Traditional learning process in solar radiation modeling usually requires historical data to perform regularization using training and cross-validation approaches. However, in applications where no historical data are available, regularization cannot be performed using traditional techniques. This paper presents a hierarchical Bayesian framework with the extended Kalman filter (Bayesian-EKF) to perform regularization in sequential learning of the artificial neural network (ANN) for solar radiation modeling. A highly stochastic time series for daily solar radiation, the global horizontal irradiance (GHI), is modeled based on different meteorological variables including temperature (T), relative humidity (RH), wind speed (WS), and sunshine duration (SSD). A comparison is made with well-known methods including the ANN-based nonlinear autoregressive with exogenous inputs neural network (NARX-NN) and Wiener filter-based multivariate linear regression (MLR). The method is validated on test data using coefficient of determination (R2) and root mean squared error (RMSE). The proposed technique effectively estimates the noise components in the data and achieves superior performance as compared to the traditional learning processes of NARX-NN and MLR. Moreover, it is more robust to statistical outliers in the data and does not require prior history for training and cross-validation. In the presence of the outliers, the performance of the NARX-NN degrades from R2 = 94.73% to R2 = 85.85% but there is virtually no difference in the case of Bayesian-EKF. Over and above, MLR performs better than NARX-NN but worse than Bayesian-EKF.

References

References
1.
Dong
,
Z.
,
Yang
,
D.
,
Reindl
,
T.
, and
Walsh
,
W. M.
,
2013
, “
Short-Term Solar Irradiance Forecasting Using Exponential Smoothing State Space Model
,”
Energy
,
55
, pp.
1104
1113
.
2.
Tolabi
,
H. B.
,
Moradi
,
M. H.
, and
Ayob
,
S. B. M.
,
2014
, “
A Review on Classification and Comparison of Different Models in Solar Radiation Estimation
,”
Int. J. Energy Res.
,
38
(
6
), pp.
689
701
.
3.
Yacef
,
R.
,
Benghanem
,
M.
, and
Mellit
,
A.
,
2012
, “
Prediction of Daily Global Solar Irradiation Data Using Bayesian Neural Network: A Comparative Study
,”
Renewable Energy
,
48
, pp.
146
154
.
4.
Voyant
,
C.
,
Darras
,
C.
,
Muselli
,
M.
,
Paoli
,
C.
,
Nivet
,
M.-L.
, and
Poggi
,
P.
,
2014
, “
Bayesian Rules and Stochastic Models for High Accuracy Prediction of Solar Radiation
,”
Appl. Energy
,
114
, pp.
218
226
.
5.
Mohammadi
,
K.
,
Shamshirband
,
S.
,
Anisi
,
M. H.
,
Alam
,
K. A.
, and
Petković
,
D.
,
2015
, “
Support Vector Regression Based Prediction of Global Solar Radiation on a Horizontal Surface
,”
Energy Convers. Manage.
,
91
, pp.
433
441
.
6.
Mohandes
,
M. A.
,
2012
, “
Modeling Global Solar Radiation Using Particle Swarm Optimization (PSO)
,”
Sol. Energy
,
86
(
11
), pp.
3137
3145
.
7.
Behrang
,
M. A.
,
Assareh
,
E.
,
Noghrehabadi
,
A. R.
, and
Ghanbarzadeh
,
A.
,
2011
, “
New Sunshine-Based Models for Predicting Global Solar Radiation Using PSO (Particle Swarm Optimization) Technique
,”
Energy
,
36
(
5
), pp.
3036
3049
.
8.
Salcedo-Sanz
,
S.
,
Casanova-Mateo
,
C.
,
Pastor-Sánchez
,
A.
, and
Sánchez-Girón
,
M.
,
2014
, “
Daily Global Solar Radiation Prediction Based on a Hybrid Coral Reefs Optimization—Extreme Learning Machine Approach
,”
Sol. Energy
,
105
, pp.
91
98
.
9.
Mostafavi
,
E. S.
,
Ramiyani
,
S. S.
,
Sarvar
,
R.
,
Moud
,
H. I.
, and
Mousavi
,
S. M.
,
2013
, “
A Hybrid Computational Approach to Estimate Solar Global Radiation: An Empirical Evidence From Iran
,”
Energy
,
49
, pp.
204
210
.
10.
Ahmad
,
M. J.
, and
Tiwari
,
G. N.
,
2009
, “
Evaluation and Comparison of Hourly Solar Radiation Models
,”
Int. J. Energy Res.
,
33
(
5
), pp.
538
552
.
11.
Ahmad
,
M. J.
, and
Tiwari
,
G. N.
,
2011
, “
Solar Radiation Models—A Review
,”
Int. J. Energy Res.
,
35
(
4
), pp.
271
290
.
12.
Yadav
,
A. K.
,
Malik
,
H.
, and
Chandel
,
S. S.
,
2014
, “
Selection of Most Relevant Input Parameters Using WEKA for Artificial Neural Network Based Solar Radiation Prediction Models
,”
Renewable Sustainable Energy Rev.
,
31
, pp.
509
519
.
13.
Yadav
,
A. K.
, and
Chandel
,
S. S.
,
2015
, “
Solar Energy Potential Assessment of Western Himalayan Indian State of Himachal Pradesh Using J48 Algorithm of WEKA in ANN Based Prediction Model
,”
Renewable Energy
,
75
, pp.
675
693
.
14.
Lu
,
N.
,
Qin
,
J.
,
Yang
,
K.
, and
Sun
,
J.
,
2011
, “
A Simple and Efficient Algorithm to Estimate Daily Global Solar Radiation From Geostationary Satellite Data
,”
Energy
,
36
(
5
), pp.
3179
3188
.
15.
Quesada-Ruiz
,
S.
,
Linares-Rodríguez
,
A.
,
Ruiz-Arias
,
J. A.
,
Pozo-Vázquez
,
D.
, and
Tovar-Pescador
,
J.
,
2015
, “
An Advanced ANN-Based Method to Estimate Hourly Solar Radiation From Multi-Spectral MSG Imagery
,”
Sol. Energy
,
115
, pp.
494
504
.
16.
Yadav
,
A. K.
, and
Chandel
,
S. S.
,
2014
, “
Solar Radiation Prediction Using Artificial Neural Network Techniques: A Review
,”
Renewable Sustainable Energy Rev.
,
33
, pp.
772
781
.
17.
MacKay
,
D. J. C.
,
1992
, “
Bayesian Interpolation
,”
Neural Comput.
,
4
(
3
), pp.
415
447
.
18.
MacKay
,
D. J. C.
,
1992
, “
A Practical Bayesian Framework for Backpropagation Networks
,”
Neural Comput.
,
4
(
3
), pp.
448
472
.
19.
Dougherty
,
E. R.
,
1998
,
Random Processes for Image and Signal Processing
,
Wiley-IEEE Press
,
Piscataway, NJ
.
20.
Nando
,
D. F.
,
Mahesan
,
N.
, and
Andrew
,
G.
,
1998
, “
Hierarchical Bayesian-Kalman Models for Regularisation and ARD in Sequential Learning
,” Department of Engineering, Cambridge University, Report No. CUED/F-INFENG/TR 307.
21.
Kalman
,
R. E.
,
1960
, “
A New Approach to Linear Filtering and Prediction Problems
,”
Trans. ASME–J. Basic Eng.
,
82
(
1
), pp.
35
45
.
22.
Jacobs
,
O. L. R.
,
1993
,
Introduction to Control Theory
,
Oxford University Press
,
Oxford, UK
.
23.
Maybeck
,
P. S.
,
1979
,
Stochastic Models, Estimation, and Control
, Vol.
1
,
Academic Press
,
New York
.
24.
Chauvin
,
Y.
, and
Rumelhart
,
D. E.
,
1995
,
Backpropagation: Theory, Architectures, and Applications
,
Lawrence Erlbaum Associates
,
Hillsdale, NJ
.
25.
Bishop
,
C.
,
2007
,
Pattern Recognition and Machine Learning
,
Springer
,
Cambridge, UK
.
26.
Jazwinski
,
A. H.
,
1969
, “
Adaptive Filtering
,”
Automatica
,
5
(
4
), pp.
475
485
.
27.
Rencher
,
A. C.
, and
Christensen
,
W. F.
,
2012
,
Methods of Multivariate Analysis
,
3rd ed.
,
Wiley
,
Hoboken, NJ
.
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