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Nat Hazards (2015) 77:1055–1068 DOI 10.1007/s11069-015-1641-x ORIGINAL PAPER

Potential of support vector regression for solar radiation prediction in Nigeria Lanre Olatomiwa • Saad Mekhilef • Shahaboddin Shamshirband Dalibor Petkovic

•

Received: 5 November 2014 / Accepted: 29 January 2015 / Published online: 8 February 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract In this paper, the accuracy of soft computing technique in solar radiation prediction based on series of measured meteorological data (monthly mean sunshine duration, monthly mean maximum and minimum temperature) taking from Iseyin meteorological station in Nigeria was examined. The process, which simulates the solar radiation with support vector regression (SVR), was constructed. The inputs were monthly mean maximum temperature (Tmax), monthly mean minimum temperature (Tmin) and monthly mean sunshine duration ( n). Polynomial and radial basis functions (RBF) are applied as the SVR kernel function to estimate solar radiation. According to the results, a greater improvement in estimation accuracy can be achieved through the SVR with polynomial basis function compared to RBF. The SVR coefficient of determination R2 with the polynomial function was 0.7395 and with the radial basis function, the R2 was 0.5877. Keywords Nigeria

SVR Solar radiation Sunshine hour Soft computing methodologies

L. Olatomiwa S. Mekhilef Power Electronics and Renewable Energy Research Laboratory (PEARL), Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia e-mail: [email protected] L. Olatomiwa Department of Electrical and Electronic Engineering, Federal University of Technology, PMB 65, Minna, Nigeria S. Shamshirband (&) Department of Computer System and Technology, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia e-mail: [email protected] D. Petkovic Department for Mechatronics and Control, Faculty of Mechanical Engineering, University of Nisˇ, Aleksandra Medvedeva 14, 18000 Nis, Serbia

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Abbreviations ANFIS Adaptive neuro-fuzzy inference system GA Genetic algorithm ACO Ant colony optimization PSO Particle swarm optimization DE Differential evolution Monthly mean global solar radiation (MJ/m2/day) H n Monthly mean sunshine duration (h) RBF Radial basis function RMSE Root-mean-square error Coefficient of determination R2 SVR Support vector regression Monthly mean maximum temperature (°C) Tmax Monthly mean minimum temperature (°C) Tmin

1 Introduction Abundant energy potential from solar radiation incident on earth’s surfaces has been seen to play important role in meeting the ever-growing energy demand of the world (Ming et al. 2014; Akikur et al. 2013; Azoumah et al. 2011; Bajpai and Dash 2012; Hasan et al. 2012). Among the various available renewable resources of the earth, solar energy has attracted enormous attention not only because it sustainable, but also abundant and environmental friendly (Akikur et al. 2013). Its wide applications can results in abatement of prevalent global warming, because it does not emit CO2 or any other greenhouse gas emissions that causes global warming. Therefore, long-term knowledge of solar insolation incident on earth’s surface at any particular locations is essential for its application such as agricultural, hydrological, ecological as well as electricity production. In electricity production, it is needed in designing and prediction of energy output of solar conversion system. Solar radiation data are best obtained from measurements taken remotely at a particular location using various solar radiation measuring instruments, but these data are limited in many metrological stations around the world, and this is sometimes due to high cost of calibration and maintenance of this equipment (Hunt et al. 1998). Over the years, numerous methods for estimating solar radiation on horizontal plane have been developed, and common among them are the empirical models (Angstrom 1924; Hargreaves and Samani 1982; Bristow and Campbell 1984; Besharat et al. 2013; Halawa et al. 2014), satellite-derived model (Pinker et al. 1995) and stochastic algorithm model (Hansen 1999; Mellit 2008). The most widely used model is empirical models that relate solar radiation with other routinely measured meteorological parameters from similar locations having same geographical coordinates. Parameters such as air temperature, sunshine duration hour and relative humidity have been widely used. Many literatures have adjudged sunshine duration, maximum and minimum temperature relations as best correlation for solar radiation prediction (Besharat et al. 2013; Trnka et al. 2005; Chen and Li 2013; Wu et al. 2007). However, at instances where sunshine duration data seem limited or inaccessible, frequently measured maximum and minimum temperatures alone have also been prove to produce good results (Hargreaves and Samani 1982; Bristow and Campbell 1984; Liu et al. 2009). Although the satellite-based methods looks promising for the estimation of solar radiation over a large

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region, its main drawback is its cost and lack of sufficient historical data because it is relatively new application. In Nigeria, several works have been carried on solar radiation predictions using the conventional empirical models, some of which can be found in these following references (Ezekwe and Ezeilo 1981; Sambo 1986; Akpabio and Etuk 2003; Fagbenle 1993; and Ajayi et al. 2014). Nevertheless, due to necessity of accurate and reliable solar radiation, artificial and computational intelligence techniques have been broadly applied to estimate solar radiation in many regions around the world. Al-Alawi and Hinai(1998) predicted solar radiation for a location with no availability of measured data. They used monthly mean daily values of temperature, pressure, relative humidity, sunshine duration hours and wind speed as inputs for artificial neural networks (ANN) technique to predict global solar radiation. They compared the results with empirical methods model and found more accuracy for ANN-based model. Mellit et al. (2006) employed the combination of neural and wavelet network to forecast daily solar radiation for photovoltaic (PV) sizing application. In their study, wavelets served as activation function. Their results of the forecast demonstrated the more favorable performance of the approach compared to other neural network models. In Jiang (2009), an ANN model was developed to estimate monthly mean daily solar radiation for eight typical cities in China. The achieved results were compared to those of conventional empirical models. The statistical analysis results indicated a good correlation between estimated values by the ANN model and the actual data with higher accuracy than other empirical models. Behrang et al. (2011) applied particle swarm optimization (PSO) technique to estimate monthly mean daily global solar radiation on a horizontal surface for 17 cities in different regions of Iran. Their results showed better performance of PSO-based models compared to the traditional empirical models. Mohandes (2012) employed PSO algorithm to train ANN in order to model the monthly mean daily global solar radiation values in Saudi Arabia. Different parameters such as month number, sunshine duration, latitude, longitude and altitude of the location were considered as inputs. The developed hybrid PSO–ANN model showed a better performance compared to back-propagation trained neural network (BP-NN). Benghanem et al. (2009) developed six ANN-based models to estimate horizontal global solar radiation at Al-Madinah in Saudi Arabia. They utilized different combinations of input parameters consisting sunshine hours, ambient temperature, relative humidity and the day of year. Their results showed that the model with higher accuracy is dependent upon sunshine duration and air temperature. Ramedani et al. (2014) employed support vector regression (SVR) technique to develop a model for the prediction of global solar radiation in Tehran, Iran. They found more superiority for SVR technique than other compared techniques. In another study, Ramedani et al. (2014) performed a comparative investigation between fuzzy linear regression (FLR) and support vector regression (SVR) techniques to predict global solar radiation in Tehran, Iran. They found that SVR approach enjoys superior performance compared to FLR. Also, in some studies, different techniques were combined to propose a hybrid approaches with more accuracy. Srivastava et al. (2012) proposed a hybrid approach which includes hidden Markov models and generalized fuzzy models to estimate solar irradiation in India. They assessed the influence of different meteorological parameters for the estimation of solar radiation using the developed model. Generally, support vector machine (SVM) is a type of soft computing technique that has gained importance in environmental related applications (Ornella and Tapia 2010; Jain et al. 2009). Basically, there are two fundamental classes of support vector machines: (a) support vector classification (SVC) and (b) support vector regression (SVR). SVC is a learning framework utilizing a high-dimensional peculiarity space (Ananthakrishnan et al.

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2013; Ye et al. 2009; Rajasekaran et al. 2008; Yang et al. 2009), while SVR focused around measurable learning hypothesis and structural risk minimization rule and has been effectively utilized for nonlinear frameworks (Wei et al. 2013; Zhang et al. 2013). The correctness of an SVM model is to a great extent relies on the determination of its model parameters. Although organized strategies for selecting parameters are important, model parameter alignment also needs to be made. In this work, radial basis function (rbf)-based SVR and polynomial basis function (poly)based SVR are applied for the estimation of global solar radiation at Iseyin meteorological station in Nigeria. The input parameters are monthly mean minimum and maximum temperature and the monthly mean sunshine duration hours, all measured at the station. The main objective of this study is to evaluate the potentials and performances of SVR for solar radiation prediction. The motivation behind this investigation is centered upon the significance of reliable solar radiation data in many applications including agricultural productions, hydrological and ecological studies as well as assessments and prediction of energy output of solar systems. Also, in most cases, the solar radiation data are not readily available due to several issues. The choices of methodology center on its simplicity, reliability, efficient computationally capability, ease of adaptability to optimization and other adaptive techniques as well as its adaptability in handling parameters that are more complex.

2 Materials and methods 2.1 Descriptions of study site and data set Difficulties and uncertainty involved in the measurement of global solar radiation where the measuring equipment is available or in most cases not available have resulted in the development of algorithms for its estimation from other available routinely measured metrological data (e.g., temperature, relative humidity, wind speed, relative humidity and sunshine hour). In Nigeria, many of the government-owned meteorological stations have no record of solar radiation data, even where the record is available there are some missing days or month without record, and this is sometimes due to improper calibration of measuring equipment employed. The data set used in this study were measured at a metrological station located at Iseyin, southwest Nigeria, with geographical coordinate 7.96° latitude North, 3.6° longitude East and 330 m altitude as shown in Fig. 1. Monthly average daily value of maximum temn) and solar radiation perature (Tmax), minimum temperature (Tmin), sunshine duration ( data (H) covering 21-year period (1987–2007) were obtained from Nigerian Meteorological Agency (NIMET) database in Oshodi, Lagos (NIMET 2014). According to the agency, the measured solar radiation data were recorded using Gunn–Bellani radiometer. This instrument produce a time-oriented parameter of solar radiation falling on a black body by measuring volume of the liquid distilled in a calibrated tube (Ajayi et al. 2014; McCulloch and Wangati 1967). The measured solar radiation in millimeter is thereafter converted to MJ/m2/day with the application of a conversion factor (1.1364) as proposed by Sambo in reference (Sambo 1986). In the case of sunshine duration measurement, Campbell–Stokes sunshine recorder was used, while minimum and maximum dry bulb thermometers were used to measure both maximum and minimum temperatures at the station. Figure 2a–d shows the station monthly distribution of solar radiation, sunshine duration, minimum and maximum temperature, respectively. Annual mean solar radiation of the

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Fig. 1 Map of Nigeria showing the location of the study site

(b)

38

8

Mean sunshine hours (h)

36 34 32 30 28 26 24 22 20

7 6 5 4 3 2 1 SEP

OCT

NOV

DEC

SEP

OCT

NOV

DEC

JUL

AUG

Months

(d) 38

Months

36 34 32 30 28 26 24 22 AUG

JUL

JUN

APR

MAY

MAR

FEB

20 JAN

DEC

NOV

SEP

OCT

AUG

JUL

JUN

MAY

APR

FEB

MAR

Mean Maximum Temp.(0C)

24 24 23 23 22 22 21 21 20 20 19 JAN

Mean Minumun Temp.(0C)

JUN

APR

Months

(c)

MAY

FEB

MAR

JAN

DEC

NOV

SEP

OCT

JUL

AUG

JUN

APR

MAY

FEB

MAR

0 JAN

Mean solar radiaon (Kwh/m2/d

(a)

Months

Fig. 2 Distribution of monthly average daily solar radiation (a), sunshine duration hour minimum temperature (b), maximum temperature (c) in case study station

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studied site is 16.34 MJ/m2/day, while the annual mean bright day sunshine hour found to be 5.5 h, with highest value (7 h) in November and lowest (3.2 h) in August. The monthly mean daily maximum temperature ranges between 27.4 °C in August and 35.5 °C in February, while the minimum value ranged from 20.3 °C in January to 23.6 °C in March. 2.2 Empirical model relations of solar radiation and other metrological data Different regression correlations that relate the clearness index ration (Kt = HG/Ho) to relative sunshine duration ( n=N and air temperature (Tmax, Tmin) as derived from both Angstrom-Page (1924) and Hargreaves and Samani models (1982) were used for the analysis of the various input parameter to the SVR model. where Kt is the ratio of the monthly mean daily solar radiation on the horizontal surface (HG) to the monthly mean daily extraterrestrial solar radiation (Ho), while ( n=N is the ratio The mathematical of monthly mean sunshine hour ( n to the monthly daylight hour (N). expressions for Ho and N are given by Eq. (1) and Eq. (2) (Allen et al. 1998). H0 ¼

24 Isc ðws sin;sind þ cos;cosdsinws Þdr p

ð1Þ

2 N ¼ ws 15

ð2Þ

360ð284 þ dÞ d ¼ 23:4sin 365

ð3Þ

where

ws ¼ cos1 ð tan ; tan dÞ 360d dr ¼ 1 þ 0:033cos 365

ð4Þ ð5Þ

Isc is solar constant (4.9212) in MJ/m2/day, [ is the latitude of the site under consideration, dr is the inverse relative distance of the sum to earth, d is the solar declination, ws is the hour angle and d is the day number range from 1 (January 1st) to 365 or 366 (December and clear-sky ran=N) 31st). Extraterrestrial radiation (H0), relative sunshine duration ( diation (Hso) were also used as indicator for data quality control, where Hso is the fraction of extraterrestrial radiation falling on earth surface on clear-sky days ðn ¼ N Þ, as expressed in Eq. (6) (al 1998). Hso ¼ ð0:75 þ 2 105 zÞH0

ð6Þ

where z is the site altitude in meters. For this study z = 330, hence Hso ¼ 0:7566Ho . Since temperature-based model could be useful in a situation where sunshine duration data are not available, the model consider for this work is in line with Hargreaves and Samani model (1982), as expressed in Eq. (7). HG ¼ aDT 0:5 Ho

ð7Þ

where DT is the difference between monthly mean maximum (Tmax) and minimum temperature (Tmin) in °C, and a is an empirical which usually varies according to different regions where the site is located.

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2.3 Support vector regression application The main principle of SVR is to do the data correlation through nonlinear mapping. If a way of computing the inner product in a feature space is available directly as a function to the original input points, it is possible to build a nonlinear learning machine, which is known as a direct computation method of a kernel function, denoted by K. The flexibility of the SVR is in regard to the kernel functions that implicitly convert the data to a higher-dimensional feature space. Results in the higher-dimensional feature space correspond to the results of the original, lower-dimensional input space. There are some methods that employ nonlinear kernels for regression problems. These include the radial basis function and polynomial-based function. The main benefit of the radial basis function is computationally efficient since radial basis function training needs only the solution of a set of linear equations instead of the lengthy and computationally demanding quadratic programming problem. Therefore, the radial basis function and polynomial with parameter r are adopted in this study. The nonlinear radial basis kernel function is defined as (Yang et al. 2009; Vapnik and Vapnik 1998; Vapnik 2000): 1 2 ð8Þ Kðx; xi Þ ¼ exp 2 x xi r where x and xi are vectors in the input space, i.e. vectors of features computed from training or test samples. In this study, the following polynomial kernel function was also used (Yang et al. 2009; Vapnik and Vapnik 1998; Vapnik 2000): d ð9Þ Kðx; yÞ ¼ xT y þ c where x and y are vectors of features computed from training or test samples, and c is a constant making a trade-off for the influence of higher-order versus lower-order terms in the polynomial. The capability of the SVR to make good estimations is dependent on input parameter selection. In this study, the monthly values of Tmin, Tmax and n during the period (1987–2007) were used for generating the SVR model. To get more reliable evaluation and comparison, SVR model is tested by evaluating a data set that was not used during the training process; 70 % of the data are used for training and 30 % of the data for testing purpose. The statistical parameters (minimum value, maximum value, mean and standard deviation) for data sets are calculated and given in Table 1. 2.4 ANFIS-based methodology In this study, adaptive neuro-fuzzy inference system (ANFIS) that is functionally equivalent to the first-order Sugeno fuzzy model (Jang 1993) was used to validate the result Table 1 Statistical parameters for data sets Variable

Statistical parameters Min

Max

Mean

Standard deviation

Variation coefficient

Tmin

18

33.7

21.7

1.311

1.719

Tmax

22.8

37.1

31.6

2.839

8.065

1.3

8.4

5.5

1.443

2.083

n

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of the proposed SVR model. ANFIS serves as a basis for constructing a set of fuzzy ‘If– Then’ rules with an appropriate membership function to generate any stipulated input– output pairs. A typical rule set with a fuzzy ‘If–Then’ rule can be expressed as (Petkovic´ and C´ojbasˇic´ 2012); if x is A then f1 ¼ p1 x þ t

ð10Þ

The ANFIS architecture for three inputs x, y and z used for the validation of the proposed model is shown in Fig. 3. The nodes at the same layer have similar functions, while the output of the ith node in layer l is denoted as Ol,i. The first layer consists of input variable membership functions (MFs) and supplies the input values to the next layer. Every node i is an adaptive node with a node function: O1;i ¼ lðx; y; zÞi

for i ¼ 1; 2;

ð11Þ

where x, y = input to the ith node; l(x, y, z) = membership functions.The MFs can be described by bell-shaped function (Petkovic´ and C´ojbasˇic´ 2012) f ðx; a; b; cÞ ¼

1þ

1 xc2b

ð12Þ

a

where {a, b, c} is the parameter set. The second layer (membership layer) multiplies incoming signals from the first layer and sends the product out. Each node output represents the firing strength of a rule or weight like: O2;i ¼ wi ¼ lðxÞi lðxÞiþ1 ;

i ¼ 1; 2:

ð13Þ

The third layer (i.e., the rule layer) is non-adaptive where every node i calculates the ratio of the rule’s firing strength to the sum of all rules’ firing strengths like: wi O3;i ¼ wi ¼ ; i ¼ 1; 2: ð14Þ w1 þ w2 The outputs of this layer are known as normalized firing strengths or normalized weights. The fourth layer (i.e., the defuzzification layer) provides the output values resulting from the inference of rules, where every node i is an adaptive node with node function:

Fig. 3 ANFIS structure with three inputs, one output and two rules

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O4;i ¼ wi fi ¼ wi ðpi x þ qi x þ ri Þ

ð15Þ

where {pi, qi, ri} is the parameter set and this layer is referred to as consequent parameters. The fifth layer (i.e., the output layer) sums up all the inputs coming from the fourth layer and transforms the fuzzy classification results into a crisp (binary). The single node in the fifth layer is not adaptive, and this node computes the overall output as the summation of all incoming signals. P wi fi X i P w i fi ¼ ð16Þ O5;i ¼ wi i i

The proposed approach is based on the combination of particle swarm optimization (PSO), genetic algorithm (GA), differential evolution (DE) algorithm, ant colony optimization (ACO) and adaptive-network-based fuzzy inference system (ANFIS). The PSO, GA, DE and ACO are employed to improve the performance of ANFIS, adjusting the membership functions and reducing the prediction deviation. The ANFIS forecasts allow to predict the solar radiation. 2.5 Model performance evaluation To assess the success of the SVR models and other soft computing technique, some statistical indicators were examined as follows: 1.

2.

Root-mean-square error (RMSE) (Willmott and Matsuura 2005) vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uP un u ðPi Oi Þ2 t ; RMSE ¼ i¼1 n Coefficient of determination (R2) (Willmott and Matsuura 2005) n 2 P Oi Oi Pi Pi i¼1 R2 ¼ P n n P Oi Oi Pi Pi i¼1

ð17Þ

ð18Þ

i¼1

where Pi and Oi are known as the experimental and predicted values, respectively, while Pi and Qi are the mean value of Pi and Oi, respectively, and n is the total number of test data.

3 Results and discussion 3.1 SVR model analysis At the beginning, the SVR network was trained with measured data by above-presented experimental procedure. After training process, the SVR networks were tested to determine the solar radiation. Based on the experiments, the input parameters (monthly mean value of minimum temperature, maximum temperature and sunshine duration hour) and output

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(solar radiation) are collected and defined for the learning techniques. For the experiments, 70 % of the data were used for sample training and the remaining 30 % for testing. We analyzed the SVR model for solar radiation estimation based on the three inputs, monthly mean minimum temperature, monthly mean maximum temperature and monthly mean sunshine duration hours. Radial basis and polynomial functions were applied as the kernel functions for SVR prediction of solar radiation. The five parameters associated with two kernels are C, e, c, d and t. SVR model accuracy largely depends on model parameter selection; therefore, in selecting user-defined parameters (i.e., C, e, c, d and t), a quite number of trials were carried out with different combinations of C and d for polynomial kernels and C and t for radial basis function kernels. Table 2 provides the optimal values of user-defined parameters for this data set with polynomial and radial basis kernels for SVR. Results of the SVR prediction models for solar radiation are shown in Fig. 4a, b for training and testing phase, respectively. In these figures, predicted solar radiation values are plotted against the observed, in the form of scatter plots. The performances of the SVR models for solar radiation prediction in defined region have been appraised via the well-known statistical indicators of the root-mean-squared error (RMSE) and coefficient of determination (R2). Performance results of proposed models are summarized in Table 3. Figure 4 shows that it is obvious that for these models, prediction results slightly differ for training and testing phase. General pattern is preserved, and it seems that prediction results for RBF kernel are more sensitive to change in testing phase than for polynomial kernel function. This observation can be easily checked from Table 3. As to the prediction results, SVR-polynomial had the smallest RMSE of 1.510218 in testing phase. Radial basis function has RMSE = 1.905994, ANFIS-ACO has RMSE = 1.5485, ANFIS-DE has RMSE = 1.601, ANFIS-GA has RMSE = 1.7696, ANFIS-PSO has RMSE = 1.8015, and ANFIS has RMSE = 2.0628 in testing phase Finally, Fig. 5 shows comparative forecasting of solar radiation by two SVR techniques during validation step. It can be observed that SVR with polynomial basis function has better forecasting abilities for solar radiation prediction than SVR with radial basis function. 3.2 Performance analysis In order to evaluate the performance of the proposed method, experimental work was carried out to determine the significance of each independent variable on the output. Root-mean-square error (RMSE) and coefficient of determination (R2) served to evaluate the differences between the predicted and actual values for both SVRs models. Table 3 compares the single radial basis SVR and polynomial basis SVR model with several soft computing methods like ANFIS-ACO (adaptive neuro-fuzzy inference system with ant colony optimization), ANFIS-DE (adaptive neuro-fuzzy inference system with

Table 2 User-defined parameters for the optical lens doublet system Radial basis function

Polynomial basis function

C

t

e

C

t

e

c

d

400

0.01

0.4

100

0.1

0.4

0.03

2

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Fig. 4 a Scatter plots of training data and predicted values using ANFIS model and b scatter plots of tested data and predicted values using ANFIS model

differential evolution), ANFIS-GA (adaptive neuro-fuzzy inference system with genetic algorithm), ANFIS-PSO (adaptive neuro-fuzzy inference system with particle swarm optimization) and ANFIS. The results in Table 3 indicate that the polynomial basis SVR model has the best capabilities of estimating the solar radiation according to testing model.

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1066 Table 3 Performance statistics of the SVR model compares to other methodologies

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RMSE

R2

Training

1.363988

0.7703

Testing

1.510218

0.7395

Training

1.141814

0.8425

Testing

1.905994

0.5877

Training

1.3505

0.7748

Testing

1.5485

0.7313

Training

1.3503

0.7748

Testing

1.601

0.6926

Training

1.2769

0.7987

Testing

1.7696

0.6354

Training

1.1157

0.8463

Testing

1.8015

0.5666

Training

1.0699

0.8586

Testing

2.0628

0.5468

SVR-polynomial

SVR-radial

ANFIS-ACO

ANFIS-DE

ANFIS-GA

ANFIS-PSO

ANFIS

Fig. 5 Forecasting of solar radiation by two SVR techniques

Experimental data SVR_RBF predicon

23

SVR_Poly predicon

Solar radiaon

21 19 17 15 13 11 9

0

10

20

30

40

50

Data samples

4 Conclusion The accuracy of support vector regression (SVR) methodology in the estimation of solar radiation using climatic variables was presented in this study. The study indicated that modeling of solar radiation is possible through the use of SVR technique. The SVR model used inputs Tmin, Tmax and n from the Iseyin station in Nigeria during the period 1987–2007. It had the

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following statistical characteristics: SVR with polynomial basis function has RMSE = 1.510218 and R2 = 0.739, SVR with radial basis function has RMSE = 1.905994 and R2 = 0.5877, ANFIS-ACO has RMSE = 1.5485 and R2 = 0.7313, ANFIS-DE has RMSE = 1.601 and R2 = 0.6926, ANFIS-GA has RMSE = 1.7696 and R2 = 0.6354, ANFIS-PSO has RMSE = 1.8015 and R2 = 0.5666, and ANFIS has RMSE = 2.0628 and R2 = 0.5468 in testing phase. Because of these, it can be concluded that SVR with polynomial basis outperforms SVR with radial basis function and other methodologies in solar radiation estimation of the studied site. Acknowledgments The authors would like to thank the Ministry of Higher Education, Malaysia, and the Bright Spark Unit of University of Malaya, Malaysia, for providing the enabling environment and financial support under the Grant No. UMRG project RP015D-13AET. The authors also want to appreciate the effort of Nigerian Meteorological Agency (NIMET) for providing the required data for this research.

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Hunt L, Kuchar L, Swanton C (1998) Estimation of solar radiation for use in crop modelling. Agric For Meteorol 91(3):293–300 Jain P, Garibaldi JM, Hirst JD (2009) Supervised machine learning algorithms for protein structure classification. Comput Biol Chem 33(3):216–223 Jang J-S (1993) ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23(3):665–685 Jiang Y (2009) Computation of monthly mean daily global solar radiation in China using artificial neural networks and comparison with other empirical models. Energy 34(9):1276–1283 Liu X et al (2009) Evaluation of temperature-based global solar radiation models in China. Agric For Meteorol 149(9):1433–1446 McCulloch J, Wangati F (1967) Notes on the use of the gunn bellani radiometer. Agric Meteorol 4(1):63–70 Mellit A (2008) Artificial Intelligence technique for modelling and forecasting of solar radiation data: a review. Int J Artif intell Soft Comput 1(1):52–76 Mellit A, Benghanem M, Kalogirou S (2006) An adaptive wavelet-network model for forecasting daily total solar-radiation. Appl Energy 83(7):705–722 Ming T et al (2014) Fighting global warming by climate engineering: Is the Earth radiation management and the solar radiation management any option for fighting climate change? Renew Sustain Energy Rev 31:792–834 Mohandes MA (2012) Modeling global solar radiation using Particle Swarm Optimization (PSO). Sol Energy 86(11):3137–3145 NIMET (2014) Nigerian Meteorological Agency. htttp://www.nimet.gov.ng Ornella L, Tapia E (2010) Supervised machine learning and heterotic classification of maize (Zea mays L.) using molecular marker data. Comput Electron Agric 74(2):250–257 ´ ojbasˇic´ Zˇ (2012) Adaptive neuro-fuzzy estimation of autonomic nervous system parameters Petkovic´ D, C effect on heart rate variability. Neural Comput Appl 21(8):2065–2070 Pinker R, Frouin R, Li Z (1995) A review of satellite methods to derive surface shortwave irradiance. Remote Sens Environ 51(1):108–124 Rajasekaran S, Gayathri S, Lee T-L (2008) Support vector regression methodology for storm surge predictions. Ocean Eng 35(16):1578–1587 Ramedani Z et al (2014a) Potential of radial basis function based support vector regression for global solar radiation prediction. Renew Sustain Energy Rev 39:1005–1011 Ramedani Z et al (2014b) A comparative study between fuzzy linear regression and support vector regression for global solar radiation prediction in Iran. Sol Energy 109:135–143 Sambo A (1986) Empirical models for the correlation of global solar radiation with meteorological data for northern Nigeria. Sol Wind Technol 3(2):89–93 Srivastava S, Bhardwaj S, Sastri O (2012) A novel hybrid model for solar radiation prediction. In International conference on emerging trends in electrical engineering and energy management (ICETEEEM), 2012. IEEE Trnka M et al (2005) Global solar radiation in Central European lowlands estimated by various empirical formulae. Agric For Meteorol 131(1):54–76 Vapnik V (2000) The nature of statistical learning theory. Springer, Berlin Vapnik VN, Vapnik V (1998) Statistical learning theory, vol 2. Wiley, New York Wei Z et al (2013) A dynamic particle filter-support vector regression method for reliability prediction. Reliab Eng Syst Saf 119:109–116 Willmott CJ, Matsuura K (2005) Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance. Clim Res 30(1):79 Wu G, Liu Y, Wang T (2007) Methods and strategy for modeling daily global solar radiation with measured meteorological data—a case study in Nanchang station, China. Energy Convers Manag 48(9): 2447–2452 Yang H et al (2009) Localized support vector regression for time series prediction. Neurocomputing 72(10):2659–2669 Ye Q, Zhang Z, Law R (2009) Sentiment classification of online reviews to travel destinations by supervised machine learning approaches. Expert Syst Appl 36(3):6527–6535 Zhang L et al (2013) Iterated time series prediction with multiple support vector regression models. Neurocomputing 99:411–422

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Potential of support vector regression for solar radiation prediction in Nigeria Lanre Olatomiwa • Saad Mekhilef • Shahaboddin Shamshirband Dalibor Petkovic

•

Received: 5 November 2014 / Accepted: 29 January 2015 / Published online: 8 February 2015 Ó Springer Science+Business Media Dordrecht 2015

Abstract In this paper, the accuracy of soft computing technique in solar radiation prediction based on series of measured meteorological data (monthly mean sunshine duration, monthly mean maximum and minimum temperature) taking from Iseyin meteorological station in Nigeria was examined. The process, which simulates the solar radiation with support vector regression (SVR), was constructed. The inputs were monthly mean maximum temperature (Tmax), monthly mean minimum temperature (Tmin) and monthly mean sunshine duration ( n). Polynomial and radial basis functions (RBF) are applied as the SVR kernel function to estimate solar radiation. According to the results, a greater improvement in estimation accuracy can be achieved through the SVR with polynomial basis function compared to RBF. The SVR coefficient of determination R2 with the polynomial function was 0.7395 and with the radial basis function, the R2 was 0.5877. Keywords Nigeria

SVR Solar radiation Sunshine hour Soft computing methodologies

L. Olatomiwa S. Mekhilef Power Electronics and Renewable Energy Research Laboratory (PEARL), Department of Electrical Engineering, Faculty of Engineering, University of Malaya, 50603 Kuala Lumpur, Malaysia e-mail: [email protected] L. Olatomiwa Department of Electrical and Electronic Engineering, Federal University of Technology, PMB 65, Minna, Nigeria S. Shamshirband (&) Department of Computer System and Technology, Faculty of Computer Science and Information Technology, University of Malaya, 50603 Kuala Lumpur, Malaysia e-mail: [email protected] D. Petkovic Department for Mechatronics and Control, Faculty of Mechanical Engineering, University of Nisˇ, Aleksandra Medvedeva 14, 18000 Nis, Serbia

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Abbreviations ANFIS Adaptive neuro-fuzzy inference system GA Genetic algorithm ACO Ant colony optimization PSO Particle swarm optimization DE Differential evolution Monthly mean global solar radiation (MJ/m2/day) H n Monthly mean sunshine duration (h) RBF Radial basis function RMSE Root-mean-square error Coefficient of determination R2 SVR Support vector regression Monthly mean maximum temperature (°C) Tmax Monthly mean minimum temperature (°C) Tmin

1 Introduction Abundant energy potential from solar radiation incident on earth’s surfaces has been seen to play important role in meeting the ever-growing energy demand of the world (Ming et al. 2014; Akikur et al. 2013; Azoumah et al. 2011; Bajpai and Dash 2012; Hasan et al. 2012). Among the various available renewable resources of the earth, solar energy has attracted enormous attention not only because it sustainable, but also abundant and environmental friendly (Akikur et al. 2013). Its wide applications can results in abatement of prevalent global warming, because it does not emit CO2 or any other greenhouse gas emissions that causes global warming. Therefore, long-term knowledge of solar insolation incident on earth’s surface at any particular locations is essential for its application such as agricultural, hydrological, ecological as well as electricity production. In electricity production, it is needed in designing and prediction of energy output of solar conversion system. Solar radiation data are best obtained from measurements taken remotely at a particular location using various solar radiation measuring instruments, but these data are limited in many metrological stations around the world, and this is sometimes due to high cost of calibration and maintenance of this equipment (Hunt et al. 1998). Over the years, numerous methods for estimating solar radiation on horizontal plane have been developed, and common among them are the empirical models (Angstrom 1924; Hargreaves and Samani 1982; Bristow and Campbell 1984; Besharat et al. 2013; Halawa et al. 2014), satellite-derived model (Pinker et al. 1995) and stochastic algorithm model (Hansen 1999; Mellit 2008). The most widely used model is empirical models that relate solar radiation with other routinely measured meteorological parameters from similar locations having same geographical coordinates. Parameters such as air temperature, sunshine duration hour and relative humidity have been widely used. Many literatures have adjudged sunshine duration, maximum and minimum temperature relations as best correlation for solar radiation prediction (Besharat et al. 2013; Trnka et al. 2005; Chen and Li 2013; Wu et al. 2007). However, at instances where sunshine duration data seem limited or inaccessible, frequently measured maximum and minimum temperatures alone have also been prove to produce good results (Hargreaves and Samani 1982; Bristow and Campbell 1984; Liu et al. 2009). Although the satellite-based methods looks promising for the estimation of solar radiation over a large

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region, its main drawback is its cost and lack of sufficient historical data because it is relatively new application. In Nigeria, several works have been carried on solar radiation predictions using the conventional empirical models, some of which can be found in these following references (Ezekwe and Ezeilo 1981; Sambo 1986; Akpabio and Etuk 2003; Fagbenle 1993; and Ajayi et al. 2014). Nevertheless, due to necessity of accurate and reliable solar radiation, artificial and computational intelligence techniques have been broadly applied to estimate solar radiation in many regions around the world. Al-Alawi and Hinai(1998) predicted solar radiation for a location with no availability of measured data. They used monthly mean daily values of temperature, pressure, relative humidity, sunshine duration hours and wind speed as inputs for artificial neural networks (ANN) technique to predict global solar radiation. They compared the results with empirical methods model and found more accuracy for ANN-based model. Mellit et al. (2006) employed the combination of neural and wavelet network to forecast daily solar radiation for photovoltaic (PV) sizing application. In their study, wavelets served as activation function. Their results of the forecast demonstrated the more favorable performance of the approach compared to other neural network models. In Jiang (2009), an ANN model was developed to estimate monthly mean daily solar radiation for eight typical cities in China. The achieved results were compared to those of conventional empirical models. The statistical analysis results indicated a good correlation between estimated values by the ANN model and the actual data with higher accuracy than other empirical models. Behrang et al. (2011) applied particle swarm optimization (PSO) technique to estimate monthly mean daily global solar radiation on a horizontal surface for 17 cities in different regions of Iran. Their results showed better performance of PSO-based models compared to the traditional empirical models. Mohandes (2012) employed PSO algorithm to train ANN in order to model the monthly mean daily global solar radiation values in Saudi Arabia. Different parameters such as month number, sunshine duration, latitude, longitude and altitude of the location were considered as inputs. The developed hybrid PSO–ANN model showed a better performance compared to back-propagation trained neural network (BP-NN). Benghanem et al. (2009) developed six ANN-based models to estimate horizontal global solar radiation at Al-Madinah in Saudi Arabia. They utilized different combinations of input parameters consisting sunshine hours, ambient temperature, relative humidity and the day of year. Their results showed that the model with higher accuracy is dependent upon sunshine duration and air temperature. Ramedani et al. (2014) employed support vector regression (SVR) technique to develop a model for the prediction of global solar radiation in Tehran, Iran. They found more superiority for SVR technique than other compared techniques. In another study, Ramedani et al. (2014) performed a comparative investigation between fuzzy linear regression (FLR) and support vector regression (SVR) techniques to predict global solar radiation in Tehran, Iran. They found that SVR approach enjoys superior performance compared to FLR. Also, in some studies, different techniques were combined to propose a hybrid approaches with more accuracy. Srivastava et al. (2012) proposed a hybrid approach which includes hidden Markov models and generalized fuzzy models to estimate solar irradiation in India. They assessed the influence of different meteorological parameters for the estimation of solar radiation using the developed model. Generally, support vector machine (SVM) is a type of soft computing technique that has gained importance in environmental related applications (Ornella and Tapia 2010; Jain et al. 2009). Basically, there are two fundamental classes of support vector machines: (a) support vector classification (SVC) and (b) support vector regression (SVR). SVC is a learning framework utilizing a high-dimensional peculiarity space (Ananthakrishnan et al.

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2013; Ye et al. 2009; Rajasekaran et al. 2008; Yang et al. 2009), while SVR focused around measurable learning hypothesis and structural risk minimization rule and has been effectively utilized for nonlinear frameworks (Wei et al. 2013; Zhang et al. 2013). The correctness of an SVM model is to a great extent relies on the determination of its model parameters. Although organized strategies for selecting parameters are important, model parameter alignment also needs to be made. In this work, radial basis function (rbf)-based SVR and polynomial basis function (poly)based SVR are applied for the estimation of global solar radiation at Iseyin meteorological station in Nigeria. The input parameters are monthly mean minimum and maximum temperature and the monthly mean sunshine duration hours, all measured at the station. The main objective of this study is to evaluate the potentials and performances of SVR for solar radiation prediction. The motivation behind this investigation is centered upon the significance of reliable solar radiation data in many applications including agricultural productions, hydrological and ecological studies as well as assessments and prediction of energy output of solar systems. Also, in most cases, the solar radiation data are not readily available due to several issues. The choices of methodology center on its simplicity, reliability, efficient computationally capability, ease of adaptability to optimization and other adaptive techniques as well as its adaptability in handling parameters that are more complex.

2 Materials and methods 2.1 Descriptions of study site and data set Difficulties and uncertainty involved in the measurement of global solar radiation where the measuring equipment is available or in most cases not available have resulted in the development of algorithms for its estimation from other available routinely measured metrological data (e.g., temperature, relative humidity, wind speed, relative humidity and sunshine hour). In Nigeria, many of the government-owned meteorological stations have no record of solar radiation data, even where the record is available there are some missing days or month without record, and this is sometimes due to improper calibration of measuring equipment employed. The data set used in this study were measured at a metrological station located at Iseyin, southwest Nigeria, with geographical coordinate 7.96° latitude North, 3.6° longitude East and 330 m altitude as shown in Fig. 1. Monthly average daily value of maximum temn) and solar radiation perature (Tmax), minimum temperature (Tmin), sunshine duration ( data (H) covering 21-year period (1987–2007) were obtained from Nigerian Meteorological Agency (NIMET) database in Oshodi, Lagos (NIMET 2014). According to the agency, the measured solar radiation data were recorded using Gunn–Bellani radiometer. This instrument produce a time-oriented parameter of solar radiation falling on a black body by measuring volume of the liquid distilled in a calibrated tube (Ajayi et al. 2014; McCulloch and Wangati 1967). The measured solar radiation in millimeter is thereafter converted to MJ/m2/day with the application of a conversion factor (1.1364) as proposed by Sambo in reference (Sambo 1986). In the case of sunshine duration measurement, Campbell–Stokes sunshine recorder was used, while minimum and maximum dry bulb thermometers were used to measure both maximum and minimum temperatures at the station. Figure 2a–d shows the station monthly distribution of solar radiation, sunshine duration, minimum and maximum temperature, respectively. Annual mean solar radiation of the

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Fig. 1 Map of Nigeria showing the location of the study site

(b)

38

8

Mean sunshine hours (h)

36 34 32 30 28 26 24 22 20

7 6 5 4 3 2 1 SEP

OCT

NOV

DEC

SEP

OCT

NOV

DEC

JUL

AUG

Months

(d) 38

Months

36 34 32 30 28 26 24 22 AUG

JUL

JUN

APR

MAY

MAR

FEB

20 JAN

DEC

NOV

SEP

OCT

AUG

JUL

JUN

MAY

APR

FEB

MAR

Mean Maximum Temp.(0C)

24 24 23 23 22 22 21 21 20 20 19 JAN

Mean Minumun Temp.(0C)

JUN

APR

Months

(c)

MAY

FEB

MAR

JAN

DEC

NOV

SEP

OCT

JUL

AUG

JUN

APR

MAY

FEB

MAR

0 JAN

Mean solar radiaon (Kwh/m2/d

(a)

Months

Fig. 2 Distribution of monthly average daily solar radiation (a), sunshine duration hour minimum temperature (b), maximum temperature (c) in case study station

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studied site is 16.34 MJ/m2/day, while the annual mean bright day sunshine hour found to be 5.5 h, with highest value (7 h) in November and lowest (3.2 h) in August. The monthly mean daily maximum temperature ranges between 27.4 °C in August and 35.5 °C in February, while the minimum value ranged from 20.3 °C in January to 23.6 °C in March. 2.2 Empirical model relations of solar radiation and other metrological data Different regression correlations that relate the clearness index ration (Kt = HG/Ho) to relative sunshine duration ( n=N and air temperature (Tmax, Tmin) as derived from both Angstrom-Page (1924) and Hargreaves and Samani models (1982) were used for the analysis of the various input parameter to the SVR model. where Kt is the ratio of the monthly mean daily solar radiation on the horizontal surface (HG) to the monthly mean daily extraterrestrial solar radiation (Ho), while ( n=N is the ratio The mathematical of monthly mean sunshine hour ( n to the monthly daylight hour (N). expressions for Ho and N are given by Eq. (1) and Eq. (2) (Allen et al. 1998). H0 ¼

24 Isc ðws sin;sind þ cos;cosdsinws Þdr p

ð1Þ

2 N ¼ ws 15

ð2Þ

360ð284 þ dÞ d ¼ 23:4sin 365

ð3Þ

where

ws ¼ cos1 ð tan ; tan dÞ 360d dr ¼ 1 þ 0:033cos 365

ð4Þ ð5Þ

Isc is solar constant (4.9212) in MJ/m2/day, [ is the latitude of the site under consideration, dr is the inverse relative distance of the sum to earth, d is the solar declination, ws is the hour angle and d is the day number range from 1 (January 1st) to 365 or 366 (December and clear-sky ran=N) 31st). Extraterrestrial radiation (H0), relative sunshine duration ( diation (Hso) were also used as indicator for data quality control, where Hso is the fraction of extraterrestrial radiation falling on earth surface on clear-sky days ðn ¼ N Þ, as expressed in Eq. (6) (al 1998). Hso ¼ ð0:75 þ 2 105 zÞH0

ð6Þ

where z is the site altitude in meters. For this study z = 330, hence Hso ¼ 0:7566Ho . Since temperature-based model could be useful in a situation where sunshine duration data are not available, the model consider for this work is in line with Hargreaves and Samani model (1982), as expressed in Eq. (7). HG ¼ aDT 0:5 Ho

ð7Þ

where DT is the difference between monthly mean maximum (Tmax) and minimum temperature (Tmin) in °C, and a is an empirical which usually varies according to different regions where the site is located.

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2.3 Support vector regression application The main principle of SVR is to do the data correlation through nonlinear mapping. If a way of computing the inner product in a feature space is available directly as a function to the original input points, it is possible to build a nonlinear learning machine, which is known as a direct computation method of a kernel function, denoted by K. The flexibility of the SVR is in regard to the kernel functions that implicitly convert the data to a higher-dimensional feature space. Results in the higher-dimensional feature space correspond to the results of the original, lower-dimensional input space. There are some methods that employ nonlinear kernels for regression problems. These include the radial basis function and polynomial-based function. The main benefit of the radial basis function is computationally efficient since radial basis function training needs only the solution of a set of linear equations instead of the lengthy and computationally demanding quadratic programming problem. Therefore, the radial basis function and polynomial with parameter r are adopted in this study. The nonlinear radial basis kernel function is defined as (Yang et al. 2009; Vapnik and Vapnik 1998; Vapnik 2000): 1 2 ð8Þ Kðx; xi Þ ¼ exp 2 x xi r where x and xi are vectors in the input space, i.e. vectors of features computed from training or test samples. In this study, the following polynomial kernel function was also used (Yang et al. 2009; Vapnik and Vapnik 1998; Vapnik 2000): d ð9Þ Kðx; yÞ ¼ xT y þ c where x and y are vectors of features computed from training or test samples, and c is a constant making a trade-off for the influence of higher-order versus lower-order terms in the polynomial. The capability of the SVR to make good estimations is dependent on input parameter selection. In this study, the monthly values of Tmin, Tmax and n during the period (1987–2007) were used for generating the SVR model. To get more reliable evaluation and comparison, SVR model is tested by evaluating a data set that was not used during the training process; 70 % of the data are used for training and 30 % of the data for testing purpose. The statistical parameters (minimum value, maximum value, mean and standard deviation) for data sets are calculated and given in Table 1. 2.4 ANFIS-based methodology In this study, adaptive neuro-fuzzy inference system (ANFIS) that is functionally equivalent to the first-order Sugeno fuzzy model (Jang 1993) was used to validate the result Table 1 Statistical parameters for data sets Variable

Statistical parameters Min

Max

Mean

Standard deviation

Variation coefficient

Tmin

18

33.7

21.7

1.311

1.719

Tmax

22.8

37.1

31.6

2.839

8.065

1.3

8.4

5.5

1.443

2.083

n

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of the proposed SVR model. ANFIS serves as a basis for constructing a set of fuzzy ‘If– Then’ rules with an appropriate membership function to generate any stipulated input– output pairs. A typical rule set with a fuzzy ‘If–Then’ rule can be expressed as (Petkovic´ and C´ojbasˇic´ 2012); if x is A then f1 ¼ p1 x þ t

ð10Þ

The ANFIS architecture for three inputs x, y and z used for the validation of the proposed model is shown in Fig. 3. The nodes at the same layer have similar functions, while the output of the ith node in layer l is denoted as Ol,i. The first layer consists of input variable membership functions (MFs) and supplies the input values to the next layer. Every node i is an adaptive node with a node function: O1;i ¼ lðx; y; zÞi

for i ¼ 1; 2;

ð11Þ

where x, y = input to the ith node; l(x, y, z) = membership functions.The MFs can be described by bell-shaped function (Petkovic´ and C´ojbasˇic´ 2012) f ðx; a; b; cÞ ¼

1þ

1 xc2b

ð12Þ

a

where {a, b, c} is the parameter set. The second layer (membership layer) multiplies incoming signals from the first layer and sends the product out. Each node output represents the firing strength of a rule or weight like: O2;i ¼ wi ¼ lðxÞi lðxÞiþ1 ;

i ¼ 1; 2:

ð13Þ

The third layer (i.e., the rule layer) is non-adaptive where every node i calculates the ratio of the rule’s firing strength to the sum of all rules’ firing strengths like: wi O3;i ¼ wi ¼ ; i ¼ 1; 2: ð14Þ w1 þ w2 The outputs of this layer are known as normalized firing strengths or normalized weights. The fourth layer (i.e., the defuzzification layer) provides the output values resulting from the inference of rules, where every node i is an adaptive node with node function:

Fig. 3 ANFIS structure with three inputs, one output and two rules

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O4;i ¼ wi fi ¼ wi ðpi x þ qi x þ ri Þ

ð15Þ

where {pi, qi, ri} is the parameter set and this layer is referred to as consequent parameters. The fifth layer (i.e., the output layer) sums up all the inputs coming from the fourth layer and transforms the fuzzy classification results into a crisp (binary). The single node in the fifth layer is not adaptive, and this node computes the overall output as the summation of all incoming signals. P wi fi X i P w i fi ¼ ð16Þ O5;i ¼ wi i i

The proposed approach is based on the combination of particle swarm optimization (PSO), genetic algorithm (GA), differential evolution (DE) algorithm, ant colony optimization (ACO) and adaptive-network-based fuzzy inference system (ANFIS). The PSO, GA, DE and ACO are employed to improve the performance of ANFIS, adjusting the membership functions and reducing the prediction deviation. The ANFIS forecasts allow to predict the solar radiation. 2.5 Model performance evaluation To assess the success of the SVR models and other soft computing technique, some statistical indicators were examined as follows: 1.

2.

Root-mean-square error (RMSE) (Willmott and Matsuura 2005) vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ uP un u ðPi Oi Þ2 t ; RMSE ¼ i¼1 n Coefficient of determination (R2) (Willmott and Matsuura 2005) n 2 P Oi Oi Pi Pi i¼1 R2 ¼ P n n P Oi Oi Pi Pi i¼1

ð17Þ

ð18Þ

i¼1

where Pi and Oi are known as the experimental and predicted values, respectively, while Pi and Qi are the mean value of Pi and Oi, respectively, and n is the total number of test data.

3 Results and discussion 3.1 SVR model analysis At the beginning, the SVR network was trained with measured data by above-presented experimental procedure. After training process, the SVR networks were tested to determine the solar radiation. Based on the experiments, the input parameters (monthly mean value of minimum temperature, maximum temperature and sunshine duration hour) and output

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(solar radiation) are collected and defined for the learning techniques. For the experiments, 70 % of the data were used for sample training and the remaining 30 % for testing. We analyzed the SVR model for solar radiation estimation based on the three inputs, monthly mean minimum temperature, monthly mean maximum temperature and monthly mean sunshine duration hours. Radial basis and polynomial functions were applied as the kernel functions for SVR prediction of solar radiation. The five parameters associated with two kernels are C, e, c, d and t. SVR model accuracy largely depends on model parameter selection; therefore, in selecting user-defined parameters (i.e., C, e, c, d and t), a quite number of trials were carried out with different combinations of C and d for polynomial kernels and C and t for radial basis function kernels. Table 2 provides the optimal values of user-defined parameters for this data set with polynomial and radial basis kernels for SVR. Results of the SVR prediction models for solar radiation are shown in Fig. 4a, b for training and testing phase, respectively. In these figures, predicted solar radiation values are plotted against the observed, in the form of scatter plots. The performances of the SVR models for solar radiation prediction in defined region have been appraised via the well-known statistical indicators of the root-mean-squared error (RMSE) and coefficient of determination (R2). Performance results of proposed models are summarized in Table 3. Figure 4 shows that it is obvious that for these models, prediction results slightly differ for training and testing phase. General pattern is preserved, and it seems that prediction results for RBF kernel are more sensitive to change in testing phase than for polynomial kernel function. This observation can be easily checked from Table 3. As to the prediction results, SVR-polynomial had the smallest RMSE of 1.510218 in testing phase. Radial basis function has RMSE = 1.905994, ANFIS-ACO has RMSE = 1.5485, ANFIS-DE has RMSE = 1.601, ANFIS-GA has RMSE = 1.7696, ANFIS-PSO has RMSE = 1.8015, and ANFIS has RMSE = 2.0628 in testing phase Finally, Fig. 5 shows comparative forecasting of solar radiation by two SVR techniques during validation step. It can be observed that SVR with polynomial basis function has better forecasting abilities for solar radiation prediction than SVR with radial basis function. 3.2 Performance analysis In order to evaluate the performance of the proposed method, experimental work was carried out to determine the significance of each independent variable on the output. Root-mean-square error (RMSE) and coefficient of determination (R2) served to evaluate the differences between the predicted and actual values for both SVRs models. Table 3 compares the single radial basis SVR and polynomial basis SVR model with several soft computing methods like ANFIS-ACO (adaptive neuro-fuzzy inference system with ant colony optimization), ANFIS-DE (adaptive neuro-fuzzy inference system with

Table 2 User-defined parameters for the optical lens doublet system Radial basis function

Polynomial basis function

C

t

e

C

t

e

c

d

400

0.01

0.4

100

0.1

0.4

0.03

2

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Fig. 4 a Scatter plots of training data and predicted values using ANFIS model and b scatter plots of tested data and predicted values using ANFIS model

differential evolution), ANFIS-GA (adaptive neuro-fuzzy inference system with genetic algorithm), ANFIS-PSO (adaptive neuro-fuzzy inference system with particle swarm optimization) and ANFIS. The results in Table 3 indicate that the polynomial basis SVR model has the best capabilities of estimating the solar radiation according to testing model.

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1066 Table 3 Performance statistics of the SVR model compares to other methodologies

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RMSE

R2

Training

1.363988

0.7703

Testing

1.510218

0.7395

Training

1.141814

0.8425

Testing

1.905994

0.5877

Training

1.3505

0.7748

Testing

1.5485

0.7313

Training

1.3503

0.7748

Testing

1.601

0.6926

Training

1.2769

0.7987

Testing

1.7696

0.6354

Training

1.1157

0.8463

Testing

1.8015

0.5666

Training

1.0699

0.8586

Testing

2.0628

0.5468

SVR-polynomial

SVR-radial

ANFIS-ACO

ANFIS-DE

ANFIS-GA

ANFIS-PSO

ANFIS

Fig. 5 Forecasting of solar radiation by two SVR techniques

Experimental data SVR_RBF predicon

23

SVR_Poly predicon

Solar radiaon

21 19 17 15 13 11 9

0

10

20

30

40

50

Data samples

4 Conclusion The accuracy of support vector regression (SVR) methodology in the estimation of solar radiation using climatic variables was presented in this study. The study indicated that modeling of solar radiation is possible through the use of SVR technique. The SVR model used inputs Tmin, Tmax and n from the Iseyin station in Nigeria during the period 1987–2007. It had the

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following statistical characteristics: SVR with polynomial basis function has RMSE = 1.510218 and R2 = 0.739, SVR with radial basis function has RMSE = 1.905994 and R2 = 0.5877, ANFIS-ACO has RMSE = 1.5485 and R2 = 0.7313, ANFIS-DE has RMSE = 1.601 and R2 = 0.6926, ANFIS-GA has RMSE = 1.7696 and R2 = 0.6354, ANFIS-PSO has RMSE = 1.8015 and R2 = 0.5666, and ANFIS has RMSE = 2.0628 and R2 = 0.5468 in testing phase. Because of these, it can be concluded that SVR with polynomial basis outperforms SVR with radial basis function and other methodologies in solar radiation estimation of the studied site. Acknowledgments The authors would like to thank the Ministry of Higher Education, Malaysia, and the Bright Spark Unit of University of Malaya, Malaysia, for providing the enabling environment and financial support under the Grant No. UMRG project RP015D-13AET. The authors also want to appreciate the effort of Nigerian Meteorological Agency (NIMET) for providing the required data for this research.

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