Global Solar Radiation Modeling Using Polynomial … Solar Radiation Modeling Using Polynomial...

Applied Mathematical Sciences, Vol. 8, 2014, no. 8, 367 - 378

HIKARI Ltd, www.m-hikari.com

http://dx.doi.org/10.12988/ams.2014.39516

Global Solar Radiation Modeling

Using Polynomial Fitting

Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh

Fundamental and Applied Sciences Department

Universiti Teknologi PETRONAS, Bandar Seri Iskandar

31750 Tronoh, Perak Darul Ridzuan, Malaysia

Copyright © 2014 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh. This is an open

access article distributed under the Creative Commons Attribution License, which permits unrestricted

use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper studies the use of polynomial fitting to model the global solar radiation that

was measured by using solarimeter at Universiti Teknologi PETRONAS (UTP), located

in Perak, Malaysia. The measured solar radiation data was filtered by using fitting and

smoothing methods. The polynomial data fitting method was used for data smoothing

and was tested by using different degrees of polynomial curve fittings. The error

measurement was calculated by using the root mean square error (RMSE) and by

determining the 2R value. The polynomial fittings were carried out by using MATLAB.

Keywords: solar radiation; smoothing; curve fitting; polynomial

I. INTRODUCTION

Solar radiation prediction and forecasting are importance in the generation of

electricity and to provide the alternative energy [1]. The demand for energy is

increasing exponentially as the human population increases dynamically [2, 3, 4, 5]. In

Karim et al. [2], wavelet transform has been used to compress the solar radiation data

and to develop a new mathematical model for solar radiation data forecasting and

prediction. Their work utilized two types of wavelets namely Meyer wavelets and

Symlet 6 wavelets. Wu and Chan [1] have proposed a novel hybrid model to predict the

This work is supported by UniversitiTeknologi PETRONAS through its Short Term Internal Research

Funding (STIRF) No. 35/2012.

368 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh

hourly solar radiation data collected at Nanyang Technological University, Singapore.

They use (Autoregressive and Moving Average (ARMA) and Time Delay Neural

Network (TDNN). Their method gives better prediction with higher accuracy. Besides,

polynomial fitting have been use in computer graphics and geometric modeling. For

example, Khan [6] has used quadratic Bezier curve for video fitting and Sun et al. [7]

have use cubic spline interpolation in 2D animation. Sulaiman et al. [8] has analyst the

irradiation data using time series technique. They remove the deterministic component

by using Fourier analysis and they utilized the ARMA models in order to analysis the

stochastic component of the irradiation data. They conclude that the large variance of

the residual distribution is typical at the place where we have cloudy climate such as

Malaysia. Genc et al. [9] studied the use of cubic spline functions to analyze the solar

radiation in Izmir, Turkey. They conclude that cubic spline regression provides a more

accurate description of the relationship between total solar radiation data and the time of

the day as compared to linear regression model. Two good sources on solar radiation

data modeling are Khatib et al. [10] and Sen [11].

Motivated by the works of Wu and Chan [1] and Karim et al. [2], in this paper the

authors will use polynomial data fitting to predict the solar radiation data with good

prediction accuracy at minimal error. The best solar radiation model is obtained by

using quadratic polynomial fitting.

II. POLYNOMIAL DATA FITTING

The polynomial data fitting or regression model determination is based on the steps as

elaborated in this section.

For the given observation data ,,...,2,1,, Niyx ii the regression model is described as

.,...,2,1, Nixfy iii

(1)

where f is a regression function and i are zero-mean independent random error with a

common variance 2 . The main idea in the regression analysis is to construct a

mathematical model for f and estimate it based on the noisy data. The function f can be

approximate either by using polynomial of degree N , smoothing spline, Gaussian

function or wavelet based method (Karim and Kong [12], Karim [13]).

There exist various types of data fitting or data regression [14, 15]. It can be either the

polynomial based or non-polynomial.

Let nnxaxaxaaxf ...2

210 where n is a positive integer and the degree of the

polynomial. Now, say 1 nN then we may fit the data by using least square approach.

This can be achieved by defining the error of fitting model:

Global solar radiation modeling 369

iii xfye

niniii xaxaxaay ...2

210 (2)

Taking sum square of the error in (2) gives us

2

0

2210

0

2 ...

N

i

niniii

N

i

i xaxaxaayeS (3)

To obtain least square fitting, the sum of error in (3) must be minimized. Hence,

.,...,1,0,0 nia

S

i

(4)

Eq. (4) will give the following set of equations (in matrix forms):

CBA (5)

where

,

221

2432

132

2

ni

ni

ni

ni

niiii

niiii

niii

xxxx

xxxx

xxxx

xxxN

B

,2

1

0

na

a

a

a

A

and .2

ini

ii

ii

i

yx

yx

yx

y

C

Eq. (5) can be solved by using any numerical methods for example the Cholesky’s

method, Gaussian elimination etc. If 1n , the least square fitting in (2) is called linear

regression method (or linear fitting) given by

xaay 10 (6)

The system of equations in (5) become

.1

0

2

ii

i

ii

i

yx

y

a

a

xx

xN (7)

The solution for (7) is given by


,2

11

2

1111

2

0

n

i

i

n

i

i

n

i

ii

n

i

i

n

i

i

n

i

i

xxn

yxxyx

a

.2

11

2

1111

n

i

i

n

i

i

n

i

i

n

i

i

n

i

ii

xxn

yxyxn

a

Now, from the basic statistics, we have the following equations:

,__

1

yxnyxS

n

i

iixy

2

1

2 xnxS

n

i

ixx

,1_

n

x

x

n

i

i

n

y

y

n

i

i 1

_

Thus, we can rewrite 0a and 1a into the following simplify form:

,_

1

_

0 xaya xx

xy

S

Sa 1 (8)

By using the same idea, least square data fitting for any degree can be obtained. The

processes for data fitting are involving several steps. It was summarized as follows:

1. Input solar radiation data

2. Apply curve fitting method (using Matlab)

3. Calculate RMSE and 2R value

4. Repeat Step 2 and 3 with other degree of polynomial

5. Compare all the results and find the best fitting model

6. Use the best model for solar radiation prediction and forecasting

III. SOLAR RADIATION DATA COLLECTION

The emitted solar radiation is the electromagnetic radiation that is emitted by the sun

in the wavelength region of 280 nm to 4000 nm [2].The intensity of solar radiation received outside earth’s atmosphere is 1353 W/m

2, which is commonly referred to as

solar constant. This value varies after passing through the atmosphere and the amount received on the surface of the earth fluctuates, based on the meteorological conditions and also time of the day. At UTP, the data is monitored and measured by using sun tracking system equipped with solarimeters, as shown in Figure 1. The system is placed on a tower, located outside the UTP Solar Lab, and data is captured by using computerized data acquisition system. The averaged measured data sets for the month of January 2011 are as shown in Figure 2. [2]. The system has the ability to capture solar radiation data every minute and data resolution can be adjusted according to the requirements. The diffuse solar radiation can also be measured by using a semi-circle ring place above one of the solarimeters. The tracking system allows the measurement of solar radiation received directly by facing the measuring sensors directly towards the sun, as it is a 2-axis tracking system.


Figure 1.Sun tracking system Figure 2.Daily average of measured solar

equipped with solarimeter radiation data in UTP, Malaysia.

IV. DATA FITTING MODEL FRAMEWORK

This section will gives framework for solar radiation data fitting. Figure 3 shows the

framework.

Figure 3.Data Fitting Framework

Polynomial (Regression) Fitting Model

Data Simulation in Matlab

Validation and Error Measurement.

Choose the best fitting model with

smaller RMSE and higher 2R (inspect

the fitting graphs)

Proposed Data Fitting Model

Further Research

UTP Solar Radiation Data Gathering


V. RESULTS AND DISCUSSION

The data fitting framework in Section IV above will be use for our purpose. We apply

polynomial fitting (regression) starting with degree 1 (linear) until degree 6 (sextic). All

the polynomial coefficients are calculated based on 95% confidence interval. Table 1

summarized all the polynomial fitting results. Figs. 4 (a) until 4 (f) show the examples

of polynomial fitting for solar radiation data.

Table 1. Polynomial fitting

Polynomial Fitting

Statistics

RMSE 2R

xaaxf 10

9.423,553.3 10 aa

291.7 0.0096

27

i

i

i xaxf

2

0

,1.132,7.231 10 aa

844.42 a

105.9 0.8746

i

i

i xaxf

3

0

,5.187,7.372 10 aa

1157.0,704.9 12 aa

97.85 0.8975


Table 1 (Con’t)

i

i

i xaxf

4

0

,156.4,69.60 10 aa

,504.1,81.19 32 aa

02892.04 a

60.87 0.9621

i

i

i xaxf

5

0

,3.173,4.137 10 aa

,138.5,09.59 32 aa

,1735.04 a

002066.05 a

48.82 0.9767

i

i

i xaxf

6

0

,3.169,7.137 10 aa

,964.4,81.57 32 aa

,1621.04 a

,00171.05 a

66 10231.4 a

50.02 0.9767

(a)


(b)

(c)

(d)


(e)

(f)

From Figs. 4 (a) until 3 (f), it can clearly be seen that once the degree of the polynomial

is increasing, the fitting graphs will starting to wiggle. Among the entire fitting model,

quadratic, cubic and quartic polynomials seem to give better results as compare with the

other fitting model. There is trade-off between less RMSE and higher 2R value. For

polynomial fitting with degree are quadratic, cubic and quartic, the value of RMSE and 2R can be obtained in Table 1. From the table, Polynomial fitting with quartic degree

gives better 2R (0.8975) and RMSE is 60.87. But by detail inspection to the figure, we

can see that at both end of the graphs, the fitting model looks starting to wiggles. From

Wu and Chan [1] and the main results in Al-Sadah et al. [16], we believe the best model

Figure 4. Various polynomial fitting (a) linear (n=1)

(b) quadratic (n=2) (c) cubic (n=3)

(d) quartic (n=4) (e) quintic (n=5) and (f) sextic (n=6)


for the polynomial fitting for data in UTP is quadratic polynomial. This is due to the

fact that, the quadratic fitting gives better indication to the solar radiation data compare

with cubic and quartic polynomial fitting. Even though it RMSE is 105.9 and 2R is

0.8746, but from statistical point of view, the quadratic fitting give 87.46% indication to

the variance of the original data. Thus the following quadratic polynomial fitting in (9)

can be used to predict the amount of solar radiation received in UTP.

i

i

i xaxf

2

0

(9)

with

,1.132,7.231 10 aa .844.42 a

Where 1x data corresponds to the solar radiation at 7 am 2x corresponds to the solar

radiation data at 8 am and so on.

CONCLUSIONS

In this paper the solar radiation data fitting by using the polynomial fit method has been

discussed in details. After the data has been smoothen, the model for solar radiation can

be use to predict or forecast the receive amount of solar radiation in UTP for a certain

month. One of the applications of the polynomial fit model can be to determine the

optimum system sizing for the solar electricity generating system. Usually there is a

need to do a proper system sizing in terms of the number of PV panels required and also

the storage size. From the numerical results, the fitting model with second degree order

gives better results without any wiggle at both end points of the graph and the value of

RMSE is 105.9 and 2R value is 0.8746.

ACKNOWLEDGMENT

The authors will like to acknowledge Universiti Teknologi PETRONAS (UTP) for the

financial support received in the form of a research grant: Short Term Internal

Research Funding (STIRF) No. 35/2012.

REFERENCES

[1] Wu, J. and Chan, C.K. Prediction of hourly solar radiation using a novel hybrid

model of ARMA and TDNN. Solar Energy 85:808-817, 2011.

[2] Karim, S.A.A., Singh, B.S.M., Razali, R. and Yahya, N. Data Compression

Technique for Modeling of Global Solar Radiation. In Proceeding of 2011 IEEE

International Conference on Control System, Computing and Engineering

(ICCSCE) 25-27 November 2011, Holiday Inn, Penang, pp. 448-35,(2011a).


[3] Karim, S.A.A., Singh, B.S.M., Razali, R., Yahya, N. and Karim, B.A. Solar

Radiation Data Analysis by Using Daubechies Wavelets. In Proceeding of 2011

IEEE International Conference on Control System, Computing and Engineering

(ICCSCE) 25-27 November 2011, Holiday Inn, Penang, pp. 571-574,(2011b).

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Compression Solar Radiation data using Haar and Daubechies Wavelets. In

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[14] Wang, Y. Smoothing Splines: Methods and Applications (Chapman & Hall/CRC

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[15] Hansen, P.C., Pereyra, V. and Scherer, G. Least Squares Data Fitting with

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Received: September 15, 2013

Global Solar Radiation Modeling Using Polynomial … Solar Radiation Modeling Using Polynomial...

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