Upload
lykhanh
View
218
Download
2
Embed Size (px)
Citation preview
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
370 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh
,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.
Global solar radiation modeling 371
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
372 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh
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
Global solar radiation modeling 373
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)
Global solar radiation modeling 375
(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)
376 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh
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).
Global solar radiation modeling 377
[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).
[4] Karim, S.A.A., S Karim, S.A.A., Singh, B.S.M., Razali, R. and Karim, B.A.A.
Compression Solar Radiation data using Haar and Daubechies Wavelets. In
Proceeding of Regional Symposium on Engineering and Technology 2011,
Kuching, Sarawak, Malaysia, 21-23 November 2011, pp. 168-174,(2011c).
[5] Karim, S.A.A., Singh, B.S.M, Karim, B.A., Hasan, M.K., Sulaiman, J., Josefina, B.
Janier., and Ismail, M.T. (2012). Denoising Solar Radiation Data Using Meyer
Wavelets. AIP Conf. Proc. 1482: 685-690. http://dx.doi.org/10.1063/1.4757559.
[6] Khan, M.A. A new method for video data compression by quadratic Bezier curve
fitting. Signal, Image and Video Processing (SIViP), Vol. 6, No. 1: 19-24, 2012.
[7] Sun, N., Ayabe, T. and Okumura, K. An Animation Engine with Cubic Spline
Interpolation. In International Conference on Intelligent Information Hiding and
Multimedia Signal Processing, pp. 109-112. 2008.
[8] Sulaiman, M.Y., Hlaing Oo, W.M., Wahab, A.M. and Sulaiman, M.Z. “Analysis of
Residuals in daily Solar Radaiation Time Series”, Renewable Energy, Vol. 29, pp.
1147-1160,(1997).
[9] Genc, A., Kinaci, I., Oturanc, G., Kurnaz, A., Bilir, S. and Ozbalta, N. Statistical
Analysis of Solar Radiation Data Using Cubic Spline Functions. Energy Sources,
Part A: Recovery, Utilization, and Environmental Effects. 24:12 1131-1138. (2002)
[10] Khatib, T., Mohamed, A. and Sopian, K. A review of solar energy modeling
techniques, renewable and Sustainable Energy Reviews 16:2864-2869, 2012.
[11] Sen, Z.. Solar Energy Fundamentals and Modeling Techniques. Atmosphere,
Environment, Climate Change and Renewable Energy. Springer-Verlag London
Limited. (2008)
[12] Karim, S.A.A and Kong, V.P. Gaussian Scale-Space and Discrete Wavelet
Transform for Data Smoothing. International Conference on Electrical, Control and
Computer Engineering Pahang, Malaysia, June 21-22, 2011. pp. 344-348,(2011).
[13] Karim, S.A.A. Data Interpolation, Smoothing and Approximation using Cubic
Spline and Polynomial. Book manuscript.
[14] Wang, Y. Smoothing Splines: Methods and Applications (Chapman & Hall/CRC
Monographs on Statistics & Applied Probability), Chapman and Hall/CRC, 2012.
378 Samsul Ariffin Abdul Karim and Balbir Singh Mahinder Singh
[15] Hansen, P.C., Pereyra, V. and Scherer, G. Least Squares Data Fitting with
Applications,The Johns Hopkins University Press (December 5, 2012).
[16] Al-Sadah, F.H., Ragab, F.M., Arshad, M.K. Hourly solar radiation over Bahrain,
Energy (15) (5), 395-402, 1990.
Received: September 15, 2013