, long-term performance is determined by a comparison of actual deviation term by term and is useful to care for outliers in the data. The RMSE values are always positive and zero in the ideal case. The RMSE gives a short-term performance of the correlations by allowing a term by term comparison of the actual deviation and is useful to care for unexpected values in the data. The R values are always less than one and one in the ideal case. The R is useful to measure if the model is good or NOT.
6. RESULTS and DISCUSSION
6.1.Extraterrestrial and Ground Measured Solar Radiation
The geographical and astronomical parameters of the measurement site used for the analysis are summarized in Table 3.
Table 3: Parameters of the measurement site
SN Parameter Value
1 Latitude (φ) in degrees 13.33
2 Longitude (λ) in degrees 39.30
3 Local time (Tloc) in hours 00:00 - 23:00
4 Time zone in hours GMT + 3
5 Number of days (nd) Between 1 for 1st of January and 365 for 31st of December
The statistics of the collected data are summarized in Table 4.
Table 4: Measured data statistics
Indicator Global Irradiance (Wm-2) Solar Duration (hr)
Count 76616.00 76616.00
Mean 253.49 0.05
Std 347.55 0.07
Max 1408.07 0.17
Min 1.05 0.00
The calculated extraterrestrial solar radiation and the re-sampled measured solar radiation data are shown in Fig. 1. The measured radiation is observed to drop during the wet season between day 150 and 250.
Figure 1: Daily extraterrestrial and measured radiation
6.2.Estimated Solar Radiation
The relative daily solar radiation and sunshine hours of measured data and corresponding predictions of the SDF model are shown in Fig. 2.
Figure 2: Relative solar radiation and sunshine duration
The relative daily sunshine hour, ̅ / ̅ (i.e the clearness index) provides information on the atmospheric conditions and the level of available solar radiation at the Earth’s surface. The classifications on the clearness index vary based on location and season according to the dominant cloud formation. The seasonal variation can be roughly described by three ranges, namely cloudy sky (0 ≤ ̅ / ̅ < 0.3), scattered clouds (0.3 ≤ ̅ / ̅< 0.7), and fair weather (0.7 ≤ ̅ / ̅≤ 1.0) [20]- [25]. Accordingly, the prevailing sunshine conditions presented in this study area are mainly scattered clouds and fair weather. The constants (a = -0.02 and b = 0.82) have a physical meaning.
Usually, a is supposed to measure the fraction of extraterrestrial radiation on cloudy days and sum of a and b is supposed to measure the fraction of radiation received on clear days [16]- [21].
The average daily global solar radiation values estimated using the selected SDF model and corresponding coefficients given by H/Ho = -0.02 + 0.82 (n/N) compared to the parallel measured values are shown in Fig. 3.
Figure 3: Ground measured and Angstrom-Prescott model estimated solar radiation
The parameters for the selected GRNN model during the analysis are summarized in Table 5.
Table 5: Parameters for the GRNN model SN Parameter Description
1 Input features Astronomical and Sun-Earth angle parameters 2 Target output Measured radiation
3 Normalized input features [-2.5, 2.5]
4 (x_train, y_train) 70 %
5 (x_test, y_test) 30%
6 Std [0.005, 0.015]
The Python code was run for different learning rates by varying the standard deviation (Std) for the normalized input data. The Mean Square Error (MSE) that minimized the actual target and output of the network was used to compare the performance of the GRNN method. As can be seen from Fig. 4 the error overshoot at the start and after Std = 0.008 points of the training and subsequently receded to a lower value as the Std become 0.008 to show superior performance. From this figure, it is possible to see that the estimation match is good, although not perfect for a wide range of solar radiation values. There are points that seem to diverge from the regressed line especially on the lower and higher radiation measurements. This might arise due to the presence of an incorrect data point, or because the data is far from other training points. The latter is the case here since the data used is not representative of all input space. Analysis of the scatter plot of training data in this figure clearly shows the case. The addition of points that span the whole data space will improve the generalization capability of the proposed neural network. As a result, global solar radiation can be related to the model output by HGRNN = 5 + 0.74 (H). The final trained neural network output after a test data was presented is given in Fig. 5.
Figure 4: GRNN training and test
Figure 5: Ground measured and GRNN model estimated solar radiation
Estimation of Daily Solar Radiation Using SDF and ANN Regression Models
6.3.Statistical Tests of Models
The results of the statistical tests for validation and comparison are summarized in Table 5.
Table 5: The statistical test results of estimations
Indicator SDF GRNN
MAE 0.81 1.51
RMSE 1.07 1.84
R 1.00 0.98
In MAE and RMSE cases, the smaller the value, the better is the model’s performance. However, R should approach one as closely as possible for better model performance [22]. Therefore, from Table 5 the models achieved a very good R-value. In addition, both MAE and RMSE values are acceptable. This means that the estimations obtained using the models and their coefficients of regression are compatible with the mean measured values. The GRNN model achieved lower R with higher MAE and RMSE values. This indicates the estimations obtained from this model are in a lower agreement with the mean measured values compared to the SDF model. The deviation between the models can be minimized in line with a large number of studies that had applied ANN for the estimation of solar radiation more accurate [26]-[28].
7. CONCLUSIONS
The objective of this study was to evaluate the appropriateness and effectiveness of SDF and ANN models for the estimation of daily global solar radiation on a horizontal surface. To select the most appropriate model for Mekelle University solar measurement site, estimation models have been collected from the literature and data (irradiance and bright sunshine hours) and Sun- Earth parameters were collected. The performance of the employed models was evaluated and compared on the basis of the statistical gauges (MAE, RMSE, and R). According to the results, estimation of daily mean global solar radiation can be done with reasonable accuracy using H/Ho
= -0.02 + 0.82 (n/N) and HGRNN = 5 + 0.74 (H) respectively. The results proved that the proposed models are able to predict the nature of available solar radiation with their own advantages and drawbacks. As such, further study is recommended for thorough treatment and possible adjustment in the input features in order to consolidate the results of the study for effective estimations of both models in the study area and elsewhere with comparable climatic conditions.
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ACKNOWLEDGMENT
The authors would like to record their gratitude to Mesele Hayelom (Ph.D. student of Enpe project) for the support and crucial contribution in delivering data leading to this article.
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